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abstract: 'Vision science, particularly machine vision, has been revolutionized by introducing large-scale image datasets and statistical learning approaches. Yet, human neuroimaging studies of visual perception still rely on small numbers of images (around 100) due to time-constrained experimental procedures. To apply statistical learning approaches that integrate neuroscience, the number of images used in neuroimaging must be significantly increased. We present BOLD5000, a human functional MRI (fMRI) study that includes almost 5,000 distinct images depicting real-world scenes. Beyond dramatically increasing image dataset size relative to prior fMRI studies, BOLD5000 also accounts for image diversity, overlapping with standard computer vision datasets by incorporating images from the Scene UNderstanding (SUN), Common Objects in Context (COCO), and ImageNet datasets. The scale and diversity of these image datasets, combined with a slow event-related fMRI design, enable fine-grained exploration into the neural representation of a wide range of visual features, categories, and semantics. Concurrently, BOLD5000 brings us closer to realizing Marr’s dream of a singular vision science — the intertwined study of biological and computer vision.'
author:
- |
Nadine Chang^1^, John A. Pyles^1^, Abhinav Gupta^1^,\
Michael J. Tarr^1^, Elissa M. Aminoff^2[\*]{}^
title: |
**BOLD5000**\
A public fMRI dataset of 5000 images
---
1\. Carnegie Mellon University 2. Fordham University [\*]{}corresponding author: Elissa Aminoff ([email protected])
Data Records {#sec:datarec .unnumbered}
============
We have publicly released BOLD5000 online. As seen in Table \[data\], we have provided a comprehensive list of collected data and the various stages of analyzed data we have made available. All relevant links and information can be found at <http://BOLD5000.org>.
Usage Notes {#usage-notes .unnumbered}
===========
The goal of this publicly available dataset is to enable joint research from various communities including neuroscience and computer vision. While this paper shows a glimpse of the richness in our data, we hope that our unique, large-scale, interdisciplinary dataset will provide new opportunities for more integrated analysis from various disciplines.
Discussion {#discussion .unnumbered}
==========
Acknowledgements {#acknowledgements .unnumbered}
================
NC participated in stimulus selection, stimulus pre-processing, stimulus analysis, experimental design, collecting fMRI data, t-SNE data analysis, writing the manuscript, public distribution of the data, creating the website, and consulted on the remaining sections of the project.
JAP developed and tested the MRI protocols, participated in experimental design, collecting fMRI data, the MRI processing pipeline, writing the manuscript, public distribution of the data, and consulted for the remaining sections of the project.
AG helped conceive the original project and and write the manuscript. AG consulted regarding stimulus selection, stimulus analysis, experimental design, and t-SNE data analysis.
MJT helped conceive the original project and write the manuscript. MJT consulted regarding stimulus selection, experimental design, data analysis, and public distribution.
EMA helped conceive the original project and participated in stimulus selection, experimental design, fMRI pre-processing data, general fMRI data analysis (GLM, ROIs), specific subsequent data analysis (design validation, data validation, representational similarity analysis), writing the manuscript, public distribution of the data and consulted on the remaining sections of the project.
We thank Scott Kurdilla for his patience as our MRI technologist throughout all data collection. We would also like to thank Austin Marcus for his assistance in various stages of this project, Jayanth Koushik for his assistance in AlexNet feature extractions, and Ana Van Gulick for her assistance with public data distribution and open science issues.
This dataset was collected with the support of NSF Award BCS-1439237 to Elissa M. Aminoff and Michael J. Tarr, ONR MURI N000141612007 and Sloan, Okawa Fellowship to Abhinav Gupta, and NSF Award BSC-1640681 to Michael Tarr.
Finally, we thank our participants for their participation and patience, without them this dataset would not have been possible.
Competing Financial Interests {#competing-financial-interests .unnumbered}
=============================
The author(s) declare no competing financial interests.
Data Citations {#sec:dataCite .unnumbered}
==============
Brain, Object, Landscape Dataset [BOLD5000](http://bold5000.org) (2018)
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---
abstract: 'This paper presents an extension of the OpenAI Gym for robotics using the Robot Operating System (ROS) and the Gazebo simulator. The content discusses the software architecture proposed and the results obtained by using two Reinforcement Learning techniques: Q-Learning and Sarsa. Ultimately, the output of this work presents a benchmarking system for robotics that allows different techniques and algorithms to be compared using the same virtual conditions.'
author:
- 'Iker Zamora, Nestor Gonzalez Lopez, Víctor Mayoral Vilches,'
- Alejandro Hernández Cordero
bibliography:
- 'references.bib'
title: 'Extending the OpenAI Gym for robotics: a toolkit for reinforcement learning using ROS and Gazebo'
---
Introduction
============
Reinforcement Learning (RL) is an area of machine learning where a software agent learns by interacting with an environment, observing the results of these interactions with the aim of achieving the maximum possible cumulative reward. This imitates the trial-and-error method used by humans to learn, which consists of taking actions and receiving positive or negative feedback.\
In the context of robotics, reinforcement learning offers a framework for the design of sophisticated and hard-to-engineer behaviors [@kober2013reinforcement]. The challenge is to build a simple environment where this machine learning techniques can be validated, and later applied in a real scenario.\
OpenAI Gym [@brockman2016openai] is a is a toolkit for reinforcement learning research that has recently gained popularity in the machine learning community. The work presented here follows the same baseline structure displayed by researchers in the OpenAI Gym ([gym.openai.com](https://gym.openai.com)), and builds a gazebo environment on top of that. OpenAI Gym focuses on the episodic setting of RL, aiming to maximize the expectation of total reward each episode and to get an acceptable level of performance as fast as possible. This toolkit aims to integrate the Gym API with robotic hardware, validating reinforcement learning algorithms in real environments. Real-world operation is achieved combining Gazebo simulator [@koenig2006gazebo], a 3D modeling and rendering tool, with ROS [@quigley2009ros] (Robot Operating System), a set of libraries and tools that help software developers create robot applications.\
As benchmarking in robotics remains an unsolved issue, this work aims to provide a toolkit for robot researchers to compare their techniques in a well-defined (API-wise) controlled environment that should speed up development of robotic solutions.
Background
==========
Reinforcement Learning has taken an increasingly important role for its application in robotics [@rlintro]. This technique offers robots the ability to learn previously missing abilities [@kormushev2013reinforcement] like learning hard to code behaviours or optimizing problems without an accepted *closed* solution.\
The main problem with RL in robotics is the high cost per-trial, which is not only the economical cost but also the long time needed to perform learning operations. Another known issue is that learning with a real robot in a real environment can be dangerous, specially with flying robots like *quad-copters*. In order to overcome this difficulties, advanced robotics simulators like Gazebo have been developed which help saving costs, reducing time and speeding up the simulation.\
The idea of combining learning in simulation and in a real environment was popularized by the Dyna-architecture (Sutton, 1990), prioritized sweeping (Moore and Atkeson,1993), and incremental multi-step Q-Learning (Pengand Williams, 1996) in reinforcement learning. In robot reinforcement learning, the learning step on the simulated system is often called “mental rehearsal” [@kober2013reinforcement]. Mental reharsal in robotics can be improved by obtaining information from the real world in order to create an accurate simulated environment. Once the training is done, just the resulting policy is transferred to the real robot.\
Architecture
============
The architecture consits of three main software blocks: OpenAI Gym, ROS and Gazebo (Figure \[fig:diagram\]). Environments developed in OpenAI Gym interact with the Robot Operating System, which is the connection between the Gym itself and Gazebo simulator. Gazebo provides a robust physics engine, high-quality graphics, and convenient programmatic and graphical interfaces.\
![Simplified software architecture used in OpenAI Gym for robotics. \[fig:diagram\]](GymDiagram2_3.png){width="48.00000%"}
The physics engine needs a robot definition[^1] in order to simulate it, which is provided by ROS or a gazebo plugin that interacts with an autopilot in some cases (depends on the robot software architecture). The Turtlebot is encapsulated in ROS packages while robots using an autopilot like Erle-Copter and Erle-Rover are defined using the corresponding autopilot. Our policy is that every robot needs to have an interface with ROS, which will mantain an organized architecture.\
Figure \[fig:diagram\] presents the a simplified diagram of the software architecture adopted in our work. Our software structure provides similar APIs to the ones presented initially by OpenAI *Gym*. We added a new collection of environments called *gazebo* where we store our own gazebo environments with their corresponding assets. The needed installation files are stored inside the *gazebo* collection folder, which gives the end-user an easier to modify infrastructure.\
Installation and setup consits of a ROS catkin workspace containing the ROS packages required for the robots (e.g.: Turtlebot, Erle-Rover and Erle-Copter) and optionally the appropriate autopilot that powers the logic of the robot. In this particular case we used the APM autopilot thereby the source code is also required for simulating Erle-Rover and Erle-Copter. Robots using APM stack need to use a specific plugin in order to communicate with a ROS/Gazebo simulation.
Environments and Robots
=======================
We have created a collection of six environments for three robots: Turtlebot, Erle-Rover and Erle-Copter. Following the desing decisions of OpenAI Gym, we only provide an abstraction for the environment, not the agent. This means each environment is an independent set of items formed mainly by a *robot* and a *world*.\
Figure \[fig:turtlesilly\] displays an environment created with the Turtlebot *robot* which has been provided with a LIDAR sensor using and a *world* called *Circuit*. If we wanted to test our reinforcement learning algorithm with the Turtlebot but this time using positioning information, we would need to create a completely new environment.\
![Turtlebot in a virtual environment learning how to navigate autonomously using a LIDAR \[fig:turtlesilly\]](turtlec1.png){width="48.00000%"}
The following are the initial environments and robots provided by our team at Erle Robotics. Potentially, the amount of supported robots/environments will will grow over time.\
**Turtlebot.** TurtleBot combines popular off-the-shelf robot components like the iRobot Create, Yujin Robot’s Kobuki, Microsoft’s Kinect and Asus’ Xtion Pro into an integrated development platform for ROS applications. For more information, please see [turtlebot.com](http://turtlebot.com) .\
The following are the four environment currently available for the Turtlebot:
- *GazeboCircuitTurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].a) : A simple circuit with a diagonal wall, which increases the complexity of the learning.
- *GazeboCircuit2TurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].b) : A simple circuit with straight tracks and 90 degree turns. Note that the third curve is a left turn, while the others are right turns.
- *GazeboMazeTurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].c) : A complex maze with different wall shapes and some narrow tracks.
- *GazeboRoundTurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].d) : A simple oval shaped circuit.
[0.2]{} ![Environments available using Turtlebot robot and a LIDAR sensor. \[fig:turtlebotEnvs\]](env1.png "fig:"){width="0.95\linewidth"}
[0.2]{} ![Environments available using Turtlebot robot and a LIDAR sensor. \[fig:turtlebotEnvs\]](env2.png "fig:"){width="0.95\linewidth"}
[0.2]{} ![Environments available using Turtlebot robot and a LIDAR sensor. \[fig:turtlebotEnvs\]](env3.png "fig:"){width="0.95\linewidth"}
[0.2]{} ![Environments available using Turtlebot robot and a LIDAR sensor. \[fig:turtlebotEnvs\]](env4.png "fig:"){width="0.95\linewidth"}
**Erle-Rover.** A Linux-based smart car powered by the APM autopilot and with support for the Robot Operating System [erlerobotics.com/blog/erle-rover](http://erlerobotics.com/blog/erle-rover/) .\
![Erle-Rover learning to avoid obstacles in a big maze with wide tracks and 90 degree left and right turns, an environment called *GazeboMazeErleRoverLIDAR-v0*.\[fig:apm1\]](apm1.png){width="48.00000%"}
**Erle-Copter.** A Linux-based drone powered by the open source APM autopilot and with support for the Robot Operating System [erlerobotics.com/blog/erle-copter](http://erlerobotics.com/blog/erle-copter/) .
![Erle-Copter learning to avoid obstacles in ErleRobotics office without ceiling, an environment called *GazeboOfficeErleCopterLIDAR-v0*.\[fig:apm2\]](apm2.png){width="48.00000%"}
Results
=======
We have experimented with two Reinforcement Learning algorithms, Q-Learning and Sarsa. The turtlebot has been used to benchmark the algorithms since we get faster simulation speeds than robots using and autopilot. We get around 60RTF (Real Time Factor), which means 60 times normal simulation speed and 30RTF when we launch the visual interfaze. This benchmarks have been made using a i7 6700 CPU and non-GPU laser mode.
![Turtlebot with LIDAR in *[GazeboCircuit2TurtlebotLIDAR-v0]{} environment.\[fig:c2result\]*](turtlec2_new.png){width="48.00000%"}
We will use *GazeboCircuit2TurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].b & \[fig:c2result\]) to perform the benchmarking. This environment consists of a simple straight lined circuit with five right turns and one left turn. We will use just a LIDAR sensor to train the Turtlebot, no positioning or other kind of data will be used. Both algorithms will use the same hyperparameters and exact environment. The actions and rewards forming the environment have been adapted to get an optimal training performance.\
**Actions**
- Forward: v = 0.3 m/s
- Left/Right: v = 0.05 m/s , w = +-0.3 rad/s
Taking only three actions will lead to a faster learning, as the ’Q’ function will fill its table faster. Obstacle avoidance could be performed with just two turning actions, but the learnt movements would be less practical.\
Left and right turns have a small linear velocity just to accelerate the learning process and avoid undesired behaviours. Setting the linear velocity to zero, the robot could learn to turn around itself constantly as it would not crash and still earn positive rewards. As Atkeson and Schaal point out:
Reinforcement learning approaches exploit such model inaccuracies if they are beneficial for the reward received in simulation.
This could be avoided changing the reward system, but we found the optimal values for this environment are the following.\
**Rewards**
- Forward: 5
- Left/Right: 1
- Crash: -200
Forward actions take five times more reward than turns, this will make the robot take more forward actions as they give more reward. We want to take as many forward actions as possible so that the robot goes forward in straight tracks, which will lead to a faster and more realistic behaviour.\
Left and right actions are rewarded with 1, as they are needed to avoid crashes too. Setting them higher would result in a zigzagging behaviour.\
Crashes earn very negative rewards for obvious reasons, we want to avoid obstacles.\
**Q-Learning**\[qlearning\]\
Q-Learning[@watkins1992q] is an Off-Policy algorithm for Temporal Difference learning. Q-Learning learns the optimal policy even when actions are selected according to an exploratory or even random policy [@sarsavsqlearn2]. $$Q(s_t,a_t) \leftarrow Q(s_t,a_t) + \alpha [r_{t+1} + \gamma max{\alpha}, Q(s_{t+1}, a_{t})-Q(s_t,a_t)]$$
Let’s see how the Turtlebot learns using Q-Learning in *GazeboCircuit2TurtlebotLIDAR-v0* environment. Simplified code is presented below:\
``` {.python basicstyle="\footnotesize" language="Python"}
env = gym.make('GazeboCircuit2TurtlebotLIDAR-v0')
qlearn = qlearn.Qlearn(alp=0.2,gam=0.9,eps=0.9)
for x in range(3000):
observation = env.reset()
state = ''.join(map(str, observation))
for i in range(1500):
action = sarsa.chooseAction(state)
observation, reward, done = env.step(action)
nextState = ''.join(map(str, observation))
qlearn.learn(state, action, reward,
nextState)
if not(done):
state = nextState
else:
break
```
After selecting environment we want to test, we have to initialize Q-learn with three parameters. Small changes in these hyperparameters can result in substantial changes in the learning of our robot. Those parameters are the following:\
- :. Setting it to 0 means the robot will not learn and a high value such as 0.9 means that learning can occur quickly.
- :. A factor of 0 will make the agent consider only current rewards, while a factor approaching 1 will make it strive for a long-term high reward.
- :. Used to randomize decisions, setting a high value such as 0.9 will make 90% of the actions to be stochastic. An intersting technique is to set an epsilon decay, where the agent starts taking more randon actions (exploration phase) and ends in an exploitation phase where all or most of the actions performed are selected from the learning table instead of being random.
The selected initial hyperparameters are, $\alpha=0.2$, $\gamma=0.9$ and $\epsilon=0.9$. In this example we use the epsilon decay technique, being the decay $\epsilon*0.9986$ every episode until it reaches our minimum epsilon value, 0.05 in this case.\
We want to run the simulation for 3000 episodes. Each episode the simulation will be resetted and the robot will start again from its initial position. Every episode we try to make a maximum of 1500 iterations, which means the robot has not crashed. Every iteration we choose an action, take a step (execute an action for a short time or distance) and receive feedback. That feedback is called *observation* and it is returns the next state to be taken and the received reward.\
To sum up, every episode the robot tries to take as many steps as possible, learning every step from the obtained rewards.\
The following graph shows the results obtained through 3000 episodes.
![Cumulated reward graph obtained from the monitoring of *GazeboCircuit2TurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].b) environment using Q-Learning. The blue line prints the whole set of readings while the red line shows an approximation to the averaged rewards. \[fig:qlearnplot\]](newplot_qlearn.png){width="48.00000%"}
We get decent results after 1600 episodes. Cumulated rewards around 2000 or higher usually mean the robot did not crash or gave more than two laps.\
**Sarsa**\
Sarsa [@rummery1994line] is an On-Policy algorithm for Temporal Difference Learning. The major difference between it and Q-Learning, is that the maximum reward for the next state is not necessarily used for updating the Q-values. Instead, a new action, and therefore reward, is selected using the same policy that determined the original action [@sarsavsqlearn2]. $$Q(s_t,a_t) \leftarrow Q(s_t,a_t) + \alpha [r_{t+1} + \gamma Q(s_{t+1}, a_{t+1})-Q(s_t,a_t)]$$
Let’s now compare how the Turtlebot learns using Sarsa in *GazeboCircuit2TurtlebotLIDAR-v0* environment. Simplified code is presented below\
``` {.python basicstyle="\footnotesize" language="Python"}
env = gym.make('GazeboCircuit2TurtlebotLIDAR-v0')
sarsa = sarsa.Sarsa(alp=0.2,gam=0.9,eps=0.9)
for x in range(3000):
observation = env.reset()
state = ''.join(map(str, observation))
for i in range(1500):
action = sarsa.chooseAction(state)
observation, reward, done = env.step(action)
nextState = ''.join(map(str, observation))
nextAction = sarsa.chooseAction(nextState)
sarsa.learn(state, action, reward,
nextState, nextAction)
if not(done):
state = nextState
else:
break
```
After selecting environment we want to test, we have to initialize Sarsa with three parameters: alpha, gamma and epsilon. They work in the same way as Q-learn, so we used the same as in the previous Q-leaning test \[qlearning\]. The selected initial hyperparameters are, $\alpha=0.2$, $\gamma=0.9$ and $\epsilon=0.9$ with $\epsilon*0.9986$ epsilon decay.\
We want to run the simulation for 3000 episodes. Every episode we try to make a maximum of 1500 iterations, which means the robot has not crashed. Every iteration we choose an action, take a step (execute an action for a short time or distance) and receive feedback. That feedback is called *observation* and it is used to build the next action to be taken.\
Since we are using Sarsa (on-policy) and not Q-learn (off-policy), we need to choose another action before we learn. This is done by selecting a new action from the previously built next state.\
As explained in the Q-Learning section, every episode the robot tries to take as many steps as possible, learning every step from the obtained rewards.\
The following graph shows the results obtained through 3000 episodes.
![Cumulated reward graph obtained from the monitoring of *GazeboCircuit2TurtlebotLIDAR-v0* (Figure \[fig:turtlebotEnvs\].b) environment using Sarsa. The blue line prints the whole set of readings while the red line shows an approximation to the averaged rewards.\[fig:sarsaplot\]](newplot_sarsa.png){width="48.00000%"}
We get decent results after 2500 episodes. Cumulated rewards around 2000 or higher usually mean the robot did not crash or gave more than two laps.\
**Benchmarking**\
The learning in Q-Learning occurs faster than in Sarsa, this happens because Q-Learning is able to learn a policy even if taken actions are chosen randomly. However, Q-learning shows more risky moves (taking turns really close to walls) while in Sarsa we see a smoother general behaviour. The major difference between Sarsa and Q-Learning, is that the maximum reward for the next state is not necessarily used for updating the Q-values (learning table). Instead, a new action, and therefore reward, is selected using the same policy that determined the original action [@sarsavsqlearn2]. This is how Sarsa is able to take into account the control policy of the agent during learning. It means that information needs to be stored longer before the action values can be updated, but also means that our robot is going to take risky actions much frequently [@sarsavsqlearn]. This smoother behaviour where forward actions are being exploited in straight tracks leads to higher maximum cumulated rewards. We get values near 3500 in Sarsa while just get cumulated rewards around 2500 in Q-Learning. Running Sarsa for more episodes will cause to get higher average rewards.\
The table below provides a numerical comparison of Q-Learning and Sarsa representing the average reward value over 200 consecutive episodes. From the data, one can tell that learning occurs much faster using the Q-Learning technique:
\[table:benchmark\]
**Episode interval** **Q-Learning** **Sarsa**
---------------------- ---------------- -----------
0-200 -114 -124
200-400 -79 -98
400-600 72 -75
600-800 212 -43
800-1000 239 -43
1000-1200 282 -6
1200-1400 243 55
1400-1600 439 65
1600-1800 676 104
1800-2000 503 127
2000-2200 510 361
2200-2400 345 164
2400-2600 776 698
2600-2800 805 550
2800-3000 685 240
: Average reward value over a 200 episode interval in 3000 episode long tests using Q-Learning and Sarsa.[]{data-label="table:benchmark"}
. Although the variance presented in Figures \[fig:qlearnplot\] & \[fig:sarsaplot\] is high[^2], the averaged plots and the table above show that out robot has learnt to avoid obstacles using both algorithms.\
Iterating 2000 episodes more will not make the variance disappear, as we have almost reached the best behaviour possible in this highly discretized environment. Laser values are discretized so that the learning does not take too long. We take only 5 readings with integer values, which are taken uniformly from the 270º lasers horizontal field of view. To sum up, using a simple reinforcement learning technique and just a LIDAR as an input, we get quite decent results.\
Future directions
=================
The presented toolkit could be further improved in the following directions:
- Support more autopilot solutions besides APM such as PX4 or Paparazzi.
- Speed up simulation for robots using autopilots. Currently, due to limitations of the existing implementation, the simulation is set to normal(real) speed.
- Pull apart environments and agents. Testing different robots in different environments (not only the ones built specifically for them) would make the toolkit more versatile.
- Provide additional tools for comparing algorithms.
- Recommendations and results in *mental reharsal* using the presented toolkit.
This research has been funded by Erle Robotics.
[^1]: Unified Robot Description Format (URDF)
[^2]: common thing in this scenario
|
---
abstract: |
The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the [*general*]{} CFT ${}_2$ with $c>1$. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit $z \to 1$, which was unknown until now. In this study, we computed it in general by studying the pole structure of the [*fusion matrix*]{} (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value $\frac{c-1}{12}$ if the total Liouville momentum exceeds beyond the [*BTZ threshold*]{}. This might be interpreted as a black hole formation in AdS${}_3$.
As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench.
We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general $n$-th Renyi entropy after a local quench and out of time ordered correlators.
bibliography:
- 'LC.bib'
---
YITP-18-106\
[[ **Light Cone Bootstrap in General 2D CFTs\
and\
Entanglement from Light Cone Singularity** ]{}]{}\
Yuya Kusuki
[*Center for Gravitational Physics,\
Yukawa Institute for Theoretical Physics (YITP), Kyoto University,\
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan.* ]{} 2em
Introduction & Summary
======================
Two-dimensional conformal field theories (2D CFTs) have an infinite dimensional symmetry, the so-called Virasoro symmetry. This symmetry leads to the fact that 2D CFTs are only specified by a central charge, spectrum of primary operators, and operator product expansion (OPE) coefficients of the primary operators in the spectrum. Owing to this simplification, 2D CFTs offer ideal avenues for exploring quantum field theories. They are also investigated as the key to understanding quantum mechanics in AdS by using the AdS/CFT correspondence. However, despite several decades of studies, no criterion has been devised to classify CFTs and only a few models (for example, minimal models) are classified; in other words, we do not know how to identify which CFT data are consistent with the modular invariance.
One important tool to determine which CFT data are consistent is [*conformal bootstrap*]{}, or equivalently, the crossing symmetry [@Belavin1984; @Ferrara1973; @Polyakov1974; @Rattazzi2008]. The conformal bootstrap equation originates from the OPE associativity. $${\frac}{1}{x_{12}^{2\D_1}x_{34}^{2\D_2}}\sum_{p}P^{11,22}_{\tau_p,l_p} g^{11,22}_{\tau_p,l_p}(u,v)={\frac}{1}{{\left(x_{14}x_{23} \right)}^{\D_1+\D_2}}{\left({\frac}{x_{24}}{x_{12}} \right)}^{\D_{12}}{\left({\frac}{x_{13}}{x_{12}} \right)}^{\D_{12}}\sum_{p}P^{12,12}_{\tau_p,l_p} g^{12,12}_{\tau_p,l_p}(v,u),$$ where $u,v$ are the cross ratios, and $g^{ij,kl}_{\tau,l}(u,v)$, $P^{ij,kl}_{\tau,l}$ are the conformal blocks and their coefficients. This equation relates different OPE coefficients, and therefore, nontrivial requirements for CFT data can be obtained using the conformal bootstrap equation. In particular, the limit $u,1-v \to 0$ (or equivalently, $z, \bar{z} \to 0$) relates high-energy physics (i.e. information about the spectrum at large conformal dimensions) to the vacuum contribution. From this [*high-low temperature duality*]{}, [^1] we can evaluate, for example, the density of states at large conformal dimensions, the so-called Cardy formula [@Cardy1986a]. Apart from this famous consequence, we also have several results from this duality [@Kraus2016; @Cardy2017; @Hikida2018; @Romero-Bermudez2018; @Brehm2018].
Recently, another kinematic limit, [*light cone limit*]{} $u \ll v \ll 1$ (or equivalently $z\ll 1-\bar{z} \ll 1$), has attracted considerable interests in higher-dimensional CFTs [@Fitzpatrick2013a; @Komargodski2013; @Alday2015; @Kaviraj2015; @Kaviraj2015a; @Alday2017a; @Simmons-Duffin2017; @Alday2017; @Sleight2018]. Unlike the limit $z,\bar{z} \to 0$, the light cone limit relates the vacuum blocks to the OPE at large spin. Therefore, the light cone bootstrap reveals the structure of the OPE in the large spin limit. In fact, the light cone bootstrap imposes a condition such that in the CFT${}_{d\geq 3}$ with two scalar operators $\phi_A$ and $\phi_B$, there must exist operators with twist [@Fitzpatrick2013a; @Komargodski2013] $$\tau= \D_A+\D_B+2n+\gamma_{AB}(n,l) \ \ \ \ \ \text{for any } n\in {\mathbb{Z}}_{\geq0},$$ and in particular, the anomalous dimension $\gamma_{AB}(n,l) \to 0$ as $l \to \infty$. In this way, the light cone limit provides substantial information about the spectrum.
In this paper, we give the following results:\
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In 2D CFTs, the task of solving the light cone bootstrap is much more difficult than that in CFT${}_{\geq 3}$, as pointed out in [@Fitzpatrick2014]. The most difficult point is to evaluate the limit $z\to1$ of the Viraosoro conformal blocks (which we will call [*light cone limit Virasoro blocks*]{} in the following). At present, there is no exact asymptotic formula for the light cone limit of Virasoro blocks, let alone their explicit form. However, in this study, we accomplished to provide the asymptotic form of the light cone limit Virasoro blocks in any unitary CFT with $c>1$. The method for accomplishing this is to study the structure of poles in the fusion matrix $ {\bold F}_{\a_s, \a_t} $, which are invertible fusion transformations between $s$ and $t$-channel conformal blocks. $$\begin{aligned}
{\mathcal{F}}^{21}_{34}(h_{\a_s}|z)=\int_{{\mathbb{S}}} \dd \a_t {\bold F}_{\a_s, \a_t}
\left[
\begin{array}{cc}
\a_2 & \a_1 \\
\a_3 & \a_4\\
\end{array}
\right]
{\mathcal{F}}^{23}_{14}(h_{\a_t}|1-z),
\end{aligned}$$ where $h_i=\a_i(Q-\a_i)$ and $c=1+6Q^2$. Consequently, we can explicitly provide the light cone singularity of the conformal blocks. We then summarize the results, but before that, we will introduce some notations. In this paper, we consider two types of Viraosoro conformal blocks. One is the Virasoro block for the correlator $\braket{O_B(\infty)O_B(1)O_A(z)O_A(0)}$ in the $O_A(z)O_A(0)$ OPE channel,
$${\mathcal{F}}^{AA}_{BB}(h_p|x) \equiv \parbox{\paw}{\usebox{\boxpa}},$$
which we call [*AABB blocks*]{}. The other type is [*ABBA blocks*]{}, which are defined as $${\mathcal{F}}^{BA}_{BA}(h_p|x) \equiv \parbox{\pdw}{\usebox{\boxpd}}.$$ The light cone singularities for the ABBA blocks is given by $$\label{eq:ABBAs}
\begin{aligned}
{\mathcal{F}}^{BA}_{BA}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} \left\{
\begin{array}{ll}
(1-z)^{4h_A-2h_B-2Q\a_A} ,& \text{if } \a_A<{\frac}{Q}{4}\ \text{and } \ \a_A<\a_B ,\\
(1-z)^{2h_B-2Q\a_B} ,& \text{if } \a_B<{\frac}{Q}{4}\ \text{and } \ \a_B<\a_A ,\\
(1-z)^{{\frac}{c-1}{24}-2h_B} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ The asymptotics of the AABB blocks are given by $$\label{eq:AABBs}
\begin{aligned}
{\mathcal{F}}^{AA}_{BB}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} \left\{
\begin{array}{ll}
(1-z)^{-2\a_A\a_B} ,& \text{if } \a_A+\a_B<{\frac}{Q}{2}\ ,\\
(1-z)^{{\frac}{c-1}{24}-h_A-h_B} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ This is one of the main results presented in this paper. One interesting point is that we can find a transition of the blocks. However, in general, we cannot observe this transition in the behaviour of a four-point function because the approximation by only one (or few) block contribution appears if considering the limit $z,\bar{z} \to 0$ of the t-channel expansion. If one wants to find this transition in a correlator, then one has to find a quantity that can be evaluated by only one block contribution and in the limit $z \to 1$. In fact, it is given by the [*light cone limit*]{} $$\braket{O_B(\infty) O_B(1) O_A(z,\bar{z}) O_A(0)} {\xrightarrow[\text{light cone limit}]{}} {\mathcal{F}}^{AA}_{BB}(0|z\to0)\overline{{\mathcal{F}}^{AA}_{BB}(0|\bar{z}\to1)}.$$ Therefore, we can detect the transition in the light cone limit. We want to emphasize that the transition point of the AABB blocks is characterised by the BTZ threshold $\a_{\text{BTZ}}={\frac}{Q}{2}$, where $h_{\text{BTZ}}=\a_{\text{BTZ}}(Q-\a_{\text{BTZ}})={\frac}{c}{24}$. Therefore, we expect that this [*light cone transition*]{} captures some important properties of holographic CFTs. In fact, one gravity interpretation of this transition can be obtained by the light cone bootstrap as we will explain later.
The detailed derivations of these singularities are provided in Appendix \[app:FM\]. In fact, this conclusion is supported by numerical computations [@Kusuki2018b; @Kusuki2018] and proven in large $c$ CFTs [@Kusuki2018a]. We intend to emphasize that the fusion matrix approach does not rely on the assumption $c \to \infty$, unlike the Zamolodchikov monodromy method [@Kusuki2018a]; therefore, the conclusion (\[eq:AABBs\]), (\[eq:ABBAs\]) holds true not only for large $c$ CFTs but also for any unitary CFT with $c>1$. Note that our formula breaks down for the CFT data ($c<1$) of the minimal model, as explained in the final paragraph of Appendix \[subapp:FM\].
In addition, we discuss the sub-leading terms of the blocks in the light cone limit. In Appendix \[subapp:sub\], we investigate the other poles (sub-leading poles) in the fusion matrix and identify them as the sub-leading terms. In fact, the sub-leading singularities perfectly match the HHLL Virasoro blocks’ singularities in the heavy-light limit.\
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In the main sections of this paper, we will apply this explicit form of light cone singularity to study the light cone bootstrap. One of the main objectives is to understand the AdS${}_3$/CFT${}_2$ duality (for example, one concrete question is how Virasoro blocks can be derived from AdS gravity). Many recent developments pertaining to conformal blocks helps uncover information about holographic CFTs [@Fitzpatrick2013; @Fitzpatrick2015; @Fitzpatrick2017; @Alkalaev2015; @Hijano2015; @Hijano2015a; @Alkalaev2016; @Asplund2015; @Roberts2015; @Alkalaev2016b; @Chen2016; @Alkalaev2016a; @Hulik2017; @Chen2017; @Alkalaev2017; @Maxfield2017; @Alkalaev2018; @Hikida2018c; @Banerjee2018; @Hikida2018a; @Belavin2018; @Hulik2018; @Chen2018] . Similarly, it is expected that the light cone singularities revealed in this paper also lead to some interesting predictions in holographic CFTs. In fact, we found that there must be a universal long-distance interaction between two objects at large spin for any unitary CFT with $c>1$ and without extra conserved current. Particularly, we found that the total twist of two particles at large spin is saturated by the BTZ threshold ${\frac}{c-1}{12}$ if the total Liouville momentum $\a_A+\a_B$ increases beyond the BTZ threshold $\a_{\text{BTZ}}={\frac}{Q}{2}$.\
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Apart from the light cone bootstrap, the light cone limit Viraosoro blocks also appear in many different situations. In the rest of this paper, we discuss [*entanglement*]{} using the explicit asymptotic form of the light cone limit Virasoro blocks. In Section \[subsec:MILC\], we first consider the Renyi entanglement entropy for a special setup with a Lorentz boosted interval and an unboosted interval, as shown in Figure \[fig:AB\]. In this case, the light cone limit naturally appears in the calculation. As a result, we found a Renyi phase transition as the replica number was varied. This implies that when we try to evaluate entanglement using the Renyi entropy, we must use the limit $n \to 1$ carefully. Note that the transition point $n_*$ is always above $n=1$; therefore, this Renyi phase transition does not contradict with the derivation of the holographic entanglement entropy formula [@Lewkowycz2013]. In Section \[subsec:DMI\], we consider the Renyi entanglement entropy for doubled CFTs, which was introduced by [@Hartman2013; @Asplund2015a]. In this setup, the light cone limit non-trivially appears in the calculation, but is similar to that in Section \[subsec:MILC\]. Therefore, the transition at $n_*$ can also be found in its Renyi entanglement entropy. We also predict that maximal scrambling is characterized by the lowest bound on the light cone singularity of a particular correlator; on the other hand, the quasi particle picture comes from the upper bound. It means that the light cone limit gives us information about scrambling. In section \[subsec:2ndREE\], we evaluate the dynamics of the 2nd Renyi entropy after a local quench. The result leads to the prediction that in a unitary CFT with $c>1$ and no extra currents, the heavier the operator used to create a local quench, the larger the 2nd Renyi entropy becomes; however, if its dimension exceeds the value ${\frac}{c-1}{32}$, the 2nd Renyi entropy is saturated. This might be related to instability and thermalisation. (See also [@Caputa2014; @He2014; @Numasawa2016; @Caputa2017; @He2017; @Guo2018], which reveled the growth of the entanglement entropy after a local quench in other setups.)\
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Using the Zamolodchikov recursion formula, we can obtain the upper bound of the Regge limit singularity in the Virasoro blocks using the light cone singularity. In particular, above the BTZ threshold $\a_A+\a_B>\a_{\text{BTZ}}$, the asymptotic form of the Regge limit Virasoro blocks can be given by a universal formula. We will explain this in Section \[sec:Regge\]. From this result, we can predict, for example, the late time behaviour of out of time ordered correlators (OTOCs) [^2] and general $n$-th Renyi entropy after a local quench.\
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Besides these main contents, we give the light cone modular bootstrap in Appendix \[app:modular\] and also we will rewrite our light cone bootstrap in this context. We give the details of the Zamolodchikov recursion relation in Appendix \[app:recursion\].
At the end of this section, we would like to emphasize the interesting points of our results.
Beyond test mass limit
: In the light cone limit, we can calculate Virasoro conformal blocks beyond the test mass limit ${\frac}{h_L}{c}\ll1$. It might give a key to understanding dynamics of [*multiple*]{} deficit angle in AdS${}_3$.
BTZ thershold
: The light cone singularity undergoes a transition at the BTZ thershold. This suggests that this [*light cone transition*]{} captures information about holographic CFTs in some sense, like the Hawking-Page transition. (One interpretation is obtained by the light cone bootstrap.)
Liouville CFT & 2+1 Gravity
: Interestingly, our result from the bootstrap equation suggests that in a particular regime, 2+1 gravity is non-trivially related to Liouville CFT (see Figure \[fig:potential\]), which cannot be observed in the test mass limit. We expect that this gives new insights into the relation between Liouville CFT and 2+1 gravity (see the end of Appendix \[subapp:relation\]).
Light Cone Bootstrap
====================
Let us consider a unitary CFT with two primary operators $O_A$ and $O_B$ in the following. In general, the operators appearing in the OPE between $O_A$ and $O_B$ have large anomalous dimensions originating from its interactions. However, the crossing symmetry implies that the large spin limit simplifies the structure of the OPE. In $d\geq 3$ unitary CFTs, the large $l$ primary operators in the OPE have a twist $$\label{eq:twist3d}
\tau_n=\tau_A+\tau_B+2n \ \ \ \ \ \text{ for any } n\in{\mathbb{Z}}_{\geq 0}.$$ This is proved in [@Alday2007; @Fitzpatrick2013a; @Komargodski2013] using the light cone bootstrap.
The key point is the existence of a twist gap between the vacuum and minimal twist. The unitarity imposes the following bounds on a twist spectrum (except for vacuum): $$\begin{aligned}
\tau \geq\left\{
\begin{array}{ll}
{\frac}{d-2}{2} ,& \text{if } l=0 ,\\
d-2 ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ This leads to the separation of the identity conformal block from the other contributions in the bootstrap equation as the global blocks $g_{\tau,l}^{ij,kl}(u,v)$ have the singularity $u^{{\frac}{\tau}{2}} v^{{\frac}{1}{2}\D_{ij}}$ with $\D_{ij}=\D_i-\D_j$ at $u,v \to 0$. In particular, in the light cone limit, the identity provides the dominant contributions to one hand side ($t$-channel expansion) in the bootstrap equation as follows: $$u^{-{\frac}{1}{2}(\D_A+\D_B)} \underset{u\ll v\ll1}{\sim} v^{-{\frac}{1}{2}(\D_A+\D_B)} u^{-{\frac}{1}{2}\D_{12}}\sum_{\tau,l} P_{\tau,l}g_{\tau,l}(v,u),$$ where the sum is taken over the primary operators of twist $\tau=\D-l$ and spin $l$ in the OPE between $O_A$ and $O_B$, and parameters $u,v$ are the cross ratios defined by $$u={\left({\frac}{x_{12} x_{34}}{x_{24}x_{13}} \right)}^2, \ \ \ \ v={\left({\frac}{x_{14} x_{23}}{x_{24}x_{13}} \right)}^2,$$ with $x_{ij}=x_i-x_j$. In this equation, it can be observed that, to match the $v$-dependence of both sides, there must exist the operators with twist (\[eq:twist3d\]).
However, in 2D CFTs, the twist bound is given by $\tau\geq0$, which means that the identity contribution cannot be separated from the other contributions. Therefore, the above process does not work in 2D CFTs. Even though there are many contributions to the zero-twist part, we can incorporate the contributions into the Virasoro conformal blocks; thus, using the Virasoro algebra, we can investigate the large spin structure even in 2D CFTs. From this background, it is interesting to investigate what is predicted from the light cone bootstrap in 2D CFTs. We will discuss it in this section. [^3]
Before moving on to the light cone bootstrap, we will interpret this statement in terms of AdS. The operators at large $l$ correspond to states with large angular momentum in AdS, and thus two-particle states at large $l$ are orbiting a common centre with a large angular momentum. At this stage, it is naturally expected that at large $l$, these two particles are well separated, as the interactions between two objects become negligible at large angular momentum. It means that the anomalous dimension of the two-object state should vanish (see Figure \[fig:binding\]).
Lowest Twist Operator at Large $l$
----------------------------------
In 2D CFTs, the conformal bootstrap equation can be given in terms of Virasoro conformal blocks as $$\label{eq:bootstrap}
\sum_p C_{12p}C_{34p} {\mathcal{F}}^{21}_{34}(h_p|z)\overline{{\mathcal{F}}^{21}_{34}}(\bar{h}_p|\bar{z})=\sum_p C_{14p}C_{23p} {\mathcal{F}}^{23}_{14}(h_p|1-z)\overline{{\mathcal{F}}^{23}_{14}}(\bar{h}_p|1-\bar{z}),$$ where $C_{ijk}$ are OPE coefficients and ${\mathcal{F}}^{ij}_{kl}(h_p|z)$ are conformal blocks, which are usually expressed using the Feynman diagram as follows:
$${\mathcal{F}}^{ji}_{kl}(h_p|z) \equiv \parbox{\pcw}{\usebox{\boxpc}}.$$
We are interested in the light cone limit $z \ll 1-\bar{z} \ll 1$ of this equation. In the following, we assume that there are no additional continuous global symmetries apart from the Virasoro symmetry. Under this assumption, the light cone limit of the left hand side of (\[eq:bootstrap\]) can be approximated using the vacuum Virasoro block. As the global block expansion is more useful to illustrate the asymptotics of the right hand side in the light cone limit, we re-express the right hand side using global blocks. As a result, the bootstrap equation in the light cone limit reduces to $${\mathcal{F}}^{AA}_{BB}(0|z)\overline{F^{AA}_{BB} (0|\bar{z})}\simeq {\left(1-z \right)}^{-h_A-h_B} {\left(1-\bar{z} \right)}^{-\bar{h}_A-\bar{h}_B} \sum_{\tau,l}P_{\tau,l}g_{\tau,l}(z,\bar{z}),$$ where $g_{\tau,l}(z,\bar{z})$ is the global block and $P_{\tau,l}$ is the conformal block coefficient. In this limit, we are interested in the large $l$ global blocks, which are given by a simple approximated form as $$g_{\tau,l}(z,\bar{z}) \sim 2^{\tau+2l} (1-\bar{z})^{{\frac}{\tau}{2}} z^{{\frac}{1}{2}\D_{AB}}{\sqrt{{\frac}{l}{\pi}}}K_{\D_{AB}}(2l{\sqrt{z}}),$$ where $\D=h+\bar{h}$, $\D_{ij}=\D_i-\D_j$, and $K_\D (x)$ are modified Bessel functions (see more details in [@Fitzpatrick2013a]).
To proceed further, we have to determine the behaviour of the Virasoro blocks in the limit $\bar{z} \to 1$. Although no exact closed form of the Virasoro blocks is available, if we restrict ourselves to the heavy-light limit, the HHLL Virasoro blocks make it possible to study the bootstrap equation discussed in [@Fitzpatrick2014]. The HHLL Virasoro blocks in the limit $\bar{z} \to 1$ lead to $$\begin{aligned}
{\mathcal{F}}^{LL}_{HH}(0|z)\overline{F^{LL}_{HH} (0|\bar{z})}
&=(1-z)^{h_L(\d-1)}{\left({\frac}{1-(1-z)^\d}{\d} \right)}^{-2h_L} (1-\bar{z})^{\bar{h}_L(\bar{\d}-1)}{\left({\frac}{1-(1-\bar{z})^{\bar{\d}}}{\bar{\d}} \right)}^{-2\bar{h}_L} \\
&{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
z^{-2h_L}{\left(1-\bar{z} \right)}^{\bar{h}_L(\bar{\d}-1) },
\end{aligned}$$ where $\d={\sqrt{1-{\frac}{24}{c}h_H}}$ and $\bar{\d}={\sqrt{1-{\frac}{24}{c}\bar{h}_H}}$. For simplicity, we assume the external operators to be $h_i\geq\bar{h}_i$, and therefore $\tau_i=2 \bar{h_i}$. When comparing the $z$ dependence of the left- and right-hand sides, one can find that there must be an infinite number of large $l$ contributions on the right-hand side to reproduce the singularity $z^{-2h_L}$ on the left-hand side. Moreover, to reproduce the singularity ${\left(1-\bar{z} \right)}^{\bar{h}_L(\bar{\d}-1) }$ on the left-hand side, there must be a contribution from an infinite number of operators having increasing spin with $\tau \to \d \tau_L+\tau_H$. This is discussed in [@Fitzpatrick2014].
In the above, we restricted our investigation to the HHLL limit because it is extremely difficult to study the Virasoro blocks in general. Nevertheless, the light cone limit interestingly simplifies the structure of the Virasoro blocks. We can find the simplification in the large $c$ limit achieved using the recursion relation [@Kusuki2018; @Kusuki2018b] and monodromy method [@Kusuki2018a]. In fact, we can also evaluate the light cone limit of [*general*]{} Virasoro conformal blocks using the fusion matrix. Here, we will only present the results from the fusion matrix and present the detailed calculation in Appendix \[app:FM\], as the calculation is complicated.
In the following, we introduce notations usually found in Liouville CFTs. $$\label{eq:notations}
c=1+6Q^2, \ \ \ \ \ Q=b+{\frac}{1}{b}, \ \ \ \ \ h_i=\a_i(Q-\a_i).$$ We denote $\a_i$ as [*Liouville momenta*]{}. We have to mention that although we use the notations in Liouville CFTs, we never use relations that only hold in Liouville CFTs. The Liouville parameters are introduced for convenience. According to the result (\[eq:FMresult2\]) in Appendix \[app:FM\], the light cone singularity is given by
$$\label{eq:LCsingularity}
\begin{aligned}
{\mathcal{F}}^{AA}_{BB}(0|z)\overline{F^{AA}_{BB} (0|\bar{z})} {\xrightarrow[z \ll 1-\bar{z} \ll 1]{}} &\left\{
\begin{array}{ll}
z^{-2h_A}{\left(1-\bar{z} \right)}^{ -2\bar{\a}_A \bar{\a}_B } ,& \text{if} \ \ \ \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2} , \\
z^{-2h_A}{\left(1-\bar{z} \right)}^{{\frac}{c-1}{24} -\bar{h}_A-\bar{h}_B } ,& \text{ otherwise},\\
\end{array}
\right.\\
\end{aligned}$$
where $\a_i=Q{\frac}{1-{\sqrt{1-{\frac}{24}{c-1}h_i}}}{2}$ and $\bar{\a}_i=Q{\frac}{1-{\sqrt{1-{\frac}{24}{c-1}\bar{h}_i}}}{2}$. To reproduce this light cone singularity from the right-hand side of the bootstrap equation, we must always have the operator in the OPE with twist. $$\label{eq:tlowest}
\begin{aligned}
\tau_{\text{lowest}}&=\left\{
\begin{array}{ll}
\tau_A+\tau_B -4 \bar{\a}_A \bar{\a}_B ,& \text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2} ,\\
{\frac}{c-1}{12} ,& \text{otherwise },\\
\end{array}
\right.\\
\end{aligned}$$ where we can re-express the Liouville momenta $\bar{\a}_i$ using the twist $\tau_i$ as $\bar{\a}_i=Q{\frac}{1-{\sqrt{1-{\frac}{12}{c-1}\tau_i}}}{2}$. In particular, if we expand $\a_A$ in $\tau_{\text{lowest}}$ at small ${\frac}{h_A}{c}$, the result exactly matches $\tau_{\text{lowest}} \to \d \tau_A+\tau_B$. This twist $\tau_{\text{lowest}}$ gives the lower bound for the twist in the OPE at large $l$ because if there exists an operator with twist lower than $\tau_{\text{lowest}}$, the singularity arising from that twist is never reproduced by the left-hand side of the bootstrap equation.
An interesting point is that if the total Liouville momentum $\tau_A+\tau_B$ increases beyond the BTZ momentum threshold $\a_{\text{BTZ}}={\frac}{Q}{2}$ [^4], the lowest twist in the OPE is given by a universal value ${\frac}{c-1}{12}$. Moreover, this universal twist equals the BTZ mass threshold. In other words, the total twist gradually increases unless the total Liouville momentum exceeds the BTZ mass threshold, and the total twist is saturated by the value of that threshold if the total Liouville momentum increases beyond it. This behaviour of the total twist is shown in Figure \[fig:total\]. Here, one might face a contradiction to the thermalisation of the HHLL Virasoro blocks. It is well known that the HHLL vacuum block can be interpreted as a two-point function on a thermal background if the mass of the heavy particle exceeds the BTZ threshold $h_H>{\frac}{c}{24}$. To explore this, let us consider the expression (\[eq:LCsingularity\]). Considering the test mass limit ${\frac}{h_A}{c} \to 0$, the threshold $\bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2}$ can be approximated using an inequality $h_B<{\frac}{c}{24}$. It means that the saturation of the singularity occurs exactly at the BTZ threshold $h_B={\frac}{c}{24}$, as expected from the analysis of the HHLL Virasoro blocks. However, the singularity of the thermal correlator is considerably different from (\[eq:LCsingularity\]). That is, above the BTZ threshold, the HHLL vacuum Virasoro block leads to the singularity $$\begin{aligned}
{\mathcal{F}}^{LL}_{HH}(0|z)
&=(1-z)^{h_L(\d-1)}{\left({\frac}{1-(1-z)^\d}{\d} \right)}^{-2h_L} \\
&{\xrightarrow[z \to 1]{}}
{\left(1-z \right)}^{-h_L },
\end{aligned}$$ which is obviously different from the singularity in (\[eq:LCsingularity\]). This is perhaps due to the test mass limit ${\frac}{h_A}{c} \to 0$; here, the light cone limit could not commute with the test mass limit $h_L/c\to0$. It means that the back reaction of a probe actually results in a non-negligible universal interaction with a heavy particle in AdS${}_3$. Note that one can find the agreement of the transition points between HHLL blocks and light cone limit singularity, which might imply that the transition of the light cone singularity is also related to thermalisation, instability, or black hole formation. In fact, the assumptions $c \to \infty$ and no extra conserved current cause a CFT to be irrational, which is expected to be chaotic. Therefore, the assumptions might be appropriate for thermalisation to occur. It would be interesting to explore this issue further.
![The left figure shows the $h_A=h_B$ dependence of the total twist in the OPE between $O_A$ and $O_B$, and the right figure shows the $h_B$ dependence with $h_A \ll h_B$. From both figures, we can find saturation above $\a_A+\a_B={\frac}{Q}{2}$. In particular, for $h_A\ll h_B$, the transition (saturation) occurs at the BTZ mass threshold ${\frac}{c-1}{24}$. A similar phenomenon can be seen in the HHLL Virasoro blocks, known as [*thermalisation*]{}.[]{data-label="fig:total"}](total1.pdf "fig:"){width="7.0cm"} ![The left figure shows the $h_A=h_B$ dependence of the total twist in the OPE between $O_A$ and $O_B$, and the right figure shows the $h_B$ dependence with $h_A \ll h_B$. From both figures, we can find saturation above $\a_A+\a_B={\frac}{Q}{2}$. In particular, for $h_A\ll h_B$, the transition (saturation) occurs at the BTZ mass threshold ${\frac}{c-1}{24}$. A similar phenomenon can be seen in the HHLL Virasoro blocks, known as [*thermalisation*]{}.[]{data-label="fig:total"}](total2.pdf "fig:"){width="7.0cm"}
We want to emphasize that our result (\[eq:tlowest\]) holds not only for large $c$ CFTs but also for any unitary CFT with $c>1$ and no extra conserved currents. Note that the absence of the first condition $c>1$ destroys the light cone OPE structure (see the final paragraph of Appendix \[subapp:FM\]), and the second condition is used to approximate the light cone limit of a correlator using the vacuum Virasoro block. Therefore, rational CFTs are an exception in our statement.
Finally, we would like to mention that the fact that the lowest twist is saturated by ${\frac}{c-1}{12}$ is consistent with the result from the light cone modular bootstrap. By using the modular symmetry, it is easy to show that there must be an infinite number of large spin primaries with twist accumulating to ${\frac}{c-1}{12}$ in any unitary CFT with $c>1$ and without extra currents (see Appendix \[app:modular\] or [@Collier2016]). However, our statement (\[eq:tlowest\]) is a little different because our result implies that there must be an infinite number of operators with twist ${\frac}{c-1}{12}$ at $ l \to \infty$ not only in the CFT, but also in the [*OPE*]{}. In this sense, our conclusion is more interesting than the result from the light cone modular bootstrap on a torus. Possibly, we could show that the operator with twist ${\frac}{c-1}{12}$ predicted from the modular bootstrap actually comes from the OPE between two operators with heavy total Liouville momentum ($\bar{\a}_A+\bar{\a}_B>{\frac}{Q}{2}$).
Large $l$ Spectrum of Twist {#subsec:spectrum}
---------------------------
In the previous section, we derived the lowest twist at large $l$, but we would also like to determine the twist spectrum. For this purpose, we need not only the leading conformal blocks but also the sub-leading contributions to the blocks. We omit the details of the calculation and only present the results; interested readers can refer to Appendix \[app:FM\], particularly \[subapp:sub\] .
As the simplest example, we first consider the heavy-light limit. In the limit $c \to \infty$ with ${\frac}{h_H}{c} ,h_L$ fixed, the light cone asymptotics of the conformal blocks is given as (\[eq:subHHLL\]) $${\mathcal{F}}^{LL}_{HH}(h_p|z){\xrightarrow[z\to1]{}} \sum_{n \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_n (1-z)^{\d(h_L+n)-h_L},$$ where ${\mathcal{P}}_n$ are some constants. Therefore, the left-hand side of the bootstrap equation is $${\mathcal{F}}^{LL}_{HH}(0|z)\overline{F^{LL}_{HH} (0|\bar{z})}
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
z^{-2h_L}{\left(\sum_{n \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_n (1-\bar{z})^{\d(\bar{h}_L+n)-\bar{h}_L} \right)}.$$ To reproduce the $\bar{z}$ dependence of each term on the left-hand side, there must be at least one primary operator with twist. $$\tau_n=\d(\tau_L+2n)+\tau_H \ \ \ \ \ \text{for any } n\in{\mathbb{Z}}_{\geq 0}.$$ Let us consider the case when the condition ${\frac}{h_L}{c} \ll 1$ is relaxed. In this case, the light cone limit of a four-point function is given as follows. If $\bar{\a}_A \bar{\a}_B<{\frac}{Q}{2}$, $${\mathcal{F}}^{AA}_{BB}(0|z)\overline{F^{AA}_{BB} (0|\bar{z})}
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
z^{-2h_A}{\left(\sum_{n \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_n (1-\bar{z})^{-2\bar{\a}_A \bar{\a}_B +n{\left(1-{\frac}{2}{Q}(\bar{\a}_A+\bar{\a}_B) \right)}} \right)};$$ otherwise, $$\label{eq:c/24}
{\mathcal{F}}^{AA}_{BB}(0|z)\overline{F^{AA}_{BB} (0|\bar{z})}
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
z^{-2h_A}(1-\bar{z})^{{\frac}{c-1}{24}-\bar{h}_A-\bar{h}_B}.$$ As a result, the light cone bootstrap imposes a condition that there must exist operators with twist. [^5]
$$\label{eq:generaltn}
\begin{aligned}
\tau_n&=\left\{
\begin{array}{ll}
\tau_A+\tau_B -4\bar{\a}_A \bar{\a}_B +2n{\left(1-{\frac}{2}{Q}(\bar{\a}_A+\bar{\a}_B) \right)} \ \ \ (\text{any } n \in {\mathbb{Z}}_{\geq0}) ,& \text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2} ,\\
{\frac}{c-1}{12} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$
Note that an infinite tower comes from an infinite number of particular poles in the fusion matrix (see Appendix \[subapp:sub\]). This infiniteness relies on the assumption $c \to \infty$; therefore, when we exceed the semiclassical limit, there is only a finite tower of operators in the OPE, unlike (\[eq:generaltn\]) in large $c$ CFTs.
In fact, for general unitary CFTs with $c>1$, the light cone limit of a four- point function can be approximated as follows: $${\mathcal{F}}^{AA}_{BB}(0|z)\overline{F^{AA}_{BB} (0|\bar{z})}
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
z^{-2h_A} {\left(\sum_{\substack{m,n \in {\mathbb{Z}}_{\geq0}\\ \text{where} \\ \bar{\a}_A+\bar{\a}_B+Q_{m,n}<{\frac}{Q}{2} }}
{\mathcal{P}}_{m,n} (1-\bar{z})^{-2\bar{\a}_A \bar{\a}_B+\m_{m,n}} \right)}, \ \ \ \ \ \text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2},$$ where the correction $\m_{m,n}$ to the highest singular power law is defined using the Liouville notation (\[eq:notations\]) and the notations $\w_{m,n} \equiv {\frac}{2}{Q}(\bar{\a}_A+\bar{\a}_B+Q_{m,n})<1$, $Q_{m,n}=mb+{\frac}{n}{b}$ as $$\m_{m,n}={\left(mb+{\frac}{n}{b} \right)} {\left((1-\w_{m,n}+m)b+{\frac}{1-\w_{m,n}+n}{b} \right)}.$$ On the other hand, if $\bar{\a}_A+\bar{\a}_B>{\frac}{Q}{2}$, the light cone asymptotics for a correlator in general unitary CFTs are the same as those in (\[eq:c/24\]). The details of the calculation are given before equation (\[eq:fullAABB\]). We can thus conclude that in any 2D CFT with $c>1$ and no extra conserved currents, there must be operators with twist $$\label{eq:general twist}
\begin{aligned}
\tau_{m,n}&=\left\{
\begin{array}{ll}
\tau_A+\tau_B -4\bar{\a}_A \bar{\a}_B +2\m_{m,n} \ \ \ (\text{any } m,n \in {\mathbb{Z}}_{\geq0} \text{ s.t. } \bar{\a}_A+\bar{\a}_B+Q_{m,n}<{\frac}{Q}{2} ) ,& \text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2} ,\\
{\frac}{c-1}{12} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ One can find that $\m_{m,n}$ differ by non-integer from one another, which means that each twist belongs to a different conformal family. We would like to emphasize that this large spin twist spectrum can be described by the fusion rule of [*Liouville CFT*]{}, which is explained in Appendix \[subapp:relation\].
This twist spectrum is considerably different from the prediction using the AdS interpretation, $\tau_n=\tau_A+\tau_B+2n$. It means that the interactions between $O_A$ and $O_B$ never vanish even at large $l$. In other words, there is a universal anomalous dimension only in 2D CFTs. The reason is that gravitational interactions in AdS${}_3$ create a deficit angle, and their effect can be detected even at infinite separation. When focusing on the test mass limit ${\frac}{h_A}{c} \to 0$, this long-distance effect can be interpreted clearly in AdS [@Fitzpatrick2014]. That is, the existence of a deficit angle in AdS${}_3$ leads to an energy shift $$\D_A \to \D_A {\sqrt{1-8G_N M}} =\a \D_A,$$ where we used the dictionary, $\D_B=M$ and $c={\frac}{3}{2G_N}$. This energy shift is universal and never vanishes even at large angular momentum; therefore, it is natural that the corresponding twist in CFT${}_2$ is also shifted by $$\tau_{\text{total}}=\tau_A+\tau_B \to \a\tau_A+\tau_B.$$ For general unitary CFTs with $c>1$, we can conclude that there is a universal binding energy between two objects at large $l$. $$\label{eq:Ebound}
\begin{aligned}
E_{\text{binding}}&{\xrightarrow[l \to \infty]{}}\left\{
\begin{array}{ll}
-4\bar{\a}_A \bar{\a}_B ,& \text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2} ,\\
{\frac}{c-1}{12}-\tau_A-\tau_B ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ This universal binding energy only exists in AdS${}_3$ and vanishes in AdS${}_{d\geq4}$, as shown in Figure \[fig:binding\]. This form of the binding energy below the BTZ threshold could be natural to some extent because similar to the gravity theory or electromagnetic theory, this form is characterized by the product of two Liouville momenta, where the Liouville momentum behaves like a [*charge*]{} (see Figure \[fig:potential\]). It would be interesting to see how this binding energy is obtained from the calculation in AdS${}_3$, determine why the Liouville momentum essentially appears in the expression of the binding energy and understand the physical meaning of this binding energy in general unitary CFTs with $c>1$. Further, we intend to understand what leads to saturation of the binding energy. We expect that this saturation is related to thermalisation and black hole formation.
![This figure shows the implications of the light cone bootstrap on the nature of AdS. In AdS${}_{d\geq 4}$, the interactions between two objects become negligible at large angular momentum. On the other hand, in AdS${}_3$, there exists a universal binding energy $-4\bar{\a}_A\bar{\a}_B$ even at large angular momentum. If the total Liouville momentum is above the BTZ threshold $\a_{\text{BTZ}}$, the binding energy is given by ${\frac}{c-1}{12}-\tau_A-\tau_B$. []{data-label="fig:binding"}](binding.pdf){width="10.0cm"}
![Potential form in AdS$_3$ between two particles is similar to the form of Coulomb potential. Moreover, this form with Liouville momenta implies that the dynamics of the multiple deficit angle at large angular momentum is completely captured by Liouville CFT.[]{data-label="fig:potential"}](potential.pdf){width="12.0cm"}
Entanglement and Light Cone Limit
=================================
The light cone limit is useful in understanding entanglement, which is discussed in this section.
One useful measure of entanglement is entanglement entropy, which is defined as $$S_A=-\tr \rho_A \log \rho_A,$$ where $\rho_A$ is a reduced density matrix for a subsystem $A$, obtained by tracing out its complement. We will also discuss its generalization, called the Renyi entropy, which is defined as $$S_A^{(n)}={\frac}{1}{1-n} \log \tr \rho_A^n,$$ and the limit $n\to1$ of the Renyi entropy defines the entanglement entropy $S_A$. Here, to characterize entanglement precisely, we measure the Renyi entropy for two disconnected intervals $A \cup B$. In particular, we consider the Renyi entropy in the light cone limit in Section \[subsec:MILC\]. This has not been extensively studied as the explicit form of a four-point function in the light cone limit was unknown until our earlier studies [@Kusuki2018a; @Kusuki2018; @Kusuki2018b]. [^6]
The entanglement entropy for two disconnected intervals, or equivalently, the mutual information, is also useful to probe how entanglement spreads. The entanglement entropy for two disconnected intervals $S_{A\cup B}$ does not measure the entanglement of $A$ with $B$ but measures the entanglement of $A \cup B$ with its complement. Nevertheless, we can determine the entanglement of $A$ with $B$ from $S_{A\cup B}$ because a strong entanglement between $A$ and $B$ means that $A \cup B$ cannot be entangled with the rest; thus, one can find that $S_{A\cup B}$ is small if $A$ is highly entangled with $B$.
For example, Calabrese and Cardy studied the entanglement entropy for disconnected intervals to conclude that, after a global quench, entanglement spreads as if correlations were carried by free quasiparticles [@Calabrese2005a; @Calabrese2009a]; this finding was refined in [@Asplund2015a] as the quasiparticle picture is only valid under some assumptions.
Interestingly, the light cone limit also appears in the study on the dynamics of entanglement as in [@Asplund2015a]. Based on this fact, we will discuss the dynamics of entanglement after a global quench in Section \[subsec:DMI\], \[subsec:DMI2\] and after a local quench in Section \[subsec:2ndREE\].
Mutual Information in Light Cone Limit {#subsec:MILC}
--------------------------------------
In this section, we consider the light cone limit of the Renyi entanglement entropy $S^{(n)}_{A\cup B}$ for two intervals $A$ and $B$, or the Renyi mutual information, which is given by \[eq:MI\] I\^[(n)]{}(A : B)=S\^[(n)]{}\_[A]{}+S\^[(n)]{}\_[B]{}-S\^[(n)]{}\_[AB]{} . Let us choose $A$ and $B$ to be $[x_1,x_2]$ and $[x_3,x_4]$ in the 2D Lorentzian spacetime ${\bf R}^{1,1}$. Note that the Lorentzian time $t$ and space $x$ are related to the complex coordinate as $z=x+it_E=x-t$. In terms of the complex coordinate, the intervals are specified by the twist operators at $$\begin{aligned}
z_1 &= \bar{z}_1 = x_1, \ \ z_3 = \bar{z}_3 = x_3, \ \ z_4 = \bar{z}_4 = x_4, \notag \\
z_2 &= x_2 -(-t) , \ \ \bar{z}_2 = x_2 + (-t),\end{aligned}$$ where $x_1 < x_2 < x_3 < x_4$ and $t> 0$. Here, we consider a simple case where only $x_2$ goes away from the $t = 0$ slice. The cross ratios are given by $$\begin{aligned}
z
&= \frac{z_{12} z_{34}}{z_{13} z_{24}}
= \frac{(x_{21} - t ) x_{43}}{ x_{31} (x_{42} + t )}
=\frac{x_{21}^{-} x_{43}^{-}}{x_{31}^{-} x_{42}^{-} } , \notag \\
\bar{z}
&= \frac{\bar{z}_{12} \bar{z}_{34}}{\bar{z}_{13} \bar{z}_{24}}
= \frac{(x_{21} + t ) x_{43}}{ x_{31} (x_{42} - t)}
=1-{\frac}{x_{41}^+ x_{32}^+}{x_{31}^+ x_{42}^+},\end{aligned}$$ where the light-cone coordinate $x_j^\pm=t_j \pm x_j$. In this setup, the light cone limit $z \ll 1-\bar{z} \ll 1$ can be interpreted physically as the limit where the interval $A$ is infinitely boosted; the Cauchy surface containing the intervals becomes singular (see Figure \[fig:AB\]). That is, the light cone limit corresponds to x\^[-]{}\_[21]{} x\^[+]{}\_[32]{} 1.
![Setup of mutual information $I(A,B)$ between two intervals $A$ and $B$ when $A$ is infinitely boosted.[]{data-label="fig:AB"}](AB.pdf){width="8cm"}
For simplicity, we assume the interval $B$ to be large ($x_4 \to \infty$ or $x_4 \gg x_3$). In this case, the cross ratios $z$, $\bar{z}$ are quite simple. $$\begin{aligned}
z=\frac{ x_{21}^- }{ x_{31} } , \ \
\bar{z} =1- \frac{ x_{32}^+ }{ x_{31} } .\end{aligned}$$ Thus, if we boost $A$ to become almost null, the $n$-th Renyi mutual information (\[eq:MI\]) can be computed using the four-point function of twist operators $$\begin{aligned}
I^{(n)}(A:B) &= \frac{1}{n-1} \log \left[ {\frac}{ \braket{\sigma_n(x_4) \bar{\sigma}_n(x_3)\sigma_n(x_2) \bar{\sigma}_n(x_1)} }{\braket{\sigma_n(x_4) \bar{\sigma}_n(x_3)} \braket{\sigma_n(x_2) \bar{\sigma}_n(x_1) }} \right]\\
&= \frac{1}{n-1} \log \left[ |z|^{4 h_{n}} G (z, \bar{z}) \right],
\end{aligned}$$ where $ G (z, \bar{z}) =\braket{\sigma_n(\infty) \bar{\sigma}_n(1)\sigma_n(z,\bar{z}) \bar{\sigma}_n(0)} $. The conformal dimension of the twist operator can be written as $h_n =\frac{c}{24} {\left(n-{\frac}{1}{n} \right)}$; therefore, the $n$-th Renyi mutual information is represented as $$\label{eq:generalMI}
\begin{aligned}
I^{(n)}(A:B) = \frac{c}{12}{\left(1+{\frac}{1}{n} \right)} \log {\frac}{x_{31}}{x_{32}^+} + {\frac}{1}{n-1}\log {\Biggl[ {\left|z\right|}^{4h_n} {\left|1-z\right|}^{4h_n} G(z, \bar{z}) \Biggr]}.
\end{aligned}$$
The $n$-th Renyi mutual information in CFTs defined by a complex scalar boson compactified on a torus is calculated in [@Caputa2017]. When the radius of a torus $\eta={\frac}{p}{q}$ is rational, I\^[(n)]{}(A,B)= [(1+ )]{} [( )]{} -(2pq) . We expect that, in any rational CFT, this could be generalized into the following form: I\^[(n)]{}(A,B)= [(1+ )]{} [( )]{} -d\_[tot]{}, where $d_{tot}=1/s_{00}$ is the total quantum dimension of the (seed) CFT. When $\eta$ is irrational, we obtain the double logarithmic divergent term. I\^[(n)]{}(A:B) = [(1+ )]{} [( )]{} - [( )]{} - + (2).
In general, the function $G(z, \bar{z})$ is nontrivial, but we can approximate it in the light cone limit as [^7] $$\label{eq:approximation}
G(z, \bar{z}) {\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n}(0|z)\overline{{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})},$$ where the conformal blocks are defined in a CFT with central charge $nc$. (not $c$ !) Therefore, we can apply our light cone limit conformal blocks for calculating the $n$-th Renyi mutual information. For simplicity, we first assume a large $c$ limit. [^8] The vacuum block with twist operators is given as $$\begin{aligned}
{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n}(0|z)\overline{{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})}
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}} &\left\{
\begin{array}{ll}
z^{-2h_n}{\left(1-\bar{z} \right)}^{ -{\frac}{c}{12n}{\left(1-n \right)}^2 } ,& \text{if} \ \ \ h_n<{\frac}{nc}{32} , \\
z^{-2h_n}{\left(1-\bar{z} \right)}^{{\frac}{c}{24n}{\left(2-n^2 \right)} } ,& \text{ otherwise}.\\
\end{array}
\right.\\
\end{aligned}$$ Inserting this vacuum block into the function $G(z,\bar{z})$ in (\[eq:generalMI\]) leads to the following result: $$\label{eq:MIresult}
\begin{aligned}
I^{(n)}(A:B)
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}} &\left\{
\begin{array}{ll}
\frac{c}{12} {\left(1-\frac{1}{n} \right)} \log {\left( \frac{x_{31}}{x_{32}^+} \right)} ,& \text{if} \ \ \ n<n_* \equiv2 , \\
\frac{c}{12} {\left(1-\frac{1}{n} \right)} \log {\left( \frac{x_{31}}{x_{32}^+} \right)}-{\frac}{c}{24}{\frac}{(n-2)^2}{n(n-1)}\log {\left( \frac{x_{31}}{x_{32}^+} \right)} ,& \text{ otherwise}.\\
\end{array}
\right.\\
\end{aligned}$$ In particular, the mutual information is given by taking the limit $n \to 1$ as I(A:B) 0. This result is consistent with the holographic calculation as in [@Kusuki2017]. We intend to emphasize that the additional logarithmic divergent term appears in the $n$-th Renyi mutual information for $n>n_*$. In many cases, to calculate the entanglement entropy (for example, the replica method), we implicitly assume that the Renyi entropy is analytic in $n$. However, we find an exception of this assumption in the light cone limit. Therefore, we have to consider this exception if we use the replica method to evaluate the entanglement entropy. We emphasize that this assumption does not contradict with the derivation of the Ryu–Takayanagi formula in [@Lewkowycz2013], as our result for the Renyi entropy is analytic in the vicinity of $n=1$.
We expect that this phase transition arises from only the light cone limit $z \ll 1-\bar{z} \ll 1$ (and $c>1$) and not the large $c$ limit. Following (\[eq:FMresult1\]) (or (\[eq:FMresult2\])) in Appendix \[subapp:FM\], we can immediately obtain the Renyi mutual information for CFTs with finite $c$ as $$I^{(n)}(A:B)
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
{\frac}{c}{12}{\left({\left(1+{\frac}{1}{n} \right)}-{\frac}{s_n}{n-1} \right)} \log {\left( \frac{x_{31}}{x_{32}^+} \right)},$$ and the function $s_n$ is given by $$\begin{aligned}
s_n& = \left\{
\begin{array}{ll}
2\a_n(Q-2\a_n) ,& \text{if } 2\a_n<{\frac}{Q}{2}\ ,\\
{\frac}{Q^2}{4} ,& \text{otherwise } ,\\
\end{array}
\right.\\
\end{aligned}$$ where $nc=1+6Q^2$ and $\a_n={\frac}{Q}{2}{\left(1-{\sqrt{1-{\frac}{c}{nc-1}{\left(n-{\frac}{1}{n} \right)}}} \right)}$, which satisfies $h_n=\a_n(Q-\a_n)$. The transition point for general $c$ is given by a more complicated form than $n_*=2$ in (\[eq:DMI\]) as follows: $$n_*={\frac}{3}{2c} {\left({\sqrt{1+{\frac}{16}{9}c^2}}-1 \right)},$$ which satisfies $n_*{\xrightarrow[c \to \infty]{}} 2 $ as expected. The $c$ dependence of $n_*$ is shown in Figure \[fig:point\].
![$c$ dependence of $n_*$. The classical limit $c \to \infty$ matches $n_*=2$.[]{data-label="fig:point"}](point.pdf){width="8.0cm"}
This shows that the transition point is located between $1<n_*<2$. Therefore, deduction of entanglement from the Renyi mutual information has to be done carefully.
In particular, we obtain the limit $n \to 1$ of the Renyi mutual information as $$I^{(n)}(A:B){\xrightarrow[n \to 1]{}} {\frac}{c^2}{12} {\frac}{n-1}{c-1}.$$ This suggests that in CFTs with $c>1$ (and no conserved primary currents), the mutual information in the light cone limit vanishes as in holographic CFTs. On the other hand, if we consider a CFT with $c=1$, the mutual information becomes ill-defined. This is natural because the equation (\[eq:FMresult1\]) (or (\[eq:FMresult2\])) is valid only if $c>1$, as mentioned in Appendix \[subapp:FM\]. We attribute this to the same reason why the quasiparticle picture breaks down if we assume no extended symmetry algebra and $c>1$, as explained in [@Asplund2015a] (see also Section \[subsec:DMI\], \[subsec:DMI2\]). We can also find similar Renyi phase transitions as the replica number $n$ varies in other situations [@Metlitski2009; @Belin2013; @Belin2015; @Belin2017; @Dong2018].
![The left figure shows the $n$ dependence of the coefficient of the Renyi mutual information (\[eq:MIresult\]). It can be observed that the $n \to \infty$ limit of the Renyi mutual information approaches $I^{(2)}(A:B)$. The right figure shows the $n$ dependence of the coefficient of $\del^2_n I^{(n)}(A:B)$. One can see that $\del^2_n I^{(n)}(A:B)$ becomes discontinuous at $n=n_*$.[]{data-label="fig:MIplot"}](MItrans1.pdf "fig:"){width="7.0cm"} ![The left figure shows the $n$ dependence of the coefficient of the Renyi mutual information (\[eq:MIresult\]). It can be observed that the $n \to \infty$ limit of the Renyi mutual information approaches $I^{(2)}(A:B)$. The right figure shows the $n$ dependence of the coefficient of $\del^2_n I^{(n)}(A:B)$. One can see that $\del^2_n I^{(n)}(A:B)$ becomes discontinuous at $n=n_*$.[]{data-label="fig:MIplot"}](MItrans2.pdf "fig:"){width="7.0cm"}
It would be interesting to point out that (\[eq:MIresult\]) leads to the fact that the $(n>2)$-th Renyi mutual information in the light cone limit is bounded by the 2nd Renyi mutual information as follows: $$I^{(2)}(A:B)<I^{(n)}(A:B) \ \ \ \ \text{if} \ \ n>2,$$ and in particular, [^9] $$\lim_{n \to \infty} I^{(n)}(A:B) = I^{(2)}(A:B).$$ These properties are depicted in Figure \[fig:MIplot\].
Dynamics of Renyi Mutual Information in Large $c$ CFTs {#subsec:DMI}
------------------------------------------------------
The dynamics of quantum information has attracted the attention of many research communities. In this sense, we are also interested in the propagation of entanglement. Fortunately, as discussed in [@Asplund2015a], entanglement memory could be characterised using light cone singularity; therefore, we might be able to study entanglement memory using our conformal blocks in the light cone limit. In this section, we explain this in detail and apply our conformal blocks to the calculation of the entanglement entropy.
For realizing the objective stated above, we consider the Renyi entropy in a doubled CFT. That is, we will consider the thermofield double state in the doubled system, $$\ket{\text{TFD}}=\sum_n{\mathrm{e}^{-{\frac}{\b}{2} H}} \ket{n}_1 \ket{n}_2,$$ as an entangled state. Two states $\ket{n}_1$ and $\ket{n}_2$ respectively exist in CFT${}_1$ and CFT${}_2$. We label the coordinates of each CFT as $(t_1,x_1)$ and $(t_2,x_2)$ and consider two disconnected intervals (see the left of Figure \[fig:TFD\]) $$A=[0,L]_1 \cup[D+L,D+2L]_2.$$ Let us choose the total Hamiltonian acting on the doubled CFT as $$H_{\text{tot}}=H_1+H_2.$$ Thus, the thermofield-double state (TFD) has a non-trivial time dependence.
The Renyi entropy in a doubled CFT can be given by a four-point function with twist operators on a thermal cylinder of periodicity $\beta$. The twist operators are put on the endpoints of $A$ with a shift $i{\frac}{\beta}{2}$ for operators in two different copies of the CFT. The time dependence is obtained by considering the analytic continuation $t \to it$ of the insertion points of the twist operators (see the right of Figure \[fig:TFD\]),
$${\left({\frac}{2\pi}{\beta} \right)}^{8h_n} {\left|w_1 w_2 w_3 w_4\right|}^{2h_n}\braket{\sigma_n(w_1,\bar{w}_1) \bar{\sigma}_n(w_2,\bar{w}_2) \sigma_n(w_3,\bar{w}_3) \bar{\sigma}_n(w_4,\bar{w}_4) },$$
where the insertion points are given by $$\begin{aligned}
w_1 &= {\mathrm{e}^{\frac{2\pi}{\beta}(-t+i\beta/4)}} , & \bw_1 &= {\mathrm{e}^{\frac{2\pi}{\beta}(t-i\beta/4)}},\notag\\
w_2 &= {\mathrm{e}^{\frac{2\pi}{\beta}(L-t+i\beta/4)}} , & \bw_2 &= {\mathrm{e}^{\frac{2\pi}{\beta}(L + t - i\beta/4)}},\\
w_3 &= {\mathrm{e}^{\frac{2\pi}{\beta}(D+2L+t-i\beta/4)}} , & \bw_3 &= {\mathrm{e}^{\frac{2\pi}{\beta}(D+2L-t+i\beta/4)}},\notag\\
w_4 &= {\mathrm{e}^{\frac{2\pi}{\beta}(D+L+t-i\beta/4)}} , & \bw_4 &= {\mathrm{e}^{\frac{2\pi}{\beta}(D+L-t+i\beta/4)}}\notag \ .\end{aligned}$$
![The left figure shows two intervals in the doubled CFT. The entanglement entropy in this setup can be given by a four-point function with twist operators on a thermal cylinder of periodicity $\beta$, as sketched in the right figure.[]{data-label="fig:TFD"}](TFD1.pdf "fig:"){width="7.0cm"} ![The left figure shows two intervals in the doubled CFT. The entanglement entropy in this setup can be given by a four-point function with twist operators on a thermal cylinder of periodicity $\beta$, as sketched in the right figure.[]{data-label="fig:TFD"}](TFD2.pdf "fig:"){width="7.0cm"}
The Renyi mutual information can be obtained in a simpler manner as follows: $$\begin{aligned}
I^{(n)}_A &= \frac{1}{n-1} \log \left[ {\frac}{\braket{\sigma_n(w_1,\bar{w}_1) \bar{\sigma}_n(w_2,\bar{w}_2) \sigma_n(w_3,\bar{w}_3) \bar{\sigma}_n(w_4,\bar{w}_4) } }{\braket{\sigma_n(w_1,\bar{w}_1) \bar{\sigma}_n(w_2,\bar{w}_2)}\braket{ \sigma_n(w_3,\bar{w}_3) \bar{\sigma}_n(w_4,\bar{w}_4) }} \right]\\
&= \frac{1}{n-1} \log \left[ |z|^{4 h_{n}} G (z, \bar{z}) \right],
\end{aligned}$$ where $ G (z, \bar{z}) =\braket{\sigma_n(\infty) \bar{\sigma}_n(1)\sigma_n(z,\bar{z}) \bar{\sigma}_n(0)} $ and $z={\frac}{w_{12}w_{34}}{w_{13} w_{24}}$.
In general, a four-point function non-trivially depends on the CFT data, and its explicit form is not known, except for a few CFTs. Therefore, to proceed further, we focus on the high-temperature limit $\b \to 0$ in the following, which simplifies the four-point function as it corresponds to some OPE limits. That is, the cross ratio is given by $$\begin{aligned}
z &\simeq {\mathrm{e}^{-{\frac}{2\pi}{\b}(D+2t)}} {\xrightarrow[\b \to 0]{}} 0,\\
\bar{z} &\simeq {\mathrm{e}^{-{\frac}{2\pi}{\b} {\left( D+2L+t -\text{max}(D+2L-t,t)-\text{max}(D,2t) \right)}}} {\xrightarrow[\b \to 0]{}} 0 \ \text{or} \ 1.
\end{aligned}$$ Outside the range ${\frac}{D}{2}<t<{\frac}{D+2L}{2}$, the anti-holomorphic cross ratio approaches $0$; thus, the function $G(z,\bar{z})$ becomes $$G(z,\bar{z}) {\xrightarrow[\b \to 0]{}} {\left|z\right|}^{-4h_n}.$$ As a result, we obtain the Renyi mutual information as $$I_A {\xrightarrow[\b \to 0]{}} 0.$$ On the other hand, in the range ${\frac}{D}{2}<t<{\frac}{D+2L}{2}$, the anti-holomorphic cross ratio is given by $$1-\bar{z}\simeq {\mathrm{e}^{-{\frac}{2\pi}{\b}\text{min}(D+2L-2t,2t-D)}} {\xrightarrow[\b\to0]{}}0.$$ In particular, this cross ratio satisfies $z \ll 1-\bar{z} \ll 1$, and therefore, this limit corresponds to the light cone limit. Thus, the calculation of the Renyi mutual information in the doubled CFT reduces to the same form as given in Section \[subsec:MILC\]. The function $G(z,\bar{z})$ is approximated as $$\begin{aligned}
G(z,\bar{z})
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}} &\left\{
\begin{array}{ll}
z^{-2h_n}{\left(1-\bar{z} \right)}^{ -{\frac}{c}{12n}{\left(1-n \right)}^2 } ,& \text{if} \ \ \ h_n<{\frac}{nc}{32} , \\
z^{-2h_n}{\left(1-\bar{z} \right)}^{{\frac}{c}{24n}{\left(2-n^2 \right)} } ,& \text{ otherwise},\\
\end{array}
\right.\\
\end{aligned}$$ and, therefore, the Renyi mutual information is $$\label{eq:DMI}
\begin{aligned}
I^{(n)}_A
{\xrightarrow[z \ll 1-\bar{z} \ll 1]{}} &\left\{
\begin{array}{ll}
\frac{c}{12} {\left(1-\frac{1}{n} \right)} {\frac}{2\pi}{\b}\text{min}(D+2L-2t,2t-D) ,& \text{if} \ \ \ n<n_* \equiv2 , \\
{\Biggl[\frac{c}{12} {\left(1-\frac{1}{n} \right)} -{\frac}{c}{24}{\frac}{(n-2)^2}{n(n-1)} \Biggr]}{\frac}{2\pi}{\b}\text{min}(D+2L-2t,2t-D) ,& \text{ otherwise}.\\
\end{array}
\right.\\
\end{aligned}$$ As a result, one can again see the Renyi phase transition as the replica number is varied. Therefore, one has to take care when trying to predict the behaviour of the entanglement entropy using the Renyi entropy.
We have to mention that from (\[eq:DMI\]), the mutual information is obtained by taking the limit $n \to 1$ as $$I_A=0.$$ Here, the mutual information vanishes for all times. It means that entanglement scrambles maximally, which contradicts with the quasiparticle behaviour shown in, for example, rational CFTs.
Renyi Mutual Information beyond Large $c$ {#subsec:DMI2}
-----------------------------------------
As mentioned in Section \[subsec:MILC\], we can generalize the calculation of the Renyi mutual information to general $c$ under some assumptions. This is extensively discussed in this section.
For simplicity, we again assume that there are no extra currents. Even in such a case, orbifoldisation leads to the ${\mathbb{Z}}_n$ current; therefore, we should approximate a correlator with twist operators, instead of (\[eq:approximation\]), as follows: $$G(z, \bar{z}) {\xrightarrow[z \ll 1-\bar{z} \ll 1]{}}
{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n}(0|z)\overline{{{\mathcal{F}}^{\text{Vir}^n/{\mathbb{Z}}_n}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})},$$ where ${\mathcal{F}}^{\text{Vir}^n/{\mathbb{Z}}_n}$ is the conformal block defined by current algebra $\text{Vir}^n/{\mathbb{Z}}_n$ and not just Virasoro algebra.
From the crossing symmetry, we can obtain $$G(z,\bar{z})=G(1-z,1-\bar{z}){\xrightarrow[\bar{z} \to 1]{}} (1-\bar{z})^{-2h_n}.$$ Therefore, we have the upper bound of the singularity of ${\mathcal{F}}^{\text{Vir}^n/{\mathbb{Z}}_n}$ as $$\lim_{\bar{z}\to1}\overline{{{\mathcal{F}}^{\text{Vir}^n/{\mathbb{Z}}_n}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})} \lsim(1-\bar{z})^{-2h_n},$$ In addition, we can also deduce the lower bound as $$\begin{aligned}
\lim_{\bar{z}\to1}\overline{{{\mathcal{F}}^{\text{Vir}^n/{\mathbb{Z}}_n}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})}
\gsim \lim_{\bar{z}\to1}\overline{{\mathcal{F}}^{\sigma_n \bar{\sigma}_n}_{ \bar{\sigma}_n \sigma_n} (0|\bar{z})}.
\end{aligned}$$ The light cone singularity of the Virasoro block with any $c>1$ is given in Appendix \[subapp:FM\], $$\begin{aligned}
{\mathcal{F}}^{AA}_{BB}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} (1-\bar{z})^{s_n-2h_n},
\end{aligned}$$ where the function $s_n$ is defined as $$\begin{aligned}
s_n& = \left\{
\begin{array}{ll}
2\a_n(Q-2\a_n) ,& \text{if } 2\a_n<{\frac}{Q}{2}\ ,\\
{\frac}{Q^2}{4} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ Here, $nc=1+6Q^2$ and $\a_n={\frac}{Q}{2}{\left(1-{\sqrt{1-{\frac}{c}{nc-1}{\left(n-{\frac}{1}{n} \right)}}} \right)}$, which satisfies $h_n=\a_n(Q-\a_n)$. In conclusion, the light cone limit of a correlator $G(z,\bar{z})$ is bounded by $$z^{-2h_n} (1-\bar{z})^{s_n-2h_n}
\lsim G(z,\bar{z})
\lsim z^{-2h_n} (1-\bar{z})^{-2h_n}.$$
As a result, the Renyi mutual information in the doubled CFTs satisfies the following inequalities: $$\label{eq:Ibound}
\begin{aligned}
{\left(\frac{c}{12} {\left(1+\frac{1}{n} \right)}-{\frac}{s_n}{n-1} \right)} {\frac}{2\pi}{\b}\text{min}(D+2L-2t,2t-D)
&\leq I^{(n)}(A:B)\\
&\leq \frac{c}{12} {\left(1+\frac{1}{n} \right)} {\frac}{2\pi}{\b}\text{min}(D+2L-2t,2t-D).
\end{aligned}$$ In fact, rational CFTs saturate the upper bound (\[eq:Ibound\]), and hence, their mutual information is universal. This shows that entanglement does not scramble and quasiparticle behaviour can be observed in rational CFTs. On the other hand, holographic CFTs appear to saturate the lower bound. In this context, we can state that if the Renyi mutual information, in theory, saturates the lower bound, then it shows maximal scrambling. This is the main conclusion of this section.
2nd Renyi Entropy after Local Quench {#subsec:2ndREE}
------------------------------------
At the end of this section, we discuss another application of light cone singularity. In fact, the light cone limit also appears if one investigates the dynamics of the Renyi entanglement entropy after a local quench. The process is as follows: We consider the locally excited state $\ket{\Psi}$, which is defined by acting with a local operator $O(x)$ on the CFT vacuum $\ket{0}$ in the following manner,[^10] \[lopw\] =[\^[-H-iHt]{}]{}O(x), where ${\mathcal{N}}$ is the normalization factor. The infinitesimally small parameter $\ep>0$ provides UV regularization as the truly localized operator has infinite energy. We choose the subsystem $A$ to be the half-space and induce excitation in its complement, thus creating additional entanglements between them. The main quantity of interest is the growth of entanglement entropy compared to the vacuum: \[difs\] S\^[(n)]{}\_A(t)=S\^[(n)]{}\_A()-S\^[(n)]{}\_A(). In fact, this quantity can also be calculated analytically using twist operators [@Nozaki2014] as $$\label{eq:defREE}
\Delta S^{(n)}_A=\frac{1}{1-n}\log \frac{{\langle O^{\otimes n}O^{\otimes n}\sigma_n \bar{\sigma_n} \rangle}}{{\langle O^{\otimes n}O^{\otimes n}\rangle}{\langle \sigma_n \bar{\sigma_n}\rangle}},$$ where the operator $O^{\otimes n}$ is defined on the cyclic orbifold CFT ${\mathcal{M}}^n/{\mathbb{Z}}_n$, using the operators in the seed CFT ${\mathcal{M}}$ as $$O^{\otimes n} = O \otimes O \otimes \cdots \otimes O.$$ The local excitation $O$ is separated by a distance $l$ from the boundary of $A$, as shown in the left of Figure \[fig:pos\].
![(Left) The positions of operators in the replica computation (\[eq:defREE\]). (Right) The equivalence between $(a)$ and $(b)$ explains the relation between a correlator with twist operators in an orbifold theory and a replica manifold. The equivalence between $(b)$ and $(c)$ can be obtained using a conformal map $w=z^n$.[]{data-label="fig:pos"}](position.pdf "fig:"){width="5.0cm"} ![(Left) The positions of operators in the replica computation (\[eq:defREE\]). (Right) The equivalence between $(a)$ and $(b)$ explains the relation between a correlator with twist operators in an orbifold theory and a replica manifold. The equivalence between $(b)$ and $(c)$ can be obtained using a conformal map $w=z^n$.[]{data-label="fig:pos"}](LQ.pdf "fig:"){width="9.0cm"}
For simplicity, we move the framework from $(a)$ to $(c)$ (through $(b)$) as shown in the right of Figure \[fig:pos\] using a conformal map $w=z^n$. We focus on $n=2$ in the following. In the late-time limit $t \gg l$, we can approximate the cross ratio as $$z\simeq1-{\frac}{\e^2}{4t^2}, \ \ \ \ \ \bar{z}\simeq{\frac}{\e^2}{4t^2},$$ which is just the light cone limit. [^11]
Using this cross ratio, we can re-express the correlator as $$\frac{{\langle O^{\otimes n}O^{\otimes n}\sigma_n \bar{\sigma_n} \rangle}}{{\langle O^{\otimes n}O^{\otimes n}\rangle}{\langle \sigma_n \bar{\sigma_n}\rangle}}
={\left|z\right|}^{4h_O}{\left|1-z\right|}^{4h_O}G(z,\bar{z}).$$ The light cone limit of the four-point function can be approximated as [^12] $$G(z,\bar{z}){\xrightarrow[1-z,\bar{z} \ll 1]{}} (1-z)^{s_O-2h_O}\bar{z}^{ -2h_O} ,$$ where the function $s_O$ is defined as $$\begin{aligned}
s_O& = \left\{
\begin{array}{ll}
2\a_O(Q-2\a_O) ,& \text{if } h_O<{\frac}{c-1}{32}\ ,\\
{\frac}{Q^2}{4} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ Therefore, the growth of the $2$nd Renyi entropy after a light local quench ($h_O\leq{\frac}{c-1}{32}$) is given by $$\D S_A^{(2)}(t){\xrightarrow[{\frac}{t}{\e} \to \infty]{}}4\a_O(Q-2\a_O)\log{\frac}{t}{\e}.$$ In particular, if expanding this at small ${\frac}{h_O}{c}$, the result reduces to $$\D S_A^{(2)}{\xrightarrow[{\frac}{h_O}{c}\ll1]{}}4h_O\log{\frac}{t}{\e}.$$ This result in the light limit is consistent with the result in [@Caputa2014a]. The growth for a heavy local quench ($h_O\geq{\frac}{c-1}{32}$) is more interesting, that is, it has the following universal form: $$\label{eq:univREE}
\D S_A^{(2)}(t){\xrightarrow[{\frac}{t}{\e} \to \infty]{}}{\frac}{Q^2}{2} \log{\frac}{t}{\e}.$$ These results are consistent with the numerical results in [@Kusuki2018b].
We can, therefore, conclude that the $2$nd Renyi entropy after a local quench undergoes a phase transition as the conformal dimension of the local quench is varied, if we restrict ourselves to a unitary (compact) CFT with $c>1$ and no extra conserved currents. That is, in one of the phases, the entropy is monotonically increasing in $h_O$, and in the other phase, it is saturated by the universal form (\[eq:univREE\]), as shown in Figure \[fig:2ndREE\]. We intend to emphasize that at least when $n=2$, the Renyi entropy after a local quench can be explicitly given without other assumptions except that $c>1$ and there are no extra currents (and discrete spectra or, equivalently, compactness). Unfortunately, we did not find this saturation when studying the entanglement entropy ($n=1$ Renyi entropy); therefore, we could not determine how to relate this saturation to the dynamics of the entanglement. Note that, in fact, we can generalize this result to any replica number $n$, as explained in the next section.
![The $h_O$ dependence of the coefficient of the growth of the $2$nd Renyi entropy after a local quench. This dependence might imply that in a CFT with $c>1$ and no extra currents, the heavier the operator used to create a local quench, the larger is the entropy growth; however, if its dimension exceeds the value ${\frac}{c-1}{32}$, then the entropy is saturated by (\[eq:univREE\])[]{data-label="fig:2ndREE"}](2ndREE.pdf){width="10.0cm"}
Regge Limit Universality {#sec:Regge}
========================
In 2D CFTs, the Regge limit is defined by the limit $z, \bar{z}\to 0$ after picking up a monodromy around $z=1$ as $(1-z) \to {\mathrm{e}^{-2\pi i}}(1-z)$. This limit is obviously different from the light cone limit; nevertheless, we can contribute to studies on the Regge limit using our light cone limit conformal blocks.
For this purpose, we introduce the elliptic form of the Virasoro blocks as follows: $${\mathcal{F}}^{21}_{34}(h_p|z)=\Lambda^{21}_{34}(h_p|q)H^{21}_{34}(h_p|q),\ \ \ \ \ \ q(z)={\mathrm{e}^{-\pi {\frac}{K(1-z)}{K(z)}}},$$ where $K(z)$ is the elliptic integral of the first kind and the function $\Lambda^{21}_{34}(h_p|q)$ is a universal prefactor given by $$\label{eq:preF}
\Lambda^{21}_{34}(h_p|q)=(16q)^{h_p-\frac{c-1}{24}}z^{\frac{c-1}{24}-h_1-h_2}(1-z)^{\frac{c-1}{24}-h_2-h_3}
(\theta_3(q))^{\frac{c-1}{2}-4(h_1+h_2+h_3+h_4)}.$$ The function $H^{21}_{34}(h_p|q)$ can be calculated recursively (see Appendix \[app:recursion\]). For simplicity, we express the function $H^{21}_{34}(h_p|q)$ using a series expansion form as $$H^{21}_{34}(h_p|q)=\sum_{n\in {\mathbb{Z}}_{\geq0}} c_n q^{n},$$ where $c_0=1$. In our recent studies [@Kusuki2018; @Kusuki2018b], we determined that the series coefficients could be expressed as $$c_n \sim\xi^n n^{\a} {\mathrm{e}^{A{\sqrt{n}}}} \ \ \ \ \ \ \ \ \text{for large $n\gg c$},$$ where $$\label{eq:signxi}
\begin{aligned}
\xi&=\left\{
\begin{array}{ll}
\d_{n,\text{even}} \times \sgn{\Biggl[{\left(h_A-{\frac}{c-1}{32} \right)}{\left(h_B-{\frac}{c-1}{32} \right)} \Biggr]} ,& \text{for AABB blocks } ,\\
1 ,& \text{for ABBA blocks }.\\
\end{array}
\right.\\
\end{aligned}$$ The values of $A$ and $\a$ are constants in $n$, depending on $h_A, h_B$, and $c$ (whose explicit forms are given in [@Kusuki2018; @Kusuki2018b; @Kusuki2018a] or (\[eq:AABBA\]) (\[eq:AABBal\]), (\[eq:ABBAA\]), (\[eq:ABBAal\]) in Appendix \[app:recursion\]). Note that the series coefficients for the AABB blocks vanish if $n$ is odd, which is described by $\d_{n,\text{even}}$ in (\[eq:signxi\]).
![The relation between cross ratio $z$ and elliptic nome $q$.[]{data-label="fig:nome"}](nome.pdf){width="12.0cm"}
The key point is that the Regge limit corresponds to the limit $q \to i, \bar{q} \to 0$ in terms of the elliptic nome (see Figure \[fig:nome\]). Considering that the series coefficients for $h_{A,B} >{\frac}{c-1}{32}$ are always non-negative and $z \to 1$ corresponds to $q \to 1$ and $z \to 0$ with $(1-z) \to {\mathrm{e}^{-2\pi i}} (1-z)$ corresponds to $q \to i$, we obtain the relationship between the light cone singularity and Regge singularity as follows: $$\lim_{z \to 1} {\left|H^{21}_{34}(h_p|q)\right|} \gsim \lim_{\substack{z \to 0\\ \text{with}\\ (1-z)\to{\mathrm{e}^{-2\pi i }}(1-z) }} {\left|H^{21}_{34}(h_p|q)\right|},$$ where we only focus on the cases $(h_1,h_2,h_3,h_4)=(h_A,h_A,h_B,h_B)$ or $(h_A,h_B,h_B,h_A)$; however, we expect that this relationship can be generalized for any pairs with $h_{1,2,3,4}>{\frac}{c-1}{32}$. Comparing (\[eq:preF\]) with (\[eq:AABBs\]), one can find that the main singularity in the light cone limit only comes from the prefactor (\[eq:preF\]), and, therefore, the function $H^{21}_{34}(h_p|q)$ does not contribute to the singularity. Therefore, we can conclude that $$H^{21}_{34}(h_p|q){\xrightarrow[\substack{z \to 0\\ \text{with}\\ (1-z)\to{\mathrm{e}^{-2\pi i }}(1-z) }]{}} O(\log z).$$ This conclusion leads to the Regge singularity of a four-point function as follows: $$\label{eq:Regge}
G(z,\bar{z}) {\xrightarrow[\substack{z \to 0\\ \text{with}\\ (1-z)\to{\mathrm{e}^{-2\pi i }}(1-z) }]{}} z^{{\frac}{c-1}{24}-h_1-h_2} \bar{z}^{-h_1-h_2},
\ \ \ \ \ \text{if } h_{1,2,3,4}>{\frac}{c-1}{32}.$$
One application of this result is the evaluation of the Renyi entropy after a local quench. As explained in Section \[subsec:2ndREE\], its growth can be analytically calculated using a four-point correlator with twist operators (\[eq:defREE\]). In Section \[subsec:2ndREE\], we moved the framework from an orbifold CFT to a seed CFT using a conformal map to evaluate the growth; however, we can also calculate the growth using just (\[eq:defREE\]) itself. In this calculation, we have to calculate the Regge limit of a four-point function, instead of the light cone limit. This result (\[eq:Regge\]) leads to the conclusion that the Renyi entropy after a local quench shows universality if $n>n_* \text{ and } h_O>{\frac}{c-1}{32}$. This aspect is discussed in detail elsewhere [@Kusuki2018b]. Note that the Regge limit also appears in the evaluation of OTOCs.
Discussion
==========
The light cone structure of Virasoro conformal blocks allows us to access information about 2D CFTs. For example, the large-spin spectrum can be derived from only the vacuum Virasoro conformal block through the light cone bootstrap equation. Based on this background, an important task was to examine the light cone singularity of the Virasoro blocks in general. In this paper, we reveal the light cone structure of general Virasoro blocks by investigating the fusion matrix (or the crossing kernel). Interestingly, light cone singularity undergoes a phase transition as the external operator dimensions are varied. This fact leads to a richer structure of the spectrum at large spin; however, the physical interpretation of the transition is presently unclear.
At this stage, the most important future works are to understand the bulk interpretation of [*universality of twist*]{} and [*transition at the BTZ threshold*]{}. For this purpose, it might need to resolve the following problems:
> i\) poor understanding of the relationship between Liouville CFT and $2+1$ dimensional gravity (see [@Krasnov2000; @Krasnov2001; @Jackson2015; @Caputa2017b]),
>
> ii\) dynamics of multiple deficit angles.
The reason for considering (i) is because the transition point is characterised by the total Liouville momentum, instead of the total mass.
We expect that this transition would be related to thermalisation or black-hole formation. There are two reasons for this:
> i\) The transition point is characterised by the [*BTZ threshold*]{}, similar to the thermalisation of the HHLL Virasoro blocks.
>
> ii\) The transition can only be found in a CFT with $c>1$ and no conserved extra currents, which is expected to be chaotic.
In the first place, there is limited information as to why the BTZ threshold appears as the transition point. We intend to examine if the BTZ threshold is related to the creation of black hole. It is interesting to explore the relationships between the transition, creation of a black hole, and thermalisation.
The light cone bootstrap equation suggests that there must be a universal binding energy even at large angular momentum. Moreover, this binding energy becomes larger if the total Liouville momentum increases beyond the BTZ threshold and the total twist is saturated by ${\frac}{c-1}{12}$ owing to the strong interaction above the BTZ threshold. It would be interesting to reproduce this binding energy at large spin from the calculation in AdS gravity.
The light cone singularity also reveals the entanglement structure. From our studies on entanglement in various setups, we found that the transition of the light cone singularity often caused discontinuousness of the derivative of the Renyi entropy in $n$. It is not possible to physically explain why the light cone limit destroys the assumption that the Renyi entropy is analytic in $n$. Apart from the transition in $n$, when considering the Renyi entropy after a local quench, we found that the Renyi entropy became large on increasing the conformal dimension of the operator used to create a quench, unless the dimension exceeded ${\frac}{c}{32}$. Above the threshold, the Renyi entropy is saturated. We expect that it is related to the saturation of entanglement in some way. It would be interesting to explore this topic in future.
One interesting future work is to generalize the analytic bootstrap program [@Alday2017a; @Simmons-Duffin2017] to two-dimensional CFTs. We believe that our result may contribute to this progress. It would be interesting to explore this issue further.
Acknowledgments {#acknowledgments .unnumbered}
===============
Especially, the author would like to express special thanks to Henry Maxfield for a very helpful discussion. We are grateful to Tadashi Takayanagi for his fruitful discussions and comments. We also thank Masamichi Miyaji, Nilay Kundu, and Pawel Caputa for useful conversations about the subject of this study, and Jared Kaplan, Luis Fernando Alday, Nikita Nemkov and Taro Kimura for giving us useful comments. YK is supported by the JSPS fellowship. YK is grateful to the conference “Strings and Fields 2018” in YITP, and the conference “Recent Developments in Gauge Theory and String Theory” in Keio U.
Light Cone Limit from Fusion Matrix {#app:FM}
===================================
Leading Term in Light Cone Limit {#subapp:FM}
--------------------------------
In the following, we introduce the notations usually found in Liouville CFTs. $$c=1+6Q^2, \ \ \ \ \ Q=b+{\frac}{1}{b}, \ \ \ \ \ h_i=\a_i(Q-\a_i).$$ Note that we can relate the parameter $\eta_i$ appearing in [@Kusuki2018a] to $\a_i$ as $\a_i=Q\eta_i$.
In this appendix, we show the asymptotic form of the conformal blocks in the limit $z\to 1$. The key point is that there are invertible fusion transformations between $s$ and $t$- channel conformal blocks [@Teschner2001a] as follows: [^13] $$\label{eq:fusiontrans}
\begin{aligned}
{\mathcal{F}}^{21}_{34}(h_{\a_s}|z)=\int_{{\mathbb{S}}} \dd \a_t {\bold F}_{\a_s, \a_t}
\left[
\begin{array}{cc}
\a_2 & \a_1 \\
\a_3 & \a_4\\
\end{array}
\right]
{\mathcal{F}}^{23}_{14}(h_{\a_t}|1-z),
\end{aligned}$$ where the contour ${\mathbb{S}}$ runs from ${\frac}{Q}{2}$ to ${\frac}{Q}{2}+ i\infty$. The kernel $ {\bold F}_{\a_s, \a_t} $ is called the [*crossing matrix*]{} or [*fusion matrix*]{}. The explicit form of the fusion matrix is given in [@Ponsot1999; @Teschner2001a]as follows: $$\label{eq:crossing}
\begin{aligned}
{\bold F}_{\a_s, \a_t}
\left[
\begin{array}{cc}
\a_2 & \a_1 \\
\a_3 & \a_4\\
\end{array}
\right]
={\frac}{N(\a_4,\a_3,\a_s)N(\a_s,\a_2,\a_1)}{N(\a_4,\a_t,\a_1)N(\a_t,\a_3,\a_2)}
\left\{
\begin{array}{cc|c}
\a_1 & \a_2 & \a_s \\
\a_3 & \a_4 & \a_t \\
\end{array}
\right\}_b,
\end{aligned}$$ where the function $N(\a_3,\a_2,\a_1)$ is $$N(\a_3,\a_2,\a_1)={\frac}{\G_b(2\a_1)\G_b(2\a_2)\G_b(2Q-2\a_3)}{\G_b(2Q-\a_1-\a_2-\a_3)\G_b(Q-\a_1-\a_2+\a_3)\G_b(\a_1+\a_3-\a_2)\G_b(\a_2+\a_3-\a_1)},$$ and $ \left\{
\begin{array}{cc|c}
\a_1 & \a_2 & \a_s \\
\a_3 & \a_4 & \a_t \\
\end{array}
\right\}_b$ is the Racah–Wigner coefficient for the quantum group $U_q(sl(2,{\mathbb{R}}))$, which is given by [^14] $$\label{eq:6j}
\begin{aligned}
&\left\{
\begin{array}{cc|c}
\a_1 & \a_2 & \a_s \\
\a_3 & \bar{\a_4} & \a_t \\
\end{array}
\right\}_b\\
&= {\frac}{S_b(\a_1+\a_4+\a_t-Q)S_b(\a_2+\a_3+\a_t-Q)S_b(\a_3-\a_2-\a_t+Q)S_b(\a_2-\a_3-\a_t+Q)}{S_b(\a_1+\a_2-\a_s)S_b(\a_3+\a_s-\a_4)S_b(\a_3+\a_4-\a_s)}\\
&\times {\left|S_b(2\a_t)\right|}^2 \int^{2Q+i \infty}_{2Q-i \infty} \dd u
{\frac}{S_b(u-\a_{12s})S_b(u-\a_{s34})S_b(u-\a_{23t})S_b(u-\a_{1t4})}{S_b(u-\a_{1234}+Q)S_b(u-\a_{st13}+Q)S_b(u-\a_{st24}+Q)S_b(u+Q)},
\end{aligned}$$ where we have used the notations $\bar{\a}=Q-\a$, $\a_{ijk}=\a_i+\a_j+\a_k$ and $\a_{ijkl}=\a_i+\a_j+\a_k+\a_l$. The functions $\G_b(x)$ and $S_b(x)$ are defined as $$\G_b(x)= {\frac}{\G_2(x|b,b^{-1})}{\G_2{\left({\frac}{Q}{2}|b,b^{-1} \right)}}, \ \ \ \ \ S_b(x)={\frac}{\G_b(x)}{\G_b(Q-x)},$$ $\G_2(x|\w_1,\w_2)$ is the double gamma function, $$\log \G_2(x|\w_1,\w_2)={\left({\frac{\partial}{\partial t}}\sum^{\infty}_{n_1,n_2=0} {\left(x+n_1 \w_1+n_2\w_2 \right)}^{-t} \right)}_{t=0}.$$ Note that the function $\G_b(x)$ is introduced such that $\G_b(x)=\G_{b^{-1}}(x)$ and satisfies the following relationship: $$\G_b(x+b)={\frac}{{\sqrt{2 \pi}}b^{bx-{\frac}{1}{2}}}{\G(bx)}\G_b(x).$$ By substituting the explicit form of the Racah–Wigner coefficients (\[eq:6j\]) into (\[eq:crossing\]), we can simplify the expression for the fusion matrix into $$\label{eq:crossing2}
\begin{aligned}
&{\bold F}_{\a_s, \a_t}
\left[
\begin{array}{cc}
\a_2 & \a_1 \\
\a_3 & \a_4\\
\end{array}
\right]\\
&={\frac}{\G_b(Q+\a_2-\a_3-\a_t)\G_b(Q-\a_2+\a_3-\a_t)\G_b(2Q-\a_1-\a_4-\a_t)\G_b(\a_1+\a_4-\a_t)}{\G_b(2Q-\a_1-\a_2-\a_s)\G_b(\a_1+\a_2-\a_s)\G_b(Q+\a_3-\a_4-\a_s)\G_b(Q-\a_3+\a_4-\a_s)}\\
&\times {\frac}{\G_b(Q-\a_2-\a_3+\a_t)\G(-Q+\a_2+\a_3+\a_t)\G_b(\a_1-\a_4+\a_t)\G_b(-\a_1+\a_4+\a_t)}{\G_b(\a_1-\a_2+\a_s)\G_b(-\a_1+\a_2+\a_s)\G_b(Q-\a_3-\a_4+\a_s)\G_b(-Q+\a_3+\a_4+\a_s)}\\
&\times {\left|S_b(2\a_t)\right|}^2 {\frac}{\G_b(2Q-2\a_s)\G_b(2\a_s)}{\G_b(2Q-2\a_t)\G_b(2\a_t)} \\
&\times \int^{2Q+i \infty}_{2Q-i \infty} \dd u
{\frac}{S_b(u-\a_{12s})S_b(u-\a_{s34})S_b(u-\a_{23t})S_b(u-\a_{1t4})}{S_b(u-\a_{1234}+Q)S_b(u-\a_{st13}+Q)S_b(u-\a_{st24}+Q)S_b(u+Q)}.
\end{aligned}$$
The conformal blocks have the following simple asymptotic form: $${\mathcal{F}}^{21}_{34}(h_{\a_s}|z) {\xrightarrow[z\to 0]{}} z^{h_s-h_1-h_2}(1+O(z)).$$ Naively substituting this asymptotics into the fusion transformation (\[eq:fusiontrans\]) leads to $${\mathcal{F}}^{21}_{34}(h_{\a_s}|z) {\xrightarrow[z\to 1]{}} (1-z)^{{\frac}{c-1}{24}-h_2-h_3}.$$ Interestingly, this reproduces a part of the asymptotic formulas (\[eq:AABBs\]) and (\[eq:ABBAs\]). However, we cannot straightforwardly understand how the transition seen in (\[eq:AABBs\]) and (\[eq:ABBAs\]) can be reproduced using the transformation (\[eq:fusiontrans\]).
The key to understanding how the transition is derived from the expression of the fusion matrix is that the function $\G_b(x)$ has poles at $x=-{\left(mb+{\frac}{n}{b} \right)}$ for $n,m \in {\mathbb{Z}}_{\geq0}$, and therefore, the fusion matrix also has poles. The following shows the poles in the first and second lines of the expression (\[eq:crossing2\]):
\[[*Poles of the Numerator*]{}\] $$\label{eq:poles}
\begin{aligned}
\a_t^{(1)}&=\pm(\a_2-\a_3)+Q+Q_{m,n}\\
\a_t^{(2)}&=\pm(\a_1-\a_4)-Q_{m,n}\\
\a_t^{(3)}&=-(\a_1+\a_4)+2Q+Q_{m,n}\\
\a_t^{(4)}&=(\a_2+\a_3)-Q-Q_{m,n}\\
\a_t^{(5)}&=\a_1+\a_4+Q_{m,n}\\
\a_t^{(6)}&=-(\a_2+\a_3)+Q-Q_{m,n}\\
\end{aligned}$$ where we define $Q_{m,n}=mb+{\frac}{n}{b}$ for $n,m \in {\mathbb{Z}}_{\geq0}$. The real values of $\a_i$ always satisfy $0 \leq \Re \a_i \leq {\frac}{Q}{2}$ for the operators in unitary CFTs, and the value $Q_{m,n}$ is always positive by definition. Therefore, each value of $\Re \a_t^{(i)}$ exists only in a particular domain as shown in Figure \[fig:alpha\]. One can find that only $\a_t^{(5)}$ and $\a_t^{(6)}$ can cross the contour ${\mathbb{S}}$ of the integral over $\a_t$ when $\a_1+\a_4<{\frac}{Q}{2}$. As a result of this crossing, the contour ${\mathbb{S}}$ is deformed as shown in Figure \[fig:contour\]. [^15]
![The domain of $\Re \a_t^{(i)}$. From this figure, we can find that only $\a_t^{(5)}$ and $\a_t^{(6)}$ can cross the contour ${\mathbb{S}}$ of the integral over $\a_t$.[]{data-label="fig:alpha"}](alpha.pdf){width="15.0cm"}
![The contour ${\mathbb{S}}$ deformed by the pole $\a_t^{(5)}$.[]{data-label="fig:contour"}](contour.pdf){width="10.0cm"}
This deformation leads to another possibility that the conformal block has the asymptotics as follows: $$\begin{aligned}
{\mathcal{F}}^{21}_{34}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} \left\{
\begin{array}{ll}
(1-z)^{h_1+h_4-h_2-h_3-2\a_1 \a_4} ,& \text{if } \a_1+\a_4<{\frac}{Q}{2} ,\\
(1-z)^{-2\a_2\a_3} ,& \text{if } \a_2+\a_3<{\frac}{Q}{2} .\\
\end{array}
\right.\\
\end{aligned}$$ If we set three parameters as $$\kappa_1=h_1+h_4-h_2-h_3-2\a_1 \a_4, \ \ \ \ \
\kappa_2=-2\a_2\a_3, \ \ \ \ \
\kappa_3={\left({\frac}{c-1}{24}-h_1-h_4 \right)},$$ then we can show the following inequalities: $$\begin{aligned}
\kappa_3-\kappa_1&={\left(\a_1+\a_4-{\frac}{Q}{2} \right)}^2>0,\\
\kappa_3-\kappa_2&={\left(\a_2+\a_3-{\frac}{Q}{2} \right)}^2>0,\\
\kappa_1-\kappa_2&=(\a_1+\a_4-\a_2-\a_3)(1-\a_1-\a_2-\a_3-\a_4),
\end{aligned}
\ \ \ \ \
\begin{aligned}
&\text{if }\a_1+\a_4<{\frac}{Q}{2},\\
&\text{if }\a_2+\a_3<{\frac}{Q}{2},\\
&\text{if }\a_1+\a_4<{\frac}{Q}{2}, \a_2+\a_3<{\frac}{Q}{2}.\\
\end{aligned}$$ Therefore, the leading contribution of the ABBA blocks in the limit $z \to 1$ is given by $$\label{eq:FMresult1}
\begin{aligned}
{\mathcal{F}}^{BA}_{BA}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} \left\{
\begin{array}{ll}
(1-z)^{4h_A-2h_B-2Q\a_A} ,& \text{if } \a_A<{\frac}{Q}{4}\ \text{and } \ \a_A<\a_B ,\\
(1-z)^{2h_B-2Q\a_B} ,& \text{if } \a_B<{\frac}{Q}{4}\ \text{and } \ \a_B<\a_A ,\\
(1-z)^{{\frac}{c-1}{24}-2h_B} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ Further, the asymptotics of the AABB blocks is given by $$\label{eq:FMresult2}
\begin{aligned}
{\mathcal{F}}^{AA}_{BB}(h_{\a_s}|z)& {\xrightarrow[z\to 1]{}} \left\{
\begin{array}{ll}
(1-z)^{-2\a_A\a_B} ,& \text{if } \a_A+\a_B<{\frac}{Q}{2}\ ,\\
(1-z)^{{\frac}{c-1}{24}-h_A-h_B} ,& \text{otherwise } .\\
\end{array}
\right.\\
\end{aligned}$$ Interestingly, these results exactly match our previous results obtained using the monodromy method in [@Kusuki2018a] and numerical results in [@Kusuki2018; @Kusuki2018b]. We intend to emphasize that the above calculation based on the fusion matrix does not rely on the large $c$ limit. Therefore, we expect that the asymptotic form derived from the monodromy method with $c \to \infty$ in [@Kusuki2018a] can be generalized to any unitary CFTs with $c>1$ by replacing the factor $c$ by $c-1$ .
One might doubt this result because it contradicts with the singularities appearing in the minimal models. In the first place, when one of the external operators corresponds to a degenerate operator, the decomposition of the $s$-channel in terms of the $t$-channel is not a continuum, but is discrete. That is, in such a case, the crossing kernel needs to be written as a linear combination of delta functions. In fact, if the external operator is degenerate, $S_b$ or $\G_b$ in the denominator of the expression (\[eq:crossing2\]) diverges, and therefore, the crossing kernel vanishes for general $\a_t$; however, these divergences can be cancelled only if $\a_t$ takes particular values. As a result, the $s$-channel is decomposed as a discrete sum of the $t$-channels, instead of an integral over the continuum spectrum. This concept is explained in greater detail in [@Hadasz2005; @EsterlisFitzpatrickRamirez2016]. We have to mention that this never happens for unitary CFTs with $c>1$ because if the central charge is larger than one, then the conformal dimensions of degenerate operators are negative.
Sub-leading Terms in Light Cone Limit {#subapp:sub}
-------------------------------------
In Section \[subsec:spectrum\], we tried to derive the spectrum of twist. For this, we need the sub-leading contributions of the conformal blocks in the light cone limit. In this appendix, we first give the sub-leading terms in the semiclassical limit using the fusion matrix and then generalize it.
We expect that the sub-leading terms come from the poles (\[eq:poles\]) with non-zero $n,m$. As an example, let us consider the contributions from the poles $\a_t^{(5)}$ with non-zero $n,m$ in the semiclassical limit. If $\a_1+\a_4+Q_{m,n}<{\frac}{Q}{2}$ is satisfied, the corresponding term labelled $m,n$ also contributes to the conformal blocks as sub-leading terms. Its singularity is given by $$(1-z)^{h_1+h_4-h_2-h_3-2\a_1\a_4+\m_{m,n}},$$ where $\m_{m,n}$ is a positive constant defined by $$\m_{m,n}={\left(mb+{\frac}{n}{b} \right)} {\left((1-\w_{m,n}+m)b+{\frac}{1-\w_{m,n}+n}{b} \right)},$$ and $\w_{m,n} \equiv {\frac}{2}{Q}(\a_1+\a_4+Q_{m,n})<1$. If taking the limit $b \to 0$ with $h_1, h_4/c$ fixed, the sub-leading terms are given by $\m_{m,0}$ as $\m_{m,n>0} \gg \m_{m,0}$, and $\w$ can be approximated as $\w_{m,0} \simeq 1-\d$ with $\d={\sqrt{1-{\frac}{24}{c}h_4}}$. Thus, we can approximate $\m_{m,0}$ as $$\m_{m,0} {\xrightarrow[b\to0]{}} m\d.$$ This leads to the sub-leading contributions to the conformal blocks as follows: $$\label{eq:subHHLL}
{\mathcal{F}}^{LL}_{HH}(h_p|z){\xrightarrow[z\to1]{}} \sum_{m \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_m (1-z)^{\d(h_L+m)-h_L},$$ where ${\mathcal{P}}_m$ are some constants. In fact, the HHLL Virasoto blocks have the same form. $$\label{eq:fullHHLL}
\begin{aligned}
{\mathcal{F}}^{LL}_{HH}(h_p|z)&= (1-z)^{h_L(\d-1)}{\left({\frac}{1-(1-z)^\d}{\d} \right)}^{h_p-2h_L} {}_2F_1(h_p,h_p,2h_p|1-(1-z)^\d)\\
&{\xrightarrow[z\to1]{}} \sum_{m \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_m (1-z)^{\d(h_L+m)-h_L},
\end{aligned}$$ which can be easily checked using the series expansion of ${\left(1-x \right)}^{-2h_L}$. Therefore, we can conclude that the sub-leading contributions come from the poles with non-zero $m,n$, as shown in Figure \[fig:contour2\].
![This figure shows from where the sub-leading contributions to the blocks are obtained.[]{data-label="fig:contour2"}](contour2.pdf){width="9.0cm"}
Note that now we take the limit $b \to 0$; therefore, an infinite number of poles $\a_1+\a_4+Q_{m,0}$ for any $m \in {\mathbb{Z}}_{\geq0}$ satisfy the inequality $$\a_1+\a_4+Q_{m,0}<{\frac}{Q}{2},$$ when the inequality $\a_1+\a_4<{\frac}{Q}{2}$ is satisfied. Thus, an infinite number of poles contribute to the conformal blocks as shown in (\[eq:fullHHLL\]) in the semiclassical limit. The condition ${\frac}{h_1}{c} \ll1$ can immediately be relaxed to non-perturbative ${\frac}{h_1}{c}$ as $$\begin{aligned}
{\mathcal{F}}^{AA}_{BB}(h_p|z)&{\xrightarrow[z\to1]{}} \left\{
\begin{array}{ll}
\sum_{m \in {\mathbb{Z}}_{\geq0} } {\mathcal{P}}_m (1-z)^{-2\a_A \a_B +m{\left(1-{\frac}{2}{Q}(\a_A+\a_B) \right)}},& \text{if } \a_A+\a_B<{\frac}{Q}{2},\\
(1-z)^{{\frac}{c-1}{24}-h_A-h_B}, & \text{otherwise} .\\
\end{array}
\right.\\
\end{aligned}$$
For general unitary CFTs with finite $c>1$, there are only finite poles satisfying the inequality $\a_1+\a_4+Q_{m,n}<{\frac}{Q}{2}$. Therefore, the conformal blocks can be expressed using a finite sum as $$\label{eq:fullAABB}
{\mathcal{F}}^{AA}_{BB}(h_p|z){\xrightarrow[z\to1]{}} \sum_{\substack{m,n \in {\mathbb{Z}}_{\geq0}\\ \text{where} \\ \a_A+\a_B+Q_{m,n}<{\frac}{Q}{2} }}
{\mathcal{P}}_{m,n} (1-z)^{-2\a_A \a_B+\m_{m,n}}, \ \ \ \ \ \text{if } \a_A+\a_B<{\frac}{Q}{2},$$ and $${\mathcal{F}}^{AA}_{BB}(h_p|z){\xrightarrow[z\to1]{}} (1-z)^{{\frac}{c-1}{24}-h_A-h_B}, \ \ \ \ \ \text{otherwise },$$ where ${\mathcal{P}}_{m,n}$ are some constants. Note that the above process can straightforwardly be applied to the ABBA blocks. That is, the result can be obtained by just adding $\m_{m,n}$ to the power of its singularity.
Many studies on the fusion matrix approach remain to be carried out. The most important one is identifying the logarithmic corrections to the light cone limit of the conformal blocks. This correction is related to the other anomalous dimension different from (\[eq:Ebound\]), which also appears in higher dimensions.
Light Cone Modular Bootstrap {#app:modular}
============================
Light Cone Modular Bootstrap {#light-cone-modular-bootstrap}
----------------------------
The modular invariance imposes the following condition on the torus partition function: $$\label{eq:modular}
Z(\tau,\bar{\tau})=Z{\left(-{\frac}{1}{\tau},-{\frac}{1}{\bar{\tau}} \right)},$$ where the torus partition function has the Virasoro character decomposition in CFTs, $$Z(\tau,\bar{\tau})=\sum_{i,j} D_{i,j} \chi_{h_i}(\tau)\bar{\chi}_{\bar{h}_j}(\bar{\tau})
=\int \dd h \dd \bar{h} \ \ \rho(h,\bar{h}) \chi_h(\tau)\bar{\chi}_{\bar{h}}(\bar{\tau}).$$ Here, function $D_{i,j}$ denotes the degeneracy of primary operators of weight $(h_i,\bar{h}_j)$, and function $\rho (h,\bar{h})$ is defined as $\rho (h,\bar{h})=\sum_{i,j} D_{i,j} \delta(h-h_i) \delta(\bar{h}-\bar{h}_j)$. If we limit ourselves to CFTs with $c>1$, the Virasoro character has the following simple form: $$\chi_0(\tau)={\frac}{q^{-{\frac}{c-1}{24}}}{\eta(\tau)}(1-q), \ \ \ \ \chi_h(\tau)={\frac}{q^{h-{\frac}{c-1}{24}}}{\eta(\tau)},$$ where $q={\mathrm{e}^{2\pi i \tau}}$ and $\eta(\tau)$ represents the Dedekind eta function.
Here, we focus on CFTs with $c>1$ and without extra currents. The partition function of such CFTs in the limit $q \to 0$ (or equivalently $\tau \to i \infty$) can be approximated by the vacuum contribution: $$Z(\tau,\bar{\tau}) {\xrightarrow[q \to 0]{}} \chi_0(\tau) \chi_0(\bar{\tau}).$$ If we define the [*large spin spectrum*]{} as $$\rho (\infty, \bar{h}) \equiv \lim_{\tau \to i\infty} \int \dd h \ \ \rho(h,\bar{h}) {\frac}{\chi_h {\left(-{\frac}{1}{\tau} \right)}}{\chi_0 (\tau)},$$ then we can factor out the $\bar{\tau}$ and $\bar{z}$ dependence in the modular bootstrap equation as $$\label{eq:chiral modular}
\chi_0 (\bar{\tau})=\int \dd \bar{h} \ \ \rho(\infty,\bar{h}) \bar{\chi}_{\bar{h}}{\left(-{\frac}{1}{\bar{\tau}} \right)}.$$ The S-transformation of the vacuum character can be expressed as [@Zamolodchikov2001] $$\label{eq:chiral modular2}
\chi_0 (\bar{\tau})=\int_{{\frac}{c-1}{24}}^{\infty} \dd \bar{h} \ \ S(\bar{h},0) \bar{\chi}_{\bar{h}}{\left(-{\frac}{1}{\bar{\tau}} \right)},$$ where the S-matrix is $$\begin{aligned}
S(\bar{h},0)=2{\sqrt{2}}{\left(\bar{h}-{\frac}{c-1}{24} \right)}^{-{\frac}{1}{2}} \sinh {\left(2 \pi b {\sqrt{\bar{h}-{\frac}{c-1}{24}}} \right)} \\
\times \sinh {\left(2 \pi b^{-1} {\sqrt{\bar{h}-{\frac}{c-1}{24}}} \right)}.
\end{aligned}$$ Upon comparing (\[eq:chiral modular\]) and (\[eq:chiral modular2\]), we find that the large spin spectrum is given by $$\label{eq:large spin spectrum}
\rho(\infty,\bar{h})=S(\bar{h},0).$$ In particular, the asymptotic behavior of this spectrum reproduces the Cardy formula [@Cardy1986a], $$\rho(\infty,\bar{h}) {\xrightarrow[\bar{h} \to \infty]{}} {\mathrm{e}^{2\pi {\sqrt{{\frac}{c-1}{6} {\left(\bar{h}-{\frac}{c-1}{24} \right)}}}}}$$ The result (\[eq:large spin spectrum\]) suggests that such CFTs have a universal spectrum at large spin.
On the other hand, if we assume that there are only primary operators with bounded spin in a CFT, then we can show that the lowest-lying operator in such a CFT must have the dimension $h_{min}={\frac}{c-1}{24}$, in that this CFT is non-compact (see [@Collier2016] for more details).
The cosmic censorship conjecture requires the condition ${\left|J\right|} \leq M$, which is translated in terms of conformal dimensions as $$\min \{h,\bar{h} \} \geq {\frac}{c}{24}.$$ In particular, a state with $\min \{h,\bar{h} \} = {\frac}{c}{24}$ corresponds to an extremal BTZ black hole (e.g., Strominger–Vafa black hole [@Strominger1996]). In this sense, the large spin spectrum (\[eq:large spin spectrum\]) corresponds to BTZ black holes.
Relation between Conformal and Modular Bootstrap {#subapp:relation}
------------------------------------------------
The counterpart of the light cone modular bootstrap (\[eq:chiral modular\]) can be given by
$$\label{eq:bootstrapAABB}
\overline{{\mathcal{F}}^{AA}_{BB}}(0|\bar{z})=\int \dd \bar{h} \ \ \rho_{AB} (\infty,\bar{h}) \overline{{\mathcal{F}}^{AB}_{AB}}(\bar{h}|1-\bar{z}),$$
where we define the [*large spin spectral density*]{}, $$\rho_{AB} (\infty,\bar{h}) \equiv \lim_{z \to 0} \int \dd h \ \ \rho_{AB} (h,\bar{h}) {\frac}{ {\mathcal{F}}^{AB}_{AB}(h|1-z)}{{\mathcal{F}}^{AA}_{BB}(0|z)}.$$ The counterpart of (\[eq:chiral modular2\]) is given by the fusion matrix, $$\label{eq:bootstrapAABB2}
\begin{aligned}
&{\mathcal{F}}^{AA}_{BB}(0|z)\\
&=
-2\pi i \sum_{\substack{\a_{n,m}<{\frac}{Q}{2} \\ n,m \in {\mathbb{Z}}_{\geq0}}}\ \text{Res}{\left(
{\bold F}_{0, \a_t}
\left[
\begin{array}{cc}
\a_A & \a_A \\
\a_B & \a_B\\
\end{array}
\right]
{\mathcal{F}}^{AB}_{AB}(h_{\a_t}|1-z);\a_{n,m} \right)}\\
&+
\int_{{\frac}{Q}{2}+0}^{{\frac}{Q}{2}+i \infty} \dd \a_t {\bold F}_{0, \a_t}
\left[
\begin{array}{cc}
\a_A & \a_A \\
\a_B & \a_B\\
\end{array}
\right]
{\mathcal{F}}^{AB}_{AB}(h_{\a_t}|1-z),
\end{aligned}$$ where $\a_{n,m}\equiv\a_A+\a_B+mb+nb^{-1}$. Since the asymptotic behavior of the Virasoro blocks is given by $${\mathcal{F}}^{ji}_{kl}(h_p|z) {\xrightarrow[z \to 0]{}} z^{h_p-h_i-h_j}$$ if we take the limit $\bar{z} \to 1$ in (\[eq:bootstrapAABB\]), then we obtain the twist spectrum and the structure constant at large spin from the fusion matrix (\[eq:bootstrapAABB2\]), which is the so-called [*light-cone bootstrap*]{}. [^16] In fact, upon comparing (\[eq:bootstrapAABB\]) and (\[eq:bootstrapAABB2\]), we find that the twist spectrum in the OPE between $O_A$ and $O_B$ at large spin can be expressed as $$\label{eq:LCresult}
\begin{aligned}
\bar{h}&=\left\{
\begin{array}{ll}
\{ \bar{\a}_{n,m}(Q-\bar{\a}_{n,m}) \ \ \ \text{for $n,m \in {\mathbb{Z}}_{\geq 0}$ s.t. $\bar{\a}_{n,m}<{\frac}{Q}{2}$} \} \cup {\lbrack{\frac}{c-1}{24},\infty \rbrack}, \\
\hspace{5cm} (\text{if } \bar{\a}_A+\bar{\a}_B<{\frac}{Q}{2}) ,\\ \\
{\lbrack {\frac}{c-1}{24},\infty \rbrack}, \hspace{3.7cm} ( \text{if } \bar{\a}_A+\bar{\a}_B\geq{\frac}{Q}{2}) ,\\
\end{array}
\right.\\
\end{aligned}$$ where $\bar{\a}_{n,m}\equiv\bar{\a}_A+\bar{\a}_B+mb+nb^{-1}$.
This spectrum has a simple interpretation. If we consider the analytic continuation of the Liouville OPE to $\a\notin ]0,Q[$, we can obtain the following fusion rules [@Ribault2014], $$\label{eq:fusion rule}
\begin{aligned}
{\mathcal{V}}_{\a_A} \times {\mathcal{V}}_{\a_B} =
\sum_{\substack{\a_{n,m}<{\frac}{Q}{2} \\ n,m \in {\mathbb{Z}}_{\geq0}}} {\mathcal{V}}_{\a_{n,m}}
+ \int^{{\frac}{Q}{2}+i \infty}_{{\frac}{Q}{2}+0} \dd \a \ {\mathcal{V}}_{\a},
\end{aligned}$$ where ${\mathcal{V}}_\a$ denotes a primary operator characterized by a conformal dimension $h=\a(Q-\a)$. Upon inspecting this fusion rule, we can straightforwardly observe that the twist spectrum (\[eq:LCresult\]) exactly matches the spectrum originating from the fusion between primary operators in the (extended) Liouville CFT (\[eq:fusion rule\]). Therefore, we can conclude that the twist spectrum at large spin approaches that of the [*(extended) Liouville CFT*]{}. This is the Virasoro counterpart of the statement that the twist spectrum at large spin in higher-dimensional CFTs ($d\geq3$) approaches the spectrum of double-trace states in a [*generalized free theory*]{}.
The Liouville CFT sometimes appears when we study chiral solutions to pure 3D gravity, which can describe spinning two-particle states in AdS${}_3$ [@Hulik2017; @Hulik2018]. This spinning two-particle solution leads to the energy $\bar{h}=\bar{\a}_{0,0}(Q-\bar{\a}_{0,0})$. In this sense, we can reproduce the large spin twist spectrum from the bulk. This solution is described by the Liouville gravity, therefore, it is naturally expected that at the quantum level, the spectrum of this two-particle state can be described by the fusion rules of the extended Liouville CFT (\[eq:fusion rule\]), and therefore, we can conclude that the twist spectrum of this two-particle state is given as $\bar{h}=\bar{\a}_{n,m}(Q-\bar{\a}_{n,m}), \ \ \ n,m\in {\mathbb{Z}}_{\geq 0} \ \ \text{s.t.} \ \ \bar{h}<{\frac}{c-1}{24}$.
Zamolodchikov Recursion Relation {#app:recursion}
================================
The Zamolodchikov recursion relation [@Zamolodchikov1987; @Zamolodchikov1984] is one of the tools used to calculate the conformal blocks numerically and has been recently receiving much attention [@Chen2017; @Ruggiero2018] because it effectively encompasses the conformal blocks beyond the known regimes or limits. We provide a brief explanation herein. [^17]
For simplicity, we will decompose the conformal blocks into two parts as $${\mathcal{F}}^{21}_{34}(h_p|z)=\Lambda^{21}_{34}(h_p|q)H^{21}_{34}(h_p|q),\ \ \ \ \ \ q(z)={\mathrm{e}^{-\pi {\frac}{K(1-z)}{K(z)}}},$$ where $K(x)$ is the elliptic integral of the first kind and the function $\Lambda^{21}_{34}(h_p|q)$ is a universal prefactor given by $$\label{eq:pre}
\Lambda^{21}_{34}(h_p|q)=(16q)^{h_p-\frac{c-1}{24}}z^{\frac{c-1}{24}-h_1-h_2}(1-z)^{\frac{c-1}{24}-h_2-h_3}
(\theta_3(q))^{\frac{c-1}{2}-4(h_1+h_2+h_3+h_4)}.$$ The function $H^{21}_{34}(h_p|q)$ can be calculated recursively using the following relation: $$H^{21}_{34}(h_p|q)=1+\sum^\infty_{m=1,n=1}\frac{q^{mn}R_{m,n}}{h_p-h_{m,n}}H^{21}_{34}(h_{m,n}+mn|q),$$ where $R_{m,n}$ is a constant in $q$, which is defined by $$\label{eq:Rmn}
R_{m,n}=2{\frac}{
\substack{m-1\\ \displaystyle{\prod} \\p=-m+1\\p+m=1 (\text{mod } 2) \ } \
\substack{n-1\\ \displaystyle{\prod} \\q=-n+1\\q+n=1 (\text{mod } 2) }
{\left(\la_2+\la_1-\la_{p,q} \right)}{\left(\la_2-\la_1-\la_{p,q} \right)}{\left(\la_3+\la_4-\la_{p,q} \right)}{\left(\la_3-\la_4-\la_{p,q} \right)}}
{\substack{
\substack{m \\ \displaystyle{\prod} \\k=-m+1 } \ \
\substack{n \\ \displaystyle{\prod} \\l=-n+1 }\\
(k,l)\neq(0,0), (m,n)
}
\la_{k,l}}.$$ In the above expressions, we used the notations $$\begin{aligned}
&c=1+6{\left(b+{\frac}{1}{b} \right)}^2, \hspace{16ex} h_i={\frac}{c-1}{24}-\l_i^2,\\
&h_{m,n}={\frac}{1}{4}{\left(b+{\frac}{1}{b} \right)}^2-\lambda_{m,n}^2, \hspace{10ex} \lambda_{m,n}={\frac}{1}{2} {\left({\frac}{m}{b}+nb \right)}.
\end{aligned}$$ Unfortunately, this recursion process is too complicated for calculating the Virasoro blocks analytically; however, from the viewpoint of numerical computations, this recursion relation is much more useful than the BPZ method [@Belavin1984]. One of the recent results involving the recursion relation is presented in our previous papers [@Kusuki2018b; @Kusuki2018]. It reveals the general solutions to this recursion relation via numerical computations. For simplicity, we re-express the function $H^{21}_{34}(h_p|q)$ as H\^[21]{}\_[34]{}(h\_p|q)=1+\_[k=1]{}\^c\_k(h\_p) q\^[k]{}, and the corresponding recursion relation as $$\label{eq:ck}
c_k(h_p) = \sum_{i=1}^k \sum_{\substack{m=1, n=1\\mn=i}} \frac{R_{m,n}}{h_p - h_{m,n}} c_{k-i}(h_{m,n}+mn).$$ For this series expansion form, our previous numerical computations suggest that the solution $c_n$ for large $n$ takes the simple [*Cardy-like*]{} form of $$\label{eq:cn}
c_n \sim\xi_n n^{\a} {\mathrm{e}^{A{\sqrt{n}}}} \ \ \ \ \ \ \ \ \text{for large $n\gg c$},$$ where $$\label{eq:xin}
\begin{aligned}
\xi_n&=\left\{
\begin{array}{ll}
\d_{n,\text{even}} \times \sgn{\Biggl[{\left(h_A-{\frac}{c-1}{32} \right)}{\left(h_B-{\frac}{c-1}{32} \right)} \Biggr]}^{{\frac}{n}{2}} ,& \text{for } {\mathcal{F}}^{AA}_{BB}(h_p|z) ,\\
1 ,& \text{for } {\mathcal{F}}^{BA}_{BA}(h_p|z) .\\
\end{array}
\right.\\
\end{aligned}$$ The parameters $A$ and $\a$ are non-trivial, depending on the external conformal dimensions and the central charge.
> [*AABB blocks:*]{} $$\label{eq:AABBA}
> \begin{aligned}
> A&=\left\{
> \begin{array}{ll}
> 2\pi{\sqrt{{\frac}{c-1}{24}-h_A-h_B+2\a_A\a_B}} ,& \text{if } h_A.h_B>{\frac}{c-1}{32} ,\\
> \pi{\sqrt{{\frac}{c-1}{24}-2h_A}} ,& \text{if } h_B>{\frac}{c-1}{32}>h_A ,\\
> 0 ,& \text{if } {\frac}{c-1}{32}>h_A.h_B ,\\
> \end{array}
> \right.\\
> \end{aligned}$$
>
> $$\label{eq:AABBal}
> \begin{aligned}
> \a&=\left\{
> \begin{array}{ll}
> 2(h_A+h_B)-{\frac}{c+5}{8} ,& \text{if } h_A.h_B>{\frac}{c-1}{32} ,\\
> 4(h_A+h_B)-{\frac}{c+9}{4} ,& \text{otherwise } .\\
> \end{array}
> \right.\\
> \end{aligned}$$
>
> [*ABBA blocks:*]{}\
> (For simplicity, we assume $h_B>h_A$, but it does not matter.) $$\label{eq:ABBAA}
> \begin{aligned}
> A&=\left\{
> \begin{array}{ll}
> 2\pi{\sqrt{{\frac}{c-1}{24}-4h_A+2Q\a_A}} ,& \text{if } h_A<{\frac}{c-1}{32} ,\\
> 0 ,& \text{if } h_A>{\frac}{c-1}{32} ,\\
> \end{array}
> \right.\\
> \end{aligned}$$
>
> $$\label{eq:ABBAal}
> \begin{aligned}
> \a&=\left\{
> \begin{array}{ll}
> 2(h_A+h_B)-{\frac}{c+5}{8} ,& \text{if } h_A<{\frac}{c-1}{32} ,\\
> 4(h_A+h_B)-{\frac}{c+9}{4} ,& \text{if } h_A>{\frac}{c-1}{32} .\\
> \end{array}
> \right.\\
> \end{aligned}$$
These expressions are numerically conjectured in [@Kusuki2018b; @Kusuki2018] and partly proven in a particular case in [@Kusuki2018a].
In the limit $z \to 1$, the summation can be approximated using $$\label{eq:sumcn}
\sum_n n^{\a}{\mathrm{e}^{A{\sqrt{n}}}} q^n \sim (1-z)^{-{\frac}{A^2}{4\pi^2}}{\left(\log (1-z) \right)}^{2\a+{\frac}{3}{2}}.$$ Inserting the expression for $A$ (\[eq:AABBA\]), (\[eq:ABBAA\]) into this approximation form, we can reproduce the conclusions (\[eq:FMresult1\]), (\[eq:FMresult2\]) in Appendix \[subapp:FM\]. In other words, our conclusions in Appendix \[app:FM\] are supported by numerical verifications and analytic calculations in a particular case. However, we have not attempted to reproduce the logarithmic dependence in (\[eq:sumcn\]) using the fusion matrix approach. We plan to take it up as our future work.
[^1]: The conformal bootstrap equation of a particular four-point function in the limit $z,\bar{z} \to 0$ is related to the modular bootstrap in the high-low temperature limit $\beta \to 0$ through a map ${\mathrm{e}^{-\beta}}={\mathrm{e}^{-\pi{\frac}{K(1-z)}{K(z)}}}$, where $K(z)$ is the elliptic integral of the first kind. For this reason, we will call the limit $z,\bar{z} \to 0$ as the [*high-low temperature limit*]{}.
[^2]: The time evolution of OTOCs at late time is also discussed in [@Caputa2016; @Caputa2017a; @Perlmutter2016; @Gu2016; @Fan2018].
[^3]: In fact, the light cone bootstrap in 2D CFTs has been studied in [@Fitzpatrick2014]; however, this study focused on only the semiclassical limit as the study relied on using HHLL Virasoro blocks. Now that we have the most general blocks in the light cone limit, we will consider more general unitary CFTs and external operators.
[^4]: The Liouville momentum $\a_{\text{BTZ}}={\frac}{Q}{2}$ leads to the conformal dimension $h={\frac}{Q^2}{4}={\frac}{c-1}{24}$. This is just the BTZ mass threshold.
[^5]: After our work appeared, similar work was done in [@Collier2018]. In [@Collier2018], this spectrum is called [*Virasoro Mean Fiald Theory*]{}. (More higher corrections are given by (\[eq:general twist\]).) Actually, this large spin twist spectrum can be described by the fusion rule of the [*Liouville CFT*]{}, as explained in Appendix \[subapp:relation\].
[^6]: This setup is holographically studied in [@Kusuki2017].
[^7]: Here, we assume that there are no extra currents, which is expected in generic holographic CFTs. We also assume that orbifoldisation does not change the essential features of the CFTs. It is nontrivial, but actually we can reproduce the holographic results under this assumption; therefore, we expect that this assumption is valid. We will discuss this topic in detail in Sections \[subsec:DMI2\].
[^8]: The twist operator has a conformal dimension of the form $c \times const.$. On the other hand, the $c$ dependence of the conformal blocks in the light cone limit appears as $(c-1)\times const.$. Therefore, the blocks with the twist operators have a complicated factor ${\frac}{c}{c-1}$, which is only simplified in the large $c$ limit.
[^9]: $S^{(2)}$ and $S^{(\infty)}$ are respectively known as [*collision entropy*]{} and [*min-entropy*]{}.
[^10]: We would like to stress that $\ep$ in (\[lopw\]) is the ultraviolet (UV) cut off of the local excitations and should be distinguished from the UV cut off (i.e. the lattice spacing) of the CFT itself.
[^11]: Actually, this limit is the double light cone limit, which is defined by the limit $1-z,\bar{z} \ll 1$ with ${\frac}{\bar{z}}{1-z}$ fixed.
[^12]: In this case, the function $G(z,\bar{z})$ is NOT defined in an orbifold theory, but just in an ordinal theory. Therefore, we do not need to consider the difficulty explained in Section \[subsec:DMI2\]. We can approximate the light cone limit of a correlator using just the Virasoro conformal block.
[^13]: A similar structure for a 1-pt function on a torus can be found in [@Teschner2003] (see also [@Hadasz2010b; @Kraus2016; @Nemkov2017]). It would be interesting to parallel our discussion for a 1-pt function on a torus.
[^14]: Ponsot–Teschner have derived a more symmetric form of the Racah–Wigner coefficient [@Teschner2014] than the traditional expression found in [@Ponsot1999; @Teschner2001a]. In this study, we used the new expression derived in [@Teschner2014].
[^15]: This deformation of the contour in a particular case was described in [@Chang2016] and we are very much grateful to Henry Maxfield for pointing out this.
[^16]: A very similar approach can be found in [@Francesco2012] (see (2.115)$\sim$(2.117)), which derives a general relation between the OPE coefficients and the fusion matrices.
[^17]: The Zamolodchikov recursion relation is also used in the conformal bootstrap [@EsterlisFitzpatrickRamirez2016; @BaeLeeLee2016; @LinShaoSimmons-DuffinWangYin2017; @CollierKravchukLinYin2017]. An earlier study [@Perlmutter2015] presents a good review of this concept and discusses the connections between various recursion relations. A generalization of the recursion relation to more general Riemann surfaces is given in [@Cho2017a].
|
---
abstract: 'The classical Cauchy–Davenport theorem implies the lower bound $n+1$ for the number of distinct subsums that can be formed from a sequence of $n$ elements of the cyclic group ${{\mathbb Z}}_p$ (when $p$ is prime and $n<p$). We generalize this theorem to a conjecture for the minimum number of distinct subsums that can be formed from elements of a multiset in ${{\mathbb Z}}_p^m$; the conjecture is expected to be valid for multisets that are not “wasteful” by having too many elements in nontrivial subgroups. We prove this conjecture in ${{{\mathbb Z}}_p^2}$ for multisets of size $p+k$, when $k$ is not too large in terms of $p$.'
address:
- |
Department of Mathematics\
University of British Columbia\
Room 121, 1984 Mathematics Road\
Vancouver, BC, Canada V6T 1Z2
- |
Université Pierre-et-Marie Curie\
Faculté de Mathématiques\
4 place Jussieu\
75005\
Paris, France
- |
Department of Mathematics\
University of British Columbia\
Room 121, 1984 Mathematics Road\
Vancouver, BC, Canada V6T 1Z2
author:
- Greg Martin
- Alexis Peilloux
- 'Erick B. Wong'
title: 'Lower bounds for sumsets of multisets in ${{\mathbb Z}}_p^2$'
---
Introduction
============
Determining the number of elements in a particular abelian group that can be written as sums of given sets of elements is a topic that goes back at least two centuries. The most famous result of this type, involving the cyclic group ${{\mathbb Z}}_p$ of prime order $p$, was established by Cauchy in 1813 [@Cauchy] and rediscovered by Davenport in 1935 [@Dav1; @Dav2]:
Let $A$ and $B$ be subsets of ${{\mathbb Z}}_p$, and define $A+B$ to be the set of all elements of the form $a+b$ with $a\in A$ and $b\in B$. Then $\#(A+B) \ge \min\{ p, \#A + \#B - 1\}$.
The lower bound is easily seen to be best possible by taking $A$ and $B$ to be intervals, for example. It is also easy to see that the lower bound of $\#A + \#B - 1$ does not hold for general abelian groups ${{\mathbf G}}$ (take $A$ and $B$ to be the same nontrivial subgroup of ${{\mathbf G}}$). There is, however, a well-known generalization obtained by Kneser in 1953 [@Kne], which we state in a slightly simplified form that will be quite useful for our purposes (see [@Nat Theorem 4.1] for an elementary proof):
Let $A$ and $B$ be subsets of a finite abelian group ${{\mathbf G}}$, and let $m$ be the largest cardinality of a proper subgroup of ${{\mathbf G}}$. Then $\#(A+B) \ge \min\{ \#{{\mathbf G}}, \#A + \#B - m\}$.
Given a sequence $A = (a_1,\dots,a_k)$ of (not necessarily distinct) elements of an abelian group ${{\mathbf G}}$, a related result involves its [*sumset*]{} $\Sigma A$, which is the set of all sums of any number of elements chosen from $A$ (not to be confused with $A+A$, which it contains but usually properly): $$\Sigma A = \bigg\{ \sum_{j\in J} a_j \colon J\subseteq \{1,\dots,k\} \bigg\}.$$ (Note that we allow $J$ to be empty, so that the group’s identity element is always an element of $\Sigma A$.) When ${{\mathbf G}}={{\mathbb Z}}_p$, one can prove the following result by writing $\Sigma A = \{0, a_1\} + \cdots + \{ 0,a_k \}$ and applying the Cauchy–Davenport theorem inductively:
\[CD pre lemma\] Let $A=(a_1,\dots,a_k)$ be a sequence of nonzero elements of ${{\mathbb Z}}_p$. Then $\#\Sigma A \ge \min\{p,k + 1\}$.
This result can also be proved directly by induction on $k$, and in fact such a proof will discover why the order $p$ of the cyclic group must be prime (intuitively, the sequence $A$ could lie completely within a nontrivial subgroup). For a formal proof, see [@DEF Lemma 2]. Again the lower bound is easily seen to be best possible, by taking $a_1=\cdots=a_k$.
It is a bit misleading to phrase such results in terms of sequences, since the actual order of the elements in the sequence is irrelevant (given that we are considering only abelian groups). We prefer to use [*multisets*]{}, which are simply sets that are allowed to contain their elements with multiplicity. If we let $m_x$ denote the multiplicity with which the element $x$ occurs in the multiset $A$, then the definition of $\Sigma A$ can be written in the form $$\Sigma A = \bigg\{ \sum_{x\in{{\mathbf G}}} \delta_x x \colon 0\le \delta_x \le m_x \bigg\},$$ where $\delta_x x$ denotes the group element $x+\cdots+x$ obtained by adding $\delta_x$ summands all equal to $x$.
When using multisets, we should choose our notation with care: the hypotheses of such results tend to involve the total number of elements of the multiset $A$ counting multiplicity, while the conclusions involve the number of distinct elements of $\Sigma A$. Consequently, throughout this paper, we use the following notational conventions:
- $|S|$ denotes the total number of elements of the multiset $S$, counted with multiplicity;
- $\#S$ denotes the number of distinct elements of the multiset $S$, or equivalently the number of elements of $S$ considered as a (mere) set.
In this notation, Lemma \[CD pre lemma\] can be restated as:
\[CD lemma\] Let $A$ be a multiset contained in ${{\mathbb Z}}_p$ such that $0\notin A$. Then $\#\Sigma A \ge \min\{p,|A| + 1\}$.
The purpose of this paper is to improve, as far as possible, this lower bound for multisets contained in the larger abelian group ${{{\mathbb Z}}_p^2}$. We cannot make any progress without some restriction upon our multisets: if a multiset is contained within a nontrivial subgroup of ${{{\mathbb Z}}_p^2}$ (of cardinality $p$), then so is its sumset, in which case the lower bound $\min\{p,|A| + 1\}$ from Lemma \[CD lemma\] is the best we can do. Therefore we restrict to the following class of multisets. We use the symbol ${\text{\bf 0}}=(0,0)$ to denote the identity element of ${{{\mathbb Z}}_p^2}$.
\[valid definition\] A multiset $A$ contained in ${{{\mathbb Z}}_p^2}$ is called [*valid*]{} if:
- ${\text{\bf 0}}\notin A$; and
- every nontrivial subgroup contains fewer than $p$ points of $A$, counting multiplicity.
The exact number $p$ in the second condition has been carefully chosen: any nontrivial subgroup of ${{{\mathbb Z}}_p^2}$ is isomorphic to ${{\mathbb Z}}_p$, and so Lemma \[CD lemma\] applies to these nontrivial subgroups. In particular, any multiset $A$ containing $p-1$ nonzero elements of a nontrivial subgroup will automatically have that entire subgroup contained in its sumset $\Sigma A$, so allowing $p$ nonzero elements in a nontrivial subgroup would always be “wasteful”.
We believe that the following lower bound should hold for sumsets of valid multisets:
\[2d conjecture\] Let $A$ be a valid multiset contained in ${{{\mathbb Z}}_p^2}$ such that $p \le |A| \le 2p-3$. Then $\#\Sigma A \ge (|A|+2-p)p$. In other words, if $|A| = p+k$ with $0\le k\le p-3$, then $\#\Sigma A \ge (k+2)p$.
It is easy to see that this conjectured lower bound would be best possible: if $A$ is the multiset that contains the point $(1,0)$ with multiplicity $p-1$ and the point $(0,1)$ with multiplicity $k+1$, then the set $\Sigma A$ is precisely $\big\{ (s,t)\colon s\in{{\mathbb Z}}_p,\, 0\le t\le k+1 \big\}$, which has $(k+2)p$ distinct elements. Conjecture \[2d conjecture\] is actually part of a larger assertion (see Conjecture \[any d conjecture\]) concerning lower bounds for sumsets in ${{\mathbb Z}}_p^m$.
One of our results completely resolves the first two cases of this conjecture:
\[conjecture true for k tiny\] Let $p$ be a prime.
1. If $A$ is any valid multiset contained in ${{{\mathbb Z}}_p^2}$ with $|A| = p$, then $\#\Sigma A \ge 2p$.
2. Suppose that $p\ge5$. If $A$ is any valid multiset contained in ${{{\mathbb Z}}_p^2}$ with $|A| = p+1$, then $\#\Sigma A \ge 3p$.
It turns out that proving part (b) of the theorem requires a certain amount of computation for a finite number of primes (see the remarks following the proof of the theorem in Section \[thm proofs section\]). Extending the conjecture to larger values of $k$ would require, by our methods, more and more computation to take care of small primes $p$ as $k$ grows. However, we are able to establish the conjecture when $p$ is large enough with respect to $k$, or equivalently when $k$ is small enough with respect to $p$:
\[conjecture true for k small in terms of p\] Let $p$ be a prime, and let $2\le k\le \sqrt{p/(2\log p+1)}-1$ be an integer. If $A$ is any valid multiset contained in ${{{\mathbb Z}}_p^2}$ with $|A| = p+k$, then $\#\Sigma A \ge (k+2)p$.
A contrapositive version of Theorem \[conjecture true for k small in terms of p\] is also enlightening:
\[main corollary\] Let $p$ be a prime, and let $2\le k\le \sqrt{p/(2\log p+1)}-1$ be an integer. Let $A$ be a multiset contained in ${{{\mathbb Z}}_p^2}\setminus \{{\text{\bf 0}}\}$ with $|A| = p+k$. If $\#\Sigma A < (k+2)p$, then there exists a nontrivial subgroup of ${{{\mathbb Z}}_p^2}$ that contains at least $p$ points of $A$, counting multiplicity.
Our methods of proof stem from two main ideas. First, we will obviously exploit the structure of ${{{\mathbb Z}}_p^2}$ as a direct sum of cyclic groups of prime order, within which we can apply the known Lemma \[CD lemma\] after using projections. Section \[direct projects section\] contains several elementary lemmas in this vein (see in particular Lemma \[sweep to the left lemma\]). It is important for us to utilize the flexibility coming from the fact that ${{{\mathbb Z}}_p^2}$ can be decomposed as the direct sum of two subgroups in many different ways. Second, our methods work best when there exists a single subgroup that contains many elements of the given multiset; however, by selectively replacing pairs of elements with their sums, we can increase the number of elements in a subgroup in a way that improves our lower bounds upon the sumset (see Lemma \[any j lemma\]). These methods, which appear in Section \[thm proofs section\], combine to provide the proofs of Theorems \[conjecture true for k tiny\] and \[conjecture true for k small in terms of p\]. Finally, Section \[conjecture section\] contains a generalization of Conjecture \[2d conjecture\] to higher-dimensional direct sums of ${{\mathbb Z}}_p$, together with examples demonstrating that the conjecture would be best possible.
Sumsets in abelian groups and direct products {#direct projects section}
=============================================
All of the results in this section are valid for general finite abelian groups and have correspondingly elementary proofs, although the last two lemmas seem rather less standard than the first few. In this section, ${{\mathbf G}}$, ${{\mathbf H}}$, and ${{\mathbf K}}$ denote finite abelian groups, and $e$ denotes a group’s identity element.
\[a la carte lemma\] Let $B_0,B_1,B_2,\dots,B_j$ be multisets in ${{\mathbf G}}$, and set $A = B_0 \cup B_1 \cup \dots \cup B_j$. For each $1\le i\le j$, specify an element $x_i \in \Sigma B_i$, and set $C = B_0 \cup \{ x_1, \dots ,x_j\}$. Then $\Sigma C \subseteq \Sigma A$.
For each $1\le i\le j$, choose a submultiset $D_i \subseteq B_i$ such that the sum of the elements of $D_i$ equals $x_i$. By definition, every element $y$ of $\Sigma C$ equals the sum of the elements of some subset $E$ of $B_0$, plus $\sum_{i\in I} x_i$ for some $I\subseteq \{1,\dots,j\}$. But then $y$ equals the sum of the elements of $E \cup \bigcup_{i\in I} D_i$, which is an element of $\Sigma A$ since $E \cup \bigcup_{i\in I} D_i \subseteq B_0 \cup \bigcup_{1\le i\le j} B_i = A$.
\[kneser bound\] Let $A_1,A_2,\dots,A_j$ be multisets in ${{\mathbf G}}$, and set $A = A_1 \cup \dots \cup A_j$. If $m$ is the largest cardinality of a proper subgroup of ${{\mathbf G}}$, then either $\Sigma A = {{\mathbf G}}$ or $\#\Sigma A \ge (\sum_{i=1}^j \# \Sigma A_i) - (j-1)m$.
Since $\Sigma A = \Sigma A_1 + \Sigma A_2 + \cdots + \Sigma A_j$ (viewed as ordinary sets), this follows immediately by inductive application of Kneser’s theorem.
For the remainder of this section, we will be dealing with groups that can be decomposed into a direct sum.
A subgroup ${{\mathbf H}}$ of ${{\mathbf G}}$ is called an [*internal direct summand*]{} if there exists a subgroup ${{\mathbf K}}$ of ${{\mathbf G}}$ such that ${{\mathbf G}}$ is the internal direct sum of ${{\mathbf H}}$ and ${{\mathbf K}}$, or in other words, such that ${{\mathbf H}}\cap {{\mathbf K}}= \{e\}$ and ${{\mathbf H}}+ {{\mathbf K}}= {{\mathbf G}}$. Equivalently, ${{\mathbf H}}$ is an internal direct summand of ${{\mathbf G}}$ if there exists a [*projection homomorphism*]{} $\pi_{{\mathbf H}}\colon {{\mathbf G}}\to {{\mathbf H}}$ that is the identity on ${{\mathbf H}}$. Note that this projection homorphism does depend on the choice of ${{\mathbf K}}$ but is uniquely determined by $\pi_{{\mathbf H}}^{-1}(e) = {{\mathbf K}}$.
\[pi commutes with Sigma lemma\] For any homomorphism $f\colon {{\mathbf G}}\to {{\mathbf H}}$, and any subset $X$ of ${{\mathbf G}}$, we have $f(\Sigma X) = \Sigma (f(X))$. In particular, if ${{\mathbf H}}$ is an internal direct summand of ${{\mathbf G}}$, then $\pi_{{\mathbf H}}(\Sigma X) = \Sigma(\pi_{{\mathbf H}}(X))$ for any subset $X$ of ${{\mathbf G}}$.
Given $y\in f(\Sigma X)$, there exists $x\in \Sigma X$ such that $f(x)=y$. Hence we can find $x_1,\dots,x_j\in X$ such that $x_1+\cdots+x_j = x$, and so $f(x_1+\cdots+x_j) = y$. But $f$ is a homomorphism, and so $f(x_1)+\cdots+f(x_j) = y$, so that $y \in \Sigma(f(X))$. This shows that $f(\Sigma X) \subseteq \Sigma(f(X))$; the proof of the reverse inclusion is similar.
\[pi split lemma\] Let ${{\mathbf G}}= {{\mathbf H}}\oplus {{\mathbf K}}$, and let $D$ and $E$ be multisets contained in ${{\mathbf H}}$ and ${{\mathbf K}}$, respectively. For any $z\in {{\mathbf G}}$, $$z \in \Sigma(D\cup E) \quad\text{if and only if}\quad \pi_{{\mathbf H}}(z) \in\Sigma D \text{ and } \pi_{{\mathbf K}}(z) \in\Sigma E.$$
Since $z = \pi_{{\mathbf H}}(z) + \pi_{{\mathbf K}}(z)$, the “if” direction is obvious. For the converse, note that $$\pi_{{\mathbf H}}(z) \in \pi_{{\mathbf H}}\big( \Sigma (D\cup E) \big) = \Sigma\big( \pi_{{\mathbf H}}(D \cup E) \big)$$ by Lemma \[pi commutes with Sigma lemma\]. On the other hand, $\pi_{{\mathbf H}}(D) = D$ and $\pi_{{\mathbf H}}(E) = \{e\}$, and so $$\pi_{{\mathbf H}}(z) \in \Sigma\big( \pi_{{\mathbf H}}(D) \cup \pi_{{\mathbf H}}(E) \big) = \Sigma \big( D \cup \{e\} \big) = \Sigma D$$ (since the sumset is not affected by whether $e$ is an allowed summand). A similar argument shows that $\pi_{{\mathbf K}}(z) \in \Sigma E$, which completes the proof of the lemma.
\[direct product lemma\] Let ${{\mathbf H}}$ and ${{\mathbf K}}$ be subgroups of ${{\mathbf G}}$ satisfying ${{\mathbf H}}\cap {{\mathbf K}}= \{e\}$. Let $D$ and $E$ be multisets contained in ${{\mathbf H}}$ and ${{\mathbf K}}$, respectively. Then $\#\Sigma(D\cup E) = \#\Sigma D \cdot \#\Sigma E$.
Notice that every element of $\Sigma(D\cup E)$ is contained in ${{\mathbf H}}+{{\mathbf K}}$; therefore we may assume without loss of generality that ${{\mathbf G}}= {{\mathbf H}}\oplus {{\mathbf K}}$. In particular, we may assume that ${{\mathbf H}}$ and ${{\mathbf K}}$ are internal direct summands of ${{\mathbf G}}$, so that the projection maps $\pi_{{\mathbf H}}$ and $\pi_{{\mathbf K}}$ exist and every element $z\in {{\mathbf G}}$ has a unique representation $z=x+y$ where $x\in {{\mathbf H}}$ and $y\in {{\mathbf K}}$; note that $x=\pi_{{\mathbf H}}(z)$ and $y=\pi_{{\mathbf K}}(z)$ in this representation.
To establish the lemma, it therefore suffices to show that $z = \pi_{{\mathbf H}}(z) + \pi_{{\mathbf K}}(z) \in \Sigma(D\cup E)$ if and only if $\pi_{{\mathbf H}}(z) \in\Sigma D \text{ and } \pi_{{\mathbf K}}(z) \in\Sigma E$; but this is exactly the statement of Lemma \[pi split lemma\].
The next lemma is a bit less standard yet still straightforward: in a direct product of two abelian groups, it characterizes the elements of a sumset that lie in a given coset of one of the direct summands.
\[orthogonal structure lemma\] Let ${{\mathbf H}}$ and ${{\mathbf K}}$ be subgroups of ${{\mathbf G}}$ satisfying ${{\mathbf H}}\cap {{\mathbf K}}= \{e\}$. Let $D$ and $E$ be multisets contained in ${{\mathbf H}}$ and ${{\mathbf K}}$, respectively. For any $y\in {{\mathbf K}}$:
1. if $y\in \Sigma E$, then $({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E) = \Sigma D + \{y\}$;
2. if $y\notin \Sigma E$, then $({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E) = \emptyset$.
As in the proof of Lemma \[direct product lemma\], we may assume without loss of generality that ${{\mathbf G}}= {{\mathbf H}}\oplus {{\mathbf K}}$. Suppose that $z$ is an element of $({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E)$. Since $z\in {{\mathbf H}}+\{y\}$, we may write $z=x+y$ for some $x\in{{\mathbf H}}$, whence $\pi_{{\mathbf K}}(z) = \pi_{{\mathbf K}}(x) + \pi_{{\mathbf K}}(y) = e+y = y$. On the other hand, since $z\in \Sigma(D\cup E)$, we see that $y \in \Sigma E$ by Lemma \[pi split lemma\]. In other words, the presence of any element $z\in ({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E)$ forces $y\in \Sigma E$, which establishes part (b) of the lemma.
We continue under the assumption $y\in \Sigma E$ to prove part (a). The inclusions $\Sigma D + \{y\} \subseteq {{\mathbf H}}+\{y\}$ and $\Sigma D + \{y\} \subseteq \Sigma(D\cup E)$ are both obvious, and so $\Sigma D + \{y\} \subseteq ({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E)$. As for the reverse inclusion, let $z \in ({{\mathbf H}}+\{y\}) \cap \Sigma(D\cup E)$ as above; then $\pi_{{\mathbf H}}(z) \in \Sigma D$ by Lemma \[pi split lemma\], whence $z = \pi_{{\mathbf H}}(z) + \pi_{{\mathbf K}}(z) = \pi_{{\mathbf H}}(z) + y \in \Sigma D + \{y\}$ as required.
Finally we can establish the lemma that we will make the most use of when we return to the setting ${{\mathbf G}}={{{\mathbb Z}}_p^2}$ in the next section.
\[sweep to the left lemma\] Let ${{\mathbf G}}= {{\mathbf H}}\oplus {{\mathbf K}}$, and let $C$ be a multiset contained in ${{\mathbf G}}$. Let $D=C\cap {{\mathbf H}}$, let $F = C \setminus D$, and let $E = \pi_{{\mathbf K}}(F)$. Then $\#\Sigma C \ge \#\Sigma D \cdot \#\Sigma E$.
Lemma \[direct product lemma\] tells us that $\#\Sigma (D\cup E) = \#\Sigma D \cdot \#\Sigma E$, and so it suffices to show that $\#\Sigma C \ge \#\Sigma (D\cup E)$. We accomplish this by showing that $$\# \big( ({{\mathbf H}}+ \{y\}) \cap \Sigma C \big) \ge \# \big( ({{\mathbf H}}+ \{y\}) \cap \Sigma (D\cup E) \big)
\label{one line at a time}$$ for all $y\in{{\mathbf K}}$.
For any $y\in{{\mathbf K}}\setminus \Sigma E$, Lemma \[orthogonal structure lemma\] tells us that $({{\mathbf H}}+ \{y\}) \cap \Sigma (D\cup E) = \emptyset$, in which case the inequality holds trivially. For any $y \in \Sigma E$, Lemma \[orthogonal structure lemma\] tells us that $({{\mathbf H}}+ \{y\}) \cap \Sigma (D\cup E) = \Sigma D + \{y\}$, and so the right-hand side of the inequality equals $\#\Sigma D$.
On the other hand, since $\Sigma E = \Sigma (\pi_{{\mathbf K}}(F)) = \pi_{{\mathbf K}}(\Sigma F)$ by Lemma \[pi commutes with Sigma lemma\], there exists at least one element $z\in \Sigma F$ satisfying $\pi_{{\mathbf K}}(z)=y$; as ${{\mathbf G}}= {{\mathbf H}}\oplus {{\mathbf K}}$, this is equivalent to saying that $z \in {{\mathbf H}}+ \{y\}$. Since $\Sigma D \subseteq {{\mathbf H}}$, we have $\Sigma D + \{z\} \subseteq {{\mathbf H}}+ \{y\}$ as well. But the inclusion $\Sigma D + \{z\} \subseteq \Sigma D + \Sigma F = \Sigma C$ is trivial, and therefore $\Sigma D + \{z\} \subseteq ({{\mathbf H}}+ \{y\}) \cap \Sigma C$; in particular, the left-hand side of the inequality is at least $\#\Sigma D$. Combined with the observation that the right-hand side equals $\#\Sigma D$, this lower bound establishes the inequality and hence the lemma.
These lemmas might be valuable for studying sumsets in more general abelian groups. They will prove to be particularly useful for studying sumsets in ${{{\mathbb Z}}_p^2}$, however, essentially because there are many ways of writing ${{{\mathbb Z}}_p^2}$ as an internal direct sum of two subgroups (which are simply lines through ${\text{\bf 0}}$).
Lower bounds for sumsets {#thm proofs section}
========================
In this section we establish Theorems \[conjecture true for k tiny\] and \[conjecture true for k small in terms of p\]; the proofs employ two combinatorial propositions which we defer to the next section. It would be possible to prove these two theorems at the same time, at the expense of a bit of clarity; however, we find it illuminating to give complete proofs of Theorem \[conjecture true for k tiny\] (the cases $|A|=p$ and $|A|=p+1$) first, as the proofs will illustrate the methods used to prove the more general Theorem \[conjecture true for k small in terms of p\]. Seeing the limitations of the proof of Theorem \[conjecture true for k tiny\] will also motivate the formulation of our main technical tool, Lemma \[any j lemma\].
Throughout this section, $A$ will denote a valid multiset contained in ${{{\mathbb Z}}_p^2}$. For any $x\in{{{\mathbb Z}}_p^2}$, we let ${\langlex\rangle}$ denotes the subgroup of ${{{\mathbb Z}}_p^2}$ generated by $x$ (that is, the line passing through both the origin ${\text{\bf 0}}$ and $x$), and we let $m_x$ denote the multiplicity with which $x$ appears in $A$, so that $|A| = \sum_{x\in{{{\mathbb Z}}_p^2}} m_x$. The fact that $A$ is valid means that $m_{\text{\bf 0}}=0$ and $\sum_{t\in{\langlex\rangle}} m_t < p$ for every $x\in{{{\mathbb Z}}_p^2}{\setminus\{{\text{\bf 0}}\}}$.
Our first lemma quantifies the notion that we can establish sufficiently good lower bounds for the cardinality of $\Sigma A$ if we know that there are enough elements of $A$ lying in one subgroup of ${{{\mathbb Z}}_p^2}$. Naturally, the method of proof is to partition $A$ into the elements lying in that subgroup and all remaining elements, project the remaining elements onto a complementary subgroup, and then use Lemma \[CD lemma\] in each subgroup separately.
\[conjecture true if enough on line lemma\] Let $A$ be any valid multiset contained in ${{{\mathbb Z}}_p^2}$. Suppose that for some $x\in{{{\mathbb Z}}_p^2}{\setminus\{{\text{\bf 0}}\}}$, $$\label{symmetric bound}
\sum_{y\in{\langlex\rangle}} m_y \ge |A| - (p-1).$$ Then $\#\Sigma A \ge (|A|+2-p)p$.
The conclusion is worse than trivial if $|A| < p-1$; also, the fact that $A$ is valid means that the left-hand side of equation is at most $p-1$, and so the lemma is vacuous if $|A| > 2p-2$. Therefore in practice the lemma will be applied only to multisets $A$ satisfying $p-1 \le |A| \le 2p-2$.
Let $D = A \cap {\langlex\rangle}$; note that $|D| \le p-1$ since $A$ is a valid multiset, and note also that $|D| = \sum_{y\in{\langlex\rangle}} m_y \ge |A| - (p-1)$ by assumption. Set $F = A \setminus D$. Choose any nontrivial subgroup ${{\mathbf K}}$ of ${{{\mathbb Z}}_p^2}$ other than ${\langlex\rangle}$, and set $E = \pi_{{\mathbf K}}(F)$. Then by Lemma \[sweep to the left lemma\], we know that $\#\Sigma A \ge \#\Sigma D \cdot \#\Sigma E$. By Lemma \[CD lemma\] and the fact that ${\text{\bf 0}}\notin D\cup E$, we obtain $$\begin{aligned}
\#\Sigma A &\ge \min \big\{ p, 1 + |D| \big\} \cdot \min \big\{ p, 1 + |E| \big\} \notag \\
&= \min \big\{ p, 1 + |D| \big\} \cdot \min \big\{ p, 1 + |A| - |D| \big\},
\label{actually special case}\end{aligned}$$ since $|E| = |F| = |A|-|D|$. The inequalities $|D| \le p-1$ and $|A| - |D| \le p-1$ ensure that $p$ is the larger element in both minima, and so we have simply $$\#\Sigma A \ge (1+|D|)(1+|A|-|D|) = \tfrac14|A|^2 + |A| + 1 - \big( |D| - \tfrac12|A| \big)^2.$$ The pair of inequalities $|D| \le p-1$ and $|A| - |D| \le p-1$ is equivalent to the inequality $\big| |D| - \tfrac12|A| \big| \le p-1- \tfrac12|A|$; therefore $$|\Sigma A| \ge \tfrac14|A|^2 + |A| + 1 - \big( p-1- \tfrac12|A| \big)^2 = (|A|+2-p)p,$$ as claimed.
This lemma alone is sufficient to establish Theorem \[conjecture true for k tiny\].
When $|A|=p$, the right-hand side of the inequality equals 1, and so the inequality holds for any $x\in A$. Therefore Lemma \[conjecture true if enough on line lemma\] automatically applies, yielding $\#\Sigma A \ge (|A|+2-p)p = 2p$ as desired. (In fact essentially the same proof gives the more general statement: if $A$ is a multiset contained in ${{{\mathbb Z}}_p^2}\setminus\{{\text{\bf 0}}\}$ but not contained in any proper subgroup, and $|A|\ge p$, then $\#\Sigma A \ge 2|A|$.)
We are assuming that $|A| = p+1$. Suppose first that there exists a nontrivial subgroup of ${{{\mathbb Z}}_p^2}$ that contains at least two points of $A$ (including possibly two copies of the same point). Choosing any nonzero element $x$ in that subgroup, we see that the inequality is satisfied, and so Lemma \[conjecture true if enough on line lemma\] yields $\#\Sigma A \ge (|A|+2-p)p = 3p$ as desired.
From now on we may assume that there does not exist a nontrivial subgroup of ${{{\mathbb Z}}_p^2}$ that contains at least two points of $A$. Since there are only $p+1$ nontrivial subgroups of ${{{\mathbb Z}}_p^2}$, it must be the case that $A$ consists of exactly one point from each of these $p+1$ subgroups; in particular, the elements of $A$ are distinct. We can verify the assertion for $p\le11$ by exhaustive computation (see the remarks after the end of this proof), so from now on we may assume that $p\ge13$.
Suppose first that all sums of pairs of distinct elements from $A$ are distinct. All these sums are elements of $\Sigma A$, and thus $\#\Sigma A \ge \binom{p+1}2 >3p$ since $p\ge13$.
The only remaining case is when two pairs of distinct elements from $A$ sum to the same point of ${{{\mathbb Z}}_p^2}$. Specifically, suppose that there exist $x_1,y_1,x_2,y_2\in A$ such that $x_1+y_1=x_2+y_2$. Partition $A = B_0 \cup B_1 \cup B_2$ where $B_1=\{x_1,y_1\}$ and $B_2=\{x_2,y_2\}$ and hence $B_0 = A \setminus \{x_1,y_1,x_2,y_2\}$; note that this really is a partition of $A$, as the fact that $x_1+y_1=x_2+y_2$ forces all four elements to be distinct. Moreover, if we define $z=x_1+y_1=x_2+y_2$, then we know that $z\ne{\text{\bf 0}}$ since $x_1$ and $y_1$ are in different subgroups.
Define $C$ to be the multiset $B_0 \cup \{ z, z \}$; by Lemma \[a la carte lemma\], we know that $\#\Sigma A \ge \#\Sigma C$. Define $D = C \cap {\langlez\rangle}$; we claim that $|D| = 3$. To see this, note that $A$ has exactly one point in every nontrivial subgroup, and in particular $A$ has exactly one point in ${\langlez\rangle}$. Furthermore, that point cannot be $x_1$ for example, since then $y_1 = z-x_1$ would also be in that subgroup; similarly that point cannot be $x_2$, $y_1$, or $y_2$. We conclude that $B_0$ has exactly one point in ${\langlez\rangle}$, whence $C$ has exactly three points in ${\langlez\rangle}$.
Now define $F = C \setminus D$, so that $|F| = |C| - |D| = (|B_0|+2)-3 = (|A|-4+2)-3 = p-4$. Let ${{\mathbf K}}$ be any nontrivial subgroup other than ${\langlez\rangle}$, and set $E = \pi_{{\mathbf K}}(F)$. The lower bounds $\#\Sigma D \ge 4$ and $\#\Sigma E \ge p-3$ then follow from Lemma \[CD lemma\]. By Lemma \[sweep to the left lemma\], we conclude that $\#\Sigma C \ge \#\Sigma D \cdot \#\Sigma E = 4(p-3) > 3p$ since $p\ge13$.
The computation that verifies Theorem \[conjecture true for k tiny\](b) for $p\le11$ should be done a little bit intelligently, since there are $10^{12}$ subsets $A$ of ${{\mathbb Z}}_{11}^2$ (for example) consisting of exactly one nonzero element from each nontrivial subgroup. We describe the computation in the hardest case $p=11$. Let us write the elements of ${{\mathbb Z}}_{11}^2$ as ordered pairs $(s,t)$ with $s$ and $t$ considered modulo $11$. By separately dilating the two coordinates of ${{\mathbb Z}}_{11}^2$ (which does not alter the cardinality of $\Sigma A$), we may assume without loss of generality that $A$ contains both $(1,0)$ and $(0,1)$. We also know every such $A$ contains a subset of the form $\{ (i,i), (j,2j), (k,3k), (\ell,4\ell) \}$ for some integers $1\le i,j,k,\ell \le 10$. Therefore the cardinality of every such $\Sigma A$ is at least as large as the cardinality of one of the subsumsets $\Sigma \big( \{ (1,0), (0,1), (i,i), (j,2j), (k,3k), (\ell,4\ell) \} \big)$.
There are $10^4$ such subsumsets, and direct computation shows that all of them have more than $33$ distinct elements except for the sixteen cases $\Sigma \big( \{ (1,0), (0,1), \pm(1,1), \pm(1,2), \pm(1,3), \pm(1,4) \} \big)$, which each contain $32$ distinct elements. It is then easily checked that any subsumset of the form $\Sigma \big( \{ (1,0), (0,1), \pm(1,1), \pm(1,2), \pm(1,3), \pm(1,4), (m,5m) \} \big)$ with $1\le m\le10$ contains more than 33 distinct elements. This concludes the verification of Theorem \[conjecture true for k tiny\](b) for $p=11$, and the cases $p\le7$ are verified even more quickly.
We now foreshadow the proof of Theorem \[conjecture true for k small in terms of p\] by reviewing the structure of the proof of Theorem \[conjecture true for k tiny\](b). In that proof, we quickly showed that the desired lower bound held if there were enough elements of $A$ in the same subgroup. Also, the desired lower bound certainly held if there were enough distinct sums of pairs of elements of $A$. If however no subgroup contained enough elements of $A$ and there were only a few distinct sums of pairs of elements of $A$, then we showed that we could find multiple pairs of elements summing to the same point in ${{{\mathbb Z}}_p^2}$. Replacing those elements in $A$ with multiple copies of their joint sum, we found that the corresponding subgroup now contained enough elements to carry the argument through.
The following lemma quantifies the final part of this strategy, where we replace $j$ pairs of elements of $A$ with their joint sum and then use our earlier ideas to bound the cardinality of the sumset from below.
\[any j lemma\] Let $A$ be any valid multiset contained in ${{{\mathbb Z}}_p^2}$, and let $z\in{{{\mathbb Z}}_p^2}{\setminus\{{\text{\bf 0}}\}}$. For any integer $j$ satisfying $$0\le j \le \tfrac12 \sum_{t\in{{{\mathbb Z}}_p^2}\setminus{\langlez\rangle}} \min\{m_t,m_{z-t}\},
\label{allows choosing pairs}$$ we have $$\#\Sigma A \ge \min\bigg\{ p, 1 + j + \sum_{y\in{\langlez\rangle}} m_y \bigg\} \min\bigg\{ p, 1 + |A| - 2j - \sum_{y\in{\langlez\rangle}} m_y \bigg\}.$$
This can be seen as a generalization of Lemma \[conjecture true if enough on line lemma\], as equation is the special case $j=0$ of this lemma.
Partition $A = B_0 \cup B_1 \cup \cdots \cup B_j$, where for each $1\le i\le j$, the multiset $B_i$ has exactly two elements, neither contained in ${\langlez\rangle}$, that sum to $z$ (the complimentary submultiset $B_0$ is unrestricted). The upper bound for $j$ is exactly what is required for such a partition to be possible; the factor of $\frac12$ arises because the sum on the right-hand side of double-counts the pairs $(t,z-t)$ and $(z-t,t)$. Then set $C$ equal to $B_0$ with $j$ additional copies of $z$ inserted. By Lemma \[a la carte lemma\], we know that $\#\Sigma A \ge \#\Sigma C$.
Now let $D$ be the intersection of $C$ with the subgroup ${\langlez\rangle}$, and let $F = C \setminus D$. Let ${{\mathbf K}}$ be any nontrivial subgroup other than ${\langlez\rangle}$, and set $E = \pi_{{\mathbf K}}(F)$. By Lemma \[sweep to the left lemma\], we know that $\#\Sigma C \ge \#\Sigma D \cdot \#\Sigma E$. However, the number of elements of $D$ (counting multiplicity) is $j$ more than the number of elements of $B_0 \cap {\langlez\rangle}$; this is the same as $j$ more than the number of elements of $A \cap {\langlez\rangle}$ (since no elements of $B_1,\dots,B_j$ lie on ${\langlez\rangle}$), or in other words $j + \sum_{y\in{\langlez\rangle}} m_y$. Similarly, the number of elements of $E$ (equivalently, of $F$) is equal to the number of elements of $B_0 \setminus {\langlez\rangle}$; this is the same as $2j$ less than the number of elements of $A \setminus {\langlez\rangle}$, or in other words $|A| - 2j - \sum_{y\in{\langlez\rangle}} m_y$. The lower bounds $\#\Sigma D \ge \min\big\{ p, 1 + j + \sum_{y\in{\langlez\rangle}} m_y \big\}$ and $\#\Sigma E \ge \min\big\{ p, 1 + |A| - 2j - \sum_{y\in{\langlez\rangle}} m_y \big\}$ then follow from Lemma \[CD lemma\]; the chain of inequalities $\#\Sigma A \ge \#\Sigma C \ge \#\Sigma D \cdot \#\Sigma E$ establishes the lemma.
We are now ready to use Lemma \[any j lemma\] to establish Conjecture \[2d conjecture\] when $|A|=p+k$, for all but finitely many primes $p$ depending on $k$. Let $H_k = 1 + \tfrac12 + \cdots + \tfrac1k$ denote the $k$th harmonic number.
\[conjecture true for p large in terms of k\] Let $k\ge2$ be any integer, and let $A$ be any valid multiset contained in ${{{\mathbb Z}}_p^2}$ such that $|A| = p+k$. If $p\ge 4(k+1)^2 H_k - 2k$, then $\#\Sigma A \ge (k+2)p$.
Using the elementary bound $H_k \le \gamma + \log(k+1)$, where $\gamma$ denotes the Euler–Mascheroni constant, we see that Theorem \[conjecture true for p large in terms of k\] holds as long as $p \ge 4(k+1)^2 (\gamma + \log(k+1))$. Theorem \[conjecture true for k small in terms of p\] can thus be readily deduced from Theorem \[conjecture true for p large in terms of k\] as follows: If $k+1 \le \sqrt{p/(2\log p+1)}$ then $p \ge 4(k+1)^2 (\tfrac14 + \tfrac12 \log p)$. In this case we certainly have $p \ge (1 + 2\log 2) (k+1)^2$, whence $\log p \ge \frac45 + 2 \log(k+1)$ and $\tfrac14 + \tfrac12 \log p \ge \gamma + \log(k+1)$.
If there are $k+1$ elements of $A$ in some nontrivial subgroup, then we are done by Lemma \[conjecture true if enough on line lemma\]. Therefore we may assume that there are at most $k$ points in each subgroup; in particular, $m_x\le k$ for all $x\in{{{\mathbb Z}}_p^2}$. We now argue that if $\Sigma A$ is small, then there must be lots of pairs of elements of $A$ that add to the same element of ${{{\mathbb Z}}_p^2}$, at which point we will be able to invoke Lemma \[any j lemma\]. We may assume that $\Sigma A \ne {{{\mathbb Z}}_p^2}$, for otherwise we are done immediately.
For each $1 \le i \le k$, we define the level set $A_i = \{x \in {{{\mathbb Z}}_p^2}: m_x \ge i\}$. Notice that $A$ can be written precisely as the multiset union $A_1 \cup A_2 \cup \cdots \cup A_k$, and so $\sum_{i=1}^k \#A_i = |A| = p+k$. Let $B_i$ be the multiset formed by the sums of pairs of elements of $A_i$ not in the same subgroup: $$B_i = \big\{ x + y \colon x,y\in A_i,\, {\langlex\rangle}\ne{\langley\rangle} \big\}.$$ Note that ${\text{\bf 0}}\notin B_i$ (the restriction ${\langlex\rangle}\ne{\langley\rangle}$ ensures that $x \ne -y$) and that every element of $B_i$ occurs with even multiplicity (the restriction ${\langlex\rangle}\ne{\langley\rangle}$ ensures that $x \ne y$). It is not hard to estimate the relative sizes of $\#A_i$ and $|B_i|$: for each $x \in A_i$ there are at most $k$ elements of $A$ lying in the subgroup ${\langlex\rangle}$. Since each such $x$ occurs with multiplicity at least $i$ in $A$, there are at most $k/i$ distinct values of $y$ excluded by the condition ${\langlex\rangle} \ne {\langley\rangle}$. Hence $|B_i| \ge \#A_i (\#A_i - k/i)$, which implies that $$\label{bound on a_i}
\#A_i \le \frac{k}{i} + \sqrt{|B_i|}.$$ Since $\sum_{i=1}^k \#A_i$ is fixed, this shows that $|B_i|$ cannot be very small on average. At the same time, $\#B_i$ cannot get very large: if $\sum_{i=1}^k \#B_i \ge (2k+1)p$, then (under our assumption that $\Sigma A \ne {{{\mathbb Z}}_p^2}$) Lemma \[kneser bound\] already yields $$\#\Sigma A \ge \sum_{i=1}^k \#\Sigma A_i - (k-1)p > \sum_{i=1}^k \#B_i - (k-1)p \ge (k+2)p.$$ where the middle inequality holds because $B_i \subseteq \Sigma A_i$. We may therefore assume henceforth that $$\label{bound on b_i}
\sum_{i=1}^k \#B_i < (2k+1)p.$$
Let us now introduce the weighted height parameter $$\label{eta def}
\eta = \max_{1\le i \le k} \left\{ \frac{i|B_i|}{2\#B_i} : \#B_i > 0 \right\}.$$ We shall show shortly that $\eta > k+1$. Assuming so, then for some $1 \le i \le k$, we have $$\frac{|B_i|}{2\#B_i} > \frac{k+1}{i},$$ so by the pigeonhole principle, there exists some $z \in B_i$ (in particular $z\ne{\text{\bf 0}}$) occurring with multiplicity greater than $2(k+1)/i$; since this multiplicity is an even integer, it must be at least $2(k+2)/i.$ For each solution $x+y = z$ corresponding to an occurrence of $z$ in $B_i$, we have by construction that $x,y \notin {\langlez\rangle}$ and $m_x, m_y \ge i$, so for this particular choice of $z$, $$\tfrac12 \sum_{t \in {{{\mathbb Z}}_p^2}\setminus{\langlez\rangle} } \min\{m_t, m_{z-t}\} \ge k+2.$$ Furthermore, $\sum_{y\in{\langlez\rangle}} m_y \le k$ by assumption. Therefore we are free to apply Lemma \[any j lemma\] with $j = (k+2) - \sum_{y\in{\langlez\rangle}} m_y,$ which gives the lower bound $$\#\Sigma A \ge \min\{p,k+3\} \min\bigg\{p,p - k - 3 + \sum_{y\in{\langlez\rangle}} m_y\bigg\} \ge (k+3)(p-k-3) \ge (k+2)p$$ (the last step used the inequality $p \ge (k+3)^2$, which certainly holds under the hypotheses of the theorem).
It remains only to verify that $\eta > k+1$. Summing the inequality over all $1\le i\le k$ yields $$p+k = \sum_{i=1}^k \#A_i \le k H_k + \sum_{i=1}^k \sqrt{|B_i|} \le kH_k + \sqrt{2\eta} \sum_{i=1}^k \sqrt{\frac{\#B_i}{i}},$$ using the definition of $\eta$. We estimate the rightmost sum using Cauchy–Schwarz together with the inequality : $$\sum_{i=1}^k \sqrt{\frac{\#B_i}{i}} \le \bigg(\sum_{i=1}^k \#B_i\bigg)^{1/2}\bigg(\sum_{i=1}^k \frac1i\bigg)^{1/2} < \sqrt{(2k+1) p H_k}.$$ Combining the previous two inequalities gives $p+k - kH_k < \sqrt{ \eta(4k+2) p H_k}$, so that $$\eta > \frac{(p+k-kH_k)^2}{(4k+2) pH_k} > \frac{p(p+2(k-kH_k))}{(4k+2) pH_k} = \frac{(p+2k)-2kH_k}{(4k+2)H_k} \ge \frac{4(k+1)^2H_k-2kH_k}{(4k+2)H_k}$$ by the hypothesis on the size of $p$. In other words, $$\eta > \frac{2(k+1)^2-k}{2k+1} = k+1+\frac1{2k+1},$$ which completes the proof of the theorem.
A wider conjecture {#conjecture section}
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As mentioned earlier, Conjecture \[2d conjecture\] is just one part of a more far-reaching conjecture concerning sumsets of multisets in ${{\mathbb Z}}_p^m$. Before formulating that wider conjecture, we must expand the definition of a valid multiset to ${{\mathbb Z}}_p^m$.
\[more valid definition\] Let $p$ be an odd prime, and let $m$ be a positive integer. A multiset $A$ contained in ${{\mathbb Z}}_p^m$ is [*valid*]{} if:
- ${\text{\bf 0}}\notin A$; and
- for each $1\le d\le m$, every subgroup of ${{\mathbb Z}}_p^m$ that is isomorphic to ${{\mathbb Z}}_p^d$ contains fewer than $dp$ points of $A$, counting multiplicity.
When $m=1$, a multiset contained in ${{\mathbb Z}}_p$ is valid precisely when it does not contain $0$; when $m=2$ and $|A| < 2p$, this definition of valid agrees with Definition \[valid definition\] for multisets contained in ${{{\mathbb Z}}_p^2}$. Note that in particular, Definition \[more valid definition\](b) implies that every valid multiset contained in ${{\mathbb Z}}_p^m$ has at most $mp-1$ elements, counting multiplicity. We now give an example showing that this upper bound $mp-1$ can in fact be achieved. Throughout this section, let $\{x_1,\dots,x_m\}$ denote a generating set for ${{\mathbb Z}}_p^m$, and let ${{\mathbf K}}_d = {\langlex_1,\dots,x_d\rangle}$ denote the subgroup of ${{\mathbb Z}}_p^m$ generated by $\{x_1,\dots,x_d\}$, so that ${{\mathbf K}}_d \cong {{\mathbb Z}}_p^d$.
\[valid example\] Let $A_1$ be the multiset consisting of $p-1$ copies of $x_1$; for $2\le j\le m$ let $A_j = \{ x_j + ax_1 \colon 0\le a\le p-1 \}$; and define $B_m = \bigcup_{j=1}^m A_j$. Then $|B_m| = (p-1) + (m-1)p = mp-1$ and ${\text{\bf 0}}\notin B_m$. To verify that $B_m$ is a valid subset of ${{\mathbb Z}}_p^m$, let ${{\mathbf H}}$ be any subgroup of ${{\mathbb Z}}_p^m$ that is isomorphic to ${{\mathbb Z}}_p^d$; we need to show that $B_m$ contains fewer than $dp$ points of ${{\mathbf H}}$.
First suppose that $x_1\notin{{\mathbf H}}$, which implies that $bx_1\notin{{\mathbf H}}$ for every nonzero multiple $bx_1$ of $x_1$. Then for each $2\le j\le m$, at most one of the elements of $A_j$ can be in ${{\mathbf H}}$, since the difference of any two such elements is a nonzero multiple of $x_1$. Therefore $|B_m\cap{{\mathbf H}}| = \ell$ for some $1\le\ell\le m-1$, and in fact all $\ell$ of these elements are of the form $x_j+ax_1$ for $\ell$ distinct values of $j$. Since no such element is in the subgroup spanned by the others, we conclude that $d\ge\ell$, and so the necessary inequality $|B_m\cap{{\mathbf H}}|=\ell\le d<dp$ is amply satisfied.
Now suppose that $x_1\in{{\mathbf H}}$. Then for each $2\le j\le m$, either all or none of the elements of $A_j$ are in ${{\mathbf H}}$. By reindexing the $x_i$, we may choose an integer $1\le\ell\le m$ such that ${{\mathbf H}}$ contains $A_1 \cup \cdots \cup A_\ell$ and is disjoint from $A_{\ell+1} \cup \cdots \cup A_m$. In particular, $|B_m\cap{{\mathbf H}}| = (p-1)+(\ell-1)p = \ell p-1$. But ${{\mathbf H}}$ contains $\{x_1,\dots,x_\ell\}$ and hence $d\ge\ell$, so that $\ell p-1\le dp-1$ as required.
We may now state our wider conjecture; Conjecture \[2d conjecture\] is the special case $q=1$ of part (a) of this conjecture.
\[any d conjecture\] Let $p$ be an odd prime. Let $m$ be a positive integer, and let $A$ be a valid multiset of ${{\mathbb Z}}_p^m$ with $|A|\ge p$. Write $|A| = qp+k$ with $0\le k\le p-1$.
1. If $0\le k\le p-3$, then $\#\Sigma A \ge (k+2)p^q$.
2. If $k=p-2$, then $\#\Sigma A \ge p^{q+1}-1$.
3. If $k=p-1$, then $\#\Sigma A \ge p^{q+1}$.
In particular, if $|A| = mp-1$ then $\Sigma A = {{\mathbb Z}}_p^m$.
We remark that the quantity $dp$ in Definition \[more valid definition\], bounding the number of elements in a valid multiset that can lie in a rank-$d$ subgroup, has been carefully chosen in light of this conjecture: by Conjecture \[any d conjecture\](c), any valid multiset $A$ with at least $dp-1$ elements counting multiplicity must satisfy $\#\Sigma A \ge p^d$. In particular, if $A$ is a valid multiset contained in a subgroup ${{\mathbf H}}< {{\mathbb Z}}_p^m$ that is isomorphic to ${{\mathbb Z}}_p^d$, then $|A|\ge dp-1$ implies that $\Sigma A = {{\mathbf H}}$. Therefore allowing $dp$ elements in such a subgroup would always be “wasteful”. Of course, the validity of Definition \[more valid definition\] for rank-$d$ subgroups depends crucially upon the truth of Conjecture \[any d conjecture\](c) for $q=d-1$.
The conjecture is restricted to multisets $A$ with $|A|\ge p$ because we already know the truth for smaller multisets, for which the definition of “valid” is simply the condition that ${\text{\bf 0}}\notin A$: when $|A|\le p-1$, the best possible lower bound is $\#\Sigma A \ge |A|+1$ as in Lemma \[CD lemma\]. We remark that Peng [@P1 Theorem 2] has proved Conjecture \[any d conjecture\](c) in the case $m=2$ and $q=1$, under even a slightly weaker hypothesis; in other words, he has shown that if $A$ is a valid multiset contained in ${{{\mathbb Z}}_p^2}$ with $|A| = 2p-1$, then $\Sigma A = {{{\mathbb Z}}_p^2}$. (We remark that Mann and Wou [@MW] have proved in the case that $A$ is actually a set—that is, a multiset with distinct elements—that $\#A = 2p-2$ suffices to force $\Sigma A = {{{\mathbb Z}}_p^2}$.) Peng considers the higher-rank groups ${{\mathbb Z}}_p^m$ as well, but the multisets he allows (see [@P2 Theorem 1]) form a much wider class than our valid multisets, and so his conclusions are much weaker than Conjecture \[any d conjecture\] for $q\ge2$. Finally, we mention that we have completely verified Conjecture \[any d conjecture\] by exhaustive computation for the groups ${{\mathbb Z}}_p^2$ with $p\le 7$ and also for the group ${{\mathbb Z}}_3^3$.
It is easy to see that all of the lower bounds in Conjecture \[any d conjecture\](a), if true, would be best possible. Given $q\ge1$ and $0\le k\le p-3$, let $A'$ be any valid multiset contained in ${{\mathbf K}}_q$ with $|A'| = qp-1$ (such as the one given in Example \[valid example\] with $m=q$), and let $A$ be the union of $A'$ with $k+1$ copies of $x_{q+1}$. Then $\Sigma A = \{ y + ax_{q+1}\colon y\in \Sigma A',\, 0\le a\le k+1\}$ and thus $\#\Sigma A' = (k+2)\#\Sigma A \le (k+2)p^q$ since $\Sigma A$ is contained in ${{\mathbf K}}_q$. Similarly, the fact that there exists a valid multiset contained in ${{\mathbf K}}_{q+1}$ with $qp+(q-1)=(q+1)p-1$ elements (such as the one given in Example \[valid example\] with $m=q+1$) shows that the lower bound in Conjecture \[any d conjecture\](c) would be best possible, since the sumset of this multiset would still be contained in ${{\mathbf K}}_{q+1}$ and thus would have at most $p^{q+1}$ distinct elements.
The lower bound in Conjecture \[any d conjecture\](b) might seem counterintuitive, especially in comparison with the pattern established in Conjecture \[any d conjecture\](a). However, we can give an explicit example showing that the lower bound $p^{q+1}-1$ for $\#\Sigma A$ cannot be increased:
\[border example\] When $p$ is an odd prime, define $B'_m$ to be the set $B_m$ from Example \[valid example\] with one copy of $x_1$ removed, so that $B'_m$ contains $x_1$ with multiplicity only $p-2$. Since $B_m$ is a valid multiset contained in ${{\mathbb Z}}_p^m$, so is $B'_m$. We have $|B'_m| = |B_m|-1 = (mp-1)-1 = (m-1)p + (m-2)$, and we claim that $-x_1\notin \Sigma B'_m$; this will imply that $\#\Sigma B'_m \le p^m-1$, and so the lower bound for $\#\Sigma A$ in Conjecture \[any d conjecture\](b) cannot be increased. (In fact it is not hard to show that every other element of ${{\mathbb Z}}_p^m$ is in $\Sigma B'_m$, and so $\#\Sigma B'_m$ is exactly equal to $p^m-1$.)
Suppose for the sake of contradition that $-x_1\in \Sigma B'_m$, and let $C$ be a submultiset of $B'_m$ such that $-x_1 = \sum_{y\in C}y$. For each $2\le j\le m$, define $\ell_j = |C\cap A_j|= \#\big( C \cap \{ x_j + ax_1\colon 0\le a\le p-1 \} \big)$. Then $$-x_1 = \sum_{y\in C} y = t x_1 + \ell_2 x_2 + \ell_3 x_3 + \cdots + \ell_m x_m$$ for some integer $t$. It follows from this equation that each $\ell_j$ must equal either $0$ or $p$. However, if $\ell_j = p$ then $$\sum_{y \in C\cap A_j} y = \sum_{0\le a\le p-1} (x_j + ax_1) = px_j + \frac{p(p-1)}2 x_1 = {\text{\bf 0}}$$ (since $p$ is odd). So in either case, if $s = |C\cap A_1|$ is the multiplicity with which $x_1$ appears in $C$, then $$-x_1 = \sum_{y\in C} y = s x_1 + \sum_{j=2}^m \sum_{y\in C\cap A_j} y = s x_1 + {\text{\bf 0}}+ \cdots + {\text{\bf 0}}.$$ This is a contradiction, however, since $s$ must lie between $0$ and $p-2$. Therefore $-x_1$ is indeed not an element of $\Sigma B'_m$, as claimed.
The line of questioning in this section turns out to be uninteresting when $p=2$: when the multiset $A$ does not contain ${\text{\bf 0}}$, the condition that no rank-$1$ subgroup of ${{\mathbb Z}}_2^m$ contain $2$ points of $A$ is simply equivalent to $A$ not containing any element with multiplicity greater than $1$. It is easy to check that if $A$ consists of any $q$ points in ${{\mathbb Z}}_2^m$ that do not lie in any subgroup isomorphic to ${{\mathbb Z}}_2^{q-1}$, then $\Sigma A$ fills out the entire rank-$q$ subgroup generated by $A$. In other words, the analogous definition of “valid” for multisets in ${{\mathbb Z}}_2^m$ would simply be a set of $q$ points that generate a rank-$q$ subgroup of ${{\mathbb Z}}_2^m$, and we would always have $\#\Sigma A = 2^{|A|} = 2^{\#A}$ for valid (multi)sets in ${{\mathbb Z}}_2^m$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The collaboration leading to this paper was made possible thanks to Jean–Jacques Risler, Richard Kenyon, and especially Ivar Ekeland; the authors also thank the University of British Columbia and the Institut d’Études Politiques de Paris for their undergraduate exchange program. The first author thanks Andrew Granville for conversations that explored this topic and eventually led to the formulation of the conjectures herein.
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|
---
author:
- 'G. Mulas'
- 'G. Malloci'
- 'C. Joblin'
- 'D. Toublanc'
date: 'Received ?; accepted ?'
title: 'Diagnostics for *specific* PAHs in the farIR: searching neutral naphthalene and anthracene in the Red Rectangle'
---
[*Context*. In the framework of the interstellar polycyclic aromatic hydrocarbons (PAHs) hypothesis, farIR skeletal bands are expected to be a fingerprint of single species in this class.\
*Aims*. We address the question of detectability of low energy PAH vibrational bands, with respect to spectral contrast and intensity ratio with “classical” Aromatic Infrared Bands (AIBs).\
*Methods*. We extend our extablished MonteCarlo model of the photophysics of specific PAHs in astronomical environments, to include rotational and anharmonic band structure. The required molecular parameters were calculated in the framework of the Density Functional Theory.\
*Results*. We calculate the detailed spectral profiles of three lowenergy vibrational bands of neutral naphthalene, and four lowenergy vibrational bands of neutral anthracene. They are used to establish detectability constraints based on intensity ratios with “classical” AIBs. A general procedure is suggested to select promising diagnostics, and tested on available Infrared Space Observatory data for the Red Rectangle nebula.\
*Conclusions*. The search for single, specific PAHs in the farIR is a challenging, but promising task, especially in view of the forthcoming launch of the Herschel Space Observatory. ]{}
Introduction
============
The presence of polycyclic aromatic hydrocarbons (PAHs) in the interstellar medium (ISM) was proposed by @leg84 and @all85, to account for the socalled “Aromatic Infrared Bands” (AIBs), a set of emission bands observed near 3.3, 6.2, 7.7, 8.6, 11.3 and 12.7 $\mu$m, in many dusty environments excited by UV photons [@leg89; @all89]. Such “classical” AIBs do not permit an unambiguous identification of any single PAH, since they just probe specific chemical bonds and not its overall structure. Indeed, despite the impressive amount of work devoted to this subject over the years, no *definitive* spectral identification of any *specific* individual member in this class exists to date. On the other hand, every single such PAH ought to show a unique spectral fingerprint in the farIR spectral region, which contains the lowfrequency vibrational modes associated with collective oscillations of the whole skeletal structure of the molecule [@zha96; @lan96; @sal99b; @job02; @mul03; @mul06; @mul06b].
In @mul06b we presented and validated a procedure to model in a systematic way the photophysics of PAHs in photon dominated regions. In that work, we computed the complete farIR emission spectrum of a sample of 20 molecules and their cations in three radiation fields, covering some typical astronomical environments in which AIBs are observed. We concluded that the main problem for the detection and identification of such bands is likely to be spectral confusion and poor contrast against a strong background continuum: perpendicular bands, which are expected to display sharp Q branches, are much favoured with respect to shallower parallel bands.
Our approach can also model molecular rotation and thus obtain the expected band rotational envelopes of both farIR emission bands [@job02] and visible absorption bands [@mal03c]. It can also estimate the contributions of fundamental and hot bands, with their distribution of anharmonic shifts, which affect the expected band profile. This makes modelling much heavier: if $\mathrm{J}_\mathrm{max}$ is the maximum angular momentum which is significantly populated, $\mathrm{J}_\mathrm{max}\left(\mathrm{J}_\mathrm{max}+1\right)$ is the number of rotational levels which must be traced, during the simulation, for each vibrational mode. Since for the molecules considered $\mathrm{J}_\mathrm{max}$ is a few hundred [@rou97; @mul98], this makes the simulation $\mathrm{J}_\mathrm{max}\left(\mathrm{J}_\mathrm{max}+1\right)~\sim~10^5$ times more demanding from a computational point of view. Furthermore, it requires the knowledge of the effective rotational constants as a function of vibrational state and of the anharmonic correction for hot bands.
To elucidate the role of spectral shape on band detectability, we present here, as a test case, a detailed calculation for three lowfrequency bands of the simplest PAH, namely neutral naphthalene (C$_{10}$H$_{8}$), and for four lowfrequency bands of the next larger PAH in the group of oligoacenes, i. e. neutral anthracene. (C$_{14}$H$_{10}$). In particular, for naphthalene we selected the lowest energy atype, btype and ctype transition, whose fundamental wavelengths are calculated to be respectively at 15.82, 27.74 and 58.56 $\mu$m; for anthracene, we selected the lowest energy atype (43.68 $\mu$m) and ctype (21.26, 26.35 and 110.23 $\mu$m) bands.
In the following Sect. \[modelling\] we present our modelling approach. Results are then presented in Sect. \[results\] and discussed in Sect. \[discussion\].
Modelling approach {#modelling}
==================
Our MonteCarlo modelling procedure is described in detail elsewhere [@mul98; @job02; @mal03c; @mul03; @mul06; @mul06b], as well as the basic molecular parameters it requires. We here use exactly the same photoabsorption crosssections and vibrational analyses of neutral napthtalene and neutral anthracene as in @mul06b, where their applicability and accuracy is discussed in detail. Since this is a proofofconcept work, we restricted ourselves to one single wellknown environment, i. e. the Red Rectangle (RR) halo, using the same radiation field (RF) previously adopted for it in @mul06, derived from the observational work of @vij05.
We here additionally studied in detail the distribution of anharmonic shifts arising from the superposition of a large number of hot bands on top of the fundamental band, in the stochastic process of PAH relaxation via IR emission. Moreover, we also calculated detailed rotational profiles obtained from statistical equilibrium.
To do this we needed the previously unavailable anharmonic correction to vibrational terms and rotational constants, and the Coriolis vibrationrotation coupling, which we computed using the <span style="font-variant:small-caps;">Gaussian03</span> quantum chemistry package [@g03]. This implements the Van Vleck perturbative approach to the above problem [e. g. @cla88], which we applied in the framework of the Density Functional Theory, with the exchangecorrelation functional B3LYP [@bec93; @ste94] and the gaussian basis set [@fri84]. While basis set convergence is not achieved yet at this level of theory, it is good enough to yield sensible results. Indeed, benchmark calculations showed that smaller basis sets are required to obtain good perturbative corrections than are needed for equilibrium geometry properties and harmonic vibrational analyses [@bar04; @bar05].
The anharmonic vibrational analysis yields the secondorder anharmonic corrections to the vibrational energy levels, which are represented, in wavenumbers, by the formula [see e. g. @bar05 and references therein]: $$\mathrm{E}_\mathrm{vib}(n_1,\ldots, n_N) = \chi_0+\!\!
\sum_i\left(n_i\!+\!\frac{1}{2}\right)\,\overline{\omega_i}
+\!\!\sum_{i,j\geq i} \chi_{ij}\,\left(n_i\!+\!\frac{1}{2}\right)
\left(n_j\!+\!\frac{1}{2}\right),$$ where $\chi_0$ and $\chi_{ij}$ are the vibrational anharmonic constants as defined in @bar95 with the corrections reported in @bar04. The values of $\chi_{ij}$ which we obtained are listed in Tables \[chinaphthalene\] and \[chianthracene\] in the Appendix for the specific bands we considered here.
Accidental near degeneracies, when they occur, lead to additional corrections (i. e. Fermi and DarlingDennison resonances), which are not systematic and must thus be evaluated, if relevant, on a case by case basis. For the three specific benchmark bands which we modelled here for neutral naphthalene, the levels from which emission is estimated to occur significantly (fundamental and first few hot bands) are unperturbed by such resonances. For bands for which they do occur, in the case of neutral naphthalene and anthracene they are smaller than anharmonic corrections anyway, so that the qualitative conclusions we may draw from our test cases remain valid.
To leading order, selection rules are the same as in the harmonic approximation, i. e. they connect levels in which a single IRactive mode changes by one quantum. The energy difference $$\Delta\mathrm{E}_k\left(n_1,\ldots,n_N\right) =
\mathrm{E}_\mathrm{vib}\left(\ldots,n_k+1,\ldots\right)-
\mathrm{E}_\mathrm{vib}\left(\ldots,n_k,\ldots\right)$$ for a transition in which only the $k^\mathrm{th}$ quantum number changes as $n_k+1 \to n_k$ is hence given by $$\Delta\mathrm{E}_k\left(n_1,\ldots,n_N\right) =
\overline{\omega_k} + 2\chi_{kk}\left(n_k+1\right) +
\sum_{i\neq k}\chi_{ik} \left(n_i+\frac{1}{2}\right),$$ which may be rewritten as $$\Delta\mathrm{E}_k\left(n_1,\ldots,n_N\right) = \Delta\mathrm{E}_k\left(0,\ldots,0\right) +
2\chi_{kk}n_k+\sum_{i\neq k}\chi_{ik}n_i.$$ This equation shows that anharmonicity produces a shift in the hot bands, with respect to the fundamental, which is proportional to the vibrational quantum numbers in the lower energy state involved in the transition. Since IR emission by PAHs results from a vibrational cascade, a given band will result in the superposition of the fundamental plus a possibly large number of hot bands, their contributions depending on the detailed statistics of the process. This in turn results in a distribution of anharmonic shifts, which convolves the pure (nonthermal) rotational envelope of the band. Both effects are sampled by our MonteCarlo procedure.
On top of this, each single transition, due to higher order perturbative effects, turns into the superposition of transitions including a distribution of nearby states. This produces a lorentzian envelope, i. e. lifetime broadening, with a width which roughly scales with the density of vibrational states at the energy at which the transition occurs. Since bands in the lowenergy modes are emitted near the end of the vibrational cascades (see Figs. \[naphthists\] and \[anthrhists\]), below or slightly above the decoupling energy [we assumed E$_\mathrm{dec}~\simeq~0.32$ eV for neutral naphthalene and 0.21 eV for neutral anthracene, @mul06b], we neglect lifetime broadening in our present simulation. Specifically, two of the three bands we modelled for naphthalene and three of the four for anthracene are emitted virtually *only* below E$_\mathrm{dec}$; only the highest energy band we considered for each molecule is emitted in a non negligible fraction at excitation energies above E$_\mathrm{dec}$. Moreover, comparison with experimental spectra of gasphase naphthalene at room temperature show a rather good correspondence of the band near $\sim$12.8 $\mu$m with a theoretical spectrum we calculated under these same assumptions [@pir06].
![Probability distribution of the excitation energies at which photons are emitted in the three bands considered of neutral naphthalene in the RR halo.[]{data-label="naphthists"}](naphthist.eps){width="\hsize"}
![Same as Fig. \[naphthists\] for the four bands considered of neutral anthracene.[]{data-label="anthrhists"}](anthrhist.eps){width="\hsize"}
As to the change of the effective rotational constants as a function of vibrational state, to leading order they have a linear dependence on vibrational quantum numbers, i. e. $$\mathrm{A}_\mathrm{eff}\left(n_1,\ldots,n_N\right) =
\mathrm{A}_\mathrm{eff}\left(0,\ldots,0\right)-\sum_i\mathrm{a}_in_i$$ and equivalent relations for $\mathrm{B}_\mathrm{eff}$ and $\mathrm{C}_\mathrm{eff}$. The $\mathrm{a}_i$, $\mathrm{b}_i$ and $\mathrm{c}_i$ constants (the socalled $\alpha$ matrix), which include three leading contributions from harmonic, anharmonic and Coriolis terms in the molecular Hamiltonian, are almost all very small for naphthalene and anthracene, producing changes in the range of a few parts in a thousand in the effective rotational constants. We report them in Tables \[vibrot\_alpha\_naphthalene\] and \[vibrot\_alpha\_anthracene\] in the Appendix.
Only in two cases, for naphthalene, the accidental near resonance of the two vibrational modes respectively at 15.82 and 15.90 $\mu$m and at 19.49 and 19.54 $\mu$m produces large Coriolis coupling terms between them, resulting in a $\sim4\%$ change in the affected rotational constant as a function of the number of quanta in that vibrational state. In the case of anthracene, the largest change in the rotational constants is less than 0.9%, again due to an accidental near resonance between the modes at 25.45 and 26.62 $\mu$m.
Results
=======
Calculated band profiles {#profiles}
------------------------
The three bands of neutral naphthalene for which we modelled the detailed rovibrational and anharmonic structure are one of each type, i. e. an atype, a btype and a ctype band respectively. They are two inplane and one outofplane (the latter being the socalled “butterfly” or “floppying”) bending modes, and each of them is the lowest frequency transition of its type. For neutral anthracene we modelled the four lowest frequency bands, which are, in order of decreasing energy, two ctype, one atype and another ctype.
Since the lowest energy butterfly mode of both molecules falls in the spectral range of the Long Wavelength Spectrograph (LWS) on board ISO, we convolved their synthetic spectra with the average resolving power of LWS with (R$\simeq$8250) and without (R$\simeq$175) the FabryPerot filter, assuming a velocity dispersion $\lesssim5$ km s$^{-1}$. Similarly, since all other bands fall in the spectral range covered by the Short Wavelength Spectrograph (SWS) of ISO, we convolved their synthetic spectra with the resolving power of SWS with (R$\simeq$30000) and without (R$\simeq$1500) the FabryPerot filter, with the same velocity dispersion. For details on SWS and LWS, see their manuals on the official ISO web page[^1]. The resulting spectra are shown in figures \[naphtha\_n\_39\] to \[anthra\_n\_66\].
![Detailed rotational structure of the band at 15.82 $\mu$m of neutral naphthalene in the RR halo. Strong Cariolis coupling with the IRinactive band at 15.90 $\mu$m causes one of the rotational constants to differ by $\sim4\%$ between the upper and lower vibrational states involved in the transition, producing a blue shaded rotational profile. The superposition of hot bands, which has a redshaded envelope, acts in the opposite direction, so that the overall profile including both effects is almost symmetric. The two panels show the band as would be seen with ISOSWS with and without the FabryPerot filter. The vertical dotted line marks the position of the origin of the fundamental band.[]{data-label="naphtha_n_39"}](naphtha_n_39_sws.eps){width="\hsize"}
![Same as Fig. \[naphtha\_n\_39\] for the naphhalene band at 27.74 $\mu$m. The rotational constants are almost unchanged in the vibrational transition, resulting in a very symmetric rotational envelope, the asymmetry being mainly due to the superposition of slightly displaced sovratones of the band. []{data-label="naphtha_n_46"}](naphtha_n_46_sws.eps){width="\hsize"}
![Same as Fig. \[naphtha\_n\_39\] for the naphthalene band at 58.56 $\mu$m. The rotational constants are almost unchanged in the vibrational transition, resulting in a very symmetric rotational envelope, with a Q branch standing out very clearly. The two panels show the band as would be seen with ISOLWS with and without the FabryPerot filter. In the lower panel, the higher resolving power allows the resolution, in the Q branch, of the separate contributions of different hot bands, with slightly different anharmonic shifts (zoomed inset). []{data-label="naphtha_n_48"}](naphtha_n_48_lws_composite.eps){width="\hsize"}
![Detailed rotational structure of the band at 21.26 $\mu$m of neutral anthracene in the RR halo. The rotational constants are almost unchanged in the vibrational transition, resulting in a very symmetric rotational envelope, with a visible Q branch whose width is due to the superposition of hot bands with different anharmonic shifts. The two panels show the band as would be seen with ISOSWS with and without the FabryPerot filter. The vertical dotted lines marks the position of the origin of the fundamental band. []{data-label="anthra_n_58"}](anthra_n_58_sws.eps){width="\hsize"}
![Same as Fig. \[anthra\_n\_58\] for the anthracene band at 26.35 $\mu$m. The rotational constants are almost unchanged in the vibrational transition, resulting in a very symmetric rotational envelope, with a sharp Q branch standing clearly out in the middle. In the lower panel, the higher resolving power allows the resolution, in the Q branch, of the separate contributions of different hot bands, with slightly different anharmonic shifts (zoomed inset). []{data-label="anthra_n_61"}](anthra_n_61_sws_composite.eps){width="\hsize"}
![Same as Fig. \[anthra\_n\_58\] for the anthracene band at 43.68 $\mu$m. The relatively large change in the rotational constants in the vibrational transition results in a visibly redshaded band. []{data-label="anthra_n_64"}](anthra_n_64_sws.eps){width="\hsize"}
![Same as Fig. \[anthra\_n\_58\] for the anthracene band at 110.23 $\mu$m. The change in rotational constants, combined with the superposition of hot bands, produces an asymmetric profile, in the middle of which a sharp Q branch clearly stands out. In the lower panel, the higher resolving power allows the resolution, in the Q branch, of the separate contributions of different hot bands, with slightly different anharmonic shifts (zoomed inset).[]{data-label="anthra_n_66"}](anthra_n_66_lws_composite.eps){width="\hsize"}
There is clearly a qualitative difference between perpendicular and parallel bands, i. e. the presence of a central Q branch containing about 20% of the total flux in the band. Such difference is very apparent in the bands at 58.56 $\mu$m of naphthalene and at 26.35 $\mu$m of anthracene: in these cases, the rotational constants are very nearly the same in the two vibrational modes involved in the transition, which makes the resulting Q branches extremely sharp, with a full width at half maximum (FWHM) of the order of $\sim$0.15 cm$^{-1}$. Moreover, in the absence of shading[^2], neither the P nor the R branch is inverted, and the Q branch sits in the gap between them, with little contamination. The superposition of the fundamental and hot bands produces a well defined pattern of anharmonic shifts, clearly visible in the zoomed inset in the lower panel of Figs. \[naphtha\_n\_48\], \[anthra\_n\_61\] and \[anthra\_n\_66\]. With the low resolution mode of LWS, the fundamental and hot bands merge in an unique, unresolved, redshaded Q branch, with a FWHM corresponding to the resolving power of the instrument. The P and R branches have a FWHM of the order of $\sim$8 cm$^{-1}$.
In the band at 21.26 of anthracene, instead, the larger difference in rotational constants between the states involved in the transition and the pattern of anharmonic shifts of the hot bands effectively masks the central Q branch, which is just barely visible in the high resolution panel of Fig. \[anthra\_n\_58\].
The parallel bands of naphthalene are resolved in only two welldefined features, which in the btype one at 27.74 $\mu$m include a split Q branch [@her91b]. The width of these partially resolved features is approximately $\sim$6 cm$^{-1}$. As to the parallel band of anthracene at 43.68 $\mu$m, relatively strong red shading and anharmonicity combine to produce a single, unresolved, asymmetric feature with an approximate FWHM of $\sim$6 cm$^{-1}$.
These differences in band shapes, which depend on the interplay of many molecular parameters, have direct consequences on their detectability.
Comparison with observations {#observations}
----------------------------
We here examine in detail the case of the RR nebula, which is one of the reference targets for the observation of AIBs and thus a prime candidate for the identification of specific PAHs in space. ISO data for this object are available in the online archive.
Under optically thin conditions, the observed spectrum of a given band produced by a given PAH, for instance neutral naphthalene, is $$\label{eq1}
\frac{dF}{d\lambda} = \frac{dF_\mathrm{cont}}{d\lambda} +
\int dr\,d\Omega\,n_\mathrm{naph.}\frac{d\mathcal{P}}{d\lambda},$$ where $\displaystyle \frac{dF_\mathrm{cont}}{d\lambda}$ is the underlying continuum spectrum, $\displaystyle \frac{d\mathcal{P}}{d\lambda}$ is the power isotropically emitted in the band by one molecule for the assumed RF, $n_\mathrm{naph.}$ is its number density and the double integral is along the line of sight and over the solid angle observed. As explained in detail in @mul06b, the IR spectrum emitted by a species in a given regime scales linearly with the intensity of the RF. Since extinction in the RR halo is negligible [@vij05], the RF will change only due to dilution with the position in the nebula and such scaling relations are valid in this case. Hence, we can factor $\displaystyle \frac{d\mathcal{P}}{d\lambda}$ as $$\frac{d\mathcal{P}}{d\lambda} =
\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\,\Lambda\left(r,\Omega\right),$$ where $\Lambda\left(r,\Omega\right)$ is an adimensional scaling factor, independent of $\lambda$, which takes into account the variation of RF intensity with position in the source and $\displaystyle \frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}$ is independent of $r$ and $\Omega$. We may thus rewrite Eq. (\[eq1\]) as $$\begin{aligned}
\label{eq2}
\frac{dF}{d\lambda} & = & \frac{dF_\mathrm{cont}}{d\lambda} +
\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}
\int dr\,d\Omega\,n_\mathrm{naph.}\,\Lambda\left(r,\Omega\right)\nonumber \\
& = & \frac{dF_\mathrm{cont}}{d\lambda}+
\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\,\Upsilon_\mathrm{naph.},\end{aligned}$$ where $\Upsilon_\mathrm{naph.}$, defined above, has the dimensions of a column density times a solid angle and is independent of $\lambda$. Classical AIBs are commonly considered as typical tracers of PAHs. Using Eq. (\[eq2\]) and integrating over one AIB, we obtain $$\begin{aligned}
%\eta_\mathrm{AIB}\,F_\mathrm{AIB}
%& = & \Upsilon_\mathrm{naph.}\,\,\mathcal{P}_\mathrm{AIB}
%& = & \Upsilon_\mathrm{naph.} \int_\mathrm{AIB}d\lambda \frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}
%\nonumber \\
%& = & \int_\mathrm{AIB}d\lambda\left(\frac{dF}{d\lambda}-\frac{dF_\mathrm{cont}}{d\lambda}\right)
\int_\mathrm{AIB}d\lambda\left(\frac{dF}{d\lambda}-\frac{dF_\mathrm{cont}}{d\lambda}\right) & = &
\Upsilon_\mathrm{naph.} \int_\mathrm{AIB}d\lambda \frac{d\mathcal{P}_\mathrm{ref}}{d\lambda} \nonumber \\
& = & \Upsilon_\mathrm{naph.}\,\,\mathcal{P}_\mathrm{AIB} \nonumber =
\eta_\mathrm{AIB}\,F_\mathrm{AIB},\nonumber\end{aligned}$$ where $\eta_\mathrm{AIB}$ is the fraction of the flux in the AIB produced by the given molecule and $\displaystyle \mathcal{P}_\mathrm{AIB}$ is the result of our model for the reference RF and the chosen band. Solving the above equation for $\Upsilon_\mathrm{naph.}$, we get $$\label{eq3}
\Upsilon_\mathrm{naph.} = \eta_\mathrm{AIB}\,\,\frac{F_\mathrm{AIB}}{\mathcal{P}_\mathrm{AIB}}.$$ Let’s now focus on another IR band of the same specific PAH. To be detectable, it must stand above the continuum by more than the noise level, i. e., for a $2\textendash\sigma$ detection: $$\frac{\displaystyle \frac{dF_\mathrm{peak}}{d\lambda}-\frac{dF_\mathrm{cont}}{d\lambda}}
{\displaystyle \frac{dF_\mathrm{cont}}{d\lambda}} \gtrsim \frac{2}{S},$$ $S$ being the signal to noise ratio, from which follows, using Eq. (\[eq2\]) and then Eq. (\[eq3\]), $$\frac{\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}
\Upsilon_\mathrm{naph.}} {\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}} =
\frac{\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}
\eta_\mathrm{AIB}\,\,\frac{F_\mathrm{AIB}}{\mathcal{P}_\mathrm{AIB}}}
{\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}} \gtrsim \frac{2}{S}.$$ In the equation above, we implicitly assumed both this latter band and the reference AIB are integrated over the same aperture on the sky. For ISO observations of the RR, this is a good assumption since RR is completely contained both in the SWS and in the LWS entrance apertures [@bre03; @men02]. Solving for $\eta_\mathrm{AIB}$, we finally obtain that the band will be detected only if $$\label{etalimit}
\eta_\mathrm{AIB} \gtrsim \frac{2}{S}\,\,\frac{\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}}
{\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}}
\,\,\frac{\mathcal{P}_\mathrm{AIB}}{F_\mathrm{AIB}}.$$ This derivation is completely general, despite the fact that we wrote it for neutral naphthalene and applies to any specific band of any specific PAH. We can now put the numbers for the three bands we studied in detail for neutral naphthalene. Upon examination of the previously calculated spectra, the strongest emission bands of neutral naphthalene in the RR are the inplane CH stretch at $\sim3.3~\mu$m and the outofplane CH bend at $\sim12.7~\mu$m; the fraction of the total IR emission of neutral naphthalene in such bands is calculated to be respectively 58.9% and 26.5%. From observations of the RR available from the online ISO database, their integrated band intensities amount to $F_\mathrm{3.3}$ = 7.35 10$^{-17}$ W cm$^{-2}$ and $F_\mathrm{12.7}$ = 3.51 10$^{-17}$ W cm$^{-2}$. From our model we have, for neutral naphthalene, $\mathcal{P}_\mathrm{3.3}$ = 2.42 10$^{-26}$ W sr$^{-1}$ and $\mathcal{P}_\mathrm{12.7}$ = 9.08 10$^{-27}$ W sr$^{-1}$, while the values of S and $\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}$ estimated from ISO archive data and those of $\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}$ calculated by our model (see Figs. \[naphtha\_n\_39\] to \[naphtha\_n\_48\]) are listed in Table \[pupamela\], along with the resulting detection limits at $\sim$2$\sigma$ level. Since $\eta_\mathrm{AIB}$ was defined as the fraction of the flux observed in a given AIB which is produced by this specific molecule, a detection limit $\eta_\mathrm{AIB}~>~1$ means that the corresponding band (1$^\mathrm{st}$ column in the table) is undetectable with the achieved S/N ratio.
[ccccccc]{} Band & S & $\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}$ & R & $\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}$ & $\eta_\mathrm{3.3}$ & $\eta_\mathrm{12.7}$\
($\mu$m) & & $\displaystyle\left(\frac{\mathrm{W}}{\mathrm{cm^2~\mu m}}\right)$ & & $\displaystyle\left(\frac{\mathrm{W}}{\mathrm{sr~\mu m}}\right)$ &\
15.82 & 25 & 4.6 10$^{-16}$ & & 7.3 10$^{-28}$ & $>1$ & $>1$\
27.74 & 45 & 1.5 10$^{-16}$ & & 1.5 10$^{-28}$ & $>1$ & $>1$\
\*[58.56]{} & \*[35]{} & \*[1.5 10$^{-18}$]{} & l & 5.4 10$^{-29}$ & 0.52 & 0.42\
& & & h & 6.5 10$^{-29}$ & 0.44 & 0.34\
The perpendicular band, with its prominent central Q branch, is a much more sensitive (about two orders of magnitude) probe for neutral naphthalene, yielding the lowest detection limit of $\eta \sim 0.4$ for both reference AIBs considered. Resolving power helps a little, since separating single hot bands yields a higher peak flux in the strongest ones.
None of these bands are detected in ISO archive data, therefore we here obtained a direct observational upper limit on $\eta$ for neutral naphthalene in the RR.
Turning to anthracene, its strongest emission bands in the RR are again the inplane CH stretch at $\sim3.3~\mu$m and the outofplane CH bend at $\sim11.3$; the fraction of the total IR emission of neutral anthracene in such bands is calculated to be respectively 37.5% and 17.2%. The integrated intensities in the former of these bands observed in the RR was given above, the one of the $\sim11.3~\mu$m band is measured to be $F_\mathrm{11.3}$ = 1.39 10$^{-16}$ W cm$^{-2}$. Our model yields, for neutral anthracene, $\mathcal{P}_\mathrm{3.3}$ = 4.7 10$^{-26}$ W sr$^{-1}$ and $\mathcal{P}_\mathrm{11.3}$ = 2.15 10$^{-26}$ W sr$^{-1}$, while the values of S and $\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}$ estimated from ISO archive data and those of $\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}$ calculated by our model (see Figs. \[anthra\_n\_58\] to \[anthra\_n\_66\]) are listed in Table \[pupamela2\], along with the resulting detection limits at $\sim$2$\sigma$ level.
[ccccccc]{} Band & S & $\displaystyle\frac{dF_\mathrm{cont}}{d\lambda}$ & R & $\displaystyle\left(\frac{d\mathcal{P}_\mathrm{ref}}{d\lambda}\right)_\mathrm{peak}$ & $\eta_\mathrm{3.3}$ & $\eta_\mathrm{11.3}$\
($\mu$m) & & $\displaystyle\left(\frac{\mathrm{W}}{\mathrm{cm^2~\mu m}}\right)$ & & $\displaystyle\left(\frac{\mathrm{W}}{\mathrm{sr~\mu m}}\right)$ &\
21.26 & 50 & 3.1 10$^{-16}$ & & 4.2 10$^{-27}$ & $>1$ & 0.44\
\*[26.35]{} & \*[90]{} & \*[1.9 10$^{-16}$]{} & l & 5.1 10$^{-28}$ & $>1$ & $>1$\
& & & h & 7.2 10$^{-28}$ & $>1$ & 0.86\
43.68 & 75 & 4.9 10$^{-17}$ & & 3.9 10$^{-28}$ & $>1$ & 0.52\
\*[110.23]{} & \*[25]{} & \*[1.5 10$^{-18}$]{} & l & 6.0 10$^{-29}$ & $>1$ & 0.33\
& & & h & 7.2 10$^{-29}$ & $>1$ & 0.27\
Classical AIBs are not the only observable quantity against which ratios, and consequently $\eta$ values, can be computed. Neutral anthracene, phenanthrene and pyrene were proposed as possible carriers of the Blue Luminescence [BL, @vij04; @vij05], a fluorescence phenomenon observed in the Red Rectangle and, subsequently, in several other astronomical sources. In a previous work, we demonstrated that phenanthrene and pyrene could be ruled out based on the failure to detect their predicted IR emission spectra [@mul06]. Anthracene remained, of the three, the only candidate compatible with available ISO observations of the Red Rectangle. The strongest constraint appeared to be the undetected longest wavelength band at $\sim$110 $\mu$m, which however contained a degree of uncertainty due to the unknown band profile. We here reproduce the comparison of the calculated band with available observations *including* the detailed modelling of band profile. This is shown in Fig. \[anthracene\_110\]. The line of reasoning leading to Eq. (\[etalimit\]) can be identically retraced substituting the integrated BL flux in the place of the integrated flux in a classical AIB [see @mul06 for a detailed derivation]. This yields $\eta_\mathrm{BL}>1$ for anthracene, i. e. it confirms that this band is undetectable with the available ISO database observations. It is however apparent (see Fig. \[anthracene\_110\]) that this band is expected to be just slightly below the detection limit, and an increase in S/N of a factor of $\sim$5 ought to reveal it, if anthracene indeed produces the observed BL.
![Comparison between the estimated IR emission spectrum of anthracene (C$_{14}$H$_{10}$) and an ISO spectrum of the RR in the wavelength range 100120 $\mu$m. Calculated spectra, under different assumptions [see @mul06 for details] are drawn in dashed, dashdotted, and continuous lines; the continuumsubtracted ISO spectrum is shown as a dotted line. The central position of the fundamental of the expected anthracene band is marked by a tick, which shows the effect of anharmonic shifts.[]{data-label="anthracene_110"}](anthracene_rr_110mu_anharmrot.eps){width="\hsize"}
Discussion and conclusions {#discussion}
==========================
The upper limits we derived are on the relative contribution of specific molecules to well defined classical AIBs, specifically at $\sim$3.3, $\sim$11.3 and $\sim12.7~\mu$m. They can be easily converted into absolute abundance limits, but this implies an assumption on their spatial distribution in the observed source and on the detailed assumed scaling of RF intensity. The terms depending on such assumptions cancel when using ratios of bands, making them more robust.
Despite obtaining very similar numerical values for the upper limits on $\eta_\mathrm{3.3}$ and $\eta_\mathrm{12.7}$ for naphthalene, their relevance is different. The $\sim3.3~\mu$m band is produced efficiently by small neutral molecules; the intensity of this band is suppressed in cations and rapidly decays with molecular size, as shown by Fig. \[frac3\_3\].
![Percentage of total IR flux emitted in the inplane CH stretch near $\sim3.3~\mu$m, as computed by our model for the sample of molecules in @mul06b, as a function of molecular size. Neutral species are represented by diamonds, cations by crosses. For each molecule, we here considered all bands in the range 3.2 to 3.37 $\mu$m, which corresponds to the observed width of the $3.3~\mu$m band in the RR.[]{data-label="frac3_3"}](frac3_3.eps){width="\hsize"}
This implies that the PAHs significantly contributing to the $\sim3.3~\mu$m band are a small subset of the whole population, namely only the small ones. The number of different species in this subset is accordingly very much smaller, thus $<40\%$ of the small PAHs (i. e. $\eta_\mathrm{3.3}$) is a much stronger constraint than $<40\%$ of all PAHs. This trend is much weaker for the flux fraction in the band at $\sim12.7~\mu$m, as shown by Fig. \[frac12\_7\]. This is a consequence of the wellknown variation in the position of the outofplane CH bend depending on specific molecular parameters (e. g. *solo*, *duo*, *trio* modes etc.), hence chemical diversity produces a large scatter which dominates over size effects. This means that the band at $\sim12.7~\mu$m is not produced preferentially by a subset of PAHs defined by size, but instead by structure.
![Same as Fig. \[frac3\_3\] for the outofplane CH bend at $\sim12.7~\mu$m. For each molecule, we here considered all bands in the range 12.3 to 13.1 $\mu$m, which corresponds to the observed width of the $12.7~\mu$m band in the RR.[]{data-label="frac12_7"}](frac12_7.eps){width="\hsize"}
As to anthracene, we obtained rather weak constraints for $\eta_\mathrm{3.3}$, somewhat stronger constraints on $\eta_\mathrm{11.3}$. However, Fig. \[frac11\_3\] shows that larger molecules can be expected to contribute a considerable fraction of the flux emitted in this band, which means that $\sim$40% is not such a stringent limit for $\eta_\mathrm{11.3}$. As for the band at $\sim12.7~\mu$m, the band at $\sim11.3~\mu$m is produced preferentially by a subset of molecules defined primarily by structure, more than by size.
![Same as Fig. \[frac3\_3\] for the outofplane CH bend at $\sim11.3~\mu$m. For each molecule, we here considered all bands in the range 10.9 to 11.7 $\mu$m, which corresponds to the observed width of the $11.3~\mu$m band in the RR.[]{data-label="frac11_3"}](frac11_3.eps){width="\hsize"}
While we applied our procedure to neutral naphthalene and anthracene, it can be generalised to any PAH for which our model is applicable and to any astronomical source for which observations of classical AIBs (or some other suitable tracer of PAHs) are available. As already mentioned, *all* PAHs have butterfly IRactive modes, which usually are the lowest frequency modes of each molecule and give rise to perpendicular bands [@mul06b]. The lowest frequency vibrational modes are almost always well separated in energy (except for highly symmetric molecules), meaning that strong Coriolis perturbations are not likely to occur for them and they ought to display Q branches which can be very sharp, since the effective momenta of inertia can be very close between the upper and lower states of the transition. In turn, according to the present calculations these Q branches ought to contain $\sim$20% of the total flux in the band, the distribution among fundamental and hot bands depending on the detailed statistics of the process and, essentially, on the vibrational modes which are left in an excited state upon decoupling. The spacing between the fundamental and various hot bands depends critically on the anharmonic vibrational constants $\chi_{ij}$, which vary with the specific molecule and band considered. If a perpendicular band is emitted preferentially when the molecule has a very low excitation energy, then relatively few hot bands will contribute to it, their Q branches remaining well resolved and sharp, and the P and R branches blending and being somewhat blurred as a result. The same will happen if a few $\chi_{ij}$ are much larger than the others, producing a welldefined sequence of sharp Q branches.
We here modelled in detail the profiles of one perpendicular band of naphthalene and three of anthracene. Among them, three display sharp Q branches (Figs. \[naphtha\_n\_48\], \[anthra\_n\_61\] and \[anthra\_n\_66\]), in which fundamental and hot bands can be separated with adequate resolving power, the other one (Fig. \[anthra\_n\_58\]) is red shaded to the point of partially washing out its structure. In this very limited sample, therefore, most perpendicular bands possess a combination of molecular parameters which yields the maximum spectral contrast. On a large sample of molecules, it can be reasonably expected that this will happen for some intense bands, producing the most favorable conditions for their detection.
In general, the average rotational energy of the molecule is expected to scale roughly with the average energy of the vibrational photons emitted [see e. g. @rou97], which implies that larger molecules ought to have a lower rotational energy which, together with decreasing rotational constants, in turn translates to narrower rotational profiles and better detectability.
The mixture of PAHs in space probably contains a wide variety of different molecules, in principle in many different hydrogenation and ionisation states. However, @lep01 [@lep03] showed that in any given environment the population of a given PAH is expected to be dominated by one or two ionisation and hydrogenation states, essentially depending on its size and physical conditions. Furthermore, chemical selection effects will favour the most stable species over the others, so that at least *some* species may be abundant enough to exceed their detection limit on *some* $\eta_\mathrm{AIB}$ (or some other PAH tracer). Moreover, such a detection limit will be pushed one or two orders of magnitude lower by the much higher sensitivity of the instruments on board the forthcoming Herschel Space Observatory and by the much reduced dust continuum emission at longer wavelengths.
A systematic study of PAH band profiles and IR flux ratios is necessary to select the most promising diagnostics and species for identification. The search for single, specific PAHs in the farIR is a challenging, but promising task.
G. Malloci acknowledges the “Ministère de la Recherche” and G. Mulas the CNRS for the financial support during their stay at CESR in Toulouse. We thank Aude Simon for helping us to effectively use the <span style="font-variant:small-caps;">Gaussian03</span> package. Part of the calculations used here were performed using CINECA and CALMIP supercomputing facilities. This work was supported by the European Research Training Network “Molecular Universe” (MRTN-CT-2004-512302).
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Computed molecular parameters from the vibrorotational analyses
===============================================================
In the following we report some of the molecular parameters obtained in the vibrorotational analyses performed for neutral naphthalene and neutral anthracene at the B3LYP/ level of theory. Tables \[chinaphthalene\] and \[chianthracene\] report, respectively, the vibrational anharmonic constants $\chi_{ij}$ for the three bands of neutral naphthalene and the four bands opf neutral anthracene considered. The changes of the effective rotational constants as a function of vibrational state are reported in Tables \[vibrot\_alpha\_naphthalene\] for naphthalene and \[vibrot\_alpha\_anthracene\] for anthracene. For consistency, we list in the tables the frequencies we previously obtained [@mul06b] using the <span style="font-variant:small-caps;">NWChem</span> package [@str03] at the same level of theory. Frequencies obtained by <span style="font-variant:small-caps;">Gaussian03</span> [@g03] are coincident within numerical errors. We give more significant digits than the actually expected accuracy to distinguish very close vibrational modes.
[cccc]{} Fundamental & 15.820 $\mu$m & 27.731 $\mu$m & 58.196 $\mu$m\
($\mu$m) &\
3.249 & -0.126 & 0.004 & 0.035\
3.250 & -0.123 & -0.018 & 0.039\
3.265 & -0.198 & 0.083 & 0.012\
3.267 & -0.198 & 0.032 & 0.018\
3.280 & -0.156 & 0.217 & -0.014\
3.284 & -0.130 & 0.175 & -0.026\
3.286 & -0.198 & 0.131 & 0.002\
3.288 & -0.178 & 0.123 & -0.008\
6.179 & -1.360 & -1.051 & -0.449\
6.278 & -0.406 & -1.057 & -0.306\
6.407 & -1.347 & -0.198 & -0.239\
6.632 & -1.163 & -0.519 & -0.309\
6.836 & -0.735 & 0.180 & -0.077\
6.842 & -0.659 & -0.563 & -0.240\
7.146 & -0.486 & 0.061 & -0.055\
7.335 & -1.522 & -0.039 & -0.332\
7.364 & -1.765 & 0.820 & -0.110\
7.875 & -0.630 & 0.704 & -0.110\
7.961 & -0.305 & 0.038 & -0.065\
8.273 & -0.406 & -1.356 & -0.327\
8.527 & 0.011 & 1.270 & 0.207\
8.551 & -0.344 & -0.482 & -0.010\
8.658 & 0.044 & 0.165 & -0.055\
8.838 & -0.382 & 0.441 & -0.229\
9.812 & 0.082 & 0.824 & -0.034\
9.906 & 0.089 & 0.099 & -0.066\
10.065 & -0.189 & -0.052 & -0.021\
10.141 & -0.140 & -0.115 & -0.126\
10.389 & -0.089 & -0.004 & -0.294\
10.601 & -0.024 & -0.068 & -0.157\
10.638 & -1.126 & 0.150 & -0.083\
11.294 & -0.221 & -0.088 & -0.457\
11.937 & -0.325 & -0.089 & -0.245\
12.545 & -0.088 & -1.264 & 0.292\
12.691 & -0.064 & -0.176 & -0.155\
12.957 & -0.335 & 0.050 & -0.190\
13.312 & 0.859 & -0.252 & -0.077\
13.937 & 0.046 & 0.087 & -0.093\
15.825 & 0.105 & -0.104 & -0.200\
15.895 & -0.226 & -0.221 & -0.463\
19.503 & 0.057 & -0.095 & 0.339\
19.536 & 0.050 & 0.254 & 0.807\
20.878 & 0.237 & 0.226 & -0.054\
21.162 & -0.225 & -0.324 & -0.349\
25.745 & 0.259 & -0.295 & -0.579\
27.737 & -0.104 & 0.236 & 0.363\
53.950 & -0.113 & -0.460 & -0.230\
58.565 & -0.200 & 0.363 & -0.258\
[cccc]{} Fundamental & a$_i$ & b$_i$ & c$_i$\
($\mu$m) &\
3.249 & 0.346 & 0.080 & 0.065\
3.250 & 0.347 & 0.077 & 0.063\
3.265 & 0.320 & 0.087 & 0.068\
3.267 & 0.323 & 0.081 & 0.064\
3.280 & 0.325 & 0.096 & 0.073\
3.284 & 0.323 & 0.093 & 0.070\
3.286 & 0.313 & 0.087 & 0.069\
3.288 & 0.313 & 0.088 & 0.068\
6.179 & 0.800 & 0.519 & 0.136\
6.278 & 0.471 & 0.387 & 0.187\
6.407 & 1.061 & 0.307 & 0.478\
6.632 & 0.661 & 0.241 & 0.303\
6.836 & 0.410 & 0.055 & -2.109\
6.842 & -0.124 & 0.302 & 2.421\
7.146 & -0.199 & 0.136 & 0.081\
7.335 & 1.918 & 0.531 & 0.395\
7.364 & 0.683 & 0.339 & 0.280\
7.875 & 0.299 & 0.161 & 0.104\
7.961 & 0.214 & 0.094 & 0.220\
8.273 & -0.055 & 0.458 & 0.324\
8.527 & -0.625 & 0.003 & 0.077\
8.551 & 0.250 & 0.098 & 0.028\
8.658 & -0.199 & 0.049 & 0.069\
8.838 & -0.488 & 0.102 & 0.324\
9.812 & 0.809 & -0.042 & 0.144\
9.906 & 0.782 & 0.009 & 0.112\
10.065 & 0.916 & 0.163 & 0.022\
10.141 & 0.954 & 0.153 & 0.017\
10.389 & 0.448 & 0.192 & 0.012\
10.601 & -2.638 & 0.159 & 0.002\
10.638 & 2.807 & 0.004 & 0.141\
11.294 & 0.546 & 0.157 & 0.005\
11.937 & 0.194 & 0.017 & -0.001\
12.545 & -2.053 & 0.193 & 0.117\
12.691 & 2.656 & 0.057 & -0.013\
12.957 & 0.333 & 0.085 & -0.029\
13.312 & 0.514 & 0.057 & 0.139\
13.937 & 0.319 & 0.107 & -0.020\
15.825 & -40.137 & -0.023 & 0.113\
15.895 & 40.437 & 0.062 & -0.045\
19.503 & -0.220 & -0.037 & -11.605\
19.536 & -0.220 & 0.061 & 11.782\
20.878 & 0.053 & 0.025 & -0.097\
21.162 & 0.692 & -0.008 & -0.104\
25.745 & 0.733 & -0.042 & -0.113\
27.737 & -1.792 & -0.050 & 0.069\
53.950 & 0.339 & 0.146 & -0.150\
58.565 & 2.085 & -0.262 & -0.249\
---------- --------------- --------------- --------------- ---------------- --------- --------------- --------------- --------------- ----------------
Fund. 21.232 $\mu$m 26.336 $\mu$m 43.060 $\mu$m 109.937 $\mu$m Fund. 21.232 $\mu$m 26.336 $\mu$m 43.060 $\mu$m 109.937 $\mu$m
($\mu$m) ($\mu$m)
3.249 -0.079 -0.063 -0.027 0.045 9.946 0.062 -0.100 0.121 -0.041
3.249 -0.078 -0.065 -0.030 0.044 10.102 -0.389 -0.628 -0.032 -0.037
3.263 -0.096 -0.033 -0.004 0.029 10.116 -0.275 -0.742 -0.035 -0.036
3.263 -0.095 -0.035 -0.007 0.030 10.407 -0.660 -0.563 -0.009 -0.055
3.282 -0.149 0.044 0.058 0.007 10.470 -0.372 -0.520 -0.008 -0.059
3.283 -0.128 0.039 0.042 0.005 10.875 -0.420 0.020 -0.284 0.007
3.284 -0.135 0.023 0.035 0.014 11.003 -1.283 -1.905 -0.057 -0.163
3.286 -0.110 0.011 0.018 0.017 11.005 -0.348 -0.140 -0.148 0.205
3.292 -0.317 0.085 0.086 -0.008 11.315 -2.030 -0.563 -0.161 -0.036
3.295 -0.299 0.087 0.081 -0.011 11.690 -0.356 -0.376 -0.030 -0.100
6.162 -0.331 -0.554 -0.338 -0.143 11.982 -1.000 -0.666 -0.035 -0.096
6.183 -0.703 -0.614 1.810 -0.184 12.563 -0.517 -0.324 0.328 0.008
6.329 -0.421 -0.520 -1.023 -0.177 13.043 -3.945 -0.560 0.021 0.197
6.486 -0.559 -0.572 -0.028 -0.119 13.173 -0.291 -3.714 0.032 0.039
6.514 -0.376 -0.471 0.081 -0.161 13.434 -0.667 0.317 0.023 -0.098
6.736 -0.137 -0.297 -0.128 -0.096 13.512 -0.712 -0.806 -0.017 -0.027
6.860 -0.295 0.048 -0.554 -0.126 13.709 -0.367 -0.668 0.031 0.035
6.864 -0.021 -0.365 0.243 -0.040 15.329 -0.554 -0.136 -0.012 -0.209
7.147 -0.144 0.118 -0.072 -0.031 15.706 0.187 0.120 0.016 -0.160
7.218 -0.192 -0.813 -0.453 -0.110 16.336 0.066 -0.027 -0.481 0.131
7.218 -0.829 -0.933 0.704 -0.098 17.109 -0.112 -0.470 -0.277 -0.088
7.434 -0.989 -0.699 -0.845 -0.140 18.710 -0.287 -0.037 -0.073 0.265
7.605 -0.008 1.095 0.019 -0.067 20.037 -0.060 -0.499 0.050 0.151
7.768 0.290 -0.460 0.512 -0.031 21.019 0.209 -0.171 0.016 -0.192
7.848 2.118 -0.500 -0.024 -0.087 21.263 0.040 -0.065 -0.030 -0.002
7.899 -0.355 -0.226 0.006 -0.135 25.448 -0.200 -0.338 0.664 -0.424
8.318 -0.272 -0.491 0.045 -0.002 25.625 0.109 -0.064 -0.244 0.915
8.473 -0.318 -0.152 0.056 -0.137 26.349 -0.065 0.171 -0.271 -0.181
8.553 -0.196 0.349 0.069 -0.060 37.412 -0.237 -0.220 -0.408 0.017
8.623 -0.535 2.072 -0.178 -0.076 43.009 -0.129 -0.554 -0.978 -0.281
8.668 -0.349 -0.198 -0.221 -0.037 43.681 -0.030 -0.271 0.024 -0.587
9.107 -0.588 -1.000 -1.709 -0.117 82.090 -0.261 -0.269 -0.828 -0.173
9.911 0.105 -0.118 0.200 -0.040 110.231 -0.002 -0.181 -0.587 -0.146
---------- --------------- --------------- --------------- ---------------- --------- --------------- --------------- --------------- ----------------
[cccc|cccc]{} Fund. & a$_i$ & b$_i$ & c$_i$ & Fund. & a$_i$ & b$_i$ & c$_i$\
($\mu$m) & & ($\mu$m) &\
3.249 & 0.166 & 0.014 & 0.014 & 9.946 & 0.433 & -0.001 & 0.027\
3.249 & 0.166 & 0.014 & 0.014 & 10.102 & 0.430 & 0.025 & 0.005\
3.263 & 0.155 & 0.015 & 0.015 & 10.116 & 0.434 & 0.024 & 0.005\
3.263 & 0.155 & 0.015 & 0.014 & 10.407 & 0.193 & 0.029 & 0.003\
3.282 & 0.161 & 0.020 & 0.018 & 10.470 & 0.072 & 0.023 & 0.001\
3.283 & 0.157 & 0.018 & 0.016 & 10.875 & 0.154 & 0.018 & 0.035\
3.284 & 0.155 & 0.018 & 0.017 & 11.003 & 0.211 & 0.031 & 0.004\
3.286 & 0.152 & 0.016 & 0.015 & 11.005 & -0.323 & 0.009 & 0.024\
3.292 & 0.176 & 0.029 & 0.024 & 11.315 & 0.401 & 0.016 & 0.000\
3.295 & 0.172 & 0.029 & 0.024 & 11.690 & 0.199 & 0.026 & 0.002\
6.162 & 0.296 & 0.106 & 0.069 & 11.982 & 0.101 & 0.024 & 0.001\
6.183 & 0.375 & 0.147 & 0.084 & 12.563 & 0.261 & 0.052 & 0.059\
6.329 & 0.206 & 0.122 & 0.082 & 13.043 & 0.141 & 0.013 & -0.005\
6.486 & 0.539 & 0.081 & 0.106 & 13.173 & 0.238 & 0.008 & -0.002\
6.514 & 0.356 & 0.086 & 0.087 & 13.434 & 0.223 & -0.004 & -0.010\
6.736 & 0.276 & 0.055 & 0.063 & 13.512 & 0.313 & 0.016 & 0.024\
6.860 & -0.009 & 0.092 & -0.239 & 13.709 & 0.092 & 0.014 & -0.002\
6.864 & 0.198 & 0.024 & 0.334 & 15.329 & 0.008 & 0.037 & -0.033\
7.147 & -0.073 & 0.036 & 0.027 & 15.706 & -1.289 & -0.003 & 0.015\
7.218 & 0.671 & 0.117 & 0.117 & 16.336 & -0.177 & 0.011 & 0.090\
7.218 & 0.784 & 0.089 & 0.079 & 17.109 & 1.376 & -0.018 & -0.015\
7.434 & 0.409 & 0.136 & 0.096 & 18.710 & -0.250 & 0.015 & 0.022\
7.605 & -0.010 & 0.095 & 0.087 & 20.037 & 0.424 & -0.019 & -0.028\
7.768 & 0.111 & 0.028 & 0.000 & 21.019 & 0.194 & -0.007 & -0.023\
7.848 & 0.082 & 0.052 & 0.062 & 21.263 & 0.013 & -0.004 & -0.023\
7.899 & 0.283 & 0.136 & 0.142 & 25.448 & -0.683 & -0.002 & -1.019\
8.318 & 0.165 & 0.009 & 0.031 & 25.625 & -0.012 & 0.020 & 1.062\
8.473 & -0.268 & 0.004 & 0.014 & 26.349 & 0.355 & -0.023 & -0.032\
8.553 & -0.180 & 0.035 & 0.037 & 37.412 & 0.261 & -0.006 & -0.034\
8.623 & 0.328 & 0.071 & -0.448 & 43.009 & 0.904 & -0.060 & -0.059\
8.668 & -0.162 & 0.017 & 0.565 & 43.681 & -1.292 & -0.026 & 0.009\
9.107 & -0.236 & 0.070 & 0.055 & 82.090 & 0.124 & 0.026 & -0.042\
9.911 & 0.391 & 0.002 & 0.030 & 110.231 & 1.647 & -0.082 & -0.091\
[^1]: `http://www.iso.vilspa.esa.es`
[^2]: A difference in the rotational constants between the upper and lower states involved in a transition leads to the inversion of some branches, with the consequent formation of band heads, and results in an overall asymmetry in the rotational profile. A band is then commonly said to be *redshaded* when it has a more extended tail on the red side, *blueshaded* in the opposite case [@her91a; @her91b]
|
---
author:
- Vytautas Paškūnas
title: 'Extensions for supersingular representations of $\operatorname{GL}_2({\mathbb {Q}_p})$'
---
Introduction
============
In this paper we study the category $\operatorname{Rep}_G$ of smooth representations of $G:=\operatorname{GL}_2({\mathbb {Q}_p})$ on ${\overline{\mathbb{F}}_{p}}$-vector spaces. Smooth irreducible ${\overline{\mathbb{F}}_{p}}$-representations of $G$ with a central character have been classified by Barthel-Livne [@bl] and Breuil [@b1]. A smooth irreducible representation $\pi$ of $G$ is supersingular, if it is not a subquotient of any principal series representation. Roughly speaking a supersingular representation is an ${\overline{\mathbb{F}}_{p}}$-analog of a supercuspidal representation.
\[A\] Assume that $p>2$ and let $\tau$ and $\pi$ be irreducible smooth representations of $G$ admitting a central character. If $\pi$ is supersingular and $\operatorname{Ext}^1_G(\tau, \pi)\neq 0$ then $\tau\cong \pi$. Moreover, if $p\ge 5$ then $\dim \operatorname{Ext}^1_G(\pi, \pi)=5$.
This answers affirmatively for $p>2$ a question of Colmez, see the introduction of [@col1]. When $p=3$ there are two cases and we can show that in one of them $\dim \operatorname{Ext}^1_G(\pi, \pi)=5$, in the other $\dim \operatorname{Ext}^1_G(\pi, \pi)\le 6$, which is the expected dimension. We note that if $\tau$ is a twist of Steinberg representation by a character or irreducible principal series then Colmez [@col1 VII.5.4] and Emerton [@em Prop. 4.2.8] prove by different methods that $\operatorname{Ext}^1_G(\tau, \pi)=0$. Our result is new when $\tau$ is supersingular or a character.
We now explain the strategy of the proof. We first get rid of the extensions coming from the centre $Z$ of $G$. Let $\zeta: Z\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be the central character of $\pi$, and let $\operatorname{Rep}_{G,\zeta}$ be the full subcategory of $\operatorname{Rep}_G$ consisting of representations with the central character $\zeta$. We show in Theorem \[centre\] that if $\operatorname{Ext}^1_G(\tau,\pi)\neq 0$ then $\tau$ also admits a central character $\zeta$. Let $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)$ parameterise all the isomorphism classes of extensions between $\pi$ and $\tau$ admitting a central character $\zeta$. We show that if $\tau\not \cong \pi$ then $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\cong \operatorname{Ext}^1_G(\tau,\pi)$ and there exists an exact sequence: $$\label{C}
0\rightarrow \operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\rightarrow \operatorname{Ext}^1_G(\pi,\pi)\rightarrow \operatorname{Hom}(Z,{\overline{\mathbb{F}}_{p}})\rightarrow 0,$$ where $\operatorname{Hom}$ denotes continuous group homomorphisms. Let $I$ be the ‘standard’ Iwahori subgroup of $G$, (see §2), and $I_1$ the maximal pro-$p$ subgroup of $I$. Since $\zeta$ is smooth, it is trivial on the pro-$p$ subgroup $I_1\cap Z$, hence we may consider $\zeta$ as a character of $ZI_1$. Let ${\mathcal H}:=\operatorname{End}_G({\operatorname{c-Ind}_{ZI_1}^{G}{\zeta}})$ be the Hecke algebra, and $\operatorname{Mod}_{{\mathcal H}}$ the category of right ${\mathcal H}$-modules. Let ${\mathcal I}: \operatorname{Rep}_{G,\zeta}\rightarrow \operatorname{Mod}_{{\mathcal H}}$ be the functor $${\mathcal I}(\kappa):=\kappa^{I_1}\cong \operatorname{Hom}_{G}( {\operatorname{c-Ind}_{ZI_1}^{G}{\zeta}},\kappa).$$ Vignéras shows in [@vig] that ${\mathcal I}$ induces a bijection between irreducible representations of $G$ with the central character $\zeta$ and irreducible ${\mathcal H}$-modules. Using results of Ollivier [@o2] we show that there exists an $E_2$-spectral sequence: $$\label{specseq1}
\operatorname{Ext}^i_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^j {\mathcal I}(\pi))\Longrightarrow \operatorname{Ext}^{i+j}_{G, \zeta}(\tau, \pi).$$ The $5$-term sequence associated to gives an exact sequence: $$\label{R}
0\rightarrow \operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau),{\mathcal I}(\pi))\rightarrow \operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\rightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)).$$ Now $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau),{\mathcal I}(\pi))$ has been determined in [@bp] and in fact is zero if $\tau\not \cong \pi$. The problem is to understand $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ as an ${\mathcal H}$-module.
We have two approaches to this. Results of Kisin [@kis1] imply that the dimension of $ \operatorname{Ext}^1_G(\pi, \pi)$ is bounded below by the dimension of $\operatorname{Ext}^1_{{\mathcal G_{{\mathbb {Q}_p}}}}(\rho, \rho)$, where $\rho$ is the $2$-dimensional irreducible ${\overline{\mathbb{F}}_{p}}$-representation of ${\mathcal G_{{\mathbb {Q}_p}}}$, the absolute Galois group of ${\mathbb {Q}_p}$, corresponding to $\pi$ under the mod $p$ Langlands, see [@b2], [@col1]. (Excluding one case when $p=3$.) Let $\mathfrak{I}$ be the image of $\operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\rightarrow\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\pi),\operatorname{\mathbb{R}}^1{\mathcal I}(\pi))$. Using and we obtain a lower bound on the dimension of $\mathfrak{I}$. By forgetting the ${\mathcal H}$-module structure we obtain an isomorphism of vector spaces: $$\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\cong H^1(I_1/Z_1,\pi),$$ where $Z_1$ is the maximal pro-$p$ subgroup of $Z$. The key idea is to bound the dimension of $H^1(I_1/Z_1,\pi)$ from above and use this to show if ${\mathcal I}(\tau)$ was a submodule of $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ for some $\tau\not \cong \pi$ then this would force the dimension of $\mathfrak{I}$ to be smaller than calculated before.
At the time of writing (an n-th draft of) this, [@kis1] was not written up and there were some technical issues with the outline of the argument in the introductions of [@col1] and [@kis], caused by the fact that all the representations in [@col1] are assumed to have a central character. Since we only need a lower bound on the dimension of $\operatorname{Ext}^1_G(\pi, \pi)$ and only in the supersingular case, we have written up the proof of a weaker statement in the appendix. The proof given there is a variation on Colmez-Kisin argument.
In order to bound the dimension of $H^1(I_1/Z_1,\pi)$ we prove a new result about the structure of supersingular representations of $G$. Let $M$ be the subspace of $\pi$ generated by $\pi^{I_1}$ and the semi-group $\bigl (\begin{smallmatrix} p^{{\mathbb{N}}} & {\mathbb{Z}_p}\\ 0 & 1\end{smallmatrix}\bigr )$. One may show that $M$ is a representation of $I$.
\[exseq1\] The map $(v,w)\mapsto v-w$ induces an exact sequence of $I$-representations: $$0\rightarrow \pi^{I_1}\rightarrow M\oplus \Pi\centerdot M\rightarrow \pi\rightarrow 0,$$ where $\Pi=\begin{pmatrix} 0 & 1\\ p & 0\end{pmatrix}$.
We show that the restrictions of $M$ and $M/\pi^{I_1}$ to $I\cap U$, where $U$ is the unipotent upper triangular matrices, are injective objects in $\operatorname{Rep}_{I\cap U}$. If $\psi:I\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character and $p>2$, using this, we work out $\operatorname{Ext}^1_{I/Z_1}(\psi, M)$ and $\operatorname{Ext}^1_{I/Z_1}(\psi, M/\pi^{I_1})$. Theorem \[exseq1\] enables us to determine $H^1(I_1/Z_1,\pi)$ as a representation of $I$, see Theorem \[main\] and Corollary \[dimH1\]. Once one has this it is quite easy to work out $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ as an ${\mathcal H}$-module in the regular case, see Proposition \[R1regp5\], without using Colmez’s work. It is also possible to work out directly the ${\mathcal H}$-module structure of $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ in the Iwahori case. However, the proof relies on heavy calculations of $\operatorname{Ext}^1_K({\mathbf 1}, \pi)$ and $\operatorname{Ext}^1_K({St}, \pi)$, where $K:=\operatorname{GL}_2({\mathbb{Z}_p})$ and ${St}$ is the Steinberg representation of $K/K_1\cong \operatorname{GL}_2({\mathbb{F}_{p}})$. So we decided to exclude it and use “stratégie de Kisin” instead.
The primes $p=2$, $p=3$ require some special attention. Theorem \[exseq1\] holds when $p=2$, but our calculation of $H^1(I_1/Z_1, \pi)$ breaks down for the technical reason that the trivial character is the only smooth character of $I$, when $p=2$. However, if $p=2$ and we fix a central character $\zeta$ then there exists only one supersingular representation (up to isomorphism) with central character $\zeta$. Hence, it is enough to show that $\operatorname{Ext}^1_G(\tau,\pi)=0$ when $\tau$ is a character, since all the other cases are handled in [@col1 VII.5.4], [@em §4]. It might be easier to do this directly.
Let ${\mathrm{Sp}}$ be the Steinberg representation of $G$. After the first draft of this paper, it was pointed out to me by Emerton that it was not known (although expected) that $\operatorname{Ext}^1_G(\eta, {\mathrm{Sp}})=0$, when $\eta:G\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character of order $2$ (all the other cases have been worked out in [@em §4], see also [@col1 §VII.4,§VII.5]). A slight modification of our proof for supersingular representations also works for the Steinberg representation. In the last section we work out $\operatorname{Ext}^1_G(\tau, {\mathrm{Sp}})$ for all irreducible $\tau$, when $p>2$. As a result of this and the results already in the literature ([@bp], [@col1], [@em]), one knows $\operatorname{Ext}^1_G(\tau, \pi)$ for all irreducible $\tau$ and $\pi$, when $p>2$. We record this in the last section.
*Acknowledgements.* The key ideas of this paper stem from a joint work with Christophe Breuil [@bp]. I would like to thank Pierre Colmez for pointing out this problem to me and Florian Herzig for a number of stimulating discussions. I would like to thank Gaëtan Chenevier, Pierre Colmez and Mark Kisin for some very helpful discussion on the “stratégie de Kisin” outlined in [@col1] and [@kis]. This paper was written when I was visiting Université Paris-Sud and IHÉS, supported by Deutsche Forschungsgemeinschaft. I would like to thank these institutions.
Notation
========
Let $G:=\operatorname{GL}_2({\mathbb {Q}_p})$, let $P$ be the subgroup of upper-triangular matrices, $T$ the subgroup of diagonal matrices, $U$ be the unipotent upper triangular matrices and $K:=\operatorname{GL}_2({\mathbb{Z}_p})$. Let ${\mathfrak{p}}:=p{\mathbb{Z}_p}$ and $$I:=\begin{pmatrix} {\mathbb{Z}_p}^{\times} & {\mathbb{Z}_p}\\ {\mathfrak{p}}& {\mathbb{Z}_p}^{\times} \end{pmatrix},\quad I_1:= \begin{pmatrix} 1+{\mathfrak{p}}& {\mathbb{Z}_p}\\ {\mathfrak{p}}& 1+{\mathfrak{p}}\end{pmatrix},
\quad K_1:=\begin{pmatrix} 1+{\mathfrak{p}}& {\mathfrak{p}}\\ {\mathfrak{p}}& 1+{\mathfrak{p}}\end{pmatrix}.$$ For $\lambda\in {\mathbb{F}_{p}}$ we denote the Teichmüller lift of $\lambda$ to ${\mathbb{Z}_p}$ by $[\lambda]$. Set $$H:=\biggl \{\begin{pmatrix} [\lambda] & 0\\ 0 & [\mu]\end{pmatrix}: \lambda, \mu\in {\mathbb{F}_{p}}^{\times}\biggr \}.$$ Let $\alpha:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be the character $$\alpha(\begin{pmatrix} [\lambda] & 0\\ 0 & [\mu]\end{pmatrix}):=\lambda\mu^{-1}.$$ Further, define $$\Pi:=\begin{pmatrix} 0 & 1\\ p & 0\end{pmatrix}, \quad s:=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}, \quad
t:= \begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}.$$ For $\lambda\in {\overline{\mathbb{F}}_{p}}^{\times}$ we define an unramified character $\mu_{\lambda}:{\mathbb {Q}_p}^{\times} \rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$, by $x\mapsto \lambda^{\operatorname{val}(x)}$.
Let $Z$ be the centre of $G$, and set $Z_1:=Z\cap I_1$. Let $G^0:=\{g\in G: \operatorname{det}g\in {\mathbb{Z}_p}^{\times}\}$ and set $G^+:=ZG^0$.
Let $\mathcal G$ be a topological group. We denote by $\operatorname{Hom}(\mathcal G, {\overline{\mathbb{F}}_{p}})$ the continuous group homomorphism, where the additive group ${\overline{\mathbb{F}}_{p}}$ is given the discrete topology. If ${\mathcal V}$ is a representation of $\mathcal G$ and $S$ is a subset of ${\mathcal V}$ we denote by $\langle \mathcal G \centerdot S\rangle$ the smallest subspace of ${\mathcal V}$ stable under the action of $\mathcal G$ and containing $S$. Let $\operatorname{Rep}_{\mathcal G}$ be the category of smooth representations of $\mathcal G$ on ${\overline{\mathbb{F}}_{p}}$-vector spaces. If $\mathcal Z$ is the centre of $\mathcal G$ and $\zeta: \mathcal Z\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character then we denote by $\operatorname{Rep}_{\mathcal G,\zeta}$ the full subcategory of $\operatorname{Rep}_{\mathcal G}$ consisting of representations with central character $\zeta$.
All the representations in this paper are on ${\overline{\mathbb{F}}_{p}}$-vector spaces.
Irreducible representations of $K$
==================================
We recall some facts about the irreducible representations of $K$ and introduce some notation. Let $\sigma$ be an irreducible smooth representation of $K$. Since $K_1$ is an open pro-$p$ subgroup of $K$, the space of $K_1$-invariants $\sigma^{K_1}$ is non-zero, and since $K_1$ is normal in $K$, $\sigma^{K_1}$ is a non-zero $K$-subrepresentation of $\sigma$, and since $\sigma$ is irreducible we obtain $\sigma^{K_1}=\sigma$. Hence the smooth irreducible representations of $K$ coincide with the irreducible representations of $K/K_1\cong \operatorname{GL}_2({\mathbb{F}_{p}})$, and so there exists a uniquely determined pair of integers $(r,a)$ with $0\le r\le p-1$, $0\le a<p-1$, such that $$\sigma\cong \operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2\otimes \operatorname{det}^a.$$ Note that $r=\dim\sigma-1$ and throughout the paper given $\sigma$, $r$ will always mean $\dim \sigma -1$. The space of $I_1$-invariants $\sigma^{I_1}$ is $1$-dimensional and so $H$ acts on $\sigma^{I_1}$ be a character $\chi_{\sigma}=\chi$. Explicitly, $$\chi(\begin{pmatrix} [\lambda] & 0\\0 & [\mu]\end{pmatrix})= \lambda^r (\lambda\mu)^a.$$ We define an involution $\sigma\mapsto \tilde{\sigma}$ on the set of isomorphism classes of smooth irreducible representations of $K$ by setting $$\tilde{\sigma}:=\operatorname{Sym}^{p-r-1}{\overline{\mathbb{F}}_{p}}^2\otimes \operatorname{det}^{r+a}.$$ Note that $\chi_{\tilde{\sigma}}=\chi_{\sigma}^s$. For the computational purposes it is convenient to identify $\operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2$ with the space of homogeneous polynomials in ${\overline{\mathbb{F}}_{p}}[x,y]$ of degree $r$. The action of $\operatorname{GL}_2({\mathbb{F}_{p}})$ is given by $$\begin{pmatrix} a & b \\ c& d\end{pmatrix} \centerdot P(x,y):= P(ax+cy, bx+dy).$$ With this identification $\sigma^{I_1}$ is spanned by $x^r$.
\[calcsym\] let $0\le j\le r$ be an integer and define $f_j\in \operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2\otimes \det^a$ by $$f_j:=\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{p-1-j}\begin{pmatrix} 1 & \lambda\\ 0 & 1\end{pmatrix} s x^r.$$ If $r=p-1$ and $j=0$ then $f_0=(-1)^{a+1}(x^r+y^r)$, otherwise $f_j=(-1)^{a+1}\bigl(\begin{smallmatrix} r\\j\end{smallmatrix}\bigr) x^jy^{r-j}$.
It is enough to prove the statement when $a=0$, since twisting the action by $\det^a$ multiplies $f_j$ by $(\det s)^a=(-1)^a$. We have $$\label{itgetskilled}
f_j=\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{p-1-j} (\lambda x+y)^r= \sum_{i=0}^r \begin{pmatrix} r\\ i\end{pmatrix} (\sum_{\lambda\in {\mathbb{F}_{p}}}
\lambda^{p-1+i-j}) x^i y^{r-i}.$$ If $a\ge 0$ is an integer then $\Lambda_a:=\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{a}$ is zero, unless $a>0$ and $p-1$ divides $a$, in which case $\Lambda_a=-1$. Note that $0^0=1$. If $a=p-1+i-j$ then $\Lambda_a \neq 0$ if and only if $i=j$ or $i-j=p-1$, which is equivalent to $r=i=p-1$ and $j=0$. This implies the assertion.
Let ${\overline{\mathbb{F}}_{p}}[[I\cap U]]$ denote the completed group algebra of $I\cap U$. Since $I\cap U\cong {\mathbb{Z}_p}$ mapping $X$ to $\begin{pmatrix} 1 & 1 \\ 0& 1\end{pmatrix} -1$ induces an isomorphism between the ring of formal power series in one variable ${\overline{\mathbb{F}}_{p}}[[X]]$ and ${\overline{\mathbb{F}}_{p}}[[I\cap U]]$. Every smooth representation $\tau$ of $I\cap U$ is naturally a module over ${\overline{\mathbb{F}}_{p}}[[I\cap U]]$, and we will also view $\tau$ as a module over ${\overline{\mathbb{F}}_{p}}[[X]]$ via the above isomorphism.
\[calcsym2\] Let $x^r\in \operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2\otimes\det^a$ then $X^r s x^r= (-1)^a r! x^r$.
We have $s x^r=(-1)^a y^r$. If $0\le i\le r$ then $X \centerdot x^{r-i}y^i= x^{r-i} (y+1)^i- x^{r-i} y^i= i x^{r-i+1}y^{i-1} +Q$, where $Q$ is a homogeneous polynomial of degree $r$, such that the degree of $Q$ in $y$ is less than $i-1$. Applying this observation $r$ times we obtain that $X^r \centerdot y^r= r! x^r.$
Irreducible representations of $G$
==================================
We recall some facts about the irreducible representations of $G$. We fix an integer $r$ with $0\le r\le p-1$. We consider $\operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2$ as a representation of $KZ$ by making $p$ act trivially. It is shown in [@bl Prop. 8] that there exists an isomorphism of algebras: $$\operatorname{End}_{G}({\operatorname{c-Ind}_{KZ}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}})\cong {\overline{\mathbb{F}}_{p}}[T]$$ for a certain $T\in \operatorname{End}_{G}({\operatorname{c-Ind}_{KZ}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}})$ defined in [@bl §3]. One may describe $T$ as follows. Let $\varphi\in {\operatorname{c-Ind}_{KZ}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}}$ be such that $\operatorname{Supp}\varphi =ZK$ and $\varphi(1)=x^r$. Since $\varphi$ generates ${\operatorname{c-Ind}_{KZ}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}}$ as a $G$-representation $T$ is determined by $T\varphi$.
\[T\]
- If $r=0$ then $$T \varphi = \Pi \varphi +\sum_{\lambda \in {\mathbb{F}_{p}}} \begin{pmatrix} 1 & [\lambda]\\ 0 & 1
\end{pmatrix} t \varphi .$$
- Otherwise, $$T \varphi = \sum_{\lambda \in {\mathbb{F}_{p}}} \begin{pmatrix} 1 & [\lambda] \\ 0 & 1
\end{pmatrix} t \varphi.$$
In the notation of [@bl] this is a calculation of $T([1, e_{\vec{0}}])$. The claim follows from the formula (19) in the proof of [@bl] Theorem 19.
It follows from [@bl Thm 19] that the map $T-\lambda$ is injective, for all $\lambda\in {\overline{\mathbb{F}}_{p}}$.
Let $\pi(r,\lambda)$ be a representation of $G$ defined by the exact sequence: $$\xymatrix{ 0 \ar[r] & {\operatorname{c-Ind}_{ZK}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}}\ar[r]^-{T-\lambda}& {\operatorname{c-Ind}_{ZK}^{G}{\operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2}}\ar[r]& \pi(r,\lambda)\ar[r]& 0.}$$ If $\eta: {\mathbb {Q}_p}^{\times}\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character then set $\pi(r,\lambda,\eta):=\pi(r,\lambda)\otimes \eta\circ \det$.
It follows from [@bl Thm.30] and [@b1 Thm.1.1] that $\pi(r,\lambda)$ is irreducible unless $(r,\lambda)=(0,\pm1)$ or $(r,\lambda)=(p-1,\pm1)$. Moreover, one has non-split exact sequences: $$\label{zero}
0 \rightarrow \mu_{\pm 1}\circ \det\rightarrow \pi(p-1,\pm 1)\rightarrow {\mathrm{Sp}}\otimes\mu_{\pm 1}\circ \det \rightarrow 0,$$ $$\label{p-1}
0 \rightarrow {\mathrm{Sp}}\otimes\mu_{\pm 1}\circ \det\rightarrow \pi(0,\pm 1)\rightarrow \mu_{\pm 1}\circ \det \rightarrow 0,$$ where ${\mathrm{Sp}}$ is the Steinberg representation of $G$, (we take as definition) and if $\lambda\in {\overline{\mathbb{F}}_{p}}^{\times}$ then $\mu_{\lambda}:{\mathbb {Q}_p}^{\times} \rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$, $x\mapsto \lambda^{\operatorname{val}(x)}$. Further, if $\lambda\neq 0$ and $(r,\lambda)\neq (0,\pm 1)$ then [@bl Thm.30] asserts that $$\label{induced}
\pi(r,\lambda)\cong {\operatorname{Ind}_{P}^{G}{\mu_{\lambda^{-1}}\otimes \mu_{\lambda}\omega^r}}.$$ It follows from [@bl Thm. 33] and [@b1 Thm 1.1] that the irreducible smooth representations of $G$ with the central character fall into $4$ disjoint classes:
- characters, $\eta\circ \det$;
- special series, ${\mathrm{Sp}}\otimes\eta\circ \det$;
- (irreducible) principal series $\pi(r, \lambda, \eta)$, $0< r\le p-1$, $\lambda\neq 0$, $(r,\lambda)\neq (p-1, \pm 1)$;
- supersingular $\pi(r, 0, \eta)$, $0\le r\le p-1$.
Supersingular representations {#supersingularrepresentations}
-----------------------------
We discuss the supersingular representations. Breuil has shown [@b1 Thm.1.1] that the representations $\pi(r,0,\eta)$ are irreducible and using the results [@bl] classified smooth irreducible representations of $G$ with a central character.
\[supersingularrep\] An irreducible representation $\pi$ with a central character is supersingular if $\pi\cong \pi(r,0,\eta)$ for some $0\le r \le p-1$ and a smooth character $\eta$.
All the isomorphism between supersingular representations corresponding to different $r$ and $\eta$ are given by $$\label{intertwine}
\pi(r,0,\eta)\cong \pi(r,0,\eta\mu_{-1})\cong \pi(p-1-r,0,\eta\omega^{r})\cong \pi(p-1-r,0,\eta\omega^{r}\mu_{-1})$$ see [@b1 Thm. 1.3]. It follows from [@bl Cor.36] that an irreducible smooth representation of $G$ with a central character is supersingular if and only if it is not a subquotient of any principal series representation.
We fix a supersingular representation $\pi$ of $G$ and we are interested in $\operatorname{Ext}^1_G(\tau,\pi)$, where $\tau$ is an irreducible smooth representation of $G$. If $\eta:G\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character, then twisting by $\eta$ induces an isomorphism $$\operatorname{Ext}^1_G(\tau,\pi)\cong \operatorname{Ext}^1_G(\tau\otimes\eta, \pi\otimes\eta).$$ Hence, we may assume that $p\in Z$ acts trivially on $\pi$, so that $\pi\cong \pi(r,0, \omega^a)$, for some $0\le r\le p-1$, and $0\le a<p-1$. It follows from [@b1 Thm. 3.2.4, Cor. 4.1.4] that $\pi^{I_1}$ is $2$-dimensional. Moreover, [@b1 Cor. 4.1.5] implies that there exists a basis $\{v_{\sigma}, v_{\tilde{\sigma}}\}$ of $\pi^{I_1}$, such that $\Pi v_{\sigma}=v_{{\tilde{\sigma}}}$, $\Pi v_{{\tilde{\sigma}}}=v_{\sigma}$ and there exists an isomorphism of $K$-representations: $$\langle K\centerdot v_{\sigma}\rangle \cong \sigma, \quad \langle K\centerdot v_{{\tilde{\sigma}}}\rangle \cong {\tilde{\sigma}},$$ where $\sigma:=\operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2\otimes \det^a$. The group $H$ acts on $v_{\sigma}$ by a character $\chi$ and on $v_{{\tilde{\sigma}}}$ by a character $\chi^s$. Explicitly, $$\label{explicitchi}
\chi(\begin{pmatrix} [\lambda] & 0\\0 & [\mu]\end{pmatrix})=\lambda^r(\lambda \mu)^a, \quad \forall \lambda,\mu\in {\mathbb{F}_{p}}^{\times}.$$
\[relations\] The following relations hold: $$\label{rel}
v_{\sigma}=(-1)^{a+1}\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{p-1-r} \begin{pmatrix} 1 & [\lambda] \\ 0 & 1\end{pmatrix} t v_{{\tilde{\sigma}}};$$ $$\label{trel}
v_{{\tilde{\sigma}}}=(-1)^{r+a+1}\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{r} \begin{pmatrix} 1 & [\lambda] \\ 0 & 1\end{pmatrix} t v_{\sigma};$$ $$\label{relX}
X^r t v_{{\tilde{\sigma}}}= (-1)^a r! v_{\sigma},
\quad X^{p-1-r}tv_{\sigma}=(-1)^{r+a}(p-1-r)! v_{{\tilde{\sigma}}}.$$
Since $tv_{{\tilde{\sigma}}}= s \Pi v_{{\tilde{\sigma}}}= s v_{\sigma}$ this is a calculation in $\operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2\otimes\det^a$, which is done in Lemmas \[calcsym\] and \[calcsym2\].
\[M\] $M:=\biggl \langle \biggl (\begin{smallmatrix} p^{{\mathbb{N}}} & {\mathbb{Z}_p}\\ 0 & 1\end{smallmatrix} \biggr ) \pi^{I_1}
\biggr \rangle$, $M_{\sigma}:= \biggl \langle \biggl ( \begin{smallmatrix} p^{2{\mathbb{N}}} & {\mathbb{Z}_p}\\ 0 & 1\end{smallmatrix} \biggr ) v_{\sigma}
\biggr \rangle$ and $M_{{\tilde{\sigma}}}:= \biggl \langle \biggl ( \begin{smallmatrix} p^{2{\mathbb{N}}} & {\mathbb{Z}_p}\\ 0 & 1\end{smallmatrix} \biggl )v_{{\tilde{\sigma}}}\biggr \rangle.$
\[Istable\] The subspaces $M$, $M_{\sigma}$, $M_{{\tilde{\sigma}}}$ are stable under the action of $I$.
We prove the statement for $M$, the rest is identical. By definition $M$ is stable under $I\cap U$. Since $I=(I\cap P^s) (I\cap U)$ it is enough to show that $M$ is stable under $I\cap P^s$. Suppose that $g_1\in I\cap P^s$, $g_2\in I\cap U$. Since $I=(I\cap U)(I\cap P^s)$ there exists $h_2\in I\cap U$ and $h_1\in I\cap P^s$ such that $g_1g_2=h_2 h_1$. Moreover, for $n\ge 0$ we have $t^{-n} (I\cap P^s) t^n \subset I$. Hence, if $v\in \pi^{I_1}$ then $(t^{-n}h_1 t^n)v\in \pi^{I_1}$ and so $$g_1 ( g_2 t^n v)= h_2 h_1 t^n v= h_2 t^n (t^{-n} h_1 t^n) v \in M, \quad \forall v\in \pi^{I_1} .$$ This implies that $M$ is stable under $I\cap P^s$.
The isomorphism $\pi(r,0, \omega^a)\cong \pi(p-r-1, 0, \omega^{r+a})$ allows to exploit the symmetry between $M_{\sigma}$ and $M_{{\tilde{\sigma}}}$. In particular, if we prove a statement about $M_{\sigma}$ which holds for all $\sigma$, then it also holds for $M_{{\tilde{\sigma}}}$ (with $\sigma$ replaced by ${\tilde{\sigma}}$).
\[Minj\] The triples $\chi\hookrightarrow M_{\sigma}$ and $\chi^s\hookrightarrow M_{\tilde{\sigma}}$ are injective envelopes of $\chi$ and $\chi^s$ in $\operatorname{Rep}_{H(I_1\cap U)}$. In particular, $M_{\sigma}^{I_1\cap U}={\overline{\mathbb{F}}_{p}}v_{\sigma}$ and $M_{{\tilde{\sigma}}}^{I_1\cap U}={\overline{\mathbb{F}}_{p}}v_{{\tilde{\sigma}}}$.
We will show the claim for $M_{\sigma}$. The relations imply that $$v_{\sigma}= (-1)^r r! (p-1-r)! X^{r+p(p-1-r)} t^2 v_{\sigma}.$$ For $n\ge 0$ define $\lambda_n:= ((-1)^r r! (p-1-r)!)^n$, $e_0:=0$ and $e_n:= r+p(p-1-r)+ p^2 e_{n-1}$. Further define $M_{\sigma, n}:=\langle (I_1\cap U) t^{2n}v_{\sigma}\rangle$. Since $t^{2n}v_{\sigma}= \lambda_1 X^{p^{2n}e_1}t^{2(n+1)}v_{\sigma}$, $M_{\sigma, n}$ is contained in $M_{\sigma, n+1}$ and hence $$M_{\sigma}=\underset{\underset{n}{\longrightarrow}}{\lim} \ M_{\sigma, n}.$$ Since $v_{\sigma}= \lambda_n X^{e_n} t^{2n} v_{\sigma}$ and $Xv_{\sigma}=0$ we obtain an isomorphism $M_{\sigma,n}\cong {\overline{\mathbb{F}}_{p}}[X]/(X^{e_n+1})$. In particular, for all $n\ge 0$ we have $M_{\sigma,n}^{I_1\cap U}= {\overline{\mathbb{F}}_{p}}v_{\sigma}$, and so $M_{\sigma}^{I_1\cap U}={\overline{\mathbb{F}}_{p}}v_{\sigma}$. Given $m\ge 0$, set $\mathcal U_m:=
\bigl (\begin{smallmatrix} 1 & {\mathfrak{p}}^m \\ 0 & 1\end{smallmatrix}\bigr )$, choose $n$ such that $e_n>p^m$ and define $M'_{\sigma,m}:= \langle (I_1\cap U)\centerdot X^{e_n+1-p^m} t^{2n}v_{\sigma}\rangle$. Then $M'_{\sigma, m}\cong {\overline{\mathbb{F}}_{p}}[X]/(X^{p^m})\cong M_{\sigma}^{\mathcal U_m} $ is an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)/\mathcal U_m}$. Since $M_{\sigma}=\underset{\longrightarrow}{\lim} \ M'_{\sigma,m}$ we obtain that $M_{\sigma}$ is an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)}$.
\[tv\_1\] Let $n\ge 0$ be an odd integer then $t^n v_{\sigma}\in M_{{\tilde{\sigma}}}$ and $t^n v_{{\tilde{\sigma}}}\in M_{\sigma}$. Hence, $t M_\sigma\subset M_{{\tilde{\sigma}}}$ and $tM_{{\tilde{\sigma}}}\subset M_{\sigma}$.
It follows from the definition that $t^2 M_{{\tilde{\sigma}}}\subset M_{{\tilde{\sigma}}}$. Hence, it is enough to consider $n=1$. Applying $t$ to we obtain $t v_{\sigma}= (-1)^a (r!)^{-1} X^{pr} t^2 v_{{\tilde{\sigma}}} \in M_{{\tilde{\sigma}}}$. If $k,m\ge 0$ are integers and $m$ even then we have $t(X^k t^m v_{\sigma})= X^{p k} t^{m} (t v_{\sigma})$ and since $tv_{\sigma}\in M_{{\tilde{\sigma}}}$ and $m$ is even we obtain $t(X^k t^m v_{\sigma})\in M_{{\tilde{\sigma}}}$. The set $\{X^k t^m v_{\sigma}: k, m\ge 0, 2\mid m\}$ spans $M_{\sigma}$ as an ${\overline{\mathbb{F}}_{p}}$-vector space. Hence, $tM_{\sigma} \subset M_{{\tilde{\sigma}}}$. The rest follows by symmetry.
\[sv\] We have $sv_{\sigma}\in M_{\sigma}$ and $ s v_{{\tilde{\sigma}}}\in M_{{\tilde{\sigma}}}$.
Since $sv_{\sigma}= s \Pi v_{{\tilde{\sigma}}}= tv_{{\tilde{\sigma}}}$ this follows from Lemma \[tv\_1\].
\[MI\] $M$ is the direct sum of its $I$-submodules $M_{\sigma}$ and $M_{{\tilde{\sigma}}}$.
Proposition \[Minj\] implies that $(M_{\sigma}\cap M_{{\tilde{\sigma}}})^{I_1}=M_{\sigma}^{I_1}\cap M_{{\tilde{\sigma}}}^{I_1}={\overline{\mathbb{F}}_{p}}v_{\sigma}\cap
{\overline{\mathbb{F}}_{p}}v_{{\tilde{\sigma}}} =0$. Hence $M_{\sigma}\cap M_{{\tilde{\sigma}}}=0$ and so it is enough to show that $M=M_{\sigma}+ M_{{\tilde{\sigma}}}$. Clearly, $M_{\sigma}\subset M$ and $M_{{\tilde{\sigma}}}\subset M$. Lemma \[tv\_1\] implies $M\subseteq M_{\sigma}+M_{{\tilde{\sigma}}}$.
We set $\pi_{\sigma}:=M_{\sigma}+\Pi\centerdot M_{{\tilde{\sigma}}}$ and $\pi_{{\tilde{\sigma}}}:=M_{{\tilde{\sigma}}}+\Pi\centerdot M_{\sigma}$.
\[G+\] The subspaces $\pi_{\sigma}$ and $\pi_{{\tilde{\sigma}}}$ are stable under the action of $G^+$.
We claim that $s\pi_{\sigma}\subseteq \pi_{\sigma}$. Now $s (\Pi M_{{\tilde{\sigma}}})= t M_{{\tilde{\sigma}}} \subset M_{\sigma}$ by Lemma \[tv\_1\]. It is enough to show that $s M_{\sigma} \subset \pi_{\sigma}$. By definition of $M_{\sigma}$ it is enough to show that $s(ut^n v_{\sigma})\in \pi_{\sigma}$ for all $u\in I_1\cap U$ and all even non-negative integers $n$. Lemma \[sv\] gives $s v_{\sigma}\in M_{\sigma}$ and if $n\ge 2$ is an even integer then $s t^n v_{\sigma} = \Pi t^{n-1}v_{\sigma} \in \Pi M_{{\tilde{\sigma}}}$ by Lemma \[tv\_1\]. Since $s (K_1\cap U)s= I_1\cap U^s$ for all $u\in K_1\cap U$, and $n\ge 0$ even, we get that $ s u t^n v_{\sigma} \in \pi_{\sigma}$. If $u\in (I_1\cap U)\setminus (K_1\cap U)$ and $n>0$ even, then the matrix identity: $$\label{trix}
\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix} 1 & \beta \\ 0 & 1\end{pmatrix}=
\begin{pmatrix} -\beta^{-1} & 1 \\ 0 & \beta \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \beta^{-1} & 1\end{pmatrix}$$ implies that $s u t^n v_{\sigma}\in M_{\sigma}$. This settles the claim. By symmetry $\pi_{{\tilde{\sigma}}}$ is also stable under $s$, and since $\pi_{\sigma}=\Pi \pi_{{\tilde{\sigma}}}$, we obtain that $\pi_{\sigma}$ is stable under $\Pi s \Pi^{-1}$. Lemma \[Istable\] implies that $\pi_{\sigma}$ is stable under $I$. Since $s$, $\Pi s \Pi^{-1}$ and $I$ generate $G^0$, we get that $\pi_{\sigma}$ is stable under $G^0$. Since $Z$ acts by a central character, $\pi_{\sigma}$ is stable under $G^+=Z G^0$. The result for $\pi_{{\tilde{\sigma}}}$ follows by symmetry.
Extensions
==========
In this section we compute extensions of characters for different subgroups of $I$.
Let $\kappa^u$, $\varepsilon$, $\kappa^l : I_1\rightarrow {\overline{\mathbb{F}}_{p}}$ be functions defined as follows, for $A=\begin{pmatrix} a & b \\ c & d\end{pmatrix} \in I_1$ we set $$\kappa^u(A)=\omega(b), \quad \varepsilon(A)=\omega(p^{-1}(a-d)), \quad \kappa^l(A)=\omega(p^{-1}c),$$ where $\omega: {\mathbb{Z}_p}\rightarrow {\overline{\mathbb{F}}_{p}}$ is the reduction map composed with the canonical embedding.
\[homi\] If $p\neq 2$ then $\operatorname{Hom}(I_1/Z_1, {\overline{\mathbb{F}}_{p}})=\langle \kappa^u,\kappa^l\rangle$. If $p=2$ then $\dim \operatorname{Hom}(I_1/Z_1, {\overline{\mathbb{F}}_{p}})=4$.
Let $\psi: I_1/Z_1\rightarrow {\overline{\mathbb{F}}_{p}}$ be a continuous group homomorphism. Since $I_1\cap U\cong I_1 \cap U^s\cong {\mathbb{Z}_p}$ there exist $\lambda, \mu\in {\overline{\mathbb{F}}_{p}}$ such that $\psi|_{I_1\cap U}=\lambda \kappa^u$ and $\psi|_{I_1\cap U^s}= \mu \kappa^l$. Then $\psi-\lambda \kappa^u -\mu\kappa^l$ is trivial on $I_1\cap U$ and $I_1\cap U^s$. The matrix identity $$\label{ssap}
\begin{pmatrix} 1 & \beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0\\ \alpha & 1\end{pmatrix}=
\begin{pmatrix} 1 & 0\\ \alpha(1+\alpha\beta)^{-1} & 1\end{pmatrix} \begin{pmatrix} (1+\alpha\beta) & \beta \\
0 & (1+\alpha \beta)^{-1}\end{pmatrix}$$ implies that $I_1\cap U$ and $I_1\cap U^s$ generate $I_1\cap \operatorname{SL}_2({\mathbb {Q}_p})$. So $\psi- \lambda \kappa^u -\mu\kappa^l$ must factor through $\operatorname{det}$. The image of $Z_1$ in $1+{\mathfrak{p}}$ under $\operatorname{det}$ is $(1+{\mathfrak{p}})^2$. If $p>2$ then $(1+{\mathfrak{p}})^2=1+{\mathfrak{p}}$ and hence $\psi= \lambda \kappa^u +\mu\kappa^l$. If $p=2$ then $\dim \operatorname{Hom}((1+{\mathfrak{p}})/(1+{\mathfrak{p}})^2, {\overline{\mathbb{F}}_{p}})=2$.
\[homip\] Assume $p>2$ then $\operatorname{Hom}((I_1\cap P)/Z_1, {\overline{\mathbb{F}}_{p}})=\langle \kappa^u,\varepsilon\rangle$ and $\operatorname{Hom}((I_1\cap P^s)/Z_1, {\overline{\mathbb{F}}_{p}})=\langle \kappa^l,\varepsilon\rangle$.
Let $\psi: (I_1\cap P)/Z_1\rightarrow {\overline{\mathbb{F}}_{p}}$ be a continuous group homomorphism. Since $I_1\cap U\cong {\mathbb{Z}_p}$ there exist $\lambda\in {\overline{\mathbb{F}}_{p}}$ such that $\psi|_{I_1\cap U}=\lambda \kappa^u$. Then $\psi-\lambda \kappa^u$ is trivial on $I_1\cap U$, and hence defines a homomorphism $(I_1\cap P)/Z_1(I_1\cap U)\cong (T\cap I_1)/Z_1\rightarrow {\overline{\mathbb{F}}_{p}}$. Since $p>2$ we have an isomorphism $(T\cap I_1)/Z_1\cong 1+p{\mathbb{Z}_p}\cong {\mathbb{Z}_p}$. Hence, there exists $\mu\in {\overline{\mathbb{F}}_{p}}$ such that $\psi=\mu \varepsilon +\lambda \kappa^u$. Conjugation by $\Pi$ gives the second assertion.
\[extI\] Let $\chi,\psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters. $\operatorname{Ext}^1_{I/Z_1}(\psi, \chi)$ is non-zero if and only if $\psi=\chi\alpha$ or $\psi=\chi\alpha^{-1}$. Moreover,
- if $p>3$ then $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \chi)=\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha^{-1}, \chi)=1$;
- if $p=3$ then $\chi\alpha=\chi\alpha^{-1}$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \chi)=2$;
- if $p=2$ then $\chi=\chi\alpha=\chi\alpha^{-1}={\mathbf 1}$ and $\dim \operatorname{Ext}^1_{I/Z_1}({\mathbf 1}, {\mathbf 1})=4$.
Since the order of $H$ is prime to $p$ and $I=H I_1$ we have $$\operatorname{Ext}^1_{I/Z_1}(\psi,\chi)\cong \operatorname{Hom}_H(\psi, H^1(I_1/Z_1, \chi)).$$ Now $H^1(I_1/Z_1, \chi)\cong \operatorname{Hom}(I_1/Z_1, {\overline{\mathbb{F}}_{p}})$, where if $\xi\in \operatorname{Hom}(I_1/Z_1, {\overline{\mathbb{F}}_{p}})$ and $h\in H$ then $[h\centerdot \xi](u)=\chi(h)\xi(h^{-1}uh)$. The assertion follows from Proposition \[homi\].
Similarly one obtains:
\[up\] Let $\chi, \psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters and let $\mathcal U=\left(\begin{smallmatrix} 1 & {\mathfrak{p}}^k\\ 0 & 1\end{smallmatrix}\right )$ for some integer $k$ then $\operatorname{Ext}^1_{H\mathcal U}(\psi, \chi)\neq 0$ if and only if $\psi=\chi\alpha^{-1}$. Moreover, $\dim \operatorname{Ext}^1_{H\mathcal U}(\chi\alpha^{-1}, \chi)=1$.
\[down\] Let $\chi, \psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters and let $\mathcal U=\bigl(\begin{smallmatrix} 1 & 0 \\{\mathfrak{p}}^k & 1\end{smallmatrix}\bigr )$ for some integer $k$ then $\operatorname{Ext}^1_{H\mathcal U}(\psi, \chi)\neq 0$ if and only if $\psi=\chi\alpha$. Moreover, $\dim \operatorname{Ext}^1_{H\mathcal U}(\chi\alpha, \chi)=1$.
\[upt\] Assume $p>2$ and let $\chi, \psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters then $\operatorname{Ext}^1_{(I_1\cap P)/Z_1}(\psi, \chi)\neq 0$ if and only if $\psi\in \{\chi, \chi\alpha^{-1}\}$. Moreover, $$\dim \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi\alpha^{-1}, \chi)= \dim \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi, \chi)=1.$$
\[downt\] Assume $p>2$ and let $\chi, \psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters then $\operatorname{Ext}^1_{(I_1\cap P)/Z_1}(\psi, \chi)\neq 0$ if and only if $\psi\in \{\chi, \chi\alpha\}$. Moreover, $$\dim \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, \chi)= \dim \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi, \chi)=1.$$
\[modX\] Let $\chi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character and let $\chi\hookrightarrow J_{\chi}$ be an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)}$, then $(J_{\chi}/\chi)^{I_1\cap U}$ is $1$-dimensional and $H$ acts on it by $\chi\alpha^{-1}$. Moreover, $\chi\alpha^{-1}\hookrightarrow J_{\chi}/\chi$ is an injective envelope of $\chi\alpha^{-1}$ in $\operatorname{Rep}_{H(I_1\cap U)}$.
Consider an exact sequence of $H(I\cap U)$-representations: $$0\rightarrow \chi\rightarrow J_{\chi}\rightarrow J_{\chi}/\chi\rightarrow 0.$$ Since $J_{\chi}$ is an injective envelope of $\chi$ in $\operatorname{Rep}_{I\cap U}$ taking $I_1\cap U$ invariants induces $H$-equivariant isomorphism $(J_{\chi}/\chi)^{I_1\cap U}\cong H^1(I_1\cap U, \chi)$. It follows from Lemma \[up\] that $\dim (J_{\chi}/\chi)^{I_1\cap U}=1$ and $H$ acts on $(J_{\chi}/\chi)^{I_1\cap U}$ via the character $\chi\alpha^{-1}$. Let $J_{\chi\alpha^{-1}}$ be an injective envelope of $\chi\alpha^{-1}$ in $\operatorname{Rep}_{H(I_1\cap U)}$, then there exists an exact sequence of $H(I_1\cap U)$-representations: $$0\rightarrow J_{\chi}/\chi\rightarrow J_{\chi\alpha^{-1}}\rightarrow Q\rightarrow 0.$$ Since $J_{\chi\alpha^{-1}}$ is an essential extension of $\chi\alpha^{-1}$, we have $J_{\chi\alpha^{-1}}^{I_1\cap U}\cong \chi\alpha^{-1}$. Hence taking $(I_1\cap U)$-invariants induces an isomorphism $Q^{I_1\cap U}\cong H^1(I_1\cap U, J_{\chi}/\chi)\cong H^2(I_1\cap U, \chi)$. Since $I_1\cap U \cong {\mathbb{Z}_p}$ is a free pro-$p$ group we have $H^2(I_1\cap U, \chi)=0$, see [@cohgal §3.4]. Hence $Q^{I_1\cap U}=0$, which implies $Q=0$.
\[extriv\] Let $\iota: J\hookrightarrow A$ be a monomorphism in an abelian category $\mathcal A$. If $J$ is an injective object in $\mathcal A$ then there exists $\sigma: A\rightarrow J$ such that $\sigma \circ \iota=\operatorname{id}$.
Since $J$ is injective the map $\operatorname{Hom}_{\mathcal A}(A,J)\rightarrow \operatorname{Hom}_{\mathcal A}(J,J)$ is surjective.
Exact sequence {#exactsequence}
==============
Let $\pi:=\pi(r, 0, \eta)$ with $0\le r\le p-1$. We use the notation of §\[supersingularrepresentations\], so that $\sigma:=\operatorname{Sym}^r{\overline{\mathbb{F}}_{p}}^2 \otimes \det^a$, with $\det^a=\eta\circ \det |_K$, and $\chi: H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ a character as in . We construct an exact sequence of $I$-representations which will be used to calculate $H^1(I_1/Z_1, \pi)$.
\[calc1\] If $r\neq 0$ then set $$w_{\sigma}:=\sum_{\lambda\in {\mathbb{F}_{p}}} \lambda^{p-r} \begin{pmatrix} 1 & [\lambda]\\ 0 & 1 \end{pmatrix} t v_{{\tilde{\sigma}}}+ (\sum_{\mu\in {\mathbb{F}_{p}}} \mu) v_{\sigma}.$$ Then $w_{\sigma}$ is fixed by $I_1\cap P^s$ and $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} w_{\sigma}= w_{\sigma} - (-1)^a r v_{\sigma}.$$
If $r=0$ then set $$w_{\sigma}:= \sum_{\lambda, \mu\in {\mathbb{F}_{p}}}\lambda\begin{pmatrix} 1 & [\mu]+p[\lambda] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}.$$ Then $$\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix} w_{\sigma}=w_{\sigma} +v_{\sigma}, \quad \begin{pmatrix} 1 & 0\\ p & 1\end{pmatrix} w_{\sigma}=
w_{\sigma}-(\sum_{\mu\in {\mathbb{F}_{p}}}\mu^2)v_{\sigma}.$$ If $\alpha\in [x]+{\mathfrak{p}}$, $\beta\in [y]+{\mathfrak{p}}$ then $$\begin{pmatrix} 1+p\alpha & 0\\ 0 & 1+p\beta\end{pmatrix} w_{\sigma}=w_{\sigma} +(x-y)(\sum_{\mu\in{\mathbb{F}_{p}}} \mu)v_{\sigma}.$$
We set $w:=w_{\sigma}$. Suppose that $r\neq 0$. Now $tv_{{\tilde{\sigma}}}= s\Pi v_{{\tilde{\sigma}}}= sv_{\sigma}$. Hence, if we identify $v_{\sigma}$ with $x^r\in \operatorname{Sym}^r {\overline{\mathbb{F}}_{p}}^2\otimes \det^a$ then Lemma \[calcsym\] applied to $j=r-1$ gives $w=-(-1)^a r x^{r-1}y$. This implies the assertion.
Suppose that $r=0$ and let $P(X):=\frac{X^p+1-(X+1)^p}{p}\in \mathbb Z[X]$, then [@serre] implies that $$\begin{split}
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} w&= \sum_{\lambda,\mu\in {\mathbb{F}_{p}}}
\lambda\begin{pmatrix} 1 & 1+[\mu]+p[\lambda] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}\\ &= \sum_{\lambda,\mu\in {\mathbb{F}_{p}}}
\lambda\begin{pmatrix} 1 & [\mu+1]+p[\lambda+P(\mu)] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}.
\end{split}$$ Hence, $$\begin{split}
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} w& =
\sum_{\lambda,\mu\in {\mathbb{F}_{p}}}
\lambda\begin{pmatrix} 1 & [\mu]+p[\lambda+P(\mu-1)] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}\\
&=\sum_{\lambda,\mu\in {\mathbb{F}_{p}}}
(\lambda-P(\mu-1))\begin{pmatrix} 1 & [\mu]+p[\lambda] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}\\
&= w- \sum_{\lambda,\mu\in {\mathbb{F}_{p}}} P(\mu-1)\begin{pmatrix} 1 & [\mu]+p[\lambda] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}\\
&= w+(-1)^a\sum_{\mu\in{\mathbb{F}_{p}}} P(\mu-1)\begin{pmatrix} 1 & [\mu] \\ 0 & 1\end{pmatrix} t v_{{\tilde{\sigma}}}\\ &= w+(\sum_{\mu\in{\mathbb{F}_{p}}} P(\mu-1))v_{\sigma},
\end{split}$$ where the last two equalities follow from , . If $p=2$ then $P(X-1)=1-X$, otherwise $P(X-1)=\sum_{i=1}^{p-1} p^{-1} \begin{pmatrix} p\\ i\end{pmatrix} X^i(-1)^{p-i}$. Hence $\sum_{\mu\in {\mathbb{F}_{p}}} P(\mu-1)=
-\sum_{\mu\in {\mathbb{F}_{p}}^{\times}} \mu^{p-1}=1$.
Now $t^2v_{\sigma}$ is fixed by $\begin{pmatrix} 1 & {\mathfrak{p}}^2\\ 0 & 1\end{pmatrix}$ and $I_1\cap P^s$, so the matrix identity $$\label{pass}
\begin{pmatrix} 1 & 0 \\ \beta & 0 \end{pmatrix} \begin{pmatrix} 1 & \alpha\\ 0 & 1\end{pmatrix}=
\begin{pmatrix} 1 & \alpha(1+\alpha\beta)^{-1} \\ 0 & 1\end{pmatrix} \begin{pmatrix} (1+\alpha\beta)^{-1} & 0 \\ \beta &
1+\alpha \beta\end{pmatrix}$$ implies that $$\begin{split}
\begin{pmatrix} 1 & 0 \\ p & 1\end{pmatrix} w&= \sum_{\lambda,\mu\in{\mathbb{F}_{p}}} \lambda
\begin{pmatrix} 1 & [\mu]+ p[\lambda-\mu^2] \\ 0 & 1\end{pmatrix}t^2 v_{\sigma}\\
&=\sum_{\lambda,\mu\in {\mathbb{F}_{p}}} (\lambda+\mu^2) \begin{pmatrix} 1 & [\mu]+ p[\lambda] \\ 0 & 1\end{pmatrix}t^2 v_{\sigma}=w-(\sum_{\mu\in {\mathbb{F}_{p}}} \mu^2)v_{\sigma}.
\end{split}$$ If $\alpha\in [x] +{\mathfrak{p}}$ and $\beta\in [y]+{\mathfrak{p}}$ then the same argument gives $$\label{pass2}
\begin{split}
\begin{pmatrix} 1+p\alpha & 0 \\ 0 & 1+p\beta\end{pmatrix} w&= \sum_{\lambda, \mu\in {\mathbb{F}_{p}}}
\lambda\begin{pmatrix} 1 & [\mu]+p[\lambda +\mu (x-y)] \\ 0 & 1\end{pmatrix} t^2 v_{\sigma}\\
&= w+(x-y)(\sum_{\mu\in {\mathbb{F}_{p}}} \mu)v_{\sigma}.
\end{split}$$
\[fix\] We have $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1\cap U} =(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}$. Moreover, let $\Delta_{\sigma}$ be the image of $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}$ in $H^1(I_1, {\mathbf 1})\cong \operatorname{Hom}(I_1, {\overline{\mathbb{F}}_{p}})$. Then the following hold:
- if either $r\neq 0$ or $p>3$ then $\Delta_{\sigma}={\overline{\mathbb{F}}_{p}}\kappa^u$;
- if $p=3$ and $r=0$ then $\Delta_{\sigma}={\overline{\mathbb{F}}_{p}}(\kappa^u+\kappa^l)$;
- if $p=2$ and $r=0$ then $\Delta_{\sigma}={\overline{\mathbb{F}}_{p}}(\kappa^u+\kappa^l +\varepsilon)$.
It follows from Proposition \[modX\] that $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1\cap U}$ is $1$-dimensional. Since $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}\neq 0$ the inclusion $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}\subseteq (M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1\cap U}$ is an equality. The image of $w_{\sigma}$ of Lemma \[calc1\] spans $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}$ and the last assertion follows from Lemma \[calc1\].
\[exseq\] The map $(v,w)\mapsto v-w$ induces an exact sequence of $I$-representations: $$0\rightarrow \pi^{I_1}\rightarrow M\oplus \Pi\centerdot M\rightarrow \pi\rightarrow 0.$$
We claim that $M\cap \Pi\centerdot M=\pi^{I_1}$. Consider an exact sequence: $$0\rightarrow \pi^{I_1}\rightarrow M\cap \Pi\centerdot M\rightarrow Q\rightarrow 0.$$ Since $M\cap \Pi\centerdot M$ is an $I_1$-invariant subspace of $\pi$, we have $(M\cap \Pi\centerdot M)^{I_1}\subseteq \pi^{I_1}$. Since $M\cap \Pi\centerdot M$ contains $\pi^{I_1}$ the inclusion is an equality. Hence, by taking $I_1$-invariants we obtain an injection $\partial: Q^{I_1}\hookrightarrow H^1(I_1, \pi^{I_1})\cong \operatorname{Hom}(I_1, {\overline{\mathbb{F}}_{p}})\oplus \operatorname{Hom}(I_1, {\overline{\mathbb{F}}_{p}})$. The element $\Pi$ acts on $H^1(I_1, \pi^{I_1})$ by $\Pi\centerdot (\psi_1, \psi_2)= (\psi_2^{\Pi}, \psi_1^{\Pi})$. Let $\Delta_{\sigma}$ (resp. $\Delta_{{\tilde{\sigma}}}$) denote the image of $(M_{\sigma}/{\overline{\mathbb{F}}_{p}}v_{\sigma})^{I_1}$ (resp. $(M_{{\tilde{\sigma}}}/{\overline{\mathbb{F}}_{p}}v_{{\tilde{\sigma}}})^{I_1}$) in $\operatorname{Hom}(I_1, {\overline{\mathbb{F}}_{p}})$. Let $\Delta$ be the image of $(M/\pi^{I_1})^{I_1}$ in $H^1(I_1, \pi^{I_1})$ so that $\Delta=\Delta_{\sigma}\oplus \Delta_{{\tilde{\sigma}}}$. By taking $I_1$-invariants of the diagram $$\xymatrix@1{ 0\ar[r] & \pi^{I_1}\ar[r]\ar[d]& M\cap\Pi\centerdot M \ar[r]\ar[d] & Q\ar[r]\ar[d] & 0\\
0\ar[r] & \pi^{I_1}\ar[r]& M \ar[r] & M/\pi^{I_1}\ar[r] & 0}$$ we obtain a commutative diagram: $$\xymatrix@1{ \;Q^{I_1}\;\ar[d]\ar@{^(->}[r]^-{\partial} & H^1(I_1, \pi^{I_1})\ar[d]^{\operatorname{id}}\\
\;(M/\pi^{I_1})^{I_1}\; \ar@{^{(}->}[r]^-{\partial} & H^1(I_1, \pi^{I_1}).}$$ and hence an injection $\partial(Q^{I_1})\hookrightarrow \Delta$. Acting by $\Pi$ we obtain an injection $\partial(Q^{I_1})\hookrightarrow \Pi\centerdot \Delta$. We claim that $\Delta\cap \Pi\centerdot \Delta=0$. We have $$\Delta\cap \Pi\centerdot \Delta=(\Delta_{\sigma}\cap \Pi\centerdot \Delta_{{\tilde{\sigma}}} )\oplus
(\Delta_{{\tilde{\sigma}}}\cap \Pi\centerdot \Delta_{\sigma}).$$ By symmetry we may assume $r<p-1$. Proposition \[fix\] applied to $M_{\sigma}$ and $M_{{\tilde{\sigma}}}$ implies that that if $r\neq 0$ then $\Delta= {\overline{\mathbb{F}}_{p}}\kappa^u\oplus {\overline{\mathbb{F}}_{p}}\kappa^u$, hence $\Pi\centerdot \Delta= {\overline{\mathbb{F}}_{p}}(\kappa^u)^{\Pi} \oplus {\overline{\mathbb{F}}_{p}}(\kappa^u)^{\Pi}=
{\overline{\mathbb{F}}_{p}}\kappa^l\oplus {\overline{\mathbb{F}}_{p}}\kappa^l$, so that $\Delta\cap \Pi\centerdot \Delta=0$. If $r=0$ then Proposition \[fix\] implies that $\Delta= {\overline{\mathbb{F}}_{p}}(\kappa^u-(\sum_{\mu\in{\mathbb{F}_{p}}} \mu^2) \kappa^l +(\sum_{\mu\in {\mathbb{F}_{p}}} \mu )\varepsilon)\oplus {\overline{\mathbb{F}}_{p}}\kappa^u$, hence $\Pi\centerdot \Delta={\overline{\mathbb{F}}_{p}}\kappa^l \oplus {\overline{\mathbb{F}}_{p}}(\kappa^l-(\sum_{\mu\in{\mathbb{F}_{p}}} \mu^2) \kappa^u -(\sum_{\mu\in {\mathbb{F}_{p}}} \mu )\varepsilon)$, again $\Delta\cap \Pi\centerdot \Delta=0$. Note that if $r=0$ then we have to apply Proposition \[fix\] to $M_{{\tilde{\sigma}}}$ with $r=p-1$, and $p-1\neq 0$. This implies that $Q^{I_1}=0$ and hence $Q=0$.
Since $G^+$ and $\Pi$ generate $G$, Proposition \[G+\] implies that $\pi_{\sigma}+\pi_{{\tilde{\sigma}}}$ is stable under the action of $G$. Since $\pi$ is irreducible we get $\pi=\pi_{\sigma}+\pi_{{\tilde{\sigma}}}$. This implies surjectivity.
\[one\] We have $M_{\sigma}\cap \Pi\centerdot M_{{\tilde{\sigma}}}=\pi_{\sigma}^{I_1}={\overline{\mathbb{F}}_{p}}v_{\sigma}$ and $M_{{\tilde{\sigma}}}\cap \Pi\centerdot M_{\sigma}=\pi_{{\tilde{\sigma}}}^{I_1}={\overline{\mathbb{F}}_{p}}v_{{\tilde{\sigma}}}$.
It is enough to show that $\pi_{\sigma}^{I_1}={\overline{\mathbb{F}}_{p}}v_{\sigma}$, since by Theorem \[exseq\] $M_{\sigma}\cap \Pi\centerdot M_{{\tilde{\sigma}}}$ is contained in $\pi^{I_1}$. Suppose not. Clearly $v_{\sigma}\in \pi_{\sigma}$, so since $\pi^{I_1}$ is $2$-dimensional, we obtain that $v_{{\tilde{\sigma}}}\in \pi_{\sigma}$. Then there exists $u_1\in M_{\sigma}$ and $u_2\in \Pi\centerdot M_{{\tilde{\sigma}}}$ such that $v_{{\tilde{\sigma}}}=u_1+u_2$. So $u_2\in \Pi\centerdot M_{{\tilde{\sigma}}}\cap (M_{\sigma}+M_{{\tilde{\sigma}}})\subset
\pi^{I_1}$ by Theorem \[exseq\]. Hence $u_2=\lambda v_{\sigma}$ for some $\lambda\in {\overline{\mathbb{F}}_{p}}$, and so $u_2\in M_{\sigma}$, and so $v_{{\tilde{\sigma}}}\in M_{\sigma}$. This contradicts $M_{\sigma}\cap M_{{\tilde{\sigma}}}=0$.
\[two\] As $G^+$-representation $\pi$ is the direct sum of its subrepresentations $\pi_{\sigma}$ and $\pi_{{\tilde{\sigma}}}$.
It follows from Theorem \[exseq\] that $\pi=\pi_{\sigma}+\pi_{{\tilde{\sigma}}}$. Now $$(\pi_{\sigma}\cap \pi_{{\tilde{\sigma}}})^{I_1}=\pi_{\sigma}^{I_1}\cap \pi_{{\tilde{\sigma}}}^{I_1}={\overline{\mathbb{F}}_{p}}v_{\sigma} \cap {\overline{\mathbb{F}}_{p}}v_{{\tilde{\sigma}}}=0.$$ Hence, $\pi_{\sigma}\cap \pi_{{\tilde{\sigma}}}=0$.
\[three\] We have $\pi\cong {\operatorname{Ind}_{G^+}^{G}{\pi_{\sigma}}}\cong {\operatorname{Ind}_{G^+}^{G}{\pi_{{\tilde{\sigma}}}}}$.
Computing $H^1(I_1/Z_1, \pi)$
=============================
We keep the notation of §\[exactsequence\] and compute $H^1(I_1/Z_1, \pi)$ as a representation of $H$ under the assumption $p>2$.
\[extXP\] Assume that $p>2$. Let $\psi,\chi: H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters. Let $N$ be a smooth representation of $(I\cap P)/Z_1$, such that $N|_{H(I_1\cap U)}$ is an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)}$. Suppose that $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\psi, N)\neq 0$ then $\psi=\chi$. Moreover, $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi, N)\cong \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi,\chi)$ is $1$-dimensional.
Suppose that we have a non-split extension $0\rightarrow N\rightarrow E\rightarrow \psi\rightarrow 0$. Since $N|_{H(I_1\cap U)}$ is injective Lemma \[extriv\] implies that the extension splits when restricted to $H(I_1\cap U)$. Hence, there exists $v\in E^{I_1\cap U}$ such that $H$ acts on $v$ by $\psi$ and the image of $v$ spans the underlying vector space of $\psi$. If $v$ is fixed by $I_1\cap T$, then since $I_1\cap T$ and $H(I_1\cap U)$ generate $I\cap P$ we would obtain a splitting of $E$ as an $I\cap P$-representation. Hence, there exists some $h\in I_1\cap T$, such that $(h-1)v\in N$ is non-zero. Since $h$ normalizes $I_1\cap U$ and $v$ is fixed by $I_1\cap U$, we obtain that $(h-1)v \in N^{I_1\cap U}$. Since $H$ acts on $v$ by $\psi$ and $T$ is abelian, we get that $H$ acts on $(h-1)v$ by $\psi$. Since $N|_{H(I_1\cap U)}$ is an injective envelope of $\chi$ we obtain that $\chi=\psi$.
By Proposition \[modX\], $N/\chi$ is an injective envelope of $\chi\alpha^{-1}$. Since $p>2$, $\chi\neq
\chi\alpha^{-1}$ and so $\operatorname{Hom}_{I\cap P}(\chi, N/\chi)= \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi, N/\chi)=0$. So applying $\operatorname{Hom}_{I\cap P}(\chi, \centerdot)$ to the short exact sequence of $(I\cap P)/Z_1$ representations $0\rightarrow \chi\rightarrow N\rightarrow N/\chi\rightarrow 0$ gives us an isomorphism $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi, N)\cong \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi,\chi)$. Lemma \[upt\] implies that these spaces are $1$-dimensional.
\[extX\] Assume that $p>2$. Let $\psi,\chi: H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be characters. Let $N$ be a smooth representation of $I/Z_1$, such that $N|_{H(I_1\cap U)}$ is an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)}$. Suppose that $\operatorname{Ext}^1_{I/Z_1}(\psi, N)\neq 0$ and let $\mathcal K$ be the kernel of the restriction map $\operatorname{Ext}^1_{I/Z_1}(\psi, N)\rightarrow
\operatorname{Ext}^1_{(I\cap P)/Z_1}(\psi, N)$ then one of the following holds:
- if $\mathcal K\neq 0$ then $\psi=\chi\alpha$;
- if $\mathcal K=0$ then $\psi=\chi$.
Moreover, $\dim\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N)=1,$ and let $R$ be the submodule of $N$, fitting in the exact sequence $0\rightarrow N^{I_1}\rightarrow R\rightarrow (N/N^{I_1})^{I_1}\rightarrow 0$, then there exists an exact sequence: $$0\rightarrow \operatorname{Hom}_I(\chi,\chi\alpha^{-2})\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi,R)\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi,N)\rightarrow 0.$$
Suppose that $\mathcal K\neq 0$ then there exists a non-split extension $0\rightarrow N\rightarrow E\rightarrow \psi\rightarrow 0$ of $I/Z_1$-representations, which splits when restricted to $I\cap P$. Hence, there exists $v\in E^{I_1\cap P}$ such that $H$ acts on $v$ by $\psi$ and the image of $v$ spans the underlying vector space of $\psi$. Let $k$ be the smallest integer $k\ge 1$ such that $v$ is fixed by $
\bigl (\begin{smallmatrix} 1 & 0 \\ {\mathfrak{p}}^k & 1 \end{smallmatrix}\bigr )$. If $k=1$ then $v$ is fixed by $I\cap U^s$. Since $I\cap U^s$ and $I\cap P$ generate $I$, we would obtain that $I$ acts on $v$ by $\psi$ and hence the extension splits. Hence, $k$ is at least $2$. Set $\mathcal U:=\bigl ( \begin{smallmatrix} 1 & 0\\ {\mathfrak{p}}^{k-1} & 1 \end{smallmatrix}\bigr )$. Our assumption on $k$ implies that $v':=\bigl ( \begin{smallmatrix} 1 & 0 \\ p^{k-1} & 1 \end{smallmatrix} \bigr )v-v\in N$ is non-zero. The matrix identity implies that $v'$ is fixed by $I_1\cap U$. Since $N^{I_1\cap U}$ is $1$-dimensional and $H$ acts on $N^{I_1\cap U}$ by $\chi$, we obtain a non-zero element in $\operatorname{Ext}^1_{H\mathcal U}(\psi, \chi)$. Lemma \[down\] implies that $\psi= \chi\alpha$. Let $\bar{v}$ be the image of $v$ in $E/N^{I_1}$. Again by Proposition \[modX\] $(N/N^{I_1})^{I_1\cap U}$ is $1$-dimensional and $H$ acts on $(N/N^{I_1})^{I_1\cap U}$ by $\chi\alpha^{-1}$. If the extension $0\rightarrow N/N^{I_1}\rightarrow E/N^{I_1}\rightarrow \psi\rightarrow 0$ is non-split, then by the same argument we would obtain a non-zero element in $\operatorname{Ext}^1_{H\mathcal U'}(\chi\alpha,\chi\alpha^{-1})$, where $\mathcal U':=\bigl (\begin{smallmatrix} 1 & 0\\ {\mathfrak{p}}^{m} & 1 \end{smallmatrix}\bigr )$, for some $m\ge 1$. This contradicts Lemma \[down\], as $p>2$ and so $\alpha$ is non-trivial. Hence we obtain an exact sequence: $$\label{yetanotherseq}
0\rightarrow \operatorname{Hom}_I(\chi\alpha,\chi\alpha^{-1})\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\chi)\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,N)\rightarrow 0.$$ If $p>3$ then $\dim \operatorname{Hom}_I(\chi\alpha,\chi\alpha^{-1})=0$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\chi)=1$. If $p=3$ then $\dim \operatorname{Hom}_I(\chi\alpha,\chi\alpha^{-1})=1$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\chi)=2$. Hence, $\dim \operatorname{Ext}^1_{I/Z_1} (\chi\alpha,N)=1$.
Assume that $\mathcal K=0$. Since we have assumed that $\operatorname{Ext}^1_{I/Z_1}(\psi,N)\neq 0$ we obtain that $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\psi, N)\neq 0$ and Lemma \[extXP\] implies that $\psi=\chi$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi,N)\le 1$. Suppose that there exists a non-split extension $0\rightarrow N\rightarrow E\rightarrow \chi\rightarrow 0$ of $I/Z_1$-representations, which remains non-split when restricted to $I\cap P$. Let $w_1$ be a basis vector of $N^{I_1\cap U}$. Lemmas \[extXP\], \[upt\] and \[homip\] imply that there exists $v\in E$ such that $H$ acts on $v$ by $\chi$ and for all $g\in I_1\cap P$ we have $g v= v+\varepsilon(g) w_1$. In particular, $v$ is fixed by $I\cap U$ and $\bigl (\begin{smallmatrix} 1 +{\mathfrak{p}}^2 & 0\\ 0 & 1+{\mathfrak{p}}^2\end{smallmatrix} \bigr )$. As before, let $k$ be the smallest integer $k\ge 1$ such that $v$ is fixed by $
\bigl (\begin{smallmatrix} 1 & 0 \\ {\mathfrak{p}}^k & 1 \end{smallmatrix}\bigr )$. We claim that $k=2$. Indeed, if $k>2$ then let $v':=\bigl ( \begin{smallmatrix} 1 & 0 \\ p^{k-1} & 1 \end{smallmatrix}\bigr )v -v$. Then $v'\in N$ is non-zero, and the matrix identity implies that $v'$ is fixed by $I_1\cap U$. Since $N^{I_1\cap U}$ is $1$-dimensional and $H$ acts on $N^{I_1\cap U}$ by $\chi$, we obtain a non-zero element in $\operatorname{Ext}^1_{H\mathcal U}(\chi, \chi)$, with $\mathcal U:=\bigl ( \begin{smallmatrix} 1 & 0\\ {\mathfrak{p}}^{k-1} & 1 \end{smallmatrix}\bigr )$. Lemma \[down\] implies that $\chi= \chi\alpha$. Since $p>2$ this cannot happen.
Consider $u:=\bigl ( \begin{smallmatrix} 1 & 0 \\ p & 1\end{smallmatrix} \bigr )v-v$. Using and the fact that $k\ge 2$ we obtain $$\begin{split}
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} u&=\begin{pmatrix} 1 & 0 \\ p(1+p)^{-1} & 1 \end{pmatrix} \begin{pmatrix} 1+p & 1 \\
0 & (1+p)^{-1}\end{pmatrix} v -v\\ &= \begin{pmatrix} 1 & 0 \\ p(1+p)^{-1} & 1 \end{pmatrix} (v+ 2w_1)-v= u+2w_1.
\end{split}$$ Since $2w_1\neq 0$ we get $u\neq 0$ and so $k=2$. By Proposition \[modX\] $(N/{\overline{\mathbb{F}}_{p}}w_1)^{I_1\cap U}$ is $1$-dimensional. This implies that $(N/{\overline{\mathbb{F}}_{p}}w_1)^{I_1\cap U}\cong (N/{\overline{\mathbb{F}}_{p}}w_1)^{I_1}$ and the image of $u$ in $N/{\overline{\mathbb{F}}_{p}}w_1$ spans $(N/{\overline{\mathbb{F}}_{p}}w_1)^{I_1\cap U}$. If we set $R:=\langle w_1, u\rangle$ then by construction we obtain that the map $\operatorname{Ext}^1_{I/Z_1}(\chi, N)\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi,N/R)$ is zero. Proposition \[modX\] implies that $(N/R)^{I_1}$ is $1$-dimensional and $H$ acts on it by a character $\chi\alpha^{-2}$. This implies the claim.
\[ext2\] Assume $p>2$ then the restriction maps $$\operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\rightarrow \operatorname{Ext}^2_{(I\cap P^s)/Z_1}(\chi,\chi),$$ $$\operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\rightarrow \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi,\chi)$$ are injective.
Consider the exact sequence of $I$-representations $0\rightarrow \chi\rightarrow {\operatorname{Ind}_{I\cap P^s}^{I}{\chi}}\rightarrow Q\rightarrow 0$. Iwahori decomposition implies that $$({\operatorname{Ind}_{I\cap P^s}^{I}{\chi}})|_{H(I_1\cap U)}\cong {\operatorname{Ind}_{H}^{H(I_1\cap U)}{\chi}},$$ and hence it is an injective envelope of $\chi$ in $\operatorname{Rep}_{H(I_1\cap U)}$. Proposition \[modX\] implies that $Q|_{H(I_1\cap U)}$ is an injective envelope of $\chi\alpha^{-1}$ in $\operatorname{Rep}_{H(I_1\cap U)}$. Since $p>2$ Lemma \[extI\] implies that $\operatorname{Ext}^1_{I/Z_1}(\chi,\chi)=0$, so using Shapiro’s lemma we obtain an exact sequence: $$\begin{split}
\operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi,\chi)\hookrightarrow \operatorname{Ext}^1_{I/Z_1}(\chi,Q)&\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\\
&\rightarrow\operatorname{Ext}^2_{(I\cap P^s)/Z_1}(\chi,\chi).
\end{split}$$ Now $\dim \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi,\chi)=1$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi,Q)=1$ by Proposition \[extX\]. This implies the result for $I\cap P^s$. By conjugating by $\Pi$ we obtain the result for $I\cap P$.
\[take\] Assume $p>2$ and let $N$ be as in Proposition \[extX\] then $\dim\operatorname{Ext}^1_{I/Z_1}(\chi,N)=1$, the natural maps $$\label{natural1}
\operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi, N),$$ $$\label{natural2}
\operatorname{Ext}^1_{I/Z_1}(\chi,N)\rightarrow \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi,N)$$ are injective and is an isomorphism.
We have an exact sequence: $$\operatorname{Ext}^1_{I/Z_1}(\chi, N)\hookrightarrow \operatorname{Ext}^1_{I/Z_1}(\chi,N/\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi,\chi).$$ Proposition \[modX\] and Lemma \[extXP\] imply that $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi,N/\chi)=0.$ The commutative diagram: $$\xymatrix@1{ \operatorname{Ext}^1_{I/Z_1}(\chi, N/\chi)\ar[r]\ar[d]^{0}&
\operatorname{Ext}^2_{I/Z_1}(\chi, \chi)\ar@{^(->}[d]^{\ref{ext2}}\\
\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi, N/\chi)\ar[r]^-{0}&
\operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi, \chi)}$$ and Corollary \[ext2\] implies that $\operatorname{Ext}^1_{I/Z_1}(\chi,N/\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi,\chi)$ is the zero map. Hence, is injective and $$\dim \operatorname{Ext}^1_{I/Z_1}(\chi, N)=\dim \operatorname{Ext}^1_{I/Z_1}(\chi,N/\chi)=1,$$ where the last equality is given by Propositions \[modX\] and \[extX\]. We know that $\operatorname{Ext}^1_{I/Z_1}(\chi, N)\neq 0$. So if is not injective, then Proposition \[extX\] gives $\chi=\chi\alpha$, but this cannot hold, since $p>2$. Since both sides have dimension $1$, is an isomorphism.
$p=3$
-----
The case $p=3$ requires some extra arguments. If you are only interested in $p\ge 5$ then please skip this subsection.
\[p3\] Assume $p=3$ and let $N$ be as in Proposition \[extX\] then the composition: $$\xymatrix@1{ \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\ar[r]^-\partial & \operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\ar[r]^-{\operatorname{Res}} & \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi\alpha,\chi)}$$ is injective, where $\partial$ is induced by a short exact sequence $0\rightarrow \chi\rightarrow N\rightarrow N/\chi\rightarrow 0$.
Since $p=3$ we have $\alpha=\alpha^{-1}$ and hence it follows from the Corollary \[take\] that $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)=1$. Corollary \[extX\] implies that the restriction map $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\rightarrow \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi\alpha, N/\chi)$ is injective. Moreover, Lemma \[extXP\] gives $\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi\alpha, N)=0$, and so the map $\partial: \operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi\alpha, N/\chi) \rightarrow \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi\alpha,\chi)$ is injective. The assertion follows from the commutative diagram: $$\xymatrix@1{ \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\ar[r]^-\partial\ar@{^(->}[d]^{\operatorname{Res}}_{\eqref{natural2}} & \operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\ar[d]^{\operatorname{Res}}\\
\;\operatorname{Ext}^1_{(I\cap P)/Z_1}(\chi\alpha, N/\chi)\; \ar@{^{(}->}[r]^-{\partial} & \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi\alpha,\chi).}$$
\[p3bis\] Assume $p=3$ and let $N$ be as in Proposition \[extX\]. Assume that $N^{K_1}\cong {\operatorname{Ind}_{HK_1}^{I}{\chi}}$ as a representation of $I$, then the composition: $$\xymatrix@1{ \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\ar[r]^-\partial & \operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\ar[r]^-{\operatorname{Res}}
& \operatorname{Ext}^2_{(I\cap P^s)/Z_1}(\chi\alpha,\chi)}$$ is zero, where $\partial$ is induced by a short exact sequence $0\rightarrow \chi\rightarrow N\rightarrow N/\chi\rightarrow 0$.
Since $p=3$ we have $\alpha=\alpha^{-1}$ and hence it follows from the Corollary \[take\] that $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)=1$. Let $\Delta$ be the image of the restriction map $$\Delta:=\operatorname{Im}(\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\rightarrow \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N/\chi)).$$ We claim that $\Delta$ is contained in the image of the natural map $$\label{map}
\operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N)\rightarrow \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N/\chi).$$ Since $p=3$ we have $\dim N^{K_1}=3$ and so the image of $N^{K_1}$ in $N/N^{I_1}$ is a $2$-dimensional $I$-stable subspace. Since it follows from Proposition \[modX\] that $(N/N^{I_1})^{I_1}$ and $((N/N^{I_1})/(N/N^{I_1})^{I_1})^{I_1}$ are $1$-dimensional we obtain an exact sequence $0\rightarrow N^{I_1}\rightarrow N^{K_1}\rightarrow R\rightarrow 0$, where where $R$ is the subspace of $N/\chi$ defined in Proposition \[extX\] (with $N/\chi$ instead of $N$). Since $N^{K_1}\cong {\operatorname{Ind}_{HK_1}^{I}{\chi}}$ we get: $$N^{K_1}|_{I\cap P^s}\cong \chi \oplus \chi\alpha \oplus \chi \cong \chi \oplus R|_{I\cap P^s}.$$ Let $\phi$ be the composition: $$\notag
\begin{split}
\operatorname{Ext}^1_{(I\cap P^s)/Z_1}&(\chi\alpha, R)\rightarrow \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N^{K_1})\rightarrow
\\&\operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N)
\rightarrow \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N/\chi).
\end{split}$$ Then we have a commutative diagram: $$\xymatrix{ \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, R)\ar@{>>}[r]^-{\ref{extX}}\ar[d]^{\operatorname{Res}} & \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\ar[d]^{\operatorname{Res}}\\
\operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, R)\ar[r]^-{\phi} & \operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N/\chi).}$$ The top horizontal arrow is surjective by Proposition \[extX\]. Hence, $\Delta$ equals to the image of $\phi\circ \operatorname{Res}$. Since the image of $\phi$ is contained in the image of we get the claim. The assertion follows from the commutative diagram: $$\xymatrix@1{ \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N/\chi)\ar[r]^-\partial\ar[d]^{\operatorname{Res}} & \operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\ar[d]^{\operatorname{Res}}\\
\operatorname{Ext}^1_{(I\cap P^s)/Z_1}(\chi\alpha, N/\chi) \ar[r]^-{\partial} & \operatorname{Ext}^2_{(I\cap P^s)/Z_1}(\chi\alpha,\chi),}$$ since the claim implies that the composition $\partial \circ \operatorname{Res}$ is the zero map.
\[done\] Assume $p=3$ let $N_{\chi}$ and $N_{\chi^s}$ be as in Proposition \[extX\] with respect to $\chi$ and $\chi^s$. Further assume that $N_{\chi^s}^{K_1}\cong {\operatorname{Ind}_{HK_1}^{I}{\chi^s}}$ as a representation of $I$, then the natural map $$\label{injExt2}
\operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi\alpha, N_{\chi})\oplus \operatorname{Ext}^2_{I/Z_1}(\chi\alpha, N_{\chi^s}^{\Pi})$$ is injective, where $N_{\chi^s}^\Pi$ denotes the twist of action of $I$ on $N_{\chi^s}$ by $\Pi$.
Applying $\operatorname{Hom}_{I/Z_1}(\chi\alpha, \centerdot)$ to the short exact sequence $0\rightarrow \chi\rightarrow N_{\chi}\rightarrow N_{\chi}/\chi\rightarrow 0$ gives a long exact sequence. Equation shows that the map $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\chi)\rightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,N_\chi)$ is surjective, which implies that $$\operatorname{Ker}(\operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi\alpha, N_{\chi}))\cong \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi}/\chi).$$ If we replace $N_{\chi}$ with $N_{\chi^s}$ and $\chi$ with $\chi^s$ the same isomorphism holds. Twisting by $\Pi$ gives: $$\operatorname{Ker}(\operatorname{Ext}^2_{I/Z_1}(\chi\alpha,\chi)\rightarrow \operatorname{Ext}^2_{I/Z_1}(\chi\alpha, N_{\chi^s}^{\Pi}))\cong \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi^s}^{\Pi}/\chi).$$ Lemma \[p3\] implies that the composition $$\operatorname{Res}\circ \partial: \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi}/\chi)\rightarrow \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi\alpha,\chi)$$ is an injection. And Lemma \[p3bis\] implies that the composition $$\operatorname{Res}\circ\partial: \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi^s}^{\Pi}/\chi)\rightarrow \operatorname{Ext}^2_{(I\cap P)/Z_1}(\chi\alpha,\chi)$$ is zero. Hence, $\partial( \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi}/\chi))\cap \partial(\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, N_{\chi}^{\Pi}/\chi))=0$ and so the map in is injective.
\[p3r0\] Assume $p=3$ and $r=0$ then $M_{{\tilde{\sigma}}}$ satisfies the assumptions of Lemma \[p3bis\].
Now $\langle (I\cap U)tv_{\sigma}\rangle =\langle I sv_{{\tilde{\sigma}}}\rangle \cong {St}|_I\cong {\operatorname{Ind}_{HK_1}^{I}{\chi^s}}$ as a representation of $I$, where ${St}\cong \operatorname{Sym}^2\overline{\mathbb F}_3^2$ is the Steinberg representation of $\operatorname{GL}_2(\mathbb{F}_3)$. Hence we have an injection ${\operatorname{Ind}_{HK_1}^{I}{\chi^s}}\hookrightarrow M_{{\tilde{\sigma}}}$. Since $M_{{\tilde{\sigma}}}|_{H(I\cap U)}$ is an injective envelope of $\chi^s$ in $\operatorname{Rep}_{H(I\cap U)}$ we obtain that $M_{{\tilde{\sigma}}}^{K_1\cap U}\cong {\operatorname{Ind}_{H(K_1\cap U)}^{H(I\cap U)}{\chi^s}}$ as a representation of $H(I\cap U)$. Hence $\dim M_{{\tilde{\sigma}}}^{K_1\cap U}=3$ and so we obtain $M_{{\tilde{\sigma}}}^{K_1\cap U}\cong M_{{\tilde{\sigma}}}^{K_1}\cong
{\operatorname{Ind}_{HK_1}^{I}{\chi^s}}$.
Using the Lemmas above we prove the main result of this section.
\[main\] Assume $p>2$ and let $\psi: H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character, such that $\operatorname{Ext}^1_{I/Z_1}(\psi,\pi_{\sigma})\neq 0$. Then $\psi\in \{\chi\alpha, \chi\}$. Moreover,
- $\dim \operatorname{Ext}^1_{I/Z_1}(\chi,\pi_{\sigma})=2$;
- if $p>3$ or $p=3$ and $r\in \{0,2\}$ then $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\pi_{\sigma})=0$;
- if $p=3$ and $r=1$ then $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\pi_{\sigma})\le 1$.
Corollary \[take\], gives injections: $$\operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\hookrightarrow \operatorname{Ext}^2_{I/Z_1}(\chi,M_{\sigma}),$$ $$\operatorname{Ext}^2_{I/Z_1}(\chi,\chi)\hookrightarrow \operatorname{Ext}^2_{I/Z_1}(\chi,\Pi\centerdot M_{{\tilde{\sigma}}}).$$ Moreover, $\operatorname{Ext}^1_{I/Z_1}(\chi,\chi)=0$. Corollary \[one\] gives a short exact sequence $0\rightarrow \chi\rightarrow M_{\sigma}\oplus \Pi\centerdot M_{{\tilde{\sigma}}}\rightarrow \pi_{\sigma}\rightarrow 0$, which induces an isomorphism: $$\operatorname{Ext}^1_{I/Z_1}(\chi,M_{\sigma})\oplus \operatorname{Ext}^1_{I/Z_1}(\chi,\Pi\centerdot M_{{\tilde{\sigma}}})\cong \operatorname{Ext}^1_{I/Z_1}(\chi,\pi_{\sigma}).$$ Corollary \[take\] implies that $\dim \operatorname{Ext}^1_{I/Z_1}(\chi,\pi_{\sigma})=2$.
Assume that $\psi\neq \chi$. From $0\rightarrow M_{\sigma}\rightarrow \pi_{\sigma}\rightarrow (\Pi\centerdot M_{{\tilde{\sigma}}})/\chi\rightarrow 0$ we obtain a long exact sequence: $$\begin{split}
\operatorname{Hom}_{I}(\psi,\chi\alpha)\hookrightarrow \operatorname{Ext}^1_{I/Z_1}(\psi,M_{\sigma})\rightarrow &\operatorname{Ext}^1_{I/Z_1}(\psi,\pi_{\sigma})\rightarrow\\
&\operatorname{Ext}^1_{I/Z_1}(\psi,(\Pi\centerdot M_{{\tilde{\sigma}}})/\chi).
\end{split}$$ If $\operatorname{Ext}^1_{I/Z_1}(\psi,M_{\sigma})\neq 0$ then Proposition \[extX\] implies $\psi=\chi\alpha$. Similarly, if $\operatorname{Ext}^1_{I/Z_1}(\psi,(\Pi\centerdot M_{{\tilde{\sigma}}})/\chi)\neq 0$ then $\psi=(\chi^s \alpha^{-1})^{\Pi}=\chi\alpha$. Hence, $\psi=\chi\alpha$ and $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \pi_\sigma)\le 1$.
If $p>3$ then Proposition \[extX\] implies that $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, M_{\sigma}/\chi)=0$. Hence the exact sequence $0\rightarrow \Pi\centerdot M_{{\tilde{\sigma}}}\rightarrow \pi_{\sigma}\rightarrow M_{\sigma}/\chi\rightarrow 0$ gives an exact sequence: $$\operatorname{Hom}_{I}(\chi\alpha,\chi\alpha^{-1})\hookrightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\Pi\centerdot M_{{\tilde{\sigma}}})\twoheadrightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\pi_{\sigma}).$$ Since $p>3$ Proposition \[extX\] implies that $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\Pi\centerdot M_{{\tilde{\sigma}}})=0$ and hence $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha,\pi_{\sigma})=0$.
Assume that $p=3$ and $r=0$ Lemmas \[done\] and \[p3r0\] give an exact sequence: $$\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \chi)\hookrightarrow \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, M_{\sigma}\oplus \Pi\centerdot M_{{\tilde{\sigma}}})\twoheadrightarrow
\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \pi_\sigma).$$ Since $p=3$ we have $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \chi)=2$ and Proposition \[extX\] gives $\dim \operatorname{Ext}^1_{I/Z_1}(\chi\alpha, M_{\sigma}\oplus \Pi\centerdot M_{{\tilde{\sigma}}})=2$. Hence $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \pi_\sigma)=0$. Since $p=3$ and $r=0$ we have $(\chi\alpha)^{\Pi}=\chi\alpha$, $\chi=\chi^s$ and since $\pi_{{\tilde{\sigma}}}=\Pi\centerdot \pi_{\sigma}$, we also obtain $\operatorname{Ext}^1_{I/Z_1}(\chi\alpha, \pi_{{\tilde{\sigma}}})=0$, which deals with the case $p=3$ and $r=2$.
\[dimH1\] Assume $p>2$ and let $\psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character. Suppose that $\operatorname{Hom}_I(\psi, H^1(I_1/Z_1, \pi))\neq 0$ then $\psi\in \{\chi, \chi^s\}$. Moreover, the following hold:
- if $p=3$ and $r=1$ then $\dim H^1(I_1/Z_1, \pi)\le 6$;
- otherwise, $\dim H^1(I_1/Z_1,\pi)=4$.
By Corollary \[two\] $\pi\cong \pi_{\sigma}\oplus \pi_{{\tilde{\sigma}}}$ as $I$-representations. The assertion follows from Theorem \[main\]. We note that if $p=3$ and $r=1$ then $\chi\alpha=\chi^s$ and $\chi^s\alpha=\chi$.
Extensions and central characters {#extandcenter}
=================================
We fix a smooth character $\zeta: Z\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ and let $\operatorname{Rep}_{G, \zeta}$ be the full category of $\operatorname{Rep}_G$ consisting of representations with central character $\zeta$. Let $V$ be an ${\overline{\mathbb{F}}_{p}}$-vector space with an action of $Z$, given by $zv=\zeta(z)v$, for all $z\in Z$ and $v\in V$. Then ${\operatorname{Ind}_{Z}^{G}{V}}$ is an object of $\operatorname{Rep}_{G, \zeta}$, moreover given $\pi$ in $\operatorname{Rep}_{G, \zeta}$ by Frobenius reciprocity we get $$\label{Frobrec}
\operatorname{Hom}_G(\pi, {\operatorname{Ind}_{Z}^{G}{V}})\cong \operatorname{Hom}_Z(\pi, V)\cong \operatorname{Hom}_{{\overline{\mathbb{F}}_{p}}}(\pi, V).$$ Hence, the functor $\operatorname{Hom}_G(\centerdot, {\operatorname{Ind}_{Z}^{G}{V}})$ is exact and so ${\operatorname{Ind}_{Z}^{G}{V}}$ is an injective object in $\operatorname{Rep}_{G, \zeta}$. Further, if $V$ is the underlying vector space of $\pi$ then we may embed $\pi\hookrightarrow {\operatorname{Ind}_{Z}^{G}{V}}$, $v\mapsto [g\mapsto g v]$. Hence, $\operatorname{Rep}_{G, \zeta}$ has enough injectives.
For $\pi_1, \pi_2$ in $\operatorname{Rep}_{G, \zeta}$ we denote $\operatorname{Ext}^1_{G,\zeta}(\pi_1,\pi_2):=\operatorname{\mathbb{R}}^1\operatorname{Hom}(\pi_1,\pi_2)$ computed in the category $\operatorname{Rep}_{G,\zeta}$.
\[centre\] Let $\pi_1$ and $\pi_2$ be irreducible representations of $G$ admitting a central character. Let $\zeta$ be the central character of $\pi_2$. If $\operatorname{Ext}^1_G(\pi_1,\pi_2)\neq 0$ then $\zeta$ is also the central character of $\pi_1$. If $\pi_1\not\cong \pi_2$ then $\operatorname{Ext}^1_{G,\zeta}(\pi_1,\pi_2)=\operatorname{Ext}^1_{G}(\pi_1,\pi_2)$. If $\pi_1\cong\pi_2$ then there exists an exact sequence: $$0\rightarrow \operatorname{Ext}^1_{G,\zeta}(\pi_1,\pi_2)\rightarrow \operatorname{Ext}^1_{G}(\pi_1,\pi_2)\rightarrow \operatorname{Hom}(Z,{\overline{\mathbb{F}}_{p}})\rightarrow 0.$$
Suppose that we have a non-split extension $0\rightarrow \pi_2\rightarrow E\rightarrow \pi_1\rightarrow 0$ in $\operatorname{Rep}_G$. For all $z\in Z$ we define $\theta_z: E\rightarrow E$, $v\mapsto z v -\zeta(z)v$. Since $z$ is central in $G$, $\theta_z$ is $G$-equivariant. If $\theta_z=0$ for all $z\in Z$ then $E$ admits a central character $\zeta$, and hence $\zeta$ is the central character of $\pi_1$ and the extension lies in $\operatorname{Ext}^1_{G,\zeta}(\pi_1,\pi_2)$. If $\theta_z\neq 0$ for some $z\in Z$ then it induces an isomorphism $\pi_1\cong \pi_2$.
We assume that $\pi_1\cong\pi_2$ and drop the subscript. Then gives $\operatorname{Hom}_G(\pi,{\operatorname{Ind}_{Z}^{G}{\zeta}})\cong\pi^*$. Fix a non-zero $\varphi\in \operatorname{Hom}_Z(\pi,\zeta)$. Since $\pi$ is irreducible we obtain an exact sequence: $$\label{seqcen}
0\rightarrow \pi\overset{\varphi}{\rightarrow} {\operatorname{Ind}_{Z}^{G}{\zeta}}\rightarrow Q\rightarrow 0.$$ Since ${\operatorname{Ind}_{Z}^{G}{\zeta}}$ is an injective object in $\operatorname{Rep}_{G,\zeta}$, and is in $\operatorname{Rep}_{G,\zeta}$ by applying $\operatorname{Hom}_G(\pi, \centerdot )$ to we obtain an exact sequence: $$\label{seq1}
\pi^*\rightarrow \operatorname{Hom}_G(\pi, Q)\rightarrow \operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\rightarrow 0.$$ If we consider as an exact sequence in $\operatorname{Rep}_G$ then by applying $\operatorname{Hom}_G(\pi, \centerdot )$ we get an exact sequence: $$\label{seq2}
\pi^*\rightarrow \operatorname{Hom}_G(\pi, Q)\rightarrow \operatorname{Ext}^1_{G}(\pi,\pi)\rightarrow \operatorname{Ext}^1_{G}(\pi, {\operatorname{Ind}_{Z}^{G}{\zeta}}).$$ Putting and together we obtain an exact sequence: $$0\rightarrow \operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\rightarrow \operatorname{Ext}^1_G(\pi,\pi)\rightarrow \operatorname{Ext}^1_{G}(\pi, {\operatorname{Ind}_{Z}^{G}{\zeta}}).$$
Let $0\rightarrow {\operatorname{Ind}_{Z}^{G}{\zeta}}\rightarrow E\rightarrow \pi\rightarrow 0$ be an extension in $\operatorname{Rep}_G$. For all $z\in Z$, $\theta_z:E\rightarrow E$ induces $\theta_z(E)\in \operatorname{Hom}_G(\pi, {\operatorname{Ind}_{Z}^{G}{\zeta}})$. Now $\theta_z(E)=0$ for all $z\in Z$ if and only if $E$ has a central character $\zeta$, but since ${\operatorname{Ind}_{Z}^{G}{\zeta}}$ is an injective object in $\operatorname{Rep}_{G,\zeta}$ Lemma \[extriv\] implies that the sequence is split if and only if $E$ has a central character $\zeta$. Now $$\label{thetapsi}
\begin{split}
\theta_{z_1z_2}(v)&= z_1z_2 v-\zeta(z_1z_2)v= z_1( z_2 v-\zeta(z_2)v)+ z_1\zeta(z_2)v-\zeta(z_1z_2)v\\&= \zeta(z_1)\theta_{z_2}(v)+\zeta(z_2)\theta_{z_1}(v).
\end{split}$$ Hence, if we set $\psi_E(z):=\zeta(z)^{-1} \theta_z(E)$, then gives $\psi_E(z_1z_2)=\psi_E(z_1)+\psi_E(z_2)$. Hence, the map $E\mapsto \psi_E$ induces an injection $\operatorname{Ext}^1_{G}(\pi, {\operatorname{Ind}_{Z}^{G}{\zeta}})\hookrightarrow \operatorname{Hom}(Z, \pi^*)$. The image of $$\operatorname{Ext}^1_G(\pi,\pi)\rightarrow \operatorname{Ext}^1_{G}(\pi, {\operatorname{Ind}_{Z}^{G}{\zeta}})\hookrightarrow \operatorname{Hom}(Z, \pi^*)$$ is $\operatorname{Hom}(Z, {\overline{\mathbb{F}}_{p}}\varphi)$, which is isomorphic to $\operatorname{Hom}(Z,{\overline{\mathbb{F}}_{p}})$.
\[CK\] Let $\pi:=\pi(r, 0, \eta)$ and $\zeta$ the central character of $\pi$. Assume that $p>2$ and $(p,r)\neq (3,1)$ then $\dim \operatorname{Ext}^1_{G, \zeta}(\pi,\pi)\ge 3$.
This follows from [@kis1 2.3.4].
At the time of writing this note, [@kis1] was not written up and there were some technical issues with the outline of the argument given in the introductions to [@col1] and [@kis]. Since we only need a lower bound on the dimension and only in the supersingular case, we have written up another proof of Proposition \[CK\] in the appendix. The proof given there is a variation of Colmez-Kisin argument.
Hecke Algebra {#hecke}
=============
Let $\zeta$ be the central character of $\pi$. Let ${\mathcal H}:=\operatorname{End}_{G}({\operatorname{c-Ind}_{ZI_1}^{G}{\zeta}})$. Let ${\mathcal I}: \operatorname{Rep}_{G,\zeta}\rightarrow \operatorname{Mod}_{{\mathcal H}}$ be the functor: $${\mathcal I}(\pi):=\pi^{I_1}\cong \operatorname{Hom}_G({\operatorname{c-Ind}_{ZI_1}^{G}{\zeta}}, \pi).$$ Let ${\mathcal T}:\operatorname{Mod}_{{\mathcal H}}\rightarrow \operatorname{Rep}_{G,\zeta}$ be the functor: $${\mathcal T}(M):=M\otimes_{{\mathcal H}} {\operatorname{c-Ind}_{ZI_1}^{G}{\zeta}}.$$ One has $\operatorname{Hom}_{{\mathcal H}}(M, {\mathcal I}(\pi))\cong \operatorname{Hom}_G({\mathcal T}(M),\pi)$. Moreover, Vignéras in [@vig Thm.5.4] shows that ${\mathcal I}$ induces a bijection between irreducible objects in $\operatorname{Rep}_{G,\zeta}$ and $\operatorname{Mod}_{{\mathcal H}}$. Let $\operatorname{Rep}^{I_1}_{G, \zeta}$ be the full subcategory of $\operatorname{Rep}_{G,\zeta}$ consisting of representations generated by their $I_1$-invariants. Ollivier has shown [@o2] that $${\mathcal I}: \operatorname{Rep}^{I_1}_{G, \zeta}\rightarrow \operatorname{Mod}_{{\mathcal H}}, \quad {\mathcal T}: \operatorname{Mod}_{{\mathcal H}}\rightarrow \operatorname{Rep}^{I_1}_{G, \zeta}$$ are quasi-inverse to each other and so $\operatorname{Mod}_{{\mathcal H}}$ is equivalent to $\operatorname{Rep}^{I_1}_{G, \zeta}$. In particular, suppose that $\tau=\langle G \centerdot \tau^{I_1}\rangle$, $\pi$ in $\operatorname{Rep}_{G, \zeta}$ and let $\pi_1:=\langle G\centerdot \pi^{I_1}\rangle \subseteq \pi$ then one has: $$\label{homit}
\begin{split}
\operatorname{Hom}_G(\tau, \pi)\cong \operatorname{Hom}_{G}(\tau, \pi_1) \cong \operatorname{Hom}_{{\mathcal H}}&({\mathcal I}(\tau), {\mathcal I}(\pi_1))\\ &\cong \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\pi))
\end{split}$$ and the natural map ${\mathcal T}{\mathcal I}(\tau)\rightarrow \tau$ is an isomorphism.
Let $J$ be an injective object in $\operatorname{Rep}_{G, \zeta}$, then the first isomorphism of implies that $J_1:=\langle G\centerdot J^{I_1}\rangle$ is an injective object in $\operatorname{Rep}^{I_1}_{G, \zeta}$. Since ${\mathcal T}$ and ${\mathcal I}$ induce an equivalence of categories between $\operatorname{Mod}_{{\mathcal H}}$ and $\operatorname{Rep}^{I_1}_{G, \zeta}$ we obtain that ${\mathcal I}(J_1)={\mathcal I}(J)$ is an injective object in $\operatorname{Mod}_{{\mathcal H}}$. Hence, gives an $E_2$-spectral sequence: $$\label{specseq}
\operatorname{Ext}^i_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^j {\mathcal I}(\pi))\Longrightarrow \operatorname{Ext}^{i+j}_{G, \zeta}(\tau, \pi)$$ The $5$-term sequence associated to gives us:
\[5term\] Let $\tau$ and $\pi$ be in $\operatorname{Rep}_{G, \zeta}$ suppose that $\tau$ is generated as a $G$-representation by $\tau^{I_1}$ then there exists an exact sequence: $$\label{5T}
\begin{split}
0\rightarrow &\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\pi))\rightarrow \operatorname{Ext}^1_{G, \zeta}(\tau, \pi)\rightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))\\
&\rightarrow \operatorname{Ext}^2_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\pi))\rightarrow \operatorname{Ext}^2_{G, \zeta}(\tau, \pi)
\end{split}$$
It is easy to write down the first two non-trivial arrows of explicitly. An extension class of $0\rightarrow {\mathcal I}(\pi)\rightarrow E\rightarrow {\mathcal I}(\tau)\rightarrow 0$ maps to the extension class of $0\rightarrow {\mathcal T}{\mathcal I}(\pi)\rightarrow {\mathcal T}(E)\rightarrow {\mathcal T}{\mathcal I}(\tau)\rightarrow 0$. Let $\epsilon$ be an extension class of $0\rightarrow \pi\rightarrow \kappa\rightarrow \tau\rightarrow 0$. We may apply ${\mathcal I}$ to get $$\label{nearlydone}
\xymatrix@1{0 \ar[r]& {\mathcal I}(\pi)\ar[r]& {\mathcal I}(\kappa)\ar[r]& {\mathcal I}(\tau)\ar[r]^-{\partial_{\epsilon}}& \mathbb R^1{\mathcal I}(\pi).}$$ The second non-trivial arrow in is given by $\epsilon\mapsto \partial_{\epsilon}$.
We are interested in when both $\pi$ and $\tau$ are irreducible. We recall some facts about the structure of ${\mathcal H}$ and its irreducible modules, for proofs see [@vig] or [@coeff §1]. As an ${\overline{\mathbb{F}}_{p}}$-vector space ${\mathcal H}$ has a basis indexed by double cosets $I_1\backslash G/Z I_1$, we write $T_g$ for the element corresponding to a double coset $I_1g Z I_1$. Given $\pi$ in $\operatorname{Rep}_{G,\zeta}$, and $v\in \pi^{I_1}$, the action of $T_g$ is given by: $$\label{Tg}
vT_g= \sum_{u\in I_1/(I_1\cap g^{-1} I_1 g)} u g^{-1} v.$$ Let $\chi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character then we define $e_{\chi}\in {\mathcal H}$ by $$e_{\chi}:=\frac{1}{|H|} \sum_{h\in H} \chi(h) T_h.$$ Then $e_{\chi}e_{\psi}=e_{\chi}$ if $\chi=\psi$ and $0$ otherwise and it follows from that $\pi^{I_1}e_{\chi}$ is the $\chi$-isotypical subspace of $\pi^{I_1}$ as a representation of $H$. The elements $T_{n_s}$, $T_{\Pi}$ and $e_{\chi}$, for all $\chi$ generate ${\mathcal H}$ as an algebra, and are subject to the following relations: $T_{\Pi}^2=\zeta(p)^{-1}$, $$\label{relTH}
e_{\chi}T_{n_s}=T_{n_s}e_{\chi^s}, \quad e_{\chi}T_{\Pi}=T_{\Pi}e_{\chi^s}, \quad \ e_{\chi}T_{n_s}^2=-e_{\chi}e_{\chi^s}T_{n_s}.$$ Note that $e_{\chi}e_{\chi^s}=e_{\chi}$ if $\chi=\chi^s$ and $e_{\chi}e_{\chi^s}=0$, otherwise. We let ${\mathcal H}^+$ be the subalgebra of ${\mathcal H}$ generated by $T_{n_s}$, $T_{\Pi} T_{n_s} T_{\Pi}^{-1}$ and $e_{\chi}$ for all characters $\chi$. One may naturally identify ${\mathcal H}^+\cong \operatorname{End}_{G^+}({\operatorname{c-Ind}_{ZI_1}^{G^+}{\zeta}})$.
\[Mrlambdaeta\] Let $0\le r \le p-1$ be an integer, $\lambda\in {\overline{\mathbb{F}}_{p}}$ and $\eta: {\mathbb {Q}_p}^{\times}\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ a smooth character, and let $\zeta$ be the central character of $\pi(r,\lambda,\eta)$ then we define ${\mathcal H}$-modules $M(r,\lambda):= \pi(r,\lambda)^{I_1}$, $M(r,\lambda, \eta):=\pi(r,\lambda, \eta)^{I_1}$.
Assume for simplicity that $\zeta(p)=1$ then it is shown in [@bp Cor. 6.4] that $M(r,\lambda, \eta)$ has an ${\overline{\mathbb{F}}_{p}}$-basis $\{v_1, v_2\}$ such that
- $v_1 e_{\chi}= v_1,\quad v_1 T_{\Pi}= v_2, \quad v_2 e_{\chi^s}= v_2, \quad v_2 T_{\Pi}=v_1$ and such that $v_1 T_{n_s}=-v_1$ if $r=p-1$ and $v_1 T_{n_s}=0$ otherwise.
- $v_2(1+T_{n_s})= \eta(-p^{-1})\lambda v_1$ if $r=0$ and $v_2 T_{n_s}= \eta(-p^{-1})\lambda v_1$ otherwise,
where $\chi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is the character $\chi(\bigl (\begin{smallmatrix} [\lambda]& 0\\ 0 & [\mu]\end{smallmatrix}\bigr))=
\lambda^r\eta([\lambda\mu])$. If $\lambda=0$ so that $\pi(r,\lambda,\eta)$ is supersingular, then $v_1=v_{\sigma}$ and $v_2=v_{{\tilde{\sigma}}}$.
\[extH\] Let $\pi$ be a supersingular representation of $G$ then
- if $r\in \{0,p-1\}$ then
- $\dim \operatorname{Ext}^1_{{\mathcal H}}(\pi^{I_1},\pi^{I_1})=1$;
- $\operatorname{Ext}^i_{{\mathcal H}}(\pi^{I_1}, \ast)=0$ for $i>1$;
- otherwise, $\dim \operatorname{Ext}^1_{{\mathcal H}}(\pi^{I_1},\pi^{I_1})=2$.
[@bp Cor. 6.7, 6.6].
We look more closely at the regular case. Let $\pi$ be supersingular with $0<r<p-1$ and assume for simplicity that $p\in Z$ acts trivially on $\pi$. For $(\lambda_1, \lambda_2)\in {\overline{\mathbb{F}}_{p}}^2$ we define an ${\mathcal H}$-module $E_{\lambda_1, \lambda_2}$ to be a $4$-dimensional vector space with basis $\{v_{\chi}, v_{\chi^s}, w_{\chi}, w_{\chi^s}\}$ with the action of ${\mathcal H}$ given on the generators $$w_{\chi} T_{n_s}= \lambda_1 v_{\chi^s}, \quad w_{\chi^s}T_{n_s}= \lambda_2 v_{\chi}, \quad
v_{\chi} T_{n_s}= v_{\chi^s} T_{n_s}=0$$ and $w_\psi T_{\Pi}=w_{\psi^s}$, $v_{\psi}T_{\Pi}= v_{\psi^s}$, $w_{\psi} e_{\psi}= w_{\psi}$, $v_{\psi} e_{\psi}= v_{\psi}$, for $\psi\in \{\chi, \chi^s\}$. Then $\langle v_{\chi}, v_{\chi^s}\rangle$ is stable under the action of ${\mathcal H}$ and we have an exact sequence: $$\label{stex}
0\rightarrow {\mathcal I}(\pi)\rightarrow E_{\lambda_1, \lambda_2}\rightarrow {\mathcal I}(\pi)\rightarrow 0$$ The extension is split if and only if $(\lambda_1,\lambda_2)=(0,0)$. It is immediate that the map ${\overline{\mathbb{F}}_{p}}^2\rightarrow \operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\pi), {\mathcal I}(\pi))$ sending $(\lambda_1, \lambda_2)$ to the equivalence class of is an isomorphism of ${\overline{\mathbb{F}}_{p}}$-vector spaces.
\[lastdino\] Let $\lambda\in {\overline{\mathbb{F}}_{p}}^{\times}$ then $$\label{lastdino1}
{\mathcal T}(E_{0,\lambda})\cong \frac{{\operatorname{c-Ind}_{KZ}^{G}{\sigma}}}{(T_{\sigma}^2)}, \quad {\mathcal T}(E_{\lambda,0})\cong \frac{{\operatorname{c-Ind}_{KZ}^{G}{{\tilde{\sigma}}}}}{(T_{{\tilde{\sigma}}}^2)}$$ where $T_{\sigma}\in \operatorname{End}_G({\operatorname{c-Ind}_{KZ}^{G}{\sigma}})$ is given by Lemma \[T\].
Let $\varphi\in {\operatorname{c-Ind}_{KZ}^{G}{\sigma}}$ such that $\operatorname{Supp}\varphi=KZ$ and $\varphi(1)$ spans $\sigma^{I_1}$. Let $\tau:= \frac{{\operatorname{c-Ind}_{KZ}^{G}{\sigma}}}{(T_{\sigma}^2)}$ and $v$ the image of $\varphi$ in $\tau$. Then $\tau=\langle G\centerdot v\rangle=\langle G\centerdot \tau^{I_1}\rangle$. And so it is enough to show that ${\mathcal I}(\tau)\cong E_{0, \lambda}$. Since $T_{\sigma}: {\operatorname{c-Ind}_{KZ}^{G}{\sigma}}\rightarrow {\operatorname{c-Ind}_{KZ}^{G}{\sigma}}$ is injective and $\pi\cong \frac{{\operatorname{c-Ind}_{KZ}^{G}{\sigma}}}{(T_{\sigma})}$, we have a an exact sequence $$\label{lastdino2}
0\rightarrow \pi\rightarrow \tau\rightarrow \pi\rightarrow 0$$ and we may identify the subobject with $T_{\sigma}(\tau)$. Now, $v$, $\Pi v$, $T_{\sigma}(v)$ and $T_{\sigma}(\Pi v)$ are linearly independent and $I_1$-invariant. Thus $\dim \tau^{I_1}\ge 4$ and since $\dim \pi^{I_1}=2$ we obtain an exact sequence of ${\mathcal H}$-modules $$\label{lastdino3}
0\rightarrow {\mathcal I}(\pi)\rightarrow {\mathcal I}(\tau)\rightarrow {\mathcal I}(\pi)\rightarrow 0$$ Hence, ${\mathcal I}(\tau)\cong E_{\lambda_1, \lambda_2}$ for some $\lambda_1, \lambda_2\in {\overline{\mathbb{F}}_{p}}$. Since $\sigma\cong \langle K\centerdot \varphi\rangle\cong \langle K\centerdot v\rangle$ and $\langle K\centerdot T_{\sigma}(v)\rangle\cong T_{\sigma}(\langle K\centerdot v\rangle)\cong \sigma$, [@coeff 3.1.3] gives $$\label{lastdino4}
v e_{\chi}= v, \quad (T_{\sigma}(v))e_{\chi}=T_{\sigma}(v), \quad v T_{n_s}= (T_{\sigma}(v))T_{n_s}=0.$$ Hence, $\lambda_1=0$. If $\lambda_2=0$ then would split and so would . Hence, $\lambda_2\neq 0$. We leave it to the reader to check that for any $\lambda\in {\overline{\mathbb{F}}_{p}}^{\times}$, $E_{0,\lambda}\cong E_{0,1}$.
\[kuku\] If $E=E_{\lambda_1,\lambda_2}$, $\lambda_1\lambda_2\neq 0$ then $\dim \operatorname{Ext}^1_{{\mathcal H}}(E, {\mathcal I}(\pi))=1$.
Applying $\operatorname{Hom}_{{\mathcal H}}(\ast, {\mathcal I}(\pi))$ to gives an exact sequence $$\label{stex2}
\begin{split}
\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\pi), {\mathcal I}(\pi))\hookrightarrow &\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\pi), {\mathcal I}(\pi))\\ &\rightarrow
\operatorname{Ext}^1_{{\mathcal H}}(E, {\mathcal I}(\pi))\rightarrow \operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\pi), {\mathcal I}(\pi))
\end{split}$$ Hence, $\dim \operatorname{Ext}^1_{{\mathcal H}}(E, {\mathcal I}(\pi))= 1 +\dim \Upsilon$, where $\Upsilon$ is the image of the last arrow in . Yoneda’s interpretation of $\operatorname{Ext}$ says that $\Upsilon\neq 0$ is equivalent to the following commutative diagram of ${\mathcal H}$-modules: $$\xymatrix{0\ar[r]& {\mathcal I}(\pi)\ar[r]\ar[d]^{=}& A\ar[r]\ar@{^(->}[d]& {\mathcal I}(\pi)\ar[r]\ar@{^(->}[d]& 0 \\
0\ar[r]& {\mathcal I}(\pi)\ar[r]& B \ar[r]& E \ar[r]& 0}$$ with $A$ non-split. Then $A\cong E_{\mu_1, \mu_2}$ for some $\mu_1, \mu_2\in {\overline{\mathbb{F}}_{p}}$. The condition $v T_{n_s}^2=0$ for all $v\in B$ is equivalent to $\mu_1\lambda_2=0$ and $\mu_2\lambda_1=0$. Since $\lambda_1\lambda_2\neq 0$ we obtain $\mu_1=\mu_2=0$ and hence a contradiction to a non-split $A$.
Main result
===========
Let $\pi$ an irreducible representation with a central character $\zeta$. A construction of [@coeff §6], [@bp §9] gives an injection $\pi\hookrightarrow \Omega$, where $\Omega$ is in $\operatorname{Rep}_{G,\zeta}$ and $\Omega|_{K}$ is an injective envelope of $\operatorname{soc}_K \pi$ in $\operatorname{Rep}_{K,\zeta}$.
\[Opi\] If $\pi\cong \pi(r, 0, \eta)$ with $0<r<p-1$ then $\Omega^{I_1}\cong E_{\lambda_1, \lambda_2}$ with $\lambda_1\lambda_2\neq 0$. Otherwise, $\Omega^{I_1}\cong \pi^{I_1}$.
Let $\sigma$ be an irreducible smooth representation of $K$ and $\operatorname{Inj}\sigma$ injective envelope of $\sigma$ in $\operatorname{Rep}_{K, \zeta}$. If $\sigma=\chi\circ \det$ or $\sigma\cong {St}\otimes \chi\circ \det$ then $\dim (\operatorname{Inj}\sigma)^{I_1}=\dim \sigma^{I_1}=1$ and $\dim (\operatorname{Inj}\sigma)^{I_1}=2$ otherwise, [@coeff 6.4.1, §4.1]. If $\pi$ is either a character, special series, a twist of unramified series or $\pi\cong \pi(0,0, \eta)$ then $\operatorname{soc}_K \pi$ is a direct summand of $({\mathbf 1}\oplus {St})\otimes \chi\circ \det$. Hence, $$\Omega^{I_1}=(\operatorname{soc}_K \Omega)^{I_1}=(\operatorname{soc}_K \pi)^{I_1} \subseteq \pi^{I_1}\subseteq \Omega^{I_1}$$ and so $\pi^{I_1}\cong \Omega^{I_1}$. If $\pi$ is a tamely ramified principal series, which is not a twist of unramified principal series, then $\dim \pi^{I_1}=2$ and $\operatorname{soc}_K \pi$ is irreducible, so $\dim \Omega^{I_1}=2$. Finally, if $\pi\cong \pi(r, 0, \eta)$ with $0< r< p-1$ then it follows from [@coeff 6.4.5] that $\Omega^{I_1}\cong E_{\lambda_1, \lambda_2}$ with $\lambda_1\lambda_2\neq 0$.
\[chop00\] Let $\pi, \tau$ be irreducible representations of $G$ with a central character, and let $\zeta$ be the central character of $\pi$. Suppose that $\operatorname{Ext}^1_G(\tau,\pi)\neq 0$. If $$\label{mapdelta}
\operatorname{Ext}^1_{G, \zeta}(\tau, \pi)\rightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\operatorname{\mathbb{R}}^1{\mathcal I}(\pi))$$ is not surjective then $\tau\cong \pi\cong \pi(r,0, \eta)$ with $0<r<p-1$.
We note that Proposition \[centre\] implies that $\zeta$ is the central character of $\tau$. Since $\Omega|_K$ is an injective object in $\operatorname{Rep}_{K, \zeta}$, $\Omega|_{I_1}$ is an injective object in $\operatorname{Rep}_{I_1,\zeta}$. Hence, $\operatorname{\mathbb{R}}^1{\mathcal I}(\Omega)=0$ and we have an exact sequence: $$0\rightarrow {\mathcal I}(\pi)\rightarrow {\mathcal I}(\Omega)\rightarrow {\mathcal I}(\Omega/\pi)\rightarrow \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\rightarrow 0.$$ Assume $\pi\cong \pi(r, 0,\eta)$, $0<r<p-1$. Let $\partial\in \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\mathbb R^1{\mathcal I}(\pi))$ be non-zero. Suppose that $\tau\not \cong \pi$ then $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau),{\mathcal I}(\pi))=0$, [@bp 6.5], Lemma \[Opi\] implies ${\mathcal I}(\Omega)/{\mathcal I}(\pi)\cong {\mathcal I}(\pi)$. So we have a surjection $$\label{chop000}
\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\Omega/\pi))\twoheadrightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)).$$ Further, we have an isomorphism $$\label{aberhallo}
\operatorname{Hom}_G(\tau,\Omega/\pi)\cong \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\Omega/\pi)).$$ Choose $\psi\in \operatorname{Hom}_{G}(\tau, \Omega/\pi)$ mapping to $\partial$ under the composition of and . Since $\tau$ is irreducible, by pulling back the image of $\psi$ we obtain an extension $0\rightarrow \pi\rightarrow E_{\psi}\rightarrow \tau\rightarrow 0$ inside of $\Omega$. By construction, maps the class of this extension to $\partial$.
If $\pi\not \cong \pi(r, 0,\eta)$ with $0<r<p-1$ then Lemma \[Opi\] says that ${\mathcal I}(\Omega/\pi)\cong \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ and arguing as above we get that is surjective.
\[chop1\] Let $\pi$, $\tau$ be irreducible representations of $G$ with a central character, and suppose that $\pi$ is supersingular with a central character $\zeta$. If $\operatorname{Ext}^1_G(\tau,\pi)\neq 0$ and $\tau\not\cong \pi$ then $$\operatorname{Ext}^1_{G}(\tau,\pi)\cong \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\mathbb R^1{\mathcal I}(\pi)).$$
Proposition \[centre\] implies that the central character of $\tau$ is $\zeta$ and $\operatorname{Ext}^1_G(\tau, \pi)\cong \operatorname{Ext}^1_{G, \zeta}(\tau, \pi)$. By [@bp Cor.6.5], $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau),{\mathcal I}(\pi))=0$. The assertion follows from Propositions \[5term\], \[chop00\].
\[obv\] Let $\pi$ and $\tau$ be supersingular representations of $G$ with the same central character. Suppose that $\pi^{I_1}\cong \tau^{I_1}$ as $H$-representations then $\pi\cong \tau$.
It follows from the explicit description of supersingular modules $M(r,0,\eta)$ of ${\mathcal H}$ in §\[hecke\] or [@coeff Def.2.1.2] that ${\mathcal I}(\tau)\cong{\mathcal I}(\pi)$ as ${\mathcal H}$-modules. Hence, $\tau \cong {\mathcal T}{\mathcal I}(\tau)\cong {\mathcal T}{\mathcal I}(\pi)\cong \pi$.
\[R1regp5\] Let $\pi=\pi(r,0,\eta)$ with $0<r<p-1$, and let $\zeta$ be the central character of $\pi$. Assume that $p\ge 5$ then $\operatorname{\mathbb{R}}{\mathcal I}^1(\pi)\cong {\mathcal I}(\pi)\oplus {\mathcal I}(\pi)$.
Corollary \[three\] implies that we have an isomorphism of ${\mathcal H}^+$-modules $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\cong \operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})\oplus \operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{{\tilde{\sigma}}})$. Let $v\in \operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})$ it follows from Theorem \[main\] that $ve_{\chi}=v$. Since $0<r<p-1$ we have $\chi\neq \chi^s$ and so $ve_{\chi^s}=0$. Since $T_{n_s}\in {\mathcal H}^+$, $vT_{n_s}\in \operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})$ and hence $vT_{n_s}= vT_{n_s}e_{\chi}=ve_{\chi^s}T_{n_s}=0$. So $T_{n_s}$ kills $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})$ and by symmetry it also kills $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{{\tilde{\sigma}}})$. Theorem \[main\] gives $\dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})=2$. If we chose a basis $\{v, w\}$ of $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{\sigma})$ then $\{vT_{\Pi}, w T_{\Pi}\}$ is a basis of $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi_{{\tilde{\sigma}}})$. And it follows from the explicit description of $M(r,0,\eta)$ in §\[hecke\] that $\langle v, vT_{\Pi}\rangle$ is stable under the action of ${\mathcal H}$ and is isomorphic to $M(r,0,\eta)$.
\[R1regp3\] Let $\pi$ and $\zeta$ be as in Proposition \[R1regp5\] and let $\tau$ be an irreducible representation of $G$ with a central character $\zeta$. Assume $p>2$ and $\tau\not \cong \pi$ then $\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))=0$.
Assume that $\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))\neq 0$ if $p\ge 5$ then Proposition \[R1regp5\] implies that ${\mathcal I}(\tau)\cong {\mathcal I}(\pi)$, and hence $\tau\cong \pi$. Assume that $p=3$ then the assumption $0<r<p-1$ forces $r=1$. Corollary \[dimH1\] implies that $\tau^{I_1}\cong \chi\oplus \chi^s$ as an $H$-representation, where $\chi$ is as in . It follows from Lemma \[obv\] that $\tau$ cannot be supersingular. Since $\chi\neq \chi^s$ we get that $\tau$ is a principal series representation. Corollary \[chop1\] implies that $\operatorname{Ext}^1_G(\tau,\pi)\neq 0$. Let $\eta$ be one of the characters $\omega\circ \det$, $\mu_{-1}\circ \det$, $\omega\mu_{-1}\circ \det$. Since $p=3$ and $r=1$, gives $\pi\cong \pi\otimes\eta$. Twisting by $\eta$ gives $\operatorname{Ext}^1_G(\tau\otimes \eta, \pi)\neq 0$, and hence $\operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau\otimes \eta), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))\neq 0$. Since $p>2$ [@bl Thm 34, Cor 36] imply that $\tau\not \cong \tau\otimes \eta$ and so ${\mathcal I}(\tau)\not \cong {\mathcal I}(\tau\otimes \eta)$ as ${\mathcal H}$-modules. This implies that $\dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ is at least $4\times 2=8$, which contradicts Corollary \[dimH1\].
\[result\] Assume that $p>2$ and let $\tau$ and $\pi$ be irreducible smooth representations of $G$ admitting a central character. Suppose that $\pi$ is supersingular and $\operatorname{Ext}^1_G(\tau, \pi)\neq 0$ then $\tau\cong \pi$.
If $0< r<p-1$ the assertion follows from Corollary \[chop1\] and Proposition \[R1regp3\]. Suppose that $r\in \{0,p-1\}$. Let $\mathfrak I$ be the image of $\operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\rightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\pi), \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))$. Then it follows from Propositions \[CK\], \[5term\] and Lemma \[extH\] that $\dim \mathfrak I\ge 3-1=2$. Hence, ${\mathcal I}(\pi)\oplus {\mathcal I}(\pi)$ is a submodule of $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$. By forgetting the action of ${\mathcal H}$ we obtain an isomorphism of vector spaces $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\cong H^1(I_1/Z_1, \pi)$. Corollary \[dimH1\] implies that $\dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)=4$. Since $\dim {\mathcal I}(\pi)=2$ we obtain $$\label{derived0}
\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\cong {\mathcal I}(\pi)\oplus {\mathcal I}(\pi).$$ Corollary \[chop1\] implies the result.
We note that the proof in the regular case $0<r<p-1$ is purely representation theoretic and makes no use of Colmez’s functor. The Iwahori case $r\in\{0, p-1\}$ could also be done representation theoretically. One needs to work out the action of ${\mathcal H}$ on $H^1(I_1/Z_1, \pi)$. This can be done, but it is not so pleasant, in particular $p=3$ requires extra arguments.
\[pupsi\] Let $\pi\cong \pi(r,0, \eta)$ with $0<r<p-1$, then $$\dim {\mathcal I}(\Omega/\pi) \centerdot e_{\chi}T_{n_s}\ge 1, \quad \dim {\mathcal I}(\Omega/\pi) \centerdot e_{\chi^s}T_{n_s}\ge 1.$$
We have an exact sequence of $K$-representations: $$0\rightarrow \pi^{K_1}\rightarrow \Omega^{K_1}\rightarrow (\Omega/\pi)^{K_1}$$ Since $\Omega|_{K}\cong \operatorname{Inj}\sigma \oplus \operatorname{Inj}{\tilde{\sigma}}$, we have $\Omega^{K_1}\cong \operatorname{inj}\sigma \oplus \operatorname{inj}{\tilde{\sigma}}$, where $\operatorname{inj}$ denotes an injective envelope in the category $\operatorname{Rep}_{K/K_1}$, [@coeff 6.2.4]. In [@bp 20.1, §16] we have determined the $K$-representation $\pi^{K_1}\cong \pi^{K_1}_{\sigma}\oplus \pi^{K_1}_{{\tilde{\sigma}}}$. It follows from the description and [@bp 3.4, 3.5] that $\pi^{K_1}_{\sigma}$ is isomorphic to the kernel of $\operatorname{inj}\sigma \twoheadrightarrow {\operatorname{Ind}_{I}^{K}{\chi}}$. Hence, $(\Omega/\pi)^{K_1}$ contains ${\operatorname{Ind}_{I}^{K}{\chi}}\oplus {\operatorname{Ind}_{I}^{K}{\chi^s}}$ as a subobject and so $(\Omega/\pi)^{I_1}$ contains $V:=({\operatorname{Ind}_{I}^{K}{\chi}}\oplus {\operatorname{Ind}_{I}^{K}{\chi^s}})^{I_1}.$ Moreover, $V$ is stable under the action of $T_{n_s}$, and $\dim V e_{\chi} T_{n_s}=\dim V e_{\chi^s} T_{n_s}=1$, [@coeff 3.1.11]. This yields the claim.
\[gotchabound\] Let $\pi\cong \pi(r,0, \eta)$ with $0<r<p-1$. If $p\ge 5$ then $$\dim \operatorname{Ext}^1_{K/Z_1}(\sigma, \pi)\le 2, \quad \dim \operatorname{Ext}^1_{K/Z_1}({\tilde{\sigma}}, \pi)\le 2.$$ If $p=3$ then $$\dim \operatorname{Ext}^1_{K/Z_1}(\sigma, \pi)\le 3, \quad \dim \operatorname{Ext}^1_{K/Z_1}({\tilde{\sigma}}, \pi)\le 3.$$
We have $\operatorname{Hom}_{K/Z_1}(\sigma, \pi)\cong \operatorname{Hom}_{K/Z_1}(\sigma, \Omega)$, since by construction $\operatorname{soc}_K \Omega \cong \operatorname{soc}_K \pi$. Moreover, since $\Omega|_K$ is injective in $\operatorname{Rep}_{K, \zeta}$ we have $\operatorname{Ext}^1_{K/Z_1}(\sigma, \Omega)=0$. Hence, $$\label{gotcha1}
\operatorname{Hom}_{K/Z_1}(\sigma, \Omega/\pi)\cong \operatorname{Ext}^1_{K/Z_1}(\sigma, \pi).$$ It follows from [@coeff 4.1.5] that if $\kappa$ is any smooth $K$-representation then one has $$\label{gotcha2}
\operatorname{Hom}_{K/Z_1}(\sigma, \kappa)\cong \operatorname{Ker}( {\mathcal I}(\kappa)e_{\chi} \overset{T_{n_s}}{\longrightarrow} {\mathcal I}(\kappa)e_{\chi^s}).$$ Now Lemma \[pupsi\], and imply that $$\label{gotcha3}
\dim \operatorname{Ext}^1_{K/Z_1}(\sigma, \pi)\le \dim {\mathcal I}(\Omega/\pi)e_{\chi}- 1= \dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)e_{\chi}.$$ It follows from Theorem \[main\] that if $p\ge 5$ then $\dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)e_{\chi}=2$ and if $p=3$ then $\dim \operatorname{\mathbb{R}}^1{\mathcal I}(\pi)e_{\chi}\le 3$. The same proof also works for ${\tilde{\sigma}}$.
\[apchibound\] Let $\pi\cong \pi(r,0, \eta)$ with $0<r<p-1$. If $p\ge 5$ then $$\dim \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)\le 3.$$ If $p=3$ then $$\dim \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)\le 4.$$
Recall that we have an exact sequence: $$\label{apchi1}
0\rightarrow {\operatorname{c-Ind}_{KZ}^{G}{\sigma}}\overset{T}{\rightarrow} {\operatorname{c-Ind}_{KZ}^{G}{\sigma}}\rightarrow \pi\rightarrow 0.$$ Applying $\operatorname{Hom}_{G}(\ast, \pi)$ to gives an exact sequence $$\label{apchi2}
\operatorname{Hom}_G({\operatorname{c-Ind}_{KZ}^{G}{\sigma}}, \pi)\hookrightarrow \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)\rightarrow \operatorname{Ext}^1_{G, \zeta}({\operatorname{c-Ind}_{KZ}^{G}{\sigma}}, \pi).$$ We may think of this exact sequence first as Yoneda Exts in $\operatorname{Rep}_{G, \zeta}$, but since $\operatorname{Rep}_{G, \zeta}$ has enough injectives Yoneda’s $\operatorname{Ext}^n$ is isomorphic to $\operatorname{\mathbb{R}}^n \operatorname{Hom}\cong \operatorname{Ext}^n_{G, \zeta}$. For any $A$ in $\operatorname{Rep}_{G, \zeta}$ we have $$\operatorname{Hom}_G({\operatorname{c-Ind}_{KZ}^{G}{\sigma}}, A)\cong \operatorname{Hom}_{K/Z_1}(\sigma, \mathrm F A),$$ where $\mathrm F: \operatorname{Rep}_{G,\zeta}\rightarrow \operatorname{Rep}_{K, \zeta}$ is the restriction. The functor $\mathrm F$ is exact and maps injectives to injectives, hence $$\label{apchi3}
\operatorname{Ext}^1_{G, \zeta}({\operatorname{c-Ind}_{KZ}^{G}{\sigma}}, A)\cong \operatorname{Ext}^1_{K/Z_1}(\sigma, \mathrm F A).$$ Now , and Proposition \[gotchabound\] give the assertion.
The same proof gives:
\[wanted\] Let $n\ge 1$ and $\tau=\frac{{\operatorname{c-Ind}_{KZ}^{G}{\sigma}}}{(T^n)}$ or $\tau=\frac{{\operatorname{c-Ind}_{KZ}^{G}{{\tilde{\sigma}}}}}{(T^n)}$. If $p\ge 5$ then $\dim \operatorname{Ext}^1_{G, \zeta}(\tau, \pi)\le 3$; if $p=3$ then $\dim \operatorname{Ext}^1_{G, \zeta}(\tau, \pi)\le 4$.
\[equality\] Assume $p>2$ and $\pi\cong \pi(r, 0, \eta)$ supersingular. If $(p,r)\neq (3,1)$ then $\dim \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)=3$.
Proposition \[CK\] or §\[app\] gives $\dim \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)\ge 3$. If $0<r<p-1$ then equality follows from Proposition \[apchibound\]. If $r=0$ or $r=p-1$ then $\operatorname{Ext}^2_{{\mathcal H}}({\mathcal I}(\pi), {\mathcal I}(\pi))=0$ and $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\pi),{\mathcal I}(\pi))$ is $1$-dimensional by Lemma \[extH\]. Hence, and give $\dim \operatorname{Ext}^1_{G, \zeta}(\pi, \pi)=3$.
For future use we record the following:
Assume $p>2$ and $\pi\cong \pi(r, 0, \eta)$ supersingular. Let $0\rightarrow {\mathcal I}(\pi)\rightarrow E \rightarrow {\mathcal I}(\pi)\rightarrow 0$ be a non-split extension of ${\mathcal H}$-modules. If $(p,r)\neq (3,1)$ then $\dim \operatorname{Ext}^1_{G, \zeta}({\mathcal T}(E), \pi)\le 3$.
If $(p,r)\neq (3,1)$ then we have $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)\cong {\mathcal I}(\pi)\oplus {\mathcal I}(\pi)$ and so $\dim \operatorname{Hom}_{{\mathcal H}}(E, \operatorname{\mathbb{R}}^1{\mathcal I}(\pi))=2$. So if $\dim \operatorname{Ext}^1_{{\mathcal H}}(E, {\mathcal I}(\pi))\le 1$ then allows us to conclude. If $r=0$ or $r=p-1$ the latter may be deduced from Lemma \[extH\]. If $0<r<p-1$ and $E\cong E_{\lambda_1, \lambda_2}$ with $\lambda_1\lambda_2\neq 0$ then this is given by Lemma \[kuku\]. If $\lambda_1\lambda_2=0$ then ${\mathcal T}(E)\cong \frac{{\operatorname{c-Ind}_{KZ}^{G}{\sigma}}}{(T^2)}$ or $\tau=\frac{{\operatorname{c-Ind}_{KZ}^{G}{{\tilde{\sigma}}}}}{(T^2)}$ and the assertion is given by Corollary \[wanted\].
Non-supersingular representations {#non}
=================================
We compute $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)$, when $\pi$ is the Steinberg representation of $G$ or a character and $\tau$ is an irreducible representation of $G$ under the assumption $p>2$. The results of this paper combined with [@bp] give all the extensions between irreducible representations of $G$, when $p>2$. We record this below. A lot of cases have been worked out by different methods by Colmez [@col1] and Emerton [@em]. The new results of this section are determination of $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathrm{Sp}})$, where ${\mathrm{Sp}}$ is the Steinberg representation, and showing that if $\eta: G\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character of order $2$ then $\operatorname{Ext}^1_G(\eta, {\mathrm{Sp}})=0$.
\[H1Sp\] Assume $p>2$ and let $\psi:H\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character. Suppose that $\operatorname{Ext}^1_{I/Z_1}(\psi,{\mathrm{Sp}})\neq 0$ then $\psi={\mathbf 1}$ the trivial character. Moreover, $\dim \operatorname{Ext}^1_{I/Z_1}({\mathbf 1}, {\mathrm{Sp}})=2$.
It follows from that $\pi(p-1,1)\cong {\operatorname{Ind}_{P}^{G}{{\mathbf 1}}}$. By restricting to $I$ we obtain an exact sequence of $I$-representations: $$\label{exseqSteinberg}
0\rightarrow {\mathbf 1}\rightarrow {\operatorname{Ind}_{I\cap P^s}^{I}{{\mathbf 1}}}\oplus {\operatorname{Ind}_{I\cap P}^{I}{{\mathbf 1}}}\rightarrow {\mathrm{Sp}}\rightarrow 0.$$ If we set $M:= {\operatorname{Ind}_{I\cap P^s}^{I}{{\mathbf 1}}}$ then ${\operatorname{Ind}_{I\cap P}^{I}{{\mathbf 1}}}\cong M^{\Pi}$, and $M|_{H(I\cap U)}\cong {\operatorname{Ind}_{H}^{H(I\cap U)}{{\mathbf 1}}}$ is an injective envelope of ${\mathbf 1}$ in $\operatorname{Rep}_{H(I\cap U)}$. So is an analog of Theorem \[exseq\]. The proof of Theorem \[main\] goes through without any changes. For $p=3$ we note that $M^{K_1}\cong {\operatorname{Ind}_{HK_1}^{I}{{\mathbf 1}}}$ and hence $M$ satisfies the assumptions of Lemma \[done\].
Let $\omega:{\mathbb {Q}_p}^{\times}\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ be a character, such that $\omega(p)=1$ and $\omega|_{{\mathbb{Z}_p}^{\times}}$ is the reduction map composed with the canonical embedding.
\[Rst\] Assume $p>2$ then $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathbf 1})\cong M(p-3,1, \omega)$ and $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathrm{Sp}})\cong M(p-1,1)$,
Recall gives an exact sequence $$\label{zerobis}
0 \rightarrow {\mathbf 1}\rightarrow \pi(p-1,1)\rightarrow {\mathrm{Sp}}\rightarrow 0.$$ Applying ${\mathcal I}$ to we get an exact sequence: $$0\rightarrow \operatorname{\mathbb{R}}^1{\mathcal I}({\mathbf 1}) \rightarrow \operatorname{\mathbb{R}}^1{\mathcal I}(\pi(p-1,1))\rightarrow \operatorname{\mathbb{R}}^1{\mathcal I}({\mathrm{Sp}}).$$ Now [@bp Thm.7.16] asserts that $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi(p-1,1))\cong M(p-3,1,\omega)\oplus M(p-1,1)$. Now $H$ acts on $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathbf 1})$ and $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi(p-1,1))$ via $h\mapsto T_{h^{-1}}$. It follows from Definition \[Mrlambdaeta\] that $$M(p-1,1)\cong {\mathbf 1}\oplus {\mathbf 1}, \quad M(p-3,1, \omega)\cong \alpha\oplus \alpha^{-1}$$ as $H$-representations. Propositions \[homi\], \[extI\] imply that $$\operatorname{\mathbb{R}}^1{\mathcal I}({\mathbf 1})\cong H^1(I_1/Z_1,{\mathbf 1})\cong \operatorname{Hom}(I_1/Z_1,{\overline{\mathbb{F}}_{p}})=\langle \kappa^u,\kappa^l\rangle\cong \alpha\oplus\alpha^{-1}$$ as $H$-representations. Since $p>2$ we get $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathbf 1})\cong M(p-3,1, \omega)$. Then $M(p-1,1)$ is a $2$-dimensional submodule of $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathrm{Sp}})$. However, Proposition \[H1Sp\] implies that $\operatorname{\mathbb{R}}^1{\mathcal I}({\mathrm{Sp}})$ is $2$-dimensional, so the injection is an isomorphism.
\[extmod1Sp\] Let $M$ be an irreducible ${\mathcal H}$-module. If $\operatorname{Ext}^1_{{\mathcal H}}(M,{\mathcal I}({\mathbf 1}))$ or $\operatorname{Ext}^1_{{\mathcal H}}(M,{\mathcal I}({\mathrm{Sp}}))$ is non-zero then $M\in \{{\mathcal I}({\mathbf 1}), {\mathcal I}({\mathrm{Sp}})\}$. Moreover, $$\dim \operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}({\mathbf 1}),{\mathcal I}({\mathrm{Sp}}))=\dim \operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}({\mathrm{Sp}}),{\mathcal I}({\mathbf 1}))=1.$$ If $p>2$ then $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}({\mathbf 1}),{\mathcal I}({\mathbf 1}))$ and $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}({\mathrm{Sp}}),{\mathcal I}({\mathrm{Sp}}))$ are zero, and if $p=2$ then both spaces are $1$-dimensional.
Recall that gives an exact sequence: $$\label{p-1bis}
0 \rightarrow {\mathrm{Sp}}\rightarrow \pi(0,1)\rightarrow {\mathbf 1}\rightarrow 0.$$ Applying ${\mathcal I}$ we obtain an exact sequence: $$\label{Sptriv}
0\rightarrow {\mathcal I}({\mathrm{Sp}})\rightarrow M(0,1)\rightarrow {\mathcal I}({\mathbf 1})\rightarrow 0.$$ If $\operatorname{Ext}^1_{{\mathcal H}}(M,{\mathcal I}({\mathrm{Sp}}))\neq 0$ and $M\not \cong {\mathcal I}({\mathbf 1})$ then from we obtain that $\operatorname{Ext}^1_{{\mathcal H}}(N,M(0,1))\neq 0$, and [@bp Cor.6.5] implies that $M$ is either a subquotient of $M(0,1)$ or a subquotient of $M(p-1,1)$. Hence $M\cong {\mathcal I}({\mathrm{Sp}})$. Using one can deal in the same way with $\operatorname{Ext}^1_{{\mathcal H}}(N,{\mathcal I}({\mathbf 1}))$. Since ${\mathcal I}({\mathbf 1})$ and ${\mathcal I}({\mathrm{Sp}})$ are one dimensional, one can verify the rest by hand using the description of ${\mathcal H}$ in terms of generators and relations given in .
Let $\pi$ and $\tau$ be irreducible representations of $G$ admitting the same central character $\zeta$. Assume that $\pi$ is not supersingular. When $p>2$ for given $\pi$ we are going to list all $\tau$ such that $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\neq 0$. If one is interested in $\operatorname{Ext}^1_G(\tau,\pi)$ then this can be deduced from Proposition \[centre\]. If $\eta:G\rightarrow {\overline{\mathbb{F}}_{p}}^{\times}$ is a smooth character then $\operatorname{Ext}^1_{G,\zeta}(\tau\otimes \eta, \pi\otimes\eta)\cong \operatorname{Ext}^1_{G,\zeta}(\tau, \pi)$. Hence, we may assume that $\pi$ is ${\mathbf 1}$, ${\mathrm{Sp}}$ or $\pi(r,\lambda)$ with $\lambda\neq 0$ and $(r,\lambda)\neq(0,\pm 1)$, $(r,\lambda)\neq (p-1,\pm 1)$. Recall if $\lambda\neq 0$ and $(r,\lambda)\neq (0,\pm 1)$ then [@bl Thm.30] asserts that $$\label{inducedbis}
\pi(r,\lambda)\cong {\operatorname{Ind}_{P}^{G}{\mu_{\lambda^{-1}}\otimes \mu_{\lambda}\omega^r}}.$$ It follows from that if $\lambda\neq \pm 1$ then $\pi(0,\lambda)\cong \pi(p-1,\lambda)$. Hence, we may assume that $1\le r \le p-1$. Propositions \[5term\] and \[chop00\] gives us an exact sequence: $$\label{nearly}
\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau), {\mathcal I}(\pi))\hookrightarrow\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\twoheadrightarrow \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)).$$
Let $\pi$, $\tau$ and $\zeta$ be as above. Assume that $p>2$ and $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\neq 0$. Let $d$ be the dimension of $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)$.
- if $\pi\cong {\mathbf 1}$ then one of the following holds:
- $\tau\cong {\mathrm{Sp}}$, and $d=1$;
- $p\ge 5$, $\tau\cong \pi(p-3,1,\omega)\cong {\operatorname{Ind}_{P}^{G}{\omega\otimes \omega^{-1}}}$ and $d=1$;
- $p=3$, $\tau\cong {\mathrm{Sp}}\otimes \omega\circ \det$ and $d=1$;
- if $\pi\cong {\mathrm{Sp}}$ then $\tau\cong {\mathbf 1}$ and $d=2$;
This follows from , Lemma \[extmod1Sp\] and Proposition \[Rst\]. We note that if $p=3$ then $\pi(p-3,1,\omega)$ is reducible, but has a unique irreducible subobject isomorphic to ${\mathrm{Sp}}\otimes \omega\circ \det$.
For the sake of completeness we also deal with $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)$ when $\pi$ is irreducible principal series. We deduce the results from [@bp §8], but they are also contained in [@col1] and [@em].
Let $\pi$, $\tau$ and $\zeta$ be as above. Assume that $p>2$, $\pi\cong \pi(r,\lambda)$ with $1\le r \le p-1$, $\lambda\in {\overline{\mathbb{F}}_{p}}^{\times}$ and $(r,\lambda)\neq (p-1,\pm 1)$. Then $$\operatorname{Ext}^1_{G,\zeta}(\pi(r,\lambda),\pi(r,\lambda))\cong \operatorname{Hom}({\mathbb {Q}_p}^{\times}, {\overline{\mathbb{F}}_{p}}).$$ In particular, $\dim \operatorname{Ext}^1_{G,\zeta}(\pi(r,\lambda),\pi(r,\lambda))=2$. Moreover, suppose that $\tau\not \cong \pi$ and $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\neq 0$. Let $d$ be the dimension of $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)$ then one of the following holds:
- if $(r,\lambda)=(p-2,\pm 1)$ then such $\tau$ does not exist;
- if $(r,\lambda)=(p-3, \pm 1)$ (hence $p\ge 5$) then $\tau\cong {\mathrm{Sp}}\otimes \omega^{-1}\mu_{\pm 1}\circ \det$ and $d=1$;
- otherwise, $\tau\cong \pi(s,\lambda^{-1}, \omega^{r+1})$, where $0\le s\le p-2$ and $s\equiv p-3-r\pmod{p-1}$, and $d=1$.
Note that if $\pi=\pi(r, \lambda)$ is as in (iii) and we write $\pi\cong {\operatorname{Ind}_{P}^{G}{\psi_1\otimes\psi_2\omega^{-1}}}$, then it follows from that $\pi(s,\lambda^{-1}, \omega^{r+1})\cong{\operatorname{Ind}_{P}^{G}{\psi_2\otimes\psi_1\omega^{-1}}}$.
The first assertion follows from [@bp Cor.8.2]. Assume that $\tau\not \cong \pi$ then it follows from [@bp Cor.6.5, 6.6, 6.7] that $\operatorname{Ext}^1_{{\mathcal H}}({\mathcal I}(\tau),{\mathcal I}(\pi))=0$. Hence, implies that $\operatorname{Ext}^1_{G,\zeta}(\tau,\pi)\cong \operatorname{Hom}_{{\mathcal H}}({\mathcal I}(\tau),\operatorname{\mathbb{R}}^1{\mathcal I}(\pi))$. The assertions (i),(ii) and (iii) follow from [@bp Thm.7.16], where $\operatorname{\mathbb{R}}^1{\mathcal I}(\pi)$ is determined. The difference between (ii) and (iii) is accounted for by the fact that if $r=p-3$ then $s=0$ and if $\lambda=\pm 1$ then $\pi(s,\lambda^{-1},\omega^{r+1})\cong
\pi(0,\pm 1, \omega^{p-2})$, which is reducible, but has a unique irreducible submodule isomorphic to ${\mathrm{Sp}}\otimes \omega^{-1}\mu_{\pm 1}\circ \det$.
Lower bound on $\dim \operatorname{Ext}^1_G(\pi, \pi)$ {#app}
======================================================
Let ${\mathbb F}$ be a finite field of characteristic $p>2$ and $W({\mathbb F})$ the ring of Witt vectors. Let $0\le r\le p-1$ be an integer and set $$\pi(r):= \frac{{\operatorname{c-Ind}_{KZ}^{G}{\operatorname{Sym}^r {\mathbb F}^2}}}{(T)}.$$ We note that the endomorphism $T$ is defined over ${\mathbb F}$, see [@bl Prop 1]. In this section, we bound the dimension of $\operatorname{Ext}^1_{G}(\pi(r), \pi(r))$ from below, using the ideas of Colmez and Kisin. Let $L$ be a finite extension of $W({\mathbb F})[1/p]$ and ${\mathcal O}$ the ring of integers in $L$. Let ${\mathcal G}_{{\mathbb {Q}_p}}$ be the absolute Galois group of ${\mathbb {Q}_p}$. Let $\operatorname{Rep}_{{\mathcal O}} G$ be the category of ${\mathcal O}[G]$-modules of finite length, with the central character, and such that the action of $G$ is continuous for the discrete topology. Let $\operatorname{Rep}_{{\mathcal O}}{\mathcal G}_{{\mathbb {Q}_p}}$ be the category of ${\mathcal O}[{\mathcal G}_{{\mathbb {Q}_p}}]$-modules of finite length, such that the action of ${\mathcal G}_{{\mathbb {Q}_p}}$ is continuous for the discrete topology. Colmez in [@col1] has defined an exact functor $${\mathbf{V}}: \operatorname{Rep}_{{\mathcal O}} G\rightarrow \operatorname{Rep}_{{\mathcal O}} {\mathcal G}_{{\mathbb {Q}_p}}.$$ Set $\rho(r):={\mathbf{V}}(\pi(r))$, then $\rho(r)$ is an absolutely irreducible $2$-dimensional ${\mathbb F}$-representation of ${\mathcal G}_{{\mathbb {Q}_p}}$, uniquely determined by the following: $\det \rho =\omega^{r+1}$; the restriction of $\rho$ to inertia is isomorphic to $\omega_2^{r+1}\oplus \omega_2^{p(r+1)}$, where $\omega_2$ is the fundamental character of Serre of niveau $2$. In the notation of [@b2], $\rho(r)=\operatorname{ind}\omega_2^{r+1}$. We note that since, $\pi(r)$ and $\rho(r)$ are absolutely irreducible, the functor ${\mathbf{V}}$ induces an isomorphism: $$\label{isohom}
\operatorname{Hom}_G(\pi(r),\pi(r))\cong \operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}(\rho(r), \rho(r))\cong {\mathbb F}.$$ Let $\eta: {\mathcal G}_{{\mathbb {Q}_p}}\rightarrow {\mathcal O}^{\times}$ be a crystalline character lifting $\zeta:=\omega^r$ the central character of $\pi(r)$. We consider $\eta$ as a character of the centre of $G$, $Z(G)\cong {\mathbb {Q}_p}^{\times}$ via the class field theory. To simplify the notation we set $\pi:=\pi(r)$ and $\rho:=\rho(r)$. Let $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ be the full subcategory of $\operatorname{Rep}_{{\mathcal O}} G$, such that $\tau$ is an object in $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ if and only if the central character of $\tau$ is equal to (the image of) $\eta$, and the irreducible subquotients of $\tau$ are isomorphic to $\pi$. We note that $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ is abelian.
For $\tau$ and $\kappa$ in $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ we let $\operatorname{Ext}^1_G(\kappa, \tau)$ be the Yoneda $\operatorname{Ext}^1$ in $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$, so an element of $\operatorname{Ext}^1_{G}(\kappa, \tau)$ can be viewed as an equivalence class of an exact sequence $$\label{explain}
0\rightarrow \tau\rightarrow E\rightarrow \kappa\rightarrow 0,$$ where $E$ lies in $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$. Applying ${\mathbf{V}}$ to we get an exact sequence $0\rightarrow {\mathbf{V}}(\tau)\rightarrow {\mathbf{V}}(E)\rightarrow {\mathbf{V}}(\kappa) \rightarrow 0$. Hence, a map $$\label{VExt}
\operatorname{Ext}^1_{G}(\kappa, \tau)\rightarrow \operatorname{Ext}^1_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), {\mathbf{V}}(\tau)).$$ A theorem of Colmez [@col1 VII.5.3] asserts that is injective, when $\tau=\kappa=\pi$.
\[Ext1inj\] Let $\tau$ and $\kappa$ be in $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ then ${\mathbf{V}}$ induces an isomorphism, and an injection respectively: $$\operatorname{Hom}_{G}(\kappa, \tau)\cong \operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), {\mathbf{V}}(\tau)),$$ $$\operatorname{Ext}^1_G(\kappa, \tau)\hookrightarrow \operatorname{Ext}^1_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), {\mathbf{V}}(\tau)).$$
We may assume that $\tau\neq 0$ and $\kappa\neq 0$. We argue by induction on $\ell(\tau)+\ell(\kappa)$, where $\ell$ is the length as an ${\mathcal O}[G]$-module. If $\ell(\tau)+\ell(\kappa)=2$ then $\tau\cong \kappa\cong \pi$ and the assertion about $\operatorname{Ext}^1$ is a Theorem of Colmez cited above, the assertion about $\operatorname{Hom}$ follows from . Assume that $\ell(\tau)>1$ then we have an exact sequence: $$\label{exactseq1}
0\rightarrow \tau'\rightarrow \tau\rightarrow \pi\rightarrow 0.$$ Since ${\mathbf{V}}$ is exact we get an exact sequence: $$\label{exactseq2}
0\rightarrow {\mathbf{V}}(\tau')\rightarrow {\mathbf{V}}(\tau)\rightarrow {\mathbf{V}}(\pi)\rightarrow 0.$$ Applying $\operatorname{Hom}_G(\kappa, \centerdot)$ to and $\operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), \centerdot)$ to we obtain two long exact sequences, and a map between them induced by ${\mathbf{V}}$. With the obvious notation we get a commutative diagram: $$\xymatrix{0\ar[r]& A^0\ar[r]\ar[d]^{\cong}& B^0\ar[r]\ar[d]& C^0\ar[r]\ar[d]^{\cong}& A^1\ar[r]\ar@{^(->}[d]& B^1\ar[r]\ar[d]&
C^1\ar@{^(->}[d]\\
0\ar[r]& \mathcal{A}^0\ar[r]& \mathcal{B}^0\ar[r]& \mathcal{C}^0\ar[r]& \mathcal{A}^1\ar[r]& \mathcal{B}^1\ar[r]& \mathcal{C}^1.}$$ The first and third vertical arrows are isomorphisms, fourth and sixth injections by induction hypothesis. This implies that the second arrow is an isomorphism, and the fifth is an injection. Hence, $$\operatorname{Hom}_{G}(\kappa, \tau)\cong \operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), {\mathbf{V}}(\tau)),$$ $$\operatorname{Ext}^1_G(\kappa, \tau)\hookrightarrow \operatorname{Ext}^1_{{\mathcal G}_{{\mathbb {Q}_p}}}({\mathbf{V}}(\kappa), {\mathbf{V}}(\tau)).$$
If $\ell(\tau)=1$ and $\ell(\kappa)>1$ then one may argue similarly with $\operatorname{Hom}_G( \centerdot, \tau)$ and $\operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}(\centerdot, {\mathbf{V}}(\tau))$.
From now on we assume that $(p,r)\neq (3,1)$. Let $\operatorname{Rep}_{{\mathcal O}}^{\pi,\eta}{\mathcal G}_{{\mathbb {Q}_p}}$ be the full subcategory of $\operatorname{Rep}_{{\mathcal O}}{\mathcal G}_{{\mathbb {Q}_p}}$, with objects $\rho'$, such that there exists $\pi'$ in $\operatorname{Rep}_{{\mathcal O}}^{\pi,\eta} G$ with $\rho'\cong {\mathbf{V}}(\pi')$. Lemma \[Ext1inj\] implies that ${\mathbf{V}}$ induces an equivalence of categories between $\operatorname{Rep}_{{\mathcal O}}^{\pi, \eta} G$ and $\operatorname{Rep}_{{\mathcal O}}^{\pi,\eta} {\mathcal G}_{{\mathbb {Q}_p}}$. In particular, $\operatorname{Rep}_{{\mathcal O}}^{\pi,\eta} {\mathcal G}_{{\mathbb {Q}_p}}$ is abelian. We define three deformation problems for $\rho$, closely following Mazur [@mazur]. Let $D^{u}$ be the universal deformation problem; $D^{\omega\eta}$ the deformation problem with the determinant condition, so that we consider the deformations with determinant equal to $\omega \eta$, [@mazur §24]; $D^{\pi,\eta}$ a deformation problem with the categorical condition, so that we consider those deformations, which as representations of ${\mathcal O}[{\mathcal G}_{{\mathbb {Q}_p}}]$ lie in $\operatorname{Rep}_{{\mathcal O}}^{\pi,\eta} {\mathcal G}_{{\mathbb {Q}_p}}$, [@mazur §25], [@rama]. Since $\rho$ is absolutely irreducible, the functors $D^{u}$, $D^{\omega\eta}$, $D^{\pi, \eta}$ are (pro-)representable by complete local noetherian ${\mathcal O}$-algebras $R^{u}$, $R^{\omega\eta}$, $R^{\pi, \eta}$ respectively. By the universality of $R^u$ we have surjections $R^u\twoheadrightarrow R^{\omega\eta}$ and $R^u\twoheadrightarrow R^{\pi, \eta}$.
For $\rho'$ in $\operatorname{Rep}_{{\mathbb F}} {\mathcal G_{{\mathbb {Q}_p}}}$ we set $h^i(\rho'):=\dim_{{\mathbb F}} H^i({\mathcal G}_{{\mathbb {Q}_p}}, \rho')$. Let $V$ be the underlying vector space of $\rho$, the ${\mathcal G}_{{\mathbb {Q}_p}}$ acts by conjugation on $\operatorname{End}_{{\mathbb F}} V$. We denote this representation by $\mathrm{Ad}(\rho)$, in particular $\mathrm{Ad}(\rho)\cong \rho\otimes\rho^*$. Local Tate duality gives $$h^2(\rho\otimes\rho^*)=h^0(\rho\otimes\rho^* \otimes \omega)=\dim \operatorname{Hom}_{{\mathcal G}_{{\mathbb {Q}_p}}}(\rho,\rho\otimes \omega).$$ Now [@b1 Lem. 4.2.2] implies that $\rho\cong \rho\otimes \omega$ if and only if $p=2$ or $(p,r)=(3,1)$. Since both cases are excluded here, we have $h^2(\mathrm{Ad}(\rho))=0$. Since $\rho$ is absolutely irreducible $h^0(\rho\otimes\rho^*)=1$. The local Euler characteristic gives: $$4=\dim \rho\otimes\rho^*=-h^0(\rho\otimes\rho^*)+h^1( \rho\otimes\rho^*)-h^2(\rho\otimes\rho^*)$$ and so $h^1(\mathrm{Ad}(\rho))=5$. Since $p>2$ the exact sequence of ${\mathcal G}_{{\mathbb {Q}_p}}$-representations: $$0\rightarrow \mathrm{Ad}^0(\rho)\rightarrow \mathrm{Ad}(\rho)\overset{trace}{\rightarrow} {\mathbb F}\rightarrow 0$$ splits. Hence $h^1(\mathrm{Ad}^0(\rho))=3$ and $h^2(\mathrm{Ad}^0(\rho))=0$. It follows from [@mazur2] that $R^u\cong {\mathcal O}[[t_1,\ldots, t_5]]$ and $R^{\omega\eta}\cong {\mathcal O}[[t_1,t_2, t_3]]$.
Inverting $p$ we get surjections $R^u[1/p]\twoheadrightarrow R^{\omega\eta}[1/p]$ and $R^u[1/p]\twoheadrightarrow R^{\pi, \eta}[1/p]$, and hence closed embeddings $$\operatorname{Spec}R^{\omega\eta}[1/p]\hookrightarrow \operatorname{Spec}R^u[1/p],\quad \operatorname{Spec}R^{\pi,\eta}[1/p]\hookrightarrow \operatorname{Spec}R^u[1/p].$$ Let $x\in \operatorname{Spec}R^{\omega\eta}[1/p]$ be a closed point with residue field $E$. Specializing at $x$ we obtain a continuous $2$-dimensional $E$-representation $V_x$ of ${\mathcal G}_{{\mathbb {Q}_p}}$. Suppose that $V_x$ is crystalline, and if $\lambda_1, \lambda_2$ are eigenvalues of $\varphi$ on $D_{crys} (V_x^*)$ then $\lambda_1\neq \lambda_2$ and $\lambda_1\neq \lambda_2 p^{\pm 1}$ then Berger-Breuil in [@bergerbreuil] associate to $V_x$ a unitary $E$-Banach space representation $B_x$ of $G$. Choose a $G$-invariant norm $\|\centerdot\|$ on $B_x$ defining the topology and such that $\|B_x\|\subseteq |E|$ and let $B^0_x$ be the unit ball with respect to $\|\centerdot\|$. Berger has shown in [@berger] that $B^0_x\otimes_{{{\mathcal O}_E}} {\mathbb F}\cong \pi$ as $G$-representations. The constructions in [@bergerbreuil] and [@col1] are mutually inverse to one another. This means $$V_x\cong E\otimes_{{{\mathcal O}_E}}\underset{\leftarrow}{\lim}\ \mathbf V(B_x^0/\varpi_E^n B_x^0).$$ Hence, every such $x$ also lies in $\operatorname{Spec}R^{\pi,\eta}[1/p]$. A Theorem of Kisin [@kis1 1.3.4] asserts that the set of crystalline points, satisfying the conditions above, is Zariski dense in $\operatorname{Spec}R^{\omega\eta}[1/p]$. Since $\operatorname{Spec}R^{\omega\eta}[1/p]$ and $\operatorname{Spec}R^{\pi, \eta}[1/p]$ are closed subsets of $\operatorname{Spec}R^u[1/p]$, we get that $\operatorname{Spec}R^{\omega\eta}[1/p]$ is contained in $\operatorname{Spec}R^{\pi, \eta}[1/p]$. Since $R^{\omega\eta}[1/p]$ is reduced we get a surjective homomorphism $R^{\pi, \eta}[1/p]\twoheadrightarrow R^{\omega\eta}[1/p]$. Let $I$ be the kernel of $R^u\twoheadrightarrow R^{\pi,\eta}$ and let $a\in I$. The image of $a$ in $R^{\pi,\eta}[1/p]$ is zero, hence $a$ maps to $0$ in $R^{\omega\eta}[1/p]$. Since $R^{\omega\eta}$ is $p$-torsion free, the map $R^{\omega\eta}\rightarrow R^{\omega\eta}[1/p]$ is injective, and hence the image of $a$ in $R^{\omega\eta}$ is zero. So the surjection $R^u\twoheadrightarrow R^{\omega\eta}$ factors through $R^{\pi,\eta}\twoheadrightarrow R^{\omega\eta}$. Let ${\mathfrak m}_{\pi,\eta}$ and ${\mathfrak m}_{\omega\eta}$ be the maximal ideals in $R^{\pi,\eta}$ and $R^{\omega\eta}$ respectively. Then we obtain a surjection: $$\label{surjectionD}
D^{\pi, \eta}({\mathbb F}[\varepsilon])^*\cong \frac{{\mathfrak m}_{\pi, \eta}}{\varpi_L R^{\pi,\eta}+{\mathfrak m}^2_{\pi,\eta}}\twoheadrightarrow
\frac{{\mathfrak m}_{\omega\eta}}{\varpi_L R^{\omega\eta}+{\mathfrak m}^2_{\omega\eta}}\cong D^{\omega\eta}({\mathbb F}[\varepsilon])^*,$$ where ${\mathbb F}[\varepsilon]$ is the dual numbers, $\varepsilon^2=0$, and star denotes ${\mathbb F}$-linear dual. It follows from that $\dim_{{\mathbb F}}D^{\pi, \eta}({\mathbb F}[\varepsilon])\ge \dim_{{\mathbb F}}D^{\omega\eta}({\mathbb F}[\varepsilon])=3$. Now $D^u({\mathbb F}[\varepsilon])\cong \operatorname{Ext}^1_{{\mathbb F}[{\mathcal G}_{{\mathbb {Q}_p}}]}(\rho, \rho)$, [@mazur §22] and so $D^{\pi, \eta}({\mathbb F}[\varepsilon])$ is isomorphic to the image of $\operatorname{Ext}^1_{G, \zeta}(\pi, \pi)$ in $\operatorname{Ext}^1_{{\mathbb F}[{\mathcal G}_{{\mathbb {Q}_p}}]}(\rho, \rho)$ via , where $\operatorname{Ext}^1_{G,\zeta}(\pi,\pi)$ is Yoneda $\operatorname{Ext}$ in the category of smooth ${\mathbb F}$-representations of $G$ with central character $\zeta$. Now, [@col1 VII.5.3] implies that the map $\operatorname{Ext}^1_{G, \zeta}(\pi, \pi)\rightarrow \operatorname{Ext}^1_{{\mathbb F}[{\mathcal G}_{{\mathbb {Q}_p}}]}(\rho, \rho)$ is an injection. We obtain:
Let $\pi$ be as above and assume that $(p,r)\neq (3,1)$ then $\dim_{{\mathbb F}}\operatorname{Ext}^1_{G,\zeta}(\pi,\pi)\ge 3$.
[19]{} <span style="font-variant:small-caps;">L. Barthel and R. Livné</span>, ‘Irreducible modular representations of $\operatorname{GL}_2$ of a local field’, *Duke Math. J.* 75 (1994) 261-292. <span style="font-variant:small-caps;">L. Berger and C. Breuil</span>, ’Sur quelques représentations potentiellement cristallines de $\operatorname{GL}_2({\mathbb {Q}_p})$’, to appear in *Astérisque*. <span style="font-variant:small-caps;">L. Berger</span>, ’Représentations modulaires de $\operatorname{GL}_2({\mathbb {Q}_p})$ et représentations galoisiennes de dimension 2’, to appear in *Astérisque*. <span style="font-variant:small-caps;">C. Breuil</span>, ‘Sur quelques représentations modulaires et $p$-adiques de $\operatorname{GL}_2({\mathbb {Q}_p})$ I’, *Compositio Mathematica* 138 (2) (2003), 165-188. <span style="font-variant:small-caps;">C. Breuil</span>, ’Sur quelques représentations modulaires et $p$-adiques de $\operatorname{GL}_2({\mathbb {Q}_p})$ II’, *J. Inst. Math. Jussieu* 2, (2003), 1-36. <span style="font-variant:small-caps;">C. Breuil and V. Paškūnas</span>, ‘Towards mod $p$ Langlands correspondence for $\operatorname{GL}_2$’, to appear in *Memoirs of Amer. Math. Soc.* <span style="font-variant:small-caps;">P. Colmez</span>, ‘Représentations de $\operatorname{GL}_2({\mathbb {Q}_p})$ et $(\phi,\Gamma)$-modules’, to appear in *Astérisque*. <span style="font-variant:small-caps;">M. Emerton</span>, ‘Ordinary parts of admissible representations of $p$-adic reductive groups II: derived functors’, to appear in *Astérisque*. <span style="font-variant:small-caps;">M. Kisin</span>, ‘The Fontaine-Mazur conjecture for $\operatorname{GL}_2$’, *J.A.M.S.* 22(3) (2009) 641-690. <span style="font-variant:small-caps;">M. Kisin</span>, ’Deformations of $G_{{\mathbb {Q}_p}}$ and $\operatorname{GL}_2({\mathbb {Q}_p})$ representations’, to appear in *Astérisque*. <span style="font-variant:small-caps;">B. Mazur</span>, ‘Deforming Galois representations’, *Galois groups over $\mathbb Q$* (Berkley, CA, 1987), 385-437, Springer, New York, 1989, Eds. Y. Ihara, K. Ribet, J.-P. Serre. <span style="font-variant:small-caps;">B. Mazur</span>, ’An introduction to the deformation theory of Galois representations’, *Modular forms and Fermat’s last theorem* (Boston, MA, 1995), 243–311, Springer, New York, 1997, Eds. G. Cornell, J.H. Silverman, G. Stevens. <span style="font-variant:small-caps;">R. Ollivier</span>, ‘Le foncteur des invariants sous l’action du pro-$p$-Iwahori de $\operatorname{GL}_2(F)$’, *J. reine angew. Math.* 635 (2009) 149-185. <span style="font-variant:small-caps;">V. Paškūnas</span>, ‘Coefficient systems and supersingular representations of $\operatorname{GL}_2(F)$’, *Mémoires de la SMF* 99 (2004). <span style="font-variant:small-caps;">R. Ramakrishna</span>, ’On a variation of Mazur’s deformation functor’, *Compositio* 87 269-286, 1993. <span style="font-variant:small-caps;">J.-P. Serre</span>, *Corps Locaux*, Act.Sci.Ind. 1296, Paris 1962. <span style="font-variant:small-caps;">J.-P. Serre</span>, *Cohomologie Galoisienne*, Lecture Notes in Mathematics 5, Springer 1964, Eds A.Dold and B.Eckmann. <span style="font-variant:small-caps;">M.-F. Vignéras</span>, ‘Representations modulo $p$ of the $p$-adic group $\operatorname{GL}(2,F)$’, *Compositio Math.* 140 (2004) 333-358.
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abstract: |
With the development of Internet of Things (IoT), IoT intelligence becomes emerging technology. “Curse of Dimensionality” is the barrier of data fusion in edge devices for the success of IoT intelligence. Deep learning has attracted great attention recently, due to the successful applications in several areas, such as image processing and natural language processing. However, the success of deep learning benefits from GPU computing. A Linguistic Attribute Hierarchy (LAH), embedded with Linguistic Decision Trees (LDTs) can represent a new attribute deep learning. In contrast to the conventional deep learning, an LAH could overcome the shortcoming of missing interpretation by providing transparent information propagation through the rules, produced by LDTs in the LAH. Similar to the conventional deep learning, the computing complexity of optimising LAHs blocks the applications of LAHs.
In this paper, we propose a heuristic approach to constructing an LAH, embedded with LDTs for decision making or classification by utilising the distance correlations between attributes and between attributes and the goal variable. The set of attributes is divided to some attribute clusters, and then they are heuristically organised to form a linguistic attribute hierarchy. The proposed approach was validated with some benchmark decision making or classification problems from the UCI machine learning repository. The experimental results show that the proposed self-organisation algorithm can construct an effective and efficient linguistic attribute hierarchy. Such a self-organised linguistic attribute hierarchy embedded with LDTs can not only efficiently tackle ‘curse of dimensionality’ in a single LDT for data fusion with massive attributes, but also achieve better or comparable performance on decision making or classification, compared to the single LDT for the problem to be solved. The self-organisation algorithm is much efficient than the Genetic Algorithm in Wrapper for the optimisation of LAHs. This makes it feasible to embed the self-organisation algorithm in edge devices for IoT intelligence.
address:
- 'School of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield, UK'
- 'Smart Perception Lab, Milton Keynes, UK'
author:
- Hongmei He
- Zhenhuan Zhu
title: 'A Heuristically Self-Organised Linguistic Attribute Deep Learning in Edge Computing For IoT Intelligence'
---
IoT Intelligence in Edges,Linguistic Attribute Deep Learning ,Linguistic Decision Tree ,Semantics of a Linguistic Attribute Hierarchy (LAH) ,Distance Correlation Clustering ,Self-Organisation of an LAH
Introduction {#sec:Intro}
============
The Internet-of-Things (IoT) provides us with a large amount of sensor data. However, the data by themselves do not provide value unless we can turn them into actionable and/or contextualized information. Big data analysis allows us to gain new insights by batch-processing and off-line analysis. Currently, a microprocessor-based sensor node can support many channels (e.g. a Microchip processor can support up to 49 channel inputs [@datasheet2017]). Real-time sensor data analysis and decision-making is preferably automated on-board of IoT devices, which will make IoT intelligence towards reality.
Although the computing capability of a microprocessor has improved very much, the ‘curse of dimensionality’ is still a big challenge in data driven machine intelligence, as the computing complexity of designed model function increases as the increasing of input space. Blum and Rivest [@Blum1992] have proved that training a 2-layer, 3-nodes and n inputs neural network is NP-Complete. Obviously, the big barrier of blocking the applications of deep-learning is the computing complexity, although it shows great attractive on solving complex nonlinear problems. With the strong capability of GPU, deep learning for 2-20 depth networks is successful (e.g. Google AlphaGo). To save the cost, an edge device of IoT systems may not need to equip with GPUs, if the machine intelligence algorithm inside the device is efficient enough. Hence, the performance improvement of computational intelligence is continuous work, especially for creating effective and efficient computing model in edge devices to implement IoT intelligence.
Linguistic decision tree (LDT), a probabilistic tree, has been well used for decision making and classification problems. Given an input space of n attributes, each of which can be described with limit labels. An LDT consists of a set of attribute nodes and a set of edges. An edge, linking from an attribute node, represents a label expression that describes the attribute node (see Section \[sec:LDT-LS\]). However, the branch number of a decision tree exponentially increases as the number of input attributes increases. This shortcoming of an LDT greatly blocks its applications.
A hierarchical model could help overcome the “Curse of Dimensionality” in fuzzy rule-based learning [@Raju1991]. Campello and Amaral [@Campello2006] provided a cascade hierarchy of sub-models, using fuzzy relational equations [@Pedrycs1993], and unilaterally transformed the cascade models into the mathematically equivalent non-hierarchical one. Lawry and He[@Lawry2008] proposed a linguistic attribute hierarchy (LAH). It is a hierarchy of LDTs, each of which has a reduced input space, and represents different functions in the problem space. However, as the relationship between inputs and the output in the whole problem space could be strong non-linear and uncertain, different LAHs will have different performance for the problem to be solved. All of the research in [@Campello2006; @Lawry2008] neither investigated the performance of the proposed hierarchies, nor studied how the hierarchies can be constructed optimally.
He and Lawry [@He2009a; @He2009b] investigated cascade LAHs, and developed a genetic wrapper algorithm (GWA) to optimise the cascade LAHs. It was shown that a cascade hierarchy can derive much less rules, compared to the single decision tree. However, the accuracy tends to drop as decision threshold increases. Later, they also investigated the optimisation of generic LAHs [@He2014]. The optimisation of LAHs using the GWA is NP-complete, as the branch number of a decision tree is increasing exponentially as the input attribute number increases, and the convergency of GWA directly affects the speed of optimisation. Hence, the evolutionary algorithm takes very long time to optimise LAHs, even if the evolution takes a small number of iterations. When the attribute number is over 60, the time of evolution process on a PC is not acceptable.
Another issue of hierarchical approaches, including generic deep learning, is the lack of linguistic interpretability, especially neural networks have been viewed as a black box. This is because that intermediate variables, being arbitrarily introduced to interconnect the sub-models, do not present any meanings in a real system, and hence it is difficult to give a clear intuitive interpretation [@Campello2006]. However, in an LAH, information propagation is completed via the linguistic rules extracted from LDTs. Hence, a transparent hierarchical decision making or classification can be provided.
In a distributed system, we may get partial information from different information resources, based on which, an initial estimate/decision could be done locally. Correspondingly, a collective decision is required. How do these information resources make contribution to the final collective decision? In an LAH, an intermediate variable is equivalent to the initial decision in terms of partial information [@He2009a; @He2009b]. Hence, the structure of the hierarchy determines the process of decision or fusion process. He et. al. [@He2015a] further proposed an off-line optimisation of sensor arrangement for task-oriented sensor fusion, aligning with the structure of an self-organised wireless sensor network.
Mitra et al.[@Mitra2002] proposed a fuzzy knowledge-based network, whose structure can be automatically optimised by using linguistic rules of a fuzzy decision tree, thus to help in reducing the complexity of network training for a specific task. An LAH is equivalent to a forward neural network, in which, each neuron is the function of lower layer attributes, but represented by an LDT. The structure of an LAH is different to that of a classic multi-layer feed-forward neural network, in which, all input attributes are in the input layer. In an LAH, input attributes can feed a neuron (e.g. an LDT) at any layer, but for the simplicity of network structure, they cannot repeatedly feed more than one neuron.
The information propagation from bottom to top of an LAH through the LDTs in the LAH provides a new approach to attribute deep learning. In fact, a neural network or any other machine learning models can be used, instead of an LDT in the LAH. For example, a cascade of CMACs has been proposed in [@He2015b]. However, LAH embedded with linguistic decision trees could provide good interpretation for decision making or classification.
He et al. [@He2017] used an LAH to interpret the process of semantic attribute deep learning for spam detection. The LAH was constructed manually in terms of the semantics of attributes. For IoT intelligence or edge computing, the onboard adaptive sensor fusion is a critical challenge, due to the limit of resources in edge devices (e.g. computing capacity, memory and even power supply). Hence, efficiently automatic constructing a linguistic attribute hierarchy becomes necessary for this purpose.
In this research, a self-organised LAH (SOLAH), embedded with LDTs is proposed for the deep learning of attribute semantics in decision making or classification. A distance correlation-based clustering algorithm is proposed to decompose attributes to several clusters, and a linguistic attribute hierarchy is constructed with the produced clusters in terms of the average distance correlation of cluster members to the goal variable. The LDT, fed by a cluster of attributes with lower distance correlation, will be placed in the lower layer of the LAH. The preliminary experiments are conducted on the SMS spam database from UCI machine learning repository [@Lichman2013]. A set of databases will be used to validate the performance of the SOLAHs. The experimental results are compared with that of the single LDTs for different databases.
Existing work in Deep Learning
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Recently, Deep Learning has been put a lot of attention by researchers. It uses multiple processing layers of computational models to learn representations of data with multiple levels of abstraction. Each successive layer is fed by the outputs of previous layer, forming a hierarchy of attributes from bottom to top. Deep learning uses the back-propagation algorithm to learn internal parameters that are used to compute the representation in each layer from the representation in the previous layer, thus to find the hidden structure or information in large data sets. Various deep learning architectures such as deep neural networks, convolutional deep neural networks, deep belief networks and recurrent neural networks have been developed. Deep learning has been successfully applied in the areas of speech recognition [@Miao2015; @ZhangZ2017; @Zhang2019], image processing [@HeK2016], object detection [@Girshick2014; @Ren2017], drug discovery [@Gawehn2016] and genomics [@Park2015], etc. Especially, deep convolutional nets (ConvNets) have brought about breakthroughs in processing images [@Chen2016], video [@Karpathy2014], speech [@Zhang2019] and audio [@Lee2009], whereas recurrent nets have shone light on sequential data such as text and speech [@LeCun2015].
ConvNets are constructed layer by layer for data processing. A classic application of a ConvNet is in image processing, utilising the properties of a colour image, consisting of the RGB channels of pixel values. Since the early 2000s, ConvNets have been successfully applied for the detection, segmentation and recognition of objects and regions based on images, for example, face recognition [@Taigman2014].
Druzhkov and Kustikova [@Druzhkov2016] did a survey on deep learning methods for image classification and object detection, covering autoencoders, restricted Boltzmann machines and convolutional neural networks. For example, Szegedy et al. [@Szegedy2013] used deep neural networks to solve the problem of object detection for both classifying and precisely locating objects of various classes; Simonyan et al. [@Simonyan2013] used deep fisher networks for the classification of large-scale images from ImageNet (<http://image-net.org/>); Krizhevsky et al. [@Krizhevsky2012] trained a large deep convolutional neural network to classify the 1.3 million high-resolution images in the LSVRC-2010 ImageNet training set into the 1000 different classes; and He et al. [@HeY2014] proposed an unsupervised feature learning framework, Deep Sparse Coding, which extends sparse coding to a multi-layer architecture for visual object recognition tasks.
Acoustic modeling is another area, where large, deep neural networks have been successfully applied. Like ImageNet, the massive quantities of existing transcribed speech data provide rich resources for deep learning. Dahl [@Dahl2015] provided a brief review of the deep neural net approach to large vocabulary speech recognition (LVSR) in his thesis. Mohamed et al.[@Mohamed2012] showed that hybrid acoustic models of pre-trained deep neural networks, instead of Gaussian mixture models (GMMs), could greatly improve the performance of a small-scale phone recognition; By using rectified linear units and dropout, Dahl et al. [@Dahl2013] further improve the model for a large vocabulary voice search task. A combination of a set of deep learning techniques has led to more than 1/3 error rate reduction over the conventional state-of-the-art GMM-HHM (Hidden Markov Model) framework on many real-world LVCSR tasks [@Yu2015].
The qualitative properties of text data, very different to that of other modality data, provide critical challenges in use of machine learning. Recently Socher et al. [@Socher2013] developed recursive deep models for semantic compositionality over a sentiment treebank, and improved the accuracy. Natural language understanding is another exploration of deep learning application, which could make a large impact over the next few years [@LeCun2015].
The property of compositional hierarchies of some signals is well exploited by deep neural networks through composing lower-level features to abstract the high-level one. For example, an image can be represented by a hierarchy from local edges, motifs, parts, to objects. Similarly, speech and text also have a hierarchy from sounds to phones, phonemes, syllables, words and sentences. In other words, this properties promote the capacities of deep neural networks. However, the interpretability of the decision making process is still an issue. Also, we cannot always explicitly see the semantics of higher-level features in other application domains.
With rapid progress and significant successes in a wide spectrum of applications, deep learning is being applied in many safety-critical environments. Ching et al. [@Ching2018] forecasted that deep learning enabling changes at both bench and bedside with the potential to transform several areas of biology and medicine, although the limited amount of labelled data for training presents problems as well as legal and privacy constraints on work with sensitive health records. However, deep neural networks (DNNs) have been recently found vulnerable to well-designed input samples called adversarial examples, and adversarial perturbations are imperceptible to human but can easily fool DNNs in the testing/deploying stage [@Yuan2019]. The transparency of LAH allows the observation of the decision process. He et al. [@He2017] have investigated the effect of different input attributes on different positions in a linguistic attribute hierarchy. This might indicate a linguistic attribute deep learning could provide good transparency to help people to defend adversarial perturbations to IoT Intelligence, which is out of the scope of this research.
GPU has become the necessary hardware facility for the research on deep learning due to its complexity. Justus et al. [@Justus2018] analysed various factors that influence the complexity of deep learning, and divided these factors to three categorises: (1) Layer features, such as activate function, optimiser of weights (e.g. Gradient Descent) and number of training samples; (2) layer specific features, including four subcategories: i. Multi- Layer Perception features (e.g. number of inputs, number of neurons and number of layers), ii. Convolutional features (e.g. matrix size, kernel size, stride size, input padding, input depth and output depth, etc.); iii. Pooling features (e.g. matrix size, stride size, input padding), iv. Recurrent features (e.g. Recurrence type, bi-directionality); (3) Hardware Features (e.g. GPU technology, GPU count, GPU memory, GPU memory bandwidth, GPU clock speed, GPU cord count, GPU peak performance, Card Connectivity). The first two categories of factors determine the performance of the function represented by a deep learning model, while the GPU feature directly affect the efficiency of the computing. It is hard to express the polynomial relationship between the computing cost and all the various factors, or use a big O(\*) to represent the computing complexity.
Linguistic attribute deep learning with a LAH
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A linguistic attribute hierarchy (LAH)
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The process of aggregation of evidence in multi-attribute decision making or classification based on attributes $x_1,..., x_n$ can be represented with a functional mapping $y = f(x_1,...,x_n)$, where $y$ is the goal variable. However, this mapping is often dynamic and uncertain, and it is difficult to find a mathematic equation to precisely describe the mapping function. An attribute hierarchy represents the function $f$ with a hierarchy of sub-functions, each of which represents a new intermediate attribute. The set of original attributes $\{x_1,..., x_n\}$ is categorised into $m$ clusters $s_1,..., s_m$. When a cluster of attributes is used to make initial decision, an new intermediate attribute is produced by the function of the clustering attributes. One or more intermediate attributes can be combined with another cluster of attributes to make further decision in next level, or multiple intermediate attributes can be directly used to make decision in next level. As there are $m$ attribute clusters, at least $m$ subfunctions are produced. Namely, $z_i=G_i(s_i)$ for $i=1,...,m$ and $n-1\geq m$, as the maximal level of hierarchy with $n$ input attributes is the cascade hierarchy with $n-1$ subfunctions. The mapping function $f$ is represented by a new function $F$ of the intermediate attributes $z_1,..,z_\tau$ and/or a cluster ($s_i$) of input attributes. The intermediate attributes can be represented by subfunctions of $G_1...G_{\tau}$, fed by lower level of attribute set, which could include intermediate attributes and/or another cluster $s_j$ of input attributes. Hence, $y=f(x_1,...,x_n)$ = $F(z_1,...,z_{\tau}, s_i)$ = $F(G_1(S_1)$,...,$G_{\tau}(S_{\tau}), s_i)$. A linguistic decision tree (LDT) can explicitly model both the uncertainty and vagueness in a mapping system, and linguistic rules extracted from the LDT often implies our knowledge of aggregation. When an LAH uses LDTs to represent the functional mappings between parent and child attribute nodes, the semantic information will be transparently propagated through the hierarchy from bottom to top. Assume the goal variable $y\in \Omega_y$, and a set $\mathcal{L}_y$ of labels can be used to describe the goal variable. An introduced intermediate attribute $z$ will represent an approximate of the goal variable $y$. Namely, $z\in \Omega_y$, and the label set $\mathcal{L}_y$ can be used to describe $z$. Hence, an LAH can present a hierarchical decision making, and it can provide transparent linguistic interpretation, which helps perform semantic attribute deep learning.
For examples, assume five attributes are fed to an LAH. Fig. \[fig:LAH\] (a) shows a cascade LAH, embedded with 4 LDTs, through which, the decision information is cascaded from the bottom model $LDT_1$ to the top model $LDT_4$; Fig. \[fig:LAH\] (b) illustrates a general LAH, embedded with 3 LDTs, through which, the final decision is made by $LDT_3$, based on the two intermediate attributes from $LDT_1$ and $LDT_2$ in the bottom of the LAH, and the additional attribute $x_5$. The intermediate attributes $z_1$ and $z_2$ represent an approximate of the goal variable $y$, $z_1$ is decided by $x_1$ and $x_2$, and $z_2$ is decided by $x_3$ and $x_4$.
A Linguistic decision tree based on Label Semantics {#sec:LDT-LS}
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An LDT [@Qin2005a; @LawryJ2008] is a probabilistic tree under the framework of label semantics [@Lawry2006]. Label semantics introduces two fundamental and interrelated measures: Appropriateness measure ($\mu_L(x)$) and mass assignment ($\nu_x$), where, $\mu_L(x)$ quantifies how a label $L$ is appropriate to describe x based on agent’s knowledge of the current labeling conventions, shown in the collected samples or evidences; $\nu_x$ quantifies how a particular subset of labels is appropriate to describe $x$. The particular subset of all and only the labels that are appropriate to describe $x$ is called focal set, denoted as $F$. Given a set of labels $L = \{s, m, l\}$, the focal set can be $F$ = $\{\{s\}$, $\{s, m\}$, $\{m\}$, $\{m, l\}$, $\{l\}\}$.
An LDT consists of a node set $V$ and an edge set $E$. A node $v \in V$ is associated to an attribute, and an edge $e\in E$ from a node is a focal set, which is appropriate to describe the attribute, indicated by the node. A path from the top node to a leaf of the LDT is called a branch, denoted as $B$. The branch length ($l$) is the number of nodes on the branch $B$. Each branch provides a conjunction of focal sets: $F_1\wedge$ ... $\wedge F_l$, with the inference of a set of mass assignments conditional to the branch $B$, denoted as $\nu_y(F|B)$, for each focal set $F\in \mathcal{F}_y$, where, $\mathcal{F}_y$ is a set of focal sets that are appropriate to describe different values distributed in the domain of the goal variable $y$. The semantics of an LDT is described in Definition \[def:linguistic\].
\[def:linguistic\] The rule derived from a branch $B_i$ in an LDT is presented as:
$$\begin{aligned}
\label{eq:rule}
F_{{i}_{1}}\wedge ... \wedge F_{i_l} \rightarrow F:\nu_y(F|B_i),\end{aligned}$$
where, $F_{i_k}$ is the $k$-th focal set on branch $B_i$, and $F_{i_k}\in \mathcal{F}_{x_{i_k}}$. Given attribute values $\vec{x}=(x_1, \ldots,
x_n)$. The mass assignment $\nu_y$ can be obtained using Jeffrey’s rule [@Jeffrey1990]:
$$\begin{gathered}
\label{eq:Jeffery}
\nu_y(F)=\sum_{i=1}^{\beta} \mu_{B_i}(\vec{x})\nu_y(F|B_i), F\in \mathcal{F}_y\end{gathered}$$
where $\beta$ is the branch number of the LDT, and $\nu_y(F|B_i)$ is equivalent to the conditional probability $p(F|B_i)$. $x_{i_j}$ denotes the $j$-th attribute on the $i$-th branch. Assume a focal set $F_{i_j}$ is appropriate to describe attribute $x_{i_j}$ on the branch $B_i$. The appropriateness measure of $\vec{x}$ on branch $B_i$ is the product of all mass assignments on focal sets that are appropriate to describe the corresponding nodes on the branch, as expressed in Formula (\[eq:pr-branch\]). $$\begin{gathered}
\label{eq:pr-branch}
\mu_{B_i}(\vec{x}) = \prod_{j=1}^l \nu_{x_{i_j}}(F_{i_j}).\end{gathered}$$
The semantics of a linguistic attribute hierarchy
=================================================
When an LAH is represented a hierarchy of LDTs, the information is propagated through the LDTs from information sources (e.g. sensors) in the bottom to the decision variable on the top. Therefore, the rules can be derived as conditional expressions through Formula \[eq:rule\] in the label semantics framework. The output variable (either an intermediate attribute or a goal variable) of each LDT in the LAH can be calculated with Formula (\[eq:Jeffery\]). This provides an approach to quantifying the degree of our belief how each focal set $F\in \mathcal{F}_y$ is appropriate to describe the goal, given partial input attributes and/or the previous results of decision or classification (intermediate attributes) in lower level.
Now, we use the LAH in Fig. \[fig:LAH\] (b) as an example to demonstrate the upwards information propagation through an LAH. In Fig. \[fig:LAH\] (b), two LDTs ($LDT_1$, $LDT_2$) are located in the bottom of the LAH. $y=f(z_1,z_2, x_5)= f(g_1(x_1,x_2),g_2(x_3,x_4),x_5)$. The mappings $g_1$, $g_2$, and $f$ are represented in the form of linguistic decision trees (i.e. $LDT_1$, $LDT_2$ and $LDT_3$). $LDT_1$ provides the function of quantifying our belief of $z_1$ on goal labels in terms of input attributes $x_1$ and $x_2$, $LDT_2$ presents the function of quantifying our belief of $z_2$ on goal labels in terms of input attributes $x_3$ and $x_4$, and $LDT_3$ offers the function of quantifying our final belief of the goal variable $y$ on its labels in terms of lower level believes of $z_1$ and $z_2$, as well as another input attribute $x_5$ from information source (e.g. sensors). Assume the input attributes are clustered to $\kappa$ subsets ${s_1,...s_\kappa}$, which can feed the intermediate attributes or/and the goal variable in an LAH. As an intermediate attribute $z_i \in \Omega_y$ is the approximate of the goal variable $y$, it can be estimated by a decision making model (e.g. LDT), fed with $s_i$. Hence, the decision making model can be trained with the samples of ($S_i,y$). Given the values of all input attributes $\vec{x}=(x_1, \ldots, x_n)$, the mass assignments of all intermediate attributes and the goal variable can be estimated with Formula (\[eq:Jeffery\]) through all the LDTs in the LAH. Namely, the decision information is propagated through the bottom LDTs to the top LDT in the LAH. The semantics of an LAH is defined as Definition \[def:HLinguistics\].
\[def:HLinguistics\] The semantics of an LAH is the synthetisation of rules extracted from the branches in the LAH, which are allocated by the given sample, $\vec{x}=(x_1, \ldots, x_n)$. Assume $k$ outputs (i.e. $k$ intermediate attributes) of LDT($t_1$) ... LDT($t_k$) are the inputs of LDT($t_i$), the rule will be $B^{t_1} \wedge ... \wedge B^{t_k} \rightarrow B^{t_i}$, where $B^{t_1}$ ... $B^{t_k}$ can be derived in the form of Formula (\[eq:rule\]), by an LDT, either directly based on information sources (i.e. the input attributes), or based on the intermediate attributes from the lower level of LDTs.
For the instance of Fig. \[fig:LAH\] (b), given all the input attribute values, $\vec{x}$ = $\{x_1$, $x_2$, $x_3$, $x_4$, $x_5\}$ in all samples. For generality, we use $\ell$ to denote the label expression, associated to an edge in an LDT. For the special case, the edges of an LDT are associated to a focal set, $\ell$ represents a focal set $F$. Hence, the semantics of each decision tree can be described as:
For $LDT_1(x_1,x_2)$, as it is fed with two attributes, the maximum branch length is 2. Hence, the rule corresponding to the branch $B^1_{i}$ can be: $$\begin{aligned}
\ell^1_{{i}_{1}}\wedge \ell^1_{{i}_2} \rightarrow
F:\nu_y(F|B^1_{i}), F\in \mathcal{F}_y.\nonumber\end{aligned}$$
Similarly, for $LDT_2(x_3,x_4)$, the rule corresponding to the branch $B^2_{j}$ can be: $$\begin{aligned}
\ell^2_{{j}_1}\wedge \ell^2_{{j}_2} \rightarrow
F:\nu_y(F|B^2_{j}), F\in \mathcal{F}_y.\nonumber\end{aligned}$$
For $LDT_3(z_1,z_2, x_5)$, the maximum branch length is 3. Hence, the rule corresponding to the branch $B^3_{k}$ can be: $$\begin{aligned}
\ell^3_{{k}_1}\wedge \ell^3_{{k}_2} \wedge \ell^3_{{k}_3} \rightarrow
F:\nu_y(F|B^3_{k}), F\in \mathcal{F}_y.\nonumber\end{aligned}$$
The synthetic semantics of the LAH is: $$\begin{aligned}
(\ell^1_{{i}_{1}}\wedge \ell^1_{{i}_2}) \vee (\ell^2_{{j}_{1}}\wedge \ell^2_{{j}_2})& \rightarrow \ell^3_{{k}_1}\wedge \ell^3_{{k}_2}\wedge \ell^3_{{k}_3} \nonumber\\
&\rightarrow F:\nu_y(F|B^3_{k}), F\in \mathcal{F}_y.\nonumber\end{aligned}$$
Construction of Linguistic Attribute Hierarchy
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The basic idea for the construction of a Linguistic Attribute Hierarchy includes two steps: (1) Decomposition of attributes with a distance correlation based clustering algorithm;\
(2) Self-organisation of a linguistic attribute hierarchy in terms of the distance correlation between clustered attributes and the goal variable.
Attribute decomposition based on their distance correlation
-----------------------------------------------------------
### Distance correlation
Distance correlation can be used to statistically measure the dependence between two random variables or vectors, which could have different dimensions. It is zero if and only if the random variables are statistically independent. Assume $(x_i, y_i), i= 1, 2, ..., n$ be a set of samples. The $n$ by $n$ distance matrices $(a_{i,j} )$ and $(b_{i,j})$ present the all pairwise distances for $x$ and $y$ in all the samples, respectively. $$\begin{aligned}
a_{i,j} = \parallel x_{i}-x_{j}\parallel;
b_{i,j} = \parallel y_{i}-y_{j}\parallel.\end{aligned}$$ where $|| \bullet ||$ denotes Euclidean norm. Then take all doubly centered distances: $$\begin{aligned}
A_{i,j} = a_{i,j}-\bar{a}_{i.}-\bar{a}_{.j}+\bar{a}_{..},
B_{i,j} = b_{i,j}-\bar{b}_{i.}-\bar{b}_{.j}+\bar{b}_{..},\end{aligned}$$ where, $\bar{a}_{i.}$ is the $i$-th row mean, $\bar{a}_{.j}$ is the $j$-th column mean, and $\bar{a}_{..}$ is the grand mean of the distance matrix for the $x$ samples. The notation is similar for the $b$ values on the $y$ samples. The squared sample distance covariance (a scalar) is simply the arithmetic average of the products $A_{i,j}$ and $B_{i,j}$: $$\begin{aligned}
dCov_n^2 (x,y)=\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n A_{i,j}B_{i,j}.\end{aligned}$$ The sample distance variance is the square root of $$\begin{aligned}
dVar_{n}^{2}(x)= dCov_{n}^{2}(x,x) = \frac {1}{n^{2}}\sum _{i,j}A_{i,j}^{2},\end{aligned}$$ The distance correlation of two random variables can be calculated through dividing their distance covariance by the product of their distance standard deviations, as Fomula (\[eq:crown\]). $$\begin{aligned}
\label{eq:crown}
dCorr(x,y)=\frac {dCov(X,Y)}{\sqrt {dVar(x)\ dVar(y)}}.\end{aligned}$$ Distance correlation has the properties:
- $0\leq dCorr_{n}(x,y)\leq 1$, $0\leq dCorr(x,y)\leq 1$;
- The distance correlation matrix is symmetric (i.e. $dCorr(i,j) = dCorr(j,i)$), and $dCorr(i,i)$=1.
- $dCorr(x,y)=0$, if and only if $x$ and $y$ are independent;
- $dCorr_{n}(x,y)=1$ and $dCorr(x,y)=1$ indicates that the linear subspaces spanned by $x$ and $y$ samples respectively almost surely have an identical dimension [@Szekely2009].
### Clustering attributes based on distance correlation
As we do not need to consider the distance correlation between an attribute and itself, we set the diagonal of the distance correlation matrix to zeros for the convenience of computing. First, we find the maximum value in the distance correlation matrix $dCorr$. The maximum value is denoted as $d_{max}$, the column of the maximum value in the matrix is denoted as $i_{max}$. Here we set a value $\alpha$ as the range of distance correlation difference in a cluster. Namely, the distance correlation of attributes in a cluster, correlating to attribute $i_{max}$ will be in $(d_{max}-\alpha, d_{max}]$. A feasible approach to setting the value of $\alpha$ is: $$\begin{aligned}
(max(dCorr)-min(dCorr))/k,\end{aligned}$$
Usually, $min(dCorr)$ is not a zero, and $k$ is the preset number of clusters. Secondly, all relevant columns and rows, where the identified cluster members in the cluster are located, are set to zeros. The process is repeated with starting to find next attribute with largest distance correlation in the unvisited attributes for the next cluster, until all elements of $dCorr$ are zeros. Algorithm \[alg:dCorrClustering\] provides the pseudo-code of the clustering algorithm, where the produced clusters are saved in $\mathcal{S}$.
Initialise($\mathcal{S})$; t=0; $t$ = $t$+1; $\alpha=(max(dCorr)-min(dCorr))/k$; $[d_{max},i_{max}]=max(dCorr)$; $T \leftarrow i_{max}$; $T \leftarrow find\_i(d_{max}> dCorr_{i,i_{max}}\geq d_{max}-\alpha)$); $S_t \leftarrow T$; $clearCols(dCorr, T$); $clearRows(dCorr, T$);
Self-organisation of linguistic attribute hierarchy
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Once attributes are clustered based on their distance correlation, we can construct a linguistic attribute hierarchy in terms of the distance correlation between clustered attributes and the goal variable. If an attribute has stronger distance correlation with decision variable, this indicates the attribute has stronger linear correlation to the goal variable, namely, the mapping function between the attribute and the goal could be more linear. In other words, it is easier to make decision in terms of the attribute. Our preliminary experimental results show that attributes, which have stronger distance correlation to the goal variable, should be fed to the LDT in higher layer of the linguistic attribute hierarchy. The average of distance correlation between attributes in a cluster and the goal variable can be calculated with Formula (\[eq:average-dCorr\]): $$\begin{aligned}
\label{eq:average-dCorr}
\overline{dCorr}(s) = \sum_{x\in s} dCorr(x,y)/t, t=|s|.
\end{aligned}$$ where, $x$ is the attributes in cluster $s$, $y$ is the goal variable, and $t$ is the size of cluster (i.e. number of attributes in cluster $s$).
The basic idea of constructing an LAH based on the produced clusters is that:
- calculating average distance correlation for each produced cluster;
- Sort all average distance correlations between clusters and decision variable;
- The LDT fed by the cluster with lower average distance correlation will be at lower level of the LAH;
- The LDTs fed by the clusters that have similar average values of distance correlation will be at the same level of the LAH. A threshold $\theta$ is set for the assessment of clusters at the same level. The outputs of all LDTs at the same level will be the partial inputs of the next LDT in the next level of LAH;
- The LDT fed by the cluster with the highest average distance correlation will be on the top of the LAH.
Assume $\overline{dCorr}(s_i)$ represents the average distance correlation of cluster $s_i$ to the goal variable, and $\theta$ represents the maximum difference of distance correlation values of clusters at the same level to the goal variable. Namely, if $$\begin{aligned}
|\overline{dCorr}(s_i)- \overline{dCorr}(s_j)|<\theta,\end{aligned}$$ then the LDTs fed with $s_i$ and $s_j$ will be at the same level in the LAH.
Algorithm \[alg:SOLAH\] provides the pseudo-code of the self-organisation of LAH, where $dCorrXY$ is the vector of distance correlation of all attributes to the goal variable, $\mathcal{S}$ is the set of clusters produced by the distance correlation-based clustering algorithm above, $\mathcal{R}$ is the set of all the average distance correlations for cluster set $\mathcal{S}$ with $K$ clusters to the goal variable, $Z$ is the index of an intermediate attribute, which starts from $n+1$, the input attribute number, $\mathcal{I}$ is used to save the indices of intermediate attributes and the corresponding clusters at the same level, $\mathcal{H}$ is used to save the hierarchy to be constructed. For all clusters, if $R_i-R_a <\theta$, save the intermediate attribute index $Z$ to $\mathcal{I}$, and append $Z$ and the cluster $S_{t_i}$ to the hierarchy $\mathcal{H}$, else, start a new level, append current $Z$, other intermediate attributes saved in $\mathcal{I}$ and corresponding cluster $S_{t_i}$ to the hierarchy $\mathcal{H}$, and save current $Z$ to $\mathcal{I}$. If more than one LDTs at the same level in the LAH, and there is no further cluster to be constructed, then the outputs of those LDTs will be the inputs of the top LDT in the LAH. If the LDT constructed with the last cluster does not share the same level with other LDTs, then it will be the top LDT of LAH.
= avDistCorr($dCorrXY,\mathcal{S}$); $Z = n$; $K$=|$\mathcal{S}$|; = sort($\mathcal{R}$); %increasingly $\mathcal{I} = \phi$;$\mathcal{H} = \phi$; $i$ = 0, $a$ = 0; $H_i = \phi$; $i=i$+1; $Z = Z$+1, $\mathcal{I} \leftarrow(Z)$; $\mathcal{H} = \leftarrow (Z, \mathcal{S}_{t_i}$); $a = i$; % starting a new level $\mathcal{H} \leftarrow(Z, \mathcal{I}, \mathcal{S}_{t_i}$); $\mathcal{I}=Z$; $H \leftarrow(Z+1,I)$; $Y = Z$+1; $Y = Z$;
Training of a linguistic attribute hierarchy
--------------------------------------------
In 2009, He and Lawry [@He2009a; @He2009b] first time proposed using the domain and labels of the goal variable to describe intermediate attributes. This made the hierarchy training became possible, and the hierarchical decision making became meaningful. In [@He2014], He and Lawry developed a non-recursively post-order traversal algorithm to train a given LAH, where all LDTs are trained with LID3 in a bottom-up way. Intermediate attribute values are estimated with the trained LDTs, and then they are input to next level LDTs. For example, in Fig. \[fig:LAH\] (b), $LDT_1$, $LDT_2$ and $LDT_3$ are trained in turn, where, the intermediate variables $z_1$ and $z_2$ use the corresponding $y$ values in all training points, as the goal values of $LDT_1$ and $LDT_2$ respectively. After $LDT_1$ and $LDT_2$ are trained, the intermediate variables $z_1$ and $z_2$ for the training samples can be estimated by $LDT_1$ and $LDT_2$, respectively. The estimated values of $z_1$ and $z_2$, the values of $x_5$, and the values of the goal variable $y$ in the training samples are used to train the $LDT_3$, and finally the goal variable $y$ can be estimated by $LDT_3$. Here, we introduce a recursive postorder implementation of the LAH training algorithm (see Algorithm \[alg:LAH-Train\]).
$\mathcal{H}$ is a global variable, indicating the LAH to be trained. Initially, $v$ is the root of the LAH to be trained, and all leaves in the LAH are tagged as visited, as they are the input attributes from sensor nodes. $v.ch$ represents the child set of node $v$, indicating the attributes that feed to the node $v$. $S$ is used to save the attributes that are not on the branch from the top root to the current node $v$, and $S\setminus v$ is to remove $v$ from set $S$.
return; $postorder$($v.ch_i$, $S\setminus v$); $LDT_v$=$LID3(v, S)$; $z_v=LDT_v(\mathcal{D})$; $\mathcal{D} \leftarrow z_v$;
The LID3 algorithm for LDT training
-----------------------------------
LID3 was used to train an LDT with a given database [@Qin2005a; @He2009a]. It is an update of the classic ID3 algorithm [@Quinlan1986], through combining label semantics. The training process is conducted through selecting the attribute that obtains the maximum information gain to extend a branch. The functions used in LID3 have been formulated in [@Qin2005a; @He2009a], such as information entropy of a branch ($E$), expected entropy ($EE$) when an attribute node $x$ is added to a branch, and information gain (IG). Here, a recursive algorithm implementation of LID3 is proposed (Algorithm \[alg:LID3\]). A threshold $\vartheta$ is setup to stop the branch extension when the conditional probability $P(C|B)$ reaches the threshold $\vartheta$. The maximum of branch length is the number of attributes that feed the LDT. Initially, $T = \phi$ (empty), hence, current node point $B_v = \phi$ as well. For each branch, the conditional probabilities for all classes $C\in \mathcal{C}$ are calculated. The most informative attribute is selected as the next node to be extended to the existing LDT, and all focal sets that are appropriated to describe the attribute in the domain of $\Omega_x$ will be appended to the tree $T$. The process is continued until the maximum conditional probability arrives the specified threshold.
$P(C_i|B_v)$ = conditionProb($B_v$); return; $IG(B_v,x) = E(B_v)-EE(B_v, x)$; $\hat{x} = argmax_{x\in S}(\{IG\})$; $B_{\hat{v}} = B_v+x$ $T \leftarrow {F\in \mathcal{F}_{\hat{x}}}$; $S = S\setminus \hat{x}$; $LID3 (B_{\hat{v}}, S)$;
Experiments and Evaluation
==========================
Experiment methodologies
------------------------
All databases, used for the experiments, are from UCI machine learning repository [@Lichman2013]. The experiments are conducted in two stages:
\(1) A case study will be done on the benchmark database of Message Spams. This experiment is to demonstrate the use of developed approach for self-organisation of a linguistic attributes hierarchy, given the data.
\(2) A set of databases will be tested with the SOLAHs. The performance are used to validate the developed approach to self-organising LAHs by comparing with the performance obtained with the single LDT.
[**Experimental environment:**]{} The experiments are carried out on a laptop with 64-bit Windows 10 and x64-based processor with Intel (R) Core (TM) i5-4210U CPU @1.7GHZ 2.4GHZ, 8GB memory. The LAH training algorithm is implemented in C++.
[**Discretisation:**]{} For simplicity, all attributes are expressed with three labels, except binary variables, decision or goal variables, which are expressed with the labels as they have. All neighbouring fuzzy intervals are overlapping with 50%.
[**Mass assignments of attributes:**]{} In terms of Label Semantics, if $n$ labels are used to describe a continuous variable $x$, then there will be $2n-1$ focal sets that are appropriate to describe $x \in \Omega_x$. For discrete variables, the mass assignment on a focal set that contains only one label is 1, but the mass assignment on a focal set that contains two successive labels is 0.
[**Ten-fold cross validation:**]{} 90% of data is used for training, 10% is used for test. Therefore, data is partitioned to ten parts equally.
[**Performance measure:**]{} The performance is measured with:
\(1) *Accuracy:* $\mathcal{A} = \frac{TP+TN}{N}$, where, $TP$ is the number of true positive estimates and $TN$ is the number of true negative estimates, and $N$ is the number of samples in the database.
\(2) *Area under ROC curve ($AUR$)*: A ROC curve is used to measure how well the classifier separates the two classes without reference to a goal threshold. The area under the ROC curve has been formalized in [@He2014].
\(3) *Rule number $\beta$:* As defined in [@He2014], the rule number of an LAH is the sum of branch numbers, extracted from all LDTs in the LAH. Since a probability threshold was introduced during the process of LDT training, a branch training may stop earlier. Hence, the actual branch number in an LDT may be much less than a full LDT. (4) *The running time $t$:* it is the time, spending on the process of ten-fold crossing validation, including training, testing, and overhead for data splitting and exchanging.
A case study on a benchmark database
------------------------------------
To demonstrate the developed approach, the first experiment is carried out on the benchmark database, ’SMSSpamCollection’. The performance for the LAH that was constructed by the proposed approach, the LAH in [@He2017], obtained manually, and the single LDT, is evaluated and compared.
### The database
The SMSSpamCollection database [@Almeida2011], has 5574 raw messages, including 747 spams. The two sets of features, extracted for the research [@He2016; @He2017], are used for the experiments. Table \[tab:SMS-20\] shows the set of 20 features, and Table \[tab:SMS-14\] presents the set of 14 features.
$x$ key word $x$ key-word
----- ---------- ----- ---------------------------------------
0 urgent 10 stop
1 congrat 11 click
2 ! 12 Text,Txt
3 WIN/WON 13 sex
4 Offer 14 girl
5 Award 15 cash
6 Prize 16 free
7 Call 17 0p, 1p, ..., 9p
8 Reply 18 EURO, GBP, pound,$\mathcal{L}$, \$, €
9 Send 19 price
: The set of 20 features [@He2016][]{data-label="tab:SMS-20"}
$x$ key word $x$ key-word
----- ------------- ----- ---------------------------------------
0 urgent 7 call
1 congrat 8 Reply/send
2 ! 9 stop/click
3 WIN/WON 10 text/txt
4 Offer 11 cash/price
5 Award/Prize 12 free/0p...9p
6 sex/girl 13 EURO, GBP, pound,$\mathcal{L}$, \$, €
: The set of 14 features [@He2017][]{data-label="tab:SMS-14"}
### Experimental results on SMS-20
The six clusters obtained by the proposed distance correlation -based clustering algorithm are: $S_1$:{$x_7$,$x_1$,$x_4$,$x_{18}$}, $S_2$:{$x_0$,$x_{19}$}, $S_3$:{$x_{15}$,$x_{16}$}, $S_4$:{$x_6$,$x_2$,$x_3$,$x_8$}, $S_5$:{$x_{9}$,$x_{11}$}, $S_6$:{$x_5$,$x_{10}$,$x_{12}$,$x_{13}$,$x_{14}$,$x_{17}$}.
$S_6$ has the lowest average distance correlation to the decision variable. Hence, the LDT fed by $S_6$ should be at the bottom of the SOLAH, and produces an intermediate attribute $z_{20}$. Here $z$ is used to represent an intermediate attribute. $z_{20}$ with set $S_4$ feeds an LDT in the second level according to the average distance correlation of $S_4$ to decision variable. Sets $S_5$ and $S_4$ have similar average distance correlations to decisional variable. Hence, they sit at the same level in the SOLAH, and their outputs $z_{21}$ and $z_{22}$ feed the third level of LDT constructed with set $S_1$. Sets $S_2$ and $S_3$ have similar average distance correlations to the decision variable. Hence they are at the same level. As they are at the top level, $S_2$, $S_3$ and the output $z_{23}$ of the LDT at the third level will be an input of the top LDT. Figs. \[fig:LAH-SMS20\] and \[fig:HS-2\] illustrate the SOLAH and the best $LAH_m$, manually produced in [@He2017]. The two LAHs have the same levels and the same number of LDTs. But the compositions are different.
![The SOLAH on SMS-20[]{data-label="fig:LAH-SMS20"}](LAH-SMS20.eps){width="3.5in"}
![A LAH\_m on SMS-20[@He2017][]{data-label="fig:HS-2"}](HS-2.eps){width="3.5in"}
Table \[tab:LAH-SMS20-perf\] shows the performance for the SOLAH, the $LAH_m$ in [@He2017] and the single LDT fed by the whole feature vector (denoted as $LDT(\vec{x})$. It can be seen that the accuracy $A$ of $SOLAH$ is very close to the performance of $LAH_m$ and $LDT(\vec{x})$, and the performance of the AUC is slightly smaller than that of $LAH_m$, but better than that of $LDT(\vec{x})$. However, $SOLAH$ has the smallest branch number $beta$ and running time $T$, which is the time of ten-fold crossing validation on a solution, measured in milliseconds (ms).
LAHs A $AUC$ $\beta$ $T$(ms)
------------------------------------- ---------- ---------- --------- ---------
$SOLAH$, Fig. \[fig:LAH-SMS20\] 0.936491 0.946402 197 101766
$LAH_m$[@He2017], Fig. \[fig:HS-2\] 0.948511 0.949233 267 143718
$LDT(\bar{x})$ 0.956225 0.895815 775 1339234
: The performance of solutions[]{data-label="tab:LAH-SMS20-perf"}
### Experimental results on SMS-14
For this experiment, the cluster number is preset to 4. But the clusters obtained by the DCC algorithm are: $S_1$:{$x_6$,$x_11$}, $S_2$:{$x_9$,$x_{4}$,$x_8$,$x_{10}$}, $S_3$:{$x_{5}$,$x_{0}$,$x_{1}$,$x_{2}$,$x_{3}$,$x_{12}$,$x_{13}$}. Set $S_1$ has the lowest average distance correlation to the decision variable, thus the LDT fed by $S_1$ is placed at the bottom of the SOLAH. The LDT fed by set $S_1$ and the output of the first level at the second level of the SOLAH, and the LDT fed by set $S_3$ sits at the top level of the LAH. Figs. \[fig:LAH-SMS14\] and \[fig:HH3\] illustrate the SOLAH and the $LAH_m$ with the best $AUC$, which was obtained by the attribute composition 3 in [@He2017]. The two LAHs have the same levels, and the attributes are decomposed to 3 clusters. But the compositions of LAHs are different. In the $LAH_m$, all LDTs fed by the clusters of input attributes are placed at the bottom of the LAH. Hence a new LDT, fed by all intermediate attributes, which were produced by the bottom level of LDTs, is added on the top of the LAH. $SOLAH$ has higher level than $LAH_m$, and all LDTs fed by the attribute clusters are cascaded.
![The SOLAH on SMS-14[]{data-label="fig:LAH-SMS14"}](LAH-SMS14-3.eps){width="3.2in"}
![A LAH on SMS-14, obtained manually in [@He2017][]{data-label="fig:HH3"}](HH3.eps){width="3.2in"}
Table \[tab:LAH-SMS14\] shows the performance for the SOLAH, the $LAH_m$, constructed with composition 3 manually in [@He2017] and the single LDT ($LDT(\vec{x})$). All solutions in Table \[tab:LAH-SMS14\] obtained similar performance in accuracy and the area under ROC. All performance values of $SOLAH$ are in between $LAH_m$ and $LDT(\bar{x})$.
LAHs A $AUC$ $\beta$ $T$(ms)
------------------------------------- ---------- ---------- --------- ---------
$SOLAH$, Fig. \[fig:LAH-SMS20\] 0.955328 0.943391 339 101766
$LAH_m$[@He2017], Fig. \[fig:HS-2\] 0.951202 0.957466 130 79719
$LDT(\bar{x})$ 0.961607 0.924536 583 613704
: The performance of solutions[]{data-label="tab:LAH-SMS14"}
### Impact of the preset cluster number
Similar to classic $k$-means clustering algorithm, the preset cluster number $k$ is important for the construction of LAHs, and thus has important impact on the performance of decision making or classification. Here, the relationships between $k$ and structures of LAHs and between $k$ and performance are observed. It should be noticed that the real cluster number may be different to the preset cluster number, as the present cluster number decides the range ($\alpha$) of distance correlation of a cluster $C$, which is formed with the attributes that have the distance correlations in the range $\alpha$ to an attribute $x$, which has the largest distance correlation in the distance correlation matrix. The rest attributes could have larger distance correlations than the members except attribute $x$ in cluster $C$. Table \[tab:LAH-SMS14-1\] shows the solutions for $k$= 2...10. It can be seen that the real cluster numbers are different to the preset cluster number $k$ in some cases. The cluster numbers varies from 3 to 6. All SOLAHs have similar performance. The SOLAH constructed with 3 clusters obtained the highest accuracy, but it has more branches. The SOLAHs constructed with 4 clusters have the highest $AUC$ values. The clusters obtained when $k=7$ is the same as that when $k$=8, which is happened when $k=9,10$ as well. In Table \[tab:LAH-SMS14-1\], $\beta$ denotes the branch number, $\iota$ denotes the level of the hierarchy.
$k$ $K$ A $AUC$ $\beta$ $\iota$ $T$(ms)
------ ----- ------- ------- --------- --------- ---------
2 3 0.956 0.946 339 4 221547
3 4 0.937 0.953 184 4 101875
4 4 0.936 0.950 154 4 84093
5 6 0.932 0.949 159 5 91375
6 6 0.932 0.948 154 6 89828
7,8 6 0.932 0.945 108 6 63313
9,10 6 0.932 0.941 130 5 76859
: The properties of LAHs on SMS14[]{data-label="tab:LAH-SMS14-1"}
Similarly, Table \[tab:LAH-SMS20-K\] lists the performance of SOLAHs constructed with various $k$ values from 2 to 10. The AUC values of all SOLAHs are very close. When the preset $k=2$, the SOLAH was constructed with two clusters. It obtained the highest accuracy, but lowest AUC value and the largest branch number. When the preset $k$=9,10, the same cluster sets were produced, with which, the SOLAH was produced. It has the lowest accuracy, but the smallest branch number. For all preset $k$=3...6, different sets of 5 clusters were produced, with which, four different SOLAHs were produced. They have the same level, and their performance values ($A$ and AUC) are very close. When the preset $k$=5,6,7, the produced SOLAHs have the same performance values ($A$ and AUC). But they have different branch numbers.
$k$ $K$ A $AUC$ $\beta$ $\iota$ $T$(ms)
------ ----- ------- ------- --------- --------- ---------
2 2 0.957 0.933 540 3 467328
3 5 0.934 0.950 189 5 111016
4 5 0.949 0.941 203 5 114265
5 5 0.936 0.946 187 5 101594
6 5 0.936 0.946 197 5 107766
7 6 0.936 0.946 204 5 115750
8 7 0.929 0.940 163 6 93891
9,10 7 0.891 0.944 157 7 104938
: The properties of LAHs on SMS20[]{data-label="tab:LAH-SMS20-K"}
From the data on both SMS20 and SMS14, it can be seen that a suitable cluster size $K$ will have a good trade-off on accuracy performance and time complexity (branch number).
Validation on some benchmark databases
--------------------------------------
### The 12 benchmark databases
The experiments on 12 benchmark databases from UCI machine learning repository are conducted for validating the proposed approach of constructing LAHs. Table \[tab:datasets\] provides the basic properties of the 12 data sets, including, database name, attribute number ($n$), goal state number ($N_g$), total sample number ($N$), class distribution ($N_c$).
Datasets $n$ $N_g$ $N$ $N_c$
------------------- ----- ------- ----- -----------------------
Beast Cancer (BC) 9 2 286 201,85
Wisconsin BC 9 2 569 357,212
Ecoli 7 8 336 143,77,52,35,20,5,2,2
Glass 9 6 214 70,17,76,13,9,29
Heart-C 13 2 303 165,138
Heart-statlog 13 2 270 150,120
Hepatitis 19 2 155 70,85
Wine 13 3 178 59,71,48
Liver Disorders 6 2 346 146,200
Diabetes 8 2 768 500,268
Ionoshpere 33 2 351 231,120
Sonar 60 2 208 97,111
: Database properties[]{data-label="tab:datasets"}
### Comparisons with LDT
Table \[tab:comparisonLDT\] shows the performance of SOLAHs and corresponding single LDTs on the 12 benchmark data sets. It can be seen that the SOLAH achieved dominated better performance in accuracy $A$ and $AUC$ than the single LDT for most databases. For databases, Ecoli and Liver, SOLAHs achieve similar accuracy to LDTs, but their performance in $AUC$ is better than that of LDTs. The SOLAHs for most databases have less branches than the single LDT trained by the same database.
------------ ------ ------ --------- ----- --------- ----- ------ ------ --------- ------
$A$ $\beta$ $T$ $\iota$ $K$ $A$ $\beta$ $T$
BreastC. 0.78 0.84 985 9 4 3 0.70 0.64 3201 66
WBC 0.97 0.98 1100 36 3 2 0.95 0.96 1029 45
Ecoli 0.85 0.93 1103 15 4 3 0.85 0.89 3337 12
Glass 0.81 0.91 5124 48 3 2 0.68 0.83 9433 156
Heart-c 0.81 0.81 1452 24 5 4 0.76 0.78 1345 40
Heart-s. 0.79 0.83 972 11 5 4 0.77 0.80 1053 24
Hepatitis 0.92 0.94 4278 47 4 6 0.79 0.82 1677 45
Wine 1.00 0.99 929 6 5 5 0.94 0.98 1309 26
Liver 0.56 0.60 226 3 3 2 0.57 0.56 1585 24
Diabetes 0.76 0.82 1018 33 3 2 0.74 0.79 14865 667
Ionosphere 0.87 0.71 4393 14 5 9 0.87 0.87 12141 6953
Sonar 0.73 0.80 26328 52 6 22 0.67 0.73 9253 4968
------------ ------ ------ --------- ----- --------- ----- ------ ------ --------- ------
: Performance comparison between constructed LAHs and the single LDTs for different databases, the unit of time (T) is second[]{data-label="tab:comparisonLDT"}
As databases Ionosphere and Sonar have 33 and 60 attributes, respectively, the ten-fold crossing validation of a single LDT on the two databases is not acceptable. Therefore, the two-fold crossing validation of single LDTs on the two high dimension databases are performed. For comparison, the two-fold crossing validation of the constructed LAHs on the two databases are carried out as well. The results are provided in Table \[tab:comparison\]. For database Ionosphere, the LAH has the same accuracy as the LDT, but the area under ROC obtained by the SOLAH is smaller than that by the single LDT. Namely, the half size of data may not be enough to train the SOLAH. However, the running times of SOLAHs for the two databases are 14s and 52s respectively, which are much less than that of LDTs for the two databases. The database Sonar provides a very interesting case: the SOLAH has 26328 branches, while the single LDT has only 9253 branches. Regarding the branch number, the complexity of the SOLAH model is worse than the single LDT. But the running time for SOLAH is 52s, while 4968s for LDT. This is because that the computing time is not only related to branch number, but also related to branch length. The computing complexity can be $O(\beta\times l)$. It can be seen that the SOLAH for the Sonar database has 22 layers, which means that at least 22 LDTs are embedded in the SOLAH. There are total 60 input attributes, plus 22 intermediate attributes, the average input attributes of an LDT in the SOLAH is 3.68. Hence, the average length of branch is 3,68. We can roughly say that the running time of the SOLAH is proportional to $26328\times 3.68$ = 96887, but the single LDT has 4968 branches, the length of which is 60, then the running time of the single LDT is proportional to $4968\times 60$ = 298080, which is much larger than the figure for the SOLAH.
In order to observe the performance improvement of the LAH that is trained by 90% of data when ten-fold crossing validation is applied, Table \[tab:highDims\] particularly lists the performance of the SOLAHs with ten-fold crossing validation on the two high dimensional databases. The performance in both accuracy and the area under ROC is improved, especially for the database, Ionosphere, the performance in accuracy and the area under ROC are improved very much, compared to the SOLAH with two-fold crossing validation. Moreover, the running times of the SOLAHs for the two databases are 143s and 507s, respectively, which are much shorter than that of the single LDT with two-fold crossing validation in Table \[tab:comparisonLDT\].
Databases $A$ $AUC$ $\beta$ $T(ms)$ $\iota$ $K$
------------ ------ ------- --------- --------- --------- -----
Ionosphere 0.95 0.98 7717 143 5 9
Sonar 0.78 0.84 33820 507 6 22
: Performance of SOLAHs on the two high dimensional databases[]{data-label="tab:highDims"}
### Comparisons with other machine learning algorithms
The average accuracy and standard deviations of SOLAH for ten runs on the 12 data sets from UCI machine learning repository are further compared with the optimal LAH, obtained by a Genetic Algorithm Wapper in [@He2014], as well as three well-known machine learning algorithms, such as C4.5, Naive Bayes (NB) and Neural Networks (NN). The performance of these algorithms WAS evaluated with WEKA [@WF00] by Qin and Lawry in [@Qin2005a], where, WEKA [@WF00] was used to generate the results of J48 (C4.5 in WEKA) unpruned tree, Naive Bayes and Neural Networks with default parameter settings for ten runs of 50%-50% splitting training and test data.
Table \[tab:comparison\] shows the performance of SOLAHs and the existing results in literature for the 12 data sets. SOLAH wins 4 data sets, which is next to OptLAH, which wins 5 data sets. SOLAH obtains comparable performance in accuracy, compared to other algorithms, and even achieves better performance than the LAHs, optimised by GAW, for some tested data sets.
$Databases$ C4.5 N.B. N.N. OptLAH SOLAH
------------- ---------------- --------------------- --------------------- --------------------- -----------------------
BreastC. $69.16\pm4.14$ $71.26\pm2.96$ $66.50\pm3.48$ $71.77\pm2.06$ ${\bf75.17}\pm4.30 $
WBC $94.38\pm1.42$ $96.28\pm0.73$ $94.96\pm0.80$ ${\bf96.67}\pm0.20$ $94.85\pm0.25 $
Ecoli $78.99\pm2.23$ $85.36\pm2.42$ $82.62\pm3.18$ $84.02\pm0.92$ ${\bf86.11}\pm0.91 $
Glass $64.77\pm5.10$ $45.99\pm7.00$ $64.30\pm3.38$ ${\bf71.31}\pm1.75$ $71.26\pm8.08 $
Heart-c $75.50\pm3.79$ ${\bf84.24}\pm2.09$ $79.93\pm3.99$ $82.81\pm4.25$ $78.38\pm3.50 $
Heart-s. $75.78\pm3.16$ $84.00\pm1.68$ $78.89\pm3.05$ ${\bf84.85}\pm2.31$ $78.15\pm1.05 $
Hepatitis $76.75\pm4.68$ $83.25\pm3.99$ $81.69\pm2.48$ ${\bf94.84}\pm1.01$ $85.15\pm9.12 $
Ionosphere $89.60\pm2.13$ $82.97\pm2.51$ $87.77\pm2.88$ $89.80\pm1.63$ ${\bf92.12}\pm4.03$
Liver $65.23\pm3.86$ $55.41\pm5.39$ ${\bf66.74}\pm4.89$ $58.46\pm0.76$ $56.16\pm0.90 $
Diabetics $72.16\pm2.80$ $75.05\pm2.37$ $74.64\pm1.41$ ${\bf76.07}\pm1.33$ $75.09\pm1.39 $
Sonar $70.48\pm0.00$ $70.19\pm0.00$ ${\bf81.05}\pm0.00$ $74.81\pm4.81$ $75.72\pm3.74 $
Wine $88.09\pm4.14$ $96.29\pm2.12$ $96.85\pm1.57$ $97.58\pm0.27$ ${\bf98.17} \pm2.95 $
: Average accuracies (%) and standard deviations obtained by C4.5, NB, NN, OptLAH and SOLAHs[]{data-label="tab:comparison"}
Conclusions
===========
In this paper, the contribution of the innovative research are summarised as below:
\(1) We proposed heuristic algorithm to construct a linguistic attribute hierarchy for semantic attribute deep learning in decision making or classification. The self-constructed linguistic attribute hierarchy provides a new form of deep learning, in contrast to conventional deep learning.
\(2) The proposed algorithm for the self-organisation of an LAH is much more efficient than meta-heuristic algorithm, and the self-organised linguistic attribute hierarchy can obtain the fusion performance better than or comparable to the single linguistic decision tree, fed with the full set of attributes.
\(3) The most important is that the heuristical self-organisation of such linguistic attribute hierarchy can effectively solve the ’curse of dimensionality’ in machine learning, which is critical challenge in the implementation of IoT intelligence. Hence, the research results will promote a wider of applications of the linguistic attribute hierarchy in big data analysis and IoT intelligence.
\(4) A linguistic attribute hierarchy, embedded with linguistic decision trees, will provide a transparent hierarchical decision making or classification. Hence, it could help us to look insight of the decision making process for different purposes (e.g. the effect of adversary samples in decision making).
\(5) Comparing with other machine learning in literature, the self-organised LAH obtains comparable performance on the tested data sets, and even achieved better performance than the optimal LAHs, obtained by GAW for some tested data sets.
We will implement the LAH on an embedded system (e.g. a Raspberry Pi system) for a specific task, and will further improve the algorithms and develop new algorithms to construct high efficient and effective linguistic attribute hierarchy, embedded with other machine learning models for decision making or classification in future.
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Appendix {#appendix .unnumbered}
========
14 attribute-based LAHs for k = 2..10 (Figs. \[fig:LAH14-2\]-\[fig:LAH14-9\])
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![LAH based on SMS14 with $k$ = 2[]{data-label="fig:LAH14-2"}](LAH_SMS14_k2.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 3[]{data-label="fig:LAH14-3"}](LAH_SMS14_k3.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 4[]{data-label="fig:LAH14-4"}](LAH_SMS14_k4.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 5[]{data-label="fig:LAH14-5"}](LAH_SMS14_k5.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 6[]{data-label="fig:LAH14-6"}](LAH_SMS14_k6.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 7 and $k$ = 8.[]{data-label="fig:LAH14-7"}](LAH_SMS14_k7.eps){width="3.0in"}
![LAH based on SMS14 with $k$ = 9 and $k$=10.[]{data-label="fig:LAH14-9"}](LAH_SMS14_k9.eps){width="3.0in"}
LAHs based on 20 attributes for k=1,..,10 (Figs. \[fig:LAH20-2\]-\[fig:LAH20-9\])
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![LAH for k=2.[]{data-label="fig:LAH20-2"}](LAH_SMS20_k2.eps){width="3.0in"}
![LAH for k=3[]{data-label="fig:LAH20-3"}](LAH_SMS20_k3.eps){width="3.0in"}
![LAH for k=4.[]{data-label="fig:LAH20-4"}](LAH_SMS20_k4.eps){width="3.0in"}
![LAH for k=5.[]{data-label="fig:LAH20-5"}](LAH_SMS20_k5.eps){width="3.0in"}
![LAH for k=6.[]{data-label="fig:LAH20-6"}](LAH_SMS20_k6.eps){width="3.0in"}
![LAH for k=7.[]{data-label="fig:LAH20-7"}](LAH_SMS20_k7.eps){width="3.0in"}
![LAH for k=8.[]{data-label="fig:LAH20-8"}](LAH_SMS20_k8.eps){width="3.0in"}
![LAH for $k$=9 and $k$=10[]{data-label="fig:LAH20-9"}](LAH_SMS20_k9.eps){width="3.0in"}
|
---
author:
- 'P. D’Aquino, J. F. Knight, S. Kuhlmann, and K. Lange'
title: Real closed exponential fields
---
Introduction
============
A *real closed field* is an ordered field in which every non-negative element is a square, and every odd degree polynomial has a root.
Tarski’s celebrated elimination of quantifiers [@Tarski] shows that the axioms for real closed fields generate the complete theory of the ordered field of reals, so this theory is decidable.
\[Integer part\] An integer part of an ordered field $R$ is a discretely ordered subring $Z$ such that for each $r\in R$, there exists $z\in Z$ with $z\leq r < z+1$.
If $R$ is Archimedean, then $\mathbb{Z}$ is the unique integer part. In general, the integer part for $R$ is not unique. Shepherdson [@S] showed that a discrete ordered ring $Z$ is an integer parts of some real closed fields if and only if $Z$ satisfies a fragment of Peano Arithmetic called [*Open Induction*]{}. Open Induction is the first order theory, in the language $\mathcal{L}=\{+, \cdot, <, 0, 1\}$, of discretely ordered commutative rings with a multiplicative identity $1$ whose set of non-negative elements satisfies, for each quantifier-free formula $\Phi(x, y)$, the associated induction axiom $I(\Phi)$: $$(\forall y)[\Phi(0, y) \ \&\ (\forall x)[\Phi(x, y)\rightarrow \Phi(x+1, y)]\rightarrow (\forall x)\ \Phi(x, y)].$$
We now consider real closed fields. In [@MR], Mourgues and Ressayre proved the following.
Every real closed field has an integer part.
In an extended abstract [@R], Ressayre outlines the proof for an analogue of Theorem A for exponential real closed fields:
\[D:EXP\] A real closed exponential field is a real closed field $R$ endowed with an isomorphism of ordered groups:
$(R,+,0,<) \rightarrow (R^{>0},\cdot ,1,<)$
$x\mapsto 2^x$
where $(R,+,0,<) $ is the additive group of $R$ and $(R^{>0},\cdot ,1,<)$ is the multiplicative group of positive elements of $R$. That is, $2^x$ satisfies the following axioms:
*[1.]{} $2^{x+y} = 2^x2^y$,*
*[2.]{} $x < y$ implies $2^x < 2^y$,*
*[3.]{} for all $x > 0$, there exists $y$ such that $2^y = x$; i.e., $\log(x)$ is defined.*
We require also the following:
*[4.]{} $2^1 = 2$,*
*[5.]{} for all $x\in R$, $x>n^2$ implies $2^x>x^n$ ($n\geq 1$).*
We now consider integer parts closed under exponentiation in real closed exponential fields.
\[Exponential integer part\] Let $R$ be a real closed exponential field. An *exponential integer part* is an integer part $Z$ such that for all positive $z\in Z$, we have $2^z\in Z$.
We observe that $\mathbb{Z}$ is an exponential integer part for any Archimedean real closed field. In an extended abstract [@R], Ressayre outlined the proof for the analogue of Theorem A for real closed exponential fields.
\[Ressayre\] \[Ressayre\]
If $R$ is a real closed exponential field, then $R$ has an exponential integer part.
In this paper we revisit Ressayre’s extended abstract, providing the details of the proofs, and focusing on the complexity of the construction. In §\[S:AlgPrelim\], we give the necessary algebraic preliminaries. In §\[S:MRconst\], we briefly outline Mourgues and Ressayre’s construction of an integer part for a real closed field. In §\[S:Rconst\], we provide the details of Ressayre’s construction of an exponential integer part for a real closed exponential field. The construction is canonical with respect to a given real closed field, a residue field section, and a well ordering of the elements of the real closed field. In §\[S:compEIP\], we look at the complexity of Ressayre’s construction. We produce a low real closed exponential field $R$, with a $\Delta^0_3$ residue field section $k$ and a $\Delta^0_3$ ordering $<$ of type $\omega+\omega$, so that Ressayre’s construction applied to these inputs is not completed in the least admissible set. Since our $R$ is recursively saturated, there is another exponential integer part that is $\Sigma^0_2$. In general, for a countable real closed exponential field, we may use $\Sigma$-saturation (an old notion due to Ressayre), to produce an exponential integer part $Z$ such that $\omega_1^{(R,Z)} = \omega_1^Z$; i.e., $(R,Z)$ is an element of a fattening of the least admissible set over $R$.
Algebraic preliminaries {#S:AlgPrelim}
=======================
In this section, we give some algebraic background for the construction Mourgues and Ressayre. We recall the natural valuation on an ordered field $R$.
\[Archimedean equivalence\]
For $x,y\in R^{\times}:=R\setminus \{0\}$, $x\sim y$ iff there exists $n\in \mathbb N$ such that $n|x| \geq |y|$ and $n |y|\geq |x|$, where $|x|:= \mbox{ maxÊ} \{ x, -x \}.$ We denote the equivalence class of $x\in R$ by $v(x)$.
\[Value group\] The [*value group of $R$*]{} is the set of equivalence classes $v(R^{\times})=\{v(x)\mid x\in R^{\times}\}$ with multiplication on $v(R^{\times})$ defined to be $v(x)v(y)=v(xy)$. We endow $v(R^{\times})$ with the order
$v(x)<v(y)$ if $(\forall n\in \mathbb N)[\, n |x|<|y|\, ]$.
By convention, we let ${v(0)}< v(R^{\times})$.
Under the given operation and ordering, $v(R^{\times})$ is an ordered Abelian group with identity $v(1)$. Moreover, the map $x\mapsto v(x)$ is a [*valuation*]{}, i.e. it satisfies the axioms $v(xy)=v(x)v(y)$ and $v(x+y)\leq \mbox{ max } \{ v(x), v(y)\} $.
If $R$ is a real closed field, then the value group $v(R^{\times})$ is divisible [@EP Theorem 4.3.7]. An Abelian group $(G, \cdot)$ is [*[divisible]{}*]{} if for all $g\in G$ and $0\not=n\in \mathbb N$, $g^{\frac{1}{n}}\in G$. Note that a divisible Abelian group $(G, \cdot)$ is a $\mathbb{Q}$-vector space when scalar multiplication by $q\in \mathbb{Q}$ is defined to be $g^q$. This observation motivates the following definition.
\[Generating set\] Let $(G, \cdot)$ is a divisible Abelian group. We say that $B$ is a *generating set* if each element of $G$ can be expressed as a finite product of rational powers of elements of $B$. We denote the Abelian Group generated by a set $B\subset R$ by ${\langle {B}\rangle_{\mathbb{Q}}}$.
\[Value group section\] A [*value group section*]{} is the image of an embedding of ordered groups $t:v(R^{\times}) \hookrightarrow R^{>0}$ such that $v(t(g))=g$ for all $g\in v(R^{\times})$.
If $R$ is real closed field, there are subgroups of $(R^{>0},\cdot)$ that are value group sections (see [@Kap Theorem 8]). Note that we use the term “value group section” to refer to the image of the described embedding, not the embedding itself. In [@KL], it is shown that for a countable real closed field $R$, there is a value group section $G$ that is $\Delta^0_2(R)$. Moreover, this is sharp. There is a computable real closed field $R$ such that the halting set $K$ is computable relative to every value group section.
\[Valuation ring\]
The *valuation ring* is the ordered ring $$\mathcal O_v:= \{ x\in R: v(x)\leq 1\}$$ of finite elements.
The valuation ring has a unique maximal ideal $$\mathcal M_v:=\{ x\in R: v(x)<1\}$$ of infinitesimal elements.
\[Residue field\]
The *residue field* is the quotient $\mathcal O_v / \mathcal M_v$.
The residue field $k$ is an ordered field under the order induced by $R$. It is Archimedean, so it is isomorphic to a subfield of $\mathbb R$. We denote the residue of $x\in \mathcal O_v$ by $\overline{x}.$
\[Residue field section\] A [*residue field section*]{} is the image of an embedding of ordered fields $\iota :k \hookrightarrow R$ such that $\overline{\iota (c)}=c$ for all $c\in k$.
If $R$ is a real closed field, then $k$ is real closed [@EP Theorem 4.3.7] and residue field sections exist [@Kap Theorem 8]. To construct a residue field section, we look for a maximal real closed Archimedean subfield. In [@KL], the second and fourth authors proved the following result on the complexity of residue field sections.
\[residue\]
For a countable real closed field $R$, there is a residue field section that is $\Pi^0_2(R)$.
Proposition \[residue\] is sharp in the following sense.
There is a computable real closed field with no $\Sigma^0_2$ residue field section.
We let $k(G)$ denote the field generated by $k\cup G$.
\[$k((G))$\]Let $k$ be an Archimedean ordered field and $G$ an ordered Abelian group.
1. The field $k((G))$ of generalized series is the set of formal sums $s = \sum_{g\in G} a_g g$ with $a_g\in k$ and $\mbox{Supp}(s):=\{g\in G: s_g\not=0\}$ is an anti-wellordered subset of $G$.
2. The *length* of $s$ is the order type of $Supp(s)$ under the reverse ordering. Later, we may write $s = \sum_{i<\alpha} a_i g_i$, where $g_i\in G$ with $g_i > g_j$ for $i < j < \alpha$, and $a_i\in k^{\times}$. Under this notation, the length of $s$ is $\alpha$.
3. For $s = \sum_{g\in S} a_g g$ and $t = \sum_{g\in T} b_g g$ in $k((G))$ where $Supp(s)\subset S$ and $Supp(t)\subset T$, the *sum* $s+t$ and the *product* $s\cdot t$ are defined as for ordinary power series.
1. In $s+t$, the coefficient of $g$ is $a_g+b_g$.
2. In $s\cdot t$, the coefficient of $g$ is the sum of the products $a_{g'}b_{g''}$, where $g = g'\cdot g''$.
4. $k((G))$ is ordered anti-lexicographically by setting $s>0$ if $a_g>0$ where $g=:\mbox{max(Supp(s))}$.
For a proof that $k((G))$ is a totally ordered field, see [@F Chapter VIII, Theorem 10]. If $k$ is real closed and $(G, \cdot)$ is an ordered divisible Abelian group, then $k((G))$ is real closed by [@EP Theorem 4.3.7]. The field $k((G))$ carries a canonical valuation $v: k((G))^{\times}\rightarrow G$, defined by $s\mapsto \mbox{ max(supp}(s))$, with value group $G$. Given a subset $X\subset G$, we set $$k((X))=\{s\in k((G))\mid Supp(s)\subset X\}.$$ We let $G^{\le 1}=\{g\in G\mid g\le 1\}$, and similarly define $G^{<1}$ and $G^{>1}$. The valuation ring is the ring of finite elements $k((G^{\leq 1}))$, its valuation ideal is the ideal of infinitesimals $k((G^{< 1}))$, and the residue field is $k$. The canonical additive complement to the valuation ring is $k((G^{> 1}))$, the ring of purely infinite series. The group of positive units of $k((G^{\leq 1}))$ is denoted by $\mathcal U_v^{>0}$, and consists of series $s$ in the valuation ring with coefficient $a_g>0$ for $g=1$. In this setting the following decompositions of the additive and multiplicative groups of $k((G))$ will be useful $$(k((G)), +)= k((G^{\leq 1}))\oplus k((G^{> 1})) \mbox{ and } (k((G))^{>0}, \cdot )= \mathcal U_v^{>0}\cdot G .$$
Truncation-closed embeddings
----------------------------
\[Truncation-closed subfield\]
Let $F$ be a subfield of $k((G))$. We say that $F$ is truncation closed if whenever $s = \sum_{g\in G} a_g g \in F$ and $h\in G$, the restriction $s_{<h}=\sum_{g<h} a_g g$ also belongs to $F$.
Mourgues and Ressayre [@MR] showed that every real closed field has an integer part in the following way. In [@MR Lemma 3.2], Mourgues and Ressayre observed the following.
\[Mourgues and Ressayre\]
If $F$ is a truncation closed subfield of $k((G))$ and $Z_F$ consists of the elements of the form $t+z$, where $t\in F\cap k((G^{>1}))$ and $z\in\mathbb{Z}$, then $Z_F$ is an integer part for $F$.
If $s\in F$, we have $s = t + t'$, where $t\in k((G^{>1}))$ and $t'\in k((G^{\leq 1}))$. Take $z\in\mathbb{Z}$ such that $z\leq t' < z+1$. Then $t+z\leq s < t+z+1$.
In [@MR Corollary 4.2], they showed the following restatement of Theorem A.
Let $R$ be a real closed field with value group $G$ and residue field $k$. Then there is an order (valuation) preserving isomorphism $\delta$ from $R$ onto a truncation closed subfield $F$ of $k((G))$. Thus $\delta^{-1}(Z_F)$ is an integer part for $R$.
We refer to $\delta$ as a “development function” $\delta$.
Exponential integer parts
-------------------------
In [@R] Ressayre imposes a further condition on $\delta$ which ensures that the truncation integer part is also closed under exponentiation. The following is a rephrasing of Theorem B and of [@R Theorem 4].
Let $(R,2^x)$ be an exponential real closed field. Fix a residue field section $k\subset R$. Then there is a value group section $G\subset R^{>0}$ and a truncation closed embedding $\delta:R \hookrightarrow k((G))$ fixing $k$ and $G$, and such that $$\label{crucial} \delta (\log(G)) = \delta(R)\cap k((G^{>1})).$$
We argue that if $\delta$ satisfies condition (\[crucial\]) then the truncation integer part $\delta^{-1}(Z_F)$ is an exponential integer part of $R$. The exponential function $2^x$ defined on $R$ induces an exponential function on $F=\delta (R)$ by setting: $2^y=\delta (2^x)$ where $y=\delta (x)$ for $x\in R$. So, it suffices to show the following lemma, which appears in [@BKK Proposition 5.2].
\[L:expIP\] $Z_F$ is an exponential integer part of $F$ with respect to the induced exponential function.
Let $z\in Z_F$ and $z>0$, then $z= a+ y$ where $y\in F\cap k((G^{>1}))$ and $a\in \mathbb Z$. We compute $2^z=2^a2^y$. If $y=0$ then $a>0$, so $2^z\in \mathbb N\subset Z_F$. If $y\not= 0$ then $y>0$, and $2^y>1$. We now show that $2^y\in G$. By (\[crucial\]) $y=\delta (\log (g))$ for some $g\in G$. Then $2^y=\delta (2^{\log g})=\delta (g)=g$, as required. Therefore, $2^y\in G^{>1}$, and so $2^z=2^a2^y$ belongs to $k((G^{>1}))$, and also to $F=\delta (R) $. So, $2^z\in F\cap k((G^{>1}))\subset Z_F$.
Development Triples {#S:MRconst}
===================
Mourgues and Ressayre prove Theorem A$'$ by showing how to extend a partial embedding $\phi$ from a subfield $A$ of $R$ onto a truncation closed subfield $F$ of $k((G))$ to be defined to some $r\in R-A$ while preserving truncation closure.
\[Development triple\]
Suppose $R$ is a real closed field, with residue field $k$. We say that $(A,H,\phi)$ is a [*development triple with respect to $R$ and $k$*]{} if
1. $A$ is a real closed subfield of $R$ containing $k$,
2. $H\subset A$ is a value group section for $A$, and
3. $\phi$ is a order preserving isomorphism from $A$ onto a truncation closed subfield of ${k(( {H})) }$ such that $\phi\upharpoonright k(H)$ is the identity.
**Notation**. We write $(A', H', \phi')\supseteq(A,H, \phi)$, if $A'\supseteq A$, $H'\supseteq H$, and $\phi'\supseteq \phi$.
Given a development triple $(A, H, \phi)$ and an element $r\in R-A$, we want to obtain a development triple $(A', H', \phi')\supset (A, H, \phi)$ with $r\in A'$. We use the following definitions to describe $\phi'(r)$.
Let $\alpha$ be an ordinal. The [*development of $r\in R$ over $(A, H, \phi)$ of length $\alpha$*]{} is an element $t_\alpha\in{k(( {H})) }$ satisfying:
- $t_0= 0$ if $\alpha=0$, and otherwise,
- $t_\alpha={\sum_{i<\alpha}a_i{g_i}}$ where $$(\forall \beta<\alpha)(\exists \hat{r}_\beta\in A)[ \phi(\hat{r}_\beta)={\sum_{i<\beta}a_i{g_i}}\ \&\ {g_\beta}=v(r- \hat{r}_\beta)\in G ]$$
It is straightforward to prove the next lemma.
\[L:devuniq\] Let $(A, H, \phi)$ be a development triple, $r\in R$, and $\alpha$ an ordinal for which $t_\gamma$ exists. Then,
1. $t_\gamma$ is unique and, for all $\beta\le \gamma$, $t_\beta=(t_\gamma)_{<\beta}$.
2. There is a development $t_\alpha$ of $r$ over $(A, H, \phi)$ of maximum length $\alpha$.
Lemma \[L:devuniq\] allows us to make the following definition.
The [*maximum development of $r$ over $(A, H, \phi)$*]{} is the unique development of $r$ over $(A, H, \phi)$ of maximum length $\alpha$.
\[O:GammainH\] Let $(A, H, \phi)$ be a development triple. Let $t_\alpha$ be the maximum development of $r\in R-A$ over $(A, H, \phi)$. Let The following statements are equivalent: (i) $\Gamma(r)\subset H$, (ii) $\Gamma(r)$ has no least element, and (iii) $t_\alpha\not\in\phi(A)$.
Also, if $t_\alpha\not\in\phi(A)$, then $\alpha$ is a limit ordinal or 0.
Since these observations will not be used in the remainder of the paper, we omit the proof. We now restate the key lemma of Theorem A by Mourgues and Ressayre [@MR] in the language of development triples.
\[T:MRext\] Suppose $(A, H, \phi)$ is a development triple with respect to a real closed field $R$ and $r\in R-A$. There is a development triple $(A', H', \phi')$ extending $(A, H, \phi)$ such that $r\in A'$. Moreover, if the maximum development of $r$ over $(A, H, \phi)$ is $t_\alpha\in{k(( {H})) }$, then
We present Mourgues and Ressayre’s construction in the framework of development triples, because we will need development triples with additional properties to examine the complexity of the exponential case, which is our main goal.
We use the following theorem to construct the development triple $(A', H', \phi')$ extending $(A, H, \phi)$.
\[T:cutsextend\] Let $A\subset A'$ and $B\subset B'$ be real closed fields such that there is an order preserving isomorphism $\phi$ from $A$ onto $B$. If we have $a\in A'-A$ and $b\in B'-B$ such that $$(\forall x\in A)[x<a \iff \phi(x)<b],$$ then there is a unique order preserving isomorphism $\phi'\supset\phi$ from onto $RC(B\cup\{b\})$ with $\phi'(a)=b.$
We have two cases: the immediate transcendental case where $t\not\in\phi(A)$ and the value transcendental case where $t\in\phi(A)$, as seen in [@Kap]. Note there is no residue transcendental case because $k$ is a residue field section for $R$, not just $A$. In both cases, we choose $\phi'(r)$ so that the cut of $r$ over $A$ is the same as the cut of $\phi'(r)$ over $\phi(A)$.
\[P:cut\] Let $(A, H, \phi)$ be a development triple with respect to $R$ and let $r\in R-A$. Suppose $r$ has development $t=t_\alpha$ over $(A, H, \phi)$. If $t\not\in\phi(A)$, then for all $x\in A$, $$x< r \text{ (in $R$)} \iff \phi(x)< t \text{ (in ${k(( {H})) }$)}$$ Similarly, if there is some $r'\in A$ such that $\phi(r')=t$, then for all $x\in A$, $$x< r \text{ (in $R$)} \iff \phi(x)< t+\epsilon g \text{ (in ${k(( {H'})) }$)}$$ where $g=|r-r'|$ and $\epsilon=\pm 1$ such that $g=|r-r'|=\epsilon (r-r')$ and $H'={\langle {H\cup\{g\}}\rangle_{\mathbb{Q}}}$.
The proof of Lemma \[P:cut\] can be found in Lemma 3.3, p. 191 of [@VMM] (see also Theorem 6.2, p. 85 of [@KKMZ]). Lemma \[P:cut\] can also be proved using Observation $\ref{L:cofinal}$, whose proof we omit.
\[L:cofinal\] For an ordered set $B$, $b\in B$, and $C\subset B$, let $C_{<b}=\{c\in C\mid c<b\}$ and let $C_{>b}=\{c\in C\mid c>b\}$.
Let $(A, H, \phi)$ be a development triple with respect to $R$ and let $r\in R-A$. Suppose $r$ has development $t=t_\alpha$ over $(A, H, \phi)$. Let $\hat{A}=\{\hat{r}_\beta\mid \beta<\gamma\}$. Let $t'=t$ if $t\not\in\phi(A)$ and let $t'= t+\epsilon g $ as in Lemma \[P:cut\] if $t\in\phi(A)$. At least one of the following two statements holds.
1. The set $\hat{A}_{<r}$ is cofinal in $A_{<r}$, and $\phi(\hat{A}_{<r})$ is cofinal in $\phi(A)_{<t'}$.
2. The set $\hat{A}_{>r}$ is coinitial in $A_{>r}$, and $\phi(\hat{A}_{>r})$ is coinitial in $\phi(A)_{>t'}$.
Let $t'$ be the development $t$ of $r\in R-A$ over $(A, H, \phi)$ if $t\not\in\phi(A)$, and let $t'=t+\epsilon g$ as in Lemma \[P:cut\] if $t\in\phi(A)$. By Lemma \[P:cut\] and Theorem \[T:cutsextend\], there is a unique order preserving isomorphism $\phi'\supset \phi$ from $RC(A\cup \{r\})$ onto $RC(\phi(A)\cup \{t'\})$ with $\phi'(r)=t'$. Moreover, by the definition of development, the proper truncations of $t'$ are all in $\phi(A)$. The following lemma says that the range of $\phi'$ is truncation closed.
\[Mourgues-Ressayre\]\[Development\] \[development lemma\]
Let $F$ be a truncation closed subfield of $k({G})$, and let $t\in k((G)) - F$, where all proper initial segments of $t$ are in $F$. Then the real closure of $F(t)$ is also a truncation closed subfield of $k((G))$.
Thus, in both the immediate transcendental case and the value transcendental case, we have defined a development triple $(A'=RC(A\cup \{r\}), H', \phi')$ extending $(A, H, \phi)$ with $\phi'(r)=t'$.
We have proven Theorem \[T:MRext\].
\[O:immedtrans\] Let $(A', H', \phi')$ be the development triple extending $(A, H, \phi)$ such that $r\in A'$ for $r\in R-A$ constructed in Theorem \[T:MRext\]. In the immediate transcendental case where $t\not\in\phi(A)$, $H'=H$, whereas in the value transcendental case where $t\in\phi(A)$, $H$ is a proper subgroup of the group $H'$ given in Lemma $\ref{P:cut}$.
We will use the next notion extensively in the section on exponential case.
1. The triple $(A', H', \phi')$ is a [*value group preserving extension*]{} of $(A, H, \phi)$ if $(A', H', \phi')$ extends $(A, H, \phi)$ and $H'=H$.
2. A triple $(A, H, \phi)$ is [*maximal*]{} if $(A, H, \phi)$ admits no proper value group preserving extension.
The next observation follows immediately from Observation \[O:GammainH\]. We will use the equivalence of the first two statements often.
\[O:maximality\] Let $(A, H, \phi)$ be a development triple. The following are equivalent.
1. $(A, H, \phi)$ is maximal.
2. For all $r\in R-A$, the development of $r$ over $(A, H, \phi)$ is in $\phi(A)$.
3. $(\forall r\in R-A)[\, \Gamma(r)\not\subset H].$
4. For all $r\in R-A$, $H$ is not a value group section for $RC(A\cup\{r\})$.
($1\implies 2$) Suppose $(A, H, \phi)$ is maximal. Given $r\in R-A$. Let $t$ be the development of $r$ over $(A, H, \phi)$. If $t\not\in\phi(A)$, by Theorem \[T:MRext\] and Observation \[O:immedtrans\], there is a development triple $(RC(A\cup\{r\}), H, \phi')$ with $\phi'(r)=t$ properly extending $(A, H, \phi)$, contradicting the maximality of $(A, H, \phi)$.
($2\implies 3$) If there is an $r\in R-A$ with $\Gamma(r)\subset H$, we have that the development $t$ of $r$ over $(A, H, \phi)$ satisfies $t\not\in\phi(A)$ by Observation \[O:GammainH\].
($3\implies 4$) Let $r\in R-A$. Since $\Gamma(r)\not\subset H$, there is an $r'\in A$ such that $v(r-r')\not\in H$. Since $r-r'\in RC(A\cup\{r\})$, $H$ is not a value group for $ RC(A\cup\{r\})$.
($4\implies 1$) Suppose there is some $(A', H', \phi')$ extending $(A, H, \phi)$ and $H'\not=H$. Let $r\in A'-A$ with $v(r)\in H'-H$. Then, $H$ is not a value group section for $RC(A\cup\{r\})$.
Note that $(k, \{1\}, id)$ is a maximal development triple as is any triple of the form $(R, G, \delta)$ with respect to a real closed field $R$.
\[MRcan\] Given a real closed field $R$, a residue field section $k$, and a well ordering of $R=(r_i)_{i<\lambda}$, there is a canonical development triple $(R, G, \delta)$ with respect to $R$ and $k$.
We construct $(R, G, \delta)$ from a chain of development triples $(R_i,G_i,\delta_i)_{i<\lambda}$. Once $R_i$ is defined for $i<\lambda$, we let ${m(i)}<\lambda$ be the least ordinal such that $r_{m(i)}\in R-R_i$. We define $(R_0,G_0,\delta_0)$ to be $(k,\{1\},id)$. Let $j<\lambda$. Given development triples $(R_i, G_i, \delta_i)_{i<j}$, we define $(R_j, G_j, \delta_j)$ by induction as follows. If $j$ is a limit ordinal, we let $R_j=\cup_{i<j} R_i$, $G_j=\cup_{i<j} G_i$, and $\delta_j=\cup_{i<j} \delta_i$. Note that $(R_j, G_j, \delta_j)$ is a development triple since it is a union of a chain of development triples. If $j=l+1$ is a successor ordinal, we take $(R_j, G_j, \delta_j)\supset(R_l, G_l, \delta_l)$ so that $r_{m(l)}\in R_j$ using Theorem \[T:MRext\].
Similarly, one prove that there is a canonical maximal development triple extension by using Theorem \[T:MRext\] and Observation \[O:immedtrans\] and checking whether some $r\in R$ can be added to a triple using Lemma \[O:maximality\].
\[Maximal\]
Let $(A,H,\phi)$ be a development triple with respect to $R$ and $k$. Given a well ordering of $R=(r_i)_{i<\lambda}$, there is a canonical development triple $(A,H,\phi')$ extending $(A,H,\phi)$ that is maximal.
Exponential integer parts {#S:Rconst}
=========================
To show that every real closed exponential field has an exponential integer part, Ressayre lets the value group section do most of the work. Below, we define a special kind of development triple, with the added features we want for the group.
\[Dyadic development triple\]
Let $R$ be a real closed exponential field, and let $k$ be a residue field section. Let $(A, H, \phi)$ be a development triple with respect to $R$ and $k$. Then $(A,H,\phi)$ is a *dyadic development triple* for $R$ and $k$ if $$\phi(\log H)=\phi(A)\cap {k(( {H^{>1}})) }.$$
Equivalently, $(A, H, \phi)$ is dyadic if
1. for all $r\in H$, $\log r\in A$ and $\phi(\log r)\in {k(( {H^{>1}})) }$, and
2. for all $r\in A$, if $\phi(r)\in{k(( {H^{>1}})) }$, then $2^r\in H$.
Rephrasing Lemma \[L:expIP\] in terms of this terminology, if $R$ has a dyadic triple $(R, G, \delta)$, then $R$ has an exponential integer part. So, proving Theorem B$'$ is equivalent to showing that every real closed exponential field $R$ with a residue field section $k$ has a dyadic triple $(R, G, \delta)$ with respect to $R$ and $k$.
Extending dyadic triples
------------------------
Most of the work in the proof of Theorem B$'$ is showing how to extend one dyadic and maximal triple to another such triple.
\[Main Technical Lemma\] \[main technical\] Suppose $(A,H,\phi)$ is a dyadic and maximal triple, and $y\in R - A$. Then there is a dyadic and maximal triple $(A',H',\phi')\supseteq (A,H,\phi)$ such that $y\in A$.
Without loss of generality, we may suppose that $y$ is positive, $v(y)>1$, and $v(y)\not\in H$. We can take $v(y)\not\in H$ by Lemma \[O:maximality\] (3). We may further suppose that $y>0$, since otherwise we could replace $y$ by $-y$, and we may suppose that $v(y)>1$, since otherwise we could replace $y$ by $y^{-1}$.
We will obtain the required dyadic triple $(A',H',\phi')$ as the union of a chain of maximal development triples $(B_i,H_i,\phi_i)$ with the following features.
1. $H_0\supseteq H$ is a value group section for $RC(\{\log_i(y)\mid i\in\omega\})$ where $\log_0 y=y$ and $\log_{i+1} y=\log_i(\log y)$ for all $i\in \omega$.
2. If $r\in H_i$, then $\log(r)\in B_i$ and $\phi_i(\log(r))\in {k(( {H_i^{>1}})) }$.
3. If $r\in B_i$ and $\phi_i(r)\in{k(( {H_i^{>1}})) }$, then $2^r\in H_{i+1}$.
We begin by defining the development triple $(B_0, H_0,\phi_0)$. We first define a sequence $(y_i)_{i\in\omega}$ of elements in $R$ and describe some of their properties.
\[C:yinf\] Let $(A, H, \phi)$ be a dyadic and maximal triple. Let $y=y_0\in R-A$ have the following properties for $i=0$: $$\label{E:v(y_i)}
y_i>0 \ \&\ v(y_i)>1\ \&\ v(y_i)\not\in H.$$ Given $y_i$, let $p_{i+1}$ be the development of $\log y_i$ over $(A, H, \phi)$. We inductively assume that $y_i$ satisfies (\[E:v(y\_i)\]). Then,
1. $(\exists r'_{i+1}\in A)[\phi(r'_{i+1})=p_{i+1}]$,
2. $y_{i+1} := |\log(y_i) - r'_{i+1}|$ satisfies (\[E:v(y\_i)\]), and
3. $p_{i+1}\in{k(( {H^{>1}})) }$.
Suppose inductively that $y_i$ satisfies (\[E:v(y\_i)\]). Let $p_{i+1}$ be the development of $\log(y_i)$ over $(A, H, \phi)$. Since $(A, H, \phi)$ is maximal, there exists some $r'_{i+1}\in A$ such that $\phi(r'_{i+1})=p_{i+1}$. By definition of (maximum) development, we have that $v(y_{i+1})\not\in H$, and, in particular, $v(y_{i+1})\not=1$. Suppose for a contradiction that $v(y_{i+1})<1$ or $y_{i+1}=0$. We have $\log(y_i) = r'_{i+1} + \pm y_{i+1}$, and $r_{i+1}' = s + s' $, where $\phi(s)\in{k(( {H^{>1}})) }$ is the truncation of $\phi(r_{i+1}' )$ so that $\phi(s')\in{k(( {H^{\le 1}})) }$. So, $y_i = 2^{s}2^{s'}2^{\pm y_{i+1}}$. Since $v(s')\le 1$, we have $2^{s'}$ equals some $c$ with $v(c)=1$. If $v(y_{i+1})<1$ or $y_{i+1}=0$, then $2^{\pm y_{i+1}} = (1 + d)$, where $d$ is $0$ or $v(d)<1$. Since $(A,H,\phi)$ is dyadic and $\phi(s)\in{k(( {H^{>1}})) }$, we have $2^{s}\in H$. Then, $v(y_i)=2^{s}$ , contradicting our assumption that $v(y_i)\not\in H$. So, $y_{i+1}\not=0$ and $v(y_{i+1})>1$. Since $v(y_{i+1})<v(g)$ for all $g\in Supp(\phi(r'_{i+1}))$, we see that $\phi(r'_{i+1})=p_{i+1}\in{k(( {H^{>1}})) }$.
\[y\_iincr\]
For all $i,n \in\omega$, $(y_{i+1})^n < y_i$. Hence, $v(y_i)\not= v(y_j)$ for $i\not=j$.
From the definition of $y_{i+1}$, we see that $y_{i+1} < \log(y_i)$, so $y_{i+1}^n < \log(y_i)^n$. Since $v(y_i)>1$, $\log(y_i)^n < 2^{\log(y_i)}=y_i$ by property (5) of Definition \[D:EXP\].
Let $H_{0,n}={\langle {H\cup\{y_i\mid i\in\omega\}}\rangle_{\mathbb{Q}}}$. Let $H_0=\cup_{n\in\omega} H_{0,n}$.
For each $n$, $v(y_n)\not\in H_{0,n}$. Hence, $H_0\supset H$ is a value group section for $RC(A\cup H_0)$.
The statement is clear for $n = 0$. We show the statement for $n+1$. We assume for a contradiction that $v(y_{n+1})\in H_{0, n}$, i.e., $y_{n+1}=cgy_0^{q_0}\cdot \ldots\cdot y_n^{q_n}$, where $c\in R$, $v(c)=1$, $g\in G$, and $q_i\in \mathbb{Q}$. Taking logs, we obtain $$\log(y_{n+1}) = \log(c)+\log(g) + q_0\log(y_0)+\ldots+q_n\log(y_n).$$ Recall that, by definition, $\log(y_i) = r'_{i+1}+\epsilon_{i+1} y_{i+1}$, where $\phi(r'_{i+1})$ is the development of $\log(y_i)$ over $(A,H,\phi)$ and $\epsilon_{i+1} = \pm 1$. Then, by subsitution and rearranging terms, we have that $ \epsilon_{n+2}y_{n+2}$ equals $$\log(c) + [\log(g) + q_0r'_1+\ldots + q_nr'_{n+1}-r'_{n+2}]+[q_0\epsilon_1y_1 + \ldots + q_n\epsilon_{n+1}y_{n+1}]$$ We have $v(\log c)=1$, $v(\log(g) + q_0r'_1+\ldots + q_nr'_{n+1}-r'_{n+2})\in H^{>1}$, and $v(q_0\epsilon_1y_1 + \ldots + q_n\epsilon_{n+1}y_{n+1})=v(y_1)$ by Lemmas \[C:yinf\] and \[y\_iincr\]. Thus, $v(y_{n+2})$ is either in $H$ or equals $v(y_1)$, contradicting either Lemma \[C:yinf\] or Lemma \[y\_iincr\].
\[O:logh\] If $h\in H_0$, then the development of $\log h$ over $(A, H, \phi)$ is in ${k(( {H_0^{>1}})) }$.
Let $h\in H_0$ with $h=g\prod_{i=0}^n y_{l_i}^{q^i}$. So, $$\log h=\log g+\sum_{i=0}^n q_i(r'_{i+1}+\pm y_{i+1})$$ where $q_i\in \mathbb{Q}$. The developments of $\log g$ and $r'_{i+1}$ are in ${k(( {H^{>1}})) }$ since $(A, H, \phi)$ is dyadic and by construction. Since $v(y_{i+1})>1$, $\log(h)$ has a development in ${k(( {H_0^{>1}})) }$.
By Theorem \[T:MRext\] and Lemma \[Maximal\], we obtain $B_0$ and $\phi_0$ such that $(B_0, H_0, \phi_0)$ is maximal and extends $(A, H, \phi)$.
We define $H_1={\langle {H_0\cup\{2^r\mid r\in B_0 \ \&\ \ \phi_0(r)\in {k(( {H_0^{>1}})) }\}}\rangle_{\mathbb{Q}}}$. As was the case for $H_0$, for all $h\in H_1$, $\phi_0(\log h)\in {k(( {H_0^{>1}})) }$. The next lemma ensures that $H_1$ is a value group section for $RC(B_0\cup H_1)$. Given $(B_i,H_i,\phi_i)_{i<\alpha}$ such that $(B_i,H_i,\phi_i)$ is maximal and $\phi_i(h)\in{k(( {H_i^{>1}})) }$ for all $h\in H_i$, we define the triple $(B_\alpha,H_\alpha, \phi_\alpha)$. If $\alpha=j+1$, we let $H_{j+1}={\langle {H_j\cup\{2^r\mid r\in B_j \ \&\ \ \phi_j(r)\in {k(( {H_j^{>1}})) }\}}\rangle_{\mathbb{Q}}}$. By Theorem \[T:MRext\] and Lemma \[Maximal\], we obtain $B_{j+1}$ and $\phi_{j+1}$ such that $(B_{j+1}, H_{j+1}, \phi_{j+1})$ is maximal and extends $(B_j,H_j,\phi_j)$.
For $\alpha$ a limit ordinal, we let $H_\alpha = \cup_{i<\alpha} H_{i}$. If $(\cup_{i<\alpha} B_{i}, H_\alpha, \cup_{i<\alpha} \phi_{i})$ is not maximal, then we use Lemma \[Maximal\] to find a maximal triple $(B_\alpha, H_\alpha, \phi_\alpha)$ extending $(\cup_{i<\alpha} B_{i}, H_\alpha, \cup_{i<\alpha} \phi_{i})$. If $(\cup_{i<\alpha} B_{i}, H_\alpha, \cup_{i<\alpha} \phi_{i})$ is maximal, then we set $B_\alpha=\cup_{i<\alpha} B_{i}$ and $\phi_\alpha=\cup_{i<\alpha} \phi_{i}$. In this case, $(B_\alpha, H_\alpha, \phi_\alpha)$ is the desired dyadic triple $(A',H',\phi')$.
The analogue of the proof of Lemma \[O:logh\] shows the following.
\[L:Halphalog\]
For all $\alpha$, if $h\in H_\alpha$, then $\phi_\alpha(\log h)\in {k(( {H_\alpha^{>1}})) }$.
The next lemma shows that $H_\alpha$ is a value group section for $RC(\cup_{i<\alpha} B_i \cup H_\alpha)$.
\[L:valimp=\]
For all $\alpha$, if $h,h'\in H_\alpha$ and $v(h)=v(h')$, then $h = h'$.
If $v(h)=v(h')$, then $h = ch'$, for some $c\in R^{>0}$ with $v(c)=1$. By Lemma \[L:Halphalog\], we have $\phi_\alpha(\log h), \phi_\alpha(\log h')\in {k(( {H_\alpha^{>1}})) }$. Since $\log(h) = \log(c) + \log(h')$, we must have $\phi_\alpha(\log h)=\phi_\alpha(\log h')$ and $\log(c) = 0$, so $c = 1$.
Since $R$ is a set, there exists some limit ordinal $\lambda$ such that $B_\lambda = \cup_{i<\lambda} B_{i}$ and $\phi_\lambda=\cup_{i<\lambda} \phi_{i}$. Then $(B_\lambda,H_\lambda,\phi_\lambda)$ is a dyadic and maximal triple extending $(A, H, \phi)$ and for which $y\in B_\lambda$, as required for Proposition \[main technical\].
\[L:canRconst\] Given a real closed exponential field $R$, a residue field section $k$, and a well ordering of $R=(r_i)_{i<\lambda}$, there is a canonical dyadic triple $(R,G,\delta)$ with respect to these data.
The proof is the same as in Lemma \[MRcan\], except that we use the following corollary at limit steps in our construction.
\[cor\] Suppose $(A, H,\phi)$ is the union of a chain of dyadic triples. Then there is a dyadic and maximal triple $(A',H',\phi')$ extending $(A, H,\phi)$.
The triple $(A, H,\phi)$ may not be maximal. By Lemma \[Maximal\], we extend $(A, H,\phi)$ to a maximal triple $(\hat{A}, H,\hat{\phi})$. If $\hat{A} = R$, then $(R, H,\hat{\phi})$ is a dyadic triple. If not, take $i<\lambda$ least such that $r_i\in R-\hat{A}$. By Proposition \[main technical\], we can extend $(\hat{A}, H,\hat{\phi})$ to a dyadic and maximal triple $(A',H',\phi')$.
Recursive saturation, Barwise-Kreisel Compactness, and $\Sigma$-saturation
==========================================================================
For our example illustrating the complexity of Ressayre’s Construction, we shall use recursive saturation and a version of Compactness for computable infinitary sentences. We also describe a different method for producing exponential integer parts. For this we need $\Sigma$-saturation, a kind of saturation for infinitary formulas. Recursive saturation has already come up in connection with integer parts. In [@DKS], it was shown that a countable real closed field has an integer part satisfying Peano arithmetic if and only if the real closed field is Archimedean or recursively saturated.
### Recursive saturation
Recursive saturation was defined by Barwise and Schlipf [@BS].
\[Recursive saturation\]
A structure $\mathcal{A}$ is *recursively saturated* if for all tuples $\overline{a}$ in $\mathcal{A}$ and all c.e. sets of formulas $\Gamma(\overline{a},x)$, if every finite subset of $\Gamma(\overline{a},x)$ is satisfied in $\mathcal{A}$, then some $b\in\mathcal{A}$ satisfies all of $\Gamma(\overline{a},x)$.
Countable recursively saturated structures can be expanded as follows.
\[Barwise-Schlipf\] \[Barwise-Schlipf\]
Let $\mathcal{A}$ be a countable recursively saturated $L$-structure. Let $\Gamma$ be a c.e. set of sentences, in a language $L'\supseteq L$. If the consequences of $\Gamma$ in the language $L$ are true in $\mathcal{A}$, then $\mathcal{A}$ can be expanded to a model of $\Gamma$.
In [@MM] Macintyre and Marker considered the complexity of recursively saturated models. We shall need the following result.
\[Macintyre-Marker\] \[MM\]
Suppose $E$ is an enumeration of a countable Scott set $\mathcal{S}$. Let $T$ be a complete theory in $\mathcal{S}$. Then $T$ has a recursively saturated model $\mathcal{A}$ such that $D^c(\mathcal{A})\leq_T E$.
The next result may be well-known. The proof will be obvious to anyone familiar with the proof of Theorem \[Barwise-Schlipf\].
\[B-S complexity\]
Suppose $\mathcal{A}$ is a countable recursively saturated structure, say with universe $\omega$, and let $\Gamma$ be a c.e. set of finitary sentences, in an expanded language, such that the consequences of $\Gamma$ are all true in $\mathcal{A}$. Then $\mathcal{A}$ can be expanded to a model $\mathcal{A}'$ of $\Gamma$ such that $D^c(\mathcal{A}')$ is computable in the jump of $D^c(\mathcal{A})$.
\[Proof Sketch\]
We carry out a Henkin construction, as Barwise and Schlipf did, and we observe that the jump of $D^c(\mathcal{A})$ is sufficient. We make a recursive list of the sentences $\varphi(\overline{a})$ in the expanded language, with names for the elements of $\omega$. We also make a recursive list of the c.e. sets $\Gamma(\overline{a},x)$. At each stage $s$, we have put into $D^c(\mathcal{A}')$ a c.e. set $\Sigma_s(\overline{a})$ of sentences involving a finite tuple of constants, such that the consequences in the language of $\mathcal{A}$ are true in $R$ of the constants $\overline{a}$. At stage $s+1$, we consider the next sentence $\varphi(\overline{a})$. We add $\varphi(\overline{a})$ to $\Sigma_s(\overline{a})$ if our consistency condition is satisfied, and otherwise we add the negation. Then we consider the next c.e. set $\Gamma(\overline{a},x)$. To check consistency, we see if the consequences of adding this, with some new constant $e$ for $x$, are true of $\overline{a}$. Then we look for $b$ such that for $b = x$, the consequences are satisfied by $\overline{a},b$.
### Compactness for infinitary logic
Kripke-Platek set theory ($KP$) differs from $ZFC$ in that the power set axiom is dropped, and the separation and replacement axioms are restricted to formulas with bounded quantifiers. An *admissible set* is a model of $KP$ that is *standard*; i.e., the epsilon relation is the usual one and the model forms a transitive set. If $A$ is an admissible set, and $B\subseteq A$, then $B$ is *$\Sigma_1$ on $A$* if it is defined by an existential formula, possibly with parameters. A set is *$A$-finite* if it is an element of $A$. The least admissible set is $A = L_{\omega_1^{CK}}$. In this case, a set $B\subseteq\omega$ is $\Sigma_1$ on $A$ if it is $\Pi^1_1$, and it is $A$-finite if it is hyperarithmetical. For a countable language $L$, there are uncountably many formulas of $L_{\omega_1\omega}$. For a countable admissible set $A$, the *admissible fragment $L_A$* consists of the $L_{\omega_1\omega}$ formulas that are elements of $A$. In the case where $A$ is the least admissible set, the $L_A$-formulas are essentially the computable infinitary formulas.
\[Barwise Compactness\]\[T:BarwiseCompact\]
Let $A$ be a countable admissible set, and let $L$ be an $A$-finite language. Suppose $\Gamma$ is a set of $L_A$-sentences that is $\Sigma_1$ on $A$. If every $A$-finite subset of $\Gamma$ has a model, then $\Gamma$ has a model.
As a special case, we have the following.
\[Barwise-Kreisel Compactness\]
Let $L$ be a computable language. Suppose $\Gamma$ is a $\Pi^1_1$ set of computable infinitary $L$-sentences. If every hyperarithmetical subset of $\Gamma$ has a model, then $\Gamma$ has a model.
Ressayre’s notion of $\Sigma_A$-saturation, defined in [@R0], [@R1] is associated with Barwise Compactness. We start with an admissible set $A$. Some people omit the axiom of infinity from $KP$. Then $L_\omega$ qualifies as an admissible set, and we get recursive saturation as a special case, where $A = L_\omega$. Ressayre worked independently of Barwise and Schlipf, and the first version of his definition, in [@R0], was actually earlier.
Suppose $A$ is a countable admissible set and let $L$ be an $A$-finite language. An $L$-structure $\mathcal{A}$ is *$\Sigma_A$-saturated* if
1. for any tuple $\overline{a}$ in $\mathcal{A}$ and any set $\Gamma$ of $L_A$-formulas, with parameters $\overline{a}$ and free variable $x$, if $\Gamma$ is $\Sigma_1$ on $A$ and every $A$-finite subset is satisfied, then the whole set is satisfied.
2. let $I$ be an $A$-finite set, and let $\Gamma$ be a set, $\Sigma_1$ on $A$, consisting of pairs $(i,\varphi)$, where $i\in I$ and $\varphi$ is an $L_A$-sentence. For each $i$, let $\Gamma_i = \{\varphi:(i,\varphi)\in\Gamma\}$. Similarly, if $\Gamma'\subseteq\Gamma$, let $\Gamma'_i = \{\varphi:(i,\varphi)\in\Gamma'\}$. If for each $A$-finite $\Gamma'\subseteq\Gamma$, there is some $i$ such that all sentences in $\Gamma'_i$ are true in $\mathcal{A}$, then there is some $i$ such that all sentences in $\Gamma_i$ are true in $\mathcal{A}$.
\[P:SigmaSat\]
A countable structure $\mathcal{A}$ is $\Sigma_A$-saturated iff it lives in a fattening of $A$, with no new ordinals.
Countable $\Sigma$-saturated models have the property of expandability.
\[Ressayre\]\[T:SigmaSat\]
Suppose $\mathcal{A}$ is a countable $\Sigma_A$-saturated $L$-structure. Let $L'\supseteq L$, and let $\Gamma$ be a set of $L'_A$-sentences, $\Sigma_1$ on $A$, s.t. the consequences of $\Gamma$, in the language $L$, are all true in $\mathcal{A}$. Then $\mathcal{A}$ has an expansion satisfying $\Gamma$. Moreover, we may take the expansion to be $\Sigma_A$-saturated.
Complexity of integer parts
---------------------------
In [@KL], the second and fourth authors studied the complexity of some parts of the construction of Mourgues and Ressayre. In particular, they proved the following result, which we shall use later.
For a countable real closed field $R$, there is a residue field section $k$ that is $\Pi^0_2(R)$.
.
Complexity of exponential integer parts {#S:compEIP}
=======================================
We now turn to our main new result. We show that there is a real closed exponential field with a residue field section and a well ordering, all arithmetical, such that Ressayre’s construction is not completed in $L_{\omega_1^{CK}}$.
Complexity of Ressayre’s construction {#SS:compR}
-------------------------------------
We turn to the complexity of Ressayre’s construction of an exponential integer part for a real closed exponential field. Let $R$ be a countable real closed exponential field. By Lemma \[L:canRconst\], given a fixed residue field section $k$ and a well ordering $\prec$ of the elements of $R$, then Ressayre’s construction of an exponential integer part is canonical. To fix notation, let $(R_0, G_0, \delta_0)$ be the development triple with $R_0=k$, $G_0=\{1\}$, and $\delta_0=id$. Let $y$ be the $\prec$-first element of $R-k$, adjusted so that $y$ is positive and infinite. We will focus in this section on the chain of development triples $(B_j,H_j,\delta_j)_{j<\zeta}$ leading to the first non-trivial maximal and dyadic triple $(R_1, G_1, \delta_1)$ with $y\in R_1$. The chain $(B_j,H_j,\phi_j)_{j<\zeta}$ is defined in Proposition \[main technical\] and each element extends $(R_0, G_0, \delta_0)$. We recall that this chain satisfies the following conditions:
1. $H_0={\langle {\{y_i = \log^i(y)\mid i\in\omega\}}\rangle_{\mathbb{Q}}}$,
2. $H_{j+1}={\langle {H_j\cup\{2^r\mid r\in B_j \ \&\ \ \phi_j(r)\in {k(( {H_j^{>1}})) }\}}\rangle_{\mathbb{Q}}}$,
3. for limit $j$, $H_j = \cup_{j' < j} H_{j'}$,
4. for all $j$, $B_j$ is maximal for $H_j$, obtained by applying Lemma \[Maximal\]
5. The length of the chain is the first limit ordinal $\zeta$ such that $\cup_{j<\zeta} B_j$ is maximal for $H_{\zeta} = \cup_{j<\zeta} H_j$.
Each step is sufficiently effective that the whole construction is constructible. However, we would like to know whether the entire construction can be completed in $L_{\omega_1^{CK}}$, the hyperarithmetical universe. There are two possible sources of complexity, of which the ordinals required for the construction play an important role.
1. An object constructed at some step in the chain of development triples needed to arrive at the first non-trivial maximal dyadic triple may not be hyperarithmetical. Such objects include the lengths of developments in the triples.
2. The length of the chain of development triples needed to arrive at the first non-trivial maximal dyadic triple may not be hyperarithmetical.
We produce an example of a hyperarithmetical real closed field for which Ressayre’s construction cannot be completed in $L_{\omega_1^{CK}}$. Let $\mathcal{C}$ be the chain of development triples leading to the first non-trivial maximal dyadic triple for our example. We show that even if all objects in the construction of $\mathcal{C}$, including the lengths of all developments, are hyperarithmetical, then the length of $\mathcal{C}$ is not hyperarithmetical.
\[offcharts\]
There is a low real closed exponential field $R$, with a $\Delta^0_3$ residue field section $k$ and a $\Delta^0_3$ ordering $<$ of type $\omega+\omega$, such that Ressayre’s construction, even of the first non-trivial maximal and dyadic triple $(R_1,G_1,\delta_1)$, is not completed in $L_{\omega_1^{CK}}$.
To prove Theorem \[offcharts\], we use recursive saturation together with Barwise Compactness. We begin by fixing the particular real closed exponential field $R$ cited in Theorem \[offcharts\].
There is a recursively saturated real closed exponential field $R$ such that $D^c(R)$ is low.
\[Proof of Lemma\]
By the low basis theorem, there is a low completion $K$ of $PA$. Let $\mathcal{S}$ be the Scott set consisting of sets representable with respect to $K$. There is an enumeration $E$ $\mathcal{S} = Rep(K)$ such that $E\leq_T K$. In $\mathcal{S}$, we find a completion $T$ of the set of axioms for real closed exponential fields. By Theorem \[MM\], of Macintyre and Marker [@MM], there is a recursively saturated model $R$ of $T$ such that $D^c(R)\leq_T E$. This is the real closed exponential field $R$ that we want.
We now choose the residue field section $k$ for $R$ cited in Theorem \[offcharts\]. We do not ask that it respect the exponential function.
There is a $\Delta^0_3$ residue field section $k$ for $R$.
There is a residue field section $k$ that is $\Pi^0_2(R)$. Since $R$ is low, $k$ is $\Delta^0_3$.
To prove Theorem \[offcharts\], we construct a $\Delta^0_3$ well ordering $\prec$ of $R$ so that Ressayre’s construction in Lemma \[L:canRconst\] either produces a non-hyperarithmetical object in $(B_j,H_j,\delta_j)_{j<\zeta}$ or so that this chain of development triples leading up to $(R_1, G_1, \delta_1)$ has length greater $\omega_1^{CK}$, i.e., $\zeta\ge \omega_1^{CK}$. We will apply Barwise Compactness (Theorem \[T:BarwiseCompact\]) to particular set of set of sentences $\Gamma$. Let $\Gamma$ consist of the following sentences.
1. An infinitary sentence $\psi$ characterizing the $\omega$-models of $KP$.
An $\omega$-model has the feature that each element of the definable element $\omega$ has only finitely many elements, a fact that we can express using a computable infinitary sentence. An $\omega$-model of $KP$ contains the hyperarithmetical sets. In particular, we have the real closed exponential field $R$ and the residue field section $k$, with the indices we have chosen for them.
2. A finitary sentence $\varphi_{\prec}$ saying of a new symbol $\prec$ that it is a $\Delta^0_3$ ordering of $R$ of order type $\omega+\omega$.
The sentence $\varphi_{\prec}$ states that there exists an element of $\omega$ that is a $\Delta^0_3$ index for $\prec$, and $(R, \prec)$ is isomorphic to the ordinal $\omega+\omega$. Necessarily, the isomorphism will be $\Delta^0_3$, so it is an element of any model of $KP$.
3. A sentence $\varphi_\alpha$, for each computable limit ordinal $\alpha$, saying that if for all $\beta<\alpha$, the triples $(B_j,H_j,\gamma_j)$ are in $L_\alpha$ for all $j \leq \beta$, then $B_\beta\not= \cup_{j < \beta} B_j$.
Note that we identify an element of $k((H_i))$ of length $<\alpha$ with a decreasing function from $\alpha$ to $k\times H_j$.
We must show that every hyperarithmetical subset of $\Gamma$ has a model. For a computable ordinal $\alpha$, let $\Gamma_\alpha$ consist of $\psi$, $\varphi_{\prec}$ and $\varphi_\beta$ for Each hyperarithmetical subset of $\Gamma$ is included in one of the sets $\Gamma_\alpha$. So, to show $\Gamma$ is consistent, we must show each $\Gamma_\alpha$ is consistent for all $\alpha<\omega_1^{CK}$. In other words, for each $\alpha<\omega_1^{CK}$, we must show that there is a $\Delta^0_3$ ordering $\prec_\alpha$ on $R$, of order type $\omega+\omega$, so that when Ressayre’s construction is run according to the well ordering $\prec_\alpha$, if all triples $(B_j, H_j, \gamma_j)$ are in $L_\alpha$ for $j<\alpha$, then the length of the chain of development triples $(B_j,H_j,\phi_j)$ leading to the first nontrivial maximal and dyadic triple is greater than $\alpha$.
Special elements
----------------
Let $\alpha$ be a computable ordinal, and we fix a path through $\mathcal{O}$. To show that $\Gamma_\alpha$ has a model, we use some special elements, which we name by constants $y$, $y_i$, $i\in\omega$, $c_\beta$, for $\beta<\alpha$ either $0$ or a limit ordinal, and $c_{\beta,i}$ for all $\beta < \alpha$ and all $i\in\omega$. We first state some properties that we would like for the constants. We define all of the constants in terms of $y$, $c_0$, and $c_\beta$ for limit $\beta<\alpha$. Assuming that $(B_j, H_j, \gamma_j)$ are in $L_\alpha$ for $j\le \beta$, we want $c_\beta\in B_\beta - \cup_{\gamma<\beta} B_\beta$. To assure this, we specify a development that we want for $c_\beta$, in terms of constants $c_{\gamma+1,i}$ for $\gamma<\beta$, which we want in $H_{\gamma+1} - H_\gamma$.
In order to obtain a model of $\Gamma_\alpha$ with these constants, we give a c.e. set of finitary axioms, partially describing our constants. Since $R$ is recursively saturated, we then apply Theorem \[Barwise-Schlipf\] of Barwise and Schlipf to get an expansion $R_\alpha$ satisfying these finitary axioms. Finally, we define a $\Delta_3^0$ well ordering $\prec_\alpha$ so that when Ressayre’s construction is run using $R_\alpha, k$, and $\prec_\alpha$ the constants have all of the desired properties. Hence, $R_\alpha$, $k$, $\prec_\alpha$ and $L_{\omega_1^{CK}}$ witness the truth of $\Gamma_\alpha$.
### Descriptions of the constants
Set $y$ to be a positive and infinite element of $R$, and set $y_i = \log^i(y)$, so that all the $y_i$ are positive and infinite and satisfy $$y_0 > y_1 > y_2 > y_3 \ldots$$ We will define the well ordering $\prec_\alpha$ so that $y_i\in H_0$ for all $i\in\omega$ when Ressayre’s construction is run according to $\prec_\alpha$. Furthermore, we will define $\prec_\alpha$ and ensure the constants satisfy certain properties so that they will be assigned particular developments.
We want $c_0$ to have the development $\sum_{1\le i<\omega} y_i$, which is a development in ${k(( {H_0^{>1}})) }$ if $y_i\in H_0$ for all $i<\omega$. The description of $c_0 = c_{0,1}$ is
$y_1 < c_0 < 2y_1$, $y_2 < c_0 - y_1 < 2y_2$, etc.\
and\
$c_{0, i}=c_0-\sum_{j=1}^{i-1} y_j$.
We defined $c_{0,i}$ so that if $c_0$ is assigned the development $\sum_{1\le i<\omega} y_i$, then $c_{0,i}$ will have the development $\sum_{i \le j<\omega} y_j$.
Let $\gamma\le \alpha$ be a successor ordinal with $\gamma=\beta+1$. We define $c_{\gamma, j}$ to be $c_{\beta+1,j}=2^{c_{\beta,j+1}}$ for $0<j<\omega$.
Let $\gamma\le\alpha$ be a limit ordinal where the notation for $\gamma$ in our fixed path through $\mathcal{O}$ gives a sequence of successor ordinals $(\gamma_i)_{i\in\omega}$ converging to $\gamma$. The description of $c_{\gamma}=c_{\gamma, 1}$ is
$c_{\gamma_1,1} < c_\gamma < 2c_{\gamma_1,1}$, $c_{\gamma_2,2} < c_\gamma - c_{\gamma_1,1} < 2c_{\gamma_2,2}$, etc., and\
$c_{\gamma, i}=c_\gamma-\sum_{j=1}^{i-1} c_{\gamma_j, j}$.
This completes our description of any element $c_{\gamma, i}$ for $\gamma\le\alpha<\omega_1^{CK}$ and $0<i\in\omega$.
Suppose $\gamma\le\alpha$ is a limit ordinal. We want $c_{\gamma, i}$ to be assigned the development $\sum_{ i\le j<\omega} c_{\gamma_j,j}$. In order for $\sum_{ i\le j<\omega} c_{\gamma_j,j}$ to be a development under Ressayre’s construction with $\prec_\alpha$, we will need to ensure that each element $c_{\gamma_j,j}$ is a member of the value group section $G_1$ and that for all nonzero $j\in\omega$. The next lemma shows that the latter condition holds. Later, we will choose the well ordering $\prec_\alpha$ on $R$ carefully to ensure that the former condition holds as well.
The descriptions of the constants $c_{\beta, i}$ for $\beta\le\alpha$ and $i\in\omega$ imply that for all $\beta\le\alpha$ $$\label{E:y>rbeta}
y_0>c_{\beta,1}>y_1>c_{\beta,2}>y_2>c_{\beta,3}>y_3>c_{\beta,4}> \ldots$$
From the description of $c_{0, i}$, we can see that (\[E:y>rbeta\]) holds for $\beta=0$. Let $\gamma\le \alpha$ be a successor ordinal with $\gamma=\beta+1$. We inductively assume that the descriptions for the elements $c_{\beta, k}$ imply the ordering in (\[E:y>rbeta\]). By applying a power of $2$ to the inequalities in (\[E:y>rbeta\]) and the definition of $c_{\gamma, i}$, we obtain the ordering $$y_0>c_{\gamma,1}>y_1>c_{\gamma,2}>y_2>c_{\gamma,3}>y_3>c_{\gamma,4}> \ldots$$
Let $\gamma\le\alpha$ be a limit ordinal where the notation for $\gamma$ in our fixed path through $\mathcal{O}$ gives a sequence of successor ordinals $(\gamma_i)_{i\in\omega}$ converging to $\gamma$. Moreover, by induction, we have that the descriptions of the $(c_{\gamma_i, i})_{i\in\omega}$ imply that
$$y_0>c_{\gamma_1,1}>y_1>c_{\gamma_2,2}>y_2>c_{\gamma_3,3}>y_3>c_{\gamma_4,4}> \ldots$$
By the description of $c_{\gamma, i}$, we have that $$y_0>c_{\gamma, 1}>c_{\gamma_1,1}>y_1>c_{\gamma, 2}>c_{\gamma_2,2}>y_2>c_{\gamma, 3}>c_{\gamma_3,3}>y_3> \ldots,$$ completing the induction.
For the given computable ordinal $\alpha$, we may take the set of finitary sentences describing the constants to be computably enumerable. Since $R$ is recursively saturated, we get an expansion $R_\alpha$ of $R$ with special elements $c_{\beta, i}\in R$ satisfying the appropriate sentences for $\beta\le\alpha$ and $i\in\omega$ by Theorem \[Barwise-Schlipf\]. We may take $R_\alpha$ to be $\Delta_2^0$ since $R$ is low and the oracle $\Delta^0_2$ can determine whether an element satisfies a given description of some $c_{\beta}$.
The ordering
------------
We now describe a $\Delta^0_3$ well ordering $\prec_\alpha$ of $R$ such that when Ressayre’s construction is run on $R$, $k$, and $<_\alpha$, if $(B_j, H_j, \gamma_j)$ are in $L_\alpha$ for $j<\alpha$, then the development chain leading to the first nontrivial maximal and dyadic triple has length greater than $\alpha$. Moreover, we construct $\prec_\alpha$ so that $(R, \prec_\alpha)$ has order type $\omega+\omega$. We set $y$ to be the $\prec_\alpha$-least element of $R$. The special elements $c_\beta$ for $\beta\le \alpha$ will make up the remainder of the initial segment of type $\omega$, and the other elements will make up the remaining segment of type $\omega$. Two elements in the same $\omega$ segment are ordered according to the standard type $\omega$ ordering on their codes. Since $R_\alpha$ is $\Delta_2^0$, we can use $\Delta_3^0$ to determine, for a given $r\in R_{\alpha}$, whether there exists some $\beta\le\alpha$ such that $r=c_\beta$, i.e., whether $r$ should be placed in the intial $\omega$ segment or the latter. Hence, $\prec_\alpha$ is $\Delta_3^0$.
We now run Ressayre’s construction on $R$, $k$, and $\prec_\alpha$ as described. For the remainder of the section, we let $(B_i, H_i, \phi_i)_{i<\zeta}$ be the chain of development triples in this construction leading up to the first maximal and nontrivial dyadic triple $(R_1, G_1, \delta_1)$. We want to show that $\zeta>\alpha$.
### Lemmas about the constants
We want to show that the constants $c_\beta$ get the developments we want for them. The following lemmas are useful.
For all $\beta\leq\alpha$, for all $h\in H_\beta^{>1}$, there is some $i$ such that $h > y_i$.
We proceed by induction on $\beta$. Since $y$ is the $\prec_\alpha$-least element of $R$, $H_0$ equals ${\langle {y_i\mid i<\omega}\rangle_{\mathbb{Q}}}$. So, $h\in H_0$ is a finite product of rational powers of the $y_i$. Let $i$ be least such that there is a factor $y_i^{q_i}$. Since $h\in H_0^{>1}$, $q_i$ must be positive. Then $h > y_{i+1}$. Suppose the statement holds for $\beta$, and $h \in H_{\beta+1}^{>1}$. By construction, we may assume that $h = 2^r$, where $\phi_\beta(r)\in {k(( {H_\beta^{>1}})) }$ has a positive initial coefficient. Say $w(r) = h' > y_i$. Then $h > 2^{y_i} > y_i$. Finally, suppose the statement holds for $\gamma < \beta$, where $\beta$ is a limit ordinal. Since $H_\beta = \cup_{\gamma<\beta} H_\gamma$, the statement holds for $H_\beta$.
\[L:logh\] For all $\beta\le\alpha$, if $h\in H_\beta$ and $h\not=1$, then there is some $\gamma\le \alpha$ with $\gamma=0$ or $\gamma<\beta$ such that $\log h\in B_\gamma$ and $\delta_\gamma(\log h)\in {k(( {H_\gamma^{>1}})) }$.
We prove the lemma by induction on $\beta\le\alpha$. If $h\in H_0$, then $h=\prod_{i=0}^ny_{l_i}^{q_i}$ with all $q_i\in\mathbb{Q}$ nonzero and ${l_{i}}<{l_{i+1}}$ for $0\le i<n$. Then $\log h=\sum_{i=0}^n\ {q_i}\log y_{l_i}$. Since $\log y_{l_i}=y_{l_i+1}$, $\log h\in B_0$ and $\delta_0(\log h) \in {k(( {H_\gamma^{>1}})) }$.
Suppose the statement holds for all $\lambda<\beta$. If $\beta$ is a limit ordinal, $h\in H_\beta$ implies $h\in H_\lambda$ for some successor ordinal $\lambda<\beta$, so the statement holds by induction. Suppose $\beta=\lambda+1$ and $h\in H_\beta-H_\lambda$. By construction, $h=h'{\prod_{i=0}^{n}} {2^{t_i}}$ where $h'\in H_\lambda$ and $t_i\in B_\lambda$ and $\delta_\lambda(t_i)\in {k(( {H_\lambda^{>1}})) }$ for all $1\le i\le n$. Then, $\log h= \log h'+\sum_{i=0}^n t_i$ has the desired features by induction.
To show that the elements $c_{\beta,i}$ get the developments we want for them, we must show that other elements cannot compete for these developments.
\[L:Ind\] Suppose Ressayre’s construction is run on a well ordering $\prec$ such that $y$ is the first element and the elements $c_{\beta}$ for $\beta\le \alpha$ form the initial $\omega$ segment. For all $ \beta, \gamma\le \alpha$, the following statements hold.
1. \[lv\] If $\gamma$ is a limit ordinal greater than $\beta$, then $c_{\gamma,i}$ has no development over $H_\beta$.
2. \[sv\] If $\gamma$ is a successor ordinal greater than $\beta$, then $c_{\gamma,i}$ has no valuation in $H_\beta$.
3. \[ld\] If $\beta$ is $0$ or a limit ordinal, then $c_{\beta,i}\in B_\beta$ and $c_{\beta,i}$ has the development $\sum_{ i\le j<\omega} c_{\beta_j,j}\in {k(( {H_\beta^{>1}})) }$ where the sequence $(\beta_i)_{i\in\omega}$ is defined as follows. If $\beta=0$, then $\beta_i=i$ for all $i\in\omega$. If $\beta$ is a limit ordinal, $(\beta_i)_{i\in\omega}$ is the sequence of successor ordinals converging to $\beta$ given by the notation for $\beta$ in the fixed path through $\mathcal{O}$.
4. \[sd\] If $\beta$ is a successor ordinal, then $c_{\beta,i}$ is in $H_\beta^{>1}$.
We begin with the case where $\beta=0$. Clearly, $c_0$ will be assigned the development $\sum_{1\le i<\omega} y_i$ if it is the first element after $y$ in $\prec$. However, $c_0$ may not be the first such element; there may be finitely many other $c_\beta$ before $c_0$. Statements \[lv\] and \[sv\] imply that these finitely many $c_\beta$ would not interfere with assigning the development $\sum_{ 1\le i<\omega} y_i$ to $c_0$. Hence, Statements \[lv\] and \[sv\] give Statement \[ld\].
We begin by showing for all $\gamma>0$ and all $i\in\omega$ that $c_{\gamma,i}$ has no valuation in $H_0$. Suppose otherwise, and let $\gamma$ be the first ordinal witnessing the failure. If $\gamma$ is a limit ordinal, then the valuation of $c_{\gamma,i}$ is the same as that of $c_{\gamma_i,i}$, where $\gamma_i$ is a smaller successor ordinal. So, we may suppose that $\gamma = \lambda+1$ for some $\lambda$. The element $c_{\lambda+1,i}$ was defined to be $2^{c_{\lambda,i+1}}$. Since $c_{\lambda+1,i}$ has a valuation in $H_0$, we have that $c_{\lambda+1,i}$ equals $cy_0^{q_0}y_1^{q_1}\cdots y_n^{q_n}$, where $c$ is finite and the $q_i\in\mathbb{Q}$. Then, taking logs, we have $c_{\lambda,i+1} = \log(c) + q_0y_1 + q_1y_2 + \cdots + q_ny_{n+1}$. We see that $c_{\lambda,i+1}$ has valuation equal to some $y_i$, which is in $H_0$, since both $c_{\lambda+1,i}$ and $c_{\lambda,i+1}$ are infinite. We must have $\lambda = 0$ and $\gamma = 1$, since otherwise we have reached a contradiction. If we use a different ordering, putting $c_0$ first after $y$, then $c_0$ would be assigned the desired development. Then, $c_{0,i+1}$ would be in $B_0$ and $c_{1,i} = 2^{c_{0,i+1}}$ would be in $H_1 - H_0$ by Ressayre’s construction. Therefore, $c_{1,i}$ has no valuation in $H_0$. So, Statements \[lv\], \[sv\],\[ld\], and \[sd\] hold when $\beta=0$.
Suppose $\beta\le\alpha$ is arbitrary and that the statements in Lemma \[L:Ind\] hold for all $\lambda<\beta$ and all $\gamma\le\alpha$.
We begin by proving Statement \[sd\]. If $\beta=\lambda+1$ is a successor ordinal, we have that $c_{\beta, i}=2^{c_{\lambda, i+1}}$. If $\lambda$ is itself a successor ordinal, we have that ${c_{\lambda, i+1}}\in H_\lambda^{>1}$ by Statement \[sd\] for $\lambda$ of the induction hypothesis. Since ${c_{\lambda, i+1}}\in H_\lambda^{>1}$, $c_{\beta, i}=2^{c_{\lambda, i+1}}\in H_\beta^{>1}$ by construction. If $\lambda$ is a limit ordinal, we have that $c_{\lambda,i+1}\in B_\lambda$ is assigned a development in ${k(( {H_\lambda^{>1}})) }$ by Statement \[ld\] for $\lambda$ of the induction hypothesis. Again, by construction, $c_{\beta, i}=2^{c_{\lambda, i+1}}\in H_\beta^{>1}$ as desired.
We now show that Statements \[lv\] and \[sv\] hold for $\beta$ and all $\gamma\le \alpha$ by induction on $\gamma$. Given some ordinal $\gamma'$, additionally suppose that Statements \[lv\] and \[sv\] hold for all $\gamma<\gamma'$ with respect to $\beta$. First, suppose $\gamma'=\gamma+1$ is a successor ordinal, and suppose for a contradiction that $c_{\gamma', i}=2^{c_{\gamma, i+1}}$ has a valuation in $H_\beta$. By construction of $H_\beta$, we have that $$c_{\gamma', i}=2^{c_{\gamma, i+1}}=ch2^{b_1}\cdots2^{b_n}$$ where $c$ is finite, $h\in H_{\lambda}$, and $b_j\in B_\lambda$ so that $\phi_\lambda(b_j)\in {k(( {H_\lambda^{>1}})) }$ for some ordinal $\lambda<\beta$. Taking logs of both sides, we have that $$c_{\gamma, i+1}=\log(c) + \log h+b_1+\ldots+b_n.$$ By Lemma \[L:logh\], $\log h\in B_\lambda$ and $\phi_\lambda(\log h)\in {k(( {H_{\lambda}^{>1}})) }$. Thus, $v(c_{\gamma, i+1})\in H_\lambda$. If $\gamma$ is a successor ordinal, this would contradict Statement \[sv\] of the inductive hypothesis with respect to $\lambda$. So, suppose $\gamma$ is a limit ordinal. Consider the sequence $({\gamma_j})_{j\in\omega}$ of successor ordinals given by the notation for $\gamma$ such that $\lim_{j\rightarrow\infty}\gamma_j=\gamma$. Let $l$ be the least natural number such that $\gamma_l>\lambda$. We have $c_{\gamma_j, j}\in H_\lambda^{>1}$ for $j<l$ by Statement \[sd\] of the inductive hypothesis. Thus, $$c_{\gamma, i+1}-\sum_{i+1\le j<l}c_{\gamma_j, j}=\log(c) + \log h+b_1+\ldots+b_n-\sum_{i+1\le j<l}c_{\gamma_j, j}$$ The left hand side of the equation has the same valuation as $c_{\gamma_l, l}$ by definition of $c_{\gamma, i+1}$. The right hand side of the equation consists of elements whose developments are in ${k(( {H_\lambda^{>1}})) }$ and the finite element $\log c$. Thus, $v(c_{\gamma_l, l})\in H_\lambda^{>1}$, contradicting Statement \[sv\] of the inductive hypothesis applied to $\lambda$. This completes the case where $\gamma'$ is a succesor ordinal.
Second, suppose that $\gamma'$ is a limit ordinal, and suppose for a contradiction that $c_{\gamma', i}$ has a development over $H_\beta$. Consider the sequence $({\gamma'_j})_{j\in\omega}$ of successor ordinals given by the notation for $\gamma'$ such that $\lim_{j\rightarrow\infty}\gamma'_j=\gamma'$. Let $l$ be the least natural number such that $\gamma'_l>\beta$. By Statement \[sd\] of the induction hypothesis, we have that $c_{\gamma'_j, j}\in H_\beta^{>1}$ for all $\gamma'_j<\beta$. If $\beta=\gamma'_j$, we also have $c_{\gamma'_j, j}\in H_\beta^{>1}$ by the proof above of Statement \[sd\] for $\beta$. Since $c_{\gamma', i}$ has a development over $H_\beta$, the difference $c_{\gamma', i}-\sum_{i\le j<l}c_{\gamma'_j, j}$ does as well. Thus, $c_{\gamma', i}-\sum_{i\le j<l}c_{\gamma'_j, j}$ has a valuation in $H_\beta$. Since $c_{\gamma', i}-\sum_{i\le j<l}c_{\gamma'_j, j}$ has the same valuation as $c_{\gamma'_l, l}$ by definition, $v(c_{\gamma'_l, l})\in H_\beta$. Since $\gamma'_l>\beta$ is a successor ordinal less than $\gamma'$, this contradicts Statement \[sv\] of the induction hypothesis with respect to $\beta$. This completes our induction on $\gamma'$. We have proved Statements \[lv\] and \[sv\] for $\beta$ and all $\gamma\le\alpha$.
We finally prove Statement \[ld\] for $\beta$. Suppose $\beta$ is a limit ordinal. By Statements \[ld\] and \[sd\] of the induction hypothesis, we have that all $c_{\lambda, i}$ for $\lambda<\beta$ receive their desired developments in $B_\lambda$. In particular, $c_{\beta_j, j}\in H_\beta^{>1}$ for all $j\in\omega$. We have that no element $c_{\gamma,k}$ for $\gamma>\beta$ has a development over $H_\beta$ by Statements \[lv\] and \[sv\] for $\beta$. Since the elements $c_{\gamma}$ are the only elements that could come before $c_\beta$ in the initial $\omega$ segment of the well ordering $\prec$, the element $c_{\beta, i}$ will enter $B_\beta$ and $\phi_\beta(c_{\beta, i})=\sum_{i\le j<\omega} c_{\beta_j, j}\in {k(( {H_\beta^{>1}})) }$. Thus, Statement \[ld\] holds for $\beta$. This completes the proof of Lemma \[L:Ind\].
We now show that $\Gamma_\alpha$ is consistent. The formulas $\psi$ and $\varphi_\prec$ are satisfied by $R, k$, and $\prec_\alpha$ by construction. We now show $\varphi_\lambda$ holds for each limit ordinal $\lambda< \alpha$. If
If $(B_j, H_j, \gamma_j)$ is not in $L_\lambda$ for some $j<\lambda$, then $\varphi_\lambda$ holds trivially. Otherwise, by Lemma \[L:Ind\] Statements \[lv\] and \[sv\], there is an element of namely $c_j=c_{j, 1}$, so $\varphi_\lambda$ is satisfied. Thus, $\Gamma_\alpha$ is consistent.
We are in a position to apply Barwise Compactness. By Theorem \[T:BarwiseCompact\], we obtain an $\omega$-model of $KP$ with $R$ and $k$ as elements, and a $\Delta^0_3$ ordering $\prec$ of type $\omega+\omega$ such that if Ressayre’s construction is run on $R$, $k$, and $\prec$, producing a chain of development triples $(B_i, H_i, \phi_i)_{i<\zeta}$ leading to the first non-trivial maximal and dyadic triple $(R_1, G_1, \delta_1)$, then either some triple $(B_j, H_j, \gamma_j)$ for $j<\zeta$ is not in $L_{\omega_1^{CK}}$, or else the length of the chain $\zeta$ is noncomputable. This completes the proof of Theorem \[offcharts\]. Hence, Ressayre’s construction on $R$, $k$, and $\prec$ cannot be completed in $L_{\omega_1^{CK}}$.
Although Ressayre’s construction may not be carried out in $L_{\omega_1^{CK}}$, we can use $\Sigma$-saturation obtain an exponential integer part in a fattening of $L_{\omega_1^{CK}}$.
\[fatteningEIP\]
Let $R$ be a hyperarithmetical real closed exponential field. There is an exponential integer part $Z$ such that $(R,Z)$ lives in a fattening of $L_{\omega_1^{CK}}$.
Since $R$ is hyperarithmetical, it is trivially $\Sigma_A$-saturated. Let $\Gamma$ be the natural set of sentences saying that $Z$ is an exponential integer part. By Theorem B, $R$ has an exponential integer part, so the consequences of $\Gamma$ are true in $R$. Therefore, by Theorem \[T:SigmaSat\], there is an exponential integer part $Z$ such that $(R,Z)$ is $\Sigma_A$-saturated. This means that $(R,Z)$ lives in a fattening of $L_{\omega_1^{CK}}$, with no non-computable ordinals by Proposition \[P:SigmaSat\].
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---
abstract: |
We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if $\lambda_1=\lambda_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $\lambda_1\neq \lambda_2$, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.
*Keywords:* Random graphs, configuration model, first passage percolation, competing growth, coexistence, continuous-time branching process.
MSC 2010 classification: 60K35, 05C80, 90B15.
author:
- 'Daniel Ahlberg[^1]'
- 'Maria Deijfen[^2]'
- 'Svante Janson[^3]'
date: 8 November 2017
title: 'Competing first passage percolation on random graphs with finite variance degrees[^4]'
---
plus1pt minus1pt
Introduction
============
Fix $n\geq 1$ and let $(d_1,\ldots,d_n)$ be a sequence of positive integers that may depend on $n$. Consider a graph with $n$ vertices and degrees $(d_1,\ldots,d_n)$ generated by the configuration model, that is, equip each vertex $i\in\{1,\ldots,n\}$ with $d_i$ half-edges, and pair half-edges uniformly at random to create edges. For all half-edges to find a partner we assume that the total degree $\sum d_i$ is even. Assign independently to each edge $e$ in the resulting graph two independent exponentially distributed passage times $X_1(e)$ and $X_2(e)$ with parameter $\lambda_1$ and $\lambda_2$, respectively. At time 0, two uniformly chosen vertices are infected with infections type 1 and type 2, respectively, and the infections then spread via nearest neighbors: When a vertex becomes type 1 (2) infected, the time that it takes for the infection to traverse an edge $e$ emanating from the vertex is given by $X_1(e)$ ($X_2(e)$). If the other end point of the edge $e$ is still uninfected at that time, it becomes type 1 (2) infected and remains so forever. It also becomes immune to the other infection type.
In this paper we study the above competing growth process on a random graph generated from a given degree sequence subject to the regularity conditions stated below. These conditions ensure that the graph contains a giant component occupying all but a vanishing fraction of the vertices as $n\to\infty$, and hence that almost all vertices will [w.h.p[.=1000]{}]{} be infected when the process terminates. The question that we will be interested in is the outcome of this competition. Specifically, will both types occupy a strictly positive fraction of the vertices in the limit as $n\to\infty$? We show that the answer is yes if and only if $\lambda_1=\lambda_2$. This question has previously been studied for the configuration model with constant degrees [@regular] and infinite variance degrees [@winner]; see the end of this section for a summary of earlier work.
Given a degree sequence $(d_1^{{\scriptscriptstyle}(n)},d_2^{{\scriptscriptstyle}(n)},\ldots,d_n^{{\scriptscriptstyle}(n)})$ with $\sum d_i^{{\scriptscriptstyle}(n)}$ even, write $D_n$ for the degree of a vertex chosen uniformly at random, so that $${\operatorname{\mathbb P{}}}(D_n=k)=\#\{i:d_i=k\}/n.$$ Our assumptions on the (sequence of) degree sequences are the following:
- $(D_n)_{n\ge1}$ converges in distribution to a random variable $D$ with ${{\mathbb E}}[D^2]<\infty$, and $${{\mathbb E}}[D_n^2]\to{{\mathbb E}}[D^2];$$
- $d_i\geq 2$ for all $i$, and ${\operatorname{\mathbb P{}}}(D>2)>0$.
Assumption (A1) could equivalently be formulated as the sequence of empirical distributions being uniformly square integrable and converging to a probability distribution $(p_d)_{d\in\mathbb{N}}$ on the positive integers. One standard example in which (A1) is satisfied is when $(d_1, d_2,\ldots, d_n)$ are independent realizations of a random variable $D$ with finite variance. By increasing a randomly chosen degree by 1, if necessary, we can make sure that the total degree is even. If we condition on the sequence $(D_i)_{i=1}^n$ and assume that ${{\mathbb E}}[D^2]<\infty$, ${\operatorname{\mathbb P{}}}(D\ge2)=1$ and ${\operatorname{\mathbb P{}}}(D>2)>0$, then (A1) and (A2) hold [w.h.p[.=1000]{}]{} and thus our results, as stated below, apply.
A graph generated by the configuration model may contain self-loops and multiple edges, but the assumption (A1) implies that the probability of obtaining a simple graph is bounded away from 0 as $n\to\infty$; see [@AngHofHol16; @simpleI; @simpleII]. Furthermore, it is well-known that conditioning on the resulting graph being simple yields a uniform sample among simple graphs with the specified degree sequence; see [@Remco_book Chapter 7]. Hence our results apply also for such a uniformly chosen simple graph.
Let $D^*$ be a size biased version of $D$, that is, ${\operatorname{\mathbb P{}}}(D^*=d)=d{\operatorname{\mathbb P{}}}(D=d)/{{\mathbb E}}[D]$. The threshold for the occurrence of a (unique) giant component in the graph is given by ${{\mathbb E}}[D^*-1]=1$; see [@SvanteMalwina; @MR-95]. This can be seen by exploring the components in the graph via nearest neighbors, starting from a uniformly chosen vertex. As $n\to\infty$, this exploration can be approximated by a branching process and, by construction of the graph, the offspring distribution of explored vertices in the second and later generations is given by $D^*-1$. The relative size of the giant component is given by the survival probability in the approximating branching process; see [@SvanteMalwina; @MR-98]. Condition (A2) above implies that the survival probability is 1, so that the asymptotic fraction of vertices in the giant component is 1.
Now consider the competition process described above. Write $N_i(n)$ for the total number of type $i$ infected vertices when the process terminates, and $\bar{N}_i(n)=N_i(n)/n$ for the corresponding fraction. Note that, since the giant component spans all but a vanishing fraction of the vertices, we have that $\bar{N}_1(n)+\bar{N}_2(n){\overset{\mathrm{p}}{{\longrightarrow}}}1$, and it is therefore enough to consider $\bar{N}_1(n)$. Furthermore, by symmetry, we may assume that $\lambda_1\leq \lambda_2$. The following is our main result.
\[th:main\] Assume that the degree sequence satisfies (A1) and (A2).
- If $\lambda_1=\lambda_2$, then $\bar{N}_1(n){\overset{\mathrm{d}}{{\longrightarrow}}}V$, where $V$ is a continuous random variable with a strictly positive density on $(0,1)$.
- If $\lambda_1<\lambda_2$, then $\bar{N}_1(n){\overset{\mathrm{d}}{{\longrightarrow}}}0$.
Starting with two given infected vertices, e.g. vertices 1 and 2, or several infected vertices of each type (fixed in number as $n\to\infty$) gives the same results, except that the distribution of the limiting fraction $V$ will depend on the degrees of the initially infected vertices. Moreover, the theorem extends to a fixed number of competing types larger than two, in which case all types of maximal strength each conquer a positive fraction of the vertices.
The assumption $d_i\geq 2$ ensures that the giant component comprises almost all vertices. Weakening this condition to ${{\mathbb E}}[D^*-1]>1$ gives a graph where the giant component may contain a smaller fraction of the vertices. The competition process can be analyzed also on such a graph and the non-trivial case then arises when both initial vertices belong to the giant component. We believe that our methods apply also in this case, but it would require dealing with a conditioning on both initial vertices being in the giant component. Establishing a version of Theorem \[th:main\] in that case would make it applicable also for e.g. the Erdős–Renyi graph and the generalized random graph analyzed in [@BDM-L]. These models give simple graphs with random degrees and, conditionally on the degrees, the graph is uniform on the set of all simple graphs with those given degrees.
**Outline of the proof**
In the proof below we establish that there is an initial phase where the outcome of the competition is determined, followed by a phase that lasts until close to the end, and where the fractions of two types are essentially constant. An important tool in the proof is a standard technique for exploring the graph and the evolution of the infections simultaneously. A vertex is detected when it is reached by the infection and the half-edges attached to the vertex are then declared active, of type 1 or 2 depending on the type of the vertex. A half-edge remains active until it is opened for infection. A partner half-edge is then chosen and, if the vertex of this half-edge is still uninfected at that time, this leads to infection transfer and activation of new half-edges. The process can be defined in continuous time or in discrete steps by observing it only at the time points when an edge is opened; see Section 2 for a more detailed description. Write $S^{{\scriptscriptstyle}(i)}_k$ for the number of active type $i$ half-edges after $k$ steps in this process and $S_k=S^{{\scriptscriptstyle}(1)}_k+S^{{\scriptscriptstyle}(2)}_k$ for the total number active half-edges. Define $M_k$ to be the fraction of active type 1 half-edges among all active half-edges, more precisely defined by $$\label{eq:Mk}
M_k:=\begin{cases}
\frac{S^{{\scriptscriptstyle}(1)}_k}{S_k} & \text{if } S_k>0;\\
M_{k-1} & \text{if }S_k=0.
\end{cases}$$
In a key step we show that, if $\lambda_1=\lambda_2$, then $M_k$ is a martingale. We then give an estimate of its quadratic variation which implies that $M_k$ is essentially constant for $k\geq \nu_n$ for any sequence of integers $\nu_n\to\infty$. The probability that a newly infected vertex is infected by type 1 is hence roughly constant for $k\geq \nu_n$ and equal to $M_{\nu_n}$. The initial stages of the competition, on the other hand, can be approximated by a branching process and asymptotic results on branching processes imply that $M_{\nu_n}$ converges to a continuous random variable $V\in(0,1)$ if $\lambda_1=\lambda_2$, and to 0 if $\lambda_1<\lambda_2$. This yields Theorem \[th:main\](a). The proof of Theorem \[th:main\](b) is completed by letting the weaker type 1 infection spread with the same larger intensity $\lambda_2$ as the type 2 infection for $k\geq\nu_n$. The fraction of type 1 vertices among infected vertices for $k\geq \nu_n$ in such a process is close to 0 by the above results, and the type 1 infection clearly captures even fewer vertices in the original process.
The rest of the paper is organized so that the exploration process is described in more detail in Section 2, along with the initial branching process approximation. The results on $M_k$, specifying the evolution of the infections during the main phase, are then given in Section 3. Theorem \[th:main\] is proved in Section 4. Finally some directions for future work are described in Section 5.
**Previous work**
Competition on the configuration model has previously been studied in the case when the degree distribution follows a power-law with exponent $\tau\in(2,3)$, that is, when the mean degree is finite, but the variance infinite. In that case one of the types occupies all but a finite number of vertices as $n\to\infty$, and both types have a positive probability of winning, regardless of the values of the intensities; see [@winner]. The process has also been studied on random regular graphs generated by the configuration model with constant degree; see [@regular]. Our results generalize the results in [@regular] when the competition starts from fixed initial sets. However, the results in [@regular] also cover the case with growing initial sets, and give precise quantifications of the asymptotic number of vertices of each type.
In the present work, as well as in [@regular; @winner], the passage times are assumed to be exponential. The model can of course be defined analogously for passage times with arbitrary distributions. It has been analyzed in [@fixspeedI; @fixspeedII] for configuration graphs with power-law exponent $\tau\in(2,3)$ and constant passage times, so that all randomness comes from the underlying graph. When the types have different speed, the faster type occupies all but a vanishing fraction of the vertices, while when the speeds are the same, the types may or may not occupy positive fractions depending on the specific choice of the two initial vertices. A slightly different competition process with constant passage times is analyzed in [@Cooper], and the present competition process is analyzed on preferential attachment graphs in [@prefatt].
Finally, we mention that competing first passage percolation with exponential passage times has previously been studied on $\mathbb{Z}^d$. In that setting coexistence may occur for equal strength competitors, whereas the case of unequal strength remains to be fully resolved; see [@2tRich] for a survey and references.
The initial phase
=================
In this section we first define the exploration of the graph and the flow of infection in more detail. We then describe a branching process approximation of the number of active half-edges of the two types in the early stages of the growth. This leads to a characterization of the limiting behavior of a continuous time version of $M_k$ (defined in ) at the end of the initial phase.
**The exploration process**
To describe the exploration process, fix $\lambda_1,\lambda_2>0$, possibly different. At time 0 we start with the vertices and the attached half-edges. The pairing of the half-edges however is hidden and is revealed during the process. Each half-edge is throughout the process classified as either *free* or *paired*, and a free half-edge is in turn labeled as *active* of either type 1 or 2, or *inactive*. The initial set of active type $i$ half-edges consists of the half-edges attached to the uniformly chosen initial type $i$ vertex, while all other half-edges are inactive. Since the initially infected vertices are chosen randomly, the initial numbers $a_1$ and $a_2$ of active type 1 and type 2 half-edges, respectively, are random. However, we condition on them in the sequel, and hence assume that they are given numbers. The sets of half-edges are now updated inductively in continuous time as follows, with ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ denoting the number of active half-edges of type $i$ at time $t$.
1. \[sj1\] Each active half-edge of type $i=1,2$ infects with intensity ${\lambda}_i$, that is, it is equipped with an exponential clock with intensity ${\lambda}_i$, and infects when the clock rings. When a half-edge $q$ infects, it picks a partner $r$ uniformly at random from all free half-edges distinct from $q$. Let $x$ and $y$ be the vertices that $q$ and $r$, respectively, are attached to. Then $q$ and $r$ go from free to paired and form an edge $xy$.
2. \[sjold\] If $y$ is already infected, nothing more happens. In this case, $r$ was also active (of the same type as $q$ or not), and the number of active half-edges decreases by 2.
3. \[sjnew\] If $y$ is not infected, it becomes infected by the same type as $x$, and all remaining half-edges at $y$ become active of this type. This means that, if $q$ has type $i$ and $y$ has degree $d_y$, then the number ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ of active half-edges of type $i$ increases by $d_y-2$, while the number of active half-edges of the other type does not change.
A discrete version of the process is obtained by observing the continuous time process at the times half-edges are paired. In each step $k$ of the discrete time process, an active half-edge $q$ is chosen at random, with probability proportional to ${\lambda}_i$ where $i$ is its type. The chosen half-edge infects as in \[sj1\]–\[sjnew\] above. In both cases, if there are no remaining active half-edges, the infections have stopped, but it still remains to complete the graph. We then join any two uniformly chosen half-edges, that is, we choose a uniform matching of the remaining half-edges. The number of active type $i$ half-edges after $k$ steps is denoted by ${S^{{\scriptscriptstyle}(i)}}_k$. Throughout, quantities related to discrete time processes will be denoted by standard roman letter, while quantities related to processes in continuous time will be denoted by calligraphy letter. For instance ${\mathcal M}_t$ denotes the continuous time version of $M_k$, defined in , that is, ${\mathcal M}_t={\mathcal{S}^{{\scriptscriptstyle}(1)}}_t/({\mathcal{S}^{{\scriptscriptstyle}(1)}}_t+{\mathcal{S}^{{\scriptscriptstyle}(2)}}_t)$.
**Branching process approximation**
We now describe how the early evolution of ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ ($i=1,2$) can be coupled with two independent branching processes. Stronger results in this direction have been obtained in [@one_fpp]. However, we only need the coupling up to some time $t_n\to\infty$ (without further restrictions on $t_n$). This is fairly easy to establish and we therefore describe it here.
Our aim is to prove the following result on the fraction of active type 1 half-edges in the initial phase.
\[prop:initial\] There exists a deterministic sequence of integers $t_n\to \infty$ such that ${\mathcal M}_{t_n}{\overset{\mathrm{d}}{{\longrightarrow}}}V$ as $n\to\infty$, where $V$ is a continuous random variable with strictly positive density on $(0,1)$ if $\lambda_1=\lambda_2$ and $V\equiv 0$ if $\lambda_1<\lambda_2$.
We now consider the initial phase of the continuous time exploration process when $t$ is so small that rather few vertices have been infected. First consider the general case with ${\lambda}_1,{\lambda}_2>0$, possibly different, and the process described by \[sj1\]–\[sjnew\] above. In order to study the initial phase, we introduce the corresponding process where half-edges in \[sj1\] are drawn with replacement, that is, the half-edge $r$ is chosen uniformly at random from the set of *all* half-edges, independently of previous picks. In this version we do not have to keep track of the actual sets of active half-edges, only their numbers, which we denote by ${\mathcal{B}^{{\scriptscriptstyle}(1)}}_t$ and ${\mathcal{B}^{{\scriptscriptstyle}(2)}}_t$. Moreover, we pretend that the chosen half-edge and its vertex are not used before, so we ignore \[sjold\] and always update ${\mathcal{B}^{{\scriptscriptstyle}(1)}}$ and ${\mathcal{B}^{{\scriptscriptstyle}(2)}}$ as in \[sjnew\]. This means that ${\mathcal{B}^{{\scriptscriptstyle}(1)}}$ and ${\mathcal{B}^{{\scriptscriptstyle}(2)}}$ are two independent continuous time Markov branching processes with intensities ${\lambda}_1$ and ${\lambda}_2$, respectively, and the same offspring distribution $D_n^*-1$, where $D_n^*$ is the size-biased distribution corresponding to the empirical distribution $D_n$, that is, ${\operatorname{\mathbb P{}}}(D_n^*=d):=d{\operatorname{\mathbb P{}}}(D_n=d)/{{\mathbb E}}[D_n]$. Of course, we take ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_0=a_i$.
Furthermore, define ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ to be a branching process defined as ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t$ but with the offspring distribution changed to $D^*-1$. Thus ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$, unlike ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ and ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t$, does not depend on $n$. Since ${{\mathbb E}}[D^*-1]={{\mathbb E}}[D(D-1)]/{{\mathbb E}}[D]<\infty$, there is no explosion, and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ is a.s. finite for all $t$. Specifically, for every fixed $T<\infty$, the process ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}$ has a.s. only a finite number of births (infections) in $[0,T]$. Moreover, since $D_n{\overset{\mathrm{d}}{{\longrightarrow}}}D$ and ${{\mathbb E}}[D_n]\to{{\mathbb E}}[D]<\infty$, we have that $D_n^*{\overset{\mathrm{d}}{{\longrightarrow}}}D^*$. It follows that, for every fixed $T<\infty$, we can couple ${\mathcal{B}^{{\scriptscriptstyle}(i)}}$ and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}$ such that they agree with probability $1-o(1)$ each time an individual gets offspring at a time $t\leq T$, that is, [w.h.p[.=1000]{}]{} ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ for all $t\le T$.
Now return to the actual exploration process. We can obtain it from the version with replacement by accepting a selected half-edge $r$ if it is free, and otherwise resampling. Moreover, we also check if the accepted half-edge already is active, and then we apply \[sjold\] instead of \[sjnew\]. During a fixed time interval $[0,T]$, the process ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ has a.s. only finitely many births and thus, since ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ [w.h.p[.=1000]{}]{} on this interval, the number of births in $[0,T]$ for ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t$ is $O_p(1)$. Furthermore, the number of half-edges that are paired in $[0,T]$ is $O_p(1)$, and so is the number of half-edges that are declared active in $[0,T]$. Hence, at each of the $O_p(1)$ births in $[0,T]$, the probability that a paired or active half-edge is picked in the process ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t$ is $o(1)$. Consequently, [w.h.p[.=1000]{}]{}, only free inactive half-edges are selected in ${\mathcal{B}^{{\scriptscriptstyle}(i)}}_t$ for $t\le T$ and the process then agrees completely with ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ for $t\le T$.
We have shown that the processes ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t$ and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ can be coupled (for $i=1,2$ simultaneously) such that, for every fixed $T$, we have that ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_t={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ for $t\le T$. Let $$\tau_n:=\inf{\ensuremath{\bigl\{t\ge0:{\mathcal{S}^{{\scriptscriptstyle}(i)}}_t\neq{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t \text{ for some $i\in{\ensuremath{\{1,2\}}}$}\bigr\}}}.$$ It follows that ${\operatorname{\mathbb P{}}}(\tau_n\le T)\to0$ for every fixed $T$, that is, $\tau_n{\overset{\mathrm{p}}{{\longrightarrow}}}\infty$. This implies that there is a deterministic sequence $t_n\to\infty$ such that ${\operatorname{\mathbb P{}}}(\tau_n\le t_n)\to0$. In other words, [w.h.p[.=1000]{}]{} $$\label{x=z}
{\mathcal{S}^{{\scriptscriptstyle}(1)}}_t={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_t
\quad\text{and}\quad
{\mathcal{S}^{{\scriptscriptstyle}(2)}}={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_t\quad
\text{for } t\le t_n.$$ Fix such a sequence $t_n\to\infty$ where, for later use, we pick the sequence such that each $t_n$ is an integer. For the proof of Theorem \[th:main\], it will be useful to adjust the sequence slightly to ensure that the number of vertices that have been infected at time $t_n$ is small. Thus, let ${\mathcal N}_t$ be the number of edges identified in the exploration process at time $t$; this equals the number of times that \[sj1\] has been performed. Also let ${\widehat{\mathcal{N}}}_t$ be the analogous quantity for the process ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_t\cup{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_t$. With the coupling above, we have ${\mathcal N}_t={\widehat{\mathcal{N}}}_t$ for $t<\tau_n$, and hence [w.h.p[.=1000]{}]{} ${\mathcal N}_{t_n}={\widehat{\mathcal{N}}}_{t_n}$. We may assume, by decreasing $t_n$ if necessary, that ${\widehat{\mathcal{N}}}_{t_n}\le n{^{1/3}}$ [w.h.p]{}.
We also define a related sequence of integers $\nu_n$ such that, in the discrete time exploration process, the branching process approximation remains valid beyond step $\nu_n$. To do this, note that ${\widehat{\mathcal{N}}}_{t_n}{\overset{\mathrm{a.s.}}{{\longrightarrow}}}\infty$ as [${n\to\infty}$]{}, since $t_n\to\infty$. Hence, ${\widehat{\mathcal{N}}}_{t_n}{\overset{\mathrm{p}}{{\longrightarrow}}}\infty$ and ${\mathcal N}_{t_n}{\overset{\mathrm{p}}{{\longrightarrow}}}\infty$, and thus there exists a deterministic sequence $\nu_n$ of integers such that $\nu_n\to\infty$ and [w.h.p[.=1000]{}]{} $$\label{nun}
n{^{1/3}}\ge{\widehat{\mathcal{N}}}_{t_n}= {\mathcal N}_{t_n}\ge \nu_n.$$
Finally note that, by our assumptions, $D_n^*\ge2$ and thus $D^*\ge 2$ so that $D^*-1\ge1$. This means that the branching processes ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t$ never decrease. In particular, they never become extinct, and therefore ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_t\to\infty$ a.s. as [${t\to\infty}$]{}.
With the above coupling at hand we can prove Proposition \[prop:initial\].
Suppose first that ${\lambda}_1={\lambda}_2$. The branching processes ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_t$ and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_t$ are independent and have the same offspring distribution, but possibly different initial values $a_1$ and $a_2$. If we restrict to integer values of $t$, we obtain two independent Galton–Watson processes ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k$ and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_k$ with the same offspring distribution. Moreover, this offspring distribution has a finite mean $m>1$, since, by assumption, ${{\mathbb E}}[D^2]<\infty$ and thus ${{\mathbb E}}[D^*]<\infty$ (in fact we have $m=e^{{{\mathbb E}}[D^*-1]}$). By the Seneta–Heyde theorem [@Heyde] (see also [@AthreyaNey Theorem I.10.3]) there exists a derministic sequence $c_k$ such that ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_k/c_k\to W_i$ a.s., where $W_i\in(0,\infty)$ is a random variable, and thus $$\frac{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k}{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k+{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_k} {\overset{\mathrm{a.s.}}{{\longrightarrow}}}V$$ for some random variable $V\in(0,1)$. By [@AthreyaNey Theorem II.5.2] and the subsequent remark, the variable $W_i$ ($i=1,2$) is continuous with strictly positive density on $(0,\infty)$ and hence $V$ is continuous with strictly positive density on $(0,1)$. Since $t_n\to\infty$, and we have assumed that $t_n\in{\mathbb N}$, it follows that $$\label{Szlim}
\frac{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{t_n}}{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{t_n}+{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_{t_n}} {\overset{\mathrm{a.s.}}{{\longrightarrow}}}V \in(0,1)$$ as [${n\to\infty}$]{}. Alternatively, we can use the continuous-time version of the Seneta–Heyde theorem by Cohn [@Cohn] to directly arrive at . Since ${\mathcal{S}^{{\scriptscriptstyle}(i)}}_{t_n}={\widehat{\mathcal{B}}^{{\scriptscriptstyle}(i)}}_{t_n}$ [w.h.p[.=1000]{}]{} by , it follows from that $$\label{Sxlim}
{\mathcal M}_{t_n}= \frac{{\mathcal{S}^{{\scriptscriptstyle}(1)}}_{t_n}}{{\mathcal{S}^{{\scriptscriptstyle}(1)}}_{t_n}+{\mathcal{S}^{{\scriptscriptstyle}(2)}}_{t_n}} {\overset{\mathrm{d}}{{\longrightarrow}}}V\in(0,1),$$ and the first part of Proposition \[prop:initial\] is proved.
Now suppose that ${\lambda}_1<{\lambda}_2$. By time-scaling we may assume that ${\lambda}_1=1$ and ${\lambda}_2={\lambda}>1$. Then ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{{\lambda}t}$ and ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_t$ are two independent continuous time branching processes, with the same intensity and the same offspring distribution (with finite mean). Hence, as in the case with equal intensities, there exist $c_k$ such that a.s.$$\begin{aligned}
{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{{\lambda}k}/c_k&\to W_1 \label{ax1}
\\
{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_k/c_k&\to W_2, \label{ax2}\end{aligned}$$ where $W_1$ and $W_2$ are random variables with $W_i\in(0,\infty)$ a.s. Furthermore, $c_{k+1}/c_k\to m>1$. For any fixed $j\ge0$, we have for large enough $k$ that ${\lambda}k\ge k+j$, and thus ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{k+j}\le{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{{\lambda}k}$. Hence, by , a.s. $$\limsup_{\ensuremath{{k\to\infty}}}\frac{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k}{c_k}
=
\limsup_{\ensuremath{{k\to\infty}}}\frac{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{k+j}}{c_{k+j}}
\le
\limsup_{\ensuremath{{n\to\infty}}}\frac{{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_{{\lambda}k}}{c_k}\cdot\frac{c_k}{c_{k+j}}
=W_1 m^{-j}.$$ Since $W_1<\infty$, $m>1$ and $j\ge0$ is arbitrary, it follows that $\limsup_{\ensuremath{{k\to\infty}}}{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k/c_k=0$ a.s. and thus, recalling from that ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_k/c_k\to W_2>0$, that ${\widehat{\mathcal{B}}^{{\scriptscriptstyle}(1)}}_k/{\widehat{\mathcal{B}}^{{\scriptscriptstyle}(2)}}_k{\overset{\mathrm{a.s.}}{{\longrightarrow}}}0$. Hence, and hold with $V\equiv 0$.
The deterministic phase
=======================
In this section we show that the fraction $M_k$ of active type 1 half-edges among all active half-edges remains roughly constant after the initial phase in the exploration process for equal intensities. At the very end of the process, when most half-edges have already been paired, this might fail, but we show that the fraction is indeed constant during the main part of the process. Here we will work mainly in discrete time, and then connect to continuous time in the proof of Theorem \[th:main\]. We denote the total number of edges in the graph by $N$, that is, $$N=\frac{1}{2}\sum_id_i;$$ this is the total number of steps in the discrete time exploration process.
\[prop:main\] Assume that $\lambda_1=\lambda_2=1$ and let $\nu_n$ be defined as in . As [${n\to\infty}$]{} we have for any ${\varepsilon}>0$ that $$\label{lx1}
\sup_{\nu_n\le k\le (1-{\varepsilon})N}{\bigl|M_k-M_{\nu_n}\bigr|} {\overset{\mathrm{p}}{{\longrightarrow}}}0.$$
Proposition \[prop:main\] is valid for any sequence $\nu_n\to\infty$ with $\nu_n\le(1-{\varepsilon})N$. However, we will apply it to the sequence $\nu_n$ defined in and therefore formulate it for this. The idea is that the branching process approximation in Section 2 remains valid beyond step $\nu_n$ in the discrete process, and Proposition \[prop:main\] then ensures that the proportion of type 1 vertices does not change after that.
The key observation in the proof of Proposition \[prop:main\] is that $M_k$ is a martingale when $\lambda_1=\lambda_2$. We then show that the second moment assumption implies that the contribution to the quadratic variation of this martingale during the range $\nu_n$ to $(1-{\varepsilon})N$ is vanishingly small. With this at hand it is not hard to show .
\[LM\] If $\lambda_1=\lambda_2$, then $(M_k)_{k=0}^N$ is a martingale.
Recall that $S_k$ denotes the total number of active half-edges after $k$ steps. Define ${\Delta}S_k=S_{k+1}-S_k$, and similarly for other sequences.
Let ${\mathcal F}_k$ be the ${\sigma}$-field generated by all events up to step $k$. Next, reveal whether a new vertex is infected in step $k$, and if so, the identity (and thus the degree) of the new infected vertex (however, we do not yet reveal the classification of the involved half-edges). Let ${\mathcal F}_k^+\supset {\mathcal F}_k$ denote the ${\sigma}$-field generated by the events revealed so far. If a new node of degree $d$ is infected, then ${\Delta}S_k=d-2$, and ${\Delta}{S^{{\scriptscriptstyle}(1)}}_k$ is either $d-2$ or 0, with conditional probabilities (given ${\mathcal F}_k^+$) $M_k$ and $1-M_k$, respectively. Hence, in this case, $${{\mathbb E}}\left[{\Delta}{S^{{\scriptscriptstyle}(1)}}_k\mid{\mathcal F}_k^+\right] = M_k(d-2)$$ and thus $${{\mathbb E}}\left[{S^{{\scriptscriptstyle}(1)}}_{k+1}\mid{\mathcal F}_k^+\right] = {S^{{\scriptscriptstyle}(1)}}_k+M_k(d-2)
= M_k(S_k+d-2)=M_kS_{k+1};$$ Hence, ${{\mathbb E}}{\bigl(M_{k+1}\mid{{\mathcal F}^+}_k\bigr)}=M_k$. If no new vertex is infected, and $S_k>0$, then ${\Delta}S_k=-2$. Since the two paired half-edges are then both drawn uniformly at random (without replacement) from the active half-edges, each one of them has (conditional) probability $M_k$ of being of type 1. Hence $${{\mathbb E}}\left[{\Delta}{S^{{\scriptscriptstyle}(1)}}_k\mid{\mathcal F}_k^+\right] = -2M_k$$ and thus $${{\mathbb E}}\left[{S^{{\scriptscriptstyle}(1)}}_{k+1}\mid{\mathcal F}_k^+\right] = {S^{{\scriptscriptstyle}(1)}}_k-2M_k= M_k(S_k-2)=M_kS_{k+1}.$$ Consequently, if $S_k>2$, so that $S_{k+1}>0$, then ${{\mathbb E}}\left[M_{k+1}\mid{{\mathcal F}^+}_k\right]=M_k$. If $S_k=2$, so that $S_{k+1}=0$, or if $S_k=S_{k+1}=0$, then $M_{k+1}=M_k$ by definition. Hence, in all cases ${{\mathbb E}}\left[M_{k+1}\mid{{\mathcal F}^+}_k\right]=M_k$, and thus ${{\mathbb E}}\left[M_{k+1}\mid{\mathcal F}_k\right]=M_k$.
In order to obtain a bound on the quadratic variation of (a stopped version of) $M_k$, we need to show that $S_k$ grows at least linearly in $k$ throughout the range $\nu_n$ to $(1-{\varepsilon})N$.
\[LC\] If $\lambda_1=\lambda_2$, then, for every ${\varepsilon}>0$ there exists $c>0$ such that [w.h.p[.=1000]{}]{} $S_k\ge ck$ whenever $\nu_n\le k\le (1-{\varepsilon})N$.
Assume that $\lambda_1=\lambda_2=1$. The total set of active half-edges then evolves as in a one-type process with a single unit rate infection type. We consider a continuous time representation of such a process, inspired by [@SvanteMalwina]. As in our continuous time exploration process, each half-edge is throughout classified as *free* or *paired*, and free half-edges are labeled as *active* or *inactive*. All half-edges are assigned independent unit rate exponential life lengths and, to start the growth, two vertices are chosen uniformly at random and their half-edges are declared active, while all other half-edges are inactive. The process then evolves in that an active half-edge $q$ is chosen uniformly at random and, when the life length of a free half-edge $r\neq q$ (active or inactive) expires, then $q$ and $r$ are paired. The vertex to which $r$ is attached becomes infected (if it was not infected already) and its remaining half-edges are activated. This procedure is repeated until there are no active half-edges left. It is straightforward to verify that the process is equivalent to the two-type growth process with equal rates once types are ignored, and we furthermore ignore the time scales. Note that, in the original continuous time process, the growth is slow in the beginning when there are few active half-edges, while in this version, the growth is fast in the beginning when there are many free half-edges whose life lengths compete.
We first show that a large proportion of the edges are identified in finite time.
For every ${\varepsilon}>0$ there exists $t_0=t_0({\varepsilon})$ such that the number of pairings up to time $t_0$ is at least $(1-{\varepsilon})N$ [w.h.p[.=1000]{}]{}
Note that the time of the $k$th pairing is the sum of $k$ independent exponentials with parameters $2N-1,2N-3,\ldots,2N-2k+1$. Let $\xi_1,\xi_2,\ldots,\xi_N$ be independent and exponentially distributed with parameter 2 and write $\xi_{(1)}<\xi_{(2)}<\cdots<\xi_{(N)}$ for the order statistics of the $\xi_k$’s. Due to the memoryless property $\xi_{(k)}$ is the sum of $k$ independent exponentials with parameters $2N,2N-2,\ldots,2N-2k+2$, and it follows that the time of the $k$th pairing is stochastically dominated by $\xi_{(k+1)}$. We are hence done if we show that $\xi_{(\lceil(1-{\varepsilon})N\rceil+1)}\le t_0$ [w.h.p[.=1000]{}]{} for some $t_0$ or, equivalently, that the number of $\xi_k$ that exceed $t_0$ is at most ${\varepsilon}N-1$. This however follows from the law of large numbers if we pick $t_0$ large such that ${{\mathbb P}}(\xi_k>t_0)<{\varepsilon}$.
There exists $\delta>0$ such that throughout the interval $[0,t_0]$ the proportion of uninfected vertices with degree at least $3$ is at least $\delta$ [w.h.p[.=1000]{}]{}
Fix $d\ge3$ such that $p_d>0$. Let $V_d(t)$ denote the number of vertices of degree $d$ with all half-edges with life lengths longer than $t$. Again by the (weak) law of large numbers we have that $$\Bigl|\frac{1}{n}V_d(t_0)-p_d\,e^{-dt_0}\Bigr|{\overset{\mathrm{p}}{{\longrightarrow}}}0\quad\text{as }n\to\infty.$$ The number of uninfected vertices of degree $d$ at time $t_0$ is at least $V_d(t_0)-2$, so the claim follows.
We now return to the discrete time exploration process. Recall that $\Delta S_k=S_{k+1}-S_k$ and that $\mathcal{F}_k$ is the $\sigma$-field of events determined by the process up to time $k$. After $k$ steps there are $2N-2k$ unpaired half-edges and hence $${\operatorname{\mathbb P{}}}\big(\Delta S_k=-2{\,\big|\,}\mathcal{F}_k\big)\,=\,\frac{S_k-1}{2N-2k-1}\,\le\,\frac{S_k}{2N-2k}.$$ If the active half-edge that is paired in step $k+1$ is connected to an inactive half-edge attached to a vertex with degree at least 3, then the number of active half-edges increases. The degree of the vertex of the inactive half-edge has a size biased distribution, and hence the probability that it is at least 3 is at least as large as the proportion of uninfected vertices with degree at least 3. Combining the above two claims we find that, for all $k=1,2,\ldots,(1-{\varepsilon})N$, [w.h.p[.=1000]{}]{} $${\operatorname{\mathbb P{}}}\big(\Delta S_k\ge1{\,\big|\,}\mathcal{F}_k\big)\,\ge\,\delta\Big(1-\frac{S_k}{2N-2k}\Big).$$ In particular, whenever $1\le S_k\le {\varepsilon}\delta N/4$, we have that $${\operatorname{\mathbb P{}}}\big(\Delta S_k=-2{\,\big|\,}\mathcal{F}_k\big)\le\delta/8\quad \text{and}\quad{\operatorname{\mathbb P{}}}\big(\Delta S_k\ge1{\,\big|\,}\mathcal{F}_k\big)\ge\delta/2.$$ Now, let $\zeta_1,\ldots,\zeta_N$ be i.i.d. random variables taking values $-2$ and $1$ with probability $\delta/8$ and $\delta/4+{\varepsilon}\delta/8$, respectively, and otherwise the value $0$, and define $X_k:=\sum_{j=1}^k\zeta_j$. Then, by the law of large numbers, $X_k>{\varepsilon}\delta k/16$ [w.h.p[.=1000]{}]{} for all $k\ge\nu_n$, while $X_k$ is unlikely to ever exceed ${\varepsilon}\delta N/4$. Moreover, since $\nu_n=o(\sqrt{n})$ by , the number of active half-edges is unlikely to ever hit zero in the first $\nu_n$ steps.[^5] We conclude that there is a coupling between $(S_k)_{k\ge1}$ and $(X_k)_{k\ge1}$ such that [w.h.p[.=1000]{}]{}$$S_k\ge X_k\quad\text{for all }k=1,2,\ldots,(1-{\varepsilon})N.$$ Consequently, $S_k\ge{\varepsilon}\delta k/16$ [w.h.p[.=1000]{}]{} whenever $\nu_n\le k\le(1-{\varepsilon})N$.
Fix ${\varepsilon}>0$ and $c$ as in [Lemma \[LC\]]{}, and let $\tau$ be the stopping time $\min\{k\ge\nu_n:S_k< ck\}$. Thus, by [Lemma \[LC\]]{}, [w.h.p[.=1000]{}]{} $\tau>(1-{\varepsilon})N$. Let ${\tilde M}_k:=M_{k\land \tau}$, that is, the martingale $M$ stopped at $\tau$. Then $({\tilde M}_k)_{k=0}^N$ is also a martingale. We consider the quadratic variation of this martingale.
\[LQ\] As [${n\to\infty}$]{}, $${{\mathbb E}}\left[\sum_{k=\nu_n}^{(1-{\varepsilon})N} |{\Delta}{\tilde M}_k|^2\right] \to0.$$
Throughout the proof, $C$ denotes a constant, possibly depending on ${\varepsilon}$ and $c$, that may be different on each occurrence. Let $k\in[\nu_n,(1-{\varepsilon})N]$. We may suppose that $S_k\ge ck$, since otherwise $\tau\le k$ and ${\Delta}{\tilde M}_k=0$. Then, $$\label{gDM}
{\Delta}{\tilde M}_k = {\Delta}M_k
= \frac{{S^{{\scriptscriptstyle}(1)}}_k+{\Delta}{S^{{\scriptscriptstyle}(1)}}_k}{S_{k}+{\Delta}S_{k}} - \frac{{S^{{\scriptscriptstyle}(1)}}_k}{S_k}
= \frac{S_k{\Delta}{S^{{\scriptscriptstyle}(1)}}_k-{S^{{\scriptscriptstyle}(1)}}_k{\Delta}S_k}{S_k(S_{k}+{\Delta}S_k)}.$$ If a new vertex of degree $d$ is infected at time $k+1$, then ${\Delta}{S^{{\scriptscriptstyle}(1)}}_k$ equals either $0$ or ${\Delta}S_k=d-2$. In either case, implies that $$| {\Delta}{\tilde M}_k|
\le
\frac{d-2}{S_{k}+d-2}
\le \frac{d}{S_{k}+d}
\le \frac{d}{c{k}+d}
\le C \frac{d}{k+d}.$$ If no new vertex is infected at time $k+1$, then ${\Delta}S_k=-2$ and yields (for large $k$) $$| {\Delta}{\tilde M}_k|
\le
\frac{2}{S_{k}-2}
\le \frac{2}{c{k}-2}
\le \frac{C}{k}.$$ Hence, if $d{^{(k)}}$ is the degree of the vertex infected at time $k+1$, with $d{^{(k)}}=0$ if there is no such vertex, then $$\label{ele}
{{\mathbb E}}\left[\sum_{k=\nu_n}^{(1-{\varepsilon})N} |{\Delta}{\tilde M}_k|^2\right]
\le
C {{\mathbb E}}\left[\sum_{k=\nu_n}^{(1-{\varepsilon})N} {\Bigl(\frac{d{^{(k)}}}{k+d{^{(k)}}}\Bigr)}^2\right]
+ C\sum_{k=\nu_n}^\infty \frac{1}{k^2}.$$
After step $k$, there are $2(N-k)$ free half-edges and hence, for each vertex $i$ and step $k\le(1-{\varepsilon})N$, the probability that $i$ is infected in step $k+1$, given that it has not been infected earlier, equals $d_i/(2(N-k)-1)\leq Cd_i/n$. Hence, for any $k\le(1-{\varepsilon})N$, $$\label{win}
{{\mathbb E}}\left[{\Bigl(\frac{d{^{(k)}}}{k+d{^{(k)}}}\Bigr)}^2\right]
\le C {\sum_{i=1}^n}\frac{d_i}{n}{\Bigl(\frac{d_i}{k+d_i}\Bigr)}^2
= C \frac{1}n{\sum_{i=1}^n}\frac{d_i^3}{(k+d_i)^2}
=C {{\mathbb E}}\left[\frac{D_n^3}{(k+D_n)^2}\right].$$ For any $d\ge1$, we have the estimates $${\sum_{k=1}^\infty}\frac{d^3}{(k+d)^2}
\le \sum_{k=1}^d \frac{d^3}{d^2} + \sum_{k=d+1}^\infty\frac{d^3}{k^2}
\le d^2 + \frac{d^3}d=2d^2$$ and $$\sum_{k=\nu_n}^\infty \frac{d^3}{(k+d)^2}
\le \sum_{k=\nu_n}^\infty\frac{d^3}{(k+1)^2}
\le \frac{d^3}{\nu_n}.$$ Hence, $$\label{auf}
\sum_{k=\nu_n}^\infty \frac{D_n^3}{(k+D_n)^2}
\le {2D_n^2\land\frac{D_n^3}{\nu_n}}.$$ By assumption, $D_n{\overset{\mathrm{d}}{{\longrightarrow}}}D$ and $\nu_n\to\infty$, and thus $2D_n^2\land\nu_n{^{-1}}{D_n^3} \le \nu_n{^{-1}}D_n^3{\overset{\mathrm{p}}{{\longrightarrow}}}0$. Furthermore, $D_n^2$ is uniformly integrable, and thus so is $2D_n^2\land\nu_n{^{-1}}{D_n^3}$. Consequently, we have by that $$\label{sof}
{{\mathbb E}}\left[\sum_{k=\nu_n}^\infty \frac{D_n^3}{(k+D_n)^2}\right]
\le
{{\mathbb E}}\left[2D_n^2\land\frac{D_n^3}{\nu_n}\right]\to0.$$ The proposition now follows from , and .
Since ${\tilde M}_k-{\tilde M}_{\nu_n}$, with $k\ge\nu_n$, is a martingale, Doob’s inequality and [Lemma \[LQ\]]{} yield $${{\mathbb E}}\left[\sup_{\nu_n\le k\le (1-{\varepsilon})N}{\bigl|{\tilde M}_k-{\tilde M}_{\nu_n}\bigr|}^2\right]
\le 4 {{\mathbb E}}\left[{\bigl|{\tilde M}_{{\lfloor(1-{\varepsilon})N\rfloor}}-{\tilde M}_{\nu_n}\bigr|}^2\right]
=4 {{\mathbb E}}\left[\sum_{k=\nu_n}^{{\lfloor(1-{\varepsilon})N\rfloor}-1} |{\Delta}{\tilde M}_k|^2\right]\to0.$$ Hence, $ \sup_{\nu_n\le k\le (1-{\varepsilon})N}{\bigl|{\tilde M}_k-{\tilde M}_{\nu_n}\bigr|} {\overset{\mathrm{p}}{{\longrightarrow}}}0$, and follows since by [Lemma \[LC\]]{}, [w.h.p[.=1000]{}]{} $\tau>(1-{\varepsilon})N$ and thus $M_k={\tilde M}_k$ for $k\le(1-{\varepsilon})N$.
Proof of Theorem \[th:main\]
============================
We can now prove Theorem \[th:main\] by combining Proposition \[prop:initial\] and Proposition \[prop:main\].
First assume that $\lambda_1=\lambda_2$. Fix $\varepsilon>0$ and let the sequences $\nu_n$ and $t_n$ be as in Propositions \[prop:initial\] and \[prop:main\]. Recall from the paragraph preceding that ${\mathcal N}_t$ denotes the number of steps (pairings of half-edges) that have been performed at time $t$ in the continuous time exploration process. By definition, we have that ${\mathcal M}_{t_n}=M_{{\mathcal N}_{t_n}}$ and, by , that ${\mathcal N}_{t_n}\geq \nu_n$ w.h.p. Hence, by Proposition \[prop:main\], $$\sup_{\nu_n\le k\le (1-{\varepsilon})N}{\bigl|M_k-{\mathcal M}_{t_n}\bigr|} {\overset{\mathrm{p}}{{\longrightarrow}}}0.$$ Furthermore, by Proposition \[prop:initial\], the fraction ${\mathcal M}_{t_n}$ converges in distribution to a continuous random variable with support on $(0,1)$. Since a vertex that is infected in step $k+1$ in the discrete time exploration process is infected by type 1 independently with probability $M_k$, it follows from the law of large numbers that the fraction of type 1 vertices among all vertices that are infected at steps $k\in[\nu_n,(1-{\varepsilon})N]$ converges in distribution to $V$.
Recall from that $\nu_n\leq n^{1/3}$ by definition. Hence the number of vertices that are infected before step $\nu_n$ does not exceed $n^{1/3}$. The number of vertices that are infected after step $(1-{\varepsilon})N$ w.h.p. does not exceed ${\varepsilon}({{\mathbb E}}[D]+1)n$, since $N\leq ({{\mathbb E}}[D]+1)n$ w.h.p. The asymptotic fraction of vertices infected for $k\in[\nu_n,(1-{\varepsilon})N]$ is hence at least $1-C{\varepsilon}$. Since ${\varepsilon}>0$ is arbitrary, part (a) of the theorem follows.
To prove part (b), assume that $\lambda_1<\lambda_2$ and consider a modified version of the process where, after time $t_n$, the weaker type 1 infection spread with the same larger intensity $\lambda_2$ as the type 2 infection. To generate this process, we independently equip each half-edge $h$ with two independent Poisson processes ${\mathcal P}^{{\scriptscriptstyle}(1)}_h$ and ${\mathcal P}^{{\scriptscriptstyle}(2)}_h$, both with rate $\lambda_2$, and let $\check{{\mathcal P}}^{{\scriptscriptstyle}(1)}_h$ denote a thinned version of ${\mathcal P}^{{\scriptscriptstyle}(1)}_h$ where each point is kept with probability $\lambda_1/\lambda_2$, so that $\check{{\mathcal P}}^{{\scriptscriptstyle}(1)}_h$ is a Poisson process with rate $\lambda_1$. The process is then generated by letting the possible infection times for an active type 1 or 2 half-edge $h$ be specified by $\check{{\mathcal P}}^{{\scriptscriptstyle}(1)}_h$ and ${\mathcal P}^{{\scriptscriptstyle}(2)}_h$, respectively, up until time $t_n$, and by ${\mathcal P}^{{\scriptscriptstyle}(1)}_h$ and ${\mathcal P}^{{\scriptscriptstyle}(2)}_h$ after that time. The original process can be generated by using the thinned process $\check{{\mathcal P}}^{{\scriptscriptstyle}(1)}_h$ for type 1 throughout the whole time course. The corresponding discrete time processes are defined by observing the continuous time processes at the times of pairings.
Let $\check{{{\mathcal S}}}^{{\scriptscriptstyle}(i)}_t$ denote the number of active type $i$ half-edges at time $t$ in the modified process, and similarly for other quantities. The above construction provides a coupling of the original process and the modified process where $\check{{{\mathcal S}}}^{{\scriptscriptstyle}(i)}_t={{\mathcal S}}^{{\scriptscriptstyle}(i)}_t$ for $t\leq t_n$ and $i=1,2$, while $\check{{{\mathcal S}}}^{{\scriptscriptstyle}(1)}_t\geq {{\mathcal S}}^{{\scriptscriptstyle}(1)}_t$ and $\check{{{\mathcal S}}}^{{\scriptscriptstyle}(2)}_t\leq {{\mathcal S}}^{{\scriptscriptstyle}(2)}_t$ for $t>t_n$. It follows that $\check{{\mathcal M}}_t={\mathcal M}_t$ for $t\leq t_n$ and $\check{{\mathcal M}}_t\geq {\mathcal M}_t$ for $t>t_n$. Analogously, if ${\mathcal V}^{{\scriptscriptstyle}(i)}_i$ denotes the set of infected vertices of type $i$ at time $t$, we have that $\check{{\mathcal V}}^{{\scriptscriptstyle}(1)}_t\supseteq {\mathcal V}^{{\scriptscriptstyle}(1)}_t$ and $\check{{\mathcal V}}^{{\scriptscriptstyle}(2)}_t\subseteq {\mathcal V}^{{\scriptscriptstyle}(2)}_t$ for all $t$. Hence the number of type 1 infected vertices is at least as large in the modified process as in the original process, and it will suffice to show that the fraction of type 1 infected vertices in the modified process converges to 0.
The modified process has equal intensities for the infection types after time $t_n$, that is, after step ${\mathcal N}_{t_n}$ in the discrete time process. By , we have ${\mathcal N}_{t_n}\geq \nu_n$ [w.h.p[.=1000]{}]{} and it then follows from Proposition \[prop:main\] that $$\sup_{{\mathcal N}_{t_n}\le k\le (1-{\varepsilon})N}{\bigl|\check{M}_k-\check{{\mathcal M}}_{t_n}\bigr|} {\overset{\mathrm{p}}{{\longrightarrow}}}0.$$ Up to time $t_n$, on the other hand, type 1 spreads with a strictly smaller intensity and thus, by Proposition \[prop:initial\], the fraction $\check{{\mathcal M}}_{t_n}$ converges to 0 in probability. By the same arguments as in the proof of part (a), this yields that the fraction of type 1 infected vertices in the modified process converges to 0, as desired.
Further work
============
We have studied competing first passage percolation on the configuration model with finite variance degrees and exponential edge weights, and shown that both infection types occupy positive fractions of the vertex set if and only if they spread with the same intensity. There are several natural extensions of this work. One would be to investigate the scaling of the number of vertices of the losing type when the intensities are different. The results in [@regular] contain results in this direction for random regular graph and we conjecture that the results would be similar for finite variance graphs. Specifically we conjecture that, when $\lambda_1<\lambda_2$, the number of vertices occupied by type 1 is of the order $n^{\lambda_1/\lambda_2}$. In contrast to the case when the degree variance is infinite, treated in [@winner], the winner hence does not take it all, but the loosing type also grows to infinity with $n$.
In [@regular], also more general initial conditions are considered, where the initial number of one or both types may grow with $n$. This could also be done in our case and, in addition, one could consider initial sets where the vertices are chosen based on degree. Is it for instance possible for a weaker type to capture a positive fraction of the vertices if it can start from one or more high degree vertices, while the stronger type starts from a vertex with small degree?
Another extension would be to study more general passage time distributions, possibly different for the two types. Also in the general case, the initial growth of the types can be approximated by branching processes, but these are then not Markovian. A reasonable guess is that the possibility for both types to occupy positive fractions of the vertex set is determined by the relation between the Malthusian parameters of these branching processes, as discussed in [@fixspeedI].
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[^1]: Department of Mathematics, Stockholm University; [[email protected]]{}
[^2]: Department of Mathematics, Stockholm University; [[email protected]]{}
[^3]: Department of Mathematics, Uppsala University; [[email protected]]{}
[^4]: The authors are grateful to Olle Nerman for pointing out some relevant references. This work was in part supported by the Swedish Research Council (grants 237-2013-7302 and 2016-04442 DA, MD) and by the Knut and Alice Wallenberg Foundation (SJ).
[^5]: Indeed, either $S_k$ exceeds $2\nu_n$ before reaching zero, which is good, or the probability of pairing two active half-edges is at most $2\nu_n/(N-2\nu_n)$ in each of these step, so the claim follows from the union bound.
|
---
abstract: 'A coarse grained description of a two-dimensional prey-predator system is given in terms of a 3-state lattice model containing two control parameters: the spreading rates of preys and predators. The properties of the model are investigated by dynamical mean-field approximations and extensive numerical simulations. It is shown that the stationary state phase diagram is divided into two phases: a pure prey phase and a coexistence phase of preys and predators in which temporal and spatial oscillations can be present. The different type of phase transitions occuring at the boundary of the prey absorbing phase, as well as the crossover phenomena occuring between the oscillatory and non-oscillatory domains of the coexistence phase are studied. The importance of finite size effects are discussed and scaling relations between different quantities are established. Finally, physical arguments, based on the spatial structure of the model, are given to explain the underlying mechanism leading to oscillations.'
address: ' Département de Physique Théorique, Université de Genève, CH 1211 Genève 4, Switzerland.'
author:
- Tibor Antal and Michel Droz
title: 'Phase Transitions and Oscillations in a Lattice Prey-Predator Model'
---
=10000
[2]{}
Introduction {#sec:intro}
============
The dynamics of interacting species has attracted a lot of attention since the pioneering works of Lotka [@lotka] and Volterra [@volterra]. In their independent studies, they showed that simple prey-predator models may exhibit limit cycles during which the populations of both species have periodic oscillations in time. However, this behavior depends strongly on the initial state, and it is not robust to the addition of more general non-linearities or to the presence of more than two interacting species [@montrol]. In many cases the system reaches a simple steady-state.
A better understanding of the properties of such oscillations is clearly desirable, as such population cycles are often observed in ecological systems and the underlying causes remain a long-standing open question [@Nature]. One of the best documented example concerns the Canadian lynx population. This population was monitored for more than hundred years (starting in 1820) from different regions of Canada. It was observed that the population oscillates with a period of approximately 10 years and that this synchronization was spatially extended over areas of several millions of square kilometers [@lynx]. Several attempts were made to explain these facts (climatic effects, relations with the food-web, influence if the solar cycle) without success. More recently, Blasius et al. [@Nature] introduced a deterministic three level vertical food-chain model. The three coupled nonlinear differential equations defining the model contain eight free parameters and two unknown nonlinear functions. The authors showed that an ad-hoc choice of the free parameters and nonlinear functions explains the experimental data for the Canadian lynx.
In such mean-field type models, it is assumed that the populations evolve homogeneously, which is obviously an oversimplification. An important question consists in understanding the role played by the local environment on the dynamics [@may]. There are many examples in equilibrium and nonequilibrium statistical physics showing that, in low enough dimensions, the local aspects (fluctuations) play a crucial role and have some dramatic effects on the dynamics of the system. Accordingly, a lot of activities have been devoted during the past years to the study of extended prey-predator models. The simplest spatial generalization are the so called two patches models, where the species follow the conventional prey predator rules within each patches, and can migrate from one patch to the other [@jansen]. Other works have found that the introduction of stochastic dynamics plays an important role [@bradshaw], as well as the use of discreet variables, which prevent the population to become vanishingly small.
These ingredients are included in the so called individual based lattice models, for which each lattice site can be empty or occupied by one individual of a given species or two individuals belonging to different species. It was recognized that these models give a better description of the oscillatory behavior than the usual Lotka-Volterra (L-V) equations. Indeed, the oscillations in these lattice models are stable against small perturbations of the prey and predator densities, and they do not depend on the initial state. It was also found (in two dimensional systems) that the amplitude of the oscillations of global quantities decreases with increasing system size, while the oscillations persist on local level. It was argued that coherent periodic oscillations are absent in large systems (although, [@tome] do not discard this possibility). In [@lipowski] Lipowski et al. state that this is only possible above a spatial dimension of 3. In [@provata] Provata et al. emphasize that the frequency of the oscillations are stabilized by the lattice structure and that it depends on the lattice geometry. In some papers, the stationary phase diagram was also derived , and different phases were observed as a function of the model parameters, such as an empty phase, a pure prey phase, and an oscillatory region of coexisting preys and predators. In [@tome], a coexistence region without oscillations and a domain of the control parameter space for which the stationary states depend strongly upon the initial condition, were found.
However, in all the above works no systematic finite size studies have been performed, allowing to draw firm conclusions on the phase diagram of the models as a function of their sizes. It is known [@chopard], that in ecological problems the fact that a system has a finite size is more relevant than in most of the cases encountered in statistical physics, for which one concentrates on the thermodynamic limit. Particularly, the size dependence of the amplitude of the oscillations, as well as a detailed description of the critical behavior near the phase transitions have not been investigated. Another relevant question is how much the stationary phase diagrams of these prey-predator models have some generic properties or how much they depend upon the details of the models.
The goal of this paper is to study a simple models of prey-predators on a two-dimensional lattice for which some of the above questions could be answered. Our model is based on a coarse-grained description in the sense that a given cell models a rather large part of a territory and thus can contain many preys or predators. Moreover, predators cannot leave without preys in a given cell. Those are the main differences between our model and Satulovsky and Tomé (ST) model [@tome]. Nevertheless, it turns out that the stationary state phase diagram of the two models are quite different.
Our model is defined in Sec. \[sec:model\]. Although governed by only two control parameters, this model exhibits a rich phase diagram. Two different phases are observed: a pure prey phase, and a coexistence phase of preys and predators in which an oscillatory and a non-oscillatory region can be distinguished. In some limiting cases the model can be mapped onto another well known nonequilibrium model: the [*contact process*]{} (CP) [@contact]. In Sec. \[sec:mean-field\] the properties of our model are analyzed in dynamical one and two-points mean-field approximations and no undamped oscillatory behavior is found. In Sec. \[sec:mc\], extensive Monte-Carlo simulations are performed. It is shown that, as a function of the values of the control parameters, two types of continuous nonequilibrium phase transitions towards a prey absorbing state are present. The system size dependence of the amplitude of the oscillations is studied and several scaling relations between the amplitude of the oscillations and the correlation length are obtained. In Sec. \[sec:discuss\] an underlying mechanism responsible for the spatial oscillations is proposed, which leads to a qualitative explanation of the properties of the phase diagram. In particular, we show that the spatially extended aspect of the problem is crucial to have an oscillatory region. Finally, conclusions are drawn in Sec. \[sec:conc\].
The model {#sec:model}
=========
Our system models preys and predators living together in a two dimensional territory. This territory is divided into square cells, and each of them can contain several preys and predators. In this coarse-grained description in which each cell represents a rather large territory, one can assume that each cell containing some predator will also contain some preys. Hence, a three state representation is made. Each cell of the two-dimensional square lattice (of size $L \times L$, with periodic boundary condition), labeled by the index $i$, can be at time $t$, in one of the three following states: $\sigma_i = 0, 1, 2$. A cell in state 0, 1 or 2 corresponds respectively to a cell which is empty, occupied by preys or simultaneously occupied by preys and predators. The dynamics of the system is defined as a continuous time Markov process. The transition rates for site $i$ are
- $0 \to 1$ at rate $\lambda_a (n_{i,1}+n_{i,2})/4$,
- $1 \to 2$ at rate $\lambda_b (n_{i,2})/4$,
- $2 \to 0$ at rate 1,
where $n_{i,\sigma}$ denotes the number of nearest neighbor sites of $i$ which are in the state $\sigma$. 4 is the coordination number of this two dimensional square lattice.
The first two processes model the spreading of preys and predators. The two control parameters, $\lambda_a$ and $\lambda_b$, characterize a particular prey-predator system. The reason for considering the sum, $n_{i,1}+n_{i,2}$, in the first rule is simply that all the neighboring cells of $i$ containing some prey, (hence $\sigma_i=1$ or $2$), will contribute to the prey repopulation of cell $i$. The third process represents the local depopulation of a cell due to too greedy predators. It can be interpreted as the local extinction of the species or as the moving of them to neighboring occupied sites. Spontaneous disappearance of a prey state ($\sigma_i$: $1 \to 0$) or that of the predators alone ($\sigma_i$: $2 \to 1$) is forbidden. These assumptions are reasonable because the occurrence of these processes is improbable. The rate of the third process is chosen to be 1, which sets the time scale. As a consequence, $\lambda_a$ and $\lambda_b$ are dimensionless quantities.
The above dynamical rules are an extension of the contact process model (CP) [@contact] introduced as a description of epidemic spreading. The CP is a 2-state model, $\sigma_i = 0, 1$; the status 0 and 1 represent respectively the healthy and the infected individuals. The CP dynamical rules are
- $0 \to 1$ at rate $\lambda (n_{i,1})/4$,
- $1 \to 0$ at rate 1 .
An epidemic survives for $\lambda>\lambda_{CP}^*$ = $1.6488(1)$ [@grassberger] and disappears for $\lambda<\lambda_{CP}^*$. The transition towards this absorbing state is of second order and belongs to the directed percolation (DP) universality class [@DP].
Our model differs from most of the lattice models previously investigated by the fact that on each site, each species may be represented by several individuals rather than just one. In the previously investigated models the spreading rate of the preys is simply proportional to $n_{i,1}$. Under this assumption, our model reduces essentially to the ST model, in which the control parameters are defined as $c=(1+\lambda_a+\lambda_b)^{-1}$ and $p=c(\lambda_b-\lambda_a)/2 $.
It is worth discussing first the behavior of our model in two limiting cases. In the $\lambda_a \to \infty$ limit the proportion of empty cells is negligible since the empty cells are reoccupied by preys instantly after their extinction. Hence, the lattice is completely covered by preys and the $\sigma=2$ sites behave as the infected species in the CP. Namely, when decreasing $\lambda_b$ the predator density is decreasing continuously and vanishes at the CP critical value $\lambda_b^*(\lambda_a=\infty)$ = $\lambda_{CP}^*$. One can think of the $\lambda_b \to \infty$ limit in similar terms. In this case, the proportion of the prey cells ($\sigma=1$) should be negligible since the high productivity of the predators, while the prey-predator cells should behave as the infected species in the CP. This is indeed the case if $\lambda_a > \lambda_{CP}^*$, but when $\lambda_a$ gets smaller than $\lambda_{CP}^*$, the prey density increases again instead of being zero, as we shall see later.
Mean-field analysis. {#sec:mean-field}
====================
Although apparently simple, there is no way to solve analytically the model defined above. However, analytic solutions can be obtained by making some approximations. The simplest one is the one-point mean-field approximation in which all spatial fluctuations are neglected. Thus, the system is characterized by the densities of prey, $a$, and predator, $b$, sites a=[1L\^2]{} \_i ( \_[\_i, 1]{} + \_[\_i, 2]{} ) , b=[1L\^2]{} \_i \_[\_i, 2]{} , which values satisfy the $0\le b \le a \le 1$ conditions by definition. In terms of these densities, the mean-field dynamical equations read: = \_a a(1-a)-b \[mfa\] and = \_b b(a-b)-b \[mfb\] Note, that for $b=0$ ($a=b=0$) initial condition the predator (and prey) densities remains 0.
The (\[mfa\],\[mfb\]) equations clearly differ from the usual L-V ones. The main difference lies in the interaction terms as, although a larger prey density increases the predator growth rate, the rate of the predated preys only depends on the predator density. This is a simple consequence of the fact that there are no pure predator sites without preys in this model. Thinking of a real prey-predator system it makes sense, as a predator has to consume a certain amount of preys in a given time to survive, independently of the number of preys around it. The $(1-a)$ term in the first equation plays the role of a simple Verhlaust factor which assures an upper limit for the prey density $(a\le 1)$, and similarly the $(a-b)$ term in the second equation do not let the density of predators exceed that of the preys.
The stationary states are obtained by setting the left hand sides of Eqs. (\[mfa\], \[mfb\]) to zero. Contrary to the simplest L-V equations, qualitatively different stationary states are obtained varying the parameters, $\lambda_a$ and $\lambda_b$, as illustrated on Fig. \[fig:phase\_mf\].
For $0 \le \lambda_b \le 1$ and $\lambda_a>0$, the stationary state is a pure prey absorbing state $a^s=1, b^s=0$. For $\lambda_a=0$ the stationary state is also a prey state, $b^s=0$, however, the value of $a^s$ depends upon the initial state.
In the rest of the plane $(\lambda_a, \lambda_b)$, the stationary solution is a\^s= and b\^s=a\^s- , which describes a coexistence of preys and predators (coexistence phase).
For $\lambda_b\gg1$ the $a$ and $b$ densities are approximately the same a\^s = b\^s + O() = { 1 - + O() & \_a > 1\
O() & \_a = 1\
O() & \_a < 1\
. and as a function of $\lambda_a$ they show a mean field CP behavior as it is expected from the argument given in Sec. \[sec:model\].
In the $\lambda_a\gg1$ limit (and for $\lambda_b>1$) the system is ”full of preys”, namely a\^s = 1 - (1-) + O() \[aapp\] and the predator density reads b\^s = (1-)(1-) + O() \[bapp\] and, as expected, its $\lambda_b$ dependence agrees with the prediction of the mean-field approximation for CP. This approximation predicts a second order phase transition along the whole $\lambda_b=1$ line, as in the $\lambda_b \to 1$ limit $a$ and $b$ approach linearly the values $1$ and $0$ respectively a\^s &=& 1 - + O((\_b-1)\^2)b\^s &=& \_a + O((\_b-1)\^2) .
The behavior of the densities is rather surprising at the $\lambda_a=0$ boundary of the coexistence phase. For $0<\lambda_a \ll 1$ and for $\lambda_b>1$ a\^s = + \_a( + ) + O(\_a\^2) , \[a:la<<1\] while the stationary solution, $a^s$, for $\lambda_a = 0$ depends on the initial state. Thus the mean field approximation predicts a discontinuity of the prey density along this boundary. However, the density $b^s = a^s - \lambda_b^{-1}$ is proportional to $\lambda_a$ and continuous in $\lambda_a = 0$.
Important quantities are the fluctuations of the prey and the predator densities (mean square deviations), which are normalized to be size independent for large systems \_= L\^2 (-)\^2 , = a b , \[defchi\] and $\langle \rangle$ means the time average in the stationary state. For $\lambda_a, \lambda_b\gg1$ the majority of the sites are in state $2$, with a few holes in it, hence one can suppose that the holes are independent. Consequently, the number of the holes follows a Poisson distribution, from which the average hole number equals to the mean square deviation. There are $L^2(1-a)$ holes made of sites in the state $\sigma_i = 0$ and $L^2(1-b)$ holes made of sites in the states $\sigma_i = 0$ or $1$. Thus \_a 1-a\^s \_b 1-b\^s , \[appchi\] which is in good agreement with the simulations in a region (non-oscillatory part) of the coexistence phase (see Fig. \[fig:chi\]).
The stability of the stationary state can be analyzed by linear stability. One has to investigate the eigenvalues, $\epsilon_{1,2}$, of the Jacobian matrix related to the mean field equations (\[mfa\], \[mfb\]) at the stationary densities . ( \_a a & \_b a\
\_a b & \_b b\
) |\_s= ( \_a(1-2a\^s) & -1\
\_b a\^s -1 & 1 - \_b a\^s ) .
It turns out that the real parts of the eigenvalues are always negative, assuring the stability of the solutions. This mean field approximation do not predict limit cycles, which would correspond to having an eigenvalue, $\epsilon$, with a zero real part. However, in some part of the coexistence phase the imaginary part is nonzero, so the stationary solution is approached in spirals (poles), instead of straight lines (nodes) (see Fig. \[fig:phase\_mf\] and \[fig:flow\]), as it was also observed in the ST model [@tome]. Note, that an unexpected node region appears for $\lambda_b>10$. One can consider the presence of poles as a hint for the appearance of oscillations beyond the mean-field approximation. Notice, that in this pole case, the damped oscillations are strong along the $\lambda_b$ axes (i.e. for $\lambda_a \ll 1$). The strength of them can be characterized by the ratio of the imaginary and real part of the eigenvalues, which has a singularity in the $\lambda_a \to 0$ limit. Using (\[a:la<<1\]), we obtain || = 4 \_a\^[-1/2]{} + O (\_a\^[1/2]{}) for $\lambda_a \ll 1$ and $\lambda_b>1$. In this limit one can derive an expression also for the frequency, $\omega$, of the damped oscillations = | () | = 2 \_a\^[1/2]{} + O (\_a\^[3/2]{}) .
The mean field results can be interpreted in the following way. The approximation predicts two distinct phases: the pure prey phase and the coexistence one. It also gives some hints for a possible presence of oscillations in some parts of the coexistence phase. The phase boundaries of the two phases are described by two lines: the $\lambda_b=1$ and the $\lambda_a=0$. Several quantities show a power-law behavior close to these boundaries, like $b$ and $1-a$ at the $\lambda_b=1$ boundary, and $b$, $\omega$ and the strength of the damped oscillations at the $\lambda_a=0$ boundary. This implies that the transitions are of second order, and the predator density, $b$, seems to be a good candidate for the order parameter. The order parameter goes to zero at the phase boundaries as $b \sim (\lambda_b-1)^\beta$ and $b \sim \lambda_a^\beta$ with a mean-field exponent $\beta=1$.
We performed also a pair approximation, in which the nearest neighbor correlations are also considered as parameters. It turns out that the results differ only quantitatively from that of the one point approximation. Contrary to [@tome], our system does not show limit cycle behavior on the pair approximation level either.
Monte-Carlo simulations. {#sec:mc}
========================
On general grounds, one expects that the fluctuations will play an important role in low dimensions. Our model is supposed to describe a two dimensional world and accordingly, we have performed extensive Monte-Carlo (MC) simulations for systems of sizes $L \times L$, $L$ varying between 100 and 1000. We used periodic boundary conditions.
Although our model is formulated as a continuous-time process, an equivalent (at least for not very short times) discrete time formulation is more suitable for numerical simulations. In one elementary time step one lattice site is chosen randomly and its state evolves according to the rules defined is Sec. \[sec:model\] using rescaled rates (all less than 1) as transition probabilities. One MC step is defined as the time needed such that all the sites have been, on the average, visited once. In this paper we always use the original time units defined by the model, which can be obtained simply by rescaling the time measured in MC steps.
For sufficiently large system, the stationary state does not depend on the initial conditions. Usually we filled up the lattice completely with preys as an initial state and put a few predators on it. To obtain the stationary phase diagram and the stationary values of the quantities of interest the number of MC steps performed varied in the range $10^6$ to $10^5$ for systems of linear size $L=200$ to 1000 respectively.
The corresponding phase diagram is depicted on Fig. \[fig:phase\]. Two different phases are present as a function of the two control parameters $\lambda_a$ and $\lambda_b$: a pure prey phase, a prey and predator coexistence phase with an oscillatory and a non-oscillatory region. In the oscillatory region, oscillations with a well defined frequency were observed in the prey and the predator densities (see Fig. \[fig:dens\_time\]). Although theoretically possible, we never observed an empty lattice absorbing state. The reason for that is simply that even one surviving prey fill up the system with preys in the absence of predators. As Fig. \[fig:phase\_t\] shows, the locations of the different regions of the phase space differ essentially from those obtained for the ST model.
The phase boundaries of the prey phase (see Fig. \[fig:phase\]) were obtained in the following way. Simulations were started at parameter values for which the coexistence is maintained practically forever (up to the maximal number of MC steps investigated), and we decreased one of the parameter values by $\Delta \lambda$. If the predators were still alive after a given time, $\Delta t$, we decreased the parameter further. The extinction of the predators defines the phase boundary. $\Delta \lambda$ was chosen to be in the range 0.005 to 0.04, while $\Delta t = 3\times 10^4$ MC steps. The result was very similar with $\Delta t = 10^4$ and $5\times 10^4$ MC steps. The definition of the boundary between the oscillatory and the non-oscillatory region of the coexistence phase will be described later.
On Fig. \[fig:phase\], the boundary of the prey phase is displayed for different system sizes ($L=100,..,1000$). Apparently, in the $\lambda_a>\lambda_b$ regime the size dependence is negligible, but relevant for $\lambda_a<\lambda_b$. Note that this strong size dependence of the boundary coincides with the presence of oscillations.
Decreasing $\lambda_b$ at any fixed value of $\lambda_a$, a second order phase transition takes place between the coexistence and the prey absorbing phases along a transition line $\lambda_b^*(\lambda_a)$. As for the mean field case, the predator density is considered to be the order parameter. As $\lambda_b \to \lambda_b^*(\lambda_a)$, the order parameter, $b$, and $1-a$ go to zero as b \~1-a \~(\_b - \_b\^\*(\_a))\^[\_1]{} . \[dp\_op\] As seen on Fig. \[fig:phase\], the values of $\lambda_b^*(\lambda_a)$ obtained by fitting the data with Eq. \[dp\_op\] are in very good agreement with the phase boundary obtained previously. Fitting the data leads $\beta_1 \approx 0.58(1)$ (with satisfactory precision for $\lambda_a > 0.3$; see Fig. \[fig:dens\_dp\]).
In the same limit the fluctuations of the predator density also follow a power law behavior, $\chi_b \sim (\lambda_b - \lambda_b^*(\lambda_a))^{\gamma_1}$. The exponent has been determined to a good precision as $\gamma_1\approx 0.35(3)$ for several values of $\lambda_a$ between 1 and 50. The same behavior has been obtained (only for $\lambda_a=1$ and 3) for the prey fluctuations, $\chi_a$, with the exponent $\gamma_1\approx 0.35(5)$. The critical behavior seems to be the same when the transition line is crossed while decreasing $\lambda_a$ at fixed values of $\lambda_b$. The two exponents, $\beta_1$ and $\gamma_1$, are compatible with those obtained for DP in $2+1$ dimension [@jensen]. Thus we conclude, that this absorbing state phase transition belongs to the DP universality class, as expected on general grounds [@DPconj]. Note that DP type phase transition in a similar lattice prey-predator system has already been observed in 1 dimension [@lipowska].
For $\lambda_a \to 0$, the transition line, $\lambda_b^*(\lambda_a)$, ends in a special point, ($\lambda_a=0,\lambda_b^T\approx 5.0(3)$), where all the three phases meet. For $\lambda_b>\lambda_b^T$, the MC results for the finite size dependent phase boundary suggest us that the transition happens at $\lambda_a^* = 0$ in the $L\to\infty$ limit. Approaching this transition line, $\lambda_a \to 0$, the prey density, $a$, does not go to 1 but to a finite value depending on $\lambda_b$ (see Fig. \[fig:dens\_a\]). However, according to the results depicted on Fig. \[fig:dens\_b\], the predator density, $b$, always goes to zero in this limit as a power of $\lambda_a$ with an exponent $\beta_2 \approx 1$. Surprisingly, this second order phase transition to the prey absorbing phase does not belong to the DP universality class. The presence of power law behavior, however, confirms that for an infinite system the transition occurs at $\lambda_a^* = 0$ for $\lambda_b > \lambda_b^T$. This means that for this range of $\lambda_b$, and for any arbitrary small $\lambda_a$, the coexistence of the species is possible providing that the system is large enough.
For $\lambda_b>\lambda_b^T$ the fluctuations of the two densities, $\chi_a$ and $\chi_b$, behaves similarly. For a given $\lambda_b$, there is a clear crossover at $\lambda_a^O(\lambda_b)$ from a mean field like behavior to a regime where the correlations are more important. For $\lambda_a > \lambda_a^O(\lambda_b)$ the behavior of $\chi_a$ and $\chi_b$ agrees with that predicted by mean-field theory, reflecting the fact that in this range of $\lambda_a$ the dominant behavior comes from the noise. As the $\lambda_a < \lambda_a^O(\lambda_b)$ condition coincides with the presence of oscillations, the crossover point, $\lambda_a^O(\lambda_b)$, is taken as the definition of the border between the oscillatory and the non-oscillatory region.
After a proper normalization, the relative fluctuations collapse on a single curve for $\lambda_a<\lambda_a^O(\lambda_b)$ (see Fig. \[fig:chi\]). Namely, K\_1(\_b) , where the numerical factor, $K_1(\lambda_b)$, depends only on $\lambda_b$. However, the precision of the simulation results was not satisfactory enough to obtain the functional form of $K_1(\lambda_b)$ (and of the forthcoming $K_i(\lambda_b)$ for $i=2$ ,3 and 4 either).
The simulation showed that $\chi_\rho$ ($\rho=a$ or $b$) is size independent as it was expected from its definition (\[defchi\]). As a consequence, the deviation from the average density, $\sigma = \sqrt{\chi_\rho}/L$ [@tome], is smaller for larger systems and evidently scales with $1/L$. Certainly, this deviation increases with the intensity of the oscillations. The above finite size behavior is in agreement with the results of earlier simulations which claimed that the oscillations in the global densities disappear with increasing system size [@boccara]. Our simulations predict more pronounced oscillations for smaller $\lambda_a$ and for larger $\lambda_b$.
The oscillations have to show up also in the correlation functions, C\_a(i, ) &=& (1 - \_[\_j(t), 0]{}) (1 - \_[\_[j+i]{}(t+), 0]{}) , C\_b(i, ) &=& \_[\_j(t), 2]{} \_[\_[j+i]{}(t+), 2]{} , where $j+i$ labels a lattice site distant of $i$ lattice spacing from the site $j$. $C_\rho$ ($\rho=a$ or $b$) depends only on $i$ and $\tau$ due to the homogeneity of the system in space and time. For $\tau=0$ the correlation function, $C_\rho(i)$, obtained numerically could be fitted by an exponential $C_\rho(i) \sim \exp(-i/\xi_\rho)$. In the oscillatory region the correlation lengths of preys and predators differ only through a $\lambda_b$ dependent factor, $\xi_a\approx K_2(\lambda_b)\xi_b$, and they turned out to be proportional to the fluctuations of the prey density, $\xi_a \approx \sqrt{2\chi_a}$. It means that a more correlated system displays stronger oscillations. The reason for that is simply that the dynamics of the different sites shows some synchronization within a correlation length, which results in larger oscillations (see Sec. \[sec:discuss\] for more details).
In order to determine the characteristic frequency, $\omega_\rho(\lambda_a, \lambda_b)$, and the amplitude, $A_\rho(\lambda_a, \lambda_b)$ ($\rho=a$ or $b$), of the oscillations, we measured the Fourier spectrum of the time dependent densities S\_() = \_[T]{} [1 T]{} | \_[t=1]{}\^T (t) (it) |\^2 . \[defps\] The presence of oscillations is reflected as a peak at nonzero frequency in the Fourier spectrum. Extracting this peak from a background noise, enable us to define $A_\rho$ and $\omega_\rho$ as the zeroth and the first momentum of this distribution. This analysis shows clearly that the frequency of the oscillations is independent of the system size (see Fig. \[fig:freq\]), and is the same for preys and predators. Moreover, for a wide range of the parameters in the oscillatory phase the frequency, $\omega=\omega_a=\omega_b$, is well approximated by $\lambda_a/2$. This linear behavior differs from the mean field prediction.
In the oscillatory region the oscillations are present for arbitrary large systems, however, their amplitude decreases with increasing system size, as $1/L^2$. At this point it is important to emphasize that this fact does not imply that only small oscillations are present in large systems. Indeed, for a large system the amplitude of the oscillations can be made larger by decreasing $\lambda_a$. On the other hand, when increasing $\lambda_a$ the amplitude goes to zero as a power law which makes difficult to define a phase boundary for the oscillations in this way. However, there is a simple scaling relation between the amplitude and the correlation length for the preys in the oscillatory region \_a\^2 2 \_a L\^2 A\_a , \[xca\] as it can be observed on Fig. \[fig:scale\]. The analogous expression for the predators is slightly more complicated \_b\^2 ()\^2 K\_3(\_b)\_b K\_4(\_a) L\^2 A\_b , \[xcb\] with appropriate $K_3(\lambda_b)$ and $K_4(\lambda_a)$ values.
Another quantity which characterizes the oscillations is the time dependent local correlations, $C_\rho(\tau) = C_\rho(i=0, \tau)$. A similar investigation was made in [@provata] with time dependent correlations of the average local densities. In the oscillatory region $C_\rho(\tau)$ displays damped, size independent oscillations. More precisely, the time correlations are size independent for any $L>L_c(\lambda_a, \lambda_b)$, while for any $L<L_c$ the system evolves to the prey absorbing state. Clearly, this critical system size is proportional to the correlation length, $L_c\sim\xi$. The size independence of $C(\tau)$ is a simple consequence of the fact that areas which are further than $\xi$ apart are uncorrelated. The investigation of the time dependent correlations, however, provides a rather ambiguous way to define the boundary of the oscillatory region. Indeed, one can observe local oscillations everywhere in the coexistence phase simply because, due to the cyclic dominance nature of the model, each site has to evolve in a loop ($\sigma=0 \to 1 \to 2 \to 0 \dots$). Thus, according to the value of the damping factor, it is somehow arbitrary to decide if the state is oscillatory or not.
It is worth noting, that at some particular values of $\lambda_b$ ($\lambda_b=10$ or 20) and for small $\lambda_a$ values ($\lambda_a <0.2$ or 0.4 respectively), where the correlation length is comparable to the system size ($L\sim 500$), the system can evolve to a stripe like state. In this state 3 stripes of size $L$, made of predator, prey and empty cells, are drifting through the system. However, for given $\lambda_a$ and $\lambda_b$ values, this behavior disappears when increasing the size of the system.
The comparison of the MC results with the mean-field prediction shows that the later gives a qualitatively correct description of the phase diagram (see Fig. \[fig:phase\_mf\]), as well as of the discontinuity in the prey density, $a$, along the $\lambda_a=0$ boundary.
Discussion {#sec:discuss}
==========
A qualitative understanding of the phase diagram is possible. If the birth rates are much larger than the death rate ($\lambda_a \gg 1$ and $\lambda_b \gg 1$) the system is full of preys and predators ($a \approx b \approx 1$), while for small values of $\lambda_b$ the system evidently reaches the pure prey absorbing state.
As already discussed in Sec. \[sec:model\], in the $\lambda_a \to \infty$ the system is full of preys ($a \to 1$) and the predators behave like the infected species in the CP. It means that they could survive only for $\lambda_b > \lambda_{CP}^*$, where a DP like second order transition occurs. This is in agreement with the mean field results and with the simulation for $\lambda_a=100$ (see Fig. \[fig:dens\_dp\]).
One can also derive an approximate formula for the position of the phase boundary between the non-oscillatory and the prey phase $\lambda_b^*(\lambda_a)$. For $\lambda_a\gg 1$, the system is almost full of preys ($a \approx 1$) and, in some sense, the dynamics of the predators is close to that of the CP. The predators die at rate 1 and spread at rate $\lambda_b$, but they cannot enter into the empty sites. One can introduce an effective $\tilde{\lambda_b}$ and describe the process as a CP, namely, the predators can enter any neighboring site at this rate. As the number of empty sites is proportional to leading order to $1/\lambda_a$, the effective parameter should be $\tilde{\lambda_b} = \lambda_b - c/\lambda_a$, where $c$ is a fitting parameter. As this CP displays a phase transition at $\tilde{\lambda_b} = \lambda_{CP}^*$, in terms of the original parameter the transition occurs at $\lambda_b^*(\lambda_a) = \lambda_{CP}^* + c/\lambda_a$. This conjecture is in excellent agreement with the simulation data for $\lambda_a>0.5$ with $c=1.28(3)$ (see Fig. \[fig:phase\]).
For $\lambda_b \gg 1$ the new prey sites are usually immediately occupied by predators as well. However, with a small but finite probability, a predator site can disappear before the predators spread to the new born prey site, and in this way, a prey site can be left alone and grow (similarly to the Eden model [@eden]). This rare event is negligible when the predator density is large enough and a prey island cannot grow for long periods of time. Practically this is the case for $\lambda_a > \lambda_{CP}^*$. In this case, the number of prey sites is negligible small, and the predator sites behave as the infected species in the CP. One can see on the Fig. \[fig:dens\_a\] and \[fig:dens\_b\], that for $\lambda_b=100$ the two densities ($a \approx b$) are equal to that of the CP if $\lambda_a > \lambda_{CP}^*$. However, in the vicinity of $\lambda_{CP}^*$ the densities are low, which allows an isolated prey island to grow for a long time. If $\lambda_a < \lambda_{CP}^*$ the predator islands are shrinking and, if $\lambda_a$ and $L$ are not too small, they can survive until a growing prey island reaches one of them. At this moment, the predators invade very quickly the prey territory and increase their population size (see Fig. \[fig:conf100\]). These new predator sites start to die out leaving a few prey sites alone, and the whole procedure starts again. This mechanism insures the survival of the predators much bellow the CP critical density and results in oscillations in the population sizes.
For $\lambda_b > \lambda_b^T$, but not too large, the qualitative picture is slightly different. As one can observe on Fig. \[fig:conf5\], groups of predators are wandering through the system towards prey-dense areas. If two fronts of predators meet they usually stop moving and the local population of predators starts shrinking. The oscillations are maintained in a somewhat similar way than for the $\lambda_b \gg 1$ case: these predators can only survive if the preys become dense around them. This is more probable for larger values of $\lambda_a$, and it is also clear that the predators have a better chance to survive in larger systems.
According to the above statements, the key point in the underlying mechanism of oscillations is the existence of blocked predator islands which are located and trapped in sparse prey areas. Indeed, blocked predators in sparse prey areas result in growing prey populations; however, the resulting dense prey population allows predators to move and predate again. This mechanism drives back the system to the beginning of the loop. Clearly, predators can only be trapped in sparse prey areas if $\lambda_a$ is smaller or of the order of the death rate, 1. This explains the location of the oscillatory region. Note, that the above argument is based on the spatial nature of the system, suggesting that the spatially extended character is fundamental for the existence of such prey-predator type of oscillations.
This mechanism also provides a qualitative understanding of the key properties of the system. The trapped predators can invade the prey area only when the preys are dense enough again, which takes a time proportional to $1/\lambda_a$, and leads to $\omega\sim\lambda_a$. According to the simulations the correlation length, $\xi$, increases with decreasing $\lambda_a$. Indeed, as $\lambda_a$ decreases, the trapped predators have to wait longer to escape, hence fewer groups of predators survive. This increases the distance between the groups, resulting in larger prey islands, which average size is proportional to $\xi_a$.
When the correlation length is of the order of the system size, there are islands of preys of typical size L, extruding the predators out of the system. Hence, the condition $\xi_a\sim L$ characterizes the phase boundary between the oscillatory and the prey phase. On the other hand, a correlation length of order one ($\xi_a\sim 1$), means that the noise dominates the system. Thus, $\xi_a\sim 1$ characterizes the boundary between the oscillatory and the non-oscillatory region of the coexistence phase.
As shown by the study of the time dependent correlations, domains separated by a distance larger than $\xi_a$ oscillate asynchronously around a constant value with the same frequency, $\omega(\lambda_a, \lambda_b)$. According to this picture, one can derive a more quantitative description for the oscillatory region. Let us assume that for $1\ll \xi_a \ll L$, the global densities of each species can be written as the sum of local coarse-grained densities at a typical length scale $\xi_a$. Moreover, we assume that all these local densities oscillate with the same frequency but a different phase, $\alpha_l$. In general, the amplitude of the local oscillations should depend on the parameters $\lambda_a$ and $\lambda_b$. However, as one can observe on Fig. \[fig:conf100\] and \[fig:conf5\], the predators can only enter an almost fully dense prey area and the predator fronts leave an almost empty field behind them. Hence, as suggested by the numerical simulations, everywhere in the oscillatory region, the local amplitude for the prey density can be considered as a constant, $d$. Thus a(t) = a\^s + d ()\^2 \_[l=1]{}\^[()\^2]{} (t + \_l) . Supposing that the $\alpha_l$ values change much more slowly than $\omega$, $a(t)$ shows a simple $\sin$ behavior for long periods of time (see Fig. \[fig:dens\_time\]). Thus, for $a(t)$ one can derive the value of the density fluctuations, $\chi_a$, and the amplitude of these oscillations, $A_a$, using (\[defchi\]) and (\[defps\]), and take the average over all the possible $\alpha_l$ configuration taken from a flat distribution. This procedure reproduces the result of Eq. (\[xca\]) up to a multiplicative factor in front of the correlation length.
Conclusions {#sec:conc}
===========
We have studied a two dimensional prey-predator model, (size $L \times L$), which exhibits a rich stationary state phase diagram. A particular attention has been payed to the study of finite size effects, and we were able to draw clear cut conclusions concerning the behavior of the model both for $L$ finite as well as for the limit $L \to \infty$.
Three kinds of stationary states can be reached according to the values of the control parameters : a pure prey state, and two coexisting prey-predator ones with and without oscillations. Two different kinds of second order transitions were found when going into the prey absorbing phase. The transition between oscillatory and non-oscillatory coexistence phase is, in fact, a cross-over between two asymptotic regimes characterized by a very small and a large correlation length respectively. In the oscillatory regime, scaling relations were established between several physical quantities.
A qualitative explanation for the existence of such oscillatory regime is given, pointing out the crucial role of the spatial extension of the system. Indeed, the frequency of the oscillations is determined locally due to the dynamics related to blocked predator islands in sparse prey areas. Regions of linear size $\xi_a$ oscillate with the same frequency but with different phases. This explains the decreasing amplitude of oscillations with increasing system size. On the other hand, slowly changing phases result periodic oscillations in the overall prey density for long periods of time. Moreover, for suitable choices of the control parameters one can have synchronized oscillations with finite amplitude across arbitrary large systems. Thus we think that our simple model could offer a qualitative explanation for the behavior of the lynx population problem described in Sec. \[sec:intro\].
We thank Z. Rácz and G. Szabó for helpful discussions. This work has been partially supported by the Swiss National Foundation.
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---
abstract: 'We present a perturbation analysis of the semiclassical Wigner equation which is based on the interplay between configuration and phase spaces via Wigner transform. We employ the so-called harmonic approximation of the Schrödinger eigenfunctions for single-well potentials in configuration space, to construct an asymptotic expansion of the solution of the Wigner equation. This expansion is a perturbation of the Wigner function of a harmonic oscillator but it is not a genuine semiclassical expansion because the correctors depend on the semiclassical parameter. However, it suggests the selection of a novel ansatz for the solution of the Wigner equation, which leads to an efficient regular perturbation scheme in phase space. The validity of the approximation is proved for particular classes of initial data. The proposed ansatz is applied for computing the energy density of a a quartic oscillator on caustics (focal points). The results are compared with those derived from the so-called classical approximation whose principal term is the solution of Liouville equation with the same initial data. It turns out that the results are in good approximation when the coupling constant of the anharmonic potential has certain dependence on the semiclassical parameter.'
author:
- |
E.K. Kalligiannaki[^1] & G.N. Makrakis [^2] [^3]\
Department of Applied Mathematics,\
University of Crete, 71409 Heraklion, Crete,\
Greece
title: |
Perturbation solutions of the\
semiclassical Wigner equation
---
Introduction
=============
#### WKB solutions and geometrical optics.
We consider the oscillatory initial value problem
$$\label{Schro}
\left\{
\begin{array}{l}
i\eps \frac{{\partial \psie }}{{\partial t}}= \Hop \psie(x,t),\ \ x\in
\R,\ t>0 \\
\psie(x,0)=\psie_0(x) \ , \ \ \eps<<1 \ ,
\end{array}
\right.$$
where $\Hop = -\frac{\eps^2}{2}\partial^{2}_{x} + V(x)$ is the usual quantum mechanical Schrödinger operator. High frequency waves that satisfy (\[Schro\]) with oscillatory initial data $$\label{wkb0}
\psie_0(x)=A_0(x)\exp(i S_0(x)/\eps) \ ,$$ have been traditionally studied by geometrical optics. This method departs from the construction of WKB approximate solutions in the form ([@BABU], [@BLP], [@KO]) $$\label{wkb}
\psie (x,t) = A(x,t)\exp(i S(x,t)/\eps) \ ,$$ where the phase function $S(x,t)$ and the amplitude $A(x,t)$ are usually assumed to be real-valued functions, although extensions of the method for complex-valued phases have been also developed.
Substituting (\[wkb\]) into (\[Schro\]), and retaining terms of order $O(\eps)$ and $O(1)$ with respect to $\eps$, we obtain the Hamilton-Jacobi equation for the phase
$$\label{eikonal}
\partial_{t}S + (\partial_{x}S)^{2}/2 +V =0 \ , \ \ \ \ S(x, t=0)= S_{0}(x) \ ,$$
and the transport equation for the amplitude
$$\label{transport}
\partial_{t}(A^2) +\partial_{x}\left(A^2\partial_{x}S\right) =0
\ , \ \ \ \ A(x,t=0)=A_{0}(x) \ .$$
The system (\[eikonal\])-(\[transport\]) is integrated byreduction to a system of ordinary differential equations along bicharacteristics as follows. We define the Hamiltonian function $$H(x,k)= k^{2}/2 + V(x) \ ,$$ and construct the characteristics $(\tilde{x}(t;q),
\tilde{k}(t;q))$, as the trajectories of the Hamiltonian system
$$\begin{aligned}
\label{hamsy}
\frac{d\tilde{x}}{dt}=H_{k}(\tilde{x}, \tilde{k})=\tilde{k} \ , \
\ \ \ \ \frac{d\tilde{k}}{dt}=-H_{x}(\tilde{x}, \tilde{k})=V'(\tilde{x})\end{aligned}$$
with initial conditions $$\tilde{x}(t=0;q)= q \ , \ \ \ \ \ \ \tilde{k}(t=0;q)= S'_{0}(q) $$ The projection of the characteristics $\tilde{x}=\tilde{x}(t;q)$ on the configuration space are the physical rays of geometrical optics. Then, the phase function $S$ is obtained by integration of the ordinary differential equation $$\frac{dS}{dt}= \partial_{t}S + \partial_{x}S\frac{dx}{dt}=-H(\tilde{x}(t;q), \tilde{k}(t;q)) + \tilde{k}^{2}(t;q)/2=\tilde{k}^{2}(t;q) -V(\tilde{x}(t;q)) \ ,$$ with the initial condition $$S(\tilde{x}(t=0;q))=S_{0}(q) \ .$$
Furthermore, the amplitude $A$ is calculated by applying the divergence theorem in a ray tube, and is given by $$A(\tilde{x}(t;q),t)= \frac{A_{0}(q)}{\sqrt{J(t;q)}} \ ,$$ where $$J(t;q)= \frac{\partial \tilde{x}(t;q)}{\partial q}$$ is the Jacobian of the ray transformation $q \mapsto\tilde{x}(t;q)$.
The nonlinear Hamilton-Jacob equation has, in general, multivalued solutions. This means that singularities may be formatted in finite time, when the Jacobian of the ray transformation vanishes. The set $\mathcal {C}=\{x= \tilde{x}(t;q) :J(t;q)=0 \} $ of these singularities is the caustic of the ray field. On the caustics the amplitude becomes infinite. Therefore, near such singularities the WKB method fails to predict the correct wave field. in the sense that the method cannot describe the correct scales. These scales are predicted either from analytical solutions of model problems, or from uniform asymptotic expansions for the solution of the Schrödinger equation. Such solutions show that the amplitude of the intensity of the wave field increases with the frequency $\eps^{-1}$, but, for any fixed large frequency, it remains bounded with respect to space variables on the caustics.
Assuming that the multivalued function $S$ is known away from caustics, and using boundary layer techniques and matched asymptotic expansions [@BABU], [@BK], it has been possible to constructed uniform asymptotic expansions near simple caustics. However these analytical techniques are very complicated since the matching procedure depends on the form of the particular caustic.
A different group of uniform methods valid near caustics, is based on integral representations of the solutions in phase-space. The basic methods in this category are the Maslov’s canonical operator[@MF]. [@MSS], and the Lagrangian integrals (Kravtsov-Ludwig method [@KRA], [@Lu], [@KO]), which can be considered as special cases of Fourier integral operators [@Du], [@TRE].
All the above described techniques assume an ansatz for the solution in the configuration space, which for the final determination requires the knowledge of the multivalued phase functions, or, geometrically, of the Lagrangian manifold generated by the bicharacteristics of the Hamiltonian system in phase space.
An alternative approach is based on the use of the Wigner transform. This is a function defined on phase space as the Fourier transform of the two-point correlation of the wave function. It was introduced by Wigner [@Wig] for specific purposes in quantum thermodynamics, and recently it has been successfully used in semiclassical analysis for the reformulation of wave equations as non local equations in phase space, and the study of of homogenization problems in high-frequency waves [@GMMP].
#### The Wigner equation.
The Wigner transform of a function $f \in L^2$, is defined as $$\label{wignerd}
\We[f](x,k)=\frac{1}{2\pi\eps}\int_\R e^{-\frac{i}{\eps}k\xi} f(x+\frac{\xi}{2})\overline{f}(x-\frac{\xi}{2}) d\xi$$ and for a pair of functions $ f, g\in L^2(\R)$, the (cross) Wigner transform is defined as $$\label{wignerfg}
\We[f,g](x,k)=\frac{1}{2\pi\eps}\int_\R e^{-\frac{i}{\eps}k\xi} f(x+\frac{\xi}{2})\overline{g}(x-\frac{\xi}{2}) d\xi \ .$$
By its definition $\We[f](x,k)$ is a real , square integrable, but not necessarily non-negative function in phase space. For this reason it is not a pure probability distribution function, but it is exactly this property that makes the Wigner function a powerful tool in the study of wave and quantum interference phenomena.
Some of the most important relations of Wigner functions $$\label{wignfunc}
\We(x,k):=\We[\psie](x,k)$$ with the wave function $\psie(x)$ (this in general depends also on time $t$), that are useful for the computation of physical quantities both in classical wave propagation and in quantum mechanics, are the following
1\) The integral of $\We$ with respect to the momentum $k$ gives the energy density $$\label{ampl}
\eta^{\eps}(x):= |\psie(x)|^{2}=\int_{\R} \We(x,k)dk \ ,$$ while the first moment with respect to the momentum gives the energy flux, $$\mathcal{F^{\eps}}(x) := \frac{\eps}{2i}\Bigl(\cpsie
\partial_{x}\psie -\psie \partial_{x} \cpsie \Bigr)=\int_{\R} k \We(x,k)dk$$ and its integral over the whole phase space, gives the total energy $$\int_{\R}\int_{\R}dxdk \We(x,k)=\| \psie\|_{L^2} \ .$$
2\) The Wigner transform of a WKB function $$\psie (x) = A(x)\exp(i S(x)/\eps) \ ,$$ considered as a generalized function, as $\eps \rightarrow 0$, has the weak limit $$\label{wkblimit}
\We[\psie](x,k)\rightharpoonup |A(x)|^{2}\delta\bigl(k- S'(x)\bigr) \ ,$$ where $\delta$ denotes the Dirac distribution [@LP], [@PR].
The evolution equation of the Wigner function $\We(x,k,t):=\We[\psie](x,k,t)$ corresponding to the solution $\psie(x,t)$ of (\[Schro\]) has the form[^4]
$$\label{tql}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\We(x,k,t)+\Le\We(x,k,t)=0, (x,k)\in\R^2, t>0\\
\We(x,k,t)|_{t=0}=\We_0(x,k)
\end{array}
\right.$$
where $\We_0(x,k)$ is the Wigner transform of the initial data $\psie_0(x)$. and $\Le$ is the pseudo differential operator defined by ([@BA], [@FAI])
$$\label{Les}
\Le \cdot := -\frac{2}{\eps}\sin\Bigl(\frac{\eps}{2} \Lambda \Bigr)H(x,k) \cdot$$
For a general Hamiltonian $H(x,k)$ the operator $\Lambda$ is the commutator $$\Lambda:=\frac{\partial}{\partial
x_H}\frac{\partial}{\partial k}-\frac{\partial}{\partial
x}\frac{\partial}{\partial k_H} \ ,$$ and $\sin\Bigl(\frac{\eps}{2} \Lambda \Bigr)$ acts on the product of two functions $f(x,k)$ and $g(x,k)$ by the formula $$\begin{aligned}
&\sin\Bigl(\frac{\eps}{2} \Lambda \Bigr)f(x,k) g(x,k)=\\
&=\frac{1}{2i\pi^2\eps^2}\int dk'dk''dx'dx''\left[f(x',k')g(x'',k'')-g(x',k')f(x'',k'')\right]\\
&\times \exp\left(-\frac{2i}{\eps}\left(k(x'-x'')+
k'(x''-x)+k''(x-x')\right)\right) \ .\end{aligned}$$
We also introduce for later use (see eq ($\ref{eig2}$) in Section 2) the pseudo-differential operator $$\label{Mes}
\Me \cdot :=\cos\Bigl(\frac{\eps}{2} \Lambda \Bigr)H(x,k)\cdot$$ which is known in physics’ literature as Baker’s cosine bracket, by ([@CFZ], [@FAI], [@TH]), and $\cos\Bigl(\frac{\eps}{2} \Lambda \Bigr)$ acts on the product of two functions $f(x,k)$ and $g(x,k)$ as follows $$\begin{aligned}
&\cos\Bigl(\frac{\eps}{2} \Lambda \Bigr)f(x,k) g(x,k)=\\
&=\frac{1}{2\pi^2\eps^2}\int dk'dk''dx'dx''\left[f(x',k')g(x'',k'')+g(x',k')f(x'',k'')\right]\\
&\times\exp\left(-\frac{2i}{\eps}\left(k(x'-x'')+
k'(x''-x)+k''(x-x')\right)\right) \ .\end{aligned}$$
In the particular case of the usual quantum mechanical Hamiltonian $H(x,k)=k^2/2 + V(x)$, by using (\[Les\]), the operator $\Le$ may also be written in the more standard form $$\Le=k\partial_x-\Theta^{\eps}[V]$$ where the operator $\Theta^{\eps}[V]$ is expresses as the convolution $$\Theta^{\eps}[V]\We(x,k,t):=Z^{\eps}(x,k)*_k\We(x,k,t) \ ,$$\[convtheta\] of the Wigner function with the kernel $$\label{zeta}
Z^{\eps}(x,k)=\frac{1}{i\eps}\frac{1}{2\pi}\int_{\R}e^{-iky}\left(V(x+\frac{\eps}{2}y)-
V(x-\frac{\eps}{2}y)\right)dy \ .$$ This is a non-local operator which the action of the potential on the evolution of Wigner function.
From and(\[convtheta\]) (\[zeta\]) we can write the action of $\Theta^{\eps}[V]$ as a pseudo-differential operator $$\label{theta}
\Theta^{\eps}[V]\We(x,k,t)=\frac{i}{2\pi\eps}\int_{\R}\int_{\R}e^{i(k-\xi)y}\We(x,\xi,t)\left[V(x+\frac{\eps}{2}y)-V(x-\frac{\eps}{2}y)\right]dyd\xi \ .$$
Now we observe that if the potential function $V$ is smooth enough, using its Taylor expansion we may rewrite quantum Liouville or Wigner operator $\Le$ in the form of an infinite order differential operator
$$\label{Le}
\Le= k\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial k}-
\sum_{j=1}^{\infty}\eps^{2j}\left(\frac{i}{2}\right)^{2j}\frac{V^{(2j+1)}(x)}{(2j+1)!}\frac{\partial^{(2j+1)}}{\partial k^{2j+1}} \ .$$
Thus the Wigner equation (\[tql\]) is written in the form of an infinite order ingular perturbation equation $$\label{wignereql}
\partial_{t}\We + k\partial_{x}\We- V'(x) \partial_{k}\We = \sum_{m=1}^{\infty} \alpha_m
\eps^{2m} V^{(2m+1)} (x)
\partial_{k}^{2m+1} \We(x,k,t) \ ,$$ where $\alpha_m = (-1)^m/2^{2m} (2m+1)!,\ \ m=0,1, \dots$, and $V^{(2m+1)} (x) =d^{2m+1} V(x)/dx^{2m+1}$.
The form (\[wignereql\]) of the Wigner equation shows that that this equation is a combination of the classical transport (Liouville) operator in the left hand side, with a dispersion operator of infinite order in he right hand side. Roughly speaking, this combination suggests that the phase space evolution results from the interaction between the classical transport of the Lagrangian manifold generated by the Hamiltonian and a non-local dispersion of energy from the manifold into the whole phase space. This picture is consistent with the fact that in the classical limit $\eps =0$ the dispersion mechanism disappears. Then, the solution of the Wigner equation converges weakly to the so called Wigner measure [@LP], which satisfies the Liouville equation of classical mechanics. This solution is an always well-defined semiclassical measure, and in the absence of caustics it completely retrieves the results of the WKB method
However, it has been shown in [@FM1], [@FM2] that in the case of multi-phase optics and caustic formation, the limit Wigner measure, although still well-defined as semiclassical measure on phase space, is not the appropriate tool for the computation of energy densities at a fixed point of configuration space, because; (a) it cannot be expressed as a distribution with respect to the momentum for a fixed space-time point, and thus it cannot be used to compute the amplitude of the wavefunction, on caustics, and (b) it is unable to “recognise” the correct scales of the wavefield near caustics. It has been also shown in [@FM1] that approximate Airy-type solutions of the Wigner equation can, produce reasonable solutions for multiphase problems, and at least for simple caustics,
Therefore the study of asymptotic solutions of the Wigner equation for small $\eps$ is promising for understanding the structure solutions, and for computing energy densities, in multiphase geometrical optics through integration of the Wigner function.
Several asymptotic solutions of the Wigner equation have been proposed in the recent past. Steinrück [@ST] and Pulvirenti [@PU], have constructed distributional asymptotic expansions near the solution of the classical Liouville equation, by expanding the initial data in a distributional series with respect to the small parameter. However, Heller [@EH] has noted that such expansions are not physically appropriate for studying the evolution of singular initial conditions, and he has proposed a different expansion where the first order term is the solution of a classical Liouville equation with an effective potential. The use of modified characteristics and effective potentials aims to include indirectly some quantum phenomena and it is a popular technique in physics and quantum chemistry for the treatment of the quantum Liouville equations (see, e.g., the review paper [@HWL]). It has led to reasonable numerical results, and, somehow, it can be used as an alternative of quantum hydrodynamics (Bohm equations) and of the technique of Gaussian beams. In the same direction, Narcowich [@FN1] proposed a different expansion near the classical Liouville equation without expanding the initial data with respect to the semiclassical parameter, which allows him to avoid the distributional expansions. In a rather different direction, the uniform Airy type asymptotic approximation of the Wigner function that was proposed by Berry [@BE], has been used in the works of Filippas & Makrakis [@FM1], [@FM2], to contract novel asymptotic solution of the Wigner equation in the presence of simple caustics.
#### Outline of the paper.
This paper aims to the understanding of asymptotic solutions of Wigner equation, by adopting a new strategy for the construction of asymptotic expansions, that exploits the interplay between configuration and phase spaces via Wigner teansform. Our approach has been motivated by the general idea of using spectral expansions in the construction of high-frequency solutions, which has been developed in [@BLP], Ch. 4, for Schrödinger equations. Eigenfunction expansions can be considered as ‘exact’’ solutions of the Cauchy problem ($\ref{Schro}$), which in contrary to the WKB solutions, do not face caustic problems. When transferred to the phase space by Wigner transform they give corresponding expansions which are “exact” solutions of the Wigner equation.
In this respect, our strategy is the following:
\(1) We construct the Moyal eigenfunctions of the Wigner equation (Section 2.1) and their asymptotic expansions in terms of the Moyal functions of a harmonic oscillator, which arises from the so-called harmonic approximation of Schrödinger eigenfunctions (Section 2.2). We assume a single well potential $V(x)$, so that the Schrödinger spectrum be purely discrete, and spectral information for the Wigner equation be also available.
\(2) We transform the eigenfunction expansion of the Schrödinger equation to the phase space, and we construct the solution of the Wigner equation as a series of phase-space Moyal eigenfunctions (Section 3.1). Then, we construct a [*formal*]{} asymptotic expansion (harmonic expansion) for the solution of Wigner equation using the expansions of Moyal eigenfunction derived in the first step, and finally
\(3) We propose an [*ansatz*]{} for the solution of the Wigner equation and we develop a regular-perturbation scheme in phase space, for computing the sought for asymptotic expansion for the solution of the Wigner equation (Section 3.2).
In Section 4 we present the so-called classical expansion, where the solution of the Wigner equation is expressed as a perturbation of the solution of classical Liouville equation.
As an application, in Section 5 we apply the proposed scheme for a quartic (anharmonic) oscillator, and we compute the wave amplitude at the beaks of the cusps generated by the oscillator through integration of the approximate Wigner functions with respect to the momentum. The predictions of the harmonic and the classical expansions agree at the singular points provided that a certain relation between the small semiclassical parameter and the coupling constant of the potential holds.
Aproximmation of Moyal eigenfunctions
=====================================
The Moyal eigenfunctions in phase space
---------------------------------------
It is known that, in principle, the spectrum of the quantum Liouville operator can be determined from the spectrum of the corresponding Schrödinger operator $\Hop$ (see, e.g. [@MA], [@SP]), and, in general, someone anticipates the formula $$\sigma(\Le)=\{\frac{i}{\eps}(E-E'), \ \ E, \ E' \in \sigma(\Hop)\} \ ,$$ to hold. In fact, this relation holds for the discrete spectrum $$\sigma_p(\Le)=\{\frac{i}{\eps}(\Ee_n-\Ee_m), \ \ \Ee_n, \Ee_m \in \sigma_p(\Hop)\} \ \ .$$ A similar formula holds for the point spectrum of the cosine bracket operator $\Me$ (eq ($\ref{Mes}$) below), that is $$\sigma_p(\Me)=\{\frac{1}{2}(\Ee_n+\Ee_m), \ \ \Ee_n, \Ee_m \in \sigma_p(\Hop)\} \ .$$
However, these formulae are not in general true for the absolutely and singular continuous spectrum. These spectral questions have been studied first by Spohn [@SP], and later by Antoniou et al [@ASS], who have proved the negative result $$\sigma_{sc,ac}(\Le)\neq \{\frac{i}{\eps}(E-E'), \ \ E, \ E' \in \sigma_{sc,ac}(\Hop)\} \ ,$$ where $\sigma_{sc,ac}$ denote the singular and absolutely continuous spectrum respectively.
In order to avoid the complications arising from the continuous spectrum (a;though this pertains to the most interesting cases of scattering problems), we consider operators $\Hop$ with purely discrete spectrum $\sigma(\Hop)=\sigma_p(\Hop)$, therefore $\sigma(\Le)=\sigma_p(\Le) $ and $\sigma(\Me)=\sigma_p(\Me).$ When the potential $V(x)\in L^1_{loc}(\R)$ is bounded below and $ \lim_{|x|\rightarrow \infty}V(x)=\infty $ it is known that $\Hop$ has purely discrete spectrum ( [@RSIV], [@BS]), and therefore the operators $\Le, \Me$ have also purely discrete spectrum. We denote by $\Ee_n$ and $\ue_{n}(x)$ the eigenvalues and the eigenfunctions of $\Hop$, that satisfy $\Hop \ue_{n}(x)=\Ee_n\ue_{n}(x) \ , \ \ n=1,2, \dots \ .$ It is known that $\ue_{n}$ form a complete orthonormal basis in $L^2(\R)$ .
The Moyal eigenfunctions, were introduced by Moyal [@MO], for the purposes of a concrete statistical study of quantum mechanics, and they are defined as the cross-Wigner transform (\[wignerfg\]) of the Schrödinger eigenfunctions $\ue_{n} \ , \ue_{m} \ ,\ \ \ n,m=0,1,2,\dots$.
$$\label{phinm}
\Phinm(x,k):=\We[\ue_{n},\ue_{m}](x,k)= \frac{1}{\pi\eps}\int_{\mathbf{R}}e^{-i\frac{2k}{\eps}\sigma}
\ue_{n}(x+\sigma)\overline{\ue_{m}}(x-\sigma)d\sigma \ .$$
For these functions the following theorem holds [@MO], [@TH].
Let $\Hop$ has purely discrete spectrum $\{\Ee_n\}_{n=0,1,2,\dots}$ with complete orthonormal system of eigenfunctions $\{\ue_{n}(x)\}_{n=0,1,\dots}$ in $ L^2(\R) $. Then, the functions $\{\Phi^{\eps}_{nm}\}_{ n,m=0,1,\dots} $ form a complete orthonormal basis in $L^2(\R_{xk}^2)$, and they are common eigenfunctions of the operators $\Le$ and $\Me$ with eigenvalues $\lambda_{nm}=\frac{i}{\eps}\left(\Ee_n-\Ee_m\right)$ and $\mu_{nm}=\frac{1}{2}\left(\Ee_n+\Ee_m\right)$, respectively.
Therefore, $\Phinm(x,k)$ satisfy the eigenvalue problems $$\label{eig1}
\Le\Phi^{\eps}_{nm}(x,k)=\lambda_{nm}\Phi^{\eps}_{nm}(x,k)$$
$$\label{eig2}
\Me\Phi^{\eps}_{nm}(x,k)=\mu_{nm}\Phi^{\eps}_{nm}(x,k) \ ,$$
in phase space $L^2(\R_{xk}^2)$
[**Remark 1.**]{} It is very important to note that for the computation of Moyal functions, directly in phase space, we need both eigenvalue problems, (\[eig1\]) and (\[eig2\]), as it has been explained in [@CFZ], [@KP]. Moreover, there is no evolution equation in phase space which corresponds to the eigenvalue equation (\[eig2\]) and which could be deduced from Schrödinger formulation , as is the case for the quantum Liouville equation. This means that (\[eig2\]) cannot result naturally from some initial value problem governing the Wigner function. In order to derive the second eigenvalue equation directly from phase space, Fairlie & Manogue [@FM] extended the Wigner function by introducing an imaginary time variable $s$, thus constructing a second initial value problem, with time derivative $i\partial_s$ and space operator $\Me$, for the extended Wigner function. Both the mathematical role and the physical content of this new function are still to be understood.
[**Remark 2.**]{} It is also important to note that it is not possible to compute the limits of the Moyal eigenfunctions $\Phi^{\eps}_{nm}$ as $\eps\to 0$, for any $n,m$ independent of each other, a situation which can be somehow considered as a consequence of the Bohr-Sommerfeld quantisation rule. This situation is a fundamental obstruction for the computation of the limit of the phase-space eigenfunction expansion of the Wigner function the $\eps\to 0$, from which one would expect to obtain an analogous generalised expansion of the solution of the classical Liouville equation. What can be evaluated is the classical limit $\eps\to 0$, when $\ n,m \to \infty$ and $n-m=$ constant. For integrable Hamiltonians, Berry & Balazs [@BE] [@BB] have computed the classical limit of Moyal functions $\Phinm$ in the case $n=m$, which reads as $\Phi^0_{nn}(I,\theta)=\delta\left(H(I)-(E^0_n)\right)$, where $H(I)$ is the Hamltonian in action angle variables $(I,\theta)$. In the “simplest” case of the the harmonic oscillator, Ripamonti [@RN] and Truman & Zhao [@TZ] have given independent proofs for the classical limit of the corresponding Moyal eigenfunctions $\Phinm$ for all $n,m=0,1,\dots$, based on the asymptotics of Laguerre polynomials. Finally, a formal computation in [@WB] shows that the classical limit of Moyal eigenfunctions, in terms of action-angle variables , and for all $n \neq m$, reads as $$\Phi^0_{nm}(I,\theta)=e^{-i\frac{E^0_{nm}}{\gamma_{nm}}\theta}\delta\left(H(I)-\frac12(E^0_n+E^0_m)\right) \ .$$ where $ E^0_{n}=lim_{\eps \rightarrow 0} \frac{\Ee}{\eps}$, $ E^0_{nm}=E^0_{n}-E^0_{m}$, and $\gamma_{nm}=H'\left(H^{-1}(\frac12(E^0_n+E^0_m))\right)$. Also from this formal computation becomes evident the necessity of both eigenvalue equations (\[eig1\]) and (\[eig2\]).
Harmonic approximation of Moyal eigenfunctions
----------------------------------------------
We proceed now to the construction an asymptotic expansion of the Moyal eiegenfunctions, for small $\eps$, starting from the so-called harmonic approximation of the eigenfunctions of the Schrödinger operator in the configuration space.
It is known that the eigenfunctions and eigenvalues of the Schrödinger operator $\Hop$ can be approximated by the corresponding ones of an appropriate harmonic oscillator, provided that the potential $V(x)$ satisfies the following conditions ([@HS], [@BS])
\(i) $V \in C^{\infty}(\R) $\
(ii) $V\geq 0, \textrm{ for some } R>0
\mathop {\inf }\limits_{|x| > R}V(x) >0 \ ,$\
(iii) $V' \textrm{ has finite number of zeros } \{x^{(\alpha)}\}_{\alpha=1}^\kappa \ ,$\
(iv)$ \textrm{ for each }x^{(\alpha)}\ \ V''(x^{(\alpha)})>0 \ ,$\
(v) $V \textrm{ polynomially bounded } |V(x)|\le c(1+|x|^m) \ .$
For simplicity we consider only the case $\kappa=1$, we adopt the normalization $x^{(1)}=0$ and, without loss of generality, we also assume that $V(0)=0$. Although asymptotics of eigenvalues and eigenfunctions are also available for multiple wells, in order to deal with this case it is necessary to consider detailed information on the decay of the eigenfunctions (see, e.g., [@CODUSE] and the references therein) and take account of tunnelling effects for the Wigner function which is a rather complicated task [@BAVO].
Then, the eigenvalues $\Ee_n$ and the eigenfunctions $\ue_n(x),\ n=0,1,\dots \ ,$ have the asymptotic expansions $$\label{ass1}
\frac{2}{\eps}\Ee_n=e_n+\sum\limits_{l=1}^{N}a_n^{(l)}\eps^l+O\left(\eps^{N+1}\right) \ ,$$ and $$\label{ass2}
\eps^{\frac14}\ue_n(x)\sim \psi_{n}(\frac{x}{\sqrt{\eps}})+ \sum\limits_{l=1}^{\infty} \eps^{\frac{l}{2}}\psi_n^{(l)}(\frac{x}{\sqrt{\eps}}) \ , \ \ \psi_n^{(l)}\in L^2(\R), n=0,1,\dots,\ \ l=1,2,\dots \ ,$$ respectively, where $e_n$, $\psi_n(x)$ are the eigenvalues and eigenfunctions of the harmonic oscillator $\Hop_h:=-\triangle+x^2 $, $$\begin{array}{l}
e_n=2n+1,\ \ \\
\psi_n(x)=(2^nn!\sqrt{\pi})^{-\frac12}e^{-\frac{x^2}{2}}H_n(x),\
n=0,1,2,..
\end{array}$$ $H_n(x)$ being the Hermite polynomials [@TH]. Hence we refer to the expansions (\[ass1\]), (\[ass2\]) as the harmonic approximation.
Note that (\[ass2\]) is understood in the sense that $$||[U_\eps^{-1}\ue_n-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}||_{L^2(\R)}=O(\eps^{(N+1)/2}) \ .$$ where $U_\eps$ is a unitary dilation of $f\in L^2(\R)$ defined by $$\left[U_{\eps}f\right](x):=\eps^{-\frac14}f\left(\frac{x}{\sqrt{\eps}
}\right),\ x\in \R, \ .$$
The coefficients $a_n^{(l)},\psi_n^{(l)}, n=0,1,\dots \ , \ l=1,2,\dots \ ,$ can be computed in closed form by the Rayleigh-Schrödinger perturbation technique [@RSIV].
By substituting the asymptotic expansions (\[ass2\]) of the eigenfunctions $\ue_n$ into th definition of the Moyal eigenfunctions $\Phinm(x,k),\ n,m=0,1,\dots$, we get the formal expansions
$$\label{asf1}
\Phinm(x,k)\sim \Psinm(x,k) + \sum\limits_{l=1}^{\infty}\eps^{\frac{l}{2}}\Znm^{\eps,(l)}(x,k) \\$$
The first term of the expansion the Moyal eigenfunction of the harmonic oscillator $V_h(x)=x^2/2$, that is $$\Psinm(x,k):=\frac{1}{\pi\eps}\int_{\mathbf{R}}e^{-i\frac{2k}{\eps}\sigma}
\psie_{n}(x+\sigma)\overline{\psie_{m}(x-\sigma)}d\sigma,\
n,m=0,1,2,\dots$$ with $\psi^{\eps}_n(x)=\left[U_{\eps}\psi_n\right](x)$, and $Z_{nm}^{\eps,(l)}$ given by $$\label{znm}
\begin{array}{l}
Z_{nm}^{\eps,(l)}(x,k)=[U_{\eps,(xk)}\tZnm^{(l)}](x,k)\\
\tZnm^{(l)}(\cx,\ck)=\sum\limits_{\mu=0}^{l}W[\psi_n^{(\mu)},\psi_m^{(l-\mu )}](\cx,\ck) \ .
\end{array}$$ Here $W[f,g](x,k)=\frac{1}{\pi}\int_{\R}e^{-i2k\sigma}f(x+\sigma)\overline{g}(x-\sigma)d\sigma$ is the cross-Wigner transform for $\eps=1$, and $U_{\eps,(xk)}$ is a dilation of $f\in L^2(\R_{xk}^2)$ in phase space, which is defined by $$\label{dilph}
\left[U_{\eps,(xk)}f\right](x,k):=\eps^{-1}f\left(\frac{x}{\sqrt{\eps}},\frac{k}{\sqrt{\eps}
}\right) \ , \ \ (x,k) \in \R_{xk}^2 \ ,$$ and it has the property $\|\left[U_{\eps,(xk)}f\right]\|_{L^2}=\frac{1}{\sqrt{\eps}}\|f\|_{L^2}$. Since the first term of the expansion (\[ass1\]) pertains to the harmonic oscillator, we refer to it as the [*harmonic expansion (or harmonic approximation)*]{} of the Moyal eigenfunctins.
For simplifying the formulae, in the sequel we introduce the notation ${\widetilde f}(\cx,\ck)=[U^{-1}_{\eps,(xk)}f](\cx,\ck)$.
The asymptotic expansion (\[asf1\]) of $\Phinm(x,k)$ is written in the scaled variables $(\cx,\ck)$ as $$\label{asf2}
\left[U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}\right](\cx,\ck) \sim \Psi_{nm}(\cx,\ck) + \sum\limits_{l=1}^{\infty}\eps^{\frac{l}{2}}\tZnm^{(l)}(\cx,\ck)\\$$ since $ \Psinm(x,k)=[U_{\eps,(xk)}\Psi_{nm}](x,k)$, where $\Psi_{nm}(x,k)$, are the cross-Wigner transform of Hermite functions $\psi_{n}$, $$\Psi_{nm}(x,k):=\frac{1}{\pi}\int_{\R}e^{-i2k\sigma}
\psi_{n}(x+\sigma)\overline{\psi_{m}(x-\sigma)}d\sigma.$$
Furthermore, by substituting the expansion (\[asf1\]) into the eigenvalue equations (\[eig1\]) and (\[eig2\]), and equating the coefficients of same powers of $\eps$ as it is customary in regular perturbation schemes, we expect to obtain a hierarchy of equations for the coefficient functions $Z_{nm}^{\eps,(l)}$. This procedure is quite cumbersome since the operators $\Le$ and $\Me$ depend also on the parameter $\eps$. The first step is to rescale the eigenvalue problems by using the transform $U_{\eps,(xk)}$ and then we use the properties of the potential to expand appropriately the phase space pseudo-differential operators.
Applying the transform $U_{\eps,(xk)}$ onto the eigenvalue problems (\[eig1\]), (\[eig2\])[^5], we get , $$\label{eig11}
{\widetilde L}^{\eps}U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}=\frac{i}{\eps}(\Ee_n-\Ee_m
)U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}$$ and $$\label{eig22}
{\widetilde M}^{\eps}U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}(x,k)=\frac12(\Ee_n+\Ee_m )U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}(x,k) \ .$$ The operators ${\widetilde L}^{\eps}$ and ${\widetilde M}^{\eps}$ are derived from $\Le$ and $\Me$ are derived by conjugation with the phase-space dilation
$${\widetilde L}^{\eps}:=[U_{\eps,(xk)}]^{-1} \Le [U_{\eps,(xk)}] \ ,$$
$${\widetilde M}^{\eps}:= [U_{\eps,(xk)}]^{-1} \Me [U_{\eps,(xk)}].$$
Note that for smooth potential $V(x)$, we can also use the expansions $$\Le= k\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial k}-
\sum_{j=1}^{\infty}\eps^{2j}\left(\frac{i}{2}\right)^{2j}\frac{V^{(2j+1)}(x)}{(2j+1)!}\frac{\partial^{(2j+1)}}{\partial k^{2j+1}}$$ and $$\Me= -\frac{\eps^2}{8}\triangle_{xk}+H(x,k)
+\sum_{j=1}^{\infty}\eps^{2j}\left(\frac{i}{2}\right)^{2j}\frac{V^{(2j)}(x)}{(2j)!}
\partial^{(2j)}_k+\frac{\eps^2}{8}\partial^2_k \ ,$$ to get the corresponding expansions of ${\widetilde L}^{\eps}$ and ${\widetilde M}^{\eps}$. These read as $$\label{aplt}
{\widetilde L}^{\eps}=
L_h+(\cx-\frac{1}{\sqrt{\eps}}V'(\sqrt{\eps}\cx))\pk-\sum\limits_{j=1}^{\infty}
\eps^j
\left(\frac{i}{2}\right)^{2j}\frac{1}{(2j+1)!}\frac{V^{(2j+1)}(\sqrt{\eps}\cx)}{\sqrt{\eps}}\pkj$$
$$\label{apmt}
\frac{1}{\eps}{\widetilde M}^{\eps}=
M_h+(\frac{1}{\eps}V(\sqrt{\eps}\cx)-\frac{\cx^2}{2})+\sum\limits_{j=1}^{\infty}
\eps^j\left(\frac{-1}{4}\right)^j\frac{1}{(2j)!}V^{(2j)}(\sqrt{\eps}\cx)\pkkj
+\frac{1}{8}\pkk$$
where $$L_h=\ck \px-\cx\pk \ , \ \ M_h=-\frac18 \triangle_{\cx\ck} +\frac{\ck^2}{2}+\frac{\cx^2}{2} \ ,$$ are the dilations of the operators $$\label{hlm}
\LH:=\Le=k\frac{\partial}{\partial x}-x\frac{\partial}{\partial k} \ , \ \ \ \MH:=\Me =-\frac18 \triangle_{ x k} +\frac{ k^2}{2}+\frac{ x^2}{2}$$ pertaining to the harmonic oscillator $V_h(x)=x^2/2$.
Using the smoothness assumptions of the potential we can further approximate ${\widetilde L}^{\eps}$ and ${\widetilde M}^{\eps}$ by
$$\label{apl}
{\widetilde L}^{\eps}\sim L^{\eps} := L_h+\sum\limits_{\nu=1}^{\infty} \eps^{\frac{\nu}{2}}
\B_{\nu} ( \cx,\pk )$$
and $$\label{apm}
\frac{1}{\eps}{\widetilde M}^{\eps}\sim \frac{1}{\eps}M^{\eps}:= M_h+\sum\limits_{\nu=1}^{\infty}
\eps^{\frac{\nu}{2}}\Gamma_{\nu}( \cx,\pk)$$ where $$\begin{array}{l}
\B_{\nu}( \cx,\pk)=- V^{(\nu+2)}(0)\sum\limits_{j=0}^{[(\nu-1 )
/2]+1}
\left(\frac{i}{2}\right)^{2j}\frac{1}{(2j+1)!}\frac{\cx^{\nu+1-2j}}{(\nu+1-2j)!}\pkj,\
\ \nu\geq 1,\\
\Gamma_{\nu} ( \cx,\pk )=V^{( \nu+2
)}(0)\sum\limits_{j=0}^{[\nu/2]+1}\left(\frac{i}{2}\right)^{2j}
\frac{1}{(2j)!}\frac{\cx^{\nu+2-2j}}{(\nu+2-2j)!}\pkkj,\ \ \nu\geq
1 \ ,
\end{array}$$ with $V^{(j)}(0)\equiv \frac{\partial^{j} V}{\partial x^{j}}(0)$.
Note that for polynomial potential $V(x)$, the above expansions are finite and exact, that is $L^{\eps}\equiv {\widetilde
L}^{\eps}$ and $M^{\eps}\equiv {\widetilde M}^{\eps}$.
Now, by substituting the expansions of $ {\widetilde L}^{\eps}, \ {\widetilde M}^{\eps}$ , and also the expansions (\[ass1\]) of the of eigenvalues $\Ee_{n}$ and of the eigenfunctions $U^{-1}_{\eps,(xk)}\Phi^{\eps}_{nm}$ (\[asf2\]), into the scaled eigenequations (\[eig11\]), (\[eig22\]), we obtain the following hierarchy of non homogeneous problems for the correctors $\tZnm^{(l)}\ n,m=0,1,\dots, \ l\geq
1$ $$\begin{aligned}
\label{zeig}
\left[L_h-\frac{i}{2}(e_n-e_m)\right]\tZnm^{(l)}(\cx,\ck)=B_{nm}^{(l)}(\cx,\ck), \\
\left[M_h-\frac14(e_n+e_m)\right]\tZnm^{(l)}(\cx,\ck)=G_{nm}^{(l)}(\cx,\ck) \nonumber\end{aligned}$$ where the right hand sides of (\[zeig\]) are given by
$$\begin{array}{l}
B_{nm}^{(l)}(\cx,\ck)=-\sum\limits_{\nu=1}^{l}\B_{\nu}(\cx,\pk)\tZnm^{(l-\nu)}(\cx,\ck)
+\frac{i}{2}\sum\limits_{\nu=1}^{[l/2]}(a_n^{(\nu )}-a_m^{(\nu )})\tZnm^{(l-2\nu)}(\cx,\ck) \ , \\
G_{nm}^{(l)}(\cx,\ck)=-\sum\limits_{\nu=1}^{l}\Gamma_{\nu}(\cx,\pk)\tZnm^{(l-\nu)}(\cx,\ck)
+\frac14\sum\limits_{\nu=1}^{[l/2]}(a_n^{(\nu )}+a_m^{(\nu )})\tZnm^{(l-2\nu)}(\cx,\ck) \ , \\
\tZnm^{(0)}(\cx,\ck):= \Psi_{nm}(\cx,\ck) \ .
\end{array}$$
It can be shown by direct computation that the functions $\tZnm^{(l)}(\cx,\ck)$ ( $n,m=0,1\dots$ and $l=0,1\dots$.) given by (\[znm\]) , satisfy the equations (\[zeig\]).
The asymptotic expansion for the Moyal functions satisfies the $L^2$-estimate $$\label{est1}
||\left[U_{\eps,(xk)}^{-1}\Phi^{\eps}_{nm}\right]-\sum
\limits_{l=0}^{N}\eps^{\frac{l}{2}}\tZnm^{(l)}||_{L^2(\R^2)}=O(\eps^{(N+1)/2}) \ .$$ The proof is straightforward by using known estimates for the harmonic approximation of $ \ue_n$ (see Appendix A1).
The harmonic expansion of he Wigner function
============================================
In this section we use the harmonic expansions for the Moyal eigenfunctions which were constructed in the previous section, for the construction of an asymptotic expansion of the time-dependent Wigner function $\We(x,k,t)$ (recall the initial value problem (\[tql\]). The first term of this expansion is the Wigner function for the harmonic oscillator, and for this reason we call it [*harmonic expansion*]{} of the Wigner function. This asymptotic expansion suggests a harmonic ansatz which is then used for the construction of asymptotic expansions of the Wigner equation through a regular perturbation scheme directly in the phase space
First we apply the Wigner transform onto the eigenfunction series solving the problem (\[Schro\]) and we obtain an eigenfunction series of the Wigner function, in terms of the Moyal eigenfunctions. Then, we proceed formally and we approximate the coefficients and the Moyal eigenfunctions by their harmonic approximations. It is important to note that this expansion is “quasi-asymptotic”, since in general, the coefficient depend on the small parameter $\eps$ (and for this reason it is not a genuine semiclassical expansion).
The eigenfunction expansion of the Wigner function $\We(x,k,t)$
---------------------------------------------------------------
Applying the Fourier method, we write the solution $\psie(x,t)$ of the initial value problem ($\ref{Schro}$) for the Schrödinger equation as an eigenfunction series, in terms of the eigenfunctions $\ue_{n}$ of the operator $\Hop$ (see Section 2), This reads as follows $$\label{exppsi}
\psie(x,t)=\sum_{n=0}^{\infty}\Anoe
\ue_{n}(x)e^{-i\frac{\Ee_n}{\eps}t} \ .$$
The coefficients $\Anoe$ are given as the projection of initial data onto the eigenfunctions $$\label{exppsicoef}
\Anoe=(\psie_0,\ue_{n})_{L^2(\R)}$$
By taking the Wigner transform $$\We (x,k,t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-ik\sigma}
\psie \left(x+\frac{\eps\sigma}{2},t\right)\cpsie \left(x-\frac{\eps\sigma}{2}
,t\right)d\sigma \ ,$$ of (\[exppsi\]), and using the definition (\[phinm\]) of the Moyal eigenfunctions $\Phi^{\eps}_{nm}$, we obtain the following eigenfuction expansion of the Wigner function $$\label{expw}
\We(x,k,t)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}A_{nm}^{\eps}(t) \Phi^{\eps}_{nm}(x,k) \ , \\$$ where $$\label{expwcoef1}
A_{nm}^{\eps}(t)=\frac{(\We_0,\Phinm )_{L^2(\R^2_{xk})}}{\parallel\Phinm\parallel_{L^2(\R^2)}^2}e^{-i\frac{\Ee_n-\Ee_m}{\eps}t}=
\eps (\We_0,\Phinm )_{L^2(\R^2_{xk})}e^{-i\frac{\Ee_n-\Ee_m}{\eps}t}$$
It is easy to see that the coefficients (\[expwcoef1\]) are related to coefficients (\[exppsicoef\]) by the relation $$\label{expwcoef2}
A_{nm}^{\eps}(t)=\Anoe\overline{\Amoe}e^{-i\frac{\Ee_n-\Ee_m}{\eps}t} \ .$$
The coefficients $A_{nm}^{\eps}(t),\ \ n,m=0,1\dots$ are approximated by combining (\[exppsicoef\]) with the asymptotic expansions (\[ass1\]) of the eigenvalues $\Ee_n$ and (\[ass2\]) of the Schrödinger eigenfunctions $\ue_{n}$, and they have the expansion $$\label{anm}
A_{nm}^{\eps}(t)\sim
A_{h,nm}^{\eps}(t)+\sum\limits_{j=1}^{\infty}\eps^{\frac{j}{2}}\Delta_{j,nm}^{\eps}(t)$$ where $$A_{h,nm}^{\eps}(t)=
\eps(\We_0,\Psinm )_{L^2(\R^2_{xk})}e^{-i\frac{e_n-e_m}{2}t} \ ,$$ and $$\begin{array}{l}
\Delta_{1,nm}^{\eps}(t)=e^{-i\frac{e_n-e_m}{2}t}\left( [U_{\eps,(xk)}^{-1} \We_0 ],\tZnm^{(1)}\right)_{L^2(\R^2_{\cx \ck})} \ ,\\
\Delta_{2,nm}^{\eps}(t)=e^{-i\frac{e_n-e_m}{2}t}\left[\left( [U_{\eps,(xk)}^{-1} \We_0 ],\tZnm^{(2)}\right)_{L^2(\R^2_{\cx \ck})}+(-it)(a_n^{(1)}-a_m^{(1)}) \left( [U_{\eps,(xk)}^{-1} \We_0 ],\Psi_{nm}\right)_{L^2(\R^2_{\cx \ck})}\right] \ ,\\
\Delta_{3,nm}^{\eps}(t)=e^{-i\frac{e_n-e_m}{2}t}\left[\left( [U_{\eps,(xk)}^{-1} \We_0 ],\tZnm^{(3)}\right)_{L^2(\R^2_{\cx \ck})}+(-it)(a_n^{(1)}-a_m^{(1)}) \left( [U_{\eps,(xk)}^{-1} \We_0 ],\tZnm^{(1)}\right)_{L^2(\R^2_{\cx \ck})}\right] \ \dots \ ,
\end{array}$$
Furthermore, by substituting the approximations (\[anm\]) and (\[asf1\]) of the coefficients $A_{nm}^{\eps}(t)$ and the eigenfunctions $\Phinm$, respectively, into the eigenfunction expansion (\[expw\]), we obtain the following expansion of the Wigner function $$\label{qlas1}
\We(x,k,t)\sim \We_h(x,k,t)+\sum_{l=1}^{\infty} \eps^{l/2} Z^{\eps,(l)}(x,k,t) \ ,$$ IBy direct computation we see that the function $$\label{asexpw}
\We_h(x,k,t)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}A_{h,nm}^{\eps}(t)\Psinm(x,k) \ .$$ satisfies the initial value problem $$\label{hl}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\We_h(x,k,t)+\LH\We_h(x,k,t)=0, (x,k)\in\R^2, t>0 \\\
\We(x,k,t)|_{t=0}=\We_0(x,k) \ ,
\end{array}
\right.$$ with (recall (\[hlm\])) $$\LH=k\frac{\partial}{\partial x}-x\frac{\partial}{\partial k} \ ,$$ which governs the evolution of the Wigner function for the harmonic oscillator, with the same initial data $\We_0(x,k)$ (recall the problem (\[tql\]) governing $\We$).
The harmonic ansatz
-------------------
We pretend now that we do not know anything about the Schrödinger formulation, and we want to use the harmonic expansion (\[asexpw\]) as an approximate solution (harmonic ansatz) to construct an approximate solution of the Wigner equation (\[tql\]).
In order to construct the equations for the coefficient $Z^{\eps,(l)}$, we apply the dilation $ U^{-1}_{\eps,(xk)}$ defined by (\[dilph\]), both on the problem (\[tql\]) and on the expansion (\[qlas1\]). In the new variables $(\cx,\ck) $ the Wigner equation becomes $$\label{ww}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\tWe(\cx,\ck,t)+{\widetilde
L}^{\eps}\tWe(\cx,\ck,t)=0\\
\tWe(x,k,t)|_{t=0}=\tWe_0(x,k) \ ,
\end{array}
\right.$$ while the expansion reads $$\label{qlas2}
\tWe(\cx,\ck,t)\sim \tWe_h(\cx,\ck,t)+ \sum_{l=1}^{\infty}\eps^{l/2}{\widetilde Z}^{\eps,(l)}(\cx,\ck,t) \ .$$
Substituting the transformed expansion (\[qlas2\]) of $\tWe(\cx,\ck,t)$ and the expression (\[aplt\]) of the operator ${\widetilde L}^{\eps}$ into (\[ww\]), and then equating coefficients of same powers of $\eps$, we obtain the following initial value problems $$\label{wh}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\tWe_h(\cx,\ck,t)+L_h\tWe_h(\cx,\ck,t)=0 \\
\tWe_h(\cx,\ck,t)|_{t=0}=\tWe_0(\cx,\ck)\ ,
\end{array}
\right.$$ for the “harmonic term” $\tWe_h(\cx,\ck,t)$, and the following hierarchy of problems $$\label{zl}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}{\widetilde Z}^{\eps,(l)}(\cx,\ck,t)+L_h{\widetilde Z}^{\eps,(l)}(\cx,\ck,t)=D^{\eps,(l)}(\cx,\ck,t),\ l\geq 1\\
{\widetilde Z}^{\eps,(l)}(\cx,\ck,t)|_{t=0}=0 \ ,
\end{array}
\right.$$ for the higher-order coefficients${\widetilde Z}^{\eps,(l)}(\cx,\ck,t),\ l\geq 1$. The right hand side $D^{\eps,(l)}(\cx,\ck,t$ of (\[zl\]) is given by $$D^{\eps,(l)}(\cx,\ck,t)=-\B_{l}(\cx,\pk)\tWe_h(\cx,\ck,t)-
\sum_{\nu=1}^{l-1}\B_{\nu}(\cx,\pk){\widetilde Z}^{\eps,(l-\nu)}(\cx,\ck,t)$$
[**Remark 3 .**]{} The initial data of the f problem for $\We_h$ are the same with those of the original problem (\[tql\]), and therefore the higher order problems have zero initial data but they are forced from lower orders. By this choice we avoid to expand the initial functions $\tWe_0$ with respect to the small parameter $\eps$, which for a WKB initial wave function $\psie_0$, leads to a distributional expansion (compare with the distributional expansion constructed by Pulvirenti [@PU]). A consequence of our choice is that the coefficients ${\widetilde Z}^{\eps,(l)}$ are $\eps$-dependent, and therefore the expansion (\[qlas1\]) is not a genuine semi-classical expansion (see also the comments in [@FN1] for such type of expansions). Other choices of the initial data for the harmonic problem would be either the Airy approximation of $\tWe_0$ which has been proposed by Berry [@BE] (see also [@FM1]), or a wavepacket expansion in phase space, that can be derived by applying the Wigner transform on the FBI transform of the initial wavefunction..These two alternative choices are still open to investigation.
The problems (\[wh\]) and(\[zl\]) can be integrated by the method of characteristics. We demote bt $g_h^t(q,p)$ the Hamiltonian flow of harmonic oscillator. This flow is the solution of the Hamiltonian system $$\frac{d\xi}{dt}=\eta \ , \ \ \ \ \frac{d\eta}{dt}=-\xi \ ,$$ with initial conditions $$\xi(t=0;q,p)=q,\ \ \eta(t=0;q,p)=p \ ,$$ and it is given by $$\begin{aligned}
g_h^t(q,p)&=\left(g_{h,1}^t(q,p),g_{h,2}^t(q,p)\right)\\
&=\left(\cx(q,p,t):=q\cos(t)+p\sin(t),\ck(q,p,t):=p\cos(t)-q\sin(t)\right) \ .\end{aligned}$$ The inverse flow is $$\begin{aligned}
g_h^{-t}(\cx,\ck)&=\left(g_{h,1}^{-t}(\cx,\ck),g_{h,2}^{-t}(\cx,\ck)\right)\\
&=\Bigl(q(\cx,\ck,t):=\cx \cos(t)-\ck \sin(t), p(\cx,\ck,t):=\ck \cos(t)+\cx \sin(t)\Bigr)\end{aligned}$$ Then, the solutions of the problems (\[wh\]) and (\[zl\]) are given by the formulae $$\label{solwh}
\tWe_h(\cx,\ck,t)=\We_0(q(\cx,\ck,t),p(\cx,\ck,t))$$ and $$\label{solzh}
{\widetilde Z}^{\eps,(l)}(\cx,\ck,t)=\int_{0}^{t}D^{\eps,(l)}(q(\cx,\ck,t-s),p(\cx,\ck,t-s)),s)ds$$ It is important to note that the dependence of $\tWe_h(\cx,\ck,t)$ and ${\widetilde Z}^{\eps,(l)}(\cx,\ck,t)$ on $\eps$, comes only from the dependence of initial function $\We_0$ on $\eps$.
The validity of expansion (\[qlas1\]) depends crussialy on the properties of the initial data $\We_0$. Recall here that the required properties of the potential have been stated at the beginning of Section 2.1.
In order to understand the speciality of the WKB initial data, we will now consider two classes of initial data in phase space. The first class consists of those data whose $U^{-1}_{\eps,xk}$ dilation (recall eq (\[dilph\])) is an $\eps$-independent function (in the scaled variables), and in this case we can prove the validity of the expansion in $L^2$ norm. The second class consists of initial data which are the Wigner transform of a certain WKB initial wave function, and in this case we can prove an estimate for the remainder in a weighted $L^2$ norm.
The proofs of the theorems stated below are straightforward applications of the technique proposed by Bouzouina & Robert [@BR]. who intro cued it for proving a novel (Egorov-type) estimate for the remainder in the $L^2$ operator norm for the semiclassical expansion of the evolution of quantum observables. The proofs are given in the Appendix A2.
\[thm1\] Let $\tWe$ the solution of the initial value problem (\[ww\]) with $\eps$-independent initial data $\widetilde{f}_0(\cx,\ck)\in\Sh(\R^2)$ . Then, for any $N\in \N$, $$\label{expthm2}
\tWe(\cx,\ck,t)= \tWe_h(\cx,\ck,t)+\sum_{l=1}^N \eps^{l/2} {\widetilde Z}^{\eps, (l)}(\cx,\ck,t)+R^{N+1}(\cx,\ck,t)$$ where $ {\widetilde Z}^{\eps, (l)}$ are given by formulas (\[solzh\]). Moreover the following bound for the remainder holds $$\label{remthm2}
\parallel R^{N+1}(t)\|_{L^2(\R^2_{\cx\ck})} \le C_{N}e^{t} \eps^{(N+1)/2} $$.
$ \hfill \Box$
An interesting application of Theorem 2 is in the case where the initial data $\widetilde{f}_0(\cx,\ck)$ are the dilated Wigner transform of a coherent state $$\psie_0(x)=\left(\frac{1}{\pi\eps}\right)^{1/4}e^{\frac{i k_0 x}{\eps}} e^{-\frac{(x-x_0)^2}{2\eps}} \ ,$$ which is given by $$\widetilde{f}_0(\cx,\ck)= [U_{\eps,(xk)}^{-1} \We_0](\cx,\ck)=\frac{1}{\pi}e^{-(\cx-\cx_0)^2}e^{-(\ck-\ck_0)^2} \ ,$$ where $\cx_0=\frac{x_0}{\sqrt{\eps}},\ \ck_0=\frac{k_0}{\sqrt{\eps}}$. Then, $\widetilde{f}_0(\cx,\ck)\in \Sh(\R^2)$and it is independent of $\eps$.
For a WKB initial wavefunction $$\psie_0(x)=\alpha_0(x)e^{i\frac{S_0(x)}{\eps}} \in L^2(\R) \ ,$$ the dilated Wigner function $\widetilde{f}_0(\cx,\ck)=\widetilde{f}^{\eps}_0(\cx,\ck)$ is $\eps-$dependent, and an $L^2$ estimate of the error leads to negative powers of $\eps$, a fact which is expected since the weak limit, as $\eps \rightarrow 0$, of the Wigner transform of a WKB state, is a Dirac distribution. Thus the estimate of the remainder we provide here is based on a weighted $L^2$ norm with a Gaussian weight $$r^{\eps}(\cx,\ck)=e^{-\frac{\cx^2+\ck^2}{\eps^2}} \ .$$ In this case the theorem reads as follows.
\[thm2\] Let $\tWe$ the solution of the initial value problem (\[ww\]) with initial data $\widetilde{f}^{\eps}_0(\cx,\ck)= [U^{-1}\We_0](\cx,\ck)$, where $\We_0$ is the Wigner transform of a WKB initial wavefunction $\psie_0(x)=\alpha_0(x)e^{i\frac{S_0(x)}{\eps}}$, with $$\alpha_0(x)=e^{-x^2/2} \ , \ \ S_0(x)=x^2/2 \ \ \ \text{or}\ \ S_0(x)=x \ .$$ Then for all $N \in \N$, $$\tWe(\cx,\ck,t)=\tWe_h(\cx,\ck,t)+ \sum_{l=1}^N \eps^{l/2} {\widetilde Z}^{\eps,(l)}(\cx,\ck,t)+R^{\eps,N+1}(\cx,\ck,t)$$ where $ {\widetilde Z}^{\eps,(l)}$ are given by formulas (\[solzh\]). Moreover the following bound for the remainder holds $$\|R^{\eps,N+1}(t)\|_ {L^2_{r^{\eps}}}\le C_{N}e^{t} \eps^{(N+1)/2} \ ,$$ where $\|\cdot \|_ {L^2_{r^{\eps}}}$ denotes the $L^2_{r^{\eps}} $ norm, $$\| f \|_ {L^2_{r^{\eps}}} =\left(\int_{\R^2} |f(\cx,\ck)|^2 r^{\eps}(\cx,\ck) d\cx d\ck\right)^{1/2} \ ,$$ with $r^{\eps}(\cx,\ck)= e^{-\frac{\cx^2+\ck^2 }{\eps^2}}$.
$ \hfill \Box$
The classical expansion of the Wigner function
==============================================
When the potential is smooth,the quantum Liouville operator $\Le$, can be expanded in the form (see [@HWL]) $$\Le=\Lc-\sum_{j=1}^{\infty}\eps^{2j}\Theta_j (x,\frac{\partial}{\partial k}) \ ,$$ where $$\Lc\equiv k\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial k} \ ,$$ is the classical Liouville operator, and $$\Theta_j (x,\frac{\partial}{\partial k})\equiv \left(\frac{i}{2}\right)^{2j}\frac{V^{(2j+1)}(x)}{(2j+1)!}\frac{\partial^{(2j+1)}}{\partial k^{2j+1}} \ .$$
We observe that formally, as $\eps\rightarrow 0$, the operator $\Le$ reduces to the classical operator $\Lc$. Therefore, it is plausible to assume the expansion [@ST] $$\label{qlas3}
\We(x,k,t)\sim \We_c(x,k,t)+ \sum_{l=1}^{\infty}\eps^{2l}Z_c^{\eps,(l)}(x,k,t) \ .$$ where $\We_c(x,k,t)$ satisfies the classical problem $$\label{wc}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\We_c(x,k,t)+\Lc\We_c(x,k,t)=0 \\
\We_c(x,k,t)|_{t=0}=\We_0(x,k) \ .
\end{array}
\right.$$ For this reason, in the sequel we will refer to the expansion(\[qlas3\]) as the [*classical approximation*]{}. This expansion has been proposed by Steinrück [@ST] for the case of $\eps$-independent initial data, and it has been later studied by F. Narcowich [@FN1], and recently rigorously by Pulvirenti [@PU] for a class of WKB initial data (see also the related studies[@AR], [@ROB]).
Substituting (\[qlas3\]) into the problem (\[tql\]), and separating powers of $\eps$ as it is customary in regular perturbations, we find that while $Z_c^{\eps,(l)}$ satisfy the hierarchy of problems $$\label{zcl}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}Z_c^{\eps,(l)}(x,k,t)+\Lc Z_c^{\eps,(l)}(x,k,t)=\Theta^{(l)}(x,k,t),\ l\geq 1\\
Z_c^{\eps,(l)}(x,k,t)|_{t=0}=0 \ ,
\end{array}
\right.$$ where $$\Theta^{(l)}(x,k,t)=\sum_{j=1}^{l}\Theta_{j} (x,\frac{\partial}{\partial k})Z_c^{\eps,(l-j)}(x,k,t) \ ,$$ and $Z_c^{\eps,(0)}\equiv \We_c$.
The solutions of problems (\[wc\]) and (\[zcl\]) are constructed by the method of characteristics, and they are given by
$$\label{clsol}
\begin{array}{l}
\We_c(x,k,t)=\We_0( q(x,k,-t), p(x,k,-t))\\
\\
Z_c^{\eps,(l)}(x,k,t)=\int_0^t \Theta^{(l)}(q(x,k,t-s) ,p(x,k,t-s) ,s)ds
\end{array}$$
where $q(x,k,t)\ ,\ p(x,k,t))$ are the bicharacteristics associated to the potential $V$, that is the solutions of the Hamiltonian system $$\label{hams}
\left\{
\begin{array}{l}
\frac{dq}{dt}=p,\qquad \frac{dp}{dt}=-V'(q)\\
\\
q(0)=x,\qquad p(0)=k \ .
\end{array}
\right.$$
We must emphasise here that in the classical expansion (\[qlas3\]). the initial data are propagated along the bicharacteristics associated to the potential $V$, in contrary to the harmonic approximation developed in Section 3, where the initial data are propagated along the bicharacteristics of the approximating harmonic oscillator with potential $V_h$.
[**Remark 4.** ]{}(on the structure of the classical expansion)
(a)The expansion (\[qlas3\]) is not a genuine semiclassical expansion because its coefficients depend on the small parameter $\eps$.
\(b) The classical expansion is of the multiplicative type $$\We(x,k,t)= \We_c(x,k,t)\Bigl(1 +\sum_{\ell \ge 1} \eps^{\ell}\We_{\ell}(x,k,t)\Bigr) \ ,$$ which is not the case for the harmonic expansion.
\(c) As $\ \eps \rightarrow 0$, all terms of the classical expansion concentrate near the Langrangian manifold $\Lambda_{t}=\{\ p(x,k,t) = S'_{0}(q(x,k,t)\ \}$ generated by the Hamiltonian flow. Therefore, in the case that the ray field has caustics, the first term $\ \We_c(x,k,t)\ $ of the expansion behaves much as a Dirac distribution,and it is not efficient in computing energy densities for fixed position in configuration space (cf [@FM1]).
[**Remark 5.** ]{}(on the applicability of the expansion)
It follows from the construction of the approximations that the harmonic expansion is expected to be applicable at least in a region of width $O(\sqrt{\eps})$ near the bottom of the potential well, and the classical expansion near the Lagrangian manifold, both for short time. For this reason, in the example of the next section we compare the two expansions near the potential well of an anharmonic (quartic) oscillator.
Example: Anharmonic oscillator and caustics
============================================
As an application of the developed expansions, we combine the harmonic and the classical expansions of the Wigner function with property (\[ampl\]), in order to estimate the amplitude $|\psie|$ of the wavefunction for the quartic oscillator with (anharmonic) potential $$V(x)=x^2/2+\mu x^4/4 \ , \qquad \mu >0 \ ,$$ and WKB initial data (\[wkb0\]) of the Gauss-Fresnel type, that is $$a_0(x)=e^{-\frac{x^2}{2}},\ S_0(x)=\frac{x^2}{2} \ .$$
The corresponding Wigner equation in phase space is
$$\label{ql}
\left\{
\begin{array}{l}
\frac{\partial}{\partial t}\We(x,k,t)+\Le\We(x,k,t)=0, (x,k)\in\R^2, t>0\\
\We(x,k,t)|_{t=0}=\We_0(x,k)
\end{array}
\right.$$
where $$\Le\equiv k\frac{\partial}{\partial x}-(x+\mu
x^3)\frac{\partial}{\partial k}+ \frac{\eps^2}{4}\mu x
\frac{\partial^{(3)}}{\partial k^3} \ ,$$ with initial data $$\We_0(x,k)=\frac{1}{\sqrt{\pi}
\eps}e^{-x^2}e^{-\frac{(k-x)^2}{\eps^2}} \ .$$
First we compute the bicharacteristics and the rays for the corresponding harmonic oscillator (with potential $V_h(x)=x^2/2$), and also the bicharacteristics of the quartic oscillator.
#### Bicharacteristics, rays and caustics.
The bicharacteristics of the harmonic oscillator are easily computed from the Hamiltonian system $$\left\{
\begin{array}{l}
\frac{dx}{dt}=k,\ \ x(0)=q \\
\frac{dk}{dt}=-V'_{h}(x)=-x\ \ k(0)=p
\end{array}
\right.$$ and they are given by $$(x_h (q,p,t),k_h (q,p,t))=g_h^t(q,p)=(q\cos(t)+p\sin(t),p\cos(t)-q\sin(t)) \ .$$ The inverse bicharactericts are $$(q_h(x,k,t),p_h(x,k,t))=g_h^{-t}(x,k)=(x \cos(t)-k \sin(t),k \cos(t)+x \sin(t)) \ .$$
From the condition $p =S_0'(q)=q$, we obtain the equations of the rays $$\tilde{x}_h=\tilde{x}_h(t;q)=q(\cos(t)+\sin(t))$$ and by solving the equation $$J(q,t)=\frac{\partial \tilde{x}_h}{\partial q}=(\cos(t)+\sin(t))=0$$ with respect to $q=q(t)$, we find the caustics, which for harmonic oscillator is a sequence of focal points (Figure 1) $$(x_\nu, t_\nu)=(0, \nu\pi-\frac{\pi}{4}),\ \ \nu=1,2,...$$
![[*Rays & caustic Harmonic oscillator $V_h(x)=x^2/2$*]{}](rays_caush){width="50.00000%"}
The bicharacteristics $(x_{V}(q,p,t), k_{V}(q,p,t))=g^{t}_{V}(q,p)$ for the quartic oscillator are found from the corresponded Hamiltonian system, and they are given by $$x_V(q,p,t)=A(q,p)sd(\Gamma(q,p)t+C(q,p),B(q,p)), \qquad
k_V(q,p,t)=\frac{\partial x_V}{\partial t}(q,p,t)$$ where $$\begin{array}{l}
A(q,p)= \frac{\sqrt{c(q,p)}}{( 2\mu c(q,p)+1 )^{1/4}} \ , \qquad \qquad
B^2(q,p)=\frac{\sqrt{2\mu c(q,p)+1}-1}{2\sqrt{2\mu c(q,p)+1}} \ ,\\
\\
\Gamma(q,p)=(2\mu c(q,p)+1)^{1/4} \ , \qquad \qquad
C(q,p)=sd^{-1}(\frac{q}{A(q,p)},B(q,p))
\end{array}$$ with $$c(q,p)=p^2+q^2+\mu\frac{q^4}{2} \ ,$$ and $sd(a,b)=\frac{sn(a,b)}{dn(a,b)}$,$sn,\ dn$ are the Jacobi elliptic functions.
Unfortunately it is not possible to obtain an analytical formula for the caustic. However, by considering the rays $\tilde{x}_V=\tilde{x}_V(t;q)=x_V(q,p=S'_{0}(q),t)$ and solving numerically the equation $J(q,t)=\partial_{q}\tilde{x}_V=0$, which is available in explicit form, we have observed that the caustic consists of a family of cusps with beaks at the focal points of the corresponding harmonic oscillator (Figure 2). We have checked analytically this observation by proving that the focal points of harmonic oscillator are indeed zeros of the Jacobian for the quartic oscillator.
![[*Rays & caustic Quartic oscillator $V(x)=x^2/2 +\mu x^4/4$*]{}](rays_causq){width="50.00000%"}
Since the expressions of the bicharacteristics for the anharmonic oscillator are very complicated, the analytical computation of coefficient $Z_c^{\eps,(l)}(x,k,t)$ of the classical expansion isimpossible. Nevertheless, it is possible to compute the classical expansion of the Wigner function approximately, by using an approximation of the characteristics for small values of the coupling constant $\mu$. Indeed, by the method of multiple scales, for $\mu$ being the small parameter, we solve the ordinary differential equation $$\ddot{x}+ x +\mu x^3=0 \ , \ \ x(0)=q \ , \ \ \dot x(0)=p \ ,$$ which, is equivalent to the Hamiltonian system for the quartic oscillator, and we get the following approximation of the bicharacteristics $$\label{quartbich}
\left\{
\begin{array}{l}
x_a(q,p,t)=q\cos(\omega t)+p\sin(\omega t)+O(\mu)\\
k_a(q,p,t)=\dot x_a(q,p,t))=p\omega\cos(\omega t)-q\omega\sin(\omega t)+O(\mu) \ ,
\end{array}
\right.$$ and also the approximation of the inverse bicharacteristics $$\left\{
\begin{array}{l}
q_a(x,k,t)=x_a(x,k,-t)=x\cos(\omega t)-k\sin(\omega t)+O(\mu)\\
p_a(x,k,t)=x_a(x,k,-t)=k\omega\cos(\omega t)+x\omega\sin(\omega t)+O(\mu) \ ,
\end{array}
\right.$$ where $\omega= \omega(q,p;\mu)=1+\frac38\mu(q^2+p^2)+O(\mu^2)$ is the approximate angular velocity of the quartic oscillator.
#### Amplitude via harmonic approximation.
The harmonic expansion of $\We(x,k,t)$ reads $$\nonumber
\We(x,k,t)\sim \We_h(x,k,t)+
\sum_{j=1}^{\infty}\eps^{j}Z^{\eps,(2j)}(x,k,t)$$ where $$\begin{aligned}
\We_h(x,k,t)&=&\We_0(q_h(x,k,t),p_h(x,k,t))= \nonumber \\
&=&\frac{1}{\sqrt{\pi} \eps}e^{-(x\cos(t)-k\sin(t))^2}e^{-\frac{(k(\cos(t)+\sin(t))-x(\sin(t)-\cos(t)))^2}{\eps^2}} \ .\end{aligned}$$ Integrating $\We_h$ with respect to $k$ (recall (\[ampl\])), we compute the principal contribution of $\We(x,k,t)$ to the amplitude $$\begin{aligned}
|\psie_h(x,t)|^2 \approx \int_{\R}\We_h(x,k,t)dk
=\frac{1}{\sqrt{\eps^2\sin^2(t)+(\cos(t)+\sin(t))^2}}\nonumber\\
\times exp\left(-x^2\frac{(-2\sin^2(t)+(\cos(t)+\sin(t))^2)^2}{\eps^2\sin^2(t)+(\cos(t)+\sin(t))^2}\right) \ .\end{aligned}$$
At the focal points we have $$|\psie_h(x=0,t_\nu)|^2 \approx \frac{\sqrt{2}}{\eps} \ .$$
The coefficient $Z^{\eps,(2)}(x,k,t)$ are computed in Appendix A3, in two different ways which lead to the same approximation up to the order $O(\eps^2)$. The contribution of this term in the amplitude $|\psie|^2$ at the focal points, is also computed in the Appendx A3, and it is given by $$\eps\int_{\R}Z^{\eps,(2)}(0, k,t_{\nu})d k=\eps\frac{1}{\sqrt{\eps}}\int_{\R}Z^{\eps,(2)}(0,\ck,t_{\nu})d\ck=\frac{\sqrt{2}}{\pi\eps}\mu (\beta+\beta^{\eps})$$ where $$\beta=\frac{\pi}{8} ( \mu-1/4)-3 \ , \qquad \beta^{\eps}=\frac{17}{2} \eps -(\frac{3\pi}{16}( \mu-1/4)+3)\eps^2 \ .$$ By continuing the computation to higher orders, it turns out that the contribution of the subsequent terms of the expansion is also of the order $O({1}/{\eps})$.
#### Amplitude via classical approximation.
The classical expansion (\[qlas3\]) of $\We(x,k,t)$ reads $$\begin{array}{l}
\We(x,k,t)\sim \We_c(x,k,t)+
\sum_{l=1}^{\infty}\eps^{2l}Z_c^{\eps,(l)}(x,k,t)\\
\end{array}$$ The leading term $\We_c(x,k,t)$ and the subsequent coefficients $Z_c^{\eps,(l)}(x,k,t)$ are calculated using (\[clsol\]) and the approximate characteristics (\[quartbich\]), and they are given by $$\begin{array}{l}
\We_c(x,k,t)\sim \We_a(q_a(x,k,t), p_a(x,k,t)):= \We_0(q_a(x,k,t), p_a(x,k,t)) \ , \\
\\
Z_c^{\eps,(l)}(x,k,t)= \int_0^t \Theta^{(l)}(q_a(x,k,s-t), p_a(x,k,s-t),s)ds \ .
\end{array}$$
Integrating $\We_c$ with respect to $k$, we compute the principal contribution of $\We(x,k,t)$ to the amplitude
$$\begin{aligned}
|\psie(0,t_\nu)|^2\approx&\int_{\R}\We_a(0,k,t_{\nu})dk\equiv \int_{\R}\We_0(q_a(0,k,t_{\nu}),p_a(0,k,t_{\nu}))dk \nonumber\\
= &\frac{1}{\sqrt{\pi}\eps}\int_{\R}e^{-\frac{k^2}{2}(1-\mu\gamma k^2)^2}e^{-(\frac{\mu}{\eps})^2\alpha^2k^6}dk\\
=&\frac{1}{\sqrt{\pi}\eps^{2/3}}\int_{\R}e^{-\frac{\eps^{2/3}\xi^2}{2}(1-\mu\gamma \eps^{2/3}\xi^2)^2}e^{-\mu^2\alpha^2\xi^6}d\xi \nonumber \\
=&\frac{1}{\sqrt{\pi}\eps^{2/3}\mu^{1/3}}\int_{\R}e^{-\frac{\eps^{2/3}y^2}{2\mu^{2/3}}(1-\mu^{1/3}\gamma
\eps^{2/3}y^2)^2}e^{-\alpha^2y^6}dy \ , \nonumber\end{aligned}$$
where $\gamma,\ \alpha$ are constants independent of $\mu$ and $\eps$.
Hence $$|\psie(0,t_{\nu})|^2\sim O\left(\frac{1}{\eps^{2/3}\mu^{1/3}}\right),\ \textrm{for}\ \mu\sim \eps^{1-\delta},0\le \delta\le 1 \ ,$$
and
$$|\psie(0,t_{\nu})|^2\sim O\left(\frac{1}{\eps}\right),\ \textrm{for}\ \mu\sim \eps^{1+\delta}, \ \ \delta \ge 0 \ .$$
These estimates have the same order with the corresponding one which was computed from the harmonic expansion if we choose $\mu=O(\eps^{1+\delta})$. Therefore, if we allow the dependence of the coupling constant $\mu$ on the parameter $\eps$, the result shows that for this particular relation between the frequency and the strength of anharmonicity of the potential, both expansions predict the same semiclassical effect, at least at the focal points. However, from the above analysis we also expect agreement of the predicted wave fields everywhere. Moreover, and more important, this picture suggests that different expansions should be used according to the actual relation of the parameters $\eps$ and $\mu$, but it is still open and quite difficult to determine precise criteria for transferring from one approximation to the other.
Finally, it is interesting to remark that the constructed expansions, at least for the example of quartic oscillator, are related as follows $$\begin{aligned}
\We(x,k,t) \xrightarrow{{\eps<<1}} \We_c(x,k,t)+\sum \eps^{2l} Z_c^{\eps,(l)}\ \xrightarrow{{\mu<<1,O(\eps)}} \We_h(x,k,t)+ \sum \eps^l Z^{\eps,(2l)} \ .\end{aligned}$$
Discussion
==========
We have constructed a couple of approximations of the Wigner function for the Schrödinger equation with oscillatory initial data. The first one, which we call the harmonic approximation, has the form $$\We(x,k,t)\sim \We_h(x,k,t)+
\sum_{j=1}^{\infty}\eps^{j}Z^{\eps,(2j)}(x,k,t)$$ where the principal term $\We_h$ is the Wigner function of a harmonic oscillator associated to the harmonic approximation of the potential $V(x)$. The second one, which has been used in quantum mechanics long time ago. has the multiplicative form $$\We(x,k,t)= \We_c(x,k,t)\Bigl(1 +\sum_{\ell \ge 1} \eps^{\ell}\We_{\ell}(x,k,t)\Bigr) \ ,$$ where $\We_c$ is the limit Wigner distirbution which is the solution of the Liouville equation of classical mechanics. For the construction of both approximations we choose the initial data for the principal terms to be the initial Wigner function $\We(x,k,t=0)=\We_0(x,k)$, that is $\We_h(x,k,t=0)=\We_0(x,k)$ and $\We_c(x,k,t=0)=\We_0(x,k)$ . This choice allows us,, after an appropriate scaling of phase space coordinates, to apply a regular perturbation scheme, which, however, has the consequence that the constructed expansions are not genuine semiclassical expansions, because the correctors depend on the small parameter. The expansions are used in the computation of the wave amplitude for a quartic oscillator with WKB data of the Gauss-Fresnel type. They both give reasonable symptomatic approximations at the focal points (caustic) of the oscillator for certain dependence of the coupling constant of the anharmonic potential with the small semiclassical parameter of the Schrödinger equation.
#### Acknowledgements.
EKK has been partially supported by the Research grant 88735, University of Crete (Programme: Graduate fellowships “Heraclitus”, funded by the Greek Ministry of Education). GNM has been partially supported by the Archimedes Center for Modeling, Analysis & Computation (ACMAC), Crete, Greece (grant FP7-REGPDT-2009-1). GNM would like to thank R. Littlejohn (Berkeley), R. Schubert (Bristol) and A. Athanassoulis (Leicester) for helpful discussions.
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Appendices {#appendices .unnumbered}
==========
Appendix A1:Proof of estimate (\[est1\]) {#appendix-a1proof-of-estimate-est1 .unnumbered}
----------------------------------------
For the proof of the estimate (\[est1\]) we start from the representation
$$\begin{aligned}
\left[U_{\eps,(xk)}^{-1}\Phi^{\eps}_{nm}\right](\cx,\ck)
&=&\eps
\frac{1}{\pi\eps}\int_{\R}e^{-i\frac{2\ck}{\sqrt{\eps}}\sigma}
\ue_{n}(\sqrt{\eps}\cx+\sigma)\overline{\ue_{m}(\sqrt{\eps}\cx-\sigma)}d\sigma \nonumber \\
&=&\frac{1}{\pi}\int_{\R}e^{-i2\ck
y}\left[U_{\eps}^{-1}\ue_n\right](\cx+y)\left[U_{\eps}^{-1}\ue_m\right](\cx-y)dy \ ,
\end{aligned}$$
and we use the identity $$\begin{array}{l}
\left[U_{\eps}^{-1}\ue_n\right](\eta)\left[U_{\eps}^{-1}\ue_m\right](\xi)-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}
\sum\limits_{\mu=0}^{l}\psi_n^{(\mu)}(\eta)\psi_m^{(l-\mu)}(\xi)\\
=\left(\left[U_{\eps}^{-1}\ue_n\right](\eta)-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}(\eta)\right)\left(\left[U_{\eps}^{-1}\ue_m\right](\xi)-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}(\xi)\right)\\
+\left(\left[U_{\eps}^{-1}\ue_n\right](\eta)-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}(\eta)\right)\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}(\xi)\\
+\left(\left[U_{\eps}^{-1}\ue_m\right](\xi)-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}(\xi)\right)\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}(\eta)\\
+\sum\limits_{l=N+1}^{2N}\eps^{\frac{l}{2}}\sum\limits_{\mu=l-N}^{N}\psi_n^{(\mu)}(\eta)\psi_m^{(l-\mu)}(\xi) \ .
\end{array}$$ Then, we have $$\begin{array}{l}
||\left[U_{\eps,(xk)}^{-1}\Phi^{\eps}_{nm}\right]-\sum \limits_{l=0}^{N}\eps^{\frac{l}{2}}\tZnm^{(l)}||^2_{L^2(\R^2)}\\
=\int_{\R}\int_{\R}|\left[U_{\eps,(xk)}^{-1}\Phi^{\eps}_{nm}\right](\cx,\ck)-\sum \limits_{l=0}^{N}\eps^{\frac{l}{2}}\tZnm^{(l)}(\cx,\ck)|^2d\cx d\ck\\
=\int_{\R}\int_{\R}|\frac{1}{\pi}\int_{\R}e^{-i2\ck
y}\left(\left[U_{\eps}^{-1}\ue_n\right](\cx+y)\left[U_{\eps}^{-1}\ue_m\right](\cx-y)-
\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\sum\limits_{\mu=0}^{l}\psi_n^{(\mu)}(\cx+y)\psi_m^{(l-\mu)}(\cx-y)\right)dy|^2d\cx d\ck\\
=\int_{\R}\int_{\R}|\left[U_{\eps}^{-1}\ue_n\right](\cx+y)\left[U_{\eps}^{-1}\ue_m\right](\cx-y)-
\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\sum\limits_{\mu=0}^{l}\psi_n^{(\mu)}(\cx+y)\psi_m^{(l-\mu)}(\cx-y)|^2d\cx dy\\
=\int_{\R}\int_{\R}|\left[U_{\eps}^{-1}\ue_n\right](\eta)\left[U_{\eps}^{-1}\ue_m\right](\xi)-
\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\sum\limits_{\mu=0}^{l}\psi_n^{(\mu)}(\eta)\psi_m^{(l-\mu)}(\xi)|^2d\eta d\xi\\
\le ||[U_\eps^{-1}\ue_n
]-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}||^2_{L^2(\R)}
||[U_\eps^{-1}\ue_m]-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}||^2_{L^2(\R)}\\
+||[U_\eps^{-1}\ue_n]-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_n^{(l)}||^2_{L^2(\R)}
\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}||\psi_m^{(l)}||^2_{L^2(\R)}
+||[U_\eps^{-1}\ue_m]-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}||^2_{L^2(\R)}
\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}||\psi_n^{(l)}||^2_{L^2(\R)}\\
+\sum\limits_{l=N+1}^{2N}\eps^{\frac{l}{2}}\sum\limits_{\mu=l-N}^{N}||\psi_n^{(\mu)}||^2_{L^2(\R)}
||\psi_m^{(l-\mu)}||^2_{L^2(\R)} \ .
\end{array}$$
Now since $\psi_n^{(\mu)}\in L^2(\R), \ \mu=0,1,\dots,\
\ n=0,1,\dots$ and $$||[U_\eps^{-1}\ue_m]-\sum\limits_{l=0}^{N}\eps^{\frac{l}{2}}\psi_m^{(l)}||^2_{L^2(\R)}=O(\eps^{(N+1)/2}) \ ,$$ we obtain the desired estimate.
Appendix A2: Proof of Theorems 2 & 3 {#appendix-a2-proof-of-theorems-2-3 .unnumbered}
------------------------------------
In the sequel we denote by $\|\cdot \|$ the norms $\|\cdot \|_ {L^2_{r^{\eps}}}$ or $\|\cdot \|_ {L^2}$ and we write them explicitly when the distinction is necessary. The constants $C_{N}$ are of the generic form $cN!N^{\alpha}$, with $\alpha>0 \ , \ \ c>0$ (see also [@BR]).
For the proof of the Theorems \[thm1\] and \[thm2\] we need the following lemmas and propositions.
\[faa\](Faa di Bruno formula, [@CO] , [@BR], [@LERN]).
Let $f:\R^2\rightarrow \R,\ g:\R^2\rightarrow \R^2$ smooth enough functions. For any multi-index $\nu \in \N^2$ and $z=(\cx,\ck)\in \R^2$ , $$\partial^{\nu}\left(f\circ g \right)(z)=\sum_{0\neq |\gamma|\le |\nu|,\ \gamma\in \N^2} \left[ (\partial^{\gamma}f)\circ g \right](z) B_{\nu,\gamma}\left[\partial g\right](z)$$ where $$B_{\nu,\gamma}\left[\partial g\right]= \nu!\sum_ {\alpha_{\beta}} \prod_{\beta\neq 0} \frac{1}{\alpha_{\beta} !} \left(\frac{\partial^\beta g_1}{\beta!}\right)^{\alpha_{\beta_1}}\left(\frac{\partial^\beta g_2}{\beta!}\right)^{\alpha_{\beta_2}}$$ with $\beta=(\beta_1,\beta_2),\ \alpha_{\beta}=(\alpha_{\beta_1}, \alpha_{\beta_2})$ $$\quad \sum \alpha_{\beta_1}=\gamma_1,\ \sum \alpha_{\beta_2}=\gamma_2,\ \sum \beta_1\alpha_{\beta_1}=\nu_1,\ \sum \beta_2\alpha_{\beta_2}=\nu_2 .$$
\[hflow\] For the Hamiltonian flow $g_h^t=g_h^t(\cx,\ck)$ of the harmonic oscillator, and for all $(\cx,\ck)$ and $t>0$, hold
1. $$\label{hflow1}
\begin{array}{l}
|\partial_{\ck} g_{h,i}^{\pm t}|\le 1, \quad |\partial_{\cx} g_{h,i}^{\pm t}|\le 1 \ , \\
\\
\partial^{\nu}_{\ck} g_{h,i}^{\pm t}=\mathbb{O}_{\nu\times \nu},\quad \partial^{\nu}_{\cx} g_{h,i}^{\pm t}=\mathbb{O}_{\nu\times \nu},\ \ i=1,2, \ \nu=2,3,\dots \ .
\end{array}$$
where $\mathbb{O}_{\nu\times \nu}$ denotes the ${\nu\times \nu}$ zero matrix.
2. $$\label{hflow2}
|g_{h,i}^{\pm t}(\cx,\ck)|^{\nu}\le( |\cx|+|\ck|)^{\nu},\ i=1,2, \ \nu=1,2,\dots$$
3. $$\label{hflow3}
\left|B_{\nu,\gamma}\left[\partial g_h^{-t}\right]\right|\le C_{\nu,\gamma}$$
where $B_{\nu,\gamma}$ are defined in the the previouss Lemma 1, and
4. For all $f \in \Sh(\R^2)$ $$\label{hflow4}
\|f(g_h^{-t})\|=\|f\|$$
The proof of Lemma 2 is based on a straightforward computation.
\[nest\]
1. For all $\widetilde{f}_0(\cx,\ck)\in \Sh(\R^2)$ and $m\ge 1$ holds $$\|(|\cx|+|\ck|)^{m}\partial^{\beta}\widetilde{f}_0\|_{L^2}\le C_{m,\beta}$$
2. For $\widetilde{f}^{\eps}_0(\cx,\ck)$ as in Theorem \[thm2\] holds
$$\|(|\cx|+|\ck|)^{m}\partial^{\beta}\widetilde{f}^{\eps}_0\|_ {L^2_{r^{\eps}}}\le \eps^{m+1}C_{\beta}$$
where $\beta=(\beta_1,\beta_2)\in \N^2,\ \ \partial^{\beta}f= \partial^{\beta_1}_{\cx}\partial^{\beta_2}_{\ck} f$
[Proof of Lemma 3:]{}
The first part of Lemma 3 is immediate, since $\widetilde{f}_0(\cx,\ck)\in \Sh(\R^2)$. The proof of the second part, relies on direct computation using the explicit form of $\widetilde{f}^{\eps}_0(\cx,\ck)$. We show the details for the case $\alpha_0(x)=e^{-x^2/2}$ and $S_0(x)=x^2/2$. For the case of $S_0(x)=x$ we proceed similarly. For $$\widetilde{f}^{\eps}_0(\cx,\ck)=\frac{1}{\pi}e^{-\eps\cx^2}e^{-\frac{(\ck-\cx)^2}{\eps}}$$ and for any $\beta=(\beta_1,\beta_2)\in \N^2$, $\beta\neq 0$, we have $$\begin{array}{l}
\partial^{\beta}\widetilde{f}^{\eps}_0(\cx,\ck)=(\sqrt{\eps})^{-|\beta|}\frac{(-2i)^{\beta_2}2^{-|\beta|}e^{i|\beta|\pi/2}}{\sqrt{\pi}}e^{-\eps\cx^2}e^{-\frac{(\ck-\cx)^2}{\eps}}\\
\times \sum_{j=0}^{\beta_1} \left( {\begin{matrix}
\beta_1 \\
j \\
\end{matrix}} \right) (-2)^je^{-ij\pi/2} H_j(\sqrt{\eps}\cx)H_{|\beta|-j}\left(\frac{\ck-\cx}{\sqrt{\eps}}\right)
\end{array}$$ where $H_j$ are the Hermite polynomials.
The term that dominates $\partial^{\beta}\widetilde{f}^{\eps}_0(\cx,\ck)$ for small values of $\eps$, is
$$G_{\beta}^{\eps}(\cx,\ck)=\eps^{-|\beta|}c_{\beta}e^{-\eps\cx^2}e^{-\frac{(\ck-\cx)^2}{\eps}}(\ck-\cx)^{|\beta|} \ ,$$ with $c_{\beta}=(-2i)^{\beta_2}2^{-|\beta|}e^{i|\beta|\pi/2}/\sqrt{\pi}$.
Therefore it is enough to prove a bound for the term $(|\cx|+|\ck|)^{m}G_{\beta}^{\eps}$. We have
$$\|(|\cx|+|\ck|)^{m}\partial^{\beta}\widetilde{f}^{\eps}_0\|_ {L^2_{r^{\eps}}}\le
\sum_{j=1}^{m} \||\cx|^j|\ck|^{m-j}\partial^{\beta}\widetilde{f}^{\eps}_0\|_ {L^2_{r^{\eps}}}
\sim\sum_{j=1}^{m} \||\cx|^j|\ck|^{m-j}G_{\beta}^{\eps} \|_ {L^2_{r^{\eps}}} \ ,$$ and $$\begin{array}{l}
\| |\cx|^j|\ck|^{m-j}G_{\beta}^{\eps} \|^2_ {L^2_{r^{\eps}}}\\
= c_{\beta}^2\eps^{-2|\beta|}\int_{\R}\int_{\R}
|\cx|^{2j}|\ck|^{2(m-j)}(\ck-\cx)^{2|\beta|}e^{-2\eps\cx^2}e^{-\frac{2(\ck-\cx)^2}{\eps}}e^{-\frac{\cx^2+\ck^2 }{\eps^2}}d\cx d\ck\\
=c_{\beta}^2\eps^{-2|\beta|+2|\beta|+2m+2}\int_{\R}\int_{\R}
|\cx|^{2j}|\ck|^{2(m-j)}(\ck-\cx)^{2|\beta|}e^{-2\eps^3\cx^2}e^{-2\eps(\ck-\cx)^2}
e^{-(\cx^2+\ck^2)}d\cx d\ck\\
=\eps^{2m+2}c_{\beta}^2c^{\eps} \ , \\
\end{array}$$ where $c^{\eps}\to c < \infty$, as $\eps \to 0$. This concludes the proof of $$\|(|\cx|+|\ck|)^{m}\partial^{\beta}\widetilde{f}^{\eps}_0\|_ {L^2_{r^{\eps}}} \le \eps^{m+1}C_{\beta}$$ $\blacksquare$
\[bound1\]
1. For all $f\in \Sh(\R^2)$ and $\nu_j \in \N,\ \ j=1,2\dots$ holds $$\label{Bn}
\|\B_{\nu_1}\left[\B_{\nu_2}\left[\dots\B_{\nu_j}\left[f(g_h^{-t_{j+1}})\right] \dots \right](g_h^{-(t_1-t_{2})}, t_2 )\right](g_h^{-(t-t_1)} ,t_1 ) \|_{L^2}\le C_{\nu_1,\dots,\nu_j}$$
2. For $\widetilde{f}^{\eps}_0(\cx,\ck)$ as in Theorem · \[thm2\], and $\nu_j \in \N,\ \ j=1,2\dots$ holds
$$\label{Bn2}
\|\B_{\nu_1}\left[\B_{\nu_2}\left[\dots\B_{\nu_j}\left[f(g_h^{-t_{j+1}})\right] \dots \right](g_h^{-(t_1-t_{2})}, t_2 )\right](g_h^{-(t-t_1)} ,t_1 ) \|_ {L^2_{r^{\eps}}}\le C_{\nu_1,\dots,\nu_j}$$
[Proof of Proposition 1:]{}
Recall that the operators $\B_{\nu_j}$ are given by the formula $$\B_{\nu_j}= - V^{(\nu_j+2)}(0)\sum\limits_{\lambda_j=0}^{[(\nu_j-1 )/2]+1} c_{\lambda_j,\nu_j} \cx^{\nu_j+1-2\lambda_j}\pklj \ ,$$ with $c_{\lambda_j,\nu_j}=\left(\frac{i}{2}\right)^{2\lambda_j}\frac{1}{(2\lambda_j+1)!(\nu_j+1-2\lambda_j)! } \ .$
By Lemma \[hflow\], the left hand side of (\[Bn\]), (\[Bn2\]) reads as $$\| \B_{\nu_1}\left[\B_{\nu_2}\left[\dots\B_{\nu_j}\left[f(g_h^{-t_{j+1}})\right] \dots \right](g_h^{-(t_1-t_{2})}, t_2 ) \right] \parallel \ ,$$ and hence it is enough to estimate this quantity. First we prove (\[Bn\]) for all $j$. We give the details only for the cases $j=1$ and $j=2$. For $j=1$ we have, $$\begin{aligned}
\| \B_{\nu_1}\left[f(g_h^{-t_{1}})\right]\|_{L^2}& \le |V^{(\nu_1+2)}(0)| \sum\limits_{\lambda_j=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1} \|\cx^{\nu_1+1-2\lambda_1}\pkl f(g_h^{-t_{1}})\|_{L^2} \\&\le \sum\limits_{\lambda_j=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1} \sum_{0\neq |\gamma|\le \nu_1}
C'_{\nu_1,\gamma} \|\cx^{\nu_1+1-2\lambda_1} \left[ (\partial^{\gamma }f)\circ g_h^{-t} \right](z)\left|B_{\nu_1,\gamma}\left[\partial g_h^{-t}\right]\right|\|_{L^2} \\&
\le \sum\limits_{\lambda_j=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1}\sum_{0\neq |\gamma|\le \nu_1}
C_{\nu_1,\gamma} \| (g_{h,1}^{-t})^{\nu_1+1-2\lambda_1} \partial^{\gamma }f(z)\| _{L^2}\\&
=\sum\limits_{\lambda_j=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1} \sum_{0\neq |\gamma|\le \nu}
C_{\nu_1,\gamma} \| (|\cx|+|\ck| )^{\nu_1+1-2\lambda_1} \partial^{\gamma }f(z)\|_{L^2}\end{aligned}$$ where in the first step we used Faa di Bruno formula, and the then (\[hflow2\]),(\[hflow3\]).
The first part of Lemma \[nest\] ensures that $$\| (|\cx|+|\ck| )^{\nu_1+1-2\lambda_1} (\partial^{\gamma }f(z)\|_{L^2} \le C_{\lambda_1,\nu_1,\gamma} \ ,$$ thus $$\begin{aligned}
\| \B_{\nu_1}\left[f(g_h^{-t_{1}})\right]\|_{L^2}
\le|V^{(\nu_1+2)}(0)| \sum\limits_{\lambda_1=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1} C_{\lambda_1,\nu_1} \le C_{\nu_1} \ ,\end{aligned}$$ which proves (\[Bn\]) for $j=1$.
For $j=2$, we have
$$\begin{aligned}
\|& \B_{\nu_1}\left[\B_{\nu_2}\left[f(g_h^{-t_{2}})\right](g_h^{-(t_1-t_{2})})\right]\| \le \\ & \le |V^{(\nu_1+2)}(0)| \sum\limits_{\lambda_j=0}^{[(\nu_1-1 )/2]+1} c_{\lambda_1,\nu_1} \sum_{|\beta^1|\le 2\lambda_1+1} c_{\beta}\|(|\cx|+|\ck|)^{\nu_1+1-2\lambda_1}\partial^{\beta}\left[\B_{\nu_2}\left[f(g_h^{-t_{2}})\right]\right]\| \\
& \le |V^{(\nu_1+2)}(0)| |V^{(\nu_2+2)}(0)| \sum\limits_{\lambda_1=0}^{[(\nu_1-1 )/2]+1} \sum\limits_{\lambda_2=0}^{[(\nu_2-1 )/2]+1} c_{\lambda_1,\nu_1} c_{\lambda_2,\nu_2} \\
&\sum_{|\beta^1|\le 2\lambda_1+1} c_{\beta}\|(|\cx|+|\ck|)^{\nu_1+1-2\lambda_1}\partial^{\beta}\left[ \cx^{ \nu_2+1-2\lambda_2}\partial_{\ck}^{2\lambda_2+1}\left[f(g_h^{-t_{2}})\right]\right]\| \\
&\le\prod_{i=1,2} |V^{(\nu_i+2)}(0)| \sum\limits_{\lambda_1=0}^{\nu_1^{\ell} }\sum\limits_{\lambda_2=0}^{\nu_2^{\ell}} \sum_{|\beta^1|\le u^1}\sum_{\alpha_1=0}^{\beta_1}\sum_{|\beta^2|\le u^2} C(\nu_1,\nu_2,\lambda_1,\beta^1,\beta^2,\alpha_1)\|(|\cx|+|\ck|)^{\gamma_1}\partial^{\beta^2}f\|\end{aligned}$$
where $$\nu_i^{\ell}= [(\nu_i-1 )/2]+1,\ \beta^i\in \N^2,\ u^1=\lambda_1+1,\ u^2=(\beta^1_1-\alpha_1,\beta^1_2+\lambda_2+1) \ ,$$ and $$\gamma_1=\nu_1+\nu_2+2-2\lambda_1-2\lambda_2-\alpha_1 \ .$$ Therefore $$\| \B_{\nu_1}\left[\B_{\nu_2}\left[f(g_h^{-t_{2}})\right](g_h^{-(t_1-t_{2})})\right]\|_{L^2} \le C_{\nu_1,\nu_2} \ .$$ In the same way we can prove it for the general case for all $j\ge 3$, by applying successively the Faa di Bruno formula, the Leibniz formula and using Lemma \[hflow\], to get $$\begin{aligned}
\label{bb}
&\|\B_{\nu_1}\left[\B_{\nu_2}\left[\dots\B_{\nu_j}\left[f(g_h^{-t_{j+1}})\right] \dots \right](g_h^{-(t_1-t_{2})}, t_2 )\right](g_h^{-(t-t_1)} ,t_1 ) \|_{L^2} \le \\
& \le\prod_{i=1}^{j} |V^{(\nu_i+2)}(0)| \sum\limits_{\lambda_1=0}^{\nu_1^{\ell} }\sum_{|\beta^1|\le u^1} \dots \sum\limits_{\lambda_j=0}^{\nu_j^{\ell}}\sum_{\alpha_{j-1} } \sum_{|\beta^j|\le u^j} C(\nu_i,\lambda_i,\beta^i,\alpha_i)\|(|\cx|+|\ck|)^{\gamma_{j-1}}\partial^{\beta^j}f\|_{L^2}\le C_{\nu_1,\dots,\nu_j}\nonumber\end{aligned}$$ with $$\nu_i^{\ell}= [(\nu_i-1 )/2]+1,\ \beta^i\in \N^2\ ,$$ and $$\gamma_i=\sum_{k=1}^{i}(\nu_k+1-2\lambda_k)-\sum_{k=1}^{i-1}\alpha_k \ .$$
For proving the second part we follow the same procedure as before, by using the second part of Lemma \[nest\]. To proceed we observe that $\|(|\cx|+|\ck|)^{\gamma_{j-1}}\partial^{\beta^j}f\|_ {L^2_{r^{\eps}}}\le \eps^{\gamma_{j-1}+1}C_{\beta^j}$. This estimate and the fact that ${\gamma_{j-1}}$ in (\[bb\]) is always non-negative, ensure that $$\begin{aligned}
\|\B_{\nu_1}\left[\B_{\nu_2}\left[\dots\B_{\nu_j}\left[f(g_h^{-t_{j+1}})\right] \dots \right](g_h^{-(t_1-t_{2})}, t_2 )\right](g_h^{-(t-t_1)} ,t_1 ) \|_ {L^2_{r^{\eps}}} \le C_{\nu_1,\dots,\nu_j} \ , \end{aligned}$$ which ends the proof of the proposition. $\blacksquare$
[Proof of Theorem \[thm1\]:]{}
The $(N+1)$-order remainder (\[remthm2\]) of the asymptotic expansion (\[expthm2\]), that is $$R^{N+1}(\cx,\ck,t):=\tWe(\cx,\ck,t)-\sum_{l=0}^N \eps^{l/2} {\widetilde Z}^{\eps,(l)}(\cx,\ck,t),\ \ N=0,1,\dots \ ,$$ solves, for aany $N$, the initial value problem $$\begin{aligned}
\label{erN}
\frac{\partial}{\partial t}{R^{N+1}}(\cx,\ck,t)+L_hR^{N+1}(\cx,\ck,t)=-\sum_{\nu=1}^{N+1} \B_{\nu} \left[ R^{N+1-\nu}\right](\cx,\ck,t)\\
R^{N+1}(\cx,\ck,t)|_{t=0}=0\end{aligned}$$ where $R^{0}=\tWe_h \ \ \text{and} \ \ \B_{\nu}:= \B_{\nu}(\cx,\pk)$.
According to Dunhamel’s principle the solution of problem (\[erN\]), is given by the formula $$R^{N+1}(\cx,\ck,t)=-\int_0^t \sum_{\nu=1}^{N+1}\eps^{\nu/2} \B_{\nu}\left[R^{N+1-\nu}\right](g_h^{-(t-s)}(\cx,\ck),s)ds,\ \ N=0,1,\dots$$ Applying the above formula successively for each $R^{N+1-\nu}$, we have $$\begin{aligned}
&R^{N+1}(\cx,\ck,t)=\eps^{\frac{N+1}{2}} \left\{ -\int_0^tdt_1 \B_{N+1}\left[\widetilde{f}_0(g_h^{-t_1})\right]\left(g_h^{-(t-t_1)} ,t_1\right) \right.\\
&+\int_0^tdt_1\int_0^{t_1}dt_2 \B_{N }\left[\B_1\left[\widetilde{f}_0(g_h^{-t_2})\right]\left(g_h^{-(t_1-t_2)},t_2\right)\right]\left(g_h^{-(t-t_1)} ,t_1\right)\\
&+\int_0^tdt_1\int_0^{t_1}dt_2 \B_{N-1 }\left[\B_2\left[\widetilde{f}_0(g_h^{-t_2})\right]\left(g_h^{-(t_1-t_2)},t_2\right)\right]\left(g_h^{-(t-t_1)} ,t_1\right)dt_1dt_2\\
&-\int_0^tdt_1\int_0^{t_1}dt_2\int_0^{t_2}dt_3 \B_{N-1 }\left[\B_1\left[\B_1\left[\widetilde{f}_0(g_h^{-t_3})\right]\left(g_h^{-(t_2-t_3)},t_3\right)\right]\left(g_h^{-(t_1-t_2)},t_2\right)\right]\left(g_h^{-(t-t_1)} ,t_1\right) \\
&+\dots\\
&\left.+(-1)^N \int_0^tdt_1\int_0^{t_1}dt_2\dots \int_0^{t_N}dt_{N+1}\B_1\left[\B_1\left[\dots \B_1\left[\widetilde{f}_0(g_h^{-t_{N+1}})\right]\dots \right]\left(g_h^{-(t_1-t_2)},t_2\right)\right]\left(g_h^{-(t-t_1)} ,t_1 \right) \right\}\end{aligned}$$ where $\widetilde{f}_0(\cx,\ck)$ are the initial data of the problem (\[wh\]).
Hence $$\begin{aligned}
&\|R^{N+1}(\cx,\ck,t)\|_{L^2}\le \eps^{\frac{N+1}{2}} \left\{ \int_0^tdt_1\|\B_{N+1}\left[f_0(g_h^{-t_1})\right](g_h^{-(t-t_1)} ,t_1)\|_{L^2} \right. \\
&+\int_0^tdt_1\int_0^{t_1}dt_2\| \B_{N }\left[\B_1\left[f_0(g_h^{-t_2})\right](g_h^{-(t_1-t_2)},t_2)\right](g_h^{-(t-t_1)} ,t_1)\|_{L^2} \\
&+\dots \\
&+ \left. \int_0^tdt_1\int_0^{t_1}dt_2\dots \int_0^{t_N}dt_{N+1}\|\B_1\left[\B_1\left[\dots \B_1\left[ f_0(g_h^{-t_{N+1}})\right]\dots \right](g_h^{-(t_1-t_2)},t_2)\right](g_h^{-(t-t_1)} ,t_1 ) \|_{L^2}\right\}\end{aligned}$$
Since $\widetilde{f}_0(\cx,\ck)\in\Sh(\R^2)$ and $\eps$-independent, the first part of Proposition \[bound1\] implies that every term in the right hand side of the above inequality is bounded, and therefore we get $$\begin{aligned}
\|R^{N+1}(\cx,\ck,t)\|_{L^2}&\le \eps^{\frac{N+1}{2}} C_{N} \left\{ \int_0^tdt_1 +\int_0^tdt_1\int_0^{t_1}dt_2 + \dots +\int_0^tdt_1\dots\int_0^{t_N}dt_{N+1}
\right\} \ ,\\
&= \eps^{\frac{N+1}{2}} C_{N} \sum_{k=1}^{N+1} \frac{t^k}{k!} = \eps^{\frac{N+1}{2}} C_{N} e^t \ ,\end{aligned}$$ which ends the proof. $\blacksquare$
[Proof of Theorem \[thm2\]:]{}
For $\alpha_0(x)=e^{-x^2/2}$ and $S_0(x)=x^2/2$, we have $$\widetilde{f ^{\eps}}_{0}(\cx,\ck)=\frac{1}{\sqrt{\pi}}e^{-\eps\cx^2}e^{-\frac{(\ck-\cx)^2}{\eps}} \ ,$$ while for $S_0(x)=x$ we have $$\widetilde{f^{\eps}}_{0}(\cx,\ck)=\frac{1}{\sqrt{\pi}}e^{-\eps\cx^2}e^{-\frac{(\ck-1)^2}{\eps}} \ .$$ In both cases $\widetilde{f^{\eps}}_0(\cx,\ck) \in \Sh(\R^2)$. So we proceed similarly to the proof of Theorem 2 and we use the second part of Proposition \[bound1\], to obtain
$$\begin{aligned}
\|R^{N+1}(\cx,\ck,t)\|_ {L^2_{r^{\eps}}}&\le \eps^{\frac{N+1}{2}} C_{N} \left\{ \int_0^tdt_1 +\int_0^tdt_1\int_0^{t_1}dt_2 + \dots +\int_0^{t_N}dt_{N+1}
\right\}\Rightarrow\\
&= \eps^{\frac{N+1}{2}} C_{N} e^t \ ,\end{aligned}$$
which ends the proof of Theorem \[thm2\]. $\blacksquare$
Appendix A3: Expansion of $Z^{\eps,(2)}$ {#appendix-a3-expansion-of-zeps2 .unnumbered}
=========================================
In this appendix we compute the coefficient $Z^{\eps,(2)}$ of the harmonic expansion in the case of the quartic oscillator . This computation is performed by solving the problem (\[zl\]) with $\ell=2$, that is $$\begin{aligned}
\label{ivpz2}
\Bigl(\frac{\partial}{\partial t}+L_h\Bigr){\widetilde Z}^{\eps,(2)}(\cx,\ck,t)=D^{(2)}(\cx,\ck,t) \nonumber \\
{\widetilde Z}^{\eps,(2)}(\cx,\ck,t)|_{t=0}=0 \ ,\end{aligned}$$ in two different ways.
#### First way:
We expand ${\widetilde Z}^{\eps,(2)}$ with respect to the Moyal eigenfunctions $\Psi_nm$ of the harmonic oscillator, $${\widetilde Z}^{\eps,(2)}(\cx,\ck,t)=\sum_{n}\sum_{m} z_{nm}^{\eps}(t)\Psi_{nm}(\cx,\ck) \ ,$$ and we substitute this series into (\[ivpz2\]), together with the eigenfunction series (\[qlas2\]) of $\tWe_h$, which appears in the $D^{(2)}(\cx,\ck,t)$. Then we use the orthogonality of $\Psi_{nm}$ to derive a hierarchy of equations for the coefficients $z_{nm}^{\eps}(t) \ , \ \ n,m=0,1,\dots \ .$
These equations can be easily integrated because of the polynomial type of the potential and the special form of Moyal eigenfunctions (Laguerre polynomials), and, after a long and cumbersome computation, we get $$\begin{aligned}
{\widetilde Z}^{\eps,(2)}(\cx,\ck,t)&=-\frac{\mu}{4\pi}Re\left(\sum_{n}\sum_{m} z_{nm}^{\eps}(t)e^{-i(n-m)t}\Psinm(\cx,\ck)\right)+ \nonumber \\
&+\frac{3\mu}{4\pi}tIm\left(\sum_{n}\sum_{m} A_{h,nm,o}^{\eps}(n^2+n)e^{-i(n-m)t}\Psinm(\cx,\ck)\right) \ , \nonumber \\\end{aligned}$$ where $$\begin{aligned}
z_{nm}^{\eps}(t)&=-\frac{1}{8}(e^{i4t}-1)A_{h,n(m+4),o}^{\eps}((m+1)(m+2)(m+3)(m+4))^{1/2}+\\
&+\frac{1}{8}(e^{-i4t}-1)A_{h,n(m-4),o}^{\eps}(m(m-1)(m-2)(m-3))^{1/2}-\\
&-\frac{1}{4}(e^{i2t}-1)A_{h,n(m+2),o}^{\eps}((4m+6)(m+1)(m+2))^{1/2}+\\
&+\frac{1}{4}(e^{-i2t}-1)A_{h,n(m-2),o}^{\eps}((4m+2)m(m-1))^{1/2} \ ,\\\end{aligned}$$ with $$A_{h,nm,o}^{\eps}=(\tWe_0,\Psi_{nm})_{L^2(\R^2_{\cx,\ck})} \ .$$
Then we have $$\begin{aligned}
\int_{\R}{\widetilde Z}^{\eps,(2)}(\cx,\ck,t)d\ck&=
-\frac{\mu}{4\pi}\sqrt{\eps}\frac{1}{|z^{\eps}| |1-u^{\eps}|}e^{-\cx^2}e^{-2\cx^2Re\left(\frac{u^{\eps}}{1-u^{\eps}}\right)}\times \\
&\times Re\left(-\frac{1}{8}(e^{i4t}-1)(\bar{w}^{\eps})^2g^{\eps}_1(\cx,\ck,t)+\frac{1}{8}(1-e^{i4t})g^{\eps}_2(\cx,\ck,t)-\right.\\
&\left.-\frac{1}{2}(e^{i2t}-1)\bar{w}^{\eps}g^{\eps}_3(\cx,\ck,t)-(e^{-i2t}-1)e^{i2t}g^{\eps}_4(\cx,\ck,t)\right)+\\
&+\frac{3\mu}{4\pi}t\sqrt{\eps}\frac{1}{|z^{\eps}| |1-u^{\eps}|}e^{-\cx^2}e^{-2\cx^2Re\left(\frac{u^{\eps}}{1-u^{\eps}}\right)}
Im\left(g^{\eps}_5(\cx,\ck,t)\right)\end{aligned}$$ where $$u^{\eps}=-w^{\eps}e^{i2t} \ , \ \ \ w^{\eps}=\frac{1-z^{\eps}}{z^{\eps}},\ z^{\eps}=\frac12(1+\eps-i) \ .$$ and $$\begin{aligned}
g^{\eps}_1(\cx,\ck,t)&=3\frac{(u^{\eps})^2}{(1-u^{\eps})^{2}}+12\cx^2\frac{(u^{\eps})^2}{(1-u^{\eps})^{3}}+4\cx^4\frac{(u^{\eps})^2}{(1-u^{\eps})^{4}}+
6\frac{u^{\eps}}{(1-u^{\eps})}+12\cx^2\frac{u^{\eps}}{(1-u^{\eps})^{2}}+3\\
g^{\eps}_2(\cx,\ck,t)&=\frac{3}{(1-u^{\eps})^{2}}+12\cx^2\frac{1}{(1-u^{\eps})^{3}}+\cx^4\frac{1}{(1-u^{\eps})^{4}}\\
g^{\eps}_3(\cx,\ck,t)&=6\frac{(u^{\eps})^2}{(1-u^{\eps})^{2}}+24\cx^2\frac{(u^{\eps})^2}{(1-u^{\eps})^{3}}+8\cx^4\frac{(u^{\eps})^2}{(1-u^{\eps})^{4}}+
9\frac{u^{\eps}}{(1-u^{\eps})}+18\cx^2\frac{u^{\eps}}{(1-u^{\eps})^{2}}+3\\
g^{\eps}_4(\cx,\ck,t)&=3\frac{u^{\eps}}{(1-u^{\eps})^{2}}+12\cx^2\frac{u^{\eps}}{(1-u^{\eps})^{3}}+4\cx^4\frac{u^{\eps}}{(1-u^{\eps})^{4}}+
\frac{5}{2(1-u^{\eps})}+5\cx^2\frac{1}{(1-u^{\eps})^{2}}\\
g^{\eps}_5(\cx,\ck,t)&=\frac32\frac{(u^{\eps})^2}{(1-u^{\eps})^{2}}+6\cx^2\frac{(u^{\eps})^2}{(1-u^{\eps})^{3}}+2\cx^4\frac{(u^{\eps})^2}{(1-u^{\eps})^{4}}+\frac{3u^{\eps}}{(1-u^{\eps})}+3\cx^2\frac{u^{\eps}}{(1-u^{\eps})} \ .\end{aligned}$$
At the focal points $(x_{\nu},t_{\nu})=(0,\nu\pi-\pi/4),\ \nu=1,2,...$, $$|1-u^{\eps}_{\nu}|=|1-i\frac{1-\bar{z}^{\eps}}{\bar{z}^{\eps}}|=\frac{1}{|z^{\eps}|}\frac{\eps}{\sqrt{2}} \ ,$$ and thus we get $$\int_{\R}{\widetilde Z}^{\eps,(2)}(0,\ck,t_{\nu})d\ck=\frac{\sqrt{2}}{\pi\eps^{3/2}}\mu (\beta+\beta^{\eps}) \ ,$$ with $$\beta^{\eps}=\left( -\eps^2(\frac{3\pi}{16}( \mu-1/4)+3)+\eps\frac{17}{2}\right) \ \ \text{and}\ \
\beta=\frac{\pi}{8} ( \mu-1/4)-3 \ .$$
#### Second way:
By Dunhamel’s principle, the solution of (\[ivpz2\]) is given by $${\widetilde Z}^{\eps,(2)}(\cx,\ck,t)=\int_{0}^{t}D^{(2)} (q(\cx,\ck,t-s ),p ( \cx,\ck,t-s ),s )ds \ ,$$ where $$D^{(2)}(\cx,\ck,s)=-\mu\left[\frac14\cx \pkkk -\cx^3 \pk\right]\We_h(\cx,\ck,s)=-\B_2(\cx,\pk){\widetilde W}^{\eps}_{h}(\cx,\ck,t) \ .$$
With the aid of symbolic computations with MAPLE, for any $(\cx,\ck,t)$ we obtained the expression $$\begin{aligned}
{\widetilde Z}^{\eps,(2)}(\cx,\ck,t)&=\mu\We_h(\cx,\ck,t)\times \nonumber \\
&\times\left[\frac{1}{\eps^3}F_1(\cx,\ck,t)+\frac{1}{\eps^2}F_2(\cx,\ck,t)+\frac{1}{\eps}F_3(\cx,\ck,t)+\right. \nonumber\\
&+\left. F_4(\cx,\ck,t)+\eps F_5(\cx,\ck,t)+ \eps^2 F_6(\cx,\ck,t)+\eps^3F_7(\cx,\ck,t)\right] \ , \nonumber \\\end{aligned}$$ where $$\begin{aligned}
F_1(\cx,\ck,t)=&2[\ck c(t)+\cx s(t)]^3(\cx f_1(t)+\ck f_2(t))\\
F_2(\cx,\ck,t)=&-8[\ck c(t)+\cx s(t)]f_3(t)\\
F_3(\cx,\ck,t)=&-2[\ck c(t)+\cx s(t)]\left(\ck^3f_4(t)+3\cx\ck^2f_5(t)+3\cx^2\ck f_6(t)+\cx^3f_7(t)+\right.\\
&\left.+3[\ck c(t)+\cx s(t)][\cx\cos(t)-\ck\sin(t)](\ck f_8(t)+\cx f_9(t))\right)\\\end{aligned}$$ and $$\begin{aligned}
F_4(\cx,\ck,t)=&8\left((\cx\cos(t)-\ck\sin(t))f_{10}(t)+[\ck c(t)+\cx s(t)]f_{11}(t)\right)\\
F_5(\cx,\ck,t)=&2(\cx\cos(t)-\ck\sin(t)) 3[\ck c(t)+\cx s(t)](\cx f_{12}(t)+\ck f_{13}(t))+\\
&+2(\cx\cos(t)-\ck\sin(t))(\cx^3f_{14}(t)+3\cx^2\ck f_{15}(t)+3\cx \ck^2f_{16}(t)+\ck^3 f_{17}(t)))\\
F_6(\cx,\ck,t)=&-8(\cx\cos(t)-\ck\sin(t))f_{18}(t)\\
F_7(\cx,\ck,t)=&-2(\cx\cos(t)-\ck\sin(t))^3(\cx f_{19}(t)+\ck f_{20}(t))\\\end{aligned}$$ with $c(t)=\cos(t)+\sin(t),\ s(t)=\sin(t)-\cos(t)$, and $f_j(t)$ being also nonlinear combinations of harmonic functions of time $t$.
Returning to the variables $(x,k)$, we obtain
$$\begin{aligned}
\label{expz2}
Z^{\eps,(2)}(x,k,t)&=\mu\We_H(x,k,t)\times\left[\frac{1}{\eps^5}F_1(x,k,t)+\frac{1}{\eps^{5/2}}F_2(x,k,t)+\frac{1}{\eps^3}F_3(x,k,t)\right. \nonumber \\
&\left.+\frac{1}{\eps^{1/2}}
F_4(x,k,t) +\frac{1}{\eps}F_5(x,k,t)+\eps^{3/2}F_6(x,k,t)+\eps F_7(x,k,t)\right] \ . \nonumber \\\end{aligned}$$
The integration of the expansion (\[expz2\]), which is a rather long and complicated computation, leads to the same result.
[^1]: E-mail:[email protected]
[^2]: E-mail:[email protected]
[^3]: Also: Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Crete, Greece
[^4]: This equation is referred in the context of classical wave propagation as the Wigner equation [@PR] and in quantum mechanics as the quantum Liouville equation [@MA]
[^5]: this amounts to the change of variables $(x,k)\rightarrow (
\cx=\frac{x}{\sqrt{\eps}},\ \ck=\frac{k}{\sqrt{\eps}}$)
|
---
abstract: 'The small group of $\lambda$ Bootis stars comprises late B to early F-type stars, with moderate to extreme (up to a factor 100) surface underabundances of most Fe-peak elements and solar abundances of lighter elements (C, N, O, and S). The main mechanisms responsible for this phenomenon are atmospheric diffusion, meridional mixing and accretion of material from their surroundings. Especially spectroscopic binary (SB) systems with $\lambda$ Bootis type components are very important to investigate the evolutionary status and accretion process in more details. For HD 210111, also $\delta$ Scuti type pulsation was found which gives the opportunity to use the tools of asteroseismology for further investigations. The latter could result in strict constraints for the amount of diffusion for this star. Together with models for the accretion and its source this provides a unique opportunity to shed more light on these important processes. We present classification and high resolution spectra for HD 210111. A detailed investigation of the most likely combinations of single star components was performed. For this, composite spectra with different stellar astrophysical parameters were calculated and compared to the observations to find the best fitting combination. HD 210111 comprises two equal (within the estimated errors) stars with $T_\mathrm{eff}$=7400K, log$g$=3.8dex, \[M/H\]=$-$1.0dex and $v \sin i$=30kms$^{-1}$. This result is in line with other strict observational facts published so far for this object. It is only the third detailed investigated $\lambda$ Bootis type SB system, but the first one with a known IR-excess.'
date: 'Accepted 2011 October 13. Received 2011 October 5; in original form 2011 April 8'
title: 'HD 210111: a new $\lambda$ Bootis type SB system'
---
\[firstpage\]
Stars: chemically peculiar – stars: binaries: spectroscopic – stars: variables: delta Scuti – stars: individual: HD 210111
Introduction {#intro}
============
The group of $\lambda$ Bootis stars comprise of true Population I, late B to early F-type stars, with moderate to extreme (up to a factor 100) surface underabundances of most Fe-peak elements and solar abundances of lighter elements (C, N, O, and S). Only a maximum of about 2% of all objects in the relevant spectral domain are found to be $\lambda$ Bootis type stars (@Pau02).
@Mic86 suggested that the peculiar chemical abundances on the stellar surfaces are due to accretion of circumstellar material that is mixed in the shallow convection zone of the star by the joint action of gravitational settling and radiative acceleration. It naturally explains why the anomalous abundance pattern is similar to that found in the gas phase of the interstellar medium (ISM), where refractory elements like iron and silicon have condensed into dust grains.
@Kam02 and @Mar09 developed a model which is based on the interaction of the star with its local ISM environment. Different levels of underabundance are produced by different amounts of accreted material relative to the photospheric mass. The small fraction of this star group is explained by the low probability of a star-cloud interaction and by the effects of meridional circulation, which washes out any accretion pattern a few million years after the accretion has stopped. The hot end of this model is due to strong stellar winds for stars with $T_\mathrm{eff}$$>$12000K whereas the cool end at about 6500K is defined by convection which prevents the accreted material to manifest at the stellar surface.
Up to now, there are at least eight double-lined spectroscopic binary (SB2) systems with suspected $\lambda$ Bootis candidates known (@Pau02). A detailed abundance analysis was done only for HD84948 and HD171948 so far (@Hei02a and @Ili02). Both SB2 systems consist of true $\lambda$ Bootis type objects. The analysis of such SB2 systems is very important for the above described model because it shows that both components of the system are $\lambda$ Bootis type stars. It is clearly in contradiction with the suggested scenario by [@Far04] who suggested that two unresolved solar abundant stars mimic a metal-weak single star spectrum. The limitations and the inconsistencies of this scenario with accurate photometric measurements was already discussed by @Stu06.
In this paper we present the detection of HD 210111 (HR 8437, HIP 109306) being a true $\lambda$ Bootis type SB system. New high resolution and classification spectroscopy is analysed and discussed in the context of already published data. This object, also showing $\delta$ Scuti type pulsation, is especially interesting because an IR excess was already detected which makes it an ideal test case for the accretion model.
Observations {#obs}
============
The high resolution spectra, used for the abundance and stellar parameter estimation (Fig. \[hrs\]), were taken from the UVES Paranal Observatory Project (@Bag03). In total, 24 spectra from the night of 07.07.2002 are available which were observed with a slit width of 0.5Å, resulting in a spectral resolution of about 80000. The final, averaged, spectrum was normalized using standard IRAF routines[^1].
We compared the line profiles of the averaged spectrum to those of the individual spectra to exclude a misinterpretation due to an incorrect merging of the data. The line profiles due to the SB2 nature are clearly visible in all spectra.
Additional high resolution spectra were obtained at the 1.5 meter telescope (20./21.10.2009) at the Cerro Tololo Inter-American Observatory and the 2.15 meter telescope (05./06.09.2011) at the Complejo Astron[ó]{}mico El Leoncito (CASLEO) with the Echelle de Banco Simmons (EBASIM) Spectrograph. The integration times were set to ten minutes and one hour, respectively. The spectral resolutions are comparable to that of UVES but the signal-to-noise ratio is much lower. As supplement, a classification resolution (R$\approx$500) spectrum (Fig. \[class\]) was obtained at the Southern Astrophysical Research (SOAR) 4.1 meter telescope at Cerro Pachon using the Goodman Spectrograph in the night of 26./27.07.2010.
In a first step, all the spectroscopic data have been reduced using standard IRAF routines. The standard reduction includes: bias subtraction, flat-fielding, cosmic ray cleaning and wavelength calibration.
{width="85mm"}
. \[class\]
Target characteristics {#target}
======================
HD 210111 (HR 8437, HIP 109306) is one of the southern bright (V=6.83mag) $\lambda$ Bootis prototype stars. It is classified as kA2hA7mA2Vas $\lambda$ Boo with peculiar hydrogen lines by @Gra88. In Fig. \[class\] we present a new high signal-to-noise classification resolution spectrum of the target. Together with the appropriate MK standard stars, it shows the uniqueness of the overall spectral features.
The high resolution spectrum (R$\approx$50000) presented by @Hol91 indicate asymmetric line profiles which were not discussed in that paper. It is interesting to note that HD 183324 (HR 7400), another well established member of the $\lambda$ Bootis group, shows similar line profile characteristics as HD 210111 according to their presented spectra. This star might also be an undetected SB2 system. On the other hand, @Hol95 observed a clean, unperturbed rotationally broadened K line. This can be understood within the framework of a small radial velocity difference between the two binary components, the broadening of the K line and the spectral resolution. Even in our spectrum, this line (as well as the hydrogen lines) are not affected by the SB2 characteristics. In the catalogue of @Gre99 this star is marked as a suspected binary system. @Far04 suspected that HD 210111 could be an undetected SB system on the basis of a cross-correlation of three spectra (R$\approx$28000) observed in 1993 and 1994. However, this material was not sufficient to draw any decisive conclusions.
The $\delta$ Scuti type pulsation of this object was detected in 1994 and a global observational campaign was reported by @Bre06. In total, they found thirteen statistically significant pulsation frequencies with very small photometric amplitudes in the visual. It is well known that diffusion and accretion affect the pulsation frequencies of stars at the upper main sequence (@Tur02). In $\lambda$ Bootis stars, the opacity in the metal bump will be significantly lowered. However, only little direct pulsational excitation from Fe-peak elements was found, but effects due to settling of helium along with the enhancement of hydrogen are important. Nevertheless, the structure of the star is changed and thus the frequencies of the excited modes. Since the effects are rather small, new satellite based observations with Convection, Rotation and Planetary Transits (CoRoT) or Microvariability & Oscillations of Stars (MOST) are clearly needed.
A significant IR-excess from IRAS data due to a circumstellar disk was detected for HD 210111 (@Pau03). Since we find that both components of this system are very similar with an equal luminosity (Sect. \[redan\]), the conclusions about the characteristic dust temperature, the fractional dust luminosity, and the radiative equilibrium distance of the above mentioned paper are still valid.
The overall characteristics of this star makes it an excellent test case for the published models explaining the $\lambda$ Bootis phenomenon. With the tools of asteroseismology further insights of the stellar atmospheres should be possible whereas a detailed analysis of its environment will allow to understand the source as well as the mechanism of accretion.
![Upper curve: A part of the averaged UVES spectrum. Lower curve: The difference between the best fit synthetic composite spectrum from a wide variety of single star spectrum combinations and the observed spectrum.[]{data-label="hrs"}](mn11_0765_fig2.eps){width="85mm"}
Analysis {#redan}
========
The ATLAS9 model atmospheres, including the treatment of convection by @Can96, were calculated with scaled solar abundances using pretabulated opacity distribution functions taken from the Vienna New Model Grid of Stellar Atmospheres (NEMO) (@Hei02b). All models are scaled to the solar abundance values as listed by @Gre98, except for C, N and O for which the values of @Asp05 were used. The microturbulent velocity $\xi$ was set to 2.0kms$^{-1}$. This is a typical value used for previous abundances studies of $\lambda$ Bootis and A-type stars. Changing $\xi$ to a value of 3.5kms$^{-1}$ (see @Hei02a) alters the spectral broadening by an amount which is negligible compared to the rotational broadening. The atomic transition parameters for the spectrum synthesis have been taken from the Vienna Atomic Line Database (VALD; @Kup99).
Starting values for an effective temperature, surface gravity and \[M/H\] were based on the results by @Stu93. The continuum flux was calculated with the program SYNTH (@Pis92). In order to derive a “composite” spectrum, the individual spectrum of each component was shifted by an arbitrary relative radial velocity, weighted by the continuum flux ratios and added.
First of all, we estimated the rotational velocity $v \sin i$ by inspecting a composite spectrum with several different values for both components. A value of $v \sin i$$\approx$30kms$^{-1}$ for both components fitted the observations best and was set, accordingly, for the further analysis. The heuristic error of the rotational velocity is $\approx$5kms$^{-1}$. Prior published values of the rotational velocity for HD 210111 range from 50 to 60kms$^{-1}$ on the basis of classification to high resolution spectroscopy (@Sol01). Besides the broadening due to the instrumental profile, a radial velocity difference of 50kms$^{-1}$ at 4200Å is needed to convolve two single line profiles with a $v \sin i$ of 30kms$^{-1}$ to a “composite” one with 60kms$^{-1}$. This value can be set as upper limit of the orbital total radial velocity amplitude for all published data.
As next step, the radial velocities were manually determined and set to $-$23 and +18kms$^{-1}$, respectively. Due to the high resolution of the spectrum and the wide separation of the components, these values are unambiguous with an accuracy better than 1kms$^{-1}$. The averaged mean radial velocity of $-$5kms$^{-1}$ is in the same range as the three measurements ($-$5.13$\pm$0.59kms$^{-1}$) published by @Gre99. These observations were made in a time interval of 434 days. Such a rather constant mean radial velocity could point towards a very long orbital period of this spectroscopic binary system.
The published values for the effective temperature range from 7400 to 7900K whereas the log$g$ ones are between 3.75 and 3.90dex (@Pau02). We calculated all combinations of stellar atmospheres with 7000$<$$T_\mathrm{eff}$$<$8000K, 3.7$<$log$g$$<$4.0dex and \[M/H\]=\[+0.0,$-$0.5,$-$1.0,$-$1.5\]. A possible much cooler component would have been detected by enhanced X-ray fluxes from the ROSAT measurements (@Hue98). The composite spectra were semi-automatically compared to the observed spectrum in the wavelength region between 4500 and 5500Åincluding the most prominent unblended metallic lines. In addition, the K, D, and the lines in the red spectral region were used. The ten best fitting composite spectra deduced via a SIMPLEX method described by @Gra03 were manually inspected and re-fitted.
@Boh99 presented time series spectroscopy of HD 210111 finding non-radial pulsation (NRP) of this object with a period of 49 minutes and a peak-to-peak amplitude of 2.5% of the continuum in the mean-absolute-deviation. The corresponding radial velocity amplitude of such kind of variation is below 3kms$^{-1}$ depending on the detected pulsation mode (@Kis02). Our reported absolute radial velocity difference of 41kms$^{-1}$ is more than one order of magnitude larger than that and can not be explained by NRP.
Finally, we found that a composite spectrum with equal components fitted the observed spectrum best. In Fig. \[hrs\] the observed spectrum in the wavelength region from 5425 to 5470Å together with the difference to the synthetic one is shown. The fit of the line profiles and depths is better than 1% which is very satisfactory. Changing the effective temperatures by $\pm$100K and the surface gravities by $\pm$0.05dex already significantly (3$\sigma$ of the estimated error due to the signal-to-noise ratio) decreases the quality of the fit. For the metallicity we can only state that the best fifteen composite spectra always comprise two components with \[M/H\]=$-$1.0dex. No combination with at least one solar abundant component is able to fit the observed line profiles in a satisfactory way.
Finally we conclude that HD 210111 consists of two similar (within the estimated errors) stars with $T_\mathrm{eff}$=7400K, log$g$=3.8dex, \[M/H\]=$-$1.0dex and $v \sin i$=30kms$^{-1}$. This result perfectly matches the strict limitations of the photometric 7-color Geneva, Str[ö]{}mgren $uvby$ and $\Delta$a measurements presented in @Stu06.
Within the usable wavelength range, we mainly find Fe and Si lines as well as a few Mg, Si, Cr and Ni lines. For these elements, the best-fit spectra with an abundance of $-$1.0dex compared to the Sun in each of the components agree with the observed line profiles which is in line with the result by @Stu93.
For a more detailed abundance analysis, additional data, especially with a large separation of the two components are needed.
We fitted a composite synthetic spectrum with the above listed parameters to the H$\beta$ line profiles of the available high resolution observations. Besides the known inadequateness of fitting the hydrogen line core correctly, the normalization of the echelle spectra is quite problematic because the line spreads over two consecutive orders. The classification resolution spectrum, on the other hand, allows only a rough determination and check of the parameter space. The fits themselves result in a good agreement. However, we are not able to use the H$\beta$ line profiles for further improvements via an iterative method.
From the parallax measurement of the final release of the Hipparcos catalogue (@Leu07), we derived a distance of 78$\pm$4pc for HD 210111. With the absolute bolometric magnitude of the Sun ($M_{\rm Bol}$)$_{\odot}$=4.75mag (@Cay96) and the bolometric correction taken from @Dri00, the luminosity (logL$_{\ast}$/L$_{\odot}$) was calculated. As the next step, we used the post-MS evolutionary tracks and isochrones from @Cla95 to estimate the mass and age. The models were calculated with solar abundances. That is justified because the stellar abundance is restricted to the surface only.
Using the derived astrophysical parameters, an age of almost 1Gyr for this system and a mass of 1.9M$_{\odot}$ for both components is deduced. The errors are about 10%. This is in line with the ages published by @Ili95 and @Pau02.
Its brightness and close distance to the Sun makes HD 210111 to an excellent candidate for follow-up observations in the NIR and IR not only to study its surrounding environment, i.e. the circumstellar disk characteristics, but also to understand the accretion process at such evolved ages.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the financial contributions of the Austrian Agency for International Cooperation in Education and Research (WTZ CZ-10/2010 and HR-14/2010). UH acknowledges support from the Swedish National Space Board. This paper was partially supported by PIP0348 by CONICET. Data from the UVES Paranal Observatory Project (ESO DDT Program ID 266.D-5655) were used for this paper. This research has made use of the model atmosphere grid NEMO and the Vienna Atomic Line Database VALD. Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Minist[é]{}rio da Ci[é]{}ncia, Tecnologia, e Inovaç[ã]{}o (MCTI) da Rep[ú]{}blica federativa do Brasil, the US National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU).
Asplund M. et al., 2005, in Barnes T.G. III, Bash F.N., eds, ASP Conf. Ser. Vol. 336, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, Astron. Soc. Pac., San Francisco, p. 25 Bagnulo S. et al., 2003, The Messenger, 114, 10 Bohlender D. A., Gonzalez J.-F., Matthews J. M., 1999, A&A, 350, 553 Breger M. et al., 2006, A&A, 455, 673 Canuto V. M., Goldman I., Mazzitelli I.. 1996, ApJ, 473, 550 Cayrel de Strobel G., 1996, A&AR, 7, 243 Claret A., 1995, A&AS, 109, 441 Drilling J. S., Landolt A. U., 2000, in Cox A.N., ed, Allen’s Astrophysical Quantities, 4$^{\rm th}$ edition, Springer, New York, p. 381 Faraggiana R. et al., 2004, A&A, 425, 615 Gray R. O., 1988, AJ, 95, 220 Gray R. O. et al., 2003, AJ, 126, 2048 Grenier S. et al., 1999, A&AS, 137, 451 Grevesse N., Sauval A. J., 1998, Space Science Reviews, 85, 161 Heiter U., 2002, A&A, 381, 959 Heiter U. et al., 2002, A&A, 392, 619 Holweger H., Rentzsch-Holm I., 1995, A&A, 303, 819 Holweger H., St[ü]{}renburg S., 1991, A&A, 252, 255 H[ü]{}nsch M., Schmitt J. H. M. M., Voges W., 1998, A&AS, 132, 155 Iliev I. Kh., Barzova I. S., 1995, A&A, 302, 735 Iliev I. Kh. et al., 2002, A&A, 381, 914 Kamp I., Paunzen E., 2002, MNRAS, 335, L45 Kiss L. L.,Derekas A., Alfaro E. J., Biro I. B., Csak B.,Garrido R., Szatmary K., Thomson J. R., 2002, A&A, 394, 97 Kupka F., Piskunov N., Ryabchikova T. A., Stempels H. C., Weiss W. W., 1999, A&AS, 138, 119 Martinez-Galarza J. R., Kamp I., Su K. Y. L., G[á]{}sp[á]{}r A., Rieke G., Mamajek E. E., 2009, AJ, 694, 165 Michaud G., Charland Y., 1986, ApJ, 311, 326 Paunzen E., Iliev I. Kh., Kamp I., Barzova I. S., 2002, MNRAS, 336, 1030 Paunzen E., Kamp I., Weiss W. W., Wiesemeyer H., 2003, A&A, 404, 579 Piskunov N. N., 1992, in Glagolevsky Yu.V., Romanjuk I.I., eds, Stellar Magnetizm, St. Petersburg: Nauka, p. 92 Solano E., Paunzen E., Pintado O. I., Varela J., 2001, A&A, 374, 957 St[ü]{}tz Ch., Paunzen E., 2006, A&A, 458, L17 St[ü]{}renburg S., 1993, A&A, 277, 139 Turcotte S., 2002, Aerts C., Bedding T.R., Christensen-Dalsgaard J., eds, ASP Conf. Ser. Vol. 259, Radial and Nonradial Pulsations as Probes of Stellar Physics, Astron. Soc. Pac., San Francisco, p. 258 van Leeuwen F., 2007, A&A, 474, 653
\[lastpage\]
[^1]: Available from http://iraf.noao.edu/
|
---
author:
- 'Rumen S. Bachev'
date: 'Received ...; accepted ...'
title: 'Quasar optical variability: searching for interband time delays'
---
Introduction
============
Although variability is an extensively studied and rather common feature of quasars, little is currently known about the exact nature of the processes driving the flux changes. Variations are observed in practically all energy bands, often with time delays between them. This fact suggests a common cause (process) connecting otherwise spatially separated regions (presumably parts of an accretion disk), where most of the corresponding wavelengths come from. Since the time lags are rather short for the expected distances, it is the speed of light that is perhaps the only way to casually connect these separate regions. Several competing ideas, often leading to different signs of the time lags between the bands have been discussed in the literature (Czerny et al. 2008):
- [**Reprocessing**]{} of the central high-energy continuum into lower energy bands from the outer (and colder) regions of an accretion disk (Krolik et al. 1991). For this mechanism to work, the central X-ray emission must “see” the outer parts – either because of elevation of the central source above the disk surface, of warping, or of flaring of the disk (or all three). Thus, a time lag between the variable X-rays and the lower wavelength (e.g. UV, optical, IR) will be observed with a relation between the lag and the wavelength, roughly scaled as $\tau_{\rm \lambda} \sim \lambda^{4/3}$ (Sect. 4) for a standard accretion disk (Shakura & Syunyaev 1973). Therefore, if the reprocessing is responsible for the most of the optical variability, a delay between every two optical bands (say $B$ and $R$-bands) will be expected, with the shorter wavelengths leading (a positive lag). However, if the variable X-rays come not from the center, but from another location instead (e.g. from local flares above the disk surface), the sign of the lags can be either positive or negative, depending on the radial distance at which these flares predominantly occur. In this case, a connection between the signs of the lags and the fundamental quasar parameters could be searched, as the radial distance of the X-ray flares is probably governed by the structure of a corona above the disk, which in turn should depend on the quasar parameters.
- [**A weak blazar component**]{}, often assumed to exist even in radio-quiet sources (Czerny et al. 2008, and the references therein). Such a component can produce variable optical/IR synchrotron radiation, and respectively – variable X-rays via the synchrotron self-Compton mechanism. These X-rays can, in turn, be reprocessed in a disk into optical/UV photons, leading to a complicated interband time-delay picture, in which the redder (synchrotron) variations will probably lead the bluer (reprocessed) ones.
- [**Disk fluctuations**]{}, drifting inward. This mechanism is sometimes invoked (Arévalo et al. 2008) to explain cases where the long-wavelength variation are leading. However, the viscous timescale at the optical band generation distance is much longer than the timescales of a few days, discussed in this paper.
The situation can be very complicated, as several of the effects mentioned above can work in combination, or of course, an entirely different mechanism can be responsible for the interband relations. In either case, studying the lags between the continuum changes in different wavelengths for a large sample of quasars may help to clarify the situation, and therefore, to create a better picture of the quasar central engines.
Sergeev et al. (2005) studied the light curves of 14 nearby Seyfert galaxies, observed on $\sim$60–150 epochs in 4 broad-band filters and found positive delays for most of the cases, varying between a fraction of a day and a few days. Their results are broadly consistent with the reprocessing model. Recently, similar results were reported by Liu et al. (2008) and Arévalo et al. (2008). Here we extend their works, applying a similar technique to a larger sample of quasars, optically monitored at the Wise observatory (Giveon et al. 1999) in two colors for several years. The sample is described in more detail in the next section. In Sect. 3 we describe the method we used to find the time delay between the bands, i.e. the interpolation cross-correlation function ($ICCF$) method. Next, the results for the time lags are presented and compared with the predictions of the reprocessing model. Finally, we discuss the reliability of the lag estimates and different implications for the central engine models of quasars, by comparing the deviations from the reprocessing model predictions with various quasar characteristics.
The sample
==========
This work analyses publicly available light curve data for a sample of 42 Palomar-Green (Schmidt & Green 1983), mostly radio-quiet quasars, monitored at the Wise observatory in two observers-frame colors ($B$ and $R$-bands), Giveon et al. (1999). This sample was chosen mostly because of the high accuracy of the CCD photometry, typically around 0.01 mag, in comparison with earlier, photographic plate based monitoring campaigns (Giveon et al., 1999, and the references therein). The time span of the monitoring was about 7 year, with a typical average monitoring interval of $\sim40$ days, though ranging significantly. The best sampled objects were observed on about 80 separate epochs, while the least sampled – on only about 25 epochs (see Sect. 5.1.1 for a discussion on sampling issues). All the objects are nearby, typically of $z\simeq0.2$, which insures that the analysis applies to almost the same rest-frame wavelength region. As an additional argument for using this sample is that many of the objects have reverberation-mapping central mass estimates (and respectively – Eddington ratios), so one can study possible relations between the time lags and the accretion parameters. For the rest of the objects, due to the good optical spectra available, the “size–luminosity” (Kaspi et al. 2005) relation can be applied for the same purpose. The sample is presented in Table 1. The object name, the red shift and the number of observational epochs are given in the first three columns, followed by the measured time lags between $B$ and $R$-bands, $\tau_{\rm obs}$ (see Sect. 4 for details). Next columns show the black hole mass, the accretion rate (in Eddington units) and the expected lag for the simple reprocessing model (Sect. 4).
The $ICCF$ method
=================
In order to study the wavelength – time delay dependence we performed a linear-interpolation cross-correlation ($ICCF$) analysis (Gaskell & Sparke 1986) between the light curves of the two bands. The maximum of $ICCF(\tau)$ is assumed to give the time delay between the bands. The interpolation between the photometric points is necessary due to the unevenly sampled data and is one of the frequently used methods. In our particular case, the magnitude values for every second day of the interpolated light curve were later on used for the cross-correlation analysis. This time interval of 2 days seems like a reasonable choice, since it is shorter than the majority of the real data intervals, but is not too short so a huge number of “artificial” data points to influence the analysis. Such a 2-day interval leads to an additional ($rms$) uncertainty of $\sigma_{\rm int}\simeq0.6$ days in the peak position, which however is generally smaller than the expected errors of other nature (see Sect. 5.1).
Other methods described in the literature (e.g. discrete $CCF$, Edelson & Krolik 1988; $z$-transformed $CCF$, Alexander 1997) do not seem to give significantly different results when applied to the same data (White & Peterson 1994; see also Fig. 3). The weighted delays of the top 80% of the $ICCF$ are typically 1.5 – 2 times larger for this sample (see also Sergeev et al., 2005, for similar results). They correlate significantly (0.93) with the peak values, which are used throughout this paper.
Results
=======
Figure 1 shows on the same scale the $ICCF$s around the zero lag for all 42 objects. The $ICCF$ peak there is typically the most prominent one, except for a few cases where additional (similar or even higher) maxima are present at significant distances from $\tau=0$ (e.g. PG 1012+008). For such peculiar cases, only the “central” maximum is used for the analysis.
[lcccccc]{}
Object & z & N & $\tau_{\rm obs}$ & $\log M_{\sun}$ & $\log \dot m$& $\tau_{\rm exp}$\
PG 0026+129 & 0.142 & 72 & 4 & 7.83 & 0.05 & 4.1\
PG 0052+251 & 0.155 & 76 & $-$4 & 8.75 & $-$0.82 & 8.6\
PG 0804+761 & 0.100 & 88 & 12 & 8.35 & $-$0.30 & 7.0\
PG 0838+770 & 0.131 & 29 & 30 & 7.99 & $-$0.49 & 3.5\
PG 0844+349 & 0.064 & 66 & 6 & 7.76 & $-$0.48 & 2.5\
PG 0923+201 & 0.190 & 25 & 16 & 9.09 & $-$1.13 & 11.6\
PG 0953+414 & 0.239 & 60 & 14 & 8.49 & $-$0.20 & 9.4\
PG 1001+054 & 0.161 & 26 & $-$6 & 7.65 & $-$0.02 & 2.9\
PG 1012+008 & 0.185 & 23 & $-$64 & 8.07 & $-$0.32 & 4.5\
PG 1048+342 & 0.167 & 31 & 8 & 8.24 & $-$0.69 & 4.4\
PG 1100+772 & 0.313 & 47 & 16 & 9.11 & $-$0.75 & 16.0\
PG 1114+445 & 0.144 & 25 & 2 & 8.42 & $-$0.93 & 4.8\
PG 1115+407 & 0.154 & 25 & 10 & 7.51 & $-$0.14 & 2.2\
PG 1121+422 & 0.234 & 26 & $-$20 & 7.86 & $-$0.23 & 3.5\
PG 1151+117 & 0.176 & 23 & 28 & 8.44 & $-$0.80 & 5.4\
PG 1202+281 & 0.165 & 38 & 6 & 8.46 & $-$1.05 & 4.7\
PG 1211+143 & 0.085 & 24 & 16 & 7.83 & 0.05 & 4.1\
PG 1226+023 & 0.158 & 45 & $-$16 & 8.88 & $-$0.01 & 19.6\
PG 1229+204 & 0.064 & 45 & 0 & 8.00 & $-$0.80 & 2.8\
PG 1307+085 & 0.155 & 30 & 12 & 8.54 & $-$0.65 & 7.2\
PG 1309+355 & 0.184 & 32 & $-$4 & 8.16 & $-$0.42 & 4.7\
PG 1322+659 & 0.168 & 28 & 0 & 8.08 & $-$0.41 & 4.2\
PG 1351+640 & 0.087 & 35 & 0 & 8.66 & $-$1.06 & 6.3\
PG 1354+213 & 0.300 & 26 & 2 & 9.46 & $-$0.49 & 33.3\
PG 1402+261 & 0.164 & 28 & $-$2 & 7.85 & 0.02 & 4.1\
PG 1404+226 & 0.098 & 28 & $-$16 & 6.71 & 0.23 & 0.9\
PG 1411+442 & 0.089 & 29 & 14 & 7.87 & $-$0.54 & 2.8\
PG 1415+451 & 0.114 & 30 & 2 & 7.80 & $-$0.58 & 2.4\
PG 1416$-$129& 0.086 & 25 & 6 & 8.92 & $-$1.12 & 9.0\
PG 1427+480 & 0.221 & 29 & 12 & 7.98 & $-$0.34 & 3.8\
PG 1444+407 & 0.267 & 27 & $-$2 & 8.16 & $-$0.12 & 6.0\
PG 1512+370 & 0.371 & 36 & $-$2 & 9.17 & $-$0.87 & 15.9\
PG 1519+226 & 0.137 & 33 & 26 & 7.78 & $-$0.31 & 2.9\
PG 1545+210 & 0.266 & 43 & 12 & 9.17 & $-$0.93 & 15.1\
PG 1613+658 & 0.129 & 64 & 0 & 8.95 & $-$1.46 & 7.3\
PG 1617+175 & 0.114 & 56 & 4 & 8.73 & $-$0.88 & 8.0\
PG 1626+554 & 0.133 & 30 & 0 & 8.37 & $-$0.94 & 4.4\
PG 1700+518 & 0.292 & 54 & 6 & 8.89 & $-$0.45 & 14.3\
PG 1704+608 & 0.371 & 40 & 40 & 9.20 & $-$0.77 & 17.9\
PG 2130+099 & 0.061 & 79 & 2 & 7.81 & $-$0.37 & 2.9\
PG 2233+134 & 0.325 & 33 & $-$12 & 8.15 & 0.36 & 8.5\
PG 2251+113 & 0.323 & 43 & 2 & 9.04 & $-$0.62 & 15.7\
Figure 2 presents the distribution of the $rest~frame$ time delays for all the objects from the sample. When all of them are included, the average $\tau_{\rm rest}$ is +3.2 days ($\pm4.2$ at 95% confidence interval) with a standard deviation of 13.2 days, and the median is +2.7 days. The $t$-test gives for the null hypothesis (assuming that the mean value for the parent population is zero) gives a $p$-value of 0.13, so the null hypothesis cannot be rejected at a significant enough level (meaning that the average time delay for the sample cannot be statistically distinguished from zero). However, the situation changes if one takes into account the presence of highly deviating delay values in the distribution (e.g. PG 1012+008 with $\tau_{\rm rest}\simeq-54$ days, Figure 2). Two more tests, based on the median value, reject the idea that the median delay is zero at least at 95% level. First, the sign test, based on counting the number of values above and below the median, gives a $p$-value of 0.02. Second, very similar are the results from the signed rank test, which is based on comparing the average ranks of values above and below the median. These contradicting results from the mean and the median tests indicate that omitting deviating values is perhaps justified. Once one such value is omitted (PG 1012+008, see above), the sample mean becomes +4.6 ($\pm3.2$ at 95% conf. interval) days ($p=0.006$), and the median – +3.5 ($p=$0.012 and 0.006 for the tests mentioned above, respectively). The remaining objects’ distribution resembles Gaussian (shown in Figure 2), although the K-S test rejects the idea, strictly speaking.
In a case that the reprocessing from a standard accretion disk is primarily responsible for the optical variations and respectively – for the interband time lags, one would expect a wavelength dependent delay between the bands, which can be expressed (following Frank et al. 2002) as follows:
$\tau_{\rm B-R} \simeq 5 \dot m^{1/3}M_{\rm 8}(\lambda_{\rm R, 5000}^{4/3} - \lambda_{\rm B, 5000}^{4/3}) $ \[days\],
where $\dot m$ is the accretion rate in Eddington units, $M_{\rm 8}$ is the central mass in $10^{8}$ Solar masses, and $\lambda_{\rm R, 5000}$, $\lambda_{\rm B, 5000}$ are the average wavelengths of the bands, measured in units of 5000Å[^1]. This expression is obtained under the assumption that the disk emits mostly due to a viscous heating and the reprocessed radiation is only a small addition to the emitted flux. Thus, the disk rings will reprocess most effectively radiation of a wavelength close to the maximum of their Planck curves. Therefore, knowing the disk radial temperature distribution (Frank et al. 2002), one can approximately assess the wavelength dependence of the time lag. Another important assumption is that the reprocessed hard (X-ray) radiation comes from a location very close to the center. If this were not the case, but the X-rays come from e.g. a jet base elevated high above the disk instead, an additional geometrical factor of $\sim\cos(\theta)$ $shortens$ the lag between $B$ and $R$-bands, due to the decrease of the path difference. Table 1 (the last column) presents the expected time lags, calculated based on the accretion parameters ($M$ and $\dot m$) from the previous two columns, adopted from Kaspi et al. (2005).
Discussion
==========
Sources of scatter
------------------
### Photometric errors and sampling influence
In order to get an idea about how much different uncertainties affect the time lags (the position of the $ICCF$ peak), we performed two tests to the light curve data of a well sampled object, PG 0804+761, with a clearly defined positive lag, to see if the photometric errors associated with the data points and the sampling can alter significantly the result. Figure 3 (left panel) shows a number of $ICCF$s of the light curves with a random noise, appropriately added to mimic the expected photometric error. One sees that the effect is not significant, leading to a $\sigma_{\rm phot}\simeq0.4$ days uncertainty of the peak position. The sampling, as expected, can affect more significantly the peak position. This object is observed on $N=88$ epochs, while many others – in less than 30. In order to study the effects of the scarce sampling, we randomly retrieved a sub sample of the original data points, which was further used for the interpolation and the cross-correlation analysis. The results for a number of simulations are shown in Figure 3, middle and right panels for 50% and 25% sub samples, respectively. One may get the impression of a significant scatter of the peak position, although it is due mostly to the different peak values (the maxima of the $ICCF$s are not normalized); the peak position itself varies a little – $\sim$1.3 days ($rms$) for the 50%-th sub sample and $\sim$3.3 days for the 25%-th one. Therefore, the photometric error and the sampling effects cannot alter significantly the lag results, at least for the well-sampled objects. Yet, for the very scarcely observed objects, the uncertainties of the peak position might be significant. Based on the simulations described above, we found appropriate a very tentative error assessment approach, taking into account all sources of errors: $\sigma_{\rm \tau} \simeq \sqrt{\sigma_{\rm int}^2 + \sigma_{\rm phot}^{2} + 25(2-\log(N))^{2}}$, where we adopt $\sigma_{\rm int}=0.6$ (Sect. 3) and $\sigma_{\rm phot}=0.4$ days. This expression may work reasonably well for our sample (where $N<100$), but should by no means be considered universal; see Gaskell & Peterson (1987) for more details on the error issues.
Note, that the statistical errors of the cross-correlation functions are very small due to the large number of interpolated magnitude values used and their effect on the peak position is negligible. If the observed time lags are as a result of reprocessing, the errors estimated above, following this simple approach, appear to be significantly underestimated for a number of reasons in comparison with the observed scatter (Fig. 4, see also Sect. 5.2).
### Other sources of scatter
Although the observed and the calculated lags appear to be broadly consistent for the well-sampled objects (Sect. 5.2), the scatter is significant. Except for the errors, described in Sect. 5.1.1, there are also several other factors that may contribute to the scatter:
- **$ICCF$ method limitations**. It is not clear to what extent replacing large missing parts of the real light curve with straight lines, randomly or systematically alters the lag results. Note, that the errors discussed in Sect. 5.1.1 are not the errors introduced by the interpolation itself.
- **Correlated photometric errors**. If the photometric errors of $B$ and $R$-bands happen to be correlated to some extent (due to variable seeing, atmospheric transparency changes, reduction errors, low-level variability of comparison stars, etc. common problems), leading to a common offset of both $B$ and $R$-magnitudes for a given epoch of observation, the time lag between the bands will naturally decrease (as an absolute value), as these effects will lead to an increased correlation between the data sequences at zero lag.
- **Accretion parameters’ uncertainties**. Uncertainties in the black hole mass and accretion rate (possibly systematic!) naturally increase the scatter.
- **Central source location**. The height of the central irradiating source above the disk is another unknown that may well vary from object to object, as well as in time (Sect. 5.2). Furthermore, nothing guarantees that this source is located along the central axis; it can simply be an active region, located somewhere above the disk, even closer to the R-band emitting parts than to the B-band such (Sect. 1).
- **Emission lines’ contribution**. The broad emission lines (H$\alpha$ and H$\beta$) can also contribute to the scatter, as they may fall into the $B$ and $R$-bands and are expected to vary with a lag behind the continuum. For most of the objects, however, the broad-line response times are significant, typically $\sim$100 days, which is much longer than the average continuum lags. Also, for most of the objects, the red shift is high enough to move H$\alpha$ out of the $R$-band, but around $z\simeq0.2$ H$\beta$ enters this band.
Quasar properties
-----------------
Except for the uncertainties of different nature (see above), the scatter between the expected and the observed lags can also be attributed to the possibility that the reprocessing may not be the primary driver of the optical variability for some of the objects. As mentioned before, a weak blazar component can also play a role, in addition to other (possibly unknown) mechanisms. If so, one may expect to be able to differentiate between the quasars, for which the reprocessing is responsible for the observed lags from those, for which this mechanism is different, based on other quasar properties, such as radio loudness, X-ray spectral index, continuum colors, optical spectra, etc. The idea is that these different mechanisms may leave their signatures on the observed quasar appearance. We tested the normalized time lag difference against various quasar observables, including luminosity, accretion rate, equivalent widths of H$\beta$ and \[OIII\] lines and their ratio, FeII/H$\beta$ ratio, radio power, X-ray to optical continuum index[^2], as well as CIV$\lambda$1549 equivalent width and shift[^3], but found no significant correlations, except for some tendencies. The most interesting cases are shown in Figure 4. The top-left panel shows the influence of the number of observational epochs – one sees that for $N \ga 35$ the scatter reduces significantly, meaning that for many objects, the number of the observations may be simply not high enough to reveal the true time lags. Based on this observation, all the remaining panels of this figure show separately the objects for which $N>35$ (as filled squares) from the remaining, undersampled objects (as crosses). Interestingly, this tendency is not that strong for the higher-mass objects ($\log M_{\rm BH}>8.5$), for which the scatter appears to be generally small for all $N$ (only in the top-left panel the separation is based on $\log M_{\rm BH}$). The radio-loudness does not seem to play a significant role in the scatter (middle-left), meaning probably that a possible blazar component does not contribute significantly to the optical variability for this sample. A weak tendency for the bluer object to have a delay between the bands shorter than expected, even for the well-sampled objects, is shown in the bottom-left panel (Figure 4). It is not clear how to explain this effect, if real at all, but it may be connected to the way continuum is generated (i.e. the exact accretion disk structure and properties, being perhaps different from the standard model). The top-right panel shows the influence of the of the width of the broad lines (H$\beta$). The scatter appears to be most significant for the narrower-line (FWHM$\la$3000 kms$^{-1}$) objects, which may have implications for possible quasar population differences (e.g. Sulentic et al. 2000). The soft X-ray spectral index panel (middle-right) reveals an interesting possibility that the optical variations of the harder ($\alpha_{\rm X}\ga 1.8$) objects are perhaps less likely to be attributed to reprocessing in a $standard$ accretion disk, which may have implications for the origin of the soft X-ray excess. Finally, the bottom-right panel demonstrates that, indeed, the emission lines around $z\simeq0.2$ may also play a role in the continuum time lags, as the scatter appears to be the largest there (see Sect. 5.1.2 for more details).
If we take into account only the best sampled objects ($N \ga 35$), the average normalized difference between the observed and the expected time lags, $\overline{(\tau_{\rm obs}-\tau_{\rm exp})/\tau_{\rm exp}}$, is $-0.32$, slightly less than the expected value of zero. Should there be no systematic errors involved in the estimation of $\tau_{\rm exp}$, and in a case that the reprocessing is mainly responsible for the lags, another possibility emerges, i.e. the irradiating source is located above the disk, as mentioned previously. The value of $-0.32$ corresponds roughly to an elevation angle of $\simeq45$ deg., meaning a height above the disk in order of the distances at which the optical continuum is generated, i.e. 100 – 1000 gravitational radii (Arévalo et al. 2008), raising the possibility that the jet base is the primary source of irradiating X-rays (see however Czerny & Janiuk 2007, for a warm absorber interpretation). This interpretation, however, due to the large scatter, cannot be justified statistically with the available optical data, so it has to be considered only as an interesting possibility.
Conclusions
===========
In this paper we analyze the light curves and study the time lags between the optical continuum bands for a large sample of quasars. In spite of the significant scatter, we show that the lags are broadly consistent with the reprocessing model, according to which the optical variations are largely due to the reprocessing of the central X-ray radiation in a surrounding thin accretion disk. There are also some indications that the central X-ray irradiating source may be located at some height (a few hundred Schwarzschild radii) above the disk plane, representing perhaps a (failed) jet base. The deviations of the observed lags from the expectations, assuming the reprocessing model, do not seem to correlate significantly with any other quasar properties and are probably due mostly to the scarce sampling. The paper also demonstrates that the broad-band optical monitoring of quasars could be a powerful tool to study the central engines, provided the light curve is well sampled. A small robotic telescope (e.g. 0.4–0.6m), dedicated to monitoring a large sample of brighter quasars once every 2–3 days, with an accuracy of 0.02 mag or better in 3 filters, might be able to clarify the role of reprocessing in quasar variability.
Alexander T., 1997, in “Astronomical Time Series”, Eds. D. Maoz, A. Sternberg, and E.M. Leibowitz, (Dordrecht: Kluwer), p. 163 Arévalo P., Uttley P., Kaspi S., et al., 2008, MNRAS, to appear Bachev R., Marziani P., Sulentic J. W., et al., 2004, ApJ, 617, 171 Czerny B., Janiuk A., 2007, A&A 464, 167 Czerny B., Siemiginowska A., Janiuk A., Gupta A. C., 2008, MNRAS 386, 1557 Edelson R. A., Krolik J. H., 1988, ApJ, 333, 646 Frank, J., King, A., Raine, D.: 2002, “Accretion Power in Astrophysics”, Cambridge University Gaskell C. M., Sparke L. S., 1986, ApJ, 305, 175 Gaskell C. M., Peterson B. M., 1987, ApJS 65, 1 Giveon U., Maoz D., Kaspi S., et al., 1999, MNRAS 306, 637 Kaspi S., Maoz D., Netzer H., et al.; 2005, ApJ 629, 61 Kellermann K. I., Sramek R., Schmidt M., et al., 1989, AJ 98, 1195 Krolik J. H., Horne K., Kallman T. R., et al., 1991, ApJ, 371, 541 Liu H. T., Bai J. M., Zhao X. H., Ma L., 2008, ApJ 677, 884 Sergeev S. G., Doroshenko V. T., Golubinskiy Yu. V., et al., 2005, ApJ 622, 129 Shakura N. I., Syunyaev R. A., 1973, A&A 24, 337 Schmidt M., Green R. F., 1983, ApJ, 269, 352 Sulentic J.W., Marziani P., Dultzin-Hacyan D., 2000, ARA&A, 38, 521 Sulentic J.W., Bachev R., Marziani P., et al., 2007, ApJ 666, 757 White R. J., Peterson B. M., 1994, PASP 106, 879
[^1]: This expression applies to the quasars’ rest-frame. Due to the similar way the times and the wavelengths are affected by the red shift, for the observer’s frame $\tau$ $reduces$ by a factor of $(1+z)^{1/3}$, which is only a few percent for this sample, and is much less than the expected errors.
[^2]: Data are taken from Giveon et al. (1999) and Kaspi et al. (2005)
[^3]: Data are taken from Bachev et al. (2004) and Sulentic et al. (2007)
|
---
abstract: 'Energy, water, health, transportation and emergency services act as backbones for our society. Aiming at high degrees of efficiency, these systems are increasingly automated, depending on communication systems. However, this makes these prone to cyber attacks, resulting in data leaks, reduced performance or even total system failure. Beyond a survey of existing vulnerabilities, we provide an experimental evaluation of targeted uplink jamming against ’s air interface. Primarily, our implementations of smart attacks on the , the as well as on the radio access procedure are outlined and tested. In exploiting the unencrypted resource assignment process, these attacks are able to target and jam specific UE resources, effectively denying uplink access. Evaluation results reveal the criticality of such attacks, severely destabilizing , while minimizing attacker exposure. Finally we derive possible mitigations and recommendations for stakeholders, which serve to improve the robustness of mission critical communications and enable the design of resilient next generation mobile networks.'
author:
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bibliography:
- 'References.bib'
title: 'Towards Resilient 5G: Lessons Learned from Experimental Evaluations of LTE Uplink Jamming'
---
Introduction
============
Nowadays, societies heavily depend on services provided by so-called , including energy, water, health, transportation, public safety and communication systems. To improve efficiency and management of such infrastructures, comprehensive automation is pursued, necessitating an integration of . Due to the lengthy and costly deployment of dedicated wired networks, harnessing (public) mobile communication technologies is widely regarded a suitable approach. Yet, ubiquitous connectivity eases access not only for authorized users, but also for malicious third parties. Cyber attacks on communication networks can severely degrade functionality and system stability. An attack, e.g. aimed at the power grid’s infrastructure, could trigger events leading to outages or blackouts [@smartgrid_security]. Two major ways of disrupting a ’s wireless communication can be distinguished. On one side, so-called barrage jammers may be employed, which essentially obscure all user signals in a certain frequency range by using wideband noise. Such jammers are very effective in disrupting services and straightforward to implement. Yet, they can be detected and located with low effort, allowing authorities to stop the attack and hold attackers accountable. On the other side, smart jammers exploit inherent system properties such as protocol flaws, requiring less power. Thus, highly precise, covert attacks are enabled, only affecting target devices instead of an entire area or frequency band. Subsequently, attacks cannot be recognized as easily. However, this class of jammers requires more advanced technological skills and knowledge. Hence, smart jammers are of major interest for groups with sufficient resources and technical capacities, which want to perform attacks, while staying hidden. Examples include hostile intelligence services or well-funded terrorist groups. Employed efficiently, smart jammers provide the means to secretly undermine critical, public infrastructures which depend on wireless communications.
It has to be emphasized that this publication is not intended as guideline for such groups, but rather serves to indicate vulnerabilities and derive mitigation strategies. Therefore, the main contributions of this paper can be summarized as below:
- General overview of possible attack vectors against current cellular mobile communication networks,
- Design of a smart jammer for issuing attacks on the and , including corresponding evaluation results,
- Recommendations for improving communication systems, focusing on developments.
The remainder of this paper is structured as follows: Section \[sec:attacks\] provides a survey of key jammer properties and known attacks. It is followed by an overview of related work (Sec. \[sec:relwork\]). Next, we introduce our smart uplink jammer (Sec. \[sec:concept\]) and present evaluation results (Sec. \[sec:results\]). Recommendations for increased resiliency are given in Section \[sec:recomm\]. Finally, Section \[sec:conclusion\] provides a summary and an outlook on future work.
Jammer Properties, Possible Attacks\
and Mitigations {#sec:attacks}
====================================
This section provides an overview of key jammer properties and possible attack vectors in infrastructures.
Jammer Properties
-----------------
To disrupt radio communications, barrage jammers allow configuration of the used power level, bandwidth and duty cycle. Higher power levels increase the affected area and boost the impact on user signals, yet make the attacker more vulnerable to detection. Duty cycles indicate periods, in which the jammer is active. In contrast to barrage jammers, which are limited to the above mentioned properties, smart jammers provide the following additional parameters:
**Synchronization**: Synchronizing jamming to target signals maximizes an attack’s impact, while minimizing visibility.
**Eavesdropping**: For the same reasons, gathering information about a target system before initiating an attack is beneficial. In , this task is facilitated by the lack of and encryption. Thus, subscribers may be de-anonymized [@rupprecht-19-layer-two] or locations can be tracked [@gutiguti]. In case of stationary , e.g. in Smart Grids, a high level of channel awareness could be obtained and utilized by the attacker.
**Placement**: Due to their small form factor, smart jammers allow for arbitrary placement. Preferable locations may be close to the for downlink or near the for uplink jamming. Thus, lower power levels can be employed, minimizing attack visibility. In contrast, high powered barrage jammers may be of significant size.
**Directionality**: Smart jammers may utilize highly directional antennas, focusing on specific targets. This increases range while decreasing power consumption, visibility and size.
Attack Vectors
--------------
can be attacked at the physical layer, physical channel and protocol level. Examples of each category are given below, summarized by Figure \[fig:attack\_vectors\].
{width="100.00000%"}
### Physical Layer Attacks
\
**Cyclic Prefix Attack**: As a key component of the signal, the cyclic prefix prevents inter symbol interference and enables frequency domain equalization. To degrade radio performance, noise can be emitted during cyclic prefix transmission. Such an attack is considered highly effective as no specific mitigation is known, but it requires near perfect knowledge of the channel.
**Synchronization Attack**: In the course of a synchronization attack, the is prevented from receiving the / (down-/uplink) by shifting the symbol timing peak. Here, the main challenge is locking onto the target signal.
**Reference Signal Attack**: Reference signal (pilot) attacks (/) cause faulty channel estimations. Thereby, diverging conditions are assumed, limiting physical channel throughput. As downlink pilot signals are sparse, an attack requires a minimal duty cycle and transmission power. However, precise synchronization based on an analysis of channel conditions is required.
### Physical Channel Attacks
\
**Jamming**: This method corrupts the channel format indicator. First, the jammer synchronizes to the cell, to receive and decode the . Next elements are calculated and noise or a modulated signal is transmitted on top of them. If the corrupted is higher than the actual one, the tries to decode non-existent . In case the corrupted is lower, the overlooks . For both scenarios, the first subframe slot’s symbols are expected in the wrong location, causing errors in the corresponding transport blocks.
**Jamming**: The carries configuration as well as downlink user data. Hence, it allows targeting one or multiple . Precise synchronization and decoding are required to locate resources of a particular .
**Paging Jamming**: Here jamming is modified, by targeting paging channel . The channel informs idle of pending downlink data. If respective are jammed, the is never notified of transmissions and remains idle. This attack reduces a jammer’s duty cycle to predictable paging periods, relaxing decoding requirements.
**Jamming**: The purest version of jamming is achieved by using a barrage of white noise on this channel. It is straightforward to implement, enabling throughout an entire cell.
**Jamming**: The is located at the edge of the uplink bandwidth. Thus, it can be identified and jammed with low effort, affecting many at once. The attack can lead to misinterpretation of received signals as scheduling requests, (negative) acknowledgments, or precoding matrices, drastically reducing up- and downlink throughput. While some control data may be multiplexed into the , use of the enables higher cell capacities and data rates (e.g. via ). Another advantage for the adversary is that the power limit is commonly lower than the power limit in order to reduce inter-cell interference.
**UE Targeted Uplink Jamming**: Since the entire uplink (both and ) is slaved to the downlink with delay, attackers can identify and target a particular ’s . This is due to the nature of the , which relies on obfuscation via mask rather than proper encryption. An implementation and evaluation of such a jammer is detailed in Section \[sec:concept\]. Given a decoder, a target device’s can be found via its , obtainable by combining de-anonymization [@gutiguti] with tools like C3ACE [@Falkenberg2017c].
### Protocol Attacks
\
**Cell Barring Spoofing**: Rogue mimic real cells, barring from connecting to specific cells by transmitting *cellBarred* and *intraFreqReselection* flags. Thus, will ignore any cells on that frequency. Also, requires to ban cells from which or are not received within a defined time frame. Hence, corrupting these blocks leads to barred cells for up to as well [@3gpp_rel14_ts_36.304].
**Reject Spoofing**: An identical rogue cell is created by synchronizing to a legitimate cell and mimicking its relevant parameters such as ID, tracking area, etc. Next, target are forced from their original cell, and the rogue cell rejects subsequent reattachment attempts. Depending on the specific implementation, may treat the as generally barred. Thus, this attack can be very efficient, yet requires significant effort.
Related Work {#sec:relwork}
============
In recent years, several publications have dealt with the analysis of different vulnerabilities. An overview of various attack vectors against the air interface is given in [@lichtman_overview], estimating potential jammer to signal power ratios. Hussain et al. propose LTEInspector [@lteinspector], which combines a symbolic model checker with a cryptographic protocol verifier. It is applied to analyze several different attacks, most of which are validated using a testbed set-up. Though commercial are used for verification, evaluation is mainly focused on protocol implementations and core network signaling. Cyclic prefix and pilot jamming are evaluated for both up- and downlink in [@cp_pilot_jamming]. Measurements indicate that is more resilient against these types of attack than . A method for disrupting the is introduced in [@pcfich_jamming], using simulations for validation. However, real-world measurements would be required for a comprehensive assessment. Also, will mitigate the threat by removing the . Labib et al. use radio frequency spoofing to impair synchronization and cell selection [@labib_cell_selection; @labib_cell_sync]. Based on experiments with a software , enhancements to the aforementioned techniques are proposed. Simulations are performed in [@lichtman_pucch] to evaluate jamming against different formats. Possible mitigations are discussed, e.g. foregoing transmissions at the cost of limiting system capacity. and de-anonymization ( catching) attacks are evaluated against commercial in a laboratory by [@mjolsnes_imsi_catcher]. Yet, the employed is not commercial grade. Rao et al. provide physical layer measurements on resilience, considering attacks on different features of the downlink signal [@Rao2017LTEPL]. However, the same open-source and implementations are evaluated against each other. Thus, results cannot be generalized to commercial equipment.\
In comparison to related work, this publication evaluates further aspects of the air interface, considering the impact on end-to-end connectivity of commercial grade equipment.
Targeted Uplink Jammering:\
Concept, Implementation and Evaluation Scenario {#sec:concept}
===============================================
Within this section, we introduce our approaches to jamming and describe the scenario and measurement setup.
Smart Uplink Jamming Concepts
-----------------------------
### PUSCH/PUCCH Jammer
Our jammer, implemented on top of srsLTE [@gomez2016srslte], disturbs the and respectively. To perform the attack, we synchronize the device with a cell, decode the and transmit on , originally assigned to the target . Building on the implementation in srsLTE, the jammer imitates the victim device. Upon activation, the jammer jumps to the Connected State immediately, utilizing the provided target . Afterwards, the jammer is able to transmit on top of the ’s assigned . No other radio access or scheduling requests are issued and remains nearly unchanged.
###
Besides the above described jammers, an attack on the radio access procedure is devised, referred to as . It targets the first message of one or several . The attack is structured as shown in Figure \[fig:prattle\]: 1) The jammer continuously monitors the for Radio Access Response Grants. 2) Connection Requests, identified by grants, are jammed repeatedly. 3) Due to this attack, the fails the complete procedure and retries until a maximum number of attempts (6 to 200) is reached. After that, the corresponding cell is to be treated as barred for up to [@3gpp_rel14_ts_36.304]. In the following, however, only the former two attacks are evaluated.
![Principle of the PRATTLE attack[]{data-label="fig:prattle"}](img/prattle.pdf){width="1\columnwidth"}
### Critical Infrastructure Jamming Scenario
To evaluate the developed uplink jammer’s effectiveness, we consider a Smart Grid scenario. Such possess well-defined requirements, as specified by e.g. the ’s standard 61850 [@61850-5:TC57:IEC], which are extremely challenging regarding reliability and latency. Moreover, cyber attacks on power grid infrastructures may be of severe consequences, endangering all dependent systems. Attackers profit from the static nature of grid assets, allowing for optimal target localization and jammer placement. We specifically consider 61850 based systems, regularly transmitting measurement data between Smart Grid substations.
![Experimental setup for evaluating the uplink jammer[]{data-label="fig:setup"}](img/setup.pdf){width="1\columnwidth"}
Experimental Measurement Setup
------------------------------
Measurements are performed within our laboratory as shown in Figure \[fig:setup\], in isolation from public radio networks. For this purpose, the under test is placed in a shielding box and connected to the via a box-internal antenna. uplink and jammer signal are combined and fed to the . For synchronization, the downlink signal of a commercial, -based Amarisoft base station is sent to and jammer. An Ettus Research N210 / 2x2 serves as radio frequency frontend. The supports dynamic power control, status displays, an for retrieving diagnostic data, information on active and identities. It is configured with a bandwidth of and parametrized to emulate an observed real world cell. Transmit and receive gain of the are adjusted to achieve stable, near optimal radio link conditions. , and functionalities are provided by an NG40 Virtual . The jammer itself utilizes the Ettus Research B210 platform, which is recommended for use with srsLTE. As victim , we employ a Sierra Wireless Air Prime EM7455 (Qualcomm Snapdragon X7 LTE Modem, Cat. 6) and a Huawei ME909s-120 (HiSilicon Balong 711, Cat. 4), as both devices enable an automated evaluation process. Additional qualitative evaluations with an LG G5 smartphone (Qualcomm X12 LTE) confirm the results. Measurements are conducted for and at different jammer gain levels, with two implementations (i.e. ). The transmission of measurement values according to 61850, is replicated with our purpose-built traffic generator, offering a higher degree of flexibility than standard tools. Initial experiments show that the limits the ’s capability of mitigating jamming-induced, degraded , as the aims to reduce the ’s transmission power for minimizing interference between users. Further, our evaluations indicate high sensitivity towards variations of set-up parameters.
Evaluation Results {#sec:results}
==================
For evaluation, measurement values are transmitted every via packets. Jammer gain is increased from to with a step size of . At every gain level $20$ runs of duration each are performed. The jammer is active for between $t_{start}=\SI{6}{s}$ and $t_{end}=\SI{45}{s}$. We employ the following measurement procedure: 1) start of , and traffic generation/reception servers, 2) initiation of the ’s packet stream, 3) jammer start, 4) jammer stop, 5) recovery period of the - link, 6) waiting for the traffic generator’s termination packet (if the link is re-established) and the receiver’s report.
### PUCCH vs. PUSCH jamming
Figure \[fig:pusch\_pucch\] gives exemplary results of both tested at three different jammer gain levels. Repeated runs show similar behavior. For low gains (about $\SI{2}{dB}$), it can be noted that jamming does not affect the connection. Applying high gains, retransmissions occur without loss of packets. However, for jamming retransmissions occur, reducing throughput. Packets are dropped as the is unable to drain its transmit buffer. Boosting gain further reduces throughput to zero. -only jamming has a low impact as the shifts control data to the .
![Comparison of the number of received packets under the impact of and jamming, considering three jammer gain levels for two commercial modems[]{data-label="fig:pusch_pucch"}](img/PUSCH_vs_PUCCH.pdf){width="1\columnwidth"}
### PUSCH jamming for different UE implementations
The behavior of different under jamming is compared in Figure \[fig:pusch\_overview\]. It illustrates the total number of successfully received packets for different jammer gains. Both start to lose packets from gains of approximately . Yet, at higher gain levels the two devices recover slightly, with the Huawei increasing throughput between - and the Sierra Wireless in the range of -. This is caused by the detecting the worsening channel conditions. It therefore grants additional resources and requests the to use more robust modulation and coding schemes. Furthermore, the Huawei recovers at a gain of , which is explained in more detail in the following paragraph. Overall the Huawei modem shows better resilience, achieving higher throughput for most configurations. However, in contrast to the Sierra Wireless , it frequently crashes, requiring manual restarts.
![ jammer impact under different jammer gains[]{data-label="fig:pusch_overview"}](img/Sierra_vs_Huawei.pdf){width="1\columnwidth"}
The following analysis provides additional details on the Huawei ’s reaction to jamming. Figure \[fig:pusch\_details\] shows its uplink throughput over time for selected gains. Again, runs with the same parameter set exhibit similar behavior. The top part of Figure \[fig:pusch\_details\] (gain: ) shows strongly fluctuating throughput starting at , due to jamming induced link degradation. At ( respectively) the loses connectivity and attempts to reconnect using its previous . Thus, the jammer is able to continue its attack as soon as the reconnects. After failing the radio access procedure several times, the enters a back off period of (in this case) . With higher gains (middle of Figure \[fig:pusch\_details\]), the is forced from the cell immediately after radio access ($\ll\SI{1}{s}$), as its signal is significantly weaker than the jammer’s. The bottom part of Figure \[fig:pusch\_details\] is specific to the Huawei device. At a very high jammer gain of the connection is disrupted in a way, which causes the to assume a complete connection loss. Hence, it starts an entirely new connection, utilizing a different and sending an Connection Request instead of an Reestablish Request. In this way the shakes off the jammer, achieving stable transmission after .
Repeating experiments with the Sierra Wireless show behavior similar to the Huawei modem. However, the Sierra Wireless handset does not recover in any case. Also, the ’s back off period increases with the number of failures (up to: Sierra , Huawei ), i.e. at higher jammer gains. In several cases it even refuses to connect for up to . This indicates that the cell barring timer is used to exclude the from its list of selectable cells.
![Detailed behavior under jamming[]{data-label="fig:pusch_details"}](img/PUSCH_Huawei.pdf){width="1\columnwidth"}
Uplink Jamming Mitigations
--------------------------
An uplink jamming attack may be identified by the on basis of up- and downlink channel conditions diverging significantly from each other as well as from historical data. Our evaluations using the Huawei device already point to a possible mitigation strategy. It was shown that switching the allows to render the attack ineffective, requiring the attacker to re-identify the . Hence, new, unpredictable should be assigned on both connection and reconnection attempts. Yet, since control channels in LTE are not encrypted, this is not a sustainable option. Beyond that, should assign more robust modulation and coding schemes more quickly and allow critical a higher transmit power.
Recommendations for\
5G Standardization and Deployment {#sec:recomm}
=================================
In the following we provide recommendations for 5G stakeholders organized by standardization and deployment aspects.
Standardization of 5G Mobile Networks
-------------------------------------
As a major aspect of new radio, beamforming increases resilience against jamming. However, there are still several shortcomings, which should be addressed by future releases. Though finally supports encryption and integrity protection of $>0$ and [@3gpp_rel15_ts_38.331; @3gpp_rel15_ts_33.501], null encryption is still acceptable [@3gpp_rel15_ts_33.501]. Even if encryption is applied, the air interface is not secured until after the successfully completes the entire attach procedure [@3gpp_rel15_ts_33.501].
### Mitigating Cell Barring and Reject Spoofing Attacks
According to the standard [@3gpp_rel15_ts_38.331] intra-frequency barring is still applied without verifying broadcasted barring information. Such cell impersonation attacks could be prevented through the use of verification schemes for cell configuration (either for all or those required by ). Therefore, the following mechanism may be employed, using signatures applicable to both and : 1) Operators generate public/private key pairs of reasonable size (e.g. ). 2) Operators provision public keys in the . 3) Operators sign hashes of concatenated and scheduled contents with their private keys 4) Base stations broadcast the signature in dedicated . 5) receive all periodic , tentatively applying their settings, then verify the signature using the public key. 6) If verification succeeds, settings are committed (including intra-frequency cell barring), otherwise the settings are ignored and the cell is treated as malicious. While this approach prolongs the cell search, it precludes barring spoofing and reject attacks.
### Mitigating Uplink Jamming Attacks
As in , the 5G is not encrypted. Hence findings still apply (with few limitations). A simple means of increasing the computational load for the attacker is scrambling the entire with a sequence derived from the destination , rather than just scrambling the (as in ). To compensate for limited entropy, an hopping scheme can be employed at base stations by periodically reconfiguring the , once encryption for 1 [@3gpp_rel15_ts_38.331] is established. Thus, the probability of an attacker learning the during its lifetime is reduced.\
Besides, some general measures can be used to improve the resilience of . Providing the standard in a machine readable formate would facilitate (automatic) identification of critical flaws. Lightweight, mutual authentication and encryption for the air interface should be introduced. Open, less complex, single-purpose protocols, such as Internet protocols, should be preferred to highly flexible solutions with many optional, potentially vulnerable extensions. Also, legacy compatibility with insecure mechanisms and protocols is to be avoided.
5G System Operation for Critical Infrastructures
------------------------------------------------
To identify irregularities in system operation, 5G mobile network operators should deploy advanced monitoring systems. Rogue and impersonated can be identified with this method. Beamforming antennas allow to increase , suppress interference signals and restrict jammer placement options, thus enhancing resilience. Distributed beamforming techniques such as provide more robust radio links by receiving signals from multiple . Secure reallocation schemes for temporary mitigate de-anonymization. Disallowed or limited null-ciphering protects user data and improves overall security.
Prior to deployment, operators ought to check coverage to ensure are offered high signal strength connections via more than one , providing mobile network redundancy. Location specific connection properties should be considered during installation, e.g. directed antennas may be employed under line-of-sight conditions. Operators should procure , which support higher schemes for spatial redundancy. Locking into tracking areas mitigates rogue with different area codes. Also, cards without voice services prevent de-anonymization through call-repetition and downgrade attacks, i.e. forcing into insecure and slow 2G/3G cells. utilities and mobile network providers can improve attack detection by continuously exchanging performance data, e.g. via secure . Also, appropriate layer 4 protocols help reduce the impact of jamming.
Conclusion and Outlook {#sec:conclusion}
======================
Within this paper we provide an overview of relevant cyber attacks on mobile communication networks, endangering the stability of . In particular, we present our concept for disrupting uplink communications on the of a Smart Grid infrastructure. Corresponding experimental evaluations demonstrate the attack’s effectiveness. The jammer is shown to significantly reduce throughput, forcing the to back off or even crash. Moreover, our analysis also indicates potential mitigations such as the application of scrambled or encrypted . Recommendations to mobile network and operators as well as to standardization bodies are derived, revealing a path towards secure and resilient . Key improvements involve beamforming and massive .\
In future work, we aim at broadening the evaluation to further equipment. Also, the public radio access termination attack, described in Section \[sec:attacks\], is to be implemented and analyzed in detail. For mitigation, we plan to develop a robust, lightweight encryption/scrambling strategy to counter eavesdropping, which serves as basis for attacks such as those presented here.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by the federal state of Northrhine-Westphalia and the “European Regional Development Fund” (EFRE) 2014-2020 via the “CPS.HUB/NRW” project under grant number EFRE-0400008, and has been carried out within the Franco-German Project *BERCOM* (FKZ: 13N13741), co-funded by the German Federal Ministry of Education and Research (BMBF). The authors thank Stefan Monhof and Robert Falkenberg for valuable discussions.
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---
abstract: 'In this paper, we study the problem of hyperspectral pixel classification based on the recently proposed architectures for compressive whisk-broom hyperspectral imagers without the need to reconstruct the complete data cube. A clear advantage of classification in the compressed domain is its suitability for real-time on-site processing of the sensed data. Moreover, it is assumed that the training process also takes place in the compressed domain, thus, isolating the classification unit from the recovery unit at the receiver’s side. We show that, perhaps surprisingly, using distinct measurement matrices for different pixels results in more accuracy of the learned classifier and consistent classification performance, supporting the role of information diversity in learning.'
author:
- 'Mohammad Aghagolzadeh, Hayder Radha[^1][^2]'
bibliography:
- 'refs.bib'
title: On Hyperspectral Classification in the Compressed Domain
---
Hyperspectral imaging, remote sensing, compressive whisk-broom sensing, pixel classification.
Introduction {#sec:intro}
============
Recently, there has been a surge toward compressive architectures for hyperspectral imaging and remote sensing [@spm14_cs]. This is mainly due to the increasing amount of hyperspectral data that is being collected by high-resolution airborne imagers such as NASA’s AVIRIS[^3] and the fact that a large portion of data is discarded during compression or during feature mining prior to learning [@hi_review]. It has been noted in [@fow14] that many of the proposed compressive architectures are based on the spatial mixture of pixels across each frame and correspond to physically costly or impractical operations while most existing airborne hyperspectral imagers employ scanning methods to acquire a pixel or a line of pixels at a time. To address this issue, practical designs of compressive *whisk-broom* and *push-broom* cameras were suggested in [@fow14]. In this work, we tackle the problem of hyperspectral pixel classification based on compressive whisk-broom sensors; i.e. each pixel is measured at a time using an individual random measurement matrix. Extension of the presented analysis for the compressive push-broom cameras is straightforward.
To set this work apart from existing efforts that have also focused on the problem of classification from the compressive hyperspectral data, such as [@fow09], we must mention two issues with the typical indirect approach of applying the classification algorithms to the *recovered* data: $a$) the sensed data cannot be decoded at the sender’s side (airborne device) due to the heavy computational cost of compressive recovery, making on-site classification infeasible, $b$) the number of measurements (per pixel) may not be sufficient for a reliable signal recovery. It has been established that classification in the compressed domain would succeed with far less number of random measurements than it is required for a full data recovery [@svm_cs]. However, the compressive framework of [@svm_cs] corresponds to using a fixed projection matrix for all pixels which limits the *measurement diversity* that has been promoted by several recent studies for data recovery and learning [@fow09j; @globsip13; @few].
Rather than devising new classification algorithms, this work is focused on studying the relationship between the camera’s sensing mechanism, namely the employed random measurement matrix, and the common Support Vector Machine (SVM) classifier. It must be emphasized that the general problem of classification based on compressive measurements has been addressed for the case where a fixed measurement matrix is used [@sp_cs; @svm_cs]. However, our aim is to study the impact of *measurement diversity* on the learned classifier. In particular, we investigate two different sensing mechanisms that were introduced in [@fow14] [^4]:
- [**FCA-based sensor:**]{} A Fixed Coded Aperture (FCA) is used to modulate the dispersed light before it is collected at the linear sensor array. This case corresponds to using a fixed measurement matrix for each pixel and a low-cost alternative to the DMD system below.
- [**DMD-based sensor:**]{} A Digital Micromirror Device (DMD) is used to modulate the incoming light according to an arbitrary pattern that is changed for each measurement. Unlike the previous case, DMD adds the option of sensing each pixel using a different measurement matrix. Both cases are illustrated in Figure \[fig:S\].
![FCA-based versus DMD-based sensing. Here, rows represent pixels and columns represent spectral bands.[]{data-label="fig:S"}](true.png){width="\linewidth"}
Complete data
![FCA-based versus DMD-based sensing. Here, rows represent pixels and columns represent spectral bands.[]{data-label="fig:S"}](fix.png){width="\linewidth"}
FCA-sensed data
![FCA-based versus DMD-based sensing. Here, rows represent pixels and columns represent spectral bands.[]{data-label="fig:S"}](var.png){width="\linewidth"}
DMD-sensed data
SVM has been shown to be a suitable classifier for hyperspectral data [@hi_review]. Specifically, we employ an efficient linear SVM classifier with the exponential loss function that gives a smooth approximation to the hinge-loss. To train the classifier in the compressed domain, we must *sketch* the SVM loss function using the acquired measurements for which we employ some of the techniques developed in [@sp_cs]. Furthermore, given that the sketched loss function gives a close approximation to the true loss function and that the learning objective function is smooth, it is expected that the learned classifier is close to the ground-truth classifier based on the complete hyperspectral data (which is unknown). As it has been discussed in [@r21], recovery of the classifier is of independent importance in some applications.
This paper is organized as follows. In the Section \[sec:two\] we present the learning algorithm that gets the compressive measurements as input and produces a linear pixel classifier in the signal domain. Section \[sec:simo\] contains the simulation results and their analysis. We conclude the paper in Section \[sec:conclusion\].
Problem Formulation and the Proposed Framework {#sec:two}
==============================================
Overview of SVM for spectral pixel classification {#subsec:problem}
-------------------------------------------------
In a supervised hyperspectral classification task, a subset of pixels are labeled by a specialist who may have access to the side information about the imaged field such as being physically present at the field for measurement. The task of learning is then to employ the labeled samples for tuning the parameters of the classification machine to predict the pixel labels for a field with similar material compositions. Note that, for subpixel targets, an extra stage of spectral unmixing is required to separate different signal sources involved in generating a pixel’s spectrum [@r10]. For simplicity, we assume that the pixels are homogeneous (consist of single objects).
Recall that most classifiers are inherently composed of binary decision rules. Specifically, in multi-categorical classification, multiple binary classifiers are trained according to either One-Against-All (OAA) or One-Against-One (OAO) schemes and voting techniques are employed to combine the results [@r11]. In a OAA-SVM classification problem, a decision hyperplane is computed between each class and the rest of the training data, while in a OAO scheme, a hyperplane is learned between each pair of classes. As a consequence, most studies focus on the canonical binary classification. Similarly in here, our analysis is presented for the binary classification problem which can be extended to multi-categorical classification.
In the linear SVM classification problem, we are given a set of training data points (corresponding to hyperspectral pixels) $x_j\in\mathbb{R}^d$ for $j=1,2,\dots,n$ and the associated labels $z_j\in \{-1,+1\}$. The inferred class label for $x_j$ is $\mbox{sign}(x_j^T w - b)$ that depends on the classifier $w\in \mathbb{R}^d$ and the bias term $b\in \mathbb{R}$. The classifier $w$ is the normal vector to the affine hyperplane that divides the training data in accordance with their labels. When the training classes are inseparable by an affine hyperplane, *maximum-margin soft-margin* SVM is used which relies on a *loss function* to penalize the amount of misfit. For example, a widely used loss function is $\ell(r)=\left(\mbox{max}\{0,1-r\}\right)^p$ with $r=z_j(x_j^T w-b)$. For $p=1$, this loss function is known as the hinge loss, and for $p=2$, it is called the squared hinge loss or simply the quadratic loss. The optimization problem for soft-margin SVM becomes[^5] $$(w^*,b^*)=\arg\min_{w,b} \frac{1}{n}\sum_{j=1}^n\ell(z_j(x_j^T w-b)) + \frac{\lambda}{2} \|w\|^2_2
\label{eq:svm_primal_soft}$$ In this paper, we use the smooth exponential loss function, which can be used to approximate the hinge loss while retaining its margin-maximization properties [@exploss]: $$\ell(z)=e^{-\gamma z}
\label{eq:exp_loss}$$ where $\gamma$ controls the smoothness. We use $\gamma=1$.
SVM in the compressed domain
----------------------------
Let $y_j=\Phi_j x_j\in \mathbb{R}^{d'}$ denote the low-dimensional measurement vector for pixel $j$ where $d'\leq d$ is size of the photosensor array in the compressive whisk-broom camera [@fow14]. As explained in [@single_pixel], a DMD architecture can be used to produce a $\Phi_j$ with random entries in the range $[0,1]$ or random $\pm 1$ entries, resulting in a sub-Gaussian measurement matrix that satisfies the isometry conditions with a high probability [@simple]. Recall that the measurement matrix $\Phi_j$ is fixed in a FCA-based architecture while it can be distinct for each pixel in a DMD-based architecture.
As noted in [@sp_cs], the orthogonal projection onto the row-space of $\Phi_j$ can be computed as $P_j=\Phi_j^T (\Phi_j\Phi_j^T)^{-1}\Phi_j$. Consequently, an (unbiased) estimator for the inner product $x_j^T w$ (assuming a fixed $x_j$ and $w$) based on the compressive measurements would be $y_j^T (\Phi_j\Phi_j^T)^{-1} \Phi_j w$. As a result, the soft-margin SVM based on the compressive measurements can be expressed as: $$\hat{w}^*=\arg\min_{w}
\frac{1}{n}\sum_{j=1}^n\ell(z_j y_j^T (\Phi_j\Phi_j^T)^{-1} \Phi_j w)
+ \frac{\lambda}{2} \|w\|^2_2
\label{eq:svm_primal_rand}$$ (we have omitted the bias term $b$ for simplicity).
We must note that the formulation in (\[eq:svm\_primal\_rand\]) is different from what was suggested in [@svm_cs] for a fixed measurement matrix. In particular, we solve for $\hat{w}^*$ in the $d$-dimensional space. Meanwhile, the methodology in [@svm_cs] would result in the following optimization problem: $$\tilde{w}^*=\arg\min_{w} \frac{1}{n}\sum_{j=1}^n\ell(z_j y_j^T w) + \frac{\lambda}{2} \|w\|^2_2
\label{eq:svm_primal_rand2}$$ which solves for $\tilde{w}^*$ in the low-dimensional column-space of $\Phi$. Also note that, in the case of fixed measurement matrices, (\[eq:svm\_primal\_rand\]) and (\[eq:svm\_primal\_rand2\]) correspond to the same problem with the relationship $\hat{w}^*= \Phi^T (\Phi\Phi^T)^{-1} \tilde{w}^*$ (because of the $\ell_2$ regularization term which zeros the components of $\hat{w}^*$ which lie in the null-space of $\Phi$). In other words, (\[eq:svm\_primal\_rand\]) represents a generalization of (\[eq:svm\_primal\_rand2\]) for the case when the measurement matrices are not necessarily the same. This allows us to compare the two cases of $a$) having a fixed measurement matrix and $b$) having a distinct measurement matrix for each pixel, which is the subject of this paper. For simplicity, assume that each $\Phi_j$ consists of a subset of $d'$ rows from a random orthonormal matrix, or equivalently $\Phi_j\Phi_j^T=I_{d'}$; thus, $P_j=\Phi_j^T\Phi_j$. Also assume that, in the case of DMD-based sensing, each $\Phi_j$ is generated independently of the other measurement matrices.
Following the recent line of work in the area of randomized optimization, for example [@r19], we refer to the new loss $\ell(z_j x_j^T \Phi_j^T (\Phi_j\Phi_j^T)^{-1} \Phi_j w)$ as the *sketch* of the loss, or simply the *sketched loss* to distinguish it from the true loss $\ell(z_j x_j^T w)$. Similarly, we refer to $\hat{w}^*$ as the *sketched classifier* as opposed to the ground-truth classifier $w^*$.
![Linear SVM classification —depicted for $d=2$ for illustration. Small arrows represent each $\Phi_j\in\mathbb{R}^{1\times d}$.[]{data-label="fig:F"}](clusterfuck2_star.eps){width="\linewidth"}
FCA-sensed data
![Linear SVM classification —depicted for $d=2$ for illustration. Small arrows represent each $\Phi_j\in\mathbb{R}^{1\times d}$.[]{data-label="fig:F"}](clusterfuck_star.eps){width="\linewidth"}
DMD-sensed data
Figure \[fig:F\] depicts the two cases of using a fixed measurement matrix (FCA-sensed data) and distinct measurement matrices (DMD-sensed data) for training a linear classifier. It is helpful to imagine that, in the sketched problem, each $x_j$ is multiplied with $P_j w$ (the projection of $w$ onto the column-space of $\Phi_j$) since $y_j^T\Phi_j w=(P_j x_j)^T w= x_j^T (P_j w)$. As shown in Figure \[fig:F\] (left) with $P_j=P$ for all $j$, there is a possibility that $w^*$ would nearly align with the null-space of the random low-rank matrix $P=\Phi^T\Phi$. For such $P$, any vector $P w$ may not well discriminate between the two classes and ultimately result in the classification failure. Figure \[fig:F\] (right) depicts the case when a distinct measurement is used for each point. When $\Phi_j$ is symmetrically distributed in the space and $n$ is large, there is always a bunch of $\Phi_j$’s that nearly align with $w^*$ whereas other $\Phi_j$’s can be nearly orthogonal to $w^*$ or somewhere between the two extremes. This intuitive example hints about how measurement diversity pays off by making the optimization process more stable with respect to the variations in the random measurements and the separating hyperplane.
Simulations {#sec:simo}
===========
Handling the bias term
----------------------
It is not difficult to see that employing a distinct $\Phi_j$ for each data vector $x_j$ necessitates having distinct values of bias $b_j$ (for each $\Phi_j$). Note that in the case of fixed measurement matrix, i.e. when $\Phi_j=\Phi$ for all $j$, bias terms would be all the same and linear SVM works normally as noted in [@svm_cs]. However, using a customized bias term for each point would clearly result in overfitting and the learned $\hat{w}^*$ would be of no practical value. Furthermore, the classifier cannot be used for prediction since the bias is unknown for the new input samples. In the following, we address these issues.
First, let $\mathcal{S}$ denote a set of $k$ distinct measurement matrices, i.e. $\mathcal{S}=\{\Phi^{(1)},\Phi^{(2)},\dots,\Phi^{(k)}\}$. Instead of using an arbitrary measurement matrix for each pixel, we draw an entry from $\mathcal{S}$ for each pixel. Given that $n\gg k$, each element of $\mathcal{S}$ is expected be utilized for more than once. This allows us to learn the bias for each outcome of measurement matrix (without the overfitting issue). Note that $k$ signifies the degree of measurement diversity: $k=1$ refers to the least diversity, i.e using a fixed measurement matrix, and measurement diversity is increased with $k$. The new optimization problem becomes: $$\begin{aligned}
\left( \hat{w}^*, b_1^*,\dots,b_k^* \right)
&=& \arg\min_{w,b_1,\dots,b_k} \frac{\lambda}{2} \|w\|^2_2 + \nonumber\\
&& \frac{1}{n}\sum_{j=1}^n\ell(z_j y_j^T \Phi^{(t_j)} w + b_{t_j})
\label{eq:svm_khaf}\end{aligned}$$ where $t_j$ randomly (uniformly) maps each $j\in\{1,2,\dots,n\}$ to an element of $\{1,2,\dots,k\}$. The overfitting issue can now be restrained by tuning $k$; reducing $k$ results in less overfitting. In our simulations, we use $k\geq\ceil*{d/d'}$ to ensure that $\mathcal{S}$ spans $\mathbb{R}^d$ with a probability close to one.
For prediction, the corresponding bias term is selected from the set $\{b_1^*,b_2^*,\dots,b_k^*\}$.
Results
-------
The dataset used in this section is the well-known Pavia University dataset [@r18.1] which is available with the ground-truth labels[^6][^7]. For each experiment, we perform a 2-fold cross-validation with $1000$ training and $1000$ testing samples. As discussed earlier, multi-categorical SVM classification algorithms typically rely on pair-wise or One-Against-One (OAO) classification results. Hence, we evaluate the sketched classifier on a OAO basis by reporting the pair-wise performances in a table . Finally, since the measurement operator is random and subject to variation in each experiment, we repeat each experiment for $1000$ times and perform a worst-case analysis of the results.
Consider the case where a single measurement is made from each pixel, i.e. $d'=1$ and $\Phi_j\in\mathbb{R}^{1\times d}$ is a random vector in the $d$-dimensional spectral space. Clearly, this case represents an extreme scenario where the signal recovery would not be reliable and classification in the compressed domain becomes crucial, even at the receiver’s side where the computational cost is not of greatest concern. For performance evaluation, we are interested in two aspects: ($a$) the prediction accuracy over the test dataset, ($b$) the recovery accuracy of the classifier (with respect to the ground-truth classifier) —whose importance has been discussed in [@r21].
We define the classification accuracy as the minimum (worst) of the True Positive Rate (sensitivity) and the True Negative Rate (specificity). Figure \[fig:A\] shows an instance of the distribution of the classification accuracy for a pair of classes over $1000$ random trials. As it can be seen, in the presence of measurement diversity, classification results are more consistent (reflected in the low variance of accuracy). Due to the limited space, we only report the worst-case OAO accuracies (i.e. the minimum pair-wise accuracies among $1000$ trials) for the Pavia scene. The results for the case of one-measurement-per-pixel ($d'=1$) are shown in Tables \[tab:pavia\_fca\] and \[tab:pavia\_dmd\]. Similarly, the results for the case of $d'=3$ (which is equivalent to the sampling rate of a typical RGB color camera) are shown in Tables \[tab:pavia\_fca\_3\] and \[tab:pavia\_dmd\_3\]. Note that the employed SVM classifier is linear and would not result in perfect accuracy (i.e. accuracy of one) when the classes are not linearly separable. To see this, we have reported ground-truth accuracies in Table \[tab:pavia\_gt\].
![Distributions of the classification accuracy (Asphalt vs. Meadows) for the Pavia University dataset ($d'=1$).[]{data-label="fig:A"}](accu2.eps){width="\linewidth"}
FCA measurement
![Distributions of the classification accuracy (Asphalt vs. Meadows) for the Pavia University dataset ($d'=1$).[]{data-label="fig:A"}](accu1.eps){width="\linewidth"}
DMD measurement
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------ --------
Asphalt 0.45 0.38 0.42 0.36 0.44
Meadow 0.48 0.48 0.41 0.47
Gravel 0.44 0.44 0.44
Trees 0.42 0.53
Soil 0.44
: One FCA measurement per pixel: worst-case classification accuracies (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_fca"}
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------ --------
Asphalt 0.71 0.64 0.79 0.60 0.71
Meadow 0.72 0.61 0.46 0.73
Gravel 0.79 0.60 0.44
Trees 0.69 0.79
Soil 0.60
: One DMD measurement per pixel: worst-case classification accuracies (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_dmd"}
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------ --------
Asphalt 0.61 0.80 0.94 0.63 0.86
Meadow 0.67 0.82 0.50 0.62
Gravel 0.94 0.62 0.54
Trees 0.89 0.93
Soil 0.66
: Three FCA measurements per pixel: worst-case classification accuracies (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_fca_3"}
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------ --------
Asphalt 0.91 0.76 0.96 0.87 0.84
Meadow 0.90 0.82 0.57 0.91
Gravel 0.95 0.82 0.49
Trees 0.93 0.96
Soil 0.80
: Three DMD measurements per pixel: worst-case classification accuracies (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_dmd_3"}
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------ --------
Asphalt 1.00 0.97 0.97 1.00 0.94
Meadow 0.99 0.96 0.89 0.99
Gravel 1.00 1.00 0.86
Trees 0.98 1.00
Soil 0.99
: Ground-truth accuracies for the Pavia scene.[]{data-label="tab:pavia_gt"}
To measure the classifier recovery accuracy, we compute the cosine similarity, or equivalently the correlation, between $\hat{w}^*$ and $w^*$: $$C(\hat{w}^*,w^*)=\frac{\left<\hat{w}^*,w^*\right>}{\|\hat{w}^*\|_2 \|w^*\|_2}$$ In Tables \[tab:pavia\_err\_fca\_3\] and \[tab:pavia\_err\_dmd\_3\], we have reported the average recovery accuracy for the case of three-measurements-per-pixel (i.e. $d'=3$).
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------- --------
Asphalt 0.051 0.055 0.113 0.056 0.048
Meadow 0.100 0.033 0.019 0.077
Gravel 0.122 0.064 0.050
Trees 0.017 0.123
Soil 0.031
: Three FCA measurements per pixel: average recovery accuracy (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_err_fca_3"}
Classes Meadow Gravel Trees Soil Bricks
--------- -------- -------- ------- ------- --------
Asphalt 0.164 0.189 0.483 0.129 0.132
Meadow 0.468 0.147 0.140 0.380
Gravel 0.617 0.272 0.197
Trees 0.102 0.582
Soil 0.128
: Three DMD measurements per pixel: average recovery accuracy (1000 trials) for the Pavia scene.[]{data-label="tab:pavia_err_dmd_3"}
Conclusion {#sec:conclusion}
==========
In the field of ensemble learning, it has been discovered that the diversity among the base learners enhances the overall learning performance [@r20]. Meanwhile, our aim has been to exploit the diversity that can be efficiently built into the sensing system. Both measurement schemes of pixel-invariant (measurement without diversity) and pixel-varying (measurement with diversity) have been suggested as practical designs for compressive hyperspectral cameras [@fow14]. The presented analysis indicates that employing a DMD would result in more accurate recovery of the classifier and a more stable classification performance compared to the case when an FCA is used. Meanwhile, for tasks that only concern class prediction (and not the recovery of the classifier), FCA is (on average) a suitable low-cost alternative to the DMD architecture.
[^1]: M. Aghagolzadeh and H. Radha are with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48823 USA e-mail: {aghagol1, radha}@msu.edu.
[^2]: This work was supported by NSF Awards CCF-1117709 and CCF-1331852
[^3]: <http://aviris.jpl.nasa.gov>
[^4]: For more details regarding the physical implementation of compressive whisk-broom sensors, we refer the reader to [@fow14] which illustrates conceptual schematics of whisk-broom and push-broom cameras.
[^5]: *Discussion*: Similar results can be obtained using the dual form. Recent works have shown that advantages of the dual form can be obtained in the primal as well [@r12]. As noted in [@r12], the primal form convergences faster to the optimal parameters $(w^*,b^*)$ than the dual form. For the purposes of this work, it is more convenient to work with the primal form of SVM although the analysis can be properly extended to the dual form.
[^6]: <http://www.ehu.eus/ccwintco/>
[^7]: The Indian Pines dataset was not included due to the small size of the image which is not sufficient for a large-scale cross-validation study.
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---
abstract: 'We compute the two-loop QCD helicity amplitudes for the process $e^+e^- \to q\bar q g$. The amplitudes are extracted in a scheme-independent manner from the coefficients appearing in the general tensorial structure for this process. The tensor coefficients are derived from the Feynman graph amplitudes by means of projectors, within the conventional dimensional regularization scheme. The actual calculation of the loop integrals is then performed by reducing all of them to a small set of known master integrals. The infrared pole structure of the renormalized helicity amplitudes agrees with the prediction made by Catani using an infrared factorization formula. We use this formula to structure our results for the finite part into terms arising from the expansion of the pole coefficients and a genuine finite remainder, which is independent of the scheme used to define the helicity amplitudes. The analytic result for the finite parts of the amplitudes is expressed in terms of one- and two-dimensional harmonic polylogarithms.'
---
IPPP/02/28\
DCTP/02/56\
CERN-TH/2002-122\
hep-ph/0206067\
June 2002
2.5cm
1.cm [L.W. Garland]{}$^a$, [T. Gehrmann]{}$^b$, [E.W.N. Glover]{}$^a$, [A. Koukoutsakis]{}$^a$ and [E. Remiddi]{}$^c$ .7cm [*$^a$ Department of Physics, University of Durham, Durham DH1 3LE, England*]{} .4cm [*$^b$ Theory Division, CERN, CH-1211 Geneva 23, Switzerland*]{} .4cm [*$^c$ Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, I-40126 Bologna, Italy*]{}
2.0cm
[*Email-addresses:* ]{}[email protected], [[email protected]]{},\
, [[email protected]]{},\
Introduction
============
The three-jet production rate in electron–positron annihilation [@petra; @ellis] and related event shape observables were measured to a very high precision at LEP, where they were used in particular for the determination of the strong coupling constant $\alpha_s$. At present, the error on the extraction of $\alpha_s$ from these data is dominated by the uncertainty inherent in the theoretical next-to-leading order (NLO) calculation [@ert1; @ert2; @kn; @gg; @cs] of the jet observables (see [@bethke] for a review). The planned TESLA [@tesla] linear $e^+e^-$ collider will allow precision QCD studies at energies even higher than at LEP. Given the projected luminosity of TESLA, one again expects the experimental errors to be well below the uncertainty of the NLO calculation.
The calculation of next-to-next-to-leading order (NNLO), i.e. ${\cal
O}(\alpha_s^3)$, corrections to the three-jet rate in $e^+e^-$ annihilation has been considered as a highly important project for a long time [@kunszt]. In terms of matrix elements, it requires the computation of three contributions: the tree level $\gamma^* \to 5$ partons amplitudes [@tree5p1; @tree5p2; @tree5p3], the one-loop corrections to the $\gamma^* \to
4$ partons amplitudes [@onel4p1; @onel4p2; @onel4p3; @onel4p4], and the two-loop (as well as the one-loop times one-loop) corrections to the $\gamma^* \to
3$ partons matrix elements. In a previous publication [@3jme], we have derived both the interference of the tree and two-loop matrix elements and the self-interference of the one-loop amplitudes averaged over all external helicities. In the present work, we extend this calculation to compute the two-loop helicity amplitudes for the process $e^+e^- \to q\bar q g$.
The most precisely measured observables related to $e^+e^- \to 3$ jets are the jet production rate itself and a number of event-shape variables. The calculation of these phenomenologically most relevant applications, which also dominate the extraction of $\alpha_s$, at NNLO accuracy requires only the helicity averaged squared matrix element at the two-loop level derived in [@3jme]. Nevertheless, the helicity amplitudes presented here are interesting for a number of reasons:
- Oriented event-shape observables, which measure the spatial orientation of the final-state jets relative to the direction of the incoming beams require, even for unpolarized beams [@osland], the calculation of the polarization tensor of the virtual photon mediating the interaction. This polarization tensor can be recovered from the helicity amplitudes.
- Likewise, to determine the direction of the decay leptons in the crossed process, $V+1$ jet production at unpolarized hadron colliders, it is necessary to compute the polarization tensor of the vector boson.
- Polarization of the beams is an important option for the future linear $e^+e^-$ collider TESLA [@tesla], thus providing a direct measurement of event-shape observables in polarized $e^+e^-$ annihilation.
- NNLO predictions for $(V+1)$-jet production at the RHIC polarized proton–proton collider and for $(2+1)$ jet production at a currently discussed polarized upgrade of the HERA collider do require the calculation of the two-loop helicity amplitudes. These observables would then form part of a full NNLO determination of the polarized parton distribution functions in the proton.
- The study of formal aspects of two-loop matrix elements, such as their collinear limits or their high energy behaviour can be carried out more elegantly on the basis of the underlying helicity amplitudes.
Two-loop helicity amplitudes have up to now only been derived for $2\to 2$ bosonic scattering processes with all external legs on-shell: for $gg \to \gamma \gamma$ [@m5], $\gamma \gamma \to
\gamma \gamma$ [@m6; @m10] and $gg \to gg$ [@m0; @m9]. The latter calculation also confirmed earlier results for the squared two-loop $gg \to gg$ matrix element [@m4]. In the above calculations, which were all carried out within dimensional regularization [@dreg1; @dreg2; @hv] with $\dd=4-2\e$ space-time dimensions, two different methods were used to access the helicity structure of the matrix element: explicit contraction with the external polarization vectors [@m5; @m6; @m0; @m9] or projection onto the individual components of the Lorentz-invariant decomposition of the amplitude [@m10]. Once these are applied to expose the helicity structure, one is left with the task of computing a large number of two-loop integrals. Using integration-by-parts [@hv; @chet1; @chet2] and Lorentz-invariance [@gr] identities, these can be reduced [@laporta] to a small number of so-called master integrals, which were derived for massless on-shell two-loop four-point functions in [@onshell1; @onshell2; @onshell3; @onshell4; @onshell5; @onshell6]. If an explicit contraction with the external polarization vectors is performed, one also has to compute two-loop integrals over the $(\dd-4)$ dimensional subspace of loop momenta, which reduce however to simple vacuum diagrams [@m9]. For $2\to 2$ scattering processes with external fermions and all external legs on-shell ($e^+e^- \to e^+e^-$, $q\bar q \to q'\bar q'$, $q\bar q
\to q\bar q$, $q \bar q \to gg$, $q\bar q \to g \gamma$ and $q\bar q \to \gamma
\gamma$), only the squared, helicity-averaged two-loop matrix elements were computed so far [@m1; @m2; @m3; @m8].
The method employed here to extract the two-loop helicity amplitudes for $e^+e^- \to q\bar q g$ is similar to the approach of [@m10] by applying projections on all components of the Lorentz-invariant decomposition of the amplitude. Using this approach, the corresponding one-loop helicity amplitudes were derived in [@gg]. The master integrals relevant in the present context are massless four-point functions with three legs on-shell and one leg off-shell. The complete set of these two-loop integrals was computed in [@mi], while earlier partial results had been presented in [@num; @smirnov][^1]. The master integrals in [@mi] are expressed in terms of two-dimensional harmonic polylogarithms (2dHPLs). The 2dHPLs are an extension of the harmonic polylogarithms (HPLs) of [@hpl]. All HPLs and 2dHPLs that appear in the divergent parts of the planar master integrals have weight $\leq 3$ and can be related to the more commonly known Nielsen generalized polylogarithms [@nielsen; @bit] of suitable arguments. The functions of weight 4 appearing in the finite parts of the master integrals can all be represented, by their very definition, as one-dimensional integrals over 2dHPLs of weight 3, hence of Nielsen’s generalized polylogarithms of suitable arguments according to the above remark. A table with all relations is included in the appendix of [@mi]. Numerical routines providing an evaluation of the HPLs [@grnum1] and 2dHPLs [@grnum2] are available.
After carrying out ultraviolet renormalization of the amplitudes in the $\overline{{\rm MS}}$ scheme, one is left with poles which are purely of infrared origin. The infrared pole structure of the amplitudes can be predicted using Catani’s infrared factorization formula [@catani]. We use this formalism to present the infrared poles and the finite parts of the helicity amplitudes in a compact form.
This paper is structured as follows: in Section \[sec:method\], we outline the calculational method used to derive the helicity amplitudes and discuss the techniques used to extract the ultraviolet and infrared pole structure. We also elaborate on the relation to previous work. The two-loop helicity amplitudes are computed (in the Weyl–van der Waerden formalism, which is briefly described in the Appendix) in Section \[sec:helicity\]. Finally, Section \[sec:conc\] contains a discussion of the results and conclusions.
Method
======
\[sec:method\]
Notation {#subsec:defs}
--------
We consider the production of a quark–antiquark–gluon system in electron–positron annihilation, $$e^+(p_5) + e^-(p_6) \to \gamma^* (p_4) \longrightarrow q(p_1) + \bar q (p_2) + g(p_3)\; .$$ It is convenient to define the invariants $$\sab = (p_1+p_2)^2\;, \qquad \sac = (p_1+p_3)^2\;, \qquad
\sbc = (p_2+p_3)^2\;,$$ which fulfil $$p_4^2 =(p_1+p_2+p_3)^2 = \sab + \sac + \sbc \equiv \sabc \; ,$$ as well as the dimensionless invariants $$x = \sab/\sabc\;, \qquad y = \sac/\sabc\;, \qquad z = \sbc/\sabc\;,$$ which satisfy $x+y+z=1$.
The renormalized amplitude $|{\cal M}\rangle$ can be written as $$\label{eq:Mdef}
|{\cal M}\rangle = V^\mu \S_\mu(q;g;\bar q)\; ,$$ where $V^\mu$ represents the lepton current and $\S_\mu$ denotes the hadron current. In a previous paper [@3jme], we have considered the unpolarized decay process $$\gamma^* (p_4) \longrightarrow q(p_1) + \bar q (p_2) + g(p_3)\; .$$ for which the amplitude is obtained from Eq. (\[eq:Mdef\]) by replacing the lepton current by the polarization vector of the virtual photon $\epsilon_4^\mu$. The hadron current may be perturbatively decomposed as $$\S_\mu(q;g;\bar q) =
\sqrt{4\pi\alpha}e_q \sqrt{4\pi\alpha_s} \; {\bom
T}^{a}_{ij}\,\left(\S^{(0)}_\mu(q;g;\bar q)
+ \left(\frac{\alpha_s}{2\pi}\right) \S^{(1)}_\mu(q;g;\bar q)
+ \left(\frac{\alpha_s}{2\pi}\right)^2 \S^{(2)}_\mu(q;g;\bar q)
+ {\cal O}(\alpha_s^3)\right),
\label{eq:renorme}$$ where $e_q$ denotes the quark charge, $a$ is the adjoint colour index for the gluon and $i$ and $j$ are the colour indices for quark and antiquark. $\alpha_s$ is the QCD coupling constant at the renormalization scale $\mu$, and the $\S^{(i)}_\mu$ are the $i$-loop contributions to the renormalized amplitude. Renormalization of ultraviolet divergences is performed in the $\overline{{\rm MS}}$ scheme.
The general tensor {#subsec:genten}
------------------
The most general tensor structure for the hadron current $\S_\mu(q;g;\bar q)$ is $$\begin{aligned}
\S_\mu(q;g;\bar q) &=&
\bar u(p_1)\slash \!\!\! p_3 u(p_2) \left(
A_{11} \epsilon_3.p_1~p_{1\mu}
+ A_{12} \epsilon_3.p_1~p_{2\mu}
+ A_{13} \epsilon_3.p_1~p_{3\mu}
\right)
\nonumber \\
&+& \bar u(p_1)\slash \!\!\!p_3 u(p_2) \left(
A_{21} \epsilon_3.p_2~p_{1\mu}
+ A_{22} \epsilon_3.p_2 ~p_{2\mu}
+ A_{23} \epsilon_3.p_2 ~p_{3\mu}
\right)
\nonumber \\
&+& \bar u(p_1)\gamma_\mu u(p_2) \left(
B_{1} \epsilon_3.p_1
+ B_{2} \epsilon_3.p_2
\right)
\nonumber \\
&+&\bar u(p_1)\slash \!\!\!\epsilon_3 u(p_2) \left(
C_{1} p_{1\mu}
+C_{2} p_{2\mu}
+C_{3} p_{3\mu}
\right)
\nonumber \\
&+& D_{1} \bar u(p_1)\slash \!\!\!\epsilon_3 \slash \!\!\!p_3 \gamma_\mu u(p_2)
\nonumber \\
&+& D_{2} \bar u(p_1)\gamma_\mu \slash \!\!\!p_3 \slash \!\!\!\epsilon_3 u(p_2) \; ,\end{aligned}$$ where the constraint $\epsilon_3 \cdot p_3 = 0$ has been applied. All coefficients are functions of $\sac$, $\sbc$ and $\sabc$. The above tensor structure is a priori $\dd$-dimensional, since the dimensionality of the external states has not yet been specified. The hadron current is conserved and satisfies $$\S_\mu(q;g;\bar q)~p_{4}^{\mu} = 0\; ;$$ it must also obey the QCD Ward identity when the gluon polarization vector $\epsilon_3$ is replaced with the gluon momentum, $$\S_\mu(q;g;\bar q) (\epsilon_3 \to p_3) = 0.$$ These constraints yield relations amongst the 13 distinct tensor structures and applying these identities gives the gauge-invariant form of the tensor, $$\begin{aligned}
\S_\mu(q;g;\bar q)~~=&& A_{11}(\sac,\sbc,\sabc) T_{11\mu}
+ A_{12}(\sac,\sbc,\sabc) T_{12\mu}
+ A_{13}(\sac,\sbc,\sabc) T_{13\mu} \nonumber \\
&+&
A_{21}(\sac,\sbc,\sabc) T_{21\mu}
+ A_{22}(\sac,\sbc,\sabc) T_{22\mu}
+A_{23}(\sac,\sbc,\sabc) T_{23\mu}\nonumber \\
&+&B(\sac,\sbc,\sabc) T_{\mu},\end{aligned}$$ where $A_{ij}$ and $B$ are gauge-independent functions and the tensor structures $T_{IJ\mu}$ and $T_{\mu}$ are given by $$\begin{aligned}
T_{1J\mu} &=&\bar u(p_1)\slash \!\!\! p_3 u(p_2)
\epsilon_3.p_1p_{J\mu}
-\frac{s_{13}}{2}
\bar u(p_1)\slash \!\!\! \epsilon_3 u(p_2)p_{J\mu}
+\frac{s_{J4}}{4}
\bar u(p_1)\slash \!\!\!\epsilon_3 \slash \!\!\!p_3 \gamma_\mu
u(p_2),\\
T_{2J\mu} &=&\bar u(p_1)\slash \!\!\! p_3 u(p_2)
\epsilon_3.p_2p_{J\mu}
-\frac{s_{23}}{2}
\bar u(p_1)\slash \!\!\! \epsilon_3 u(p_2)p_{J\mu}
+\frac{s_{J4}}{4}
\bar u(p_1)\gamma_\mu \slash \!\!\!p_3 \slash \!\!\!\epsilon_3
u(p_2),\\
T_{\mu}&=& s_{23}\left(
\bar u(p_1)\gamma_\mu u(p_2)
\epsilon_3.p_1
+\frac{1}{2}
\bar u(p_1)\slash \!\!\!\epsilon_3 \slash \!\!\!p_3 \gamma_\mu
u(p_2) \right)
\nonumber \\
&-&s_{13}\left(
\bar u(p_1)\gamma_\mu u(p_2)
\epsilon_3.p_2
+\frac{1}{2}
\bar u(p_1)\gamma_\mu \slash \!\!\!p_3 \slash \!\!\!\epsilon_3
u(p_2) \right).\end{aligned}$$ Each of the tensor structures satisfies both current conservation and the QCD Ward identity. The coefficients are further related by symmetry under the interchange of the quark and antiquark, $$\begin{aligned}
A_{21}(\sac,\sbc,\sabc) &=& - A_{12}(\sbc,\sac,\sabc),\nonumber \\
A_{22}(\sac,\sbc,\sabc) &=& - A_{11}(\sbc,\sac,\sabc),\nonumber \\
A_{23}(\sac,\sbc,\sabc) &=& - A_{13}(\sbc,\sac,\sabc),\nonumber \\
B(\sac,\sbc,\sabc) &=& B(\sbc,\sac,\sabc).\end{aligned}$$
Projectors for the tensor coefficients {#subsec:projectors}
--------------------------------------
The coefficients $A_{IJ}$ and $B$ may be easily extracted from a Feynman diagram calculation, using projectors such that $$\sum_{\rm spins} {\cal P}(X) ~\epsilon_4^\mu \S_\mu(q;g;\bar q) = X(\sac,\sbc,\sabc).$$ The explicit forms for the seven projectors in $d$ space-time dimensions are, $$\begin{aligned}
{\cal P}(A_{11}) &=&
\frac{(\sbc\sabc \dd+\sac\sab \dmb)}{2\sac^3\sab^2\dmc\sabc}\Taa
-\frac{(\sac+\sbc) \dmb}{2\sac^2\sab^2\dmc\sabc}\Tab\nonumber \\
&&
-\frac{((\sbc +\sab)\dd+2\sac)}{2\sab\sac^3\dmc\sabc}\Tac
-\frac{(\sbc\sabc\dmb+\sac\sab \dmd)}{2\sbc\sab^2\sac^2\sabc\dmc}\Tba\nonumber \\
&&
+\frac{(\sac+\sbc)\dmd}{2\dmc\sab^2\sabc\sac\sbc}\Tbb
+\frac{(\sbc+\sab)\dmd}{2\sbc\sab\sac^2\sabc\dmc}\Tbc\nonumber \\
&&
-\frac{1}{2\sac^2\sab^2\dmc}\T,\nonumber \\
{\cal P}(A_{12}) &=&
-\frac{(\sac+\sbc) \dmb}{2\sac^2\sab^2\dmc\sabc}\Taa
+\frac{\dmb(\sbc\sab \dmd+\sac\sabc \dmb)}{2\sac^2\sab^2\sbc\dmc\sabc\dmd}\Tab\nonumber \\
&&
-\frac{\dmb(\sac+\sab)}{2\sac^2\sab\sbc\dmc\sabc}\Tac
+\frac{(\dmf \dmb(\sac+\sbc)-4\sab)}{2\dmd\sab^2\sac\sbc\sabc\dmc}\Tba \nonumber \\
&&
-\frac{(\sbc\sab \dmd+\sac\sabc \dmb)}{2\sab^2\sac\sbc^2\dmc\sabc}\Tbb
+\frac{(2\sbc+(\sac +\sab) \dmb)}{2\sab\sac\sbc^2\sabc\dmc}\Tbc\nonumber \\
&&
-\frac{\dmb}{2\dmd\sab^2\sac\sbc\dmc}\T,\nonumber \\
{\cal P}(A_{13}) &=&
-\frac{((\sbc +\sab)\dd+2\sac)}{2\sab\sac^3\dmc\sabc}\Taa
-\frac{ \dmb(\sac+\sab)}{2\sac^2\sab\sbc\dmc\sabc}\Tab\nonumber \\
&&
+\frac{(\sac\sbc \dmb+\sab\sabc\dd)}{2\sac^3\sab\sbc\sabc\dmc}\Tac
+\frac{((\sab+\sbc) \dmb+2\sac)}{2\sac^2\sab\sbc\dmc\sabc}\Tba\nonumber \\
&&
+\frac{(\sac+\sab)\dmd}{2\sab\sac\sbc^2\sabc\dmc}\Tbb
-\frac{(\sac+\sab)(\sbc+\sab)\dmd}{2\sac^2\sab\sbc^2\sabc\dmc}\Tbc\nonumber \\
&&
+\frac{1}{2\sbc\sac^2\sab\dmc}\T,\nonumber \\
{\cal P}(A_{21}) &=&
-\frac{(\sbc\sabc \dmb+\sac\sab \dmd)}{2\sac^2\sab^2\sbc\dmc\sabc}\Taa
+\frac{(-4\sab+(\sac+\sbc)\dmf \dmb)}{2\dmd\sab^2\sac\sbc\sabc\dmc}\Tab\nonumber \\
&&
+\frac{(\sbc+\sab) \dmb+2\sac)}{2\sac^2\sab\sbc\dmc\sabc}\Tac
+\frac{\dmb(\sbc\sabc\dmb+\sac\sab \dmd)}{2\sab^2\sac\sbc^2\sabc\dmc\dmd}\Tba\nonumber \\
&&
-\frac{(\sac+\sbc) \dmb}{2\sab^2\sbc^2\sabc\dmc}\Tbb
-\frac{(\sbc+\sab) \dmb}{2\sac\sbc^2\sab\dmc\sabc}\Tbc\nonumber \\
&&
+\frac{\dmb}{2\dmd\sab^2\sac\sbc\dmc}\T,\nonumber \\
{\cal P}(A_{22}) &=& \frac{(\sac+\sbc)\dmd}{2\sac\sab^2\sbc\dmc\sabc}\Taa
-\frac{(\sbc\sab \dmd+\sac\sabc \dmb)}{2\sab^2\sac\sbc^2\dmc\sabc}\Tab\nonumber \\
&&
+\frac{(\sac+\sab)\dmd}{2\sac\sbc^2\sab\dmc\sabc}\Tac
-\frac{(\sac+\sbc) \dmb}{2\sab^2\sbc^2\sabc\dmc}\Tba\nonumber \\
&&
+\frac{(\sbc\sab \dmb+\sac\sabc\dd)}{2\sbc^3\sab^2\dmc\sabc}\Tbb
-\frac{(\sac\dd+\sab\dd+2\sbc)}{2\sab\sbc^3\sabc\dmc}\Tbc\nonumber \\
&&
+\frac{1}{2\sbc^2\sab^2\dmc}\T,\nonumber \\
{\cal P}(A_{23}) &=& \frac{(\sbc+\sab)\dmd}{2\sbc\sab\sac^2\sabc\dmc}\Taa
+\frac{(2\sbc+(\sac+\sab) \dmb)}{2\sac\sbc^2\sab\dmc\sabc}\Tab\nonumber \\
&&
-\frac{(\sac+\sab)(\sbc+\sab)\dmd}{2\sac^2\sab\sbc^2\sabc\dmc}\Tac
-\frac{(\sbc+\sab) \dmb}{2\sac\sbc^2\sab\dmc\sabc}\Tba\nonumber \\
&&
-\frac{((\sac +\sab)\dd+2\sbc)}{2\sab\sbc^3\sabc\dmc}\Tbb
+\frac{(\sac\sbc \dmb+\sab\sabc\dd)}{2\sac\sab\sbc^3\sabc\dmc}\Tbc\nonumber \\
&&
-\frac{1}{2\sbc^2\sac\sab\dmc}\T,\nonumber \\
{\cal P}(B) &=& -\frac{1}{2\dmc\sab^2\sac^2}\Taa
-\frac{\dmb}{2\dmd\sab^2\sac\sbc\dmc}\Tab
+\frac{1}{2\sbc\sab\dmc\sac^2}\Tac\nonumber \\
&&
+\frac{ \dmb}{2\dmd\sab^2\sac\sbc\dmc}\Tba
+\frac{1}{2\sbc^2\sab^2\dmc}\Tbb
-\frac{1}{2\sac\sab\dmc\sbc^2}\Tbc\nonumber \\
&&
+\frac{1}{2\dmd\sab^2\sac\sbc}\T.\end{aligned}$$
The perturbative expansion of the tensor coefficients {#subsec:perturbative}
-----------------------------------------------------
Each of the unrenormalized coefficients $A_{IJ}$ and $B$ has a perturbative expansion of the form $$\begin{aligned}
A_{IJ}^{{\rm un}} &=& \sqrt{4\pi\alpha}e_q \sqrt{4\pi\alpha_s} \; {\bom
T}^{a}_{ij}\,\left[
A_{IJ}^{(0),{\rm un}}
+ \left(\frac{\alpha_s}{2\pi}\right) A_{IJ}^{(1),{\rm un}}
+ \left(\frac{\alpha_s}{2\pi}\right)^2 A_{IJ}^{(2),{\rm un}}
+ {\cal O}(\alpha_s^3) \right] \;,\nonumber \\
B^{{\rm un}} &=& \sqrt{4\pi\alpha}e_q \sqrt{4\pi\alpha_s} \; {\bom T}^{a}_{ij}\,\left[
B^{(0),{\rm un}}
+ \left(\frac{\alpha_s}{2\pi}\right) B^{(1),{\rm un}}
+ \left(\frac{\alpha_s}{2\pi}\right)^2 B^{(2),{\rm un}}
+ {\cal O}(\alpha_s^3) \right] \;,\nonumber \\\end{aligned}$$ where the dependence on $(\sac,\sbc,\sabc)$ is implicit. At tree level, $$\begin{aligned}
A_{IJ}^{(0),{\rm un}}(\sac,\sbc,\sabc) &=& 0,\\
B^{(0),{\rm un}}(\sac,\sbc,\sabc)
&=& \frac{2}{s_{13}s_{23}} \;.\end{aligned}$$ The one-loop contributions can be written in terms of the one-loop box integral in $\dd=6-2\epsilon$ dimensions, Box$^6(s_{ij},s_{ik},s_{ijk})$, and the one-loop bubble, Bub$(s_{ij})$, as follows: $$\begin{aligned}
\lefteqn{A_{11}^{(1),{\rm un}}(\sac,\sbc,\sabc) =}\nonumber \\
&& N \Biggl (
-\frac{\dmd}{2(\sac+\sab)\sac} ~\Babc
-\frac{\dmd}{2\sab\sac} ~\bac
\nonumber \\ &&
\hspace{1cm}-\frac{(\dmb\sbc\sab+\dmd\sbc\sac+4\sab(\sab+\sac))}{2\sab\sac(\sac+\sab)^2} ~\bbc
\nonumber \\ &&
\hspace{1cm}-\frac{\dmd(4\sab+\dmb\sbc)}{4\sab\sac} ~\Boxyz
\Biggr)\nonumber \\
&&
+\frac{1}{N}\Biggl (
\frac{\dmd}{2(\sac+\sab)\sac} ~\Babc
+\frac{\dd}{2\sac^2} ~\bab
\nonumber \\ &&
\hspace{1cm}+\frac{(\sab+\sac)(\dd\sbc+4\sac)+2\sbc\sac}{2(\sac+\sab)^2\sac^2} ~\bbc
\nonumber \\ &&
\hspace{1cm}+\frac{\dmd\dmf}{4\sac} ~\Boxxy
+\frac{\dmb(\dd\sbc+4\sac)}{4\sac^2} ~\Boxxz
\Biggr),
\label{eq:aNLO}\\
\lefteqn{A_{12}^{(1),{\rm un}}(\sac,\sbc,\sabc) =}\nonumber \\
&&
N \Biggl (
-\frac{\dmj}{2\sab(\sbc+\sab)} ~\bac
-\frac{(\dmj\sac-4\sab)}{2\sab\sac(\sac+\sab)} ~\bbc
\nonumber \\ &&
\hspace{1cm}+\frac{(4\dmd\sab-\dmb\dmj\sac)}{4\sab\sac} ~\Boxyz
\Biggr)\nonumber \\
&&
+\frac{1}{N}\Biggl (
\frac{\dmb}{2\sbc\sac} ~\bab
+\frac{(\dmb\sab+2\dmf\sbc)}{2\sbc\sab(\sbc+\sab)} ~\bac
\nonumber \\ &&
\hspace{1cm}+\frac{\dmf(\sab+2\sac)}{2\sab\sac(\sac+\sab)} ~\bbc
\nonumber \\ &&
\hspace{1cm}+\frac{(\dmb^2\sab\sac+4\dmd\sab\sbc+2\dmd\dmf\sac\sbc)}{4\sab\sac\sbc} ~\Boxxy
\nonumber \\ &&
\hspace{1cm}+\frac{\dmf(\dmb\sab+2\dmd\sac)}{4\sab\sac} ~\Boxxz
\Biggr),\\
\lefteqn{A_{13}^{(1),{\rm un}}(\sac,\sbc,\sabc) =}\nonumber \\
&&
N \Biggl (
\frac{\dmf}{2(\sac+\sab)\sac} ~\bbc
+\frac{\dmd\dmf}{4\sac} ~\Boxyz
\Biggr)\nonumber \\
&&
+\frac{1}{N}\Biggl (
-\frac{\dmd}{\sac(\sbc+\sac)} ~\Babc
-\frac{\dmd}{2\sbc\sac} ~\bac
\nonumber \\ &&
\hspace{1cm}+\frac{(4\sab\sac^2-\dd\sab(\sac+\sbc)^2-2\dmb\sac\sbc(\sac+\sbc))}{2(\sbc+\sac)^2\sac^2\sbc} ~\bab
\nonumber \\ &&
\hspace{1cm}-\frac{(2\dmc\sac+\dd\sab)}{2\sac^2(\sac+\sab)} ~\bbc
-\frac{\dmd(\dmb\sab+4\sbc)}{4\sbc\sac} ~\Boxxy
\nonumber \\ &&
\hspace{1cm}-\frac{\dmb(\dd\sab+2\dmd\sac)}{4\sac^2} ~\Boxxz
\Biggr), \\
\lefteqn{B^{(1),{\rm un}}(\sac,\sbc,\sabc) =}\nonumber \\
&&
N \Biggl (
\frac{\dd^2-3\dd+4}{4\dmd\sac\sbc} ~\Babc
\nonumber \\ &&
\hspace{1cm}+\frac{(4\dmc\sab(\sab+\sbc)+\dmd(\dd-7)\sbc\sac)}{2\sab\sbc(\sbc+\sab)\sac\dmd}
~\bac
\nonumber \\ &&
\hspace{1cm}+\frac{(4\dmc\sab^2+\dmb(\dd-7)\sac\sbc)}{8\sab\sac\sbc} ~\Boxyz
\Biggr)\nonumber \\
&&
+\frac{1}{N}\Biggl (
\frac{(7\dd-16-\dd^2)}{4\dmd\sac\sbc} ~\Babc
+\frac{(16-5\dd)}{4\dmd\sac\sbc} ~\bab \nonumber \\ &&
\hspace{1cm}-\frac{(\sab+\dmf\sbc)}{2(\sbc+\sab)\sbc\sab} ~\bac \nonumber \\ &&
\hspace{1cm}-\frac{(4\dmc\sbc\sab+\dmd\dmf\sac\sbc+\dmb\sab\sac)}{4\sab\sac\sbc} ~\Boxxy
\Biggr)
\nonumber \\ &&
+ \Biggl \{ \sac \leftrightarrow \sbc \Biggr \}.
\label{eq:bNLO}\end{aligned}$$ Explicit expansions of the one-loop integrals around $\epsilon \sim 0$ in terms of HPLs and 2dHPLs are listed in Appendix A of [@3jme].
Similarly, the unrenormalized two-loop $A_{IJ}^{(2),{\rm un}}$ and $B^{(2),{\rm un}}$ coefficients were obtained analytically (making extensive use of the computer algebra programs MAPLE [@maple], FORM2 [@form2] and FORM3 [@form3], where the latter two are particularly well suited for handling the large-size expressions arising at intermediate stages of the calculation) in terms of a basis set of two-loop master integrals. This basis set comprises 14 planar topologies and 5 non-planar topologies. Five of the topologies require more than one master integral, so that in total 24 master integrals are needed. A more detailed discussion can be found in Ref. [@3jme]. However, we note that Laurent expansions for each of these master integrals have been derived in [@mi] by solving differential equations for the master integrals (equations that are differential with respect to the momentum scales involved in the diagram). The $\e$-expansions of $A_{IJ}^{(2),{\rm un}}$ and $B^{(2),{\rm un}}$ can therefore be obtained by directly substituting the $\e$-expansions of the individual master integrals.
Relation to previous work {#subsec:earlier}
-------------------------
We have considered the case where the correlations with the lepton current are ignored in a previous paper [@3jme]. In this instance, the squared amplitude for the process $\gamma^* \to q\bar q
g$, summed over spins, colours and quark flavours, was denoted by $$\langle{\cal M}|{\cal M}\rangle = \sum | \epsilon_4 \cdot \S(q;g;\bar q) | ^2
= {\cal T} (x,y,z)\; .$$ The perturbative expansion of ${\cal T} (x,y,z)$ at renormalization scale $\mu^2 = q^2 = s_{123}$ reads: $$\begin{aligned}
{\cal T} (x,y,z) &=& 16\pi^2\alpha\sum_q e_q^2 \alpha_s(q^2)\Bigg[
{\cal T}^{(2)} (x,y,z) +
\left(\frac{\alpha_s(q^2)}{2\pi}\right){\cal T}^{(4)} (x,y,z) \nonumber \\
&& \hspace{1.3cm}
+ \left(\frac{\alpha_s(q^2)}{2\pi}\right)^2{\cal T}^{(6)} (x,y,z)
+ {\cal O}(\alpha_s^3(q^2)) \Bigg] \;,\end{aligned}$$ where $$\begin{aligned}
\label{eq:T2}
{\cal T}^{(2)} (x,y,z) &=& \langle{\cal M}^{(0)}|{\cal M}^{(0)}\rangle
= 4 V (1-\e)\left[ (1-\e)\left(\frac{y}{z}+\frac{z}{y}\right)
+\frac{2(1-y-z)-2\e yz}{yz}\right]\;,\\
\label{eq:T4}
{\cal T}^{(4)} (x,y,z) &=&
\langle{\cal M}^{(0)}|{\cal M}^{(1)}\rangle +
\langle{\cal M}^{(1)}|{\cal M}^{(0)}\rangle \; ,\\
\label{eq:T6}
{\cal T}^{(6)} (x,y,z) &=&
\langle{\cal M}^{(1)}|{\cal M}^{(1)}\rangle +
\langle{\cal M}^{(0)}|{\cal M}^{(2)}\rangle +
\langle{\cal M}^{(2)}|{\cal M}^{(0)}\rangle \;,\end{aligned}$$ where $V=N^2-1$, with $N$ the number of colours. ${\cal T}^{(4)} (x,y,z)$ was first derived in [@ert1; @ert2] through to ${\cal
O}(\epsilon^0)$ while an explicit expression for it to all orders in $\epsilon$ was given in [@3jme]. The contribution to ${\cal T}^{(6)} (x,y,z)$ from the interference of two-loop and tree diagrams $${\cal T}^{(6,[2\times 0])} (x,y,z) =
\langle{\cal M}^{(0)}|{\cal M}^{(2)}\rangle +
\langle{\cal M}^{(2)}|{\cal M}^{(0)}\rangle \;,$$ as well as the one-loop self-interference $${\cal T}^{(6,[1\times 1])} (x,y,z) =
\langle{\cal M}^{(1)}|{\cal M}^{(1)}\rangle \;$$ were first derived in [@3jme].
It is straightforward to obtain the interference of the tree and $i$-loop amplitudes in terms of the tensor coefficients, $A_{IJ}$ and $B$. We find $$\begin{aligned}
\label{eq:square}
\lefteqn{
\langle{\cal M}^{(0)}|{\cal M}^{(i)}\rangle =}\nonumber \\ &&
\frac{V}{2}\, \Biggl\{
2 (1-\epsilon) \left((\sab \sabc+\sab \sac+\sac \sbc)-\epsilon(\sac+\sbc)(\sab+\sac)\right) A_{11}^{(i)} (\sac,\sbc,\sabc)
\nonumber \\
&&\phantom{\biggr\{}+\left(2(\sab+\sbc)^2-2\epsilon \left(\sabc \sbc +(\sab+\sbc)^2\right) + 2\epsilon^2
(\sac+\sbc)(\sab+\sbc) \right) A_{12}^{(i)}(\sac,\sbc,\sabc)\nonumber \\
&&\phantom{\biggr\{}
+2\left(\sbc-\epsilon(\sac+\sbc))(\sabc-\epsilon(\sac+\sbc)\right)A_{13}^{(i)}(\sac,\sbc,\sabc)\nonumber \\
&&\phantom{\biggr\{}+2\left(\sac^2+\sbc^2+2\sab\sabc-2\epsilon(\sabc^2-\sab\sac-\sab\sbc-\sac\sbc)+\epsilon^2(\sac+\sbc)^2\right) B^{(i)}(\sac,\sbc,\sabc)
\nonumber \\
&&\hspace{2cm}+ \{p_1 \leftrightarrow p_2 \}
\Biggr\}.\end{aligned}$$ The above relation holds for the unrenormalized as well as for the renormalized matrix element, involving the appropriate unrenormalized or renormalized tensor coefficients respectively. Similar, but more lengthy, expressions can easily be obtained for the interference of $i$- and $j$-loop amplitudes. We have checked that inserting the expressions for $A_{IJ}^{(i)}$ and $B^{(i)}$ into Eq. (\[eq:square\]) reproduces our earlier results [@3jme] at the one- and two-loop level both at the master integral level and after making an expansion in $\epsilon$.
Ultraviolet renormalization {#subsec:renorm}
---------------------------
The renormalization of the matrix element is carried out by replacing the bare coupling $\alpha_0$ with the renormalized coupling $\alpha_s\equiv \alpha_s(\mu^2)$, evaluated at the renormalization scale $\mu^2$ $$\alpha_0\mu_0^{2\e} S_\e = \alpha_s \mu^{2\e}\left[
1- \frac{\beta_0}{\e}\left(\frac{\alpha_s}{2\pi}\right)
+\left(\frac{\beta_0^2}{\e^2}-\frac{\beta_1}{2\e}\right)
\left(\frac{\alpha_s}{2\pi}\right)^2+{\cal O}(\alpha_s^3) \right]\; ,$$ where $$S_\e =(4\pi)^\e e^{-\e\gamma}\qquad \mbox{with Euler constant }
\gamma = 0.5772\ldots$$ and $\mu_0^2$ is the mass parameter introduced in dimensional regularization [@dreg1; @dreg2; @hv] to maintain a dimensionless coupling in the bare QCD Lagrangian density; $\beta_0$ and $\beta_1$ are the first two coefficients of the QCD $\beta$-function: $$\beta_0 = \frac{11 \CA - 4 T_R \NF}{6}, \qquad
\beta_1 = \frac{17 \CA^2 - 10 C_A T_R \NF- 6C_F T_R \NF}{6}\;,$$ with the QCD colour factors $$\CA = N,\qquad C_F = \frac{N^2-1}{2N},
\qquad T_R = \frac{1}{2}\; .$$
We denote the $i$-loop contribution to the unrenormalized coefficients by $A_{IJ}^{(i),{\rm un}}$ and $B^{(i),{\rm un}}$, using the same normalization as for the decomposition of the renormalized amplitude (\[eq:renorme\]); the dependence on $(\sac,\sbc,\sabc)$ is always understood implicitly. The renormalized coefficients are then obtained as $$\begin{aligned}
A_{IJ}^{(0)} &=&0,
\nonumber \\
A_{IJ}^{(1)} &=&
S_\e^{-1} A_{IJ}^{(1),{\rm un}}
, \nonumber \\
A_{IJ}^{(2)} &=&
S_\e^{-2} A_{IJ}^{(2),{\rm un}}
-\frac{3\beta_0}{2\e} S_\e^{-1}
A_{IJ}^{(1),{\rm un}} ,\end{aligned}$$ and $$\begin{aligned}
B^{(0)} &=& B^{(0),{\rm un}} ,
\nonumber \\
B^{(1)} &=&
S_\e^{-1} B^{(1),{\rm un}}
-\frac{\beta_0}{2\e} B^{(0),{\rm un}} , \nonumber \\
B^{(2)} &=&
S_\e^{-2} B^{(2),{\rm un}}
-\frac{3\beta_0}{2\e} S_\e^{-1}
B^{(1),{\rm un}}
-\left(\frac{\beta_1}{4\e}-\frac{3\beta_0^2}{8\e^2}\right)
B^{(0),{\rm un}}.\end{aligned}$$
For the remainder of this paper we will set the renormalization scale $\mu^2 = q^2$. The full scale dependence of the tensor coefficients is given by $$\begin{aligned}
A_{IJ} &=& \sqrt{4\pi\alpha}e_q \sqrt{4\pi\alpha_s} \; {\bom
T}^{a}_{ij}\, \bigg\{
\left(\frac{\alpha_s(\mu^2)}{2\pi}\right)
A_{IJ}^{(1)}
+ \left(\frac{\alpha_s(\mu^2)}{2\pi}\right)^2
\bigg[A_{IJ}^{(2)}
+\frac{3\beta_0}{2}A_{IJ}^{(1)}
\ln\left({\mu^2\over q^2}\right)
\bigg]
+ {\cal O}(\alpha_s^3) \bigg\},\nonumber \\
B &=& \sqrt{4\pi\alpha}e_q \sqrt{4\pi\alpha_s} \; {\bom
T}^{a}_{ij}\, \bigg\{
B^{(0)} + \left(\frac{\alpha_s(\mu^2)}{2\pi}\right)
\left[
B^{(1)}
+\frac{\beta_0}{2} B^{(0)} \ln\left({\mu^2\over q^2}\right)
\right]
\nonumber \\
&&+ \left(\frac{\alpha_s(\mu^2)}{2\pi}\right)^2
\bigg[B^{(2)}
+\biggl(\frac{3\beta_0}{2}B^{(1)} +\frac{\beta_1}{2}B^{(0)} \biggr)
\ln\left({\mu^2\over q^2}\right)
+\frac{3 \beta_0^2}{8} B^{(0)}
\ln^2\left({\mu^2\over q^2}\right)\bigg]
+ {\cal O}(\alpha_s^3) \bigg\}.\end{aligned}$$
Infrared behaviour of the tensor coefficients {#subsec:infrared}
---------------------------------------------
After performing ultraviolet renormalization, the amplitudes still contain singularities, which are of infrared origin and will be analytically cancelled by those occurring in radiative processes of the same order. Catani [@catani] has shown how to organize the infrared pole structure of the one- and two-loop contributions renormalized in the scheme in terms of the tree and renormalized one-loop amplitudes. The same procedure applies to the tensor coefficients. In particular, the infrared behaviour of the one-loop coefficients is given by $$\begin{aligned}
A_{IJ}^{(1)} &=& A_{IJ}^{(1),{\rm finite}},\nonumber \\
B^{(1)} &=& {\bom I}^{(1)}(\epsilon) B^{(0)} + B^{(1),{\rm finite}},\end{aligned}$$ while the two-loop singularity structure is $$\begin{aligned}
A_{IJ}^{(2)} &=& {\bom I}^{(1)}(\epsilon) A_{IJ}^{(1)}+A_{IJ}^{(2),{\rm finite}},\nonumber \\
B^{(2)} &=& \Biggl (-\frac{1}{2} {\bom I}^{(1)}(\epsilon) {\bom I}^{(1)}(\epsilon)
-\frac{\beta_0}{\epsilon} {\bom I}^{(1)}(\epsilon)
+e^{-\epsilon \gamma } \frac{ \Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}
\left(\frac{\beta_0}{\epsilon} + K\right)
{\bom I}^{(1)}(2\epsilon) + {\bom H}^{(2)}(\epsilon)
\Biggr )B^{(0)}\nonumber \\
&& + {\bom I}^{(1)}(\epsilon) B^{(1)}+ B^{(2),{\rm finite}},
\label{eq:polesa}\end{aligned}$$ where the constant $K$ is $$K = \left( \frac{67}{18} - \frac{\pi^2}{6} \right) \CA -
\frac{10}{9} T_R \NF.$$ The finite remainders $A_{IJ}^{(i),{\rm finite}}$ and $B^{(i),{\rm finite}}$ remain to be calculated.
For this particular process, there is only one colour structure present at tree level which, in terms of the gluon colour $a$ and the quark and antiquark colours $i$ and $j$, is simply $\bom{T}^{a}_{ij}$. Adding higher loops does not introduce additional colour structures, and the amplitudes are therefore vectors in a one-dimensional space. Similarly, the infrared singularity operator $\bom{I}^{(1)}(\epsilon)$ is a $1 \times 1$ matrix in the colour space and is given by $$\bom{I}^{(1)}(\epsilon)
=
- \frac{e^{\epsilon\gamma}}{2\Gamma(1-\epsilon)} \Biggl[
N \left(\frac{1}{\epsilon^2}+\frac{3}{4\epsilon}+\frac{\beta_0}{2N\epsilon}\right)
\left({\tt S}_{13}+{\tt S}_{23}\right)-\frac{1}{N}
\left(\frac{1}{\epsilon^2}+\frac{3}{2\epsilon}\right)
{\tt S}_{12}\Biggr ]\; ,\label{eq:I1}$$ where (since we have set $\mu^2 = s_{123}$) $${\tt S}_{ij} = \left(-\frac{s_{123}}{s_{ij}}\right)^{\epsilon}.$$ Note that on expanding ${\tt S}_{ij}$, imaginary parts are generated, the sign of which is fixed by the small imaginary part $+i0$ of $s_{ij}$. The origin of the various terms in Eq. (\[eq:I1\]) is straightforward. Each parton pair $ij$ in the event forms a radiating antenna of scale $s_{ij}$. Terms proportional to ${\tt S}_{ij}$ are cancelled by real radiation emitted from leg $i$ and absorbed by leg $j$. The soft singularities ${\cal O}(1/\epsilon^2)$ are independent of the identity of the participating partons and are universal. However, the collinear singularities depend on the identities of the participating partons.
Finally, the term of Eq. (\[eq:polesa\]) that involves ${\bom H}^{(2)}(\epsilon)$ produces only a single pole in $\epsilon$ and is given by $$\label{eq:htwo}
{\bom H}^{(2)}(\epsilon)
=\frac{e^{\epsilon \gamma}}{4\,\epsilon\,\Gamma(1-\epsilon)} H^{(2)} \;,$$ where the constant $H^{(2)}$ is renormalization-scheme-dependent. As with the single-pole parts of $\bom{I}^{(1)}(\epsilon)$, the process-dependent $H^{(2)}$ can be constructed by counting the number of radiating partons present in the event. In our case, there is a quark–antiquark pair and a gluon present in the final state, so that $$H^{(2)} = 2H^{(2)}_{q}+H^{(2)}_g\; ,$$ where, in the scheme: $$\begin{aligned}
H^{(2)}_g &=&
\left(\frac{1}{2}\zeta_3+{\frac {5}{12}}+ {\frac {11\pi^2}{144}}
\right)N^2
+{\frac {5}{27}}\,\NF^2
+\left (-{\frac {{\pi }^{2}}{72}}-{\frac {89}{108}}\right ) N \NF
-\frac{\NF}{4N}, \\
H^{(2)}_q &=&
\left({7\over 4}\zeta_3+{\frac {409}{864}}- {\frac {11\pi^2}{96}}
\right)N^2
+\left(-{1\over 4}\zeta_3-{41\over 108}-{\pi^2\over 96}\right)
+\left(-{3\over 2}\zeta_3-{3\over 32}+{\pi^2\over 8}\right){1\over
N^2}\nonumber \\
&&
+\left({\pi^2\over 48}-{25\over 216}\right){(N^2-1)N_F\over N}\;,\end{aligned}$$ so that $$\begin{aligned}
\label{eq:Htwo}
H^{(2)} &=&
\left(4\zeta_3+\frac{589}{432}- \frac{11\pi^2}{72}\right)N^2
+\left(-\frac{1}{2}\zeta_3-\frac{41}{54}-\frac{\pi^2}{48} \right)
+\left(-3\zeta_3 -\frac{3}{16} + \frac{\pi^2}{4}\right) \frac{1}{N^2}\nonumber \\
&&
+\left(-\frac{19}{18}+\frac{\pi^2}{36} \right) N\NF
+\left(-\frac{1}{54}-\frac{\pi^2}{24}\right) \frac{\NF}{N}+ \frac{5}{27} \NF^2.\end{aligned}$$ The factors $H^{(2)}_q$ and $H^{(2)}_g$ are directly related to those found in gluon–gluon scattering [@m4], quark–quark scattering [@m2] and quark–gluon scattering [@m3] (which each involve four partons) as well as in the quark form factor [@kl1; @qff1; @qff2; @qff3] and gluon form factor [@harlander]. We also note that (on purely dimensional grounds) one might expect terms of the type ${\tt S}_{ij}^2$ to be present in $H^{(2)}$. Of course such terms are $1 + {\cal
O}(\epsilon)$ and therefore leave the pole part unchanged, only modifying the finite remainder. At present it is not known how to systematically include these effects.
Helicity amplitudes {#sec:helicity}
===================
We can extend the results of the previous section to include $Z$ boson exchange, $$e^+(p_5) + e^-(p_6) \to \left(Z^*,\gamma^*\right) (p_4) \longrightarrow q(p_1) + \bar q (p_2) + g(p_3)\; ,$$ where the off-shell vector boson now distinguishes between left- and right-handed fermions by keeping track of the helicity of the final state quarks.[^2] A convenient method to evaluate the helicity amplitudes is in terms of Weyl–van der Waerden spinors, which is described briefly in Appendix \[sec:appa\] and in detail in [@WvdW; @six].
It is also straightforward to include the spin-correlations with the initial state by contracting the hadronic current with the lepton current $V_\mu$ for fixed helicities of the initial state electron (and positron). Using the spinor calculus of Appendix \[sec:appa\] we can express the lepton current $V_\mu$ in terms of the helicities of the incident $e^+$ and $e^-$ (with momenta $p_5$ and $p_6$ respectively). Explicitly, $$\begin{aligned}
V_\mu^\gamma(e^++,e^--) = e\sigma_\mu^{\dot AB}p_{6\dot
A}p_{5B}\frac{L^\gamma_{ee}}{s},
& ~~~~~ &
V_\mu^Z(e^++,e^--) = e\sigma_\mu^{\dot AB}p_{6\dot
A}p_{5B}\frac{L^Z_{ee}}{s-M_Z^2+i\Gamma_ZM_Z},\nonumber \\
V_\mu^\gamma(e^+-,e^-+) = e\sigma_\mu^{\dot AB}p_{5\dot
A}p_{6B}\frac{R^\gamma_{ee}}{s},
& ~~~~~ &
V_\mu^Z(e^+-,e^-+) = e\sigma_\mu^{\dot AB}p_{5\dot
A}p_{6B}\frac{R^Z_{ee}}{s-M_Z^2+i\Gamma_ZM_Z}.\nonumber \\\end{aligned}$$
The hadronic current $\S_{\mu}$ is related to the fixed helicity currents, $\S_{{\dot A}B}$, by $$\S_{\mu}(q+;g\lambda;\overline{q}-) = R^V_{f_1f_2}
\sqrt2\, \sigma_{\mu}^{{\dot A}B} \S_{{\dot A}B}(q+;g\lambda;\overline{q}-) ,$$ $$\S_{\mu}(q-;g\lambda;\overline{q}+) = L^V_{f_1f_2}
\sqrt2\, \sigma_{\mu}^{{\dot A}B} \S_{{\dot A}B}(q-;g\lambda;\overline{q}+) .$$ As in Eq. (\[eq:renorme\]), the gauge boson coupling is extracted from $\S_{{\dot A}B}$. As mentioned earlier, the left- and right-handed currents couple with a different strength when the vector boson is a $Z$.
The currents with the quark helicities flipped follow from parity conservation: $$\S_{{\dot A}B}(q-;g\lambda;\overline{q}+) =
( \S_{{\dot B}A}(q+;g(-\lambda);\overline{q}-))^*\ .$$ Charge conjugation implies the following relations between currents with different helicities: $$\S_{{\dot
A}B}(q\lambda_{q};g\lambda;\overline{q}\lambda_{\overline{q}}) =
(-1)
\S_{{\dot A}B}(\overline{q}\lambda_{\overline{q}};g\lambda;q\lambda_{q}).$$ All helicity amplitudes are therefore related to the amplitudes with $\lambda_{q} = +$ and $\lambda_{\bar q} = -$.
Explicitly, we find $$\begin{aligned}
{\S}_{\dot AB}(q+;g+;\overline{q}-)
&=&
\alpha(y,z)~
\frac{p_{1\dot AD}p_2^D p_{2B}}{\langle p_1p_3 \rangle\langle p_3{p}_2 \rangle}
+\beta(y,z)~
\frac{p_{3\dot AD} p_2^D p_{2B}}{\langle p_1p_3 \rangle\langle p_3{p}_2 \rangle}
+\gamma(y,z)~
\frac{p_{1\dot CB}p_3^{\dot C}p_{3\dot A}}{\langle p_1p_3 \rangle\langle p_3{p}_2 \rangle^*} \nonumber \\
&& +~\delta(y,z)~\frac{\langle p_1p_3\rangle^*}{\langle p_1p_3 \rangle\langle p_1{p}_2 \rangle^*}
\left(p_{1\dot AB}+p_{2\dot AB}+p_{3\dot AB}\right)
\;.
\label{eq:helamp}\end{aligned}$$ The other helicity amplitudes are obtained from $\S_{\dot
AB}(q+;g+;\bar q-)$ by the above parity and charge conjugation relations, while the coefficients $\alpha$, $\beta$ and $\gamma$ are written in terms of the tensor coefficients: $$\begin{aligned}
\alpha(y,z) &=& \frac{\sbc\sac}{4}\bigg(2 B(\sac,\sbc,\sabc) + A_{12}(\sac,\sbc,\sabc) - A_{11}(\sac,\sbc,\sabc)\bigg),\nonumber \\
\beta(y,z) &=& \frac{\sac}{4}\bigg(2\sbc B(\sac,\sbc,\sabc) + 2 (\sab+\sac)A_{11}(\sac,\sbc,\sabc) \nonumber \\&& \hspace{0.7cm}
+\sbc\big(A_{12}(\sac,\sbc,\sabc)+A_{13}(\sac,\sbc,\sabc)\big)\bigg),\nonumber \\
\gamma(y,z) &=& \frac{\sac\sbc}{4}\bigg(A_{11}(\sac,\sbc,\sabc)-A_{13}(\sac,\sbc,\sabc)\bigg) ,\nonumber \\
\delta(y,z) &=& - \frac{\sab\sac}{4}A_{11}(\sac,\sbc,\sabc).\end{aligned}$$ When the hadron tensor is contracted with $\epsilon_4^\mu$ or the lepton current $V^\mu$, the final term of Eq. (\[eq:helamp\]) vanishes[^3]. Furthermore, current conservation implies the following relation between the four helicity coefficients, $$\alpha(y,z)-\beta(y,z)-\gamma(y,z)-\frac{2\sabc}{\sab}~\delta(y,z) = 0.$$ This relation is fulfilled automatically once the tensor coefficients are inserted and does therefore not yield a further reduction of the tensor basis.
As with the tensor coefficients, the helicity amplitude coefficients $\alpha$, $\beta$ and $\gamma$ are vectors in colour space and have perturbative expansions: $$\Omega = \sqrt{4\pi\alpha} \sqrt{4\pi\alpha_s} \; \bom{T}^a_{ij}\, \left[
\Omega^{(0)}
+ \left(\frac{\alpha_s}{2\pi}\right) \Omega^{(1)}
+ \left(\frac{\alpha_s}{2\pi}\right)^2 \Omega^{(2)}
+ {\cal O}(\alpha_s^3) \right] \;,\nonumber \\$$ for $\Omega = \alpha,\beta,\gamma$. The dependence on $(y,z)$ is again implicit.
The ultraviolet and infrared properties of the helicity coefficients match those of the tensor coefficients, $$\begin{aligned}
\Omega^{(0)} &=& \Omega^{(0),{\rm un}} ,
\nonumber \\
\Omega^{(1)} &=&
S_\e^{-1} \Omega^{(1),{\rm un}}
-\frac{\beta_0}{2\e} \Omega^{(0),{\rm un}} , \nonumber \\
\Omega^{(2)} &=&
S_\e^{-2} \Omega^{(2),{\rm un}}
-\frac{3\beta_0}{2\e} S_\e^{-1}
\Omega^{(1),{\rm un}}
-\left(\frac{\beta_1}{4\e}-\frac{3\beta_0^2}{8\e^2}\right)
\Omega^{(0),{\rm un}},\end{aligned}$$ and $$\begin{aligned}
\Omega^{(1)} &=& {\bom I}^{(1)}(\epsilon) \Omega^{(0)} +
\Omega^{(1),{\rm finite}},\nonumber \\
\Omega^{(2)} &=& \Biggl (-\frac{1}{2} {\bom I}^{(1)}(\epsilon) {\bom I}^{(1)}(\epsilon)
-\frac{\beta_0}{\epsilon} {\bom I}^{(1)}(\epsilon)
+e^{-\epsilon \gamma } \frac{ \Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}
\left(\frac{\beta_0}{\epsilon} + K\right)
{\bom I}^{(1)}(2\epsilon) + {\bom H}^{(2)}(\epsilon)
\Biggr )\Omega^{(0)}\nonumber \\
&& + {\bom I}^{(1)}(\epsilon) \Omega^{(1)}+ \Omega^{(2),{\rm finite}},\end{aligned}$$ where ${\bom I}^{(1)}(\epsilon)$ and ${\bom H}^{(2)}(\epsilon)$ are defined in Eqs. (\[eq:I1\]) and (\[eq:htwo\]) respectively.
At leading order $$\alpha^{(0)}(y,z) = \beta^{(0)}(y,z) = 1\qquad
\mbox{and}\qquad \gamma^{(0)}(y,z) =0.$$ The renormalized next-to-leading order helicity amplitude coefficients can be straightforwardly obtained to all orders in $\epsilon$ from the tensor coefficients using Eqs. (\[eq:aNLO\])–(\[eq:bNLO\]). For practical purposes, they are needed through to ${\cal O}(\epsilon^2)$ in evaluating the infrared-divergent one-loop contribution to the two-loop amplitude, while only the finite piece is needed for the one-loop self-interference. They can be decomposed according to their colour structure as follows: $$\Omega^{(1),{\rm finite}}(y,z) =
N\, a_{\Omega}(y,z) + \frac{1}{N}\, b_{\Omega}(y,z) + \beta_0\, c_{\Omega}(y,z)
\;.
\label{eq:oneloopamp}$$ The expansion of the coefficients through to $\e^2$ yields HPLs and 2dHPLs up to weight 4 for $a_{\Omega}$, $b_{\Omega}$ and up to weight 3 for $c_{\Omega}$. The explicit expressions are of considerable size, such that we only quote the $\e^0$-terms here (although these have been known already for a long time [@gg]). The expressions through to ${\cal O}(\e^2)$ can be obtained in FORM format from the authors. An example of the size and structure of those coefficients can be found in [@3jme], where we explicitly list the helicity-averaged one-loop times one-loop and tree times two-loop matrix elements. The one-loop coefficients read: $$\begin{aligned}
a_{\alpha}(y,z) &=&
- \frac{7}{4}
- \frac{\pi^2}{12}
+ \frac{3}{8}\H(0;z)
- \frac{1}{2} \H(0,z) \G(0;y)
- \frac{1}{2} \H(1,0;z)
- \frac{3}{8} \G(0;y)
+ \frac{1}{2} \G(1,0;y) \nonumber \\
&& - \frac{1}{4(1-z)^2} \H(0;z)
- \frac{1}{4(1-z)} \Big(1+2\H(0;z)\Big) + {\cal O} (\e) \; ,
\nonumber \\
b_{\alpha}(y,z) &=&
\frac{z^2}{2y^2} \Big(
\H(0;z) \G(1 - z;y)
+ \H(1;z) \G( - z;y)
- \G( - z,1 - z;y)
\Big)
+ \frac{z}{2y} \Big(
- \H(0;z)\nonumber \\
&&
+ 2 \H(0;z) \G(1 - z;y)
- \H(1;z)
+ 2 \H(1;z) \G( - z;y)
+ \G(1 - z;y)
- 2 \G( - z,1 - z;y)
\Big)\nonumber \\
&&
+ \frac{1}{2y (1-z)}\H(0;z)
-\frac{1}{2y} \H(0;z)
+ \frac{1}{4(1-z)^2}\H(0;z)
+ \frac{1}{4(1-z)} \Big(
1
+ 2 \H(0;z)
\Big)
+ \frac{7}{4}
- \frac{3}{4} \H(0;z)\nonumber \\
&&
+ \frac{1}{2} \H(0;z) \G(1 - z;y)
+ \frac{1}{2} \H(0,1;z)
- \frac{3}{4} \H(1;z)
+ \H(1;z) \G( - z;y)
- \frac{1}{2} \H(1;z) \G(0;y)\nonumber \\
&&
+ \frac{3}{4} \G(1 - z;y)
+ \frac{1}{2} \G(1 - z,0;y)
- \G( - z,1 - z;y)
+ \frac{1}{2} \G(0,1 - z;y)
- \frac{1}{2} \G(1,0;y)+ {\cal O} (\e)\;,
\nonumber \\
c_{\alpha}(y,z) &=& -\frac{1}{4} \H(0;z) - \frac{1}{4} \G(0;y)
+ \frac{i\pi}{2} + {\cal O} (\e)\;, \nonumber \\
a_{\beta}(y,z) &=&
- \frac{3}{2}
- \frac{\pi^2}{12}
+ \frac{3}{8} \H(0;z)
- \frac{1}{2} \H(0;z) \G(0;y)
- \frac{1}{2} \H(1,0;z)
- \frac{3}{8} \G(0;y)
+ \frac{1}{2} \G(1,0;y)\nonumber \\
&&
+\frac{1}{4(1-z)} \H(0;z)+ {\cal O} (\e)\; ,
\nonumber \\
b_{\beta}(y,z) &=&
\frac{z(1-z)}{2y^2} \Big(
- \H(0;z) \G(1 - z;y)
- \H(1;z) \G( - z;y)
+ \G( - z,1 - z;y)
\Big)
+ \frac{z}{2y} \Big(
- \H(0;z)\nonumber \\
&&
+ 2\H(0;z) \G(1 - z;y)
- \H(1;z)
+ 2\H(1;z) \G( - z;y)
+ \G(1 - z;y)
- 2\G( - z,1 - z;y)
\Big)\nonumber \\
&&
+ \frac{1}{2y} \Big(
- 2\H(0;z) \G(1 - z;y)
+ \H(1;z)
- 2\H(1;z) \G( - z;y)
- \G(1 - z;y)
+ 2\G( - z,1 - z;y)
\Big)\nonumber \\
&&
+ \frac{z}{2(y+z)^2} \Big(
- \H(1;z)
+ \G(1 - z;y)
\Big)
+ \frac{z}{2(y+z)}
- \frac{1}{4(1-z)} \H(0;z)
+ \frac{1}{2(y+z)} \Big(
\H(1;z)\nonumber \\
&&
- \G(1 - z;y)
\Big)
+ \frac{3}{2}
- \frac{3}{4} \H(0;z)
+ \frac{1}{2} \H(0;z) \G(1 - z;y)
+ \frac{1}{2} \H(0,1;z)
- \frac{3}{4} \H(1;z)\nonumber \\
&&
+ \H(1;z) \G( - z;y)
- \frac{1}{2} \H(1;z) \G(0;y)
+ \frac{3}{4} \G(1 - z;y)
+ \frac{1}{2} \G(1 - z,0;y)
- \G( - z,1 - z;y)\nonumber \\
&&
+ \frac{1}{2} \G(0,1 - z;y)
- \frac{1}{2} \G(1,0;y)
\Big)+ {\cal O} (\e) \;,
\nonumber \\
c_{\beta}(y,z) &=& -\frac{1}{4} \H(0;z) - \frac{1}{4} \G(0;y)
+ \frac{i\pi}{2} + {\cal O} (\e) \;,\nonumber \\
a_{\gamma}(y,z) &=& -\frac{1}{4} + \frac{1}{4(1-z)^2} \H(0;z)
+ \frac{1}{4(1-z)} \Big( 1-\H(0;z)\Big) + {\cal O} (\e)
\;,\nonumber \\
b_{\gamma}(y,z) &=&
\frac{1}{4} + \frac{z}{2y^2}\Big(
- \H(0;z) \G(1 - z;y)
- \H(1;z) \G( - z;y)
+ \G( - z,1 - z;y)
\Big)
- \frac{1}{2y(1-z)} \H(0;z)\nonumber \\
&&
+ \frac{1}{2y} \Big(
\H(0;z)
+ \H(1;z)
- \G(1 - z;y)
\Big)
+ \frac{z}{2(y+z)^2} \Big(
\H(1;z)
- \G(1 - z;y)
\Big)
- \frac{z}{2(y+z)}\nonumber \\
&&
+ \frac{1}{2(y+z)} \Big(
- \H(1;z)
+ \G(1 - z;y)
\Big)
- \frac{1}{4(1-z)^2}\H(0;z)
+ \frac{1}{4(1-z)}\Big(
- 1
+ \H(0;z)
\Big) + {\cal O} (\e)\; ,
\nonumber \\
c_{\gamma}(y,z) &=& 0 \; .
\label{eq:omega1}\end{aligned}$$ It should be noted that these finite pieces of the one-loop coefficients can equally well be written in terms of ordinary logarithms and dilogarithms, see [@ert1; @gg]. The reason to express them in terms of HPLs and 2dHPLs here is their usage in the infrared counter-term of the two-loop coefficients, which cannot be fully expressed in terms of logarithmic and polylogarithmic functions.
The finite two-loop remainder is obtained by subtracting the predicted infrared structure (expanded through to ${\cal O}(\epsilon^0)$) from the renormalized helicity coefficient. We further decompose the finite remainder according to the colour structure, as follows: $$\begin{aligned}
\Omega^{(2),{\rm finite}}(y,z) &=&
N^2 A_\Omega(y,z) + B_\Omega(y,z) + \frac{1}{N^2} C_\Omega(y,z)
+ N\NF D_\Omega(y,z) \nonumber \\ &&
+ \frac{\NF}{N} E_\Omega(y,z) + \NF^2 F_\Omega (y,z)
+ \NFZ \left(\frac{4}{N}-N\right) G_\Omega(y,z) \; ,
\label{eq:twoloopamp}\end{aligned}$$ where the last term is generated by graphs where the virtual gauge boson does not couple directly to the final-state quarks. This contribution is denoted by $\NFZ$ and is proportional to the charge weighted sum of the quark flavours. In the case of purely electromagnetic interactions we find, $$N_{F,\gamma} = \frac{\sum_q e_q}{e_q}\; .$$ Including $Z$-interactions, the same class of diagrams yields not only a contribution from the vector component of the $Z$, which for the right-handed quark amplitude is given by $$N_{F,Z} = \frac{\sum_q \left(L^Z_{qq}+R^Z_{qq}\right)}{2R^Z_{qq}}\; ,$$ but also a contribution involving the axial couplings of the $Z$ [@kk]. This contribution vanishes if summed over isospin doublets. The large mass splitting of the third quark family induces a non-vanishing contribution from this class of diagrams, which can however not be computed within the framework of massless QCD employed here, but can only be obtained within an effective theory with large top-quark mass. In contrast to the vector contribution from these diagrams, which is finite, one could expect divergences in the axial vector contribution, which would be cancelled by the single unresolved limits of the corresponding axial contributions to four-parton final states [@onel4p1; @onel4p2]. Results from the four-parton final states show that this axial contribution is numerically very small [@menloparc].
The helicity coefficients contain HPLs and 2dHPLs up to weight 4 in the $A,B,C,G$-terms, up to weight 3 in the $D,E$-terms (which do moreover contain only a limited subset of purely planar master integrals) and up to weight 2 in the $F$-term. The size of each helicity coefficient is comparable to the size of the helicity-averaged tree times two-loop matrix element quoted in [@3jme]. We do therefore only quote the $A$- and $D$-terms of each coefficient, which form the leading colour contributions, and which turn out to be numerically dominant, approximating the full expressions to an accuracy of about 20%. The complete set of coefficients in FORM format can be obtained from the authors.
These leading colour terms are: $$\begin{aligned}
\lefteqn{A_{\alpha}(y,z) = }\nonumber \\
&& \frac{1}{48y(1-z)} \Big[
\pi^2
- 13 \H(0;z)
+ 6 \H(1,0;z)
+ 6 \G(1,0;y)
\Big]
- \frac{1}{48y(1-y-z)} \Big[
\pi^2
- 13 \H(0;z)
+ 6 \H(1,0;z)
\nonumber \\ &&
+ 6 \G(1,0;y)
\Big]
- \frac{z}{16(1-y)^2} \G(0;y)
- \frac{z}{16(1-y)}
+ \frac{z}{12(1-y-z)^2} \Big[
- \frac{5\pi^2}{6}
- 5 \H(0;z) \G(0;y)
- 5 \H(1,0;z)
\nonumber \\ &&
+ 5 \G(1,0;y)
\Big]
+ \frac{z}{16(1-y-z)} \Big[
\frac{14\pi^2}{3}
- 11 \H(0;z)
+ 28 \H(0;z) \G(0;y)
+ 28 \H(1,0;z)
+ 11 \G(0;y)
\nonumber \\ &&
- 28 \G(1,0;y)
\Big]
+ \frac{z^2}{16(1-y-z)^2} \Big[
\frac{11\pi^2}{6}
+ 11 \H(0;z) \G(0;y)
+ 11 \H(1,0;z)
- 11 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{1}{3(1-y)} \G(0;y)
+ \frac{1}{48(1-z)^2} \Big[
- \frac{\pi^2}{6}
+ \pi^2 \big(3 \H(0;z)
+3 \H(1;z)
- \G(1 - z;y)
+ \G(0;y)\big)
+ 6\zeta_3
\nonumber \\ &&
- \frac{355}{6} \H(0;z)
- 6\H(0;z) \G(1 - z,0;y)
+ 10 \H(0;z) \G(0;y)
+ 45 \H(0,0;z)
+ 12 \H(0,0;z) \G(0;y)
\nonumber \\ &&
+ 18 \H(0,1,0;z)
- \H(1,0;z)
- 6 \H(1,0;z) \G(1 - z;y)
+ 6 \H(1,0;z) \G(0;y)
+ 12 \H(1,0,0;z)
+ 18 \H(1,1,0;z)
\nonumber \\ &&
+ 6 \G(1 - z,1,0;y)
- 6 \G(0,1,0;y)
\Big]
+ \frac{1}{72(1-z)}\Big[
\pi^2 \big(
- 8
+ 9 \H(0;z)
+ 9 \H(1;z)
- 3 \G(1 - z;y)
\nonumber \\ &&
+ 3 \G(0;y)
\big)
+ 18\zeta_3
- \frac{277}{4}
- 65 \H(0;z)
- 18 \H(0;z) \G(1 - z,0;y)
+ 39 \H(0;z) \G(0;y)
+ 81 \H(0,0;z)
\nonumber \\ &&
+ 36 \H(0,0;z) \G(0;y)
+ 54 \H(0,1,0;z)
- 48 \H(1,0;z)
- 18 \H(1,0;z) \G(1 - z;y)
+ 18 \H(1,0;z) \G(0;y)
\nonumber \\ &&
+ 36 \H(1,0,0;z)
+ 54 \H(1,1,0;z)
+ 18 \G(1 - z,1,0;y)
+ 15 \G(0;y)
- 18 \G(0,1,0;y)
- 9 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{1}{48(1-y-z)^2} \Big[
- 2\pi^2 \H(1;z)
- \pi^2 \G(0;y)
+ 12 \zeta_3
- 6 \H(1,0;z) \G(0;y)
- 12 \H(1,1,0;z)
- 6 \G(0,1,0;y)
\Big]
\nonumber \\ &&
+ \frac{1}{48(1-y-z)} \Big[
- 4\pi^2 \H(1;z)
- 2\pi^2 \G(0;y)
+ 24 \zeta_3
- 13 \H(0;z)
- 12 \H(1,0;z) \G(0;y)
- 24 \H(1,1,0;z)
\nonumber \\ &&
- 20 \G(0;y)
- 12 \G(0,1,0;y)
\Big]
+ \frac{\pi^2}{288} \Big[
- \frac{928}{3}
- 5 \H(0;z)
+ 12 \H(0;z) \G(1 - z;y)
+ 36 \H(0;z) \G(0;y)
\nonumber \\ &&
- 12 \H(0;z) \G(1;y)
+ 24 \H(0,1;z)
+ 24 \H(1;z) \G(1 - z;y)
- 24 \H(1;z) \G( - z;y)
- 12 \H(1;z) \G(1;y)
\nonumber \\ &&
+ 24 \H(1,0;z)
+ 12 \H(1,1;z)
- 44 \G(1 - z;y)
+ 12 \G(1 - z,0;y)
- 24 \G(1 - z,1;y)
+ 24 \G( - z,1 - z;y)
\nonumber \\ &&
- 24 \G(0,1 - z;y)
+ 49 \G(0;y)
- 24 \G(0,1;y)
+ 12 \G(1,1 - z;y)
+ 44 \G(1;y)
- 36 \G(1,0;y)
+ 24 \G(1,1;y)
\Big]
\nonumber \\ &&
+ \frac{\zeta_3}{72} \Big[
317
- 18 \H(0;z)
+ 90 \H(1;z)
- 72 \G(1 - z;y)
- 18 \G(0;y)
- 18 \G(1;y)
\Big]
+ \frac{11\pi^4}{360}
+ \frac{1}{72} \Big[
- \frac{89959}{144}
\nonumber \\ &&
+ \frac{2149}{12} \H(0;z)
- 66 \H(0;z) \G(1 - z,0;y)
- 18 \H(0;z) \G(1 - z,1,0;y)
+ 36 \H(0;z) \G( - z,1 - z,0;y)
\nonumber \\ &&
- 36 \H(0;z) \G(0,1 - z,0;y)
- 66 \H(0;z) \G(0;y)
+ 126 \H(0;z) \G(0,0;y)
- 18 \H(0;z) \G(0,1,0;y)
\nonumber \\ &&
+ 18 \H(0;z) \G(1,1 - z,0;y)
- 3 \H(0;z) \G(1,0;y)
- 36 \H(0;z) \G(1,0,0;y)
+ \frac{23}{2} \H(0,0;z)
\nonumber \\ &&
+ 72 \H(0,0;z) \G(0;y)
+ 36 \H(0,0;z) \G(0,0;y)
+ 72 \H(0,0,1,0;z)
+ 3 \H(0,1,0;z)
- 18 \H(0,1,0;z) \G(1 - z;y)
\nonumber \\ &&
+ 36 \H(0,1,0;z) \G( - z;y)
+ 18 \H(0,1,0;z) \G(0;y)
- 18 \H(0,1,0;z) \G(1;y)
+ 36 \H(0,1,1,0;z)
\nonumber \\ &&
- 71 \H(1,0;z)
- 66 \H(1,0;z) \G(1 - z;y)
+ 18 \H(1,0;z) \G(1 - z,0;y)
+ 36 \H(1,0;z) \G( - z,1 - z;y)
\nonumber \\ &&
- 36 \H(1,0;z) \G( - z,0;y)
- 36 \H(1,0;z) \G(0,1 - z;y)
+ 96 \H(1,0;z) \G(0;y)
+ 18 \H(1,0;z) \G(1,1 - z;y)
\nonumber \\ &&
- 18 \H(1,0;z) \G(1,0;y)
+ 72 \H(1,0,0;z)
+ 36 \H(1,0,0;z) \G(0;y)
+ 72 \H(1,0,1,0;z)
\nonumber \\ &&
+ 36 \H(1,1,0;z) \G(1 - z;y)
- 36 \H(1,1,0;z) \G( - z;y)
- 18 \H(1,1,0;z) \G(1;y)
+ 36 \H(1,1,0,0;z)
\nonumber \\ &&
+ 18 \H(1,1,1,0;z)
+ 18 \G(1 - z,0,1,0;y)
+ 66 \G(1 - z,1,0;y)
+ 36 \G(1 - z,1,1,0;y)
\nonumber \\ &&
- 36 \G( - z,1 - z,1,0;y)
- 36 \G( - z,0,1,0;y)
+ 36 \G(0,1 - z,1,0;y)
+ \frac{49}{3} \G(0;y)
+ 160 \G(0,0;y)
\nonumber \\ &&
- 36 \G(0,0,1,0;y)
- 30 \G(0,1,0;y)
+ 36 \G(0,1,1,0;y)
- 18 \G(1,1 - z,1,0;y)
+ 71 \G(1,0;y)
\nonumber \\ &&
- 126 \G(1,0,0;y)
+ 54 \G(1,0,1,0;y)
- 66 \G(1,1,0;y)
+ 36 \G(1,1,0,0;y)
- 36 \G(1,1,1,0;y)
\Big]
\nonumber \\ &&
+ i\pi \Bigg\{
- \frac{11}{16(1-z)^2} \H(0;z)
+ \frac{1}{16(1-z)} \Big[
- 11
- 22 \H(0;z)
\Big]
+ 2\zeta_3
+ \frac{1}{48} \Big[
- \frac{44\pi^2}{3}
- \frac{2345}{18}
- 11 \H(0;z)
\nonumber \\ && \hspace{9mm}
- 66 \H(0;z) \G(0;y)
- 66 \H(1,0;z)
- 110 \G(0;y)
+ 66 \G(1,0;y)
\Big] \Bigg\}
\nonumber \\
\lefteqn{D_{\alpha}(y,z) =} \nonumber \\&&
\frac{1}{12y(1-z)}\H(0;z)
- \frac{1}{12y(1-y-z)}\H(0;z)
+ \frac{z}{6(1-y-z)^2} \Big[
\frac{\pi^2}{6}
+ \H(0;z) \G(0;y)
+ \H(1,0;z)
\nonumber \\ &&
- \G(1,0;y)
\Big]
+ \frac{z}{4(1-y-z)} \Big[
- \frac{\pi^2}{3}
+ \H(0;z)
- 2\H(0;z) \G(0;y)
- 2\H(1,0;z)
- \G(0;y)
+ 2\G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{4(1-y-z)^2} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
- \frac{1}{12(1-y)}\G(0;y)
\nonumber \\&&
+ \frac{1}{72(1-z)^2} \Big[
\pi^2
+ 25 \H(0;z)
- \frac{3}{2} \H(0;z) \G(0;y)
- 9 \H(0,0;z)
+ 6 \H(1,0;z)
\Big]
\nonumber \\&&
+ \frac{1}{144(1-z)} \Big[
4\pi^2
+ 38
+ 37 \H(0;z)
- 6 \H(0;z) \G(0;y)
- 36 \H(0,0;z)
+ 24 \H(1,0;z)
- 3 \G(0;y)
\Big]
\nonumber \\&&
+ \frac{1}{12(1-y-z)} \Big[
\H(0;z)
+ 2 \G(0;y)
\Big]
+ \frac{\pi^2}{72} \Big[
\frac{395}{12}
- \H(0;z)
+ 2 \G(1 - z;y)
- \G(0;y)
- 2 \G(1;y)
\Big]
\nonumber \\&&
- \frac{19}{36} \zeta_3
+ \frac{1}{144} \Big[
\frac{3661}{18}
- 25 \H(0;z)
+ 24 \H(0;z) \G(1 - z,0;y)
+ 29 \H(0;z) \G(0;y)
- 36 \H(0;z) \G(0,0;y)
\nonumber \\&&
+ 6 \H(0;z) \G(1,0;y)
- 28 \H(0,0;z)
- 36 \H(0,0;z) \G(0;y)
- 6 \H(0,1,0;z)
+ 40 \H(1,0;z)
\nonumber \\&&
+ 24 \H(1,0;z) \G(1 - z;y)
- 30 \H(1,0;z) \G(0;y)
- 36 \H(1,0,0;z)
- 24 \G(1 - z,1,0;y)
+ 53 \G(0;y)
\nonumber \\&&
- 82 \G(0,0;y)
+ 6 \G(0,1,0;y)
- 40 \G(1,0;y)
+ 36 \G(1,0,0;y)
+ 24 \G(1,1,0;y)
\Big]
\nonumber \\&&
+ i\pi \Bigg\{
\frac{1}{8(1-z)^2} \H(0;z)
+ \frac{1}{8(1-z)} \Big[
1
+ 2 \H(0;z)
\Big]
+ \frac{1}{48} \Big[
\frac{8\pi^2}{3}
- \frac{28}{3}
+ 13 \H(0;z)
+ 12 \H(0;z) \G(0;y)
\nonumber \\&& \hspace{9mm}
+ 12 \H(1,0;z)
+ 31 \G(0;y)
- 12 \G(1,0;y)
\Big] \Bigg\}
\nonumber \\
\lefteqn{A_{\beta}(y,z) =} \nonumber \\ &&
- \frac{z}{16(1-y)^2} \G(0;y)
- \frac{z}{16(1-y)}
+ \frac{z}{16(y+z)^2} \Big[
\frac{47\pi^2}{3} \H(1;z)
- \frac{47\pi^2}{3} \G(1 - z;y)
- 94 \H(0;z) \G(1 - z,0;y)
\nonumber \\&&
- 94 \H(0,1,0;z)
- 99 \H(1,0;z)
- 94 \H(1,0;z) \G(1 - z;y)
+ 94 \H(1,0;z) \G(0;y)
+ 94 \H(1,1,0;z)
\nonumber \\&&
+ 94 \G(1 - z,1,0;y)
+ 94 \G(0,1,0;y)
- 99 \G(1,0;y)
\Big]
+ \frac{z}{16(y+z)} \Big[
- \frac{47\pi^2}{3}
+ 11
+ 44 \H(0;z)
\nonumber \\&&
- 94 \H(0;z) \G(0;y)
- 94 \H(1,0;z)
- 55 \G(0;y)
+ 94 \G(1,0;y)
\Big]
+ \frac{z}{12(1-y-z)^2} \Big[
\frac{5\pi^2}{6}
- \frac{\pi^2}{2} \H(1;z)
\nonumber \\&&
- \frac{\pi^2}{4} \G(0;y)
+ 3\zeta_3
+ 5 \H(0;z) \G(0;y)
+ 5 \H(1,0;z)
- \frac{3}{2} \H(1,0;z) \G(0;y)
- 3 \H(1,1,0;z)
- \frac{3}{2} \G(0,1,0;y)
\nonumber \\ &&
- 5 \G(1,0;y)
\Big]
+ \frac{z}{12(1-y-z)} \Big[
- \frac{19\pi^2}{3}
+ 5 \H(0;z)
- 38 \H(0;z) \G(0;y)
- 38 \H(1,0;z)
- \frac{53}{4} \G(0;y)
\nonumber \\ &&
+ 38 \G(1,0;y)
\Big]
+ \frac{z^2}{8(y+z)^3} \Big[
- 11\pi^2 \H(1;z)
+ 11\pi^2 \G(1 - z;y)
+ 66 \H(0;z) \G(1 - z,0;y)
+ 66 \H(0,1,0;z)
\nonumber \\ &&
+ 33 \H(1,0;z)
+ 66 \H(1,0;z) \G(1 - z;y)
- 66 \H(1,0;z) \G(0;y)
- 66 \H(1,1,0;z)
- 66 \G(1 - z,1,0;y)
\nonumber \\ &&
- 66 \G(0,1,0;y)
+ 33 \G(1,0;y)
\Big]
+ \frac{z^2}{16(y+z)^2} \Big[
+ 22\pi^2
- 33 \H(0;z)
+ 132 \H(0;z) \G(0;y)
+ 132 \H(1,0;z)
\nonumber \\ &&
+ 33 \G(0;y)
- 132 \G(1,0;y)
\Big]
+ \frac{z^2}{16(y+z)} \Big[
11\pi^2
- 11 \H(0;z)
+ 66 \H(0;z) \G(0;y)
+ 66 \H(1,0;z)
\nonumber \\ &&
+ 11 \G(0;y)
- 66 \G(1,0;y)
\Big]
+ \frac{z^2}{48(1-y-z)^2} \Big[
- \frac{53\pi^2}{6}
- 53 \H(0;z) \G(0;y)
- 53 \H(1,0;z)
+ 53 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{16(1-y-z)} \Big[
11\pi^2
- 11 \H(0;z)
+ 66 \H(0;z) \G(0;y)
+ 66 \H(1,0;z)
+ 11 \G(0;y)
- 66 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{8(y+z)^4} \Big[
\frac{11\pi^2}{2} \H(1;z)
- \frac{11\pi^2}{2} \G(1 - z;y)
- 33 \H(0;z) \G(1 - z,0;y)
- 33 \H(0,1,0;z)
\nonumber \\ &&
- 33 \H(1,0;z) \G(1 - z;y)
+ 33 \H(1,0;z) \G(0;y)
+ 33 \H(1,1,0;z)
+ 33 \G(1 - z,1,0;y)
+ 33 \G(0,1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{8(y+z)^3} \Big[
- \frac{11\pi^2}{2}
- 33 \H(0;z) \G(0;y)
- 33 \H(1,0;z)
+ 33 \G(1,0;y)
\Big]
+ \frac{z^3}{16(y+z)^2} \Big[
- \frac{11\pi^2}{2}
\nonumber \\ &&
- 33 \H(0;z) \G(0;y)
- 33 \H(1,0;z)
+ 33 \G(1,0;y)
\Big]
+ \frac{z^3}{8(y+z)} \Big[
- \frac{11\pi^2}{6}
- 11 \H(0;z) \G(0;y)
- 11 \H(1,0;z)
\nonumber \\ &&
+ 11 \G(1,0;y)
\Big]
+ \frac{z^3}{16(1-y-z)^2} \Big[
\frac{11\pi^2}{6}
+ 11 \H(0;z) \G(0;y)
+ 11 \H(1,0;z)
- 11 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{8(1-y-z)} \Big[
- \frac{11\pi^2}{6}
- 11 \H(0;z) \G(0;y)
- 11 \H(1,0;z)
+ 11 \G(1,0;y)
\Big]
+ \frac{5}{24(1-y)} \G(0;y)
\nonumber \\ &&
+ \frac{1}{48(1-z)} \Big[
\pi^2 \big(
\frac{1}{6}
- 3 \H(0;z)
- 3 \H(1;z)
+ \G(1 - z;y)
- \G(0;y)
\big)
- 6\zeta_3
+ \frac{355}{6} \H(0;z)
\nonumber \\ &&
+ 6 \H(0;z) \G(1 - z,0;y)
- 10 \H(0;z) \G(0;y)
- 45 \H(0,0;z)
- 12 \H(0,0;z) \G(0;y)
- 18 \H(0,1,0;z)
\nonumber \\ &&
+ \H(1,0;z)
+ 6 \H(1,0;z) \G(1 - z;y)
- 6 \H(1,0;z) \G(0;y)
- 12 \H(1,0,0;z)
- 18 \H(1,1,0;z)
\nonumber \\ &&
- 6 \G(1 - z,1,0;y)
+ 6 \G(0,1,0;y)
\Big]
+ \frac{1}{8(y+z)} \Big[
- \frac{7\pi^2}{3} \H(1;z)
+ \frac{7\pi^2}{3} \G(1 - z;y)
+ 14 \H(0;z) \G(1 - z,0;y)
\nonumber \\ &&
+ 14 \H(0,1,0;z)
+ 25 \H(1,0;z)
+ 14 \H(1,0;z) \G(1 - z;y)
- 14 \H(1,0;z) \G(0;y)
- 14 \H(1,1,0;z)
\nonumber \\ &&
- 14 \G(1 - z,1,0;y)
- 14 \G(0,1,0;y)
+ 25 \G(1,0;y)
\Big]
+ \frac{1}{8(1-y-z)^2} \Big[
\frac{\pi^2}{3} \H(1;z)
+ \frac{\pi^2}{6} \G(0;y)
- 2\zeta_3
\nonumber \\ &&
+ \H(1,0;z) \G(0;y)
+ 2 \H(1,1,0;z)
+ \G(0,1,0;y)
\Big]
+ \frac{1}{24(1-y-z)} \Big[
\frac{13\pi^2}{6}
+ 10 \H(0;z) \G(0;y)
+ 13 \H(1,0;z)
\nonumber \\ &&
+ 10 \G(0;y)
- 7 \G(1,0;y)
\Big]
+ \frac{\pi^2}{288} \Big[
- \frac{946}{3}
- 5 \H(0;z)
+ 12 \H(0;z) \G(1 - z;y)
+ 36 \H(0;z) \G(0;y)
\nonumber \\ &&
- 12 \H(0;z) \G(1;y)
+ 24 \H(0,1;z)
+ 24 \H(1;z) \G(1 - z;y)
- 24 \H(1;z) \G( - z;y)
- 12 \H(1;z) \G(1;y)
\nonumber \\ &&
+ 24 \H(1,0;z)
+ 12 \H(1,1;z)
- 44 \G(1 - z;y)
+ 12 \G(1 - z,0;y)
- 24 \G(1 - z,1;y)
+ 24 \G( - z,1 - z;y)
\nonumber \\ &&
- 24 \G(0,1 - z;y)
+ 49 \G(0;y)
- 24 \G(0,1;y)
+ 12 \G(1,1 - z;y)
+ 44 \G(1;y)
- 36 \G(1,0;y)
+ 24 \G(1,1;y)
\Big]
\nonumber \\ &&
+ \frac{11\pi^4}{360}
+ \frac{\zeta_3}{72} \Big[
317
- 18 \H(0;z)
+ 90 \H(1;z)
- 72 \G(1 - z;y)
- 18 \G(0;y)
- 18 \G(1;y)
\Big]
+ \frac{1}{144} \Big[
- \frac{79987}{72}
\nonumber \\ &&
+ \frac{1735}{6} \H(0;z)
- 132 \H(0;z) \G(1 - z,0;y)
- 36 \H(0;z) \G(1 - z,1,0;y)
+ 72 \H(0;z) \G( - z,1 - z,0;y)
\nonumber \\ &&
- 72 \H(0;z) \G(0,1 - z,0;y)
- 150 \H(0;z) \G(0;y)
+ 252 \H(0;z) \G(0,0;y)
- 36 \H(0;z) \G(0,1,0;y)
\nonumber \\ &&
+ 36 \H(0;z) \G(1,1 - z,0;y)
- 6 \H(0;z) \G(1,0;y)
- 72 \H(0;z) \G(1,0,0;y)
+ 23 \H(0,0;z)
\nonumber \\ &&
+ 144 \H(0,0;z) \G(0;y)
+ 72 \H(0,0;z) \G(0,0;y)
+ 144 \H(0,0,1,0;z)
+ 6 \H(0,1,0;z)
\nonumber \\ &&
- 36 \H(0,1,0;z) \G(1 - z;y)
+ 72 \H(0,1,0;z) \G( - z;y)
+ 36 \H(0,1,0;z) \G(0;y)
- 36 \H(0,1,0;z) \G(1;y)
\nonumber \\ &&
+ 72 \H(0,1,1,0;z)
- 160 \H(1,0;z)
- 132 \H(1,0;z) \G(1 - z;y)
+ 36 \H(1,0;z) \G(1 - z,0;y)
\nonumber \\ &&
+ 72 \H(1,0;z) \G( - z,1 - z;y)
- 72 \H(1,0;z) \G( - z,0;y)
- 72 \H(1,0;z) \G(0,1 - z;y)
+ 192 \H(1,0;z) \G(0;y)
\nonumber \\ &&
+ 36 \H(1,0;z) \G(1,1 - z;y)
- 36 \H(1,0;z) \G(1,0;y)
+ 144 \H(1,0,0;z)
+ 72 \H(1,0,0;z) \G(0;y)
\nonumber \\ &&
+ 144 \H(1,0,1,0;z)
+ 72 \H(1,1,0;z) \G(1 - z;y)
- 72 \H(1,1,0;z) \G( - z;y)
- 36 \H(1,1,0;z) \G(1;y)
\nonumber \\ &&
+ 72 \H(1,1,0,0;z)
+ 36 \H(1,1,1,0;z)
+ 36 \G(1 - z,0,1,0;y)
+ 132 \G(1 - z,1,0;y)
+ 72 \G(1 - z,1,1,0;y)
\nonumber \\ &&
- 72 \G( - z,1 - z,1,0;y)
- 72 \G( - z,0,1,0;y)
+ 72 \G(0,1 - z,1,0;y)
+ \frac{8}{3} \G(0;y)
+ 320 \G(0,0;y)
\nonumber \\ &&
- 72 \G(0,0,1,0;y)
- 60 \G(0,1,0;y)
+ 72 \G(0,1,1,0;y)
- 36 \G(1,1 - z,1,0;y)
+ 160 \G(1,0;y)
\nonumber \\ &&
- 252 \G(1,0,0;y)
+ 108 \G(1,0,1,0;y)
- 132 \G(1,1,0;y)
+ 72 \G(1,1,0,0;y)
- 72 \G(1,1,1,0;y)
\Big]
\nonumber \\ &&
+ i\pi \Bigg\{
\frac{11}{16(1-z)} \H(0;z)
+ 2\zeta_3
+ \frac{1}{48} \Big[
- \frac{44\pi^2}{3}
- \frac{1751}{18}
- 11 \H(0;z)
- 66 \H(0;z) \G(0;y)
- 66 \H(1,0;z)
\nonumber \\ && \hspace{9mm}
- 110 \G(0;y)
+ 66 \G(1,0;y)
\Big]\Bigg\}
\nonumber \\
\lefteqn{D_{\beta}(y,z) =} \nonumber \\ &&
\frac{z}{4(y+z)^2} \Big[
- \frac{4\pi^2}{3} \H(1;z)
+ \frac{4\pi^2}{3} \G(1 - z;y)
+ 8 \H(0;z) \G(1 - z,0;y)
+ 8 \H(0,1,0;z)
+ 9 \H(1,0;z)
\nonumber \\ &&
+ 8 \H(1,0;z) \G(1 - z;y)
- 8 \H(1,0;z) \G(0;y)
- 8 \H(1,1,0;z)
- 8 \G(1 - z,1,0;y)
- 8 \G(0,1,0;y)
\nonumber \\ &&
+ 9 \G(1,0;y)
\Big]
+ \frac{z}{4(y+z)} \Big[
\frac{4\pi^2}{3}
- 1
- 4 \H(0;z)
+ 8 \H(0;z) \G(0;y)
+ 8 \H(1,0;z)
+ 5 \G(0;y)
- 8 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z}{6(1-y-z)^2} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
+ \frac{z}{12(1-y-z)} \Big[
\frac{7\pi^2}{3}
- 2 \H(0;z)
\nonumber \\ &&
+ 14 \H(0;z) \G(0;y)
+ 14 \H(1,0;z)
+ 5 \G(0;y)
- 14 \G(1,0;y)
\Big]
+ \frac{z^2}{2(y+z)^3} \Big[
\pi^2 \H(1;z)
- \pi^2 \G(1 - z;y)
\nonumber \\ &&
- 6 \H(0;z) \G(1 - z,0;y)
- 6 \H(0,1,0;z)
- 3 \H(1,0;z)
- 6 \H(1,0;z) \G(1 - z;y)
+ 6 \H(1,0;z) \G(0;y)
\nonumber \\ &&
+ 6 \H(1,1,0;z)
+ 6 \G(1 - z,1,0;y)
+ 6 \G(0,1,0;y)
- 3 \G(1,0;y)
\Big]
+ \frac{z^2}{4(y+z)^2} \Big[
- 2\pi^2
+ 3 \H(0;z)
\nonumber \\ &&
- 12 \H(0;z) \G(0;y)
- 12 \H(1,0;z)
- 3 \G(0;y)
+ 12 \G(1,0;y)
\Big]
+ \frac{z^2}{4(y+z)} \Big[
- \pi^2
+ \H(0;z)
\nonumber \\ &&
- 6 \H(0;z) \G(0;y)
- 6 \H(1,0;z)
- \G(0;y)
+ 6 \G(1,0;y)
\Big]
+ \frac{z^2}{12(1-y-z)^2} \Big[
\frac{5\pi^2}{6}
+ 5 \H(0;z) \G(0;y)
\nonumber \\ &&
+ 5 \H(1,0;z)
- 5 \G(1,0;y)
\Big]
+ \frac{z^2}{4(1-y-z)} \Big[
- \pi^2
+ \H(0;z)
- 6 \H(0;z) \G(0;y)
- 6 \H(1,0;z)
- \G(0;y)
\nonumber \\ &&
+ 6 \G(1,0;y)
\Big]
+ \frac{z^3}{2(y+z)^4} \Big[
- \frac{\pi^2}{2} \H(1;z)
+ \frac{\pi^2}{2} \G(1 - z;y)
+ 3 \H(0;z) \G(1 - z,0;y)
+ 3 \H(0,1,0;z)
\nonumber \\ &&
+ 3 \H(1,0;z) \G(1 - z;y)
- 3 \H(1,0;z) \G(0;y)
- 3 \H(1,1,0;z)
- 3 \G(1 - z,1,0;y)
- 3 \G(0,1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{2(y+z)^3} \Big[
\frac{\pi^2}{2}
+ 3 \H(0;z) \G(0;y)
+ 3 \H(1,0;z)
- 3 \G(1,0;y)
\Big]
+ \frac{z^3}{4(y+z)^2} \Big[
\frac{\pi^2}{2}
+ 3 \H(0;z) \G(0;y)
\nonumber \\ &&
+ 3 \H(1,0;z)
- 3 \G(1,0;y)
\Big]
+ \frac{z^3}{2(y+z)} \Big[
\frac{\pi^2}{6}
+ \H(0;z) \G(0;y)
+ \H(1,0;z)
- \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{4(1-y-z)^2} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
+ \frac{z^3}{2(1-y-z)} \Big[
\frac{\pi^2}{6}
+ \H(0;z) \G(0;y)
\nonumber \\ &&
+ \H(1,0;z)
- \G(1,0;y)
\Big]
- \frac{1}{12(1-y)} \G(0;y)
+ \frac{1}{72(1-z)} \Big[
- \pi^2
- 25 \H(0;z)
+ \frac{3}{2} \H(0;z) \G(0;y)
\nonumber \\ &&
+ 9 \H(0,0;z)
- 6 \H(1,0;z)
\Big]
+ \frac{1}{2(y+z)} \Big[
\frac{\pi^2}{6} \H(1;z)
- \frac{\pi^2}{6} \G(1 - z;y)
- \H(0;z) \G(1 - z,0;y)
- \H(0,1,0;z)
\nonumber \\ &&
- 2 \H(1,0;z)
- \H(1,0;z) \G(1 - z;y)
+ \H(1,0;z) \G(0;y)
+ \H(1,1,0;z)
+ \G(1 - z,1,0;y)
+ \G(0,1,0;y)
\nonumber \\ &&
- 2 \G(1,0;y)
\Big]
+ \frac{1}{6(1-y-z)} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
- \G(0;y)
+ \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{\pi^2}{72} \Big[
\frac{395}{12}
- \H(0;z)
+ 2 \G(1 - z;y)
- \G(0;y)
- 2 \G(1;y)
\Big]
- \frac{19}{36}\zeta_3
+ \frac{1}{144} \Big[
\frac{2977}{18}
- 10 \H(0;z)
\nonumber \\ &&
+ 24 \H(0;z) \G(1 - z,0;y)
+ 29 \H(0;z) \G(0;y)
- 36 \H(0;z) \G(0,0;y)
+ 6 \H(0;z) \G(1,0;y)
- 28 \H(0,0;z)
\nonumber \\ &&
- 36 \H(0,0;z) \G(0;y)
- 6 \H(0,1,0;z)
+ 40 \H(1,0;z)
+ 24 \H(1,0;z) \G(1 - z;y)
- 30 \H(1,0;z) \G(0;y)
\nonumber \\ &&
- 36 \H(1,0,0;z)
- 24 \G(1 - z,1,0;y)
+ 56 \G(0;y)
- 82 \G(0,0;y)
+ 6 \G(0,1,0;y)
- 40 \G(1,0;y)
\nonumber \\ &&
+ 36 \G(1,0,0;y)
+ 24 \G(1,1,0;y)
\Big]
\nonumber \\ &&
+ i\pi \Bigg\{
- \frac{1}{8(1-z)} \H(0;z)
+ \frac{1}{48} \Big[
\frac{8\pi^2}{3}
- \frac{46}{3}
+ 13 \H(0;z)
+ 12 \H(0;z) \G(0,y)
+ 12 \H(1,0;z)
+ 31 \G(0;y)
\nonumber \\ && \hspace{9mm}
- 12 \G(1,0;y)
\Big]\Bigg\}
\nonumber \\
\lefteqn{A_{\gamma}(y,z) =} \nonumber \\ &&
+ \frac{1}{48y(1-z)} \Big[
- \pi^2
+ 13 \H(0;z)
- 6 \H(1,0;z)
- 6 \G(1,0;y)
\Big]
+ \frac{1}{48y(1-y-z)} \Big[
\pi^2
- 13 \H(0;z)
\nonumber \\ &&
+ 6 \H(1,0;z)
+ 6 \G(1,0;y)
\Big]
+ \frac{z}{16(y+z)^2} \Big[
- \frac{11\pi^2}{3} \H(1;z)
+ \frac{11\pi^2}{3} \G(1 - z;y)
+ 22 \H(0;z) \G(1 - z,0;y)
\nonumber \\ &&
+ 22 \H(0,1,0;z)
+ 55 \H(1,0;z)
+ 22 \H(1,0;z) \G(1 - z;y)
- 22 \H(1,0;z) \G(0;y)
- 22 \H(1,1,0;z)
\nonumber \\ &&
- 22 \G(1 - z,1,0;y)
- 22 \G(0,1,0;y)
+ 55 \G(1,0;y)
\Big]
+ \frac{z}{16(y+z)} \Big[
\frac{11\pi^2}{3}
- 11
- 22 \H(0;z)
\nonumber \\ &&
+ 22 \H(0;z) \G(0;y)
+ 22 \H(1,0;z)
+ 33 \G(0;y)
- 22 \G(1,0;y)
\Big]
+ \frac{z}{8(1-y-z)^2} \Big[
\frac{\pi^2}{3} \H(1;z)
+ \frac{\pi^2}{6} \G(0;y)
\nonumber \\ &&
- 2\zeta_3
+ \H(1,0;z) \G(0;y)
+ 2 \H(1,1,0;z)
+ \G(0,1,0;y)
\Big]
+ \frac{z}{48(1-y-z)} \Big[
\frac{10\pi^2}{3}
+ 13 \H(0;z)
\nonumber \\ &&
+ 20 \H(0;z) \G(0;y)
+ 20 \H(1,0;z)
+ 20 \G(0;y)
- 20 \G(1,0;y)
\Big]
+ \frac{z^2}{8(y+z)^3} \Big[
\frac{22\pi^2}{3} \H(1;z)
\nonumber \\ &&
- \frac{22\pi^2}{3} \G(1 - z;y)
- 44 \H(0;z) \G(1 - z,0;y)
- 44 \H(0,1,0;z)
- 33 \H(1,0;z)
- 44 \H(1,0;z) \G(1 - z;y)
\nonumber \\ &&
+ 44 \H(1,0;z) \G(0;y)
+ 44 \H(1,1,0;z)
+ 44 \G(1 - z,1,0;y)
+ 44 \G(0,1,0;y)
- 33 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{16(y+z)^2} \Big[
- \frac{44\pi^2}{3}
+ 33 \H(0;z)
- 88 \H(0;z) \G(0;y)
- 88 \H(1,0;z)
- 33 \G(0;y)
+ 88 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{16(y+z)} \Big[
- \frac{22\pi^2}{3}
+ 11 \H(0;z)
- 44 \H(0;z) \G(0;y)
- 44 \H(1,0;z)
- 11 \G(0;y)
+ 44 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{12(1-y-z)^2} \Big[
\frac{5\pi^2}{6}
+ 5 \H(0;z) \G(0;y)
+ 5 \H(1,0;z)
- 5 \G(1,0;y)
\Big]
+ \frac{z^2}{16(1-y-z)} \Big[
- \frac{22\pi^2}{3}
\nonumber \\ &&
+ 11 \H(0;z)
- 44 \H(0;z) \G(0;y)
- 44 \H(1,0;z)
- 11 \G(0;y)
+ 44 \G(1,0;y)
\Big]
+ \frac{z^3}{8(y+z)^4} \Big[
- \frac{11\pi^2}{2}\H(1;z)
\nonumber \\ &&
+ \frac{11\pi^2}{2} \G(1 - z;y)
+ 33 \H(0;z) \G(1 - z,0;y)
+ 33 \H(0,1,0;z)
+ 33 \H(1,0;z) \G(1 - z;y)
\nonumber \\ &&
- 33 \H(1,0;z) \G(0;y)
- 33 \H(1,1,0;z)
- 33 \G(1 - z,1,0;y)
- 33 \G(0,1,0;y)
\Big]
+ \frac{z^3}{8(y+z)^3} \Big[
\frac{11\pi^2}{2}
\nonumber \\ &&
+ 33 \H(0;z) \G(0;y)
+ 33 \H(1,0;z)
- 33 \G(1,0;y)
\Big]
+ \frac{z^3}{16(y+z)^2} \Big[
\frac{11\pi^2}{2}
+ 33 \H(0;z) \G(0;y)
+ 33 \H(1,0;z)
\nonumber \\ &&
- 33 \G(1,0;y)
\Big]
+ \frac{z^3}{8(y+z)} \Big[
\frac{11\pi^2}{6}
+ 11 \H(0;z) \G(0;y)
+ 11 \H(1,0;z)
- 11 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{16(1-y-z)^2} \Big[
- \frac{11\pi^2}{6}
- 11 \H(0;z) \G(0;y)
- 11 \H(1,0;z)
+ 11 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{8(1-y-z)} \Big[
\frac{11\pi^2}{6}
+ 11 \H(0;z) \G(0;y)
+ 11 \H(1,0;z)
- 11 \G(1,0;y)
\Big]
+ \frac{1}{48(1-z)^2} \Big[
\pi^2 \big( \frac{1}{6}
- 3 \H(0;z)
\nonumber \\ &&
- 3 \H(1;z)
+ \G(1 - z;y)
- \G(0;y) \big)
- 6\zeta_3
+ \frac{355}{6} \H(0;z)
+ 6 \H(0;z) \G(1 - z,0;y)
- 10 \H(0;z) \G(0;y)
\nonumber \\ &&
- 45 \H(0,0;z)
- 12 \H(0,0;z) \G(0;y)
- 18 \H(0,1,0;z)
+ \H(1,0;z)
+ 6 \H(1,0;z) \G(1 - z;y)
\nonumber \\ &&
- 6 \H(1,0;z) \G(0;y)
- 12 \H(1,0,0;z)
- 18 \H(1,1,0;z)
- 6 \G(1 - z,1,0;y)
+ 6 \G(0,1,0;y)
\Big]
\nonumber \\ &&
+ \frac{1}{48(1-z)} \Big[
\pi^2 \big( - \frac{7}{6}
+ 3 \H(0;z)
+ 3 \H(1;z)
- \G(1 - z;y)
+ \G(0;y) \big)
+ 6\zeta_3
+ \frac{277}{6}
- \frac{493}{6} \H(0;z)
\nonumber \\ &&
- 6 \H(0;z) \G(1 - z,0;y)
+ 4 \H(0;z) \G(0;y)
+ 45 \H(0,0;z)
+ 12 \H(0,0;z) \G(0;y)
+ 18 \H(0,1,0;z)
\nonumber \\ &&
- 7 \H(1,0;z)
- 6 \H(1,0;z) \G(1 - z;y)
+ 6 \H(1,0;z) \G(0;y)
+ 12 \H(1,0,0;z)
+ 18 \H(1,1,0;z)
\nonumber \\ &&
+ 6 \G(1 - z,1,0;y)
- 10 \G(0;y)
- 6 \G(0,1,0;y)
+ 6 \G(1,0;y)
\Big]
+ \frac{1}{48(1-y-z)} \Big[
- \pi^2
+ 13 \H(0;z)
\nonumber \\ &&
- 6 \H(1,0;z)
- 6 \G(1,0;y)
\Big]
+ \frac{1}{48} \Big[
\pi^2
- \frac{277}{6}
+ 23 \H(0;z)
+ 6 \H(0;z) \G(0;y)
+ 6 \H(1,0;z)
+ 10 \G(0;y)
\nonumber \\ &&
- 6 \G(1,0;y)
\Big]
+ i\pi \Bigg\{
\frac{11}{16(1-z)^2}\H(0;z)
+ \frac{11}{16(1-z)} \Big[ 1 - \H(0;z) \Big]
- \frac{11}{16} \Bigg\}\nonumber\\
\lefteqn{D_{\gamma}(y;z) =} \nonumber \\&&
- \frac{1}{12y(1-z)}\H(0;z)
+ \frac{1}{12y(1-y-z)} \H(0;z)
+ \frac{z}{4(y+z)^2} \Big[
\frac{\pi^2}{3} \H(1;z)
- \frac{\pi^2}{3} \G(1 - z;y)
\nonumber \\ &&
- 2 \H(0;z) \G(1 - z,0;y)
- 2 \H(0,1,0;z)
- 5 \H(1,0;z)
- 2 \H(1,0;z) \G(1 - z;y)
+ 2 \H(1,0;z) \G(0;y)
\nonumber \\ &&
+ 2 \H(1,1,0;z)
+ 2 \G(1 - z,1,0;y)
+ 2 \G(0,1,0;y)
- 5 \G(1,0;y)
\Big]
+ \frac{z}{4(y+z)} \Big[
- \frac{\pi^2}{3}
+ 1
+ 2 \H(0;z)
\nonumber \\ &&
- 2 \H(0;z) \G(0;y)
- 2 \H(1,0;z)
- 3 \G(0;y)
+ 2 \G(1,0;y)
\Big]
+ \frac{z}{12(1-y-z)} \Big[
- \frac{\pi^2}{3}
- \H(0;z)
\nonumber \\ &&
- 2 \H(0;z) \G(0;y)
- 2 \H(1,0;z)
- 2 \G(0;y)
+ 2 \G(1,0;y)
\Big]
+ \frac{z^2}{2(y+z)^3} \Big[
- \frac{2\pi^2}{3} \H(1;z)
+ \frac{2\pi^2}{3} \G(1 - z;y)
\nonumber \\ &&
+ 4 \H(0;z) \G(1 - z,0;y)
+ 4 \H(0,1,0;z)
+ 3 \H(1,0;z)
+ 4 \H(1,0;z) \G(1 - z;y)
- 4 \H(1,0;z) \G(0;y)
\nonumber \\ &&
- 4 \H(1,1,0;z)
- 4 \G(1 - z,1,0;y)
- 4 \G(0,1,0;y)
+ 3 \G(1,0;y)
\Big]
+ \frac{z^2}{4(y+z)^2} \Big[
\frac{4\pi^2}{3}
- 3 \H(0;z)
\nonumber \\ &&
+ 8 \H(0;z) \G(0;y)
+ 8 \H(1,0;z)
+ 3 \G(0;y)
- 8 \G(1,0;y)
\Big]
+ \frac{z^2}{4(y+z)} \Big[
\frac{2\pi^2}{3}
- \H(0;z)
+ 4 \H(0;z) \G(0;y)
\nonumber \\ &&
+ 4 \H(1,0;z)
+ \G(0;y)
- 4 \G(1,0;y)
\Big]
+ \frac{z^2}{6(1-y-z)^2} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^2}{4(1-y-z)} \Big[
\frac{2\pi^2}{3}
- \H(0;z)
+ 4 \H(0;z) \G(0;y)
+ 4 \H(1,0;z)
+ \G(0;y)
- 4 \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{z^3}{2(y+z)^4} \Big[
\frac{\pi^2}{2} \H(1;z)
- \frac{\pi^2}{2} \G(1 - z;y)
- 3 \H(0;z) \G(1 - z,0;y)
- 3 \H(0,1,0;z)
- 3 \H(1,0;z) \G(1 - z;y)
\nonumber \\ &&
+ 3 \H(1,0;z) \G(0;y)
+ 3 \H(1,1,0;z)
+ 3 \G(1 - z,1,0;y)
+ 3 \G(0,1,0;y)
\Big]
+ \frac{z^3}{2(y+z)^3} \Big[
- \frac{\pi^2}{2}
\nonumber \\ &&
- 3 \H(0;z) \G(0;y)
- 3 \H(1,0;z)
+ 3 \G(1,0;y)
\Big]
+ \frac{z^3}{4(y+z)^2} \Big[
- \frac{\pi^2}{2}
- 3 \H(0;z) \G(0;y)
- 3 \H(1,0;z)
\nonumber \\ &&
+ 3 \G(1,0;y)
\Big]
+ \frac{z^3}{2(y+z)} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
+ \frac{z^3}{4(1-y-z)^2} \Big[
\frac{\pi^2}{6}
\nonumber \\ &&
+ \H(0;z) \G(0;y)
+ \H(1,0;z)
- \G(1,0;y)
\Big]
+ \frac{z^3}{2(1-y-z)} \Big[
- \frac{\pi^2}{6}
- \H(0;z) \G(0;y)
- \H(1,0;z)
+ \G(1,0;y)
\Big]
\nonumber \\ &&
+ \frac{1}{72(1-z)^2} \Big[
- \pi^2
- 25 \H(0;z)
+ \frac{3}{2} \H(0;z) \G(0;y)
+ 9 \H(0,0;z)
- 6 \H(1,0;z)
\Big]
+ \frac{1}{144(1-z)} \Big[
2\pi^2
- 38
\nonumber \\ &&
+ 65 \H(0;z)
- 3 \H(0;z) \G(0;y)
- 18 \H(0,0;z)
+ 12 \H(1,0;z)
+ 3 \G(0;y)
\Big]
- \frac{1}{12(1-y-z)} \H(0;z)
+ \frac{19}{72}
\nonumber \\ &&
- \frac{5}{48} \H(0;z)
- \frac{1}{48} \G(0;y)
+ i\pi \Bigg\{
- \frac{1}{8(1-z)^2}\H(0;z)
+ \frac{1}{8(1-z)} \Big[ -1 + \H(0;z) \Big]
+ \frac{1}{8} \Bigg\}\end{aligned}$$
From the $\Omega^{(1),{\rm finite}}$ and $\Omega^{(2),{\rm finite}}$, it is possible to recover the finite pieces of the helicity-averaged tree times two-loop and one-loop squared matrix elements by squaring (\[eq:helamp\]): $$\begin{aligned}
{\cal F}inite^{(2\times 0)} (x,y,z) &=& 8\, V\;
{\cal R}\Bigg[ \frac{(1-y)(1-y-z)}{yz}
\alpha^{(2),{\rm finite}}(y,z)
+ \frac{1-y}{y}\beta^{(2),{\rm finite}}(y,z) \nonumber \\
& & \hspace{1.3cm}
-\gamma^{(2),{\rm finite}}(y,z)
+ (y\leftrightarrow z)
\Bigg]\; , \nonumber \\
{\cal F}inite^{(1\times 1)} (x,y,z) &=& 4\, V\;
{\cal R}\Bigg[ (1-y-z)\left(\frac{(1-y-z)}{yz} + \frac{1}{2}\right)
\,
|\alpha^{(1),{\rm finite}}(y,z)|^2 \nonumber \\
&& \hspace{1cm} + \left( \frac{1-y-z}{2} + \frac{z}{y} \right)\,
|\beta^{(1),{\rm finite}}(y,z)|^2
+ \left( \frac{1-y-z}{2} + \frac{y}{z} \right)\,
|\gamma^{(1),{\rm finite}}(y,z)|^2 \nonumber \\
&& \hspace{1cm} +
\left(
- 3+y+z
+ \frac{2-2z}{y}
\right)\;
\alpha^{(1),{\rm finite}}(y,z)\beta^{*(1),{\rm finite}}(y,z)
\nonumber \\
&& \hspace{1cm} - (1-y-z)\;
\alpha^{(1),{\rm finite}}(y,z)\gamma^{*(1),{\rm finite}} (y,z)
\nonumber \\
&& \hspace{1cm}
- (1+y+z)\; \beta^{(1),{\rm finite}}(y,z)\gamma^{*(1),{\rm finite}}(y,z)
+ (y\leftrightarrow z) \Bigg]\;.
\label{eq:recover}\end{aligned}$$
It is important to notice that (\[eq:recover\]) corresponds, by the very nature of the Weyl–van der Waerden helicity formalism, to a scheme with external momenta and polarization vectors in four dimensions (internal states are always taken to be $\dd$-dimensional), which is sometimes called the ’t Hooft–Veltman scheme [@hv]. This scheme is different from the conventional dimensional regularization used in [@3jme], where all external momenta and polarization vectors are $\dd$-dimensional. Nevertheless, one obtains from (\[eq:recover\]) the same ${\cal F}inite^{(2\times 0)} (x,y,z)$ as in Eq. (4.17) and ${\cal F}inite^{(1\times 1)} (x,y,z)$ as in Eq. (4.25) of [@3jme], since all scheme-dependent terms are correctly accounted for by the finite contributions arising from expanding the tree level and one-loop contributions in the renormalization and infrared factorization formulae.
It should also be noted that only the ${\cal O}(\e^0)$ terms of $\Omega^{(1),{\rm finite}}$ contribute to ${\cal F}inite^{(1\times 1)} (x,y,z)$, terms subleading in $\e$ are not required, since no term is multiplied with a divergent factor. Comparing the size of these ${\cal O}(\e^0)$ terms (\[eq:omega1\]) with the size of ${\cal F}inite^{(1\times 1)} (x,y,z)$ in [@3jme], it becomes clear that the squared one-loop amplitude can be evaluated much more elegantly by squaring the finite remainders of the helicity amplitudes than by computing the squared matrix element.
Conclusions and Outlook {#sec:conc}
=======================
In this paper, we have presented analytic formulae for the one- and two-loop virtual helicity amplitudes to the process $\gamma^*\to q\bar q g$. These amplitudes have been derived by defining projectors, which isolate the coefficients of the most general tensorial structure of the matrix element at any order in perturbation theory. Once the general tensor is known, the helicity amplitudes follow in a straightforward manner – they are linear combinations of the tensor coefficients. We applied the projectors directly to the Feynman diagrams and used the conventional approach of relating the ensuing tensor integrals to a basis set of master integrals. This latter step is identical to that employed to evaluate the interference of tree- and two-loop graphs in Ref. [@3jme], apart from the fact that the projector is no longer the tree-level amplitude. As anticipated, the finite remainder from the interference of tree- and two-loop amplitudes can be reconstructed from the appropriate helicity amplitudes, with the difference between treating the external states in $d$ dimensions or four dimensions being isolated in the infrared-singular terms.
The results presented here therefore complement the earlier calculation of the interference of tree- and two-loop graphs in Ref. [@3jme]. Knowledge of the helicity amplitudes allows additional information on the scattering process. In particular, observables that require knowledge of the polarization tensor of the virtual photon, such as oriented event shapes in unpolarized $e^+e^-$ scattering or event shapes in polarized $e^+e^-$ scattering, can be described at two-loop order.
Similar results can in principle be obtained for $(2+1)$-jet production in deep inelastic $ep$ scattering or $(V+1)$-jet production in hadron–hadron collisions. However, the rather different domains of convergence of the HPLs and 2dHPLs makes this a non-trivial task, which is discussed in a separate paper [@ancont]. Nevertheless, the helicity approach will provide information on the direction of the decay leptons in $(V+1)$-jet production (with or without polarized protons). Determination of the polarized parton distribution functions in polarized electron–proton scattering will also benefit from the knowledge of the two-loop helicity amplitudes in the appropriate kinematic region.
Even though the evaluation of two-loop QCD matrix elements is becoming well established, the virtual corrections form only part of a full NNLO calculation. They must be combined with the one-loop corrections to $\gamma^*\to 4$ partons [@onel4p1; @onel4p2; @onel4p3; @onel4p4], where one of the partons becomes collinear or soft, as well as tree-level processes $\gamma^*\to 5$ partons [@tree5p1; @tree5p2; @tree5p3] with two soft or collinear partons in a way that allows all of the infrared singularities to cancel one another. This task is far from trivial, even though the factorization properties of both the one-loop, one-unresolved-parton contribution [@onel1; @onel2; @onel3; @onel4; @onel5; @onel6] and the tree-level, two-unresolved-parton contributions [@twot1; @twot2; @twot3; @twot4] have been studied. Early studies for the case of photon-plus-one-jet final states in electron–positron annihilation in [@ggamma0; @ggamma], which involves both double radiation and single radiation from one-loop graphs, indicate the feasibility of developing a numerical NNLO program implementing the experimental definition of jet observables and event-shape variables, and significant progress is anticipated in the near future.
[**Note added:**]{} After this paper was first released, part of its results were confirmed in an independent calculation using the methods described in [@muw; @w2]. In hep-ph/0207043, Moch, Uwer and Weinzierl obtain results for the full one-loop amplitude (\[eq:oneloopamp\]) and for the contributions to the two-loop amplitude (\[eq:twoloopamp\]) which are proportional to $N_F$ (i.e. the terms $D_\Omega$ and $E_\Omega$), all in agreement with the results presented here.
Acknowledgements {#acknowledgements .unnumbered}
================
EWNG thanks Adrian Signer for useful discussions. This work was supported in part by the EU Fourth Framework Programme “Training and Mobility of Researchers”, network “Quantum Chromodynamics and the Deep Structure of Elementary Particles”, contract FMRX-CT98-0194 (DG 12-MIHT).
Weyl–van der Waerden spinor calculus
====================================
\[sec:appa\]
The basic quantity is the two-spinor $\psi_{A}$ or $\psi^A$ and its complex conjugate $\psi_{\dot{A}}$ or $\psi^{\dot{A}}$. Raising and lowering of indices is done with the antisymmetric tensor $\varepsilon$, $$\begin{aligned}
\varepsilon_{AB} = \varepsilon^{AB} = \varepsilon_{\dot{A}\dot{B}} =
\varepsilon^{\dot{A}\dot{B}} =
\left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array}\right) .\end{aligned}$$ We define a antisymmetric spinorial “inner product”: $$\langle\psi_{1}\psi_{2}\rangle = \psi_{1A}\varepsilon^{BA}\psi_{2B}
= \psi_{1A}\psi^{A}_{2} = -\psi_1^A\psi_{2A}= -\langle\psi_{2}\psi_{1}\rangle,$$ and $$\langle\psi_{1}\psi_{2}\rangle^* = \psi_{1\dot{A}}\psi^{\dot{A}}_{2} .$$ Any momentum vector $k_{\mu}$ gets a bispinor representation by contraction with $\sigma^{\mu}$: $$k_{\dot A B}=\sigma^\mu_{\dot A B}k_\mu=
\left(\begin{array}{cc}
k_0+k_3 & k_1+ik_2 \\ k_1-ik_2 & k_0-k_3
\end{array}\right),$$ where $\sigma^{0}$ is the unit matrix and $\sigma_{i}$ are the Pauli matrices. Since $$\sigma^{\mu}_{\dot{A}B}\sigma^{\nu\dot{A}B} = 2g^{\mu\nu},$$ we have $$k_{\dot{A}B}p^{\dot{A}B} = 2k\cdot p .$$ For light-like vectors one can show that $$k_{\dot{A}B} = k_{\dot{A}}k_{B},$$ where $$k_A =\left(\begin{array}{c}
(k_1-ik_2)/\sqrt{k_0-k_3} \\ \sqrt{k_0-k_3}
\end{array}\right),$$ so that for light-like vectors we have $$2k\cdot p =\langle kp\rangle\langle kp\rangle^*=\left|\langle kp \rangle\right|^2 .$$ The following relation is often useful: $$\sigma^\mu_{\dot A B}\sigma_\mu^{\dot C D}=2{\delta_{\dot A}}^{\dot C}
{\delta_B}^D\ .$$
For massless spin-$\frac{1}{2}$ particles the four-spinors can be expressed in two-spinors as follows: $$\begin{aligned}
u_+(p)&=&v_-(p)=
\left(\begin{array}{c}p_B\\0\end{array}\right) ,\nonumber \\
u_-(p)&=&v_+(p)=
\left(\begin{array}{c}0\\p^{\dot B}\end{array}\right) ,\nonumber \\
\bar{u}_+(q)&=&\bar{v}_-(q)=
\left(\begin{array}{cc}0,&-iq_{\dot A}\end{array}\right) ,\nonumber \\
\bar{u}_-(q)&=&\bar{v}_+(q)=
\left(\begin{array}{cc}iq^A,&0\end{array}\right) .\end{aligned}$$ The Dirac $\gamma$ matrices now become $$\gamma^\mu=\left(\begin{array}{cc}0&-i\sigma^\mu_{\dot B A} \\
i\sigma^{\mu\dot A B}&0\end{array}\right) ,$$ so that, for example: $$\bar{u}_{+}(q)\gamma^{\mu}v_{-}(p) = q_{\dot{A}}\sigma^{\mu\dot{A}B}p_{B}~.$$
The general electroweak vertex for vector boson $V$ coupling to two fermions is denoted by $ie\delta_{ij}\Gamma_\mu^{Vf_1f_2}$, where $i$ and $j$ are the colour labels associated with the fermions $f_1$ and $f_2$ respectively. The vertex contains left- and right-handed couplings, $$\Gamma_\mu^{V,f_1f_2} =
L^V_{f_1f_2}\gamma_\mu\left(\frac{1-\gamma_5}{2}\right)
+R^V_{f_1f_2}\gamma_\mu\left(\frac{1+\gamma_5}{2}\right),$$ where for a photon, $$L^\gamma_{f_1f_2}
=R^\gamma_{f_1f_2}
=-e_{f_1} \delta_{f_1f_2},$$ and for a $Z$-boson, $$L^Z_{f_1f_2}=\frac{I_3^{f_1}-\sin^2\theta_We_{f_1}}
{\sin\theta_W\cos\theta_W}\delta_{f_1f_2},~~~~~~
R^Z_{f_1f_2}
=\frac{-\sin\theta_We_{f_1}}
{\cos\theta_W}\delta_{f_1f_2}.$$ Here, $e_f$ represents the fractional electric charge, $I_3^f$ the weak isospin and $\theta_W$ the weak mixing angle. In the Weyl–van der Waerden notation, the vertex $\Gamma_\mu^{V,f_1f_2}$ becomes, $$\Gamma_\mu^{V,f_1f_2}=
\left(\begin{array}{cc}0&-iL^V_{f_1f_2}\sigma_{\mu\dot B A} \\
iR^V_{f_1f_2}\sigma_\mu^{\dot A B}&0\end{array}\right) .$$
For the polarization vectors of outgoing gluons and photons we use the spinorial quantities $$\begin{aligned}
e^{+}_{\dot{A}B}(k) & = & \sqrt{2} \frac{k_{\dot{A}}b_{B}}{\langle bk\rangle}~, \\
e^{-}_{\dot{A}B}(k) & = & \sqrt{2} \frac{b_{\dot{A}}k_{B}}{\langle bk\rangle^{*}}~.\end{aligned}$$ The gauge spinor $b$ is arbitrary and can be chosen differently in each gauge-invariant expression. A suitable choice can often simplify the calculation.
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B.A. Kniehl and J.H. Kühn, Phys. Lett. [**B224**]{} (1989) 229. L.J. Dixon and A. Signer, Phys. Rev. Lett. [**78**]{} (1997) 811 \[arXiv:hep-ph/9609460\]; Phys. Rev. D [**56**]{} (1997) 4031 \[arXiv:hep-ph/9706285\].
T. Gehrmann and E. Remiddi, arXiv:hep-ph/0207020. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Nucl. Phys. [**B425**]{} (1994) 217 \[arXiv:hep-ph/9403226\]. D.A. Kosower, Nucl. Phys. [**B552**]{} (1999) 319 \[arXiv:hep-ph/9901201\]. D.A. Kosower and P. Uwer, Nucl. Phys. [**B563**]{} (1999) 477 \[arXiv:hep-ph/9903515\]. Z. Bern, V. Del Duca and C.R. Schmidt, Phys. Lett. [**B445**]{} (1998) 168 \[arXiv:hep-ph/9810409\]. Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, Phys. Rev. (1999) 116001 \[arXiv:hep-ph/9903516\]. S. Catani and M. Grazzini, Nucl. Phys. [**B591**]{} (2000) 435 \[arXiv:hep-ph/0007142\]. J.M. Campbell and E.W.N. Glover, Nucl. Phys. [**B527**]{} (1998) 264 \[arXiv:hep-ph/9710255\]. S. Catani and M. Grazzini, Phys. Lett. [**B446**]{} (1999) 143 \[arXiv:hep-ph/9810389\]; Nucl. Phys. [**B570**]{} (2000) 287 \[arXiv:hep-ph/9908523\]. F.A. Berends and W.T. Giele, Nucl. Phys. [**B313**]{} (1989) 595. S. Catani, in [@kunszt].
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Phys. Lett. B [**414**]{} (1997) 354 \[arXiv:hep-ph/9705305\]. A. Gehrmann-De Ridder and E.W.N. Glover, Nucl. Phys. [**B517**]{} (1998) 269 \[arXiv:hep-ph/9707224\].
[^1]: Note that an alternative approach avoiding the need to use the integration-by-parts and Lorentz-invariance identities to reduce the integrals appearing in the Feynman diagrams to a basis set has recently been proposed[@muw; @w2]. This method relies on obtaining analytic expressions for the basic topologies with arbitrary powers of the propagators and arbitrary dimensions, which can often be found in terms of nested sums involving $\Gamma$-functions. The $\Gamma$-functions can be directly expanded in $\epsilon$ and the nested sums related to multiple polylogarithms.
[^2]: Note that the full matrix element for any process should be summed over both photon and $Z$-boson exchange.
[^3]: And for this reason was omitted in Ref. [@gg].
|
---
author:
- Andre Pawlowski
- Victor van der Veen
- Dennis Andriesse
- Erik van der Kouwe
- Thorsten Holz
- Cristiano Giuffrida
- Herbert Bos
bibliography:
- 'paper.bib'
title: |
VPS: Excavating High-Level C++ Constructs from\
Low-Level Binaries to Protect Dynamic Dispatching
---
<ccs2012> <concept> <concept\_id>10002978.10003022.10003465</concept\_id> <concept\_desc>Security and privacy Software reverse engineering</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the German Research Foundation (DFG) within the framework of the Excellence Strategy of the Federal Government and the States – EXC 2092 <span style="font-variant:small-caps;">CaSa</span> – 39078197, by the United States Office of Naval Research under contracts N00014-17-1-2782 and N00014-17-S-B010 “BinRec”, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 786669 (ReAct), No. 825377 (UNICORE), and No. 640110 (BASTION). Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of any of the sponsors or any of their affiliates.
|
---
abstract: 'Sonography techniques use multiple transducer elements for tissue visualization. Signals detected at each element are sampled prior to digital beamforming. The required sampling rates are up to 4 times the Nyquist rate of the signal and result in considerable amount of data, that needs to be stored and processed. A developed technique, based on the finite rate of innovation model, compressed sensing (CS) and Xampling ideas, allows to reduce the number of samples needed to reconstruct an image comprised of strong reflectors. A significant drawback of this method is its inability to treat speckle, which is of significant importance in medical imaging. Here we build on previous work and show explicitly how to perform beamforming in the Fourier domain. Beamforming in frequency exploits the low bandwidth of the beamformed signal and allows to bypass the oversampling dictated by digital implementation of beamforming in time. We show that this allows to obtain the same beamformed image as in standard beamforming but from far fewer samples. Finally, we present an analysis based CS-technique that allows for further reduction in sampling rate, using only a portion of the beamformed signal’s bandwidth, namely, sampling the signal at sub-Nyquist rates. We demonstrate our methods on in vivo cardiac ultrasound data and show that reductions up to 1/25 over standard beamforming rates are possible.'
address: |
Technion - Israel Institute of Technology\
Dept. of Electrical Engineering\
Haifa 32000, Israel
bibliography:
- 'general.bib'
title: FOURIER DOMAIN BEAMFORMING FOR MEDICAL ULTRASOUND
---
Array Processing, Beamforming, Compressed Sensing, Speckle, Ultrasound
Introduction {#sec:intro}
============
Diagnostic ultrasound has been used for decades to visualize body structures. The overall imaging process is described as follows: An energy pulse is transmitted along a narrow beam. During its propagation echoes are scattered by acoustic impedance perturbations in the tissue, and detected by the elements of the transducer. Collected data are sampled and digitally processed in a way referred to as beamforming, which results in signal-to-noise ratio (SNR) enhancement. Such a beamformed signal forms a line in the image.
According to the classic Shannon-Nyquist theorem [@shannon1949communication], the sampling rate at each transducer element should be at least twice the bandwidth of the detected signal. Rates up to 4 times the Nyquist rate are required in order to avoid artifacts caused by digital implementation of beamforming [@szabo2004diagnostic]. Taking into account the number of transducer elements and the number of lines in an image, the amount of sampled data that needs to be digitally processed is enormous, motivating methods to reduce sampling rates. Reduction of processing rate is possible within the classical sampling framework by exploiting the fact that the signal is modulated onto a carrier and occupies only a portion of its entire baseband bandwidth. Accordingly, modern systems digitally down-sample the data at the system’s front-end. However, this does not change the sampling rate since demodulation takes place in the digital domain. In addition, the processing rate may be reduced up to 1/4 of the sampling rate, but the signal becomes complex in this setup, and the number of samples effectively is only twice smaller.
A different approach to sampling rate reduction is introduced in [@tur2011innovation]. Tur et. al. regard the ultrasound signal detected by each receiver within the framework of finite rate of innovation (FRI) [@vetterli2002sampling], modeling it as $L$ replicas of a known pulse-shape, caused by scattering of the transmitted pulse from reflectors, located along the transmitted beam. Such an FRI signal is fully described by $2L$ parameters, corresponding to the replica’s delays and amplitudes. These parameters can be extracted from a small set of the signal’s Fourier series coefficients. A mechanism, referred to as Xampling, derived in [@gedalyahu2011multichannel], extracts such a set of coefficients from $4L$ real-valued samples. This work is continued in [@wagner2012compressed], where Wagner et. al. introduce a generalized scheme, referred to as compressed beamforming, which allows to compute the Fourier series coefficients of the beamformed signal from the low-rate samples of signals detected at each element. The problem of reconstruction of the beamformed signal from a small number of its Fourier series coefficients is solved via a compressed sensing (CS) technique, while assuming a small number $L$ of replicas. This approach allows to reconstruct an image comprised of macroscopic perturbations, but cannot treat the speckle, which is of significant importance in medical imaging.
In our work we extend the notion of compressed beamforming to beamforming in frequency and show explicitly how to perform it. Beamforming in frequency exploits the low bandwidth of the beamformed signal and allows to bypass the oversampling dictated by the digital implementation of beamforming in time. We reconstruct the beamformed signal perfectly with a simple inverse discrete Fourier transform (IDFT) from a small set of its discrete Fourier transform (DFT) coefficients, that are computed from low-rate samples of individual signals. We show that beamforming in frequency allows to preserve the integrity of an image with 7 fold reduction in the number of samples used for its reconstruction.
Finally, we introduce an analysis based CS-technique [@candes2011compressed] for signal reconstruction using only a portion of the beamformed signal’s bandwidth. We demonstrate our methods on in vivo cardiac ultrasound data and show that reductions of up to 1/25 of beamforming rate and up to 1/3 of Nyquist rate are possible. The proposed reconstruction method outperforms the classic synthesis CS approach used in [@wagner2012compressed], when the same number of samples is used.
The rest of the paper is organized as follows: in Section \[sec:beam in time\], we review beamforming in time. In Section \[sec:beam in freq\] we describe the principles of frequency domain beamforming. In Section \[sec:analysis\] we describe how the reduction in sampling rate is achieved. In Section \[sec:CS\] we discuss two possible CS approaches to signal reconstruction.
Beamforming in time {#sec:beam in time}
===================
Most modern imaging systems use multiple transducer elements to transmit and receive acoustic pulses. Appropriate processing of the signals detected by the individual array elements allows to enhance the quality of the resulting image. The most commonly used technique, referred to as beamforming, increases SNR by averaging the individual signals after their alignment with appropriate time-delays. We begin by describing the beamforming process which takes place in a typical B-mode imaging cycle. Our presentation is based mainly on [@jensen1999linear] and [@wagner2012compressed].
![$M$ receivers aligned along the $x$ axis. An acoustic pulse is transmitted at direction $\theta$. The echoes scattered from perturbation in radiated tissue are received by the array elements.[]{data-label="fig:array"}](plots/array){width="4.5cm"}
Consider an array comprised of $M$ transceiver elements aligned along the $x$ axis, as illustrated in Fig. \[fig:array\]. The reference element $m_0$ is set at the origin and the distance to the $m$-th element is denoted by $\delta_m$. The image cycle begins at $t=0$, when the array transmits an energy pulse in the direction $\theta$. The pulse propagates trough the tissue at speed $c$, and at time $t\geq0$ its coordinates are $(x,z)=(ct\sin{\theta},ct\cos{\theta})$. A potential point reflector located at this position scatters the energy, such that the echo is detected by all array elements at a time depending on their locations. Denote by $\varphi_m(t;\theta)$ the signal detected by the $m$-th element and by $\hat{\tau}_m(t;\theta)$ the time of detection. It is readily seen that: $$\label{tau^m}
\hat{\tau}_m(t;\theta)=t+\frac{d_m(t;\theta)}{c},$$ where $d_m(t;\theta)=\sqrt{(ct\cos{\theta})^2+(\delta_m-ct\sin{\theta)}^2}$ is the distance traveled by the reflection. Beamforming involves averaging the signals detected by multiple receivers while compensating the differences in detection time.
Using , the detection time at $m_0$ is $\hat{\tau}_{m_0}(t;\theta)=2t$ since $\delta_{m_0}=0$. Applying an appropriate delay to $\varphi_m(t;\theta)$, such that the resulting signal $\hat{\varphi}_m(t;\theta)$ satisfies $\hat{\varphi}_m(2t;\theta)=\varphi_m(\hat{\tau}_m(t;\theta))$, we can align the reflection detected by the $m$-th receiver with the one detected at $m_0$. Denoting $\tau_m(t;\theta)=\hat{\tau}_m(t/2;\theta)$ and using , the following aligned signal is obtained: $$\begin{aligned}
\label{phim}
\hat{\varphi}_m(t;\theta)&=\varphi_m(\tau_m(t;\theta);\theta),\\ \nonumber
\tau_m(t;\theta)&=\frac{1}{2}\left(t+\sqrt{t^2-4(\delta_m/c)t\sin{\theta}+4(\delta_m/c)^2}\right).\end{aligned}$$ The beamformed signal may now be derived by averaging the aligned signals: $$\label{phi beamformed}
\Phi(t;\theta)=\frac{1}{M}\sum_{m=1}^M{\hat{\varphi}_m(t;\theta)}.$$ Ultrasound systems perform the beamforming process defined in in the digital domain, implying that the analog signals $\varphi_m(t;\theta)$ detected at the receiver elements are first sampled. Rates up to 4 times the Nyquist rate, dictated by the bandwidth of the individual signal, are required in order to improve the system’s beamforming resolution and to avoid artifacts caused by digital implementation. From now on we will denote this sampling rate as the beamforming sampling rate $f_s$.
To conclude this section we evaluate the number of samples taken at each transducer element. Our evaluation is based on the imaging setup used to acquire in vivo cardiac data. The acquisition was performed with a GE breadboard ultrasonic scanner of 64 acquisition channels. The radiated depth $r=16$ cm and the speed of the sound $c=1540$ m/sec yield a signal of duration $T=2r/c\simeq210$ $\mu$sec. The acquired signal is characterized by a narrow bandpass bandwidth of $2$ MHz, centered at the carrier frequency $f_0\approx3.1$ MHz, leading to a beamforming rate of $f_s\approx16$ MHz and $Tf_s=3360$ real-valued samples.
Beamforming in frequency {#sec:beam in freq}
========================
We now show that beamforming can be performed equivalently in the frequency domain, leading to substantial reduction in the number of samples, needed to obtain the same image quality.
We extend the notion of compressed beamforming, introduced in [@wagner2012compressed], to beamforming in frequency and show that a linear combination of the DFT coefficients of the individual signals, sampled at the beamforming rate $f_s$, yields the DFT coefficients of the beamformed signal, sampled at the same rate. We follow the steps in [@wagner2012compressed] and start from the computation of the Fourier series coefficients of the beamformed signal $\Phi(t;\theta)$.
The support of $\Phi(t;\theta)$ is limited to $[0,T]$, with $T$ defined by the transmitted pulse penetration depth. Its Fourier series coefficients are given by: $$\label{fourier coeff of beamformed 1}
c_k^s=\frac{1}{T}\int_0^T \Phi(t;\theta)\expk dt.$$ Plugging into , it can be shown that $$\label{fourier coeff of beamformed 2}
c_k^s=\frac{1}{M}\sum_{m=1}^M c_{k,m}^s,$$ where $c_{k,m}^s$ are the Fourier coefficients of $\hat{\varphi}_m(t;\theta)$. These coefficients can be written as $$\label{c k m}
c_{k,m}^s=\frac{1}{T}\int_0^T g_{k,m}(t;\theta) \varphi_m(t;\theta)dt,
\vspace{-0.3cm}$$ with $$\begin{aligned}
\label{g j m q j m}
g_{k,m}(t;\theta)=&q_{k,m}(t;\theta)\expk, \nonumber \\
q_{k,m}(t;\theta)=& I_{[|\gamma_m|,\tau_m(T;\theta))}(t) \left(1+\frac{\gamma_m^2\cos{\theta}^2}{(t-\gamma_m\sin{\theta})^2}\right)\times \\*
&\exp{\left\{i\frac{2\pi}{T}k\frac{\gamma_m-t\sin{\theta}}{t-\gamma_m\sin{\theta}}\gamma_m\right\}}, \nonumber\end{aligned}$$ where $\gamma_m=\delta_m/c$, and $I_{[a,b]}$ is the indicator function.
Our next step is to substitute $\varphi_m(t)$ by its Fourier series coefficients. Denoting the $n$-th Fourier coefficient by $\varphi_m^s[n]$ and using we can rewrite as follows: $$\label{c k m fourier}
c_{k,m}^s=\sum_n \varphi_m^s[n]Q_{k,m;\theta}[k-n],$$ where $Q_{k,m;\theta}[n]$ are the Fourier coefficients of $q_{k,m}(t;\theta)$ with respect to $[0,T)$. According to Proposition 1 in [@wagner2012compressed], $c_{k,m}^s$ can be approximated sufficiently well when we replace the infinite summation in by the finite one: $$\label{c k m approx}
c_{k,m}^s\simeq\sum_{n\in\nu(k)}\varphi_m^s[n]Q_{k,m;\theta}[k-n].$$ The set $\nu(k)$ is defined according to the decay properties of $\{Q_{k,m;\theta}[n]\}$. Equations and provide a relationship between the Fourier series coefficients of the beamformed and the individual signals. Denote by $N=\lfloor T\cdot f_s\rfloor$ the number of samples in each signal. Since all signals are sampled at a rate which is higher than their Nyquist rate, the relation between the DFT of length $N$ and the Fourier series coefficients is given by: $$\begin{aligned}
\label{DFT-Fourier series}
c_k=Nc_k^s,~~~ \varphi_m[n]=N\varphi_m^s[n],\end{aligned}$$ where $c_k$ and $\varphi_m[n]$ denote the DFT coefficients of the beamformed and individual signals respectively. Plugging into and , we get the desired relation: $$\begin{aligned}
\label{DFT beam - DFT individual}
c_k&\simeq\frac{1}{M}\sum_{m=1}^M\sum_{n\in\nu(k)}\varphi_m[n]Q_{k,m;\theta}[k-n].\end{aligned}$$ Note that in order to calculate an arbitrary set $\kappa$ of DFT coefficients of the beamformed signal, we need $\nu=\cup_{k\in\kappa}\nu(k)$ DFT coefficients of each one of the individual signals.
Applying an IDFT on $\{c_k\}_{k=1}^N$, we obtain the beamformed signal. We can now proceed to standard image generation steps which include log-compression and interpolation. To demonstrate the equivalence of beamforming in time and frequency, we applied both methods on in vivo cardiac data obtained as explained in Section \[sec:beam in time\], yielding the images shown in Fig. \[fig:time vs freq beamformed\]. As can be seen, both images are identical.
![Cardiac images constructed with different beamforming techniques. (a) Time domain beamforming. (b) Frequency domain beamforming. []{data-label="fig:time vs freq beamformed"}](plots/timeBeamformed){width="4.6cm"}
(a)
![Cardiac images constructed with different beamforming techniques. (a) Time domain beamforming. (b) Frequency domain beamforming. []{data-label="fig:time vs freq beamformed"}](plots/freqBeamformed){width="4.6cm"}
(b)
Rate Reduction by Beamforming in Frequency {#sec:analysis}
==========================================
In the previous section we showed the equivalence of beamforming in time and frequency. We next demonstrate that beamforming in frequency allows to reduce the required number of samples of the individual signals. To this end we consider two questions: 1) how many DFT coefficients of the beamformed signal do we need for its perfect reconstruction; 2) how many samples of the individual signals should be taken in order to compute these DFT coefficients?
Parametric representation {#ssec:param representation}
-------------------------
We begin by answering the first question using a parametric model for the beamformed signal. According to [@tur2011innovation; @wagner2012compressed], the beamformed signal can be modeled as a sum of a small number of replicas of the known transmitter pulse with unknowns amplitudes and delays: $$\label{beam FRI}
\Phi(t;\theta)\simeq\sum_{l=1}^L\tilde{b}_l h(t-t_l),$$ where $h(t)$ is the transmitted pulse-shape, $L$ is the number of scattering elements in direction $\theta$, $\{\tilde{b}_l\}_{l=1}^L$ are the unknown amplitudes of the reflections and $\{t_l\}_{l=1}^L$ denote the times at which the reflection from the $l$-th element arrived at the reference receiver $m_0$. Sampling both sides of at rate $f_s$ and quantizing the unknown delays $\{t_l\}_{l=1}^L$ with quantization step $1/f_s$, such that $t_l=q_l/f_s, q_l\in\mathbb{Z}$, we can rewrite as follows: $$\label{beam FRI n}
\Phi[n;\theta]\simeq\sum_{l=1}^L\tilde{b}_l h[n-q_l]=\sum_{l=0}^{N-1} b_l h[n-l],$$ where $$\begin{aligned}
\label{tilde b - b}
b_l = \left\{ \begin{array}{rl}
\tilde{b}_l &\mbox{ if $l=q_l$} \\
0 &\mbox{ otherwise}.
\end{array} \right.\end{aligned}$$ Calculating the DFT using : $$\begin{aligned}
\label{DFT BF}
c_k=\sum_{n=0}^{N-1}\Phi[n;\theta]e^{-i \frac{2\pi}{N}k n}=h_k\sum_{l=0}^{N-1} b_l e^{-i \frac{2\pi}{N}k l},\end{aligned}$$ where $h_k$ is the DFT coefficient of $h[n]$.
The transmitted pulse $h(t)$ may be modeled as a narrowband waveform, $g(t)$, modulated by a carrier at frequency $f_0$: $h(t)=g(t)\cos(2\pi f_0 t)$. When such a pulse is sampled at rate $f_s$, most of its DFT coefficients are zero, as shown in Fig. \[fig:pulseDFT\]. Obviously, implies that the only non-zero DFT coefficients are in the bandwidth of the transmitted pulse. This allows us to exploit the low bandwidth of the beamformed signal and calculate only non-zero DFT coefficients. Since $\Phi[n;\theta]$ is real, its DFT coefficients are symmetric, thus we only need to know half of the overall non-zero elements. Denote the set of non-zero DFT coefficients by $\kappa$. In typical cardiac imaging the bandwidth of $g(t)$ is equal to $2$ MHz, the modulation frequency $f_0=3.1$ MHz, and the sampling rate $f_s=16$ MHz, leading to $K=|\kappa|\approx360$. Once these $K$ coefficients are known, we can reconstruct $\Phi[n;\theta]$ by padding the elements of $\kappa$ with an appropriate number of zeros and performing an IDFT. Hence, we have shown that the number of DFT coefficients of the beamformed signal required for its perfect reconstruction is the cardinality of the set $\kappa$. This number is only $1/9$ of the overall number of DFT coefficients of beamformed signal dictated by $f_s$.
![The DFT coefficients of the transmitted pulse. The bandwidth of the waveform is $2$ MHz, $f_0=3.1$ MHz, $f_s=16$ Mhz.[]{data-label="fig:pulseDFT"}](plots/pulseDFT){width="5.0cm"}
Reduced rate sampling {#ssec:red rate samp}
---------------------
We now address the second question: how many samples of the individual signals should be taken in order to compute the set of non-zero DFT coefficients $\kappa$?
As shown in Section \[sec:beam in freq\], we need to know a set $\nu$ of the DFT coefficients of each individual signal in order to compute a set $\kappa$ of the DFT coefficients of the beamformed signal. The experimental results show that in ultrasound imaging scenario, $|\kappa|\leq|\nu|\leq1.33|\kappa|$. According to , we equivalently seek the same set $\nu$ of Fourier series coefficients of each individual signal. The above problem is addressed in [@wagner2012compressed]. A mechanism, proposed there, allows to obtain a set $\kappa$ of Fourier coefficients of the beamformed signal from $|\nu|$ samples of each of the individual signals, filtered with an appropriate kernel. Namely, the number of the samples taken from the individual signal is $|\nu|$. For the setup mentioned in Subsection \[ssec:param representation\], $|\kappa|\approx360$, implying that in the worst case we need only $|\nu|=1.33|\kappa|\approx480$ samples of the individual signal, while for beamforming in time domain we need $\approx3360$ samples as shown in Section \[sec:beam in time\]. This allows us to achieve 7-fold reduction in sampling rate without compromising image quality. The images, created by these two techniques, shown in Fig. \[fig:time vs freq 360 beamformed\], are identical.
![Cardiac images constructed with different beamforming techniques. (a) Time domain beamforming, $3360$ samples per individual signal. (b) Frequency domain beamforming using $480$ samples per individual signal (7 fold reduction in sampling rate).[]{data-label="fig:time vs freq 360 beamformed"}](plots/timeBeamformed){width="4.6cm"}
(a)
![Cardiac images constructed with different beamforming techniques. (a) Time domain beamforming, $3360$ samples per individual signal. (b) Frequency domain beamforming using $480$ samples per individual signal (7 fold reduction in sampling rate).[]{data-label="fig:time vs freq 360 beamformed"}](plots/freqB_360){width="4.6cm"}
(b)
Further Reduction via CS {#sec:CS}
========================
We showed that it is possible to reconstruct a beamformed signal perfectly from the set $\kappa$ of its non-zero DFT coefficients, computed from a small number of samples of the individual signals. We show next that further reduction in sampling rate is possible, when taking only a subset $\mu\subset\kappa$, $|\mu|=M < K=|\kappa|$, of DFT coefficients of the beamformed signal. Defining a $K$-length vector $\mathbf{c}$ with $k$-th entry $c_k/h_k, k\in\kappa$, we can rewrite in matrix form: $$\label{DFT BF matrix full}
\mathbf{c}=\mathbf{D} \mathbf{b}$$ where $\mathbf{D}$ is a $K\times N$ matrix formed by taking the set $\kappa$ of rows from an $N\times N$ DFT matrix, and vector $\mathbf{b}$ is of length $N$ with $l$-th entry $b_l$. Since from now on only subset $\mu$ is given, define an $M$-length vector $\mathbf{c_{\mu}}$ with $k$-th entry $c_k/h_k, k\in\mu$ and rewrite as follows: $$\label{DFT BF matrix}
\mathbf{c_{\mu}}=\mathbf{A} \mathbf{D} \mathbf{b}$$ where $\mathbf{A}$ is $M\times K$ measurement matrix which picks the subset $\mu$ of rows from $\mathbf{D}$, implying that $\mathbf{A D}$ is $M\times K$ matrix formed by taking the set $\mu$ of rows from an $N\times N$ DFT matrix.This property of $\mathbf{A D}$ will be used below.
Synthesis approach
------------------
Since from the signal of interest is completely defined by the unknown delays and amplitudes, a possible approach is to extract those values from the available set $\mu$ of DFT coefficients. Equation can be viewed from a CS perspective, when we assume that the number of scatterers $L$ is small, since according to it implies that the vector $\mathbf{b}$ is sparse. Hence has the form of a classic sparse synthesis model [@eldar2012compressed], where vector $\mathbf{c}_{\mu}$ has a sparse representation in $\mathbf{AD}$. The goal is to reconstruct an $L$-sparse vector $\mathbf{b}$ from its projection onto a subset of $M$ orthogonal vectors given by the rows of matrix $\mathbf{AD}$. With an appropriate choice of $L$ and subset of fourier coefficients $\mu$, such a problem can be solved using CS methodology, including $l1$ optimization and greedy algorithms. Indeed, in [@wagner2012compressed], orthogonal matching pursuit (OMP) [@tropp2007signal] shows sufficiently good performance.
The synthesis approach has a significant drawback. The assumption of a small number of reflecting elements $L$, forces us to treat only the strong reflectors located in direction $\theta$. In such a setup we essentially loose all the weak reflectors that appear as speckle, namely, granular pattern that can be seen in Fig. \[fig:time vs freq beamformed\], and carry important information in medical imaging.
Analysis approach
-----------------
To avoid loss of speckle information, imposed by the assumption of sparsity, we propose using an analysis approach [@candes2011compressed]. In this methodology we aim to reconstruct the set $\kappa$ from its subset $\mu$, while assuming that the analyzed vector $\mathbf{D}^*\mathbf{c}$ is compressible. The analysis approach can be translated into the $\textit{l}_1$ optimization problem: $$\label{l1 sol}
\min_{\mathbf{c}}\|\mathbf{D}^* \mathbf{c}\|_1 \textrm{~~~subject to~~~} \|\mathbf{A}\mathbf{c}-\mathbf{c}_\mu\|_2\leq\varepsilon.$$ According to Theorem 1.4 in [@candes2011compressed], the solution to is very accurate, if the measurement matrix $\mathbf{A}$ satisfies the restricted isometry property adapted to $\mathbf{D}$ (D-RIP) and the elements of $\mathbf{D}^* \mathbf{c}$ decay rapidly. As we mentioned before, $\mathbf{AD}$ is a partial DFT matrix, therefore, according to results in [@candes2011compressed], $\mathbf{A}$ satisfies the D-RIP.
A typical beamformed ultrasound signal is comprised of a relatively small number of strong reflections and a bunch of much weaker scattered echoes. It is, therefore, natural to assume that $\mathbf{b}$ is compressible, implying that $\mathbf{c}$ has a compressible expansion in $\mathbf{D}$. Since $\mathbf{D}$ is a partial DFT matrix, its Gram matrix is nearly diagonal, implying that $\mathbf{D}^* \mathbf{c}$ is also compressible [@candes2011compressed] and satisfies the decay requirement.
To demonstrate the proposed method, a subset $\mu$ of $100$ Fourier coefficients corresponding to the central frequency samples in the bandwidth of the transmitted pulse were chosen. To calculate $\mu$ we need at most $133$ samples per individual signal, implying 25 fold reduction in sampling rate. The result is shown in Fig. \[fig:part BW\] (a). To compare the proposed solution with the previously developed OMP based method [@wagner2012compressed], the same subset $\mu$ was used to reconstruct the beamformed signal assuming $L=25$ strong reflectors in each direction $\theta$. The resulting image is shown in Fig. \[fig:part BW\] (b).
![Cardiac images constructed from partial spectrum data with 25 fold reduction in sampling rate. (a) Modified $\textit{l}_1$ optimization solution. (b) OMP based reconstruction. []{data-label="fig:part BW"}](plots/opt100){width="4.6cm"}
(a)
![Cardiac images constructed from partial spectrum data with 25 fold reduction in sampling rate. (a) Modified $\textit{l}_1$ optimization solution. (b) OMP based reconstruction. []{data-label="fig:part BW"}](plots/OMP100){width="4.6cm"}
(b)
|
---
author:
- 'Freddy Cachazo${}^{a}$, Song He${}^{a,b}$ and Ellis Ye Yuan${}^{a,c}$'
bibliography:
- 'ScatteringEquations.bib'
title: Scattering Equations and KLT Orthogonality
---
=1
Introduction
============
Recently the tree-level S-matrix of a variety of theories has been found as an integral over the moduli space of maps from the $n$-punctured sphere into the null light cone in momentum space [@Witten:2003nn; @Roiban:2004yf; @Cachazo:2012da; @Cachazo:2012kg; @Huang:2012vt; @Cachazo:2013iaa]. In all cases a given point in the space of kinematic invariants is mapped into $(n-3)!$ points in the moduli space. In this note we point out that the system of polynomial equations that connects the two spaces is universal and independent of the spacetime dimension. In section \[sec:eq\] we derive the basic set of equations connecting the space of kinematic invariants and that of the puncture locations; we call them the [*scattering equations*]{}. In section \[sec:solve\] we give an algorithm for finding all solutions to the scattering equations which proves that the number of solutions is also $(n-3)!$ in any dimension. In order to motivate the fact that the scattering equations are indeed the backbone of massless particle scattering in any dimension, in section \[sec:factorization\] we study how general kinematic invariants can nicely be extracted and show the behavior of the equations in factorization limits.
The solutions to the scattering equations also satisfy a remarkable property which we call [*KLT orthogonality*]{}. This was conjectured in the context of four dimensional scattering by one of the authors and Geyer [@Cachazo:2012da]. KLT orthogonality states that when the standard field theoretic version of the Kawai-Lewellen-Tye (KLT) construction [@Kawai:1985xq] is interpreted as a bilinear form acting on an $(n-3)!$ dimensional vector space, the vectors made of Parke-Taylor-like functions constructed from the puncture locations evaluated on different solutions are mutually orthogonal with respect to the KLT bilinear. In section \[sec:KLT\] we make precise the elements entering the conjecture and give a proof of its validity in any dimension.
In four dimensions, KLT orthogonality was crucial for constructing a formula for gravity amplitudes from rational maps starting with the Witten-RSV formula [@Witten:2003nn; @Roiban:2004yf] for gauge theory amplitudes [@Cachazo:2012da]. It is natural to assume that given the validity of KLT orthogonality in any dimension, such a connection between gauge theory and gravity S-matrices should also exist. We end in section \[sec:discussion\] with comments on these connections and also to the high energy limit of string scattering amplitudes.
Scattering Equations
====================
\[sec:eq\]
In this section we derive the scattering equations starting with the connection between the scattering data of $n$ massless particles and maps from the $n$-punctured sphere into the null cone in $D$ dimensional momentum space. Given a set of $n$ null vectors in $D$ dimensions $\{ k_1^\mu ,k_2^\mu ,\ldots ,k_n^\mu \}$ that satisfy momentum conservation, the map is given by [@Cachazo:2013iaa] k\^\_a = \_[|z-\_a|=]{} dz \[maps\] where $p^\mu(z)$ is a collection of $D$ degree $n-2$ polynomials.
Given that all $k_a^\mu$ are null vectors it is clear that $p(\sigma_a)^2$ must vanish for all $a$. Since $p(z)^2$ is a polynomial of degree $2n-4$, knowing $n$ roots is not enough to find it. We need $n-3$ additional conditions ($p(z)^2$ is not monic). The extra conditions turn out to have a very elegant origin. They are the conditions needed for the vector $p^\mu(z)$ to be null for all $z$. In other words, $p^\mu(z)$ must be a map from $\mathbb{CP}^1$ to the null cone in (complexified) momentum space. Clearly, if $p^2(z)=0$ then its derivative must also vanish, so $p(z)\cdot p'(z)=0$. It is natural to evaluate this condition on the $n$ puncture locations $\sigma_a$. A simple exercise shows that out of the $n$ conditions p(\_a)p’(\_a)=0 \[pconstraints\]with $a$ in $\{1,2,\ldots ,n\}$, only $n-3$ are linearly independent. These $n-3$ equations are exactly the remaining equations needed to determine $p^\mu(z)$.
Alternatively, the equations (\[maps\]) can also be treated as linear equations for the coefficients of the polynomial p\^(z) = p\_0\^+p\_1\^z+ + p\^\_[n-2]{}z\^[n-2]{} whose solution is p\_\^=\_[a=1]{}\^n{}\_[a]{}\^[n-1-]{}k\_a\^,{0,1,…,n[-]{}2}, where the symbol $\{\sigma\}_{b}^m$ is defined as the symmetrized product of $m$ $\sigma$’s which do not involve $\sigma_b$ {}\_[b]{}\^m=(-1)\^m\_[{a\_i}{1,…,n}\\{b}]{}\_[a\_1]{}\_[a\_2]{}…\_[a\_m]{}. Plugging this into the constraints turns them into p(\_a)p’(\_a) =v\_a\_[ba]{}k\_ak\_b\_[m=1]{}\^[n-2]{}m{}\_b\^[n-1-m]{}\_a\^[m-1]{}=\_[ba]{}, \[deri\]with $v_a=\prod_{b\neq a}(\sigma_a-\sigma_b)$ and $s_{ab}=2k_a\cdot k_b$ are the standard kinematic invariants[^1]. Since $v_a$ can never be singular (for generic external data), we can eliminate it from the equations and obtain \[scatteringequations\] \_[ba]{}=0,a{1,…,n}. We call these the *scattering equations*, which are the main object of study in this paper.
Solving the Scattering Equations
================================
\[sec:solve\]
The scattering equations look deceivingly simple but once the denominators are removed they turn out to be an intricate system of polynomial equations in $n-3$ variables which can resist straightforward numerical algorithms. In [@Cachazo:2013iaa] it was proven that for four dimensional kinematics the total number of solutions is $(n-3)!$. In this section we provide a simple numerical algorithm for finding all solutions which makes it clear why the number of solutions is also $(n-3)!$ in any number of dimensions. The algorithm is inspired by the behavior of the equations in soft limits.
Given that only $n-3$ equations are linearly independent we select \_[ba]{}=0,a{4,5,…,n}.\[sol\] and use $\SL2C$ invariance[^2] to fix the value of $\sigma_1\to \infty$, $\sigma_2=0$ and $\sigma_3=1$.
The algorithm for finding solutions is inductive in nature: Introduce a parameter $\epsilon$ which takes values in the interval $[0,1]$ and define s\_[nb]{}() =s\_[nb]{}, b{1,2,…,n-1}. Replacing $s_{nb}$ with $s_{nb}(\epsilon )$ in the original equations (\[sol\]) gives a set of $\epsilon$-dependent equations.
The new equations evaluated at $\epsilon = 0$ are \_[b=2,ba]{}\^[n-1]{}=0,a{4,5,…,n-1}. Note that the sum starts at $b=2$ due to our gauge choice.
The last equation in (\[sol\]), i.e, for $a=n$, drops out and the new system is exactly of the form of the equations for $n-1$ particles. At first sight, one would think that the new equations cannot be interpreted exactly as physical equations since the kinematic invariants entering them are not in the kinematic space for $n-1$ particles; they are still the original $n$-particle ones. However, since none of the invariants containing $k_1^\mu$ appear in the equations (recall that $\sigma_1 \to \infty$) one can pretend that $k_1^\mu$ has been deformed so as to keep momentum conservation valid when $\epsilon k_n^\mu$ vanishes.
The inductive argument assumes that the new system with $n-4$ equations and same number of variables has been solved giving rise to $(n-4)!$ solutions. Let us label the solutions as $\{ \sigma^{(i)}_1,\sigma^{(i)}_2,\ldots ,\sigma^{(i)}_{n-1}\}$ with $i\in \{ 1,2,\ldots ,(n-4)!\}$. Of course, $\sigma^{(i)}_a$ for $a\in \{1,2,3\}$ have values independent of $(i)$ which are fixed from the start. At this point no values for $\sigma_n$ have been found.
In order to proceed, divide the interval $[0,1]$ into $M$ segments. The goal at this point is to find solutions of the system of equations at $\epsilon = 1/M$. Given that $\epsilon$ is not equal to zero, we can consider the last equation in (\[sol\]), which after the substitution $s_{nb}\to s_{nb}(\epsilon )$ remains invariant up to an overall factor of $\epsilon$, \_[b=2]{}\^[n-1]{}=0. \[rec\] It is possible to obtain values of $\sigma_n$ which are very close to the actual solutions at $\epsilon = 1/M$ by simply solving (\[rec\]) when all variables except $\sigma_n$ are taken to be evaluated on one of the $(n-4)!$ solutions found previously. Extracting the numerator of equation (\[rec\]) one finds \_[b=2]{}\^[n-1]{}s\_[nb]{}\_[c=2, cb]{}\^[n-1]{}(\_n-\^[(i)]{}\_c) = 0. This is a polynomial of degree $n-3$ in $\sigma_n$ and therefore we get $n-3$ approximate values of $\sigma_n$ for each of the $(n-4)!$ solutions to the remaining $n-1$ $\sigma$’s. Let’s denote the approximate values $\sigma_n^{(i,J)}$ with $J=1,\ldots, n-3$.
The next step is to use any standard numerical algorithm[^3] to find a solution to the full system of equations at $\epsilon = 1/M$ near each of the $(n-3)!$ points $\{ \sigma^{(i)}_1,\sigma^{(i)}_2,\ldots ,\sigma^{(i)}_{n-1}, \sigma_n^{(i,J)} \}$.
Finally, one can iterate the last part of the algorithm solving the equations at $\epsilon = m/M$ using the solutions found at $\epsilon = (m-1)/M$ as seed. In this way one can follow all $(n-3)!$ solutions to the point $\epsilon =1$ and obtain the desired answer. For more details on how this algorithm can be implemented in practice, issues that can arise and some examples including a solution for $n=15$, see appendix \[app:numericalalgorithm\].
In three and four dimensions this algorithm can be made more efficient when applied to finding solutions in particular $R$-charge sectors which are not present in higher dimensions but we will not discuss this here.
General Kinematic Invariants and Factorization
==============================================
\[sec:factorization\]
In this section we discuss some of the properties of the scattering equations which motivate their physical interpretation as being the backbone of tree level scattering of massless particles.
The most general kinematic invariants in a physical theory are constructed by taking a subset of labels ${\cal S}\subset \{1,2,\ldots ,n\}$ and computing (\_[a]{}k\^\_a)\^2. Without loss of generality one can classify the kinematic invariants by the number of elements in ${\cal S}$ and take a representative in each group given by (k\_1+k\_2++k\_m)\^2 with $m = |{\cal S}|$.
With the benefit of hindsight, let us introduce a special change of variables tailored to the study of this invariant, \[reparametrization\] \_a=\_n+u\_a,a{m+1,…,n-1}, where we regard $u_{n-1}$ as a fixed value while $\{\tau,u_{m+1},\ldots,u_{n-2}\}$ are new variables. Consider the first $m$ scattering equations \_[ba]{} = 0 a{1,…, m}. At this point it is convenient to introduce a shorthand notation $\sigma_{ab}\equiv \sigma_a-\sigma_b$ (as well as $\sigma_{a,b}$ in case the other notation may cause confusion), which will be used throughout this paper when needed to keep formulas easily readable.
Multiplying the $a^\text{th}$ equation by $\sigma_a-\sigma_n$ and after a simple manipulation we get \[leftequations\] s\_[a,2]{}++s\_[a,m]{}+(1+)s\_[a,m+1]{}++(1+)s\_[a,n-1]{}+s\_[a,n]{}=0, with $a\in\{1,\ldots,m\}$.
Adding up all these $m$ equations leads to a formula for the kinematic invariant of interest \[factorizationchannel\] (k\_1++k\_m)\^2 =(s\_[1,m+1]{}++s\_[m,n-1]{}).
This equation is the core of the connection between the physical space of kinematic invariants and the configuration of puncture locations on $\mathbb{CP}^1$. Note that on the factorization channel where $k_1^\mu+\cdots+k_m^\mu$ is on-shell, i.e. $(k_1+\cdots+k_m)^2=0$, one has two branches of solutions. One with $\tau=0$ and the other with \[branch2\] (s\_[1,m+1]{}++s\_[m,n-1]{}) = 0. Consider the solution $\tau=0$. In this branch, the scattering equations separate into two sets of scattering equations but with less particles on each. Very nicely, all the equations in (\[leftequations\]) directly become \[leftequations2\] +++=0, where $k_I^\mu=-(k_1^\mu+\cdots+k_m^\mu)$ is the new null vector.
Any subset of $m-2$ of these equations can be taken as the definition of the scattering equations for the set of $(m+1)$ null vectors $\{ k_1^\mu , k_2^\mu ,\ldots ,k_m^\mu , k_I^\mu \}$ if $\sigma_n$ is identified with $\sigma_I$. In particular, note that by adding all the equations one finds the equation corresponding to $a=I$, i.e. \_[a=1]{}\^m = 0. The second set of scattering equations can simply be obtained by repeating the same analysis after using momentum conservation in the form (k\_1+k\_2++k\_m)\^2 = (k\_[m+1]{}+k\_[m+2]{}++k\_m)\^2 and coordinates $\sigma_a = \sigma_1 + \tau v_a$.
This indicates that the original $(n-3)!$ solutions give rise to $(m-2)!\times (n-m-2)!$ singular solutions in the branch $\tau=0$. Here ‘singular’ refers to the situation when some crossratios of $\sigma_a$’s belong to the set $\{0,1,\infty\}$. It is not difficult to show that crossratios which involve two points from the set $\{1,2,\ldots ,m\}$ and two points from its complement are singular, while all others remain finite.
This is all, of course, very familiar in the study of string scattering amplitudes in which the boundary of the moduli space of Riemann surfaces is connected to the physical singularities of amplitudes [@Green:1987sp]. This has also been observed in four dimensions in the context of field theories in twistor space constructions [@Vergu:2006np; @Skinner:2010cz; @Cachazo:2012pz]. These facts are more evidence of the universality of the scattering equations.
KLT Orthogonality
=================
\[sec:KLT\]
In this section we explore a very striking property of the solutions to the scattering equations which we call KLT orthogonality. In short, the property states that“Parke-Taylor" vectors made from distinct solutions are orthogonal with respect to the Kawai-Lewellen-Tye (KLT) bilinear. The precise meaning of this terminology will be give in subsection \[kltorthogonalityproof\]. Before moving to the definitions and to the proof of the property let us prove a crucial result which at first sight seems unrelated.
Generalized Jacobian and Its Rank
---------------------------------
Let us study the Jacobian matrix associated to the scattering equations. This is a symmetric $n \times n$ matrix with entries, \_[a b]{}(\_[ca]{} )/(\_b)=
, ab,\
-\_[ca]{}\_[a c]{},a=b.
\[cases\]As mentioned above only $n-3$ of the scattering equations are linearly independent and therefore the matrix $\Phi$ has rank $n-3$. This matrix was first encountered by Cachazo and Geyer in [@Cachazo:2012da] (inspired by a formula for MHV gravity amplitudes found by Hodges in [@Hodges:2012ym]) where it played an important role in a formula for gravity amplitudes constructed from gauge theory ones using the KLT construction[^4]. This is not accidental as we will see in the next subsection.
Consider now a generalization of $\Phi_{ab}$ which is natural from its origin as the fusion of two gauge theory amplitudes, \_[ab, ab]{}, \_[a a]{}-\_[ca]{} \_[a c]{}. where $\sigma_a$ and $\sigma'_a$ are both assumed to be solutions to the scattering equations. Note that when the two sets $\{\sigma \}$ and $\{\sigma'\}$ are chosen to be the same solution then $\Psi(\{\sigma \},\{\sigma \})=\Phi(\{\sigma \})$. The matrix $\Psi_{ab}$ has also made an appearance very recently in the studies of string amplitudes [@Stieberger:2013hza].
[**Proposition 1.**]{} [*Given two solutions $\{\sigma \}$ and $\{\sigma' \}$ to the scattering equations, the matrix $\Psi(\{\sigma \},\{\sigma' \})$ has rank $n-3$ when $\{\sigma \} = \{\sigma' \}$ and rank $n-4$ when $\{\sigma \} \neq \{\sigma' \}$.*]{}
0.1in
[*Proof:*]{} Let us start by taking $\{\sigma \} \neq \{\sigma' \}$. In this case one can construct four null vectors. Explicitly, v\_1 = (
[c]{} 1\
1\
\
1\
),v\_2 = (
[c]{} \_1\
\_2\
\
\_n\
),v\_3 = (
[c]{} ’\_1\
’\_2\
\
’\_n\
),v\_4 = (
[c]{} \_1’\_1\
\_2’\_2\
\
\_n’\_n\
). The fact that $v_1$ is a null vector is trivial as the columns of $\Psi$ add up to zero. Next, we consider $v_2$, the argument is identical for $v_3$. Computing the product $\Psi\, v_2$ one finds \^n\_[b=1]{} \_[a b]{} \_b=-\^n\_[b=1]{} \_[a b]{} (\_[a]{}-\_[b]{})+\_a(\^n\_[b=1]{} \_[a b]{})=-\_[ba]{} =0, where the last equation holds by the scattering equations. Finally, for $\Psi\, v_4$ one has \^n\_[b=1]{} \_[a b]{} \_b ’\_b=\^n\_[b=1]{} s\_[a b]{}-\_a \_[ba]{}-’\_a \_[ba]{}+\_a’\_a \_[b=1]{}\^n \_[ab]{}=0, where we have used momentum conservation and the scattering equations.
This computation shows that $\Psi(\{\sigma \},\{\sigma' \})$ has four null vectors and therefore rank $n-4$. If the two solutions are taken to be the same then we loose one null vector as $v_2=v_3$. Therefore $\Psi(\{\sigma \},\{\sigma \})$ only has three null vectors given by u\_1 = (
[c]{} 1\
1\
\
1\
),u\_2 = (
[c]{} \_1\
\_2\
\
\_n\
),u\_3 = (
[c]{} \_1\^2\
\_2\^2\
\
\_n\^2\
), which ends the proof.
Proof of KLT Orthogonality {#kltorthogonalityproof}
--------------------------
We proceed to the proof of KLT orthogonality. For each solution of the scattering equations one can construct an $n!$ dimensional vector with entries labeled by a permutation $\omega\in S_n$ and defined as . The entries in a given vector are not all independent. In fact, it is obvious that two permutations related by a cyclic transformation have the same entry. Not so obvious is the fact that the set of entries with two labels, say $1$ and $n$, fixed to some positions form a basis in which all other entries can be expressed as linear combinations. The precise linear combinations are known as the Kleiss-Kuijf (KK) relations [@Kleiss:1988ne]. Even less obvious is the fact that using the scattering equations one can further fix the position of a third label, say $n-1$. These relations give rise to the Bern-Carrasco-Johanssen (BCJ) relations of gauge theory amplitudes in four dimensions [@Bern:2008qj].
The conclusion from this discussion is that all physical information is encoded in $(n-3)!$–dimensional vectors obtained from the $n!$ ones by fixing the position of $1,n-1,n$ and letting $\omega \in S_{n-3}$ permute the rest of the labels. Even after selecting the three labels we still have the choice of where to put them. We will only make use of two choices: $(1,\omega(2), \ldots , \omega(n-2), n-1,n) $ and $(1,\omega(2), \ldots , \omega(n-2), n,n-1)$. Let’s denote the corresponding $(n-3)!$–dimensional vectors by V() = , \[Vr\]and U() = . \[Ur\]
In this language the Kawai-Lewellen-Tye construction gives rise to a bilinear form S\[|\]= \^[n[-]{}2]{}\_[i=2]{}(s\_[1, (i)]{}+\^[i[-]{}1]{}\_[j=2]{} ((j), (i))\_ s\_[(j),(i)]{}), where $\alpha,\beta\in S_{n-3}$, $\theta(i,j)_\beta=1$ if the ordering of $i,j$ is the same in both sequences of labels, $\alpha(2,\ldots,n{-}2)$ and $\beta(2,\ldots,n{-}2)$, and zero otherwise (the convention we use here follows that in [@BjerrumBohr:2010ta][^5]). Given any two solutions of the scattering equations, $$\{\sigma^{(i)}_1,\sigma^{(i)}_2,\ldots ,\sigma^{(i)}_n\} \quad {\rm and} \quad \{\sigma^{(j)}_1,\sigma^{(j)}_2,\ldots ,\sigma^{(j)}_n\}$$ (where $(i),(j)$ label the choice of solutions and take values in $\{ 1,2,\ldots ,(n-3)!\}$), define two vectors, $V^{(i)}(\omega)$ and $U^{(j)}(\omega)$, obtained by evaluating (\[Vr\]) and (\[Ur\]) on the corresponding solutions. A natural inner product can then be defined as \[defscalarproduct\] (i,j):= \_[,S\_[n-3]{}]{}V\^[(i)]{}()S\[|\]U\^[(j)]{}(). Now we are ready to formally state KLT Orthogonality in the following proposition.
0.15in
[**Proposition 2 (KLT Orthogonality).**]{} [*The inner product satisfies \[proposition2\] =\_[ij]{} for any values of $i$ and $j$.*]{}
0.15in
Before proceeding to the proof we first have to connect the KLT bilinear form to the generalized Jacobian defined in the previous subsection (A similar construction has been observed and discussed in [@Stieberger:2013hza]). The starting point is to note that $$\frac{(i,j)}{(i,i)^{\frac{1}{2}}(j,j)^{\frac{1}{2}}}$$ is clearly invariant under $\SL2C\times \SL2C$ where one group acts on $\{\sigma^{(i)}\}$ while the other acts on $\{\sigma^{(i)}\}$. Let’s partially fix both $\SL2C$ redundancies with the convenient choice $\sigma^{(i)}_{n-1}=\sigma^{(j)}_{n}=\infty$ and $\sigma^{(i)}_{n}=\sigma^{(j)}_{n-1}=1$ and define K\_n({},{’}) \_[,S\_[n[-]{}3]{}]{}S\[|\],\[bilinear\] where $\sigma_{ab} = \sigma_a-\sigma_b$. The motivation for this definition is that $K_n$ appears in the numerator of after the partial gauge fixing, with the identifications $\sigma=\sigma^{(i)}$ and $\sigma'=\sigma^{(j)}$.
It is also convenient to define an auxiliary co-rank one $(n{-}2) \times (n{-}2)$ matrix $\psi^{(n)}$ with entries, \_[a b, ab]{}=, \_[a a]{}=-\_[ba]{} \_[a b]{},for $a,b=1,\ldots,n{-}2$. Since both the rows and columns of $\psi^{(n)}$ add up to zero, it is easy to see that all $(n{-}3) \times (n{-}3)$ minors of $\psi^{(n)}$ are the same, and therefore invariant under any permutations of $1,2,\ldots, n{-}2$. We denote such a minor as $\det'\psi^{(n)}$.
0.1in
[**Proposition 3.**]{} [*The two functions defined above are identical up to a sign. Explicitly, $K_n(\{\sigma \},\{\sigma'\})=(-1)^n\det'\psi^{(n)}$.*]{}
0.1in
[*Proof:*]{} Here we provide a sketch of the proof postponing all details to appendix \[app:prop2\] for the interested reader[^6].
The main observation is that both $K_n$ and $\det' \psi^{(n)}$ are rational functions of $\sigma_a$’s which can be taken to be completely unconstrained complex variables. In other words, the set $\{ \sigma \}$ is not assumed to be a solution of the scattering equations. For our purposes we choose a given variable, say $\sigma_1$, and define the functions $K_n(\sigma_1)$ and $\det' \psi^{(n)}(\sigma_1)$ by keeping all other variables fixed.
Clearly, both $K_n(\sigma_1)$ and $\det' \psi^{(n)}(\sigma_1)$ vanish as $\sigma_1\to \infty$. Therefore, it is sufficient to show that $K_n(\sigma_1)$ and $(-1)^n\det'\psi^{(n)}(\sigma_1)$ have the same poles and residues at finite values of $\sigma_1$. The only possible pole locations are at $\sigma_1=\sigma_a$ for $a=2,3,\ldots,n{-}2$. Since both functions are manifestly invariant under permutations of $2,3,\ldots,n{-}2$, we only need to consider one of the $n{-}3$ poles, e.g. that at $\sigma_1=\sigma_2$. In appendix \[app:prop2\] we show that the residues of $K_n(\sigma_1)$ and of $(-1)^n\det' \psi^{(n)}(\sigma_1)$ at $\sigma_1=\sigma_2$ indeed agree. This concludes the proof that $K_n=(-1)^n\det' \psi^{(n)}$.
0.1in
[*Proof of Proposition 2:*]{} Finally we are ready to put all the pieces together and prove KLT orthogonality. With the canonical choice $\sigma^{(i)}_{n-1}=\sigma^{(j)}_{n}=\infty$ and $\sigma^{(i)}_{n}=\sigma^{(j)}_{n-1}=1$, we have \[korthogonality\] =.In addition, one finds that the minor of $\psi$ obtained by removing the first row and column is identical to that of $\Psi(\{\sigma \},\{\sigma' \})$ after removing rows $\{1,n-1,n\}$ and columns $\{1,n-1,n\}$ in this canonical gauge. We denote these determinants by $|\psi^{(n)}|^1_1$ and $|\Psi|_{1,n-1,n}^{1,n-1,n}$ respectively. Using first Proposition 3 the following identity holds, = \[KLTorthogonal\] and using Proposition 1, we learn that the denominators are not singular since $\Psi(\{\sigma^{(k)}\},\{\sigma^{(k)}\})$ has co-rank $3$ while the numerator vanishes if $i\neq j$ since $\Psi(\{\sigma^{(i)}\},\{\sigma^{(j)}\})$ has co-rank four. This concludes our proof of KLT orthogonality.
Discussions
===========
\[sec:discussion\]
We end with some comments on possible future directions related to applications of the scattering equations and KLT orthogonality to tree–level amplitudes of massless particles in arbitrary dimensions as well as a very intriguing connection to the high energy scattering of strings.
Towards Yang-Mills and Gravity Amplitudes in Arbitrary Dimensions
-----------------------------------------------------------------
In the light of the properties discussed in this paper it is very tempting to propose the existence of formulas for Yang-Mills and gravity scattering amplitudes in any dimensions based on rational maps. The first important observation is the construction of an $\SL2C$ covariant and permutation invariant measure that restricts an integration over $\sigma_a$ to solutions to the scattering equations. This is easily achieved by noting that \_a ’(\_[ba]{} ) \_[ij]{}\_[jk]{}\_[ki]{}\_[ai,j,k]{}(\_[ba]{} ) is independent of the choice $\{i,j,k\}$ and hence permutation invariant (Recall that $\sigma_{ab} = \sigma_a-\sigma_b$).
With this observation, it is natural to propose the following formulations for gauge theory and gravity amplitudes in any dimensions M\^\_n(1,2,…,n)&=& \_a ’(\_[ba]{} ) , \[YM\]\
M\^\_n&=& \_a ’(\_[ba]{} ) E\^2({k,, }), \[gravity\] where $E(\{k,\epsilon, \sigma\})$ is a permutation invariant function of $\sigma_a$, momenta $k_a^\mu$ and polarization vectors $\epsilon_a^\mu$. Note that $\SL2C$ invariance of the integrand constraints $E$: under an $\SL2C$ transformation, $\sigma_a\to \frac{A\sigma_a+B}{C\sigma_a+D}$, $E$ must transform as E({k,, })E({k,, }) \_[a=1]{}\^n(C\_a+D)\^2. Most likely, $E$ should also be gauge invariant for each solution to the scattering equations. This possibility motivates the idea of considering the contribution from each solution as building blocks. Each block satisfies many physical properties that the full amplitude satisfy. We leave the investigation of this fascinating possibility for future work.
Some evidence suggesting that the formula for Yang-Mills must exist is the following. Firstly, similar formulas in four dimensions can be derived for individual R-charge sectors. Secondly, the BCJ fundamental relations for Yang-Mills amplitudes are known to hold in any dimension [@Bern:2008qj; @BjerrumBohr:2009rd]. Using the arguments in [@Cachazo:2012uq] one can show that $M^{\mathrm{YM}}_n$ satisfies them as a direct consequence of the scattering equations.
It is clear from KLT orthogonality and the KLT relations that, if the formula for Yang-Mills exists, so does the formula for gravity. In a nutshell, define the permutation invariant combination $\det'\Psi(\{\sigma\},\{\sigma'\})\equiv |\Psi(\{\sigma\},\{\sigma'\})|^{i j k}_{r s t}/(\sigma_{i j}\sigma_{j k}\sigma_{k i} \sigma_{r s}\sigma_{s t}\sigma_{t r})$, and note that when $\{\sigma\}=\{\sigma'\}$ it reduces to the permutation invariant Jacobian of scattering equations $\det'\Phi(\{\sigma\})\equiv \det'\Psi(\{\sigma\},\{\sigma\})$. The KLT relations for two copies of Yang-Mills amplitudes give, && \_a ’ (\_[ba]{} ) \_a ’ (\_[ba]{} )’({}, {’})E({})E({’})=&& \^[(n[-]{}3)!]{}\_[i,j=1]{}E({\^[(i)]{}})E({\^[(j)]{}})=&& \_a ’ (\_[ba]{} ) E\^2({}), where we have denoted the $i$-th solution as $\sigma^{(i)}$ for $i\in \{1,\ldots,(n{-}3)!\}$, and the last equality follows from Proposition 2.
High Energy Scattering of Strings
---------------------------------
\[sec:string\]
Recently [@Mafra:2011nv; @Mafra:2011nw] it was shown that superstring disk amplitudes can be expressed as linear combinations of $(n{-}3)!$ Yang-Mills partial amplitudes (e.g. in the $U$ basis defined in ) with coefficients encoding $\alpha'$-corrections, M\^\_n (’)&=&\_[,S\_[n[-]{}3]{}]{} \_[i< j]{} |z\_[i j]{}|\^[’ s\_[i j]{}]{} M\_n\^()&& D\^[n[-]{}3]{}\_[’]{} z \_[S\_[n[-]{}3]{}]{}M\^\_n(1,(2,,n[-]{}2), n, n[-]{}1), \[open1\]where one integrates over positions of string vertex operators on the disk boundary, with the canonical ordering $z_i<z_{i{+}1}$, and fixes the PSL$(2,\mathbb{R})$ redundancy by choosing e.g. $(z_1,z_{n{-}1}, z_n)=(0,1,\infty)$; $z_{ij}\equiv z_i-z_j$ and the “Parke-Taylor" vectors of $z$’s are in the $V$ basis. On the first line we have denoted $M^{\mathrm{YM}} (\tau)\equiv M^{\mathrm{YM}}(1,\tau(2,\cdots,n{-}2),n,n{-}1)$); on the second line, we have denoted the disk integral measure with Koba-Nileson factors as $D^{n{-}3}_{\alpha'} z$, which contains the entire $\alpha'$-dependence. Closed string tree amplitudes are given by integrals over string vertex insertions $ (z_i, \bar z_i) $ on a complex sphere. It is well known that by decomposing vertex operators the integrand can be written as the product of left-moving and right-moving open string integrands [@Kawai:1985xq], thus the closed string amplitudes are bilinear forms of two copies of Yang-Mills amplitudes, M\^\_n (’)= D\^[n[-]{}3]{}\_[’]{}z\_i D\^[n[-]{}3]{}\_[’]{}|z\_i \_[,, ,]{} M\^\_n () M\^\_n() \[closed1\]where the sum is over $\tau,\tilde\tau, \rho,\tilde\rho\in S_{n{-}3}$. Plugging into and , and using Proposition 2 we find, &&M\^\_n (1,…, n;’)= \_[z\_i<z\_[i[+]{}1]{} ]{}D\^[n[-]{}3]{}\_[’]{} z\_i \_a’ (\_[ba]{})E({}) ’ ({z}, {}),&& M\^\_n (’)=\_[\^[n[-]{}3]{}]{} D\^[n[-]{}3]{}\_[’]{} z\_i D\^[n[-]{}3]{}\_[’]{} |z\_i \_a’ (\_[ba]{}) \_a’ (\_[ba]{})&& E({})E({})’ ({z},{})’ ({|z}, {}).\[openclosed\] In , the open string tree amplitude is quite nicely given by a sum of disk-boundary integrals of $\det'\Psi$, over $(n{-}3)!$ solutions; in contrast, for closed string amplitudes, in addition to sphere integrals of two copies of $\det'\Psi$, one needs to sum over $(n{-}3)!^2$ solutions! This can be compared with gravity amplitudes, , where by KLT orthogonality one sums over only $(n{-}3)!$ terms. In particular, in maximal supergravity in four dimensions, tree amplitudes enjoy an enhanced R-symmetry $SU(8)$ due to KLT orthogonality, but it is well known that closed-string amplitudes only have $SU(4)\times SU(4)$ symmetry.
In [@Gross:1987ar] the high energy limit, *i.e.* the limit $M^2\gg 1/\alpha'$ where kinematic invariants have a typical energy scale, $s_{ab}\sim M^2$, was considered, and disk/sphere integrals can be evaluated using saddle point approximations. We observe that the saddle point equations are exactly the scattering equations we found in field theory. For disk integrals, $z$’s are real, and we have \_[a<b]{}|z\_[ab]{}|\^[’s\_[a b]{}]{}=e\^[1 2\_[ab]{} ’ s\_[a b]{}|z\_[ab]{}|]{} 0= \_[ba]{}s\_[ab]{}|z\_[ab]{}|=\_[ba]{} , for $a=1,\ldots, n$. For sphere integrals, we have $\ln |z_{ab}|^2=\ln (z_{ab}\bar z_{ab})$ instead of $\ln |z_{ab}|$. Note that all the kinematic invariants are real, the saddle point equations for $z$ and those for $\bar z$ are equivalent. In the limit $\alpha' s_{ab}\sim \alpha' M^2 \to \infty$, the sphere integral can be approximated by a sum over saddle points labeled by $i\in\{1,2,\ldots,(n{-}3)!\}$. Explicitly, we have that $ M^{\mathrm{closed}}_n$ can be written as & & \_[i,j,j=1]{}\^[(n[-]{}3)!]{} E(\^[(j)]{})E(\^[(j)]{}) +(1[M\^2 ’]{})& = & \_[i=1]{}\^[(n[-]{}3)!]{} E\^2({k,,\^[(i)]{}})+(1[M\^2 ’]{}), where on the second line we identified $\sigma^{(i)}_a=z^{(i)}_a$, $\tilde\sigma^{(i)}_a=\bar z^{(i)}_a$ using Proposition 2, and the leading term becomes a sum over $(n{-}3)!$ solutions. Hence we see that simplifications of closed string amplitudes in the high energy limit are closely related to the KLT orthogonality . It is very intriguing that the high energy limit related to $\alpha'\to \infty$ bears significant similarities with the field theory limit, $\alpha'\to 0$. It would be fascinating to further explore potential connections.
The authors would like to thank Erik Schnetter for useful discussions on the numerical algorithms. This work is supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.
*Note*: While this manuscript was being prepared for submission, the paper [@Stieberger:2013nha] appeared which has overlap with some of the results of this work.
Numerical Algorithm {#app:numericalalgorithm}
===================
In this appendix we discuss some details of the numerical implementation of the algorithm described in section 3 for finding solutions of the scattering equations.
In physical applications one is usually interested in real kinematic data, i.e., $s_{ab}\in \mathbb{R}$. In this case, it is likely that when taking values of $\epsilon$ in the interval $[0,1]$ one can come close to a singularity of the Jacobian of the equations. This is undesirable as most numerical algorithms use Newton’s method and could fail. In this case a more convenient choice is to use a unit semicircle centered at $\epsilon = 1/2$ in the complex $\epsilon$ plane. This technique also has the advantage of avoiding falling into a cycle outside the basin of attraction of the solution.
In theoretical computations one usually chooses rational kinematic data, i.e., $s_{ab}\in \mathbb{Q}$. The reason is that physical amplitudes are computed as a sum of a rational function of $\sigma$’s evaluated over all $(n-3)!$ solutions and therefore it is guaranteed to be a rational number. Using enough working precision one can rationalize and hence obtain the exact answer even though the intermediate steps were numerical. In this particular case, an additional issue can arise; using a large number of rational kinematic invariants increases the possibilities of simple linear relations arising among them and therefore the number of possible Jacobian singularities. In this case one should start with rational numbers that are unlikely to satisfy simple linear relations. For example, instead of generating rational numbers for $s_{ab}$ by using the function [`RandomInteger`]{}$[\{-100,100\}]/100$ in [<span style="font-variant:small-caps;">Mathematica</span>]{} one should use [`RandomInteger`]{}$[\{-10^6,10^6\}]/10^6$.
A More Explicit Version of the Algorithm
----------------------------------------
In order to implement algorithm in practice one can start by defining the equations as follows \_[ba]{}=0,a{4,5,…,n}. A good starting point for the algorithm is to deform the system of equations for $n$ particles down to seven points where all solutions can be easily found using, e.g., [`NSolve`]{} in <span style="font-variant:small-caps;">Mathematica</span>. This is achieved by simply taking \_a =
1 a { 1,2,…, 7}\
0a{ 8,9,…, n },
This system has $24$ solutions and the algorithm can then be applied by letting $\epsilon_8=1/10^3$ and proceeding as in section 3. This will lead to $5$ solutions for each of the original $24$ solutions at $\epsilon_8 =1$. This can then be continued until reaching the desired $(n-3)!$ solutions. It is important to mention that in this implementation, one does not have to perform each step to a very high working precision. The reason is that once $(n-3)!$ approximate solutions have been found, one can use them as seeds in a [`FindRoot`]{} procedure with the desired working precision. This is actually welcome as the function [`FindRoot`]{} can fail to reach the default precision goals for intermediate values of $\epsilon$ but this is not important as long as one can continue to get an approximate value of a solution at $\epsilon =1$.
Example
-------
Here we give as an example the result of the computation of one solution for $n=15$. The kinematic data was chosen as random rational numbers for $s_{ab}$ with $1 \leq a < b \leq 14$ given below in lexicographic order $$\left\{\frac{49}{13},-\frac{69}{26},-\frac{17}{6},-\frac{29}{25},
\frac{52}{11},\frac{55}{3},-\frac{43}{34},-5,-\frac{5}{16},-\frac{76}{45},-\frac{8}{43},
\frac{73}{45},-\frac{15}{47},-\frac{25}{3},\frac{90}{37},2,-24,\frac{35}{24},56,\right.$$
$$-\frac{49}{23},
-\frac{20}{3},\frac{1}{11},\frac{16}{41},\frac{31}{16},-\frac{31}{7},\frac{79}{19},
-\frac{49}{25},-\frac{58}{7},-\frac{10}{3},\frac{16}{7},\frac{9}{11},\frac{21}{10},\frac{71}{21},
\frac{17}{19},-\frac{47}{12},\frac{5}{24},-\frac{31}{48},-\frac{65}{19},$$
$$\frac{32}{37},-\frac{85}{41},
-\frac{35}{13},-\frac{79}{44},\frac{7}{5},-\frac{9}{5},\frac{19}{5},\frac{23}{15},-\frac{1}{2},\frac{74}{11},
-\frac{38}{43},-\frac{56}{9},-\frac{13}{4},-\frac{7}{4},-\frac{39}{35},\frac{11}{7},-\frac{3}{34},\frac{21}{22},
\frac{56}{23},$$
$$\frac{78}{35},\frac{22}{7},\frac{6}{13},-\frac{48}{25},-\frac{49}{43},\frac{4}{33},-\frac{5}{4},
-\frac{39}{23},\frac{29}{28},\frac{16}{7},-\frac{15}{8},\frac{25}{12},-\frac{21}{25},-\frac{95}{14},
-\frac{7}{5},\frac{5}{8},\frac{17}{36},\frac{31}{43},7,\frac{42}{31},\frac{33}{43},$$
$$\left. \frac{80}{23},
-\frac{39}{38},\frac{27}{17},\frac{8}{7},-\frac{11}{27},\frac{94}{27},-\frac{11}{8},9,-\frac{43}{20},
\frac{82}{33},-\frac{61}{6},\frac{10}{33} \right\} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$ and with s\_[13,14]{} = -\_[1<a<b<14]{}s\_[ab]{} The rest of the kinematic invariants, i.e., those of the form $s_{15,a}$, are obtained by using momentum conservation. The data does not satisfy any Gram determinant equations and therefore it can be taken to correspond to general dimensions. Note that working in general dimensions means that this computation is at least equivalent to that of a computation in a four dimensional $k=7$ or N${}^5$MHV sector.
Below we show the solution to $25$ digits of precision but using it as a seed one can increase the precision at very little cost.
Recall that using the $\SL2C$ invariance of the equations we choose to fix the value of the first three $\sigma$’s. Our choice is $\sigma_1\to \infty$, $\sigma_2 =0$ and $\sigma_3=1$. The remaining $12$ $\sigma$’s are: $$\begin{array}{ccl}
\sigma_4 & = & 0.5496193504318654421011817+0.3319870690807634365609033 i \\
\sigma_5 & = & 0.13759872147189287197217158+0.00723562043413197816599421 i \\
\sigma_6 & = & 0.6947313566266571926077990+0.0526139619994048265322461 i \\
\sigma_7 & = & 1.0865440087837388144794260+0.3768740374717843745501630 i \\
\sigma_8 & = & 23.33664228765061141631063-0.33658205624253639217240 i \\
\sigma_9 & = & 2.160942080177577331147289+1.124899577655694226591314 i \\
\sigma_{10} & = & 3.265453266150672900415255-3.358839499723287173276593 i \\
\sigma_{11} & = & 0.04841609220705573133055635+0.00901941920135424505351115 i \\
\sigma_{12} & = & 3.124269149584940975154808-2.943058583858387329266730 i \\
\sigma_{13} & = & 1.1676848034334310319712031-0.0732151746711586090152780 i \\
\sigma_{14} & = & -19.36286341034387769858701+4.73029434693411454422806 i \\
\sigma_{15} & = & -1.1581504995107345733001948+0.0739431466630364554298497 i. \\
\end{array}$$
Details of the Proof of Proposition 3
=====================================
\[app:prop2\]
In the proof of KLT orthogonality (Proposition 2), one important ingredient is Proposition 3, which states that \[app:prop3\] K\_n=(-1)\^n’\^[(i)]{}, where K\_n:=\_[,S\_[n-3]{}]{} S\[|\], \[ares\]and $\psi^{(n)}$ is a co-rank one $(n-2)\times(n-2)$ matrix \_[ab]{}:=
,&ab\
-\_[ca]{}\_[ac]{},&a=b
, where $a,b\in \{1,2,\ldots ,n-2\}$. And $\det'$ is any $(n-3)\times(n-3)$ minor of $\psi^{(n)}$. Here both permutations $\alpha$ and $\beta$ act on the set of labels $\{2,3,\ldots,n-2\}$. In the formula above we use the natural convention in which the label $i$ in $\beta(i)$ denotes exactly the position in the whole sequence $(1,\beta(2),\ldots,\beta(n-2))$. The reason we mention this is that below the proof requires the introduction of other permutations with different properties.
The first step in the proof of is to study all terms in with a given permutation $\alpha$. This motivates the introduction of X’\_n(1,(2),…,(n-2)):=\_[S\_[n-3]{}]{} \[jis\]so that becomes \[app:KnXn\] K\_n=\_[S\_[n[-]{}3]{}]{}.
Preliminary Simplification of $K_n$
-----------------------------------
Next we prove the identity \[app:Xn\] X’\_n(1,(2),…,(n-2))= \_[i=2]{}\^[n-2]{}\_[j=1]{}\^[i-1]{}. This is done recursively. Without loss of generality, we can fix $\alpha$ to be the identity permutation, i.e., $\alpha(i)=i$.
Firstly is trivially correct at $n=4$. When $n>4$, we assume that holds for the case $n-1$, and show that $X'_n(1,2,\ldots,n-2)$ defined by satisfies the recursion relation \[app:Xnrecursion\] X’\_n(1,2,…,n-2)=X’\_[n-1]{}(1,2,…,n-3)\_[m=1]{}\^[n-3]{}, which obviously holds for the r.h.s. of .
Clearly $\sigma'_{n-2}$ only appears in the factor multiplying $X'_{n-1}$ on the r.h.s. of , so the strategy for proving the recursion is simply to collect terms in $X'_{n}$ according to factors of the form $1/\sigma'_{m,n-2}$ for every $m\in\{1,\ldots,n-3\}$. In this operation label $n-2$ is special, therefore it is convenient to define permutations $\gamma\in S_{n-4}$ which act on the reduced label set $\{2,\ldots,n-3\}$. Moreover, we are interested in collecting terms where $n-2$ sits at a particular position, say $\beta(k) =n-2$. For terms of this form we define (i)=
(i),&i<k\
(i+1),&ik.
Next, modify the terms in $X'_n$ for which $k<n-2$ (i.e. the label $n-2$ does not sit at the end of the original sequence) as follows =(-), where the common factor on r.h.s. above is exactly the one naturally associated to the reduced sequence $(1,\gamma(2),\ldots,\gamma(n-3))$. As a result of this manipulation, every term in $X'_n$ possesses a unique factor $1/\sigma'_{m,n-2}$ for some $m\in\{1,\ldots,n-3\}$.
We first collect terms with $1/\sigma'_{1,n-2}$, which can be observed to always have the form . Since S\[2,…,n-2|n-2,(2),…,(n-3)\]=s\_[1,n-2]{}S\[2,…,n-3|(2),…,(n-3)\], they directly add up to be \[app:collection1\] X’\_[n-1]{}(1,…,n-3). Then we collect terms with $1/\sigma'_{m,n-2}$ for any $m\in\{2,\ldots,n-3\}$. Observe that for a particular sequence $(1,\gamma(2),\ldots,\gamma(n-3))$, together with a particular identification $m=\gamma(l)$ for some $l\in\{2,\ldots,n-3\}$, the contributing terms always come in pairs, with a common denominator $\sigma'_{1,\gamma(2)}\cdots\sigma'_{\gamma(n-4),\gamma(n-3)}\sigma'_{m,n-2}$, and it is easy to see that the two numerators add up to be
S\[2,…,n-2|…,(l),n-2,…\]-S\[2,&…,n-2|…,n-2,(l),…\]\
=&s\_[m,n-2]{}S\[2,…,n-3|…,(l),…\].
Note that the permutation $\gamma$ in the final object $S[2,\ldots,n-3|\gamma]$ exactly matches that associated with the denominator. Considering all possible $\gamma\in S_{n-4}$, it is straightforward to see that all terms which contain the factor $1/\sigma'_{m,n-2}$ add up to \[app:collection2\] X’\_[n-1]{}(1,…,n-3). Combining both and , we can directly see that the recursion holds, thus verifying .
Residue Computation
-------------------
In order to complete the proof of Proposition 3, simply note that both $K_n$ and $\det' \psi^{(n)}$ are rational functions of $\sigma_1$. In this part of the proof the set $\{ \sigma \}$ is not assumed to be a solution of the scattering equations and therefore $\sigma_1$ is a completely unconstrained complex variable.
A simple observation is that both $K_n(\sigma_1)$ and $\det' \psi^{(n)}(\sigma_1)$ vanish as $\sigma_1\to \infty$. Therefore, it is sufficient to show that $K_n(\sigma_1)$ and $(-1)^n\det'\psi^{(n)}(\sigma_1)$ have the same poles and residues at finite values of $\sigma_1$. Clearly, the only possible pole locations are at $\sigma_1=\sigma_a$ for $a=2,3,\ldots,n{-}2$. Since both functions are manifestly invariant under permutations of $2,3,\ldots,n{-}2$, we only need to consider one of the $n{-}3$ poles, e.g. that at $\sigma_1=\sigma_2$.
Let us explicitly compute the residue at $\sigma_1=\sigma_2$ for both $K_n(\sigma_1)$ and of $(-1)^n\det' \psi^{(n)}(\sigma_1)$.
From together with , it is easy to compute the residue of $K_n(\sigma_1)$ at $\sigma_1=\sigma_2$, which gives, \_[\_1=\_2]{} K\_n(\_1)=\_[S\_[n[-]{}4]{}]{}\_[a=3]{}\^[n[-]{}2]{}1[\_[(a[-]{}1),(a)]{}]{}\^[a[-]{}1]{}\_[b=1]{},thus we only need to prove that the residue of $\det'\psi^{(n)}(\sigma_1)$ at $\sigma_1=\sigma_2$ is the same.
Given that proving the equality between the residues amounts to proving the equality of the two functions, here we proceed by induction in the statement $K_n(\sigma_1)= (-1)^n\det' \psi^{(n)}(\sigma_1)$. In other words, we check that it is valid for $n=4$ and assume its validity for $n-1$ to show that for the case $n$ both functions have the same residue at $\sigma_1 = \sigma_2$.
The starting point, $K_4=\det'\psi^{(4)}=\frac {s_{12}}{\sigma_{12}\sigma'_{12}}$ is easily checked by inspection.
For $\det'\psi^{(n)}$, we choose to delete the first row and column and denote it as $|\psi^{(n)}|^1_1$. In the minor the only term that has the pole $\sigma_{1}=\sigma_2$ is the entry $\psi_{2,2}$, and we find \_[\_1=\_2]{}’\^[(n)]{}(\_1)=-|\^[(n)]{}(\_1=\_2)|\^[12]{}\_[12]{}, where $|\psi^{(n)}|^{12}_{12}$ denotes the minor obtained by deleting rows and columns $1$ and $2$.
Note that when evaluated at $\sigma_1=\sigma_2$, $$\psi_{1a}+\psi_{2a}=\frac1 {\sigma_{1a}}(\frac{s_{1a}}{\sigma'_{1a}}+\frac{s_{2a}}{\sigma'_{2a}}),$$ and the minor $|\psi^{(n)}(\sigma_1=\sigma_2)|^{12}_{12}$ can be rewritten as a minor of the following $n{-}3$ by $n{-}3$ matrix $\hat\psi^{(n{-}1)}$, \_[1 c]{}=\_[c 1]{}=(+),\_[ab,ab]{}=\_[ab]{},\_[aa]{}=-\_[ba,b2]{}\_[ab]{}, for $c=3,4,\ldots,n$ and $a,b=1,3,4,\ldots,n{-}2$, thus $|\psi^{(n)}(\sigma_1=\sigma_2)|^{12}_{12}=|\hat\psi^{(n{-}1)}|^1_1$. Since $|\hat\psi^{(n{-}1)}|^1_1$ is invariant under the permutations of the labels $1,3,\ldots,n{-}2$, by the induction assumption, it is the same function as $(-1)^{n-1}\hat K_{n{-}1}$ given by , in which $\hat X'_{n{-}1}$ is given by , X’\_[n[-]{}1]{}(1,3,…,[n[-]{}2]{})=\_[i=3]{}\^[n[-]{}2]{}(+\_[j=3]{}\^[i[-]{}1]{})=\_[i=3]{}\^[n[-]{}2]{}\_[j=1]{}\^[i[-]{}1]{}and its permutations on the set of $3,\ldots,n{-}2$. Thus we conclude that \_[\_1=\_2]{}’\^[(n)]{}(\_1)=(-1)\^[n]{}\_[S\_[n[-]{}4]{}]{}\_[i=3]{}\^[n[-]{}2]{}1 [\_[(i[-]{}1),(i)]{}]{}\_[j=1]{}\^[i[-]{}1]{}.
This completes the proof of Proposition 3.
[^1]: The last equality in is obtained by adding to the second expression a term $v_a \sum_{b\neq a}k_a\cdot k_b (n{-}1)\sigma_a^{n{-}2}$, which vanishes due to momentum conservation.
[^2]: To see the $\SL2C$ invariance of the equations one has to use momentum conservation.
[^3]: For example, this is implemented in [<span style="font-variant:small-caps;">Mathematica</span>]{} as the function [`FindRoot`]{}
[^4]: The diagonal terms in the Cachazo-Geyer formula can be trivially simplified to give those in (\[cases\]) by using the scattering equations.
[^5]: The precise notation used in [@BjerrumBohr:2010ta] is $S[\alpha|\beta]_1$ reflecting the fact that $1$ was chosen as a pivot.
[^6]: Stieberger and Taylor have also provided a proof of this connection in [@Stieberger:2013hza] as part of the equivalence among several formulas. Our proof differs from theirs in that it is tailored to Proposition 3.
|
---
author:
- 'Olga Boyko, Olga Martynyuk, Vyacheslav Pivovarchik '
title: On spectra of quadratic operator pencils with rank one gyroscopic linear part
---
-17mm -10mm 17truecm 23,5truecm
\[theorem\][��।������]{} \[theorem\][�����]{} \[theorem\][�।�������]{}
Abstract. The spectrum of a selfadjoint quadratic operator pencil of the form\
$\lambda^2M-\lambda G-A$ is investigated where $M\geq 0$, $G\geq 0$ are bounded operators and $A$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $G$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $G$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $A$ are given.
keywords: quadratic operator pencil, gyroscopic force, eigenvalues, algebraic multiplicity
2010 Mathematics Subject Classification : Primary: 47A56, Secondary: 47E05, 81Q10
Introduction.
=============
Quadratic operator pencils of the form $L(\lambda)=\lambda^2M-\lambda G-A$ with a selfadjoint operator $A$ bounded below describing potential energy, a bounded symmetric operator $M\geq 0$ describing inertia of the system and an operator $G$ bounded or subordinate to $A$ occur in different physical problems where, in most of cases, they have spectra consisting of normal eigenvalues (see below Definition 2.2). Usually the operator $G$ is symmetric (see, e.g. [@KO], [@Sh] and Chapter 4 in [@MP]) or antisymmetric (see [@Pumz] and Chapter 2 in [@MP]). In the first case $G$ describes gyroscopic effect while in the latter case damping forces.
The problems in which gyroscopic forces occur can be found in [@Fe], [@DD], [@Th], [@Za], [@Mo], [@Mi1], [@Mi2], [@PI]. The spectra of problems described in these papers may contain complex eigenvalues what leads to instability of the corresponding dynamical system. The gyroscopic stabilization in terms of spectral theory means, roughly speaking, that the spectrum of an operator pencil $L(\lambda)=\lambda^2M-\lambda G-A$ with selfadjoint $A$ and symmetric $M$ and $G$ can be real while the spectrum of $L_0(\lambda)=\lambda^2M-A$ contains pure imaginary eigenvalues. A sufficient condition for the pencil $L$ to have only real eigenvalues is the overdamping condition: $$\label{1.1}
(Gy,y)^2+4(My,y)(Ay,y)\geq 0$$ here and in what follows $(\cdot,\cdot)$ stands for the inner product in the corresponding Hilbert space. In [@M] pencils which satisfy (\[1.1\]) are called hyperbolic.
In the present paper we consider the case of symmetric operator $G\geq 0$, i.e. $(Gy,y)\geq 0$ for all $y\in D(G)$. As far as we know location of complex eigenvalues of the corresponding operator pancil was not considered before. Up to our knowledge, also the case of rank one operator $G$ in the pencil $L(\lambda)$ was not considered.
In Section 2 we obtain some general results on location of eigenvalues of the pencil $L$.
In Section 3 we consider the case of rank one operator $G$ (problems with such operator $G$ can be found e.g. in [@IN]). We show that the real eigenvalues are in certain sense interlaced. In case of $A>>0$ (i.e. $A\geq \epsilon I$ for some $\epsilon>0$) this interlacing is equivalent to the ’self-interlacing’ described in [@Tya] for finite dimensional (matrix) problems.
It should be mentioned that the spectra of problems described by the pencil $\lambda^2M-i\lambda K-A$ with selfadjoint $K\geq 0$, $M\geq 0$ (i.e. $(Ky,y)\geq 0$, $(My,y)\geq 0$ for all $y\in D(K)$ and $y\in D(M)$, respectively) and selfadjoint $A$ bounded below can be described also as sets of zeros of the shifted generalized Hermite-Biehler functions of the form $\omega(\lambda)=P(\lambda^2)+i\lambda Q(\lambda^2)$ where $P(\lambda^2)$, $\lambda Q(\lambda^2)$ (or $\lambda P(\lambda^2)$, $\lambda^2 Q(\lambda^2)$) are sine-type functions and $\frac{Q(z)}{P(z)}$ is essentially positive Nevanlinna function (see [@PW] or [@MP]).
Similarly, the spectra of problems described by the pencil $\lambda^2M-\lambda G-A$ with selfadjoint $G\geq 0$, $M\geq 0$ and selfadjoint $A$ bounded below can be described also as sets of zeros of the functions of the form $\omega(\lambda)=P(\lambda^2)+\lambda Q(\lambda^2)$ where $\frac{Q(z)}{P(z)}$ is again essentially positive Nevanlinna function (see the function $\omega$ in (\[yyy\]) in Section 4 as an example).
In Section 4 the results obtained in Section 3 are applied to a Sturm-Liouville boundary value problems with a boundary condition linear in the spectral parameter. Our results give necessary conditions (Theorem 4.1) for a sequence of numbers to be the spectrum of such a problem (problem (\[4.1\])–(\[4.3\]). In our future publication we will show that these conditions are also sufficient and give a method of recovering the potential of the Sturm-Liouville equation. We will also show that it is enough to know one spectrum of such a boundary value problem (problem (\[4.1\])–(\[4.3\]) to recover the potential of the Sturm-Liouville equation.
We give also in Section 4 an example of a boundary value problem the spectrum of which consists of two subsequences one of which is symmetric with respect to the real and to the imaginary axes and independent of $G$.
Abstract Results
================
Let us denote by $B(H)$ the set of bounded closed operators acting on a separable Hilbert space $H$. We deal with the following quadratic operator pencil $$L(\lambda)=\lambda^2M-\lambda G-A,$$ where $M\in B(H)$, $G\in B(H)$ and $A$ is a selfadjoint operator on $H$. Since $D(M)=D(G)=H$, the domain of the pencil is chosen as usually: $D(L(\lambda))=D(M)\cap D(G)\cap D(A)=D(A)$. Thus, it is independent of $\lambda$.
In what follows we assume the following condition to be satisfied:
: [*$M\in B(H)$, $M\geq 0$, $G\in B(H)$ and $G\geq 0$, $A=A^{*}\geq -\beta I$ ($\beta$ is a positive number); for some $\beta_1>\beta$ there exists $(A+\beta_1I)^
{-1}\in S_{\infty}$, where by $S_{\infty}$ we denote the set of compact operators on $H$; $ker A\cap ker G\cap ker M=\{0\}$.*]{}
.
(see, e.g. [@MP], Definition 1.1.3) [*1. A number $\lambda_0\in{{\bf C}}$ is said to be an eigenvalue of the pencil $L$ if there exists a vector $y_0\in D(A)$ (called an eigenvector of $L$, corresponding to $\lambda_0$ ) such that $y_0\not=0$ and $L(\lambda_0)y_0=0$. Vectors $y_1, ...,y_{m-1}$ are called associated to $y_0$ if* ]{} $$\label{@}
\mathop{\sum}\limits_{s=0}^{k}\frac{1}{s!}\left.\frac{d^sL(\lambda)}{d\lambda^s}
\right|_{\lambda=\lambda_0}y_{k-s}=0, \ \ (k=1, 2, ..., m-1).$$ [*The number $m$ is said to be the length of the chain composed of the eigen- and associated vectors $y_0, y_1, ..., y_{m-1}$.\
2. The geometric multiplicity of an eigenvalue is defined to be the number of the corresponding linearly independent eigenvectors. The algebraic multiplicity of an eigenvalue is defined to be the greatest value of the sum of the lengths of chains corresponding to linearly independent eigenvectors. An eigenvalue is called semisimple if its algebraic multiplicity equals its geometric multiplicity. An eigenvalue is called simple if its algebraic multiplicity is 1.\
3. An eigenvalue $\lambda_0$ is said to be isolated if it has some deleted neighborhood contained in the resolvent set $\rho(L)$. An isolated eigenvalue $\lambda_0$ of finite algebraic multiplicity is said to be normal.* ]{}
In the case of linear monic operator pencils $\lambda I-A$ with bounded operator $A$ this definition of a normal eigenvalue coincides with the corresponding definition in [@GK1] (Chapter I, paragraph 2) for operators.
The proof of this lemma is the same as the proof of Lemma 1.1.11 in [@MP].
\
[*4. If $A\geq 0$, then the spectrum $\sigma(L)$ of $L$ (if not empty) is located on the real axis.\
5. If $A>>0$, i.e. $A\geq\epsilon I$, $\epsilon>0$, then $\sigma (L)\subset {{\bf R}}\backslash \{0\}$.* ]{}
[[**Proof.** ]{} ]{}Since the pencil $L$ is an analytic Fredholm operator valued function, its spectrum consists of eigenvalues of finite algebraic multiplicity and either $\sigma(L)={{\bf C}}$ or all the eigenvalues are normal, see e.g., [@GGK], Chapter XI, Corollary 8.4.
Let $y_0$ be an eigenvector corresponding to the eigenvalue $\lambda_0$. Then $$(L(\lambda_0)y_0,y_0)=0, \ \ \ (y_0\not=0),$$ where $(\cdot,\cdot)$ is the inner product in $H$ and consequently, $$\label{2.1}
((\mbox{Re}\lambda_0)^2-(\mbox{Im}\lambda_0)^2)(My_0,y_0)-\mbox{Re}\lambda_0
(Gy_0,y_0)-(Ay_0,y_0)=0$$ and $$\label{2.2}
\mbox{Im}\lambda_0(2\mbox{Re}\lambda_0(My_0,y_0)-(Gy_0,y_0))=0.$$ If $\mbox{Im}\lambda_0\not=0$, then in view of Condition I the inequality $\mbox{Re}\lambda_0\geq 0$ follows from (\[2.2\]). Hence $\sigma(L)={{\bf C}}$ is impossible and assertions 1 and 2 are proved.
If $G>0$ and $\mbox{Im}\lambda_0\not=0$, then (\[2.2\]) implies $\mbox{Re}\lambda_0\not= 0$ and therefore $\mbox{Re}\lambda_0> 0$. Assertion 3 is proved.
Now let $A\geq 0$. If $\mbox{Im}\lambda_0\not=0$ then multiplying (\[2.2\]) by $\frac{\mbox{Re}\lambda_0}{\mbox{Im}\lambda_0}$ we subtract the resulting equation from (\[2.1\]) and obtain $$- ((\mbox{Re}\lambda_0)^2+(\mbox{Im}\lambda_0)^2)(My_0,y_0)-(Ay_0,y_0)=0.$$ This implies $(My_0,y_0)=(Ay_0,y_0)=0$. Since the operators $M$ and $A$ are selfadjoint we obtain $My_0=Ay_0=0$, and consequently $Gy_0=0$. This contradicts Condition I. Thus, statement 4 is proved.
If $A>>0$ then in addition to the previous arguments we have $0\notin \sigma(L)$. That means statement 5 is valid.
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[[**Proof.** ]{} ]{}1. Let $\lambda_0$ be an eigenvalue of $L$ located on $(-\infty,0)$. Let us denote by $y_0$ (one of) the corresponding eigen- and by $y_1$ the first associated vector of the chain. By Definition 2.2 $$\label{2.3}
\lambda_0^2My_1- \lambda_0Gy_1-Ay_1+2\lambda_0My_0-Gy_0=0.$$ Multiplying (\[2.3\]) by $y_0$ we obtain $$\label{2.4}
((\lambda_0^2- \lambda_0G-A)y_1,y_0)+((2\lambda_0M- G)y_0,y_0)=0.$$ Taking into account that $\lambda_0$ is real we obtain from (\[2.4\]): $$(y_1,(\lambda_0^2M- \lambda_0G-A)y_0)+((2\lambda_0M- G)y_0,y_0)=0,$$ what means $$\label{2.5}
((2\lambda_0M-G)y_0,y_0)=0.$$ Equation (\[2.5\]) is possible for $\lambda_0<0$ only if $(My_0,y_0)=(Gy_0,y_0)=0$. Since $M$ and $G$ are selfadjoint we arrive at $My_0=Gy_0=0$ and, consequently, $L(\lambda_0)y_0=-Ay_0=0$. Thus, $y_0\in\mbox{ker}M\cap \mbox{ker}G\cap\mbox{ker}A$. Then due to Conditions I we have $y_0=0$, a contradiction.
2\. Let $G>0$ on $ker A$ and let $y_0\not=0$ be an eigenvector corresponding to the eigenvalue $\lambda_0=0$ and therefore $y_0\in ker A$. Then (\[2.5\]) implies $(Gy_0,y_0)=0$, a contradiction.
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[**Lemma 2.6**]{} [*If $M>>0$ (i.e. $(My,y)\geq m ||y||^2$, $m>0$) then all (if any) the algebraically nonsimple real eigenvalues lie on the interval $[0,\frac {1}{2m}||G^{\frac{1}{2}}||^2]$.*]{}
[[**Proof.** ]{} ]{}This result follows immediately from (\[2.5\])
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.
[[**Proof.** ]{} ]{}Let $0\not=y_0\in\mbox{ker}A$ and let $y_1$ be the first associated vector of the chain. Then $$\label{2.8}
\left.\frac{dL(\lambda)}{d\lambda}\right|_{\lambda=0}y_0+L(0)y_1=
-Gy_0-Ay_1=0.$$ If $y_0\in\mbox{ker}G$ then $y_1$ can be chosen equal to 0. If $y_0\not\in\mbox{ker}G$, then (\[2.8\]) implies $$-(Gy_0,y_0)-(Ay_1,y_0)=-(Gy_0,y_0)-(y_1,Ay_0)=-(Gy_0,y_0)=0.$$ Combining the last equation with the condition $G\geq 0$ we obtain $Gy_0=0$, a contradiction. It remains to prove that in case of $y_0\in ker G$, $y_1=0$ the third vector of the chain (see Definition 2.2) does not exist. Suppose it does exist and denote it by $y_2$. Then, see (\[@\]), $$-Ay_2- Gy_1+My_0= - Ay_2+My_0=0.$$ Consequently, $$0=-(Ay_2,y_0)+(My_0,y_0)=(My_0,y_0).$$ Then $My_0=0$ and $My_0=Gy_0=Ay_0=0$ what contradicts Condition I.
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Let us introduce the following parameter-dependent operator pencil: $$L(\lambda,\eta)=\lambda^{2}M-\lambda\eta G-A.$$ It is clear that $L(\lambda,1)=L(\lambda)$ and $L(\lambda,0)=\lambda^2M-A$.
(see [@MP], Theorem 1.2.7).
*Let $\eta_0\in {{\bf C}}$ and let the domain $\Omega\subset {{\bf C}}$ be a domain which contains exactly one eigenvalue $\lambda_0$ of the pencil $L(\lambda,\eta_0)$. Denote by $m$ the algebraic multiplicity of $\lambda_0$. Then there exist numbers $\epsilon>0$ and $m_1\in{{\bf N}}$ $m_1\leq m$, such that the following assertions are true in the deleted neighborhood $0<|\eta-\eta_0|<\epsilon$:\
1. The pencil $L(\lambda ,\eta)$ possesses exactly $m_1$ distinct eigenvalues inside the domain $\Omega$. Those eigenvalues can be arranged in groups $\lambda_{sj} (\eta)$, ($s=\overline{1,l}$; $j=\overline{1,p_s}$; $\mathop{\sum}\limits _{s=1}^{l}p_s=m_1$) in such a way that the functions of the group, i.e. $\lambda_{s1},
\lambda_{s2},...,\lambda_{sp_s}$ compose a complete set of a $p_s$-valued function. In this case those eigenvalues can be presented in the form of the following Puiseux series expansion $$\lambda_{sj}(\eta)=\lambda_0+\mathop{\sum}\limits_{k=1}^{\infty}a_{sk}
(((\eta-\eta_0)^{\frac{1}{p_s}})_j)^k, \ \ (j=1,2, ..., p_s),$$ holds, where, for $j=1, 2, ..., p_s$ $$((\eta-\eta_0)^{\frac{1}{p_s}})_j=|\eta-\eta_0|^{\frac{1}{p_s}}exp\left(\frac{2\pi(j-1)+i\arg(\eta-\eta_0)}{p_s}\right).$$*
2\. A basis of the eigen-space corresponding to $\lambda_{sj}(\eta)$ can be presented in the following form $$y_{sj}^{(q)}(\eta)=y_{s0}^{(q)}+\mathop{\sum}\limits_{k=1}^{\infty}y_{sk}^{(q)}
(((\eta-\eta_0)^{\frac{1}{p_s}})_j)^k, \ \ (j=1,2, ...,p_s), \ \
(q=1,2, ..., \alpha_s),$$ where $\alpha_s$ is the geometric multiplicity of $\lambda_{sj}(\eta)$, $y_{s0}^{(q)}$ belong to the eigen-subspace of $L(\lambda,\eta_0)$ corresponding to the eigenvalue $\lambda_0$.
It should be mentioned that Theorem 2.8 is a generalization of the Weierstrass theorem on function analytic in two variables (see, e.g. [@Mh] p. 476). It was proved in [@GS] for bounded operators.
[**Remark 2.9**]{} If $\lambda_0\not=0$ is a real or pure imaginary eigenvalue of $L(\cdot, \eta_0)$ for some real $\eta_0$ and if eigenvalues of $L(\cdot,\eta)$ near $\lambda_0$ for real $\eta$ near $\eta_0$ are also real or pure imaginary, then $p_s=1$ for all $s$ in Theorem 2.8.
[**Remark 2.10**]{} If both $\eta$ and $\eta_0$ are real so that $\arg(\eta-\eta_0)$ is an integer multiple of $\pi$, then we can put $$\label{2.12}
((\eta-\eta_0)^{\frac{1}{p_s}})_j=|\eta-\eta_0|^{\frac{1}{p_s}}exp\left(\frac{2\pi(j-1)+i\delta_i\arg(\eta-\eta_0)}{p_s}\right)$$ where $\delta_i$ is an odd integer. This would change the numeration of the roots for $\eta-\eta_0<0$ if $\frac {\delta_i-1}{p_s}$ is not even integer but allows us when taking $\delta_s=p_s$ if $p_k$ is odd to have $((\eta-\eta_0)^{\frac{1}{p_s}})_1$ real for all $\eta$ in a real neighborhood of $\eta_0$.
If we additionally assume that $G$ is bounded and boundedly invertible then we can write, assuming $\lambda\not=0$, $$L(\lambda,\eta)=\lambda G^{\frac{1}{2}}(-\eta I+\lambda G^{-\frac{1}{2}}MG^{-\frac{1}{2}}-\lambda^{-1}G^{-\frac{1}{2}}AG^{-\frac{1}{2}})G^{\frac{1}{2}}.$$ Hence $$L(\lambda,\eta)=\lambda G^{\frac{1}{2}}Q(\lambda,\eta)G^{\frac{1}{2}}$$ where $$Q(\lambda,\eta)=-\eta I+\lambda G^{-\frac{1}{2}}MG^{-\frac{1}{2}}-\lambda^{-1}G^{-\frac{1}{2}}AG^{-\frac{1}{2}}.$$
For $\lambda\not=0$ we note that if $ker Q(\lambda,\eta)\not=\{0\}$ then its dimension is the geometric multiplicity of the eigenvalue $\lambda$ of the pencil $Q(\cdot,\eta)$ as well as the geometric multiplicity of the eigenvalue $\eta$ of the pencil $Q(\lambda,\cdot)$. The algebraic multiplicities can be different, but for $\lambda\in {{\bf R}}$, we have a standard problem for a self-adjoint operator with the spectral parameter $\eta$, and hence all eigenvalues of $Q(\lambda,\cdot)$ for real $\lambda$ are real and semisimple. Thus, we have
(cf. [@MP], Lemma 1.2.10)
*Assume that $G>>0$, let $\lambda_0\in {{\bf R}}\backslash \{0\}$ and let $\eta_0$ be an eigenvalue of the pencil $Q(\lambda_0,\cdot)$ with (geometric) multiplicity 1. Then there are $\epsilon>0$ and $l$ real analytic functions*
$$\label{2.17}
\eta_s(\lambda)=\eta_0+\mathop{\sum}\limits_{k=p_s}^{\infty}b_{s,k}(\lambda-
\lambda_0)^k, \ \ s=1,2,...,l, \ \ |\lambda-\lambda_0|<\epsilon,$$
where $p_s\in{{\bf N}}$, $b_{sp_s}\in{{\bf R}}\backslash\{0\}$, $b_{sk}\in{{\bf R}}$ for $k>p_s$, such that $(\eta_s(\lambda))_{s=1}^l$ represents the eigenvalues near $\eta_0$ of the pencil $Q(\lambda,\cdot)$, counted with multiplicity, for each $\lambda\in{{\bf C}}$ with $|\lambda-\lambda_0|<\epsilon$.
[[**Proof.** ]{} ]{}For real $\lambda$, the eigenvalues of the self-adjoint operator function $Q(\lambda,\cdot)$ are real. Hence the lemma immediately follows from Theorem 2.8 and Remark 2.9.
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([@MP], Lemma 1.4.1) [*Let the conditions of Lemma 2. 11 be satisfied. Let $\lambda_0>0$ and $\eta_0>0$, then in some neighborhood of ($\lambda_0, \eta_0$), i.e. in $\{(\lambda,\eta):|\lambda-\lambda_0|<\epsilon,
|\eta-\eta_0|<\delta,\epsilon>0,\delta>0\}$ all the eigenvalues are given by the following formula* ]{} $$\label{2.18}
\lambda_j(\eta)=\lambda_0+\mathop{\sum}\limits_{k=1}^{\infty}\beta_k
((\eta-\eta_0)^{\frac{1}{r}}_j)^k, \ \ (j=1,2,...,r),$$ [*where $\beta_1\not=0$ is real if $r$ is odd and real or pure imaginary if $r$ is even, $(\eta-\eta_0)^{\frac{1}{r}}_j$, ($j=1,2,...,r$) means the complete set of branches of the root*]{}.
[[**Proof.** ]{} ]{}We obtain this result immediately after inverting (\[2.17\]).
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$$\label{*****}
\lambda_j+\lambda_{-j}\geq 0.$$ [[**Proof.** ]{} ]{} We index the eigenvalues in such a way that if $\lambda_{-j}(0)<0$ then $\lambda_{j}\left(0\right)=-\lambda_{-j}(0)$ and $\lambda^2_{j}(0)\leq \lambda^2_{j+1}(0)$.
The eigenvalues of $L(\cdot,\eta)$ are piecewise analytic functions of $\eta$. The eigenvalues may loose analyticity only when they collide. This follows from the results above.
The eigenvalues located on $(-\infty, 0)$ are analytic functions of $\eta>0$ because there are no nonreal eigenvalues in the open left half-plane (see Lemma 2.4, Statement 2 ) and, consequently, for all $\eta>0$ and $\lambda\in (-\infty, 0)$ the collision happens according to (\[2.18\]) with $r=1$ and $\beta_1\in {{\bf R}}\backslash \{0\}$.
Differentiating $L(\lambda_j(\eta),\eta)y_j(\eta)=0$ with respect to $\eta$ and taking the inner product with $y_j(\eta)$ leads to $$\lambda_j^{\prime}(\eta)((2\lambda_j(\eta)M-G)y_j(\eta),y_j(\eta))-\lambda_j(\eta)(Gy_j(\eta),y_j(\eta))=0$$ and therefore $$\label{2.20}
\lambda_{j}^{\prime}(\eta)=\frac{\lambda_{j}(\eta)\left(Gy_{j}(\eta),
y_{j}(\eta)\right)}{2\lambda_{j}\left(\eta\right)
(My_{j}(\eta),y_j(\eta))-\eta\left(
Gy_{j}\left(\eta\right),y_{j}\left(\eta\right)\right)}.$$ Here $(My_j(\eta),y_j(\eta))$ and $(My_j(\eta),y_j(\eta))$ are positive and depend continuously on $\eta$. It follows from (\[2.20\]) that if $\lambda_j(\eta)<0$ then $\lambda_j^{\prime}(\eta)>0$. These negative eigenvalues do not cross the origin due to Lemma 2.7.
The denominator of (\[2.20\]) is positive for sufficiently small $\eta>0$ and $\lambda_j(\eta)>0$ while that the numerator is nonnegative. We therefore have shown that $\lambda_j(\eta)$ and $\lambda_{-j}(\eta)$ are nondecreasing, which gives $$\lambda_j(\eta)+\lambda_{-j}(\eta)\geq 0$$ for $\eta\geq 0$ small enough.
While $\eta>0$ increases, $\lambda_{j}^\prime\left(\eta\right)$ can change its sign only when the denominator in the right-hand side of (\[2.20\]) vanishes, i.e. when eigenvalues collide. If such a collision takes place on the interval $\left(0
,\infty\right)$, then the eigenvalues involved behave according to formula (\[2.18\]). Such a coalescence on the half-axis $(0
,\infty)$ is of one of the following three kinds.
The first one is the case of $r$ odd in (\[2.18\]) and $\beta_1>0$ real. In this case we identify the eigenvalue moving to the right along the real axis after the collision with the one which moved to the right along the real axis before the collision.
By a collision of the second kind we mean one which has $r$ even and $\beta_1\in{{\bf R}}\backslash\{0\}$ in (\[2.18\]). After such a collision two new real eigenvalues appear which are moving in opposite directions along the real axis, and such a collision cannot violate Theorem 2.13.
The third kind of collision has even $r$ and pure imaginary $\beta_1\not=0$. Let $\lambda_{j}(\eta)$ take part in such a collision at $\eta=\eta_0\in(0,1]$. Then a collision of the second kind indeed occurred at some $\eta\in\left(0,\eta_0,\right)$ in some point $\lambda_{\times}\in\left(\lambda_{j}(\eta_0),\infty\right)$ on the real axis. In this case the eigenvalue that has arisen after this collision and is moving to the right is identified as $\lambda_{j}(\eta)$
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[**Remark 2.14**]{} The eigenvalues for which the denominator\
$2\lambda_{j}\left(\eta\right)
(My_{j}\left(\eta\right),y_j(\eta))-\eta\left(
Gy_{j}\left(\eta\right),y_{j}\left(\eta\right)\right)$ is positive (negative) are called eigenvalues of type I (type II) in [@KO]. However, we use terms ’type I’ and ’type II’ in Section 3 differently.
[**Lemma 2.15**]{} [*Let $M\geq m I$, $m>0$, then for all $\eta\in [0,1]$ the nonreal eigenvalues (if any) lie in a bounded domain $\{\lambda: 0\leq{\rm Re}\leq \frac {\eta}{2m}||G^{\frac{1}{2}}||^2\}$, $|{\rm Im}\lambda|\leq m^{-\frac{1}{2}}\beta^{\frac{1}{2}}$.* ]{}
[[**Proof.** ]{} ]{} Let $\lambda\notin{{\bf R}}$ be an eigenvalue of $L(\cdot,\eta)$ and $y$ be the corresponding eigenvector. Then $$\lambda^2(My,y)-\eta\lambda(Gy,y)-(Ay,y)=\lambda^2(My,y)-\eta\lambda||G^{\frac{1}{2}}y||^2-(Ay,y)=0$$ and, since $\lambda$ is not real, $${\rm Re}\lambda=\frac{1}{2}\eta||G^{\frac{1}{2}}y||^2(My,y)^{-1}, \ \ {\rm Im}\lambda=\frac{1}{2}(My,y)^{-1}\sqrt{|\eta^2||G^{\frac{1}{2}}y||^4+4(My,y)(Ay.y)|}$$ what implies the statement of the lemma.
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In what follows writing [*number of eigenvalues in a domain*]{} we mean the number with account of their algebraic multiplicities.
[**Theorem 2.16**]{} [*Let the conditions of Theorem 2.13 be satisfied. If the negative eigenvalues $\lambda_{-j}$ are arranged in pairs with positive eigenvalues $\lambda_j$ such that (\[\*\*\*\*\*\]) is true then the number of positive eigenvalues not included in the pairs is $$2\kappa_A- \kappa_c,$$ where $\kappa_A$ is the number of the negative eigenvalues of the operator $A$ and $\kappa_c$ is the number nonreal eigenvalues.*]{}
[[**Proof.** ]{} ]{}According to Theorem 2.8 the total multiplicity is preserved locally. Due to Lemma 2.15 the nonreal eigenvalues do not disappear at infinity while $\eta$ grows from 0 to 1. For $\eta=0$ we have $2\kappa_A-\kappa_c=0$ and there are no positive eigenvalue not included in a pair with a negative eigenvalue.
Thus the number of the eigenvalues in the open upper half-plane may reduce only at collisions on the real axis. Let $s$ eigenvalues reach a point on the real axis at $\eta=\eta_0$ moving from the open upper half-plane. Due to the symmetry of the spectrum with respect to the real axis the same number of eigenvalues reaches the same point moving from the open lower half-plane. This results in disappearing of $s$ eigenvalues from each of open upper and lower half-planes and appearing $2s$ real eigenvalues.
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Pencils with rank one linear part
=================================
Let us consider the quadratic operator pencil $L(\lambda,\eta)$ with operators $M$, $G$, $A$ acting in the Hilbert space $H\oplus
{{\bf C}}$ and satisfying Condition I. Moreover, in this section let the following condition be valid:
[**Condition II:**]{} $$G=b\left(\matrix{0 & 0 \cr 0 & 1 }\right), \ \ \ b>0.$$
[[**Proof.** ]{} ]{}1. Let $y_1=\left(\matrix{y_{11}\cr y_{12}}\right)$ be an eigenvector of $L(\cdot,\eta_0)$ corresponding to the eigenvalue $\lambda$ and let $y_2=\left(\matrix{y_{21}\cr y_{22}}\right)$ be an eigenvector of $L(\cdot,\eta_0)$ corresponding to $-\lambda$. Then $$\label{*}
(\lambda^2M-\lambda\eta_0G-A)y_1=0,$$ $$\label{**}
(\lambda^2M+\lambda\eta_0G-A)y_2=0,$$ and consequently $$\lambda^2(y_2,My_1)-\lambda\eta_0(y_2, Gy_1)-(y_2,Ay_1)=0,$$ $$\lambda^2(My_2,y_1)+\lambda\eta_0(Gy_2,y_1)-(Ay_2,y_1)=0.$$ Taking into account that $M$, $G$ and $A$ are selfadjoint, the difference of the above equations gives $$0=(Gy_2,y_1)=by_{22}\overline{y}_{12}.$$ Then one of the factors must be zero, say $y_{12}=0$, which gives $Gy_1=0$. Hence (\[\*\]) and (\[\*\*\]) lead to $L(\pm\lambda, \eta)y_1=0$, which completes the proof of part 1.
2\. Due to the symmetry of the spectrum, see Lemma 2.3 part 2, it follows that if $i\tau$ ($\tau\in{{\bf R}}\backslash \{0\}$) is an eigenvalue of $L(\cdot,\eta_0)$, then also $-i\tau$ is an eigenvalue $L(\cdot,\eta_0)$. Let $y$ be an eigenvector of $L(\cdot,\eta_0)$ corresponding to the eigenvalue $i\tau$. Then $$-\tau^2(My,y)-i\tau\eta_0(Gy,y)-(Ay,y)=0.$$ Since $M$, $G$ and $A$ are selfadjoint, all three inner products are real. Therefore $\lambda\not=0$ and $\eta_0>0$ give $(Gy,y)=0$ and thus $Gy=0$ because $G\geq 0$. It follows $L(\pm\lambda,\eta)y=L(\lambda,\eta_0)y=0$ for all $\eta\in [0,1]$.
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[**Definition 3.2**]{} (see Definition 1.5.2 in [@MP]) [*Let $\eta_0\in (0,1]$ and $m_I(\lambda):=\mathop{min}\limits_{\eta\in (0,1]}m(\lambda,\eta)$, where $m(\lambda,\eta)$ denotes the multiplicity of an eigenvalue $\lambda$ of the pencil $L(\cdot,\eta)$.\
1. An eigenvalue $\lambda$ of $L(
\cdot,\eta_0)$ is said to be an eigenvalue of type I if $\lambda$ is an eigenvalue of the pencil $L(
\cdot,\eta)$ for each $\eta\in (0,1]$.\
2. For $\lambda\in{{\bf C}}$ let $m_0(\lambda)=dim (ker L(\lambda)\cap ker G)$. If $m_0(\lambda)>0$, then each nonzero vector in $ker L(\lambda)\cap ker G$ is called an eigenvector of type I for $L$ at $\lambda$.\
3. An eigenvalue $\lambda$ of the pencil $L(\cdot,\eta_0)$ is said to be an eigenvalue of type II $m(\lambda,\eta_0)\not=m_I(\lambda)$.* ]{}
[**Remark 3.3**]{} 1. An eigenvalue can be both of type I and type II. If this is the case for some $\eta$, then we say that $\lambda$ is an eigenvalue of the pencil $L(\cdot, \eta)$ of type I multiplicity $m_I(\lambda)$ and of type II multiplicity $m(\lambda,\eta)-m_I(\lambda)$.\
2. If $0$ is an eigenvalue of the pencil $L$, then it follows from Lemma 2.7 that $0$ is an eigenvalue of $L(\cdot,\eta)$ for all $\eta\in(0,1]$, and, if $dim ker A=n$, the algebraic multiplicity $m(0,\eta)$ is $2n$ if $Gy=0$ for any $y\in ker A$ and $2n-1$ if there exists $0\not=y\in ker A$ such that $Gy\not=0$.\
3. If $ker M\cap ker A=\{0\}$, then the pencil $L(\cdot,0)$ satisfies Condition I. Since eigenvalues $\lambda$ of type I are eigenvalues of the pencil $L(\cdot,\eta)$ for all $\eta\in(0,1]$, it follows from (\[2.18\]) that $m_I(\lambda)$ branches of the eigenvalue $\lambda$ are constant near $\eta=0$, so that $m_I(\lambda)\leq m(\lambda,0)$, whereas the remaining $m(\lambda,0)-m_I(\lambda)$ branches are not constant.
[**Lemma 3.4**]{} [*Assume that $ker M\cap ker A=\{0\}$. Then the eigenvalues of type I of $L(\cdot,\eta)$, which are independent of $\eta\in (0,1]$, are located on imaginary and real axes symmetrically with respect to $0$. If additionally $M+G\geq\epsilon I$ ($\epsilon>0$), at most finitely many of the eigenvalues of type I are on the imaginary axis.*]{}
The proofs of this and the next lemmas are the same as the proofs of Lemmas 1.5.4 and 1.5.5 in [@MP], respectively, with the only change $K$ for $G$.
[**Theorem 3.6**]{} [*Assume that $ker M\cap ker A=\{0\}$.\
1. $\lambda\not=0$ is an eigenvalue of type I of the operator pencil $L$ if and only if $\lambda$ is an eigenvalue of the pencil $L(\cdot,0)$ having an eigenvector of the form $(y_0,0)^T$, and $m_I(\lambda)$ is the dimension of the space of eigenvectors of this form.\
2. If $\lambda\not=0$ is an eigenvalue of type I of the pencil $L$, but not an eigenvalue of type II, then $\lambda$ is semisimple.\
3. If $\lambda\not=0$ is an eigenvalue of the pencil $L$ of type II, then $ker L(\lambda)$ has a basis consisting of $m_I(\lambda)$ eigenvectors of type I and one eigenvector $y_0$ with $Gy_0\not=0$ with maximal chain length $m(\lambda)-m_I(\lambda)$, that is, there is a chain $(y_j)_{j=0}^{m(\lambda)-m_I(\lambda)-1}$ of the eigenvector $y_0$ and, if $m(\lambda)-m_I(\lambda)>1$, associated vectors of $L$ at $\lambda$.\
4. If $\lambda\not=0$ is an eigenvalue of type II of the pencil $L$, then $-\lambda$ is not an eigenvalue of type II of the pencil $L$.*]{}
The proof of Theorem 3.6 is the same as the proof of Theorem 1.5.6 in [@MP] with the change of $K$ for $G$.
Let us describe location of the eigenvalues $\{\lambda_j^{II}(\eta)\}$ of type II. Let us enumerate them as follows: $$(\lambda^{II}_{1}(0))^2<(\lambda^{II}_2(0))^2<...<(\lambda^{II}_{\kappa_{II}}(0))^2<0\leq (\lambda^{II}_{\kappa_{II}+1}(0))^2<(\lambda^{II}_{\kappa_{II}+2}(0))^2<...$$ $$\lambda^{II}_{-j}(0)=-\lambda^{II}_{j}(0).$$ Here $2\kappa_{II}$ is the number of pure imaginary eigenvalues of $L(\cdot, 0)$ of type II.
[**Proposition 3.7**]{} [*The negative eigenvalues of type II preserve their order:*]{} $$...<\lambda^{II}_{-j-1}(\eta)<\lambda^{II}_{-j}(\eta)<...<\lambda^{II}_{-\kappa_{II}-1}(\eta)\leq 0.$$ [[**Proof.** ]{} ]{}We know from Lemma 2.5 that all the negative eigenvalues are semisimple. From Statement 3 of Theorem 3.6 we conclude that the eigenvalues of type II are geometrically simple (however, an eigenvalue of type II can for some value of $\eta$ coincide with an eigenvalue of type I). Since $\lambda_j^{II}(\eta)$ are continuous functions of $\eta$ we arrive at the statement of the proposition.
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[**Theorem 3.8**]{} [*Assume that $ker M\cap ker A=\{0\}$ and that $M+G>>0$. Then the eigenvalues of type II of the operator pencil $L$ possess the following properties.\
1. $L(\cdot, \eta)$ has nonreal eigenvalues of type II located symmetrically with respect to the real axis. Denote their number by $2\kappa_{II}(\eta)\geq 0$.\
2. For all $\eta\in (0,1]$ these nonreal eigenvalues of type II lie in the open right half-plane.\
3. $|\lambda_{-j}^{II}(\eta)|\notin\{\lambda^{II}_j(\eta)\}$ for all $j\geq \kappa_{II}(0)+1$ if $\lambda_{ \kappa_{II}(0)+1}>0$ and for all $j\geq \kappa_{II}(0)+2$ if $\lambda_{ \kappa_{II}(0)+1}(0)=0$.\
4. The number $n_j(\eta)$ of eigenvalues of type II in each interval $(|\lambda_{-j}^{II}(\eta)|, |\lambda_{-j-1}^{II}(\eta)|)$ ($j\geq \kappa_{II}(0)+1$ if $\lambda^{II}_{\kappa_{II}(0)+1}(0)>0$ and $j\geq \kappa_{II}(0)+2$ if $\lambda^{II}_{\kappa_{II}(0)+1}(0)=0$ ) is odd.\
5. If $\lambda^{II}_{\kappa_{II}+1}(0)>0$ the number $n_0(\eta)$ of eigenvalues of type II in the interval $(0, |\lambda^{II}_{-\kappa_{II}-1}(\eta)|)$ is even.\
6. Denote by $$\tilde{\kappa}_{II}(\eta)=\frac{1}{2}\left\{\begin{array}{c}n_0(\eta)-1+\mathop{\sum}\limits_{j=1}^{\infty}(n_j(\eta)-1) \ \ {\rm if} \ \ \lambda^{II}_{-\kappa_{II}}(0)=0, \\ n_0(\eta)+\mathop{\sum}\limits_{j=1}^{\infty}(n_j(\eta)-1), \ \ {\rm if} \ \ \lambda^{II}_{-\kappa_{II}}(0)\not=0 \end{array}\right. .$$ Then $\tilde{\kappa}_{II}(\eta)+\kappa_{II}(\eta)+\kappa_I=\kappa_A$ where $\kappa_A$ is the number of negative eigenvalues of the operator pencil $\lambda M-A$ and $2\kappa_I$ is the number (independent of $\eta$) of pure imaginary eigenvalues of $L(\cdot, \eta)$ of type I.*]{}
[[**Proof.** ]{} ]{}Statement 1 follows from Lemma 2.3.
By Lemma 2.15 we know that the nonreal eigenvalues lie in the closed right half-plane. But if such an eigenvalue is pure imaginary, then it is of type I. This means that nonreal eigenvalues of type II lie in the open right half-plane. Statement 2 is proved.
If $|\lambda_{-j}^{II}(\eta)|\in\{\lambda^{II}_j(\eta)\}$ for some $j\geq \kappa_{II}(0)+1$ if $\lambda_{ \kappa_{II}(0)+1}>0$ or some $j\geq \kappa_{II}(0)+2$ if $\lambda_{ \kappa_{II}(0)+1}=0$ then by Lemma 3.1 $\lambda^{II}_j(\eta)$ is independent of $\eta$ and therefore is of type II. This contradiction proves Statement 3.
Since the right-hand side of (\[2.20\]) is nonnegative for small $\eta>0$ and $\lambda_j(\eta)\in{{\bf R}}$ the real eigenvalues for such $\eta$ move to the right along the real axis. Therefore, the number of eigenvalues of type II in each interval $(|\lambda_{-j}(\eta)|, |\lambda_{-j-1}(\eta)|)$ ($j\geq \kappa_{II}(0)+1$ if $\lambda^{II}_{\kappa_{II}(0)+1}>0$ and $j\geq \kappa_{II}(0)+2$ if $\lambda^{II}_{\kappa_{II}(0)+1}=0$ ) is 1 for $\eta>0$ small enough. This number can increase only when a collision of the second kind of eigenvalues of type II happens on the real axis. Such a collision increases the number of the eigenvalues in such an interval by 2. Then this number can increase by 2 as the result of a collision of the second kind and decrease by 2 as the result of a collisions of the third type. Thus $n_j(\eta)$ is odd for all $\eta>0$ what proves Statement 4.
Let $\lambda^{II}_{\kappa_{II}+1}(0)>0$. Then for $\eta>0$ small enough there are no eigenvalues of type II on the interval $(0, |\lambda_{-\kappa_{II}(0)-1}|)$. This number can increase by 2 as a result of collision of the second kind and decrease by 2 as a result of collision of the third kind. Statement 5 follows.
It is clear that $\tilde{\kappa}_{II}(0)=0$ and $\kappa_{II}(0)+\kappa_{I}=\kappa_A$. The number $2\kappa_{II}(\eta)$ of nonreal eigenvalues of type II increases (decreases) by 2 at collisoons of the second (third) kind. But simultaneously $\tilde{\kappa}_{II}$ decreases (increases) by 2. Thus, the sum $\kappa_{II}(\eta)+\tilde{\kappa}_{II}(\eta)$ remains unchanged at collisions of any of the three kinds. This proves Statemens 6.
------------------------------------------------------------------------
Applications
============
1\. We consider a Sturm-Liouville problem with the gyroscopic condition at the right end $$\label{4.1}
-y^{\prime\prime}+q(x)y=\lambda^2y,$$ $$\label{4.2}
y(0)=0,$$ $$\label{4.3}
y^{\prime}(a)+\lambda \alpha y(a)=0.$$ Here $\lambda$ is the spectral parameter, the parameter $\alpha>0$ and the potential $q$ is real-valued and belongs to $L_2(0,a)$.
Let us introduce the operators $A$, $G$ and $M$ acting in the Hilbert space $H=L_2(0,a)\cup{{\bf C}}$ according to the formulae: $$A\left(
\begin{array}{c}
v(x) \\
c \\
\end{array}
\right) =\left(
\begin{array}{c}
-v^{\prime\prime}\left(x\right) +q\left( x\right) v\left( x\right) \\
v^{\prime }(a)
\end{array}
\right),$$ $$D\left( A\right) =\left\{
\begin{array}{c}
\left(
\begin{array}{c}
v\left( x\right) \\
c
\end{array}
\right): \ \ v\left( x\right) \in W_{2}^{2}\left( 0,a\right), \ \
\ \ v\left( 0\right) =0, \ \ c=v(a)
\end{array}
\right\},$$ $$M=\left(
\begin{array}{cc}
I & 0 \\
0 & 0
\end{array}
\right), \ \ G=\left(
\begin{array}{cc}
0 & 0 \\
0 & \alpha
\end{array}
\right).$$ The operator $A$ is selfadjoint and bounded below ($A\geq \beta I$) and there exists $-\beta_1<-\beta\leq ||y||^{-2}(Ay,y)$ such that $(A+\beta_1I)^{-1}$ is a compact operator (see [@MP], Proposition 2.1.1).
We associate the quadratic operator pencil $$\label{***}
L\left( \lambda \right) =\lambda ^{2}M-\lambda G-A$$ with problem (\[4.1\])–(\[4.3\]).
We identify the spectrum of problem [(\[4.1\])]{} - [(\[4.3\])]{} with the spectrum of the pencil $L\left( \lambda \right)$. It is clear that $M\geq 0$, $G\geq 0$ and $M+G>>0$. The spectrum of the pencil consists of normal eigenvalues (see Section 2 of this paper).
Let us prove that all of these eigenvalues are of type II. Suppose a pure imaginary $i\tau\not=0$ is an eigenvalue. Then by Lemma 3.1 $-i\tau$ is also an eigenvalue and according to the proof of Lemma 3.1 $c=v(a)=0$ in (\[4.3\]). Therefore, the second component of the equation $L(\lambda_0)Y=0$ gives $v^{\prime}(a)=0$ what contradicts $v(a)=0$. In the same way, one can prove that there are no symmetrically located real eigenvalues and that the possible eigenvalue at the origin is simple.
Thus, the conditions of Theorem 3.8 are satisfied and all the eigenvalues are of type II and therefore statements 1)–6) of Theorem 3.8 are valid. Thus we obtain
[**Theorem 4.1**]{} [*1. Problem (\[4.1\])–(\[4.3\]) may have nonreal eigenvalues located symmetrically with respect to the real axis. Denote their number by $2\kappa\geq 0$.\
2. All the nonreal eigenvalues lie in the open right half-plane.\
3. If we denote by $\{\lambda_j\}_{-\infty}^{\kappa_0}$ the negative eigenvalues of problem (\[4.1\])–(\[4.3\]) then $|\lambda_{k}|\notin\{\lambda_j\}_{-\infty,j\not=0}^{\infty}$ for all $-\infty<k\leq \kappa_0$.\
4. The number $n_j$ of eigenvalues in each interval $(|\lambda_{j}|, |\lambda_{j-1}|)$ ($-\infty< j\leq \kappa_0$) is odd.\
5. If $0\notin\{\lambda_{j}\}_{-\infty, j\not=0}^{\infty}$ then the number $n_0$ of eigenvalues in the interval $(0, |\lambda_{\kappa_0}|$) is even. If $0\in\{\lambda_{j}\}_{-\infty, j\not=0}^{\infty}$ then the number of eigenvalues in the interval $(0, |\lambda_{\kappa_0}|)$ is odd and $0$ is a simple eigenvalue.\
6. Denote by $$\tilde{\kappa}=\frac{1}{2}\left\{\begin{array}{c}n_0+\mathop{\sum}\limits_{j=1}^{\infty}(n_j-1) \ \ {\rm if} \ \ 0\not\in\{\lambda_j\}_{-\infty,j\not=0}^{\infty}, \\ n_0-1+\mathop{\sum}\limits_{j=1}^{\infty}(n_j-1), \ \ {\rm if} \ \ 0\in\{\lambda_j\}_{-\infty, j\not=0}^{\infty} \end{array}\right. .$$ Then $\tilde{\kappa}+\kappa=\kappa_A$ where $\kappa_A$ is the number of negative eigenvalues of the operator pencil $\lambda M-A$ or what is the same of problem*]{} $$\label{4.11}
-y^{\prime\prime}+q(x)y=\lambda y,$$ $$\label{4.12}
y(0)=y'(a)=0.$$
2\. In the second example we have eigenvalues of both types. Let $q=const>0$. Consider the following spectral problem $$\label{4.7}
-y_j^{\prime\prime}+qy_j=\lambda^2y_j, \ \ \ j=1,2,$$ $$\label{4.8}
y_1(0)=y_2(0)=0,$$ $$\label{4.9}
y_1(a)=y_2(a),$$ $$\label{4.10}
y_1^{\prime}(a)=y_2^{\prime}(a)+\lambda\alpha y_2(a).$$
Let us introduce the operators $A$, $G$ and $M$ acting in the Hilbert space $H=L_2(0,a)\oplus L_2(0,a)\oplus{{\bf C}}$ according to the formulae: $$A\left(
\begin{array}{c}
v_1(x) \\ v_2(x) \\
c
\end{array}
\right) =\left(
\begin{array}{c}
-v_1^{\prime\prime}\left(x\right) +q v_1\left( x\right) \\
-v_2^{\prime\prime}\left(x\right) +q v_2\left( x\right) \\
v_2^{\prime }(a)
\end{array}
\right),$$ $$D(A)=$$ $$\left\{
\begin{array}{c}
\left(
\begin{array}{c}
v_1\left( x\right) \\ v_2(x) \\
c
\end{array}
\right): \ \ v_1\left( x\right) \in W_{2}^{2}\left( 0,a\right), \ \ v_2\left( x\right) \in W_{2}^{2}\left( 0,a\right),
\ \ v_1\left( 0\right) = v_2(0)=0, \ \ c=v_2(a)
\end{array}
\right\},$$ $$M=\left(
\begin{array}{ccc}
I & 0 & 0 \\
0 & I & 0 \\
0 & 0 & 0
\end{array}
\right), \ \ G=\alpha \left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{array}
\right).$$
The corresponding operator pencil $L(\lambda)$ given be (\[\*\*\*\]) satisfies conditions of Theorem 3.8. Moreover, in case of $q=const$ we can find the characteristic function of this problem, i.e. the function the set of zeros of which coincides with the spectrum of problem (\[4.7\])–(\[4.9\]): $$\label{yyy}
\omega(\lambda)=\frac{\sin\sqrt{\lambda^2+q } \ a}{\sqrt{\lambda^2+q}} \ \left(2\cos \sqrt{\lambda^2+q} \ a+\alpha\lambda\frac{\sin\sqrt{\lambda^2+q} \ a}{\sqrt{\lambda^2+p}}\right).$$ The spectrum of problem (\[4.7\]) – (\[4.10\]), i.e. the set of zeros of $\omega$ consists of two subsequences $\{\lambda_j^I\}_{-\infty, j\not=0}^{\infty}$ and $\{\lambda_j^{II}\}_{-\infty, j\not=0}^{\infty}$ where $$\label{.}
\lambda_j^{I}=\pm\sqrt{\left(\frac{\pi j}{a}\right)^2-q}, \ \ \ \ \ j=\pm 1, \pm 2, ...$$ and $\{\lambda_j^{II}\}_{-\infty, j\not=0}^{\infty}$ is the set of zeros of the function $\left(2\cos \sqrt{\lambda^2+q} \ a+\alpha\lambda\frac{\sin\sqrt{\lambda^2+q} \ a}{\sqrt{\lambda^2+p}}\right)$. According to Definition 3 2 (with $\eta=\alpha$) we obtain
[**Theorem 4.2**]{} [*The zeros $\{\lambda_j^I\}_{-\infty, j\not=0}^{\infty}$ are eigenvalues of the type I of problem (\[4.7\])–(\[4.10\]) while the zeros $\{\lambda_j^{II}\}_{-\infty, j\not=0}^{\infty}$ are the eigenvalues of type II.*]{}
Analyzing the behavior of the function $\left(2\cos \sqrt{\lambda^2+q} \ a+\alpha\lambda\frac{\sin\sqrt{\lambda^2+q} \ a}{\sqrt{\lambda^2+p}}\right)$ we conclude that (for $q=const$)
\(i) $n_j=1$ for all $j$ and $n_0=0$ if $0\in\{\lambda_j\}_{-\infty, j\not=0}^{\infty}$ and $n_0=1$ if $0\notin\{\lambda_j\}_{-\infty, j\not=0}^{\infty}$.
\(ii) An eigenvalue of type I cannot coincide with an eigenvalue of type II for any value of $\alpha>0$ because if $$\label{99}
\frac{\sin\sqrt{\lambda^2+q } \ a}{\sqrt{\lambda^2+q}}=0$$ and $$\left(2\cos \sqrt{\lambda^2+q} \ a+\alpha\lambda\frac{\sin\sqrt{\lambda^2+q} \ a}{\sqrt{\lambda^2+p}}\right)=0$$ then $$\cos \sqrt{\lambda^2+q} \ a=0$$ what contradicts (\[99\]).
\(iii) In the same way we conclude that $\{-\lambda^{II}_j\}_{-\infty, j\not=0}^{\infty}\cap \{\lambda_j^{I}\}_{-\infty, j\not=0}^{\infty}=\emptyset$.
To avoid considering many cases we give explicit result for one particular case. Here we describe the eigenvalues $\{\lambda_j\}_{-\infty, j\not=0}^{\infty}=\{\lambda_j^{I}\}_{-\infty, j\not=0}^{\infty}\cup\{\lambda_j^{II}\}_{-\infty, j\not=0}^{\infty}$ identifying $\lambda_{2j-1}=\lambda_J^{II}$ and $\lambda_{2j}=\lambda_j^{I}$ ($j\in{{\bf N}}$).
Using (i)–(iii) and Theorem 3.8 we obtain
[**Corollary 4.3**]{}
*Let $q>0$ and $q^{1\over 2}a\pi^{-1}\in{{\bf N}}$. Then*
a\) $\lambda=0$ is an eigenvalue of geometric multiplicity 1 and algebraic multiplicity 2;
b\) there are $2(q^{1\over 2}a\pi^{-1}-1)$ simple pure imaginary (nonzero) eigenvalues located symmetrically with respect to the origin which can be found by (\[.\]) with $j=\pm 1, \pm 2, ..., \pm (q^{1\over 2}a \pi^{-1}-1)$;
c\) there are $2q^{1\over 2}a\pi^{-1}$ complex (neither real nor pure imaginary) eigenvalues located symmetrically with respect to the real axis in the open right half plane;
d\) real eigenvalues can be indexed such that $\lambda_j<0$ for $-\infty<j\leq -2q^{1\over 2}a\pi^{-1}-1:=\kappa_0$ and the interval $(0,|\lambda_{\kappa_0}|)$ is free of eigenvalues
e\) If we denote by $\{\lambda_j\}_{-\infty}^{\kappa_0}$ the negative eigenvalues of problem (\[4.7\])–(\[4.10\]) then $|\lambda_{2k-1}|\notin\{\lambda_j\}_{-\infty,j\not=0}^{\infty}$ for all $-\infty<2k-1\leq \kappa_0$;
f\) The number of eigenvalues in each interval $(|\lambda_{2j-1}|, |\lambda_{2j-3}|)$ ($-\infty<2j -1\leq \kappa_0$ is 2 (one of type I and one of type II);
g\) The number of eigenvalues in each the intervals $(-((\pi(j+1)a^{-1})^2-q)^{1\over 2}, -((\pi j a^{-1})^2-q)^{1\over 2})$ and in each the intervals $(((\pi ja^{-1})^2-q)^{1\over 2}, ((\pi (j+1) a^{-1})^2-q)^{1\over 2})$ for $j=q^{1\over 2}a\pi^{-1}, q^{1\over 2}a\pi^{-1}+1,...$ is 1.
[99]{} D.B. DeBra, R. H. Delp, Rigid body attitude stability and natural frequences in a circular orbit. J. Astronaut. Sci. [**8**]{} (1961), 14–17. V.I. Feodosiev. Vibrations and stability of a pipe conveying a flowing liquid. Inzh. sbornik [**10**]{} (1951), 169–170 \[in Russian\] I.C. Gohberg, M.G. Krein. Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, [**18**]{} AMS, New York, 1969. I.C. Gohberg, S. Goldberg, M.A. Kaashoek. Classes of Linear Operators, I. Operator Theory: Adv. Appl. [**49**]{} Birkhäuser, Basel, 1990. I.C. Gohberg, E.I. Sigal. An Operator generalization of the logarithmic residue theorem and Rouché’s theorem. Math. USSR Sb. [**13:1**]{} (1971), 603-625 \[in Russian\]. Ch. G. Ibadzade, I.M. Nabiev. Recovering of the Sturm-Liouville operator with non-separated boundary condition dependent of the spectral parameter. to be published in Ukrainian Math. J. \[in Russian\] A.G. Kostyuchenko, M.B. Orazov. The problem of oscillations of an elastic half- cylinder and related self-adjoint quadratic pencils. Trudy Sem. Petrovskogo, [**6**]{} (1981), 97-147 \[in Russian\]. A.S. Markus. Introduction to the Theory of Polynomial Operator pencils, Transl. Math. Monographs [**71**]{} AMS, New York, 1988. A.I. Markushevich. Theory of Functions of a Complex Variable. III, revised English edition, Prentice-Hall Inc., Englewood Cliffs, New York, 1967. A.I. Miloslavskii. On stability of linear pipes, Dinamika Sistem, Nesuschih podvizhnuyu raspredelennuyu nagruzku, Collected papers, [**192**]{} Kharkov Aviation Institute, Kharkov, (1980) 34–47 \[in Russian\]. A.I. Miloslavskii. On the instability spectrum of an operator pencil, Matem. Zametki [**49**]{} (1991), no. 4, 88–94 \[in Russian\]; English transl. in: Mathem. Notes, [**49**]{} (1991), no. 3–4, 391–395. A.A. Movchan .On a problem of stability of a pipe with a fluid flowing through it, Prikl. Mat. Mekh. [**29**]{} (1965), 760–762 \[in Russian\]; English transl.: PMM, J. Appl. Math. Mech. [**29**]{} (1965), 902-904. M. M" oller M, V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications. Birkhäuser, Cham, 2015 M.P. Paidoussis, N.T. Issid. Dynamic stability of pipes conveying fluid. J. Sound Vibrat. [**33**]{} (1974), no. 3, 267–294. V.N. Pivovarchik. On eigenvalues of a quadratic operator pencil, Funkts. Anal. Prilozhen. [**25**]{} (1989), 80-81 \[in Russian\]. V. Pivovarchik. On spectra of a certain class of quadratic operator pencils with one-dimensional linear part. Ukrainian Math. J. [**59**]{} (2007), No.5, 702-717. V.N. Pivovarchik. On spectra of quadratic operator pencils in the right half-plane. Matem. Zametki [**45**]{} (1989), No.6, 101-103 \[in Russian\]. V.N. Pivovarchik. On the total algebraic multiplicity of the spectrum in the right half-plane for a class of quadratic operator pencils, Algebra i Analys [**3**]{} (1991), No. 2, 223 - 230 \[in Russian\]; English transl.: St. Peterburg Math. J. [**3**]{} (1992), No. 2., 447-454. V. Pivovarchik, H. Woracek. Shifted Hermite-Biehler functions and their applications, Integral Equations and Operator Theory [**57**]{} (2007) 101-126. A.A. Shkalikov. Elliptic equations in Hilbert space and associated spectral problems. Trudy Sem. Petrovskogo [**14**]{} (1989) 140-224 \[in Russian\]. W. T. Thompson. Spin stabilization of attitude against gravity torque, J. Austronaut. Sci. [**9**]{} (1962), no. 1, 31 - 33. M. Tyaglov. Self-interlacing polynomials II: matrices with self-interlacing spectrum. Electron. J. Linear Algebra [**32**]{} (2017), 51–57. E.E. Zajac. The Kelvin-Tait-Chetaev theorem and extensions. J. Austronaut. Sci. [**11**]{} (1964), 46 - 49.
South-Ukrainian National Pedagogical University,\
Staroportofrankovskaya str. 26, Odesa, Ukraine, 65020\
e-mail address: vpivovarchik@gmail com\
[email protected]
|
---
abstract: 'In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.'
address:
- |
M. Hitrik, Department of Mathematics\
UCLA\
Los Angeles\
CA 90095-1555\
USA
- |
K. Krupchyk, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
P. Ola, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
L. Päivärinta, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
author:
- Michael Hitrik
- Katsiaryna Krupchyk
- Petri Ola
- Lassi Päivärinta
title: Transmission eigenvalues for operators with constant coefficients
---
Introduction
============
Let $V\in L^\infty({\mathbb{R}}^n)$ be compactly supported with ${\operatorname{supp}}(V)=\overline{\Omega}$, where $\Omega\subset {\mathbb{R}}^n$ is a bounded domain, and let $P_0(D)$ be a partial differential operator of order $m\ge 2$ with constant real coefficients
$$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ For any $\varphi\in C^\infty_0(\Omega)$, we set $$\label{eq_norm}
\|\varphi\|_{P_0}=\|P_0(D)\varphi\|.$$ Here and in what follows the notation $\|\cdot\|$ stands for the standard $L^2$-norm. The completion of $C^\infty_0(\Omega)$ with respect to the norm is denoted by $H^{P_0}_0(\Omega)$.
The [*interior transmission problem*]{} is the following degenerate boundary value problem, $$\label{eq_ITP_johd}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{P_0}_0(\Omega).
\end{aligned}$$
We say that $\lambda\in {\mathbb{C}}$ is a [*transmission eigenvalue*]{} if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}$ and $0\ne w\in L^2_{\textrm{loc}}$.
The problem arises naturally in the study of inverse scattering theory for the operator $P_0+V$, when $P_0=-\Delta$. In this context, the significance of the notion of a transmission eigenvalue is twofold. On the one hand, in reconstruction algorithms of inverse scattering theory, such as the linear sampling method of Colton and Kirsch [@CakColbook; @ColKir96], and the factorization method of Kirsch [@KirGribook], transmission eigenvalues correspond to frequencies that one needs to avoid. The fact that they form a discrete set makes this possible. On the other hand, the knowledge of the transmission eigenvalues carries information about the scatterer [@paisyl08], and in the radially symmetric case, they determine the potential completely [@CakColGint_complex], see also [@McLP] for the previous work in this direction. It is therefore natural to investigate the question of existence of transmission eigenvalues, and relate their properties to the properties of the scatterer.
The interior transmission problem was first introduced in 1988 by Colton and Monk [@ColMonk88] in connection with an inverse scattering problem for the reduced wave equation. They were led to this problem when studying the injectivity of the far field operator. The fact that the interior transmission eigenvalues form a discrete set was shown in the Helmholz case in [@vanhatkonnatI]. The problem of existence of transmission eigenvalues, however, remained unsolved until recently. When $P_0 = -\Delta$, Päivärinta and Sylvester in [@paisyl08] proved the first existence result, and soon thereafter, the existence of an infinite set of real transmission eigenvalues was established by Cakoni, Gintides, and Haddar in [@CakDroHou]. We would also like to mention recent results on transmission eigenvalues for Maxwell’s equations, as well as for the Helmholtz equation, in the presence of cavities [@CakColHous10; @CakKir; @kir07].
In this paper we consider the case of quite general operators $P_0$ with constant coefficients, for which a well developed scattering theory is available [@horbookII]. In particular, we can allow non-elliptic operators as well as operators of higher order. We then show the discreteness of the set of transmission eigenvalues, and derive sufficient conditions for the existence of an infinite set of real transmission eigenvalues. The extension to the case of general operators with constant coefficients seems natural, as in applications one frequently encounters such operators which could be either non-elliptic, or elliptic of higher order. We also illustrate the techniques developed by applying them in two cases of physically significant operators, namely the biharmonic operator and the Dirac system in ${\mathbb{R}}^3$.
Finally, we point out that in view of the non-selfadjoint nature of the problem , a study of the distribution of transmission eigenvalues in the complex plane would be both interesting and natural. So far, only in the case when $P_0=-\Delta$, the existence of complex transmission eigenvalues has been shown in the very recent paper [@CakColGint_complex], assuming that the index of refraction is constant and sufficiently close to one. See also the numerical computation of complex transmission eigenvalues given in [@ColMonkSun].
The structure of the paper is as follows. In Section 2 we formulate the problem and state a compact embedding lemma that is needed when proving the discreteness of the set of transmission eigenvalues in Section 3, which then follows by analytic Fredholm theory. In Section 4 sufficient conditions for the existence of transmission eigenvalues are obtained, and the existence of an infinite set of transmision eigenvalues is studied in Section 5. In Section 6 we study the reformulation of the problem which is modeled on the reduced wave, rather than the Schrödinger, equation. It is interesting that this problem poses fewer restrictions on the operator than the Schrödinger version, and thus even in the second order case we do not need to assume the ellipticity. Section 7 is devoted to the study of two examples from physics, namely, the biharmonic operator and Dirac system. In the last two Sections 8 and 9 we explain the relation of transmission eigenvalues to scattering theory, relying upon a generalized Rellich theorem.
Interior transmission eigenvalues: The Schrödinger case {#sec_Schrod}
=======================================================
As described in the introduction, let $V\in L^\infty({\mathbb{R}}^n)$ be compactly supported with ${\operatorname{supp}}(V)=\overline{\Omega}$, where $\Omega\subset {\mathbb{R}}^n$ is a bounded domain, and let $P_0(D)$ be a partial differential operator of order $m\ge 2$ with constant real coefficients, $$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ The following estimate [@hor55 Theorem 2.1], [@horbookII Theorem 10.3.7] $$\|\varphi\|\le C_{\Omega}\|P_0(D)\varphi\|, \quad\varphi\in C^\infty_0(\Omega),$$ where $C_\Omega$ is a constant depending on the domain $\Omega$, implies that is a norm on $C^\infty_0(\Omega)$, and as already mentioned, completion of $C^\infty_0(\Omega)$ with respect to the norm is denoted by $H^{P_0}_0(\Omega)$; see also [@jac88].
Consider the interior transmission problem $$\label{eq_ITP}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{P_0}_0(\Omega).
\end{aligned}$$
We say that $\lambda\in {\mathbb{C}}$ is a [*transmission eigenvalue*]{} if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)$. It suffices to require that $v\ne 0$. Indeed, assume that $w=0$. Then $v\in H^{P_0}_0(\Omega)$ and extending $v$ by zero to the complement of $\Omega$, we get $(P_0-\lambda)v=0$ in ${\mathbb{R}}^n$. Since every constant coefficient differential operator is injective $\mathcal{E}'({\mathbb{R}}^n)\to \mathcal{E}'({\mathbb{R}}^n)$, we have $v=0$.
Throughout this work we shall assume that $V$ is real-valued and such that $V\ge\delta>0$ a.e. in $\overline{\Omega}$.
We will use also the following equivalent characterization of the space $H^{P_0}_0(\Omega)$. Let $$\tilde P_0(\xi)=(\sum_{|\alpha|\ge 0}|P_0^{(\alpha)}(\xi)|^2)^{1/2}, \quad P_0^{(\alpha)}(\xi)=D^{\alpha} P_0(\xi),\quad \alpha\in {\mathbb{N}}^{n},$$ and define on $C^\infty_0(\Omega)$ the norm $$\label{eq_norm_2}
\|\varphi\|_{\tilde P_0}^2=\sum_{|\alpha|\ge 0}\|P_0^{(\alpha)}(D)\varphi\|^2.$$
The estimate [@hor55 Lemma 2.8] $$\label{eq_horm}
\|P_0^{(\alpha)}(D)\varphi\|\le C_{\Omega,\alpha}\|P_0(D)\varphi\|,\quad\varphi\in C^\infty_0(\Omega),$$ yields that the norms and are equivalent on $H^{P_0}_0(\Omega)$
The space $H^{P_0}_0(\Omega)$ is isometrically imbedded into the space $B_{2,\tilde P_0}({\mathbb{R}}^n)$ via zero extension to the complement of $\Omega$. Here $$B_{2,\tilde P_0}({\mathbb{R}}^n)=\{u\in S'({\mathbb{R}}^n):\tilde P_0\hat u\in L^2({\mathbb{R}}^n)\}$$ is a Banach space with the norm $$u\mapsto\|\tilde P_0 \hat u\|_{L^2}$$ introduced in [@horbookII Definition 10.1.1]. The proof of the discreteness of the set of transmission eigenvalues depends on the following result.
\[prop\_comp\] Assume that $\tilde P_0(\xi)\to\infty$ when $|\xi|\to \infty$. Then the imbedding $$H^{P_0}_0(\Omega)\hookrightarrow L^2(\Omega)$$ is compact.
The claim follows immediately from [@horbookII Theorem 10.1.10].
A condition under which we can apply this result is found with the help of the set $$\label{eq_Lambda_p_0}
\Lambda(P_0)=\{\xi\in{\mathbb{R}}^n:\lambda\mapsto P_0(\lambda\xi+\eta)\text{ is constant } \forall \eta\in {\mathbb{R}}^n\}$$ Then [@horbookII Proposition 10.2.9] implies that $\tilde P_0(\xi)\to\infty$ as $|\xi|\to \infty$ provided that $\Lambda(P_0)=\{0\}$, i.e. there are no hidden variables in $P_0(\xi)$.
Discreteness of the set of transmission eigenvalues
===================================================
\[thm\_equivalence\] A complex number $\lambda$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^{P_0}_0(\Omega)$ satisfying $$\label{eq_teq}
(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda)u=0\quad \text{in}\quad \mathcal{D}'(\Omega).$$
Assume that $\lambda$ is a transmission eigenvalue. Then there exists a non-trivial solution $(v,w)\ne 0$ of . Define $u=v-w\in H^{P_0}_0(\Omega)$. Then $(P_0-\lambda)u=Vw$ and thus, $$(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda)u=(P_0+V-\lambda)w=0.$$
To see the reverse implication, we first claim that $$(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda)=(P_0-\lambda)\frac{1}{V}(P_0+V-\lambda).$$ Indeed, $$\begin{aligned}
&(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda)u-(P_0-\lambda)\frac{1}{V}(P_0+V-\lambda)u\\
&=(P_0-\lambda)\frac{1}{V}(P_0-\lambda)u+\frac{V}{V}(P_0-\lambda)u-(P_0-\lambda)\frac{1}{V}(P_0-\lambda)u\\
&-(P_0-\lambda)\frac{V}{V}u=0.\end{aligned}$$ Let $0\ne u\in H^{P_0}_0(\Omega)$ be a solution of . Denote now $$w=\frac{1}{V}(P_0-\lambda)u\quad \text{and}\quad v=\frac{1}{V}(P_0+V-\lambda)u.$$ Then $$(P_0-\lambda)v=0, \quad (P_0+V-\lambda)w=0\quad \text{in}\quad \Omega$$ and $$v-w=\frac{1}{V}(P_0+V-\lambda-P_0+\lambda)u=u\in H^{P_0}_0(\Omega).$$
Set $$T_\lambda=(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda).$$ Let $\lambda\in {\mathbb{C}}$. Then for $\varphi,\psi\in C^\infty_0(\Omega)$, we define a sesquilinear form $$B_\lambda(\varphi,\psi)=\langle T_\lambda\varphi,\psi \rangle_{L^2(\Omega)}=
\langle\frac{1}{V}(P_0-\lambda)\varphi ,(P_0+V-\lambda)\psi \rangle_{L^2(\Omega)}$$ which extends uniquely to a continuous sesquilinear form on $H^{P_0}_0(\Omega)\times H^{P_0}_0(\Omega)$.
\[lem\_coercive\] Assume that either $P_0(\xi)$ or $-P_0(\xi)$ is bounded from below on ${\mathbb{R}}^n$. Then there exists $\lambda_0\in {\mathbb{R}}$ such that the sesquilinear form $B_{\lambda_0}(u,v)$ is coercive on $H^{P_0}_0(\Omega)\times H^{P_0}_0(\Omega)$.
Let $\varphi\in C^\infty_0(\Omega)$ and set $C_0=\inf_\Omega1/V>0$. Then $$\label{eq_est_b}
\begin{aligned}
B_\lambda(\varphi,\varphi)&=\langle(P_0-\lambda)\frac{1}{V}(P_0-\lambda)\varphi ,\varphi \rangle_{L^2(\Omega)}+\langle (P_0-\lambda)\varphi,\varphi \rangle_{L^2(\Omega)}\\
&\ge C_0\langle(P_0-\lambda)^2\varphi ,\varphi \rangle_{L^2(\Omega)}+\langle (P_0-\lambda)\varphi,\varphi \rangle_{L^2(\Omega)}\\
&= C_0\|P_0\varphi\|^2-2C_0\lambda \langle P_0\varphi,\varphi \rangle_{L^2(\Omega)} \\ &\quad\quad\quad+ \lambda^2C_0\|\varphi\|^2-\lambda\|\varphi\|^2+ \langle P_0\varphi,\varphi \rangle_{L^2(\Omega)}.
\end{aligned}$$ To estimate the last term in the right hand side of we write $$\label{eq_est_b_4}
\langle P_0\varphi,\varphi \rangle_{L^2(\Omega)}\le \|P_0\varphi\|\|\varphi\|\le \frac{C_0}{2}\|P_0\varphi\|^2+\frac{2}{C_0}\|\varphi\|^2.$$ For the second term in the right hand side of , assuming, to fix the ideas, that $P_0(\xi)$ is bounded from below on ${\mathbb{R}}^n$, we get $$\label{eq_est_b_2}
\langle P_0\varphi,\varphi \rangle_{L^2(\Omega)}=(2\pi)^{-n}\int_{{\mathbb{R}}^n} P_0(\xi)|\hat\varphi(\xi)|^2d\xi\ge -C\|\varphi\|^2.$$ Now combining , and , we can find $\lambda_0\in {\mathbb{R}}$ with $|\lambda_0|$ large enough such that $$B_{\lambda_0}(\varphi,\varphi)\ge \frac{C_0}{2}\|P_0\varphi\|^2=\frac{C_0}{2}\|\varphi\|^2_{P_0}, \quad \forall \varphi\in C^\infty_0(\Omega).$$ This establishes the coercivity of $B_{\lambda_0}$ on $H^{P_0}_0(\Omega)\times H^{P_0}_0(\Omega)$.
Let us denote by $H^{-P_0}(\Omega)$ the dual of the space $H_0^{P_0}(\Omega)$. Notice that $H^{-P_0}(\Omega)$ can be viewed as a subspace of the space of distributions $\mathcal{D}'(\Omega)$.
\[thm\_discret\_1\] Assume that $\tilde P_0(\xi)\to\infty$ when $|\xi|\to \infty$ and that the symbol $\pm P_0(\xi)$ is bounded from below on ${\mathbb{R}}^n$, for one of the choices of the sign. Then the set of transmission eigenvalues is discrete.
The proof is based on application of the analytic Fredholm theory to the holomorphic family of operators $$T_\lambda:{\mathbb{C}}\to \mathcal{L}(H^{P_0}_0(\Omega),H^{-P_0}(\Omega)), \quad
T_\lambda=(P_0+V-\lambda)\frac{1}{V}(P_0-\lambda).$$ Since by Lemma \[lem\_coercive\] there exists $\lambda_0\in {\mathbb{R}}$ such that the sesquilinear form $B_{\lambda_0}$, corresponding to the operator $T_{\lambda_0}$, is coercive and bounded on $H^{P_0}_0(\Omega)\times H^{P_0}_0(\Omega)$, by an application of the Lax–Milgram lemma we conclude that $T_{\lambda_0}:H^{P_0}_0(\Omega)\to H^{-P_0}(\Omega)$ is invertible.
Now, for any $\lambda$ the operator $$\label{eq_perturbation}
T_\lambda-T_{\lambda_0}=-P_0\frac{\lambda-\lambda_0}{V}-\frac{\lambda-\lambda_0}{V}P_0-(\lambda-\lambda_0)+\frac{\lambda^2-\lambda_0^2}{V}:H^{P_0}_0(\Omega)\to H^{-P_0}(\Omega)$$ is compact. Indeed, the last two terms in the right hand side of are compact because $$-(\lambda-\lambda_0)+\frac{\lambda^2-\lambda_0^2}{V}:L^2(\Omega)\to L^2(\Omega)$$ is bounded and by Lemma \[prop\_comp\] the embedding $H^{P_0}_0(\Omega)\hookrightarrow L^2(\Omega)$ is compact. The second term in the right hand side of defines the operator $-\frac{\lambda-\lambda_0}{V}P_0$. This operator is compact as a composition of the bounded operators $$P_0:H^{P_0}_0(\Omega)\to L^2(\Omega),\quad
-\frac{\lambda-\lambda_0}{V}:L^2(\Omega)\to L^2(\Omega)$$ with the compact inclusion $L^2(\Omega)$ into $H^{-P_0}(\Omega)$. The first term in the right hand side of defines the operator $-P_0\frac{\lambda-\lambda_0}{V}$. This is a bounded operator $$-P_0\frac{\lambda-\lambda_0}{V}:L^2(\Omega)\to H^{-P_0}(\Omega).$$ Since the inclusion $H^{P_0}_0(\Omega)$ to $L^2(\Omega)$ is compact, we conclude the compactness of $-P_0\frac{\lambda-\lambda_0}{V}$ as an operator from $H^{P_0}_0(\Omega)$ to $H^{-P_0}(\Omega)$. Thus also $T_\lambda - T_{\lambda _0}$ is compact.
Hence, the operator $T_{\lambda}$ is Fredholm as the sum of an invertible operator and a compact operator. Since it is invertible at $\lambda=\lambda_0$, the analytic Fredholm theory guarantees that $T_\lambda$ is invertible except for a discrete set of values $\lambda$.
The assumptions of Theorem \[thm\_discret\_1\] are satisfied automatically when the operator $P_0$ is elliptic.
The condition that $\pm P_0(\xi)$ is bounded from below on ${\mathbb{R}}^n$, for one of the choices of the sign, cannot be removed completely. Indeed, let $P_0 = D_{x_1} + \Delta _{x'}$, $x=(x_1,x')\in {\mathbb{R}}^n$. Then if $(v,w)$ is a pair of transimssion eigenfunctions for a transmission eigenvalue $\lambda$ on some domain $\Omega$, then $v'(x) = e^{i\mu x_1} v(x)$ and $w'(x) = e^{i\mu x_1} w(x)$ will be transmission eigenfunctions for the transmission eigenvalue $\lambda + \mu$. Hence the set of transmission eigenvalues is either empty or ${\mathbb{C}}$.
Notice that $\text{ind}(T_\lambda)=0$, since $T_\lambda$ is the sum of an invertible operator and a compact operator. It follows that $T_\lambda$ fails to be invertible if and only if $T_\lambda$ is not injective.
Existence of transmission eigenvalues {#sec_exis}
======================================
In this section we study the question of existence of transmission eigenvalues. Let us write the operator $T_\lambda$ in the following form, $$\label{eq_T_lambda}
T_\lambda=(P_0+\frac{V}{2}-\lambda)\frac{1}{V}(P_0+\frac{V}{2}-\lambda)-\frac{V}{4}.$$
\[lem\_T(tau)\_self-adjoint\] The operator $T_\lambda$ given by , equipped with the domain $$\mathcal{D}(T_\lambda)=\{u\in H^{P_0}_0(\Omega): P_0(\frac{1}{V}P_0-\frac{\lambda}{V})u\in L^2(\Omega)\},$$ is an unbounded self-adjoint operator on $L^2(\Omega)$, for any $\lambda\in {\mathbb{R}}$.
Let $A_\lambda=\frac{1}{V^{1/2}}(P_0+\frac{V}{2}-\lambda)$. Then $A_\lambda$ with the domain $\mathcal{D}(A_\lambda)=H_0^{P_0}(\Omega)$ is a densely defined closed operator. Its $L^2$–adjoint is given by $A_\lambda^*=(P_0+\frac{V}{2}-\lambda)\frac{1}{V^{1/2}}$ with the domain $\mathcal{D}(A_\lambda^*)=\{u\in L^2(\Omega):P_0\frac{1}{V^{1/2}}u\in L^2(\Omega)\}$. Then the result follows from the fact that $T_\lambda=A_\lambda^*A_\lambda-\frac{V}{4}$.
For future reference, let us remark that the form domain of the semibounded self-adjoint operator $T_\lambda$ is $H^{P_0}_0(\Omega)$, for any $\lambda\in {\mathbb{R}}$. Let us also recall the corresponding quadratic form $$B_\lambda(u,u)=
\langle\frac{1}{V}(P_0-\lambda)u ,(P_0+V-\lambda)u \rangle_{L^2(\Omega)}, \quad u\in H^{P_0}_0(\Omega).$$ From Lemma \[lem\_coercive\], we know that there exists $\lambda_0\in {\mathbb{R}}$ such that $B_{\lambda_0}$ is coercive on $H^{P_0}_0(\Omega)$.
Similarly to [@paisyl08], we have the following result.
\[thm\_existence1\] If for some $\lambda\in {\mathbb{R}}$, there is $u\in H^{P_0}_0(\Omega)$ such that $$\label{eq_existence2}
B_\lambda(u,u) \le 0$$ then there exists a transmission eigenvalue $\lambda^*\in (\lambda_0, \lambda]$, if $\lambda_0<\lambda$, or $\lambda^*\in [\lambda, \lambda_0)$, if $\lambda_0>\lambda$.
The operator $T_\lambda$ has a compact resolvent acting on $L^2(\Omega)$ because the canonical embedding of its domain $\mathcal{D}(T_\lambda)$ to $L^2(\Omega)$ is compact. The facts that $T_\lambda$ depends continuously on $\lambda$, $T_\lambda$ is self-adjoint with compact resolvent and the form domain of $T_\lambda$ does not depend on $\lambda$ imply, through an application of the variational principle, that the eigenvalues of $T_\lambda$ depend continuously on $\lambda$ for $\lambda\in {\mathbb{R}}$. See [@paisyl08] for the details of this argument.
The hypothesis yields that $T_\lambda$ has at least one non-positive eigenvalue. Since $T_{\lambda_0}$ is positive definite, the lowest eigenvalue, which is a continuous function of $\lambda \in {\mathbb{R}}$, must pass through zero for some $\lambda^*\in (\lambda_0, \lambda]$, if $\lambda_0<\lambda$, or $\lambda^*\in [\lambda, \lambda_0)$, if $\lambda_0>\lambda$. Hence, the operator $T_\lambda^*$ is non-injective and therefore, $\lambda^*$ is a transmission eigenvalue.
We define the multiplicity of a transmission eigenvalue $\lambda$ to be the multiplicity of zero as an eigenvalue of $T_\lambda$. Since the self-adjoint operator $T_\lambda$, $\lambda\in {\mathbb{R}}$, has a compact resolvent, the multiplicity of $\lambda$ is finite.
As in [@paisyl08 Lemma 14], we get the following result on existence of more than one transmission eigenvalues.
\[lem\_existence\_p\] If there exists $\lambda\in {\mathbb{R}}$ and a p-dimensional subspace $V^p\subset H^{P_0}_0(\Omega)$ such that $$\label{eq_existence3}
B_\lambda (u,u) \le 0$$ for all $u\in V^p$, then there exist $p$ transmission eigenvalues, counting with multiplicity.
The next result tells us that if the potential is strong enough then the transmission eigenvalues exist.
Let $V(x)\ge \delta>0$. If there exists a $p$-dimensional subspace $V^p\subset H^{P_0}_0(\Omega)$ such that $$\delta\ge 2\frac{\|P_0u\|}{\|u\|}$$ for all $0\ne u\in V^p$, then there exist $p$ transmission eigenvalues, counting with multiplicity.
We have $$\begin{aligned}
B_\lambda(u,u)&=\langle \frac{1}{V}(P_0-\lambda)u,(P_0-\lambda)u \rangle + \langle (P_0-\lambda)u,u \rangle\\
&\le \frac{1}{\delta}\|P_0u\|^2-\frac{2\lambda}{\delta}\langle P_0 u,u \rangle+\frac{\lambda^2}{\delta}\|u\|^2+\langle P_0 u,u \rangle-\lambda\|u\|^2. \end{aligned}$$ Choosing $ \lambda=\delta/2 $, we get $$B_\lambda(u,u)
\le \frac{1}{\delta}\|P_0u\|^2-\frac{\delta}{4}\|u\|^2.$$ The claim follows by an application of Lemma \[lem\_existence\_p\].
Existence of infinitely many transmission eigenvalues {#sec_exis_inf}
=====================================================
Now assume, as before, that $P_0(D)$ is a partial differential operator of degree $m\ge 2$ with constant real coefficients and $\tilde P_0(\xi)\to\infty$ when $|\xi|\to \infty$ and the symbol $\pm P_0(\xi)$ is bounded from below on ${\mathbb{R}}^n$, for one of the choices of the sign. Let $\varepsilon >0$ and consider an open ball $B_\varepsilon(0)\subset {\mathbb{R}}^n$ of radius $\varepsilon$ centered at the origin. Consider the following interior transmission problem for the ball $B_\varepsilon(0)$ and an arbitrary constant potential $\delta>0$, $$\label{eq_ITP_ball}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad B_\varepsilon(0),\\
(P_0+\delta-\lambda)w=0 \quad &\text{in} \quad B_\varepsilon(0),\\
v-w \in H^{P_0}_0(B_\varepsilon(0)).
\end{aligned}$$ Notice that as the operator $P_0$ has constant coefficients and $\delta$ is a constant, if there exists a transmission eigenvalue $\lambda(\varepsilon)$ for the ball $B_\varepsilon(0)$, then $\lambda(\varepsilon)$ is a transmission eigenvalue for an arbitrary ball in ${\mathbb{R}}^n$ of radius $\varepsilon$.
Existence of an infinite discrete set of real transmission eigenvalues when $P_0=-\Delta$ in ${\mathbb{R}}^n$, $n=2,3$, was obtained in [@CakDroHou]. The proof relies on an explicit computation in the case of a constant potential when $\Omega$ is a ball, combined with a specific analytic framework of generalized eigenvalue problems.
Following the approach of [@CakDroHou], we have the following conditional result on existence of an infinite set of real transmission eigenvalues for general operators. In the proof we give a direct argument relying upon Lemma \[lem\_existence\_p\].
\[prop\_infty\] Let $V\in L^\infty({\mathbb{R}}^n)$ be real-valued compactly supported with ${\operatorname{supp}}(V)=\overline{\Omega}$ and assume that $$V\ge \delta>0\quad \text{a.e. in} \ \overline{\Omega}.$$ Furthermore, assume that for any $\varepsilon >0$, there exists a real transmission eigenvalue for . Then the problem has an infinite set of real transmission eigenvalues.
For every $p\in {\mathbb{N}}$, there exists $\varepsilon >0$ small enough such that $\Omega$ contains $p$ disjoint balls $B^1_\varepsilon,\dots,B^p_\varepsilon$ of radius $\varepsilon$, i.e. $\overline{B^i_\varepsilon}\subset \Omega$, $i=1,\dots,p$, and $\overline{B^i_\varepsilon}\cap \overline{B^j_\varepsilon}=\emptyset$ for $i\ne j$. Let $\lambda(\varepsilon)\in {\mathbb{R}}$ be a transmission eigenvalue for . Then it is a transmission eigenvalue for each of the balls $B^i_\varepsilon$ with potential $\delta$. Thus, Proposition \[thm\_equivalence\] implies that there are $0\ne u^i(\varepsilon)\in H^{P_0}_0(B^i_\varepsilon)$ such that $$(P_0+\delta-\lambda(\epsilon))\frac{1}{\delta}(P_0-\lambda)u^i(\varepsilon)=0 \quad \text{in}\quad \mathcal{D}'(B^i_{\varepsilon}), \quad i=1,\dots,p.$$ The extension by zero $\tilde u^i$ of $u^i(\varepsilon)$ to the whole of $\Omega$ is in $H^{P_0}_0(\Omega)$. Moreover, the functions $\tilde u^1,\dots, \tilde u^p$ form an orthogonal system in $H^{P_0}_0(\Omega)$, since they have disjoint supports. This implies that $$\langle \frac{1}{\delta}(P_0-\lambda(\varepsilon))\tilde u^i,(P_0+\delta-\lambda(\varepsilon))\tilde u^j \rangle_{L^2(\Omega)}=0, \quad i,j=1,\dots,p.$$ Set $V^p=\textrm{span}\{\tilde u^1,\dots,\tilde u^p\}$. Let $u\in V^p$. Then $$\begin{aligned}
B_{\lambda(\varepsilon)}(u,u)&=\langle\frac{1}{V}(P_0-\lambda(\varepsilon))u,(P_0+V-\lambda(\varepsilon))u\rangle_{L^2(\Omega)}\\
&=\langle\frac{1}{V}(P_0-\lambda(\varepsilon))u,(P_0-\lambda(\varepsilon))u\rangle_{L^2(\Omega)}+\langle\frac{1}{V}(P_0-\lambda(\varepsilon))u,V u\rangle_{L^2(\Omega)}\\
&\le\langle\frac{1}{\delta}(P_0-\lambda(\varepsilon))u,(P_0-\lambda(\varepsilon))u\rangle_{L^2(\Omega)}+\langle(P_0-\lambda(\varepsilon))u,u\rangle_{L^2(\Omega)}\\
&=\langle\frac{1}{\delta}(P_0-\lambda(\varepsilon))u,(P_0+\delta-\lambda(\varepsilon))u\rangle_{L^2(\Omega)}=0.\end{aligned}$$ Hence, it follows from Lemma \[lem\_existence\_p\] that problem has $p$ transmission eigenvalues, counting with multiplicity. As $p$ is arbitrary, the result follows.
The generalized acoustic problem
================================
Let $V\in L^\infty({\mathbb{R}}^n)$ be compactly supported in ${\mathbb{R}}^n$ with ${\operatorname{supp}}(V)=\overline{\Omega}$, where $\Omega\subset {\mathbb{R}}^n$ is a bounded domain, and $P_0(D)$ be a partial differential operator of degree $m\ge 2$ with constant real coefficients.
As in physics, while the problem $$(-\Delta-\lambda +V)u=0,$$ models the time-independent Schrödinger equation, the equation $$(-\Delta-\lambda(1+V))u=0$$ describes acoustic wave propagation with refractive index $1+V$. In this section, we study the interior transmission problem for the latter, where $-\Delta$ is replaced by a general $P_0$. This problem has the following form, $$\label{eq_ITP_Helm}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0-\lambda(1+ V))w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{P_0}_0(\Omega).
\end{aligned}$$
We say that $\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)$. It suffices to require that $v\ne 0$. Notice that $\lambda=0$ is always a transmission eigenvalue for .
We shall assume that $V$ is real-valued and such that $V\ge\delta>0$ a.e. in $\overline{\Omega}$.
As in Proposition \[thm\_equivalence\], one can show that $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^{P_0}_0(\Omega)$ satisfying $$T_\lambda u=(P_0-\lambda(1+V))\frac{1}{V}(P_0-\lambda)u=0\quad \text{in}\quad \mathcal{D}'(\Omega).$$
Define a sesquilinear form $$B_\lambda(\varphi,\psi)=\langle T_\lambda \varphi,\psi \rangle_{L^2(\Omega)}=
\langle \frac{1}{V}(P_0-\lambda)\varphi,(P_0-\lambda(1+V))\psi\rangle_{L^2(\Omega)}, \quad \varphi,\psi\in C^\infty_0(\Omega),$$ which extends uniquely to a continuous sesquilinear for on $H^{P_0}_0(\Omega)\times H^{P_0}_0(\Omega)$. Then $B_0$ is coercive in the sense that there exists $C_0>0$ such that $$B_0(\varphi,\varphi)\ge C_0\|\varphi\|^2_{P_0}, \quad \varphi\in H^{P_0}_0(\Omega).$$ Arguing as in Theorem \[thm\_discret\_1\], we get the following result.
\[thm\_disc\_Helm\] Assume that $\tilde P_0(\xi)\to\infty$ when $|\xi|\to \infty$. Then the set of transmission eigenvalues is discrete.
Remark that the multiplicity of each transmission eigenvalue except $\lambda=0$ is finite.
Notice that the assumption in Theorem \[thm\_discret\_1\] that the symbol $\pmÊP_0(\xi)$ is bounded from bellow on ${\mathbb{R}}^n$, for one of the choices of the sign, is not needed in the Helmholtz case. In particular, the results of this section are applicable when $P_0=D_{x_1}^2 + \Delta _{x'}$ is the wave operator on ${\mathbb{R}}^n$.
An inspection of the arguments of Sections \[sec\_exis\] – \[sec\_exis\_inf\] shows that the results there remain valid in the Helmholtz case. In particular, Theorem \[thm\_existence1\] and Lemma \[lem\_existence\_p\] hold true in the Helmholtz case as they stand. To formulate an analog of Proposition \[prop\_infty\], consider the following interior transmission problem for an open ball $B_\varepsilon(0)\subset {\mathbb{R}}^n$ of radius $\varepsilon>0$ centered at the origin, and an arbitrary constant potential $\delta>0$, $$\label{eq_ITP_Helm_ball}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad B_\varepsilon(0),\\
(P_0-\lambda(1+\delta))w=0 \quad &\text{in} \quad B_\varepsilon(0),\\
v-w \in H^{P_0}_0(B_\varepsilon(0)).
\end{aligned}$$
\[prop\_helm\_inf\_n\] Assume that for any $\varepsilon >0$ and $\delta>0 $, there exists a non-zero real transmission eigenvalue for . Then there exists an infinite set of real transmission eigenvalues for .
Examples
========
The biharmonic operator
-----------------------
As an example of a higher order operator to which our conditional existence results apply, we shall consider the biharmonic operator in ${\mathbb{R}}^3$, which arises, e.g., in the study of thin elastic plates.
Let $P_0=\Delta^2$ on ${\mathbb{R}}^3$. Then the problem has an infinite set of real transmission eigenvalues.
When proving this result, we shall apply Proposition \[prop\_helm\_inf\_n\]. It therefore suffices to prove that the problem has a non-trivial solution for arbitrary $\varepsilon>0$ and $\delta>0$. Without loss of generality, we may assume that $\delta$ is sufficiently small. Let $k=\lambda^{1/4}$, $k_\delta=(\lambda(1+\delta))^{1/4}$, and $$\label{eq_rho}
\rho=\frac{k_\delta}{k}=(1+\delta)^{1/4}=1+\frac{\delta}{4}+\mathcal{O}(\delta^2), \quad 0<\delta\ll 1.$$ Then writing $$\Delta^2 - k^4 = (\Delta - k^2) (\Delta + k^2) = (\Delta + k^2) (\Delta - k^2),$$ and for $\Delta^2 - k_\delta^4$ similarly, one sees that a reasonable ansatz for $v$ and $w$ in is $$v(x) = a_1j_0 ( kr) + a_2j_0 (ikr), \ w(x) = a_3 j_0 ( k_\delta r) + a_4j_0 (ik_\delta r),$$ where $j_0$ is the spherical Bessel function of order zero, and $a_j$ are constants. Our boundary conditions now take form $$\label{biharmoninen_vakioreuna}
\left(\frac{d}{dr}\right) ^l (a_1j_0 ( kr) + a_2j_0 (ikr) - a_3 j_0 ( k_\delta r) - a_4j_0 (ik_\delta r)) |_{r=\varepsilon}= 0, \quad l=0,\,1,\, 2,\, 3.$$ Using the known asymptotics for $j_0 ( k\varepsilon)$ and $j_0 ( ik\varepsilon)$ and their derivatives, as $k\to\infty$, one sees that the deteminant of the linear system is $$\begin{aligned}
d=&\begin{vmatrix}
\frac{\sin (k\varepsilon)}{k\varepsilon} & \frac{e^{k\varepsilon}}{2k\varepsilon}+ \mathcal{O}(e^{-k\varepsilon}) & -\frac{\sin (k_\delta\varepsilon)}{k_\delta\varepsilon}
& -\frac{e^{k_{\delta}\varepsilon}}{2k_\delta\varepsilon}+ \mathcal{O}(e^{-k_\delta\varepsilon}) \\
\frac{\cos( k\varepsilon)}{\varepsilon} +\mathcal{O}(\frac{1}{k}) & \frac{e^{k\varepsilon}}{2\varepsilon}+ \mathcal{O}(\frac{e^{k\varepsilon}}{k})
& -\frac{\cos( k_\delta\varepsilon)}{\varepsilon} +\mathcal{O}(\frac{1}{k})
& -\frac{e^{k_{\delta}\varepsilon}}{2\varepsilon}+ \mathcal{O}(\frac{e^{k_\delta\varepsilon}}{k_\delta}) \\
-\frac{k\sin (k\varepsilon)}{\varepsilon} + \mathcal{O}(1) & \frac{ke^{k\varepsilon}}{2\varepsilon} + \mathcal{O}(e^{k\varepsilon}) & \frac{k_\delta\sin (k_\delta\varepsilon)}{\varepsilon} +\mathcal{O}(1)
& -\frac{k_\delta e^{k_{\delta}\varepsilon}}{2\varepsilon}+ \mathcal{O}(e^{k_\delta\varepsilon})\\
-\frac{k^2\cos( k\varepsilon)}{\varepsilon} +
\mathcal{O}(k)
& \frac{k^2e^{k\varepsilon}}{2\varepsilon} + \mathcal{O}(ke^{k\varepsilon})
& \frac{k_\delta^2\cos( k_\delta\varepsilon)}{\varepsilon} +\mathcal{O}(k)
&
-\frac{k_\delta^2 e^{k_{\delta}\varepsilon}}{2\varepsilon} + \mathcal{O}(k_\delta e^{k_\delta\varepsilon})
\end{vmatrix}\\
=
&\frac{1}{4\varepsilon^4}e^{(k+k_\delta)\varepsilon}k^2\frac{\rho-1}{\rho} \left(d_1+\mathcal{O}(\frac{1}{\delta k})\right),\end{aligned}$$ where $d_1$ is given by $$\begin{vmatrix} (\rho-1)\sin(k\varepsilon) -(1+\rho)\cos(k\varepsilon) & (\rho-1)\sin(k_\delta \varepsilon) +(\rho+1)\cos (k_\delta \varepsilon)\\
-(\rho^2+\rho+2)\cos (k\varepsilon) +(\rho^2+\rho)\sin(k\varepsilon) & (2\rho^2+\rho+1)\cos(k_\delta\varepsilon) -(\rho+1)\sin(k_\delta\varepsilon)
\end{vmatrix}$$ Using , we get $$d_1=4\sin((\rho-1)k\varepsilon)+\mathcal{O}(\delta).$$ Hence, to show that the linear system has a non-trivial solution, it suffices to check that the function $$\sin((\rho-1)k\varepsilon)+\mathcal{O}(\delta)+ \mathcal{O}(\frac{1}{\delta k})$$ has real zeros, for $k$ large enough. The latter is clear, however, from the periodicity of the function $$k\mapsto \sin((\rho-1)k\varepsilon).$$ We may also notice that the minimal period of this function is $$\frac{8\pi}{\delta \varepsilon}+\mathcal{O}(\frac{1}{\varepsilon}).$$ This completes the proof.
The Dirac system
----------------
Our approach generalizes also to many systems. We demonstrate this here by carrying out the analysis in the case of the Dirac system.
The free Dirac operator in ${\mathbb{R}}^3$ is given by the $4\times 4$ matrix $$\mathcal{L}_0(D)=\begin{pmatrix} 0 & \sigma\cdot D\\
\sigma\cdot D & 0
\end{pmatrix},$$ where $D=-i\nabla$ and $\sigma=(\sigma_1,\sigma_2,\sigma_3)$ is a vector of Pauli matrices with $$\sigma_1=\begin{pmatrix}
0& 1\\
1& 0
\end{pmatrix},\quad
\sigma_2=\begin{pmatrix}
0& -i\\
i& 0
\end{pmatrix},\quad
\sigma_3=\begin{pmatrix}
1& 0\\
0& -1
\end{pmatrix}.$$ The most important property of the Dirac operator is the following one, $$\mathcal{L}_0(D)^2=-\Delta I_4,$$ where $I_4$ is the $4\times 4$ identity matrix.
Let $\Omega\subset {\mathbb{R}}^3$ be a bounded domain in ${\mathbb{R}}^3$ with a connected $C^\infty$-smooth boundary. It is known [@NakTsu00] that when equipped with the domain $$\mathcal{D}(\mathcal{L}_0)=\{\begin{pmatrix} u_+\\
u_-
\end{pmatrix}\in L^2(\Omega)^2\times L^2(\Omega)^2: \ u_+\in H_0^1(\Omega)^2, \sigma\cdot Du_-\in L^2(\Omega)^2
\},$$ the Dirac operator $\mathcal{L}_0$ is self-adjoint on $L^2(\Omega)^4$.
Notice that $$H_0^1(\Omega)^2\times H^1(\Omega)^2\subset \mathcal{D}(\mathcal{L}_0).$$ However, in general, $\mathcal{D}(\mathcal{L}_0)$ is strictly larger than the Sobolev space $H_0^1(\Omega)^2\times H^1(\Omega)^2$, see [@Sch95] for the discussion and a precise example.
Let $V(x)$ be an Hermitian $4\times4$-matrix-valued function whose entries belong to $L^\infty({\mathbb{R}}^3)$. An application of the Kato-Rellich theorem shows that the operator $\mathcal{L}_0(D)+V$ is self-adjoint on $\mathcal{D}(\mathcal{L}_0)$. Assume that $0$ is not in the spectrum of $\mathcal{L}_0(D)+V$. Then it was shown in [@NakTsu00; @salotzou] that for any $f\in H^{1/2}(\p \Omega)^2$, the boundary value problem $$\label{eq_bvp_D}
\begin{aligned}
&(\mathcal{L}_0(D)+V)u=0,\quad \text{in}\quad \Omega,\\
&u_+=f,\quad \text{on}\quad \p \Omega,
\end{aligned}$$ has a unique solution $u\in H^1(\Omega)^4$. The set of the Cauchy data for is given by $$\{(u_+|_{\p \Omega},u_-|_{\p \Omega}):u\in H^1(\Omega)^4 \ \text{is a solution of } (\mathcal{L}_0(D)+V)u=0 \text{ in }\Omega\}.$$
Assume now that ${\operatorname{supp}}(V)=\overline{\Omega}$. The interior transmission problem for the Dirac operator is the following boundary value problem, $$\label{eq_ITP_Helm_D}
\begin{aligned}
(\mathcal{L}_0(D)-\lambda I_4)v=0 \quad \text{in}\quad \Omega,\\
(\mathcal{L}_0(D)-\lambda(I_4+V))w=0 \quad \text{in}\quad \Omega, \\
v-w\in H^1_0(\Omega)^4.
\end{aligned}$$ We say that $\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)^4$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)^4$. It suffices to require that $v\ne 0$.
Notice that $\lambda=0$ is a transmission eigenvalue and the space of functions $$\{v\ne 0:\mathcal{L}_0(D)v=0\}$$ is infinite dimensional.
The standard electromagnetic potential for the Dirac operator given by $$V=\mathcal{L}_0(A)+Q,$$ with $$Q=\begin{pmatrix} q_+I_2 & 0\\
0 & q_-I_2
\end{pmatrix},\quad A=(a_1,a_2,a_3)\in L^\infty(\Omega;{\mathbb{R}}^3),\quad
q_\pm \in L^\infty(\Omega;{\mathbb{R}})$$ is included in the setup above.
Throughout this section, we shall assume that $V(x)$ is an Hermitian positive-definite $4\times 4$-matrix valued function, i.e. there is a constant $c_V>0$ such that $$\langle V(x)\eta,\eta \rangle\ge c_V|\eta|^2,\quad \forall x\in \overline{\Omega},\quad \forall \eta\in {\mathbb{C}}^4,$$ where $\langle \cdot,\cdot \rangle$ is the inner product in ${\mathbb{C}}^4$. Moreover, we shall assume that the entries of $V(x)$ belong to $C^\infty(\overline\Omega)$. Thus, the entries of the inverse matrix $V^{-1}(x)$ also belong to $C^\infty(\overline\Omega)$.
Arguing as in the earlier sections, we see that the following characterization of transmission eigenvalues holds: $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^1_0(\Omega)^4$ satisfying $$T_\lambda u=(\mathcal{L}_0-\lambda(I_4+V))V^{-1}(\mathcal{L}_0-\lambda I_4)u=0 \quad \text{in}\quad \mathcal{D}'(\Omega)^4.$$ Here $$T_\lambda =A-\lambda B+\lambda^2C,$$ where $$\begin{aligned}
A&=\mathcal{L}_0V^{-1}\mathcal{L}_0,\\
B&=V^{-1}\mathcal{L}_0+\mathcal{L}_0V^{-1}+\mathcal{L}_0,\\
C&=1+V^{-1}.\end{aligned}$$
The operator $A$ is symmetric and positive on $L^2(\Omega)^4$, when equipped with the domain $C^\infty_0(\Omega)^4$, in the sense that there is $d=d_{V,\Omega}>0$ such that $$\langle A\varphi,\varphi \rangle_{L^2(\Omega)^4}\ge
d\|\varphi\|_{L^2(\Omega)^4}^2,\quad \varphi\in C^\infty_0(\Omega)^4.$$
Let $\varphi\in C^\infty_0(\Omega)^4$. Then we get $$\begin{aligned}
\langle A\varphi,\varphi \rangle_{L^2(\Omega)^4}&=\langle V^{-1}\mathcal{L}_0\varphi,\mathcal{L}_0\varphi \rangle_{L^2(\Omega)^4}\ge c_V\|\mathcal{L}_0\varphi\|_{L^2(\Omega)^4}^2\ge d\|\varphi\|_{L^2(\Omega)^4}^2.\end{aligned}$$ Here the last inequality follows from the estimate [@horbookII Theorem 10.3.7] $$\|\mathcal{L}_0\varphi\|_{L^2(\Omega)^4}\ge c_\Omega \|\varphi\|_{L^2(\Omega)^4},\quad \varphi\in C^\infty_0(\Omega)^4,\quad c_\Omega>0.$$
The claim follows.
The second order operator $A$ is uniformly strongly elliptic in the sense that there is $c>0$ such that $$\label{eq_ellipticity_D}
\langle \sigma(A)(x,\xi)\eta,\eta\rangle\ge c|\xi|^2|\eta|^2,\quad x\in \overline{\Omega}, \quad \xi\in {\mathbb{R}}^3,\quad \eta\in {\mathbb{C}}^4,$$ where $\sigma(A)$ is the principal symbol of $A$.
In view of homogeneity of it suffices to prove it for $|\xi|=1$ and $|\eta|=1$. Let $\eta=(\eta_+,\eta_-)^T\in {\mathbb{C}}^4$. Then since $V^{-1}$ is an Hermitian positive-definite matrix valued function, we have $$\begin{aligned}
\langle \sigma(A)(x,\xi)\eta,\eta\rangle&=
\langle V^{-1}\begin{pmatrix} 0 & \sigma\cdot \xi\\
\sigma\cdot \xi & 0
\end{pmatrix}\eta, \begin{pmatrix} 0 & \sigma\cdot \xi\\
\sigma\cdot \xi & 0
\end{pmatrix}\eta
\rangle\\
&\ge c_V\begin{vmatrix}
(\sigma\cdot\xi)\eta_-\\
(\sigma\cdot\xi)\eta_+
\end{vmatrix}^2_{{\mathbb{C}}^4}>0. \end{aligned}$$ The latter inequality follows from the fact that $$\sigma\cdot\xi=\begin{pmatrix}
\xi_3 & \xi_1-i\xi_2\\
\xi_1+i\xi_2& -\xi_3
\end{pmatrix}, \quad
\det(\sigma\cdot\xi)=-|\xi|^2=-1.$$ This proves .
\[prop\_self\_D\] The operator $A$, equipped with the domain $$\mathcal{D}(A)=H^1_0(\Omega)^4\cap H^2(\Omega)^4,$$ is a positive self-adjoint operator on $L^2(\Omega)^4$.
We shall consider the Friedrichs extension of $A$ on $C_0^\infty(\Omega)^4$, denoted also by $A$, which has the domain $$\mathcal{D}(A)=\mathcal{D}(Q)\cap \mathcal{D}(A_{\textrm{max}}).$$ Here $$Q(\varphi,\varphi)=\langle A\varphi,\varphi \rangle_{L^2(\Omega)^4}$$ is the quadratic form associated with the operator $A$. The domain $\mathcal{D}(Q)$, also the form domain of $A$, is the completion of $C^\infty_0(\Omega)^4$ with respect to the norm $|\!|\!|\varphi|\!|\!|=\sqrt{Q(\varphi,\varphi)}$. The maximal realization $A_{\textrm{max}}$ of the operator $A$ is defined by $$\mathcal{D}(A_{\textrm{max}})=\{u\in L^2(\Omega)^4:Au\in L^2(\Omega)^4\}.$$
Let us now show that $$\label{eq_form_dom_D}
\mathcal{D}(Q)=H^1_0(\Omega)^4.$$ Indeed, it is easy to see that the norm $|\!|\!|\cdot|\!|\!|$ is equivalent to the following norm $$\label{eq_norm_compl_D}
\|\mathcal{L}_0\cdot\|_{L^2(\Omega)^4}+ \|\cdot\|_{L^2(\Omega)^4}.$$ Since $\|\mathcal{L}_0\cdot\|_{L^2(\Omega)^4}\le C\|\nabla\cdot\|_{L^2(\Omega)^2}$, we have $$H^1_0(\Omega)^4\subset \mathcal{D}(Q).$$ On the other hand, it follows from [@NakTsu00 Proposition 4.2] that the completion of $C^\infty(\overline{\Omega})^4$ with respect to the norm is the space $$\mathcal{H}(\Omega)=\{u\in L^2(\Omega)^4:\mathcal{L}_0u\in L^2(\Omega)^4\}.$$ Thus, $$\mathcal{D}(Q)\subset \mathcal{H}(\Omega).$$ It is shown in [@NakTsu00 Proposition 4.6] that the trace map $$\tau: C^\infty(\overline\Omega)^4\to C^\infty(\p \Omega)^4, \quad
u\mapsto u|_{\p \Omega},$$ extends uniquely to a bounded map on $\mathcal{H}(\Omega)$. It follows from [@NakTsu00 Proposition 4.6] that for any $u\in \mathcal{D}(Q)$, $\tau u=0$. Hence, [@NakTsu00 Proposition 4.10] implies that $u\in H^1_0(\Omega)^4$. This proves .
Hence, $$\mathcal{D}(A)=\{u\in H^1_0(\Omega)^4:Au\in L^2(\Omega)^4\}.$$ As the operator $A$ is strongly elliptic and $\Omega$ has a smooth boundary, by elliptic regularity, see for instance [@Grubbbook Section 7.5], $\mathcal{D}(A)=H^1_0(\Omega)^4\cap H^2(\Omega)^4$.
It follows from Proposition \[prop\_self\_D\] that for any $\lambda\in {\mathbb{R}}$, the operator $T_\lambda$, equipped with the domain $\mathcal{D}(A)$ is a self-adjoint operator on $L^2(\Omega)^4$, and the form domain of $T_\lambda$ is $H^1_0(\Omega)^4$.
The set of transmission eigenvalues for is discrete.
First note that the operator $$-\lambda B+\lambda^2 C: \mathcal{D}(A)\to L^2(\Omega)$$ is compact. Hence, the operator $T_\lambda:\mathcal{D}(A)\to L^2(\Omega)$ is Fredholm of index $0$, invertible at $\lambda=0$. Thus, by analytic Fredholm theory, $$T^{-1}_\lambda:L^2(\Omega)\to \mathcal{D}(A), \quad \lambda\in {\mathbb{C}},$$ is a meromorphic family of operators with residues of finite rank. This proves the claim.
As before, we see that the multiplicity of a transmission eigenvalue $\lambda\in {\mathbb{R}}$ is finite.
Let $V$ be a matrix-valued potential as above. Then there exists an infinite set of real transmission eigenvalues for .
First notice that Proposition \[prop\_helm\_inf\_n\] continues to be valid for . It is therefore sufficient to prove the existence of transmission eigenvalues for the following problem, $$\label{eq_ITP_ball_D}
\begin{aligned}
(\mathcal{L}_0(D)-\lambda I_4)v=0 \quad \text{in}\quad B_\varepsilon(0),\\
(\mathcal{L}_0(D)-\lambda(1+\delta)I_4)w=0 \quad \text{in}\quad B_\varepsilon(0), \\
v-w\in H^1_0(B_\varepsilon(0))^4.
\end{aligned}$$ Here $\varepsilon>0$, $\delta>0$ and $B_\varepsilon(0)\subset{\mathbb{R}}^3$ is an open ball of radius $\varepsilon$ centered at the origin.
When considering , we let $0\ne \lambda\in {\mathbb{R}}$ and study $$(\mathcal{L}_0(D)-\lambda I_4)v=0,$$ where $$v=\begin{pmatrix}v_+\\
v_-
\end{pmatrix},\quad
v_\pm=\begin{pmatrix}
v_\pm^1\\
v_\pm^2
\end{pmatrix}.$$ Then we have $$\begin{aligned}
-\lambda v_++\sigma\cdot D v_-&=0,\\
\sigma\cdot D v_+-\lambda v_-&=0. \end{aligned}$$ As $$\sigma\cdot D=\begin{pmatrix}
D_3 & D_1-iD_2\\
D_1+iD_2 & -D_3
\end{pmatrix},$$ we get $$\begin{aligned}
&v_+^1=\frac{1}{\lambda}(D_3v_-^1 + (D_1-iD_2)v_-^2)\\
&v_+^2=\frac{1}{\lambda}((D_1+iD_2)v_-^1 -D_3v_-^2),\\
&
\begin{pmatrix}
-\Delta-\lambda^2 & 0\\
0 & -\Delta-\lambda^2
\end{pmatrix} \begin{pmatrix}
v_-^1\\
v_-^2
\end{pmatrix}=0.\end{aligned}$$ Considering the equation $$(\mathcal{L}_0(D)-\lambda(1+\delta)I_4)w=0,$$ similarly, we obtain $$\begin{aligned}
&w_+^1=\frac{1}{\lambda(1+\delta)}(D_3w_-^1 + (D_1-iD_2)w_-^2)\\
&w_+^2=\frac{1}{\lambda(1+\delta)}((D_1+iD_2)w_-^1 -D_3w_-^2),\\
&
\begin{pmatrix}
-\Delta-\lambda^2(1+\delta)^2 & 0\\
0 & -\Delta-\lambda^2(1+\delta)^2
\end{pmatrix} \begin{pmatrix}
w_-^1\\
w_-^2
\end{pmatrix}=0.\end{aligned}$$
Notice that to prove the existence of real transmission eigenvalues for the problem , it suffices to restrict our attention to solutions $v,w$ of such that $v_-^1=v_-^2=f(r)$ and $w_-^1=w_-^2=g(r)$, $r=|x|$, are spherically symmetric solutions of the following interior transmission problem, $$\label{eq_transmis_bessel_D}
\begin{aligned}
(-\Delta-\lambda^2)f=0\quad \text{in}\quad B_\varepsilon(0),\\
(-\Delta-\lambda^2(1+\delta)^2)g=0 \quad \text{in}\quad B_\varepsilon(0),\\
f-g=0\quad \text{on}\quad \p B_\varepsilon(0),\\
\p_{r}f=\frac{\p_{r}g}{1+\delta}\quad \text{on}\quad \p B_\varepsilon(0).
\end{aligned}$$ It is clear then that for such solutions, the boundary conditions $$\begin{aligned}
v_+^1=w_+^1\quad \text{on}\quad \p B_\varepsilon(0),\\
v_+^2=w_+^2\quad \text{on}\quad \p B_\varepsilon(0).\end{aligned}$$ are satisfied.
Since $$(\Delta+\lambda^2)f(r)=f''(r)+\frac{2}{r}f'(r)+\lambda^2f(r)=0,$$ $f$ must be of the form $$f(x)=c_0j_0(\lambda r),$$ where $j_0$ is the spherical Bessel function of order zero and $c_0$ is a constant. In the same way, $$g(x)=c_1j_0(\lambda(1+\delta) r),$$ The boundary conditions in require that $$\begin{aligned}
&c_0j_0(\lambda\varepsilon)=c_1j_0(\lambda(1+\delta)\varepsilon),\\
& c_0j'_0(\lambda\varepsilon)=c_1j'_0(\lambda(1+\delta)\varepsilon).\end{aligned}$$ A nontrivial solution of this system exists if and only if $$\label{eq_det_D}
\det\begin{pmatrix} j_0(\lambda\varepsilon) & -j_0(\lambda(1+\delta)\varepsilon)\\
j'_0(\lambda\varepsilon) & -j'_0(\lambda(1+\delta)\varepsilon)
\end{pmatrix}=0.$$ Since $$j_0(r)=\frac{\sin r}{r},\quad j'_0(r)=\frac{\cos r}{r}+\mathcal{O}(1/r^2),$$ implies that $$\label{eq_det_2_D}
\sin(\lambda\delta\varepsilon)+\mathcal{O}(1/\lambda)=0, \quad \lambda\to\infty.$$ The existence of an infinite set of values $\lambda$ such that holds is clear as $\sin(\lambda\delta\varepsilon)$ is a periodic function taking positive and negative values. Each such $\lambda$ is a transmission eigenvalue for and this completes the proof.
Generalized Rellich theorem
===========================
In the last two sections, which do not depend on the material in Section 6 and 7, we would like to explain the connection between interior transmission eigenvalues and scattering theory. It is going to be provided by a generalization of the classical Rellich theorem, proved in [@Hormander73].
Let us start by summarizing the basic features of general scattering theory following [@horbookII Chapter 14]. Let $P_0$ be a partial differential operator in ${\mathbb{R}}^n$ of order $m\ge 2$ with constant real coefficients, $$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ Assume that $\Lambda(P_0)=\{0\}$, see , and that $P_0=P_0(D)$ is simply characteristic, i.e. $$\sum_{|\alpha|\le m}|P_0^{(\alpha)}(\xi)|\le C(\sum_{|\alpha|\le 1}|P_0^{(\alpha)}(\xi)|+1),\quad C>0 .$$ Examples are hypoelliptic operators [@horbookII Chapter 11] and operators of real principal type [@horbookI Chapter 8].
In order to describe mapping properties of the boundary values of the $L^2$-resolvent of $P_0$, we follow [@horbookII Chapter 14] and introduce the following Banach spaces. Define $$\begin{aligned}
B&=\{v\in L^2({\mathbb{R}}^n):\|v\|_B=\sum_{j=1}^\infty R_j^{1/2}(\int_{\Omega_j}|v|^2dx)^{1/2}<\infty \},\\
B^*&=\{u\in L^2_{\textrm{loc}}({\mathbb{R}}^n):\|u\|_{B^*}=\sup_{j> 0} R_j^{-1/2}(\int_{\Omega_j}|u|^2dx)^{1/2}<\infty \},\\\end{aligned}$$ where $$\begin{aligned}
R_0=0,&\quad R_j=2^{j-1}, \quad j=1,2,\dots,\\
\Omega_j=\{x\in {\mathbb{R}}^n:&R_{j-1}<|x|<R_j\}, \quad j=1,2,\dots.\end{aligned}$$ The space $B^*$ is the dual of $B$ and we have $$L^2_\delta\subset B\subset L^2\subset B^*\subset L^2_{-\delta}, \quad \delta> 1/2.$$ The space of $C^\infty_0$-functions is dense in $B$ but not in $B^*$. Its closure in $B^*$ is denoted by $\stackrel{\circ}{B^*}$. Then $u\in L^2_{\textrm{loc}}({\mathbb{R}}^n)$ belongs to $\stackrel{\circ}{B^*}$ if and only if $$\int_{|x|<R}|u|^2dx/R\to 0, \quad R\to\infty.$$ We also define the Sobolev space version of $B^*$, associated to $P_0(D)$, $$B^*_{P_0}=\{u\in B^*:P^{(\alpha)}_0(D)u\in B^*, \forall \alpha\}.$$
Let $Z(P_0)$ be the (necessarily finite) set of critical values of $P_0$, i.e. $$Z(P_0)=\{\lambda:\exists \xi\in {\mathbb{R}}^n \text{ s.t. } \nabla P_0(\xi)=0, P_0(\xi)=\lambda\}.$$ For $z\in \overline{{\mathbb{C}}^{\pm}}\setminus Z(P_0)$, the resolvent $R_0(z)=(P_0-zI)^{-1}$ of the simply characteristic operator $P_0$ extends to a continuous map $$R_0(z):B\to B^*_{P_0}.$$ Here $\overline{{\mathbb{C}}^{\pm}}=\{z:\pm \textrm{Im}\ z\ge 0\}$. For $\lambda\in {\mathbb{R}}\setminus Z(P_0)$, the boundary values of the resolvent are given by $$R_0(\lambda\pm i0)f=\lim_{\varepsilon\to 0^+}F^{-1}((P_0-\lambda\mp \varepsilon i)^{-1}F(f)),$$ where $F$ stands for the Fourier transformation.
We say that $u\in B^*$ is outgoing (incoming) if $u=R(\lambda+i0)f$ ($u=R(\lambda-i0)f$), $f\in B$ and $\lambda\in {\mathbb{R}}\setminus Z(P_0)$. If $u$ is outgoing or incoming then $(P_0-\lambda)u=f$.
For $\lambda\notin Z(P_0)$, the level set $M_\lambda=\{\xi\in {\mathbb{R}}^n:P_0(\xi)=\lambda\}$ is an $(n-1)$-dimensional $C^\infty$ submanifold of ${\mathbb{R}}^n$.
It is known that $u$ is both outgoing and incoming, i.e. $u=R(\lambda+i0)f=R(\lambda-i0)f$, if and only if $F(f)=\hat f\equiv 0$ on $M_\lambda$.
We shall consider multiplicative perturbations of $P_0$ given by $V\in L^\infty({\mathbb{R}}^n)$ with compact support. Such perturbations satisfy the short range condition introduced in [@horbookII Section 14.4], i.e. $$V:B^*_{P_0}\to B$$ is compact.
In order to define the scattering amplitude, we recall the following fundamental result [@horbookII Theorem 14.6.8].
Assume that $\lambda\in {\mathbb{R}}\setminus Z(P_0)$. If $u\in B^*_{P_0}$ satisfies $$(P_0+V-\lambda)u=0,$$ then $$\label{eq_Lippmann-Schwinger}
u=u_{\pm}-R_0(\lambda\mp i0)Vu$$ where $$\hat u_{\pm}=v_\pm\delta(P_0-\lambda)=v_{\pm}\frac{dS}{|P'_0|} \quad\text{and} \quad v_\pm\in L^2(M_\lambda,dS).$$ The map $v_-\mapsto v_+$ is a continuos bijection which extends to a unitary map $$\Sigma_\lambda: L^2(M_\lambda,\frac{dS}{|P'_0|})\to L^2(M_\lambda,\frac{dS}{|P'_0|}), \quad \Sigma_{\lambda}(v_-)=v_+.$$
We call $v_-$ the incoming wave and $v_+$ the outgoing wave. The unitary map $\Sigma_\lambda$ is the scattering matrix for the energy $\lambda$ and $$A_{\lambda}=I-\Sigma_\lambda$$ is the scattering amplitude.
The following result is well-known and its proof is included for completeness only.
\[lem\_A\_lambda\] Assume that $\lambda\in {\mathbb{R}}\setminus Z(P_0)$ and $u\in B^*_{P_0}$ satisfies the equation $
(P_0+V-\lambda)u=0.
$ Then the scattering amplitude $A_\lambda$ can be expressed through the Fourier transform of $Vu$ as follows, $$A_\lambda v_-(\xi)=2\pi i\hat{Vu}(\xi),\quad \xi\in M_\lambda.$$
By [@horbookII Theorem 14.6.8], we get the following Lippmann-Schwinger equation for $u$, $$u=u_{\pm}-R_0(\lambda\mp i0)Vu$$ where $$\hat u_{\pm}=v_\pm\delta(P_0-\lambda)=v_{\pm}\frac{dS}{|P'_0|} \quad\text{and} \quad v_\pm\in L^2(M_\lambda,dS).$$ Thus, $$u_+-u_-=(R_0(\lambda- i0)-R_0(\lambda+ i0))Vu.$$ Applying the Fourier transform, we have $$\begin{aligned}
(v_+-v_-)\frac{dS}{|P'_0|}&=\big(\frac{1}{P_0(\xi)-\lambda+i0}-\frac{1}{P_0(\xi)-\lambda-i0}\big)\hat{Vu}\\
&=2\pi i\delta(P_0(\xi)-\lambda)\hat{Vu}=2\pi i \hat{Vu}\frac{dS}{|P'_0|}\end{aligned}$$ that proves the claim.
In order to describe the connection between transmission eigenvalues and the scattering amplitude, we shall now review the generalized Rellich theorem. Recall that a classical theorem of Rellich states that if $v$ satisfies $(\Delta+k^2)v=0$ for $|x|>R_0$ and $v(x)|x|^{(n-1)/2}\to 0$ as $x\to\infty$, then $v(x)=0$ for $|x|\ge R_0$.
A far-reaching generalization of this result to broad classes of differential operators with constant coefficients has been given in [@Hormander73]. Let us now state a much simplified version of [@Hormander73 Corollary 3.2]. See also [@AgmonHorm76].
\[cor\_Rellich\] Assume that $\lambda$ is not a critical value of $P_0(\xi)$, $\xi\in {\mathbb{R}}^n$, and that there is a factorization $$P_0(\zeta)-\lambda=cP_{1}^{m_1}(\zeta)\cdots P_{k}^{m_k}(\zeta), \quad c\in {\mathbb{R}},$$ for which every factor $P_i(\zeta)$ has real coefficients and is algebraically irreducible over ${\mathbb{C}}^n$. Assume furthermore that each $P_i(\zeta)$ has a non-empty set of real zeros. If $u\in \mathcal{S}'\cap L^2_{loc}$ is a solution of $$\label{eq_irreduce}
(P_0(D)-\lambda)u=f,$$ with $f\in L^2_{comp}$ and $u\in \stackrel{\circ}{B^*}$, then $u$ has a compact support and $$ch{\operatorname{supp}}(u)= ch {\operatorname{supp}}(f),$$ where $ch$ stands for the convex hull.
If $P_0(\zeta)-\lambda$ has an irreducible factor which has no simple real zero, then it was furthermore proved in [@Hormander73] that for any integer $N$ one can find $u\in L^\infty\cap C^\infty$ satisfying with $f$ being compactly supported and $u(x)=o(|x|^{-N})$ but $u$ not compactly supported.
To illustrate the main ideas involved in the proof of Theorem \[cor\_Rellich\], for the convenience of the reader, we shall include a proof of the special case when $k=1$, $m=1$ and $c=1$. In doing so we shall follow [@Hormander73] closely.
Set $P_\lambda(\zeta)=P_0(\zeta)-\lambda$, $\zeta\in {\mathbb{C}}^n$. The set of real zeros of the polynomial $P_\lambda(\xi)$ is equal to $M_{\lambda}=\{\xi\in {\mathbb{R}}^n:P_\lambda(\xi)=0\}$. By the hypothesis of the theorem, $M_{\lambda}\ne\emptyset$ is an $(n-1)$-dimensional submanifold of ${\mathbb{R}}^n$.
Let $\xi_0\in M_\lambda$. Assume, as we may, that $\p_{\xi_n}P_0(\xi_0)\ne 0$, and write $\xi=(\xi',\xi_n)\in {\mathbb{R}}^{n}$, $\xi'\in {\mathbb{R}}^{n-1}$. By the implicit function theorem, there is an analytic function $g:{\mathbb{R}}^{n-1}\to {\mathbb{R}}$, $g(\xi')=\sum a_\alpha(\xi'-\xi_0')^{\alpha}$, defined locally near $\xi_0'$, such that $$M_\lambda=\{(\xi',g(\xi')):\xi'\in {\mathbb{R}}^{n-1}\},$$ locally near $\xi_0\in M_\lambda$. Hence, the series $g(\zeta')=\sum a_\alpha(\zeta'-\xi_0')^{\alpha}$ also converges for $\zeta'\in {\mathbb{C}}^{n-1}$ near $\xi_0'$.
Let $M_\lambda^{{\mathbb{C}}}$ be the zero set of $P_\lambda$ in ${\mathbb{C}}^n$, i.e. $$M_\lambda^{{\mathbb{C}}}=\{\zeta\in {\mathbb{C}}^n:P_\lambda(\zeta)=0\}.$$ Then let us show that locally near $\xi_0$, we have $$M_\lambda^{{\mathbb{C}}}=\{(\zeta',g(\zeta')):\zeta'\in {\mathbb{C}}^{n-1}\}.$$ Indeed, $
P_\lambda(\xi',g(\xi'))=0
$ for any $\xi'\in {\mathbb{R}}^{n-1}$ near $\xi'_0$ and, hence, since $P_\lambda$ and $g$ are analytic, we get that $P_\lambda(\zeta',g(\zeta'))=0
$ for any $\zeta'\in {\mathbb{C}}^{n-1}$ near $\xi'_0$.
Now since $u\in \stackrel{\circ}{B^*}$, the Fourier transform $\hat f$ vanishes on $M_\lambda$ (cf. [@horbookII Theorem 14.3.6]). Thus, by the analyticity of $\hat f$ and $g$, we have that $\hat f$ vanishes identically on an open neighborhood of $\xi_0$ in $M_\lambda^{{\mathbb{C}}}$.
Denote by $$A=\{\zeta\in {\mathbb{C}}^n:\hat f(\zeta)=0\}$$ the set of the complex zeros of $\hat f$. Notice that the sets $A$ and $M_\lambda^{{\mathbb{C}}}$ are analytic. As we have assumed that $P_\lambda(\zeta)$ is algebraically irreducible over ${\mathbb{C}}^n$, the set $M_\lambda^{{\mathbb{C}}}$ is algebraically irreducible, and hence it is analytically irreducible as well, [@Tre60]. Moreover, $A\cap M_\lambda^{{\mathbb{C}}}$ contains an open neighborhood of $\xi_0$ in $M_\lambda^{{\mathbb{C}}}$. Then it follows from [@Chirka Corollary 2, Section 5.3] that $$\label{eq_zeros}
M_\lambda^{{\mathbb{C}}}\subset A.$$ Thus, the function $$\frac{\hat f(\zeta)}{P_\lambda(\zeta)}$$ is entire in ${\mathbb{C}}^n$. An application of [@horbookI Theorem 7.3.2] shows that there exists $v\in L^2_{comp}$ such that $$P_\lambda(D)v=f.$$ Hence, $$P_\lambda(D)(u-v)=0,$$ and since $u\in \stackrel{\circ}{B^*}$, [@horbookI Theorem 7.1.27] implies that $u=v$.
Injectivity of the scattering amplitude and transmission eigenvalues
====================================================================
In this section we shall assume that the operator $P_0$ is hypoelliptic and $\lambda\in {\mathbb{R}}$ is such that $P_0-\lambda$ satisfies the assumptions of Theorem \[cor\_Rellich\]. Let $V\in L^\infty({\mathbb{R}}^n)$ be real-valued compactly supported in ${\mathbb{R}}^n$ with ${\operatorname{supp}}(V)=\overline{\Omega}$, where $\Omega\subset {\mathbb{R}}^n$ is a bounded convex domain which is of class $C^\infty$, and $V\ge\delta>0$ a.e. in $\Omega$.
Consider the interior transmission problem, $$\label{eq_ITP_2}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{P_0}_0(\Omega).
\end{aligned}$$
There is a solution $(v,w)$ of with $0\ne v\in B^*$ if and only if the scattering amplitude $A_\lambda$ is not injective.
Assume that $A_\lambda$ is not injective. Then there exists a solution $u\in B^*_{P_0}$ of $(P_0+V-\lambda)u=0$ in ${\mathbb{R}}^n$ such that $u=u_--u_s$ with $u_-\ne 0$ and the scattered wave $$u_s:=R_0(\lambda+i0)Vu$$ being both incoming and outgoing. By the mapping properties of the resolvent $R_0(\lambda+i0):B\to B^*_{P_0}$, we see that $u_-\in B^*_{P_0}$. Now since the scattered wave $u_s$ is both incoming and outgoing by Theorem [@horbookII Theorem 14.3.6] $u_s\in \stackrel{\circ}{B^*}$. Moreover, $(P_0-\lambda)u_s=Vu$. Now Theorem \[cor\_Rellich\] implies that $u_s$ has compact support and ${\operatorname{supp}}(u_s)\subset\overline{\Omega}$, thanks to the convexity of $\Omega$. Since $u,u_s\in L^2_{\textrm{loc}}({\mathbb{R}}^n)$ and $P_0$ is hypoelliptic, by hypoelliptic regularity [@horbookII Theorem 11.1.8], we have that $u_s\in B_{2,\tilde P_0}^{loc}({\mathbb{R}}^n)$. As ${\operatorname{supp}}(u_s)\subset\overline{\Omega}$, a regularization argument shows that $u_s$ can be approximated by a sequence of $C_0^\infty(\Omega)$-functions, so that $u_s|_{\Omega}\in H^{P_0}_0(\Omega)$. Now setting $v=u_-|_\Omega\ne 0$ and $w=u|_\Omega$, we get a nontrivial solution to .
Assume conversely that the problem admits a non-trivial solution $(v,w)$ with $0\ne v\in B^*$. Then by , we have $\hat v=v_-dS$ with $v_-\in L^2(M_\lambda, dS)$. Since $P_0$ is hypoelliptic, the surface $M_\lambda$ is compact, and thus, implies that $v\in B^*_{P_0}$. As $v-w\in H^{P_0}_0(\Omega)$, we get that $w\in B^*_{P_0}$ and $v-w\in \stackrel{\circ}{B^*}$. Now $(P_0-\lambda)(v-w)=Vw$ and [@horbookII Theorem 14.3.6] yields that the Fourier transform $\hat{Vw}=0$ on $M_\lambda$. Hence, applying Lemma \[lem\_A\_lambda\], we get that $A_\lambda v_-=0$.
The convexity assumption on $\Omega$ can be removed if we require that $P_0$ is elliptic and ${\mathbb{R}}^n\setminus\overline{\Omega}$ is connected.
Acknowledgements
================
We would like to thank David Colton and Fioralba Cakoni for providing us with some useful references. The research of M.H. was partially supported by the NSF grant DMS-0653275 and he is grateful to the Department of Mathematics and Statistics at the University of Helsinki for the hospitality. The research of K.K. was financially supported by the Academy of Finland (project 125599). The research of P.O. and L.P. was financially supported by Academy of Finland Center of Excellence programme 213476.
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|
[Roots of 3-manifolds and cobordisms]{}
C. Hog-Angeloni[^1], S. Matveev[^2]
Introduction
============
Given a set of [*simplifying moves*]{} on 3-manifolds, we apply them to a given 3-manifold $M$ as long as possible. What we get is a [*root*]{} of $M$. For us, it makes sense to consider three types of moves: compressions along 2-spheres, proper discs and proper annuli having boundary circles in different components of $\partial M$. Our main result is that for the above moves the root of any 3-manifold exists and is unique. The same result remains true if instead of manifolds we apply the moves to 3-cobordisms of the type $(M,\partial_-M,\partial_+M)$. The only difference between moves on manifolds and moves on cobordisms is that one boundary circle of every annulus participating in a compression of a cobordism must lie in $\partial_-M$ while the other in $ \partial_+M$. We can also restrict ourselves to considering compressions along only spheres or only spheres and discs. The existence and uniqueness in the first case is well-known and essentially comprise the content of the Milnor theorem on unique decomposition of a 3-manifold into a connected sum. For the second case our result is close to theorems about [*characteristic compression bodies*]{} and about [*cores*]{} of irreducible manifolds, presented by F. Bonahon [@Bo] and S. Matveev [@Ma], respectively.
We use Kneser existence [@Kn], but perhaps the proof of the uniqueness part is easier with the method we are developing.
We point out that considering roots of cobordisms was motivated by the paper [@Ga] of R. Gadgil, which is interesting although the proof of his main theorem seems to be incomplete.
We thank C. Gordon, W. Metzler, E. Pervova, C. Petronio, and S. Zentner for helpful discussions.
The final version of the paper has been written by the second author during his stay at MPIM Bonn. He thanks the institute for hospitality and financial support.
Moves, roots, and complexity
=============================
We introduce several moves on 3-manifolds. In this paper all 3-manifolds are assumed to be orientable.
1. [*$S$-move (compression along a 2-sphere)*]{}. Let $S$ be a 2-sphere in a 3-manifold $M$. Then we cut $M$ along $S$ and fill the two spheres arising in this way with two 3-balls.
2. [*$D$-move (compression along a disc)*]{}. Let $D$ be a proper disc in $M$. Then we cut $M$ along $D$.
3. [*$A$-move (compression along an annulus)*]{}. Let $A$ be an annulus in $M$ such that its boundary circles lie in different components of $\partial M$. Then we cut $M$ along $A$ and attach two plates $D_1^2\times I, D_2^2\times I$ by identifying their base annuli $\partial D_1^2\times I,
\partial D_2^2\times I$ with the two copies of $A$, which appear under cutting.
Let $F$ be a proper surface in a 3-manifold $M$ such that $F$ is either a sphere, or a disc, or an annulus. Then $F$ is called [*essential*]{}, if one of the following holds:
1. $F$ is a sphere not bounding a ball;
2. $F$ is a disc such that the circle $\partial D$ is nontrivial in $\partial M$;
3. $F$ is an incompressible annulus having boundary circles in different components of $\partial M$.
If $F$ is essential, then the corresponding $F$-move (i.e. the compression of $M$ along $F$) is also called essential.
\[rm:incompressible\] Later on under essential surface we will understand either an essential sphere, or an essential disc, or an essential annulus. The condition that the boundary circles of any essential annulus $A$ must lie in different components of $\partial M$ guarantees us that $A$ is boundary incompressible.
Roots and complexity {#roots}
--------------------
Let $M$ be a 3-manifold. Then a 3-manifold $N$ is called a [*root*]{} of $M$, if
1. $N$ can be obtained from $M$ by essential compressions along spheres, discs, and annuli.
2. $N$ admits no further essential compressions.
\[th:roots\] For any compact 3-manifold $M$ the root of $M$ exists and is unique up to homeomorphism and removing disjoint 3-spheres and balls.
We postpone the proof of the theorem to Section \[proof\]. Note that the condition on boundary circles of compression annuli to lie in different components of $\partial M$ is essential. Below we present an example of a 3-manifold $M$ with two incompressible boundary incompressible annuli $A,B\subset M$ such that $\partial M$ is connected and compressions of $M$ along $A$ and along $B$ lead us to two different 3-manifolds admitting no further essential compression, i.e. to two different “roots”.
[**Example.**]{}
*Let $Q$ be the complement space of the figure eight knot. We assume that the torus $\partial Q$ is equipped with a coordinate system such that the slope of the meridian is (1,0). Choose two pairs $(p,q)$, $(m,n)$ of coprime integers such that $|q|,|n|\geq 2$ and $|p|\neq |m|$. Let $a$ and $b$ be corresponding curves in $\partial Q$. Then the manifolds $Q_{p,q}$ and $Q_{m,n}$ obtained by Dehn filling of $Q$ are not homeomorphic. By \[Th\], they are hyperbolic.*
Consider the thick torus $X=S^1\times S^1 \times I$ and locate its exterior meridian $\mu=S^1\times \{\ast\}\times \{ 1\}$ and interior longitude $\lambda= \{\ast\}\times S^1\times \{ 0\}$. Then we attach to $X$ two copies $Q',Q''$ of $Q$ as follows. The first copy $Q'$ is attached to $X$ by identifying an annular regular neighborhood $N(a)$ of $a$ in $\partial Q$ with an annular regular neighborhood $N(\mu)$ of $\mu$ in $\partial X$. The second copy $Q''$ is attached by identifying $N(b)$ with $N(\lambda)$. Denote by $M$ the resulting manifold $Q'\cup X\cup Q''$.
Since $Q$ is hyperbolic, $M$ contains only two incompressible boundary incompressible annuli $A$ and $B$, where $A$ is the common image of $N(a)$ and $N(\mu)$, and $B$ is the common image of $N(b)$ and $N(\lambda)$. It is easy to see that compression of $M$ along $A$ gives us a disjoint union of a punctured $Q'_{p,q}$ and a punctured $Q''$ while the compression along $B$ leaves us with a punctured $Q'$ and a punctured $Q''_{m,n}$. After filling the punctures (by compressions along spheres surrounding them), we get two different manifolds, homeomorphic to $Q _{p,q}\cup Q$ and $Q_{m,n}\cup Q$. Since their connected components (i.e. $Q_{p,q},
Q_{m,n}, Q$) are hyperbolic, they are irreducible, boundary irreducible and contain no essential annuli. Hence $Q
_{p,q}\cup Q$ and $Q_{m,n}\cup Q$ are different roots of $M$.
Let $M$ be a compact 3-manifold. Let us apply to it essential $S$-moves as long as possible. It follows from Kneser finiteness theorem [@Kn] that the number of possible moves is bounded by a constant depending on $M$ only. Denote by $s(M)$ the maximum of these numbers taken over all chains of essential $S$-moves.
The following notion will be the main inductive parameter in our proofs.
\[compl\] Let $M$ be a 3-manifold. Then the complexity ${\bf c}(M)$ of $M$ is the pair $(g^{(2)}(\partial M),s(M))$, where $g^{(2)}(M)=\sum_i g^2(F_i)$, $g(F_i)$ is the genus of a component $F_i\subset \partial M$, and the sum is taken over all components of $\partial M$. The pairs are considered in lexicographical order.
The use of complexity as an inductive parameter is justified by the following fact.
\[lm:decrease\] Each essential $S,D,A $-move strictly decreases ${\bf c}(M)$.
If an essential $D$-move cuts $\partial M$ along a nonseparating curve $\ell$ on some component $F $ of $\partial M$, then it strictly decreases $g(F)$ and hence $c(M)$. If the move turns $F$ into two components $F', F''$, then $g(F)=g(F')+g(F'')$ and, since $\ell$ is nontrivial and thus $g(F')$, $g(F'')\neq 0$, we have $g^2(F)>g^2(F')+g^2(F'')$. This implies that $c(M)$ is decreased again. The case of the $A$-move is similar.
As follows from the definition of $s(M)$, each essential $S$-move strictly decreases $s(M)$. The boundary of $M$ remains the same, hence so does $g^{(2)}(M)$.
\[rem:A-increase\] It is easy to show that inessential $S$- and $D$-moves preserve the complexity. However, an inessential $A$-move can increase it, but only at the expense of $s(M)$ ($g^{(2)}(M)$ cannot increase). For example, if an annulus $A$ cuts off a $D^2\times I$ from $M$, then the corresponding move results in the appearing of an additional component of the type $S^2\times I$.
Equivalence of essential surfaces
---------------------------------
Throughout this section, surface means sphere or disc or annulus.
Let $M$ be a 3-manifold and $F, G$ be two essential surfaces in $M$. Then $F,G$ are [*equivalent*]{} (we write $F\sim
G$) if there exists a finite sequence of essential surfaces $ X_1, X_2,
\dots , X_n$ such that the following holds:
1. $F=X_1$ and $X_n=G$;
2. For each $i,1\leq i<n,$ the surfaces $X_i$ and $X_{i+1}$ are disjoint.
\[lm:all equiv\] Let $M$ be a 3-manifold not homeomorphic to $S^1\times S^1\times I$. Then any two essential surfaces in $M$ are equivalent.
Let $F,G$ be two essential surfaces in $M$ in general position. Then the number of curves (circles and arcs) in the intersection of $F$ and $G$ will be denoted by $\#(F\cap G)$. Arguing by induction, we may assume that any two essential surfaces $F,G$ with $\#(F\cap G)<n$ are equivalent. The base of the induction is evident: if $\#(F\cap G)=0$, then $F\sim G$ by definition. Let $F,G$ be two essential surfaces such that $\#(F\cap G)=n$.
[*Case 1.* ]{} Suppose that $F\cap G$ contains a circle $s$ which is trivial in $F$. By an innermost circle argument we may assume that $s$ bounds a disc $D\subset F$ such that $D\cap G=s$. Compressing $G$ along $D$, we get a two-component surface $G'$ such that one component is a sphere, the other is homeomorphic to $G$, and $\#(F\cap G')=n-1$. Since $G$ is an interior connected sum of the components of $G'$, at least one of them (denote it by $X$) is essential and thus $F\sim X$ by the inductive assumption. On the other hand, $X$ can be shifted away from $G$ by a small isotopy. It follows that $X\sim G$ and thus $F\sim G$.
[*Case 2.* ]{} Suppose that $F\cap G$ does not contain trivial circles, but contains an arc $a$ which is trivial in $F$. By an outermost arc argument we may assume that $a$ cuts off a disc $D\subset F$ from $F$ such that $D\cap G=a$. Compressing $G$ along $D$, we get a two-component surfaces $G'$ such that one component is a proper disc, the other is homeomorphic to $G$, and $\#(F\cap G')=n-1$. Since $G$ is an interior boundary connected sum of the components of $G'$, at least one of them (denote it by $X$) is essential and thus equivalent to $F$ by the inductive assumption. On the other hand, $X$ can be shifted away from $G$ by a small isotopy. It follows that $G\sim X$ and thus $F\sim G$.
[*Case 3.*]{} Suppose that $F$ and $G$ are annuli such that $F\cap G$ consists of circles parallel to the core circles of $F$ and $G$. Then one can find two different components $A, B$ of $\partial M$ such that a circle of $\partial F$ is in $A$ and a circle of $\partial G$ is in $B$. Denote by $s$ the first circle of $F\cap G$ we meet at our radial way along $F$ from the circle $\partial F\cap A$ to the other boundary circle of $F$. Let $F'$ be the subannulus of $F$ bounded by $\partial F\cap A$ and $s$, and $G'$ the subannulus of $G$ bounded by $s$ and $\partial G\cap B$. Then the annulus $F'\cup G'$ is essential and is isotopic to an annulus $X$ such that $\#(X\cap F)<n$ and $\#(X\cap G)=0$, see Fig. \[newannulus\] (to get a real picture, multiply by $S^1$). It follows that $F\sim G$.
[*Case 4.*]{} Let $F$ and $G$ be annuli such that $F\cap G$ consists of more than one radial segments, each having endpoints in different components of $\partial F$ and different components of $\partial G$.
[*Case 4.1.*]{} Suppose that there are two neighboring segments $s_1,s_2 \subset F\cap G\subset F$ such that $G$ crosses $F$ at $s_1,s_2$ in opposite directions. Denote by $D$ the quadrilateral part of $F$ between them. Then we cut $G$ along $s_1,s_2$ and attach to it two parallel copies of $D$ lying on different sides of $F$. We get a new surface $G'$ consisting of two disjoint annuli, at least one of which (denote it by $X$) is essential, see Fig. \[lick\] to the left. Since $\#(X\cap
F)=n-2$ and, after a small isotopy of $X$, $\#(X\cap
G)=\emptyset$, we get $F\sim X\sim G$.
[*Case 4.2.*]{} Suppose that at all segments $G$ crosses $F$ in the same direction (say, from the left to the right). Let $s_1,
s_2$ be two neighboring segments spanned by a quadrilateral part $D\subset F$ between them. Then $s_1,s_2$ decompose $G$ into two strips $L_1,L_2$ such that $L_1$ approaches $s_1$ from the left side of $F$ and $s_2$ from the right side. Then the annulus $L_1\cup D$ is isotopic to an annulus $X$ such that $\#(X\cap F)\leq n-1$ and $\#(X\cap G)=1$, see Fig. \[lick\] to the right. Since $X$ crosses $F$ one or more times in the same direction, it is essential. Therefore, $F\sim X\sim G$.
[*Case 5.*]{} This is the last logical possibility. Suppose that $F$ and $G$ are annuli such that $F\cap G$ consists of one radial segment. Denote by $G'$ the relative boundary $\partial_{rel}(N)=\Cl(N\cap
\Int M)$ of a regular neighborhood $N$ of $F\cup G$ in $M$. Then $G'$ is an annulus having boundary circles in different components of $\partial M$.
[*Case 5.1.*]{} If $G'$ is incompressible, then we put $X=G'$.
[*Case 5.2.*]{} If $G'$ admits a compressing disc $D$, then the relative boundary of a regular neighborhood $N$ of $G'\cup D$ consists of a parallel copy of $G'$ and two proper discs $D_1,D_2$. If at least one of these discs (say, $D_1$) is essential, then we put $X=D_1$.
[*Case 5.3.*]{} Suppose that discs $D_1, D_2$ are not essential. Then the circles $\partial D_1, \partial D_2$ bound discs $D_1',D_2'$ contained in the corresponding components of $\partial M$. We claim that at least one of the spheres $S_1=D_1\cup D_1',S_2=D_2\cup D_2'
$ (denote it by $X$) must be essential. Indeed, if both bound balls, then $M$ is homeomorphic to $S^1\times S^1\times I$, contrary to our assumption.
In all three cases 5.1-5.3 $X$ is disjoint to $F$ as well as to $G$. Therefore, $F\sim X\sim G$.
Proof of the main theorem {#proof}
=========================
Let $F$ be a sphere, a disc or an annulus in a 3-manifold $M$. It is convenient to denote by $C_F(M)$ the result of the $F$-move, i.e. the manifold obtained by compressing $M$ along $F$.
\[commonroot\] If $F$ is a sphere or a disc or an essential annulus, then any root of $M_F=C_F(M)$ is a root of $M$. If $F$ is an inessential annulus, then $M_F$ and $M$ have at least one common root.
It is convenient to decompose the proof into four steps.
1. If $F$ is essential, then any root of $M_F$ is a root of $M$ by definition of the root.
2. If $F$ is an inessential sphere, then $M_F$ is a union of $M$ and a disjoint 3-sphere. Therefore, all roots of $M$ and $M_F$ are the same.
3. Let $F$ be an inessential disc. Then its boundary circle bounds a disc $D\subset \partial M$. Choose a 2-sphere $S$ inside $ M$ which is parallel to the 2-sphere $F\cup D$. Then the manifold $M_F$ is obtained from the manifold $M_S=C_S(M)$ by puncturing (cutting off a ball $V\subset M_S$). We claim that any root $R$ of $M_F = M_S\setminus \Int V\subset M_S$, which can be obtained from $M_F$ by successive compressions along essential subsurfaces, is a root of $M_S$. Indeed, we simply compress $M_S$ along the same surfaces and get either $R$ (if one of those subsurfaces is a sphere surrounding $V$) or a punctured $R$ (if the puncture survives all compressions). One more compression along a sphere surrounding the puncture is sufficient to convert the punctured $R$ to $R$ (modulo disjoint 3-spheres and balls, which are irrelevant). It follows from (1), (2), and (3) that any root of $M_F$ is a root of $M$.
4. Let $F$ be an inessential (i.e. compressible) annulus and $D $ a compressing disc for $F$ such that $D\cap F=\partial D$ is a core circle of $F$. Denote by $N$ a regular neighborhood of $F\cup D$ in $M$. Then the relative boundary $\partial_{rel}N=\Cl(\partial N\cap \Int M)$ consists of a parallel copy of $F$ and two proper discs $D', D''$. Denote by $S$ a 2-sphere in $C_F(M)$ composed from a copy of $D$ and a core disc of one of the attached plates, see Fig. \[ddas\]. Then the manifolds $C_{D''}(C_{D'}(M))$ and $ C_S(C_F(M))$ are homeomorphic. Applying (1) - (3), we conclude that any root of the manifold $C_{D''}(C_{D'}(M))= C_S(C_F(M))$ is a root of both $M$ and $C_F(M)$.
Proof of Theorem 1. [Existence.]{} Let us apply to $M$ essential $S,D,A$-moves in arbitrary order as long as possible. By Lemma \[lm:decrease\], each move strictly decreases the complexity. Since every set of pairs of nonnegative integers has a minimal pair, the process stops and we get a root.
[Uniqueness.]{} Assume the converse: suppose that there is a 3-manifold having two different roots. Among all such manifolds we choose a manifold $M$ having minimal complexity. Then there exist two sequences of essential moves producing two different roots. Denote by $C_F$ and $C_G$ the first moves of the sequences, where $F,G$ are essential surfaces in $M$. By Lemma \[lm:all equiv\], there are essential surfaces $ X_1, X_2, \dots , X_n$ such that $F=X_1$, $X_n=G$, and that the surfaces $X_i$ and $X_{i+1}$ are disjoint for all $i,1\leq i<n$. We may begin the construction of a root starting with the compression along any of them. Evidently, for at least two neighboring surfaces $X_k, X_{k+1}$ the roots thus obtained are different. For convenience, we rename $X_k,X_{k+1}$ by $F,G$ thus getting two disjoint surfaces such that $C_F(M)$ and $C_G(M)$ have different roots. Then $F$ is a subsurface of $M$ and of $M_G=C_G(M)$ while $G$ is a subsurface of $M$ and of $C_F(M)$. Denote by $N$ the manifold, obtained from $M$ by compressions along both surfaces $F,G$. Of course, it coincides with $C_G(C_F(M))$ and $ C_F(C_G(M))$.
We claim that the complexity of $N$ is strictly less than the one of $M$. Indeed, if $F$ is either a sphere or a disc, then $c(N)\leq c(M_G)$ (since compression along a sphere or a disc does not increase complexity) while $c(M_G)<c(M)$ by Lemma \[lm:decrease\]. Suppose that $F$ is an annulus. Then $g^{(2)}(\partial N)$ is no greater than $g^{(2)}(\partial M_F)$, since no compression move increases the genus of the boundary. On the other hand, since $F$ is essential, then $g^{(2)}(\partial M_F)<
g^{(2)}(\partial M)$, which implies $c(N)<c(M)$.
Using the inductive assumption, we may conclude that $N$ has a unique root. The same is true for $M_F$ and $M_G$, since by Lemma \[lm:decrease\] their complexities are also smaller than $c(M)$. Now we have:
1. $M_F$ and $N$ have the same root (since they have a common root by Lemma \[commonroot\]).
2. $M_G$ and $N$ have the same root (same reason);
3. Hence $M_F$ and $M_G$ have the same root, which is a contradiction.
Other roots
===========
[Roots of cobordisms.]{} Recall that a [*3-cobordism*]{} is a triple $(M,\partial_-M,\partial_+M)$, where $M$ is a compact 3-manifold and $\partial_-M$, $\partial_+M$ are unions of connected components of $\partial M$ such that $\partial_-M\cap \partial_+M=\emptyset$ and $\partial_-M\cup \partial_+M=\partial M$. One can define $S$- and $D$-moves on cobordisms just in the same way as for manifolds. The $A$-move on cobordisms differs from the one for manifolds only in that one boundary circle of $A$ must lie in $\partial_-M$ while the other in $\partial_+M$.
\[th:rootsco\] For any compact 3-cobordism $(M,\partial_-M,\partial_+M)$ its root exists and is unique up to homeomorphisms of cobordisms and removing disjoint 3-spheres and balls.
The proof of this theorem is the same as the proof of Theorem \[th:roots\].
[$S$-Roots of manifolds.]{} We define an $S$-root of $M$ as a manifold which can be obtained from $M$ by essential $S$-moves and does not admit any further essential $S$-moves.
\[th:S-roots\] For any compact 3-manifold $ M $, its $S$-root exists and is unique up to homeomorphism and removing disjoint 3-spheres.
This theorem is actually equivalent to the theorem on the unique decomposition into a connected sum of prime factors. Indeed, the $S$-root of $M$ coincides with the union of the irreducible prime factors of $M$.
[$(S,D)$-Roots of manifolds.]{} An $(S,D)$-root of $M$ is a manifold which can be obtained from $M$ by essential $S$- and $D$- moves and does not admit any further essential $S$-moves and $D$-moves.
\[th:SD-roots\] For any compact 3-manifold $ M $ its $(S,D)$-root exists and is unique up to homeomorphism and removing disjoint 3-spheres and balls.
For irreducible manifolds this theorem can be deduced from the theorem of F. Bonahon [@Bo] on characteristic compression bodies as well as from [@Ma], where $D$-roots of irreducible manifolds had been considered under the name [*cores*]{}.
Our way for proving Theorem \[th:roots\] works also for $S$- and $(S,D)$-roots. All we need is to forget about discs and annuli in the first case and about annuli in the second. This makes the proof significantly shorter.
[999]{} F. Bonahon, Cobordism of automorphisms of surfaces, Ann. Éc. Norm. Sup. 83 (1983), 237-270. S. Gadgil, On the Andrews-Curtis conjecture and algorithms from topology, arXiv:math.GT/0108116.
H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten. Jahresbericht der Deut. Math. Verein, 28:248-260, 1929.
S. Matveev Algorithmic topology and classification of 3-manifolds, Springer ACM-monographs, V. 9 (2003), 480 pp.
[^1]: Partially supported by the INTAS Project “CalcoMet-GT” 03-51-3662
[^2]: Partially supported by the INTAS Project “CalcoMet-GT” 03-51-3662 and the RFBR grant 05-01-0293-a
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---
address: |
Department of Physics, University of California at San Diego\
9500 Gilman Drive, La Jolla, CA 92093
author:
- Dan Pirjol
title: Rare radiative B decays in perturbative QCD
---
Introduction
============
Rare radiative decays of $B$ hadrons have been extensively studied as a possible way of probing possible new physics effects. Most effort has been concentrated on the theoretically cleaner inclusive decays, which can be analyzed in an $1/m_b$ expansion with the help of an operator product expansion.
On the other hand, exclusive rare $B$ decays are sensitive to long-distance QCD effects and the relevant amplitudes depend on details of the hadronic bound states. This makes their theoretical description considerably more model-dependent, and a full description is still lacking.
In the following we describe a systematic treatment of exclusive decays involving one very energetic on-shell photon emitted from an internal quark line. In this situation the internal light quark moves very close to the light cone, and an expansion in powers of the small parameter $\Lambda/E_\gamma$ becomes useful[@ope]. This is analogous to the twist expansion for exclusive processes involving only light quarks[@BroLe].
The simplest process which can be studied along these lines is the leptonic radiative decay[@KPY] $B\to \gamma\ell \nu_\ell$, which is discussed in Sec. 2. A basic ingredient of the method is the light-cone description of a heavy-light meson, which is introduced in Sec. 2.1. In Sec. 2.2 we argue that the general structure of the leading-twist $B\to \gamma\ell \nu_\ell$ form factors is given by a convolution of the $B$ light-cone wave function with a hard scattering amplitude $T_H$.
In Sec. 3 we describe an application of this method to the calculation[@GP] of the weak annihilation contribution to the weak radiative decay $B\to \rho\gamma$. Taking into account all possible long distance amplitudes compatible with SU(3) symmetry, we present in Sec. 3.1 a few methods for constraining the unitarity triangle using isospin violating effects in $B\to \rho(\omega)\gamma$ decays[@bound].
Leading twist calculation of the $B\to\gamma \ell\nu_\ell$ decay
================================================================
Heavy mesons light-cone wavefunctions
-------------------------------------
The most general form for the Bethe-Salpeter wave function of a heavy-light meson $B(v)$ can be written in terms of 4 scalar functions $\psi_i(v\cdot z,z^2)$ ($z=x-y$) $$\label{1}
\psi_{\alpha\beta}(z,v) = \langle 0|Tb_\alpha (x) \bar q_\beta(y)|
\bar B(v)\rangle =
\left\{ (
\psi_1 - \frac12 ({\mbox{$\not{\hspace{-1.03mm}z}$}}{\mbox{$\not{\hspace{-1.03mm}v}$}}- {\mbox{$\not{\hspace{-1.03mm}v}$}}{\mbox{$\not{\hspace{-1.03mm}z}$}})\psi_2
+ {\mbox{$\not{\hspace{-1.03mm}v}$}}\psi_3 + {\mbox{$\not{\hspace{-1.03mm}z}$}}\psi_4 )\right\}_{\alpha\beta}\,.$$ In the heavy quark limit the structure of the wavefunction is further constrained by the condition ${\mbox{$\not{\hspace{-1.03mm}v}$}}\psi = \psi$. This reduces the number of independent structures to 2, which can be taken as $\psi_{1,2}$ $$\label{2}
\psi_{\alpha\beta}(z,v) = \left\{\frac{1+{\mbox{$\not{\hspace{-1.03mm}v}$}}}{2}
(\psi_1 + [{\mbox{$\not{\hspace{-1.03mm}z}$}}- {\mbox{$\not{\hspace{-1.03mm}v}$}}(v\cdot z) ]\psi_2 )
\gamma_5\right\}_{\alpha\beta}\,.$$ The momentum space $B$ meson wavefunction is defined as \[3\] \_(k) d\^4 z e\^[ikz]{}\_(v,z). In the physical applications to be discussed in the following, this wavefunction is convoluted with a scattering amplitude $T_H(k)$ which does not depend on one light-cone component of the relative momentum $k_\mu$ in the bound state (e.g. $k_-$). Therefore, this component can be integrated over, which effectively puts $z$ on the light cone ($z_+=0$). We defined here light-cone coordinates as $k_\pm \equiv k^0\pm k^3$, which can be projected out by dotting into the light-cone basis vectors $n=(1,0,0,1)$ and $\bar n=(1,0,0,-1)$. This gives $k_+=n\cdot k,
k_- = \bar n\cdot k$.
It is convenient to write the wavefunction (\[2\]) as a linear combination of light-cone projectors in spinor space $\Lambda_+ = \frac14
{\mbox{$\not{\hspace{-1.03mm}n}$}}\overline{{\mbox{$\not{\hspace{-1.03mm}n}$}}}\,,
\Lambda_- = \frac14 \overline{{\mbox{$\not{\hspace{-1.03mm}n}$}}}{\mbox{$\not{\hspace{-1.03mm}n}$}}$ $$\label{4}
\psi_{\alpha\beta}(z_+=0,z_-,z_\perp = 0) = \left\{\frac{1+{\mbox{$\not{\hspace{-1.03mm}v}$}}}{2}
(\Lambda_+ \psi_+(z_-) + \Lambda_- \psi_-(z_-))
\gamma_5\right\}_{\alpha\beta}\,.$$ with $\psi_\pm(z_-) = \psi_1 \mp \frac12 z_-\psi_2$. \[When used to compute a physical amplitude, only one of these functions will contribute to leading twist (e.g. $\psi_+$).\] This gives the most general expression for the $B$ wavefunction in the heavy quark limit, which is often found in the literature [@BBNS2; @BeFe] $$\label{5}
\psi_{\alpha\beta}(v\cdot z,z^2) = \frac{1+{\mbox{$\not{\hspace{-1.03mm}v}$}}}{2}
\left\{(\psi_+ - \frac{{\mbox{$\not{\hspace{-1.03mm}z}$}}}{2v\cdot z}
(\psi_+ - \psi_- ))
\gamma_5\right\}_{\alpha\beta}\,.$$ However, in the following we will use the form (\[4\]) which has a simple interpretation in the parton model. With the kinematics adopted above, the wavefunction $\psi_+(k_+)$ gives the probability to find the light quark in the $B$ meson carrying momentum $k_+$. Its moments are related to matrix elements of local operators $$\langle k^N_+ \rangle \equiv
\int_0^\infty dk_+ k_+^N \psi_+(k_+) =
\langle 0|\bar q \Lambda_+ {\mbox{$\not{\hspace{-1.03mm}n}$}}\gamma_5 (in\cdot D)^N h_v|\bar B(v)\rangle\,.$$
The first two moments can be expressed in terms of known $B$ hadronic parameters: $\langle k^0_+\rangle =
f_B m_B$ and $\langle k_+\rangle = \frac43
\bar\Lambda f_B m_B$, with $\bar \Lambda = m_B-m_b \simeq 350$ MeV the binding energy of the $b$ quark in a $B$ meson. The average of $k^{-1}_+$ will play an important role in the following. Although it cannot be related to the matrix element of a local operator, it is possible to give a model-independent lower bound on its magnitude. Using the normalization conditions for $N=0,1$ and the positivity of the distribution function $\psi_+$ one finds[@KPY] $$\label{bound}
\int_0^\infty dk_+ \frac{\psi_+(k_+)}{k_+} \geq
\frac{3}{4\bar\Lambda} f_B m_B\,.$$ For more specific (but less model-independent) predictions, some model has to be adopted for the distribution function $\psi_+(k_+)$. We used an ansatz inspired by a quark model with harmonic oscillator potential $\psi_+(k_+) = Nk_+
\exp\left(-\frac{1}{2\omega^2}(k_+-a)^2\right)$. The width parameter $\omega$ is varied as $\omega = 0.1-0.3$ MeV and $a$ is constrained from the normalization conditions. Using $\bar\Lambda = 0.3-0.4$ MeV gives $a=0.05-0.5$ MeV.
Leading twist form factors in $B\to\gamma \ell \nu_\ell$
--------------------------------------------------------
The simplest hard photon process which can be analyzed using an expansion in $1/E_\gamma$ is the radiative leptonic decay $B\to\gamma \ell \nu_\ell$. This proceeds through the weak annihilation of the spectator quark in the $B$ meson, as depicted in the quark diagrams in Fig. \[fig:tree\].
Interest in this decay was sparked by the observation[@BuGoWy] that its branching ratio is enhanced relative to that for the leptonic decay $B\to \ell\nu_\ell$ with $\ell=e,\mu$. Although adding one photon introduces a factor of $\alpha=\frac{1}{137}$ in the rate, it also removes the helicity suppression factor $(m_\ell/m_B)^2$; the overall effect is an enhancement in the rate of the leptonic radiative mode. Later work[@1] investigated this decay using a variety of approaches, both in connection with the prospect of constraining $f_B$ and/or $|V_{ub}|$ and in the context of the weak radiative decays $B\to \rho(\omega)\gamma$.
The amplitude for $B\to\gamma\ell\nu_\ell$ is parameterized by two formfactors $f_{V,A}(E_\gamma)$ defined as $$\begin{aligned}
\label{fV}
& &\frac{1}{\sqrt{4\pi\alpha}} \langle \gamma(q,\varepsilon)|
\bar q\gamma_\mu b|\bar B(v)\rangle =
i\epsilon_{\mu\alpha\beta\delta} \varepsilon^{*\alpha} v^\beta
q^\delta f_V(E_\gamma)\\
\label{fA}
& &\frac{1}{\sqrt{4\pi\alpha}} \langle \gamma(q,\varepsilon)|
\bar q\gamma_\mu\gamma_5 b|\bar B(v)\rangle =
\left[ q_\mu (v\cdot \varepsilon^*) - \varepsilon^*_\mu (v\cdot q)
\right] f_A(E_\gamma)\,.\end{aligned}$$ Computing the contribution of the two diagrams in Fig. \[fig:tree\] with the $B$ wavefunction (\[4\]) one finds for the formfactors (\[fV\]), (\[fA\]) $$\label{fVA}
f_V(E_\gamma) = f_A(E_\gamma) = \frac{f_B m_B}{2E_\gamma}
\left(Q_q R - \frac{Q_b}{m_b}\right) + {\cal O}(\Lambda^2/E_\gamma^2)$$ where $Q_q,Q_b$ are the spectator- and heavy-quark electric charges and the hadronic parameter $R$ describes the photon coupling to the light quark. This is given by a convolution of the leading twist $B$ meson wavefunction with a hard-scattering amplitude $T_H(E_\gamma,k_+)$ $$\label{R}
R(E_\gamma) = \int_0^\infty
d k_+ \frac{\psi_+(k_+)}{k_+} T_H(E_\gamma,k_+)$$
The hard scattering amplitude $T_H(E_\gamma,k_+)$ is given to one-loop order by[@KPY] $$\label{TH}
T_H(E_\gamma,k_+) = 1 + \frac{\alpha_s C_F}{4\pi}\left(
-\log^2\left(\frac{2E_\gamma}{k_+}\right) +
\frac52\log\left(\frac{2E_\gamma}{k_+}\right)
-\frac{4\pi^2}{3}\right)\,.$$ Using the model distribution function described in Sec. 2.1 one finds at tree level $R(E_\gamma)=2-4$ GeV$^{-1}$, while the lower bound (\[bound\]) predicts $R > 2.1$ GeV$^{-1}$ (corresponding to $\bar\Lambda = 350$ MeV).
We briefly discuss in the following a few interesting consequences of these results.
a\) To leading twist the formfactors $f_{V,A}(E_\gamma)$ scale like $1/E_\gamma$, with a proportionality coefficient which is independent on the heavy quark flavor (up to $1/m_b$ corrections). This property can be used to determine the CKM matrix element $|V_{ub}|$ from a comparison of the $B\to \gamma \ell \nu_\ell$ and $D\to \gamma \ell \nu_\ell$ photon spectra[@KPY].
b\) The form factors of the vector and axial current are equal to leading twist. This is in contrast to their behaviour in the low $E_\gamma$ region, where they receive contributions from intermediate states with different quantum numbers[@BuGoWy] ($J^P=1^-$ for $f_V$ and $J^P=1^+$ for $f_A$).
The equality of the form factors (\[fVA\]) is a particular case of a symmetry relation analogous to those discussed for the soft components of semileptonic formfactors in[@french]. The gluon couplings of a quark moving close to the light cone possess a higher symmetry[@DuGri]. This can be formalized by going over to an effective theory, which should include in addition to the soft gluon modes (LEET[@DuGri]), also collinear gluons. A complete discussion including collinear gluons has been given only recently[@BFPS].
There is however an important difference between the status of the symmetry relation $f_V=f_A$ among the leptonic radiative formfactors, and the analogous symmetry relations for the soft semileptonic formfactors[@french]. While the latter receive symmetry breaking corrections from hard gluon exchange (which have been computed recently[@BeFe]), the former relation is not changed by such effects, as checked by explicit calculation to one-loop order[@KPY].
Hadronic amplitudes similar to $R$ in (\[R\]) appear in many physical quantities involving long-distance effects induced by hard photon or gluon emission from weak annihilation topologies[@BDS]. \[When computed in the quark model, this quantity appears as the inverse of the constituent quark mass $R\to 1/m_q$.\] In the following section we study such an important application, to long-distance effects in exclusive penguin induced decays $b\to d\gamma$.
Long-distance contributions to the $B\to \rho\gamma$ decay
==========================================================
An important class of weak radiative decays are those mediated by the penguin mechanism $b\to s(d)\gamma$ $$\label{penguin}
{\cal H}_{\rm eff} = \frac{-4G_F}{\sqrt2}V_{tb} V^*_{ts} C_7
\frac{em_b}{16\pi^2}F^{\mu\nu}
\bar s\sigma_{\mu\nu} P_R b + {\cal H}_{\rm l.d.}\,,$$ with small long-distance contributions expected from time-ordered products of the weak nonleptonic Hamiltonian with the minimal electromagnetic coupling (denoted as ${\cal H}_{\rm l.d.}$).
We show in Fig. \[fig:ld\] the different quark diagrams responsible for long-distance contributions to a typical weak radiative decay $B\to V\gamma$. In the SU(3) limit any such amplitude can be written as a linear combination[@GP] of the amplitudes in Fig. \[fig:ld\] with CKM factors $V_i$ (for each photon helicity $\lambda$) $$\label{sum}
A(B\to V\gamma_\lambda) = \sum_{q=u,c,t} V_{qb} V_{qs}^*
\sum_{{\cal M}_{i,q}=A,E,P_q,PA_q} c_{i,q} {\cal M}_{i,q\lambda}\,.$$ This is analogous to a similar decomposition of nonleptonic $B$ decay amplitudes into graphic amplitudes[@GHLR], and is equivalent to a more usual SU(3) analysis in terms of reduced matrix elements.
The long-distance amplitudes appearing in (\[sum\]) are notoriously difficult to calculate. They have been estimated using various methods such as QCD sum rules[@KSW], vector meson dominance[@vmd], perturbative QCD[@pqcd] and Regge methods[@regge]. A few sample results are tabulated in Table 1, separately for the two helicities of the photon $\lambda=L,R$. These calculations show that the dominant long-distance amplitude in $b\to d\gamma$ decays comes from the weak annihilation graph $A$ in Fig. \[fig:ld\](a).
It is therefore rather fortunate that the weak annihilation amplitude can be computed in an esentially model-independent way. In the factorization approximation, one finds[@GP] $$\label{ALR}
A_{L,R} = -\frac{G_F}{\sqrt2}(C_2 +\frac{C_1}{N_c})
em_\rho f_\rho \left( f_B + E_\gamma (f_V \pm f_A)\right)$$ with $f_{V,A}(E_\gamma)$ the radiative leptonic form factors defined in (\[fV\]), (\[fA\]), and $C_1(m_b)=-0.29$, $C_1(m_b)=1.13$ are Wilson coefficients in the weak nonleptonic Hamiltonian. Nonfactorizable corrections to this result arise from hard gluons connecting the initial and final state quarks, but appear only at higher twist[@GP]. Furthermore, to leading order of a twist expansion for the radiative leptonic formfactors $f_{V,A}$ (\[fVA\]), the weak annihilation amplitude couples predominantly to left-handed photons. A similar suppression of the right-handed helicity amplitudes is noted for all other long-distance contributions (see Table 1), which has implications for proposals to search for new physics through photon helicity effects in $b\to s\gamma$ decays[@newp].
Photon helicity $|P_{t\lambda}|$ $|P_{c\lambda}|$ $|P_{u\lambda}|$ $|A_\lambda|$ $|E_\lambda|$
----------------- ------------------ ------------------ ------------------ --------------- ---------------
$\lambda=L$ $1.8$ $0.16$ $0.03$ $ 0.6$ $ 0.05$
$\lambda=R$ $0$ $0.04$ $0.007$ $0.07$ $0.007$
: Estimates of the short-distance and long-distance amplitudes in $B\to \rho\gamma$ decays (in units of $10^{-6}$ GeV). The estimates of the $WA$ and $W$-exchange amplitudes $A_\lambda$ and $E_\lambda$ used $R=2.5$ GeV$^{-1}$. The penguin type amplitudes $P_u$ and $P_c$ have been estimated using vector meson dominance.\[Table1\]
The main motivation for measuring exclusive weak $B$ radiative decays is connected with the possibility of extracting the CKM matrix element $|V_{td}|$. The weak annihilation amplitude $A$ introduces the most significant theoretical uncertainty in such a determination. Keeping only the leading long-distance amplitude, one finds for the ratio of charge-averaged rates $$\frac{{\cal B}(B^\pm \to \rho^\pm\gamma)}{{\cal B}(B^\pm \to
K^{*\pm}\gamma)} = \left|\frac{V_{td}}{V_{ts}}\right|^2 R_{SU(3)}
\left(1 -
\varepsilon_A\left|\frac{V_{ub}V_{ud}^*}{V_{td}V_{tb}^*}\right|
\cos\alpha \cos\phi_A + {\cal O}(\varepsilon_A^2)\right)$$ with $\varepsilon_A e^{i\phi_A} = A_L/P_{tL}$ and $R_{SU(3)}=0.76\pm
0.22$ an SU(3) breaking parameter in the short-distance amplitude[@LCSR]. Using the estimates of Table 1 one finds $\varepsilon_A\simeq 0.3$, which gives an uncertainty of about 15% in this determination of $|V_{td}|$ from long-distance effects. The amplitude $A$ induces also a direct CP asymmetry in charged $B$ decay $A_{CP}=
2\left|\frac{V_{ub}V_{ud}^*}{V_{tb}V_{td}^*}\right|\varepsilon_A
\sin\alpha \sin\phi_A$, which can be as large as 30% for optimal values of the weak and strong phases $\alpha,\phi_A$.
Constraining the CKM matrix with exclusive weak radiative $B$ decays {#subsec:wpp}
--------------------------------------------------------------------
Interference between the e.m. penguin and long-distance amplitudes can produce isospin breaking effects in $B^\pm\to \rho\gamma$ decays. Some of the latter contribute with a different weak phase than the former, which led to the suggestion to use isospin breaking in these decays in order to extract information about CKM parameters[@AliBr].
A more careful treatment[@bound] shows that such an approach could receive contaminations from additional long-distance effects with the [*same*]{} weak phase as the short-distance amplitude. These amplitudes can be related by SU(3) symmetry to isospin-breaking effects in $B\to K^*\gamma$ decays. Experimental data on these modes have been recently reported by the CLEO[@CLEO], BaBar[@babar] and Belle[@belle] collaborations, which find $$\begin{aligned}
\label{data1}
{\cal B}(B^\pm \to K^{*\pm}\gamma) &=& (3.76^{+0.89}_{-0.83}\pm 0.28)
\times 10^{-5}\qquad (CLEO)\\
& & (2.87\pm 1.20^{+0.55}_{-0.40})\times 10^{-5}\qquad
(BELLE)\nonumber\\
\label{data2}
{\cal B}(B^0 \to K^{*0}\gamma) &=& (4.55^{+0.72}_{-0.68}\pm 0.34)
\times 10^{-5}\qquad (CLEO)\\
& &(4.94\pm 0.93^{+0.55}_{-0.52})\times 10^{-5}\qquad (BELLE)
\nonumber\\
& & (5.2\pm 0.82\pm 0.47)\times 10^{-5}\qquad (BABAR)\nonumber\end{aligned}$$ The long-distance amplitudes responsible for the difference in rate between charged and neutral $B\to K^*\gamma$ modes are shown in Figs. \[fig:ld\](c), (e), and contain a charm loop or a gluon penguin in which the photon attaches to the spectator quark.
The preliminary data (\[data1\]), (\[data2\]) indicate that these effects could be significant. It has been proposed[@bound] therefore to eliminate them by combining $B\to \rho\gamma$ with $B\to K^*\gamma$ data, by forming the combined ratio $$\begin{aligned}
\label{Rrho}
R_\rho &\equiv& \frac{{\cal B}(B^\pm \to \rho^\pm\gamma)}
{2{\cal B}(B^0 \to \rho^0\gamma)}\cdot
\frac{{\cal B}(B^0 \to K^{*0}\gamma)}
{{\cal B}(B^\pm \to K^{*\pm}\gamma)}\\
&=& 1 - 2\varepsilon_{PA_c} -
2 \varepsilon_A
\left|\frac{V_{ub}V_{ud}^*}{V_{tb}V_{td}^*}\right|\cos\alpha
\cos\phi_A + {\cal O}(\varepsilon_i^2)\,.\nonumber\end{aligned}$$ We expanded here to linear order in the ratios of long-/short-distance amplitudes $\varepsilon_i$. The residual contamination from the (OZI-suppressed) penguin-annihilation amplitude $\varepsilon_{PA_c}$ Fig. \[fig:ld\](d) can be expected to be very small. An upper bound on its size can be given in terms of experimental data on $B_s$ decays as $|\varepsilon_{PA_c}|^2 \leq
2\Gamma(B_s\to \rho^0\gamma)/\Gamma(B^\pm\to K^{*\pm}\gamma)$.
Any measurement of the ratio (\[Rrho\]) different from 1 can be translated into a constraint on the CKM factors in the last term. The cleanest approach involves extracting the factor $\left|\frac{V_{ub}V_{ud}^*}{V_{tb}V_{td}^*}\right|\varepsilon_A$ from a combination of data in $B\to\gamma\ell \nu_\ell$ and $B\to
K^*\gamma$ decays[@bound]. This results into a bound on the weak phase $\alpha$ which excludes values around $\alpha=90^\circ$.
A simpler (although less model-independent) approach would combine theoretical calculations of $\varepsilon_A$ using (\[ALR\]), in order to constrain the combination of CKM parameters $\left|\frac{V_{ub}V_{ud}^*}{V_{tb}V_{td}^*}\right|\cos\alpha$. A typical region in the $(\rho,\eta)$ plane which can be excluded in this way is shown in Fig. \[fig:ckm\]. This largely overlaps with the area presently favored by global fits of the unitarity triangle, and can therefore be expected to be useful in further constraining it, as more data on weak radiative $B$ decays become available.
Acknowledgments {#acknowledgments .unnumbered}
===============
It has been a pleasure collaborating with Ben Grinstein, Yuval Grossman, G. Korchemsky and Tung-Mow Yan on the issues discussed here. I am grateful to the organizers of the PPP workshop for the invitation to give a talk and to the Center for Theoretical Sciences, R. O. C. for financial support. This research was supported by the DOE grant DOE-FG03-97ER40546.
[99]{}
Another class of decays which can be described systematically corresponds to diagrams with one far offshell photon $q^2 \gg \Lambda^2$ emitted from an internal quark line. The internal quark propagator is contracted to a point and the decay amplitudes are expanded in powers of $1/q^2$ with the help of an OPE[@operefs].
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---
abstract: 'The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to Matsumura. Employing the properties of tableau complexes given by Knutson, Miller and Yong, we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations.'
---
$^1$Department of Mathematics\
Sichuan University, Chengdu, Sichuan 610064, P.R. China\
$^{2}$Center for Combinatorics, LPMC\
Nankai University, Tianjin 300071, P.R. China
$^[email protected], $^[email protected]
Introduction
============
Let $S_n$ denote the set of permutations on $\{1,2,\ldots,n\}$. The double Grothendieck polynomials $\G_{w}(\x,\y)$ indexed by permutations $w\in S_n$ were introduced by Lascoux and Schützenberger [@LaSh] as polynomial representatives of the equivariant $K$-theory classes of structure sheaves of Schubert varieties in the flag manifold. These polynomials were originally defined based on the isobaric divided difference operators. Several combinatorial models have been developed to generate $\G_{w}(\x,\y)$, see, for example, [@BR; @FoKi; @KnMi; @KnMi2; @Le].
On the other hand, tableau formulas for $\mathfrak{G}_{w}(\x,\y)$ have been found for specific families of permutations. Based on the algebraic geometry of matrix Schubert varieties, Knutson, Miller and Yong [@KnMiYo-1] showed that for a 2143-avoiding permutation $w$ (also called a vexillary permutation), $\mathfrak{G}_{w}(\x,\y)$ can be generated by flagged set-valued Young tableaux. A permutation $w=w_1w_2\cdots w_n\in S_n$ is 2143-avoiding if there do not exist indices $1\le i_1<i_2<i_3<i_4\le n$ such that $w_{i_2}<w_{i_1}<w_{i_4}<w_{i_3}$. Set-valued Young tableaux were introduced by Buch [@Bu] in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Restricting to semistandard Young tableaux (namely, set-valued Young tableaux with each set containing a single integer), the Knutson-Miller-Yong formula specializes to the tableau formula for the Schubert polynomial $\mathfrak{S}_{w}(\x)$ of 2143-avoiding permutations due to Wachs [@Wa].
By introducing the structure of tableau complexes and utilizing the tools of commutative algebra, Knutson, Miller and Yong [@KnMiYo-2] found two other tableau formulas of $\G_w(\x,\y)$ for 2143-avoiding permutations in terms of semistandard Young tableaux and limit set-valued Young tableaux, respectively. A limit set-valued Young tableau is an assignment of finite nonempty subsets of positive integers to the squares of a Young diagram such that one can pick out an integer from each square to form a semistandard Young tableau.
Recently, Matsumura [@Matsumura-2] provided a tableau formula of $\G_w(\x,\y)$ for 321-avoiding permutations. A permutation $w=w_1w_2\cdots w_n$ is 321-avoiding if there do not exist indices $i_1<i_2<i_3$ such that $w_{i_1}>w_{i_2}>w_{i_3}$. To a 321-avoiding permutation $w$, one can associate a skew Young diagram, denoted $\sigma(w)$. Matsumura [@Matsumura-2] showed that for a 321-avoiding permutation $w$, $\mathfrak{G}_{w}(\x,\y)$ can be generated by flagged set-valued tableaux of shape $\sigma(w)$. This formula generalizes the tableau formula for the single Grothendieck polynomial $\G_w(\x)$ of a 321-avoiding permutation given by Anderson, Chen and Tarasca [@Anderson]. When restricted to semistandard Young tableaux, it specializes to the formula for the double Schubert polynomial $\S_w(\x,\y)$ of a 321-avoiding permutation obtained by Chen, Yan and Yang [@Chen].
In this paper, we introduce the structure of set-valued Rothe tableaux. Let $D(w)$ be the Rothe diagram of a permutation $w$. A set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the sets in each row are [*weakly decreasing*]{} and the sets in each column are [*strictly increasing*]{}. As defined by Buch [@Bu], for two finite nonempty sets $A$ and $B$ of positive integers, $A<B$ if $\max A<\min B$, and $A\leq B$ if $\max A\leq \min B$. It was noticed by Billey, Jockusch and Stanley [@BiSt] that when $w$ is a 321-avoiding permutation, $D(w)$ is a skew Young diagram after a reflection about a vertical line. In this case, each row in a set-valued Rothe tableau of shape $D(w)$ is weakly increasing after a reflection about a vertical line, and thus becomes a set-valued Young tableau. Hence set-valued Rothe tableaux can be viewed as a generalization of set-valued Young tableaux from Young diagrams to Rothe diagrams.
Our main objective is to establish set-valued Rothe tableau formulas of $\G_w(\x,\y)$ for a new family of permutations, namely, 1432-avoiding permutations. A permutation $w=w_1w_2\cdots w_n$ is 1432-avoiding if there do not exist indices $i_1<i_2<i_3<i_4$ such that $w_{i_1}<w_{i_4}<w_{i_3}<w_{i_2}$. When restricted to 321-avoiding permutations, we show that one of our formulas coincides with the formula of Matsumura [@Matsumura-2]. It should be noted that Stankova [@St] proved that the number of 1432-avoiding permutations in $S_n$ is equal to the number of 2143-avoiding permutations in $S_n$.
In order to state our results, we recall some definitions and notation. The Rothe diagram $D(w)$ of a permutation $w\in S_n$ can be viewed as a geometric configuration of the inversions of $w$. Consider an $n\times n$ square grid, where we use $(i,j)$ to denote the square in row $i$ and column $j$. Here the rows are numbered from top to bottom and the columns are numbered from left to right. For $1\leq i\leq n$, put a dot in the square $(i, w_i)$. Then the Rothe diagram $D(w)$ consists of the squares $(i,j)$ such that there is a dot in row $i$ that is to the right of $(i,j)$, and there is a dot in column $j$ that is below $(i,j)$. For example, Figure \[Rothe\](a) is the Rothe diagram of $w=426315$.
(260,65)
(0,0)(30,0)(60,0)(0,10)(30,10)(60,10) (0,20)(30,20)(60,20)(0,30)(30,30)(60,30) (0,40)(30,40)(60,40)(0,50)(30,50)(60,50) (0,60)(30,60)(60,60)
(0,0)(0,30)(0,60)(10,0)(10,30)(10,60) (20,0)(20,30)(20,60)(30,0)(30,30)(30,60) (40,0)(40,30)(40,60)(50,0)(50,30)(50,60) (60,0)(60,30)(60,60)
(55,35)(45,5) (25,25)(5,15) (35,55) (15,45)
(0,30)[(1,0)[10]{}]{}(0,40)[(1,0)[10]{}]{} (0,20)[(1,0)[10]{}]{} (0,50)[(1,0)[30]{}]{}(0,60)[(1,0)[30]{}]{} (0,20)[(0,1)[40]{}]{}(10,20)[(0,1)[40]{}]{} (20,50)[(0,1)[10]{}]{}(30,50)[(0,1)[10]{}]{}
(20,30)[(1,0)[10]{}]{}(20,40)[(1,0)[10]{}]{} (20,30)[(0,1)[10]{}]{}(30,30)[(0,1)[10]{}]{}
(40,30)[(1,0)[10]{}]{}(40,40)[(1,0)[10]{}]{} (40,30)[(0,1)[10]{}]{}(50,30)[(0,1)[10]{}]{}
(100,0)(130,0)(160,0) (100,10)(130,10)(160,10) (100,20)(130,20)(160,20) (100,30)(130,30)(160,30) (100,40)(130,40)(160,40) (100,50)(130,50)(160,50) (100,60)(130,60)(160,60)
(100,0)(100,30)(100,60) (110,0)(110,30)(110,60) (120,0)(120,30)(120,60) (130,0)(130,30)(130,60) (140,0)(140,30)(140,60) (150,0)(150,30)(150,60) (160,0)(160,30)(160,60)
(155,35)(145,5) (135,25)(105,15) (145,55) (115,45)
(100,30)[(1,0)[10]{}]{}(100,40)[(1,0)[10]{}]{} (100,20)[(1,0)[10]{}]{} (100,50)[(1,0)[30]{}]{}(100,60)[(1,0)[30]{}]{} (100,20)[(0,1)[40]{}]{}(110,20)[(0,1)[40]{}]{} (120,50)[(0,1)[10]{}]{}(130,50)[(0,1)[10]{}]{}
(120,30)[(1,0)[10]{}]{}(120,40)[(1,0)[10]{}]{} (120,30)[(0,1)[10]{}]{}(130,30)[(0,1)[10]{}]{}
(140,30)[(1,0)[10]{}]{}(140,40)[(1,0)[10]{}]{} (140,30)[(0,1)[10]{}]{}(150,30)[(0,1)[10]{}]{}
(103,52)[1]{}(103,42)[2]{}(103,32)[3]{}(103,22)[4]{} (113,52)[1]{}(123,52)[1]{}
(121,32)[3]{} (125,32)[2]{}(141,32)[2]{} (145,32)[1]{}
(200,0)(230,0)(260,0)(200,10)(230,10)(260,10) (200,20)(230,20)(260,20)(200,30)(230,30)(260,30) (200,40)(230,40)(260,40)(200,50)(230,50)(260,50) (200,60)(230,60)(260,60)
(200,0)(200,30)(200,60)(210,0)(210,30)(210,60) (220,0)(220,30)(220,60)(230,0)(230,30)(230,60) (240,0)(240,30)(240,60)(250,0)(250,30)(250,60) (260,0)(260,30)(260,60)
(255,35)(245,5) (225,25)(205,15) (235,55) (215,45)
(200,30)[(1,0)[10]{}]{}(200,40)[(1,0)[10]{}]{} (200,20)[(1,0)[10]{}]{} (200,50)[(1,0)[30]{}]{}(200,60)[(1,0)[30]{}]{} (200,20)[(0,1)[40]{}]{}(210,20)[(0,1)[40]{}]{} (220,50)[(0,1)[10]{}]{}(230,50)[(0,1)[10]{}]{}
(220,30)[(1,0)[10]{}]{}(220,40)[(1,0)[10]{}]{} (220,30)[(0,1)[10]{}]{}(230,30)[(0,1)[10]{}]{}
(240,30)[(1,0)[10]{}]{}(240,40)[(1,0)[10]{}]{} (240,30)[(0,1)[10]{}]{}(250,30)[(0,1)[10]{}]{}
(203,52)[**1**]{}(203,42)[**2**]{}(203,32)[**3**]{}(203,22)[**4**]{} (213,52)[**1**]{}(223,52)[**1**]{}
(221,32)[**2**]{} (225,32)[1]{}(241,32)[2]{} (245,32)[**1**]{} (25,-10)[[(a)]{}]{}(125,-10)[[(b)]{}]{} (225,-10)[[(c)]{}]{}
As aforementioned, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly decreasing and the columns are strictly increasing. For example, Figure \[Rothe\](b) depicts a set-valued Rothe tableau for $w=426315$. We say that a set-valued Rothe tableau is flagged by a vector $\mathbf{f}=(f_1,f_2,\ldots,f_n)$ of nonnegative integers if every integer in row $i$ does not exceed $f_i$. Let $\SVRT(w,\f)$ denote the set of set-valued Rothe tableaux of shape $D(w)$ flagged by $\f$.
For a set-valued Rothe tableau $T$ and a square $B=(i,j)$ of $T$, we use $T(B)$ or $T(i,j)$ to denote the set filled in $B$. Write $|T|=\sum_{B\in D(w)}|T(B)|$. Let $\ell(w)$ denote the length of $w$, or equivalently, $\ell(w)=|D(w)|$. For two variables $x$ and $y$, we adopt the following notation as used by Fomin and Kirillov [@FoKi]: $$x\oplus y=x+y-xy.$$ For a square $(i,j)$ of $D(w)$, define $$\begin{aligned}
\label{statistic}
m_{ij}(w)=|\{(i,k)\in D(w)\,|\,k\leq j\}|.\end{aligned}$$ Throughout this paper, we use the following specific flag $$\f_0=(1,2,\ldots,n).$$ Our main result can be stated as follows.
\[main\] For a permutation $w\in S_n$, we have the following equivalent statements.
- $w$ is a 1432-avoiding permutation.
- $\mathfrak{G}_{w}(\x,\y)$ has the following set-valued Rothe tableau formula: $$\begin{aligned}
\label{CFG-R}
\mathfrak{G}_{w}(\x,\y)=\sum_{T\in {\rm SVRT}(w,\f_0)}(-1)^{|T|-\ell(w)} \prod_{(i,j)\in D(w)}\prod_{t\in T({i,j})}(x_t\oplus y_{m_{ij}(w)+i-t}).\end{aligned}$$
Setting $y_i=0$ in , we obtain a tableau formula for single Grothendieck polynomials of 1432-avoiding permutations.
\[coro\] Let $w\in S_n$ be a 1432-avoiding permutation. Then $$\mathfrak{G}_{w}(\x)=\sum_{T\in {\rm SVRT}(w,\f_0)}(-1)^{|T|-\ell(w)}
\prod_{(i,j)\in T}\prod_{t\in T({i,j})}x_t.$$
The double Schubert polynomial $\mathfrak{S}_{w}(\x,\y)$ can be obtained from $\mathfrak{G}_{w}(\x,\y)$ by extracting the monomials of the lowest degree and then replacing $y_i$ by $-y_i$. Let ${\rm SRT}(w,\f)$ be the set of single-valued Rothe tableaux of shape $D(w)$ flagged by $\f$. In other words, ${\rm SRT}(w,\f)$ consists of the Rothe tableaux in ${\rm SVRT}(w,\f)$ such that the set filled in each square contains exactly one integer. We have the following tableau formulas for double and single Schubert polynomials.
\[coro\] Let $w\in S_n$ be a 1432-avoiding permutation. Then $$\begin{aligned}
\mathfrak{S}_{w}(\x,\y)&=\sum_{T\in {\rm SRT}(w,\f_0)}
\prod_{(i,j)\in D(w)}\prod_{t\in T({i,j})}(x_t-y_{m_{ij}(w)+i-t}),\\[5pt]
\mathfrak{S}_{w}(\x)&=\sum_{T\in {\rm SRT}(w,\f_0)}
\prod_{(i,j)\in D(w)}\prod_{t\in T({i,j})}x_t.\end{aligned}$$
Furthermore, by introducing the structure of Rothe tableau complexes and employing the properties of tableau complexes given by Knutson, Miller and Yong [@KnMiYo-2], we also find two alternative tableau formulas of $\G_w(\x,\y)$ for 1432-avoiding permutations. One is given in terms of single-valued Rothe tableaux, and the other is given in terms of limit set-valued Rothe tableaux. A limit set-valued Rothe tableau is an assignment of finite nonempty subsets of positive integers to the squares of a Rothe diagram such that one can pick out an integer from each square to form a single-valued Rothe tableau. Figure \[Rothe\](c) illustrates a limit set-valued Rothe tableau, where the integers in boldface form a single-valued Rothe tableau.
Let ${\rm LSVRT}(w, \f)$ denote the set of limit set-valued Rothe tableaux of shape $D(w)$ flagged by $\f$. Then we have the following two alternative tableau formulas of $\G_w(\x,\y)$ for 1432-avoiding permutations.
\[3-1\] Let $w\in S_n$ be a 1432-avoiding permutation.
- For each square $B=(i,j)\in D(w)$, set $$E_B=\bigcup_{T\in {\rm SRT}(w, \f_0)}T(i,j).$$ Then $$\begin{aligned}
\label{3-1-1}
\mathfrak{G}_{w}(\x,\y)= \sum_{T\in {\rm LSVRT}(w, \f_0)} \prod_{B=(i,j)\in D(w)}
\prod_{t\in T(i,j)}(x_t\oplus y_{m_{ij}(w)+i-t})\nonumber\\[5pt]
\cdot\prod_{t\in E_B\setminus T(i,j)}(1-x_t)(1-y_{m_{ij}(w)+i-t}).\end{aligned}$$
- Given $T\in {\rm SRT}(w,\f_0)$ and a square $B\in D(w)$, let $Y_{T,B}$ be the set of positive integers $m$ such that $m$ is larger than the (unique) integer in $T(B)$ and replacing the integer in $T(B)$ by $m$ still yields a Rothe tableau in ${\rm SRT}(w, \f_0)$. Then $$\begin{aligned}
\label{3-1-2}
\mathfrak{G}_{w}(\x,\y)= \sum_{T\in {\rm SRT}(w, \f_0)}\ \prod_{B=(i,j)\in D(w)}
\prod_{t\in T(i,j)} (x_t\oplus y_{m_{ij}(w)+i-t})\nonumber\\[5pt]
\cdot\prod_{t\in Y_{T,B}}(1-x_t)(1-y_{m_{ij}(w)+i-t}).\end{aligned}$$
Proof of Theorem \[main\] {#sec3}
=========================
In this section, we aim to prove Theorem \[main\]. For simplicity, let $$-x \oplus y=-(x \oplus y)=-(x+y-xy).$$ Denote $$\begin{aligned}
\label{rw}
G_w(\x,\y)&=\sum_{T\in {\rm SVRT}(w,\f_0)}(-1)^{|T|-\ell(w)} \prod_{(i,j)\in T}\prod_{t\in T({i,j})}(x_t\oplus y_{m_{ij}(w)+i-t})\nonumber\\[6pt]
&=(-1)^{\ell(w)}\sum_{T\in {\rm SVRT}(w,\f_0)} \prod_{(i,j)\in T}\prod_{t\in T({i,j})}(-x_t\oplus y_{m_{ij}(w)+i-t})
\end{aligned}$$ to be the right-hand side of . We finish the proof of Theorem \[main\] by separately proving the following two theorems.
\[prop1\] If $w$ is a 1432-avoiding permutation, then $\G_w(\x,\y)=G_w(\x,\y)$.
\[prop2\] If $w$ contains a 1432 pattern, then $\G_w(\x,\y)\neq G_w(\x,\y)$.
We use the opportunity here to explain that when $w$ is a 321-avoiding permutation, Theorem \[prop1\] specializes to the tableau formula for $\mathfrak{G}_{w}(\x,\y)$ due to Matsumura [@Matsumura-2]. To describe the tableau formula in [@Matsumura-2], let $f(w)=(f_1,f_2,\ldots,f_k)$ (respectively, $f^c(w)=(f_1^c,f_2^c,\ldots,f_{n-k}^c)$) be the increasing arrangement of the positions $i$ such that $w_i>i$ (respectively, $w_i\leq i$). Moreover, let $h(w)=(w_{f_1},w_{f_2},\ldots,w_{f_k})$ and $h^c(w)=(w_{f_1^c},w_{f_2^c},\ldots,w_{f_{n-k}^c})$. It can be shown that $w$ is 321-avoiding if and only if the sequences $h(w)$ and $h^c(w)$ are both increasing [@ErLi]. One may associate a skew shape $\sigma(w)=\lambda/ \mu$ to $w$ by letting $$\begin{aligned}
\label{lmu}
\lambda_i=w_{f_k}-k-(f_i-i), \ \ \ \ \ \ \mu_i=w_{f_k}-k-(w_{f_i}-i),\end{aligned}$$ where $1\leq i\leq k$. For a square $\alpha$ of $\sigma(w)$, let $r(\alpha)$ and $c(\alpha)$ denote the row index and the column index of $\alpha$, respectively.
Let $w\in S_n$ be a 321-avoiding permutation. Then $$\begin{aligned}
\label{MJAP}
\mathfrak{G}_{w}(\x,\y)=\sum_{T}(-1)^{|T|-\ell(w)}
\prod_{\alpha\in \sigma(w)}\prod_{t\in T(\alpha)}(x_t\oplus y_{\lambda_{r(\alpha)}
+f_{r(\alpha)}-c(\alpha)-t+1}),\end{aligned}$$ where $T$ ranges over set-valued Young tableaux of shape $\sigma(w)$ flagged by $f(w)$.
We show that for a 321-avoiding permutation $w$, the right-hand side of is equal to $G_w(\x, \y)$ as defined in . As observed in [@BiSt], after deleting the empty rows indexed by $f^c(w)$ and the empty columns indexed by $h(w)$ and then reflecting the resulting diagram about a vertical line, $D(w)$ coincides with the above defined skew shape $\sigma(w)$. For example, for $w=312465$, we see that $f(w)=(1,5), f^c(w)=(2,3,4,6)$ and $h(w)=(w_{f_1},w_{f_2})=(3,6)$. So the corresponding shew diagram $\sigma(w)$ is as illustrated in Figure \[skew-shape\].
(180,65) (0,0)(30,0)(60,0)(0,10)(30,10)(60,10) (0,20)(30,20)(60,20)(0,30)(30,30)(60,30) (0,40)(30,40)(60,40)(0,50)(30,50)(60,50) (0,60)(30,60)(60,60)
(0,0)(0,30)(0,60)(10,0)(10,30)(10,60) (20,0)(20,30)(20,60)(30,0)(30,30)(30,60) (40,0)(40,30)(40,60)(50,0)(50,30)(50,60) (60,0)(60,30)(60,60)
(25,55)(5,45) (15,35)(35,25) (55,15) (45,5)
(0,50)[(1,0)[20]{}]{}(0,60)[(1,0)[20]{}]{} (0,50)[(0,1)[10]{}]{}(10,50)[(0,1)[10]{}]{}(20,50)[(0,1)[10]{}]{}
(40,10)[(0,1)[10]{}]{}(50,10)[(0,1)[10]{}]{} (40,10)[(1,0)[10]{}]{}(40,20)[(1,0)[10]{}]{}
(-5,45)[(1,0)[70]{}]{}(-5,35)[(1,0)[70]{}]{} (-5,25)[(1,0)[70]{}]{}(-5,5)[(1,0)[70]{}]{}
(25,-5)[(0,1)[70]{}]{}(55,-5)[(0,1)[70]{}]{}
(130,20)[(1,0)[10]{}]{}(130,30)[(1,0)[10]{}]{} (130,20)[(0,1)[10]{}]{}(140,20)[(0,1)[10]{}]{}
(150,30)[(1,0)[20]{}]{}(150,40)[(1,0)[20]{}]{} (150,30)[(0,1)[10]{}]{}(160,30)[(0,1)[10]{}]{} (170,30)[(0,1)[10]{}]{}
(145,2)[$\sigma(w)$]{}
Therefore, each set-valued Rothe tableau $T\in\SVRT(w,\f_0)$ can be viewed as a set-valued (skew) Young tableau of shape $\sigma(w)$ flagged by $f(w)$. For a square $(i,j)\in D(w)$, assume that $\alpha$ is the corresponding square of $\sigma(w)$. Then we need to show that $$\begin{aligned}
\label{rbr}
\lambda_{r(\alpha)}+f_{r(\alpha)}-c(\alpha)+1=m_{ij}(w)+i.
\end{aligned}$$ It is not hard to check that $$\begin{aligned}
r(\alpha)&=i-|\{t\,|\,w_t\le t<i\}|,\\[5pt]
c(\alpha)&=w_{f_k}-j-|\{t\,|\,w_t>t,w_t>j\}|+1.\end{aligned}$$ Then by , we have $$\begin{aligned}
\label{hj}
\lambda_{r(\alpha)}+f_{r(\alpha)}-c(\alpha)+1
&=w_{f_k}-k+r(\alpha)-c(\alpha)+1\nonumber\\[5pt]
&=j-k+|\{t\,|\,w_t>t,w_t>j\}|+i-|\{t\,|\,w_t\le t<i\}|\nonumber\\[5pt]
&=j-|\{t\,|\,t< w_t\le j\}|-|\{t\,|\,w_t\le t<i\}|+i,\end{aligned}$$ where, at the last step, we used the relation $$\begin{aligned}
k-|\{t\,|\,w_t>t,w_t>j\}|&=|\{t\,|\,w_t>t\}|-|\{t\,|\,w_t>t,w_t>j\}|\\[5pt]
&=|\{t\,|\,t< w_t\le j\}|.\end{aligned}$$ Since $w$ is 321-avoiding, it is easy to check that if there exists some integer $t$ such that $t<w_t\le j$, then $t<i$. Moreover, if $w_t\le t<i$, then $w_t\le j$. Thus we have $$\begin{aligned}
\label{gf}
&j-|\{t\,|\,t< w_t\le j\}|-|\{t\,|\,w_t\le t<i\}|\nonumber\\[5pt]
&=j-(|\{t\,|\,t< w_t\le j,t<i\}|+|\{t\,|\,w_t\le t<i,w_t\le j\}|)\nonumber\\[5pt]
&=j-|\{t\,|\,t<i,w_t\le j\}|=|\{t\,|\,t\ge i,w_t\le j\}|\nonumber\\[5pt]
&=|\{(i,k)\in D(w)\,|\,k\leq j\}|\nonumber\\[5pt]
&=m_{ij}(w).\end{aligned}$$ Combining and yields . This completes the proof.
Isobaric divided difference operator
------------------------------------
Before proving Theorem \[prop1\] and Theorem \[prop2\], we recall some basic definitions. Given a permutation $w=w_1w_2\cdots w_n\in S_n$, the length $\ell(w)$ of $w$ is equal to the number of inversions of $w$, namely, $$\ell(w)=|\{(w_i,w_j)\,|\, 1\leq i<j\leq n, w_i>w_j\}|.$$ It is clear that $\ell(w)=|D(w)|$. Let $s_i$ denote the simple transposition interchanging $i$ and $i+1$. Notice that $ws_i$ is the permutation obtained from $w$ by swapping $w_i$ and $w_{i+1}$. The divided difference operator $\partial_i$ on the ring $\mathbb{Z}[\x]=\mathbb{Z}[x_1,x_2,\ldots, x_n]$ of polynomials with integer coefficients is defined by $$\partial_i f(\x)=\frac{f(\x)
-s_if(\x)}{x_i-x_{i+1}},$$ where $f(\x)\in \mathbb{Z}[\x]$ and $s_if(\x)$ is obtained from $f(\x)$ by interchanging $x_i$ and $x_{i+1}$. One can then define the isobaric divided difference operator $\pi_i$ as $$\pi_i f(\x)=\partial_i (1-x_{i+1})f(\x).$$
The double Grothendieck polynomial $\mathfrak{G}_{w}(\x,\y)$ for $w\in S_n$ can be defined as follows. For the longest permutation $w_0=n \,(n-1)\cdots 1$, set $$\mathfrak{G}_{w_0}(\x,\y)=\prod_{i+j\leq n}(x_i+y_j-x_iy_j).$$ For $w\neq w_0$, choose a simple transposition $s_i$ such that $\ell(ws_i)=\ell(w)+1$, and let $$\begin{aligned}
\label{GD}
\mathfrak{G}_{w}(\x,\y)=\pi_i \mathfrak{G}_{ws_i}(\x,\y),
\end{aligned}$$ where the operator $\pi_i$ only acts on the $\x$-variables. Note that is independent of the choice of the simple transposition $s_i$, since the operator $\pi_i$ satisfies the Coxeter relations $\pi_i\pi_{i+1}\pi_i=\pi_{i+1}\pi_{i}\pi_{i+1}$ and $\pi_i\pi_{j}=\pi_j\pi_i$ for $|i-j|>1$. If we set $y_i=0$ for $i\geq 1$, then $\mathfrak{G}_{w}(\x,\y)$ reduces to the single Grothendieck polynomial $\mathfrak{G}_{w}(\x)$.
The double Schubert polynomial $\mathfrak{S}_{w}(\x, \y)$ can be produced by a similar procedure [@LaSh-S; @Ma]. Set $$\mathfrak{S}_{w_0}(\x,\y)=\prod_{i+j\leq n}(x_i-y_j).$$ For $w\neq w_0$, choose a simple transposition $s_i$ such that $\ell(ws_i)=\ell(w)+1$, and let $\mathfrak{S}_{w}(\x,\y)=\partial_i \mathfrak{S}_{ws_i}(\x,\y)$. By definition, it is easily seen that $\mathfrak{S}_{w}(\x,\y)$ can be obtained from $\mathfrak{G}_{w}(\x,\y)$ by taking the lowest degree homogeneous component and then replacing $y_i$ by $-y_i$ for $i\geq 1$. Analogously, putting $y_i=0$, $\mathfrak{S}_{w}(\x,\y)$ reduces to the single Schubert polynomial $\mathfrak{S}_{w}(\x)$. For combinatorial constructions of Schubert polynomials, see for example [@As; @AsSe; @BeBi; @BeSo; @BiSt; @FoRe; @FK2; @FoSt; @LLS; @WeYo; @Wi].
Proof of Theorem \[prop1\]
--------------------------
The key idea is to show that, when $w$ is 1432-avoiding, $G_w(\x,\y)$ is compatible with the isobaric divided difference operator, which allows us to finish the proof by induction. Such an idea was first used by Wachs [@Wa] to establish the tableau formula for the Schubert polynomials of 2143-avoiding permutations. Matsumura [@Matsumura-3] and Matsumura and Sugimoto [@Matsumura-4] extended this idea to reprove the Knutson-Miller-Yong set-valued tableau formula for the Grothendieck polynomials of 2143-avoiding permutations. Our technique can be viewed as a generalization of that in [@Matsumura-3; @Matsumura-4] from Young diagrams to Rothe diagrams.
The longest permutation $w_0=n\cdots 2 1$ is 1432-avoiding. Since $D(w_0)$ is a staircase Young diagram with $n-i$ squares in row $i$, there is only one tableau $T_0$ of shape $D(w_0)$ flagged by $\f_0$, that is, every square in the $i$-th row of $T_0$ is filled with $\{i\}$. For each square $(i,j)$ of $D(w_0)$, one has $m_{ij}(w_0)=j$. Thus, $$G_{w_0}(\x,\y)=(-1)^{|T_0|-\ell(w_0)}\prod_{i+j\leq n}(x_i\oplus y_j)=\prod_{i+j\leq n}(x_i\oplus y_j),$$ which agrees with $\mathfrak{G}_{w_0}(\x,\y)$.
We now consider a 1432-avoiding permutation $w\neq w_0$. Let $r$ be the first ascent of $w$, that is, the smallest position such that $w_r<w_{r+1}$. Lemma \[tp\] claims that $ws_r$ is also 1432-avoiding. Moreover, we will prove that $$G_w(\x,\y)=\pi_r G_{ws_r}(\x,\y),$$ see Theorem \[main-s\]. This allows us to give a proof of Theorem \[prop1\] by induction.
\[tp\] Let $w\neq w_0$ be a 1432-avoiding permutation, and $r$ be the first ascent of $w$. Then $ws_r$ is a 1432-avoiding permutation.
Write $w'=ws_r=w'_1w'_2\cdots w'_n$. Suppose otherwise that $w'$ has a subsequence that is order isomorphic to $1432$. Since $w$ is 1432-avoiding and $r$ is the first ascent, any subsequence of $w'$ that is order isomorphic to 1432 must be of the form $w'_iw'_rw'_{r+1}w'_j$, where $i<r$ and $j>r+1$. Since $w'_i$ is the smallest element in this subsequence, we have $w'_i<w'_{r+1}$. Noticing that $w'_i=w_i$ and $w'_{r+1}=w_r$, we see that $w_i<w_r$. However, since $r$ is the first ascent, we must have $w_i>w_r$, leading to a contradiction. This completes the proof.
\[main-s\] Let $w\neq w_0$ be a 1432-avoiding permutation, and $r$ be the first ascent of $w$. Then $$\begin{aligned}
\label{main-se}
G_{w}(\x,\y)=\pi_r G_{ws_r}(\x,\y).\end{aligned}$$
In the rest of this subsection, we present a proof of Theorem \[main-s\], which can be outlined as follows. We first define an equivalence relation on the two sets $\SVRT(ws_r,\f_0)$ and $\SVRT(w,\f_0)$. For an equivalence class $C$ of $\SVRT(ws_r,\f_0)$, let $$\begin{aligned}
G_{ws_r}(C; \x,\y)=(-1)^{\ell(ws_r)}\sum_{T\in C} \prod_{(i,j)\in D(w)}\prod_{t\in T({i,j})}(-x_t\oplus y_{m_{ij}(w)+i-t})\label{class-g}\end{aligned}$$ denote the polynomial generated by the Rothe tableaux in $C$. In Theorem \[mm-1\], we deduce a formula for $G_{ws_r}(C; \x,\y)$. Similarly, write $G_{w}(C'; \x,\y)$ for the polynomial generated by the Rothe tableaux in an equivalence class $C'$ of $\SVRT(w, \f_0)$. We also obtain an expression for $G_{w}(C'; \x,\y)$, see Theorem \[mm-2\]. Finally, we establish a bijection $\Phi$ between the set of equivalence classes of $\SVRT(ws_r,\f_0)$ and the set of equivalence classes of $\SVRT(w, \f_0)$. The formulas given in Theorems \[mm-1\] and \[mm-2\] allow us to conclude that for any equivalence class $C$ of $\SVRT(ws_r,\f_0)$, $$\pi_r G_{ws_r}(C; \x,\y)=G_w(\Phi(C); \x,\y).$$ This leads to a proof of Theorem \[main-s\].
Unless otherwise stated, we always assume that $w\neq w_0$ is a 1432-avoiding permutation, and that $r$ is the first ascent of $w$. For $T\in\SVRT(ws_r, \f_0)$, let $$\begin{aligned}
E(T)=\{B\in D(ws_r)\,|\,\{r,r+1\}\cap T(B)\neq\emptyset\},\end{aligned}$$ that is, $E(T)$ is the subset of squares of $T$ containing at least one of $r$ and $r+1$. It should be noted that the definition of $E(T)$ for $T\in\SVRT(ws_r, \f_0)$ depends only on $r$, which has nothing to do with the first ascent of $ws_r$.
Given two Rothe tableaux $T,T'\in\SVRT(ws_r, \f_0)$, we say that $T$ is equivalent to $T'$, denoted $T\sim T'$, if $E(T)=E(T')$ and for every square $B\in D(ws_r)$, $$T(B)\setminus \{r,r+1\}=T'(B)\setminus \{r,r+1\}.$$
The equivalence relation on the set $\SVRT(w, \f_0)$ is defined in the same manner. Let $\SVRT(ws_r, \f_0)/\hspace{-.1cm}\sim$ and $\SVRT(w, \f_0)/\hspace{-.1cm}\sim$ denote the sets of equivalence classes of $\SVRT(ws_r, \f_0)$ and $\SVRT(w, \f_0)$, respectively.
Given a Rothe tableau $T$ in $\SVRT(ws_r, \f_0)$ or $\SVRT(w, \f_0)$, since the columns of $T$ are strictly increasing, each column of $T$ contains at most two squares in $E(T)$. Let $P(T)$ be the subset of $E(T)$ such that a square $B\in E(T)$ belongs to $P(T)$ if the column of $T$ containing $B$ has only one square (i.e., $B$) in $E(T)$. Let $Q(T)=E(T)\setminus P(T)$, namely, the subset of $E(T)$ such that a square $B\in E(T)$ belongs to $Q(T)$ if the column containing $B$ has exactly two squares in $E(T)$. Evidently, $T\sim T'$ if and only if $$P(T)=P(T')\quad \text{and}\quad Q(T)=Q(T').$$
Let $C$ be an equivalence class of $\SVRT(ws_r, \f_0)$ or $\SVRT(w, \f_0)$, and let $T$ be any given Rothe tableau in $C$. For $i\ge1$, let $P(T,i)$ be the set of squares of $P(T)$ in row $i$, and let $b_i(T)=|P(T,i)|$. Clearly, $P(T,i)$ is empty unless $i\geq r$. Moreover, let $$\begin{aligned}
\label{ellr}
\ell_i(T)=m_{ip_i}(ws_r)+i-r-1,\end{aligned}$$ where $(i,p_i)$ is the leftmost square in $P(T,i)$. To state the formula for $G_{ws_r}(C; \x,\y)$ or $G_{w}(C; \x,\y)$, we need to define a polynomial $h(C,i;\x,\y)$. Set $h(C,i;\x,\y)=1$ if $b_i(T)=0$, and for $b_i(T)\geq 1$, let $$\begin{aligned}
h(C,i;\x,\y)=&\sum_{k=0}^{b_i(T)}\, \prod_{j=1}^{k} (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k+1}^{b_i(T)} (-x_{r}\oplus y_{\ell_i(T)+j})\nonumber\\[5pt]
&+\sum_{k=1}^{b_i(T)}\, \prod_{j=1}^k (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k}^{b_i(T)} (-x_{r}\oplus y_{\ell_i(T)+j}).\label{hc}\end{aligned}$$ Note that $h(C,i;\x,\y)$ is independent of the choice of the Rothe tableau $T$ in $C$. As will be seen in the proof of Theorem \[mm-1\], $h(C,i;\x,\y)$ records the contributions of the integers $r$ and $r+1$ in the squares of $P(T,i)$ ($i\ge r+1$) summed over all the Rothe tableaux in $C$.
\[mm-1\] Let $w\neq w_0$ be a 1432-avoiding permutation, and $r$ be the first ascent of $w$. Assume that $C\in\SVRT(ws_r, \f_0)/\hspace{-.1cm}\sim$ and $T$ is any given Rothe tableau in $C$. Then, $$\begin{aligned}
G_{ws_r}(C; \x,\y)=&(-1)^{\ell(ws_r)} \left(\prod_{(i,j)\in D(ws_r)}\prod_{t\in T(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(ws_r)+i-t})\right)\nonumber \\[5pt]
&\ \ \ \ \ \cdot\left(\prod_{j=1}^{b_r(T)}(-x_r\oplus y_{\ell_r(T)+j})\right)\cdot H_C(\x,\y)\cdot J_C(\x,\y).\label{gggs}\end{aligned}$$ In the above expression , $$H_C(\x,\y)=\prod_{ i>r+1}h(C,i;\x,\y),$$ and $$J_C(\x,\y)=\prod_{(i,j)\in Q^+(T)}(-x_r\oplus y_{m_{ij}(ws_r)+i-r})(-x_{r+1}\oplus y_{m_{ij}(ws_r)+i-r}),$$ where $Q^{+} (T)$ denotes the subset of $Q(T)$ consisting of the squares containing $r$.
Although the formula for $G_{ws_r}(C,\x,\y)$ in looks a bit complicated, it will be clear from the proof that each factor in appears naturally. In fact, for two different Rothe tableaux $T,T'\in C$, $T$ and $T'$ can be possibly different only in the squares of $P(T)$. Thus the first factor is the contribution of the integers other than $r$ and $r+1$. We shall show that the second factor is the contribution of $r$ in $P(T,r)$, $J_C(\x,\y)$ is the contribution of $r$ and $r+1$ in $Q(T)$, and $H_C(\x,\y)$ is the contribution of $r$ and $r+1$ in $P(T,i)$ with $i\ge r+1$ summed over all the Rothe tableaux in $C$.
To prove Theorem \[mm-1\], we need two lemmas concerning the configuration of the squares in the sets $P(T)$ and $Q(T)$.
\[right\] Let $w\neq w_0$ be a 1432-avoiding permutation. Assume that $T$ is a Rothe tableau in $\SVRT(w, \f_0)$ or $\SVRT(ws_r, \f_0)$, and $(i,j)\in P(T)$. Then there do not exist two squares $(i,k),(h,k)\in Q(T)$ such that $k>j$ and $h<i$.
We only give a proof for the case when $T\in\SVRT(w, \f_0)$. The same argument applies to the case when $T\in \SVRT(ws_r, \f_0)$. Suppose to the contrary that there exist two squares $(i,k),(h,k)\in Q(T)$ such that $k>j$ and $h<i$, see Figure \[a\] for an illustration.
(100,60) (50,0)[(1,0)[10]{}]{}(50,10)[(1,0)[10]{}]{} (50,0)[(0,1)[10]{}]{}(60,0)[(0,1)[10]{}]{} (80,20)[(1,0)[10]{}]{}(80,30)[(1,0)[10]{}]{} (80,20)[(0,1)[10]{}]{}(90,20)[(0,1)[10]{}]{}
(80,0)[(1,0)[10]{}]{}(80,10)[(1,0)[10]{}]{} (80,0)[(0,1)[10]{}]{}(90,0)[(0,1)[10]{}]{}
(20,5)(33,5)(46,5)(20,25)(33,25)(46,25) (10,4)[$i$]{}(10,23.5)[$h$]{}
(55,34)(55,40)(55,46)(85,34)(85,40)(85,46) (54,52)[$j$]{}(83.5,51.5)[$k$]{}
Since $(i,j),(h,k)\in D(w)$, it follows that $w_h>j$ and $j$ appears after $w_h$ in $w$. This implies that $(h,j)\in D(w)$. Keep in mind that each of the sets $T(i,j)$, $T(i,k)$ and $T(h,k)$ contains at least one of the integers $r$ and $r+1$. Since the rows of $T$ are weakly decreasing and the columns of $T$ are strictly increasing, we see that $r\in T(h,k)$, $r+1\in T(i,j)$. This forces that $T(h,j)=\{r\}$, and hence $(i,j)\in Q(T)$, which contradicts the assumption that $(i,j)\in P(T)$.
It should be noted that Lemma \[right\] is valid for any permutation since the pattern avoidance condition is not required in the proof.
\[left\] Let $w\neq w_0$ be a 1432-avoiding permutation. Assume that $T$ is a Rothe tableau in $\SVRT(w, \f_0)$ or $\SVRT(ws_r, \f_0)$, and $(i,j)\in P(T)$. If $i>r$, then there do not exist two squares $(i,k),(h,k)\in Q(T)$ such that $h>i$ and $k<j$.
We only give a proof for $T\in \SVRT(w, \f_0)$, and the arguments for $T\in \SVRT(ws_r, \f_0)$ can be carried out in the same manner. Suppose otherwise that there exist two squares $(i,k)$ and $(h,k)$ in $Q(T)$ where $i<h$ and $j>k$, as illustrated in Figure \[b\].
(100,60) (50,0)[(1,0)[10]{}]{}(50,10)[(1,0)[10]{}]{} (50,0)[(0,1)[10]{}]{}(60,0)[(0,1)[10]{}]{}
(50,20)[(1,0)[10]{}]{}(50,30)[(1,0)[10]{}]{} (50,20)[(0,1)[10]{}]{}(60,20)[(0,1)[10]{}]{}
(80,20)[(1,0)[10]{}]{}(80,30)[(1,0)[10]{}]{} (80,20)[(0,1)[10]{}]{}(90,20)[(0,1)[10]{}]{}
(20,5)(33,5)(46,5)(20,25)(33,25)(46,25) (10,4)[$h$]{}(10,23.5)[$i$]{}
(55,34)(55,40)(55,46)(85,34)(85,40)(85,46) (54,52)[$k$]{}(83.5,53)[$j$]{}
Notice that both the sets $T(i,j)$ and $T(i,k)$ contain $r$, while the set $T(h,k)$ contains $r+1$. We have the following two claims.
Claim 1: $w_s<k$ for any $i<s<h$. Suppose otherwise that there exists some $i<s<h$ such that $w_s>k$. Then the square $(s,k)$ belongs to $D(w)$. Since $r\in T(i,k)$ and $r+1\in T(h,k)$, it follows that $r<\min T(s,k)<r+1$, leading to a contradiction.
Claim 2: $k<w_h\leq j$. Since $(h,k)$ is a square in $D(w)$, it is clear that $k<w_h$. Suppose otherwise that $w_h>j$. It follows from Claim 1 that $j$ must appear in $w$ after the position $h$. This implies that $(h,j)\in D(w)$. Since $r\in T(i,j)$ and $r+1\in T(h,k)$, we must have $T(h,j)=\{r+1\}$. This implies that $(i,j)\in Q(T)$, contradicting the assumption that $(i,j)\in P(T)$.
By Claim 2 and the fact that $w_i>j$, we see that $w_iw_h k$ forms a decreasing subsequence of $w$. Since $w$ is 1432-avoiding, we have $w_t> k$ for any $1\leq t<i$. Thus, for any $1\leq t< i$, the square $(t,k)$ belongs to $D(w)$. Keep in mind that each integer in row $i$ of $T$ cannot exceed $i$ and the columns of $T$ are strictly increasing. So we have $T(t,k)=\{t\}$ for $1\leq t\leq i$. In particular, we have $T(i,k)=\{i\}$. Since $r\in T(i,k)$, we must have $i=r$, contradicting the assumption that $i>r$. This completes the proof.
Based on Lemmas \[right\] and \[left\], we can now give a proof of Theorem \[mm-1\].
*Proof of Theorem \[mm-1\].* Assume that $T'\in \SVRT(ws_r,\f_0)$ is a Rothe tableau in the equivalence class $C$. Then $T'$ differs from $T$ only possibly in the squares of $P(T)$. Note that if $P(T,i)$ is nonempty, then we must have $i\geq r$. Moreover, since the integers appearing in $r$-th row of $T'$ cannot exceed $r$, it follows that for any square $B\in P(T,r)$, $T'(B)$ does not contain $r+1$. Thus, for $B\in P(T,r)$, $r\in T(B)=T'(B)$ and $r+1\notin T(B)=T'(B)$.
Before we proceed, we give an illustration of the configuration of the squares in the first $r+1$ rows of $D(w)$ and $D(ws_r)$, which will be helpful to analyze the contributions of the integer $r$ in the squares of $P(T,r)$. Notice that $D(w)$ is obtained from $D(ws_r)$ by deleting the square $(r,w_r)$ and then moving each square in row $r$, that lies to the right of $(r,w_r)$, down to row $r+1$. Since $r$ is the first ascent of $w$, the first $r+1$ rows of $D(w)$ and $D(ws_r)$ are as depicted in Figure \[zao\], where the square $(r,w_r)$ of $D(ws_r)$ is signified by a symbol $\heartsuit$.
(270,60) (0,0)[(1,0)[20]{}]{}(20,0)[(0,1)[20]{}]{} (20,20)[(1,0)[20]{}]{}(40,20)[(0,1)[10]{}]{} (40,30)[(1,0)[20]{}]{}(60,30)[(0,1)[10]{}]{} (60,40)[(1,0)[30]{}]{}(90,40)[(0,1)[10]{}]{} (90,50)[(-1,0)[90]{}]{}(0,0)[(0,1)[50]{}]{}
(30,0)[(1,0)[10]{}]{}(30,0)[(0,1)[10]{}]{} (40,0)[(0,1)[10]{}]{}(30,10)[(1,0)[10]{}]{}
(50,0)[(1,0)[10]{}]{}(50,10)[(1,0)[10]{}]{} (50,0)[(0,1)[10]{}]{}(60,0)[(0,1)[10]{}]{}
(70,0)[(1,0)[20]{}]{}(70,10)[(1,0)[20]{}]{} (70,0)[(0,1)[10]{}]{}(80,0)[(0,1)[10]{}]{} (90,0)[(0,1)[10]{}]{}
(100,0)[(1,0)[10]{}]{}(100,10)[(1,0)[10]{}]{} (100,0)[(0,1)[10]{}]{}(110,0)[(0,1)[10]{}]{}
(-9,15)(-5.5,15)(-2,15)(-17,13)[ $r$]{}
(95,45)(65,35) (45,25)(115,5) (25,15)
(160,0)[(1,0)[20]{}]{}(180,0)[(0,1)[20]{}]{} (180,20)[(1,0)[20]{}]{}(200,20)[(0,1)[10]{}]{} (200,30)[(1,0)[20]{}]{}(220,30)[(0,1)[10]{}]{} (220,40)[(1,0)[30]{}]{}(250,40)[(0,1)[10]{}]{} (250,50)[(-1,0)[90]{}]{}(160,0)[(0,1)[50]{}]{}
(180,10)[(1,0)[10]{}]{}
(190,10)[(1,0)[10]{}]{}(190,20)[(1,0)[10]{}]{} (190,10)[(0,1)[10]{}]{}(200,10)[(0,1)[10]{}]{}
(210,10)[(1,0)[10]{}]{}(210,20)[(1,0)[10]{}]{} (210,10)[(0,1)[10]{}]{}(220,10)[(0,1)[10]{}]{}
(230,10)[(1,0)[20]{}]{}(230,20)[(1,0)[20]{}]{} (230,10)[(0,1)[10]{}]{}(240,10)[(0,1)[10]{}]{} (250,10)[(0,1)[10]{}]{}
(260,10)[(1,0)[10]{}]{}(260,20)[(1,0)[10]{}]{} (260,10)[(0,1)[10]{}]{}(270,10)[(0,1)[10]{}]{}
(151,15)(155.5,15)(158,15)(143,13)[ $r$]{}
(255,45)(225,35) (205,25)(275,15) (185,5)
(182,12)[$\heartsuit$]{}
Obviously, the first $w_r-1$ squares in the $r$-th row (respectively, ($r+1$)-th row) of $T$ are filled with the set $\{r\}$ (respectively, $\{r+1\}$). This implies that each square in the ($r+1$)-th row of $D(ws_r)$ belongs to $Q(T)$ and the set $P(T, r+1)$ is empty. Therefore, the contribution of the $r$’s in squares of $P(T,r)$ to $G_{ws_r}(C; \x,\y)$ is $$\begin{aligned}
\label{ft}
\prod_{(r,j)\in P(T,r)}(-x_r\oplus y_{m_{rj}(ws_r)}).\end{aligned}$$
On the other hand, the contribution of the $r$’s and $(r+1)$’s in squares of $P(T,i)$ for $i>r+1$ to $G_{ws_r}(C; \x,\y)$ is $$\begin{aligned}
F_C(\x,\y)=&\sum_{T'\in C}\ \prod_{i> r+1}\prod_{(i,j)\in P^+(T',i)}(-x_r\oplus y_{m_{ij}(ws_r)+i-r})\nonumber\\[5pt]
&\ \ \cdot\prod_{(i,j)\in P^-(T',i)} \ (-x_{r+1}\oplus y_{m_{ij}(ws_r)+i-r-1}),\label{tie}\end{aligned}$$ where $P^+(T',i)$ (respectively, $P^-(T',i)$) denotes the subset of $P(T',i)$ consisting of squares containing $r$ (respectively, $r+1$). Moreover, the contribution of the $r$’s and $(r+1)$’s in squares of $Q(T)$ to $G_{ws_r}(C; \x,\y)$ is $$\begin{aligned}
\label{Qcontri}
R_C(\x,\y)=\prod_{(i,j)\in Q^+(T)}(-x_r\oplus y_{m_{ij}(ws_r)+i-r})\prod_{(i,j)\in Q^-(T)}(-x_{r+1}\oplus y_{m_{ij}(ws_r)+i-r-1}),\end{aligned}$$ where $Q^{+}(T)$ (respectively, $Q^-(T)$) denotes the subset of $Q(T)$ consisting of the squares containing $r$ (respectively, $r+1$). Consequently, we obtain that $$\begin{aligned}
G_{ws_r}(C; \x,\y)=&(-1)^{\ell(ws_r)}\left( \prod_{(i,j)\in D(ws_r)}\prod_{t\in T(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(ws_r)+i-t})\right)\nonumber \\[5pt]
&\ \ \cdot\prod_{(r,j)\in P(T,r)}(-x_r\oplus y_{m_{rj}(ws_r)})\cdot F_C(\x,\y)\cdot R_C(\x,\y).\label{kong}\end{aligned}$$
Comparing with , in order to complete the proof, we need to show that $$\begin{aligned}
\label{fit}
\prod_{(r,j)\in P(T,r)}(-x_r\oplus y_{m_{rj}(ws_r)})&=\prod_{j=1}^{b_r(T)}(-x_r\oplus y_{\ell_r(T)+j}),\\[5pt]
F_C(\x,\y)&=\prod_{ i>r+1}h(C,i;\x,\y)=H_C(\x,\y),\label{f1}\\[5pt]
R_C(\x,\y)&=J_C(\x,\y).\label{f2fs}\end{aligned}$$
Let us first prove . To this end, we show that if there are two squares $(r,j_1)$ and $(r,j_2)$ in $P(T,r)$ with $j_1<j_2$ and there exists a square $(r,j)\in D(ws_r)$ for some $j_1<j<j_2$, then $(r,j)\in P(T,r)$. It suffices to prove the following claim.
[**Claim.**]{} For $(r,j)\in P(T, r)$, there do not exist squares $(r,k),(h,k)\in Q(T)$ such that $h>r+1$ and $k<j$.
To verify this claim, we construct a Rothe tableau $\overline{T}$ from $T$ such that $\overline{T}\in \SVRT(w, \f_0)$. Let $R$ be the set of squares of $D(ws_r)$ in row $r$ that are strictly to the right of $(r,w_r)$. Define $\overline{T}$ to be the tableau obtained from $T$ by deleting the square $(r,w_r)$ together with $T(r,w_r)$, and then moving each square $B$ in $R$, together with $T(B)$, down to row $r+1$. By construction, it is easy to check that $\overline{T}\in\SVRT(w, \f_0)$. Note that $(r,j)\in P(T)$ if and only if $(r+1,j)\in P(\overline{T})$. Applying Lemma \[left\] to $\overline{T}$, we see that if $(r+1,j)\in P(\overline{T})$, then there do not exist squares $(r,k),(h,k)\in Q(\overline{T})$ with $h>r+1$ and $k<j$. Since $Q(T)=Q(\overline{T})$, we conclude the claim.
By the above claim, the configuration of the squares of $P(T)$ and $Q(T)$ in the $r$-th row of $D(ws_r)$ is as illustrated in Figure \[anm11\], where the squares in $Q^+(T)$ (respectively, $Q^-(T)$) are marked with a $\ast$ (respectively, $\star$). In view of the definition $m_{i,j}(ws_r)$ in as well as the definition $\ell_r(T)$ in , we see that holds.
(185,45) (-10,25)[(1,0)[10]{}]{}(-10,25)[(0,1)[10]{}]{} (0,25)[(0,1)[10]{}]{}(-10,35)[(1,0)[10]{}]{}
(-10,15)[(1,0)[10]{}]{}(-10,15)[(0,1)[10]{}]{} (0,15)[(0,1)[10]{}]{}(-10,25)[(1,0)[10]{}]{}
(20,25)[(1,0)[10]{}]{}(20,25)[(0,1)[10]{}]{} (30,25)[(0,1)[10]{}]{}(20,35)[(1,0)[10]{}]{}
(20,15)[(1,0)[10]{}]{}(20,15)[(0,1)[10]{}]{} (30,15)[(0,1)[10]{}]{}(20,25)[(1,0)[10]{}]{}
(30,25)[(1,0)[10]{}]{}(30,25)[(0,1)[10]{}]{} (40,25)[(0,1)[10]{}]{}(30,35)[(1,0)[10]{}]{}
(60,25)[(1,0)[10]{}]{}(60,25)[(0,1)[10]{}]{} (70,25)[(0,1)[10]{}]{}(60,35)[(1,0)[10]{}]{}
(80,25)[(1,0)[10]{}]{}(80,25)[(0,1)[10]{}]{} (90,25)[(0,1)[10]{}]{}(80,35)[(1,0)[10]{}]{}
(110,25)[(1,0)[10]{}]{}(110,25)[(0,1)[10]{}]{} (120,25)[(0,1)[10]{}]{}(110,35)[(1,0)[10]{}]{}
(130,25)[(1,0)[10]{}]{}(130,25)[(0,1)[10]{}]{} (140,25)[(0,1)[10]{}]{}(130,35)[(1,0)[10]{}]{}
(35,23)[(1,-1)[2]{}]{}(37,21)[(1,0)[46]{}]{} (83,21)[(1,-1)[2]{}]{}(85,19)[(1,1)[2]{}]{} (85,19)[(1,1)[2]{}]{}(87,21)[(1,0)[46]{}]{} (133,21)[(1,1)[2]{}]{} (71,10)[$P(T,r)$]{}
(160,25)[(1,0)[10]{}]{}(160,25)[(0,1)[10]{}]{} (170,25)[(0,1)[10]{}]{}(160,35)[(1,0)[10]{}]{}
(160,-10)[(1,0)[10]{}]{}(160,-10)[(0,1)[10]{}]{} (170,-10)[(0,1)[10]{}]{}(160,0)[(1,0)[10]{}]{}
(190,25)[(1,0)[10]{}]{}(190,25)[(0,1)[10]{}]{} (200,25)[(0,1)[10]{}]{}(190,35)[(1,0)[10]{}]{}
(190,-10)[(1,0)[10]{}]{}(190,-10)[(0,1)[10]{}]{} (200,-10)[(0,1)[10]{}]{}(190,0)[(1,0)[10]{}]{}
(5,28)[$\cdots$]{}(5,18)[$\cdots$]{} (175,28)[$\cdots$]{}(175,-7)[$\cdots$]{} (95,28)[$\cdots$]{}
(-7,28)[$\ast$]{}(-7,18)[$\star$]{} (23,28)[$\ast$]{}(23,18)[$\star$]{}
(163,-7)[$\star$]{}(163,28)[$\ast$]{} (193,-7)[$\star$]{}(193,28)[$\ast$]{}
We next prove . For $i>r$, by Lemma \[right\] and Lemma \[left\], the configuration of the squares of $P(T)$ and $Q(T)$ must be as illustrated as in Figure \[anm\]. In particular, every square in row $i$ of $D(ws_r)$ that lies between the leftmost square and the rightmost square of $P(T,i)$ must belong to $P(T,i)$.
(200,70) (0,25)[(1,0)[10]{}]{}(0,25)[(0,1)[10]{}]{} (10,25)[(0,1)[10]{}]{}(0,35)[(1,0)[10]{}]{}
(0,50)[(1,0)[10]{}]{}(0,50)[(0,1)[10]{}]{} (10,50)[(0,1)[10]{}]{}(0,60)[(1,0)[10]{}]{}
(30,25)[(1,0)[10]{}]{}(30,25)[(0,1)[10]{}]{} (40,25)[(0,1)[10]{}]{}(30,35)[(1,0)[10]{}]{}
(30,50)[(1,0)[10]{}]{}(30,50)[(0,1)[10]{}]{} (40,50)[(0,1)[10]{}]{}(30,60)[(1,0)[10]{}]{}
(60,25)[(1,0)[10]{}]{}(60,25)[(0,1)[10]{}]{} (70,25)[(0,1)[10]{}]{}(60,35)[(1,0)[10]{}]{}
(80,25)[(1,0)[10]{}]{}(80,25)[(0,1)[10]{}]{} (90,25)[(0,1)[10]{}]{}(80,35)[(1,0)[10]{}]{}
(110,25)[(1,0)[10]{}]{}(110,25)[(0,1)[10]{}]{} (120,25)[(0,1)[10]{}]{}(110,35)[(1,0)[10]{}]{}
(130,25)[(1,0)[10]{}]{}(130,25)[(0,1)[10]{}]{} (140,25)[(0,1)[10]{}]{}(130,35)[(1,0)[10]{}]{}
(65,23)[(1,-1)[2]{}]{}(67,21)[(1,0)[31]{}]{} (98,21)[(1,-1)[2]{}]{}(100,19)[(1,1)[2]{}]{} (100,19)[(1,1)[2]{}]{}(102,21)[(1,0)[31]{}]{} (133,21)[(1,1)[2]{}]{} (88,10)[$P(T,i)$]{}
(160,25)[(1,0)[10]{}]{}(160,25)[(0,1)[10]{}]{} (170,25)[(0,1)[10]{}]{}(160,35)[(1,0)[10]{}]{}
(160,0)[(1,0)[10]{}]{}(160,0)[(0,1)[10]{}]{} (170,0)[(0,1)[10]{}]{}(160,10)[(1,0)[10]{}]{}
(190,25)[(1,0)[10]{}]{}(190,25)[(0,1)[10]{}]{} (200,25)[(0,1)[10]{}]{}(190,35)[(1,0)[10]{}]{}
(190,0)[(1,0)[10]{}]{}(190,0)[(0,1)[10]{}]{} (200,0)[(0,1)[10]{}]{}(190,10)[(1,0)[10]{}]{}
(15,53)[$\cdots$]{}(15,27)[$\cdots$]{} (175,27)[$\cdots$]{}(175,2)[$\cdots$]{} (95,28)[$\cdots$]{}
(3,28)[$\star$]{}(3,53)[$\ast$]{} (33,28)[$\star$]{}(33,53)[$\ast$]{}
(163,3)[$\star$]{}(163,28)[$\ast$]{} (193,3)[$\star$]{}(193,28)[$\ast$]{}
Assume that $T'$ is a Rothe tableau in $C$. Keep in mind that $P(T',i)=P(T,i)$. Then we have the following two cases.
Case 1: In $T'$, the first $k$ $(0\leq k\leq b_i(T))$ squares in $P(T,i)$ contain $r+1$, and the remaining $b_i(T)-k$ squares in $P(T,i)$ contain $r$. In this case, running over the Rothe tableaux $T'$ in $C$, the integers $r$ and $r+1$ in $P(T,i)$ contribute $$\begin{aligned}
\label{QF-1}
\sum_{k=0}^{b_i(T)}\prod_{j=1}^{k} (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k+1}^{b_i(T)} (-x_{r}\oplus y_{\ell_i(T)+j}).\end{aligned}$$
Case 2: In $T'$, the first $k-1$ $(1\leq k\leq b_i(T))$ squares in $P(T,i)$ contain $r+1$, the $k$-th square contains both $r$ and $r+1$, and the remaining $b_i(T)-k$ squares in $P(T,i)$ contain $r$. In this case, running over the Rothe tableaux $T'$ in $C$, the integers $r$ and $r+1$ in $P(T,i)$ contribute $$\begin{aligned}
\label{QF-2}
\sum_{k=1}^{b_i(T)}\prod_{j=1}^{k} (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k}^{b_i(T)} (-x_{r}\oplus y_{\ell_i(T)+j}).\end{aligned}$$ Combining and , we see that Case 1 and Case 2 together contribute the factor $h(C,i;\x,\y)$ as defined in to the summation $F_C(\x,\y)$ in . Running over the row indices $i$ with $i>r+1$ yields .
Finally, we verify . For each $(i,j)\in Q^+(T)$, we use $(i',j)$ to denote the square in $Q^-(T)$ that lie in the same column as $(i,j)$. Then we have $$\begin{aligned}
\label{HD-1}
R_C(\x,\y)=\prod_{(i,j)\in Q^+(T)}(-x_r\oplus y_{m_{ij}(ws_r)+i-r})
(-x_{r+1}\oplus y_{m_{i'j}(ws_r)+i'-r-1}).\end{aligned}$$ Write $w'=ws_r$. We assert that $w'_t<j$ for $i<t<i'$. Suppose otherwise that $w'_t>j$. Since the square $(i',j)\in D(ws_r)$, we see that $(t,j)\in D(ws_r)$. Thus we have $r<\min T(t,j)\leq \max T(t,j)<r+1$, leading to a contradiction. This verifies the assertion. By the definition of $m_{ij}(w)$ in , it is easy to see that $$m_{ij}(w')=|\{(i,k)\in D(w')\,|\,k\le j\}|=|\{t>i\,|\,w'_t\le j\}|.$$ Therefore, by the above assertion, we obtain $$m_{ij}(ws_r)=m_{i'j}(ws_r)+i'-i-1,$$ and so that $$\begin{aligned}
\label{HD-2}
m_{i'j}(ws_r)+i'-r-1=m_{ij}(ws_r)+i-r.\end{aligned}$$ Putting into , we arrive at the equality in . This completes the proof.
The following theorem provides a formula for the polynomial $G_{w}(C'; \x,\y)$ for an equivalence class $C'\in\SVRT(w, \f_0)/\hspace{-.1cm}\sim$.
\[mm-2\] Let $w\neq w_0$ be a 1432-avoiding permutation, and $r$ be the first ascent of $w$. Assume that $C'\in\SVRT(w, \f_0)/\hspace{-.1cm}\sim$ and $T'$ is any given Rothe tableau in $C'$. Then, $$\begin{aligned}
G_{w}(C'; \x,\y)=&(-1)^{\ell(w)} \left(\prod_{(i,j)\in D(w)}\prod_{t\in T'(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(w)+i-t})\right)\nonumber \\[5pt]
&\ \cdot h(C',r+1;\x,\y)\cdot H_{C'}(\x,\y)\cdot J_{C'}(\x,\y),\label{gggg}\end{aligned}$$ where $${H}_{C'}(\x,\y)=\, \prod_{ i>r+1}{h}(C',i;\x,\y),$$ and $$J_{C'}(\x,\y)=\prod_{(i,j)\in Q^+(T')}(-x_r\oplus y_{m_{ij}(w)+i-r})(-x_{r+1}\oplus y_{m_{ij}(w)+i-r}).$$
[*Sketch of the proof.*]{} The proof is nearly the same as the arguments for Theorem \[mm-1\]. The only difference is to notice that $P(T',r)$ is empty and that the squares in $P(T',r+1)$ contributes the factor $h(C',r+1;\x,\y)$.
To finish the proof of Theorem \[main-s\], we need a one-to-one correspondence $\Phi$ between the two sets of equivalence classes: $$\Phi:\ \SVRT(ws_r,\f_0)/\hspace{-.1cm}\sim\ \longrightarrow\ \SVRT(w,\f_0)/\hspace{-.1cm}\sim.$$
[**Construction of the bijection $\Phi$**:]{} Assume that $C\in\SVRT(ws_r, \f_0)/\hspace{-.1cm}\sim$ and $T\in C$. Let $T'=\overline{T}\in\SVRT(w, \f_0)$ be the Rothe tableau as constructed in the proof of . That is, $T'$ is the Rothe tableau obtained from $T$ by deleting the square $(r,w_r)$ together with $T(r,w_r)$, and then moving each square $B$ in $R$, together with $T(B)$, down to row $r+1$, where $R$ is the set of squares of $D(ws_r)$ in row $r$ that are strictly to the right of $(r,w_r)$. Let $C'\in\SVRT(w, \f_0)/\hspace{-.1cm}\sim$ be the equivalence class containing $T'$. It is clear that $C'$ is independent of the choice of $T$. Set $\Phi(C)=C'$.
The inverse of $\Phi$ can be described as follows. Let $C'\in\SVRT(w, \f_0)/\hspace{-.1cm}\sim$ and $T'\in C'$. Let $T''$ be the Rothe tableau defined by setting $T''(B)=T'(B)$ if $B\in D(w)\setminus P(T',r+1)$, and setting $$\begin{aligned}
T''(B)=(T'(B)\setminus\{r,r+1\})\cup \{r\} \ \ \text{if $B\in P(T',r+1)$}.\end{aligned}$$ Notice that $T''\in\SVRT(w, \f_0)$. We define $T$ as the Rothe tableau obtained from $T''$ by adding the square $(r,w_r)$ filled with the set $\{r\}$, and then moving each square $B$ of $T''$ (together with the set $T''(B)$), which is to the right of the square $(r+1,w_r)$, up to row $r$. By construction, it is easily checked that $T\in\SVRT(ws_r, \f_0)$. Let $C$ be the equivalence class in $\SVRT(ws_r, \f_0)/\hspace{-.1cm}\sim$ containing $T$. Set $\Phi^{-1}(C')=C$.
Based on Theorem \[mm-1\] and Theorem \[mm-2\], we can establish the following relation.
\[equivalence\] Let $w\neq w_0$ be a 1432-avoiding permutation, and $r$ be the first ascent of $w$. For each equivalence class $C\in\SVRT(ws_r, \f_0)/\hspace{-.1cm}\sim$, we have $$\begin{aligned}
\label{F-3}
\pi_r G_{ws_r}(C; \x,\y)=G_w(\Phi(C); \x,\y).\end{aligned}$$
The proof of Theorem \[equivalence\] requires the following property concerning the operator $\pi_r$ due to Matsumura [@Matsumura-2].
\[Matsu\] For an arbitrary sequence $(a_1,a_2,\ldots, a_m)$ of positive integers, $$\begin{aligned}
\pi_r((x_r\oplus y_{a_1})\cdots (x_r\oplus y_{a_m}))=&\sum_{k=1}^m
\ \prod_{j=1}^{k-1}(x_r\oplus y_{a_j})\prod_{j=k+1}^{m}(x_{r+1}\oplus y_{a_j})\nonumber\\[5pt]
& -\sum_{k=1}^{m-1}\
\prod_{j=1}^{k}(x_r\oplus y_{a_j})\prod_{j=k+1}^{m}(x_{r+1}\oplus y_{a_j}).
\label{ppqq-1}\end{aligned}$$ Furthermore, the expression in is symmetric in $x_r$ and $x_{r+1}$.
[*Proof of Theorem \[equivalence\].*]{} Assume that $T$ is any given Rothe tableau in $C$. The polynomial $h(C,i;\x,\y)$ defined in has the following reformulation: $$\begin{aligned}
h(C,i;\x,\y)=&\sum_{k=1}^{b_i(T)+1}\, \prod_{j=1}^{k-1} (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k+1}^{b_i(T)+1} (-x_{r}\oplus y_{\ell_i(T)+j-1})\\[5pt]
&+\sum_{k=1}^{b_i(T)}\, \prod_{j=1}^k (-x_{r+1}\oplus y_{\ell_i(T)+j-1})
\prod_{j=k+1}^{b_i(T)+1} (-x_{r}\oplus y_{\ell_i(T)+j-1}).\end{aligned}$$ Hence $(-1)^{b_i(T)} h(C,i;\x,\y)$ coincides with the right-hand side of by setting $m=b_i(T)+1$ and setting $a_j=\ell_i(T)+j-1$ for $1\leq j\leq m$, and then exchanging the variables $x_r$ and $x_{r+1}$. It follows from Lemma \[Matsu\] that $$\begin{aligned}
\label{hpi}
h(C,i;\x,\y)&=(-1)^{b_i(T)}\cdot\pi_r
\left(\prod_{j=1}^{b_i(T)+1}(x_r\oplus y_{\ell_i(T)+j-1})\right)\nonumber\\
&=-\pi_r
\left(\prod_{j=1}^{b_i(T)+1}(-x_r\oplus y_{\ell_i(T)+j-1})\right),\end{aligned}$$ which is a symmetric polynomial in $x_r$ and $x_{r+1}$.
On the other hand, if a polynomial $f(\x)$ is symmetric in $x_r$ and $x_{r+1}$, then for any polynomial $g(\x)$, it is easily checked that $$\pi_r (f(\x) g(\x))= f(\x)\pi_r g(\x).$$ Therefore, applying $\pi_r$ to the formula of $G_{ws_r}(C; \x,\y)$ in Theorem \[mm-1\], we obtain that $$\begin{aligned}
\label{assd-1}
\pi_r G_{ws_r}(C; \x,\y)=&(-1)^{\ell(ws_r)} \left(\prod_{(i,j)\in D(ws_r)}\prod_{t\in T(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(ws_r)+i-t})\right)\nonumber\\[5pt]
&\ \cdot H_C(\x,\y)\cdot J_C(\x,\y) \cdot \pi_r \left(\prod_{j=1}^{b_r(T)}(-x_r\oplus y_{\ell_r(T)+j})\right).\end{aligned}$$
Let $T'\in\Phi(C)$ be any given Rothe tableau in the equivalent class of $\Phi(C)$. By the construction of $\Phi$, it is easy to see that $$\begin{aligned}
\label{assd-2}
\prod_{(i,j)\in D(w)}\prod_{t\in T(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(ws_r)+i-t})=\prod_{(i,j)\in D(w)}\prod_{t\in T'(i,j)\atop t\neq r,r+1}(-x_t\oplus y_{m_{ij}(w)+i-t}).\end{aligned}$$ Again, by the construction of $\Phi$, it is also clear that for $i>r+1$, $$b_i(T)=b_i(T')\ \ \ \ \text{and}\ \ \ \ \ell_i(T)=\ell_i(T'),$$ which imply that $$\begin{aligned}
H_C(\x,\y)=H_{C'}(\x,\y).\end{aligned}$$ Moreover, since $Q(T)=Q(T')$ and $m_{ij}(ws_r)=m_{ij}(w)$ for any $(i,j)\in Q^+(T)$, one has $$\begin{aligned}
J_C(\x,\y)=J_{C'}(\x,\y).\end{aligned}$$ Still, by the construction of $\Phi$, we see that $$b_r(T)=b_{r+1}(T')+1\ \ \ \ \text{and}\ \ \ \ \ell_r(T)=\ell_{r+1}(T')-1.$$ So, by , we have $$\begin{aligned}
\label{jfz}
\pi_r
\left(\prod_{j=1}^{b_{r}(T)}(-x_r\oplus y_{\ell_{r}(T)+j})\right)&=
\pi_r
\left(\prod_{j=1}^{b_{r+1}(T')+1}(-x_r\oplus y_{\ell_{r+1}(T')+j-1})\right)\nonumber\\[5pt]
&=-h(C',r+1;\x,\y).\end{aligned}$$
Substituting – into , we see that $\pi_r G_{ws_r}(C; \x,\y)=G_{w}(\Phi(C); \x,\y)$. This completes the proof.
By Theorem \[equivalence\] and the bijection $\Phi$, we arrive at a proof of Theorem \[main-s\]. Using induction on the length of $w$, we reach a proof of Theorem \[prop1\].
Proof of Theorem \[prop2\] {#sec4}
--------------------------
In this subsection, we confirm Theorem \[prop2\] by proving the following statement.
\[main3\] If $w$ contains a 1432 pattern, then $$\begin{aligned}
\label{aognb}
\S_{w}(\x)\neq\sum_{T\in {\rm SRT}(w,\f_0)}
\prod_{(i,j)\in D(w)}\prod_{t\in T({i,j})}x_t.\end{aligned}$$
By Corollary \[coro\], if $w$ is a 1432-avoiding permutation, then $\mathfrak{S}_{w}(\x)$ must equal the right-hand side of . Hence Theorem \[main3\] implies Theorem \[prop2\].
To finish the proof of Theorem \[main3\], we recall the balanced labeling model of Schubert polynomials given by Fomin, Greene, Reiner and Shimozono [@FoRe]. To a square $(i,j)$ in the Rothe diagram $ D(w)$, the associated hook $H_{i,j}(w)$ is the collection of squares $(i',j')$ of $D(w)$ such that either $i'=i$ and $j'\geq j$, or $i'\geq i$ and $j'=j$.
A labeling $L$ of $D(w)$ is an assignment of positive integers into the squares of $D(w)$ such that each square receives exactly one integer. We use $L(i,j)$ to denote the label in the square $(i,j)\in D(w)$. A labeling $L$ is called balanced if for every square $(i,j)\in D(w)$, the label $L(i,j)$ remains unchanged after rearranging the labels in the hook $H_{i,j}(w)$ so that they are weakly increasing from right to left and from top to bottom. Figure \[Rotheppp\] illustrates two balanced labelings for the permutation $w=25143$.
(300,55) (80,0)(105,0)(130,0)(80,10)(105,10)(130,10) (80,20)(105,20)(130,20)(80,30)(105,30)(130,30) (80,40)(105,40)(130,40)(80,50)(105,50)(130,50)
(80,0)(80,25)(80,50)(90,0)(90,25)(90,50) (100,0)(100,25)(100,50)(110,0)(110,25)(110,50) (120,0)(120,25)(120,50)(130,0)(130,25)(130,50)
(95,45)(125,35)(85,25) (115,15)(105,5)
(80,30)[(1,0)[10]{}]{}(80,40)[(1,0)[10]{}]{}(80,50)[(1,0)[10]{}]{} (80,30)[(0,1)[20]{}]{}(90,30)[(0,1)[20]{}]{}
(100,30)[(1,0)[20]{}]{}(100,40)[(1,0)[20]{}]{} (100,30)[(0,1)[10]{}]{}(110,30)[(0,1)[10]{}]{}(120,30)[(0,1)[10]{}]{}
(100,10)[(1,0)[10]{}]{}(100,20)[(1,0)[10]{}]{} (100,10)[(0,1)[10]{}]{}(110,10)[(0,1)[10]{}]{}
(83,42)[1]{}(83,32)[2]{}(103,32)[2]{}(113,32)[2]{} (103,12)[1]{}
(160,0)(185,0)(210,0)(160,10)(185,10)(210,10) (160,20)(185,20)(210,20)(160,30)(185,30)(210,30) (160,40)(185,40)(210,40)(160,50)(185,50)(210,50)
(160,0)(160,25)(160,50)(170,0)(170,25)(170,50) (180,0)(180,25)(180,50)(190,0)(190,25)(190,50) (200,0)(200,25)(200,50)(210,0)(210,25)(210,50)
(175,45)(205,35)(165,25) (195,15)(185,5)
(160,30)[(1,0)[10]{}]{}(160,40)[(1,0)[10]{}]{}(160,50)[(1,0)[10]{}]{} (170,30)[(0,1)[20]{}]{}(170,30)[(0,1)[20]{}]{}
(180,30)[(1,0)[20]{}]{}(180,40)[(1,0)[20]{}]{} (180,30)[(0,1)[10]{}]{}(190,30)[(0,1)[10]{}]{}(200,30)[(0,1)[10]{}]{}
(180,10)[(1,0)[10]{}]{}(180,20)[(1,0)[10]{}]{} (180,10)[(0,1)[10]{}]{}(190,10)[(0,1)[10]{}]{}
(163,42)[1]{}(163,32)[2]{}(183,32)[2]{}(193,32)[1]{} (183,12)[3]{}
A balanced labeling of $D(w)$ is said to be column strict if no column contains two equal labels. Let $\CSBL(w,\f_0)$ denote the set of column strict balanced labelings of $D(w)$ such that $L(i,j)\leq i$ for each square $(i,j)\in D(w)$. Fomin, Greene, Reiner and Shimozono [@FoRe] showed that $$\begin{aligned}
\label{Relate-Ba-2}
\mathfrak{S}_{w}(\x)=\sum_{L\in \CSBL(w,\f_0)} \prod_{(i,j)\in D(w)} x_{L(i,j)}.\end{aligned}$$
We are now in a position to give a proof of Theorem \[main3\].
[*Proof of Theorem \[main3\].*]{} Assume that $w$ is a permutation that contains a pattern 1432. Recall that ${\rm SRT}(w,\f_0)$ is the set of single-valued Rothe tableaux of shape $D(w)$ flagged by $\f_0$. By definition, it is clear that a Rothe tableau in $\SRT(w,\f_0)$ is a column strict balanced labeling, and hence belongs to $\CSBL(w,\f_0)$. In view of , to prove , it suffices to show that there exits a balanced labeling in $\CSBL(w,\f_0)$ that does not belong to $\SRT(w,\f_0)$. We next construct such a balanced labeling $L$ in $\CSBL(w,\f_0)$.
Suppose that the subsequence $w_{i_1}w_{i_2}w_{i_3}w_{i_4}$ of $w$ has the same relative order as the pattern 1432, that is, $w_{i_1}<w_{i_4}<w_{i_3}<w_{i_2}$. Since $w_{i_3}>w_{i_4}$, there exists at least one square in the $i_3$-th row of $D(w)$. Let $(i_3,j)$ be the rightmost square in this row. Let $$S=\{(i,j)\,|\,(i,j)\in D(w), i_1\leq i\leq i_3\}$$ be the subset of $D(w)$ consisting of the squares in column $j$ lying between row $i_1$ and row $i_3$. We classify $S$ into two subsets according to whether a square $(i,j)\in S$ is the rightmost square in the row or not. Let $S_1\subseteq S$ consists of square $(i,j)\in S$ such that $(i,j)$ is the rightmost square in row $i$. Clearly, $S_1$ is nonempty since it contains the square $(i_3,j)$. Let $S_2=S\backslash S_1$ be the complement. Since $w_{i_2}>w_{i_3}$, we see that the two squares $(i_2, j),(i_2, w_{i_3})$ belong to $D(w)$. Hence $(i_2, j)\in S_2$, and so $S_2$ is also nonempty.
Let us use an example in Figure \[Exam\] to illustrate the sets $S_1$ and $S_2$. In this example, $w=1\,4\,5\,9\,6\,10\,7\,8\,2\,3$ and the subsequence $w_1w_6w_7w_9$ forms a 1432-pattern. The rightmost square of $D(w)$ in the $i_3$-th row is the square $(7,3)$, and so we have $$S=\{(i,3)\,|\, i=2,3,4,5,6,7\}.$$ Moreover, the squares belonging to $S_1$ and $S_2$ are signified with $\spadesuit$ and $\clubsuit$ in Figure \[Exam\](a), respectively.
(250,105) (0,0)(50,0)(100,0)(0,10)(50,10)(100,10) (0,20)(50,20)(100,20)(0,30)(50,30)(100,30) (0,40)(50,40)(100,40)(0,50)(50,50)(100,50) (0,60)(50,60)(100,60)(0,70)(50,70)(100,70) (0,80)(50,80)(100,80)(0,90)(50,90)(100,90) (0,100)(50,100)(100,100)
(0,0)(0,50)(0,100)(10,0)(10,50)(10,100) (20,0)(20,50)(20,100)(30,0)(30,50)(30,100) (40,0)(40,50)(40,100)(50,0)(50,50)(50,100) (60,0)(60,50)(60,100)(70,0)(70,50)(70,100) (80,0)(80,50)(80,100)(90,0)(90,50)(90,100) (100,0)(100,50)(100,100)
(5,95)(15,15) (25,5)(35,85) (45,75)(55,55) (65,35)(75,25) (85,65)(95,45)
(10,20)[(1,0)[20]{}]{}(10,30)[(1,0)[20]{}]{} (10,40)[(1,0)[20]{}]{}(10,50)[(1,0)[20]{}]{} (10,60)[(1,0)[20]{}]{}(10,70)[(1,0)[20]{}]{} (10,80)[(1,0)[20]{}]{}(10,90)[(1,0)[20]{}]{}
(10,20)[(0,1)[70]{}]{}(20,20)[(0,1)[70]{}]{} (30,20)[(0,1)[70]{}]{}
(50,60)[(1,0)[30]{}]{}(50,70)[(1,0)[30]{}]{}
(50,60)[(0,1)[10]{}]{}(60,60)[(0,1)[10]{}]{} (70,60)[(0,1)[10]{}]{}(80,60)[(0,1)[10]{}]{}
(60,40)[(1,0)[20]{}]{}(60,50)[(1,0)[20]{}]{}
(60,40)[(0,1)[10]{}]{}(70,40)[(0,1)[10]{}]{} (80,40)[(0,1)[10]{}]{}
(-7,93)(-7,43) (-7,33)(-7,13)
(22, 83)[[$\spadesuit$]{}]{}(22, 73)[[$\spadesuit$]{}]{} (22, 63)[[$\clubsuit$]{}]{}(22, 53)[[$\spadesuit$]{}]{} (22, 43)[[$\clubsuit$]{}]{}(22, 33)[[$\spadesuit$]{}]{}
(45,-10)[(a)]{}
(150,0)(200,0)(250,0)(150,10)(200,10)(250,10) (150,20)(200,20)(250,20)(150,30)(200,30)(250,30) (150,40)(200,40)(250,40)(150,50)(200,50)(250,50) (150,60)(200,60)(250,60)(150,70)(200,70)(250,70) (150,80)(200,80)(250,80)(150,90)(200,90)(250,90) (150,100)(200,100)(250,100)
(150,0)(150,50)(150,100)(160,0)(160,50)(160,100) (170,0)(170,50)(170,100)(180,0)(180,50)(180,100) (190,0)(190,50)(190,100)(200,0)(200,50)(200,100) (210,0)(210,50)(210,100)(220,0)(220,50)(220,100) (230,0)(230,50)(230,100)(240,0)(240,50)(240,100) (250,0)(250,50)(250,100)
(155,95)(165,15) (175,5)(185,85) (195,75)(205,55) (215,35)(225,25) (235,65)(245,45)
(160,20)[(1,0)[20]{}]{}(160,30)[(1,0)[20]{}]{} (160,40)[(1,0)[20]{}]{}(160,50)[(1,0)[20]{}]{} (160,60)[(1,0)[20]{}]{}(160,70)[(1,0)[20]{}]{} (160,80)[(1,0)[20]{}]{}(160,90)[(1,0)[20]{}]{}
(160,20)[(0,1)[70]{}]{}(170,20)[(0,1)[70]{}]{} (180,20)[(0,1)[70]{}]{}
(200,60)[(1,0)[30]{}]{}(200,70)[(1,0)[30]{}]{}
(200,60)[(0,1)[10]{}]{}(210,60)[(0,1)[10]{}]{} (220,60)[(0,1)[10]{}]{}(230,60)[(0,1)[10]{}]{}
(210,40)[(1,0)[20]{}]{}(210,50)[(1,0)[20]{}]{}
(210,40)[(0,1)[10]{}]{}(220,40)[(0,1)[10]{}]{} (230,40)[(0,1)[10]{}]{}
(143,93)(143,43) (143,33)(143,13) (143,53)
(172, 83)[[$\diamondsuit$]{}]{}(172, 73)[[$\diamondsuit$]{}]{} (172, 53)[[$\diamondsuit$]{}]{}
(195,-10)[(b)]{}
Let $i_0$ be the smallest row index such that: (1) the square $(i_0,j)\in S_1$; (2) there exists a square in $S_2$ lying above $(i_0,j)$. Such an row index exists since the $i_3$-th row satisfies the above conditions. Let $S'=\{(i,j)\in S_1\,|\,i\le i_0\}$ be the subset of $S_1$ including the squares above $(i_0,j)$. In the example in Figure \[Exam\], we see that $i_0=5$ and the squares of $S'$ are signified with the symbol $\diamondsuit$.
Assume that $|S'|=k$ and $(r_1,j), \ldots, (r_k,j)$ are the squares of $S'$, where $r_1<\cdots<r_k=i_0$. Note that $r_1>i_1$. This is because $w_{i_1}$ is the smallest element of $\{w_{i_1},w_{i_2},w_{i_3},w_{i_4}\}$ and thus the square $(i_1,j)\notin D(w)$.
We now construct a balanced labeling $L$ of $D(w)$ as follows. If a square $(s,t)$ of $D(w)$ is not contained in $S'$, then we set $L(s,t)=s$. For the squares $(r_1,j), \ldots, (r_k,j)$ of $S'$, we set $L(r_1,j)=i_1$ and $L(r_{p},j)=r_{p-1}$ for $p=2,\ldots,k$. For the permutation in Figure \[Exam\], the labeling $L$ is given in Figure \[Exams\], where the integers in $S'$ are written in boldface.
(100,105) (0,0)(50,0)(100,0)(0,10)(50,10)(100,10) (0,20)(50,20)(100,20)(0,30)(50,30)(100,30) (0,40)(50,40)(100,40)(0,50)(50,50)(100,50) (0,60)(50,60)(100,60)(0,70)(50,70)(100,70) (0,80)(50,80)(100,80)(0,90)(50,90)(100,90) (0,100)(50,100)(100,100)
(0,0)(0,50)(0,100)(10,0)(10,50)(10,100) (20,0)(20,50)(20,100)(30,0)(30,50)(30,100) (40,0)(40,50)(40,100)(50,0)(50,50)(50,100) (60,0)(60,50)(60,100)(70,0)(70,50)(70,100) (80,0)(80,50)(80,100)(90,0)(90,50)(90,100) (100,0)(100,50)(100,100)
(5,95)(15,15) (25,5)(35,85) (45,75)(55,55) (65,35)(75,25) (85,65)(95,45)
(10,20)[(1,0)[20]{}]{}(10,30)[(1,0)[20]{}]{} (10,40)[(1,0)[20]{}]{}(10,50)[(1,0)[20]{}]{} (10,60)[(1,0)[20]{}]{}(10,70)[(1,0)[20]{}]{} (10,80)[(1,0)[20]{}]{}(10,90)[(1,0)[20]{}]{}
(10,20)[(0,1)[70]{}]{}(20,20)[(0,1)[70]{}]{} (30,20)[(0,1)[70]{}]{}
(50,60)[(1,0)[30]{}]{}(50,70)[(1,0)[30]{}]{}
(50,60)[(0,1)[10]{}]{}(60,60)[(0,1)[10]{}]{} (70,60)[(0,1)[10]{}]{}(80,60)[(0,1)[10]{}]{}
(60,40)[(1,0)[20]{}]{}(60,50)[(1,0)[20]{}]{}
(60,40)[(0,1)[10]{}]{}(70,40)[(0,1)[10]{}]{} (80,40)[(0,1)[10]{}]{}
(-7,93)(-7,43) (-7,33)(-7,13)
(13,82)[2]{}(13,72)[3]{} (13,62)[4]{}(13,52)[5]{} (13,42)[6]{}(13,32)[7]{} (13,22)[8]{}
(23,82)[[**1**]{}]{}(23,72)[[**2**]{}]{} (23,62)[4]{}(23,52)[[**3**]{}]{} (23,42)[6]{}(23,32)[7]{} (23,22)[8]{}
(53,62)[4]{}(63,62)[4]{}(73,62)[4]{}
(63,42)[6]{}(73,42)[6]{}
By the construction of $L$, it is not hard to check that $L$ is a column strict balanced labeling in $\CSBL(w, \f_0)$. Moreover, the entries in the $j$-th column of $L$ are not increasing. So $L$ does not belong to $\SRT(w,\f_0)$. This completes the proof.
Rothe tableau complexes {#sec5}
=======================
In this section, we prove the tableau formulas in Theorem \[3-1\]. To do this, we investigate the structure of Rothe tableau complexes, which is a specific family of the tableau complexes as introduced by Knutson, Miller and Yong [@KnMiYo-2]. Using Theorem \[main\] and the properties of tableau complexes established in [@KnMiYo-2], we obtain two alternative tableau formulas for the Grothendieck polynomials of 1432-avoiding permutations, as given in Theorem \[3-1\].
Let us proceed with a brief review of the Hilbert series of the Stanley-Reisner ring of a simplicial complex, see [@MiSt; @Stan] for more detailed information. An (abstract) simplicial complex $\Delta$ on a finite vertex set $V$ is a collection of subsets of $V$ such that if $\sigma\in \Delta$ and $\tau\subseteq \sigma$, then $\tau\in \Delta$. Each subset $\sigma\in \Delta$ is called a face of $\Delta$. A face $\sigma$ is called a facet of $\Delta$ if $\sigma$ is not a subset of any other faces. Clearly, $\Delta$ is determined by its facets.
Let $\mathds{k}[\mathbf{t}]$ be the ring of polynomials over a field $\mathds{k}$ in the variables $t_v$ where $v\in V$. The Stanley-Reisner ideal $I_{\Delta}$ is the ideal generated by the monomials corresponding to the subsets of $V$ that are not faces of $\Delta$, namely, $$I_\Delta=\left<\prod_{v\in \tau}t_v\,|\, \text{$\tau\subseteq V$, but $\tau\not\in \Delta$}\right>.$$ The Stanley-Reisner ring of $\Delta$, denoted $\mathds{k}[\Delta]$, is the quotient ring $\mathds{k}[\mathbf{t}]/ I_\Delta$. The Hilbert series $H(\mathds{k}[\Delta]; \mathbf{t})$ of $\mathds{k}[\mathbf{t}]/ I_\Delta$ is equal to the sum of monomials not belonging to $I_{\Delta}$. It is well known [@MiSt; @Stan] that $H(\mathds{k}[\Delta]; \mathbf{t})$ has the following formula: $$H(\mathds{k}[\Delta]; \mathbf{t})=
\frac{K(\mathds{k}[\Delta];\mathbf{t})}{\prod_{v\in V}(1-t_v)},$$ where $$K(\mathds{k}[\Delta];\mathbf{t})=\sum_{\sigma\in \Delta}\prod_{v\in \sigma}t_v\prod_{v\not\in \sigma }(1-t_v).$$ The numerator $K(\mathds{k}[\Delta];\mathbf{t})$ is called the $K$-polynomial of $\mathds{k}[\Delta]$.
We now restrict attention to the $K$-polynomials of tableau complexes introduced in [@KnMiYo-2]. Let $X$ and $Y$ be two finite sets. A map $f$ from $X$ to $Y$ is called a tableau, which can be viewed as an assignment of elements of $Y$ to elements of $X$ such that each $x\in X$ receives exactly one element of $Y$. A tableau $f$ can also be identified with the following set $$\{(x\mapsto y)\,|\, \text{$x\in X$ and $f(x)=y$}\}\subseteq X\times Y$$ of ordered pairs. Let $U$ be a subset of tableaux from $X$ to $Y$, and let $E\subseteq X\times Y$ be a set of ordered pairs such that $f\subseteq E$ for each $f\in U$. The tableau complex corresponding to $U$ and $E$, denoted $\Delta_E(X\xrightarrow{U}Y)$, can be defined as follows. Let us first define a simplex $\Delta_E$. For each pair $(x\mapsto a)\in E$, write $(x\arrownot\mapsto y)=E\setminus \{(x\mapsto y)\}$ for the complement of $\{(x\mapsto y)\}$, and let $$V=\{(x\arrownot\mapsto y)\,|\, (x\mapsto y)\in E\}.$$ Denote by $\Delta_E$ the simplex with vertex set $V$, that is, $\Delta_E$ is the collection of all of the subsets of $V$.
Let $F \subseteq V$ be a face of $\Delta_E$. Assume that $F$ has $k$ vertices $ (x_1\arrownot\mapsto y_1),\ldots, (x_k\arrownot\mapsto y_k)$. Then $F$ can be identified with the following subset of $E$: $$E\setminus \{(x_i \mapsto y_i)\,|\, 1\leq i\leq k\}.$$ On the other hand, each subset of $E$ can be viewed as a set-valued tableau from $X$ to $Y$, that is, a map that assigns each element of $X$ with a subset of $Y$. To be more specific, for a subset $A$ of $E$, the corresponding set-valued tableau is defined by assigning $x\in X$ with the subset $\{y\in Y\,|\,
(x \mapsto y)\in A\}$. So the face $F$ of $\Delta_E$ can also be identified with a set-valued tableau such that for $x\in X$, $$F(x)=\{y\in Y\,|\, (x \mapsto y)\in E, \text{$(x \mapsto y)\neq (x_i \mapsto y_i)$ for $1\leq i\leq k$}\}.$$ From now on, a face $F$ of $\Delta_E$ can be identified either with a subset of $E$ or with a set-valued tableau from $X$ to $Y$, which will not cause confusion from the context. By the definition of $\Delta_E$, a vertex $(x\arrownot\mapsto y)\in V$ belongs to $F$ if and only if the pair $(x\mapsto y) $ does not belong to $F$.
Recall that $U$ is a set of tableaux from $X$ to $Y$ such that $f\subseteq E$ for each $f\in U$. So each tableau $f$ in $U$ is a face of $\Delta_E$. The tableau complex $\Delta_E(X\xrightarrow{U}Y)$ is defined as the subcomplex of $\Delta_E$ such that the facets of $\Delta_E(X\xrightarrow{U}Y)$ are the tableaux in $U$. This means that a set-valued tableau $F\subseteq E$ is a face of $\Delta_E(X\xrightarrow{U}Y)$ if and only if $F$ contains some tableau $f\in U$.
When $X$ and $Y$ are further endowed with partially ordered structures, Knutson, Miller and Yong [@KnMiYo-2] found three different expressions for the $K$-polynomial of a tableau complex.
\[KMY\] Let $X$ and $Y$ be two finite posets. For each $x\in X$, let $Y_x$ be a totally ordered subset of $Y$. Let $\Psi$ be a set of pairs $(x,x')$ in $X$ with $x<x'$. Let $U$ be the set of tableaux $f\colon X\rightarrow Y$ such that
- $f(x)\in Y_x$;
- $f$ is weakly order preserving, that is, if $x\leq x'$, then $f(x)\leq f(x')$;
- if $(x,x')\in \Psi$, then $f(x)<f(x')$.
Set $E=\bigcup_{f\in U}f$. Let $\mathbf{t}=\{t_{(x\arrownot\mapsto a)}\,|\, (x\arrownot\mapsto a)\in V\}$ be the set of variables corresponding to the vertices of the tableau complex $\Delta=\Delta_E(X\xrightarrow{U}Y)$. Then, $\Delta$ is homeomorphic to a ball or a sphere. Moreover, the corresponding $K$-polynomial has the following expressions.
- Let $U_1$ be the set of set-valued tableaux $F\subseteq E$ such that every tableau $f\subseteq F$ lies in $U$. Then, $$\label{KMY-1}
K(\mathds{k}[\Delta];\mathbf{t})=\sum_{F\in U_1} (-1)^{|F|-|X|} \prod_{x\in X}\prod_{a\in F(x)}\left(1-t_{(x\arrownot\mapsto a)}\right).$$
- Let $U_2$ be the set of set-valued tableaux $F\subseteq E$ each containing some tableau $f\in U$. Then, $$\label{KMY-2}
K(\mathds{k}[\Delta];\mathbf{t})=\sum_{F\in U_2} \prod_{x\in X}\left(\prod_{a\in F(x)}\left(1-t_{(x\arrownot\mapsto a)}\right)\prod_{a\in E(x)\setminus F(x)}t_{(x\arrownot\mapsto a)}
\right).$$
- Given a tableau $f\in U$ and $x\in X$, let $Y_f(x)$ be the set of $y\in Y$ such that $f(x)< y$ and moving the label on $x$ from $f(x)$ up to $y$ still yields a tableau in $U$. Then, $$\label{KMY-3}
K(\mathds{k}[\Delta];\mathbf{t})=\sum_{f\in U}
\prod_{x\in X}\left(\left(1-t_{(x\arrownot\mapsto f(x))}\right)\prod_{a\in Y_f(x)}t_{x\arrownot\mapsto a}\right).$$
We now consider the specific tableau complex such that the facets are the single-valued Rothe tableaux in $\SRT(w, \f_0)$. To be consistent with the aforementioned notation, let $X=D(w)$ and $Y$ be the set of positive integers. Set $U=\SRT(w, \f_0)$ and $$E=\bigcup _{T\in \SRT(w, \f_0)} T.$$ We denote the above defined tableau complex by $\Delta(w)=\Delta_E(X\xrightarrow{U}Y)$, and call $\Delta(w)$ the Rothe tableau complex for $w$.
Using Theorem \[main\] and Theorem \[KMY\], we can now give a proof of Theorem \[3-1\].
[*Proof of Theorem \[3-1\]*]{}. We define a partial ordering on $D(w)$ as follows. For two distinct squares $B$ and $B'$ of $D(w)$, we use $B\rightarrow B'$ to represent that either $B$ and $B'$ are in the same row and $B$ lies to the right of $B'$, or $B$ and $B'$ are in the same column and $B$ lies above $B'$. Define $B<B'$ if there exists a sequence $(B=B_1,B_2,\ldots,B_k=B')$ of squares of $D(w)$ such that $$B=B_1\rightarrow B_2\rightarrow \cdots \rightarrow B_k=B'.$$
For each square $B=(i,j)$ of $D(w)$, let $Y_B=\{1,2,\ldots,i\}$. Moreover, we set $\Psi$ to be the set of pairs $(B,B')$ with $B<B'$ such that $B$ and $B'$ are in the same column of $D(w)$. Now we see that the tableaux satisfying the conditions $(a)$, $(b)$ and $(c)$ in Theorem \[KMY\] are exactly the single-valued Rothe tableaux in $\SRT(w,\f_0)$. Recall that the set $U_1$ defined in Theorem \[KMY\] consists of the set-valued tableaux $F\subseteq E$ such that every tableau in $F$ lies in $U$. Clearly, $F\subseteq E$ is a set-valued tableau satisfying that every tableau contained in $F$ lies in $U$ if and only if $F$ is a set-valued Rothe tableau in $\SVRT(w,\f_0)$. Thus we have $U_1=\SVRT(w, \f_0)$. Replacing $t_{x\arrownot\mapsto a}$ with $x=(i,j)\in D(w )$ by $$\frac{x_a}{y_{m_{ij}(w)+i-a}}$$ and then replacing $x_t$ by $1-x_t$ and $y_t$ by $\frac{1}{1-y_t}$, the $K$-polynomial $ K(\mathds{k}[\Delta];\mathbf{t})$ in becomes $$\begin{aligned}
\sum_{T\in \SVRT(w, \f_0)} (-1)^{|T|-\ell(w)} \prod_{(i,j)\in D(w)}\prod_{t\in T(i,j)}
(x_t\oplus y_{m_{ij}(w)+i-t}),
\end{aligned}$$ which agrees with the formula in Theorem \[main\]. Making the same substitutions in and , we are led to and respectively. This completes the proof.
[**Acknowledgments.**]{} This work was supported by the 973 Project and the National Science Foundation of China.
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---
abstract: 'Nuclei of early-type dwarf galaxies (dEs) are usually younger than the galaxy main body, and such discrepancy in age has been a puzzle. To explore the origin of young nuclei in dEs, we study a sample of dEs having compact star-forming blobs that are visually similar to dEs’ nuclei but by far bluer. We find that (1) the compact star-forming blobs have a typical stellar mass of one percent of the host galaxy stellar mass; (2) some of the blobs are positioned slightly off from the center of the galaxies; (3) the H$\alpha$ equivalent width measured from the publicly available Sloan Digital Sky Survey fiber spectroscopy shows their formation ages being an order of few Mega-year; and (4) their emission line metallicities, 12+log(O/H), are as high as the solar value, while the underlying galaxies have the typical stellar populations of dEs, i.e., log(Z/Z$_{\sun}$)$\sim$$-0.8$. Based on the results, we argue that the central star-forming blobs can provide a caught-in-the-act view of nuclei formation in dEs, and discussing possible formation mechanisms of young nuclei in old dEs. We particularly propose that these off-centered compact star-forming regions may act as seeds of nuclei as proposed in the ‘wet migration’ scenario of [@Guillard16].'
author:
- Sanjaya Paudel
- 'Suk-Jin Yoon'
title: Starbursting Nuclei in Old Dwarf Galaxies
---
Introduction
============
From the ground- and space-based imaging of early-type dwarf galaxies (dEs), it has been shown that the central nuclei are common in these galaxies [@Cote06; @Lisker07; @Brok14]. In particular, around a stellar mass of 10$^{9}$ M$_{\sun}$, the nucleation fraction reaches $\gtrsim$ 90% [@Turner12; @Brok14; @Janssen19]. These central nuclei have typical sizes of a few to a few tens of parsecs with a mass of nearly 1% of their hosts [@Cote06; @Rossa06].
High-resolution imaging from the Hubble Space Telescope observations have revealed that dE nuclei are slightly bluer than the rest part of their host galaxies [@Lotz04; @Cote06]. Moreover, detailed stellar population studies from integrated light spectroscopy have shown that they are, on average, younger than the host galaxies’ stellar halos [@Chilingarian09; @Paudel11; @Guerou15; @Kacharov18]. The nuclei of dEs have size and mass ranges similar to those of bright globular clusters and ultra-compact dwarf galaxies, and thus it is natural to speculate that these objects are the remnant nuclei of tidally disrupted dEs [@Bekki03; @Paudel10; @Mieske13; @Seth14; @Vogge16].
Nuclei of dEs follow the same scaling relation of a galaxy’s mass and its central black hole mass, extending down to the low-mass systems where nuclei are common [@Ferrarese06]. In a number of dwarf galaxies, both black holes and nuclei are present at the galactic center [@Jiang11; @Reines13]. Whether black holes and nuclei form via the same mechanism or either of them serve as a seed for the other is still an open question.
The correlation between the nuclear mass and the host galaxy mass suggests a link between the formation histories of the nuclei and their galaxies, that is, their coevolution. Since the nuclei of galaxies sit at the depth of the galactic potential well, various phenomena such as supermassive black holes, active galactic nuclei, central starbursts, and extreme stellar densities may be associated with the nuclei [@Ferrarese06; @Boker10; @Seth06]. One topic of much recent interest is whether all dEs started with central nuclei and coevolved together, or nuclei formed at the substantially later part of host galaxy evolution [@Cen01; @Bekki07; @Guillard16]. In terms of size, luminosity, and overall frequency, dE nuclei are quite similar to the nuclear star clusters of late-type spirals, allowing us to speculate that dEs are environmentally transformed objects [@Cote06; @Georgiev14].
In this context, a detailed understanding of how nuclear clusters formed may reveal the general mechanism that drives the formation of massive central objects of all types. Two basic scenarios have been suggested: (1) nuclear clusters form via mergers of multiple globular clusters accreted through dynamical friction [@Tremaine75; @Oh00; @McLaughlin06; @Boker10], and (2) nuclear clusters form in situ from gas channeled into the centers of galaxies [@Milosavljevic04; @Bekki07]. However, in practice, these two formation scenarios do not need to be mutually exclusive. More recently, [@Guillard16] proposed a ‘wet migration’ scenario, where a massive star cluster forms near to the center, and it migrates to the center via a combination of interactions with other substructures and dynamical friction. In this work, we present a study of a sample of dEs that possess star-forming blobs near the galaxies’ isophotal center. We expect that the study of the central star-forming objects can provide a caught-in-the-act view of nuclei formation in low-mass galaxies.
The Sample
==========
The six galaxies analyzed in this work are drawn from the sample of our extended catalog of dEs in the nearby universe (z $<$ 0.01) prepared from the Sloan Digital Sky Survey (SDSS) database [@Paudel14]. We select dEs by visual inspection of both the SDSS color images and spectra [see @Paudel14], and these six galaxies in particular draw our attention. Unlike other dEs that contain relatively old or at least non-star-forming nuclei [@Paudel11], they possess a compact star-forming region near the center with no other nuclei, as we can see in the top panels of Figure \[samplefig\]. These compact star-forming regions mimic young nuclei similar to the star-forming nuclei found in gas-rich spiral galaxies [e.g., NGC 4395; @Georgiev14].
We know that some dEs indeed possess central star formation [e.g., @Lisker06; @Urich]. They are commonly known as the blue-centered early-type dwarf (dE\_bc), which possess extended blue regions (see Figure \[samplefig\], bottom panels). A standard definition of the nucleus of a dwarf galaxy is the existence of a luminosity excess over the main stellar distribution in the core region [@Janssen19]. In this work, we consider the presence of a compact star-forming blob in the core region as a young blue nucleus. This definition may not agree with the conventional definition for relatively old nuclei found in dEs. We are particularly interested in dEs with a compact star-forming blobs because they may be progenitors of young nuclei.
We provide the global properties of our sample galaxies in Table \[sampletab\]. We have chosen a redshift range of less than 0.005 or distances less than $\sim$20 Mpc. We use this distances limit because we intend to better resolve the compact star-forming regions. In the top panels of Figure \[samplefig\], we can see the prominent blue nuclei at the center, contrasting with the red-looking halos of the galaxies. For comparison, we also show some blue-centered dEs of [@Lisker06] in the bottom panels. They show more extended star-forming regions at the center, which are not as blue as the cores of our sample galaxies. Our sample galaxies are morphologically typical of dEs, and their $g-r$ colors ranging 0.8$\sim$1.0 ensure that they are at the red sequence of dwarf galaxies in the color$-$magnitude diagram [@Lisker08].
All galaxies but one are located at the outskirts of the Virgo cluster and have low relative line-of-sight radial velocities with respect to the cluster center galaxy, M87, which makes them likely members of the cluster. We thus consider a mean distance of the Virgo cluster (16.5 Mpc; [@Mei07]) as the distance to our galaxies. One, dE6, is located near to NGC 3945 group at a sky-projected distance of 780 kpc from the group center, and the relative line-of-sight radial velocity is as low as 155 kms$^{-1}$. We consider a distance of NGC 3945 (21.6 Mpc; [@Theureau07]) as the distance to dE6. The data show that none of these are located in a highly dense environment.
Data analysis
=============
Imaging
-------
The structural and photometric properties of host galaxies and their central nuclei are derived using the archival images available at the SDSS-III Data Archive Server [@Abazajian09]. The basic reduction procedures, such as bias-subtraction, flat-field-correction, sky-subtraction, and flux-calibration, were already performed in the archival images. To create an image mosaic with a desired field of view, we use a tool provided by the SDSS-III[^1]. We subtract the sky-background counts, following the same scheme as in our previous publication [@Paudel14].
For image analysis, we make extensive use of the $i$-band images, with which we derive the galaxies’ structural parameters, and use them as the references for other band images. To extract the galaxies’ major-axis light profiles, we use the IRAF $ellipse$ task. We first mask all non-related background and foreground objects manually, before running the $ellipse$ task. In the cases of the sizably off-centered nuclei, we also mask them. During the ellipse fit, the center, position angle, and ellipticity are allowed to vary.
The observed galaxy light profile is fitted to a Sérsic function using a $\chi^{2}$-minimization scheme. We skip the central 5 in the galaxy light profile to avoid contamination from nuclear light. The results of the best-fit Sérsic parameters (i.e., Sérsic indices and effective radii) are listed in Table \[sampletab\], columns 7 and 8, respectively. We show the best-fit model subtracted residual images in the upper panels of Figure \[resfig\]. The centers of the galaxies are obtained by averaging the centers of the ellipses from $R_{e}$/2 to $R_{e}$. In this range, the ellipse fit shows the center is already stable. To derive the centers of the nuclei, we use the $g$-band residual images with which we calculate their centroids. In the bottom panels of Figure \[resfig\], we show the $g-r$ color maps of our galaxies, where the black ellipses represent the best-fit ellipses at the half-light radius. We also identify the centers of the galaxies with black cross marks. The distance between the galaxy center and nuclei are listed in Table \[nuctab\], column 5.
We finally measure the total light in the nuclei from the residual images after subtracting the model galaxy light in the SDSS $g-$, $r-$, and $i-$band images. We derive the stellar masses of the nuclei from the $r$-band magnitudes and the mass-to-light ratios obtained from $g-r$ based on [@Bell03].
Spectroscopy
------------
We retrieve the optical spectra of the six dEs from the SDSS data archive. The SDSS spectra were observed with the spectrograph fed with the 3 fibers that correspond to a physical diameter of 240 pc at the distance of the Virgo cluster and 315 pc at the distance of NGC 3945. While carefully matching the positions of nuclei and the fibers, we found that the SDSS fibers were always positioned at the nuclei, rather than the galaxies’ geometric centroids (Figure \[resfig\], top panels).
The optical spectra of these galaxies show strong emission-lines and resemble a typical spectrum of a star-forming HII region. However, we attempt to derive the properties of the underlying old stellar populations using available information in the absorption features as well as the properties of star formation based on the emission-lines. We first fit the observed galaxy spectrum in a wavelength range from 4000 to 7000Å with simple stellar population models from [@Vazdekis10]. In doing so, prominent emission-line regions are masked. For this purpose, we use a publicly available full-spectrum fitting tool, the University of Lyon Spectroscopic Analysis Software [ULySS @Koleva09]. The fit is not satisfactory in the cases of dE1 and dE6, which have the strongest H$\alpha$ emission. We caution that the inferred parameters of the underlying old stellar populations should be considered qualitative. To derive them more precisely, the extraction of galaxy spectrum should be done significantly away from the central young nuclei [@Paudel11], where there is no contamination from the emission-lines.
Finally, we measure the emission-line fluxes by analyzing the stellar-continuum-subtracted spectra. In the cases of dE1 and dE6, we directly measure the emission-line fluxes from their original SDSS spectra because, as mentioned above, the fitting of the old stellar population is not satisfactory. We derive oxygen abundances, 12 + log(O/H), using the O3N2 method [@Marino13]. The H$\alpha$ emission-line flux is converted to a star-formation rate (SFR) using the calibration of [@Kennicutt98]. We find that the Balmer decrement ($c$ = H$\alpha$/H$\beta$) ranges from 3.0 to 3.5, which indicates a mild effect of dust reddening. The estimated values of 12 + log(O/H), SFR and $c$ are listed in Table \[nuctab\], columns 7, 8, and 9, respectively.
Results
=======
The observed colors of nuclei are significantly blue and they all have a negative value of $g-r$. Given that the nuclei have a narrow range of the $M_r$ magnitude ($-12.7$$\sim$$-13.5$), it is not surprising that their stellar masses of nuclei are similar ($\sim$5$\times$10$^{6}$ M$_{\sun}$). Note, however, that the estimated values of stellar mass are rather uncertain since we use a single optical color ($g-r$) to calculate the mass-to-light ratio [@Zhang17].
In Figure \[resfig\], the position of the galaxy center (black cross mark) does not always overlap with the bluest position of the map, suggesting some of the nuclei are off-centered. A half of our sample have the galactocentric distances of the nuclei that are larger than the observational point-spread functions (see Table \[nuctab\], columns 5 and 6).
From H$\alpha$, we can get a rough estimate of the star-formation age. For this, we use the Starburst99 model [@Leitherer99], assuming an instantaneous burst of star formation and a solar value of metallicity. For equivalent width EW(H$\alpha$) higher than 100 Å, the ages of star formation are estimated to be less than 10 Myr. Simple estimation of the stellar mass accumulated over the periods of star formation, assuming a solar value of metallicity, shows that the total stellar mass is two orders of magnitude less compared to the photometrically calculated stellar mass. There are many reasons for this discrepancy. As mentioned above, the use of a single optical color to calculate the mass-to-light ratio may overestimate the stellar mass. A detailed analysis of spectral energy distributions with longer baseline wavelength is required. Besides, the subtraction of underlying host galaxy light might not be perfect. In particular, due to the lack of sufficient spatial resolution, it is not trivial to extrapolate the central light from the model that is mainly fit with the outer part of the galaxies.
Analysis of the SDSS fiber spectroscopic data reveals that the nuclei have relatively high emission-line metallicity (12 + log(O/H) $>$ 8.4 dex), with a median 8.59. The solar value is 8.66 dex [@Pilyugin03]. In Figure \[masmet\], we show a relation between 12 + log(O/H) and the $r$-band absolute magnitude. The comparison sample galaxies are the SDSS star-forming galaxies of the redshift range of 0.01$\sim$0.02 [@Duc14], and our galaxies significantly deviate. The derived metallicity of the underlying old stellar population seems to be consistent with that of typical dEs [@Paudel11] with a median value of log(Z/Z$_{\sun}$) being $-$0.8. This implies that the gas is about five times more metal-rich than the underlying old stellar population.
Discussion
==========
Star Formation
--------------
The SFR derived from the SDSS fiber spectra is, on average, $\sim$0.03 M$_{\sun}$/yr. As we pointed out above, the 3 fiber mainly samples the central region, particularly the nucleus. We also measure the global SFRs of the galaxies from the FUV images observed with the Galaxy Evolution Explorer [@Martin05]. Interestingly, the FUV-based global SFRs are consistent with the SDSS fiber SFRs. This indicates that the star-formation in the galaxies is indeed concentrated within the nuclei region and the contribution from the outer part is minimal.
Examining the literature for radio observations of these dEs, we find that dE5 (UGCA 298) was detected by the ALFA-ALFA survey. It has a gas mass of 1.7$\times$10$^{7}$ M$_{\sun}$, approximately 10 times lower than its stellar mass. Assuming an HI consumption fraction of 10%, the star-formation activity will last only a few Myr with the current SFR (0.025 M$_{\sun}$ yr$^{-1}$). It is thus very likely that dE5’s star formation is temporal and at its final phase before turning into a true red and dead dE.
The underlying halo stellar population is clearly old, and another critical question to address is how the star-forming gas is acquired. All of the dEs we study here are located significantly away from the cluster/group centers (see Table \[sampletab\], column 11). Given the radial velocities are very similar to their host cluster/group central galaxies, these dEs might be the backsplash ones [@Jaffe15]. They have radially gone through the cluster/group core quickly and remained as backsplash galaxies at the outskirt of the cluster/group, where they acquire the gas leading to star formation [@Almeida14]. But the observed high metallicity does not favor this hypothesis. If they could have acquired fresh gas from the outskirts of a cluster/group, the observed emission-line metallicity would be much lower.
It has also been argued that star-formation activity can be rejuvenated from recycled gas that flows into the galactic center [@Boselli08]. That may explain the observed relatively high metallicity of these young star-forming blobs. A more quantitative conclusion can be drawn from detailed abundance analyses for old stellar populations that are not contaminated by the emission lines from gas. The absorption line indices are currently hard to measure precisely from the SDSS spectra because they are dominated by emission lines.
Formation of Nuclei
-------------------
We have identified six relatively old dEs that host a compact star-forming blob near (or at) the center. Could these star-forming blobs be future nuclei as typically seen in dEs? Detailed studies of the stellar population properties of dE nuclei have revealed that the majority of them are younger than the halos of their host galaxies, but the origin of the young age remains elusive [@Paudel11]. Two main scenarios have been proposed: (1) in situ formation of nuclei in an early epoch of galaxy evolution [@Bergh86; @Milosavljevic04], and (2) reactivated star formation by the fueling of star-forming gas later on [@Bekki07].
It is also possible that the entire nuclear structure is formed in a later phase of galaxy evolution with a burst of star formation via the sinking of star-forming gas. That creates younger nuclei at the centers of galaxies. In this vein, @Guillard16 proposed a ‘wet migration’ scenario, where a massive star cluster could form a little away from the center of a galaxy and migrate to the center via a combination of interactions with other substructures and dynamical friction. Indeed, we have identified a few exemplary cases of such off-centered compact star-forming regions that might act as seeds of nuclei as proposed in the wet migration scenario.
We thank Florent Renaud and Rory Smith for fruitful discussions and comments on the draft version of this paper. S.P. acknowledges support from the New Researcher Program (Shinjin grant No.2019R1C1C1009600) through the National Research Foundation of Korea. S.-J.Y. acknowledges support by the Mid-career Researcher Program (No.2019R1A2C3006242) and the SRC Program (the Center for Galaxy Evolution Research; No.2017R1A5A1070354) through the National Research Foundation of Korea. This study is based on the archival images and spectra from the Sloan Digital Sky Survey (the full acknowledgment can be found at http://www.sdss.org/collaboration/credits.html).
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{width="18cm"}
{width="18cm"} {width="18cm"}
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-------- ------------- -------------- ---------- -------- ------- --------- ----- ------------------------------------------------- ----------------- ------- -------------- ------- -- --
Galaxy R.A. Dec. M$_{B}$ z $g-r$ $R_{e}$ n FUV (SFR) M$^{*}$ R R.V. Env.
(h:m:s) (d:m:s) (mag) (mag) (kpc) (mag)[[**(log\[M$_{\sun}\,$yr$^{-1}$\])**]{}]{} log(M$_{\sun}$) (Mpc) (kms$^{-1}$)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
DE1 11:17:17.02 +16:19:37.65 $-$15.60 0.0036 0.89 12.5 1.1 19.7 ($-$2.11) 8.31 5.4 204 Virgo
DE2 11:30:26.21 +17:19:56.69 $-$15.02 0.0039 0.94 06.4 1.5 19.9 ($-$2.62) 8.19 4.5 114 Virgo
DE3 12:24:22.18 +21:09:35.95 $-$16.20 0.0030 0.81 12.7 1.3 19.0 ($-$1.93) 8.27 2.5 384 Virgo
DE4 12:40:50.32 +04:31:32.99 $-$15.64 0.0025 1.01 11.7 1.0 19.1 ($-$1.85) 8.20 2.3 534 Virgo
DE5 12:46:55.40 +26:33:51.38 $-$16.00 0.0027 0.94 09.1 2.2 18.8 ($-$1.70) 8.35 4.2 474 Virgo
DE6 11:53:49.00 +60:52:09.42 $-$15.23 0.0041 0.96 08.2 1.2 19.9 ($-$2.52) 8.20 0.7 155 N3945
-------- ------------- -------------- ---------- -------- ------- --------- ----- ------------------------------------------------- ----------------- ------- -------------- ------- -- --
\[sampletab\]
-------- --------- -------- -------------- ----------- ------ ------------- ------------------------------------------ ----- ------------------ ------- ------------- -------------------
Galaxy M$_{r}$ $g-r$ log(M$_{*}$) $\Delta$R FWHM 12+log(O/H) SFR $c$ EW(H$_{\alpha}$) Age Age log(Z/Z$_{\sun}$)
(mag) (mag) (M$_{\sun}$) () () (dex) [[**(log\[M$_{\sun}$yr$^{-1}$\])**]{}]{} (Å) (Myr) (Gyr) (dex)
(1) (2) (3) (4) (5) (5) (7) (8) (9) (10) (11) (12) (13)
DE1 $-$13.3 $-$0.4 6.70 3.2 0.8 8.59 $-$1.99 3.4 319 5.04 $-$ $-$
DE2 $-$12.5 $-$0.3 6.45 0.8 1.0 8.53 $-$2.20 3.0 111 6.21 0.4$\pm$0.1 $-$0.6$\pm$0.02
DE3 $-$12.7 $-$0.5 6.40 2.2 1.1 8.61 $-$2.16 3.3 124 6.13 1.2$\pm$0.1 $-$1.3$\pm$0.05
DE4 $-$13.1 $-$0.3 6.69 0.7 1.2 8.40 $-$2.31 3.4 83 6.24 0.7$\pm$0.1 $-$0.5$\pm$0.02
DE5 $-$13.5 $-$0.5 6.72 0.3 0.9 8.60 $-$1.60 3.5 199 5.61 9.8$\pm$1.1 $-$1.0$\pm$0.02
DE6 $-$13.1 $-$0.5 6.56 1.3 1.3 8.62 $-$1.90 3.0 384 4.89 $-$ $-$
-------- --------- -------- -------------- ----------- ------ ------------- ------------------------------------------ ----- ------------------ ------- ------------- -------------------
\[nuctab\]
[^1]: https://dr12.sdss.org/mosaics/
|
---
author:
- Jakub Mielczarek
title: Inflationary power spectra with quantum holonomy corrections
---
Introduction
============
Effects of the quantum nature of space at the Planck scale predicted by loop quantum gravity (LQG) [@Ashtekar:2004eh] can be studied by introducing appropriate modifications at the level of the classical Hamiltonian. This so-called effective approach enables to relate some quantum gravitational phenomena with the realm of classical physics, which proved to be especially fruitful in the cosmological context, known as loop quantum cosmology (LQC) [@Bojowald:2006da; @Ashtekar:2011ni].
As discussed in Ref. [@Bojowald:2012qya], the effective approach in quantum gravity is conceptually similar to the effective approach in solid state physics. Namely, while calculations based on many-body Hamiltonian are extremely difficult to execute, there is a whole range of effective models enabling explanation of macroscopic phenomena in terms of atomic-scale physics. As an example, relevant for our further discussion, let us refer to the nature of refractive index $n$. In the effective model of frequency dependence of $n$, one considers a single atomic dipole interacting with electromagnetic plane wave. By virtue of homogeneity of a sample the formula for $n(\omega)$, characterizing macroscopic bulk, can be derived. The formula depends on known microscopic quantities as electron mass and elementary charge, but also contains some characteristic frequencies which cannot be derived from the model. The unknown values can be either fixed experimentally or derived from quantum mechanical computations.
In LQG, there exists an analogue of the many-body Hamiltonian in solid state physics. Action of this so-called Hamiltonian constraint on the spin network states, describing the state gravitational field, is however not fully understood yet. When these difficulties are overcome it will be possible to study some macroscopic or mesoscopic gravitational configurations numerically. This is in analogy to computations performed within condensed matter physics or quantum chemistry. Meanwhile, the effective approach, competitive with the first-principle computations, can be utilized.
Construction of effective models is facilitated by certain assumptions regarding symmetries of space. Here, we focus on homogeneous and isotropic background geometry[^1] described by the flat Freidmann-Robertson-Walker (FRW) metric on which inhomogeneities are considered perturbatively. Such setup is sufficient to study the generation of primordial perturbations during the phase of slow-roll inflation.
Incorporation of LQG effects into cosmological models is performed by taking into account two types of corrections: inverse volume corrections and holonomy corrections. Both corrections reflect discrete nature of space at the Planck scale, however in a different manner. While strength of inverse volume corrections depends on volume element, holonomy corrections are sensitive to energy density. In case of inverse volume corrections, equations of motion for perturbations were derived in Ref. [@Bojowald:2008jv]. Based on this, corrections to the inflationary power spectra were derived in Ref. [@Bojowald:2010me]. The corrections were shown to be consistent with the 7-year WMAP data [@Bojowald:2011hd; @Bojowald:2011iq].
In this paper we focus on derivation of holonomy corrections to inflationary powers spectrum. As already mentioned, holonomy corrections are sensitive to energy density of matter. The characteristic energy scale, at which holonomy corrections are becoming important is $$\rho_c = \frac{3 m_{Pl}^2}{8\pi \gamma^2 L^2},$$ where the Planck mass $m_{Pl} =1.22 \cdot 10^{19}$ GeV, $\gamma \sim \mathcal{O}(1)$ is the so-called Barbero-Immirzi parameter and $L$ is a length scale of the order of the Planck scale. Therefore, the critical energy density is expected to be of the order of the Planck energy density, $\rho_c \sim \rho_{Pl}$, where $ \rho_{Pl} \equiv m_{Pl}^4$.
Holonomy corrections are modifying Friedmann equation into the following form [@Ashtekar:2006rx; @Ashtekar:2006wn] $$H^2 = \frac{8\pi}{3 m_{Pl}^2} \rho \left(1-\frac{\rho}{\rho_c} \right),
\label{Friedmann}$$ where $H$ is a Hubble factor and $\rho$ is energy density of the matter content. Positivity of the left hand side of this equation implies that energy density of matter is bounded from above, $\rho \leq \rho_c$. This leads to resolution of singularity problem of homogeneous cosmological modes. The big bang singularity is replaced by non-singular *bounce*, which merges contracting and expanding phases [@Ashtekar:2006rx; @Bojowald:2007zza].
Models of inflation are typically constructed with the use of scalar fields. In the simplest case it can be a single scalar field $\varphi$, the so-called inflaton field [@Linde:1983gd]. The single scalar is sufficient to construct a reliable model of the inflationary phase. Energy density of this field expresses as $$\rho = \frac{\dot{\varphi}^2}{2}+V(\varphi),$$ where $V(\varphi)$ is a potential term. Equation of motion governing evolution of $\varphi$ is not a subject of holonomy corrections and takes the standard form: $$\ddot{\varphi}+3H\dot{\varphi}+V_{,\varphi}=0.
\label{KGeq}$$
The holonomy corrections can be also introduced into equations governing evolution of cosmological perturbations. In particular, it was found that, while holonomy corrections are present, equation of motion for the Mukhanov variable $v$ is[@Cailleteau:2011kr]: $$\frac{d^2}{d\tau^2} v-\Omega \nabla^2 v - \frac{z^{''}_S}{z_S} v = 0.
\label{MukhanovEQ}$$ Here $\tau$ is a conformal time defined as $d\tau = dt /a$. Moreover, the holonomy correction function $$\Omega = 1-2\frac{\rho}{\rho_c}
\label{OmegaDef}$$ and $$z_S = a \frac{\dot{\varphi}}{H}.$$ It is clear that, while $\rho \ll \rho_{c}$ the classical expression with $\Omega=1$ is correctly recovered.
Based on the Mukhanov variable $v$, perturbations of curvature $\mathcal{R} = \frac{v}{z}$ can be derived. The quantity $\mathcal{R}$ is a key object characterizing scalar perturbations, allowing for computation of the scalar power spectrum.
The equation (\[MukhanovEQ\]) was originally derived by considering requirements of anomaly freedom for the scalar perturbations [@Cailleteau:2011kr]. Later, it was shown that this equation can also be obtained from the lattice loop quantum cosmology [@WilsonEwing:2011es; @WilsonEwing:2012bx]. The equation (\[MukhanovEQ\]) can be seen as a result of discretization of space for homogeneous cubic cells with the lattice spacing $L$. Cosmological consequences of equation (\[MukhanovEQ\]) have not been studied in details yet. As an interesting application, power spectra from a matter bounce was found [@WilsonEwing:2012pu].
For tensor modes (gravitational waves), holonomy corrected version of the equation is [@Cailleteau:2012fy] $$\frac{d^2}{d\tau^2}h_{i} +
2\left( \mathcal{H}-\frac{1}{2\Omega} \frac{d\Omega}{d\tau} \right)\frac{d}{d\tau}h_{i}
-\Omega \nabla^2h_{i}= 0,
\label{TensorEOM1}$$ where $i=\otimes, \oplus$ corresponds to two polarizations of gravitational waves. This equation can be rewritten into the form $$\frac{d}{d\tau^2}u-\Omega \Delta u-\frac{z_T ^{''}}{z_T}u=0,
\label{TensorEOM2}$$ where $z_T = a/\sqrt{\Omega}$ and $u =\frac{a h_{\otimes, \oplus}}{\sqrt{16 \pi G} \sqrt{\Omega}}$. The equation (\[TensorEOM1\]) differs from the equation of motion for tensor modes with holonomy corrections originally derived in Ref. [@Bojowald:2007cd]. This is because the original derivation has not been based on anomaly freedom Hamiltonian. However, taking into account the issue of anomaly freedom became possible thanks to analysis of scalar perturbations with holonomy corrections performed in Ref. [@Cailleteau:2011kr].
So far, the equation (\[TensorEOM1\]) was applied to study generation of tensor perturbation across the cosmic bounce [@WilsonEwing:2012pu; @Linsefors:2012et]. Nevertheless, there is a whole aggregation of previous analyses performed with the use of the original equation for tensor modes with holonomy corrections (See e.g. [@Copeland:2008kz; @Grain:2009kw; @Mielczarek:2010bh; @Mielczarek:2008pf]). There were also earlier attempts to study holonomy corrections for scalar perturbations. However, they were not consistent with the requirement of anomaly freedom. In particular, the studies for scalar perturbations were performed in Ref. [@Artymowski:2008sc]. There is also an alternative approach to incorporate loop quantum corrections to cosmological perturbations developed in Refs. [@Agullo:2012sh; @Agullo:2012fc; @Agullo:2013ai].
It is worth mentioning at this point that because we consider model with the scalar matter, the vector modes are not activated and identically equal zero [@Mielczarek:2011ph].
For both tensor and scalar perturbations the deformation factor $\Omega$ is placed in front of Laplace operator. Therefore, it can be considered as an effective speed of light squared. Namely, by neglecting the cosmological factor and assuming the plane wave solution $v \propto e^{i( {\bf k \cdot x} -\omega \tau)}$, we find the following dispersion relation $\omega^2 = \Omega k^2$, based on which the phase velocity $v_{ph} =
\frac{\omega}{k}= \sqrt{\Omega}$. Therefore, the refractive index $$n \equiv \frac{1}{v_{ph}} = \frac{1}{\sqrt{\Omega}}.$$ While $\rho \rightarrow \rho_c/2$ ($\Omega \rightarrow 0$) the refractive index becomes infinite, and speed of propagation tends to zero. As discussed in Ref. [@Mielczarek:2012tn] this can be associated with the state of *asymptotic silence*. At the energy densities $\rho \in ( \rho_c/2, \rho_c]$ the refractive index is purely imaginary. As discussed in Ref. [@Linsefors:2012et] this not necessarily means that space is opaque for the propagation of waves. The waves are not only evanescent in this region, but can be amplified as well. Behavior observed from the numerical computations differs with the intuition gained from *e.g* analysis of waves in plasma with frequencies lower than the plasma frequency. Furthermore, as discussed in Refs. [@Bojowald:2011aa; @Mielczarek:2012pf] the region of negative $\Omega$ can be associated with the change of metric signature from Lorentzian to Euclidean one.
However, in our calculations of inflationary power spectra, we restrict ourselves to the regime where $\Omega>0$. Therefore, the interesting behavior in vicinity of $\Omega=0$ and at the negative values of $\Omega$ will not be relevant. We will come back to the issue of evolution of modes in the $\Omega \leq 0$ in our further studies.
Slow-roll inflation
===================
During the slow-roll roll inflation Universe underwent an almost exponential expansion. The deviation from the exponential (de Sitter) growth of the scale factor is parametrized by the slow-roll parameters, which are much smaller than unity. The slow-roll roll inflation is characterized by gradual decreasing of $\varphi$ in a potential $V(\varphi)$. In this regime, energy density of the scalar field is dominated by its potential energy, therefore $\dot{\varphi}^2 \ll V(\varphi)$. Because of that, the modified Friedmann equation (\[Friedmann\]) can be approximated by $$H^2 \simeq \frac{8\pi}{3 m_{Pl}^2} V \left(1-\frac{V}{\rho_c} \right).
\label{FriedmannSR}$$ Furthermore, flatness of the potential implies that $\ddot{\varphi}$ in equation (\[KGeq\]) can be neglected, such that $$3H\dot{\varphi}+V_{,\varphi}\simeq 0.
\label{KGSR}$$ Using (\[KGSR\]) to eliminate $\dot{\varphi}$ from the condition $\dot{\varphi}^2 \ll V(\varphi)$ and by using (\[FriedmannSR\]), one can define [@Artymowski:2008sc] $$\epsilon := \frac{m_{Pl}^2}{16 \pi} \left( \frac{V_{,\varphi}}{V} \right)^2 \frac{1}{(1-V/\rho_c)}
= - \frac{\dot{H}}{H^2} \frac{1}{(1-V/\rho_c)},
\label{SRepsilon}$$ such that $\epsilon \ll 1$ for the slow-roll inflation.
By differentiating the slow-roll equation $\dot{\varphi} \simeq - \frac{V_{,\varphi}}{3H}$ we find $$\ddot{\varphi} = - \frac{V_{,\varphi\varphi} \dot{\varphi} }{3H}+\frac{V_{,\varphi} \dot{H} }{3H^2}.$$ Because $|\ddot{\varphi}| \ll | V_{,\varphi} |$, the absolute value of $$\frac{\ddot{\varphi}}{V_{,\varphi}} \simeq \frac{1}{3}\eta - \frac{1}{3} \epsilon \left( 1- \frac{V}{\rho_c} \right)
\label{eqvpvp}$$ has to be much smaller than unity. Following Ref. [@Artymowski:2008sc] let us introduce the second slow-roll parameter $$\eta := \frac{m_{Pl}^2}{8 \pi} \left( \frac{V_{,\varphi\varphi}}{V} \right) \frac{1}{(1-V/\rho_c)},$$ satisfying $|\eta| \ll 1$ for $|\ddot{\varphi}| \ll | V_{,\varphi} |$. Based on (\[eqvpvp\]) we can also define $$\delta := \eta - \epsilon \left( 1- \frac{V}{\rho_c} \right) = - \frac{\ddot{\varphi}}{H\dot{\varphi}},
\label{deltaSR}$$ satisfying $\delta \ll 1$.
While studying cosmological perturbations it is convenient to work with the conformal time $\tau \equiv \int \frac{dt}{a}$. Here, it is defined such that $\tau \in (-\infty,0)$. Based on the definition of conformal time and integrating by parts, we find $$\tau = \int \frac{dt}{a} = \int \frac{da}{a^2 H} = -\frac{1}{Ha}-\int \frac{1}{a} \frac{\dot{H}}{H^2} dt
= -\frac{1}{Ha}+\tau \epsilon \left(1-\frac{V}{\rho_c} \right),$$ where in the last equality we applied (\[SRepsilon\]). This enables us to write expression for the time dependence of the scale factor $$a = - \frac{1}{H \tau} \frac{1}{\left[1- \epsilon\left(1- \frac{V}{\rho_c}\right)\right]}.$$
In the slow-roll regime the $\Omega$ function, defined in Eq. \[OmegaDef\], is approximated by $$\Omega \simeq 1 - 2 \delta_H,$$ where for the later convenience we introduced parameter $$\delta_H := \frac{V}{\rho_c}.$$ This parameter reflects deviation from the classical slow-roll inflation due to holonomy corrections. In the classical limit, which corresponds to $\rho_c \rightarrow \infty$, $\delta_H$ goes to zero. In what follows we will consider only linear corrections in $\delta_H$. This is because $\delta_H$ is expected to be a very small quantity. The assumption that the slow-roll regime takes place in the Lorentzian regime ($\Omega > 0$) implies that $\delta_H < 1/2$. One can however motivate that $\delta_H \ll 1/2$ unless the critical energy density $\rho_c$ is not much smaller than the Planck energy density. As an example, let us consider model with a massive potential $V(\varphi)=\frac{1}{2} m^2 \varphi^2$. Taking the inflaton mass $m \sim 10^{-6}\ m_{Pl}$ and value of the scalar field $\varphi \sim m_{Pl}$ in agreement with cosmological observations one can estimate that $V(\varphi) \sim 10^{-12} \rho_{Pl}$. Therefore, for $\rho_c \sim \rho_{Pl}$ one can expect that $\delta_H$ has the extremely small value $\delta_H \sim 10^{-12}$. On the other hand if $\rho_c \sim 10^{-12}
\rho_{Pl}$ or smaller, the holonomy corrections are becoming observationally relevant and allow to constraint models with low critical energy density.
Here, we keep terms linear in both slow-roll parameters and $\delta_H$ as well as the mixed terms $\mathcal{O}(\epsilon \delta_H)$ and $\mathcal{O}(\eta \delta_H)$. Contribution from the second order expansion in the slow-roll parameters is not taken into account. However, in case the $\delta_H$ is extremely small, as estimated above, the terms $\mathcal{O}(\epsilon^2)$ will dominate contributions from $\mathcal{O}(\epsilon \delta_H)$. The derived expressions will therefore have practical application only to the regime where $\rho_{c} \in (\sim 0.01, \sim 1/2)$[^2], where the lower limit comes from estimating values of the slow-roll parameters. The estimated range overlaps with the domain which can be probed with use of currently available observational data. Furthermore, the theoretical predictions performed here will set a stage for more comprehensive considerations of the second order expansion in the slow-roll parameters. This will extend a range of testable values of $\rho_c$, of course if it is allowed by observational data.
Normalization of modes
======================
In this section we will present some possible choices of the short scale normalizations for the perturbations with holonomy corrections. In must be stressed that we do not explore any representative class of states. However, the considered normalizations seem to be the most reliable and best physically motivated. Because of this ambiguity, the choice of the normalization is the weakest point of a whole construction of the model of generation of primordial perturbations during the inflationary phase. This concerns also the case without quantum gravitational corrections. Therefore, here we pay a lot of attention to this issue.
Performing the Fourier transform $v({\bf x},\tau) = \int \frac{d^3 x}{(2\pi)^{3/2}} v_k(\tau) e^{i{\bf k \cdot x}} $ of the equations of modes for tensor and scalar perturbations we find $$\frac{d}{d\tau^2}v_k+\Omega k^2 v_k-\frac{z^{''}}{z}v_k=0,
\label{modeeq}$$ where $k^2 = {\bf k \cdot k}$, and expression on $z$ depends on whether scalar or tensor mods are studied. In both cases $\frac{z^{''}}{z} \approx \mathcal{H}^2 \simeq \frac{1}{\tau^2}$. Based on this, one can define super-horizontal limit when $ \sqrt{\Omega}k \ll \mathcal{H}$ and short scale limit when $\sqrt{\Omega} k \gg \mathcal{H}$. In the super-horizontal limit the $\Omega k^2 v_k$ factor in Eq. \[modeeq\] can be neglected and an approximate solution $v_k = c_1 z + c_2\int^{\tau}\frac{d\tau'}{z}$ can be found. Because the physical amplitudes of perturbations are proportional to the ratio $v_k/z$, it is clear that amplitudes are “frozen” at the super-horizontal scales. This process beings when $ \sqrt{\Omega}k \approx
\mathcal{H}$[^3]. In the short scale limit, the factor $\frac{z^{''}}{z}u_k$ in Eq. \[modeeq\] can be neglected and the equation for modes reduces to $$\frac{d^2}{d\tau^2} v_k+\Omega(\tau) k^2 v_k \approx 0.
\label{UVEOMdef}$$ For the slow-roll inflation, the $\Omega$ is only weakly dependent on $\tau$ and solution to equation (\[UVEOMdef\]) can be found by applying the WKB approximation. We find that $$v_k = \frac{c_1}{\sqrt{2k\sqrt{\Omega}} }e^{-ik\int^{\tau}\sqrt{\Omega(\tau')}d\tau'}+
\frac{c_2}{\sqrt{2k\sqrt{\Omega}}}e^{+ik\int^{\tau}\sqrt{\Omega(\tau')}d\tau'},
\label{WKBHolo}$$ which is superposition of plane waves traveling forward ($e^{-i k \int^{\tau}\sqrt{\Omega(\tau')}d\tau'}$) and backward ($e^{+i k \int^{\tau}\sqrt{\Omega(\tau')}d\tau'}$) in time. Validity of the WKB approximation requires that $$\frac{1}{2}\left| \frac{\Omega'}{\Omega} \right| \frac{1}{k\sqrt{\Omega}} \ll 1.$$ Because we are in the short scale limit $\sqrt{\Omega}k \gg \frac{1}{|\tau|}$ and $$\frac{\Omega^{'}}{\Omega} = - \frac{4\epsilon }{\tau} \delta_H,
\label{OmegaPrimeOmega}$$ the condition of validity of the WKB approximation simplifies to $\epsilon \delta_H < 0$. Because both $\epsilon$ and $\delta_H$ are smaller than unity for the considered slow-roll evolution, the WKB approximation (\[WKBHolo\]) holds.
Canonical commutation relation between quantum field $\hat{v}$ and its conjugated momenta requires Wronskian condition $$W(v_k,v_k') \equiv v_k \frac{d v^{*}_k}{d\tau}-v^{*}_k \frac{d v_k}{d\tau} = i
\label{Wronskian}$$ to be satisfied. This is the usual way the modes are normalized and also the place where quantum mechanics enters into description of primordial perturbations. The Wronskian condition applied to solution (\[WKBHolo\]) leads to relation $$|c_1|^2-|c_2|^2=1.$$ The initial four numbers ($c_1, c_2 \in \mathbb{C}$) parametrizing solution (\[WKBHolo\]) are therefore reduced to three. Because the total phase is physically irrelevant, the family of normalized solutions in the short scale limit is characterized by two real numbers. Their values have to be fixed by hand. The obtained solutions are used to normalize general solutions to the equations of motion.
It is worth stressing at this point that while considering the short scale limit $ \sqrt{\Omega}k \ll \mathcal{H}$ one has to be cautious about the limit $k\rightarrow \infty$. Such limit can be performed only formally because for $k > \frac{a}{l_{Pl}}$ the classical description of space is expected to be no more valid due to quantum gravitational effects. We do not consider such *trans-Planckian modes* here.
$\Omega-$deformed Minkowski vacuum
----------------------------------
For the particular choice $c_1=1$, solution (\[WKBHolo\]) contains incoming modes only: $$v_k = \frac{1}{\sqrt{2k\sqrt{\Omega}}}e^{-ik\int^{\tau}\sqrt{\Omega(\tau')}d\tau'}.
\label{WKBHoloMink}$$ This solution reduces to the so-called Minkowski (Bunch-Davies) vacuum $$v^{M}_k \equiv \frac{e^{-i k \tau }}{\sqrt{2k}}$$ in the classical limit ($\Omega \rightarrow 1$), which has been extensively used to normalize cosmological perturbations.
Based on (\[WKBHoloMink\]) we define $\Omega-$deformed Minkowski vacuum to be $$v^{\Omega M}_k \equiv \frac{e^{-ik\int^{\tau}\sqrt{\Omega(\tau')}d\tau'}}{\sqrt{2\sqrt{\Omega}k}}
\approx v^{M}_k\left[1+ \left(\frac{1}{2}+ik\tau \right) \delta_H +\mathcal{O}(\delta_H^2)\right],
\label{OmegaMdefExp}$$ where in the second equality we neglected time variation of $\Omega$.
Bojowald-Calcagni normalization
-------------------------------
Another possibility of normalizing modes was proposed in Ref. [@Bojowald:2010me] for the case of perturbations with inverse volume corrections. The proposal made by Bojowald and Calcagni was that in the short scale limit the solution to (\[UVEOMdef\]) can be written up to the first order in $\delta_H$ as follows $$v_{k}^{BC}= v^M_k(1+y(k,\tau) \delta_H),
\label{vBC1}$$ where $y(k,\tau)$ is some unknown function[^4]. By plugin in (\[vBC1\]) to (\[UVEOMdef\]) we find the following equation for the function $y$: $$y^{''}-2(2\mathcal{H}\epsilon+i k )y'+(4i \mathcal{H}\epsilon k-2\epsilon \mathcal{H}^2-2\delta_H k^2 )y-2k^2 = 0,
\label{BCEOMy1}$$ where we used relations $$\begin{aligned}
\delta_H' &=& -2\epsilon \mathcal{H} \delta_H, \\
\delta_H^{''} &=& -2\epsilon \mathcal{H}^2 \delta_H.\end{aligned}$$ The equation (\[BCEOMy1\]) requires certain simplifications. Firstly, because we are interested in the first order correction in $ \delta_H$ we can skip the factor $-2\delta_H k^2y$ in (\[vBC1\]), which would generate higher order contribution. Secondly, because we are looking for the short scale solution ($\sqrt{\Omega}k \gg \mathcal{H}$) the factor $-2\epsilon \mathcal{H}^2y$ can be ignored as well. This second approximation turns out to be beneficial while searching for analytic solution to the equation of motion for $y$. The reduced equation (\[BCEOMy1\]) is now $$y^{''}-2(2\mathcal{H}\epsilon+i k )y'+4i \mathcal{H}\epsilon k y -2k^2 = 0.
\label{BCEOMy2}$$ For $\epsilon = 0$, solution to this equation is $$y = i k \tau +c_1+c_2 e^{2 i k\tau},$$ where $c_1$ and $c_2$ are constants of integration. Because the $e^{2 i k\tau}$ factor would lead to outgoing modes we fix $c_2=0$. Value of the factor $c_1$ can be determined by considering the case $\epsilon \neq 0$. Let us now find solution in the form $y=a+bx$. By applying it to (\[BCEOMy2\]), where $\mathcal{H} = - \frac{1}{\tau}$[^5], we find special solution $y=a+bx$ with $a=\frac{1}{1+2\epsilon}$ and $b=\frac{i}{1+2\epsilon}$ to equation (\[BCEOMy2\]). Requirement of analytic continuity of the solution (in respect to $\epsilon$) between the cases $\epsilon=0$ and $\epsilon\neq0$ fixes the value of $c_1$. In consequence, we obtain $$v^{BC}_k = v^{M}_k\left[1+ \frac{1}{1+2\epsilon} \left(1+ik\tau \right) \delta_H +\mathcal{O}(\delta_H^2)\right]
\simeq v^{M}_k\left[1+\left(1+ik\tau \right) \delta_H\right].$$ It is worth noticing a slight difference between this case and predictions of the $\Omega-$deformed Minkowski vacuum (\[OmegaMdefExp\]). In contrast to that case, the method presented in this subsection does not utilize the Wronskian condition in order to normalize the mode functions. This may have advantages if we have reason to suppose that the Wronskian condition is deformed but the form of deformation is not known.
Deformed Wronskian condition
----------------------------
Let us suppose that indeed the Wronskian condition is deformed due to presence of $\Omega$. Such deformation can come from the fact that, because $\Omega$ is present in equations of motion, inner product must differ from the classical one. We will discuss this issue in more details in the next section, while here wa assume the classical Wronskian condition (\[Wronskian\]) is deformed to $$W_{\Omega}(v_k,v_k')= v_k \frac{d v^{*}_k}{d\eta}-v^{*}_k \frac{d v_k}{d\eta} = i f(\Omega),$$ where $f(\Omega)$ is some function of $\Omega$, defined such that $\lim_{\Omega \rightarrow1} =1$. In case we have no hints what the functional form of $f(\Omega)$ is we can investigate a power-low parametrization $$f(\Omega) = \Omega^n.$$ In this case, the counterpart of (\[OmegaMdefExp\]) is $$v_{k}^{(n)} \equiv \frac{e^{-i \sqrt{\Omega}k \tau }}{\sqrt{2k \Omega^{1/2-n}}}
=v_{M}\left[1+ \left(\frac{1}{2}-n+ik\tau \right) \delta_H +\mathcal{O}(\delta_H^2)\right].$$ As we see, for $n=-\frac{1}{2}$, the normalization from the deformed Wronskian condition overlaps with Bojowald-Calcagni normalization up to the first order in $\delta_H$: $$v_{k}^{(-1/2)} \simeq v^{BC}_k.$$
Inner product and the Wronskian condition
=========================================
Let us now address the issue of validity of the Wronskian normalization in presence of holonomy corrections in more details.
For the a pair of fields $\phi_1, \phi_2$ satisfying Klein-Gordon equation $(\Box-m^2)\phi =0$, the inner product is [@Wald1994] $$\langle \phi_1 | \phi_2 \rangle := i \int_{\Sigma} d^3x \sqrt{q} n^{\mu}
\left( \phi_2^* \partial_{\mu} \phi_1 - \phi_1 \partial_{\mu} \phi_2^* \right),
\label{InnerProduct}$$ where $n^{\mu} $ is a future-direction unit ($g_{\mu\nu} n^{\mu} n^{\mu} =-1$) vector normal to Cauchy surface $\Sigma$ and $q$ is a determinant of the spatial metric on $\Sigma$. To remind, the Cauchy surface is a spatial hypersurface at which the initial conditions are imposed. The inner product (\[InnerProduct\]) is defined such that it does not depend on the choice of a Cauchy surface: $$\langle \phi_1 | \phi_2 \rangle_{\Sigma_1} = \langle \phi_1 | \phi_2 \rangle_{\Sigma_2}.
\label{IPCauchyInd}$$ The proof is direct and employs a Gauss law, Klein-Gordon equation and vanishing of $\phi $ at spatial infinities (See e.g. Ref. [@Ford:1997hb]). It is also worth noticing, that the inner product (\[InnerProduct\]) is not positive-definite.
In the case studied in this paper, the Klein-Gordon equations for scalar ad tensor perturbations are deformed with respect to the classical one. Therefore, in general, one could expect that (\[InnerProduct\]) is not a good scalar product because the condition (\[IPCauchyInd\]), requiring the Klein-Gordon equation to be satisfied, is not fulfilled. This can imply that the Wronskian condition (\[Wronskian\]), resulting from normalization of modes with use of (\[InnerProduct\]), is deformed.
However, if we manage to find an effective metric $ g^{eff}_{\mu\nu}$ which leads to holonomy deformations of the equations of perturbations, then the proof the condition (\[IPCauchyInd\]) would remain in force, and the inner product (\[InnerProduct\]) can be used. In what follows we show that such construction is possible for tensor perturbations.
For any component $\phi$ of the tensor perturbations, the equation of motion is $$\frac{d^2}{d\tau^2}\phi+2\left( \mathcal{H}-\frac{1}{2\Omega} \frac{d\Omega}{d\tau} \right)\frac{d}{d\tau}\phi
-\Omega \nabla^2\phi= 0.
\label{KGefftau}$$ In the coordinate time ($dt=a d\tau$) this equation can be written as $$\ddot{\phi}+3H \dot{\phi}- \frac{\dot{\Omega}}{\Omega} \dot{\phi}- \frac{\Omega}{a^2}\nabla^2 \phi=0.
\label{KGefft}$$ The classical Klein-Gordon equation $\Box\phi =0$ (the tensor modes are massless) on the FRW background is recovered by taking $\Omega=1$. It can be proved by direct calculation, that the holonomy corrected equations for tensor modes can be derived from the wave equation $\Box\phi =0$ at the effective FRW background given by the line element $$ds^2_{eff} = g^{eff}_{\mu\nu}dx^{\mu} dx^{\nu} = -\sqrt{\Omega} N^2 dt^2
+\frac{a^2}{\sqrt{\Omega}} \delta_{ab}da^adx^b, \label{effectivmetric}$$ where $N$ is a lapse function. In particular for the coordinate time ($N=1$) we have $$\begin{aligned}
\Box \phi &=& \nabla^{\mu} \nabla_{\mu} \phi = \frac{1}{\sqrt{-g}} \partial_{\mu}(\sqrt{-g} g^{\mu\nu} \partial_{\nu} \phi ) \nonumber \\
&=& - \frac{1}{\sqrt{\Omega}} \ddot{\phi}+ \frac{\sqrt{\Omega}}{a^2}\nabla^2 \phi
- \frac{1}{\sqrt{\Omega}}\left(3H- \frac{\dot{\Omega}}{\Omega} \right)\dot{\phi},
\label{Boxeff}\end{aligned}$$ where we used $g_{\mu\nu} = g^{eff}_{\mu\nu}$. By equating (\[Boxeff\]) to zero and multiplying by $-\sqrt{\Omega}$, the equation (\[KGefft\]) is recovered.
It is worth noticing that the effective metric (\[effectivmetric\]) is conceptually similar to *dressed metric* approach to quantum fields on quantum spaces [@Ashtekar:2009mb; @Agullo:2012sh]. In our case, quantum gravitational effects are “dressing” the FRW metric leading to the effective metric (affected by $\Omega$ terms), which is felt by test fields.
Now we can check if the Wronskian condition derived based on (\[InnerProduct\]) holds the classical form. We will be interested in the form of the Wronskian condition for the field $$u = \frac{a}{\sqrt{\Omega}} \phi,$$ which, as can be seen by substituting to (\[KGefftau\]), fulfills equation $$\frac{d^2}{d\tau^2}u-\Omega \nabla^2 u- \frac{z^{''}}{z}u=0,$$ where $z=a/\sqrt{\Omega}$. Because the Cauchy hypersurface is spatial, based on $g_{\mu\nu} n^{\mu} n^{\mu} =-1$, we find $n^0 = \frac{1}{N \Omega^{1/4}}$ and $n^a=0$, which gives us $n^{\mu}\partial_{\mu}=\frac{1}{N \Omega^{1/4}} \partial_{t}$. Then, for the conformal time ($N=a$), the inner product of two fileds $\phi$ is $$\begin{aligned}
\langle \phi | \phi \rangle &=& i \int_{\Sigma} d^3x \sqrt{q} \left( \phi^* n^{\mu}\partial_{\mu} \phi - \phi n^{\mu}\partial_{\mu} \phi^* \right)
\nonumber \\
&=& i \int_{\Sigma} d^3x \frac{a^3}{\Omega^{3/4}} \frac{1}{a \Omega^{1/4}} \frac{\Omega}{a^2}
\left( u^* \frac{du}{d\tau} - u\frac{du^*}{d\tau} \right) \nonumber \\
&=& i \int_{\Sigma} d^3x \left( u^* \frac{du}{d\tau} - u\frac{du^*}{d\tau} \right) \nonumber \\
&=&-i \int_{\Sigma} d^3x W(u,u') = \int_{\Sigma} d^3x = V_0 =1,\end{aligned}$$ where the classical Wronskian condition (\[Wronskian\]) was used to get the proper normalization $\langle \phi | \phi \rangle =1$. Here, we assumed that the spatial integration is restricted to $V_0$, or equivalently the spatial topology is compact and has coordinate volume $V_0$. This volume can be conventionally fixed to one. Alternatively the field $\phi$ can be rescaled by $\phi \rightarrow \frac{1}{\sqrt{V_0}} \phi $ to compensate the contribution from the spatial integration over $V_0$.
In summary, for the tensor modes, the inner product (\[InnerProduct\]) is properly defined and normalization condition $\langle \phi | \phi \rangle =1$ leads to the classical Wronskian condition (\[Wronskian\]). The $\Omega-$deformed vacuum seems to therefore be the right choice for the tensor modes. It remains to show if the similar construction can be performed for the scalar modes as well.
Tensor power spectrum
=====================
In this section we will compute inflationary tensor power spectrum with holonomy corrections. Starting point for our considerations is the equation $$\frac{d^2}{d\tau^2}u_k+\Omega k^2 u_k- \frac{z_T ^{''}}{z_T}u_k=0,$$ where $k^2 = {\bf k \cdot k}$. Having solutions for $u_T$ and $z_T$, tensor power spectrum can be found from the definition $$\mathcal{P}_T(k) = 64 \pi G \frac{k^3}{2\pi^2} \left| \frac{u_k}{z_T} \right|^2.
\label{TensorPower}$$
With use of $z_T = a/\sqrt{\Omega}$, expression for the effective mass term can be written as $$m^2_{eff} \equiv -\frac{z_T ^{''}}{z_T}=- \frac{a^{''}}{a}+ \frac{a^{'}}{a}\frac{\Omega^{'}}{\Omega}
+\frac{1}{2}\frac{\Omega^{''}}{\Omega} -\frac{3}{4}\left(\frac{\Omega^{'}}{\Omega} \right)^2.$$ All the factors contributing to $m^2_{eff}$ can be expressed in terms of conformal time $\tau$ as well as $\epsilon, \eta$ and $\delta_H$. With use of the slow-roll conditions, these terms are: $$\begin{aligned}
\frac{a^{'}}{a} &=& - \frac{1}{\tau} \left[1+\epsilon\left(1- \delta_H\right)\right], \\
\frac{a^{''}}{a} &=& \frac{1}{\tau^2}\left[2+3\epsilon\left(1- \delta_H\right)\right], \\
\frac{\Omega^{''}}{\Omega} &=& \frac{4\epsilon }{\tau^2} \delta_H. \end{aligned}$$ The expression for $\frac{\Omega^{'}}{\Omega}$ is given in Eq. \[OmegaPrimeOmega\]. Plugin it into expression for $m^2_{eff}$ and keeping terms up to the first order in $\epsilon$ and $\delta_H$ we obtain $$m^2_{eff} = - \frac{1}{\tau^2}\left[ 2+3\epsilon\left(1- 3\delta_H\right) \right].$$ The equation for tensor modes can be therefore expressed as $$\frac{d^2 u_k}{d\tau^2} +\left[ \Omega k^2 -\left(\nu_T^2 -\frac{1}{4} \right) \frac{1}{\tau^2} \right] u_k =0,
\label{EOMTensorHankel}$$ where $$|\nu_T| = \sqrt{ \frac{9}{4}+3\epsilon \left(1-3 \delta_H \right)}
\simeq \frac{3}{2}+\epsilon \left(1-3 \delta_H \right).$$
Equation (\[EOMTensorHankel\]) reminds the standard equation for inflationary modes and it is tempting to find its analytic solution in terms of Hankel functions. This however would not be consistent because requires assumption of constancy of $\Omega$. The slow variation of $\Omega$ cannot be neglected if we already included variation of $\Omega$ in the expression for $m^2_{eff}$. To see it clearly, let us perform the following change of variables: $$\begin{aligned}
x &:=& -k\tau \sqrt{\Omega}, \\
f(x) &:=& \frac{u}{\sqrt{x}}, \end{aligned}$$ which transforms (\[EOMTensorHankel\]) into $$(1-4\epsilon \delta_H) x^2\frac{d^2f}{dx^2}+x\frac{df}{dx}+(x^2-\nu_T^2)f = 0.
\label{EquationModesF}$$ While the dependence on $\delta_H$ in $m^2_{eff}$ is contributing to $\nu_T^2$, the time dependence of $\Omega$ in front of $k^2$ generates factor $-4\epsilon \delta_H$. Because of this factor, solutions to equation (\[EquationModesF\]) are not Bessel (or equivalently Hankel) functions, what would be the case if the factor $-4\epsilon \delta_H$ is absent. Nevertheless, solutions to equation Ref. (\[EOMTensorHankel\]) can be studied numerically.
Because analytic solution to equation (\[EOMTensorHankel\]) cannot be easily found, we are forced to use another approach to find tensor power spectrum. Namely we will determine amplitude of the perturbations at the Hubble radius with use of the short scale solutions. However first, in order to approve consistency of normalization in case on non-vanishing $\Omega$ we will consider tensor power spectrum for the case with $\Omega=$ const.
$\Omega=$ const case
--------------------
As far as $\Omega$ can be considered as a constant, the effective mass term is $$m^2_{eff} \equiv -\frac{z_T ^{''}}{z_T}=- \frac{a^{''}}{a}= -\frac{1}{\tau^2}\left[2+3\epsilon\left(1- \delta_H\right)\right].$$ The equation of motion takes the form (\[EOMTensorHankel\]) with $\Omega=$const and $$|\nu_T| = \sqrt{ \frac{9}{4}+3\epsilon \left(1-\delta_H \right)}
\simeq \frac{3}{2}+\epsilon \left(1-\delta_H \right).$$ In this case, exact solution to equation (\[EOMTensorHankel\]) can be expressed in terms of Hankel functions: $$u_k = \sqrt{-\tau} \sqrt{\frac{\pi}{4}} \left[ D_1 H^{(1)}_{|\nu|}(-\sqrt{\Omega} k \tau)+
D_2 H^{(2)}_{|\nu|}(-\sqrt{\Omega} k \tau) \right].$$ The constants $D_1$ and $D_2$ were normalized such chosen such that the Wronskian condition (\[Wronskian\]) leads to relation $$|D_1|^2-|D_2|^2=1.$$ The $\Omega-$deformed Minkowski vacuum normalization is chosen by taking $D_2 = 0$ and $D_1 =e^{i\pi(2|\nu|+1)/4}$. This can be verified by considering asymptotic behavior of the Hankel function. Namely, for $x\ll 1$ $H^{(1)}_{|\nu|}(x) \approx \sqrt{\frac{2}{\pi x}} \exp \left( i(x-|\nu|\pi/2-\pi/4) \right) $. With use of this $$u_k = \sqrt{-\tau} \sqrt{\frac{\pi}{4}} e^{i\pi(2|\nu|+1)/4} H^{(1)}_{|\nu|}(-\sqrt{\Omega} k \tau) \approx
\frac{e^{-i \sqrt{\Omega}k \tau }}{\sqrt{2\sqrt{\Omega}k}} = u_k^{\Omega M},$$ for $-\sqrt{\Omega} k \tau \gg 1$.
Having the modes correctly normalized we can study the super-horizonal limit $-\sqrt{\Omega} k \tau \ll 1$. With use of approximation $H^{(1)}_{|\nu|}(x) \simeq - \frac{i}{\pi} \Gamma(|\nu|) \left(\frac{x}{2} \right)^{-|\nu|}$, which holds at $x\ll1$, we obtain $$|u_k|^2 \simeq \frac{1}{2} \frac{1}{aH} \left( \frac{k \sqrt{\Omega}}{aH} \right)^{-2|\nu|},$$ where we used $-\tau \simeq \frac{1}{aH}$. Applying it to definition (\[TensorPower\]), the tensor power spectrum from the slow-roll inflation is: $$\mathcal{P}_{T}(k) = A_T \left( \frac{k}{aH}\right)^{n_{T}},$$ where the amplitude $$A_T = \frac{16}{\pi} \left(\frac{H}{m_{\text{Pl}}} \right)^2\left(1+\delta_H \right),
\label{Attheo}$$ and the tensor spectral index $$\begin{aligned}
n_{T} =-2\epsilon \left(1-\delta_H\right).\end{aligned}$$
With use of the modified Friedmann equation (\[Friedmann\]) in the slow-roll regime $(\rho \approx V)$, one can rewrite (\[Attheo\]) into the following form $$A_T = \frac{128}{3} \frac{V}{ \rho_{Pl}}\left(1- \delta_H\right) \left(1+\delta_H \right)
= \frac{128}{3} \frac{V}{ \rho_{Pl}}+\mathcal{O}(\delta_H^2).$$ This expression is not a subject of holonomy corrections in the leading order.
$\Omega-$deformed Minkowski vacuum
----------------------------------
Let us now proceed to the proper calculations. The strategy is the following: We will use a given short scale solution and extrapolate it up to the horizon scale. A mode characterized by $k$ crosses the horizon scale when $\sqrt{\Omega} k = aH \simeq -\frac{1}{\tau}$. Above the horizon scale the modes “freeze out” and the power spectrum remains unchanged. The spectral index can be computed from the horizontal spectrum based on the formula $$n_{T} \equiv \frac{d \ln \mathcal{P}_T}{d \ln k}.
\label{nTdeff}$$
Modulus square of the $\Omega-$deformed Minkowski vacuum (\[WKBHoloMink\]) is $$\left|v^{\Omega M}_k \right|^2 =\frac{1}{2\sqrt{\Omega}k}.
\label{ModSqvOM}$$ By inserting (\[ModSqvOM\]) into the definition (\[TensorPower\]) and calculating the value at $k \sqrt{\Omega} =a H$ we find $$\mathcal{P}_{T}(k) = \frac{16}{\pi} \left(\frac{H}{m_{Pl}} \right)^2\frac{1}{\sqrt{\Omega}}
= \frac{16}{\pi} \left(\frac{H}{m_{Pl}} \right)^2(1+\delta_H)+\mathcal{O}(\delta_H^2),$$ which agrees with (\[Attheo\]).
Let us this result and to compute tensor spectral index. By using (\[nTdeff\]), we find $$n_{T} = 2 \frac{k}{H} \frac{dH}{dk}- \frac{1}{2}\frac{k}{\Omega}\frac{d\Omega}{dk}
= - 2\epsilon +\mathcal{O}(\epsilon^2\delta_H),
\label{nTOM}$$ where we used $k = -\frac{1}{\tau \sqrt{\Omega}}$, (\[SRepsilon\]) and (\[OmegaPrimeOmega\]). Here, the correction from $\delta_H$ contributes together with $\epsilon^2$ terms. Therefore, in the leading order the tensor spectral index holds its classical form.
Bojowald-Calcagni normalization
-------------------------------
Let us now compute the power spectrum for the Bojowald-Calcagni normalization. While use of condition $\sqrt{\Omega} k = aH \simeq -\frac{1}{\tau}$, the modulus square of $v^{BC}_k$ gives $$\left| v^{BC}_k \right|^2 \simeq \frac{1}{2k}(1+2\delta_H).
\label{ModSqvBC}$$ It is wort mentioning that this value does not depend on the fact that the amplitude is computed at the horizon. This is because the $i k \tau \delta_{H}$ term contributes in the second order, which is neglected. By inserting (\[ModSqvBC\]) into the definition (\[TensorPower\]) we find $$\mathcal{P}_{T}(k) = \frac{16 G}{\pi} \frac{k^2}{a^2} \Omega (1+2\delta_H)
= \frac{16}{\pi} \left(\frac{H}{m_{Pl}} \right)^2(1+2\delta_H)+\mathcal{O}(\delta_H^2).$$
Having amplitude of spectrum computed at the horizon scale, the spectral index is computed from the relation $$\begin{aligned}
n_{T} \equiv \frac{d \ln \mathcal{P}_T}{d \ln k} = 2 \frac{kHa}{ dk/d\tau} \frac{\dot{H}}{H^2}+ 2\frac{d\delta_H}{d\ln k}
= -2 \epsilon (1+\delta_H),\end{aligned}$$ where we have used expression (\[SRepsilon\]) and the fact that at the horizon $\sqrt{\Omega} k\tau=-1$.
Summing up, the tensor power spectrum with the Bojowald-Calcagni normalization can be expressed as follows $$\mathcal{P}_{T}(k) = A_T \left( \frac{k}{aH}\right)^{n_{\text{T}}},$$ where the amplitude $$A_T = \frac{16}{\pi} \left(\frac{H}{m_{Pl}} \right)^2(1+2\delta_H)
\label{AttheoBC}$$ and the tensor spectral index $$\begin{aligned}
n_{T} =-2\epsilon \left(1+\delta_H\right).
\label{nTBC}\end{aligned}$$
With use of the modified Friedmann equation (\[Friedmann\]) in the slow-roll regime $(\rho \approx V)$, one can rewrite (\[AttheoBC\]) into the following form $$A_T = \frac{128}{3} \frac{V}{ \rho_{Pl}} \left(1+\delta_H\right).$$
Scalar power spectrum
=====================
Here, for the sake of completeness we will derive equation of motion for the scalar modes in the slow-roll approximation. This equation will not be used to derive spectrum of the scalar inflationary perturbations because of the same reason as in the case of tensor modes.
Amplitude of the scalar power spectrum $$\mathcal{P}_S(k) = \frac{k^3}{2\pi^2} \left| \frac{v_k}{z_S} \right|^2
\label{ScalarPower}$$ will be calculated using the short scale solution extrapolated to the horizontal scale. While the spectral scalar is found, the spectral index will be determined by virtue of $$n_{S} \equiv \frac{d \ln \mathcal{P}_S}{d \ln k}.
\label{nSdeff}$$
Similarly as for tensor modes, the Fourier transform of the scalar perturbations fulfills equation $$\frac{d}{d\tau^2}v_k+\Omega k^2 v_k-\frac{z^{''}_S}{z_S}v_k=0,$$ where $z_S = a \frac{ \varphi'}{\mathcal{H}}$. The task now is to determine time dependance of $\frac{z^{''}_S}{z_S}$ in the slow-roll approximation.
By differentiating $z_S = a \frac{ \varphi'}{\mathcal{H}}$ with respect to conformal time and by using relation (\[SRepsilon\]) we obtain $$\frac{z'_S}{z_S} = \epsilon \left(1-\delta_H\right) \mathcal{H}+\frac{\varphi^{''} }{\varphi'}.
\label{zbSzS}$$ With use this, the expression (\[deltaSR\]) for the parameter $\delta$ can be written as $$\delta = 1- \frac{\varphi^{''} }{\varphi'\mathcal{H}} = 1+\epsilon \left(1-\delta_H\right) -\frac{z'}{z\mathcal{H}}.
\label{deltaSR2}$$ By differentiating this equality with respect to conformal time and neglecting all the non-leading contributions (*i.e.* $\delta', \epsilon', \epsilon^2, \eta'$) we obtain the following equality $$\frac{z^{''}_S}{z_S} = \left(\frac{z^{'}_S}{z_S} \right)^2 +\frac{\mathcal{H}'}{\mathcal{H}} \frac{z^{'}_S}{z_S}.
\label{zSbbzS1}$$ Combining (\[zbSzS\]) together with (\[deltaSR2\]) and (\[deltaSR\]) we find $$\frac{z^{'}_S}{z_S} = \left[1-\eta+2\epsilon(1-\delta_H)\right]\mathcal{H}.
\label{zbSzS2}$$ Furthermore $$\frac{\mathcal{H}'}{\mathcal{H}} = \mathcal{H}\left(1+\frac{\dot{H}}{H^2}\right)
= \mathcal{H}\left(1-\epsilon(1-\delta_H)\right),
\label{dHH}$$ where in the second equality we used (\[SRepsilon\]). Plugging (\[zbSzS2\]) and (\[dHH\]) to (\[zSbbzS1\]) we obtain $$\begin{aligned}
\frac{z^{''}_S}{z_S} &=& \mathcal{H}^2\left[2+5\epsilon(1-\delta_H)-3\eta \right]
= \frac{1}{\tau^2\left[1-\epsilon(1-\delta_H) \right]^2}\left[2+5\epsilon(1-\delta_H)-3\eta \right] \nonumber\\
&=& \frac{1}{\tau^2} \left[2+9\epsilon(1-\delta_H)-3\eta\right].
\label{zSbbzS2}\end{aligned}$$
Equation for scalar modes with holonomy correction in the (first order) slow-roll approximation can be therefore written as $$\frac{d^2 v_k}{d\tau^2} +\left[ \Omega k^2 -\left(\nu_S^2 -\frac{1}{4} \right) \frac{1}{\tau^2} \right] v_k =0,
\label{EOMScalarHankel}$$ where $$|\nu_S| = \sqrt{ \frac{9}{4}+9\epsilon \left(1-3 \delta_H \right)-3\eta}
\simeq \frac{3}{2}+3\epsilon \left(1-3 \delta_H \right)-\eta.$$ The classical case is correctly recovered for $\delta_H \rightarrow 0$.
$\Omega-$deformed Minkowski vacuum
----------------------------------
With use of the WKB approximation (\[WKBHoloMink\]) applied to definition (\[ScalarPower\]) we find $$\mathcal{P}_{S}(k) =
\frac{1}{\pi \epsilon} \left(\frac{H}{m_{Pl}} \right)^2 \frac{(1-\delta_H)}{\Omega^{3/2}}=
\frac{1}{\pi \epsilon} \left(\frac{H}{m_{Pl}} \right)^2 \left(1+2 \delta_H \right)+\mathcal{O}(\delta_H^2)$$ at the horizon $\sqrt{\Omega} k = aH$.
Finally, the inflationary scalar power spectrum: $$\mathcal{P}_{S}(k) = A_S\left( \frac{k}{aH}\right)^{n_{S}-1},$$ where amplitude of the scalar perturbations $$A_S = \frac{1}{\pi \epsilon} \left(\frac{H}{m_{Pl}} \right)^2 \left(1+2 \delta_H \right),
\label{Astheo}$$ and the spectral power index $$n_{S} =1+2\eta-6\epsilon +\mathcal{O}(\epsilon^2\delta_H,\epsilon\eta\delta_H).$$
As in case of the tensor modes, the $\delta_H$ correction to the spectral index is multiplied by the $\epsilon^2$ and $\epsilon\eta$ factors which are negligible in the considered order. To see it explicitly let us consider the case of massive scalar field ($V=\frac{1}{2}m^2\varphi^2$) for which $\epsilon=\eta$. Therefore $n_{S} =1-4\epsilon+\mathcal{O}(\epsilon^2\delta_H)$. Using the recent Planck fit $n_S= 0.9603\pm0.0073$ [@Ade:2013uln], we obtain $\epsilon \approx \frac{1}{4}(1-n_S) \approx 0.01$. The higher order corrections are therefor of the order $\mathcal{O}(\epsilon^2\delta_H,\epsilon\eta\delta_H)\sim 10^{-4}\delta_H$, with $|\delta_H| < \frac{1}{2}$. These terms are also typically smaller than contributions from the classical second order slow-roll expansion. With use of the present observational precision is impossible to constrain such effects.
Moreover, with use of the modified Friedmann equation (\[Friedmann\]) in the slow-roll regime $(\rho \approx V)$, one can rewrite (\[Astheo\]) into the following form $$A_S = \frac{8}{3} \frac{1}{\epsilon} \frac{V}{ \rho_{Pl}} \left(1-\delta_H \right) \left(1+2 \delta_H \right)
\approx \frac{8}{3} \frac{1}{\epsilon} \frac{V}{ \rho_{Pl}} \left(1+\delta_H \right).$$
Bojowald-Calcagni normalization
-------------------------------
The calculations can be now repeated for the case of Bojowald-Calcagni normalization. The obtained inflationary scalar power spectrum is $$\mathcal{P}_{S}(k) = A_S\left( \frac{k}{aH}\right)^{n_{S}-1},$$ where amplitude of the scalar perturbations $$A_S = \frac{1}{\pi \epsilon} \left(\frac{H}{m_{Pl}} \right)^2 \left(1+3\delta_H \right)
\label{Astheo}$$ and the spectral index $$n_{S} =1+2\eta-6\epsilon \left(1+\frac{1}{3}\delta_H \right).$$ In contrary to the previous case, the spectral index is holonomy-corrected in the leading order for the Bojowald-Calcagni normalization
For completeness, with use of the modified Friedmann equation (\[Friedmann\]) in the slow-roll regime $(\rho \approx V)$, one can rewrite (\[Astheo\]) into the following form $$A_S = \frac{8}{3} \frac{1}{\epsilon} \frac{V}{ \rho_{Pl}} \left(1-\delta_H\right) \left(1+3 \delta_H \right)
\approx \frac{8}{3} \frac{1}{\epsilon} \frac{V}{ \rho_{Pl}} \left(1+2\delta_H \right).$$
Tensor-to-scalar ratio
======================
In theoretical studies of inflation as well in confronting theoretical predictions with observations it is often useful to work with tensor-to-scalar ratio $r$. This dimensionless quantity, defined as $$r :=\frac{A_T}{A_S},$$ measures ratio between amplitudes of tensor and scalar perturbations. There is at present a huge effort to detect B-type polarization of the CMB radiation which would make determination of the amplitude of the tensor perturbations $A_T$ possible[^6]. At present, knowing the value of $A_S$ and having observational constraint on $A_T$, upper bound on the value of $r$ can be found. The strongest constraint comes from observations of the Planck satellite: $r < 0.11$ (95% CL) [@Ade:2013uln]. The theoretically predicted values of $r$ can be confronted with this bound allowing for elimination of some possible inflationary scenarios. In particular, the massive model of inflation is no more preferred in the light of the new Planck constraint [@Ade:2013uln; @Ijjas:2013vea].
Let us calculate the tensor-to-scalar ratio $r$ for the models studied in this paper. For the case with $\Omega-$deformed Minkowski vacuum normalization we obtain $$r = 16\epsilon \frac{\left(1+ \delta_H \right)}{\left(1+2 \delta_H \right)} =
16\epsilon\left(1- \delta_H \right) +\mathcal{O}(\delta_H^2).
\label{rOM}$$ Based on this and equation (\[nTOM\]) the expression for the tensor spectral index $$r \approx - 8 n_{T} \left(1-\delta_H \right).$$
As we have shown in the previous section, for the massive scalar field $\epsilon \approx 0.01$. For the classical case ($\delta_H=0$) this would give us $r = 16\epsilon = 0.16$, which is in contradiction with the Planck constraint $r < 0.11$. This reflects the mentioned disagreement between the massive scalar field model of inflation and the new Planck data. In the past, when the observational bound on the value of $r$ was weaker, the massive scalar field model of inflation was favored by the data. It is worth noticing that, by applying $\epsilon \approx 0.01$ to (\[rOM\]), together with the Planck constraint on $r$, we find that $\delta_H \gtrsim 0.3$. Therefore, presence of the quantum holonomy corrections helps to fulfill the observational bound. However, this would require the critical energy density $\rho_{\text{c}}$ to be much smaller than the Planck energy density.
For the Bojowald-Calcagni normalization we obtain $$r = 16\epsilon \frac{\left(1+2\delta_H \right)}{\left(1+3 \delta_H \right)} = 16\epsilon\left(1- \delta_H \right) +\mathcal{O}(\delta_H^2),$$ which is the same as for the $\Omega-$deformed Minkowski vacuum normalization. Finally, based on this and (\[nTBC\]) the expression for the tensor spectral index $$r \approx - 8 n_{T} \left(1-2\delta_H \right).$$
Summary
=======
In this paper we found holonomy corrections to inflationary power spectra. Such corrections reflect a discrete nature of space at the Planck scale predicted by loop quantum gravity. Calculations were performed for the slow-roll type inflation driven by a single self-interacting scalar field. The derivations were done up to the first order in the slow-roll parameters $\epsilon$ and $\eta$ as well as in the leading order in the parameter $\delta_H$, characterizing holonomy corrections.
An important issue while considering quantum fields on expanding backgrounds is a proper normalization of the modes. In our calculations we assumed that only ingoing modes are present. Short scale normalization of these modes is a subject of ambiguity due to presence of the quantum holonomy effects. We considered two, best motivated, types of normalization. The first one was based on adiabatic vacuum (WKB) approximation, while the second one was based on the method proposed by Bojowald and Calcagni in Ref. [@Bojowald:2010me].
For the first type of normalization, spectral indices are not quantum corrected in the leading order. To be precise, linear corrections in $\delta_H$ are expected. However, they are multiplied not by $\epsilon$ or $\eta$ but $\epsilon^2$ or $\eta^2$ terms. These higher order contributions were not studied systematically in this paper. Nevertheless, calculation of the holonomy-corrected inflationary spectrum including $\mathcal{O}(\epsilon^2, \eta\epsilon,\eta^2)$ terms is a natural generalization of the results presented here. This would help constraining $\delta_H$ if sufficiently accurate observational data are available. Investigation of the higher order corrections in the light of the present CMB data is, however, not possible.
As we have shown, equation of motion for tensor modes with holonomy corrections can be derived from the wave equation defined on *effective metric*, which encodes quantum gravitational effects. This observation allowed us to define a proper inner product for the tensor modes and to show that normalization of tensor modes is obtained by satisfying the classical Wronskian condition.
In this paper we focused on the region where $\Omega>0$. Much more interesting is behavior of modes in the vicinity of $\Omega=0$ and for $\Omega<0$ where the equations of modes become elliptic. The issue of imposing initial conditions at $\Omega=0$ will be a subject of the forthcoming paper [@MielczarekNEW]. Evolution of tensor modes across the region with negative $\Omega$ was addressed in Ref. [@Linsefors:2012et]. As it was shown there, tensor power spectrum is enormously amplified in the UV regime. This new behavior certainly deserves more detailed studies. Furthermore, investigation of simultaneous effects of holonomy and inverse volume corrections is now possible thanks to new results presented in Ref. [@Cailleteau:2013kqa].
I would like to thank Gianluca Calcagni for useful discussion.
[99]{}
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[^1]: The same symmetries, but for crystal lattice, were applied in the mentioned effective model of refractive index.
[^2]: Under assumption that the potential of the inflaton field is quadratic.
[^3]: It is worth noticing that this condition differs from the classical one $k \approx
\mathcal{H}$ due to presence of time dependent function $\Omega$. Furthermore, it is worth to stress that $\frac{z^{''}}{z}$ is a $\Omega$-dependent function leading corrections in expression $\frac{z^{''}}{z} \approx \mathcal{H}^2 \simeq \frac{1}{\tau^2}$. In particular, for the tensor modes $\frac{z_T^{''}}{z_T} = \mathcal{H}^2\left[2-\epsilon(1+5\delta_H)+\dots \right]$. The correction due to $\delta_H$ contributes however together with the $\epsilon$ factor, contrary to the contribution $\Omega=1-2\delta_H+\dots$ in front of the $k^2$ factor.
[^4]: In the original paper [@Bojowald:2010me] the expansion was performed not in therms of $\delta_H$ but $\delta_{Pl}$ relevant for inverse volume corrections.
[^5]: Here we use the simplified de Sitter solution instead of $\mathcal{H} = - \frac{1}{\tau}\cdot
\frac{1}{\left[1-\epsilon(1-\delta_H)]\right]}$, which is sufficient within the considered order of approximation.
[^6]: To be precise, the B-type polarization of the primordial origin was not detected yet. The B-type polarization due to gravitational lensing was recently observed for the first time by the SPTpol observatory [@Hanson:2013hsb].
|
---
abstract: |
We analyse quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space and measurements are supposed to be rank-one POVMs, possibly with additional constraints. Specifically, we consider sets of von Neumann measurements and sets of local observables. We provide upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set. The bounds exploit tools from real algebraic geometry and lead to generic results that do not only show the existence of good measurements but guarantee that almost all measurements with the same dimension characteristic perform equally well.
In particular, we show that on an $n$-dimensional Hilbert space any two states of a semi-algebraic subset can be discriminated by $k$ generic von Neumann measurements if $k(n-1)$ is larger than twice the dimension of the subset. In case the subset is given by states of rank at most $r$, we show that $k$ generic von Neumann measurements suffice to discriminate any two states provided that $k(n-1)>4r(n-r)-2$. Furthermore, we obtain corresponding results for low-rank matrix recovery of hermitian matrices and the phase retrieval problem.
author:
- Michael Kech
- 'Michael M. Wolf'
bibliography:
- 'bibliography.bib'
title: 'Quantum Tomography of Semi-Algebraic Sets with Constrained Measurements'
---
Introduction
============
Quantum state tomography, which aims at identifying quantum states from the outcomes of an experiment, is a central task in quantum information science. Full state tomography is often challenging and sometimes infeasible. However, if there is some prior information about the state under investigation, this can considerably simplify the problem: the number of measurement settings necessary to uniquely identify a given state can significantly decrease if the state is not arbitrary but is known to lie on a confined subset of state space.
Using topological properties of the measurement map and the constrained set, lower bounds on the minimal number of measurement settings necessary to discriminate any two pure states were obtained in [@heinosaari2013quantum]. Relating these topological features of the measurement map to stability properties, it was shown in [@kech] that under the premise of stability the approach of [@heinosaari2013quantum] can be generally applied. Using this result, lower bounds on the necessary number of measurement settings for several other subsets were obtained in [@kech].
The present paper deals with the issue of finding upper bounds: given a subset of state space, find a measurement scheme that can discriminate any two states of this subset with as few measurement settings as possible. This appears to be a rather hard problem in general. Already in the case of pure state quantum tomography it has received significant attention in topology [@milgram1967immersing; @mayer], quantum information science [@weigert1992pauli; @amiet1999reconstructing; @amiet1998reconstructing2; @finkelstein2004pure; @flammia2005minimal; @heinosaari2013quantum; @gross2010quantum; @mondragon2013determination; @CarmeliTeikoJussi1; @CarmeliTeikoJussi2; @flammia2005minimal; @QCS2] and sampling theory [@balan2006signal; @conca2014algebraic; @bodmann2013stable; @GrossPhaseLift].
In addition to constraining the set of states, we also restrict the set of measurements in order to capture the fact that arbitrary measurements may not be feasible in an experiment. The imposed constraints could for example be the restriction to von Neumann measurements or to local measurements when dealing with a multipartite system. The case of pure state tomography with von Neumann measurements was addressed in [@mondragon2013determination; @jaming2010uniqueness; @carmeli2015many]. In [@mondragon2013determination; @jaming2010uniqueness] it was shown that any two pure states can be discriminated by merely $4$ von Neumann measurements. This is known to be sharp for pure states of an $n$-dimensional Hilbert space if $n>4$ and [@carmeli2015many] has a special focus on the cases $n\leq 4$. The more general setting of low-rank matrix recovery with restricted measurements was considered in [@kueng2014low]. However, their focus is to determine the asymptotic behaviour, and this allows us to improve on some of their results.
We propose a method that can deal with these problems rather generally and we then apply it to different scenarios.
In this paper we neither consider the statistical aspects of quantum tomography nor the algorithmic problem of reconstructing the state from the measurement data.
*Outline.* In Section II we fix notation, introduce measurement schemes that are relevant in the following and give some preliminary results about hermitian matrices of bounded rank. Furthermore, we illustrate the connection between phase retrieval and quantum tomography.
In Section III, we propose a method to find sets of measurements that can discriminate any two states of a given subset of the state space, generalizing the approach taken in [@balan2006signal] to find frames for the phase retrieval problem. The method can be applied to all semi-algebraic subsets and it can naturally deal with constrained measurement like e.g. von Neumann measurements. Rather than giving explicit constructions, the method asserts that almost all sets of measurement that fulfil certain constraints allow for a unique identification. As a first example we apply this procedure to low-rank matrix recovery, showing that a generic frame with $m>4r(n-r)$ frame vectors can discriminate any two hermitian matrices of rank at most $r$. This generalizes [@conca2014algebraic] where the case $r=1$ was considered. In Section V, it is shown that the statement also holds when restricting to Parseval frames.
In Section IV, we prove that under a further condition the sets of measurements obtained by the method introduced in Section III fulfil the stability property introduced in [@kech]. In the scenarios where the method is feasible this condition is satisfied and therefore the stability property holds rather generally.
In Section V, we present the main result of this paper. Loosely speaking, it asserts that one can perform tomography on all semi-algebraic subsets of the state space by measuring sets of positive operator valued measures that consist exclusively of rank one operators, in particular von Neumann measurements. From this result we straightforwardly obtain Whitney type embedding results for these measurement schemes. Furthermore, we consider the problem of discriminating states of bounded rank: In [@heinosaari2013quantum; @kech] lower bounds on the number of measurement outcomes necessary to uniquely identify quantum states with bounded rank were established and these lower bounds turned out to be close to the upper bounds obtained in [@heinosaari2013quantum] where it was shown that $4r(n-r)$ measurement outcomes suffice in order to identify states of an $n$-dimensional system with rank at most $r$. However, the measurement that does realize this upper bound has a rather complicated structure. We prove that the same upper bounds as in [@heinosaari2013quantum] can be realized when measuring a positive operator valued measure which exclusively consist of rank one operators and we prove similar results for measuring sets of von Neumann measurements. Note that our results come with less measurement outcomes than the compressed sensing approach of [@gross2010quantum], however we do not provide a tractable reconstruction procedure.
Section VI deals with the problem of reconstructing states of multipartite systems from the expectation values of local observables. Just like in Section V, we first give a theorem stating that one can do tomography on all semi-algebraic subsets of the state-space by performing measurements of this type. Then we obtain Whitney type embedding results and also for the problem of identifying states of bounded rank we obtain corresponding results.
In Section VII, proofs of technical results are given.
Most of our results assert that almost all measurements have a certain property. In the appendix we present the measure with respect to which this is true.
Preliminaries
=============
Throughout $\mathcal{H}$ denotes a finite-dimensional complex Hilbert space. $H(\mathcal{H})$ denotes the real vector space of hermitian operators on $\mathcal{H}$ and $\mathcal{S}(\mathcal{H})$ denotes the set of quantum states on $\mathcal{H}$, i.e. $\mathcal{S}(\mathcal{H})=\{\varrho\in H(\mathcal{H}):\varrho\geq 0, \text{tr}(\varrho)=1\}$. We regard $H({\ensuremath{\mathcal{H}}})$ as an inner product space, equipping it with the Hilbert-Schmidt inner product. The Hilbert Schmidt norm is denoted by $\|\cdot\|_2$. By $SH({\ensuremath{\mathcal{H}}}):=\{X\in H({\ensuremath{\mathcal{H}}}):\ \|X\|_2^2=\text{tr}(X^2)=1\}$ we denote the unit sphere in $H({\ensuremath{\mathcal{H}}})$. Furthermore, for a subset $A\subseteq H({\ensuremath{\mathcal{H}}})$, $\Delta(A)$ denotes the set of differences of operators in $A$, i.e. $\Delta(A)=\{X-Y:X,Y\in A\}$. $M(m,n,\mathbb{C})\ (M(m,n,{\mathbb{R}}))$ denotes the set of complex (real) $m\times n$ matrices and we write $M(n,\mathbb{C})\ (M(n,{\mathbb{R}}))$ as shorthand for $M(n,n,\mathbb{C})\ (M(n,n,{\mathbb{R}}))$.
Constrained Measurement Schemes {#constrained-measurement-schemes .unnumbered}
-------------------------------
In quantum mechanics positive operator valued measures (POVMs) are used to describe general measurements [@holevo2011probabilistic; @busch1995operational]. For the purpose of this paper a POVM on ${\ensuremath{\mathcal{H}}}$ is a tuple $P=(Q_{1},...,Q_{m})$ of positive semidefinite operators on ${\ensuremath{\mathcal{H}}}$ such that $$\begin{aligned}
\sum_{i=1}^{m}Q_{i}={\ensuremath{\mathds{1}}}_{\mathcal{H}}.\end{aligned}$$ An element of $P$ is called an effect operator. We define the dimension of $P$ by $\dim P:=|P|$.
A whole measurement scheme might consist of measuring more than one POVM.
A measurement scheme on ${\ensuremath{\mathcal{H}}}$ is a tuple $M=(P_1,...,P_k)$ of POVMs on ${\ensuremath{\mathcal{H}}}$. We define the dimension of $M$ by $\dim M:=\dim P_1+...+\dim P_k$.
A POVM $P$ can be identified with the measurement scheme that just contains $P$. In the following we sometimes make use of this identification and regard POVMs as measurement schemes.
A POVM $P=(Q_{1},...,Q_{m})$ induces a linear map $$\begin{aligned}
h_{P}:H(\mathcal{H})&\to \mathbb{R}^{m} \\
X&\mapsto\big( \text{tr}(Q_{1}X),...,\text{tr}(Q_{m}X) \big).\end{aligned}$$ Similarly a measurement scheme $M=(P_1,...,P_k)$ induces a linear map $$\begin{aligned}
h_{M}:H(\mathcal{H})&\to \mathbb{R}^{|P_1|+...+|P_k|} \\
X&\mapsto\big( h_{P_1}(X),...,h_{P_k}(X) \big).\end{aligned}$$
\[defcomplete\] A measurement scheme $M$ is called $\mathcal{R}$-complete for a subset $\mathcal{R}\subseteq \mathcal{S}(\mathcal{H})$ if $h_{M}|_{\mathcal{R}}$ is injective.
Our main results are statements about rank one POVMs and von Neumann measurements, so let us define these terms: A POVM $P$ is called rank one POVM if all effect operators are of rank one. We denote the set of $m$-dimensional rank one POVMs on ${\ensuremath{\mathcal{H}}}$ by $\mathcal{M}_1^m({\ensuremath{\mathcal{H}}})$. In the following we implicitly assume that $m\geq \dim{\ensuremath{\mathcal{H}}}$ because otherwise $\mathcal{M}_1^m({\ensuremath{\mathcal{H}}})$ would be empty.
Later on we often use the following correspondence between linear isometries and $\mathcal{M}_1^m(\mathbb{C}^{n})$: The equations $$\begin{aligned}
M^\dagger M={\ensuremath{\mathds{1}}}_n,\ M\in M(m,n,\mathbb{C}),\end{aligned}$$ can be considered as real algebraic equations under the identification $M(m,n,\mathbb{C})\simeq {\mathbb{R}}^{2nm}$. The solution set $U(m,n)$ is the set of linear isometries $U:\mathbb{C}^n\to\mathbb{C}^m$. Note that $U(m,n)$ is non-empty if and only if $m\geq n$ and that for $n=m$ it is the set of unitaries. We write $U(n)$ as shorthand for $U(n,n)$.
Let $\{|i\rangle\}_{i\in\{1,...,m\}}$ be the standard basis of $\mathbb{C}^m$. Then, the sought correspondence is given by the map $$\begin{aligned}
\label{phi}
\begin{split}
\phi:U(m,n)&\to\mathcal{M}_{1}^m(\mathbb{C}^{n})\\
U&\mapsto (U^\dagger|1\rangle\langle 1|U,...,U^\dagger|m\rangle\langle m|U).
\end{split}\end{aligned}$$
If the effect operators of a POVM are projections on mutually orthogonal subspaces, the POVM is called von Neumann measurement. In this paper, we just deal with rank one von Neumann measurements and therefore, in the following, the term von Neumann measurement always refers to rank one von Neumann measurements. Note, that the set of rank one von Neumann measurements is precisely the set of $(\dim{{\ensuremath{\mathcal{H}}}})$-dimensional rank one POVMs.
The measurement scheme consisting of $k$ $m$-dimensional rank one POVMs on ${\ensuremath{\mathcal{H}}}$ is denoted by $\mathcal{M}^m_{1,k}({\ensuremath{\mathcal{H}}})$, i.e. $$\begin{aligned}
\mathcal{M}^m_{1,k}({\ensuremath{\mathcal{H}}})=\{(P^1,...,P^k): P^i\in\mathcal{M}_{1}^{m}({\ensuremath{\mathcal{H}}})\}.\end{aligned}$$ For $m=\dim{\ensuremath{\mathcal{H}}}$ this is the set of $k$ rank one von Neumann measurements which, we denote by $\mathcal{M}^k_{\text{vN}}({\ensuremath{\mathcal{H}}})$.
Hermitian Matrices of Bounded Rank {#hermitian-matrices-of-bounded-rank .unnumbered}
----------------------------------
In this section we prove a lemma about hermitian operators with bounded rank, which is frequently used in the following. Denote by $\mathcal{P}_r({\ensuremath{\mathcal{H}}})$ the set of hermitian operators on ${\ensuremath{\mathcal{H}}}$ with rank at most $r$, i.e. $\mathcal{P}_r({\ensuremath{\mathcal{H}}}):=\{X\in \mathcal{H}({\ensuremath{\mathcal{H}}}): \text{rank}(X)\leq r\}$. We write $\mathcal{P}_{r}^{n}$ as shorthand for $\mathcal{P}_{r}(\mathbb{C}^{n})$.
\[lemrank\] $\mathcal{P}_{r}^{n}$ is a real algebraic set of dimension $r(2n-r)$.
First note that $\mathcal{P}_{r}^{n}$ is a real algebraic set: It is given by the set of points $X\in M(n,\mathbb{C})$ for which all $(r+1)\times (r+1)$-minors vanish and that satisfy $X=X^{\dagger}$. These conditions turn into a set of real algebraic equations under the canonical identification $M(n,\mathbb{C})\simeq\mathbb{R}^{2n^2}$.
To determine the dimension of $\mathcal{P}_{r}^{n}$ consider the semi-algebraic set $V_{r}^{n}=\{(P_{1},...,P_{r}):P_i\in\mathcal{P}_{1}^{n},\ \text{tr}(P_{i}P_{j})=\delta_{ij},\ P_i\geq 0\}$ [^1]. The dimension of $V_{r}^{n}$ is given by $r(2n-r)-r$. To see this, consider the smooth and transitive action of $U(n)$ on the complex matrices $M(n,\mathbb{C})$ given by $(U,M)\to (U,UMU^\dagger)$ and let $V_D$ be the orbit of the diagonal matrix $D:=\text{diag}(r,r-1,...,1,0,...)$ under this action. Noting that the stabilizer subgroup of $D$ is $U(n-r)\times U(1)^r$ we obtain $V_D\simeq U(n)/(U(n-r)\times U(1)^r)$ by Theorem 3.62 of [@warner1971foundations]. But the semi-algebraic map $\psi:V_r^n\to V_D,\ (P_{1},...,P_{r})\mapsto \sum_{j=1}^rjP_j$ is clearly bijective. Hence we find $\dim V_r^n=\dim(U(n)/(U(n-r)\times U(1)^r))=n^2-(n-r)^2-r=r(2n-r)-r$ by Theorem 2.8.8 and Proposition 2.8.14 of [@bochnak1998real].
The semi-algebraic map $$\begin{aligned}
\label{1}
\begin{split}
\eta:{\mathbb{R}}^r\times V_{r}^{n}&\to\mathcal{P}_{r}^{n}\\
(\lambda_1,...,\lambda_r,P_{1},...,P_{r})&\mapsto \sum_{i=1}^{r} \lambda_i P_i.
\end{split}\end{aligned}$$ is clearly surjective. By Theorem 2.8.8 of [@bochnak1998real], we hence conclude that $\dim \mathcal{P}_{r}^{n}\leq\dim V_{r}^{n}+r=r(2n-r)$ and furthermore that indeed $\dim \mathcal{P}_{r}^{n}=r(2n-r)$ by noting that $\phi$ is injective if we require $\lambda_1>...>\lambda_r>0$.
\[corrank\] The set $\mathcal{D}_1:=\{X\in\mathcal{P}_{r}^{n}:\ \text{tr}(X^2)=2\}$ is a real algebraic set of dimension $r(2n-r)-1$ and the set $\mathcal{D}_2:=\{X\in\mathcal{P}_{r}^{n}:\ \text{tr}(X^2)=2,\ \text{tr}(X)=0\}$ is a real algebraic set of dimension $r(2n-r)-2$.
From the proof of Lemma \[lemrank\] it is immediate that both $\mathcal{D}_1$ and $\mathcal{D}_1$ are real algebraic sets. To determine the dimension of $\mathcal{D}_1$, one can go along the lines of the proof of Lemma \[lemrank\] and simply replace ${\mathbb{R}}^n$ by the unit sphere $S^{n-1}$ in the definition of the mapping $\eta$. Similarly, to determine the dimension $\mathcal{D}_2$, one can go along the lines of the proof of Lemma \[lemrank\] and this time replace ${\mathbb{R}}^n$ by $\{x\in S^{n-1}:\sum_{i=1}^nx_i=0\}$ in the definition of the mapping $\eta$.
Frames and Rank One POVMs {#frames-and-rank-one-povms .unnumbered}
-------------------------
Finally, we discuss the connection between pure state tomography and the phase retrieval problem in sampling theory. A finite set $F=\{v_1,...,v_m\}$ of vectors in $\mathbb{C}^n$ is called a frame if there exist constants $a,b>0$ such that $$\begin{aligned}
\label{eqframe}
a\|x\|_2^2\leq\sum_{i=1}^{m}|\langle x,v_i\rangle|^2\leq b\|x\|_2^2\ \text{for all}\ x\in\mathbb{C}^n.\end{aligned}$$ A frame $F=\{v_1,...,v_m\}$ induces a mapping $$\begin{aligned}
\begin{split}
M_F:\mathbb{C}^n\delimiter"502F30E\mathopen{{\sim}}&\to {\mathbb{R}}^m\\
[x]&\mapsto (|\langle v_1,x\rangle|^2,...,|\langle v_m,x\rangle|^2)
\end{split}\end{aligned}$$ where $x\sim y$ iff there is a $\lambda\in{\mathbb{R}}$ such that $x=e^{i\lambda}y$ [^2]. Since the task in phase retrieval is to reconstruct signals modulo phase from intensity measurements, one considers frames $F$ such that $M_F$ is injective.
To each frame $F=\{v_1,...,v_m\}$ one can associate a tuple $P_{F}=(|v_1\rangle\langle v_1|,...,|v_m\rangle\langle v_m|)$ of positive rank one operator on $\mathbb{C}^n$. $P_F$ induces a map $$\begin{aligned}
\label{map}
\begin{split}
h_{P_F}:H(\mathbb{C}^n)&\to{\mathbb{R}}^m\\
X&\to(\text{tr}(X|v_1\rangle\langle v_1|),...,\text{tr}(X|v_m\rangle\langle v_m|)).
\end{split}\end{aligned}$$ Noting that $h_{P_F}(|x\rangle\langle x|)=M_F(x)$, we conclude that $h_{P_F}|_{\mathcal{P}^n_1}$ is injective if and only if $M_F$ is injective.
A corollary of one of our main results is a statement about Parseval frames, so let us define this term. A frame $F$ is called tight frame if $a=b$ in inequality . If in addition $a=b=1$, $F$ is called Parseval frame.
The following proposition shows the well-know fact that Parseval frames correspond to rank one POVMs.
\[proppars\] Let $F$ be a Parseval frame. Then the associated set of rank one operators $P_F$ is a POVM.
Let $F=\{v_1,...,v_m\}$. Since $F$ is a Parseval frame, we obtain the following equality from inequality : $$\begin{aligned}
\sum_{i=1}^m|\langle v_i,x\rangle|^2=\|x\|_2^2.\end{aligned}$$ This can be rewritten as $$\begin{aligned}
\sum_{i=1}^m|\langle v_i,x\rangle|^2=\text{tr}(|x\rangle\langle x|\sum_{i=1}^m|v_i\rangle\langle v_i|)=\|x\|^2.\end{aligned}$$ But since this holds for all $x\in\mathbb{C}^n$ we conclude that $\sum_{i=1}^m|v_i\rangle\langle v_i|={\ensuremath{\mathds{1}}}_{\mathbb{C}^n}$: Assume $\sum_{i=1}^m|v_i\rangle\langle v_i|\neq {\ensuremath{\mathds{1}}}_{\mathbb{C}^n}$. Since $\sum_{i=1}^m|v_i\rangle\langle v_i|$ is hermitian there has to be an eigenvector $w$ of $\sum_{i=1}^m|v_i\rangle\langle v_i|$ with eigenvalue $\lambda\neq 1$. But then $\langle w|\sum_{i=1}^m|v_i\rangle\langle v_i|w\rangle=\lambda \|w\|_2^2\neq \|w\|_2^2$, a contradiction.
Note that the correspondence is given by the map $\phi$ defined in equation where the frame vectors are given by the rows of the isometry.
Let $P$ be a POVM. In pure state tomography, not $h_{P}|_{\mathcal{P}^n_1}$ is required to be injective, but $h_{P}|_{\mathcal{S}_1^n}$ where $\mathcal{S}_1^n:=\{\varrho\in\mathcal{S}(\mathbb{C}^n):\varrho^2=\varrho\}$ is the set of pure states. However, by the definition of a POVM, ${\ensuremath{\mathds{1}}}_n\in P$ and this implies that if $h_{P}|_{\mathcal{S}_1^n}$ is injective, also $h_{P}|_{\mathcal{P}_1^n}$ is injective. From this point of view, pure state quantum tomography with rank one POVMs is equivalent to phase retrieval with Parseval frames.
Method and First Example {#method}
========================
Let us begin by explaining the basic idea of the method we utilize to find one-to-one measurement schemes which originates from the approach taken in [@balan2006signal] to find frames for the phase retrieval problem.
The method essentially relies on the following observation: A measurement scheme $P:=((Q^1_1,...,Q^1_m),...,(Q^k_1,...,Q^k_m))$ is $\mathcal{R}$-complete with respect to a subset $\mathcal{R}\subseteq \mathcal{S}({\ensuremath{\mathcal{H}}})$ if and only if the equations $$\begin{aligned}
\label{constf}
\text{tr}(Q^j_i X)=0,\ \ \,\ i\in\{1,...,m-1\},j\in\{1,...,k\}\end{aligned}$$ have no solution for $X\in\Delta(\mathcal{R})-\{0\}$.
For a given subset $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$, we want to characterize non-injective measurement schemes via the equations and use the dimension theory of semi-algebraic sets to show that these have measure zero. Therefore, we consider measurement schemes that are constrained by real algebraic equalities or inequalities. In the following, the set of measurement schemes is a semi-algebraic set $\mathcal{M}$ such that for all $M\in\mathcal{M}$ we have $\dim P=m, \forall P\in M$ and $|M|=k$ where $m,k\in\mathbb{N}$ are some fixed numbers. For example, if $k=1$, this could be the restriction to the set of $m$-dimensional rank one POVMs $\mathcal{M}_1^m({\ensuremath{\mathcal{H}}})$. Furthermore, in order to ensure that the equations in fact become real algebraic equations, we have to replace $\Delta(\mathcal{R})-\{0\}$ by a suitable semi-algebraic set. We do this by constructing a semi-algebraic set $\mathcal{D}\subseteq H({\ensuremath{\mathcal{H}}})$[^3] with the following property: If there is a measurement scheme $M$ and an $X\in\Delta(\mathcal{R})-\{0\}$ with $$\begin{aligned}
\label{const}
h_M(X)=0\end{aligned}$$ then there exists $X^\prime\in\mathcal{D}$ with $$\begin{aligned}
\label{constff}
h_M(X^\prime)=0.\end{aligned}$$ If a semi-algebraic set $\mathcal{D}\subseteq H({\ensuremath{\mathcal{H}}})$ with $0\notin\mathcal{D}$ has this property, we say that $\mathcal{D}$ represents $\Delta(\mathcal{R})-\{0\}$.
The solution set of the equations characterizes the non-injective measurement schemes: Let $\tilde{\mathcal{M}}$ be the real semi-algebraic set obtained from $\mathcal{M}\times\mathcal{D}$ by imposing the equations . By construction of $\mathcal{D}$, the non-injective measurement schemes are contained in the projection of $\tilde{\mathcal{M}}\subseteq\mathcal{M}\times\mathcal{D}$ on the first factor with the canonical projection $\pi_1:\mathcal{M}\times\mathcal{D}\to\mathcal{M}$. But if $\dim \tilde{\mathcal{M}}<\dim\mathcal{M}$, we also have $\dim\pi_1( \tilde{\mathcal{M}})<\dim\mathcal{M}$ [^4] and thus the non-injective measurement schemes have measure zero in $\mathcal{M}$. Here we used the well-know fact that, for a suitably chosen measure, the measure of a semi-algebraic subset $S$ of a semi-algebraic set $A$ has measure zero in $A$ if $\dim A>\dim S$. For more details on the measure see Appendix \[appendixD\].
This approach is most efficient if the equations are transversal to $\mathcal{M}\times\mathcal{D}$. In this case $\dim \tilde{\mathcal{M}}<\dim\mathcal{M}$ is equivalent to $k(m-1)>\dim \mathcal{D}$ and thus the quality of our result is determined by how low-dimensional we can choose the semi-algebraic set $\mathcal{D}$.
To illustrate how this procedure works, let us consider the problem of low-rank matrix recovery with frames. We show that any two hermitian matrices of rank at most $r$ can be discriminated from a generic frame with $m\geq 4r(n-r)$ frame vectors. The proof we give is inspired by the proof of Theorem 3.1 in [@balan2006signal]. Let $r\in\{1,\dots,[n/2]\}$[^5].
\[frame\] Let $m\geq 4r(n-r)$. For almost all frames $F=\{v_1,..,v_{m}\}$ the map $h_{P_F}|_{\mathcal{P}_r^n}$ is injective.
Let $F=(v_1,...,v_m),\ v_i\in\mathbb{C}^n,$ and consider the equations $$\begin{aligned}
\label{eqap}
\langle v_i|X|v_i\rangle=0,\ i\in\{1,...,m\},\end{aligned}$$ in $v_i\in\mathbb{C}^n$, $X\in\Delta(\mathcal{P}_r^n)-\{0\}$. As explained above, these equations determine the subset $N$ of $F\in\mathbb{C}^{nm}\simeq \mathbb{R}^{2nm}$ for which $h_{P_F}|_{\mathcal{P}_r^n}$ fails to be injective.
Note that $\Delta(\mathcal{P}_r^n)-\{0\}=\mathcal{P}_{2r}^n-\{0\}$. Consider the algebraic set $\mathcal{D}:=\{X\in\mathcal{P}_{2r}^n:\text{tr}(X^2)=1\}$ and note that we have $\dim{\mathcal{D}}=4r(n-r)-1$ by Corollary \[corrank\]. Furthermore, $\mathcal{D}$ represents $\Delta(\mathcal{P}_r^n)-\{0\}$: Clearly $0\notin\mathcal{D}$. Next, consider a measurement scheme $M$ and $X\in\mathcal{P}_{2r}^n-\{0\}$ such that $h_M(X)=0$. But then there is $X^\prime:=\frac{X}{\|X\|_2}\in\mathcal{D}$ such that $h_M(X^\prime)=\frac{1}{\|X\|_2}h_M(X)=0$.
Under the identification $\mathbb{C}^{nm}\simeq \mathbb{R}^{2nm}$ the equations are $m$ equations on the real algebraic set $\mathbb{C}^{nm}\times\mathcal{D}$ and next we prove that imposing these equations decreases the dimension of $\mathbb{C}^{nm}\times\mathcal{D}$ by at least $m$: Note that it suffices to prove that imposing the equation on $\mathbb{C}^{nm}$, for fixed $X\in\mathcal{D}$, decreases the dimension by at least $m$. But for fixed $X\in\mathcal{D}$, the $i$-th equation of just involves the variables of the $i$-th factor in $(\mathbb{C}^{n})^m$. Thus it suffices to prove that for given $X\in\mathcal{D}$ imposing the equation $$\begin{aligned}
\label{eqnframe}
p(v):=\langle v|X|v\rangle=0,\ v\in\mathbb{C}^n,\end{aligned}$$ on $\mathbb{C}^n\simeq{\mathbb{R}}^{2n}$ decreases the dimension by at least one. But for given $X\in\mathcal{D}$ there is $v\in\mathbb{C}^n$ such that $p(v)=\langle v|X|v\rangle=\text{tr}(X| v\rangle\langle v|)\neq 0$ because $H(\mathbb{C}^n)$ has a basis of rank one operators and $X\neq 0$. Thus, is a non-trivial algebraic equation on the irreducible algebraic set $\mathbb{C}^n\simeq{\mathbb{R}}^{2n}$. But this immediately implies that does decrease the dimension [^6].
Let $\mathcal{M}$ be the algebraic subset of $\mathbb{C}^{nm}\times\mathcal{D}$ obtained by imposing the equations and denote by $\pi_1: \mathbb{C}^{nm}\times\mathcal{D}\to\mathbb{C}^{nm}$ the canonical projection on the first factor. For $m>\dim \mathcal{D}=4r(n-r)-1$, we find $\dim \pi_1(\mathcal{M})<\dim \mathbb{C}^{nm}=2nm$ since imposing the equations on $\mathbb{C}^{nm}$ decreases the dimension by at least $m$. Thus, we conclude that $\pi_1(\mathcal{M})$ has Lebesgue measure zero [^7] in $\mathbb{C}^{nm}$. Hence, the subset of $F\in\mathbb{C}^{nm}$ for which $M_F$ is injective has full Lebesgue measure. Note, that the subset of frames in $\mathbb{C}^{nm}$ has full Lebesgue measure for $m\geq n$. Choosing the measure on the set of frames to be the restriction of the Lebesgue measure, also the subset of frames for which $M_F$ is injective has full measure.
In Corollary \[parseval\] we obtain the corresponding result for Parseval frames. For $r=1$, this is the phase retrieval problem and in this case Proposition \[frame\] reproduces the main result of [@conca2014algebraic].
Let $m\ge 4n-4$. For almost all frames $F=\{v_1,..,v_{m}\}$ the map $M_F$ is injective.
Let $F=\{v_1,...,v_m\},\ v_i\in\mathbb{C}^n,$ and consider the equations $$\begin{aligned}
|\langle v_i|x\rangle|^2-|\langle v_i|y\rangle|^2=\langle v_i|(|x\rangle\langle x|-|y\rangle\langle y|)|v_i\rangle=0,\ i\in\{1,...,m\},\end{aligned}$$ in $x,y,v_i\in\mathbb{C}^n$ where $|x\rangle\langle x|-|y\rangle\langle y|\neq 0$. These equations determine the subset $N$ of $F\in\mathbb{C}^{nm}\simeq \mathbb{R}^{2nm}$ for which $M_F$ fails to be injective. It is easily seen that the equations $$\begin{aligned}
\label{opo}
\langle v_i|X|v_i\rangle=0,\ i\in\{1,...,m\},\end{aligned}$$ where $X\in\Delta(\mathcal{P}_1^n)-\{0\}$, determine the same subset $N$. But the equations are precisely the equations for $r=1$. Thus, the proof can be concluded by going along the lines of the proof of Proposition \[frame\].
Stability
=========
The measurement schemes obtained by the method presented in Section \[method\] typically come with a stability property. Let $$\begin{aligned}
\mathcal{M}(n_1,...,n_k):=\{M:=(P^1,...,P^k):\ P^i\text{ POVM with }\dim P^i=n_i\}.\end{aligned}$$ In this section we denote $\mathcal{M}(n_1,...,n_k)$ by $\mathcal{M}$. We equip $\mathcal{M}$ with the topology induced by the metric $$\begin{aligned}
d(M,M^\prime):=\|h_M-h_{M^\prime}\|=\sup_{X\in H(\mathbb{C}^n)}\frac{\|h_M(X)-h_{M^\prime}(X)\|_2}{\|X\|_2}\end{aligned}$$ where $M,M^\prime\in\mathcal{M}$.
\[defstab\] Let $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a subset. An $\mathcal{R}$-complete measurement scheme $M\in \mathcal{M}$ is stably $\mathcal{R}$-complete if there exists a neighbourhood $\mathcal{N}$ of $M$ such that every measurement scheme $M^\prime\in \mathcal{N}$ is $\mathcal{R}$-complete.
Let $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a subset and let $\mathcal{D}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a semi-algebraic set that represents $\Delta(\mathcal{R})-\{0\}$. Consider the semi-algebraic map $$\begin{aligned}
\label{psi}
\begin{split}
\psi: \mathcal{D}&\to H(\mathbb{C}^n)\\
X&\mapsto \frac{X}{\|X\|_2}.
\end{split}\end{aligned}$$ By Proposition 2.2.7 and Theorem 2.8.8 of [@bochnak1998real], $\tilde{\mathcal{D}}:=\psi(\mathcal{D})$ is semi-algebraic with $\dim\tilde{\mathcal{D}}\leq\dim\mathcal{D}$. Furthermore $\tilde{\mathcal{D}}$ clearly represents $\Delta(\mathcal{R})-\{0\}$.
\[stab\] If $\tilde{\mathcal{D}}$ is closed, every $\mathcal{R}$-complete measurement scheme $M\in\mathcal{M}$ is stably $\mathcal{R}$-complete.
Note that $\tilde{\mathcal{D}}\subseteq SH(\mathbb{C}^n)$. $SH(\mathbb{C}^n)$ is compact and thus $\tilde{\mathcal{D}}$ is compact being a closed subset of a compact set. By the continuity of the induced map $h_M$ and compactness of $\tilde{\mathcal{D}}$, $\kappa:=\min_{X\in\tilde{\mathcal{D}}}\|h_M(X)\|_2$ exists and $\kappa>0$ since $M$ is $\mathcal{R}$-complete. Now let $B(M,\kappa/2):=\{M^\prime\in \mathcal{M}:\sup_{X\in SH(\mathbb{C}^n)}\|h_M(X)-h_M^\prime(X)\|_2<\kappa/2\}$ and note that $B(M,\kappa/2)$ is open. But then $$\begin{aligned}
\min_{X\in\tilde{\mathcal{D}}}\|h_{M^\prime}(X)\|&\geq \min_{X\in\tilde{\mathcal{D}}}\|h_M(X)\||-\min_{X\in\tilde{\mathcal{D}}}\|h_{M^\prime}(X)-h_M(X)\|\\
&\geq \min_{X\in\tilde{\mathcal{D}}}\|h_M(X)\||-\max_{X\in\tilde{\mathcal{D}}}\|h_{M^\prime}(X)-h_M(X)\|\\
&\geq \kappa-\max_{X\in SH(\mathbb{C}^n)}\|h_{M^\prime}(X)-h_M(X)\|\\
&\geq \kappa-\kappa/2=\kappa/2.\end{aligned}$$ Thus all measurement schemes $M^\prime\in B(M,\kappa/2)$ are $\mathcal{R}$-complete.
Note that $\tilde{\mathcal{D}}$ need not be closed for this lemma to apply: In the situations presented in the following the conclusions solely depend on the dimension of $\tilde{\mathcal{D}}$. By Proposition 2.8.2 of [@bochnak1998real] the dimension of $\tilde{\mathcal{D}}$ coincides with the dimension of its closure $\overline{\tilde{\mathcal{D}}}$ in the norm topology on $H(\mathbb{C}^n)$. Furthermore, by Proposition 2.2.2 of [@bochnak1998real], the closure of a semi-algebraic set is semi-algebraic. Thus $\overline{\tilde{\mathcal{D}}}$ represents $\Delta(\mathcal{R})-\{0\}$ and $\dim\overline{\tilde{\mathcal{D}}}\le\dim\mathcal{D}$.
Complete Rank One POVMs
=======================
Universality of Rank One POVMs {#universality-of-rank-one-povms .unnumbered}
------------------------------
The following lemma is the main technical result of this paper. It asserts that the equations are independent when restricting to rank one POVMs. More precisely let ${\ensuremath{\mathcal{H}}}=\mathbb{C}^n$ and denote by $\{|i\rangle\}_{i\in\{1,...,n\}}$ the standard basis of $\mathbb{C}^n$.
For a fixed non-zero $X\in H(\mathbb{C}^n)$, consider the equations $$\begin{aligned}
\label{eqnblaaaa}
\begin{split}
p_i(M_1,\dots,M_k)&:=\text{tr}(M_i^\dagger|j\rangle\langle j|M_i X)=\langle j|M_iXM_i^\dagger|j\rangle=0,\\
&i\in\{1,...,k\},\ j\in\{1,...,m\},\\
q_i^{jl}(M_1,\dots,M_k)&:=\langle j|M_i^\dagger M_i|l\rangle-\delta_{jl}=0,\\
&i\in\{1,...,k\},\ j,l\in\{1,...,n\},
\end{split}\end{aligned}$$ in $(M_1,\dots,M_k)\in \Pi_{i=1}^kM(m,n,\mathbb{C})$. Under the canonical identification $M(m,n,\mathbb{C})\simeq {\mathbb{R}}^{2nm}$, these can be considered as real algebraic equations in the $2knm$ variables $(M_1,...,M_k)$.
\[LEMvN\] Let $X\in H(\mathbb{C}^n)$ with $X\neq 0$. Imposing the equations on $\Pi_{i=1}^kM(m,n,\mathbb{C})$ decreases the dimension by at least $n^2+k(m-1)$.
Regarding $X\in\mathcal{D}\subseteq H(\mathbb{C}^n)$ as an variable, the equations can be considered as equations on $\prod_{i=1}^kM(m,n,\mathbb{C})\times\mathcal{D}$. Then, Lemma \[LEMvN\] implies that imposing the equations on $\prod_{i=1}^kM(m,n,\mathbb{C})\times\mathcal{D}$ decreases the dimension by at least $n^2+k(m-1)$ for every semi-algebraic set $\mathcal{D}\subseteq H(\mathbb{C}^n)$ with $0\notin\mathcal{D}$.
Since the proof of this result is rather technical we relegate it to Section \[proof\]. Lemma \[LEMvN\] allows us to prove the main theorem of this section.
(Universality)\[THMvN\] For $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ a subset, let $\mathcal{D}$ be a semi-algebraic set that represents $\Delta(R)-\{0\}$. If $k(m-1)>\dim \mathcal{D}$, almost all measurement schemes $M\in\mathcal{M}_{1,k}^m(\mathbb{C}^n)$ are stably $\mathcal{R}$-complete.
Note that Theorem \[THMvN\] reduces the problem of finding an $\mathcal{R}$-complete rank one POVM for some subset $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$ to finding a semi-algebraic subset $\mathcal{D}\subseteq H({\ensuremath{\mathcal{H}}})$ which represents $\Delta(\mathcal{R})-\{0\}$ and in this sense Theorem \[THMvN\] guarantees the universality of rank one POVMs. Furthermore the quality of the result solely depends on the algebraic dimension of $\mathcal{D}$.
The proof of this result can be found in Section \[proof\].
From this Theorem we directly obtain a Whitney type embedding result for rank one POVMs. Essentially, it is a direct consequence of the following lemma.
\[lemdim\] Let $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$. Then $\dim(\Delta(\mathcal{R})-\{0\})\leq 2\dim\mathcal{R}$.
We can assume w.l.o.g that $\mathcal{R}$ is algebraic, because if not we can take its Zariski closure [^8]. Let $\text{Diag}(\mathcal{R}\times \mathcal{R}):=\{(X,Y)\in\mathcal{R}\times \mathcal{R}:X=Y\}$. Noting that $\text{Diag}(\mathcal{R}\times \mathcal{R})$ is an algebraic set, $\mathcal{D}:=(\mathcal{R}\times \mathcal{R})-\text{Diag}(\mathcal{R}\times \mathcal{R})$ is quasi-algebraic. But the semi-algebraic map $$\begin{aligned}
\phi:\mathcal{D}&\to\Delta(\mathcal{R})-\{0\}\\
(X_1,X_2)&\mapsto X_1-X_2\end{aligned}$$ is surjective, and thus $\dim(\Delta(\mathcal{R})-\{0\})\leq \mathcal{D}=2\dim\mathcal{R}$ by Theorem 2.8.8 of [@bochnak1998real].
\[whitney\] Let $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a subset. If $k(m-1)>2\dim \mathcal{R}$, almost all measurement schemes $M\in\mathcal{M}_{1,k}^m(\mathbb{C}^n)$ are stably $\mathcal{R}$-complete.
We can assume w.l.o.g. that $\mathcal{R}$ is algebraic because if not we can consider its Zariski closure. By the proof of Lemma \[lemdim\], $\Delta(\mathcal{R})-\{0\}$ is semi-algebraic and furthermore $\dim(\Delta(\mathcal{R})-\{0\})\leq 2\dim\mathcal{R}$. Finally, Theorem \[THMvN\] with $\mathcal{D}=\Delta(\mathcal{R})-\{0\}$ concludes the proof.
Two special cases of this Theorem may be of particular interest.
Let $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a subset. If $k(n-1)>2\dim \mathcal{R}$, almost all tuples of $k$ von Neumann measurement $M\in\mathcal{M}_{vN}^k(\mathbb{C}^n)$ are $\mathcal{R}$-complete.
This immediately follows from Corollary \[whitney\] for $m=n$.
\[corrrankone\] Let $\mathcal{R}\subseteq\mathcal{S}(\mathbb{C}^n)$ be a subset. If $m-1>2\dim \mathcal{R}$, almost all rank one POVMs $M\in\mathcal{M}_{1}^m(\mathbb{C}^n)$ are stably $\mathcal{R}$-complete.
This immediately follows from Corollary \[whitney\] for $k=1$.
Effectively we have the bound $m-1>\max\{2\dim \mathcal{R},n-2\}$ which is due to the fact that a rank one POVM on $\mathbb{C}^n$ has to be at least $n$-dimensional. If we relax this to merely requiring the POVM to be projective this shortcoming can be avoided, i.e. for projective POVMs $m-1=2\dim \mathcal{R}+1$ can be attained. This can be seen by modifying the proof of Lemma \[LEMvN\].
Rank One POVMs for States of Bounded Rank and States of Fixed Spectrum {#rank-one-povms-for-states-of-bounded-rank-and-states-of-fixed-spectrum .unnumbered}
----------------------------------------------------------------------
In this section we improve the Whitney type bounds of Corollary \[whitney\] for the cases in which the subset $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$ is given by the states of bounded rank or the states of fixed spectrum. The results we obtain in this section easily follow from theorem \[THMvN\]. Let us note that all results of this section can be immediately transferred to measurement schemes which fulfil a universality property analogous to theorem \[THMvN\].
In the following, $r\in\{1,...,[n/2]\}$. Denote by $S_r({\ensuremath{\mathcal{H}}})$ the states with rank at most $r$, i.e. $S_r({\ensuremath{\mathcal{H}}}):=\{\varrho\in \mathcal{S}({\ensuremath{\mathcal{H}}}): \text{rank}(\varrho)\leq r\}$. We write $S_r^n$ as shorthand for $S_r(\mathbb{C}^n)$.
In analogy to the proof of Proposition \[frame\], we first construct the set we use to represent $\Delta(S_r({\ensuremath{\mathcal{H}}}))-\{0\}$ and determine its dimension.
\[set\] The set $\mathcal{D}:=\{X\in\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}}):\ \text{tr}(X)=0,\ \text{tr}(X^2)=2\}$ is an algebraic set that represents $\Delta(S_r({\ensuremath{\mathcal{H}}}))-\{0\}$ and $\dim\mathcal{D}=4r(\dim{\ensuremath{\mathcal{H}}}-r)-2$.
Note that $\mathcal{S}_r({\ensuremath{\mathcal{H}}})\subseteq \mathcal{P}_r({\ensuremath{\mathcal{H}}})$ and thus $\Delta(\mathcal{S}_r({\ensuremath{\mathcal{H}}}))\subseteq \Delta(\mathcal{P}_r({\ensuremath{\mathcal{H}}}))=\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}})$. $\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}})$ is algebraic by Lemma \[lemrank\] and hence $\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}})-\{0\}$ represents $\Delta(\mathcal{S}_r({\ensuremath{\mathcal{H}}}))-\{0\}$. In fact $\Delta(\mathcal{S}_r({\ensuremath{\mathcal{H}}}))-\{0\}$ can be represented by a smaller set. Namely one can consider set $\mathcal{D}:=\{X\in\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}}):\ \|X\|_2^2=tr(X^2)=1,\ tr(X)=0\}$. Note that $\mathcal{D}$ is algebraic by Corollary \[corrank\] and that $0\notin\mathcal{D}$. The equation $tr(X)=0$ just considers the fact that states have unit trace. Next consider a measurement scheme $M$ and $X\in\Delta(\mathcal{S}_r({\ensuremath{\mathcal{H}}}))-\{0\}$ such that $h_M(X)=0$. Then, there is $X^\prime:=\frac{X}{\|X\|_2}\in\mathcal{D}$ such that $h_M(X^\prime)=0$. Hence $\mathcal{D}$ indeed represents $\Delta(\mathcal{S}_r({\ensuremath{\mathcal{H}}}))-\{0\}$. Finally, by Corollary \[corrank\], we have $\dim(\mathcal{D})=\dim(\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}}))-2=4r(n-r)-2$.
\[thmvN\] If $k(m-1)\ge 4r(n-r)-1$, almost all measurement schemes $M\in M_{1,k}^m(\mathbb{C}^n)$ are stably $\mathcal{S}_r^n$-complete.
Using the set of Lemma \[set\] to represent $\Delta(\mathcal{R})-\{0\}$, the result follows directly form Theorem \[THMvN\].
As explained in Section IV.A of [@kech], the lower bounds on the immersion dimension of complex flag manifolds of [@walgenbach2001lower] transfer to lower bounds on the dimension of $\mathcal{S}_r({\ensuremath{\mathcal{H}}})$-complete POVMs. In addition, the discussion following this explanation suggests that the upper bound on $m$ we obtain here is close to optimal.
Next, let us state some corollaries of this theorem.
\[corrvN\] If $m(n-1) \ge 4r(n-r)-1$, almost all tuples of $k$ von Neumann measurements $M\in\mathcal{M}_{\text{vN}}^m(\mathbb{C}^n)$ are stably $\mathcal{S}_r^n$-complete.
This follows from Theorem \[thmvN\] for $m=n$.
For $r=1$ this reproduces the main result of [@mondragon2013determination]. In Table \[constantspectrumbounds\] you can see how this result compares to the lower bounds of [@walgenbach2001lower] for some explicit scenarios.
[lcccccccccccc]{} l\\k & 2 & 3 & 4\
\
5 & 6/7\
\
6 & 6/7\
\
7 & 7/7 & 9/10\
\
8 & 7/7 & 9/10\
\
9 & 7/8 & 9/10 & 12/12\
\
10 & 7/8 & 10/10 & 12/13\
\[corpars\] If $m-1 \ge 4r(n-r)-1$, almost all rank one POVM $P\in\mathcal{M}_1^m(\mathbb{C}^n)$ are stably $\mathcal{S}_r^n$-complete.
This follows from \[thmvN\] for $k=1$.
The following corollary is the analogue of Proposition \[frame\] for Parseval frames.
\[parseval\] Let $m \ge 4r(n-r)$. Then, for almost all Parseval frames $F:=\{v_1,\dots,v_m\}$ the map $h_{P_F}|_{\mathcal{P}_r^n}$ is injective.
By going along the lines of the proof of Lemma \[set\], it is easily seen that $\mathcal{D}:=\{X\in\mathcal{P}_{2r}({\ensuremath{\mathcal{H}}}):\ \text{tr}(X^2)=2\}$ represents $\Delta(\mathcal{P}_r^n)-\{0\}$ and furthermore we have $\dim \mathcal{D}=4r(n-r)-1$ by Corollary \[corrank\] [^9]. Applying Theorem \[THMvN\] [^10] to the set $\mathcal{D}$ then concludes the proof.
Finally we consider states of fixed spectrum. Let $s$ [^11] be a spectrum on $\mathbb{C}^n$ and denote by $\mathcal{S}_s^n\subseteq\mathcal{S}(\mathbb{C}^n)$ the states with spectrum $s$.
Let $s$ be a spectrum on $\mathbb{C}^n$ such that the highest multiplicity of an eigenvalue in $s$ is $n-r$. Then, if $k(n-1)\ge 4r(n-r)-1$, almost all tuples of $k$ von Neumann measurements $M\in\mathcal{M}_{\text{vN}}^k(\mathbb{C}^n)$ are stably $\mathcal{S}_s^n$-complete.
This follows directly from Theorem \[thmvN\] for $m=n$ noting that $\Delta(\mathcal{S}_s^n)-\{0\}$ can be represented by the set of Lemma \[set\] [^12].
Let $s$ be a spectrum on $\mathbb{C}^n$ such that the highest multiplicity of an eigenvalue in $s$ is $n-r$. Then, if $m-1\ge 4r(n-r)-1$, almost all POVMs $P\in M_{1}^m(\mathbb{C}^n)$ are stably $\mathcal{S}_s^n$-complete.
This follows directly from Theorem \[thmvN\] for $k=1$ noting that $\Delta(\mathcal{S}_s^n)-\{0\}$ can be represented by the set of Lemma \[set\].
Complete Measurements with Local Observables
============================================
In this section we address the problem of reconstructing states of multipartite systems from the expectation values of local observables.
Let ${\ensuremath{\mathcal{H}}}=\bigotimes_{i=1}^k\mathbb{C}^{n_i}$ and let $n:=\prod_{i=1}^kn_i$. We define the set $H_{loc}({\ensuremath{\mathcal{H}}})$ of local observables on ${\ensuremath{\mathcal{H}}}$ by $$\begin{aligned}
H_{loc}({\ensuremath{\mathcal{H}}}):=\{O_1\otimes ...\otimes O_k:O_i\in SH(\mathbb{C}^{n_i})\}\subseteq H({\ensuremath{\mathcal{H}}}).\end{aligned}$$ Just like a POVM, a tuple of observables $O:=(O_1,...,O_m)\in H({\ensuremath{\mathcal{H}}})^m$, induces a linear map $h_O:H({\ensuremath{\mathcal{H}}})\to {\mathbb{R}}^m,\ X\mapsto (\text{tr}(O_1X),...,\text{tr}(O_mX))$ and hence Definition \[defcomplete\] and Definition \[defstab\] naturally generalize to finite tuples of observables.
The following theorem is the analogue of Theorem \[THMvN\] and it is the main result of this section.
\[THMlocal\](Universality) For $\mathcal{R}\subseteq \mathcal{S}({\ensuremath{\mathcal{H}}})$ let $\mathcal{D}\subseteq H({\ensuremath{\mathcal{H}}})$ be a semi-algebraic set that represents $\Delta(\mathcal{R})-\{0\}$. If $m>\dim\mathcal{D}$, almost all $O\in H_{loc}({\ensuremath{\mathcal{H}}})^m$ are stably $\mathcal{R}$-complete.
The proof of this Theorem is given in Section \[proof\].
Again, we directly obtain a Whitney type embedding result for subsets $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$ if the measurement consists of determining expectation values of local observables.
\[whitney2\] Let $\mathcal{R}\subseteq\mathcal{S}({\ensuremath{\mathcal{H}}})$ be a subset. If $m>2\dim \mathcal{R}$, almost all $O\in H_{loc}({\ensuremath{\mathcal{H}}})^m$ are stably $\mathcal{R}$-complete.
We can assume w.l.o.g. that $\mathcal{R}$ is algebraic because if not we can consider its Zariski closure. By the proof of Lemma \[lemdim\], $\Delta(\mathcal{R})-\{0\}$ is semi-algebraic and $\dim(\Delta(\mathcal{R})-\{0\})\leq 2\dim\mathcal{R}$. Finally, Theorem \[THMlocal\] concludes the proof.
Just like in the case of rank one POVMs also this measurement scheme applies to the problem of discriminating states of bounded rank or states of fixed spectrum.
\[ranklocal\] If $m \ge 4r(n-r)-1$, almost all $O\in H_{loc}({\ensuremath{\mathcal{H}}})^m$ are stably $\mathcal{S}_r({\ensuremath{\mathcal{H}}})$-complete.
Let $\mathcal{D}$ be the quasi-algebraic set of Lemma \[set\]. Then the result follows directly from Theorem \[THMlocal\].
Let $s$ be a spectrum on ${\ensuremath{\mathcal{H}}}$ such that the highest multiplicity of an eigenvalue in $s$ is $n-r$. If $m \ge 4r(n-r)-1$, almost all $O\in H_{loc}({\ensuremath{\mathcal{H}}})^m$ are stably $\mathcal{S}_s^n$-complete.
This follows directly from Corollary \[ranklocal\] noting that the set of Lemma \[set\] represents $\Delta(\mathcal{S}_s^n)-\{0\}$.
Finally, let us apply Theorem \[THMlocal\] to local Pauli observables on qubit systems. Let $\mathcal{H}=\bigotimes_{i=1}^d\mathbb{C}^2$. The set of local Pauli observables $H_\sigma({\ensuremath{\mathcal{H}}})$ on ${\ensuremath{\mathcal{H}}}$ is given by $$\begin{aligned}
H_\sigma ({\ensuremath{\mathcal{H}}}):=\{\sigma_1\otimes ...\otimes\sigma_d:\sigma_i\in SH(\mathbb{C}^{n_i})_0\}\end{aligned}$$ where $H(\mathbb{C}^{n_i})_0:=\{X\in H(\mathbb{C}^{n_i})_0:\text{tr}(X)=0\}$ is the real vector space of traceless hermitian $n_i\times n_i$ matrices and $SH(\mathbb{C}^{n_i})_0:=\{X\in H(\mathbb{C}^{n_i})_0:\|X\|_2=1\}$ is the unit sphere in $H(\mathbb{C}^{n_i})_0$.
If $m \ge 4r(2^d-r)-1$, almost all $O\in H_{\sigma}({\ensuremath{\mathcal{H}}})^m$ are stably $\mathcal{S}_r({\ensuremath{\mathcal{H}}})$-complete.
Theorem \[THMlocal\] also holds for $H_{\sigma}({\ensuremath{\mathcal{H}}})$ [^13]. The remainder of the proof is then along the lines of the proof of Corollary \[ranklocal\].
Technical Results {#proof}
=================
Proof of Lemma \[LEMvN\] {#proof-of-lemma-lemvn .unnumbered}
------------------------
Before giving the proof of Lemma \[LEMvN\] let us first explain the methods we use to compute the dimension of the relevant algebraic set.
We take advantage of the fact that the dimension of an algebraic set $V$ is given by the dimension of the tangent space at non-singular points of $V$(see Definition 3.3.3 of [@bochnak1998real]). Let us make this more precise: Let $\mathbb{R}[x_1,...,x_n]$ be the ring of real polynomials in $n$ variables and denote by $dp$ the differential of a real polynomial $p\in\mathbb{R}[x_1,...,x_n]$, i.e. $dp(y)=\sum_{i=1}^{n}\frac{\partial p}{\partial x_i}|_ydx_i$. Let $V_I$ be the real common zero locus of a set of real polynomials $I:=\{p_1,...,p_m\}\subseteq \mathbb{R}[x_1,...,x_n]$. For all $x\in V_I$, $$\begin{aligned}
\label{eqnsrank}
\sum_{i=1}^{m}\alpha_i dp_i(x)=0\end{aligned}$$ gives a system of linear equations in $\alpha_1,...,\alpha_m\in\mathbb{R}$. In the following we mainly use the following facts:
1. The rank of the system of linear equations at a non-singular point of $V_I$ is given by $n-d$ where $d$ is the dimension of $V_I$ [^14].
2. The non-singular points of $V_I$ are an algebraic subset of dimension less than $d$ by Proposition 3.3.14 of [@bochnak1998real].
.
By computing these systems of linear equations, we prove that for a given non-zero $X\in H(\mathbb{C}^n)$, imposing the equations on $\Pi_{i=1}^kM(m,n,\mathbb{C})$ decreases the dimension by at least $n^2+k(m-1)$.
First, let us state a lemma which allows us to efficiently compute the systems of linear equations for the equations . Let $A\in M(s,m,{\mathbb{R}})$, $C\in M(m,t,{\mathbb{R}})$, $B\in H(\mathbb{C}^n)$. Furthermore, identify $M(m,n,\mathbb{C})$ with $\mathbb{R}^{2mn}$ via the canonical map $\iota:M(m,n,\mathbb{C})\to\mathbb{R}^{2mn},\ Y\mapsto (\text{Re}(Y),\text{Im}(Y))$ . Then the equations $$\begin{gathered}
p^I_{lo}(Y):=\text{Im}(AYBY^\dagger C)_{lo}=0,\ \ p^R_{lo}(Y):=\text{Re}(AYBY^\dagger C)_{lo}=0,\\ l\in\{0,...,s\},o\in\{0,...,t\},\end{gathered}$$ can be considered as real algebraic equations in the variables $y^R_{jk}:=(\text{Re}(Y))_{jk}, y^I_{jk}:=(\text{Im}(Y))_{jk},\ j\in\{0,...,m\},k\in\{0,...,n\}$.
\[lemlgs\] Let $Y\in M(m,n,\mathbb{C})$ be such that $AYBY^\dagger C=0$. Then, the system of linear equations $$\begin{aligned}
L(Y):=\sum_{l=1}^{s}\sum_{o=1}^{t}\left(\alpha^R_{lo}dp^R_{lo}(Y)+\alpha^I_{lo}dp^I_{lo}(Y)\right)=0\end{aligned}$$ in $\alpha^R_{lo}\in{\mathbb{R}},\ \alpha^I_{lo}\in{\mathbb{R}}$ is equivalent to $A^TM_\alpha C^TYB+CM_\alpha^\dagger AYB=0$ where $(M_\alpha)_{lo}:=\alpha^{R}_{lo}+i\alpha^I_{lo},\ l\in\{0,...,s\},o\in\{0,...,t\}$.
Let $$\begin{aligned}
L_{jk}=(\partial_{y^R_{jk}}-i\partial_{y^I_{jk}}) \sum_{l=1}^s\sum_{o=1}^t \left(\alpha_{lo}^R\ p ^R_{lo}+\alpha_{lo}^I\ p^I_{lo}\right),\ j\in\{1,...,m\},k\in\{1,...,n\}.\end{aligned}$$ Then the system of linear equations $\{L_{jk}(Y)=0\}_{j\in\{1,...,m\},k\in\{1,...,n\}}$ is equivalent to $L(Y)=0$ since
$$\begin{aligned}
L&=\sum_{l=1}^s\sum_{o=1}^t \left(\alpha_{lo}^R\sum_{j=1}^m\sum_{k=1}^n\left((\partial_{y^R_{jk}}p^R_{lo})dy^R_{jk}
+(\partial_{y^I_{jk}}p^R_{lo})dy^I_{jk}\right)+\alpha_{lo}^I\sum_{j=1}^m\sum_{k=1}^n
\left((\partial_{y^R_{jk}}p^I_{lo})dy^R_{jk}+(\partial_{y^I_{jk}}p^I_{lo})dy^I_{jk}\right)\right)\\
&=\sum_{j=1}^m\sum_{k=1}^n\left(\left(\partial_{y^R_{jk}}\sum_{l=1}^s\sum_{o=1}^t \left(\alpha_{lo}^R\ p^R_{lo}+\alpha_{lo}^I\ p^I_{lo}\right)\right)dy^R_{jk}+\left(\partial_{y^I_{jk}}\sum_{l=1}^s\sum_{o=1}^t \left(\alpha_{lo}^R\ p^R_{lo}+\alpha_{lo}^I\ p^I_{lo}\right)\right)dy^I_{jk}\right)\\
&=\sum_{j=1}^m\sum_{k=1}^n\left(\text{Re}(L_{jk})dy_{jk}^R-\text{Im}(L_{jk})dy_{jk}^I\right).\end{aligned}$$
Let $\partial_{y_{jk}}=\partial_{y_{jk}^R}-i\partial_{y_{jk}^I}$ and note that $\partial_{y_{jk}}Y_{lm}=2\delta_{jl}\delta_{km}$, $\partial_{y_{jk}}Y^*_{lm}=0$. Then,
$$\begin{aligned}
L_{jk}(Y)&=(\partial_{y_{jk}^R}-i\partial_{y_{jk}^I}) \sum_{l=1}^s\sum_{o=1}^t \left( \alpha_{lo}^R \frac{1}{2}(AYBY^\dagger C+AY^*B^*Y^T C)_{lo}+\alpha_{lo}^I\frac{1}{2i}(AYBY^\dagger C-AY^*B^*Y^T C)_{lo}\right)\\
&=\frac{1}{2}\partial_{y_{jk}}\sum_{l=1}^s\sum_{o=1}^t \left((M_\alpha^*)_{lo} (AYBY^\dagger C)_{lo}+(M_\alpha)_{lo}(AY^*B^*Y^T C)_{lo}\right)\\
&=\sum_{l=1}^s\sum_{o=1}^t\sum_{p=1}^m\sum_{q=1}^n\left((M_\alpha^*)_{lo}A_{lp}\delta_{pj}\delta_{qk}(BY^\dagger C)_{qo}+(M_\alpha)_{lo}(AY^*B^*)_{lq}\delta_{qk}\delta_{pj}C_{po}\right)\\
&=(A^TM_\alpha^*C^TY^*B^T+CM_\alpha^T AY^*B^*)_{jk}\\
&=(A^TM_\alpha C^TYB+CM_\alpha^\dagger AYB)^*_{jk}.\end{aligned}$$
Hence $L(Y)=0$ is equivalent to $A^TM_\alpha C^TYB+CM_\alpha^\dagger AYB=0$.
Under the identification $M(m,n,\mathbb{C})\simeq {\mathbb{R}}^{2mn}$ given by the map $\iota$ defined above, also the equations $$\begin{gathered}
r^R_{lo}(Y):=\text{Re}(Y^{\dagger}Y)_{lo}-\delta_{lo}=0,\ r^I_{lo}(Y):=\text{Im}(Y^{\dagger}Y)_{lo}=0,\\ l,o\in\{1,...,n\},\end{gathered}$$ can be considered as real algebraic equations in the variables $y^R_{jk}:=(\text{Re}(Y))_{jk}, y^I_{jk}:=(\text{Im}(Y))_{jk},\ j\in\{0,...,m\},k\in\{0,...,n\}$.
\[corlgs\] Let $Y\in M(m,n,\mathbb{C})$ be such that $Y^{\dagger}Y-{\ensuremath{\mathds{1}}}_n=0$. Then, the system of linear equations $$\begin{aligned}
L(Y):=\sum_{l,o=1}^{n}\left(\gamma^R_{lo}dr^R_{lo}(Y)+\gamma^I_{lo}dr^I_{lo}(Y)\right)=0\end{aligned}$$ in $\gamma^R_{lo}\in{\mathbb{R}},\ \gamma^I_{lo}\in{\mathbb{R}}$ is equivalent to $Y(M_\gamma+M_\gamma^\dagger)=0$ where $(M_\gamma)_{lo}:=\gamma^{R}_{lo}+i\gamma^I_{lo},\ l,o\in\{1,\cdots,n\}$.
The proof of this result can be obtained by going along the lines of the proof of Lemma \[lemlgs\], so we just give the calculation that differs: $L(Y)=0$ is equivalent to $\{L_{jk}(Y)=0\}_{j\in\{1,\cdots,m\},k\in\{1,\cdots,n\}}$ where
$$\begin{aligned}
L_{jk}(Y)&=(\partial_{y_{jk}^R}-i\partial_{y_{jk}^I}) \sum_{l,o=1}^n \left( \gamma_{lo}^R \frac{1}{2}(Y^{\dagger}Y+Y^{T}Y^*)_{lo}+\gamma_{lo}^I\frac{1}{2i}(Y^{\dagger}Y-Y^{T}Y^*)_{lo}\right)\\
&=\frac{1}{2}\partial_{y_{jk}}\sum_{l,o=1}^n \left((M_\gamma^*)_{lo} (Y^{\dagger}Y)_{lo}+(M_\gamma)_{lo}(Y^{T}Y^*)_{lo}\right)\\
&=\sum_{l,o=1}^n\sum_{p=1}^m\left((M_\gamma^*)_{lo}\delta_{ko}\delta_{jp}(Y^\dagger)_{lp}+(M_\gamma)_{lo}(Y^*)_{po}\delta_{lk}\delta_{pj}\right)\\
&=(Y^*M_\gamma^*+Y^*M_\gamma^T)_{jk}\\
&=(YM_\gamma+YM_\gamma^\dagger)^*_{jk}.\end{aligned}$$
Hence $L(Y)=0$ is equivalent to $Y(M_\gamma+M_\gamma^\dagger)=0$.
Note that combining the equations of Lemma \[lemlgs\] and Corollary \[corlgs\] yields the system of linear equations $Y(M_\gamma+M_\gamma^\dagger)+A^TM_\alpha C^TYB+CM_\alpha^\dagger AYB=0$.
Let us now give the proof of Lemma \[LEMvN\].
For a given non-zero $X\in H(\mathbb{C}^n)$ and $i\in\{1,...,k\}$, consider the following equations in $(M_1,\dots,M_k)\in \prod_{i=1}^kM(m,n,\mathbb{C})$: $$\begin{aligned}
p_i^j(M_1,...,M_k):=\text{tr}(M_i^\dagger|j\rangle\langle j|M_iX)=\langle j|M_iXM_i^\dagger |j\rangle&=0,\ \ j\in\{1,...,m\},\end{aligned}$$ and $$\begin{aligned}
q_i^{jl}(M_1,...,M_i):=(M_{i}^\dagger M_{i})_{jl}-\delta_{jl}=0, \ \ j,l\in\{1,...,n\}.\end{aligned}$$ Under the canonical identification $\prod_{i=1}^kM(m,n,\mathbb{C})\simeq {\mathbb{R}}^{2knm}$, these equations can be regarded as real algebraic equations in $2knm$ variables. Let $I_i:=\{p_i^j\}_{j\in\{1,...,m\}}$ and $J_i:=\{q_i^{jl}\}_{j,l\in\{1,...,n\}}$.
We have to show that the dimension of the real common zero locus of the equations $K_k=\bigcup_{i=1}^k I_i\cup J_i $ is at most $2kmn-kn^2-k(m-1)$. Denote by $\iota_1: M(m,n,\mathbb{C})\to \Pi_{i=1}^k M(m,n,\mathbb{C}),\ M\mapsto (M,0,...)$ the inclusion in the first factor and let $\pi_i:\Pi_{i=1}^k M(m,n,\mathbb{C})\to M(m,n,\mathbb{C}) ,\ (M_1,...,M_i,...,M_k)\mapsto M_i$ be the projection on the $i$-th factor. Then we find $I_i\cup J_i=(J_1\cup I_1)\circ \iota_1\circ \pi_i$, where $(J_1\cup I_1)\circ \iota_1\circ \pi_i:=\{p\circ \iota_1\circ \pi_i:p\in J_1\cup I_1\}$. Thus, we conclude that $V_{K_k}\simeq \prod_{i=1}^k V_{K_1}$ and it suffices to reduce to $k=1$. We stick to the notation introduced in the beginning of this section and denote the algebraic set obtained from $M(m,n,\mathbb{C})$ by imposing the equations $I:=I_1$ and $J:=J_1$ by $V_{I\cup J}$.
Let us now determine the system of linear equations $L$ associated to $I\cup J$ at $U\in V_{I\cup J}$. The contribution of the $j$-th equation of $I$ to $L$ is obtained from Lemma \[lemlgs\] by choosing $A=\langle j|$, $B=X$, $C=|j\rangle$, $Y=U$ and thus the contribution of $I$ is given by $$\begin{aligned}
\sum_{j=1}^m\alpha_j^R|j\rangle\langle j|UX,\ \alpha_j^R\in{\mathbb{R}}.\end{aligned}$$ Similarly, by Corollary \[corlgs\], the contribution of $J$ to $L$ is given by, $$\begin{aligned}
U(M_\gamma+M_\gamma^\dagger)\end{aligned}$$ where $(M_\gamma)_{jk}:=\gamma_{jk}^R+i\gamma_{jk}^I, i,j\in\{1,...n\}, \gamma_{jk}^R,\gamma_{jk}^I\in{\mathbb{R}}$. Note that this just gives conditions on the hermitian part of $M_\gamma$ and define $\Gamma\in H(\mathbb{C}^n)$ by $\Gamma:=M_\gamma+M_\gamma^\dagger$.
Combining these two parts, the system of linear equations associated to the equations $I\cup J$ at $U\in V_{I\cup J}$ is equivalent to the following system of linear equations in $\alpha_1,...,\alpha_m\in{\mathbb{R}}$ and $\gamma_{kj}^R\in {\mathbb{R}}, \gamma_{kj}^I\in {\mathbb{R}},\ k,j\in\{1,...,n\},$ $$\begin{aligned}
\label{lgs1}
U\Gamma + D_\alpha UX=0\end{aligned}$$ where $D_{\alpha}=\sum_{j=1}^m\alpha_j|j\rangle\langle j|$. Observing that $\Gamma$ is uniquely determined by the equations , the rank of is at least $n^2$ and we can reduce to the anti-hermitian part of to find the remaining $m-1$ independent equations: $$\begin{aligned}
\label{lgs}
0=U\Gamma U^\dagger + D_\alpha U XU^\dagger -\left(U\Gamma U^\dagger +D_\alpha U XU^\dagger \right)^\dagger=-[U XU^\dagger,D_\alpha].\end{aligned}$$
Let $M$ be a subset of $\{1,...,m\}$ and define the diagonal projection $D_M\in M(m,\mathbb{R})$ by $\langle i|D_M|j\rangle:=\delta_{i,j}\delta_{j,M}$, where $\delta_{j,M}=1\text{ for }j\in M$ and $0$ else. The following observation is the crucial idea for the remainder of the proof: If $[U XU^\dagger,D_M]\neq 0$ for all proper subsets $M$ of $\{1,...,m\}$ then $m-1$ of the operators $\{[U XU^\dagger,D_{\{i\}}]\}_{i\in\{1,...,m\}}$ are linearly independent. To show this, assume that there are $a_j\in {\mathbb{R}},\ j\in\{1,...,m\}$, with $a_k\neq a_l$ for some $k,l$ such that $\sum_{j=1}^m a_j[U XU^\dagger,D_{\{j\}}]=0$. Since the commutativity of hermitian matrices is determined solely by their eigenspaces, we deduce $[U XU^\dagger,D_{E}]=0$, where $E:=\{j\in\{1,...,m\}:a_j=a_k\}$. But this is a contradiction since $E$ is a proper subset of $\{1,\cdots,m\}$. Hence, the only solution is $a_1=a_2=\cdots=a_m$ and this proves the claim. Thus, in this case we conclude that the solution of the system of linear equations is given by $\alpha_1=...=\alpha_m$ and hence there are $m-1$ linearly independent equations.
Next, we decompose $V_{I\cup J}$ into quasi-algebraic subsets for which the argument we just gave can be applied. Let $P[m]$ be the set of partitions of $\{1,...,m\}$. We say that a subset $S\subseteq\{1,...,m\}$ is subordinate to a partition $P\in P[m]$ if there is $M\in P$ such that $S$ is a proper subset of $M$. For given $P\in P[m]$, define the quasi-algebraic set $W_P$ to be the set of $U\in V_{I\cup J}$ such that $$\begin{aligned}
\label{eqcommu}
[D_M,U XU^\dagger ]=0,\ \forall M\in P,\end{aligned}$$ and $$\begin{aligned}
[D_N,U XU^\dagger ]\neq 0,\ \forall N\subseteq\{1,...,m\}\ \text{subordiante to }P.\end{aligned}$$ The set $V_{I\cup J}$ can clearly be decomposed into the sets $W_P$: $$\begin{aligned}
V_{I\cup J}=\bigcup_{P\in P[m]}W_P.\end{aligned}$$
Having already checked that $2mn-\dim W_P= m-1+n^2$ if $P$ is the trivial partition, we conclude the proof by showing that $2mn-\dim W_P\ge m-1+n^2$ for all non-trivial $P\in P[m]$ [^15]. In order to prove this, we first show that the rank of the system of linear equations associated to $W_P$ is at least $n^2+m-1$ for all points in $W_P$.
Let $P=\{M_1,...,M_l,M_{l+1}\}\in P[m]$ be an arbitrary non-trivial partition. Choosing $A=D_{M_j}$, $B=X$, $C=\text{id}_m$ and $Y=U$ in Lemma \[lemlgs\] yields $$\begin{aligned}
D_{M_j}M_{\beta_j}UX +M_{\beta_j}^\dagger D_{M_j}UX,\end{aligned}$$ where $M_{\beta_j}\in M(m,\mathbb{C})$ with $(M_{\beta_j})_{lo}:=\beta^R_{j;lo}+i\beta^I_{j;lo},\ l,o\in\{1,...,m\},\ \beta^R_{j;lo},\beta^I_{j;lo}\in{\mathbb{R}}$ and similarly with the roles of $A$ and $C$ exchanged. Thus, equation for $M_j$ gives the following contribution to the system of linear equations associated to $W_P$ at $U\in W_P$: $$\begin{aligned}
[D_{M_j},M_{\beta_j}-M_{\beta_j}^\dagger]UX.\end{aligned}$$
Thus, the system of linear equations associated to $W_P$ at $U\in W_P$ is equivalent to the following system of linear equations in $\alpha_1,...,\alpha_m\in{\mathbb{R}}$, $\gamma_{kj}^R\in {\mathbb{R}}, \gamma_{kj}^I\in {\mathbb{R}},\ k,j\in\{1,...,n\},$ and $\beta^R_{j;lo}\in{\mathbb{R}},\beta^I_{j;lo}\in{\mathbb{R}},\ j\in\{1,...,l+1\},l,o\in\{1,...,m\}$: $$\begin{aligned}
U\Gamma + D_\alpha UX+\sum_{k=1}^{l+1}[D_{M_k},M_{\beta_k}-M_{\beta_k}^\dagger]UX=0.\end{aligned}$$ Again, we can eliminate $\Gamma$ by reducing to the anti-hermitian part to obtain
$$\begin{aligned}
\label{lgs2}
U\Gamma U^\dagger+D_{\alpha} U XU^\dagger +&\sum_{k=1}^{l+1}[D_{M_k},M^H_{\beta_k}]UXU^\dagger-\left(U\Gamma U^\dagger+D_{\alpha} U XU^\dagger +\sum_{k=1}^{l+1}[D_{M_k},M_{\beta_k}^H]UXU^\dagger\right)^\dagger \nonumber\\
&\Leftrightarrow [U XU^\dagger ,D_{\alpha}]+\sum_{k=1}^{l+1}[U XU^\dagger ,[M_{\beta_k}^H,D_{M_k}]]=0, \end{aligned}$$
where $M_{\beta_j}^H$ is the anti-hermitian $m\times m$ matrix defined by $M_{\beta_j}^H:=M_{\beta_j}-M_{\beta_j}^\dagger$.
Conjugating with $D_{M_j}$ yields $$\begin{aligned}
[UXU^\dagger,D_{M_j}D_{\alpha}]=0,\end{aligned}$$ where we used $[UXU^\dagger,D_{M_j}]=0$ together with $D_{M_j}[M_{\beta_k}^H,D_{M_k}]D_{M_j}=D_{M_j}M_{\beta_j}^HD_{M_j}-D_{M_j}M_{\beta_j}^HD_{M_j}=0$. By construction of $W_P$, we have $[UXU^\dagger,D_{M_j}D_M]\neq 0$ for all proper subsets $M\subseteq M_j$. Since the commutativity of hermitian matrices is solely determined by their eigenspaces we conclude just like in the case of the trivial partition that $D_{M_j}D_{\alpha}\propto D_{M_j}$ for all $j\in\{1,...,l+1\}$ [^16]. Thus, if there is $U\in W_P$, the rank of at $U$ is at least $n^2+m-l-1$.
To find the remaining $l$ independent equations consider the remaining equations $$\begin{aligned}
\sum_{j=1}^{l+1}[U XU^\dagger,[M^H_{\beta_j},D_{M_j}]]=0.\end{aligned}$$
There is $i\in\{1,...,l+1\}$ with $D_{M_i}U XU^\dagger\neq 0$ because otherwise we would conclude that $U XU^\dagger= 0$ which is a contradiction since $U\in U(m,n)$ and $X\neq 0$ by assumption. Multiplying by $D_{M_k}$ from the left and $D_{M_i}$ from the right yields $$\begin{aligned}
&\sum_{j=1}^{l}D_{M_k}[U XU^\dagger,[M^H_{\beta_j}, D_{M_j}]]D_{i}=0\\
\Leftrightarrow&\sum_{j=1}^{l}[U XU^\dagger ,D_{M_k}[M^H_{\beta_j}, D_{M_j}]]D_{i}=0\\
\Leftrightarrow&[U XU^\dagger ,D_{M_k}(M^H_{\beta_{i}}-M^H_{\beta_k}) D_{i}]=0.\end{aligned}$$ For each $k\in\{1,...,l+1\}-\{i\}$ this gives at least one equation on $M^H_{\beta_k}$: First, assume $|M_{i}|=1$. Then there is $q\in\{1,\dots,m\}$ such that $M_i=\{q\}$. Furthermore, since $$\begin{aligned}
0\neq D_{M_i}U XU^\dagger=D_{M_i}U XU^\dagger D_{M_i}=\langle q|U XU^\dagger|q\rangle|q\rangle\langle q|,\end{aligned}$$ we conclude that $\langle q|U XU^\dagger|q\rangle\neq 0$. But this is a contradiction to the $q$-th equation of $I$.
Hence we can assume $|M_{i}|\geq 2$. By construction of $W_P$ there is an eigenvector $v_k\neq 0$ of $U XU^\dagger$ in the range of $D_{M_k}$ with eigenvalue $\lambda_k$ and a eigenvector $v_{i}\neq 0$ of $U XU^\dagger$ in the range of $D_{M_{i}}$ with eigenvalue $\lambda_{i}$. Since we assumed $|M_{i}|\geq 2$, by construction of $W_P$, $U XU^\dagger$ has at least two eigenvectors in the range of $D_{M_{i}}$ with different eigenvalues because otherwise there would be a proper subset of $N\subseteq M_i$ such that $[UXU^\dagger,D_N]=0$. Thus, we can choose $\lambda_{i}$ such that $\lambda_{i}\neq \lambda_k$. We then find $$\begin{aligned}
&\langle v_k|[U XU^\dagger ,D_{M_k}(M^H_{\beta_{i}}-M^H_{\beta_k}) D_{M_i}]|v_i\rangle=0\\
\Leftrightarrow&\langle v_k|(M^H_{\beta_{i}}-M^H_{\beta_k})|v_i\rangle(\lambda_k-\lambda_i)=0\\
\Leftrightarrow&\langle v_k|M^H_{\beta_{i}}|v_i\rangle-\langle v_k|M^H_{\beta_k}|v_i\rangle=0.\end{aligned}$$ But this clearly gives a non-trivial condition on $M^H_{\beta_k}$ since $M^H_{\beta_k}=M_{\beta_j}-M_{\beta_j}^\dagger$. Thus we conclude that, if there is $U\in W_P$, the rank of at $U$ is at least $m-l-1+l=m-1$ and hence the rank of the system of linear equations associated to $W_P$ at $U$ is at least $n^2+m-1$. But if $W_P$ is non-empty, it does contain a non singular-point by Proposition 3.3.14 of [@bochnak1998real]. And thus the rank of the system of linear equations associated to $W_P$ at this non-singular point is at least $n^2+m-1$. Hence $2kmn-\dim W_P\ge n^2+m-1$ by Proposition 3.3.10 of [@bochnak1998real].
Proof of Theorem \[THMvN\] {#bla .unnumbered}
--------------------------
Let $\psi$ be the map defined in . We can assume that $\mathcal{D}$ is a closed subset of $SH(\mathbb{C}^n)$ because if not we can replace it by the closure of $\psi(\mathcal{D})$ without increasing its dimension [^17]. Let $\tilde{\mathcal{M}}:=\{(U_1,...,U_k,X)\in\prod_{i=1}^kU(m,n)\times\mathcal{D}:\langle j|U_iXU_i^\dagger|j\rangle=0,\ j\in\{1,\dots,m\},i\in\{1,\dots,k\}\}$.
First, we fix the measure on $\mathcal{M}_{1,k}^m(\mathbb{C}^n)$: Let $\phi$ be the map defined in equation . We define the measure $\mu$ on $\mathcal{M}_{1,k}^m(\mathbb{C}^{n})$ to be the pushforward measure of the $2knm$-dimensional Hausdroff measure $\mu_H$ on $\prod_{i=1}^kU(m,n)\subseteq {\mathbb{R}}^{2nmk}$, i.e. $\mu(A):=\phi_*(\mu_{H})(A)=\mu_H(\phi^{-1}(A))$ for $A\subseteq\mathcal{M}_{\text{vN}}^m(\mathbb{C}^{n})$ a measurable set.
Note that $\phi$ is the quotient projection with respect to the left action of the toral goup $T:=\prod_{i=1}^kT(m),\ T(m):=\{\text{diag}(\lambda_1,...,\lambda_m):\lambda_i\in U(1)\}$ on $\prod_{i=1}^kU(m,n)$ given by $((U_1,...,U_k),(T_1,...,T_k))\mapsto((U_1,...,U_k),(T_1U_1,...,T_kU_k))$. Also note that the equations are invariant under the action of $T$ and hence $T\pi_1(\tilde{\mathcal{M}})=\pi_1(\tilde{\mathcal{M}})$ where $\pi_1:\prod_{i=1}^kU(m,n)\times \mathcal{D}\to \prod_{i=1}^k U(m,n)$ is the projection on the first factor. Thus, for $\mu_H(\pi_1(\tilde{\mathcal{M}}))=0$, we find $$\begin{aligned}
\mu\left(\phi\circ\pi_1(\tilde{\mathcal{M}})\right)&=\mu_H\left(\phi^{-1}\left(\phi\circ\pi_1(\tilde{\mathcal{M}})\right)\right)
\\&=\mu_H\left(T\pi_1(\tilde{\mathcal{M}})\right)\\
&=\mu_H\left(\pi_1(\tilde{\mathcal{M}})\right)=0.\end{aligned}$$ Hence, it suffices to prove that $\mu_H(\pi_1(\tilde{\mathcal{M}}))=0$.
Finally, for $k(m-1)>\dim\mathcal{D}$ we find $\dim \pi_1(\tilde{\mathcal{M}})\leq\dim \prod_{i=1}^kU(m,n)+\dim\mathcal{D}-m(k-1)<\dim \prod_{i=1}^kU(m,n)$ by Lemma \[LEMvN\]. So $\pi_1(\tilde{\mathcal{M}})$ has $\mu_H$-measure zero in $\prod_{i=1}^kU(m,n)$. The stability follows directly from Lemma \[stab\].
Note that by the remark after Lemma \[LEMvN\], this proof just depends on $\mathcal{D}\subseteq H(\mathbb{C}^n)$ and hence naturally extends to semi-algebraic subsets $\mathcal{R}\subseteq H(\mathbb{C}^n)$. Furthermore, this proof shows that indeed $\pi_1(\tilde{\mathcal{M}})$ has $\mu_H$-measure zero in $\prod_{i=1}^kU(m,n)$. Thus the statement of Theorem \[THMvN\] naturally also holds for Parseval frames $U\in U(n,m)$.
Proof of Theorem \[THMlocal\] {#proof-of-theorem-thmlocal .unnumbered}
-----------------------------
For a given non-zero $X\in H({\ensuremath{\mathcal{H}}})$, consider the equations $$\begin{aligned}
\label{eqnblaaaaa}
p^i((O^1_1,\dots,O^1_k),\dots,(O^m_1,&\dots,O^m_k)):=\text{tr}((O^i_1\otimes\dots\otimes O^i_k)X)=0,\ i\in\{1,...,m\},\end{aligned}$$ in $((O^1_1,\dots,O^1_k),\dots,(O^m_1,\dots,O^m_k))\in (\Pi_{i=1}^kH(\mathbb{C}^{n_i}))^m$. Under the identification $H(\mathbb{C}^{n_i})\simeq {\mathbb{R}}^{n_i^2}$, these equations can be considered as real algebraic equations in the variables $((O^1_1,\dots,O^1_k),\dots,(O^m_1,\dots,O^m_k))$. The following Lemma is the analogue of Lemma \[LEMvN\].
\[LEMlocal\] Let $X\in H({\ensuremath{\mathcal{H}}})$ be non-zero. Imposing the equations on $(\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m$ decreases the dimension by at least $m$.
The equation $p_i$ just involves the variables $(O^i_1,\dots,O^i_k)$ of the $i$-th factor of $(\Pi_{i=1}^kH(\mathbb{C}^{n_i}))^m$. Thus, it suffices to prove that, for given non-zero $X\in H({\ensuremath{\mathcal{H}}})$, imposing the equation $$\begin{aligned}
\label{abc}
p((O_1,\dots,O_k)):=\text{tr}((O_1\otimes\dots\otimes O_k)X)=0\end{aligned}$$ on $\Pi_{i=1}^kSH(\mathbb{C}^{n_i})$ decreases the dimension by at least one.
In order to see that this is true, note that there are $(O_1,\dots,O_k)\in\Pi_{i=1}^kSH(\mathbb{C}^{n_i})$ such that $\text{tr}((O_1\otimes\dots\otimes O_k)X)\neq 0$ because $\bigotimes_{i=1}^kH(\mathbb{C}^{n_i})$ has a basis of normalized local operators and $X\neq 0$. But then, the equation is a non-trivial algebraic equation on the irreducible algebraic set $\Pi_{i=1}^kSH(\mathbb{C}^{n_i})$ and thus the dimension has to decrease since for a proper algebraic subset $V$ of an irreducible algebraic set $W$ we have $\dim V<\dim W$.
By going along the lines of the this proof, it is easily seen that Lemma \[LEMlocal\] also holds when going from hermitian matrices to traceless hermitian matrices, i.e. if we replace $(\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m$ by $(\Pi_{i=1}^kSH(\mathbb{C}^{n_i})_0)^m$. Furthermore, the proof of Theorem \[THMlocal\] also holds when going from $(\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m$ to $(\Pi_{i=1}^kSH(\mathbb{C}^{n_i})_0)^m$ and considering $H_{loc,0}({\ensuremath{\mathcal{H}}}):=\{O_1\otimes ...\otimes O_k:O_i\in SH(\mathbb{C}^{n_i})_0\}$ instead of $H_{loc}({\ensuremath{\mathcal{H}}})$.
Now we can give the proof of Theorem \[THMlocal\].
Let $\psi$ be the map defined in . We can assume that $\mathcal{D}$ is a closed subset of $SH({\ensuremath{\mathcal{H}}})$ because if not we can replace it by the closure of $\psi(\mathcal{D})$ without increasing its dimension [^18]. Let $\mathcal{M}$ be the semi-algebraic set obtained from $(\Pi_{i=1}^kH(\mathbb{C}^{n_i}))^m\times \mathcal{D}$ by imposing the equations .
For $m>\dim\mathcal{D}$ we get $\dim \pi_1(\mathcal{M})<\dim (\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m$ by Lemma \[LEMlocal\].
Now consider $\theta(\pi_1(\mathcal{M}))$ where $$\begin{aligned}
\theta:(\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m&\to(H_{loc}({\ensuremath{\mathcal{H}}}))^m,\\
(O^1_1,\dots,O^1_k),\dots,(O^m_1,\dots,O^m_k)&\mapsto (O^1_1\otimes\dots\otimes O^1_k),\dots,(O^m_1\otimes\dots\otimes O^m_k).\end{aligned}$$ Note that $\theta$ is a surjective semi-algebraic map and thus $(H_{loc}({\ensuremath{\mathcal{H}}}))^m$ is semi-algebraic with $\dim((H_{loc}({\ensuremath{\mathcal{H}}}))^m)\le \dim((\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m).$ Furthermore, $\theta$ is injective when restricting to positive matrices and hence $d:=\dim (H_{loc}({\ensuremath{\mathcal{H}}}))^m=\dim(\Pi_{i=1}^kSH(\mathbb{C}^{n_i}))^m$.
Finally, since $\dim \pi_1(\mathcal{M})<d$ and $\theta$ is semi-algebraic, we have $\dim\left(\theta(\pi_1(\mathcal{M}))\right)<d$ and thus $\theta(\pi_1(\mathcal{M}))$ has zero $d$-dimensional Hausdorff measure. Stability follows directly from Lemma \[stab\].
Hausdorff Measure on Semi-Algebraic Sets {#appendixD}
========================================
The term ”almost” all used in many of the results of this paper refers to the Hausdorff measure on real affine space. In this section we define the Hausdorff measure and we prove the well-known fact that a semi-algebraic set of dimension $d$ has zero $(d+1)$-dimensional Hausdorff measure.
For a non-empty subset $A\subseteq \mathbb{R}^n$ the diameter of $S$ is defined by $\text{diam}(S):=\sup\{\|x-y\|_2:x,y\in S\}$.
Let $m\in\mathbb{R}$. For an arbitrary subset $S\subseteq{\mathbb{R}}^n$ the $m$-dimensional Hausdorff measure $\mu_H^m(S)$ is defined by (see Section 2.3 of [@morgan2008geometric]) $$\begin{aligned}
\mu_H^m(S)=\lim_{\delta\to 0}\inf\{\sum_{i=1}^\infty (\text{diam}(S_i))^m:S\subseteq\cup_{i\in\mathbb{N}}(S_i),\ \text{diam}S_i<\delta\}.\end{aligned}$$
\[propdim\] Let $m>n$. A semi-algebraic set $S$ of dimension $n$ has zero $m$-dimensional Hausdorff measure.
Every $n$-dimensional semi-algebraic set $S$ can be expressed as $S=\bigcup_{i=1}^k S_i$ for some $k\in \mathbb{N}$ where the $S_i$ are diffeomorphic to $(0,1)^{n_i}$, $n_i\leq n$ (see Proposition 2.9.10 of [@bochnak1998real]). Let us denote these diffeomorphisms by $\phi_i:(0,1)^{n_i}\to S_i$. Since $S$ is a finite union it suffices to prove that the $m$-dimensional Hausdorff measure of $S_i$ is zero for $m>n$.
For each point $p\in S_i$, there is a neighbourhood $N_p$ of $p$ such that $\phi_i|_{N_p}$ is Lipschitz. Constructing such neighbourhoods for all $p\in S_i$, we obtain an open cover of $S_i$ by the open sets $\{N_p\}_{p\in S_i}$ and since ${\mathbb{R}}^n$ is second countable there is a countable subcover $\{S_i\cap N_{p_j}\}_{j\in\mathbb{N}}$.
Finally, we just have to see that the Hausdorff measure of $N_{p_j}$ is zero for all $j\in\mathbb{N}$. But $\phi_i(N_{p_j})$ is the image of a set of zero $m$-dimensional Hausdorff measure under a Lipschitz map and thus $\phi_i(N_{p_j})$ has zero $m$-dimensional Hausdorff measure as well.
Note that this proof in particular shows that the $n$-dimensional Hausdorff measure of an $n$-dimensional semi-algebraic set does not vanish and hence it is a suitable measure for our purposes.
The set of measurement schemes always is a semi-algebraic subset $S$ of a real affine space and the measure we choose for $S$ is the $m$-dimensional Hausdorff measure where $m$ is the dimension of $S$. If we say that almost all elements of an $m$-dimensional semi-algebraic set $S$ has a certain property we mean that it fails to hold on a subset $A\subseteq S$ that has $m$-dimensional Hausdorff measure zero. We do this by showing that the algebraic dimension of $A$ is smaller than $m$ and applying Proposition \[propdim\].
[^1]: A hermitian matrix is positive semidefinite if and only if all of its principal minors are greater than of equal to zero. Thus, the equations $P_i\geq 0$ can be regarded as algebraic inequalities.
[^2]: Note that $M_F$ is also well-defined for $F=\{v_1,...,v_m\}$ with $v_i\in\mathbb{C}^n$, i.e. if we do not require $F$ to be a frame.
[^3]: Here we identify $H({\ensuremath{\mathcal{H}}})$ with $(\dim{\ensuremath{\mathcal{H}}})^2$-dimensional real affine space.
[^4]: $\pi_1$ maps semi-algebraic sets to semi-algebraic sets and does not increase the dimension. See Theorem 2.2.1 and Proposition 2.8.6 of [@bochnak1998real].
[^5]: Here $[x]:=$largest integer $i$ such that $i\leq x$.
[^6]: Every proper algebraic subset of the irreducible algebraic set ${\mathbb{R}}^{2m}$ has dimension less than $2m$.
[^7]: The Lebesgue measure on ${\mathbb{R}}^n$ is a rescaling of the $n$-dimensional Hausdorff-measure.
[^8]: The algebraic dimension is invariant under taking the Zariski closure, see Proposition 2.8.2 of [@bochnak1998real]
[^9]: Note that the definition of a representing set naturally generalizes to subsets $\mathcal{R}\subseteq H(\mathbb{C}^n)$.
[^10]: Theorem \[THMvN\] also applies in this situation. See the remark after proof of Theorem \[THMvN\] for more details.
[^11]: A spectrum on $\mathbb{C}^n$ is a multiset of $n$ increasingly ordered positive real numbers that sum up to one. We call the elements of $s$ eigenvalues.
[^12]: For more details see Lemma IV.3 of [@kech].
[^13]: See the remark after proof of Lemma \[LEMlocal\].
[^14]: See Definition 3.3.4 and Proposition 3.3.10 of [@bochnak1998real].
[^15]: Note that, depending on the choice of $X$, many of the $W_P$ might be empty. If $X={\ensuremath{\mathds{1}}}_n$, $n=m$ for instance, all $W_P$ would be empty.
[^16]: In particular, note that if $D_\alpha$ solves the system of linear equations we have $[U XU^\dagger,D_{\alpha}]=0$.
[^17]: See remark after Lemma \[stab\] for more details.
[^18]: See remark after Lemma \[stab\] for more details.
|
---
abstract: 'The out-degree distribution of citation networks is investigated. Statistical data of the number of papers cited within a paper (out-degree) for different journals in the period 1991-1999 is reported. The out-degree distribution is characterized by a maximum at intermediate out-degrees. At the left of the maximum there are strong fluctuations from journal to journal while is quite universal at the right, with two classes of journals. These two classes are associated with the existence or not of a restriction in the maximum number of pages per paper. The shape of the out-degree distribution does not change appreciable from period to period, but the average out-degree is observed to increase logarithmically with the number of published papers. These features are modeled using a recursive search model.'
address: 'International School for Advanced Studies, via Beirut 4, 34014 Trieste, Italy ([email protected])'
author:
- Alexei Vazquez
title: Statistics of citation networks
---
Introduction
============
The number of citations to a paper gives an estimate of its quality and relevance for the scientific community. This fact explains our interest to know how many times our papers are cited. Besides this absolute measure, we are also interested in the number of citations relative to other scientist works. For instance, one may look for the top-cited articles or scientists in a given period.
Two recent works have reported a statistical analysis of citation data [@laherre98; @redner98]. Lahèrre and Sornette [@laherre98] have analyzed the statistics of the number of citations to a given scientist, within the top-cited scientists over the period 1981-June 1997. They observed that the distribution of this magnitude is well fitted by a stretched exponential. On the other hand, Redner [@redner98] has studied the statistics of citations to a given paper using two different data sets. In this case, for large number of citations the data was fitted better by a power law decay with exponent 3. Latter Tsallis and Albuquerque [@tsallis00] have claimed that this data can be fitted by a single curve derived from the non-extensive formalism, with a power law decay for large number citations.
In both cases they have investigated the in-degree distribution of the citation network, where the in-degree is the number of citations to a given scientist or a paper, respectively. However, up to the author knowledge, no report has been done about the out-degree distribution, where the out-degree is the number of papers cited within a paper. This magnitude also gives information about the topology of the network. For instance, since the average in-degree and out-degree are identical, it is possible to compute the average in-degree from measurements of the out-degree distribution. The advantage of this approach is that it is easier to collect data about the out-degree distribution.
In order to investigate the out-degree distribution of the citation network data has been collected from different journals in the period 1991-1999. The analysis of the out-degree distribution reveals that the journals are divided into two classes, associated with the existence or not of a restriction in the maximum number of pages per article. Moreover, it is observed that the mean out-degree grows logarithmically with the number of published papers. Finally, the data is compared with a recursive search model [@vazquez00] obtaining some agreement. It is concluded that the recursive search is one of the mechanisms that determine the topology of citation networks.
Statistical data {#sec:data}
================
Out-degree distribution {#sec:out-degree}
-----------------------
Data of citations within a paper (out-degree $k_{ou}$) of 12 journals was analyzed. The data was collected from the Science Citation Index report [@ISI]. They were selected among the journals with more papers published per year in order to have good statistics. The out-degree distribution $P_{ou}(k_{ou})$ was computed for the periods 1991, 1991-92, 1991-93,... 1991-99 for each journal.
In all cases the out-degree distribution exhibits a maximum value $P_m$ at an intermediate out-degree $k_m$. The coordinates of the maximum $(k_m,P_m)$ does not change appreciable from period to period. The rescaled plots $P_{ou}(k_{ou})/P_m$ vs. $k_{ou}/k_m$ are shown in fig. \[fig:1\] for the periods 1991 and 1991-99, while the value of $k_m$ and $P_m$ are given in table \[tab:1\].
For $k<k_m$ there are strong fluctuations of the out-degree distribution from journal to journal. These fluctuations are not reduced for the 1991-99 period, for which more statistical data is available. Hence, the shape of the out-degree distribution in this region is non-universal.
On the contrary, for $k_{ou}>k_m$ all the curves practically overlap into two universal curves, which divide the journals into two classes. For the 1991 data there is a strong dispersion in the data and, as a consequence, the universal tendency is not so clear. Nevertheless, the dispersion is reduced for the 1991-99 data, and the universal behavior becomes more evident. Hence, there is a generic universal behavior which is obscured by the statistical dispersion. Moreover, the two universal curves can be fitted by an exponential decay with decay rates 0.4 and 1.6, respectively.
The analysis of the journals on each class reveals that this subdivision is determined by the existence or not of a restriction in the number of pages per article. One class contains the journals with a limitation in the maximum number of pages per article while the other contains the journals without this restriction. In the first class there is a faster exponential decay beyond the maximum while the exponential decay is slower for the other class, [*i.e.*]{} the restriction in the number of citations within a paper carries as a consequence a reduction in the width of the out-degree distribution, as one would expect.
The shape of the out-degree distribution around the maximum does not change to much from period to period and the coordinates of the maximum remains approximately constant. However, from fig. \[fig:1\] it can be seen that there are changes in the tail of the distribution. For 1991 there were just a few papers with $k_{ou}/k_m$ larger than 1. On the contrary, for the period 1991-99 a lot more were found. This fact is manifested an the mean out-degree $\langle k_{ou}\rangle$, which increases slowly from period to period. The plot of $\langle k_{ou}\rangle$ for each period vs. the number of papers $N$ published in that period is shown in fig. \[fig:2\].
For some journals $\langle k_{ou}\rangle$ is well fitted by a linear dependency with $N$ while for others a logarithmic growth gives a better fit to the data. These two different behaviors of $\langle k_{ou}\rangle$ vs. $N$ are not related with the subdivision into two classes of journals described above. Actually, as it is shown below, they can be obtained as the limiting cases o a single logarithmic dependency.
If the mean out-degree increases logarithmically with the number of published papers $n$ (even considering papers published before 1991) then $$\frac{d\langle k_{ou}\rangle}{dn}=\frac{b}{n},
\label{eq:1}$$ where $b$ is a constant that may change from journal to journal. Now, let $N_0$ be the number of papers published before 1991 and let $k_0$ be the mean out-degree considering the $N_0$ published before 1991. Then, integrating eq. (\[eq:1\]) from $N_0$ to $N_0+N$ one obtains that $$\langle k_{ou}\rangle=k_0+b\ln\left(1+\frac{N}{N_0}\right).
\label{eq:2}$$ For $N/N_0\ll1$ this expression can be approximated by $$\langle k_{ou}\rangle\approx k_0+b\frac{N}{N_0},
\label{eq:3}$$ while for $N/N_0\gg1$ one obtains $$\langle k_{ou}\rangle\approx k_0+b\ln\frac{N}{N_0}.
\label{eq:4}$$ Hence, the linear and logarithmic growth can be obtained as two asymptotic behaviors of eq. (\[eq:2\]) depending on the ratio $N/N_0$.
The data was fitted by the logarithmic dependency in (\[eq:2\]). For some journals (J. Chem. Phys., Phys. Rev. B, Phys. Rev. D, Biochem., Febs. Lett., Phys. Rev. Lett.) the fitting to that expression is very good obtaining the complete set of parameters $k_0$, $b$, and $N_0$. However, for the other journals, any large value of $N_0$ yield an equally good fit to the data, but always with the ratio $b/N_0$ remaining constant. This suggest that the data for these journals is in the limiting case given by eq. (\[eq:3\]), where the linear dependency is observed.
The parameters $N_0$, $b$ and $b/N_0$ resulting from the best fit are reported in table \[tab:1\]. The values of $N_0$ are small if one expect that $N_0$ is the total number of papers published before 1991. For instance for Biochem. (USA) $N_0\approx980$, which is smaller than the number of papers published per year on this journal (around 1500). Hence, $N_0$ cannot be interpreted as the total number of papers published before 1991. This fact suggest that the logarithmic dependency is not manifested in the past history of the journals.
In-degree distribution {#sec:in-degree}
----------------------
In a directed network the average in-degree and out-degree are identical. Nevertheless, we should be careful in extrapolating the results obtained here for the average out-degree. The journals have cross-references among them and, therefore, the in-degree distribution relates different journals. In principle one should group the journals in clusters in such a way that journals in different clusters have not cross-references among them. In any case the fact that the growth has been observed for all the journals analyzed here is a strong evidence that the overall average out-degree, which is equal to the overall average in-degree, increases with the number of published papers. Moreover, the growth is well fitted by the logarithmic dependency in eq. (\[eq:2\]) or its asymptotic limits in eqs. (\[eq:3\]) and (\[eq:4\]).
We can go beyond averages and make also some conclusions concerning to the in-degree distribution. If the in-degree distribution follows the power law $P_{in}(k_{in})\sim
k_{in}^{-\gamma}$ and its average grows logarithmically with the network size then the exponent $\gamma$ should be equal to 2, or at least close to it. Recently Redner [@redner98] have done a statistical analysis of the in-degree distribution of two different data sets. The first was the citation distribution of papers published in 1981 and cited between 1981-June 1997 (ISI). The second data set was the citation distribution, as of June 1997, of the papers published in Phys. Rev. D in the period 1974-1994 which were cited at least once (PRD).
In the ISI data only citations to papers published in 1981 is considered, while citation to papers published between 1982-June 1997 is not. Hence, This data set is better to investigate the process of aging in the citation network, in which very old papers are rarely cited. However, it is not good to investigate the in-degree distribution because it does not considers the citation to papers published between 1982-June 1997. On the other hand, in the PRD data one is not considering the citations to papers published in the period 1995-June 1997. However, they are only three years over a period of 23 years. Hence, with this data set one has a better approximation to the in-degree distribution.
A log-log plot of the in-degree distribution for the PRD data is shown in fig. \[fig:2a\]. The distribution shows two different regimes. For small in-degrees the distribution can be fitted by a power law with an approximate exponent 1.3. For large in-degrees the curve can be also fitted by a power law but with a larger exponent. A power law fit to this part of the curve gives the exponent $\gamma=1.9\pm0.2$. This exponent is smaller than the value 3 reported by Redner [@redner98]. In any case, this second power law, is manifested in only one decade and, therefore, is difficult to make any final statement with the present data. However, the exponent $\gamma=1.9\pm0.2$ reported here is more consistent with the out-degree distribution data analyzed above.
A power law in-degree distribution with power law exponent $\gamma=3$ will give a finite average when $N\rightarrow\infty$. For finite $N$ the mean in-degree will grow with $N$, with a tendency to saturate to the stationary value obtained when $N\rightarrow\infty$. On the other hand, if the exponent is $\gamma=2$ the mean in-degree will always grow with $N$ following a logarithmic dependency. This second picture is more consistent with the behavior of the mean out-degree as a function $N$, which increases logarithmically and in some cases linear with $N$.
Recursive search model {#sec:model}
======================
A second step will be to propose a model that explains the features described above, [*i.e.*]{} an average out-degree (in-degree) which increases logarithmically with the network size, an out-degree distribution with a maximum at intermediates out-degrees and a power law in-degree distribution with decay exponent 2. In the literature we can find two models which lead to power law in-degree distributions. They are the free-scale and the recursive search models, introduced by Barabási and co-workers [@barabasi99] and by the author [@vazquez00], respectively.
The free-scale model and its generalizations [@dorogovtsev00; @krapivsky00] lead to power law in-degree distributions with power law exponent $\gamma>2$ and, therefore, with a finite average in-degree. The exponent $\gamma=2$ can be reached asymptotically but only after a fine-tune of the parameters involved in the model. On the contrary, the recursive search model [@vazquez00] is characterized by a robust exponent $\gamma=2$ over a wide range of its parameters, leading to a logarithmic growth of the average in-degree as the network size increases. On the other hand, in the free-scale model the out-degree is fixed for all nodes. Hence, to describe the out-degree distribution observed in the citation data, one has first to make some generalization of it, which introduces some randomness in the out-degree. On the contrary, as it is shown below, the recursive search model leads to out-degree distributions that are already in good agreement with the citation data. All these elements suggest that the recursive search model is more appropriate to describe the citation network.
The main idea of a recursive search is to be connected to one node (paper) of the network and any time we get in contact with a new node follows its links, exploring in this way part of the network. This process mimics in some elementary way our recursive searches over the citation network. In order to perform an efficient search one usually follows only the links of those nodes that appear to be of good quality according to certain criteria. To take into account this effect the parameter $p$ ($0\leq p\leq1$) is introduced, which is defined as the probability that a node satisfies our quality requirements. Thus, it is assumed that each time a node satisfies our quality requirements, which happens with probability $p$, a link is created to it and the search is continued latter following its links. This search process has the limitation that in order to follow the links of a node one has to create first a link to it, a constraint which may be no realistic in some cases. However, it already captures the essential features of a recursive search.
The recursive search model is then defined by two rules: adding and walking. In the adding rule one adds a node to the network. If more than one node is present then one performs the walking rule defined as follows: if a link is created to a node in the network then with probability $p$ a link is also created to each of its nearest neighbors. This rule is performed until no link is created and in that case one goes back to the adding rule.
The model exhibits a phase transition at a threshold probability $p_c\approx0.4$ [@vazquez00]. Below $p_c$ the average in-degree approaches a stationary value as the network size (number of nodes $N$) increases. On the contrary, above the threshold the average out-degree (in-degree) increases logarithmically with $N$ and the in-degree distribution displays a robust power law with exponent $\gamma\approx2$, independent of $p$. Thus, in the range $p_c\leq p\leq1$ the in-degree distribution obtained from the model has the same features discussed above for the citation data.
For $p=1$ the average out-degree (in-degree) increases as $\langle
k_{ou}\rangle=a+b\ln N$ with $b=1$. For $p_c<p<1$ it still increases logarithmically with $N$ but with $b<1$. However, the statistical data reported in the previous section shows that $b$ is in general greater than 1, ranging from 1.2 to 5.2.
On the other hand, the comparison of the out-degree distribution obtained from the model and that reported here for different journals is shown in fig. \[fig:3\]. As it can be seen there is some agreement with the data for the class of journals with a limitation in the number of pages per article. The main features, a maximum at intermediate out-degrees, universal behavior at the right of the maximum and non-universal behavior at the left of the maximum, are also observed for the recursive search model. However, the model underestimate the out-degree distribution for large $k_{ou}$ and it does not give a good agreement with the journals without restriction in the number of pages.
Hence, the recursive search model explains the main features observed for the in-degree and out-degree distributions of the citation network. However, the model is still to simple to give a complete description, with a quantitative agreement. One can make a generalization of the model in which one starts the search at different nodes. This modification will carry as a consequence a wider out-degree distribution and probably a better agreement with the journals without a restriction in the number of pages. On the other hand, the preferential attachment [@barabasi99] is another mechanism which may comes to play. During the search one may follow preferentially those paper which are more cited, which is a measure of their quality. This will be manifested in the recursive search model as a dependence of $p$ on $k_{in}$. Work on that direction is in progress.
Summary and conclusions {#sec:summary}
=======================
In summary, empirical data of the out-degree distribution of the network of citations has been presented. The main features observed were a maximum at intermediate out-degrees, universal behavior at the right of the maximum, non-universal behavior at the left of the maximum, and an average out-degree (in-degree) which increases logarithmically with the number of published papers. These features were in part explained using a recursive search model introduced in [@vazquez00]. It is then concluded that recursive searches are one of the fundamental mechanisms in the formation of citation networks.
acknowledgments {#acknowledgments .unnumbered}
===============
I thanks Y. Moreno Vega and A. Vespignani for helpful comments and suggestions.
[99]{}
Lahèrre J. and Sornette D., Eur. Phys. J. [**2**]{} (1998) 525.
Redner S., Eur. Phys. J. B [**4**]{} (1998) 131.
Tsallis C. and de Albuquerque M. P., Eur. Phys. J. [**13**]{} (2000) 777.
Vazquez A., cond-mat/0006132; to appear in Europhys. Lett..
Science Citation Index and Journal Citation Reports (Institute for Scientific Information, Philadelfia).
Barabási A.-L. and Albert R., Science [**286**]{} (1999) 50.
Dorogovtsev S. N., Mendes J. F. F., and Samukhin A. N., Phys. Rev. Lett. [**85**]{} (2000) 4633.
Krapivsky P. L., Redner S., and Leyvraz F., Phys. Rev. Lett. [**85**]{} (2000) 4629.
Journal $k_m$ $P_m$ $N_0$ $b$ $b/N_0$
---------------------------------- ------- ------------ ------- ----- ---------
Astrophys. J.$^{\ref{eq:3}}$ 26 0.026 - - 1e-4
J. Appl. Phys.$^{\ref{eq:3}}$ 14 0.044 - 7e-5
J. Chem. Phys.$^{\ref{eq:2}}$ 28 0.024 7700 2.6 4e-4
Phys. Rev. B$^{\ref{eq:2}}$ 20 0.038 33000 2.2 7e-5
Phys. Rev. D$^{\ref{eq:2}}$ 23 0.028 5900 3.5 6e-4
Appl. Phys. Lett.$^{\ref{eq:3}}$ 13 0.079 - - 4e-5
BBRC$^{\ref{eq:3}}$ 22 0.048 - - 2e-4
Biochem. (USA)$^{\ref{eq:2}}$ 39 0.027 980 1.9 2e-3
Febs. Lett.$^{\ref{eq:2}}$ 24 0.043 7700 5.2 7e-4
J. Biol. Chem.$^{\ref{eq:3}}$ 37 0.030 - - 5e-5
PNAS$^{\ref{eq:3}}$ 31 0.039 - - 1e-4
Phys. Rev. Lett.$^{\ref{eq:2}}$ 20 (\[eq:2\]) 8300 1.2 1e-4
: Coordinates of the maximum ($k_m,P_m$) for each journal and fitting parameters obtained from the fit of the mean out-degree to either eqs. (\[eq:2\]) or (\[eq:3\]). On each case the equation used to fit the data is indicated as a supra-script.[]{data-label="tab:1"}
|
---
abstract: 'Popular parameterizations of the freeze-out conditions in relativistic heavy-ion collisions are discussed. Similarities and differences between the blast-wave model and the single-freeze-out model, both used recently to interpret the RHIC data, are outlined. A non-boost-invariant extension of the single-freeze-out model is proposed and applied to describe the recent BRAHMS data.'
address: |
$^1$ Institute of Physics, Świȩtokrzyska Academy,\
ul. Świȩtokrzyska 15, PL-25406 Kielce, Poland\
$^2$ Institute of Nuclear Physics, Polish Academy of Sciences\
ul. Radzikowskiego 152, PL-31342 Kraków, Poland
author:
- 'Wojciech Florkowski $^{1,2}$ and Wojciech Broniowski $^2$'
title: '[**Hydro-inspired parameterizations of freeze-out in relativistic heavy-ion collisions**]{}[^1] [^2]'
---
The blast-wave model of Siemens and Rasmussen {#sect:SR}
=============================================
In 1979 Siemens and Rasmussen formulated a model describing the hadron production in Ne + Na F reactions at the beam energy of 800 MeV per nucleon [@Siemens:1978pb]. The physical picture behind the model was that the fast hydrodynamic expansion of the produced hadronic matter leads to a sudden decoupling of hadrons and freezing of their momentum distributions, which retain their thermal character (although modified by the collective expansion effects) until the observation point. In their own words, Siemens and Rasmussen described the collision process as follows: “central collisions of heavy nuclei at kinetic energies of a few hundred MeV per nucleon produce fireballs of hot, dense nuclear matter; such fireballs explode, producing blast waves of nucleons and pions”. In this way, with Ref. [@Siemens:1978pb], the concept of the blast waves of hadrons and the blast-wave model itself entered the field of relativistic heavy-ion collisions.
Although the model of Siemens and Rasmussen was motivated by an earlier hydrodynamic calculation by Bondorf, Garpman, and Zimanyi [@Bondorf:1978kz], the results presented in Ref. [@Siemens:1978pb] were not obtained by solving the hydrodynamic equations but followed from the specific assumptions on the freeze-out conditions. The most important ingredient of the model was the spherically symmetric expansion of the shells of matter with constant radial velocity. With an additional assumption about the times when such shells disintegrate into freely streaming hadrons (this point will be discussed in a greater detail in Sect. \[sect:rad\]) Siemens and Rasmussen obtained the formula for the momentum distribution of the emitted hadrons [@Siemens:1978pb] $${dN \over d^3p} = Z \exp\left( -{\gamma E \over T} \right)
\left[ \left( 1 + {T \over \gamma E} \right)
{\hbox{sinh} a \over a} - {T \over \gamma E} \,\hbox{cosh}a \right].
\label{SR1}$$ In Eq. (\[SR1\]) $Z$ is a normalization factor, $E=\sqrt{m^2+p^2}$ denotes the hadron energy, $T$ is the temperature of the fireball (the same for all fluid shells), and $\gamma=(1-v^2)^{-1/2}$ is the Lorentz gamma factor with $v$ denoting the radial collective velocity (radial flow). A dimensionless parameter $a$ is defined by the equation $$a = {\gamma v p \over T}.
\label{a}$$ Small values of $v$ (and $a$) correspond to small expansion rate and, as expected, a simple Boltzmann factor is obtained from Eq. (\[SR1\]) in the limit $v \to 0$, $${dN \over d^3p} \to Z \exp\left( -{E \over T} \right).
\label{SR2}$$ The fits to the data based on the formula (\[SR1\]) gave $T$ = 44 MeV and $v$ = 0.373. Interestingly, the value of the radial flow $v$ turned out to be quite large suggesting the strong collective behavior. This was an unexpected feature summarized by the authors with the statement: “Monte Carlo studies suggest that Ne + Na F system is too small for multiple collisions to be very important, thus, this evidence for a blast feature may be an indication that pion exchange is enhanced, and the effective nucleon mean free path shortened in dense nuclear matter”.
Cooper-Frye formula {#sect:CF}
===================
Below we shall analyze the formal steps leading to Eq. (\[SR1\]). Our starting point is the expression defining the momentum distribution of particles as the integral of the phase-space distribution function $f(x,p)$ over the freeze-out hypersurface $\Sigma$, i.e., the renowned Cooper-Frye formula [@Cooper:1974mv], $$E \, {dN \over d^3p} = {dN \over dy\, d^2p_\perp} =
\int d^3\Sigma_\mu(x) p^\mu f(x,p).
\label{CF1}$$ The three-dimensional element of the freeze-out hypersurface in Eq. (\[CF1\]) may be obtained from the formula $$d^3\Sigma_\mu = \varepsilon_{\mu \alpha \beta \gamma}
{d x^\alpha \over d\alpha }
{d x^\beta \over d\beta }
{d x^\gamma \over d\gamma } d\alpha d\beta d\gamma,
\label{d3sigma}$$ where $\varepsilon_{\mu \alpha \beta \gamma}$ is the Levi-Civita tensor and $\alpha,\beta,\gamma$ are the three independent coordinates introduced to parameterize the hypersurface.
We note that for systems in local thermodynamic equilibrium we have $$E \, {dN \over d^3p} = \int d^3\Sigma_\mu(x) \, p^\mu
f_{\rm eq} \left(u_\mu(x) \,p^\mu \right),
\label{CF2}$$ where the function $f_{\rm eq}$ is the equilibrium distribution function $$f_{\rm eq}(E) = {1\over (2 \pi)^3}
\left[\exp\left({E - \mu \over T}\right) + \epsilon \right]^{-1}.
\label{eq}$$ Here the case $\epsilon = +1\,(-1)$ corresponds to the Fermi-Dirac (Bose-Einstein) statistics, and the limit $\epsilon \to 0$ yields the classical (Boltzmann) statistics. For a static fireball one finds $$d^3\Sigma_\mu = (dV,0,0,0), \quad u_\mu = (1,0,0,0),$$ and Eq. (\[CF2\]) is reduced to the formula $${dN \over d^3p} = V f_{\rm eq}(E),
\label{CF3}$$ where $V$ is the volume of the system. Eq. (\[CF3\]) agrees with Eq. (\[SR2\]) in the classical limit if the normalization constant $Z$ is taken as $$Z = {V \over (2\pi)^3} \exp\left({\mu \over T}\right).
\label{Z}$$
Spherically symmetric freeze-outs {#sect:rad}
=================================
\
For spherically symmetric freeze-outs it is convenient to introduce the following parameterization of the space-time points on the freeze-out hypersurface [@Rischke:1996em] $$x^\mu = (t,x,y,z) = \left( t(\zeta), r(\zeta) \sin\theta \cos\phi,
r(\zeta) \sin\theta \sin\phi, r(\zeta) \cos\theta \right).
\label{xmurad}$$ The freeze-out hypersurface is completely defined if a curve, i.e., the mapping $\zeta
\longrightarrow \left(t(\zeta),r(\zeta)\right)$ in the $t-r$ space is given. This curve defines the (freeze-out) times when the hadrons in the shells of radius $r$ stop to interact, see Fig. \[fig:tr\]. The range of $\zeta$ may be always restricted to the interval: $0 \leq \zeta \leq 1$. The three coordinates: $\phi \in [0,2 \pi], \theta \in
[0,\pi]$, and $\zeta \in [0,1]$ play the role of the variables $\alpha,\beta,\gamma$ appearing in Eq. (\[d3sigma\]). Hence, the element of the spherically symmetric hypersurface has the form $$\begin{aligned}
\! d^3\Sigma^\mu \!&=& \!
\left( r^\prime(\zeta),
t^\prime(\zeta) \sin\theta \cos\phi,
t^\prime(\zeta) \sin\theta \sin\phi,
t^\prime(\zeta) \cos\theta \right)
r^2(\zeta) \sin\theta \,d\theta\,d\phi\,d\zeta, \nonumber \\
\label{d3sigmarad}\end{aligned}$$ where the prime denotes the derivatives taken with respect to $\zeta$. Besides the spherically symmetric hypersurface we introduce the spherically symmetric (hydrodynamic) flow $$\begin{aligned}
u^\mu &=& \gamma(\zeta)\left( 1,
v(\zeta) \sin\theta \cos\phi,
v(\zeta) \sin\theta \sin\phi,
v(\zeta) \cos\theta \right),
\label{umurad1}\end{aligned}$$ where $\gamma(\zeta)$ is the Lorentz factor, $\gamma(\zeta) = (1-v^2(\zeta))^{-1/2}$. In a similar way the four-momentum of a hadron is parameterized as $$p^\mu = \left[E, p \sin\theta_p \cos\phi_p,
p \sin\theta_p \sin\phi_p,
p \cos\theta_p \right],
\label{pmurad}$$ and we find the two useful expressions: $$p \cdot u = \left(E - p v(\zeta) \cos\theta \right)\,\gamma(\zeta),
\label{purad}$$ $$d^3\Sigma \cdot p = \left(E r^\prime(\zeta)
- p t^\prime(\zeta) \cos\theta \right) r^2(\zeta) \sin\theta
\,d\theta \, d\phi \, d\zeta.
\label{Sigmaprad}$$ We note that the spherical symmetry allows us to restrict our considerations to the special case $\theta_p = 0$.
In the case of the Boltzmann statistics, with the help of Eqs. (\[CF1\]), (\[purad\]) and (\[Sigmaprad\]), we obtain the following form of the momentum distribution $$E {dN \over d^3p} = \int\limits_0^1 { e^{-(E \gamma-\mu)/T} \over 2 \pi^2}
\left[ E \,{\hbox{sinh}a \over a} \, {dr\over d\zeta}
+ T \,{(\hbox{sinh}a - a \hbox{cosh}a) \over a \gamma v }
{dt \over d\zeta} \right] r^2(\zeta) d\zeta.
\label{dNd3prad1}$$ Here $v, \gamma, r$ and $t$ are functions of $\zeta$, and the parameter $a$ is defined by Eq. (\[a\]). The thermodynamic parameters $T$ and $\mu$ may also depend on $\zeta$. To proceed further we need to make certain assumptions about the $\zeta$-dependence of these quantities. In particular, to obtain the model of Siemens and Rasmussen we assume that the thermodynamic parameters as well as the transverse flow velocity are constant $$T = \hbox{const}, \quad \mu = \hbox{const}, \quad
v = \hbox{const} \quad (\gamma = \hbox{const} , \quad a = \hbox{const} ).$$ Moreover, we should assume that the freeze-out curve in the $t - r$ space satisfies the condition $$dt = v \,dr, \quad t = t_0 + v r.
\label{dtvdr}$$ In this case we obtain the formula $${dN \over d^3p} = { e^{-(E \gamma-\mu)/T} \over 2 \pi^2}
\left[ \left(1+ {T \over \gamma E}\right) \,{\hbox{sinh}a \over a} \,
- {T \over \gamma E} \, \hbox{cosh}a \,\right]
\int\limits_0^1 r^2(\zeta) {dr \over d\zeta} d\zeta.
\label{dNd3prad2}$$ Equation (\[dNd3prad2\]) coincides with Eq. (\[SR1\]) if we use Eq. (\[Z\]) and make the following identification $$\int r^2(\zeta) {dr \over d\zeta} d\zeta = { r^3_{\rm max} \over 3}, \quad
V = {4 \over 3} \pi r^3_{\rm max}.
\label{V}$$ Note that the quantity $r_{\rm max}$ does not necessarily denote the maximum value of the radius of the system, see the dotted line on the left-hand-side of Fig. \[fig:tr\].
An interesting and perhaps unexpected feature of the model proposed by Siemens and Rasmussen is the relation between the times and positions of the freeze-out points, see Eq. (\[dtvdr\]) illustrated on the right-hand-side part of Fig. \[fig:tr\]. Eq. (\[dtvdr\]) indicates that the fluid elements which are further away from the center freeze-out later. Moreover, taking into account Eq. (\[dtvdr\]) in the formula (\[d3sigmarad\]) we find that the four-vector describing the hypersurface is parallel to the four-vector describing the flow, compare Eqs. (\[d3sigmarad\]) and (\[umurad1\]) giving $d^3\Sigma^\mu \sim u^\mu$ in this case. As we shall see the same features are assumed in the single-freeze-out model [@Broniowski:2001we; @Broniowski:2001uk; @Broniowski:2002nf].
It is worth to emphasize that in the hydrodynamic approach the $t-r$ freeze-out curves contain the space-like and time-like parts [^3]. The treatment of the space-like parts leads to conceptual problems since particles emitted from such regions of the hypersurface enter again the system and the hydrodynamic description of such regions (combined with the use of the Cooper-Frye formula) is inadeqate. Recently much work has been done to develop a consistent description of the freeze-out process from the space-like parts [@Bugaev:2004kq; @Csernai:2004pr]. However, very often only a quantitative argument is presented [@Teaney:2001av] that the contributions from the space-like parts are small and may be neglected compared to the contributions from the time-like regions. The choice of Siemens and Rasmussen seems to have anticipated such arguments.
Boost-invariant blast-wave model of Schnedermann, Sollfrank, and Heinz {#sect:binv}
======================================================================
The model presented above is appropriate for the low-energy scattering processes where the two nuclei completely merge at the initial stage of the collision and further expansion of the system is, to large extent, isotropic. At higher energies such a picture is not valid anymore and, following the famous paper by Bjorken [@Bjorken:1983qr], the boost-invariant and cylindrically symmetric models have been introduced to describe the collisons [^4].
The boost-invariance (symmetry with respect to the Lorentz transformations) may be incorporated in the hydrodynamic equations, kinetic equations, and also in the modeling of the freeze-out process. In the latter case, the appropriate formalism was developed by Schnedermann, Sollfrank, and Heinz [@Schnedermann:1993ws]. The ansatz for the boost-invariant, cylindrically symmetric freeze-out hypersurface has the form $$x^\mu = (t,x,y,z) =
\left( {\tilde \tau}(\zeta) \hbox{cosh } \alpha_\parallel,
\rho(\zeta) \cos\phi, \rho(\zeta) \sin\phi,
{\tilde \tau}(\zeta) \hbox{sinh } \alpha_\parallel \right).
\label{xmubinv}$$ Here, the parameter $\alpha_\parallel$ is the space-time rapidity. At $\alpha_\parallel=0$ the longitudinal coordinate $z$ is also zero and the variable ${\tilde \tau}(\zeta)$ coincides with the time coordinate $t$. Similarly to the spherical expansion discussed in Sect. \[sect:rad\], the boost-invariant freeze-out hypersurface is completely defined if the freeze-out curve $\zeta \to \left( {\tilde \tau}(\zeta), \rho(\zeta) \right)$ is given. This curve defines the freeze-out times of the cylindrical shells with the radius $\rho$. Because of the boost-invariance it is enough to define this curve at $z=0$, since for finite values of $z$ the freeze-out points may be obtained by the Lorentz transformation.
The volume element of the freeze-out hypersurface is obtained from Eq. (\[d3sigma\]), $$d^3\Sigma^\mu = \left( {d\rho \over d\zeta} \hbox{cosh }\alpha_\parallel,
{d {\tilde \tau} \over d\zeta} \cos\phi,
{d {\tilde \tau} \over d\zeta} \sin\phi,
{d\rho \over d\zeta} \hbox{sinh }\alpha_\parallel \right)
\rho(\zeta){\tilde \tau}(\zeta) d\zeta d\alpha_\parallel d\phi.
\label{d3sigmabinv}$$ Similarly to Eq. (\[xmubinv\]) the boost-invariant four-velocity field has the structure $$u^\mu = \hbox{cosh}\,\alpha_\perp(\zeta) \hbox{cosh}\,\alpha_\parallel
\left( 1 , \hbox{tanh}\,\alpha_\perp(\zeta) \cos\phi,
\hbox{tanh}\,\alpha_\perp(\zeta) \sin\phi,
\hbox{tanh}\,\alpha_\parallel \right).
\label{umubinv}$$ We note that the longitudinal flow is simply $v_z = \hbox{tanh}\, \alpha_\parallel = z/t$ (as in the one-dimensional Bjorken model), whereas the transverse flow is $v_r = \hbox{tanh}\,\alpha_\perp(\zeta)$.
With the standard parameterization of the particle four-momentum in terms of rapidity $y$ and the transverse mass $m_\perp$, $$p^\mu = \left(m_\perp \hbox{cosh} y, p_\perp \cos\varphi, p_\perp
\sin\varphi, m_\perp \hbox{sinh} y \right),
\label{pmubinv}$$ we find $$p \cdot u = m_\perp \hbox{cosh}(\alpha_\perp)
\hbox{cosh}(\alpha_\parallel-y) - p_\perp
\hbox{sinh}(\alpha_\perp) \cos(\phi-\varphi),
\label{pubinv}$$ and $$d^3\Sigma \cdot p = \left[m_\perp \hbox{cosh}(y-\alpha_\parallel)
{d\rho \over d\zeta} - p_\perp \cos(\phi-\varphi) {d {\tilde \tau}
\over d\zeta} \right]
\rho(\zeta){\tilde \tau}(\zeta) d\zeta d\alpha_\parallel d\phi.
\label{sigmapbinv}$$
For the Boltzmann statistics, with $\beta = 1/T$, the Cooper-Frye formalism gives the following momentum distribution $$\begin{aligned}
& & {dN \over dy d^2p_\perp} = \nonumber \\
& & {e^{\beta \mu} \over (2\pi)^3}
\int\limits_0^{2\pi} d\phi
\int\limits_{-\infty}^{\infty} d\alpha_\parallel \int\limits_0^1 d\zeta
\,\, \rho(\zeta) {\tilde \tau}(\zeta) \left[m_\perp
\hbox{cosh}(\alpha_\parallel-y) {d\rho \over d\zeta}
- p_\perp \cos(\phi-\varphi) {d{\tilde \tau} \over
d\zeta} \right] \nonumber \\
& & \times
\exp\left[-\beta m_\perp \hbox{cosh}(\alpha_\perp)
\hbox{cosh}(\alpha_\parallel-y)+ \beta p_\perp
\hbox{sinh}(\alpha_\perp) \cos(\phi-\varphi) \right].
\label{bmdN1}\end{aligned}$$ The form of Eq. (\[bmdN1\]) shows explicitly that the distribution $dN/(dy d^2p_\perp)$ is independent of $y$ and $\varphi$, in accordance with our assumptions of the boost-invariance and cylindrical symmetry. The integrals over $\alpha_\parallel$ and $\phi$ in Eq. (\[bmdN1\]) are analytic and lead to the Bessel functions $K$ and $I$, $$\begin{aligned}
{dN \over dy d^2p_\perp} &=&
{e^{\beta \mu} \over 2 \pi^2} m_\perp
K_1 \left[\beta m_\perp \hbox{cosh}(\alpha_\perp) \right]
I_0 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right]
\int\limits_0^1 d\zeta \,\, \rho(\zeta) {\tilde \tau}(\zeta)
{d\rho \over d\zeta}
\nonumber \\
&& - {e^{\beta \mu} \over 2 \pi^2} p_\perp
K_0\left[\beta m_\perp \hbox{cosh}(\alpha_\perp) \right]
I_1 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right]
\int\limits_0^1 d\zeta \,\, \rho(\zeta) {\tilde \tau}(\zeta)
{d{\tilde \tau} \over d\zeta}. \nonumber \\
\label{KandI}\end{aligned}$$ In the spirit of the blast-wave model of Siemens and Rasmussen we have assumed here that the radial velocity is constant, $v_r = \hbox{tanh}\,\alpha_\perp(\zeta)
= \hbox{const}$, otherwise the Bessel functions should be kept under the integral over $\zeta$.
In order to achieve the simplest possible form of the model, the common practice is to neglect the second line of Eq. (\[KandI\]). This procedure means that one assumes implicitly the freeze-out condition $d{\tilde \tau} / d\zeta = 0 \, ({\tilde \tau} = \tau = \hbox{const})$. In this case the boost-invariant blast-wave model is reduced to the formula $${dN \over dy d^2p_\perp} =
\hbox{const}\, m_\perp
K_1 \left[\beta m_\perp \hbox{cosh}(\alpha_\perp) \right]
I_0 \left[\beta p_\perp \hbox{sinh}(\alpha_\perp)\right].
\label{K1I0a}$$ where the constant has absorbed the factor $e^{\beta \mu} \tau \rho^2_{\rm max}/ (4 \pi^2)$. Eq. (\[K1I0a\]) forms the basis of numerous phenomenological analyses of the transverse-momentum spectra measured at the SPS and RHIC [@Retiere:2003kf] energies.
Resonances
==========
The main drawback of the formalism outlined above is that it neglects the effect of the decays of hadronic resonances. Such an approach may be justified at lower energies but should be improved at the relativistic energies where most of the light particles are produced in the decays of heavier resonance states. The expressions giving the rapidity and transverse momentum spectra of particles originating from two- and three-body decays of the resonances with a specified momentum distribution were worked out by Sollfrank, Koch, and Heinz [@Sollfrank:1990qz; @Sollfrank:1991xm]. Their formulae may also be used to account for the feeding from the resonances in the blast-wave model, as already proposed in Ref. [@Schnedermann:1993ws]. In other words, we wish to stress that the choice of the freeze-out hypersurface and of the flow profile are elements [*completely independent*]{} of the treatment of the resonances. Both are important with the latter being the basic ingredient in the calculation of particle abundances and the key to success of thermal models.
At the SPS and RHIC energies it is important to include not only the decays of the most common resonances such as $\eta, \rho, \omega, K^*$ or $\Delta$, but also of much heavier states. Although their contributions are supressed by the Boltzmann factor, their number increases strongly with the mass [@Broniowski:2000bj; @Broniowski:2004yh], hence their role can be easily underestimated. The effects of sequential decays of heavy resonances were first realized in statistical analyses of the ratios of hadronic abundances/multiplicities (for recent results see [@Florkowski:2001fp; @Braun-Munzinger:2001ip; @Torrieri:2004zz; @Wheaton:2004qb]) which showed that the statistical models give a very good description of the data, provided most of the hadrons appearing in the Particle Data Tables are included in the calculations.
In order to discuss the role of the sequential decays of the resonances it is convenient to start with a general formalism giving the Lorentz-invariant phase-space density of the measured particles [@Bolz:1992hc] $$\begin{aligned}
&& n_{1 }\left( x_{1},p_{1}\right) = E_1 {dN_1 \over d^3p_1 d^4x_1 } =
\label{npix1p1} \nonumber \\
&& \int \frac{d^{3}p_{2}}{E_{p_{2}}}
B\left( p_{2},p_{1}\right) \int d\tau _{2}\Gamma _{2}e^{-\Gamma _{2}\tau
_{2}} \int d^{4}x_{2}\delta ^{\left( 4\right) }
\left( x_{2}+\frac{p_{2}\tau_{2}}{m_{2}}-x_{1}\right)... \nonumber \\
&&\times \int \frac{d^{3}p_{N}}{E_{p_{N}}}B\left( p_{N},p_{N-1}\right) \int
d\tau _{N}\Gamma _{N}e^{-\Gamma _{N}\tau _{N}} \nonumber \\
&& \hspace{5mm} \times \int d\Sigma _{\mu }\left(
x_{N}\right) \,p_{N}^{\mu }\,\,\delta ^{\left( 4\right) }\left( x_{N}+\frac{%
p_{N}\,\tau _{N}}{m_{N}}-x_{N-1}\right) f_{N}\left[ p_{N}\cdot u\left(x_{N}\right)
\right]. \nonumber \\
\label{ornik1}\end{aligned}$$ Here the indices $1,2, ..., N$ label hadrons in one chain of the sequential decays. The first resonance is produced on the freeze-out hypersurface and has the label $N$. The final hadron has the label $1$, for more details see [@Broniowski:2001uk; @Broniowski:2002nf]. The function $B(k,q)$ is the probability distribution for a resonance with momentum $k$ to produce a particle with momentum $q$ in a two-body decay $$B(k,q) = {b \over 4 \pi p^*}
\delta \left( {k \cdot q \over m_R}- E^* \right).
\label{Bkq}$$ The function $B(k,q)$ satisfies the normalization condition $$\int {d^3q \over E_q} B(k,q) = b,
\label{normB}$$ where $b$ is the branching ratio for a given decay channel and $p^* (E^*)$ is the momentum (energy) of the emitted particle in the resonance’s rest frame (a generalization to three-body decays is straightforward and explained in Refs. [@Florkowski:2001fp]).
Integration of Eq. (\[ornik1\]) over all space-time positions gives the formula for the momentum distribution $$\begin{aligned}
&& E_{p_1} {dN_1 \over d^3 p_1} = \int d^{4}x_{1}\,n_{1}
\left(x_{1},p_{1}\right) = \label{npip1} \nonumber \\
&&\int \frac{d^{3}p_{2}}{E_{p_{2}}}B\left( p_{2},p_{1}\right)...\int
\frac{d^{3}p_{N}}{E_{p_{N}}}B\left( p_{N},p_{N-1}\right)
\int d\Sigma_{\mu }\left( x_{N}\right) \,p_{N}^{\mu }\,\,f_{N}
\left[ p_{N}\cdot u\left(x_{N}\right) \right]. \nonumber \\
\label{ornik2}\end{aligned}$$
Equation (\[ornik2\]) serves as the starting point to prove that for constant values of the thermodynamic parameters on the freeze-out hypersurface the ratios in the full phase-space ($4 \pi$) are the same as in the local fluid elements. In this way, a connection between the measured ratios and the local thermodynamic parameters is obtained [@Heinz:1998st]. One may also check that for the boost-invariant systems it is enough to consider the ratios at any value of the rapidity to infer the values of the thermodynamic parameters [@Broniowski:2001we; @Broniowski:2002nf].
The experimental RHIC data show, however, that the rapidity distributions are of Gaussian shape [^5] and the thermodynamic parameters vary with rapidity (the measured ${\bar p}/p$ ratio depends on $y$), hence, the system created at RHIC is, strictly speaking, not boost-invariant. In this situation the relation between the measured ratios and thermodynamic parameters is not obvious. Fortunately, the RHIC data show also a rather flat rapidity distribution and constant ratios in the rapidity range $-1 < y < 1$ [@Bearden:2003fw; @Bearden:2004yx]. In this region (the central part of the broad Gaussian) the system to a good approximation may be treated as boost-invariant and the standard analysis of the ratios may be performed to obtain the thermodynamic parameters at $y=0$.
Single-freeze-out model
=======================
The analysis of the ratios of hadron multiplicities measured at RHIC gives a typical temperature of 170 MeV. On the other hand, the analysis of the spectra based on Eq. (\[K1I0a\]) gives a lower temperature of about 100 - 140 MeV. Such a situation was observed already at the SPS energies, which motivated the introduction of the concept of two different freeze-outs.
Certainly, if the spectra contain important contributions from high lying states, the value of $T$ obtained from the blast-wave formula fitted to the spectra cannot be interpreted as the temperature of the system in the precise thermodynamic sense. First, the contributions from the resonances (feeding mostly the low-momentum region) should be subtracted from the spectra of light hadrons, giving the insight to the properties of the primordial particles. Using other words, we may argue that the calculation of the ratios should include the same number of the resonances as the corresponding calculation of the spectra.
An example of such a calculation is the single-freeze-out model formulated in Refs. [@Broniowski:2001we; @Broniowski:2001uk]. In this model the decays of the resonances as well as the transverse flow change the spectra of the primordial particles in such a way that it is possible to describe well the spectra and the ratios with a single value of the temperature. The basic effect here is that the hadronic decays lead to effective cooling of the spectra.
Similarly to the original blast-wave models discussed above, the single freeze-out model assumes a certain form of the freeze-out hypersurface in the Minkowski space. In this case it is defined by the constant value of the proper time $$\tau = \sqrt{t^2-r^2_x-r^2_y-r^2_z} = {\rm const}.
\label{tau}$$ The transverse size of the system is defined by the parameter $\rho_{\rm max}$, $$\rho=\sqrt{r_x^2+r_y^2}, \quad
\rho < \rho_{\rm max},
\label{rhomax}$$ and the velocity field at freeze-out is taken in the Hubble-like form [^6] $$u^\mu = \frac{x^\mu}{\tau} = \frac{t}{\tau} \left(1,
\frac{x}{t}, \frac{y}{t}, \frac{z}{t}\right).
\label{hubflow}$$ The natural parameterization of the freeze-out hypersurface has the form $$\begin{aligned}
t &=&\tau \cosh \alpha _{\parallel }\cosh \alpha _{\perp },\quad z=\tau
\sinh \alpha _{\parallel }\cosh \alpha _{\perp }, \nonumber \\
x &=&\tau \sinh \alpha _{\perp }\cos \phi ,\quad y =\tau \sinh \alpha
_{\perp }\sin \phi,
\label{txyz}\end{aligned}$$ which may be considered as the special case of the formula (\[xmubinv\]). Eq. (\[d3sigma\]) leads to the following expression defining the volume element $$d\Sigma^\mu(x) = u^\mu(x)\, \tau ^{3} \, {\rm sinh}(\alpha _{\perp})
{\rm cosh}(\alpha _{\perp}) \, d\alpha _{\perp}
d\alpha _{\parallel } d\phi.
\label{d3sigmaumu}$$
A very important feature of the choice (\[tau\]) - (\[hubflow\]) is that the volume element is proportional to the four-velocity field. This feature holds also in the model of Siemens and Rasmussen. In this case the treatment of the resonance is very much facilitated. In particular, Eq. (\[ornik2\]) may be rewriten in the form $$\begin{aligned}
E_{p_1} {dN_1 \over d^3 p_1}\!\!\!\!&=&\!\!\!\!\int d\Sigma \left(
x_{N}\right) \int \frac{d^{3}p_{2}}{E_{p_{2}}}
B\left( p_{2},p_{1}\right) \nonumber \\
& & ...
\int \frac{d^{3}p_{N}}{E_{p_{N}}}B\left( p_{N},p_{N-1}\right) p_{N}
\,\cdot u\left( x_{N}\right) \,f_{N}\left[ p_{N}\cdot u\left( x_{N}\right)
\right] \nonumber \\
&=&\int d\Sigma \left( x_{N}\right)
p_{1}\,\cdot u\left( x_{N}\right)
\,f_{1}\left[ p_{1}\cdot u\left( x_{N}\right) \right],
\label{ornik3}\end{aligned}$$ where we have introduced the notation $$\begin{aligned}
& & p_{i-1}\,\cdot u\left( x_{N}\right) \,f_{i-1}\left[ p_{i-1}\cdot
u\left( x_{N}\right) \right] \label{trsim}
\nonumber \\& & =\int \frac{d^{3}p_{i}}{E_{p_{i}}}B\left(
p_{i},p_{i-1}\right) p_{i}\,\cdot u\left( x_{N}\right) \,f_{i}\left[
p_{i}\cdot u\left( x_{N}\right) \right].
\label{acta1}\end{aligned}$$ In the local rest frame, the iterative procedure defined by Eq. (\[acta1\]) becomes a simple one-dimensional integral transform $$f_{i-1}\left( q\right) =
\frac{b m_R}{2 E_q p^\ast q} \int_{k_{-}(q)}^{k_{+}(q)} dk\, k \,
f_i\left( k\right),
\label{acta2}$$ where $k_\pm (q) = m_R | E^* q \pm p^* E_q |/m_1^2$. Eqs. (\[acta1\]) and (\[acta2\]) allow us to deal with a very large number of decays in the very efficient way, very similar to that used in the calculation of the hadron abundances.
Non boost-invariant single-freeze-out model
===========================================
The model described above may be generalized to the non boost-invariant version in the minimal way by the modification of the system boundaries. Introducing a dependence of the transverse size on the longitudinal coordinate $z$ (or $\alpha_\parallel$), we break explicitly the assumption of the boost-invariance. At the same time, however, the local properties of the hypersurface and flow remain unchanged allowing us to treat the resonances in the same simple way as described in the previous Section.
Since the measured rapidity distributions are approximately gaussian, it is natural to start with the gaussian ansatz for the dependence of the transverse size on the parameter $\alpha_\parallel$ and restrict the region of the integration over $\alpha_\perp$ to the interval $$0 \leq \alpha_\perp \leq
\alpha_\perp^{\rm max} \exp[- \alpha^2_\parallel/(2 \Delta^2)].$$ The original boost-invariant version is recovered in the limit $\Delta \longrightarrow \infty$. Using the values of the thermodynamic parameters obtained from the boost-invariant version of the model applied to the hadronic ratios measured at midrapidity ($T$ = 165.6 MeV and $\mu_B = 28.5$ MeV), we are left with three extra parameters ($\tau$, $\rho_{\rm max} = \tau \, \hbox{sinh} \,
\alpha_\perp^{\rm max}$, and $\Delta$), which should be fitted to the $p_\perp$-spectra collected at different values of the rapidity.
The result of such a fit to the available BRAHMS data on $\pi^+$, $\pi^-$, $K^+$, and $K^-$ production are shown in Fig. \[fig:brpt\]. The optimal values of the parameters found in the fit are: $\tau = 8.33$ fm, $\alpha_\perp^{\rm max} = 0.825$, and $\Delta = 3.33$. One can see that the model reproduces the data very well in a wide range of the transverse-momentum and rapidity. In Fig. \[fig:bry\] we show the model rapidity distributions compared to the data. Small discrepancies (of about 10%) between the model and the data may be seen for the pions at $y=0$. Note that the comparison of the rapidity distributions in Fig. \[fig:bry\] is done with a linear scale; definitely, small discrepancies may be expected for a such simplified description of the freeze-out.
It should be emphasized that the non-boost invariant version of the model presented above is not capable of describing correctly the ${\bar p}/p$ ratio. In the present framework this requires an introduction of the rapidity dependence of the baryon chemical potential.
Conclusions
===========
We have discussed several parameterizations of the freeze-out conditions in the relativistic heavy-ion collisions. We have argued that the single freeze-out model used to describe the RHIC data is a natural development of the blast-wave models worked out, among others, by Siemens, Rasmussen, Heinz, Schnedermann, and Sollfrank. The main advantage of the single-freeze-out model is that it includes all well established resonance decays, allowing us to treat the chemical and thermal freeze-out as essentially one phenomenon. In this respect, the single-freeze-out model is very similar to the original blast-wave model. Further similarities concern the shape of the freeze-out hypersurface (only the time-like parts are considered) and the strict use of the Cooper-Frye formula. Due to the limited space, we have not discussed here the variety of models where, instead of the Cooper-Fry formula, the so-called emission functions are introduced and modeled. An example of such an a approach is the Buda-Lund model [@Csanad:2004mm].
One of us (WF) acknowledges clarifying discussions with Jan Rafelski and Giorgio Torrieri.
[11]{}
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K. A. Bugaev, Phys. Rev., [**C70**]{} (2004) 034903, nucl-th/0401060.
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[^1]: Lecture presented by WF at the [*XLIV Cracow School of Theoretical Physics*]{}, Zakopane, Poland, 28 May – 6 June 2004
[^2]: Research supported in part by the Polish State Committee for Scientific Research, grant number 2 P03B 05925
[^3]: We use the convention that in the space-like (time-like) region the vector normal to the freeze-out curve is space-like (time-like).
[^4]: As is discussed in greater detail below, the data delivered by the BRAHMS Collaboration indicate that the systems produced at RHIC may be treated as boost-invariant only in the limited rapidity range $-1 < y < 1$. Moreover, the assumption about the cylindrical symmetry is valid only for the most central data.
[^5]: This feature has revived the interest in the Landau hydrodynamic model [@Landau:1953gs; @Belenkij:1956cd], see, for example, Ref. [@Steinberg:2004vy].
[^6]: For a recent attempt to connect the parameterization (\[tau\]) - (\[hubflow\]) with hydrodynamic calculation see Ref. [@Chojnacki:2004ec].
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abstract: 'Each paper should be preceded by a short Abstract of not more than 150 words, which should be written as a single paragraph and should not contain references.'
title: 'Test-beam performance of a tracking TRD prototype'
---
J.T. Shank, J.S. Whitaker,V.A. Polychronakos, V. Radeka, D. Stephani, H. Beker, R.K. Bock, M. Botlo, W.J. Willis, T. Akesson, V. Chernyatin, B. Dolgoshein, P. Nevski, L. DiCiaccio, M. Koratzinos and A. Lopez
Introduction
============
The biological effect of radiation depends on the quality of the radiation as well as on the amount of energy absorbed. Evidence suggests that this quality dependence is primarily caused by the differences in rates of energy loss [@Raby1966; @Dupont1961]. The general criteria that have been used are:
1. to terminate the iteration when the residue between iterated and experiment values is of the order of experimental errors [@Raby1966; @Appleman1959; @vanBerg1965; @Bryant1985; @Allen1977];
2. to terminate when the smoothest solution has been obtained.
For an overall appreciation of the work carried out in this field, see Refs. [@Keil1969] and [@Guignard1983].
Examples of figures and tables
==============================
Including tabular material
--------------------------
Smith tabulated the average values of mean linear energy transfer (LET) obtained by the different methods used in Ref. [@Appleman1959], and these are reproduced in Table \[tab:LET\]. Note that a table is produced with LaTeX’s `table` environment, and that the caption should be positioned *above* the tabular material.
[|p[6cm]{}|c|c|]{} Radiation & Smith$^{a)}$ & Jones\
1 MeV $\alpha$ & &\
200 kVp X-rays total & 3.25 & 1.79\
200 kVp X-rays (primary) & 2.60 & 1.48\
Table \[tab:LET\] is reproduced from the publication mentioned earlier and shows the good agreement between predictions and calculations. Comparison should be made with the decay curves shown in Figs. 6 and 7 of Ref. [@vanBerg1965], and further information is given in Section \[sec:curvature\] and Appendix \[sec:app\].
Including figures
-----------------
Figures are defined with the `figure` environment. The source for producing the figure can be included inline, as in Fig. \[fig:sincos\] which shows how the $\sin$ and $\cos$ functions evolve with the help of the `picture` environment and the `pict2e` and `curve2e` packages.
(6.6,3)(0,-1.5) (0,-1.5)[(0,3)[3]{}]{} (0,0)[(6.6,0)]{} (6.3,0.1)[$\theta$]{} (0.05,-0.3)[$0$]{} (2.9,-0.3)[$\pi$]{} (6.23,-0.3)[$2\pi$]{} (0.1,1.1)[$\cos\theta$]{} (0,1)<1,0>(1.570796,0)<1,-1> (3.1415924,-1)<1,0>(6.283185,1)<1,0> (1.65,1.1)[$\sin\theta$]{} (0,0)<1,1>(1.570796,1)<1,0> (4.712389,-1)<1,0>(6.283185,0)<1,1>
Figures can also be imported, in EPS or PDF, PNG and JPEG format, depending on whether you run `latex` or `pdflatex`. Figure \[fig:extpict\] show how to include a picture with the `\includegraphics` command of LaTeX’s `graphicx` package.
![Including a figure from an external file[]{data-label="fig:extpict"}](testpicture){width=".5\linewidth"}
Examples of equations {#sec:curvature}
---------------------
Equation \[eq:a1\] representing a straight line at an angle $\theta$, is $$n^k(h)u=\lambda h t g q_k \label{eq:a1}$$ and $$n^k(h) =k h \frac{k}{32} \label{eq:a2}$$ where:
- $\lambda$ is the distance between two consecutive sweep lines,
- $u$ is the least count.
We consider a parabola, the tangent of which is parallel to the vertical axis of the main matrix. A sufficiently good approximation to a parabola \[see Eq. \[eq:a3\]\] drawn inside the matrix can be given by the following formula:
$$n_q = \alpha q^2 \quad \alpha=\frac{\lambda^2}{3 Ru} \label{eq:a3}$$
where $R$ is the radius of curvature. We have the following relationship: $$n_q + \mu_q \text{ with } \mu_q = \alpha (2q + 1). \label{eq:a4}$$
Conclusion
==========
The theoretical considerations presented have been confirmed by their close agreement with the results of practical experiments. An account of the earlier work carried out in this field can be found in the bibliography.
It is expected that in the next few years many new results will be published, since a significant number of new experiments have recently been launched.
Acknowledgements {#acknowledgements .unnumbered}
================
We wish to thank C. Brown for his most enlightening comments on this topic.
[99]{} J.M. Raby, Biophysical aspects of radiation quality, International Atomic EnergyAgency, Technical Reports Series No. 58 (1966). J.-P. Dupont, Proc. Int. Conf. on Radiation Hazards, Columbia, 1960 (Academic Press Inc., New York, 1961), Vol. II, p. 396. H. Appleman et al., J. Med. Biol. **8** (1959) 911. E. van Berg, D. Johnson and J. Smith, Rad. Res. **5** (1965) 215. P. Bryant and S. Newman (eds.), The generation of high fields, CAS–ECFA–INFN Workshop, Frascati, 1984. (ECFA 85/91, CERN 85/07, CERN, Geneva, 1985). M.A. Allen et al., IEEE Trans. Nucl. Sci. **NS–24** (1977) 1780. E. Keil, Nucl. Instrum. Methods **100** (1969) 419. G. Guignard, CERN LEP–TH/83–38 (1983).
Bibliography {#bibliography .unnumbered}
============
I.C. Percival and D. Richards, Introduction to dynamics (Cambridge University Press, 1982).L. Garrido (ed.), Dynamical systems and chaos: Proc. Sitges 1982, Lecture Notes in Physics No. 179 (Springer, Berlin, 1983).
Construction on a flat site {#sec:app}
===========================
General considerations
----------------------
Following on ECFA recommendation, the project described in this report is based on the assumption that the machine is built close to the present CERN site, and has been taken into account in the cost estimate.
Effects on the construction
---------------------------
The general layout of the machine would be very similar to that shown in the main body of the report.
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---
abstract: |
There are several mathematical models yielding $1/f$ noise. For example, $1/f$ spectrum can be obtained from stochastic sequence of pulses having power-law distribution of pulse durations or from nonlinear stochastic differential equations. We show that a couple of seemingly different models exhibiting $1/f$ spectrum are due to the similar scaling properties of the signals. In addition, we demonstrate a connection between signals with the power-law behavior of the power spectral density generated by the nonlinear stochastic differential equations and modeled by a sequence of random different pulses. An approximation of solutions of the nonlinear stochastic differential equations by the sequence of pulses correctly reproduces the power-law parts of the probability density function and of the power spectral density. This connection provides further insights into the origin of $1/f$ noise.
[*Keywords*]{}: stochastic processes (theory), current fluctuations, stationary states
address: 'Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 12, LT-01108 Vilnius, Lithuania'
author:
- J Ruseckas and B Kaulakys
title: 'Scaling properties of signals as origin of $1/f$ noise'
---
Introduction
============
Signals having the power spectral density (PSD) at low frequencies $f$ of the form $S(f)\sim1/f^{\beta}$ with $\beta$ close to $1$ are commonly referred to as “$1/f$ noise”, “$1/f$ fluctuations”, or “flicker noise.” Power-law distributions of spectra of signals with $0.5<\beta<1.5$, as well as scaling behavior are ubiquitous in physics and in many other fields [@Scholarpedia2007; @Weissman1988; @Barabasi1999; @Gisiger2001; @Wong2003; @Wagenmakers2004; @Newman2005; @Szabo2007; @Castellano2009; @Eliazar2009; @Eliazar2010; @Perc2010; @Orden2010; @Kendal2011; @Torabi2011; @Diniz2011]. Despite the numerous models and theories proposed since its discovery more than 80 years ago [@Johnson1925; @Schottky1926], the subject of $1/f$ noise remains still open for new discoveries. Most models and theories of $1/f$ noise are not universal because of the assumptions specific to the problem under consideration. A short categorization of the theories and models of $1/f$ noise is presented in the introduction of the paper [@Kaulakys2009]. See also recent experimental investigations and physical models of $1/f$ noise in condensed matter [@Liu2009; @Dmitruk2011; @Liu2013; @Han2013; @Kuhlmann2013] and a review by Balandin [@Balandin2013].
Often $1/f$ noise is modeled as the superposition of Lorentzian spectra with a wide range distribution of relaxation times [@McWhorter1957]. An influential class of the models of $1/f$ noise involves self-organized criticality (SOC). In 1987 Bak *et al* [@Bak1987] introduced the notion of SOC with the motivation to explain the universality of $1/f$ noise. Although paper [@Bak1987] is the most cited paper in the field of $1/f$ noise problems, it was shown later on [@Jensen1989; @Kertesz1990] that the mechanism proposed in [@Bak1987] results in $1/f^{\beta}$ fluctuations with $1.5<\beta\leq2$ and does not explain the omnipresence of $1/f$ noise. The $1/f$ noise in the fluctuations of mass was first seen in a sandpile model with threshold dissipation by Ali [@Ali1995]. Maslov *et al* [@Maslov1999] studied a one-dimensional directed model of sandpiles and showed that the exponent $\beta$ is exactly $1$ for noise in the spectrum of fluctuations of mass. An example of $1/f$ noise in the classical sandpile model has been provided in [@Baiesi2006].
Another models of $1/f$ noise involve a class of maps generating intermittent signals. It is possible to generate power-laws and $1/f$-noise from simple iterative maps by fine-tuning the parameters of the system at the edge of chaos [@Procaccia1983; @Schuster1988] where the sensitivity to initial conditions of the logistic map is a lot milder than in the chaotic regime [@Costa1997]. Manneville [@Manneville1980a] showed that iterative function can produce interesting behavior, power-laws and $1/f$ PSD. In paper [@RuseckasChaos2013] a mechanism of intermittency exhibiting $1/f$ noise which occurs in nonlinear dynamical systems with invariant subspace and having the transverse Lyapunov exponent equal to zero was considered. Intermittency as a mechanism of $1/f$ noise continues to attract attention [@Laurson2006; @Pando2007; @Shinkai2012].
In many cases the physical processes can be represented by a sequence of random pulses. The mathematical way of generating power-law noise from a sequence of pulse has been discussed by Halford [@Halford1968]. The spectrum of the signal consisting of the pulse sequences which belong to the class of Markov process, was investigated in [@Heiden1969; @Shick1974]. In those models the power-law PSD appears due to power-law distribution of pulse durations. Main objection to this approach is an apparent lack of physical mechanism generating durations over several orders of magnitude [@Press1978]. On the other hand, we can point out that the diffusion process gives rise to broad distributions of lifetimes. It is known that for the unbiased random walk, the distribution of the first return times has a power-law form with the exponent $-3/2$ [@Redner2001]. Another possible mechanism leading to broad distribution of lifetimes is owing to the formation of avalanches. In many situations (for example in Barkhausen noise [@Kuntz2000], fluid invasion into disordered media [@Rost2007], dislocation avalanches in microcrystals [@Papanikolaou2012]) the volume of avalanches has power-law distribution and, in addition, the volume scales as a power-law function of the duration of the avalanche. Similar properties have also avalanches in the models of SOC [@Freeman2000; @Laurson2005; @Bartolozzi2006; @Yadav2012; @LeBlanc2013].
One more way of obtaining $1/f$ noise from a signal consisting of pulses has been presented in [@Kaulakys1998; @Kaulakys1999; @Kaulakys2000-2; @Kaulakys2005]. It has been shown that the intrinsic origin of $1/f$ noise may be a Brownian motion of the interevent time of the signal pulses, similar to the Brownian fluctuations of the signal amplitude, resulting in $1/f^{2}$ noise. Recently, the nonlinear stochastic differential equations (SDEs) generating signals with $1/f$ noise were obtained in [@Kaulakys2004; @Kaulakys2006] (see also papers [@Kaulakys2009; @Ruseckas2010; @ruseckas-2011]), starting from the point process model of $1/f$ noise. Analysis of the long-range correlated bursting signals is presented in [@Davidsen2009; @Grigolini2010; @Moloney2011; @Martin2011; @Gontis2012ACS; @Eliazar2013; @Krisponeit2013], as well.
The purpose of this paper is to show the connection between the nonlinear SDEs generating signals with $1/f$ noise and signals consisting of random pulses with the power-law distribution of pulse durations. As we will demonstrate, in both of these models $1/f$ spectrum appears due to the scaling properties of the signal. In addition, the signal generated by SDE can be approximated by rectangular pulses yielding the same distribution of signal intensity and the same power-law exponent in the PSD. Although the models generating $1/f$ noise that we consider in this paper (nonlinear SDEs and random pulses with the power-law distribution of pulse durations) are not unique, the method of their derivation from the scaling properties of the signal was not investigated before. We obtain nonlinear SDEs generating signals with $1/f$ noise starting [*not*]{} from the point process model, as has been done in [@Kaulakys2004; @Kaulakys2006], but [*from the scaling properties of the signal required to get $1/f$ noise*]{}. This approach allows us to reveal new connections between those seemingly different models.
The paper is organized as follows: In we consider nonlinear SDEs generating signals with $1/f^{\beta}$ PSD and show that such SDEs can be obtained by requiring a proper scaling. In we analyze signals consisting of random pulses with the power-law distribution of pulse durations and power-law dependence of pulse height on the pulse duration. We show that such pulses have the same scaling properties as the signal generated by SDEs in . In we produce the connection between the nonlinear SDEs modeling and the rectangular pulses series more explicitly. summarizes our findings.
Nonlinear SDE generating signals with $1/f^{\beta}$ noise from scaling {#sec:SDE}
======================================================================
Nonlinear SDEs generating signals with $1/f^{\beta}$ PSD are derived in papers [@Kaulakys2004; @Kaulakys2006]. In this Section we show these SDEs can be obtained only from the scaling properties required by $1/f^{\beta}$ PSD. This new technique reveals more directly the origin of $1/f^{\beta}$ PSD compared to the derivation starting from the point process model, as has been done in [@Kaulakys2004; @Kaulakys2006].
Pure $1/f^{\beta}$ PSD is physically impossible because the total power would be infinite. Therefore, we will consider signals with PSD having $1/f^{\beta}$ behavior only in some wide intermediate region of frequencies, $f_{\mathrm{min}}\ll f\ll f_{\mathrm{max}}$, whereas for small frequencies $f\ll f_{\mathrm{min}}$ PSD is bounded. We can obtain nonlinear SDE generating signals exhibiting $1/f$ noise using the following considerations. Wiener-Khintchine theorem relates PSD $S(f)$ to the autocorrelation function $C(t)$: $$C(t)=\int_{0}^{+\infty}S(f)\cos(2\pi ft)\rmd f\,.\label{eq:W-K}$$ If $S(f)\sim f^{-\beta}$ in a wide region of frequencies, then for the frequencies in this region the PSD has a scaling property $$S(af)\sim a^{-\beta}S(f)$$ when the influence of the limiting frequencies $f_{\mathrm{min}}$ an $f_{\mathrm{max}}$ is neglected. From the Wiener-Khintchine theorem it follows that the autocorrelation function has the scaling property $$C(at)\sim a^{\beta-1}C(t)\label{eq:auto-scaling}$$ in the time range $1/f_{\mathrm{max}}\ll t\ll1/f_{\mathrm{min}}$. The autocorrelation function can be written as [@Ruseckas2010; @Risken1996; @Gardiner2004] $$C(t)=\int\rmd x\int\rmd x'\, xx'P_{0}(x)P(x',t|x,0)\,,\label{eq:auto}$$ where $P_{0}(x)$ is the steady state probability density function (PDF) and $P(x',t|x,0)$ is the transition probability (the conditional probability that at time $t$ the signal has value $x'$ with the condition that at time $t=0$ the signal had the value $x$). The transition probability can be obtained from the solution of the Fokker-Planck equation with the initial condition $P(x',t|x,0)=\delta(x'-x)$. One of the ways to obtain the required property is for the steady state PDF to have the power-law form $$P_{0}(x)\sim x^{-\lambda}\label{eq:prob-steady}$$ and for the transition probability to have the scaling property $$aP(ax',t|ax,0) = P(x',a^{\mu}t|x,0)\,,\label{eq:prob-scaling}$$ where $\mu$ is the scaling exponent, the meaning of which will be revealed below, equation . Indeed, from equations – and a change of variables it follows $$\begin{aligned}
C(at) & = \int\rmd x\int\rmd x'\, xx'P_{0}(x)P(x',at|x,0) \\
& \sim \int\rmd x\int\rmd x'\, x^{1-\lambda}x'a^{\frac{1}{\mu}}
P(a^{\frac{1}{\mu}}x',t|a^{\frac{1}{\mu}}x,0) \\
& \sim a^{\frac{\lambda-3}{\mu}}\int\rmd u\int\rmd u'\, uu'P_{0}(u)P(u',t|u,0)\,.\end{aligned}$$
Thus, the autocorrelation function has the required property with $\beta$ given by equation $$\beta = 1 + (\lambda-3)/\mu\,.\label{eq:beta-sde-1}$$ Note, that according to equation the change of the magnitude of the stochastic variable $x\rightarrow ax$ is equivalent to the change of time scale $t\rightarrow a^\mu t$.
In order to avoid the divergence of steady state PDF the diffusion of stochastic variable $x$ should be restricted at least from the side of small values and, therefore, holds only in some region of the variable $x$, $x_{\mathrm{min}}\ll x\ll x_{\mathrm{max}}$. When the diffusion of stochastic variable $x$ is restricted, equation also cannot be exact. However, if the influence of the limiting values $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ can be neglected for time $t$ in some region $t_{\mathrm{min}}\ll t\ll t_{\mathrm{max}}$, we can expect that the scaling approximately holds in this time region.
To get the required scaling of the transition probability, the SDE should contain only powers of the stochastic variable $x$. This will be the case if the coefficient in the noise term is the power-law depending, i.e., proportional to $x^{\eta}$. The drift term then is fixed by the requirement for the steady state PDF. Thus we consider SDE [@Kaulakys2009] $$\rmd x=\sigma^{2}\left(\eta-\frac{1}{2}\lambda\right)x^{2\eta-1}\rmd t
+\sigma x^{\eta}\rmd W_t\,.\label{eq:sde}$$ Here $W_t$ is a standard Wiener process (the Brownian motion) and $\sigma$ is the white noise intensity. Note that SDE is the same as in papers [@Kaulakys2009; @Kaulakys2006], only here we obtained it from the consideration of the scaling properties, not starting from the point process model. Changing the variable $x$ in to the scaled variable $x_{\mathrm{s}}=ax$ or introducing the scaled time $t_{\mathrm{s}}=a^{2(\eta-1)}t$ and using the property of the Wiener process $\rmd W_{t_{\mathrm{s}}}=a^{\eta-1}\rmd W_t$ one gets the same resulting equation. Thus, change of the scale of the variable $x$ and change of time scale are equivalent, as in equation , and the exponent $\mu$ is $$\mu=2(\eta-1)\,.\label{eq:mu}$$ From equation it follows that the power-law exponent in the PSD of the signal generated by SDE is $$\beta=1+\frac{\lambda-3}{2(\eta-1)}\,.\label{eq:beta-sde}$$
In order to obtain a stationary process and avoid the divergence of steady state PDF the diffusion of stochastic variable $x$ should be restricted or equation should be modified. The simplest choice of the restriction is the reflective boundary conditions at $x=x_{\mathrm{min}}$ and $x=x_{\mathrm{max}}$. Another choice would be modification of equation to get rapidly decreasing steady state PDF when the stochastic variable $x$ acquires values outside of the interval $[x_{\mathrm{min}}, x_{\mathrm{max}}]$. For example, the steady state PDF $$P_{0}(x)\sim\frac{1}{x^{\lambda}}\exp\left\{
-\left(\frac{x_{\mathrm{min}}}{x}\right)^{m}
-\left(\frac{x}{x_{\mathrm{max}}}\right)^{m}\right\}$$ with $m>0$ has a power-law form when $x_{\mathrm{min}}\ll x\ll x_{\mathrm{max}}$ and exponential cut-offs when $x$ is outside of the interval $[x_{\mathrm{min}}, x_{\mathrm{max}}]$. Such exponentially restricted diffusion is generated by the SDE $$\rmd x=\sigma^{2}\left[\eta-\frac{1}{2}\lambda
+\frac{m}{2}\left(\frac{x_{\mathrm{min}}^{m}}{x^{m}}
-\frac{x^{m}}{x_{\mathrm{max}}^{m}}\right)\right]x^{2\eta-1}\rmd t
+\sigma x^{\eta}\rmd W_t\label{eq:sde-restricted}$$ obtained from equation by introducing additional terms in the drift.
The presence of the restrictions at $x=x_{\mathrm{min}}$ and $x=x_{\mathrm{max}}$ makes the scaling not exact and this limits the power-law part of the PSD to a finite range of frequencies $f_{\mathrm{min}}\ll f\ll f_{\mathrm{max}}$. Let us estimate the limiting frequencies. Taking into account the limiting values $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$, equation for the transition probability corresponding to SDE becomes $$aP(ax',t|ax,0;ax_{\mathrm{min}},ax_{\mathrm{max}})=
P(x',a^{\mu}t|x,0;x_{\mathrm{min}},x_{\mathrm{max}})\,.\label{eq:prob-scaling-bound}$$ Here $x_{\mathrm{min}}$, $x_{\mathrm{max}}$ are the parameters of the transition probability. The steady state distribution $P_{0}(x;x_{\mathrm{min}},x_{\mathrm{max}})$ has the scaling property $$aP_{0}(ax;ax_{\mathrm{min}},ax_{\mathrm{max}})=
P_{0}(x;x_{\mathrm{min}},x_{\mathrm{max}})\,.\label{eq:prob-steady-bound}$$ Inserting equations and into equation we obtain $$C(t;ax_{\mathrm{min}},ax_{\mathrm{max}})=
a^{2}C(a^{\mu}t,x_{\mathrm{min}},x_{\mathrm{max}})\,.$$ This equation means that time $t$ in the autocorrelation function should enter only in combinations with the limiting values, $x_{\mathrm{min}}t^{1/\mu}$ and $x_{\mathrm{max}}t^{1/\mu}$. We can expect that the influence of the limiting values can be neglected and the scaling holds when the first combination is small and the second large, that is when time $t$ is in the interval $\sigma^{-2}x_{\mathrm{max}}^{-\mu}\ll t\ll\sigma^{-2}x_{\mathrm{min}}^{-\mu}$. Then, using equation the frequency range where the PSD has $1/f^{\beta}$ behavior can be estimated as $$\sigma^{2}x_{\mathrm{min}}^{\mu}\ll2\pi f\ll
\sigma^{2}x_{\mathrm{max}}^{\mu}\,.\label{eq:freq-range}$$ However, numerical solutions of proposed nonlinear SDEs show that this estimation is too broad, i.e., the numerically obtained frequency region with the power-law behavior of PSD is narrower than according to equation . Note, that for $\mu=0$, i.e., $\eta=1$ the width of the frequency region is zero, and we do not have $1/f^{\beta}$ power spectral density.
![(Color online) (a) Typical signal generated by equation with reflective boundaries at $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line shows the slope $f^{-1}$. Used parameters are $\eta=2$, $\lambda=3$, $x_{\mathrm{min}}=1$, $x_{\mathrm{max}}=1000$, and $\sigma=1$.[]{data-label="fig:sde"}](fig1a "fig:"){width="33.00000%"}![(Color online) (a) Typical signal generated by equation with reflective boundaries at $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line shows the slope $f^{-1}$. Used parameters are $\eta=2$, $\lambda=3$, $x_{\mathrm{min}}=1$, $x_{\mathrm{max}}=1000$, and $\sigma=1$.[]{data-label="fig:sde"}](fig1b "fig:"){width="33.00000%"}![(Color online) (a) Typical signal generated by equation with reflective boundaries at $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line shows the slope $f^{-1}$. Used parameters are $\eta=2$, $\lambda=3$, $x_{\mathrm{min}}=1$, $x_{\mathrm{max}}=1000$, and $\sigma=1$.[]{data-label="fig:sde"}](fig1c "fig:"){width="33.00000%"}
Comparison of the numerically obtained steady state PDF and the PSD with analytical expressions for SDE with $\eta=\mu=2$ and $\lambda=3$ is presented in figure \[fig:sde\]. For the numerical solution we use the Euler-Maruyama approximation, transforming the differential equations to difference equations. We can use constant time step, however at large values of $x$ the coefficients in the equations become large and thus require a very small time step. More effective method of solution is to use a variable time step, decreasing with the increase of $x$. As in [@Kaulakys2004; @Kaulakys2006] we choose the time step in such a way that the coefficient before noise becomes proportional to the first power of $x$. Very similar numerical results one gets also by using the Milstein approximation [@Kaulakys2009]. We see good agreement of the numerical results with the analytical expressions. A numerical solution of the equations confirms the presence of the frequency region for which the power spectral density has $1/f^{\beta}$ dependence. The $1/f$ interval in the PSD in figure \[fig:sde\] is approximately between $f_{\mathrm{min}}\approx10^{0}$ and $f_{\mathrm{max}}\approx10^{3}$ and is much narrower than the width of the region $1\ll f\ll10^{6}$ predicted by equation . The width of this region can be increased by increasing the ratio between the minimum and the maximum values of the stochastic variable $x$.
As we see in figure \[fig:sde\]a, the numerical calculations exhibit a structure of the signal consisting of peaks or bursts. Analysis [@Kaulakys2009] reveals that the sizes of the bursts are approximately proportional to the squared durations of the bursts with the power-law distributions of the bursts durations and interburst time. The exponent of the PDF of the interburst time approximately equal to $-3/2$ has been obtained numerically [@Kaulakys2009] and analytically [@Gontis2012ACS].
Stochastic pulse sequences {#sec:pulses}
==========================
In this Section we consider pulse sequences with independent pulses. The shapes of the pulses are characterized by an arbitrary large set of parameters $\xi$, whereas the occurrence times of the pulses are described by a set of time moments $\{t_k\}$. The general form of the signal can be written as $$I(t)=\sum_{k}A(t-t_{k},\xi_{k})\,,\label{eq:signal}$$ where functions $A(t,\xi)$ determine the shape of individual pulses. The pulse duration $\tau$ is included in the set of parameters $\xi$ or, more generally, is a function of the parameters, $\tau(\xi)$. Inter-pulse duration is $\vartheta_{k}=t_{k+1}-t_{k}$. Such a pulse sequence is schematically shown in figure \[fig:pulses\]. We assume that: (i) the pulse sequences are stationary and ergodic; (ii) parameters $\xi$ of different pulses are independent; (iii) all pulses are described by the same function $A(t,\xi)$; (iv) the pulse parameters $\xi$ have the distribution $P(\xi)$.
![Sequence of random pulses.[]{data-label="fig:pulses"}](fig2){width="40.00000%"}
The easies way to calculate PSD of a pulse sequence is to start from the definition of the power spectral density directly [@Heiden1969]. However, in order to demonstrate connection with the previous Section we will consider the expression for the autocorrelation function of the signal. The autocorrelation function is given by the equation $$C(t)=\lim_{T\rightarrow\infty}\left\langle \frac{1}{T-t}\int_{0}^{T-t}I(t')I(t'+t)
\rmd t'\right\rangle \,,\label{eq:autocorr-pulses}$$ where $T$ the observation time interval and the brackets $\langle\cdot\rangle$ denote averaging over realizations of the pulse sequence. Using the signal we can write $$C(t)=\lim_{T\rightarrow\infty}\left\langle \frac{1}{T}\sum_{k,k'=1}^{N}
\int_{0}^{\infty}A(t',\xi_{k})A(t'+t+t_{k}-t_{k'},\xi_{k'})\rmd t'\right\rangle\,,$$ where $N$ is the number of pulses during the observation time interval $T$.
The autocorrelation function can be decomposed into two parts, the first part containing the autocorrelation of each pulse with itself and the second part containing all cross terms: $$C(t)=\lim_{T\rightarrow\infty}\left\langle \frac{1}{T}\sum_{k=1}^{N}
\int_{0}^{\infty}A(t',\xi_{k})A(t'+t,\xi_{k})\rmd t'\right\rangle +\mbox{other terms.}
\label{eq:c-intermed}$$ In many cases $1/f^{\beta}$ PSD is caused only by the first part. For example, it is known that when the pulses occur randomly as a Poisson process, the PSD of the signal depends only on the shapes of the pulses, as given by Carson’s theorem [@Carson1931]. Conditions when a sequence of randomly occurring pulses leads to $1/f^{\beta}$ noise were investigated in [@Halford1968]. Note, that when inter-pulse duration $\vartheta_{k}=t_{k+1}-t_{k}$ is uncorrelated with the duration of the pulse, different pulses may be overlapping. Even in case when the other terms in equation are nonzero, the first part can have different dependence on time $t$ and dominate for some time range $t_{\mathrm{min}}<t<t_{\mathrm{max}}$. Therefore, initially we will consider only the first part of equation . It can be written as $$C(t)=\nu\left\langle \int_{0}^{\infty}A(t',\xi)A(t'+t,\xi)\rmd t'\right\rangle \,,\label{eq:c1}$$ where $\nu$ is the mean number of pulses per unit time. Since the pulse duration $\tau$ is a function of the parameters $\xi$, the PDF of pulse durations is $$P_{\tau}(\tau')=\int\delta(\tau(\xi)-\tau')P(\xi)\rmd\xi$$ Introducing the autocorrelation function of the pulses with the same duration $\tau'$, $$C(t,\tau')=\int\rmd\xi\,\delta(\tau(\xi)-\tau')\frac{P(\xi)}{P_{\tau}(\tau')}
\int_{0}^{\tau'}\rmd t'\, A(t',\xi)A(t'+t,\xi) \,,\label{eq:c-t-tau}$$ we can write equation in the form $$C(t)=\nu \int P_{\tau}(\tau)C(t,\tau)\rmd\tau\,. \label{eq:corr}$$ If the PDF of pulse durations has a power-law form $$P_{\tau}(\tau)\sim\tau^{\rho}\label{eq:prob-tau}$$ and the autocorrelation function of the pulses with the same duration has the scaling property $$C(at,a\tau)=a^{2\gamma+1}C(t,\tau)\label{eq:c-tau-scaling}$$ then it follows that the autocorrelation function $C(t)$ has the required property with $\beta$ given by equation $$\beta=\rho+2\gamma+3\,\label{eq:beta}.$$ The meaning of the parameter $\gamma$ will be revealed below, equations and . Note, that the scaling in equation is the same as described by equation (9) in [@Kuntz2000], where the pulse area $S\sim\tau^{\gamma+1}$ is used instead of pulse duration $\tau$.
In order to avoid the divergence of the PDF of pulse durations , equation should hold only in some region of the pulse durations $\tau$, $\tau_{\mathrm{min}}\leqslant\tau\leqslant\tau_{\mathrm{max}}$. In this case the scaling cannot be exact. However, if the influence of the limiting values $\tau_{\mathrm{min}}$ and $\tau_{\mathrm{max}}$ can be neglected for time $t$ in some region $t_{\mathrm{min}}\ll t\ll t_{\mathrm{max}}$, we can expect that the scaling approximately holds for this time region.
One of the ways to get the required scaling of the autocorrelation function $C(t,\tau)$ is to consider pulses having the same shape, only stretched in height and in time. The signal consisting of such pulses was investigated in [@Halford1968]. For stretched pulses we can write $$A(t,\xi)=h(\xi)y(t/\tau(\xi))\,,$$ where $h(\xi)$ is the height of the pulse. The function $y(t_s)$ is nonzero only when $0\leq t_s\leq 1$.
From equation we obtain $$C(t,\tau)=\tau\overline{h^{2}}(\tau)c(t/\tau)\,,\label{eq:c-tau-scaling-a}$$ where $$\overline{h^{2}}(\tau')=\int\delta(\tau(\xi)-\tau')h^{2}(\xi)
\frac{P(\xi)}{P_{\tau}(\tau')}\rmd\xi$$ is the mean squared amplitude of pulses having the same duration $\tau'$ and the function $$c(t_{\mathrm{s}})=\int_{0}^{1}y(t^{\prime}_{\mathrm{s}})
y(t^{\prime}_{\mathrm{s}}+t_{\mathrm{s}})\rmd t^{\prime}_{\mathrm{s}}$$ is the autocorrelation function of pulse shapes. If the height $h$ of the pulse is a power-law function of the pulse duration $\tau$, $$h\sim\tau^{\gamma}\,,\label{eq:h}$$ then the scaling of the autocorrelation function holds. It should be noted, that even when pulse height $h$ is proportional to $\tau^{\gamma}$, the coefficient of proportionality is not necessarily constant. In particular, the sign of the pulses can be random. Only coefficient of proportionality for the average of the square of the pulse height $\overline{h^{2}}(\tau)\sim\tau^{2\gamma}$ should be constant. If pulse height is a power-law function of the pulse duration, then the change of the magnitude of the pulse height $h\rightarrow ah$ is caused by the change of pulse duration $\tau\rightarrow a^{1/\gamma}\tau$. Comparing this scaling property to equation we see that the power-law exponent $\gamma$ plays a similar role as $-1/\mu$, i.e., $$\gamma=-\frac{1}{\mu}=\frac{1}{2(1-\eta)}\,.\label{eq:gamma-eta}$$ The sign minus in equation appears because stretching the time, as in equation , is equivalent to the shortening of the pulse duration.
Now we will investigate the influence of limiting pulse durations $\tau_{\mathrm{min}}$ and $\tau_{\mathrm{max}}$. From the assumptions made above, equations , , and , we have that $P_{\tau}(\tau)\overline{h^{2}}(\tau)=B\tau^{\beta-3}$ when $\tau_{\mathrm{min}}\leqslant\tau\leqslant\tau_{\mathrm{max}}$. Here $B$ is the coefficient independent from $\tau$. For time $\tau_{\mathrm{min}}\ll t\ll\tau_{\mathrm{max}}$ we can write the autocorrelation function, according to equations and , as $$C(t)=\nu B\int_{\tau_{\mathrm{min}}}^{\tau_{\mathrm{max}}}\tau^{\beta-2}c(t/\tau)\rmd\tau
=\nu Bt^{\beta-1}\int_{1}^{\frac{\tau_{\mathrm{max}}}{t}}u^{\beta-2}c(1/u)\rmd u\,.
\label{eq:auto-approx}$$ According to paper [@Halford1968], physically reasonable pulses are square-integrable and have everywhere finite derivative of the autocovariance function with respect to time. Then for large $u$ we can approximate $c(1/u)\approx c(0)+c'(0)/u$ and get $$C(t)\approx\cases{
\nu Bt^{\beta-1}\left[\frac{c(0)}{1-\beta}+\frac{c'(0)}{2-\beta}\right]\,,& $0<\beta<1$ ,\\
\nu B\left[c(0)\ln\tau_{\mathrm{max}}+c'(0)-c(0)\ln t\right]\,, & $\beta=1$ ,\\
\nu B\left[c(0)\frac{\tau_{\mathrm{max}}^{\beta-1}}{\beta-1}+t^{\beta-1}
\frac{c'(0)}{2-\beta}\right]\,, & $1<\beta<2$ .\\}$$
Thus, for $0<\beta<2$ and $\tau_{\mathrm{min}}\ll t\ll\tau_{\mathrm{max}}$ the term containing time $t$ has the scaling property the same as in equation , the limiting values of the pulse durations $\tau_{\mathrm{min}}$ and $\tau_{\mathrm{max}}$ do not influence the scaling of the autocorrelation function. On the other hand, if $\beta>2$ then the influence of $\tau_{\mathrm{max}}$ becomes significant.
[@llll]{} Signal&$\gamma$&$\rho$&Meaning of $\rho$Constant area pulses&$-1$&$0$&Uniform distribution of pulse durationsConstant energy pulses&$-1/2$&$-1$&Pulses of constant height&$0$&$-2$&Uniform distribution of inverse durationsGeometrically similar pulses&$1$&$-4$&
As has been pointed out in [@Ruseckas2003], the condition for $1/f$ spectrum, $\rho+2\gamma+2=0$, can be easily satisfied. The power-law dependence of the pulse height on the pulse duration can occur naturally. Various cases are listed in table \[tab:power-cases\]. The value $\gamma=0$ corresponds to the pulses of constant height; $\gamma=-1$ corresponds to constant area pulses. Geometrically similar pulses have $\gamma=1$. If the energy is proportional to square of the signal, the constant energy pulses correspond to $\gamma=-1/2$. Since we have $1/f$ spectrum when $\rho=-2(\gamma+1)$, this spectrum occurs for constant area pulses ($\gamma=-1$) and uniform distribution of pulse durations ($\rho=0$) in a wide interval. For constant height pulses ($\gamma=0$) we have $1/f$ spectrum when the distribution of inverse durations $\tau^{-1}$ is uniform, that is when $P_{\tau}(\tau)\propto\tau^{-2}$.
Signal consisting of overlapping constant height pulses has PDF of Poisson distribution. On the other hand, pulses with $\gamma\neq0$ can lead to power-law tails in the PDF of the signal. Let us consider rectangular pulses with the only random parameter being the pulse duration $\tau$. Large signal intensities are due to pulses of large height, for which one can neglect the overlap between pulses. Each pulse of the height $h(\tau)$ occurs with the probability $P_{\tau}(\tau)$ and lasts for time $\tau$. Thus the PDF of the signal intensity $I=h$ is (see for analogy [@Kaulakys2005]) $$P_{I}(I)=\frac{\tau}{\langle\tau\rangle}P_{\tau}(\tau)
\left.\frac{\rmd\tau}{\rmd h}\right|_{h=I}\,.\label{eq:pdf-1}$$ If the PDF of pulse durations has the power-law $P_{\tau}(\tau)\propto\tau^{\rho}$ form and the height of the pulse depends on the pulse duration as $h\propto\tau^{\gamma}$, then from equation we obtain $P_{I}(I)\propto I^{-\lambda}$, where $$\lambda=1-\frac{2+\rho}{\gamma}\,.\label{eq:lambda-pulses}$$
For the pure $1/f$ noise $2+\rho=-2\gamma$ and we get the exponent $\lambda=3$. For the case of $f^{-\beta}$ spectrum we have the following relation between the exponent $\beta$ of the spectrum and exponent $\lambda$ of the signal PDF: $$\beta=1+\gamma(3-\lambda)\,.\label{eq:beta-pulses}$$ Taking into account equation we see that relation is the same as by equation describing the power-law spectrum of the signal generated by the nonlinear SDE . Note that PDF of the signal intensity has the same power-law exponent $\lambda=3$ also when $1/f$ noise is generated by the nonlinear SDE .
![(Color online) (a) Typical signal consisting from the constant area rectangular pulses with the uniformly distributed durations, $\rho=0$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line corresponds to the spectrum obtained using Wiener-Khintchine theorem from the autocorrelation function. The time intervals between the pulses are distributed according to Poisson process with the average $\bar{\vartheta}=5$ . The parameters used are $\tau_{\mathrm{min}}=0.01$ and $\tau_{\mathrm{max}}=100$.[]{data-label="fig:fixed-area"}](fig3a "fig:"){width="33.00000%"}![(Color online) (a) Typical signal consisting from the constant area rectangular pulses with the uniformly distributed durations, $\rho=0$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line corresponds to the spectrum obtained using Wiener-Khintchine theorem from the autocorrelation function. The time intervals between the pulses are distributed according to Poisson process with the average $\bar{\vartheta}=5$ . The parameters used are $\tau_{\mathrm{min}}=0.01$ and $\tau_{\mathrm{max}}=100$.[]{data-label="fig:fixed-area"}](fig3b "fig:"){width="33.00000%"}![(Color online) (a) Typical signal consisting from the constant area rectangular pulses with the uniformly distributed durations, $\rho=0$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) The PSD of such a signal. The dashed (green) line corresponds to the spectrum obtained using Wiener-Khintchine theorem from the autocorrelation function. The time intervals between the pulses are distributed according to Poisson process with the average $\bar{\vartheta}=5$ . The parameters used are $\tau_{\mathrm{min}}=0.01$ and $\tau_{\mathrm{max}}=100$.[]{data-label="fig:fixed-area"}](fig3c "fig:"){width="33.00000%"}
Typical signal for rectangular constant area pulses ($\gamma=-1$) is shown in figure \[fig:fixed-area\]a, the PDF of the signal is shown in figure \[fig:fixed-area\]b and the PSD in figure \[fig:fixed-area\]c. We see a good agreement of numerically obtained PSD with the analytical estimation. In figure \[fig:fixed-area\]b we can see that the PDF has a power-law tail for large signal intensities. Note that due to the overlapping of the pulses the PDF of the signal at smaller intensities is not a power-law and acquires a power-law tail only for larger intensities, where the overlap can be neglected. This is in contrast with the SDE , where the steady state PDF of the signal can have a power-law form for all values of the signal sufficiently far from limiting values $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$.
Connection between the nonlinear SDE and stochastic pulse sequences {#sec:sde-pulses}
===================================================================
As it was shown in Secs. \[sec:SDE\] and \[sec:pulses\], the signals having $1/f^{\beta}$ PSD and generated by the nonlinear SDEs have similar scaling like the signals consisting from random pulses. In this Section we approximate nonlinear SDE by a sequence of pulses and show that this approximation gives the same PDF of signal intensity and a power-law region in the PSD with the same exponent. However, other details of the approximated signal can be different: for example, the signal consisting from pulses does not exhibit the intermittent bursts characteristic for solution of SDE.
Let us consider SDE together with reflective boundaries at $x=x_{\mathrm{min}}$ and $x=x_{\mathrm{max}}$. Using Euler-Maruyama approximation with time step $\Delta t=s$ and replacing the stochastic differential equation with the difference equation we have $$x_{k+1}=x_{k}+\sigma^{2}\left(\eta-\frac{\lambda}{2}\right)x_{k}^{2\eta-1}s+\sigma x_{k}^{\eta}\sqrt{s}\varepsilon_{k}\,.$$ Here $\varepsilon_{k}$ is a Gaussian random variable with zero mean and unit variance. Variable time step [@Kaulakys2006] $$s_{k}=\frac{\kappa^{2}}{\sigma^{2}}x_{k}^{2(1-\eta)}\label{eq:hk}$$ results in the equation $$x_{k+1}=x_{k}\left(1+\frac{1}{2}\kappa^{2}(2\eta-\lambda)+\kappa\varepsilon_{k}\right)\,,\label{eq:discrete}$$ where $\kappa\ll1$ is a small parameter. The approximation of SDE becomes better with decreasing $\kappa$. It should be noted that equation has an universal form: $x_{k}$ enters only in the first power independent on the exponent $\eta$. We can estimate the steady state PDF of $x_{k}$ in $k$-space by considering equation as an Euler-Maruyama approximation of the SDE in $k$-space, $$\rmd x=\kappa^{2}\left(\eta-\frac{\lambda}{2}\right)x\rmd k+\kappa x \rmd W_{k}\,.
\label{eq:xk}$$
Taking into account reflective boundaries at $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ we get from the Fokker-Planck equation in $k$-space corresponding to equation that the steady state PDF has the power-law form, $P(x_{k})\propto x_{k}^{2(\eta-1)-\lambda}$. From this steady state PDF and equation it follows that PDF of $s_{k}$ has a power-law form $P_k(s_{k})\propto s_{k}^{\rho}$ with $$\rho=\frac{\lambda-1}{2(\eta-1)}-2\,.\label{eq:rho-x}$$ The same expression for the power-law exponent $\rho$ can be obtained from equations and .
From equation we obtain $x_{k+n}$: $$x_{k+n}=x_{k}\prod_{i=1}^{n}\left(1+\frac{1}{2}\kappa^{2}(2\eta-\lambda)
+\kappa\varepsilon_{k+i-1}\right)\approx x_{k}(1+\kappa\sqrt{n}\varepsilon+\cdots)\,,$$ where $\varepsilon$ is a Gaussian random variable with zero mean and unit variance. Here we used the fact that the sum of $n$ Gaussian variables $\varepsilon_{k+i-1}$ is a Gaussian variable with the dispersion equal to $n$. We can conclude that $x_{k+n}$ does not differ significantly from $x_{k}$ as long as $\kappa\sqrt{n}\ll1$. The maximal value of $n$ when $x_{k+n}$ is approximately equal to $x_{k}$ is $n_{\mathrm{max}}\sim1/\kappa^{2}$. The duration in which the stochastic variable $x$ does not change significantly is $$\tau(x_{k})=n_{\mathrm{max}}s_{k}=\frac{1}{\sigma^{2}}x_{k}^{2(1-\eta)}=\frac{1}{\sigma^{2}}x_{k}^{1/\gamma}\,,\label{eq:tau-x}$$ where $\gamma$ is given by . The duration $\tau$, being proportional to $s_{k}$, has the power-law PDF with the same exponent $\rho$ as the PDF of $s_{k}$: $$P_{\tau}(\tau)=\cases{
C\tau^{\rho}\,, & $\tau_{\mathrm{min}}\leq\tau\leq\tau_{\mathrm{max}}$ ,\\
0\,, & otherwise .\\}
\label{eq:pdf-tau}$$ Here $C$ is normalization coefficient and $$\tau_{\mathrm{min}}=\frac{1}{\sigma^{2}x_{\mathrm{max}}^{2(\eta-1)}}\,,\qquad\tau_{\mathrm{max}}=\frac{1}{\sigma^{2}x_{\mathrm{min}}^{2(\eta-1)}}\,.$$ From equation the value of the stochastic variable $x_{k}$ is connected with the duration $\tau$ by the relation $$x_{k}=\sigma^{2\gamma}\tau^{\gamma}\label{eq:x-tau}\,.$$
Therefore, we can approximate the signal generated by SDE by rectangular pulses of random duration $\tau$ having the PDF of durations and pulse height $h\equiv x_k$ related to the pulse duration $\tau$ by equation . The pulses are not overlapping and immediately follow each other. Although the durations of adjacent pulses obtained from the signal generated by SDE are correlated, for simplicity we will neglect this correlation. The PDF of the signal $x$ constructed as such a pulse sequence has power-law form. Using equations and we get that the power-law exponent in the PDF $P(x)$ is equal to $-\lambda$, with $\lambda$ appeared in SDE .
When pulses occur not randomly but follow each other, the other terms in equation are nonzero. However, one can check that for some range of time $t$ the first part in equation dominates. Thus the PSD of this pulse sequence has a power-law part with the exponent given by equation . Using the value of the exponent $\gamma$ from equation we get the power-law exponent in the PSD. The frequency range $\tau_{\mathrm{max}}^{-1}\ll f\ll\tau_{\mathrm{min}}^{-1}$ where PSD of the signal consisting of pulses has power-law behavior coincides with inequalities . Thus, the proposed approximation of SDE by the sequence of pulses correctly reproduces power-law parts of the PDF and the PSD of the generated signal.
![(Color online) (a) Typical signal consisting of the equal energy pulses that immediately follow each other. The PDF of pulse durations $\tau$ is given by equation with $\rho=-1$, $\tau_{\mathrm{min}}=10^{-6}$, and $\tau_{\mathrm{max}}=1$ and pulse heights are proportional to $\tau^{-1/2}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) Comparison of the PSD calculated using the signal generated by SDE with the same parameters as in figure \[fig:sde\] (black line) and using the signal consisting of pulses (gray line). The dashed (green) line shows the slope $1/f$.[]{data-label="fig:cmp"}](fig4a "fig:"){width="33.00000%"}![(Color online) (a) Typical signal consisting of the equal energy pulses that immediately follow each other. The PDF of pulse durations $\tau$ is given by equation with $\rho=-1$, $\tau_{\mathrm{min}}=10^{-6}$, and $\tau_{\mathrm{max}}=1$ and pulse heights are proportional to $\tau^{-1/2}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) Comparison of the PSD calculated using the signal generated by SDE with the same parameters as in figure \[fig:sde\] (black line) and using the signal consisting of pulses (gray line). The dashed (green) line shows the slope $1/f$.[]{data-label="fig:cmp"}](fig4b "fig:"){width="33.00000%"}![(Color online) (a) Typical signal consisting of the equal energy pulses that immediately follow each other. The PDF of pulse durations $\tau$ is given by equation with $\rho=-1$, $\tau_{\mathrm{min}}=10^{-6}$, and $\tau_{\mathrm{max}}=1$ and pulse heights are proportional to $\tau^{-1/2}$. (b) The PDF of the signal intensity. The dashed (green) line shows the power-law with the exponent $-3$. (c) Comparison of the PSD calculated using the signal generated by SDE with the same parameters as in figure \[fig:sde\] (black line) and using the signal consisting of pulses (gray line). The dashed (green) line shows the slope $1/f$.[]{data-label="fig:cmp"}](fig4c "fig:"){width="33.00000%"}
To illustrate the approximation of nonlinear SDE by a sequence of pulses, in figure \[fig:cmp\] we compare the PDF and the PSD of the signal consisting from pulses to the PDF and the PSD of the signal shown in figure \[fig:sde\]. The SDE has parameters $\eta=2$, $\lambda=3$, $x_{\mathrm{min}}=1$, $x_{\mathrm{max}}=1000$, therefore, duration of the pulses has power-law PDF with $\rho=-1$ (according to equation ) and $\tau_{\mathrm{min}}=10^{-6}$, $\tau_{\mathrm{max}}=1$. The height of each pulse is proportional to the duration of the pulse to the power of $\gamma=-1/2$, obtained from equation . This value of $\gamma$ means that each pulse has the same energy. The signal consisting of such pulses is shown in figure \[fig:cmp\]a. As one can see, this signal looks rather different from the one shown in figure \[fig:sde\]a. Large values of the signal in figure \[fig:cmp\]a do not come in the intermittent bursts as in figure \[fig:sde\]a. This difference is caused by the assumption that the durations of different pulses are uncorrelated. The PDF of the signal, shown in \[fig:cmp\]b, is the same as in figure \[fig:sde\]b. Comparison of the PSDs is shown in figure \[fig:cmp\]c. There is qualitative agreement between the PSD of the signal generated by the nonlinear SDE and the PSD of the signal consisting of pulses. The PSD of the signal consisting of pulses has a power-law part in a different range of frequencies, from $f\approx10^{1}$ up to $f\approx4\times10^{4}$. This difference from the expected range $\tau_{\mathrm{max}}^{-1} < f < \tau_{\mathrm{min}}^{-1}$ is caused by neglected other terms in equation .
Conclusions {#sec:concl}
===========
In summary, we have demonstrated the connection between the nonlinear SDEs generating signals with $1/f^{\beta}$ noise and signals consisting of random pulses with the power-law distribution of pulse durations. The exponent $\rho$ of the power-law PDF of pulse durations and the exponent $\gamma$ characterizing the dependence of the pulse height on the pulse duration are related to the parameters $\eta$ and $\lambda$ of the SDE by means of the equations and . The signal generated by SDE and corresponding signal consisting of rectangular pulses yield the same distribution of signal intensity and the same power-law exponent in the PSD . The appearance of $1/f^{\beta}$ spectrum and relationship between parameters can be obtained just by considering the scaling properties of the signals. The revealed connection between different models of $1/f$ noise provides further insights into the origin and relationship between different models of $1/f$ noise.
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|
---
abstract: 'We exhibit different examples of minimal sets for an IFS of homeomorphisms with rotation number equal to 0. It is proved that these examples are, from a topological point of view, the unique possible cases.'
address:
- 'N. Guelman, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
- 'J. Iglesias, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay'
- 'A. Portela, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
author:
- 'N. Guelman, J.Iglesias and A.Portela.'
title: 'Examples of minimal set for IFSs.'
---
Introduction.
=============
Let $f:S^{1}\to S^{1}$ be a homeomorphism. A minimal set for $f$ is an $f$-invariant closed set that is minimal (in the sense of inclusion).
Taking into account the rotation number of a homeomorphism $f:S^{1}\to S^{1}$ there exist three possibilities:
1. $f $ has a finite orbit i.e. there exists a finite minimal set,
2. $ S^1$ is minimal for $f $,
3. there is a unique minimal set that is a Cantor set.
This trichotomy can be extended to actions of groups of homeomorphisms on the circle (see for example [@n]):
Let $G$ be a subgroup of homeomorphisms of $S^1$. Then one of these three possibilities occur:
1. there exists a finite orbit of $G$,
2. $ S^1$ is minimal for $G $,
3. there is a unique minimal set that is a Cantor set.
The aim of this paper is describe any minimal set for the action of a finitely generated semigroup.
Given n maps $f_1,...,f_n$, $f_i:S^{1}\to S^{1}$, let $IFS(f_1,...,f_n)$ be the set of finite composition of $f_i$, that is any finite “word” of $f_1,....,f_n$. For any $x\in S^{1}$ we define orbit of $x$ as $O(x)=\{\phi(x): \ \phi\in
IFS(f_1,...,f_n) \}$. A set $M\subset S^{1}$ is invariant if $f_i(M)\subset M$ for any $i=1,...,n$. It is obvious that $\phi(M)\subset M$ for any $ \phi\in
IFS(f_1,...,f_n)$. A non- empty set $M$ is minimal, when it is closed, invariant and $\overline{O(x)}=M$ for any $x\in M$.
An $IFS$ can have infinite many minimal sets and it is possible that $\phi^{-1}(M)\neq M$ for an element of the family. This is a big difference with the case of a group action. There, any minimal set is invariant for the future and for the past. The aim of this paper is to describe the minimal sets of $IFS$, studied from a topological point of view, and to give examples of any possible case. We will construct examples where the rotation number of any generator is 0, so that some examples could be $C^{1}$-robust. Several authors have constructed examples of minimal sets of $IFS$. In [@br] sufficient conditions are given for an $IFS$ generated by two elements that are close to rotation so that $S^{1}$ is a minimal set. In fact, Duminy (see [@n], section 3.3) has proved that if $f,g$ are diffeomorphisms $C^2$-close to identity, the minimal set for $IFS(f,g)$ is $S^1$.
In [@s] sufficient conditions are given for an $IFS$ generated by two elements with overlap and rotation number 0 so that the minimal set is a Cantor set. These construction are important tools for the construction of our examples.
If $M\subset S^{1}$ is a closed set, we call $\mathcal{I}$ the set of connected components of the interior of $M$, $int(M)$, and $K$ the maximal Cantor set included in $M\setminus int (M)$. We will prove that we can write $M$ as the disjoint union of $ \mathcal{I}$, $ K$ and $ N$, where $N$ is a countable set.
In [@br] is proven a trichotomy for a minimal set of an $IFS$: it is finite, it is a Cantor set or it has non-empty interior. We will improve this result proving the following
\[theor\] Let $f_1,...,f_n$ be open functions, $f_i:I\to I$ where $I$ is a closed interval or $S^{1}$ and $M=\mathcal{I}\cup K\cup N$ is minimal set for the family $IFS(f_1,...,f_n)$. The possibilities for M are:
1. If $int(M)=\emptyset$ then M is finite or Cantor set.
2. If $int(M)\neq\emptyset$ then
- M is $S^1$,
- $M= \mathcal{I}\cup K\cup N$ with $ \partial \mathcal{I}\cap N\neq\emptyset$,
- $M= \mathcal{I}\cup N$, or
- $M= \mathcal{I}\cup K$.
From this theorem we have that unique cases for $ \mathcal {I}, K $ and $ N $ that are not possible are the following:
1. $K= \mathcal{I}=\emptyset$ and $N$ is an infinite set.
2. $\mathcal{I}=\emptyset$, $K\neq\emptyset$ and $N\neq\emptyset$.
3. $N\neq\emptyset$, $K\neq \emptyset $ and $\partial \mathcal{I}\cap N=\emptyset$.
Note that a particular case of $M= \mathcal{I}\cup N$, is realized when $M$ is an union of disjoint intervals (finite or infinite).
We will exhibit examples (see section \[examples\]) to show that any case of Theorem \[theor\] can be realized and we will prove that the remaining cases are not possible.
When $M= \mathcal{I}\cup K$ (see example 7), then $M$ is a symmetric Cantorval (see definition \[cantdef\]). This sets were studied in [@GN] and [@MO] and we use definition given by Z. Nitecki (see [@Ni]). We prove in Proposition \[cantorvals2\] that this is the unique case where $M$ is a symmetric Cantorval.
Examples given by Denjoy show that it is possible that a Cantor set is a minimal set for a diffeomorphism. Then, it is a natural question if any Cantor set is minimal for a diffeomorphism. As any two Cantor sets are “$C^{0}$-homeomorphic” then by conjugating a Denjoy’s example, it is easy to prove that any Cantor set is a minimal set for a $C^{0}$-homeomophism. A homeomorphism of class $C^{2}$ can not have a Cantor minimal set. Herman [@H], McDuff [@mc], Kercheval [@N], Kra [@KS] and Portela [@P] proved that there exist Cantor sets ( for example the usual triadic set) such that are not minimal sets for $C^{1}$ diffeomorphism. We will see in Subsection \[subsec1\](example (1)) , that the usual triadic set is minimal for $IFS(f,g)$ where $f$ y $g$ are $C^{\infty}$, then it is natural to ask if any Cantor set is minimal for an $ IFS$ where any element is of class $C^{1}$?\
A more general question is the following:
Given a closed set $M=\mathcal{I}\cup K\cup N$ different of (1), (2) and (3). Are there open functions $f_1,...,f_n$, $f_i:S^{1}\to S^{1}$, such that $M$ is a minimal set for the family $IFS(f_1,...,f_n)$?
In [@s1] K. Shinohara exhibited an example of an $IFS(f,g)$, where $f,g:I \rightarrow I$ are $C^1$-close to identity, with minimal set different of $I$. A natural question is in which of the possibilities of Theorem 1 is the minimal set of Shinohara’s example.
Topological Classification of closed sets.
==========================================
In this section we will give a topological description of closed sets and we will prove lemmas that will be used in the proof of the main Theorem
Recall that a set is called perfect if it is closed and any point is an accumulate one. Note that any perfect set is uncountable. Given a set $M\subset S^{1}$, we denote $M^{'}$ the set of accumulate points.
\[l5\] Let $M\subset S^{1}$, where $M$ is closed, totally disconnected and uncountable, then there exists $K\subset M$, $K$ a maximal Cantor set such that $M\setminus K$ is countable.
Let $B(x,\varepsilon )$ the ball of center $x$ and radius $\varepsilon >0$. Let $$K=\{x\in M : \ \forall \varepsilon >0 \mbox{ it holds that } B(x,\varepsilon)\cap M \mbox{ is uncountable } \}.$$ It is easy to see that $K$ is closed and totally disconnected since $M$ is. To prove that $K$ is a Cantor we have to prove that $K^{'}=K$. For that, it is enough to show that $M\setminus K$ is countable. Let $y\in M\setminus K$ then there exists $\varepsilon>0$ such that $B(y,\varepsilon)\cap M $ is countable. Let $U_y$ be an open interval with rational end points and $y\in U_y\subset B(y,\varepsilon)$ then $$M\setminus K=\bigcup_ {y\in M\setminus K}U_y\cap M.$$ Since the number of $U_y$ is countable and $U_y\cap M$ is countable, it follows that $M\setminus K$ is countable. $K$ is maximal since if there exists a Cantor set $K_1$, with $K\subsetneq K_1$ then $K_1\setminus K$ is uncountable, therefore $M\setminus K$ is uncountable, which is a contradiction.
\[principal\] Let $M\subset S^{1}$ be a closed set then $M$ can be written in a unique way as a disjoint union of $\mathcal{I}$, $K$ and $N$ where $\mathcal{I}$ is the union of the connected components of $int (M)$, $K$ is a maximal Cantor set in $M\setminus \mathcal{I}$ and $N$ is a countable set.
In the case that $int(M)=\emptyset$ we have two possibilities: If $M$ is countable, the statement is obvious; if $M$ is uncountable, by Lemma \[l5\], we have that $M=K\cup N$.\
In the case that $int(M)\neq\emptyset$, let $\mathcal{I}$ be the union of the interior of the non-trivial connected components of $M$. Then $M\setminus \mathcal{I}$ is closed and $int (M\setminus \mathcal{I})=\emptyset$. Therefore by the previous case we have that $M=\mathcal{I}\cup K\cup N$, so we have showed the existence. The unicity is a consequence of the definition of $\mathcal{I}$ and because $K$ is maximal in $M\setminus \mathcal{I}$.
\[obs1\]
Since $M$ is closed it follows that $\overline{\mathcal{I}}\subset M$ then $\partial \mathcal{I}\subset K\cup N$.
From now on, we consider $IFS(f_1,...,f_n)$ where $f_i:I\to I$ are open function, $I$ is an open interval or $S^{1}$ and $M$ is a minimal set for the family.
Next lemma is a consequence of the fact that any perfect set is uncountable.
\[l1\] Let $M$ be a minimal set with $\sharp M=\infty$. Then $M$ is uncountable.
If $M^{'}=\emptyset$ then $M$ is finite. Therefore $M^{'}\neq\emptyset$. Since $f_i:I\to I$ are open functions then $f_i(M^{'})\subset M^{'}$. Therefore, $M=M^{'}$ by minimality of $M$, so $M$ is a perfect set and it follows that it is uncountable.
Then, the case $\mathcal{I}= K=\emptyset$ and $N$ with infinite many points is not possible.
\[l2\] Suppose that a minimal set has finitely many non-trivial connected components, then the minimal set is such a union.
Let $I_1,...,I_n$ be the non-trivial connected components of $M$. Since any $f_i$ is open, then $f_i(I_j)$ is an interval for any $i$ and $j$. Therefore, $f_i(I_j)\subset \cup I_k$. It follows that $M=\cup I_k.$
Let us introduce the following
\[cantdef\] A symmetric Cantorval is a nonempty compact subset $M$ of $S^1$ (or the real line) such that:\
(1) $M$ is the closure of its interior (i.e., the nontrivial components are dense)\
(2) Both endpoints of any nontrivial component of $M$ are accumulation points of trivial (i.e., one-point) components of $M$.
\[obs2\] If $M= \mathcal{I}\cup K\cup N$ is a symmetric Cantorval, then for all $z\in K\cup N$ there exists $\{x_n\}\subset K\cup N$, $x_n\neq z$ such that $x_n\to z$.
Example 7 of section \[examples\] shows that a minimal set can be a symmetric Cantorval.
\[cantorvals\] Let $M= \mathcal{I}\cup K\cup N$. If $M$ is a symmetric Cantorval then $N=\emptyset$.
Suppose that $N\neq\emptyset$. Let $x_0\in N$ and $\varepsilon >0$ be such that $\overline{B(x_0, \varepsilon )}\cap K=\emptyset$. We will prove that $\overline{B(x_0, \varepsilon )}\cap N$ is a perfect set. It is obvious that it is closed. Given $z\in \overline{B(x_0, \varepsilon )}\cap N$ as $M$ is a symmetric Cantorval , by Remark \[obs2\], there exist $\{x_n\}\subset (K\cup N)\cap\overline{B(x_0, \varepsilon )} $, $x_n\neq z$ such that $x_n\to z$. As $\overline{B(x_0, \varepsilon )}\cap K=\emptyset$ then $x_n\in N$, so $\overline{B(x_0, \varepsilon )}\cap N$ is a perfect set. Therefore it is uncountable and this is a contradiction.
\[cantorvals2\] Let $IFS(f_1,...,f_n)$ where $f_i:I\to I$ are open function, $I$ is an open interval or $S^{1}$ and $M$ is a minimal set for the family. Then\
$M$ is a symmetric Cantorval if and only if $M=\mathcal{I}\cup K$.
($\Rightarrow$) If $M= \mathcal{I}\cup K\cup N$, by Lemma \[cantorvals\] we have that $N=\emptyset$.\
($\Leftarrow$) Given $z\in M$ and $I_0\in \mathcal{I}$ as $M$ is a minimal set there exist $\{ \phi_n \}\subset IFS(f_1,...,f_n)$ such that $\phi_n(I_0)\to z$. Then $M$ is the closure of its interior. As $\partial \mathcal{I}\subset K$ and $K$ is a Cantor set then both endpoints of any nontrivial component of $M$ are accumulation points of trivial components of $M$.
First part of the Proof of Theorem \[theor\]
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This proof has 2 steps. In the first one we will prove that the following cases are not possible. In the second one we will construct examples of the remaining cases. These examples will be shown in Section \[examples\].
As a consequence of the previous lemmas, the following cases can not occur:
- $K= \mathcal{I}=\emptyset$ and $N$ with infinite many points. Because of $M$ would be countable and this fact is not possible by Lemma \[l1\].
- $\mathcal{I}=\emptyset$ and $K\neq\emptyset$, $N\neq\emptyset$. Since $\mathcal{I}=\emptyset$, any $f_i$ is open and $K$ is a maximal Cantor set, then $f_i (K)\subset K$ for any $i$. So, for any $\phi$ in IFS we have that $\phi (K)\subset K$. Therefore $M\subset K$, which is a contradiction.
- $M=K\cup \mathcal{I}\cup N$ where $\partial \mathcal{I}\subset K$. Since $\partial \mathcal{I}\subset K$ then $K\cup \mathcal{I}$ is closed, so $S^{1}\setminus( K\cup \mathcal{I})$ is open and $N\subset S^{1}\setminus( K\cup \mathcal{I})$. Let $T$ be a connected component of $S^{1}\setminus( K\cup \mathcal{I})$ with $N\cap T\neq \emptyset$. Notice that $N\cap T$ is countable. On the other hand, for $x\in K$ there exists $\phi$ in IFS such that $\phi(x)\in T$. Then $\phi (K)\cap T$ is uncountable, which is a contradiction.
Tools for example’s construction.
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We will use the following IFS and functions for the construction of the examples.
Examples of IFS with Cantor minimal set {#subsec1}
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In this section we will construct IFS with Cantor minimal set.
1. Given an interval $I\subset S^{1}$, we consider an interval $I^{'}=[a,b]$ with $I^{'}\subsetneq I$. Let $f$ and $g$ be linear maps of angular coefficient 1/3 in $I^{'}$, as Figure \[figura1\](a) and such that $f|_{I^{c}}=g|_{I^{c}}=Id$. If $\Lambda_0=[a,b]$ and $\Lambda_{k+1}=f(\Lambda_k)\cup g(\Lambda_k)$ then $K=\cap \Lambda_k$ is a Cantor set ( the usual triadic set). It is easy to see that $K$ is a minimal set for $IFS(f,g)$.\
Later we will use the following properties:
1. $f^{-1}(K)\cap I'\subset K$ and $g^{-1}(K)\cap I'\subset K$.
2. If $x\in I^{'}$ then $\overline{O(x)}\supset K.$
2. Next example is in [@s]. Given two intervals $\widehat{I}$ and $I$, with $\widehat{I}\subsetneq I$, we consider intervals $I_{-1},I_0$, $I_1$ and the maps $f$ y $g$ as in Figure \[figura1\] (b). These maps verify that:
- $f(I_{-1})\subset I_{-1}$, $f(I_{0})\subset int( I_{-1})$, $f(I_{1})\subset int(I_{0})$.
- $g(I_{1})\subset I_{1}$, $g(I_{0})\subset int( I_{1})$, $g(I_{-1})\subset int(I_{0})$.
- $f^{'}|_{ I_{-1}\cup I_{0}\cup I_{1} },g^{'}|_{ I_{-1}\cup I_{0}\cup I_{1} }<\lambda<1.$
- $f|_{I^{c}}=g|_{I^{c}}=Id.$
If $\Lambda_0:= I_{-1}\cup I_{0}\cup I_{1}$ and $\Lambda_{k+1}:=f(\Lambda_k)\cup g(\Lambda_k)$ then $K=\cap_{k\geq 0} \Lambda_k$ is a Cantor set and it is minimal for $IFS(f,g)$.
As in the Example (1), following properties are satisfied:
1. $f^{-1}(K)\cap \widehat{I}\subset K$ and $g^{-1}(K)\cap \widehat{I}\subset K$.
2. If $x\in \widehat{I}$ then $\overline{O(x)}\supset K.$
From now on, we write $\{ f_{_{K,I}},g_{_{K,I}} \}$ when IFS generated by $f$ and $g$ has minimal set $K\subset I$, which is a Cantor set, $f$ and $g$ are the identity map in $I^{c}$ and verifies properties (a) and (b) as above.
3. Let $IFS(f)$, where $f$ is a Denjoy diffeomorphism.
Examples of IFS with an interval minimal set {#subsec2}
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Next example is an IFS whose minimal set is an interval and each generator has 0 rotation number.\
Construction and properties of the following example are in [@br]. Let $I^{' }=[a,b]$ and $I$ be two intervals with $I^{'}\subsetneq I$, we consider $T^{+}_{I^{'}I}$ and $T^{-}_{I^{'}I}$ be linear maps in $I^{'}$ (see Figure \[tmas\] (a)) and $T^{+}_{I^{'},I}$ and $T^{-}_{I^{'},I}$ are the identity map on $I^{c}$. The minimal set of $IFS(T^{+}_{I^{'}I},T^{-}_{I^{'}I})$ is the interval $I^{'}$. Let us denote $f=T^{-}_{I^{'}I}$ and $g=T^{+}_{I^{'}I}$; for any $x\in [a,b]$ we have that $f^{n}(x)\to a$ and $g^{n}(x)\to b$. Then, if $M\subset [a,b]$ is a minimal set then $\overline{O(a)}= \overline{O (b)}=M$. Suppose that $\overline{O (a)}\neq [a,b]$, let $I_0$ be the connected component of $[a,b]\setminus \overline{O (a)}$ with maximum length. Since $f([a,b])\cup g([a,b])=[a,b]$ then $f^{-1}(I_0)\cap [a,b]\neq\emptyset$ or $g^{-1}(I_0)\cap [a,b]\neq\emptyset$. In the case that $f^{-1}(I_0)\cap [a,b]\neq\emptyset$, if $f^{-1}(I_0)\subset [a,b]$ as $|f^{'}|<1$ then $|f^{-1}(I_0)|> |I_0|$ which is a contradiction. Therefore $f^{-1}(I_0)$ contains $a$ or $b$, hence $I_0$ contains $a$ or $b$ which is also a contradiction. The case $g^{-1}(I_0)\cap [a,b]\neq\emptyset$ is analogous.
Notice that for a big enough $n\in {\mbox{$\mathbb{N}$}}$, $IFS((T^{+})^{n}_{I^{'}I},(T^{-})^{n}_{I^{'}I})$ is in similar conditions than example (1) of section \[subsec1\], therefore it has a Cantor minimal set. It is obvious that any minimal set of $IFS((T^{+})_{I^{'}I},(T^{-})_{I^{'}I})$ contains a minimal set of $IFS((T^{+})^{n}_{I^{'}I},(T^{-})^{n}_{I^{'}I})$. Therefore the minimal set of $IFS((T^{+})^{n}_{I^{'}I},(T^{-})^{n}_{I^{'}I})$ and $IFS((T^{+})^{}_{I^{'}I},(T^{-})^{}_{I^{'}I})$ are different from a topological point of view.
Map to consider. {#subsec3}
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Given three intervals $J, J^{'}$ and $I$ with $J^{'}\subsetneq J$ and $J\cap I=\emptyset$. Let $h$ be a homeomorphism such that $h|_{J^{'}}=Id$ $h(J^{c})\subset I$.\
We will denote $h$ by $h_{J^{'},J,I}$.
Example’s Construction. {#examples}
=======================
In this section we construct seven examples to prove Theorem \[theor\].
[**[Example 1]{}**]{}. $M=K$ (minimal set that is a Cantor set) or $M$ is a finite set. We consider the following examples:
1. Let $IFS( f_{_{K,I}},g_{_{K,I}} )$ as in subsection \[subsec1\] examples (1) y (2).
2. Let $g$ be a Denjoy homeomorphism and we consider $IFS( g )$.
It is clear that both examples have a Cantor minimal set. The difference between (1) y (2) is that example (1) can be constructed $C^{\infty}$ and the generators of the IFS have 0 rotation number. In (2) the map $g$ can be $C^{1+\alpha}$ with $\alpha <1$ and the rotation number of $g$ is irrational. In (2) the minimal is unique but this is not true for example (1).
Let $f$ be a homeomorphism with fixed points. It is obvious that the family $IFS( f )$ has a finite minimal set.
These are examples if item (1) of Theorem \[theor\].\
Examples with $int(M)\neq\emptyset$.
[**[Example 2.]{}**]{} $M=S^{1}$.\
It is easy to find examples where the generators have irrational rotation number, but our aim is to exhibit examples of IFS with generators having 0 rotation numbers.
Let $I\subset S^{1}$ be an interval, and let $I^{'}$ and $J$ intervals such that $J\subset I^{'}\subsetneq I$. Let $H_1: S^{1}\to S^{1}$ be a homeomorphism such that $S^{1}\setminus I^{'}\subset H_1(J)$ and $H_2: S^{1}\to S^{1}$ such that $H_2(S^{1}\setminus I^{'})\subset J$. We consider $$IFS (T^{+}_{I^{'},I}, \ T^{-}_{I^{'},I} , \ H_1 , \ H_2 ).$$ The minimal set $M$ for this IFS is $M=S^{1}$. To show this, let $x\in S^{1}$. If $x\in I^{'}$ then $\overline{O(x)}\supset I^{'}$. Hence, $H_2(\overline{O(x)})\supset H_2(I^{'})\supset J^{c}$. Therefore $\overline{O(x)}=S^{1}$. If $x\notin I^{'}$ then $H_2(x)\in I^{'}$. From the previous case we have $\overline{O(x)}=S^{1}$.
[**[Example 3.]{}**]{} $M=\mathcal{I}\cup K\cup N$ (a Cantor set union intervals union countable points). In this case we have $\partial \mathcal{I}=N$.
Let $I_0\subset S^{1}$ be an interval. We consider IFS $( f_{_{K,I_0}},g_{_{K,I_0}} )$. Let $[a,b] \subset I_0$ be a minimal interval containing a $K$, let $I$ be a connected component of $K^{c}$ with $I\subset [a,b]$ and $I^{'}\subset I$. Let $J\subset S^{1}\setminus I_0$ and $J^{'}\subset J$. We consider $$IFS( f_{_{K,I_0}},\ \ g_{_{K,I_0}},\ \ h_{J^{'},J,I^{'}},\ \ T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I}).$$
We will prove that the minimal set for this IFS is $$M=K\cup \{ \phi (I^{'}): \ \ \phi \in IFS( f_{_{K,I_0}}, g_{_{K,I_0}}) \} \cup I^{'}.$$
We begin by proving that $M$ is IFS-invariant. Let $x\in M$. If $x\in K$, then $f_{_{K,I_0}}(x)\in K$ and $g_{_{K,I_0}}(x)\in K$. As $K\subset I_0$ then $x\notin J$ so $h_{J^{'},J,I^{'}}(x)\in I^{'}$. Finally, since $T^{+}_{I^{'}I}$ and $ T^{-}_{I^{'}I}$ are the identity on $K$ then $T^{+}_{I^{'}I}(x)\in K$ and $ T^{-}_{I^{'}I}(x)\in K$.\
If $x\in I^{'}$, then $f_{_{K,I_0}}(x)\in f_{_{K,I_0}}(I^{'})\subset M$. The same occurs with $g_{_{K,I_0}}(x)$. By definition of $h_{J^{'},J,I^{'}}$ we have that $h_{J^{'},J,I^{'}}(x)\in I^{'}$. Also by definition of $T^{+}_{I^{'}I}$ we have that $T^{+}_{I^{'}I}(x)\in I^{'}$ and the same occurs with $ T^{-}_{I^{'}I}$. If $x\in \phi (I^{'})$, the proof is analogous that in case $x\in I^{'}$. Now, we are going to prove that any $x\in M$ has dense orbit in $M$. Let $y\in M$ and $U$ a neighborhood of $y$ we will show that there exists $\phi \in IFS( f_{_{K,I_0}},\ \ g_{_{K,I_0}},\ \ h_{J^{'},J,I^{'}},\ \ T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I}) $ with $\phi (x)\in U$. First we take $x\in K$. If $y\in K$, it is clear that there exists $\phi \in IFS( f_{_{K,I_0}},\ \ g_{_{K,I_0}}) $ such that $\phi (x)\in U$. If $y\in I^{'}$, as $x\notin J$ then $ h_{J^{'},J,I^{'}}(x)\in I^{'}$, then there is $\phi_1\in IFS (T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I})$ such that $\phi_1\circ h_{J^{'},J,I^{'}}(x)\in U$. If $y\in \phi (I^{'})$ the proof is analogous.
Now we take $x\in I^{'}$. If $y\in K$, then by property (b) of example (1) in Section \[subsec1\], there exists $\phi \in IFS( f_{_{K,I_0}},\ \ g_{_{K,I_0}}) $ such that $\phi (x)\in U$. If $y\in I^{'}$ then there is $\phi\in IFS (T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I})$ such that $\phi (x)\in U$. If $y\in \phi (I^{'})$, we consider $U^{'}=\phi ^{-1}(U)\cap I^{'}$. There is $\phi_1\in IFS (T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I})$ such that $\phi_1(x)\in U^{'}$, therefore $\phi \phi_1(x)\in U$.
If $x\in \phi (I^{'})$, then $ h_{J^{'},J,I^{'}}(x)\in I^{'}$, and the proof is the same that case $x\in I^{'}$.
[**[Example 4.]{}**]{} $M=K\cup \mathcal{I}\cup N $ (a Cantor set union intervals union countable points). In this case we have $N \nsubseteq \partial \mathcal{I}$.
Let $I_0\subset S^{1}$ be an interval. We consider IFS $( f_{_{K,I_0}},g_{_{K,I_0}} )$. Let $[\alpha ,\beta ] \subset I_0$ be a minimal interval containing a $K$.
Let $I$ be a connected component of $K^{c}$ with $I\subset [\alpha ,\beta ]$ and $I^{'}\subset I$.
Let $I^{'}=[a,b]$, and $\varphi$ as in Figure \[varphi\], with $\varphi|_{(I^{'})^{c}}=Id$.
Let $I_1\subset [a,b]$ such that $\varphi ^{n}(I_1)\cap \overline{I_1}=\emptyset$ for any $n>0$. Notice that $\varphi ^{n}(I_1)\to b$.
Let $\widehat{I}_1$ such that $I_1\subset \widehat{I}_1$ and $\varphi (I_1)\cap\overline{ \widehat{I}_1}= \emptyset$.
Let $J\subset S^{1}\setminus I_0$ and $J^{'}\subset J$.
We consider $$IFS( f_{_{K,I_0}},\ \ g_{_{K,I_0}}, \ \ \varphi, \ \ h_{J^{'},J,I_{1}}, \ \ T^{+}_{I_{1}\widehat{I_1}}, \ \ T^{-}_{I_{1}\widehat{I_1}}
) .$$
Let us denote $\mathcal{F}= IFS( f_{_{K,I_0}}, g_{_{K,I_0}}) $
Then the minimal set is $$M=K\cup \bigcup_{n\geq 0}\varphi^{n}(I_1) \cup \bigcup_{\phi\in\mathcal{F} } \phi ( \bigcup_{n\geq 0}\varphi^{n}(I_1) ) \cup \bigcup_{\phi\in\mathcal{F} } \phi (b)\cup \{b\}.$$
The proof of minimality of $M$ is analogous of above examples.
[**[Example 5.]{}**]{} $M=K\cup \mathcal{I}\cup N $ ( $M$ is an union of intervals.)\
Let $I_0$, $I$ and $\psi$ as Figure \[intervalos\] (a) and $I_1\supset I_0$ such that $\overline{\psi^{n}(I_1)}\cap \overline{I_1}=\emptyset$. Let $J$ a interval such that $\overline{J}\subset S^{1}\setminus (\bigcup_{n\geq 0} \{ \psi (I_1) \} \cup I)$ and $J^{'}\subset J$. We consider $h_{J^{'},J,I_{0}}$ and $H$ as Figure \[intervalos\] (b). Note that $H(I_0)=I$ and $H( \bigcup_{n\geq 1} \{ \psi^n (I_0) \} \cup I )\subset \psi (I_0)$.
We consider $$IFS( T^{+}_{I_{0}I_1},\ \ T^{-}_{I_{0}I_1} , \ H , \ \psi, h_{J^{'},J,I_{0}}, ).$$
We will prove that the minimal set for this IFS is $$M=\bigcup_{n\geq 0} \{ \psi^{n} (I_0) \} \cup I.$$
We begin by proving that $M$ is IFS-invariant. Let $x\in M$. If $x\in I_0$, then $T^{+}_{I_{0}I_1}(x), T^{-}_{I_{0}I_1}(x)\in I_0$, $H(x)\in I$ and as $J\cap I_0=\emptyset$ we have that $h_{J^{'},J,I_{0}}(x)\in I_0$. If $x\in \psi^{n} (I_0) $ with $n>0$ as $\psi^{n} (I_0)\cap I_1=\emptyset$ then $T^{+}_{I_{0}I_1}$ and $ T^{-}_{I_{0}I_1}$ are the identity on $\psi^{n} (I_0) $ with $n>0$. $H(x)\in \psi (I_0)$ and $J\cap \psi (I_0)=\emptyset$ then $h_{J^{'},J,I_{0}}(x)\in I_0$. The proof is analogous that in case $x\in I$.
Now, we are going to prove that any $x\in M$ has dense orbit in $M$. Let $y\in M$ and $U$ a neighborhood of $y$, we will show that there exists $\phi \in IFS( T^{+}_{I_{0}I_1},\ \ T^{-}_{I_{0}I_1} , \ H , \ \psi, h_{J^{'},J,I_{0}}, ) $ with $\phi (x)\in U$. First we take $x\in I_0$. If $y\in I_0$, it is clear that there exists $\phi \in IFS( T^{+}_{I_{0}I_1},\ \ T^{-}_{I_{0}I_1} ) $ such that $\phi (x)\in U$. If $y\in \psi^{n} (I_0)$, as $x\notin J$ then $ h_{J^{'},J,I_{0}}(x)\in I_{0}$, then there is $\phi_1\in IFS ( T^{+}_{I_{0}I_1},\ \ T^{-}_{I_{0}I_1})$ such that $\psi^{n}\circ\phi_1\circ h_{J^{'},J,I^{'}}(x)\in U$. In the reaming cases the proof is analogous.
[**[Example 6.]{}**]{} We consider $S^{1}$ as the interval $[0,1]$ identifying $0$ with $1$.
$M=\mathcal{I}\cup N$ where $\partial \mathcal{I}\cup \{0\} = N$. (countable union of intervals union a finite number of points.
Let $\psi$ as in Figure \[varphi\] (b). Let $I$ such that $\overline{I}\cap \psi (I)=\emptyset$, therefore it holds that $\psi ^{n}(I)\to 1$ $(1=0)$ . Let $J$ such that $\overline {J}\cap \psi^{n} (I)=\emptyset$, for any $n\geq 0$ and let $J^{'}\subset J$. Consider $I^{'}\subset I$ and
$$IFS( \psi ,\ \ h_{J^{'},J,I^{'}},\ \ T^{+}_{I^{'}I},\ \ T^{-}_{I^{'}I}) .$$ It is easy to show that
$$M= \{ 0\}\cup \bigcup _{n\geq 0}\psi^{n} (I^{'}).$$ [**[Example 7.]{}**]{} $M=\mathcal{I}\cup K$ with $\partial \mathcal{I}\subset K$. (symmetric Cantorval).
Next lemma will be used in the construction of this example.
\[l33\] Let $K_1\subset [a,b]$ and $K_2\subset [c,d]$ be Cantor sets with $a,b\in K_1$ and $c,d\in K_2$ such that:
1. There exist sequences $\{I_i\}_{i\in{\mbox{$\mathbb{N}$}}}$ and $\{I^{'}_i\}_{i\in{\mbox{$\mathbb{N}$}}}$, where $I_i$ and $ I^{'}_i$ are connected components of $K_1^{c}$ with $(\cup I_i)\cup (\cup I^{'}_i) =K_{1}^{c}$ and $I_i\cap I'_k=\emptyset$ for all $i,k$.
2. There exist sequences $\{J_j\}_{j\in{\mbox{$\mathbb{N}$}}}$ and $\{J^{'}_j\}_{j\in{\mbox{$\mathbb{N}$}}}$, where $J_j$ and $ J^{'}_j$ are connected components of $K_2^{c}$ with $(\cup J_j)\cup (\cup J^{'}_j) =K_{2}^{c}$ and $J_j\cap J'_k=\emptyset$ for all $j,k$.
3. $\overline{ \{\partial I_i:\ i\in {\mbox{$\mathbb{N}$}}\}}=\overline{ \{\partial I^{'}_i:\ i\in {\mbox{$\mathbb{N}$}}\}}=K_1$ and $\overline{\{\partial J_j: \ j\in {\mbox{$\mathbb{N}$}}\}}=\overline{\{\partial J^{'}_j: \ j\in {\mbox{$\mathbb{N}$}}\}}=K_2$.
Then there exists a increasing homeomorphism $\psi:[a,b]\to [c,d]$ such that $\psi (K_1)=K_2$ and $\psi (\cup I_i)=\cup J_j$.
Let $I_{i_{0}}$ be an interval of $\{I_n \}$ with maximal length and we choose $J_{j_{0}}$ in an analogous way. Let $\psi :\overline{I}_{i_{0}}\to \overline{J}_{j_{0}}$ any increasing homeomorphism. Among all the elements of $\{I'_n \}$ to the left of $I_{i_{0}}$, let $I'_{i_{1}}$ one of maximal length.
Among all the elements of $\{I'_n \}$ to the right of $I_{i_{0}}$, let $I'_{i_{2}}$ one of maximal length.
In an analogous way we define $J^{'}_{j_{1}}$ and $J^{'}_{j_{2}}$. So define
$\psi :\overline{I^{'}}_{i_{1}}\to \overline{J^{'}}_{j_{1}}$ and $\psi :\overline{I^{'}}_{i_{2}}\to \overline{J^{'}}_{j_{2}}$ as any increasing homeomorphism.
Among all the elements of $\{I_n \}$ to the left of $I^{'}_{i_{1}}$ let $I^{}_{i_{3}}$ be one of maximal length. Let $I^{}_{i_{4}}$ be one of maximal length between $I^{'}_{i_{1}}$ and $I^{}_{i_{0}}$. let $I^{}_{i_{5}}$ be one of maximal length between $I^{}_{i_{0}}$ and $I^{'}_{i_{2}}$, and let $I^{}_{i_{6}}$ be be one of maximal length to the right of $I^{'}_{i_{2}}$. Analogously define $J_{j_{3}}$, $J_{j_{4}}$, $J_{j_{5}}$, $J_{j_{6}}$ and $\psi :\overline{I}_{i_{p}}\to \overline{J}_{j_{p}}$ for $p=3,4,5,6.$
Proceeding inductively we obtain $\psi: (\cup \overline{I}_i )\cup (\cup \overline{I^{'}}_i) \to (\cup \overline{J}_j )\cup (\cup\overline{J^{'}}_j)$ an increasing map such that if $I$ and $I^{'}$ are connected components of $K_1^{c}$ such that $I$ is on the left of $I^{'}$ then $\psi (I)$ is on the left of $\psi(I^{'})$.
Now we will prove that $\psi: (\cup \overline{I}_i)\cup( \cup \overline{I^{'}}_i) \to (\cup \overline{J}_j) \cup (\cup \overline{J^{'}}_j)$ is uniformly continuous. Suppose by contradiction that there exist $\{x_n\}$, $\{y_n\}$ and $\varepsilon_0>0$ such that $d(x_n,y_n)<1/n$ and $d(\psi(x_n),\psi(y_n))>\varepsilon_0$. Without loss of generality we can assume that $x_n<y_n$, $x_n,y_n\to x_0\in [a,b]$ and $x_n$ and $y_n$ are in different connected components of $ (\cup \overline{I}_i)\cup( \cup \overline{I^{'}}_i) $ . Since $\psi$ is continuous in $(\cup \overline{I}_i)\cup( \cup \overline{I^{'}}_i) $, we have that $x_0 \notin (\cup \overline{I}_i)\cup( \cup \overline{I^{'}}_i)$.
So, if $n$ is big enough, $x_n$ and $y_n$ are in different connected components of $K_1^c$. As $d(\psi(x_n),\psi(y_n))>\varepsilon_0$ there exists $J=(\alpha, \beta)$ connected component of $K_2^c$ such that $\psi(x_n)\leq\alpha<\beta\leq \psi(y_n)$, as $(\cup J_j)\cup (\cup J^{'}_j)=K_2^c$ then there exists a connected components $I$ of $K_1^c$ with $I\subset (\cup I_n)\cup (\cup I^{'}_n)$ and $\psi(I)=J$. But as $x_n\leq \psi^{-1}(\alpha )<\psi^{-1}(\beta)\leq y_n$ and $x_n,y_n\to
x_0$ we obtain a contradiction.
As $\overline{ \{\partial I_i:\ i\in {\mbox{$\mathbb{N}$}}\}}=\overline{ \{\partial I^{'}_i:\ i\in {\mbox{$\mathbb{N}$}}\}}=K_1$ then we can defined $\psi$ in $[a,b]$ , by continuity and we are done.
Now we will construct the example 7.
Given an interval $I\subset S^{1}$, we consider an interval $I^{'}\subsetneq I$. Let $f,g$ and $h$ linear functions with angular coefficient 1/5 in $I^{'}$ (as Figure \[figura6\] (a)) and such that $f|_{I^{c}}=g|_{I^{c}}=h|_{I^{c}}=Id$. If $\Lambda_0=I^{'}$ and $\Lambda_{k+1}=f(\Lambda_k)\cup g(\Lambda_k)\cup h(\Lambda_k)$ then $K=\cap_{k\geq 0} \Lambda_k$ is a Cantor set.
\[rk1\]
We make the following observations regarding the family $\mathcal{F}_1=IFS(f, g,h) $:
- If $J\subset I^{'} $ is a connected component of $K^{c}$ then $J=\phi (I_0)$ or $J=\phi (I_1)$ with $\phi\in \mathcal{F}_1$, where $I_0$ and $I_1$ are as Figure \[figura6\] (a).
- If $J_0=\phi_0(I_0)$ and $J_1=\phi_1(I_1)$ with $\phi_0,\phi_1 \in \mathcal{F}_1$ then $J_0\cap J_1=\emptyset$.
Therefore the set of connected components of $K^{c}$ is the union of $\{I_n\}$ and $\{J_n\}$, where $I_n=\phi_0(I_0)$ for some $\phi_0 \in \mathcal{F}_1$ and $J_n=\phi_1(I_1)$ for some $\phi_1 \in \mathcal{F}_1$. Moreover $\overline{ \{\partial I_n:\ n\in {\mbox{$\mathbb{N}$}}\}}=\overline{ \{\partial J_n:\ n\in {\mbox{$\mathbb{N}$}}\}}=K$.
- The set $K\cup (\cup_{\phi \in {\mathcal{F}_1}}\phi (I_0))$ is not connected neither is a Cantor set.
We consider an interval $[a,b]\subset I^{'}$ with $a,b\in K$ such that $I_0\subset [a,b]$, $[a,b]\cap I_1=\emptyset$ and $K\cap [a,b]$ is a Cantor set. Let $\overline{I_0}=[c,d]$ and $x_0\in K\cap (a,c)$ be such that $K\cap [a,x_0]$ is a Cantor set. By the lemma \[l33\], we can to construct $\psi :[a,b]\to [a,b]$ a increasing homeomorphism such that:
- $\psi (K\cap [a,x_0])=K\cap [a,c]$.
- If $\mathcal{H}=\{ \phi (I_0):\ \phi\in\mathcal{F}_1 \}$ then $\psi (\mathcal{H} \cap [a,x_0])\subset \mathcal{H}\cap [a,c]$.
Let $T^{+}$ be such that (see Figure\[figura6\] (b)):
$$T^{+}(x)= \left\{
\begin{array}{l}
\psi (x) \mbox{ if $x\in [a,x_0],$}\\
r_{1} (x) \mbox{ if $x\in [x_0,c]$} \mbox{, with } r_1([x_0,c])\subset [c,d] \mbox { and } r_1 \mbox{ linear}, \\
r_2 (x) \mbox{ if $x\in [c,d]$} \mbox { and } r_2 \mbox{ linear}, \\
Id \mbox{ if $x\in [a,d]^{c}.$}\\
\end{array}
\right.$$
In summary, the map $T^{+}$ satisfies the following properties: points that are in $K\cap [a,x_0]$ are mapped by $T^{+}$ in points that are in $K\cap [a,c]$. The elements of $\mathcal{H}$ included in $[a,x_0]$ are mapped by $T^{+}$ in the elements of $\mathcal{H}$ that are in $[a,c]$ and $T^{+}([x_0,d])\subset [c,d].$\
If we consider $y_0\in (d,b)$ in analogous way to $x_0$, we can to construct $T^{-}$ (in analogous way to construction of $T^{+}$) such that: points that are in $K\cap [y_0,b]$ are mapped by $T^{-}$ in point that are in $K\cap [d,b]$, the elements of $\mathcal{H}$ included in $[y_0,b]$ are mapped by $T^{-}$ in the elements of $\mathcal{H}$ that are included in $[d,b]$ and $T^{-}([c,y_0])\subset [c,d]$ and $T^{-}([c,b]^{c})=Id.$\
Moreover taking $r_1$ and $r_2$ conveniently in $T^{+}$ ( in analogous way for $T^{-}$), we can construct $T^{+}$ and $T^{-}$ such that $I_0=[c,d]$ is a minimal set for the family $IFS(T^{+}, T^{-} )$ (this construction is as the example of Subsection \[subsec2\]).
Let $IFS(f,g,h, T^{+},T^{-})$ and $M=K\cup \{ \phi (I_{0}): \ \ \phi \in IFS( f , g, h \} \cup I_{0} . $
The proof of minimality of $M$ is analogous of above examples. $\Box$
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abstract: 'Radiative strength functions (RSFs) in $^{93-98}$Mo have been extracted using the ($^3$He,$\alpha \gamma$) and ($^3$He,$^3$He$^{\prime} \gamma$) reactions. The RSFs are U-shaped as function of $\gamma$ energy with a minimum at around $E_{\gamma}=3$ MeV. The minimum values increase with neutron number due to the increase in the low-energy tail of the giant electric dipole resonance with nuclear deformation. The unexpected strong increase in strength below $E_{\gamma}=3$ MeV, here called soft pole, is established for all $^{93-98}$Mo isotopes. The soft pole is present at all initial excitation energies in the $5-8$ MeV region.'
address:
- '$^1$ Department of Physics, University of Oslo, N-0316 Oslo, Norway'
- '$^2$ Lawrence Livermore National Laboratory, L-414, 7000 East Avenue, Livermore, CA 94551, USA'
- '$^3$ North Carolina State University, Raleigh, NC 27695, USA'
- '$^4$ Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA'
- '$^5$ Department of Physics, Osmangazi University, Meselik, Eskisehir, 26480 Turkey'
- '$^6$ Department of Physics, [Å]{}bo Akademi, FIN-20500 Turku, Finland'
- '$^7$ Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA'
author:
- 'M. Guttormsen,$^{1}$[^1] R. Chankova,$^{1}$ U. Agvaanluvsan,$^{2,3}$ E. Algin,$^{2,3,4,5}$ L.A. Bernstein,$^{2}$ F. Ingebretsen,$^{1}$ T. L[ö]{}nnroth,$^{6}$ S. Messelt,$^{1}$ G.E. Mitchell,$^{3,4}$ J. Rekstad,$^{1}$ A. Schiller,$^{2}$ S. Siem,$^{1}$ A.C. Sunde,$^{1}$ A. Voinov,$^{7}$ and S. [Ø]{}deg[å]{}rd$^{1}$'
title: 'Radiative strength functions in $^{93-98}$Mo'
---
[2]{}
Introduction
============
The $\gamma$-decay of nuclei at high excitation energy tends to follow certain statistical rules. The dominating $\gamma$-transition driving factors are the number of accessible final states and the $\gamma$-ray transmission coefficient. The largest uncertainties are connected to the latter factor. In the description of this factor Blatt and Weisskopf [@blatt] included an $E_{\gamma}^{2L+1}$ dependency, where $L$ is the angular momentum transfer in the transition. In their definition of the radiative strength function (RSF), this simple energy dependence was divided out. With such a definition, the single-particle RSF (Weisskopf) estimates become independent of $\gamma$-ray energy. Various concepts of RSFs and $\gamma$ decay in the continuum are outlined in the reviews of Bartholomew [*et al.*]{} [@bart1; @bart2].
It has been well known that the RSF is not at all constant, but shows an additional $E_{\gamma}^x$ dependency with $x=1-2$ for $\gamma$ energies in the $4-8$ MeV region. Axel [@axel] argued that this feature is due to the collective giant electric dipole resonance (GEDR), which represents the essential mechanism for the $\gamma$-decay. However, the situation is more complex. Further studies [@joly; @gutt84; @igash] reveal fine structures in the RSF, which are commonly called pygmy resonances. This name does not refer to specific structures: the E1 pygmy resonance in the $E_{\gamma}=5-7$ MeV region of gold to lead nuclei could be due to neutron skin oscillations [@enders00], whereas bumps in the 3 MeV region of rare earth nuclei are now determined to be of M1 character [@schi04; @krti]. The electromagnetic character and measured strength of the latter pygmy resonance is compatible with the scissors mode. Recently [@emel0; @voin04], the RSF picture of iron isotopes has been further modified by the observation of an anomalous increase in strength at $\gamma$ energies below 4 MeV. It is clear that the present situation urges for new experimental results.
The stable molybdenum isotopes are well suited as targets for the study of nuclear properties when going from spherical to deformed shapes. In this work we perform a systematic analysis of the RSFs of the six $^{93-98}$Mo isotopes. The RSFs depend on the dynamic properties of electric charges present within these systems ($Z=42$). Since the nuclear deformation varies from spherical shapes ($\beta \sim 0$) at $N= 51$ to deformed shapes ($\beta \sim 0.2$) at $N=56$, we expect to observe effects due to shape changes. Furthermore, these nuclei reveal weak GEDR tails at low $E_{\gamma}$ making them interesting objects in the search for other weak structures in the RSF.
The Oslo Cyclotron group has developed a sensitive tool to investigate RSFs for $E_{\gamma}$ below the neutron binding energy $S_n$. The method is based on the extraction of primary $\gamma$-ray spectra at various initial excitation energies $E_i$ measured in particle reactions with one and only one charged ejectile. From such a set of primary $\gamma$-spectra, nuclear level densities and RSFs can be extracted [@hend1; @melb0; @schi0]. The level density reveals essential nuclear structure information such as thermodynamic properties and pair correlations as function of temperature. These aspects of the molybdenum isotopes will be presented in a forthcoming work. Various applications of the Oslo method have been described in Refs. [@schi1; @gutt1; @schi2; @gutt3; @voin1; @siem1; @dy03; @femo].
Experimental method
===================
The particle-$\gamma$ coincidence experiments were carried out at the Oslo Cyclotron Laboratory for $^{93-98}$Mo using the CACTUS multi-detector array. The charged ejectiles were detected with eight particle telescopes placed at an angle of 45$^{\circ}$ relative to the beam direction. An array of 28 NaI $\gamma$-ray detectors with a total efficiency of $\sim$15% surrounded the target and particle detectors.
In the present work, results from eight different reactions on four different targets are discussed. Results from two of those reactions have been reported earlier. The beam energies for the different reactions are given in parentheses:\
1) $^{94}$Mo($^3$He,$\alpha \gamma$)$^{93}$Mo (new, 30 MeV)\
2) $^{94}$Mo($^3$He,$^3$He$^{\prime}\gamma$)$^{94}$Mo (new, 30 MeV)\
3) $^{96}$Mo($^3$He,$\alpha \gamma$)$^{95}$Mo (new, 30 MeV)\
4) $^{96}$Mo($^3$He,$^3$He$^{\prime}\gamma$)$^{96}$Mo (new, 30 MeV)\
5) $^{97}$Mo($^3$He,$\alpha \gamma$)$^{96}$Mo (reported in [@emel0; @femo], 45 MeV)\
6) $^{97}$Mo($^3$He,$^3$He$^{\prime}\gamma$)$^{97}$Mo (reported in [@emel0; @femo], 45 MeV)\
7) $^{98}$Mo($^3$He,$\alpha \gamma$)$^{97}$Mo (new, 45 MeV)\
8) $^{98}$Mo($^3$He,$^3$He$^{\prime}\gamma$)$^{98}$Mo (new, 45 MeV)\
The targets were self-supporting metal foils enriched to $\sim 95$% with thicknesses of $\sim 2$ mg/cm$^2$. The experiments were run with beam currents of $\sim 2$ nA for 1–2 weeks. The reaction spin windows are typically $I\sim 2-6 \hbar$.
The experimental extraction procedure and the assumptions made are described in Refs. [@hend1; @schi0] and references therein. For each initial excitation energy $E_i$, determined from the ejectile energy and reaction $Q$ value, $\gamma$-ray spectra are recorded. Then the spectra are unfolded using the known $\gamma$-ray response function of the CACTUS array [@unfold]. These unfolded spectra are the basis for making the first generation (or primary) $\gamma$-ray matrix [@gutt0], which is factorized according to the Brink-Axel hypothesis [@axel; @brink] as $$P(E_i,E_{\gamma}) \propto \rho (E_i -E_{\gamma}) {\mathcal{T}} (E_{\gamma}).
\label{eq:axel}$$ Here, $\rho$ is the level density and ${\mathcal{T}}$ is the radiative transmission coefficient.
The $\rho$ and ${\mathcal{T}}$ functions can be determined by an iterative procedure [@schi0] through the adjustment of each data point of these two functions until a global $\chi^2$ minimum of the fit to the experimental $P(E_i,E_{\gamma})$ matrix is reached. It has been shown [@schi0] that if one solution for the multiplicative functions $\rho$ and ${\mathcal{T}} $ is known, one may construct an infinite number of other functions, which give identical fits to the $P$ matrix by $$\begin{aligned}
\tilde{\rho}(E_i-E_\gamma)&=&A\exp[\alpha(E_i-E_\gamma)]\,\rho(E_i-E_\gamma),
\label{eq:array1}\\
\tilde{{\mathcal{T}}}(E_\gamma)&=&B\exp(\alpha E_\gamma){\mathcal{T}} (E_\gamma).
\label{eq:array2}\end{aligned}$$ Consequently, neither the slope ($\alpha$) nor the absolute values of the two functions ($A$ and $B$) can be obtained through the fitting procedure.
The parameters $A$ and $\alpha$ can be determined by normalizing the level density to the number of known discrete levels at low excitation energy [@ENSDF] and to the level density estimated from neutron-resonance spacing data at the neutron binding energy $S_n$ [@IA98]. The procedure for extracting the total level density $\rho$ from the resonance energy spacing $D$ is described in Ref. [@schi0]. Here, we will only discuss the determination of parameter $B$ of Eq. (\[eq:array2\]), which gives the absolute normalization of ${\mathcal{T}}$. For this purpose we utilize experimental data on the average total radiative width of neutron resonances at $S_n$ $\langle\Gamma_{\gamma} \rangle$.
We assume here that the $\gamma$-decay in the continuum is dominated by E1 and M1 transitions. For initial spin $I$ and parity $\pi$ at $S_n$, the width can be written in terms of the transmission coefficient by [@kopecky] $$\begin{aligned}
\langle\Gamma_\gamma\rangle=\frac{1}{2\rho(S_n, I, \pi)} \sum_{I_f}&&\int_0^{S_n}{\mathrm{d}}E_{\gamma} B{\mathcal{T}}(E_{\gamma})
\nonumber\\
&&\rho(S_n-E_{\gamma}, I_f),
\label{eq:norm}\end{aligned}$$ where the summation and integration run over all final levels with spin $I_f$, which are accessible by $\gamma$ radiation with energy $E_{\gamma}$ and multipolarity E1 or M1.
A few considerations have to be made before $B$ can be determined. Methodical difficulties in the primary $\gamma$-ray extraction prevents determination of the functions ${\mathcal{T}}(E_{\gamma})$ in the interval $E_{\gamma}<1$ MeV and $\rho(E)$ in the interval $E > S_n-1$ MeV. In addition, ${\mathcal{T}}(E_{\gamma})$ at the highest $\gamma$-energies, above $E_{\gamma} \sim S_n-1$ MeV, suffers from poor statistics. For the extrapolation of $\rho$ we apply the back-shifted Fermi gas level density as demonstrated in Ref. [@dy03]. For the extrapolations of ${\mathcal{T}}$ we use an exponential form. As a typical example, the extrapolations for $^{98}$Mo are shown in Fig. \[fig:rhorsf\]. The contribution of the extrapolations of $\rho$ and ${\cal T}$ to the calculated radiative width in Eq. (\[eq:norm\]) does not exceed $15$% [@voin1]. The experimental widths $\langle\Gamma_\gamma\rangle$ in Eq. (\[eq:norm\]) are listed in Table \[tab:tab1\]. For $^{94}$Mo, this width is unknown and is estimated by an extrapolation based on the $^{96}$Mo and $^{98}$Mo values.
The total radiative strength function for dipole radiation ($L=1$) can be calculated from the normalized transmission coefficient ${\mathcal{T}}$ by $$f (E_{\gamma}) =\frac{1}{2\pi} \frac{ {\mathcal{T}}(E_{\gamma})}{ E_{\gamma}^3}.
\label{eq:fT}$$ The RSFs extracted from the eight reactions are displayed in Fig. \[fig:webrsf\]. As expected, the RSFs seem not to show any clear odd-even mass differences. The results obtained for the ($^3$He,$\alpha$) and ($^3$He,$^3$He$^{\prime}$) reactions populating the same residual nucleus reveal very similar RSFs. Also for $^{96}$Mo two different beam energies have been applied, giving very similar RSFs. Thus, the observed energy and reaction independency gives further confidence in the Oslo method.
Description of the radiative strength functions
===============================================
An inspection of the experimental RSFs of Fig. \[fig:webrsf\] reveals that the RSFs are increasing functions of $\gamma$ energy for $E_{\gamma} > 3$ MeV. This indicates that the RSFs are influenced by the tails of the giant resonances. As follows from previous work, the main contribution (about 80%) is due to the electric dipole resonance (GEDR). The magnetic resonance (GMDR) and the isoscalar E2 resonance are also present in this region.
If the GEDR is described by a Lorentzian function, one will find that the strength function approaches zero in the limit $E_\gamma\rightarrow0$. However, the $^{144}$Nd(n,$\gamma \alpha )$ reaction [@Popov] strongly suggests that $f_{{\mathrm{E}}1}$ has a finit value in this limit. Kadmenski[ĭ]{}, Markushev and Furman (KMF) have developed a model [@kad] describing this feature for the electric dipole RSF:
$$f_{\mathrm{E1}}(E_\gamma,T)=\frac{1}{3\pi^2\hbar^2c^2} \frac{0.7\sigma_{\mathrm{E1}}\Gamma_{\mathrm{E1}}^2(E_\gamma^2+4\pi^2T^2)} {E_{\mathrm{E1}}(E_\gamma^2-E_{\mathrm{E1}}^2)^2}.
\label{eq:E1}$$
The temperature $T$ depends on the final state $f$ and for simplicity we adapt the schematic form $$T(E_f)=\sqrt{U_f/a},
\label{eq:tef}$$ where the level density parameter is parametrized as $a=0.21 A^{0.87}$ MeV$^{-1}$. The intrinsic energy is estimated by $U_f=E_f-C_1-E_{\rm pair}$ with a back-shift parameter of $C_1=-6.6A^{-0.32}$ MeV [@egidy]. The pairing energy contribution $E_{\rm pair}$ is evaluated from the three point mass formula of Ref. [@doba].
Although this model has been developed for spherical nuclei, it has been successfully applied to $^{56,57}$Fe and several rare earth nuclei [@voin04; @voin1; @siem1; @dy03] assuming a constant temperature parameter $T$ in Eq. (\[eq:E1\]), i.e. one that is independent of excitation energy. In this work we assume that the temperature depends on excitation energy according to Eq. (\[eq:tef\]), which gives an increase in the RSF at low $\gamma$ energy.
In order to compare the excitation energy dependent RSF with experiments, Eq. (\[eq:E1\]) should be folded with the final excitation energies of the specific experiment giving $$\langle f_{\mathrm{E1}}(E_\gamma) \rangle=\frac{1}{E_2-E_1}\int_{E_1-E_{\gamma}}^{E_2-E_{\gamma}}{\mathrm{d}}E_f f_{\mathrm{E1}}(E_f),
\label{eq:fE1ave}$$ where the integration runs over all final excitation energies $E_f$ which are experimentally accessible by transition with given energy $E_{\gamma}$. Our data set of primary $\gamma$-ray spectra is typically taken between $E_1 \sim 4$ MeV and $E_2 \sim S_n$ initial excitation energy.
The M1 GMDR contribution to the total RSF is described by a Lorentzian form through $$f_{\mathrm{M1}}(E_\gamma)=\frac{1}{3\pi^2\hbar^2c^2} \frac{\sigma_{\mathrm{M1}}E_\gamma\Gamma_{\mathrm{M1}}^2} {(E_\gamma^2-E_{\mathrm{M1}}^2)^2+E_\gamma^2\Gamma_{\mathrm{M1}}^2}.
\label{eq:M1}$$ This approach is in accordance with numerous experimental data obtained so far, and is recommended in Ref.[@IA98].
Although of minor importance, the E2 radiation strength $f_{{\rm E}2}$ has also been included. Here, we use the same Lorentzian function as in Eq. (\[eq:M1\]), but with different resonance parameters and an additional factor $3/(5E_{\gamma}^2)$. The resonance parameters for the E1, M1 and E2 resonances are taken from the compilation of Refs. [@IA98; @atom] and are listed in Table \[tab:tab1\].
The enhanced RSF at low $\gamma$ energies has at present no theoretical explanation. Recently, the same enhancement has been observed in the iron isotopes [@emel0; @voin04]. We call this structure a soft pole in the RSF and choose a simple power law parametrization given by $$f_{\rm soft pole}=\frac{1}{3\pi^2\hbar^2c^2}{\cal{A}}E_{\gamma}^{-b},$$ where $\cal{A}$ and $b$ are fit parameters, and $E_{\gamma}$ is given in MeV.
Previously, a pygmy resonance around $E_{\gamma}\sim 3$ MeV has been reported in several rare earth nuclei [@voin1; @siem1; @dy03]. The electromagnetic character of the corresponding RSF structure is now established to be of M1 type [@schi04; @krti] and is interpreted as the scissors mode. Deformed nuclei can in principle possess this collective motion, and e.g., $^{98}$Mo with a deformation of $\beta \sim 0.18$, could eventually show some reminiscence of the scissors mode. Data on $^{94}$Mo [@frans03] and $^{96}$Mo [@frans04] show a summed $M1$ strength to mixed symmetry $1^+$ states around $\sim 3.2$ MeV in the order of $\sim 0.6 \mu_N^2$. This is about one order of magnitude lower than the $M1$ strength observed in well-deformed rare earth nuclei using the present method. This $M1$ strength is deemed too weak to cause a visible bump in our RSFs above 3 MeV.
We conclude that a reasonable composition of the total RSF is $$f=\kappa (f_{{\mathrm{E}}1} + f_{{\mathrm{M}}1} +f_{\rm soft pole})+E_{\gamma}^2f_{\mathrm{E}2},$$ where $\kappa$ is a normalization constant. Generally, its value deviates from unity for several reasons; the most important reasons are theoretical uncertainties in the KMF model and the evaluation of $B$ in Eq. (\[eq:norm\]). We use $\kappa$, ${\cal {A}}$ and $b$ as free parameters in the fitting procedure, and the results for the eight reactions are summarized in Table \[tab:tab2\].
In Fig. \[fig:fitexp\] the various contributions to the total RSF of $^{98}$Mo are shown. The main components are the GEDR resonance and the unknown low energy structure. We observe that the E1 component exhibits an increased yield for the lowest $\gamma$ energies due to the increase in temperature $T$. However, this effect is not strong enough to explain the low energy upbend.
Figure \[fig:webrsf\] shows the fit functions for all reactions and gives qualitative good agreements with the experimental data. The fitting parameters $\kappa$, ${\cal{A}}$ and $b$ are all similar within the uncertainties. It should be noted that the soft pole parameters coincide with the description of the $^{57}$Fe nucleus [@voin04] having ${\cal{A}}= 0.47 (7)$ mb/MeV, and $b=2.3(2)$.
The almost a factor-of-two higher values of the RSFs for $E_{\gamma} > 3$ MeV when going from $N=51$ to $56$ can be understood from the corresponding evolution of nuclear deformation. Following the onset of deformation the GEDR will split into two parts, where $1/3$ of its strength is shifted down in energy and $2/3$ up. Photoneutron cross sections [@atom] do not show a splitting into two separate bumps, however, the observed increase in width $\Gamma_{{\rm E}1}$ as function of neutron number (see Table \[tab:tab1\]) supports the idea of a splitting, which is well known from other more deformed nuclei. Figure \[fig:webrsf\] demonstrates that the adopted widths describe very well the variation of the RSF strength as function of mass number.
In order to test that the prominent soft pole structure is present in the whole excitation energy region, we have performed the following test. Assuming that the level density from Eq. (\[eq:axel\]) is correct, we can estimate the shape of the strength functions starting at various initial excitation energies using
$$f(E_\gamma,E_i)=\frac{1}{2\pi} \frac{{\cal N}(E_i)P(E_i,E_{\gamma})}{\rho(E_i-E_{\gamma})E_{\gamma}^3}.
\label{eq:fei}$$
Actually, $f(E_\gamma,E_f)$ would have been the proper expression to investigate, but due to technical reasons we choose $f(E_\gamma,E_i)$. One problem is that the normalization constant is only roughly known through the estimate
$${\cal N}(E_i)=\frac{\int_0^{E_i} {\mathrm{d}}E_{\gamma }\rho(E_i-E_{\gamma})
{\cal T}(E_{\gamma})}{\int_0^{E_i}{\mathrm{d}}E_{\gamma}P(E_i,E_{\gamma})}
\label{eq:nei}$$
with $E_i<S_n$. However, for the expression $f(E_\gamma,E_i)$ we are only interested in the shape of the RSFs, and an exact normalization is therefore not crucial. The evaluation assumes that eventual temperature dependent behavior of the RSF is small compared to the soft pole structure[^2].
In Fig. \[fig:rsfi68\], the RSFs for $^{96,98}$Mo are shown at various initial energies $E_i$. For comparison, the figure also includes the global RSFs (solid lines) obtained with the Oslo method (Fig. \[fig:webrsf\]). Within the error bars the data support that the soft pole is present in all the excitation bins studied.
The origin of the soft pole cannot be explained by any known theoretical model. One would therefore need to know the $\gamma$-ray multipolarity as guidence for theoretical approaches to this phenomenon. Rough estimates of the reduced strength can be obtained from $$B(XL \uparrow)= \frac{1}{8\pi ^3}\frac{L[(2L+1)!!]^2}{L+1}
\left( \frac{\hbar c}{\langle E_{\gamma} \rangle} \right)^2 I.$$ The integrated cross-section $I$ is determined by $$I= \int_{\rm 1 MeV}^{\rm 3 MeV} {\mathrm{d}}E_{\gamma} \sigma(E_{\gamma}),$$ and the average energy is given by $$\langle E_{\gamma} \rangle= I^{-1}\int_{\rm 1 MeV}^{\rm 3 MeV} {\mathrm{d}}E_{\gamma}E_{\gamma} \sigma(E_{\gamma}),$$ where $$\sigma(E_{\gamma})=3(\pi \hbar c)^2 E_{\gamma} f_{\rm softpole}(E_{\gamma}).$$ In the evaluation, we have integrated the soft pole between 1 and 3 MeV. Thus, the estimates give a lower limit for the respective $B(XL \uparrow)$ values. The correct result will of course depend on the functional form of $f_{\rm softpole}(E_{\gamma})$ below 1 MeV, however, no experimental data exist in this region and any assumption here would be highly speculative.
With the assumptions above, we get on the average a $B(E1)$ value of $0.02$ e$^2$fm$^2$, which is $0.07$ % of the sum rule for the GEDR. For $B(M1)$, we get roughly $1.8 \mu _N ^2$, which is $3-4$ times larger than the observed strength to mixed symmetry $1^+$ states around 3 MeV [@frans03; @frans04]. Finally, we obtain a $B(E2)$ value around $8000 $e$^2$fm$^4$, which is $3-8$ times larger that for the excitation to the first excited $2^+$ states in the even molybdenum isotopes. Thus, we cannot exclude any of these multipolarities, since neither of them would yield unreasonably high transition strengths.
Summary and conclusions
=======================
As expected, the observed RSFs reveal very similar shapes since they all refer to isotopes with the same nuclear charge. When going from $N=51$ to $56$ the RSF increases by almost a factor of two for $E_{\gamma}>3$ MeV, which can be understood from the change of nuclear deformation. With the onset of deformation, the increasing resonance GEDR width $\Gamma_{{\mathrm{E}}1}$ is responsible for the increasing strength.
An enhanced strength at low $\gamma$ energies is observed, which is equally strong for all isotopes and excitation energies studied. A similar enhancement has also been seen in the iron isotopes. The multipolarity of the soft pole radiation is unknown and there is still no theoretical explanation for this very interesting phenomenon.
Financial support from the Norwegian Research Council (NFR) is gratefully acknowledged. Part of this work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract W-7405-ENG-48. A.V. acknowledges support from a NATO Science Fellowship under project number 150027/432. E.A, U.A, and G.E.M acknowledge support from the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through DOE Research Grant No. DE-FG03-03-NA00076 and U.S. Department of Energy Grant No. DE-FG02-97-ER41042.
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-------------- --------------------- -------------------------- -------------------------- --------------------- -------------------------- -------------------------- --------------------- -------------------------- -------------------------- ---------------------------------
Nucleus $E_{{\mathrm{E}}1}$ $\sigma_{{\mathrm{E}}1}$ $\Gamma_{{\mathrm{E}}1}$ $E_{{\mathrm{M}}1}$ $\sigma_{{\mathrm{M}}1}$ $\Gamma_{{\mathrm{M}}1}$ $E_{{\mathrm{E}}2}$ $\sigma_{{\mathrm{E}}2}$ $\Gamma_{{\mathrm{E}}2}$ $\langle \Gamma_\gamma \rangle$
(MeV) (mb) (MeV) (MeV) (mb) (MeV) (MeV) (mb) (MeV) (meV)
$^{93}$Mo 16.59 173.5 4.82 9.05 0.86 4.0 13.91 2.26 4.99 160(20)
$^{94}$Mo 16.36 185.0 5.50 9.02 1.26 4.0 13.86 2.24 4.98 170(40)$^a$
$^{95}$Mo 16.28 185.0 5.76 8.99 1.38 4.0 13.81 2.22 4.97 135(20)
$^{96}$Mo 16.20 185.0 6.01 8.95 1.51 4.0 13.76 2.21 4.96 150(20)
$^{97}$Mo 16.00 187.0 5.98 8.92 1.58 4.0 13.71 2.19 4.95 110(15)
$^{98}$Mo 15.80 189.0 5.94 8.89 1.65 4.0 13.66 2.17 4.93 130(20)
\[tab:tab1\]
-------------- --------------------- -------------------------- -------------------------- --------------------- -------------------------- -------------------------- --------------------- -------------------------- -------------------------- ---------------------------------
: Parameters used for the radiative strength functions. The data are taken from Ref. [@IA98]. The E1 resonance parameters for the even Mo isotopes are based on photo absorption experiments [@atom], and the parameters for the odd Mo isotopes are derived from interpolations.
$^a$Estimated from systematics.
------------------------------------- ------------- -------- ----------
Reaction ${\cal{A}}$ $b$ $\kappa$
(mb/MeV)
($^3$He,$\alpha$) $^{93}$Mo 0.37(7) 2.6(3) 0.44(4)
($^3$He,$^3$He$^{\prime}$)$^{94}$Mo 0.48(5) 2.5(2) 0.36(2)
($^3$He,$\alpha$) $^{95}$Mo 0.48(6) 2.6(2) 0.39(2)
($^3$He,$^3$He$^{\prime}$)$^{96}$Mo 0.60(4) 3.2(2) 0.36(1)
($^3$He,$\alpha$) $^{96}$Mo 0.47(14) 2.7(6) 0.32(4)
($^3$He,$^3$He$^{\prime}$)$^{97}$Mo 0.47(7) 2.4(3) 0.38(3)
($^3$He,$\alpha$) $^{97}$Mo 0.30(10) 2.2(5) 0.45(5)
($^3$He,$^3$He$^{\prime}$)$^{98}$Mo 0.22(7) 2.1(5) 0.52(4)
------------------------------------- ------------- -------- ----------
: Parameters obtained from the fitting of the radiative strength function.
\[tab:tab2\]
![Measured level density $\rho$ (upper panel) and radiative transmission coefficient ${\cal T}$ (lower panel) for $^{98}$Mo. The straight lines are extrapolations needed to calculate the normalization integral of Eq. (\[eq:norm\]). The triangle in the upper panel is based on resonance spacing data at $S_n$.[]{data-label="fig:rhorsf"}](rhorsf.ps)
![Normalized RSFs for $^{93-98}$Mo. The filled and open circles represent data taken with the ($^3$He,$\alpha$) and ($^3$He,$^3$He$^{\prime}$) reactions, respectively. The filled triangles in $^{93,95}$Mo are estimates of $E1$ RSF of hard primary $\gamma$-rays [@KU95] . The solid and dashed lines are fits to the RSF data from the two respective reactions (see text).[]{data-label="fig:webrsf"}](webrsf.ps)
![Experimental radiative strength function of $^{98}$Mo compared to a model description including GEDR, GMDR and the isoscalar E2 resonance. The empirical soft pole component is used to describe the low energy part of the RSF.[]{data-label="fig:fitexp"}](fitexp.ps)
![RSFs for $^{96,98}$Mo at various initial excitation energies. The soft pole is present for all $E_i$. The solid lines display the RSFs obtained in Fig. \[fig:webrsf\].[]{data-label="fig:rsfi68"}](rsfi68.ps)
[^1]: Electronic address: [email protected]
[^2]: Simulations using the KMF model indicate a maximum 20 % effect from temperature dependence of the RSF.
|
---
abstract: 'Tomography aims to recover a three-dimensional (3D) density map of a medium or an object. In medical imaging, it is extensively used for diagnostics via X-ray computed tomography (CT). Optical diffusion tomography is an alternative to X-ray CT that uses multiply scattered light to deliver coarse density maps for soft tissues. We define and derive tomography of cloud droplet distributions via passive remote sensing. We use multi-view polarimetric images to fit a 3D polarized radiative transfer (RT) forward model. Our motivation is 3D volumetric probing of vertically-developed convectively-driven clouds that are ill-served by current methods in operational passive remote sensing. These techniques are based on strictly 1D RT modeling and applied to a single cloudy pixel, where cloud geometry is assumed to be that of a plane-parallel slab. Incident unpolarized sunlight, once scattered by cloud-droplets, changes its polarization state according to droplet size. Therefore, polarimetric measurements in the rainbow and glory angular regions can be used to infer the droplet size distribution. This work defines and derives a framework for a full 3D tomography of cloud droplets for both their mass concentration in space and their distribution across a range of sizes. This 3D retrieval of key microphysical properties is made tractable by our novel approach that involves a restructuring and differentiation of an open-source polarized 3D RT code to accommodate a special two-step optimization technique. Physically-realistic synthetic clouds are used to demonstrate the methodology with rigorous uncertainty quantification.'
author:
- |
Aviad Levis[^1]\
Viterbi Faculty of Electrical Engineering\
Technion - Israel Institute of Technology\
Haifa, 3200003, Israel\
[[email protected]]{} Yoav Y. Schechner\
Viterbi Faculty of Electrical Engineering\
Technion - Israel Institute of Technology\
Haifa, 3200003, Israel\
[[email protected]]{} Anthony B. Davis\
Jet Propulsion Laboratory\
California Institute of Technology\
Pasadena, CA, 91109\
[[email protected]]{} Jesse Loveridge\
Dept. of Atmospheric Sciences\
University of Illinois\
Champaign, IL, 61820\
[[email protected] ]{}
bibliography:
- 'references.bib'
title: 'Multi-view polarimetric scattering cloud tomography and retrieval of droplet size'
---
Introduction & Outline {#sec:introduction}
======================
Clouds play a significant role at local and global scales, affecting weather, the water cycle, solar power generation, and impacting Earth’s energy balance [@trenberth2009earth]. Moreover, uncertainties in global climate models are significantly affected by our limited understanding, and therefore modeling, of cloud dynamics and microphysics [@boucher2013clouds]. Thus, understanding, modeling, and predicting cloud properties is a key issue with worldwide socio-economic implications that is in the center of many research studies [@rosenfeld1998satellite]. Much of the current understanding relies on routine remote sensing of cloud properties such as by the MODerate resolution Imaging Spectrometer (MODIS) [@Platnick2003MODIS]. In practice, global-scale retrievals have so far been based on an individual pixel basis, using a crude approximation that clouds are plane-parallel slabs. This approximation uses a 1D radiative transfer (RT) model, which leads to biases in many retrievals [@marshak2006impact] while other retrievals simply fail [@cho2015frequency]. Convective clouds are therefore a blind spot due to their 3D nature.[^2] In its 2018 *Decadal Strategy for Earth Observation from Space* [@NAS_ESDS_2017], the National Academies of Sciences, Engineering, and Medicine have indeed identified “Clouds, Convection, and Precipitation” as one of its five top-priority Targeted Variables for NASA’s next generation of satellite missions. To bridge this gap, new technology is needed to study clouds as 3D volumetric objects, on a global scale. The [*CloudCT*]{} [@cloudct] space mission, by the European Research Council (ERC) is specifically destined to provide data and products for this goal. It will involve 10 nano-satellites orbiting in formation, thus acquiring simultaneously unique multi-view measurements of such vertically-developed 3D clouds (Fig. \[fig:cloudct\]).
Moreover, common retrieval of cloud droplet characteristics use two optical bands simultaneously [@nakajima1990determination]: a visible band, where reflected radiance increases with cloud optical thickness, and a shortwave IR (SWIR) band, where absorption by condensed water depends on cloud droplet size. To sense droplet size in 3D by CloudCT or other future missions, sensors will need to have either SWIR or polarization capability.
![Artist’s illustration of the CloudCT [@cloudct] mission: a distributed multi-view system of 10 nano-satellites orbiting the Earth in formation. Measurements acquired by the formation will enable tomographic retrievals of cloud properties.[]{data-label="fig:cloudct"}](cloudct.png){width="0.6\linewidth"}
Why polarized light?
--------------------
There is an additional caveat in common retrievals, which rely on SWIR absorption [@nakajima1990determination]. In addition to absorption, light undergoes multiple scattering in clouds. Multiple scattering diminishes sensitivity to droplet microphysics. High sensitivity to microphysics is embedded in single-scattering events. It is thus beneficial to pick-up single-scatter signals, out of the strong multiply-scattered background radiance. Polarization signals of scattered light are dominated by single-scattering events, and are thus highly sensitive to the type and size specifications of scatters. Thus in recent years, there is growing interest in polarimetric imagers for remote sensing of clouds and aerosols [@deschamps1994polder; @BreonGoloub_98; @kalashnikova2011sensitivity; @lukashin2013uncertainty; @diner2013airborne; @diner2018advances]. In turn, increased interest in polarimetric sensing capabilities has led to the development of 1D and 3D polarized (or “vector”) RT codes [@emde2015iprt; @emde2018iprt] with an aim of improving retrieval algorithms. Motivated by the CloudCT mission formulation—only the first of many to come in innovative passive cloud remote sensing—we develop herein a novel framework for 3D remote sensing of cloud properties using multi-view polarimetric measurements.
Why passive tomography?
-----------------------
From its etymology, the word “tomography” means a slice-by-slice recovery of an object’s 3D internal structure using 2D projections of cumulative density. In the computer age, this task is termed [*Computed Tomography*]{} (CT) [@KakSlaney88]. Common medical CT approaches are transmission-based X-ray CT or single-photon emission computed tomography (SPECT). There, 2D projections represent straight line-of-sight (LOS) integrals of the local X-ray opacity or nuclear marker density, respectively. In both imaging modalities, the inverse problem of recovering the medium content is [*linear*]{} [@gordon1970algebraic].
Biomedical imaging also involves CT modalities which are not based on linear projections. A prime example is Optical Diffusion Tomography (ODT) [@arridge1999optical; @boas2001imaging; @arridge2009optical], which uses non-ionizing near-infrared light. It is worth noting the work by Che et al. [@che2018inverse] which departs from physics-based approaches into the realm of machine-learning.
In ODT, a patient’s organ is surrounded by a large number of point sources of pulsed isotropic near-infrared irradiation and a large number of time-resolving omnidirectional sensors. The organ transmits radiation diffusely, with very little absorption. Anomalous 3D absorbing or vacuous regions can be detected and assayed using [*nonlinear*]{} inverse diffusion spatiotemporal analysis, that relies on very high orders of scattering. The detected radiance is blurred, yielding limited 3D spatial resolution. However, ODT can yield sufficient diagnostic information, using non-ionising radiation. Medical CT modalities generally use [*active*]{} radiation. Active methods are also used for [*local*]{} atmospheric sensing or scatterers by radar and lidar. There, a transmitter and receiver are generally collocated and signals are based on backscattering and time-resolved two-way transmission. Probing is solved per LOS using methods which are computationally relatively simple. However, the technology is expensive, horizontal sampling is generally very limited, and irradiance decays fast from the transmitter. Passive sensing is less expensive, uses minimal power, and can image wide swaths of Earth. Thus [*global coverage mandates passive imaging from space*]{}. Consequently, this paper focuses on derivation of 3D passive tomography of scatterer fields.
Passive remote sensing does not benefit from pulsed sources for echo-location. It should rely on multi-angular data. Linear CT models (analogous to medical Xray CT and SPECT) were used to study gas emission and absorption in 3D plumes in the vicinity of pollution sources [@Hashmonay1999; @Todd2001] or volcanoes [@Kazahaya2008; @Wright2008]. There, Rayleigh-scattered sunlight was transmitted through the gas to a spectrometer on a platform flying around the plume. Following the vision of Werner et al. [@warner1986liquid], Huang et al. [@huang2008determination; @huang2008cloud] used scanning microwave radiometers to reconstruct 2D slices of particle density in clouds based on its impact on local emissivity.
Linear CT was also adapted by Garay et al. [@garay2016tomographic] to characterize a smoke plume over water emanating from a coastal wild fire. There, the signal is sunlight scattered to space and detected by the Multi-angle Imaging Spectro-Radiometer (MISR) sensor [@Diner1998] at nine viewing angles. The analysis in [@garay2016tomographic] yields the direct transmission through the plume per LOS, from which linear CT analysis yields the plume density without using solar radiometers under the plume.
In general, however, retrieving atmospheric scatterer fields in 3D requires a full forward model of scattering in 3D. The model satisfies neither a direct transmission model of linear CT, nor the diffusion limit of ODT. In passive imaging of scatterers, the light source irradiating the atmosphere is the sun: uncontrolled, steady and mono-directional. Aides et al. [@AidesEtAl2013] formulated CT based on single-scattered light. Their forward model is based on sets of broken-ray paths, where light changes direction once from the sun to a sensor.
All the above atmospheric tomography methods assumed the medium to be optically thin enough for direct and once-scattered radiation to dominate the measured radiance. We depart radically from this assumption, drawing inspiration from the success of active ODT, though necessarily with a different forward model. We formulate an inverse 3D RT problem for cloud tomography utilizing multi-view multi-spectral polarimetric images. In contrast to linear CT, the image formation model is nonlinear in the microphysical and density variables. Our approach seeks an optimal fit of droplet microphysical parameters. This is based on a computational 3D polarized RT forward model, the [*vector Spherical Harmonics Discrete Ordinates Method*]{} (vSHDOM) [@evans1998spherical; @doicu2013multi]. To this effect, we generalize our demonstrated iterative inversion approach [@levis2015airborne; @HolodovskyEtAl2016; @levis2017multiple] to take advantage of polarimetric measurements.
Outline
-------
In the next section, we cover basic cloud droplet optics using Mie scattering theory and the fundamentals of polarized 3D RT. The latter yields radiance which has a clear decomposition into single- and multiply-scattered light. This decomposition supports the solution to the inverse problem at hand. We then lay out our 3D cloud tomography method where we target three basic microphysical properties, volumetrically. Necessary but tedious mathematical details are presented in the [*Appendix*]{}. Subsequently, the new 3D cloud tomographic capability is demonstrated on realistic synthetic clouds from a Large Eddy Simulation (LES) that provide ground truth for unambiguous retrieval error quantification. We conclude with a summary of our results and an outline of future developments, mostly looking toward CloudCT and other future space-based uses.
Background {#sec:background}
==========
This section describes bulk microphysical parameterization of scattering media, the polarimetric radiative transfer image formation (forward) model and the relation between them. The section also describes the coordinate systems in use (per-scatterer, imager and Earth frames). We further decompose the polarized radiance into single-scattered and high-order scattered components. These foundations are used in subsequent sections, to formulate tomographic recovery.
Scatterer microphysical properties {#subsec:microphysics}
----------------------------------
In the lower atmosphere, cloud particles are droplets of liquid water that are very nearly spherical, having radius $r$. They are however polydisperse, with a droplet size distribution denoted $n(r)$. For most remote-sensing purposes, $n(r)$ is parameterized using an [*effective radius*]{} in $\mu$m and a dimensionless [*variance*]{} [@hansen1971multiple]: $$\begin{aligned}
r_{\rm e} = \frac{ \int_0^{\infty} r^3 n(r) {\rm d}r}{\int_0^{\infty} r^2 n(r) {\rm d}r}, \quad v_{\rm e} = \frac{\int_0^{\infty} \left(r {-} r_{\rm e}\right)^2 r^2n(r) {\rm d}r}{r_{\rm e}^2\int_0^{\infty} r^2 n(r) {\rm d}r}.
\label{eq:reff_veff}
\end{aligned}$$ A commonly used parametric size distribution, having empirical support [@hansen1971multiple] is the [*Gamma*]{}-distribution (Fig. \[fig:gamma\]): $$n(r) = N \, c \, r^{(v_{\rm e}^{-1} \! {-}3)} \exp [- r / (r_{\rm e}v_{\rm e})],
\label{eq:gamma}$$ where we require $v_{\rm e} < 1/2$. Here $c = (r_{\rm e}v_{\rm e})^{(2{-}v_{\rm e}^{-1})} / \Gamma(v_{\rm e}^{-1} \! {-}2)$ is a normalization constant and $$N = \int_0^{\infty} n(r) {\rm d}r
\label{eq:N}$$ is the droplet number concentration. Let $\rho_{\rm w}$ be the density of liquid water. An important cloud characteristic is the water mass density or [*Liquid Water Content*]{} (LWC) per unit volume: $${\rm LWC} = \frac{4}{3} \pi \rho_{\rm w} \int_0^{\infty} r^3 n(r) {\rm d}r.
\label{eq:LWC}$$ It is expressed as LWC = $\nicefrac{4}{3} \, \pi \rho_{\rm w} r_{\rm e}^3 (1-v_{\rm e})(1-2v_{\rm e})$ for the Gamma distribution in .
{width="90.00000%"}
Polarized light {#subsec:polarized}
---------------
A light wave is associated with orthogonal components of a random electric wave, $E_1(t)$ and $E_2(t)$, where $t$ is time. The components’ direction unit vectors are respectively $\hat{\bf E}_1$ and $\hat{\bf E}_2$. The wave propagates in direction ${\bm \omega} = \hat{\bf E}_1 \times \hat{\bf E}_2$. It is convenient to define the polarized light state in terms of the Stokes [@hansen1971multiple] vector ${\bf I} = \left(I, Q, U, V\right)^\top$. Each component of ${\bf I}$ expresses temporal expectation: $$\begin{aligned}
I = \langle E_{1}E^*_{1} + E_{2}E^*_{2}\rangle_t, \quad
Q = \langle E_{1}E^*_{1} - E_{2}E^*_{2}\rangle_t, \\
U = \langle E_{1}E^*_{2} + E_{2}E^*_{1}\rangle_t, \nonumber \quad
V = {\rm i}\langle E_{1}E^*_{2} - E_{2}E^*_{1}\rangle_t, \nonumber
\label{eq:stokes}
\end{aligned}$$ where ${\rm i} = \sqrt{-1}$. Unpolarized intensity is $I$. The degrees of polarization (DOP) and linear polarization (DoLP) are respectively defined as the ratios $\nicefrac{\sqrt{Q^2 + U^2 + V^2}}{I},~\nicefrac{\sqrt{Q^2 + U^2}}{I}$. The angle of linear polarization (AoLP) is $\nicefrac{1}{2}\,\tan^{-1}(U/Q)$.
Single scattering of polarized light {#subsec:single_scattering}
------------------------------------
Light interaction with a single particle is described by the total extinction cross-section $s_{\rm t}(r,\lambda)$, decomposed into scattering and absorption cross-sections, respectively: $$\begin{aligned}
s_{\rm t}(r,\lambda) = s_{\rm s}(r,\lambda)
+ s_{\rm a}(r,\lambda).
\label{eq:single_crosssection}
\end{aligned}$$ In Mie scattering by spheres, introduced further on, we have $$\begin{aligned}
\label{eq:Mie_crosssection}
s_{\rm t}(r,\lambda) = \pi \, r^2 \, \mathcal{Q}_{\rm t}(2\pi \, r/\lambda), \quad
s_{\rm s}(r,\lambda) = \pi \, r^2 \, \mathcal{Q}{\rm s}(2\pi \, r/\lambda), \quad s_{\rm a}(r,\lambda) = \pi \, r^2 \, \mathcal{Q}{\rm a}(2\pi \, r/\lambda) \nonumber
\end{aligned}$$ where $\mathcal{Q}{\rm t},\mathcal{Q}{\rm s}, \mathcal{Q}{\rm a}$ are dimensionless efficiency factors, which depend on the normalized size parameter $2\pi \, r/\lambda$. In the limit $r \gg \lambda$, $\mathcal{Q}{\rm t} \approx 2$. Furthermore, when $s_{\rm s}(r,\lambda) \gg s_{\rm a}(r,\lambda)$, then $\mathcal{Q}{\rm s} \approx 2$ and $\mathcal{Q}{\rm a} \approx 0$.
Define size-weighted average over some function $a(r)$ by[^3] $$\begin{aligned}
\langle a \rangle_r = \frac{1}{N}\int_0^{\infty} a(r) n(r) {\rm d} r.
\label{eq:r_average}
\end{aligned}$$ Note that we use here an approximation, commonly used in multi-spectral remote sensing, of a single rendering with spectrally-averaged optical properties. The material optical properties can furthermore be approximated, in the absence of molecular absorption, by using a single wavelength for each spectral band. This is valid if wavelength dependencies within a spectral band are weak, a condition met when narrow bands are considered. Macroscopic optical cross-sections are then expressed as weighted averages $$\begin{aligned}
\sigma_{\rm t}(\lambda) {=} \langle s_{\rm t}(r,\lambda) \rangle_r,~
\sigma_{\rm s}(\lambda) {=} \langle s_{\rm s}(r,\lambda) \rangle_r,~
\sigma_{\rm a}(\lambda) {=} \langle s_{\rm a}(r,\lambda) \rangle_r.
\label{eq:bulk_cs}
\end{aligned}$$ Throughout the text, dependency on $\lambda$ is generally omitted for simplicity; however, it is used at specific points as needed.
Scattering, as a fraction of the overall interaction [@marshak20053d], is expressed by the dimensionless [*single scattering albedo*]{} $$\begin{aligned}
\varpi = \frac{\sigma_{\rm s}}{\sigma_{\rm t}}.
\label{eq:ssa}
\end{aligned}$$ The extinction coefficient (or optical density) is denoted by $\beta$. Following Eqs. \[\[eq:N\],\[eq:LWC\],\[eq:bulk\_cs\]\], $\beta = N \sigma_{\rm t}$ is expressed in terms of the LWC as [@chylek1978extinction] $$\begin{aligned}
\beta = \frac{{\rm LWC}}{\frac{4}{3}\pi \rho_w \langle r^3 \rangle_r} \sigma_{\rm t} = {\rm LWC} \cdot \tilde{\sigma}_{\rm t}.
\label{eq:extinct}
\end{aligned}$$ Here, $\tilde{\sigma}_{\rm t}$ is the mass extinction coefficient (in units of $\nicefrac{{\rm m}^2}{{\rm g}}$).
Let ${\bm \omega}$ and ${\bm \omega}'$ be the unitary incident and scattered ray direction vectors respectively in Fig. \[fig:gamma\]. Single-scattering geometry is defined by the local coordinate system of the incoming beam’s electric fields. As stated above, the electric field of incoming light is decomposed into components along orthogonal directions. We set them as $${\bf E}_1 \propto {\bm \omega} \times {\bm \omega}', \quad
{\bf E}_2 \propto {\bf E}_1 \times {\bm \omega}.$$ The scattering angle is $\theta = \cos^{-1}({\bm \omega} {\cdot} {\bm \omega}')$. The angular redistribution of singly-scattering light from a sphere of is defined by the $4{\times}4$ dimensionless [*Mueller*]{} matrix ${\bf P}_{\rm s}(\theta, r)$. The macroscopic [*phase matrix*]{} is the size-weighted average $$\begin{aligned}
{\bf P}(\theta) = \frac{\langle s_{\rm s}(r) {\bf P}_{\rm s}(\theta, r) \rangle_r}{\sigma_{\rm s}}.
\label{eq:bulk_p}
\end{aligned}$$ For spherical (or just randomly-oriented) particles, the phase-matrix ${\bf P}(\theta)$ takes the following symmetric form [@hansen1971multiple] $$\begin{aligned}
{\bf P} \left(\theta \right) =
\begin{bmatrix}
p_{11} \left(\theta \right) & p_{21} \left(\theta \right) & 0 & 0 \\
p_{21} \left(\theta \right) & p_{22} \left(\theta \right) & 0 & 0 \\
0 & 0 & p_{33} \left(\theta \right) & -p_{43} \left(\theta \right) \\
0 & 0 & p_{43} \left(\theta \right) & p_{44} \left(\theta \right) \\
\end{bmatrix},
\label{eq:phasemat}
\end{aligned}$$ where $p_{11}$ is the (unpolarized) scattering phase-function. In single-scattering of unpolarized incident sunlight, the DoLP of scattered light amounts to the ratio $|p_{21}| / p_{11}$.
### Rayleigh scattering
The Rayleigh model describes light scattering by particles much smaller than the wavelength. The Rayleigh phase matrix takes the following form [@chandrasekhar1950radiative] $$\begin{aligned}
{\bf P}_{\rm Rayl} \left(\theta \right) =
\begin{bmatrix}
\frac{3}{4}\left(1+\cos^2\theta\right) & -\frac{3}{4}\sin^2\theta & 0 & 0 \\
-\frac{3}{4}\sin^2\theta & \frac{3}{4}\left(1+\cos^2\theta\right) & 0 & 0 \\
0 & 0 & \frac{3}{2}\cos\theta & 0 \\
0 & 0 & 0 & \frac{3}{2}\cos\theta \\
\end{bmatrix}.
\label{eq:pol_rayl}
\end{aligned}$$ The single-scattering DoLP due to air molecules is then $$\begin{aligned}
{\rm DoLP}_{\rm Rayl}(\theta) = \frac{\sin^2\theta}{1+ \cos^2\theta}.
\label{eq:dolp_rayl}
\end{aligned}$$ According to a maximum DoLP is attained at single-scattering angle $\theta = 90^{\circ}$.
### Mie scattering
Mie theory describes how light interacts with a spherical particle of size comparable to $\lambda$ [@bohren2008absorption]. Denote $\mu{=}\cos\theta$. Mie scattering is defined in terms of complex-valued amplitude scattering functions[^4] $S_1(\mu), S_2(\mu)$, which correspond to scattering of the $E_1, E_2$ electric field components. Scattering of the Stokes vector ${\bf I}$ is described by the phase matrix ${\bf P}_{\rm Mie}(\mu)$, which is fully defined by six matrix components: $$\begin{aligned}
p^{\rm Mie}_{11} = \frac{\varrho}{2}\left(S_1 S_1^* + S_2 S_2^*\right), \nonumber \quad
p^{\rm Mie}_{12} = \frac{\varrho}{2}\left(S_1 S_1^* - S_2 S_2^*\right), \nonumber \\
p^{\rm Mie}_{22} = \frac{\varrho}{2}\left(S_1 S_1^* + S_2 S_2^*\right), \nonumber \quad
p^{\rm Mie}_{33} = \frac{\varrho}{2}\left(S_1 S_2^* + S_2 S_1^*\right), \nonumber \\
p^{\rm Mie}_{43} = \frac{\varrho}{2}\left(S_1 S_2^* - S_2 S_1^*\right), \nonumber \quad
p^{\rm Mie}_{44} = \frac{\varrho}{2}\left(S_1 S_2^* + S_2 S_1^*\right).
\label{eq:pol_mie}
\end{aligned}$$ Here, $\varrho$ is a normalization constant, set to satisfy $\frac{1}{2}\int_{-1}^1 p^{\rm Mie}_{11} (\mu ) {\rm d} \mu = 1$.
Mie scattering due to water droplets is peaked at specific angles. For a single droplet or monodisperse material, ${\bf P}^{\rm Mie}$ has sharp scattering lobes at angles that depend on the droplet’s $\nicefrac{r}{\lambda}$ ratio. A macroscopic voxel contains droplets in a range of radii $r$, smoothing the scattering lobes. The smoothing effect depends on $v_{\rm e}$ (Fig. \[fig:mie\_cloudbow\_glory\]) and, to a far lesser extent, the spectral bandwidth (Fig. \[fig:mie\_cloudbow\_glory\]). Two angular domains that stand out for remote-sensing purposes are the cloud-bow ($\theta \in [135^{\circ}, 155^{\circ}]$) and glory ($\theta \in [175^{\circ}, 180^{\circ}]$). Both domains have peaks that are sensitive to the droplet microphysical parameters, and are significantly polarized (i.e., peaks are visible in the $p_{12}^{\rm Mie}$ component). The latter fact renders these peaks distinguishable in the presence of a multiply-scattered signal component.
{width="90.00000%"}
Multiple scattering of polarized light {#subsec:multiple_scattering}
--------------------------------------
The [*Radiative Transfer Equation*]{} (RTE) [@chandrasekhar1950radiative] describes multiple scattering interactions of monochromatic partially polarized light within a medium. Transmittance between two points ${\bm x}_1, {\bm x}_2$ is $$\begin{aligned}
\label{eq:T}
T \left({\bm x}_1 {\rightarrow} {\bm x}_2\right) =
\exp{\left[-\int_{{\bm x}_1}^{{\bm x}_2}
\! \beta({\bm x}){\rm d}{\bm x}\right]}.
\end{aligned}$$ An atmospheric domain $\Omega$ has boundary $\partial \Omega$. The intersection of $\partial \Omega$ with a ray originating at point ${\bm x}$ in direction $-{\bm \omega}$ (Fig. \[fig:domain\]) is denoted ${\bm x}_0 ({\bm x},\omega)$. Denote the Stokes vector field as ${\bf I} {\left({\bm x},{\bm \omega} \right)}$. Then ${\bf I}({\bm x}_0, {\bm \omega})$ is the Stokes vector of radiation which propagates in direction ${\bm \omega}$ at boundary point ${\bm x}_0({{\bm x},\bm \omega})$. The non-emissive forward RT model [@chandrasekhar1950radiative] couples ${\bf I}{\left({\bm x},{\bm \omega} \right)}$ to a vector [*source field*]{} ${\bf J}{\left({\bm x},{\bm \omega} \right)}$ (Fig. \[fig:domain\]) by $$\begin{aligned}
&& {\bf I} {\left({\bm x},{\bm \omega} \right)}= {\bf I}({\bm x}_0, {\bm \omega}) T\left({\bm x}_0 {\rightarrow} {\bm x}\right)
+
\int_{{\bm x}_0}^{\bm x} {\bf J}({\bm x}',{\bm \omega}) \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}\right) {\rm d}{\bm x}', \label{eq:rte_integral} \\
&&{\bf J} {\left({\bm x},{\bm \omega} \right)}= \frac{\varpi({\bm x})}{4\pi} \int_{4\pi}
{\bf P}\left({\bm x},{\bm \omega}{\cdot}{\bm \omega'}\right) {\bf I} \left({\bm x},{\bm \omega'}\right) {\rm d}{\bm \omega}'.
\label{eq:J}
\end{aligned}$$ Equations \[\[eq:rte\_integral\]-\[eq:J\]\] are solved numerically, either directly with an explicit solver [@doicu2013multi] or indirectly using a Monte-Carlo path tracer [@mayer2009radiative]. We use vSHDOM [@doicu2013multi] to simulate scattered Stokes components of a realistic atmosphere, having both Mie and Rayleigh scattering due to water droplets and air molecules.
{width="75.00000%"}
Multiple scattering interactions are defined using two coordinate systems. Local [*scatterer coordinates*]{} are set by . Stokes measurements in satellites, however, are defined in [*Meridional*]{} coordinates. Let $\hat{{\bf z}}$ denote the zenith direction vector at every point on Earth. In meridian coordinates, the electric field components are defined by direction vectors $$\begin{aligned}
\hat{{\bf m}}_1 = \frac{\hat{{\bf z}} \times {\bm \omega}}{ \| \hat{{\bf z}} \times {\bm \omega}\|}, \quad
\hat{{\bf m}}_2 = {\bm \omega} \times \hat{{\bf m}}_1.
\end{aligned}$$ Each pixel-scale Stokes measurement is described by a coordinate system defined by $\hat{{\bf m}}_1$ and $\hat{{\bf m}}_2$. The transformation between the two coordinate systems amounts to a multiplication of ${\bf I}$ by a Mueller rotation matrix.
Sampling ${\bf I}{\left({\bm x},{\bm \omega} \right)}$ at the location of each camera and direction of each camera pixel yields the measured Stokes vector. A measurement $k$ is done at the camera position ${\bm x}_k$, LOS direction ${\bm \omega}_k$, and wavelength $\lambda_k$ (Fig. \[fig:domain\]). Thus, Eqs. \[\[eq:rte\_integral\]-\[eq:J\]\] yield the pixel measurement model $$\begin{aligned}
{\bf I}[k] = {\bf I} \left({\bm x}_0, {\bm \omega}_k \right) T\left({\bm x}_0 {\rightarrow} {\bm x}_k\right)
+
\int_{{\bm x}_0}^{{\bm x}_k} {\bf J}\left({\bm x}',{\bm \omega}_k\right) \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}'.
\label{eq:forward_stokes}
\end{aligned}$$
Single-scattering separation {#subsec:singscat_sep}
----------------------------
It is often convenient to separate the single-scattering contribution from the rest of the radiance field [@nakajima1988algorithms]. The solar irradiance at the top of the atmosphere (TOA) is $F_{\rm Sun}$. It is unoplarized, thus corresponds to a Stokes vector ${\bf F}_{\rm Sun}{=}\left( F_{\rm Sun},~0,~0,~0 \right)^\top$. The Sun is modeled as an ideal directional source with direction ${\bm \omega}_{\rm Sun}$. A solar ray heading to point ${\bm x}$ intersects the TOA at point ${\bm x}_{\rm Sun}$. The solar transmittance is given by $T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}\right)$. Let $\delta$ denote [*Dirac’s delta*]{}. Thus, ${\bf I}$ can be written as a sum of the [*diffuse*]{} component ${\bf I}_{\rm d}$, and direct solar component: $${\bf I} {\left({\bm x},{\bm \omega} \right)}= {\bf I}_{\rm d} {\left({\bm x},{\bm \omega} \right)}+ \delta \left({\bm \omega} {-} {\bm \omega}_{\rm Sun} \right){\bf F}_{\rm Sun} T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}\right) .
\label{eq:I_seperation}$$ Inserting \[\[eq:I\_seperation\]\] into \[\[eq:J\]\] yields $$\label{eq:J_seperation}
{\bf J} {\left({\bm x},{\bm \omega} \right)}= {\bf J}_{\rm d} {\left({\bm x},{\bm \omega} \right)}+
\frac{\varpi({\bm x})}{4\pi}
{\bf P}\left({\bm x},{\bm \omega}{\cdot}{\bm \omega}_{\rm Sun}\right){\bf F}_{\rm Sun} T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}\right),$$ where $${\bf J}_{\rm d} {\left({\bm x},{\bm \omega} \right)}= \frac{\varpi({\bm x})}{4\pi} \int_{4\pi}
{\bf P}\left({\bm x},{\bm \omega}{\cdot}{\bm \omega'}\right) {\bf I}_{\rm d} \left({\bm x},{\bm \omega'}\right) {\rm d}{\bm \omega}' .
\label{eq:J_d}$$ Consider Fig. \[fig:domain\]. Denote a [*broken-ray*]{} path of direct sunlight which undergoes single scattering at ${\bm x}'$, then reaches the camera: $${\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k.
\label{eq:broken_path}$$ It projects in direction ${\bm \omega}_k$ to pixel at ${\bm x}_k$, thus contributing to the measurement ${\bf I}({\bm x}_k, {\bm \omega}_k)$. Using Eqs. \[\[eq:rte\_integral\],\[eq:J\_seperation\]\], the single-scattered contribution from ${\bm x}'$ is $$\begin{aligned}
{\bf I}_{\rm Single}\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k \right) =
\frac{\varpi({\bm x}')}{4\pi} \beta ({\bm x}') {\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega}_{\rm Sun}\right){\bf F}_{\rm Sun} T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}'\right) T\left({\bm x}' {\rightarrow} {\bm x}_k\right).
\label{eq:single_scatter_x}
\end{aligned}$$ Thus, the entire single-scattered signal accumulates contributions along the LOS $${\bf I}_{\rm Single}\left({\bm x}_k \right) = \int_{{\bm x}_0}^{{\bm x}_k} {\bf I}_{\rm Single}\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k \right) {\rm d}{\bm x}'.
\label{eq:single_scatter}$$
Ray tracing {#subsec:raytrace}
-----------
Ray tracing computes a function over a straight line through a 3D domain. A common operation is path-integration (e.g. Eqs. \[\[eq:T\],\[eq:rte\_integral\]\]). Let $h({\bm x})$ be a continuous field. Define a grid of discrete points . Denote $h[g] = h({\bm x}_g)$. A path-integral over $h({\bm x})$ is numerically computed using an interpolation kernel $K$ $$\begin{aligned}
\label{eq:interpolation_kernel}
\int_{{\bm x}_1}^{{\bm x}_2}
\! h({\bm x}){\rm d}{\bm x} = \sum \limits_{g=1}^{N_{\rm grid}} h[g] \int_{{\bm x}_1}^{{\bm x}_2} K\left({\bm x} - {\bm x}_g\right){\rm d}{\bm x}.
\end{aligned}$$ For zero-order interpolation (i.e., voxel grid), degenerates to $$\begin{aligned}
\label{eq:numerical_integration}
\int_{{\bm x}_1}^{{\bm x}_2}
\! h({\bm x}){\rm d}{\bm x} = \sum \limits_{g=1}^{N_{\rm grid}} h[g] \ell_g \left({\bm x}_1 {\rightarrow} {\bm x}_2 \right),
\end{aligned}$$ where $\ell_g \left({\bm x}_1 {\rightarrow} {\bm x}_2 \right)$ is the intersection of the path with voxel $g$ (Fig. \[fig:domain\]). For voxel indices $g$ that do not intersect the path ${\bm x}_1 {\rightarrow} {\bm x}_2$, the value of $\ell_g \left({\bm x}_1 {\rightarrow} {\bm x}_2 \right)$ is 0.
section[Cloud Tomography]{} \[sec:tomography\] So far, we described the forward (image-formation) model, i.e., how images are formed, given cloud properties. In this work, we formulate a novel inverse tomographic problem of recovering the unknown cloud microphysical properties, volumetrically. In voxel $g$, the vector of unknown parameters is $\left({\rm LWC}[g],~r_{\rm e}[g],~v_{\rm e}[g] \right)$. The unknown microphysical parameters are concatenated to a vector of length $3N_{\rm grid}$ $${\bf \Theta} = \big(...,{\rm LWC}[g],~r_{\rm e}[g],~v_{\rm e}[g],...\big)^\top, \quad 1\leq g \leq N_{\rm grid}.
\label{eq:param_vec}$$ Neglecting circular polarization, each pixel measures a Stokes vector, ${\bf y}_{\bf I}=\big(y_{\rm I}, y_{\rm Q}, y_{\rm U}\big)$ at $N_{\lambda}$ wavelengths. Let $N_{\rm views}$ and $N_{\rm pix}$ denote the number of view points and camera pixels. The total number of Stokes measurements is thus $N_{\rm meas} {=} N_{\lambda} N_{\rm views} N_{\rm pix}$. The measurement vector of length $3N_{\rm meas}$ is expressed as $${\bf y} = \big( {\bf y}_{\bf I}[1],....,{\bf y}_{\bf I}[N_{\rm meas}]\big)^{\top}.
\label{eq:meas_vec}$$ In this section, we formulate the use of measurements ${\bf y}$ (multi-view, multi-pixel, multi-spectral, polarimetric measurements) for tomographic retrieval of ${ \bf \Theta}$ (3D volumetric cloud density and microphysics). It is worth mentioning at this point that Stokes components are not measured directly. Rather, they are computationally retrieved from measurements of different polarization states (see [*Appendix*]{} for the AirMSPI measurement model).
Polarimetric information {#subsec:information}
------------------------
To make an initial assessment of the sensitivity of polarimetric measurements, we simulate a simple homogeneous cubic cloud (Fig. \[fig:cuboid\]), parameterized by two microphysical parameters: $({\rm LWC}, r_{\rm e})$. Back-scattered Stokes measurements are taken at the TOA for angles and wavelengths sampled by the Airborne Multi-angle Spectro-Polarimetric Imager (AirMSPI) [@diner2013airborne].
![A homogeneous cubic cloud illuminated with solar radiation at a zenith angle of $15^\circ$ off-nadir. The solar azimuth angles are $\phi_0 = [0.0^\circ, 67.5^\circ]$. The outgoing Stokes vector ${\bf I}$ is simulated at AirMSPI resolution and wavelengths, with AirMSPI measuring along a North-bound track.[]{data-label="fig:cuboid"}](single_voxel.pdf){width="0.3\linewidth"}
Define $I[k], U[k], Q[k]$ as simulated Stokes components at measurement index $k$. Define a cost function for each of the Stokes components $$\begin{aligned}
\mathcal{D}_{\rm I}\left({\rm LWC}, r_{\rm e}\right) &=& \sum_{k=1}^{N_{\rm meas}} \left(I [k] - y_{\rm I}[k] \right)^2, \label{eq:minimization21} \\
\mathcal{D}_{\rm Q}\left({\rm LWC}, r_{\rm e}\right) &=& \sum_{k=1}^{N_{\rm meas}} \left(Q [k] - y_{\rm Q}[k] \right)^2, \\
\mathcal{D}_{\rm U}\left({\rm LWC}, r_{\rm e}\right) &=& \sum_{k=1}^{N_{\rm meas}} \left(U [k] - y_{\rm U}[k] \right)^2,
\label{eq:minimization22}
\end{aligned}$$ where we hold $v_{\rm e}$ constant. Equations \[\[eq:minimization21\]-\[eq:minimization22\]\] are 2D manifolds. Figure \[fig:loss\_manifold\] plots the cost manifolds for different solar azimuth angles, $\phi_0$. While there is an ambiguity between ${\rm LWC}$ and $r_{\rm e}$ when relying on $\mathcal{D}_{\rm I}$, there are better defined minima for $\mathcal{D}_{\rm Q}$ and $\mathcal{D}_{\rm U}$. This indicates that polarization measurements carry valuable information.
![Logarithm of the 2D cost manifolds for a 2-parameter homogeneous cubic cloud (Fig. \[fig:cuboid\]). Each column of plots corresponds to the cost of the different Stokes components in Eqs. \[\[eq:minimization21\]-\[eq:minimization22\]\]. Each row of plots corresponds to a different Solar azimuth angle $\phi_0$.[]{data-label="fig:loss_manifold"}](loss_manifold1.pdf){width="0.7\linewidth"}
Inverse problem formulation {#subsec:inv_formulation}
---------------------------
Denote ${\bf I}_{\bf \Theta}$ as the image formation model. Tomography can be formulated as minimization of a data-fit function. We preform $$\begin{aligned}
\hat{{\bf \Theta}} = \underset{{\bf \Theta}}{\arg\min}~ \mathcal{D} \left({\bf I}_{\bf \Theta}, {\bf y} \right) = \underset{{\bf \Theta}}{\arg\min} \left({\bf I}_{\bf \Theta} - {\bf y} \right)^{\top} {\bf \Sigma}^{-1} \left({\bf I}_{\bf \Theta} - {\bf y} \right),
\label{eq:minimization0}
\end{aligned}$$ Here ${\bf \Sigma}$ is related to the co-variance of the measurement noise. For brevity, we omit the subscript ${\bf \Theta}$ but remember that $${\bf I} {\equiv} {\bf I }_{\bf \Theta}, ~~ {\bf J} {\equiv} {\bf J}_{\bf \Theta}, ~~ \beta {\equiv} \beta_{\bf \Theta}, ~~ \varpi {\equiv} \varpi_{\bf \Theta}, ~~ {\bf P} {\equiv} {\bf P}_{\bf \Theta}, ~~ T {\equiv} T_{\bf \Theta}.
\label{eq:shorthand}$$ Assuming noise in different pixels, wavelengths and angles is uncorrelated, degenerates to $$\hat{{\bf \Theta}} = \underset{{\bf \Theta}}{\arg\min} \sum \limits_{k=1}^{N_{\rm meas}} \left({\bf I}[k] - {\bf y}_{\bf I}[k] \right)^{\top} {\bf R}^{-1} \left({\bf I}[k] - {\bf y}_{\bf I}[k] \right).
\label{eq:minimization}$$ The matrix ${\bf R}$ depends on the particular sensor technology. Description of ${\bf R}$, tailored to the AirMSPI sensor, is detailed in the [*Appendix*]{}.
We solve by a gradient-based approach. The gradient with respect to the unknown parameters ${\bf \Theta}$ is $$\begin{aligned}
{\bf \nabla}_{\bf \Theta} \mathcal{D} \left({\bf I}_{\bf \Theta}, {\bf y}\right) = 2\sum \limits_{k=1}^{N_{\rm meas}} \big( {\bf I}[k]-{\bf y}_{\bf I}[k] \big)^{\top} {\bf R}^{-1} {\bf \nabla}_{\bf \Theta} {\bf I}[k].
\label{eq:grad}
\end{aligned}$$ The term ${\bf \nabla}_{\bf \Theta} {\bf I}[k]$ is the [*Jacobian*]{} of the sensing model. Equation \[\[eq:grad\]\] is used to formulate an update rule for an iterative optimization algorithm $${\bf \Theta}_{b+1} = {\bf \Theta}_{b} - \chi_b \nabla_{\bf \Theta} \mathcal{D}\left({\bf I}_{\bf \Theta},{\bf y} \right),
\label{eq:update_rule}$$ where $b$ denotes the iteration index and $\chi_b$ is a scalar. We use L-BFGS [@zhu1997algorithm] for numerical optimization that, in particular, determines adaptively the value of $\chi_b$. One approach to computing the gradient $\nabla_{\bf \Theta} \mathcal{D}$ is the [*Adjoint RTE*]{} [@doicu2019linearizations; @martin2014adjoint]. Due to the recursive nature of the RTE, computing the gradient through the exact Jacobian ${\bf \nabla}_{\bf \Theta} {\bf I}[k]$ is computationally expensive. In the following sections, we derive a method to make the computation of the gradient tractable and efficient. We do that by approximating the Jacobian $\nabla_{\bf \Theta} {\bf I}$ in a tractable way, using a two-step iterative algorithm [@levis2015airborne; @levis2017multiple].
Iterative solution approach {#subsec:iterative_sol}
---------------------------
We formulate an iterative algorithm which alternates between two steps (See the diagram in Fig. \[fig:block\_diagram\]).
![A block diagram of the iterative algorithm. Red marks hyper-parameter. Numerical parameters of vSHDOM and L-BFGS are summarized in the .[]{data-label="fig:block_diagram"}](block_diagram2.pdf){width="0.6\linewidth"}
Starting with an initial guess, ${\bf \Theta}_0$, [Step 1]{} uses vSHDOM to compute the forward (recursive) RT equations. This renders synthetic images according to the multi-view geometry, spectral bands and spatial samples of the cameras. Keeping ${\bf I}_{\rm d}$ fixed, [Step 2]{} efficiently computes an approximate gradient with respect to ${\bf \Theta}$. The approximate gradient is fed into an L-BFGS step to update the current estimate ${\bf \Theta}_b$.\
[Step 1: RTE Forward Model]{}
The first step in the estimation approach is running the forward model in Eqs. \[\[eq:rte\_integral\]-\[eq:J\]\] using a numerical RTE solver. This requires transforming microphysical to optical properties at every voxel ($g$) and spectral band ($\lambda$): $$\label{eq:transform}
{\rm LWC}[g],~ r_{\rm e}[g],~ v_{\rm e}[g] ~ \longrightarrow ~ \beta_{\lambda}[g], ~\varpi_{\lambda}[g], ~{\bf P}_{\lambda}[g].$$ Implementing using Eqs. \[\[eq:bulk\_cs\]-\[eq:bulk\_p\]\] during each optimization iteration can be time-consuming. Therefore, define grids and , for which lookup tables $\tilde{{\bf \sigma}}_{\lambda}\left(r_{\rm e}, v_{\rm e}\right),~ \varpi_{\lambda}\left(r_{\rm e}, v_{\rm e}\right),~ {\bf P}_{\lambda}\left(r_{\rm e}, v_{\rm e}\right)$ are pre-computed. With these pre-computed tables and $\{ {\rm LWC}[g], r_{\rm e}[g], v_{\rm e}[g]\}$, vSHDOM [@doicu2013multi] renders the Stokes vector at each 3D voxel and direction. This is the forward modeling procedure. The result is the set of fields ${\bf I}{\left({\bm x},{\bm \omega} \right)}, {\bf I}_{\rm d}{\left({\bm x},{\bm \omega} \right)}, {\bf J}{\left({\bm x},{\bm \omega} \right)},
{\bf I}_{\rm Single}\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k \right)$.\
[Step 2: Approximate Jacobian Computation]{}
The forward vRTE model in depends on optical properties $\left(\beta, \varpi, {\bf P}\right)$, which themselves depend on the sought microphysics. The Jacobian at voxel $g$ is expressed by applying the chain-rule to . For example, the derivative with respect to the effective radius is $$\frac{\partial {\bf I}[k]}{\partial r_{\rm e}[g]} = \frac{\partial {\bf I}[k]}{\partial \beta[g]}\frac{\partial \beta[g]}{\partial r_{\rm e}[g]} + \frac{\partial {\bf I}[k]}{\partial \varpi[g]}\frac{\partial \varpi[g]}{\partial r_{\rm e}[g]} + \frac{\partial {\bf I}[k]}{\partial {\bf P}[g]}\frac{\partial {\bf P}[g]}{\partial r_{\rm e}[g]}.
\label{eq:chain_rule}$$ Analogously, replacing $r_{\rm e}$ in with ${\rm LWC}$ or $v_{\rm e}$ yields the respective microphysical derivatives. We proceed by expressing the derivatives $\partial \{\beta, \varpi, {\bf P} \} / \partial \{{\rm LWC}, r_{\rm e}, v_{\rm e} \}$. Afterwards, we expand and combine the derivatives $\partial {\bf I} / \partial \{\beta, \varpi, {\bf P} \}$ to express .
For each voxel, the derivatives of $\left(\beta, \varpi, {\bf P}\right)$ with respect to the microphysics are calculated using pre-computed tables $$\begin{aligned}
\frac{\partial \beta}{\partial \rm LWC} = \tilde{{\bf \sigma}}(r_{\rm e},v_{\rm e}), \quad \! \frac{\partial \beta}{\partial r_{\rm e}} = \frac{\tilde{{\bf \sigma}} (r_{\rm e} {+} \varepsilon_{r_{\rm e}}, v_{\rm e}) {-} \tilde{{\bf \sigma}} (r_{\rm e},v_{\rm e})}{\varepsilon_{r_{\rm e}}}, ~\\
\frac{\partial \varpi}{\partial \rm LWC} = 0, \quad \quad \quad \quad \frac{\partial \varpi}{\partial r_{\rm e}} = \frac{\varpi (r_{\rm e} {+} \varepsilon_{r_{\rm e}}, v_{\rm e}) {-} \varpi (r_{\rm e}, v_{\rm e})}{\varepsilon_{r_{\rm e}}}, \\
\frac{\partial {\bf P}}{\partial \rm LWC} = 0, \quad \quad \quad \quad \frac{\partial {\bf P} }{\partial r_{\rm e}} = \frac{{\bf P} (r_{\rm e} {+} \varepsilon_{r_{\rm e}}, v_{\rm e}) {-} {\bf P} (r_{\rm e},v_{\rm e})}{\varepsilon_{r_{\rm e}}}~,
\label{eq:optical_derivatives}
\end{aligned}$$ where $v_{\rm e}$ derivatives are computed analogously to the $r_{\rm e}$ derivatives. Using the shorthand notation , the overall Jacobian is given by a sum of terms $$\partial_g {\bf I}[k] = A_1 + A_2 + A_3 + A_4 + A_5 + A_6.
\label{eq:voxel_derivative}$$ The full expression for each term in Eq. \[\[eq:voxel\_derivative\]\] is given in the [*Appendix*]{}. For example, $$A_1 = - \ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right){\bf I}\left({\bm x}_0, {\bm \omega}_k \right) T\left({\bm x}_0 {\rightarrow} {\bm x}_k\right) \left[ \partial_g\beta \right]. \label{eq:A1}$$
Let us focus on the term $$\begin{aligned}
A_4 &=& \int_{{\bm x}_0}^{{\bm x}_k} \bigg\{\frac{\varpi({\bm x}')}{4\pi} \int_{4\pi}
{\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) \big[ \partial_g {\bf I} \left({\bm x}',{\bm \omega'}\right)\big] {\rm d}{\bm \omega}' \bigg\}
\beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}'. \label{eq:A4}
\end{aligned}$$ This Jacobian term stands out, because it is only term which requires computing the derivative of ${\bf I}$. This derivative is computationally expensive because ${\bf I}$ is computed recursively through the RTE \[Eqs. \[eq:rte\_integral\]-\[eq:J\]\]. In principle, a change in the microphysics of one voxel can recursively affect the radiance at every other voxel. We decompose $\partial_g {\bf I}$ using the diffuse-direct decomposition of $$\begin{aligned}
\partial_g {\bf I} \left({\bm x}',{\bm \omega'}\right) = \partial_g {\bf I}_{\rm d} \left({\bm x}',{\bm \omega'}\right)
+
\delta \left({\bm \omega}' {-} {\bm \omega}_{\rm Sun} \right){\bf F}_{\rm Sun} \big[ \partial_g T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}'\right) \big]. \label{eq:A4_separation}
\end{aligned}$$ At the core our approach for computational efficiency is the assumption that the diffuse light ${\bf I}_{\rm d}$ is less sensitive to slight changes in the microphysical properties of any single voxel $g$. Rather, ${\bf I}_{\rm d}$ is impacted mainly by bulk changes to the over-all volume. Thus, we approximate by keeping ${\bf I}_{\rm d}$ independent of ${\bm \Theta}$ for a single iteration of the gradient computation, i.e., $$\partial_g {\bf I}_{\rm d} {\approx} 0.
\label{eq:approx}$$ This bypasses the complexity of recursively computing $\partial_g {\bf I}_{\rm d}$.
It is important to note that at every iteration, the Jacobian ${\bf \nabla}_{\bf \Theta} {\bf I}[k]$ still [**is**]{} impacted by ${\bf I}_{\rm d}$. This is because ${\bf I}_{\rm d}$ affects ${\bf I}$ through Eq. \[\[eq:I\_seperation\]\], and ${\bf I}$ appears in the terms $A_1,\ldots, A_6$. As the estimated medium properties evolve through iterations, so does ${\bf I}_{\rm d}$ (in [Step 1]{}, above). We just assume during [Step 2]{} that $\partial_g {\bf I}_{\rm d}$ is negligible compared to other terms in Eq. \[\[eq:voxel\_derivative\]\].
Contrary to ${\bf I}_{\rm d}$, the single-scattered component is highly sensitive to changes in the micro-physical properties of a single voxel. We therefore include an exact treatment of single-scattering in the gradient computation (in the [*Appendix*]{}). This is the essence of our numerical optimization approach. It enables tackling multiple-scattering tomography, in practice. Simulation results presented in the following section rely on additional numerical considerations (e.g., initialization, preconditioning, convergence criteria), which are all described in the accompanying [*Appendix*]{}.
Simulations {#sec:simulations}
===========
As mentioned, real data of simultaneous spaceborne multi-angular polarimetric images of clouds does not yet exist, but a mission to supply this data is in the works. Therefore, we use careful simulations to test the approach. We simulate an atmosphere with molecular Rayleigh scattering and liquid water clouds. Rayleigh scattering is taken from the AFGL database [@anderson1986afgl] for a summer mid-latitude atmosphere. Mie tables are pre-computed for $r_{\rm e} \in [4, 25]\,\mu$m and $v_{\rm e} = 0.1$ with $N_{r_{\rm e}} = 100$. The surface is Lambertian with a water-like albedo of $0.05$. For realistic complexity, a Large Eddy Simulation (LES) model [@matheou2014large] was used to generate a cloud field. Each voxel is of size . The LES outputs [@matheou2014large] are clouds with 3D variable ${\rm LWC}$ and 1D (vertically) variable $r_{\rm e}$. A typical value [@yau1996short] of $v_{\rm e}=0.1$ was chosen. Consequently, the present recovery demonstrations recover ${\rm LWC}$ and $r_{\rm e}$ on their respective native LES grid. On the other hand, $v_{\rm e} = 0.1$ is excluded from the unknowns.
From the generated cloud field, two isolated cloudy regions are taken for reconstruction:
1. [Scene A:]{} An atmospheric domain of dimensions $0.64{\times} 0.72{\times} 20~{\rm km}^3$ with an isolated cloud (see synthetic AirMSPI nadir view in Fig. \[fig:noise\]).
2. [Scene B:]{} An atmospheric domain of dimensions $2.42{\times} 2.1{\times} 8~{\rm km}^3$ with several clouds of varying optical thickness (see synthetic AirMSPI nadir view in Fig. \[fig:nadir\_view2\]).
Synthetic measurements rendered with the spatial resolution and angular sampling of AirMSPI [@diner2013airborne], namely, $10$ m pixels and 9 viewing angles: ${\pm}70.5^\circ$, ${\pm}60^\circ$, ${\pm}45.6^\circ$, ${\pm}26.1^\circ$, and $0^\circ$ from zenith, where ${\pm}$ indicates fore- and aft-views along the northbound flight path. Solar zenith angle is $15^\circ$ from nadir in the measurement plane, i.e., $0^\circ$ solar azimuth. We simulate measurements at AirMSPI’s three polarized spectral bands: $\lambda = \left[ 0.47, 0.66, 0.865 \right]~\mu$m. The bandwidths are narrow enough ($\approx$46 nm) to render images using a single representative wavelength per band.
![[Scene A]{} synthesized Stokes image using vSHDOM, before and after the application of a realistic AirMSPI noise model. We show here the [*Bidrectional Reflectance Factor*]{} (BRF) of the nadir view at $\lambda {=}0.67\mu {\rm m}$.[]{data-label="fig:noise"}](airmspi_noise.pdf){width="0.6\linewidth"}
![[Scene B]{} synthesized Stokes using vSHDOM. We show here the BRF of the nadir view at $\lambda {=} 0.67\mu {\rm m}$.[]{data-label="fig:nadir_view2"}](nadir_view2.pdf){width="0.8\linewidth"}
Single scattering albedos for these wavelengths are all within $10^{-4}$ of unity. In other words, and in sharp contrast with the operational Nakajima–King [@nakajima1990determination] bi-spectral non-tomographic retrieval, absorption by droplets plays no role in this demonstration of tomography of cloud microphysics. The measurements are synthesized with realistic noise, according to the AirMSPI data acquisition model (see [*Appendix*]{}).
Qualitative volumetric results of the recovered LWC for [Scene A]{} are shown in Fig. \[fig:lwc1\_true\]. Scatter plot of the recovered LWC and the recovery results of $r_{\rm e}$ for [Scene A]{} are given in Fig. \[fig:scatter1\]. Analogous plots for [Scene B]{} recovery results are given in the [*Appendix*]{}.
For quantitative assessment of the recovery, we use local mean error $\epsilon$, and global bias measures [@aides2013multi] $\vartheta$: $$\begin{aligned}
\epsilon_{\rm LWC} = \frac{\|\hat{\rm LWC} {-} {\rm LWC}\|_1}{\|{\rm LWC}\|_1},~~ \vartheta_{\rm LWC} = \frac{\|\hat{\rm LWC}\|_1 {-} \|{\rm LWC}\|_1}{\|{\rm LWC}\|_1}, ~~
\epsilon_{r_{\rm e}} = \frac{\|\hat{r_{\rm e}} - r_{\rm e}\|_1 }{\|r_{\rm e}\|_1}.
\end{aligned}$$ The quantitative error measures upon convergence for the two scenes are:\
,\
.\
Using a 2.50 GHz CPU, the recovery run-time of cloud properties in [Scenes A,B]{} was ${\sim} 13$ hours and ${\sim} 10$ days, respectively.
Multi-angular tomographic retrieval enables vertical resolution of the droplet effective radius. By contrast, a homogeneous droplet radius is typically retrieved by mono-angular observations fitted to a plane-parallel homogeneous cloud model. The retrieval errors of droplet radii in the demonstrations above are significantly smaller than retrieval errors of a homogeneous droplet radius. The latter can easily exceed 50% in similar conditions to our study i.e, shallow cumuli and illumination conditions (see e.g. [@Seethala2012EvaluatingTS]).
![[Scene A]{} recovery results. \[Left\] Slices of the true cloud generated by LES. \[Right\] Slices of the cloud estimated tomographically using AirMSPI polarized bands.[]{data-label="fig:lwc1_true"}](true_lwc1.pdf "fig:"){width="0.35\linewidth"} ![[Scene A]{} recovery results. \[Left\] Slices of the true cloud generated by LES. \[Right\] Slices of the cloud estimated tomographically using AirMSPI polarized bands.[]{data-label="fig:lwc1_true"}](estimated_lwc1.pdf "fig:"){width="0.35\linewidth"}
![[Scene A]{} []{data-label="fig:scatter1"}](scatter1.pdf "fig:"){width="0.35\linewidth"} ![[Scene A]{} []{data-label="fig:scatter1"}](estimated_reff1.pdf "fig:"){width="0.35\linewidth"}
Summary & Outlook {#sec:conclusion}
=================
We derive tomography of cloud microphysics based on multi-view/multi-spectral polarimetric measurements of scattered sunlight. This novel type of tomography uses, for the first time, 3D polarized RT as the image formation model. We define a model-fitting error function and compute approximate gradients of this function to make the recovery tractable. Demonstration are done on synthetic 3D clouds, based on a Large Eddy Simulation with the effective radius assumed to vary only vertically.
Future work will address the extent to which polarimetric measurements penetrate the cloud and the relation between $r_{\rm e}$ in the [*outer shell*]{} and $r_{\rm e}$ in the cloud [*core*]{}, as defined by Forster et al. [@forster2019cloud]. Furthermore, we will relax the fixed $v_{\rm e}$ assumption that was used in the demonstrations, and thus assess full microphysical retrieval capabilities of polarization measurements. A thorough discussion on these assumptions and their applicability to real-world clouds is given in the [*Appendix*]{}. Moreover, future plans include experimental demonstration and use, while the CloudCT formation orbits.
Lastly, we note that our atmospheric tomography approach herein can be adapted to aerosols, including dense plumes of wild fire smoke, volcanic ash, and dust. Research is ongoing [@forster2019cloud] about such adaptation for satellite data as can be obtained from the multi-view imaging from MISR on Terra and a SWIR view from the collocated MODIS, as well as in the planned CloudCT [@cloudct].
Acknowledgements {#sec:acknowledgements .unnumbered}
================
We thank I. Koren, D. Rosenfeld, A. Aides, D. Diner, L. Di Girolamo, and G. Matheou for support and fruitful discussions. We acknowledge F. Evans and A. Doicu for the online vSHDOM code. The authors are grateful to the US-Israel Binational Science Foundation (BSF grant 2016325) for continuously facilitating our international collaboration. Aviad Levis work was partially supported by the Zuckerman Foundation. Yoav Schechner is a Landau Fellow supported by the Taub Foundation. His work was conducted in the Ollendorff Minerva Center (BMBF). Anthony Davis’ work was carried out at JPL/Caltech, supported by NASA’s SMD/ESD/(RST+TASNPP) and ESTO/AIST programs. Support for Jesse Loveridge’s work from JPL under contract \#147871 is gratefully acknowledged. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 810370: CloudCT).
Jacobian Derivation
===================
In Eq. \[\[eq:voxel\_derivative\]\] of the main text, the Jacobian is written as a sum of six terms $$\partial_g {\bf I}[k] = A_1 + A_2 + A_3 + A_4 + A_5 + A_6.
\label{supeq:voxel_derivative}$$ In this section we expand and describe each of these terms. Using Eqs. \[\[eq:T\]\] and \[\[eq:numerical\_integration\]\], the transmittance derivative is $$\partial_g T\left({\bm x}_1 {\rightarrow} {\bm x}_2\right)= - T\left({\bm x}_1 {\rightarrow} {\bm x}_2\right) \ell_g \left({\bm x}_1 {\rightarrow} {\bm x}_2 \right)\partial_g\beta.
\label{supeq:trans_grad}$$ Then, $$\begin{aligned}
A_1 &=& - \ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right){\bf I}\left({\bm x}_0, {\bm \omega}_k \right) T\left({\bm x}_0 {\rightarrow} {\bm x}_k\right) \left[ \partial_g\beta \right], \label{supeq:A1} \\
A_2 &=& \ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right)\int\limits^{{\bm x}_k}_{{\bm x}_0} \bigg[\frac{\partial_g \varpi} {4\pi} \int\limits_{4\pi}
{\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) {\bf I} \left({\bm x}',{\bm \omega'}\right) {\rm d}{\bm \omega}' \bigg] \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}', \\
A_3 &=& \ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right)\int\limits^{{\bm x}_k}_{{\bm x}_0} \bigg\{\frac{\varpi({\bm x}')}{4\pi} \int\limits_{4\pi}
\big[ \partial_g {\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) \big] {\bf I} \left({\bm x}',{\bm \omega'}\right) {\rm d}{\bm \omega}' \bigg\} \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}', \\
A_4 &=& \int\limits^{{\bm x}_k}_{{\bm x}_0} \bigg\{\frac{\varpi({\bm x}')}{4\pi} \int\limits_{4\pi}
{\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) \big[ \partial_g {\bf I} \left({\bm x}',{\bm \omega'}\right)\big] {\rm d}{\bm \omega}' \bigg\} \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}', \label{supeq:A4}\\
A_5 &=& \ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right) \left[ \partial_g\beta\right] \int\limits^{{\bm x}_k}_{{\bm x}_0} {\bf J}\left({\bm x}', {\bm \omega}_k \right) T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}', \\
A_6 &=& -\ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right) \left[ \partial_g\beta \right] \int\limits^{{\bm x}_k}_{{\bm x}_0} {\bf J}\left({\bm x}', {\bm \omega}_k \right) \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}'.
\label{supeq:A6}
\end{aligned}$$ Note that ${\bf I} {\left({\bm x},{\bm \omega} \right)}$ and ${\bf J} {\left({\bm x},{\bm \omega} \right)}$ are computed in [Step 1]{} and are therefor ready for use when computing $A_1,A_2,A_3,A_5$ and $A_6$. Furthermore, $\ell_g \left({\bm x}_0 {\rightarrow} {\bm x}_k \right){=}0$ for any voxel that is not on the LOS of pixel $k$. Therefore, the terms $A_1,A_2,A_3,A_5,A_6$ are computed using a single path tracing ${\bm x}_k {\rightarrow} {\bm x}_0$.
We now give special attention to $A_4$ in Eq. \[\[supeq:A4\]\]. Using the diffuse-direct decomposition of , we decompose as $$\begin{aligned}
A_4 & =& \int\limits^{{\bm x}_k}_{{\bm x}_0} \bigg\{\frac{\varpi({\bm x}')}{4\pi} \int\limits_{4\pi}
{\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) \big[ \partial_g {\bf I}_{\rm d} \left({\bm x}',{\bm \omega'}\right) \big] {\rm d}{\bm \omega}' \bigg\} \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}' \nonumber \\
& + &
\int\limits^{{\bm x}_k}_{{\bm x}_0} \bigg\{\frac{\varpi({\bm x}')}{4\pi} \int\limits_{4\pi}
{\bf P}\left({\bm x}',{\bm \omega}_k{\cdot}{\bm \omega'}\right) \delta \left({\bm \omega}' {-} {\bm \omega}_{\rm Sun} \right){\bf F}_{\rm Sun} \big[ \partial_g T\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}'\right) \big] {\rm d}{\bm \omega}' \bigg\} \beta({\bm x}') T\left({\bm x}' {\rightarrow} {\bm x}_k\right) {\rm d}{\bm x}'.
\label{supeq:A4_separation}
\end{aligned}$$ The first term in is based on $\partial_g {\bf I}_{\rm d}$, ie., a derivative of the diffuse (high order scattering) component. Herein lies a recursive complexity. In principle, a differential change in the microphysics of one voxel can recursively affect the radiance at every other voxel, and this affects all the pixels. To make calculations numerically efficient, we approximate . The approximation assumes that relative to other components in the Jacobian, ${\bf I}_{\rm d}$ is less sensitive to a differential changes in the microphysical properties at voxel $g$. Thus, is approximated by keeping ${\bf I}_{\rm d}$ independent of ${\bm \Theta}$ for a single iteration of the gradient computation, i.e, $$\partial_g {\bf I}_{\rm d} {\approx} 0 \;.
\label{supeq:dId}$$
The second term in is based on differentiation of the direct component. This is straight-forward to compute using . Consequently, using Eq. \[\[supeq:dId\]\] and the definition of ${\bf I}_{\rm Single}\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k \right)$ in , the term $A_4$ in is approximated by $$A_4 \approx \tilde{A_4} = \left[ \partial_g\beta\right] \int\limits^{{\bm x}_k}_{{\bm x}_0} \ell_g \left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' \right){\bf I}_{\rm Single}\left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_k \right){\rm d}{\bm x}'.
\label{supeq:direct_derivative}$$ The term $\ell_g \left({\bm x}_{\rm Sun} {\rightarrow} {\bm x}' \right)$ in contributes to voxels [*outside*]{} of the LOS. The integral in $\tilde{A_4}$ is computed with a [*broken-ray*]{} [@florescu2011inversion] path ${\bm x}_k {\rightarrow} {\bm x}' {\rightarrow} {\bm x}_{\rm Sun}$, as illustrated in Fig. \[fig:domain\].
Using Eqs. \[\[eq:rte\_integral\],\[supeq:trans\_grad\],\[supeq:A1\],\[supeq:A6\]\], $A_1$ and $A_6$ are combined to $$A_{1,6} = A_1 + A_6 = - \left[ \partial_g\beta\right] {\bf I} \left({\bm x}_g, {\bm \omega}_k \right).
\label{supeq:relation}$$ Overall, in our iterative procedure, we approximate the Jacobian in by $$\begin{aligned}
\partial_g {\bf I}[k] = A_{1,6} + A_2 + A_3 + \tilde{A_4} + A_5.
\label{supeq:approximate_jacobian}
\end{aligned}$$ Equations \[\[supeq:A1\]\]–\[\[supeq:direct\_derivative\]\] formulate the Jacobian in terms of a voxel grid (zero-order interpolation). However, in practice we use a trilinear interpolation kernel $K$ in , consistent with vSHDOM internal interpolation [@evans1998spherical].
Measurement Noise
=================
The inverse problem defined in the main text is formulated in terms of measured Stokes vectors \[Eq. \[eq:meas\_vec\]\]. However, Stokes vectors are not measured directly. Rather, they are derived from intensity measurements taken through filters. The raw intensity measurements are noisy. Noise is dominated by Poisson photon noise, which is independent across different raw measurements. However, the estimation of Stokes components [*from*]{} independent intensity measurements yields noise which is correlated across the components of the Stokes vector, per-pixel. In this section, we describe the synthesis model we employ to generate realistic noise in simulations. Our synthesis is based on the AirMSPI [@van2018calibration] sensor model. Furthermore, we derive the expression for ${\bf R}$, which we use in the recovery process (Eq. \[\[eq:minimization\]\] in the main text).
AirMSPI measures a modulated intensity signal at $N_{\rm sub} {=} 23$ subframes. Define a normalized frame which spans the unitless integration time interval $\psi \in [-0.5, 0.5]$. Denote the temporal center and span of each subframe as $\psi_l$ and $\Delta \psi = 1/N_{\rm sub}$, respectively (Fig. \[fig:subframe\]). Based on the sensing process described in Ref. [@van2018calibration], define the following modulation function, whose parameters are given in Table \[table:mspi\_params\]: $$M[l] = J_0[\kappa(\psi_l)] + \frac{1}{3}\left(\frac{\pi \Delta \psi}{2} \right)^2\gamma_0^2(\lambda) \bigg\{ J_2[\kappa(\psi_l)] - \cos[2(\pi \psi_l - \eta)]J_0[\kappa(\psi_l)] \bigg\},
\label{eq:Ml}$$ with $$\kappa(\psi_l) = -2 \gamma_0(\lambda) \sin(\pi \psi_l - \eta)\sqrt{1+\cot^2(\pi \psi_l - \eta)}.$$
(1,0.1809971) (0,0)[![A normalized frame spans the interval $[-0.5,0.5]$, evenly divided into $N_{\rm sub}$ subframes.[]{data-label="fig:subframe"}](subframe.pdf "fig:"){width="\unitlength"}]{} (-0.0025119,0.00700272)[(0,0)\[lt\]]{} (0,0)[![A normalized frame spans the interval $[-0.5,0.5]$, evenly divided into $N_{\rm sub}$ subframes.[]{data-label="fig:subframe"}](subframe.pdf "fig:"){width="\unitlength"}]{} (0.48146543,0.15464435)[(0,0)\[lt\]]{} (0,0)[![A normalized frame spans the interval $[-0.5,0.5]$, evenly divided into $N_{\rm sub}$ subframes.[]{data-label="fig:subframe"}](subframe.pdf "fig:"){width="\unitlength"}]{} (0.03304296,0.15545671)[(0,0)\[lt\]]{} (0.94477292,0.0058503)[(0,0)\[lt\]]{} (0,0)[![A normalized frame spans the interval $[-0.5,0.5]$, evenly divided into $N_{\rm sub}$ subframes.[]{data-label="fig:subframe"}](subframe.pdf "fig:"){width="\unitlength"}]{} (0.45242054,0.0128213)[(0,0)\[lt\]]{}
[$\gamma_0(470 {\rm nm})$]{} [$\gamma_0(660 {\rm nm})$]{} [$\gamma_0(865 {\rm nm})$]{} [$\xi(470 {\rm nm})$]{} [$\xi(660 {\rm nm})$]{} [$\xi(865 {\rm nm})$]{} [$\eta$]{}
------------------------------ ------------------------------ ------------------------------ ------------------------- ------------------------- ------------------------- ------------
4.472 3.081 2.284 1.0 0.27 0.03 0.009
: Modulation parameters [@van2018calibration] used for synthesis of AirMSPI measurements.[]{data-label="table:mspi_params"}
Here $J_0$, $J_2$ are the Bessel functions of the first kind of order 0 and 2, respectively. Denote by $\xi(\lambda)$ a wavelength-dependent ratio, which is drawn from quantum efficiencies and spectral bandwidths[^5] of each AirMSPI band (Table \[table:mspi\_params\]). Using simulated Stokes vectors derived by vSHDOM, AirMSPI measurements are synthesised as passing through two polarization analyzing filters [@van2018calibration]. As defined in Eq. \[\[eq:forward\_stokes\]\] in the main text, ${\bf I}[k]$ is the Stokes vector in pixel $k$. Correspondingly the intensity is $I[k]$, while $Q[k],U[k]$ are the polarized components. Measurements $l$ through the two filters of AirMSPI are modeled by $$\begin{aligned}
\label{supeq:mspi_meas1}
&d_0[l,k] = \xi(\lambda) \big(I[k] + M[l]Q[k] \big) \\
&d_{45}[l,k] = \xi(\lambda) \big(I[k] + M[l]U[k] \big),
\label{supeq:mspi_meas2}
\end{aligned}$$ where $M[l]$ and $\xi(\lambda)$ are given in Eq. \[\[eq:Ml\]\] and Table \[table:mspi\_params\], respectively. The units of $d$ are Watts. Define ${\bf d}[k] = \left(d_0[1,k],...,d_0[N_{\rm sub},k], d_{45}[1,k],...,d_{45}[N_{\rm sub},k] \right)^\top$. In matrix from, the transformations by Eqs. \[\[supeq:mspi\_meas1\]-\[supeq:mspi\_meas2\]\] are written using a single $46 {\times} 3$ modulation matrix ${\bf M}$ $${\bf d}[k] = {\bf M} {\bf I}[k].$$
Detection is by a camera which generates photo-electrons in each pixel well. The relation between $d_0[l,k]$ or $d_{45}[l,k]$ and the [*expected*]{} unit-less number of photo-electrons in the pixel is given by a gain $G$. The number of photo-electrons is random ([*Poissonian*]{}) around this expected value. The vector of simulated electron counts is thus synthesized by a Poisson process $${\bf e}[k] \sim Poisson \Big\{ {\rm round} \left( G \cdot {\bf d}[k] \right) \Big\} = Poisson \Big\{ {\rm round} \left( G \cdot {\bf M} {\bf I}[k] \right) \Big\}.
\label{supeq:measurement_noise}$$ The gain $G$ is chosen to let the maximum signal at each camera view (i.e. maximum over pixels, wavelengths and subframe measurements) reach the maximum [*full-well depth*]{} of $200,\!000$ electrons, consistent with AirMSPI specifications. synthesizes raw AirMSPI signals including noise (Fig. \[fig:noise\]). The synthesized AirMSPI signals, including this noise, are now used as inputs to the calculation of measured Stokes vectors in each pixel and viewpoint. The vector of electron counts ${\bf e}[k]$ in each pixel $k$ is transformed into Stokes synthetic data \[Eq. \[eq:meas\_vec\]\] using a $3 {\times} 46$ demodulation matrix ${\bf W}$ $${\bf y}_{\bf I}[k] = ({\bf M}^\top {\bf M})^{-1}{\bf M}^\top{\bf e}[k] = {\bf W} {\bf e}[k].
\label{supeq:y}$$ The vectors ${\bf y}_{\bf I}[k]$ form the data for tomographic analysis.
Our tomographic analysis takes into account the noise properties, including noise correlation. As we now show, the measurement model \[\[supeq:y\]\] yields correlated noise of different Stokes components. Thus, ${\bf R}^{-1}$ () is not diagonal. Denote the diagonal co-variance matrix of the photo-electron readings by ${\bf C}_{\rm e}^{-1}{=}\text{diag}\big({\bf e}\big)$. Let $\mathcal{I}_{46{\times}46}$ denote the [*Identity*]{} matrix. The signal is generally dominated by unpolarized multiply-scattered background light. Relative to it, the magnitude of the modulated polarization signal is small. Thus, per pixel $k$, the diagonal matrix ${\bf C}_{\rm e}^{-1}[k]$ is approximately constant with a global weight $${\bf C}_{\rm e}^{-1}[k] \approx \alpha[k] \mathcal{I}_{46{\times}46}.
\label{supeq:C}$$ Using Eqs. \[\[supeq:y\],\[supeq:C\]\] for each pixel, the Stokes co-variance matrix is $${\bf C}^{-1}[k] = {\bf M}^\top {\bf C}_{\rm e}^{-1}[k] {\bf M} \approx \alpha[k]{\bf M}^\top {\bf M}.
\label{supeq:pixel_noise}$$ A maximum-likelihood estimator corresponding to a Poisson process should have a weight $\alpha[k] \propto 1/\|{\bf e}\|_1$, to account for higher photon noise in brighter pixels. In simulations, however, we found that $\alpha[k]=1$ worked better. This is perhaps due to richer information carried by denser cloud regions, i.e. brighter pixels. Overall the expression we minimize in is $$\hat{{\bf \Theta}} = \underset{{\bf \Theta}}{\arg\min} \sum_{k=1}^{N_{\rm meas}} \left({\bf I}[k] - {\bf y}_{\bf I}[k] \right)^\top {\bf M}^\top {\bf M} \left({\bf I}[k] - {\bf y}_{\bf I}[k] \right),
\label{supeq:minimization_noise}$$ i.e. ${\bf R}^{-1} = {\bf M}^\top {\bf M}$.
Numerical considerations {#subsec:numerical}
========================
In this section we describe numerical considerations that stabilize the recovery.
Hyper-parameters
----------------
Our code requires the choice of hyper-parameters for rendering with vSHDOM [@pincus2009computational] in [Step 1]{} and optimization with [*scipy*]{} L-BFGS [@zhu1997algorithm; @scipy] in [Step 2]{}. Table \[table:numerical\_params\] summarizes the numerical parameters used in our simulations.
[$N_{\mu}$]{} [$N_{\phi}$]{} [splitting accuracy]{}
--------------- ---------------- ------------------------
8 16 0.1
: Numerical parameters. For vSHDOM parameter definitions, see Ref. [@pincus2009computational]. For L-BFGS parameter definitions, see Ref. [@scipy].[]{data-label="table:numerical_params"}
[gtol]{} [gtol]{} [maxls]{}
------------------ ------------------ -----------
$1{\rm e}^{-16}$ $1{\rm e}^{-16}$ 30
: Numerical parameters. For vSHDOM parameter definitions, see Ref. [@pincus2009computational]. For L-BFGS parameter definitions, see Ref. [@scipy].[]{data-label="table:numerical_params"}
Preconditioning
---------------
Multivariate optimization can suffer from ill-conditioning due to different scales of the sought variables. This is expected when recovered variables represent different physical quantities with different units and orders of magnitude. A preconditioning of the update rule in takes the following form $${\bf \Theta}_{b+1} = {\bf \Theta}_{b} - \chi_b {\bf \Pi}^{-1} \nabla_{\bf \Theta} \mathcal{D}\left({\bf I}_{\bf \Theta},{\bf y} \right),$$ where we apply a diagonal scaling matrix ${\bf \Pi}$ ([*Jacobi*]{} preconditioner) to scale the different physical variables $({\rm LWC}, r_{\rm e})$. Thus, ${\bf \Pi}$ takes the form $${\bf \Pi} = {\rm diag}\big(\Pi_{\rm LWC},~\Pi_{r_{\rm e}},....,\Pi_{\rm LWC},~\Pi_{r_{\rm e}}\big).$$ In our tests, we use $\Pi_{\rm LWC}$ = 15 and $\Pi_{r_{\rm e}}$ = 0.01 to scale the parameters to a similar magnitude and closer to unity upon initialization.
Initialization
--------------
The recovery is initialized by the estimation of a cloud voxel mask, which bounds the cloud 3D shape. The 3D shape bound of the cloud is estimated using [*Space-Carving*]{} [@veikherman2014clouds]. Space-carving is a geometric approach to estimate a bound to 3D shape via multi-view images. The following steps are preformed in our space-carving algorithm
1. Each image is segmented into [*potentially cloudy*]{} and [*non-cloudy*]{} pixels (we use a simple radiance threshold).
2. From each camera viewpoint, each [*potentially cloudy*]{} pixel back-projects a ray into the 3D domain. Voxels that this ray crosses are voted as potentially cloudy.
3. Voxels which accumulate “cloudy” votes in at least 8 out of the 9 AirMSPI viewpoints are marked as cloudy.
Outside of the shape bound, LWC = 0 throughout iterations. Within the estimated cloud-shape bound, the volume content is initialized as homogeneous with , and $v_{\rm e}$ = 0.1. Then, inside of the shape-bound, {LWC,$r_{\rm e}$,$v_{\rm e}$} change throughout iterations, possibly diminishing LWC to very small values.
Convergence
-----------
Our approach alternates between [Step 1]{} (RTE rendering) and [Step 2]{} (approximate gradient) until convergence (Fig. \[fig:block\_diagram\]). The convergence criteria are dictated by the L-BFGS step: at each iteration, the relative change to the forward model and its gradient are compared to the [*ftol*]{} and [*gtol*]{} parameters (see Table \[table:numerical\_params\] for values used). See [*SciPy*]{} documentation [@scipy] for exact description of the L-BFGS stopping criteria.
Qualitative Results: [Scene B]{}
================================
Qualitative volumetric results of the recovered LWC for [Scene B]{} are shown in Fig. \[fig:lwc2\_true\]. A scatter plot of the recovered LWC and the recovery results of $r_{\rm e}$ for [Scene B]{} are given in Fig. \[fig:scatter2\].
![[Scene B]{} recovery results. \[Left\] Slices of the true LES generated region. \[Right\] Slices of the estimated region.[]{data-label="fig:lwc2_true"}](true_lwc2.pdf "fig:"){width="35.00000%"} ![[Scene B]{} recovery results. \[Left\] Slices of the true LES generated region. \[Right\] Slices of the estimated region.[]{data-label="fig:lwc2_true"}](estimated_lwc2.pdf "fig:"){width="35.00000%"}
![[Scene B]{} \[Left\] Scatter plot of the estimated vs. true LWC. The fit correlation is 0.96. \[Right\] Recovery results of the 1D effective radius[]{data-label="fig:scatter2"}](scatter2.pdf "fig:"){width="35.00000%"} ![[Scene B]{} \[Left\] Scatter plot of the estimated vs. true LWC. The fit correlation is 0.96. \[Right\] Recovery results of the 1D effective radius[]{data-label="fig:scatter2"}](estimated_reff2.pdf "fig:"){width="35.00000%"}
Spatial Variation of The Effective Radius {#sec:discussion}
=========================================
In nature, generally the droplet effective radius $r_{\rm e}$ and variance $v_{\rm e}$ vary in 3D. However, operational remote sensing algorithms, which rely on 1D RT and plane-parallel cloud models, retrieve a single value for $r_{\rm e}$ (and for $v_{\rm e}$), for each cloudy pixel. This occurs both in bi-spectral [@nakajima1990determination; @EwaldEtAl_19] and polarimetric [@BreonGoloub_98; @AlexandrovEtAl_12] techniques. In these approaches, it is always uncertain which portion of the cloud the retrieved quantity corresponds to, because light penetrates into the cloud and simultaneously scatters from different depths inside it. In polarization analysis of plane-parallel cloud models, it is often assumed that the retrieved microphysical parameters correspond approximately to an optical depth of unity. At any rate, this uncertainty complicates the interpretation of retrieved values in studies which rely on them.
The mathematical approach of the paper is formulated for 3D variation of all the required fields: ${\rm LWC},r_{\rm e},v_{\rm e}$. As Fig. 3 in the main paper shows, polarization is sensitive to $r_{\rm e}$ of any voxel which scatters sunlight towards the camera. Moreover, the formulation explicitly models and seeks spatially varying microphysics, using multi-angular data. We confidently anticipate the same sensitivity to $v_{\rm e}$. The demonstrations in the simulations used a representation in which $r_{\rm e}$ varies vertically, not horizontally. This is more general than the operational methods mentioned above, yet more degenerate than full 3D heterogeneity. We now discuss the implication of such a representation.
Textbook cloud physics (e.g., [@yau1996short]) is based on the mental picture of a parcel of moist air containing a certain number of cloud condensation nuclei that is ascending vertically in the buoyancy-driven part of the convective cycle. Since temperature and pressure are strongly stratified environmental quantities, moist adiabatic thermodynamics thus predict a vertically-varying droplet size distribution, at least in the so-called “convective core” of the cloud. For the present study, this restriction of microphysical variability to the vertical dimension only applies to both $r_{\rm e}$ and $v_{\rm e}$.
There is compelling evidence that the horizontal variability $r_{\rm e}$ is indeed small over a cloud scale. This evidence comes from in-situ aircraft observations of shallow cumulus [@BlythLatham_91; @FrenchEtAl_00; @GerberEtAl_08], modelling studies [@KhainEtAl_19] and theory [@PinskyKhain_18]. However, there are also select observations of monsoonal clouds [@BeraEtAl_16] and theoretical arguments [@PinskyKhain_18] that suggest there is a sharp gradient in the droplet effective radius in the very outer shell of the clouds. If this is the case, then a representation having vertical-only variation of $r_{\rm e}$ loses validity at the outer shell. This may cause bias in retrievals based on polarimetry. The reason is that polarization signals are dominated by single-scattering, which is most likely to occur at shallow depth in the cloud.
The value of $v_{\rm e}$ can also vary significantly across different environmental conditions. This is seen in research flights including in-situ measurements [@CostaEtAl_00; @LuEtAl_08; @MartinsSilva-Dias_09; @HudsonEtAl_12; @PandithuraiEtAl_12]. Moreover, in LES simulations of shallow cumulus clouds with bin microphysics, $v_{\rm e}$ might range from 0.01 to 0.26 [@Igelvan-den-Heever_17]. The core of a cloud tends to have a low effective variance as condensation is the dominant process there [@LuSeinfeld_06; @Igelvan-den-Heever_17]. Cloud edges, in contrast, experience also evaporation and entrainment mixing, as the cloud is diluted by environmental air [@WangEtAl_11]. This tends to increase $v_{\rm e}$. If the cloud has precipitation, spatial variability of $v_{\rm e}$ increases [@MilbrandtYau_05].
These points show that, on the one hand, the approximations in the demonstrations are often reasonable. On the other hand, it is indeed worth representing cloud microphysical parameters as functions in 3D, then retrieving them in tomography, to push the frontier of cloud physics research. Retrieving a large number of degrees of freedom can be managed better by using more information from diverse sources. One option is to include additional sources of measurements, e.g., by using a combination of the AirMSPI [@diner2013airborne] and Research Spectro-Polarimeter (RSP) [@cairns1999research] airborne instruments. Another option is to introduce tailored regularization schemes, which mathematically express the natural trends of horizontal variability mentioned above. The 3D tomographic approach presented in the paper is a significant enabler for probing such questions. It offers more flexibility than current operational analyses, which are largely based on 1D RT and bulk retrieved values for a whole cloud.
[^1]: Now at the Computing and Mathematical Sciences Department, California Institute of Technology, Pasadena, CA, 91125.
[^2]: Shallow convective clouds in the planetary boundary layer are also overlooked due to their unresolved scale in low-resolution sensors.
[^3]: The size integral of is in practice terminated at $r_{\rm max}$ = 70 $\mu$m.
[^4]: For a full mathematical description, see [@bohren2008absorption].
[^5]: For the exact calculation of the ratio see Eq. \[24\] of [@van2018calibration].
|
---
abstract: |
We propose a new diagram, the Kinematic-Excitation diagram (KEx diagram), which uses the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} line ratio and the [\[O[III]{}\] $\lambda$5007]{}emission line width ($\sigma_{{[O\,{\footnotesize III}]}}$) to diagnose the ionization source and physical properties of the Active Galactic Nuclei (AGNs) and the star-forming galaxies (SFGs). The KEx diagram is a suitable tool to classify emission-line galaxies (ELGs) at intermediate redshift because it uses $only$ the [\[O[III]{}\] $\lambda$5007]{} and [[H$\beta$]{}]{} emission lines. We use the SDSS DR7 main galaxy sample and the Baldwin$-$Phillips$-$Terlevich (BPT) diagnostic to calibrate the diagram at low redshift. We find that the diagram can be divided into 3 regions: one occupied mainly by the pure AGNs (KEx-AGN region), one dominated by composite galaxies (KEx-composite region), and one contains mostly SFGs (KEx-SFG region). AGNs are separated from SFGs in this diagram mainly because they preferentially reside in luminous and massive galaxies and have high [\[O[III]{}\]]{}/[[H$\beta$]{}]{}. The separation of AGN from star-forming galaxies is even cleaner thanks to the additional 0.15/0.12 dex offset in $\sigma_{{[O\,{\footnotesize III}]}}$ at fixed luminosity/stellar mass.
We apply the KEx diagram to 7,866 galaxies at 0.3 $<$ z $<$ 1 in the DEEP2 Galaxy Redshift Survey, and compare it to an independent X-ray classification scheme using $Chandra$ observation. X-ray AGNs are mostly located in the KEx-AGN region while X-ray SFGs are mostly located in the KEx-SFG region. Almost all of Type1 AGNs lie in the KEx-AGN region. These confirm the reliability of this classification diagram for emission line galaxies at intermediate redshift. At z$\sim$2, the demarcation line between star-forming galaxies and AGNs should shift 0.3 dex higher in $\sigma_{{[O\,{\footnotesize III}]}}$ to account for evolution.
author:
- 'Kai Zhang, Lei Hao'
title: A New Diagnostic Diagram of Ionization Source for High Redshift Emission Line Galaxies
---
Introduction
============
The diagnostic diagrams are major tools to understand the nature of galaxies. They are crucial in evaluating the galaxy evolution scenarios such as the cosmic accretion and star-formation histories. The most widely-used diagram is the BPT (Baldwin, Philips & Terlevich 1981) or VO87 diagram (Veilleux & Osterbrock 1987). The advent of spectroscopic sky surveys like the SDSS and the photoionization models (Ferland et al. 1998) make the classification observationally constrained and theoretically understood. Kewley et al. (2001) used a variety of [H[II]{}]{} region photoionization models to give a theoretical star-forming galaxy boundary on the BPT diagram. The sources above this line are unlikely to be ionized by stars. Kauffmann et al. (2003) used the SDSS main galaxies sample to map their detailed distribution on the BPT diagram, and proposed that the right branch of the seagull shape distribution are all AGNs. The sources lie between the two dividing lines are called composite galaxies because their gas may be ionized by AGN and SF at the same time. Kewley et al. (2006) further proposed low ionization line criteria for separating Seyfert2s and LINERs. Other refinement of the classification are proposed by many authors (e.g., Stasińska et al. 2006; Cid Fernandes et al. 2010, 2011).
Our understanding of the BPT diagram is very comprehensive. The y-axis of the BPT diagram reflects mainly the ionization parameter while the x-axis is mostly determined by the metallicity (Storchi-Bergmann, Calzetti, & Kinney 1994, Raimann et al. 2000, Denicoló et al. 2002; Pettini & Pagel 2004; Stasińska et al. 2006; Groves et al. 2004a,b; Groves et al. 2006; Kewley & Ellison 2008). The distinguishing power of the BPT diagram relies on the fact that the AGN radiation is harder than star-forming galaxies at similar stellar mass, AGNs have higher ionization parameters, and AGNs reside exclusively in massive metal-rich galaxies (Kauffmann et al. 2003, Groves et al. 2006).
The BPT diagram, however, suffers a few limitations. It needs at least 4 lines ([\[O[III]{}\] $\lambda$5007]{}, [[H$\beta$]{}]{}, [\[N[II]{}\] $\lambda$6583]{}, [[H$\alpha$]{}]{}) to make a classification. When the strength of these 4 lines are similar, the more signal-to-noise ratio cuts imposed, the more sources are missed. The [\[N[II]{}\] $\lambda$6583]{} and [[H$\alpha$]{}]{} emission line will shift out of the optical wavelength range when the redshift is greater than 0.4, making the classification diagram futile for higher redshift sources with optical spectrum only.
With spectroscopic sky surveys pushing to higher redshift and fainter luminosities, the need for a good classification diagram for emission line galaxies (ELGs) at higher redshift is compelling. Some efforts have been made to develop diagnostic diagrams with spectral features in narrower wavelength range. Tresse et al. (1996) and Rola et al. (1997) proposed to use the of EW([\[O[II]{}\] $\lambda$3727]{}) (equivalent width of [\[O[II]{}\] $\lambda$3727]{}), EW([\[O[III]{}\] $\lambda$5007]{}) and EW([[H$\beta$]{}]{}) for galaxy classification. Stasińska et al. (2006) studied using [\[O[II]{}\] $\lambda$3727]{} for galaxy classification, and proposed a method that uses 4000Å break: $D_n(4000)$, EW([\[O[II]{}\] $\lambda$3727]{}), and EW([\[Ne[III]{}\] $\lambda$3870 ]{}) (DEW diagram) to select pure AGNs with z$<$1.3 using only the optical spectra. Trouille et al. (2011) proposed to use $g-z$, [\[Ne[III]{}\]]{}, and [\[O[II]{}\]]{} to clearly separate AGNs from star-forming galaxies at intermediate redshift. A fruitful way to push to high redshift is to retain the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} while replacing the [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} with other quantities like H band absolute magnitude (Weiner et al. 2006), [\[O[II]{}\]]{}/[[H$\beta$]{}]{} (Lamareille 2010), U-B color (Yan et al. 2011), or stellar mass (Juneau et al. 2011, 2013, Mass-Excitation diagnostic, MEx). Marocco et al. (2011) also used $D_n(4000)$ vs [\[O[III]{}\]]{}/[[H$\beta$]{}]{} for high-z galaxies classification. These methods take advantage of the fact that AGNs reside in massive, red galaxies in local universe, and are in general efficient in separating pure AGNs and star-forming galaxies. The composite galaxies, however, are mixed with Seyfert2s or star-forming galaxies on these diagrams.
The emission line velocity dispersion ($\sigma$) may trace the kinematics of different components in AGNs and star-forming galaxies. [\[O[III]{}\]]{} in AGN comes from the narrow line region, which better traces the bulge kinematics (e.g. Ho 2009). [\[O[III]{}\]]{} in star-forming galaxies mainly comes from the [H[II]{}]{} regions, which locate mainly in the disk. The kinematic of the bulge/disk is expected to be different. Catinella et al. (2010) showed that the velocity dispersion is different for bulge and disk dominated galaxies at given baryonic mass. Besides, emission lines of AGN have extra broadening due to outflows (Greene & Ho 2005; Zhang et al. 2011). In principle, we could use the width of the narrow emission lines as a proxy of the influence of bulge potential for AGNs/star-forming galaxies classification. Following the idea of simplifying the BPT diagram as introduced in last paragraph (Weiner et al. 2006; Lamareille 2010; Yan et al. 2011; Juneau et al. 2011, 2013), and the idea of different kinematics of AGN and star-forming galaxies, we propose to replace the [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} in the BPT diagram with $\sigma_{{[O\,{\footnotesize III}]}}$ (or $\sigma_{gas}$ in general) to separate AGNs from star-forming galaxies at high redshift. In Section 2, we give descriptions of the data we use. In Section 3, a new diagnostic diagram: the Kinematics-Excitation Diagram (KEx diagram) is proposed, and we explain why it works and calibrate it at z$<$0.3. Section 4 gives the calibration of the KEx diagram at 0.3$<$z$<$1, and Section 5 gives the calibration at z$\sim$2. Discussion is given in Section 6. We use a cosmology with $H_{\rm 0}$ = 70 kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}$ = 0.3, and $\Omega_{\rm \Lambda}$ = 0.7 throughout this paper.
Sample and Measurements
=======================
Low-redshift data
-----------------
We start from the main galaxy sample (Strauss et al. 2002) of the Sloan Digitial Sky Survey Data Release 7 (Abazajian et al. 2009). The sample is complete in r-band Petrosian magnitude between 15 and 17.77 over 9380 $deg^2$. We limit the redshift range to z $<$ 0.33, and there are 835,410 spectroscopic galaxies. To properly measure the emission lines, we use the scheme developed in Hao et al. (2005) to subtract the stellar absorptions.They used several hundreds SDSS low redshift pure absorption galaxies to construct the PCA eignspectra, and used the first 8 eignspectra to fit the continuum. An A-star template is added to represent young stellar population. A power law is also added when fitting AGN spectra. The continuum-subtracted line emissions are left for refined line fitting. We use 1-gaussian function to fit the [\[O[II]{}\] $\lambda$3727]{}, [[\[O[I]{}\]]{}$\lambda$6300]{}, [[H$\beta$]{}]{}, [\[O[III]{}\] $\lambda$5007]{}, [\[N[II]{}\] $\lambda$6548]{}, [[H$\alpha$]{}]{}, [\[N[II]{}\] $\lambda$6583]{}, and [\[S[II]{}\] $\lambda \lambda$6717, 6731]{} (Hereafter [\[O[II]{}\]]{}, [[\[O[I]{}\]]{}]{}, [[H$\beta$]{}]{}, [\[O[III]{}\]]{}, [\[N[II]{}\] $\lambda$6548]{}, [[H$\alpha$]{}]{}, [\[N[II]{}\] $\lambda$6583]{}, and [\[S[II]{}\]]{}) respectively. The $\sigma$ (line width, 1/2.35 Full Width at Half Maximum in [$\mathrm{km~s^{-1}}$]{}) of the gaussian profile for [\[O[III]{}\] $\lambda$5007]{} is denoted as $\sigma_{{[O\,{\footnotesize III}]}}$. The line ratio of [\[N[II]{}\] $\lambda$6583]{}/[\[N[II]{}\] $\lambda$6548]{} is fixed to 3 and their profile and center are tied to be the same. In addition, we fit [[H$\alpha$]{}]{} and [[H$\beta$]{}]{} a second time adding one broad gaussian to account for possible [[H$\alpha$]{}]{} and [[H$\beta$]{}]{} broad lines. The lower limit of $\sigma$ of the broad component is 400[$\mathrm{km~s^{-1}}$]{}. The typical FWHM of [[H$\alpha$]{}]{} broad component of Type1 AGNs is larger than 1200[$\mathrm{km~s^{-1}}$]{} (Hao et al. 2005). We regard the broad [[H$\alpha$]{}]{} component to be prominent if a F-test suggests the improvement is significant at 3$\sigma$ level. The intrinsic velocity dispersion $\sigma_{int}$ of the emission lines is obtained by subtracting the instrument resolution of $\sim56km/s$ using $\sigma_{int}^2=\sigma_{obs}^2-\sigma_{Instrument}^2$. The errors of the $\sigma$ and line strength are obtained by the MPFIT package which only includes the fitting errors (Markwardt 2009). We perform a simple test to check how well we can measure the line width. We add a gaussian to a continuum with a given equivalent width (EW). The $\sigma$ of the gaussian is the combination of emission line intrinsic width $10^{1.8}=63.1{\ensuremath{\mathrm{km~s^{-1}}}}$ (typical star-forming galaxy) and the instrumental resolution of 56[$\mathrm{km~s^{-1}}$]{}. Random errors are added according to $S/N$=3, 5, 7, 10. We fit the emission line using the MPFIT package, and measure the error of $\sigma$ by comparing the measured value with the input one. The simulation is run for 500 times. At EW=3 (typical value for $-0.5<log {[O\,{\footnotesize III}]}/{{\rm H\ensuremath{\beta}}}<1$ star-forming galaxy) and $S/N$=3, 5, 7, 10, the errors in emission line width: $\sigma$ are 0.19 dex, 0.08 dex, 0.06 dex, and 0.04 dex. For the worst case: EW=3, S/N=3, the error in $\sigma$ is 0.19 dex. For sources with higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{} and higher emission line width, the measurement are more reliable.
Intermediate-redshift data
--------------------------
Our intermediate-redshift galaxy sample is based on observations from the DEEP2 Galaxy Redshift Survey (hereafter DEEP2; Davis et al. 2003; Newman et al. 2013). [^1] The DEEP2 survey has a limiting magnitude of $R_{AB}$ = 24.1, and it covers 3.2 $deg^2$ spanning 4 separate fields on the sky. The spectra span a wavelength range of 6500-9100Å at a spectral resolution of R$\sim$5000. The DEEP2 DR4 include 52,989 galaxies. For our study, we limit the redshift range to be 0.32 $<$ z $<$ 0.82, to ensure the detection of [[H$\beta$]{}]{} and [\[O[III]{}\]]{} in the DEEP2 wavelength coverage. The sample size is cut to 12,739 galaxies. The spectra are obtained with the DEIMOS spectrograph (Faber et al. 2003) at the Keck Observatory and reduced with the pipeline[^2] developed by the DEEP2 team at the University of California Berkeley. All the DEEP2 footprints are observed by $Chandra$ Advanced CCD Imaging Spectrometer (ACIS-I) with total exposures across all four XDEEP2 fields range from $\sim$10 ks to 1.1 Ms (Goulding et al. 2012; Laird et al. 2009; Nandra et al. 2005). The intermediate-redshift data is used for calibration of the new KEx diagnostic diagram at z$<$1.
Classification using SDSS main galaxy sample
============================================
Classification
--------------
\[fig-1\] {width="18cm"}
Since the BPT diagram needs at least [\[O[III]{}\]]{}, [[H$\beta$]{}]{}, [\[N[II]{}\]]{} and [[H$\alpha$]{}]{}for a classification, it is not applicable to sources at z$>$0.4 with optical spectrum alone. We propose a new diagnostic diagram: [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs $\sigma_{{[O\,{\footnotesize III}]}}$ to diagnose the ionization source and physical properties of emission line galaxies. We call it the Kinematics-Excitation Diagram (KEx diagram) hereafter. This approach shares similar logic to the work of Yan et al (2011) and Juneau et al (2011).
The dividing lines proposed by Kewley et al. (2001, 2006) and Kauffmann et al. (2003) are used to classify the emission line galaxies into star-forming galaxies, composite galaxies, Seyfert2s, and LINERS, as shown in Figure 1. We require the signal to noise ratio to be greater than 3 for [[H$\beta$]{}]{}, [\[O[III]{}\]]{}, [[\[O[I]{}\]]{}]{}, [\[N[II]{}\]]{}, [[H$\alpha$]{}]{} and [\[S[II]{}\]]{} lines to ensure classification on the BPT diagram into sub-types. The [[\[O[I]{}\]]{}]{}/[[H$\alpha$]{}]{}diagram is shown for reference and not taken into account in the classification. How the different types of galaxies populate the KEx diagram is shown in Panel (d) of Figure 1.
We plot different types of emission line galaxies on [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs $\sigma_{{[O\,{\footnotesize III}]}}$ plot in Figure 2 separately. From left to right are star-forming galaxies, composite galaxies, LINERs and Seyfert2s. In panel (a), the star-forming galaxies cluster around the lower-left corner on the diagram, the boundary of the star-forming galaxies is clear and sharp. We derive an empirical curve to follow the boundary: $$\log {[O\,{\footnotesize III}]}/{{\rm H\ensuremath{\beta}}}=-2\times \log \sigma_{{[O\,{\footnotesize III}]}} +4.2$$
This curve can be used to separate AGNs from star-forming galaxies. The detailed distribution of BPT-classified galaxies on the KEx diagram is given in Table 1. 97% (5674/5860) of the BPT-classified Seyfert2s (above Kewley01 line) and 35% (5587/16003) of the BPT-classified composite sources lie above the new classification line and will be classified as AGNs by the KEx diagram. 98.8% of BPT-classifed SFGs are classified as KEx-SFGs. 81% of the KEx-classifed AGNs are BPT-classifed AGNs (above Kewley01 line), 90% of the KEx-classifed SFGs are BPT-classifed SFGs. For all the sources on the upper side of the line, 7.7% are classified as star-forming galaxies in traditional BPT diagram. 65.1% (10,416/16,003) of BPT composites are in the KEx-SFG region. 46.9% ((10,416+116+186)/(16003+998+5860)) of non-SF galaxies are on the KEx-SFG side. This means the new diagram is very efficient for selecting AGNs above Kewley01 line with high completeness and low contamination rate.
One may notice that the composite galaxies cluster near the SFG-AGN dividing line and clearly separate from Seyfert2s. We draw a horizontal line: $$\log {[O\,{\footnotesize III}]}/{{\rm H\ensuremath{\beta}}}=0.3$$ to cut out a region dominated by composites.. Above this line, the galaxies are mainly LINERs and Sy2s and we call it KEx-AGN region. Below the log [\[O[III]{}\]]{}/[[H$\beta$]{}]{}=0.3 line while above the SFG-AGN line, composite galaxies dominate, and we name it KEx-composite region. Precisely, there are 5,074 composites in KEx-composite region. In this region, there are only 335 Sy2s, 424 LINERs, and 882 star-forming galaxies making up 24.4% of non BPT composites in the KEx-composite region.
We can see in Figure 2 and Table 1 that 116/998 LINERs are classified as star-forming galaxies on the KEx diagram, 426/998 are KEx-composites, and 458 are KEx-AGNs. LINER like emission could be produce by Low-luminosity AGNs (Ferland & Netzer 1983; Halpern & Steiner 1983; Groves et al. 2004b; Ho 2008), post-AGB stars (Binette et al. 1994; Yan & Blanton 2012; Singh et al. 2013), fast shocks (Dopita & Sutherland 1995), photoionization by the hot X-ray-emitting gas (Voit & Donahue 1990; Donahue & Voit 1991), or thermal conduction from the hot gas (Sparks et al. 1989). Despite the ionization origins are diverse, the host galaxies of LINERs are massive, making them only weakly overlap with the star-forming galaxies on the KEx diagram. Since our KEx diagram does not include the information of low-ionization lines, the LINERs are not well separated from AGNs.
We note that only 1/3 of the BPT-classified composites galaxies are in the KEx-composite region, while most of the remaining 2/3 are in the KEx-SFGs region. Only a small fraction are in the KEx-AGN region. This may be becausethe BPT-classified composite galaxies have a diverse origin too. They could be relatively weak AGNs (Kauffmann et al. 2003; Yuan et al. 2010; Ellison et al. 2011), shock heated (e.g., Rich et al. 2014. ), or [H[II]{}]{} region with a special physical condition (Kewley et al. 2001). Trouille et al (2011) showed that the composites are most similar to AGNs in their TBT diagram and show not only photoionization properties like AGNs but also an excess X-ray emission relative to the infrared emission, indicating non stellar processes. This is a more likely scenario than shocks or varying [H[II]{}]{} region conditions. In the case of shocks, even galaxies with a lot of regions locally dominated by shocks have overall line ratios that place them in the BPT-SFG region instead of the composite region (Rich et al 2011). Some composites may have an intrinsically less luminous AGN, and that the NLR gas is moving slower, both in rotation and in outflow. This means our KEx diagram, which is successful in separating strong AGN from SFGs, may not have enough diagnostic power to pick out weak AGNs who have low contrast in both the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} ratio and the kinematics relative to SFGs. KEx diagram also suffers from the mix of composites with other populations, especially the KEx-SFG, but to a lesser degree than some of the alternative diagrams.
In Figure 3, we plot KEx-AGN, KEx-composites, and KEx-SFGs on the BPT diagram. The KEx-composites cluster around the composite region on the BPT diagram. It is possible that the KEx diagram can be used to further diagnose the real nature of the composite galaxies. Panel (b) in Figure 2 also suggest that the [\[O[III]{}\]]{} line width could potentially be used to further constrain the nature of composite galaxies. For example, broad emission line width is usually regarded as a tracer of shock (e.g., Rich et al. 2011,2014). The different line width between sub-population of composites could potentially be used to constrain the relative importance of AGN and SF processes. We leave the exploration of the power of the KEx diagram to diagnose sub-classes for future studies.
\[fig-1\] {width="18cm"}
\[fig-1\] {width="9cm"}
Why the KEx diagram works?
--------------------------
{width="16cm"}
{width="16cm"}
Comparing to the widely used BPT diagram, the KEx diagram uses $\sigma_{{[O\,{\footnotesize III}]}}$ instead of the [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} ratio as the horizontal axis to diagnose the ionizing source and physical properties of emission line galaxies. We showed that it gives very consistent result with the BPT diagram. Here we discuss the physical reasons behind the diagram and why it works. $\sigma_{{[O\,{\footnotesize III}]}}$ in principle traces the motion of the gas. To first order, this motion is determined by the gravity of the galaxy. The emission line width correlate well with the stellar velocity dispersion in almost all types of emission line galaxies (e.g, Nelson 2000; Wang & Lu 2001; Bian et al. 2006; Chen et al. 2008; Komossa et al. 2007, 2008; Dumas et al. 2007; Greene & Ho 2005; Ho 2009). In AGNs, several additional sources of line broadening may at work in addition to the stellar kinematics (Greene & Ho 2005). So the basic principle behind the KEx diagram is the different kinematics of emitting gas in AGNs and star-forming galaxies. The boundary between star-forming galaxies and AGNs on the KEx indicates there is a maximum $\sigma_{{[O\,{\footnotesize III}]}}$ for star-forming galaxies. In local star-forming galaxies, [\[O[III]{}\]]{} comes from the [H[II]{}]{} region, which is more correlated with the kinematics of the disk. It was found that the width of narrow emission lines of star-forming galaxies could efficiently trace the maximum rotation velocity of a galaxy (Rix et al. 1997; Mallén-Ornelas et al. 1999; Weiner et al. 2006; Mocz et al. 2012), so the line width reflects how fast the disk is rotating. The rotation speed is directly linked to the luminosity of the galaxy through the Tully-Fisher relation (TFR, Tully & Fisher 1977). The most luminous spiral galaxies also rotate the fastest. A reasonable hypothesis is that the galaxies near the maximum $\sigma_{{[O\,{\footnotesize III}]}}$ boundary in the KEx diagram are the most luminous ones. We plot all the star-forming galaxies in our sample on the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. R band absolute magnitude in Figure 4. The star-forming galaxies cluster in the left-down corner of the diagram, and show a clear boundary. We draw a magenta boundary curve to define the maximum luminosity a star-forming galaxy can reach at a given [\[O[III]{}\]]{}/[[H$\beta$]{}]{}. Using the Tully-Fisher relation obtained by Mocz et al. (2012), we convert the luminosity in the curve to $\sigma_{gas}$, and obtain a [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. $\sigma_{gas}$ curve, which is shown as the dashed line in Figure 2 (a). We can see that the dashed line covers the regions where SFGs and Composites locate, separating Seyfert2s from SFGs and Composites. This illustrates that SFGs and Composites are consistent with the TFR relation prediction, while AGNs are located outside the locus.
We overplot the distribution of Seyfert2s (red dots) on the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. $M_r$ (absolute petrosian magnitude) and [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. $M_*$ (stellar mass) diagrams (MEx Diagram, Juneau et al. 2011) in Figure 4. The magenta lines in the right panel are the MEx dividing lines. The sources above the upper curve are MEx-AGN and the region between the solid upper curve and dashed line is the MEx-composite region. The contours are 5, 40, 70, 95 percentiles. The stellar mass is drawn from the MPA-JHU catalog. Seyfert2s occupy the bright and massive end of Figure 4. This is consistent with previous findings that Seyfert2s reside exclusively in massive, luminous galaxies (e.g., Kauffmann et al. 2003). Even though the host of Seyfert2s are luminous and massive, the overlap between Seyfert2s and star-forming galaxies on these two plots is large compared with that on the KEx diagram. 4.2% (246/5860) of Sy2s are on the MEx-SF side, while 3.2% (186/5860) of Sy2s are on the KEx-SF side. On the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs $M_R$ plot, 7.7% (450/5960) of Sy2s lie on the SFG side of the dividing curve shown in the left panel of Figure 4. The fraction of BPT-classified Seyferts that are mis-classified as SFGs are higher on these two diagrams than on the KEx diagram.
There is an additional enhancement in $\sigma_{{[O\,{\footnotesize III}]}}$ at fixed luminosity or stellar mass for AGNs relative to SFGs. This enhancement helps AGNs separate further from SFGs on the KEx diagram. In Figure 5, we plot the median $\sigma_{gas}$ derived from five emission lines against $M_r$ and stellar mass in the left and right panels. We focus on the $\sigma_{{[O\,{\footnotesize III}]}}$ (blue line) first and discuss other emission lines later. The errors are calculated using the bootstrap method. We can see that at a given luminosity where both AGNs and SFGs cover, the $\sigma_{{[O\,{\footnotesize III}]}}$ of Seyfert2s is on average 0.15 dex higher than star-forming galaxies. And at a given stellar mass, Seyfert2s are 0.12 dex higher than star-forming galaxies. These offsets are critical to the clean separation between Seyfert2s and star-forming galaxies on the KEx diagram. Other emission lines gave similar results.
One reason of the difference in $\sigma_{{[O\,{\footnotesize III}]}}$ may be that the emission lines in Seyfert2s are produced by gas in the narrow line region which extend into the bulge and the emission line in star-forming galaxies are emitted by gas in the [H[II]{}]{} region in the disk. The matter distribution and kinematics are very different in the disk and bulge of a galaxy It is shown in Catinella et al. (2012) that at a given luminosity or baryon mass, the disk dominated galaxy show 0.1 dex smaller $\sigma_*$ (measured from the SDSS 3" fiber) than the bulge dominated galaxies. At a given mass, the bulge dominated galaxies, which are more concentrated, show a higher $\sigma_*$ than the disk dominated galaxies. Besides the difference in host galaxies, several other physical reasons may be involved in the fact that Seyfert2s have broader emission lines. Greene & Ho (2005) found that the excess [\[O[III]{}\]]{} line width relative to the stellar or lower ionization lines kinematics is about 30-40% (0.11-0.15 dex), with small variations depending on AGN luminosity, AGN Eddington ratio, SFR, etc. When considering only the core of [\[O[III]{}\]]{}, the excess in [\[O[III]{}\]]{} line width goes away (Green & Ho 2005, Komossa et al. 2008). This supports that the [\[O[III]{}\]]{} core is produced by ionized gas in bulge while another source of broadening related to AGN (such as possibly winds or outflows from accretion disk) is at work. Radio jet may play a part in broadening the emission line too (Mullaney et al. 2013).
The differences in $\sigma_{gas}$ and $\sigma_{*}$ between AGNs and SFGs are more pronounced against $M_r$ than against M\*. In particular, $\sigma_*$ is significantly higher in AGN hosts at a fixed $M_r$ but not so different at a fixed stellar mass. This means that AGN hosts have higher mass-to-light ratios, which can be interpreted as having more important bulge components. More massive bulges host more massive black holes, and therefore be detectable as Seyfert 2s down to lower Eddington ratios. Conversely, lower mass AGNs are only identified as Seyfert 2s for comparatively higher Eddington ratios and their [\[O[III]{}\]]{} line width excess could be more pronounced relative to other lines than for higher mass AGNs if the Eddington ratio is the driving factor for additional broadening (e.g., Greene & Ho 2005; Ho 2008).
In summary, the boundary between star-forming galaxies/composites and AGNs on the KEx diagram is defined by the Tully-Fisher relation of the most luminous and massive galaxies. The AGNs reside in luminous and massive galaxies and at a given luminosity/stellar mass, their $\sigma_{{[O\,{\footnotesize III}]}}$ are 0.15/0.12 dex higher than the star-forming galaxies. These effects make the KEx diagram an efficient classification tool for emission line galaxies.
KEx Diagram Calibration at 0.3$<$z$<$1
=======================================
The main purpose of the KEx diagram is for emission line galaxies classification at high redshift. We have demonstrated the KEx diagram could successfully separate emission line galaxies in local universe, mainly due to AGN occur in massive galaxies with high bugle-to-disk ratio and AGN have extra broadening due to outflow . At high redshift, the properties of galaxy and AGN host are different. $\sigma_{[O\,{\footnotesize III}]}$ could be higher because galaxies were more gas rich, there were more unstable disks with high “$\sigma$/V” (Papovich et al. 2005; Reddy et al. 2006; Tacconi et al. 2010; Shim et al. 2011) and AGN were more luminous so had higher Eddington ratios. Besides, the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} of the star-forming galaxies would be higher too. Galaxies at $z\sim1.5$ have typically higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{}ratios than z$<$0.3 galaxies (e.g., Liu et al. 2008; Brinchmann et al. 2008; Trump et al. 2011, 2012; Kewley et al. 2013a,b). The physical properties in [H[II]{}]{} region and AGN NLR could be different from the local universe as well (Kewley et al. 2013a,b). There is also evidence for AGN in relatively low mass hosts at higher redshifts (Trump et al 2011).
To calibrate, we need a sample of AGNs and star-forming galaxies that are already classified . There are several methods for calibration: Firstly, X-ray identification may act as an independent reference for classification calibration (Yan et al. 2011; Juneau et al. 2011, 2013) even though with some drawbacks. The X-ray AGNs and optical AGNs may not be the same population (Hickox et al. 2009) and the sources with reliable deep X-ray data are limited. Besides, X-rays surveys are less sensitive to moderate-luminosity AGNs in galaxies of lower stellar masses (Aird et al. 2012) or heavily obscured Compton-thick AGNs. Yan et al. (2011) estimated at [$L\mathrm{_{bol}}$]{}$> 10^{44}erg s^{-1}$, about 2/3 of the emission-line AGNs with 0.3$<$z$<$0.8 and $I_{AB}<22$ will not be detected in the 2-7 keV band in the 200 ks Chandra images due to absorption and/or scattering of the X-rays in the EGS field. We use DEEP2 data and X-ray identification for KEx calibration in Section 4.2. DEEP2 contain both star-forming galaxies and AGNs, mostly star-forming galaxies. This can help us constrain our calibration on the SF side. Secondly, If NIR spectra are available, the [\[N[II]{}\]]{} and [[H$\alpha$]{}]{} emission lines could be used for optical classification using the BPT diagram as a corner stone. The use of the BPT at higher redshift remains potentially valid, but that it needs to be further verified and the sample size is limited as well(Trump et al. 2012, Kewley et al. 2013b; Juneau et al. 2014). At z$\sim$1, Type2 AGN sample is very limited, because Type2 AGNs can only be identified using BPT diagram to z$\sim$0.4 using only optical spectrum, and the NIR spectroscopy sample which enable the BPT diagram to extend to z$\sim$1 is limited. Thirdly, there are many Type1 AGNs from SDSS in the 0.3$<$z$<$1 range, because the broad lines can be identified using bluer wavelength range, and the volume covered by SDSS is large. Type 1 AGNs can be identified by their blue color or/and broad emission lines. We use them as independently identified AGNs to perform a sanity check of the KEx classification when considering only the narrow line components in Section 4.1. We make the assumption that the Type1 and Type2 sources have identical narrow line features in the frame of unification model (Antonucci et al. 1993). Some differences in narrow lines indeed exist due to NLR stratification, outflow, or narrow line Baldwin effect (Veilleux et al. 1991; Zhang et al. 2008, 2011; Stern & Laor 2013), but the assumption is valid grossly.
Calibration using Type1 AGNs
----------------------------
Our KEx diagram uses only [[H$\beta$]{}]{} narrow component and [\[O[III]{}\]]{} emission lines so it is straightforward to use type1 AGNs to calibrate our KEx diagram. We first use a sample of low-z(z$<$0.3) type1 AGNs from SDSS DR7 main galaxy sample for testing purpose. These sources have [[H$\alpha$]{}]{} broad component with significance greater than 3$\sigma$ as described in Section 2.1. We plot these sources in the KEx diagram in the Panel (a) of Figure 6. The [\[O[III]{}\]]{}/[[H$\beta$]{}]{} only include the narrow component of [[H$\beta$]{}]{}. These sources (4624) reside mostly in the KEx-AGN and KEx-composite regions, while only 414(9%) of them are in the KEx-SFG region. We further plot a sample of intermediate-z Type1 AGNs selected from SDSS DR4 QSO catalog with 0$<$z$<$0.8 on the KEx diagram. The sources in this sample have small contamination from host galaxies in their optical light, and their properties and data reduction are described in Dong et al. (2011). The detailed analysis of narrow line properties, especially the [\[O[III]{}\]]{}line, could be found in Zhang et al. (2011, 2013b). The bolometric luminosity range of these sources is $10^{44} \sim 10^{47}$ $erg s^{-1}$. In left panel of Figure 6, we plot this sample on KEx diagram in orange. Almost all(96%) of the sources lie in the KEx-AGN region, and a small fraction of the Type1 sources(4%) lie in the KEx-composite region. Only 13(0.3%) Type1 sources are classified as KEx-SFGs. According to unification model, if these Type1 sources are viewed edge-on, almost all of them would be rightly classified as AGNs. There are few points at z>0.3 and that they may be consistent with either no shift or a small shift of 0.1 dex.
One may notice that some Type1 AGNs lie outside the low-z locus. Some sources have higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{} line ratio and some have larger $\sigma_{{[O\,{\footnotesize III}]}}$. These could be partly understood by the orientation effect. The Type1 AGNs are found to have higher ionization state than Type2 AGNs (Veilleux et al. 1991c; Schmitt et al. 2003a,b), and this is because that high-ionization lines arise from regions closer to the nuclei thus more likely to be blocked when viewed edge-on. The inclination effect may play a role in the higher width of [\[O[III]{}\]]{} emission here. [\[O[III]{}\]]{} emission line is known to show blue-wing asymmetric profile (Heckman et al. 1981; Zhang et al. 2011) and this is believed to be due to narrow line region outflows. When the outflows are viewed in a face-on orientation, we would see larger overall outflow velocity and this would lead to larger line width at the same time.
Calibration using the DEEP2 Survey
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Our intermediate-redshift galaxy sample is based on observations from the DEEP2 Galaxy Redshift Survey. Most of the galaxies in DEEP2 surveys are star-forming galaxies, as indicated in X-ray study (Goulding et al. 2012; Laird et al. 2009; Nandra et al. 2005), and there are also many X-ray AGNs in this sample. Even though using the X-ray data to calibrate the KEx suffer some problems as discussed in previous subsection and in other works (Hickox et al. 2009; Yan et al. 2011; Juneau et al. 2011, 2013), it is interesting to check if the X-ray and the KEx classification are consistent and what causes the differences An X-ray luminosity threshold: $L_{2-10keV} > 10^{42} erg s^{-1}$ is adopted. There is no star-forming galaxies with X-ray luminosity higher than this value in local universe.The sensitivity of the X-ray data will not result in mis-classification of AGNs and star-forming galaxies, even though faint sources are missed in shallow areas. However, weak AGNs with $L_{2-10keV} < 10^{42} erg s^{-1}$ exist even though they are more ambiguous to differentiate from star-forming or starbursting galaxies with X-ray emission without additional information. For DEEP2 X-ray data, the sensitivity of the shallowest data could ensure the detection of luminous X-ray sources ($L_{2-10keV} > 10^{42} erg s^{-1}$).
The [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs $\sigma_{{[O\,{\footnotesize III}]}}$ for DEEP2 are plotted in green triangles in right panel of Figure 6. We use the same method described in Section 3 for spectral fitting. We apply a S$/$N cut of 3 to [[H$\beta$]{}]{} and [\[O[III]{}\] $\lambda$5007]{}. 7,866 sources satisfy this criteria. We convert the hard 2-8 keV X-ray flux to rest-frame 2-10keV luminosities ($L_X(2-10 keV)$) by assuming a power-law spectrum with photon index ($\gamma=1.8$). The sources with $L_X(2-10 keV)>10^{42} erg s^{-1}$ are classified as X-ray AGNs, and sources with $L_X(2-10 keV)<10^{42} erg s^{-1}$ are classified as star-forming galaxies. We caution $L_X(2-10 keV)>10^{42} erg s^{-1}$ sources may be star-forming galaxies at high-z, due to higher SF activity in the early universe. Many $L_X(2-10 keV)<10^{42} erg s^{-1}$ may be AGNs but dim intrinsically or due to obscuration. The X-ray sources are plotted in pink and purple triangles in right panel of Figure 6. We can see that most of the X-ray AGNs/star-forming galaxies are consistently classified as optical AGNs/star-forming galaxies. Out of the 93 X-ray AGNs, 48 (52%) are classified as KEx-AGNs, 18(19%) are KEx-composite, and 26 (29%) are KEx-SFGs.
Out of the 83 X-ray starbursts in our sample, 19 are KEx-AGNs, 14 are KEx-composite, and 49 (59%) are KEx-SFGs. In Juneau et al. (2011), for the X-ray starbursts, 50% (8/16) are classified as MEx-SFGs, while 19% (3/16) are in the intermediate region and the remaining 31% (5/16) reside in the AGN region. The two results are consistent. On the other hand, we notice that some sources are optically classified as star-forming galaxies but have very powerful X-ray emission, indicating harboring an active nuclei. 29% of X-ray AGNs are classified as star-forming galaxies on the KEx diagram. In Juneau et al. (2011), 20% of their X-ray AGNs are classified as MEx-intermediate, and 15% are MEx-SFGs. Considering the intermediate region is mixed with star-forming region on MEx diagram, our result is consistent with theirs. Yan et al. (2011) found 25% of their X-ray AGNs reside in star-forming region of their optical classification diagram which replaces the [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} in the BPT diagram with rest-frame U-B color. This is consistent with our result too. Castello-Mor et al. (2012) studied the sources with $L_X(2-10 keV)>10^{42} erg s^{-1}$ but classified as star-forming galaxies on the BPT diagram. These sources have large thickness parameter ($T=F_{X}/F_{{[O\,{\footnotesize III}]}}$), large X-ray to optical flux ratio ($X/O>0.1$), broad [[H$\beta$]{}]{} line width, steep X-ray spectra, and display soft excess. These mis-matches illustrate neither X-ray or optical classification are complete. Different classification schemes are complementary to each other. At z$<$1, the evolution in $\sigma_{{[O\,{\footnotesize III}]}}$ and [\[O[III]{}\]]{}/[[H$\beta$]{}]{} is not large so we don’t need to shift the dividing line.
Calibration at z$\sim$2
=======================
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In this section, we discuss extrapolating the KEx diagram to redshift greater than 2, and use z$>$2 emission line galaxies to test the validation of the diagram. The $2<z<4$ epoch is critical to galaxy formation and evolution. During this epoch, the Hubble sequence was not fully established (Kriek et al. 2009; Förster Schreiber et al. 2006, 2009), but the bimodality was in place (Kriek et al. 2009). It is at this redshift that the star-formation density and AGN activity peak (e.g., Barger et al. 2001; Elbaz & Cesarsky 2003; Di Matteo et al. 2005; Hopkins 2004; Hopkins et al. 2006; Hopkins & Beacom 2006). There are thousands of emission line galaxies discovered at this redshift range, and most of them are detected using the dropouts techniques (e.g., Steidel et al. 1996, 1998, 2003, 2004; Pettini et al. 1998, 2001), or color-selection (Franx et al. 2003; Daddi et al. 2004; Kong et al. 2006 ).
At z$<$1, the KEx can be used to successfully separate AGNs and star-forming galaxies mainly due to 3 reasons: First, AGNs reside in the most luminous and most massive galaxies. Second, the Tully-Fisher relation which define the boundary between star-forming galaxies/composites and AGNs is valid. Third, the AGNs have systematic higher $\sigma_{gas}$ than star-forming galaxies at fixed luminosity and stellar mass. The first condition is likely to be true for AGNs up to z$\sim$3 (Xue et al. 2010; Mullaney et al. 2012) because the moderately luminous AGN fraction depends strongly on stellar mass but only weakly on redshift. The TFR, however, is likely to evolve with redshift.
Unlike local disk galaxies who is totally rotation-supported with $V_{rot}/\sigma=10\sim 20$(e.g. Dib, Bell & Burkert 2006), a large fraction of high redshift star-forming galaxies have velocity dispersion comparable or even larger than rotation velocity(Förster Schreiber et al. 2006, 2009; Wright et al. 2007; Law et al. 2007; Genzel et al. 2008; Cresci et al. 2009; Vergani et al. 2012). Since we use the integrated emission line profile, the increase in velocity dispersion further broaden the emission line.
Even after we have extracted the rotation velocity using the Integrated Field Spectroscopy (IFS) for these high-redshift galaxies, the derived Tully-Fisher relation is still different from the local well-defined relation. The rotation speed of z$\sim$2 galaxies is $\sim$0.2 dex higher than galaxies of similar stellar mass in local galaxies (Cresci et al. 2009; Gnerucci et al. 2011). Meanwhile, the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} of star-forming galaxies is known to be higher at high redshift (Shapley et al. 2005; Erb 2006a; Groves et al. 2006; Liu et al. 2008; Brinchmann et al. 2008; Shirazi et al. 2014). Thus, we expect that the KEx diagram, particularly the separation boundary of AGN and star-forming galaxies, evolve to higher $\sigma_{gas}$ or higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{} with redshift because the TFR and [\[O[III]{}\]]{}/[[H$\beta$]{}]{} redshift evolution, and galaxies at high redshift have larger velocity dispersions,
To explore these effects in detail and test the application of the KEx diagram at z$\sim$2, we compile a sample of Lyman-break galaxies (LBGs) and color-selected BzK galaxies at $z>2$ who have [\[O[III]{}\]]{} and [[H$\beta$]{}]{}emission line ratio and gas velocity dispersion measurements from literature. Our sample include 10 galaxies (3 are AGNs) from Kriek et al. (2007), 1 LBG from Teplitz et al. (2000), 6 LBGs from Pettini et al. (1998, 2001), and 2 gravitationally-lensed star-forming galaxies from Hainline et al. (2009). The AGN fraction of this sample is very limited because LBGs have very low AGN fraction (about 3-5%, Erb et al., 2006b, Reddy et al., 2005, Steidel et al., 2002).
Figure 7 shows the $z>2$ LBGs are not confined to the KEx-SFGs region of local galaxies. Most of them lie in the KEx-AGN or KEx-composite region. If we shift the dividing line 0.3 dex to the right, most of the galaxies are rightly classified. Out of the 0.3 shift, the evolution of TFR could contribute 0.2 dex (Cresci et al. 2009; Gnerucci et al. 2011; Vergani et al. 2012; Buitrago et al. 2013). Besides, the high redshift star-forming galaxies have higher velocity dispersion, which is not included in the evolution of TFR.
To test if AGNs still show different kinematic from star-forming galaxies at this redshift, we compile radio galaxies that are confirmed to be AGNs and see how they distribute in the KEx diagram. We use 4 radio AGNs from Nesvadba et al. (2008, 2011) and CDFS-695: A shock or/and AGN from van Dokkum et al. (2005). They clearly separate from the star-forming galaxies on the KEx diagram after shifting the boundary by 0.3 dex. We further check the [\[O[III]{}\]]{} width distribution of a sample of z$\sim$2 QSOs from Netzer et al. (2004) and plot the histogram on lower panel of Figure 7. Only one source has [\[O[III]{}\]]{} width less than 100km/s, and most of the QSOs have $\sigma_{{[O\,{\footnotesize III}]}}\sim300km/s$. There is no doubt that these QSOs would have high [\[O[III]{}\]]{}/[[H$\beta$]{}]{} ratio even though we do not have their detailed values. Therefore, they are likely to separate from the z$\sim$2 star-forming galaxies. Strong outflow driven by AGN is reported in the Radio Galaxies (Nesvadba et al. 2008, 2011), so the large line width in [\[O[III]{}\]]{} is at least partly due to the outflow as discussed in Section 3.2. Judging from our empirical results, the KEx diagram is likely to work after shifting the dividing line 0.3 dex to the right at z$\sim$2. More data is needed to better constrain the boundary.
Other Relevant Issues
=====================
Comparison with previous classification diagrams
------------------------------------------------
Tresse et al. (1996) and Rola et al. (1997) proposed to use EW([\[O[II]{}\]]{}), EW([\[O[III]{}\]]{}) and EW([[H$\beta$]{}]{}) for galaxy classification at high redshift. Stasińska et al. (2006) studied using [\[O[II]{}\]]{} for galaxy classification, and even proposed a method that uses $D_n(4000)$, EW([\[O[II]{}\]]{}), and EW([\[Ne[III]{}\]]{}) (DEW diagram) to select pure AGNs with z$<$1.3 using only optical spectrum. But different types of galaxies overlap with each other severely on these classification diagrams.
Trouille et al. (2011) proposed to use $g-z$, [\[Ne[III]{}\]]{}, and [\[O[II]{}\]]{} to clearly separate AGNs from star-forming galaxies. This method is very efficient in separating different types of galaxies, but the [\[Ne[III]{}\]]{} emission line is weak even in AGNs. Thus this diagram requires high signal-to-noise ratio spectra of galaxies for reliable classification. This is particular hard for high redshift objects which are usually faint.
Our KEx diagram has a similar logic as the Color-Excitation (CEx) and Mass-Excitation (MEx) diagrams proposed by Yan et al. (2011) and Juneau et al. (2011). The CEx diagram makes use of the fact that AGNs reside in red or green galaxies in local galaxies. But this is likely to be wrong at higher redshift (Trump et al. 2012). The MEx diagram and the one using rest-frame H-band magnitude (Weiner et al. 2006) is based on the fact that AGNs are harbored by massive galaxies. This is more robust at high redshift due to the AGN downsizing effect. Trump et al. (2012) tested the validation of these two diagrams at z$\sim$1.5 and found that the MEx remains effective at z$>$1 but CEx needs a new calibration. As discussed in Section 3.2, we can separate AGNs and star-forming galaxies because AGN reside in massive galaxies, and have $\sigma_{{[O\,{\footnotesize III}]}}$ 0.12 dex higher than star-forming galaxies of similar stellar mass as an enhancement. This enhancement makes the KEx diagram more efficient at separating AGNs from star-forming galaxies. One advantage of the KEx is the requirements for only a spectrum that covers a small spectral range to obtain all required quantities. The KEx diagram only requires [\[O[III]{}\]]{} and [[H$\beta$]{}]{} lines for a robust classification.
Comparison of different narrow line widths
------------------------------------------
In KEx diagram, we use [\[O[III]{}\]]{} emission line width for diagnostic. In order to check if different lines have different width, in Figure 5 we plot the median value of $\sigma$ against R band absolute magnitude and stellar mass for different narrow lines using galaxies from SDSS DR7. We plot the stellar velocity dispersion ($\sigma_*$) in red line for reference. The $\sigma_{*}$ is stored in the SDSS spectrum file header. 32 K and G giant stars in M67 are used as stellar templates. These stellar templates are convolved with the velocity dispersion to fit the rest-frame wavelength range 4000-7000Å by minimizing $\chi^2$. The final estimation is the mean value of the estimates given by the “Fourier-fitting” and “Direct-fitting” methods. We found that the SFGs show similar sigma for all emission lines. The Seyfert2s, however, show some systematics in line width. The [\[O[III]{}\]]{} line is broader than other low ionization lines and recombination lines. This is expected, because the [\[O[III]{}\]]{} emitting region is more concentrated due to its high ionization potential (Veilleux et al. 1991; Trump et al. 2012). Unexpectedly, [[H$\beta$]{}]{} show the smallest line width, and the discrepancy is largest at the high luminosity high stellar mass end. One possibility for this discrepancy is the Balmer absorption fitting is not perfect. The incorrect absorption fitting affects resulting emission line flux and profile. Groves et al. (2012) found significant discrepancy in [[H$\beta$]{}]{} when using CB07 for SDSS DR7 and BC03 for SDSS DR4. The [[H$\alpha$]{}]{}, which should arises from the same emitting region of [[H$\beta$]{}]{}, does not follow the behavior of [[H$\beta$]{}]{} but show the same trend as low-ionization lines because the absorption correction is much milder.
When comparing the line width of emission lines and $\sigma_*$, we found that the difference between $\sigma_*$ in star-forming galaxies and Seyfert2s of similar stellar mass is very small. At high stellar mass end, galaxies tend to have bulges, and those would contribute to increase the measured $\sigma_*$. However, if the ionized gas in SFGs still comes from the disk, then this component does not get the additional dispersion contribution from the bulge. Also, one can expect $\sigma_{gas}<\sigma_*$ because the gas is more dissipative than the stars and can slow down dynamically (Ho 2009). We leave this topic for future studies.
[\[O[III]{}\]]{}/[[H$\beta$]{}]{} evolution
-------------------------------------------
It is found that the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} in star-forming galaxies gets higher at z$\sim$1-2 (Shapley et al. 2005; Erb 2006a; Liu et al. 2008; Brinchmann et al. 2008; Hainline et al. 2009; Wright et al. 2010; Trump et al. 2011; Shirazi et al. 2014). The reason for this trend is under debate. Brinchmann et al. (2008) found that the location of star-forming galaxies in the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} versus [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} diagnostic diagram highly depends on their excess specific star formation rate relative to galaxies of similar mass. They infer that an elevated ionization parameter U is responsible for this effect, and propose that this is also the cause of higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{} in high-redshift star-forming galaxies in the BPT diagram (Brinchmann et al. 2008; Shirazi et al. 2014) . Liu et al. (2008) argue that the high [\[O[III]{}\]]{}/[[H$\beta$]{}]{} sources have higher electron densities and temperatures. It is also possible that AGN or shock increase the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} ratio (Groves et al. 2006; Wright et al. 2010). It is interesting to check how [\[O[III]{}\]]{}/[[H$\beta$]{}]{} evolves with redshift at given $\sigma_{{[O\,{\footnotesize III}]}}$ in the KEx diagram.
In the right panel of Figure 6, we plot the median [\[O[III]{}\]]{}/[[H$\beta$]{}]{} at given $\sigma_{{[O\,{\footnotesize III}]}}$ for the SDSS z$<$0.33 galaxies and the DEEP2 galaxies on the KEx diagram. The red, orange, yellow, green, blue and purple lines are median [\[O[III]{}\]]{}/[[H$\beta$]{}]{} at given $\sigma_{{[O\,{\footnotesize III}]}}$ for $z\sim0.1$, $0.3<z<0.4$, $0.4<z<0.45$, $0.5<z<0.6$, $0.6<z<0.7$, $0.7<z<0.8$ KEx-SFGs respectively. We can see in Figure 6 that the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. $\sigma_{{[O\,{\footnotesize III}]}}$ relation does not evolve from z$\sim$0.3 to z$\sim$0.8, but these galaxies have on average 0.2 dex higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{} than the local galaxies at given $\sigma_{{[O\,{\footnotesize III}]}}$. However, we note that SDSS has a fixed aperture of 3" which acquires the light from the center of the galaxy , while DEEP2 spectra are obtained through long-slits, which enable them to include light from the outskirt of the galaxy. The outskirt of the galaxy have lower metallicity and larger rotation speed than the center. This would shift the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs. $\sigma_{{[O\,{\footnotesize III}]}}$ relation in the right-up (higher [\[O[III]{}\]]{}/[[H$\beta$]{}]{}, higher $\sigma_{{[O\,{\footnotesize III}]}}$) direction as we see . So our result are consistent with no evolution in the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} vs $\sigma_{{[O\,{\footnotesize III}]}}$ relation from z=0-0.8.
Summary and Conclusion
======================
We propose a new diagram, the Kinematic-Excitation diagram (KEx diagram), using the [\[O[III]{}\]]{}/[[H$\beta$]{}]{} line ratio and the [\[O[III]{}\] $\lambda$5007]{}emission line width ($\sigma_{{[O\,{\footnotesize III}]}}$) to diagnose the emissions of the AGNs and the star-forming galaxies. The KEx diagram uses only the [\[O[III]{}\] $\lambda$5007]{} and [[H$\beta$]{}]{}emission lines, thus it is a suitable tool to classify emission-line galaxies (ELGs) at higher redshift than more traditional line ratio diagnostics because it does not require the use of the [\[N[II]{}\]]{}/[[H$\alpha$]{}]{} ratio. Using the SDSS DR7 main galaxy sample and the BPT diagnostic, we calibrate the diagram at low redshift. We find that the diagram can be divided into 3 regions: one occupied mainly by the pure AGNs (KEx-AGN region), one dominated by composite galaxies (KEx-composite region), and one contains mostly SFGs (KEx-SFG region). The new diagram is very efficient for selecting AGNs with high completeness and low contamination rate. We further apply the KEx diagram to 7,866 galaxies at 0.3 $<$ z $<$ 1 in the DEEP2 Galaxy Redshift Survey, and compare the KEx classification to an independent X-ray classification using $Chandra$ observation. Almost all Type1 AGNs at z$<$0.8 lie in the KEx-AGN region, confirming the reliability of this classification diagram for emission line galaxies at intermediate redshift. At z$\sim$2, the demarcation line between star-forming galaxies and AGNs should be shifted to 0.3 dex higher $\sigma_{{[O\,{\footnotesize III}]}}$ due to evolution
AGNs are separated from SFGs in this diagram mainly because in addition to that they preferentially reside in luminous and massive galaxies, they show 0.15/0.12 dex higher $\sigma_{{[O\,{\footnotesize III}]}}$ than star-forming galaxies at given luminosities/stellar masses. Higher $\sigma_{{[O\,{\footnotesize III}]}}$ also arise from AGN-driven broadening effects (such as winds or outflows). When we push to higher redshift, the evolution of [\[O[III]{}\]]{}/[[H$\beta$]{}]{} and the TFR result in the shift of dividing line between AGNs and SFGs. KEx needs high enough spectral resolution to measure $\sigma_{{[O\,{\footnotesize III}]}}$, and this diagnostic diagram is purely empirical now because it is hard to link ionization and kinematics theoretically. Despite the caveats, it provides a robust diagnostic of ionization source when only [\[O[III]{}\]]{} and [[H$\beta$]{}]{} are available.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Lisa Kewley, Renbin Yan, Shude Mao, Junqiang Ge, Jong-Hak Woo, Chun Lyu for helpful discussions and suggestions. We thank an anonymous referee for helpful suggestions that improve the paper significantly. The research presented here is partially supported by the 973 Program of China under grants No. 2013CB834905 and No. 2009CB824800, by the National Natural Science Foundation of China under grants No. 11073040, by the Strategic Priority Research Program “The Emergence of Cosmological Structures” of Chinese Academy of Sciences, Grant No. XDB09000000, and by Shanghai Pujiang Talents Program under grant No. 10pj1411800. K.Z. acknowledge the supports by the No. 53 China Post-doc general Fund under grants No. 2013M531232. Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The SDSS web site is http://www.sdss.org/.
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0.0cm = 0mm = 0mm
--------------- ----------------------- ------------ -------- --------------------- --------------------- --------- -------
Type Star-Forming Galaxies Composites LINERs Seyfert2s Seyfert1s Quasars DEEP2
(DR7 galaxy sample) (DR4 quasar sample)
(1) (2) (3) (4) (5) (6) (7) (8)
Total number 97,484 16,003 998 5,860 4,624 4,158 7,866
KEx-SFGs 96,322 10,416 116 186 414 13 7,024
KEx-Composite 882 5,074 424 335 718 274 344
KEx-AGN 280 513 458 5,339 3,492 3,871 498
--------------- ----------------------- ------------ -------- --------------------- --------------------- --------- -------
\
[^1]: http://deep.ps.uci.edu/
[^2]: http://astro.berkeley.edu/$\sim$cooper/deep/spec2d/
|
---
abstract: 'We show that C\*-algebras generated by irreducible representations of finitely generated nilpotent groups satisfy the universal coefficient theorem of Rosenberg and Schochet. This result combines with previous work to show that these algebras are classifiable by their Elliott invariants within the class of unital, simple, separable, nuclear C\*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. We also show that these C\*-algebras are central cutdowns of twisted group C\*-algebras with homotopically trivial cocycles.'
address:
- 'Department of Mathematics, Miami University, Oxford, Ohio'
- 'Department of Mathematics, University of Colorado - Boulder, Boulder, Colorado'
author:
- Caleb Eckhardt and Elizabeth Gillaspy
title: 'Irreducible representations of nilpotent groups generate classifiable C\*-algebras'
---
Introduction
============
This note concludes a long line of study into the $C^*$-algebras generated by irreducible representations of finitely generated nilpotent groups. Specifically, we prove that such $C^*$-algebras satisfy the universal coefficient theorem (UCT) of Rosenberg and Schochet [@Rosenberg87]. We combine this with a slew of other results to show that these algebras are classifiable by their Elliott invariant within the class $\mathcal{C}$ of unital, simple, separable, nuclear C\*-algebras with finite nuclear dimension that satisfy the UCT.
Let $G$ be a finitely generated nilpotent group and $\pi$ an irreducible unitary representation of $G.$ Let $C^*_\pi(G)$ be the $C^*$-algebra generated by $\pi(G).$ Tikuisis, White and Winter recently took the final step in a long and beautiful journey of showing that two elements of $\mathcal{C}$ with the same Elliott invariant are isomorphic [@Tikuisis15 Corollary D]. Therefore our job consists of showing that $C^*_\pi(G)\in \mathcal{C}.$
We have known for a while that $C^*_\pi(G)$ is nuclear [@Lance73] and simple [@Moore76]. Recently, the first author together with McKenney [@Eckhardt14b] showed that $C^*_\pi(G)$ has finite nuclear dimension (this work directly relied on a long list of results including [@Eckhardt14; @Matui12; @Matui14a; @Rordam04; @Winter12] – see the introduction of [@Eckhardt14b] for the full story). As pointed out in [@Eckhardt14b Theorem 4.5] the only missing ingredient to show $C^*_\pi(G)\in \mathcal{C}$ was the UCT. It was also pointed out in [@Eckhardt14b Theorem 4.6] that if $G$ is torsion free and $\pi$ is a faithful representation then $C^*_\pi(G)$ satisfies the UCT. This observation has already found success in [@Eckhardt14a] where the authors calculated the Elliott invariant of C\*-algebras generated by faithful irreducible representations of the (torsion free) unitriangular group $UT(4,{\mathbb{Z}})$, thus classifying them within $\mathcal{C}.$
Proving Theorem \[thm:main\], that $C^*_\pi(G)$ satisfies the UCT, is the main goal of this note. In the course of our investigations we noticed that $C^*_\pi(G)$ is isomorphic to a central cutdown of a twisted group C\*-algebra $C^*(G/N,\sigma)$ for some 2-cocycle $\sigma$, where $N$ is a finite index subgroup of $Z(G).$ Moreover, the cocycle $\sigma$ is homotopic to the trivial cocycle. We include this as Theorem \[thm:structure\] as it may be of independent interest and useful for K-theory calculations.
Nilpotent lemmas {#sec:nilemmas}
================
We first recall and prove necessary facts about nilpotent groups used in subsequent sections. We refer the reader to Segal’s book [@Segal83] for information about nilpotent and polycyclic groups. Throughout this section $G$ is a **finitely generated nilpotent group.** We let $Z(G)$ denote the center of $G$ and $$G_f=\{ x\in G: \textup{the conjugacy class of }x\textup{ is finite } \}$$ Clearly $Z(G)\leq G_f \leq G.$ Define the *torsion subgroup* of $G$ as $$T(G)=\{ x\in G: x\textup{ has finite order } \}.$$
In general, $T(G)$ need not be a subgroup of $G$ but it is a standard exercise to show that $T(G) \leq G$ for nilpotent $G$ (see [@Segal83 Corollary 1.B.10]).
Every finitely generated nilpotent group is *polycyclic*, that is, there is a normal series $$\label{eq:poly}
\{ e \}=G_n \trianglelefteq G_{n-1}\trianglelefteq \cdots \trianglelefteq G_1\trianglelefteq G_0=G$$ where $G_{i}/G_{i+1}$ is cyclic for each $i=0,\ldots,n-1.$ From this definition it follows that polycyclic groups are finitely generated and that subgroups of polycyclic groups are polycyclic. Moreover, for finitely generated nilpotent groups, $T(G)$ is polycyclic and therefore finite, since it must satisfy (\[eq:poly\]). From this it easily follows that $T(G)\leq G_f.$\
\
It is most likely well-known among group theorists that $Z(G)$ has finite index in $G_f$ and that $G/G_f$ is torsion free. We were not able to locate references for these facts (although they are more-or-less corollaries of Baer’s [@Baer48]) so we include the brief proofs.
\[lem:tfreecase\] Let $G$ be torsion free, finitely generated and nilpotent. Then $Z(G)=G_f.$
By [@Baer48 Lemma 3], the group $Z(G_f)$ has finite index in $G_f.$\
\
*Case 1.* $Z(G_f)=Z(G).$ By [@Malcev49], $G/Z(G)$ is torsion free. Then $G_f/Z(G_f)$ is a finite, torsion free group, i.e. $G_f=Z(G_f).$\
\
*Case 2.* $Z(G_f)\neq Z(G).$ By assumption there are elements $x\in Z(G_f)\setminus Z(G)$ and $y\in G$ such that $yxy^{-1}\neq x.$ Let $\alpha$ be the automorphism of $Z(G_f)$ induced by conjugation by $y.$ Because $x\in G_f$ we have $\alpha^n(x)=x$ for some $n>1.$ Moreover, $Z(G_f)$ is a finitely generated torsion free abelian group, since $G$ is nilpotent, torsion free, and finitely generated. Since the group generated by $Z(G_f)$ and $y$ is nilpotent it follows that the matrix for $\alpha$ is unipotent. In particular, $1$ is the only eigenvalue of $\alpha$, so $\alpha$ cannot have any periodic, non-fixed points, a contradiction to the fact that $\alpha^n(x)=x$. Therefore $Z(G)=Z(G_f)$, so Case 2 never occurs and we are in Case 1.
\[lem:findex\] Let $G$ be a finitely generated nilpotent group, then $Z(G)$ has finite index in $G_f.$
Let $x\in G_f.$ By [@Baer48 Lemma 3], there is some $n\geq1$ such that $z=x^n \in Z(G_f).$ Let $y_1, \ldots, y_k$ generate $G.$ Since $zT(G)\in (G/T(G))_f$ and $G/T(G)$ is torsion free, by Lemma \[lem:tfreecase\] it follows that $zT(G)\in Z(G/T(G))$, i.e. $$[z,y_i]\in T(G) \textup{ for }i=1, \ldots, k.$$ Then for $n\geq 1$ we have $$\label{eq:commid}
[z^{n+1},y_i]=z^n[z,y_i]z^{-n}[z^n,y_i]=[z,y_i][z^n,y_i].$$ The first equality holds in every group and the second follows because $z\in Z(G_f)$ and $[z,y_i]\in T(G)\leq G_f.$ By induction, one uses (\[eq:commid\]) to show that $[z^n,y_i]=[z,y_i]^n$ for all $1\leq i\leq k$ and $n\geq1.$
Since $[z,y_i]\in T(G)$ it follows that there is some power of $z$ such that $[z^d,y_i]=e$ for all $i=1, \ldots, k.$ In particular $x^{nd}=z^d\in Z(G).$ Therefore $G_f/Z(G)$ is a torsion group. But $G_f/Z(G)$ is polycyclic and therefore finite.
\[lem:tfree\] Let $G$ be a finitely generated nilpotent group. Then $G/G_f$ is torsion free.
Suppose $x\in G\setminus G_f$ and $x^n\in G_f$ for some $n.$ By Lemma \[lem:findex\] we have $x^m\in Z(G)$ for some $m.$ Since $G/T(G)$ is torsion free and $x^mT(G)\in Z(G/T(G))$ we have $xT(G)\in Z(G/T(G)).$ This means that for every $y\in G$ we have $yxy^{-1}x^{-1}\in T(G)$; equivalently, $yxy^{-1}=xz$ for some $z\in T(G).$ Since $T(G)$ is finite, this means $x\in G_f$, a contradiction.
Main result {#sec:main}
===========
For a group $G$ and normalized positive definite function $\phi:G\rightarrow {\mathbb{C}}$, let $(\pi_\phi,H_\phi)$ denote the GNS representation of $G$ associated with $\phi.$ Let $C^*_{\pi_\phi}(G)$ denote the $C^*$-algebra generated by $\pi_\phi(G).$ A **trace** on $G$ is a normalized, positive definite function that is constant on conjugacy classes. Notice that a trace $\tau$ on $G$ canonically induces a tracial state on $C^*(G),$ which we will also denote by the same symbol $\tau$.
The following lemma is well-known and may be found for example in [@Brown08 Proposition 4.1.9].
Let $A$ be a C\*-algebra and $\phi$ a faithful state on $A.$ Let $\alpha$ be an automorphism of $A.$ Let $u\in A\rtimes_\alpha {\mathbb{Z}}$ be the unitary implementing $\alpha$. One extends $\phi$ to a faithful state on $A\rtimes_\alpha{\mathbb{Z}}$ by setting $\phi(xu^n)=0$ when $x\in A$ and $n\neq0.$ \[lem:faithstates\]
\[lem:crossedproducts\] Let $N$ be a discrete group and $\alpha$ an automorphism of $N.$ Set $G=N\rtimes_\alpha {\mathbb{Z}}.$ Let $u\in G$ be the element implementing $\alpha$ by conjugation. Let $\tau$ be a trace on $G$ such that $\tau(x)=0$ for all $x\in G\setminus N.$ Then $$C^*_{\pi_\tau}(G)\cong C^*_{\pi_\tau}(N)\rtimes_{\textup{Ad}\pi_\tau(u)} {\mathbb{Z}}$$
Let $\sigma:C^*_{\pi_\tau}(N)\rtimes_{\textup{Ad}\pi_\tau(u)}{\mathbb{Z}}\rightarrow C^*_{\pi_\tau}(G)$ denote the surjective \*-homomorphism corresponding to the covariant representation which is the inclusion map on $C^*_{\pi_\tau}(N)$ and sends $n\in {\mathbb{Z}}$ to $\pi_\tau(u)^n.$ By Lemma \[lem:faithstates\], $\tau\circ \sigma$ defines a faithful tracial state on $C^*_{\pi_\tau}(N)\rtimes_{\textup{Ad}\pi_\tau(u)} {\mathbb{Z}}$. Therefore $\sigma$ is injective.
\[thm:obs\] Let $G$ be finitely generated and nilpotent. Let $\tau$ be a trace on $G$ such that $\tau(x)=0$ for all $x\in G\setminus G_f.$ Then $C^*_{\pi_\tau}(G)$ is isomorphic to an iterated crossed product of $C^*_{\pi_\tau}(G_f)$ by ${\mathbb{Z}}$-actions, i.e. $$C^*_{\pi_\tau}(G)\cong C^*_{\pi_\tau}(G_f)\rtimes {\mathbb{Z}}\rtimes \cdots \rtimes {\mathbb{Z}},\label{eq:4}$$ and $C^*_{\pi_\tau}(G)$ satisfies the UCT.
By Lemma \[lem:tfree\], $G/G_f$ is torsion free. Since $G/G_f$ is finitely generated and nilpotent it is isomorphic to an iterated semi-direct product of ${\mathbb{Z}}$-actions. By repeatedly using the fact that the short exact sequence of groups $0\rightarrow A \rightarrow B\rightarrow {\mathbb{Z}}\rightarrow 0$ always splits we have $$G\cong G_f\rtimes {\mathbb{Z}}\rtimes \cdots \rtimes{\mathbb{Z}}.$$ Now follows from repeated applications of Lemma \[lem:crossedproducts\]. By Lemma \[lem:findex\], the group C\*-algebra $C^*(G_f)$ is subhomogeneous, from which it follows that $C^*_{\pi_\tau}(G_f)$ is also subhomogeneous. In particular, $C^*_{\pi_\tau}(G_f)$ is Type I, so it satisfies the UCT by [@Rosenberg87 Theorem 1.17]. Finally, [@Rosenberg87 Proposition 2.7, Theorem 4.1] shows that the UCT is preserved by ${\mathbb{Z}}$-actions, implying that $C^*_{\pi_\tau}(G)$ satisfies the UCT.
\[thm:main\] Let $G$ be a finitely generated nilpotent group and $\pi$ an irreducible representation of $G.$ Then $C^*_\pi(G)$ satisfies the UCT.
Suppose first that $\pi$ is faithful on $G.$ Since $G$ is finitely generated, by [@Carey84 Proposition 5.1] there is an extreme trace $\tau$ on $G$ such that $C^*_\pi(G)\cong C^*_{\pi_\tau}(G).$ Since $\tau$ is an extreme trace and $\pi$ is faithful on $G$, it follows from [@Carey84 Theorem 4.5] that $\tau(g)=0$ if $g\not\in G_f.$ By Theorem \[thm:obs\], $C^*_{\pi_\tau}(G)$ satisfies the UCT.
If $\pi$ is not faithful, then we replace $G$ with $G/\textup{ker}(\pi)$ and $\pi$ with $\widetilde{\pi}(g\textup{ker}(\pi)):=\pi(g)$. Then $\widetilde{\pi}$ is faithful and $C^*_{\widetilde{\pi}}(G /\text{ker}(\pi)) \cong C^*_\pi(G)$, so we simply apply the above proof to $(G/\text{ker}(\pi), \widetilde{\pi})$.
Let $G$ be a finitely generated nilpotent group and $\pi$ an irreducible representation of $G.$ Then $C^*_\pi(G)$ is classified by its ordered K-theory within the class of simple, unital, nuclear C\*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem.
This is Theorem \[thm:main\] combined with [@Eckhardt14b Theorem 4.5].
A structure theorem for $C^*_\pi(G)$
====================================
In the course of proving Theorem \[thm:main\] we discovered a structure theorem (Theorem \[thm:structure\]) for $C^*_\pi(G)$ that is most likely unknown. We therefore include it as it may be of interest to those working with twisted group $C^*$-algebras and representation theory.
Let $G$ be a discrete amenable group and $N\leq Z(G).$ Let $\omega\in \widehat{N}$ (the dual group of $N$) and also denote by $\omega$ the trivial extension of $\omega$ to $G$ (i.e., $\omega(x)=0$ for $x\not\in N.$) Let $c:G/N\rightarrow G$ be a choice of coset representatives. For $t\in G$, let $t_\omega\in H_\omega$ denote the canonical image of $t$ associated with the GNS representation.
Recall that, for any $s,t\in G,$ we have ${\langle}t_\omega,s_\omega {\rangle}=\omega(s^{-1}t).$ Hence if $s^{-1}t\not\in N$, the vectors $s_\omega$ and $t_\omega$ are orthogonal. On the other hand, for $s\in G$ we have ${\langle}s_\omega, c(sN)_\omega {\rangle}=\omega(c(sN)^{-1}s)\in {\mathbb{T}}$, from which it follows that $$s_\omega=\omega(c(sN)^{-1}s)c(sN)_\omega. \label{eq:proportional}$$
We deduce that $\{ s_\omega: s\in c(G/N) \}$ is an orthonormal basis for $H_\omega.$ Define $W:H_\omega\rightarrow \ell^2(G/N)$ by $W(s_\omega)=\delta_{sN}$ for $s\in c(G/N).$ Then $W$ is unitary. Moreover, by (\[eq:proportional\]) we have that, for any $y\in G$ and $s\in c(G/N)$, $$W\pi_\omega(y)W^*(\delta_{sN})=\omega(c(ysN)^{-1}yc(sN))\delta_{ysN}. \label{eq:twist}$$ Now (as in [@Packer92]) define the 2-cocycle $\sigma:G/N\times G/N\rightarrow {\mathbb{T}}$ by $$\sigma(xN,yN)=\omega(c(xN)c(yN)c(xyN){^{-1}}). \label{eq:cocycle}$$ By an obvious adaptation of the proof of [@Eckhardt14a Proposition 2.9] and the discussion preceding it, we have the following
\[lem:twistedgroup\] Let $G$ be a discrete amenable group and $N\leq Z(G).$ Let $\omega\in \widehat{N}.$ Also let $\omega$ denote the trivial extension to $G$ and define $\sigma$ as in (\[eq:cocycle\]). Then $C^*_{\pi_\omega}(G)$ is isomorphic to the twisted group $C^*$-algebra $C^*(G/N,\sigma).$
Let $\lambda_{G/N}$ denote the left regular representation of $G$ on $\ell^2(G/N).$
Let $G$ be a discrete amenable group and $N\leq Z(G).$ Let $\omega\in \widehat{N}$ and denote by $\omega$ the trivial extension of $\omega$ to $G.$ Let $\tau$ be a trace on $G$ such that $\tau|_N=\omega|_N.$ Then $\lambda_{G/N}\otimes \pi_\tau$ is unitarily equivalent to $\pi_\omega\otimes 1_{H_\tau}.$
Writing $\prec$ to denote weak containment, it then follows that $\pi_\tau \prec \pi_\omega$; equivalently, $C^*_{\pi_\tau}(G)$ is a quotient of $C^*_{\pi_\omega}(G).$ \[lem:weakcontainment\]
This is a slight modification of the proof of Fell’s absorption principle. Define a unitary $U$ on $\ell^2(G/N)\otimes H_\tau$ by $U(\delta_{tN}\otimes \xi)=\delta_{tN}\otimes \pi_\tau(c(tN))\xi.$ For any $y,t\in G$, $$\begin{aligned}
U^*(\lambda_{G/N}(y)\otimes \pi_\tau(y))U(\delta_{tN}\otimes \xi)&=U^*(\lambda_{G/N}(y)\otimes \pi_\tau(y))(\delta_{tN}\otimes \pi_\tau(c(tN))\xi)\\
&=U^*(\delta_{ytN}\otimes \pi_\tau(yc(tN))\xi)\\
&= \delta_{ytN}\otimes \pi_\tau(c(ytN)^{-1}yc(tN))\xi\\
&=\omega(c(ytN)^{-1}yc(tN))\delta_{ytN}\otimes\xi,\end{aligned}$$ because $\tau=\omega$ on $N$, both are multiplicative on $N$, and $c(ytN)^{-1}yc(tN)\in N.$ Unitary equivalence now follows from (\[eq:twist\]).\
\
We now show weak containment. Let $1_G$ denote the trivial representation of $G$ on ${\mathbb{C}}$. Since $G/N$ is amenable, $\lambda_{G/N}$ contains an approximately fixed vector, and thus $1_G\prec \lambda_{G/N}.$ Then $$\pi_\tau\sim 1_G\otimes \pi_\tau\prec \lambda_{G/N}\otimes \pi_\tau\sim \pi_\omega\otimes 1_{H_\tau}\prec \pi_\omega.$$
The following is well-known (see for example [@Brown08 Corollary 2.5.12]).
\[lem:condexp\] Let $H\leq G$ be discrete groups. Let $C^*_r(G)$ denote the reduced group C\*-algebra of $G.$ The linear map from ${\mathbb{C}}[G]$ to ${\mathbb{C}}[H]$ defined by $E(\lambda_s)=\lambda_s$ if $s\in H$ and $E(\lambda_s)=0$ if $s\not\in H$ extends to a conditional expectation from $C^*_r(G)$ onto $C^*_r(H).$ In particular, if $\omega$ is a tracial state on $C^*_r(G)$ such that $\omega(\lambda_s)=0$ when $s\not\in H$, then $\omega\circ E=\omega.$
\[lem:repcondexp\] Let $H\leq G$ be amenable discrete groups. Let $\tau$ be a trace on $G$ that vanishes on $G\setminus H.$ Let $E:C^*(G)\rightarrow C^*(H)$ be the conditional expectation from Lemma \[lem:condexp\]. The map $E_\tau:C_{\pi_\tau}^*(G)\rightarrow C^*_{\pi_\tau}(H)$ given by $E_\tau(\pi_\tau(x))=\pi_\tau(E(x))$ extends to a well-defined $\tau$-preserving conditional expectation onto $C^*_{\pi_\tau}(H).$
Since ${\tau}$ is a tracial state, for any $x \in C^*(G)$, we have $\pi_\tau(x)=0$ if and only if ${\tau}(x^*x)=0.$ Consequently, $\pi_\tau(x)=0$ implies ${\tau}(E(x)^*E(x))\leq \tau(E(x^*x))=\tau(x^*x)=0,$ so $\pi_\tau(E(x))=0.$ Therefore $E_\tau$ is well-defined and $\tau$-preserving by Lemma \[lem:condexp\].\
\
The map $E_\tau$ is clearly idempotent so we only need to check it is contractive. This proceeds as in the case of building conditional expectations in the arena of finite von Neumann algebras. We include the proof for the convenience of the reader.\
\
For each $x\in C^*(G)$, let $x_\tau\in H_\tau$ be the canonical image of $x.$ Since $E$ is $\tau$-preserving, the map $P(x_\tau):=E(x)_\tau$ extends to an orthogonal projection on $H_\tau.$ Since $E(x)\in C^*(H)$, $$\begin{aligned}
{\Vert}\pi_\tau(E(x)) {\Vert}&=\sup_{ y,z\in C^*(H), {\tau}(y^*y)={\tau}(z^*z)=1}|{\langle}\pi_\tau(E(x))y_\tau,z_\tau {\rangle}|\\
&=\sup_{ y,z\in C^*(H),{\tau}(y^*y)={\tau}(z^*z)=1}|{\tau}(z^*E(x)y)|\\
&=\sup_{y,z\in C^*(H), {\tau}(y^*y)={\tau}(z^*z)=1}|{\langle}P(x_\tau),(zy^*)_\tau {\rangle}|\\
&=\sup_{y,z\in C^*(H), {\tau}(y^*y)={\tau}(z^*z)=1}|{\langle}x_\tau,P((zy^*)_\tau) {\rangle}|\\
&=\sup_{y,z\in C^*(H), {\tau}(y^*y)={\tau}(z^*z)=1}|{\langle}x_\tau,(zy^*)_\tau {\rangle}|\\
&\leq\sup_{y,z\in C^*(G), {\tau}(y^*y)={\tau}(z^*z)=1}|{\langle}x_\tau,(zy^*)_\tau {\rangle}|\\
&={\Vert}\pi_\tau(x) {\Vert}.\end{aligned}$$
Let $G$ be a finitely generated nilpotent group and $\tau$ an extreme trace on $G$ such that $\tau(g)=1$ implies $g=e.$ Let $N\leq Z(G)$ be a finite index subgroup of $Z(G)$. Let $\omega$ be the trivial extension of $\tau|_N$ to $G.$ Then there is a central projection $p\in C^*_{\pi_\omega}(G_f)\cap C^*_{\pi_\omega}(G)'$ such that $$C^*_{\pi_\tau}(G)\cong pC^*_{\pi_\omega}(G).$$ \[lemma:cutdown\]
Our hypotheses, combined with [@Carey84 Theorem 4.5], imply that $\tau$ vanishes on $G\setminus G_f.$ By Lemma \[lem:weakcontainment\], the representation $\pi_\omega$ weakly contains $\pi_\tau.$ Therefore the \*-homomorphism $\sigma:C^*_{\pi_\omega}(G)\rightarrow C^*_{\pi_\tau}(G)$ given by $\sigma(\pi_\omega(x))=\pi_\tau(x)$ is well-defined. From Lemma \[lem:repcondexp\] (with $H=G_f$) we have trace-preserving conditional expectations $E_\omega:C^*_{\pi_\omega}(G)\rightarrow C^*_{\pi_\omega}(G_f)$ and $E_\tau:C^*_{\pi_\tau}(G)\rightarrow C^*_{\pi_\tau}(G_f).$ By the definitions of these expectations in Lemma \[lem:repcondexp\] we have, for any $x\in C^*(G)$, $$E_\tau(\sigma(\pi_\omega(x)))=E_\tau(\pi_\tau(x))=\pi_\tau(E(x))=\sigma(\pi_\omega(E(x)))=\sigma(E_\omega(\pi_\omega(x))).$$ Equivalently, the following diagram commutes: $$\label{eq:commute}
\xymatrix{
C^*_{\pi_\omega}(G) \ar[rr]^\sigma \ar[d]^{E_\omega}&& C^*_{\pi_\tau}(G)\ar[d]^{E_\tau}\\
C^*_{\pi_\omega}(G_f) \ar[rr]^\sigma && C^*_{\pi_\tau}(G_f)}$$ By [@Carey84 Proposition 5.1] there is an irreducible representation $\pi$ of $G$ such that $\pi(x)\mapsto \pi_\tau(x)$ defines an isomorphism from $C^*_\pi(G)$ onto $C^*_{\pi_\tau}(G).$ The irreducibility of $\pi$ then implies that $\pi_\tau(z)\in {\mathbb{C}}\cdot 1_{H_\tau}$ for all $z\in Z(G)$.
Using formula we see that $\pi_\omega(n) s_\omega = \omega(n) s_\omega $ for all $s \in c(G/N)$, hence $\pi_\omega(n) = \omega(n) \cdot 1_{H_\omega}$. By Lemma \[lem:findex\], the group $N$ has finite index in $G_f.$ Therefore, $C^*_{\pi_\omega}(G_f)$ is finite dimensional by Lemma \[lem:twistedgroup\].
Since $C^*_{\pi_\omega}(G_f)$ is finite dimensional there is a projection $p\in C^*_{\pi_\omega}(G_f)\cap C^*_{\pi_\omega}(G_f)'$ such that $$\textup{ker}(\sigma)\cap C^*_{\pi_\omega}(G_f)=(1-p)C^*_{\pi_\omega}(G_f). \label{eq:kernel}$$ Let $x\in C^*_{\pi_\omega}(G).$ By Lemma \[lem:repcondexp\], the conditional expectations $E_\omega$ and $E_\tau$ preserve their respective traces. Using this fact we obtain $$\begin{aligned}
x \in \ker(\sigma) & \Longleftrightarrow \tau(\sigma(x^*x))=0 \\
& \Longleftrightarrow \tau(E_\tau(\sigma(x^*x)))=0\\
& \Longleftrightarrow E_\tau(\sigma(x^*x))=0\\
& \Longleftrightarrow\sigma(E_\omega(x^*x))=0 \quad \textup{By (\ref{eq:commute})} \\
& \Longleftrightarrow pE_\omega(x^*x)=0 \quad \textup{By (\ref{eq:kernel})}\\
& \Longleftrightarrow E_\omega(px^*xp)=0\\
& \Longleftrightarrow \omega(E_\omega(px^*xp))=0\\
& \Longleftrightarrow \omega(px^*xp)=0\\
& \Longleftrightarrow xp=0\\
& \Longleftrightarrow x\in C^*_{\pi_\omega}(G)(1-p).\end{aligned}$$ It follows that $(1-p)$ is central. Indeed, for any $x \in C^*_{\pi_\omega}(G),$ we have $(1-p)x=(x^*(1-p))^*\in C^*_{\pi_\omega}(G)(1-p)$ (it follows from the above list of equivalences that $C^*_{\pi_\omega}(G)(1-p)$ is self-adjoint). Then $(1-p)x=y(1-p)$ for some $y\in C^*_{\pi_\omega}(G).$ Consequently, $(1-p)x(1-p)=y(1-p)=(1-p)x.$ It follows that $(1-p)xp=0.$ The same argument applied to $x^*$ shows $px(1-p)=0$, in other words, that $x$ commutes with $p.$
\[thm:structure\] Let $G$ be a finitely generated nilpotent group and $\pi$ a faithful irreducible representation of $G.$ There is a torsion free, finite index subgroup $N\leq Z(G)$, a 2-cocycle $\sigma$ on $G/N$, and a central projection $p\in C^*(G/N,\sigma)$, such that $C^*_\pi(G)\cong pC^*(G/N,\sigma).$ Moreover, $\sigma$ is homotopic to the trivial cocycle.
As in the proof of Theorem \[thm:main\] there is an extreme trace $\tau$ on $G$ satisfying the hypotheses of Lemma \[lemma:cutdown\] such that $C^*_\pi(G)\cong C^*_{\pi_\tau}(G).$
Since $Z(G)$ is a finitely generated abelian group it contains a finite index torsion free subgroup $N.$ Then $\widehat{N}$ is a $d$-torus and in particular path-connected. It follows that the cocycle defined in (\[eq:cocycle\]) for any $\omega\in \widehat{N}$ is homotopic to the trivial cocycle. The conclusion now follows from Lemmas \[lem:twistedgroup\] and \[lemma:cutdown\].
Let $C^*(G/N,\sigma)$ be as in Theorem \[thm:structure\]. Since $G/N$ is amenable, [@Echterhoff10 Theorem 1.9] applies to $C^*(G/N,\sigma).$ In particular, since $\sigma$ is homotopic to the trivial cocycle, we have $K_*(C^*(G/N,\sigma))\cong K_*(C^*(G/N)).$ Since K-theory is additive over direct sums, we feel that Theorem \[thm:structure\] may be beneficial for K-theory calculations of C\*-algebras generated by irreducible representations of nilpotent groups (see for example [@Eckhardt14a] for this idea in action).
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abstract: |
We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame, and that, however, they can be given a rigorous meaning. Then we review the definitions of the canonical and Hilbert energy-momentum tensors from a Lagrangian through the principle of stationary action in a general spacetime. Using relatively elementary mathematics, we prove precise results regarding the definition of the Hilbert tensor field, its uniqueness, and its tensoriality. We recall the meaning of its covariant conservation equation. We end with a proof of uniqueness of the energy density and flux, when both depend polynomially of the fields.\
energy conservation; conservation equation; special relativity; general relativity; Hilbert tensor; variational principle
author:
- |
Mayeul Arminjon\
*Laboratory “Soils, Solids, Structures, Risks”, 3SR\
*(Grenoble-Alpes University & CNRS), Grenoble, France.**
title: 'On the definition of energy for a continuum, its conservation laws, and the energy-momentum tensor'
---
Introduction and summary
========================
The subject of this paper is wide and there is a huge literature about it. The aim of the paper is to give a unified exposition of what, in this author’s view, are the main aspects of the subject, in a relatively short space, while, nevertheless, emphasizing or precising some not widely appreciated facts and providing strict proofs of some less obvious matters (mainly the Hilbert energy-momentum tensor), using not too sophisticated mathematics.\
In non-relativistic classical physics, the concept of energy emerges when one considers the power done (the scalar product of the force by the velocity) on a mass point or a volume element. First, in the schematic case of a mass point in a time-independent potential force field , the energy of the mass point appears from the power equation as a natural conserved scalar quantity: the sum . That quantity is still relevant if the potential depends on time, but it is not constant any more. This is well known. In the more realistic case of a continuous medium subjected to internal forces and to an external force field, the energy is a volume density and it still emerges from the power done. However, in general, the local conservation of energy then appears in the form of a balance equation, though it is one in which there is no source term. That is, energy conservation means that the energy leaving or entering a given domain is exactly identified as a flux going through the boundary surface of the domain. This also is well known — see, e.g., Ref. . We illustrate the emergence of such a true conservation equation for a continuous medium in [**Section \[NG\]**]{} by examining in detail the example of a self-gravitating system of deformable media with isentropic deformation in Newtonian gravity.\
In relativistic theories (including relativistic quantum mechanics), on the other hand, the volume energy density is essentially the component of the energy-momentum-stress tensor, in what follows “the T-tensor" for brevity. The conservation-type equations verified by the T-tensor are discussed in nearly all textbooks about special or general relativity, of course. In [**Section \[T-Minkowski\]**]{}, we recall why the conservation equation verified by the T-tensor in the Minkowski spacetime (see e.g. Landau & Lifshitz or Fock [@Fock1959]) contains two true local conservation equations of the form found in the non-relativistic example of Section \[NG\]; and why one may identify the density and flux in these two equations as those of energy and spatial momentum, respectively. We also note the dependence of the energy density and the energy flux on the reference frame. That dependence is a known fact (although a scarcely mentioned one), but often that fact is not well appreciated. Our new contribution here is to show that these quantities can nevertheless be given a rigorous meaning within a theory of general reference frames and the associated space manifolds.\
As is well known, an expression of the T-tensor may be deduced when a Lagrangian is available, the latter being assumed to govern the relevant system of matter fields via the principle of stationary action (e.g. ). There are two distinct definitions of a T-tensor from a Lagrangian: (i) The so-called “canonical" or “Noether" tensor, say , is a by-product of the Euler-Lagrange equations. (ii) The “Hilbert tensor", say , is the symmetric tensor obtained as the derivative of the Lagrangian density with respect to variations of the (spacetime) metric. In [**Section \[T-tensor\]**]{} we review the definitions of the canonical and Hilbert tensors from a Lagrangian through the principle of stationary action in a general spacetime. We recall two important but seemingly not widely known cases where the “canonical tensor" is, or is not, a tensor. Then we prove precise results regarding the definition of the Hilbert tensor field (). In doing so, we formulate sufficient conditions of regularity for the bounded set in which the action is calculated; we define exact boundary conditions to be verified by the infinitesimal coordinate change; and we give a detailed derivation of the equations. We do not need to use complex notions of fibre bundles. To our knowledge, such a relatively elementary but detailed proof is not available in the literature. Next, we recall the meaning of the standard conservation equation verified by the Hilbert tensor: we argue that one actually needs [*local*]{} definitions of the energy and momentum densities and their fluxes, in short a local definition of the T-tensor, and one needs also a [*local*]{} conservation equation for the energy. We briefly discuss a recent work that proposes a solution to the latter issue. We end Section \[T-tensor\] by stating and proving precise results regarding the uniqueness and the actual tensoriality of the Hilbert tensor (). In particular, we prove that the same variational equation applies when a [*complete*]{} variation of the [*sole*]{} metric is applied: Eq. (\[delta S special-2\]), as when the variation of the metric results from a mere [*coordinate change*]{} (or diffeomorphism): Eq. (\[delta\_S\_coordinate-change-T\]) – although the meaning of these two equations is totally different, e.g. the l.h.s. of (\[delta\_S\_coordinate-change-T\]) is zero for an invariant Lagrangian. We [*prove*]{} that the variational equation (\[delta S special-2\]) characterizes the components of the Hilbert tensor field — whence follows that it is left unchanged by the addition of a four-divergence. We also prove, in detail and by relatively elementary arguments, that this is indeed a tensor. This is just stated in the literature that we consulted, except for Ref. [@GotayMarsden1992] which uses more advanced mathematics. Of course it follows basically from the invariance of the action but, in our opinion, not in a fully trivial way.\
Finally, in [**Section \[Uniqueness E\]**]{} we investigate if the energy equation is unique for a given system of fields, i.e., if the energy density and fluxes can be considered to be uniquely defined. We show that, if the energy density and its flux depend on the fields (both the matter fields and the “long-distance" fields) in a polynomial way, then they are determined uniquely. We show this by considering separately the contributions of matter (including its potential energy in the long-distance fields) and the long-distance fields.
Local energy conservation for an isentropically deformable medium in Newtonian gravity {#NG}
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Local energy balance for the matter fields
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Let us consider a deformable continuous medium, of mass density field , having a general motion (including deformation and rotation), with velocity field , with respect to some inertial frame . I.e., , where is the spatial position associated with an event in the frame , and where is the world line of a given “particle" of the medium, parameterized by the Newtonian time . The internal force field in that medium is assumed to be described by the Cauchy stress tensor field . We assume that this motion takes place in a gravitational field, with Newtonian gravity potential . Newton’s second law for a volume element of the medium writes then: \[Newton continuum\] , where ${\textcolor{noir}{\frac{\dd}{\dd t}}}$ means the “material" (or “total") derivative: for a vector, [^1] ). The power (per unit volume) is got by taking the scalar product of (\[Newton continuum\]) with the velocity . On the new l.h.s., we have , while on the r.h.s. we note that , where is the strain rate tensor. [^2] We note also that . Still, we assume that the deformation of the continuum is isentropic, which means that the power of the internal forces is stored as the rate of elastic energy: \[Isentropy\] , with the mass density of internal (elastic) energy in the continuous medium. This assumption applies, in particular, to any [*elastic medium,*]{} and also [@Fock1959] to any [*barotropic perfect fluid,*]{} as is commonly assumed in astrophysics — although a barotropic fluid is not plainly an elastic medium in the sense that it does not have a reference configuration. We thus get: . It suggests itself to put together the terms containing an exact total derivative: \[rho de\_m/dt\] . On the r.h.s., we have a source term due to the external force field, plus a flux term. On the l.h.s., we have \[Def e\_m\] . Using the continuity equation that expresses the mass conservation: \[Continuity Eqn\] , we get easily the well known fact that (for whatever scalar function , actually): \[Particular -> Balance\] . From (\[Def e\_m\]) and (\[Particular -> Balance\]), we may rewrite (\[rho de\_m/dt\]) as \[Newton Matter Energy Balance\] , with \[w\_m NG\] and \[Phi\_m NG\] . That is, we got a balance equation with an external source term on the r.h.s. The scalar field is thus the [*volume energy density of matter,*]{} including its potential energy in the gravitational field, and the spatial vector field is thus (the surface density of) the [*matter energy flux*]{}. Equation (\[Newton Matter Energy Balance\]) can be found in the literature: see Eq. (66.11) in Fock [@Fock1959]. But its detailed derivation illustrates well the emergence of a balance equation for a continuous medium.
Balance for the gravitational field and local energy conservation equation
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Now we assume that all of the matter that produces the gravitational field is indeed in the form of isentropically deformable continuous media. (Of course, the characteristics of the media may vary in space.) Thus the point-dependent mass density is just the source of the gravitational field. It therefore obeys the gravitational field equation, i.e. the Poisson equation: \[Poisson\] . By using Cartesian coordinates, for which we have , one checks easily that Eq. (\[Poisson\]) implies the following: \[Newton Grav Energy Balance\] , where \[Def w\_g\] is the [*volume energy density of the gravitational field,*]{} and \[Def Phi\_g\] is the [*gravitational energy flux*]{}. Equation (\[Newton Grav Energy Balance\]) may be termed the energy balance equation of the gravitational field. Like (\[Newton Matter Energy Balance\]), this also is a balance equation with a source term. The source term in Eq. (\[Newton Grav Energy Balance\]) is just the opposite of the source term in (\[Newton Matter Energy Balance\]). Therefore, combining (\[Newton Matter Energy Balance\]) with (\[Newton Grav Energy Balance\]), we get the [*local energy conservation equation in Newtonian gravity*]{} [@A15]: \[Local Conserv E\] , with the total energy density and the total energy flux . Equation (\[Local Conserv E\]) is the standard form for a true local conservation of energy in a continuum, the definition of the field variables $w$ and ${{{\boldsymbol{\Phi}}}}$ depending on the particular theory. It has essentially the same form as the continuity equation (\[Continuity Eqn\]). There is also a local conservation equation for momentum in Newtonian gravity, and global (integral) conservation laws can also be derived; see e.g. Refs. [@A15; @Chandra1969]. Strangely enough, however, we did not see in the literature the local Eq. (\[Local Conserv E\]) for an elastic medium or a barotropic fluid in Newtonian gravity \[thus with the definitions (\[w\_m NG\])–(\[Phi\_m NG\]) and (\[Def w\_g\])–(\[Def Phi\_g\])\]. For instance, it is not there in the references quoted in the present paper. (The component of Eq. (13) in Ref. [@Chandra1969] is just the continuity equation (\[Continuity Eqn\]), with being indeed the (Newtonian) density of mass; thus, it is not the conservation equation for the Newtonian energy, but the one for the mass.)
Local conservation equations and the energy-momentum tensor in Minkowski spacetime {#T-Minkowski}
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Recall that the energy-momentum tensor of a continuum or a system of fields is a second-order spacetime tensor field , preferably symmetric. In the Minkowski spacetime, verifies the [*local conservation equation*]{} \[True\_Cons\] (). It is easy to see that (\[True\_Cons\]) is the conjunction of two conservation equations having the standard form (\[Local Conserv E\]). One is precisely the scalar conservation equation (\[Local Conserv E\]): \[Local Conserv E-2\] , in which now (the tensor being taken in mass units as in Fock [@Fock1959]) \[Def w Phi\] (. The other conservation equation involved in (\[True\_Cons\]) and having the form (\[Local Conserv E\]) is a (spatial) vector equation: \[Local Conserv P\] , where \[Def P Sigma\] . We may integrate either of the two conservation equations (\[Local Conserv E-2\]) and (\[Local Conserv P\]) in [*any*]{} bounded spatial domain (the integrability in an unbounded domain being not guaranteed). This gives us two integral conservation equations: \[Conserv E\] , and \[Conserv P\] . Thus in Eqs. (\[Conserv E\]) and (\[Conserv P\]), the change on the l.h.s. is due to the flux through the boundary on the r.h.s.. The scalar is interpreted as the volume density of energy, and the spatial vector is interpreted as the volume density of momentum . Therefore, in view of (\[Conserv E\]) and (\[Conserv P\]), the spatial vector is interpreted as the surface density of the energy flux, and the spatial tensor is interpreted as the surface density of the momentum flux. This interpretation may be justified in several ways, notably the following two [@Fock1959]:\
— First, by examining the non-relativistic limit for a barotropic perfect fluid or an elastic solid. In the second approximation, , and in Eqs. (\[Def w Phi\]) and (\[Def P Sigma\]) have then the following expressions ([@Fock1959], Sect. 32): , , . Therefore, at the first approximation, the special-relativistic local energy conservation (\[Local Conserv E-2\]) reduces to the continuity equation (\[Continuity Eqn\]), and at the second approximation it expresses the conservation of that rest-mass energy corrected by adding the conserved Newtonian energy \[Eq. (\[Local Conserv E\]) with \]. Also, at the first approximation, the special-relativistic momentum conservation (\[Local Conserv P\]) reduces to the Newtonian momentum conservation equation in the absence of external field: \[Local Conserv P-Newt-Free\] . — Second, by recognizing in and , for the electromagnetic field, the usual definition of the electromagnetic energy density and the Poynting vector from the relevant expression of ${{{\boldsymbol{T}}}}$ ([@Fock1959], Sect. 33).\
One does not use the symmetry of the tensor to derive Eqs. (\[Local Conserv E-2\])–(\[Local Conserv P\]) and (\[Conserv E\])–(\[Conserv P\]). If that symmetry is true, it implies that is true generally: the density of energy flux is equal to times the density of momentum. The same Eqs. (\[Conserv E\]) and (\[Conserv P\]) apply also to Newtonian gravity, as follows from Eqs. (\[Local Conserv E\]) and the gravitational extension [@A15; @Chandra1969] of (\[Local Conserv P-Newt-Free\]).\
As is easy to check, under a purely spatial change of the chart (coordinate system): \[purely-spatial-change\] ),, the energy density \[Eq. (\[Def w Phi\])$_1$\] is an invariant scalar, while \[Eq. (\[Def w Phi\])$_2$\] and \[Eq. (\[Def P Sigma\])$_1$\] transform indeed as spatial vectors; and \[Eq. (\[Def P Sigma\])$_2$\] transforms indeed as a spatial tensor. One may give a rigorous geometric meaning to such “spatial" objects by defining a relevant [*space manifold*]{} $\mathrm{M}_\mathrm{F}$, as follows [@A44]. In a [*general*]{} spacetime, one can formally define a [*reference frame*]{} as being an equivalence class of charts having the same domain of definition (an open subset of the spacetime manifold ) and exchanging by a coordinate change (“transition map") having the form (\[purely-spatial-change\]). Let $P_S: \mathbb{R}^4 \rightarrow \mathbb{R}^3, {\bf X}:= (x^\mu )\mapsto {\bf x}:= (x^j )$, be the “spatial projection". The elements (points) of the space manifold $\mathrm{M}_\mathrm{F}$ are the [*world lines*]{}, each of which is the set of events that have a given spatial projection ${\bf x}$ in some chart $\chi: \mathrm{U} \rightarrow \mathbb{R}^4,\ X\mapsto {\bf X}$, belonging to the class $\mathrm{F}$. I.e., a world line $l$ is an element of $\mathrm{M}_\mathrm{F}$, iff there is a chart $\chi \in \mathrm{F}$ and a triplet ${\bf x} \in P_S(\chi (\mathrm{U}))$, such that $l$ is the set of all events $X$ in the domain U, whose spatial coordinates are ${\bf x}$: \[l-in-M-by-P\_S\] l := {X; P\_S((X))=[**x**]{}}. It results easily from (\[purely-spatial-change\]) that (\[l-in-M-by-P\_S\]) holds true then in any chart $\chi' \in \mathrm{F}$, of course with the transformed spatial projection triplet ${\bf x}'= {\bf f}({\bf x}):= (f ^j({\bf x}))$ [@A44]. For any chart $\chi \in \mathrm{F}$, one defines the “associated chart" as the mapping which associates, with a world line $l\in \mathrm{M}_\mathrm{F}$, the constant triplet of the spatial coordinates of the events $X\in l$: \[def-chi-tilde\] : \^3,l X l, P\_S((X))=[**x**]{}.\
The set is endowed with a natural structure of three-dimensional [*differentiable manifold,*]{} of which the basic atlas is made of the associated charts $\widetilde{\chi }$, where $\chi $ is any chart belonging to the reference frame $\mathrm{F}$ [@A44]. The “spatial" objects defined above: the scalar , the vectors , , and the tensor , are simply and rigorously tensor fields on the manifold . (Of course they have in general, in addition, a dependence on the time coordinate $x^0$, thus they are, strictly speaking, one-parameter families of tensor fields on .) Fixing a reference frame in this sense can be done, for instance, by choosing one local coordinate system (chart $\chi $) on the spacetime, with its domain of definition $\mathrm{U}$: the corresponding reference frame is then the equivalence class of this chart. As soon as one has fixed a reference frame, then Eqs. (\[Local Conserv E-2\]) and (\[Local Conserv P\]), as well as Eqs. (\[Conserv E\]) and (\[Conserv P\]), are coordinate-free equations on the space manifold . In particular, the bounded spatial domain is an open subset of the manifold , having a regular boundary , so that the divergence theorem applies. (See Appendix \[Domains\] for a precise definition of the needed regularity.)\
On the other hand, if one makes a general coordinate change, for which the change in the spatial coordinates depends on the time coordinate (already if one makes a Lorentz transformation transforming the Cartesian system into another one, but with a non-zero “boost"), then (\[Def w Phi\]) defines completely different quantities and , as compared with the initial ones. The same is true for and as defined by (\[Def P Sigma\]). This means that [*there is one definition of the energy and momentum (and their fluxes) per reference frame.*]{} It is not specific to special relativity. Indeed the energy depends on the reference frame. This is true in non-relativistic physics (e.g. [@A15]) — as may be checked here on the fact that and defined in Eqs. (\[w\_m NG\])–(\[Phi\_m NG\]) involve the velocity ${\bf v}$ that depends on the inertial frame, whereas are Galilean invariants. It is also true in relativistic physics, and also in a general spacetime, and be it for the classical or the quantum-mechanical energy [@A51].
Definition of the energy-momentum tensor from a Lagrangian {#T-tensor}
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Lagrangian and stationary action principle
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We assume that the equations of motion for some “matter fields" derive from a Lagrangian through the [*principle of stationary action*]{} in a general spacetime:
\[delta\_S=0\] , . Here, is the boundary, assumed smooth, of some bounded open set in the spacetime, and is the action: in some chart whose domain of definition contains , it writes \[S general\] , where , is the local expression of the field in the chart , and , the ’s being the components of the metric tensor in the chart ; note that is an open subset of . Thus, the field has real components (or complex components for a complex field, with replaced by ). At this stage we do not need to know the exact geometric nature of the fields: whether they are scalars, vectors, more general tensors, or otherwise. We just assume that, on changing the chart: , the local expression of each of them has some definite transformation law, say , [^3] and that the Lagrangian is then invariant under the coordinate change: \[L invariant\] . That invariance has to be true at least when the chart belongs to some well-defined class and implies that the same invariance is valid for the action (\[S general\]). In this section, we shall consider the usual case that [*all*]{} charts (in the atlas of the spacetime manifold) are allowed, i.e., we shall discuss generally-covariant theories. However, it also makes sense to consider instead the class associated with a particular (“privileged") reference frame. Thus, the Lagrangian is a smooth real function , where is the coordinate vector specifying the spacetime position, , and also for . These five vectors of specify the values that may be taken at by the local expression of the field and its partial derivatives. This means that, in the expression (\[S general\]) of the action, one makes the substitution \[Assign q\^A q\^A\_mu\] . Note also that , with , is the invariant four-volume element on the spacetime \[thus for a Lorentzian metric on the four-dimensional spacetime\].\
The stationarity (\[delta\_S=0\]) is equivalent to the Euler-Lagrange equations {see e.g. ; is defined from a Gateaux derivative, as with Eq. (\[Def delta S\]) below}. In a general spacetime, the latter equations write [@Leclerc2006b]: \[E-L general\] in , with the implicit assignment (\[Assign q\^A q\^A\_mu\]). The domain of definition of the coordinate system now has to contain not only but also the boundary , because the derivation of (\[E-L general\]) needs to use the divergence theorem.
The “canonical" (or “Noether") T-tensor
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We shall give only a very brief account (see e.g. ). This object has the following expression in a given chart: \[tau tensor\] . When and does not depend explicitly on the spacetime position, this object occurs naturally from the derivation of the Euler-Lagrange equations (\[E-L general\]), which imply that it verifies the desired local conservation equation . However, such an independence happens in practice only in a flat spacetime. Moreover, in fact, this object is not necessarily a tensor — even in a flat spacetime, cf. the case of the electromagnetic field [@Leclerc2006b]: being the field tensor and the 4-potential, we have \[Canonical\_em\] . (Henceforth, indices are raised or lowered with the spacetime metric.) On the r.h.s., everything, [*but*]{} , is tensorial, hence is not a tensor; i.e., does not transform as a $(1\ 1)$ tensor for general coordinate changes. Of course this does not mean that there is no energy-momentum tensor for the electromagnetic field. (The Hilbert tensor indeed does the job, see e.g. .) But it proves that the “canonical tensor" is not necessarily a tensor. This is not often noted, e.g. it is not in Refs. , probably because (\[Canonical\_em\]) does behave as a $(1\ 1)$ tensor for linear coordinate changes, as are the Lorentz transformations to which one often restricts oneself in special relativity. In a general spacetime, is a tensor for a scalar field [@Leclerc2006b] — and also for the Dirac field [@A48].
Hilbert’s variational definition of the T-tensor {#Hilbert Tensor}
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While following the line of the classic derivation by Landau & Lifshitz , we will include many mathematical details which appear necessary in that derivation and that, for the most part, we did not find in the literature that we consulted. By this, we do not mean the geometric formulation of the physical fields as sections of appropriate fibre bundles, which has been implemented in Ref. [@GotayMarsden1992], among others, and rather extensively in Ref. [@ForgerRoemer2004] — and which we will not need. \[See the remarks following the definition (\[S general\]) of the action.\] Instead, we mean the precise definition of “the variation of the action under an infinitesimal diffeomorphism", the regularity of the boundary and the exact boundary conditions, and a clear derivation of the main formulas.\
One considers a given chart and one imposes a small change to it: . As we shall see, the domain of definition of must include the closure of the bounded open set in which one computes the action (\[S general\]), and we must assume that at the events which do not belong to . (Alternatively, one may regard the mapping defined in coordinates by as a diffeomorphism of the spacetime manifold , which coincides with the identity map for .) Thus, with any smooth vector field that vanishes if , and . That is, we change the chart for a new chart given by \[Def chi\_epsilon\] ,. (The vector field is the local expression of in the chart .) After such a coordinate change, the local expressions of the fields change, each according to its specific transformation behaviour, and the domain also changes, so both the integrand and the integration domain change in the action (\[S general\]), which thus takes [*a priori*]{} a different value. One seeks to calculate the first-order term, as , in the variation of the action, being considered as a functional of the field . This amounts to calculate the Gateaux derivative of in the direction of the vector field : \[Def delta S\] . The bounded open set is assumed to be an [*open domain*]{} in the sense of in Appendix \[Domains\], in particular it has a smooth boundary . Because is an open domain, the assumption “ if " means exactly that the support of is included in the closure ( to ). [^4] Since , this implies that all derivatives of also vanish if . It follows that the corresponding change in , as determined to the first order in {Eq. (94.2) in Ref. }: \[delta g\] , also vanishes if . \[Here, the ’s are the components of the inverse of the metric’s component matrix .\] Thus, in particular, on . [^5]
One assumes moreover that the matter Lagrangian depends on the matter fields and the metric and their first-order derivatives, but not on their higher-order derivatives. The same applies then to the “Lagrangian density" , i.e., the latter is a smooth real function . To calculate the action (\[S general\]), in the Lagrangian the values (\[Assign q\^A q\^A\_mu\]) are assigned to and . In addition, now one assigns the values and to and respectively. At this stage, it is usually admitted that the change in the action (\[S general\]), , is given by \[delta\_S\_coordinate-change-0\] , without a justification nor a precise definition of , , etc. {Actually, the step (\[delta\_S\_coordinate-change-0\]) and some later intermediate steps are even skipped by Landau & Lifshitz .} However, recall that here not only the integrand but also the integration domain change in the action (\[S general\]), so the first-order variation of the integral involves [*a priori*]{} more than just the first-order approximation of the integrand. Let us be more precise. Denote provisionally the list of all fields (matter fields and metric) by . Applying the definition (\[S general\]) with the new chart (\[Def chi\_epsilon\]), we write \[S\_epsilon\] , with , , , where is the local expression of the field in the chart . The integral (\[S\_epsilon\]) has a form that is well known, in particular, in continuum mechanics. The expression of its derivative is also well known: \[d S\_epsilon/d epsilon\] , where , with , is the “velocity field" at “time" . In particular, we get from (\[Def chi\_epsilon\]) and these definitions that . Therefore, applying the divergence theorem in (\[d S\_epsilon/d epsilon\]) and since on (hence on , because ), we get \[d S\_epsilon/d epsilon-2\] & &\
& = & Hence, defining simply the variations of the fields to the first order in : \[Def dphi dphi\_mu\] , and remembering that the ’s are the matter fields for , while is the metric field , we see that Eq. (\[d S\_epsilon/d epsilon-2\]), together with the definition (\[Def delta S\]), proves the “obvious" Eq. (\[delta\_S\_coordinate-change-0\]). We see then from Eq. (\[d S\_epsilon/d epsilon\]) that Eq. (\[delta\_S\_coordinate-change-0\]) is in general [*false*]{} if the boundary condition on is not valid.\
If we write , and if we remember that is the local expression of the field in the chart (\[Def chi\_epsilon\]), that follows the flow of the vector field at small values of , we recognize (at least in the case that is a tensor field) the definition of the Lie derivative [@Doubrovine1982] — or rather, of its opposite. That is, \[d phi Lie\] . We note also that \[d\_mu dphi/deps =d/deps d\_mu phi\] , hence \[dphi\_mu = d\_mu dphi\] . In practical terms, can be computed in two steps : first, one computes the difference between the local expressions in the initial and the modified chart at points that [*correspond*]{} together through the transition map, i.e. . Second, one uses a first-order Taylor expansion to find the difference , i.e., at one and the same point. For instance, this gives indeed the expression (\[delta g\]) for the metric . As another example, consider a vector field , with components in the initial chart. We find without difficulty \[delta V\^mu\] , where the second equality occurs due to the symmetry of the Christoffel symbols (i.e., due to the fact that the Levi-Civita connection has no torsion). This does coincide with the Lie derivative definition (\[d phi Lie\]).\
In a last step, let us assume that the matter fields obey the Euler-Lagrange equations (\[E-L general\]). Just like for the derivation of the latter equations from the stationarity of the action (\[S general\]): because in view of (\[dphi\_mu = d\_mu dphi\]), one may transform the second term in (\[delta\_S\_coordinate-change-0\]) and use the divergence theorem to make a surface term appear in it; and that surface term vanishes because on \[as can be checked for a vector on (\[delta V\^mu\]) and for the contravariant metric tensor on (\[delta g\])\]. It then follows from (\[E-L general\]) that the first two terms in (\[delta\_S\_coordinate-change-0\]) cancel one another, thus [^6] \[delta\_S\_coordinate-change-1\] , . In the same way, because , one may transform the second term in (\[delta\_S\_coordinate-change-1\]) and use the divergence theorem to make a surface term appear in it, and that surface term vanishes because on . One thus gets : \[delta\_S\_coordinate-change\] . We have thereby proved the following:
#### Theorem 1. {#Theorem 1}
[*Assume that the bounded subset of the spacetime is an open domain in the sense of . In the domain of some chart , define a one-parameter family of new charts by (\[Def chi\_epsilon\]), associated with a smooth vector field that is defined over and vanishes for . Assume that the smooth matter Lagrangian has the form . Then:*]{}\
(i) [*The first-order variation of the action, defined in Eq. (\[Def delta S\]), is given by Eq. (\[delta\_S\_coordinate-change-0\]). In this equation, the variations and , with are defined by Eq. (\[Def dphi dphi\_mu\]) or equivalently by Eqs. (\[d phi Lie\]) and (\[dphi\_mu = d\_mu dphi\]).*]{}\
(ii) [*If the matter fields obey the Euler-Lagrange equations (\[E-L general\]), then is given by Eq. (\[delta\_S\_coordinate-change\]) right above.* ]{}\
Equation (\[delta\_S\_coordinate-change\]) leads one to define an object (usually called “Hilbert energy-momentum tensor", though not in Ref. ) by its components : \[Hilbert tensor\] The symmetry of this object: , follows from the symmetry of the metric [*and*]{} the invariance of under general coordinate changes. [^7] As shown by : For the object whose components are defined by Eq. (\[Hilbert tensor\]), we have “on shell" for any regular bounded open set and for any coordinate change such that vanishes for : \[delta\_S\_coordinate-change-T\] . If the Lagrangian function is [*invariant under general coordinate changes,*]{} then the action in Eq. (\[S general\]) is invariant too, hence the change given by Eq. (\[delta\_S\_coordinate-change\]) or (\[delta\_S\_coordinate-change-T\]) is zero for any possible coordinate change. Assume, moreover, that the object given by (\[Hilbert tensor\]) turns out to be indeed a tensor. (This tensorial character does not seem to be proved in the literature, but see Subsect. \[Unique Hilbert tensor\] below.) Then, using the expression (\[delta g\]) of in terms of the vector field , and since by assumption the latter vanishes on , one gets from (\[delta\_S\_coordinate-change-T\]) : \[Semicolon\_Cons\] .
Is Eq. (\[Semicolon\_Cons\]) a true conservation equation?
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In contrast with (\[True\_Cons\]) \[, with partial derivatives\], Eq. (\[Semicolon\_Cons\]) \[with covariant derivatives\] “does not generally express any conservation law whatever" . Fock [@Fock1959] used similar words: he noted that the four scalar equations contained in (\[Semicolon\_Cons\]) “do not by themselves lead to conservation laws". To explain it quickly, the presence of covariant derivatives gives to Eq. (\[Semicolon\_Cons\]) the form of (\[True\_Cons\]) [*plus source terms,*]{} which are the terms linear in the T-tensor itself (that involve the connection coefficients). Nevertheless, Eq. (\[Semicolon\_Cons\]) can be rewritten in the form of (\[True\_Cons\]) after introducing some “pseudo-tensor of the gravitational field" . But the definition of is not unique. And behaves as a tensor only for linear coordinate transformations. As a result, it is generally agreed that Eq. (\[Semicolon\_Cons\]) can lead only (under special assumptions, e.g. an asymptotically flat spacetime) to [*global*]{} conservation laws, see e.g. . However, in order to be able to investigate the energy balance in any spatial domain, one would need to know uniquely the relevant energy density and its flux. And one would need that they obey a true and local conservation equation. (This is indeed the case in most fields of physics, e.g. in mechanics, thermodynamics, electrodynamics, chemistry, ..., as well as in Newtonian gravitation — as shown in Sect. \[NG\] — and also in several alternative relativistic theories of gravitation in a flat spacetime, e.g. [@Rosen1963; @LogunovMestvirishvili1989; @Petry2014], including a preferred-frame scalar theory with a mechanism for gravity [@A35].)\
What is thus lacking in theories based on Eq. (\[Semicolon\_Cons\]), which include general relativity and its numerous variants or extensions, is not merely an exact local concept of the [*gravitational*]{} energy. As we recalled, in special relativity the local conservation equation (\[True\_Cons\]) for the energy-momentum tensor can be rewritten as two local conservation equations of the type (\[Local Conserv E\]): a scalar one for the energy and a vector one for the three-momentum. We believe that the local concept of energy is indissolubly bound with the existence of a true local conservation equation of the type (\[Local Conserv E\]), as it exists both in non-relativistic physics and in special relativity. Since, as we recalled, the rewriting of Eq. (\[Semicolon\_Cons\]) as an exact local conservation equation having the form (\[True\_Cons\]) is neither tensorial nor unique, we consider that Eq. (\[Semicolon\_Cons\]) does not provide an exact local concept for any form of [*material energy,*]{} either: assuming the definition of the Hilbert tensor is unique (that is proved in Subsect. \[Unique Hilbert tensor\]), one could define the material energy density as, say, . But another one could prefer to choose , and a third one, . In the absence of an exact local conservation equation of the type (\[True\_Cons\]), nobody can tell which choice is correct. Another point is of course that, for a given bounded spatial domain , the time evolution of, say, the integral of \[the l.h.s. of Eq. (\[Conserv E\])\] is not equal to a flux through the boundary , but is also affected by source terms depending on the gravitational field, for which there is no univoque definition of the energy-momentum tensor.\
However, according to Padmanabhan [@Padmanabhan2015], there exists a suitable definition for what we will call an [*energy current*]{} four-vector of the gravitational field, such that the total energy current is conserved: \[Conserv calP\] , where the four-vector field , with components \[Matter current-GR\] , is the matter energy current associated with the matter distribution, for the observer(s) having the four-velocity field . {The plus sign is with the $(+ - - -)$ signature that we use. This four-vector should not be confused with the three-vector in Eq. (\[Def P Sigma\])$_1$. The gravitational current of Ref. [@Padmanabhan2015] also depends on .} Since, in coordinates adapted to the reference fluid defined by , we have [@A44; @Cattaneo1958] \[Vmu\] , we note that, in any such adapted coordinates: . In particular, in the Minkowski spacetime, and taking for the four-velocity field of some inertial reference frame, the gravitational current vanishes, hence the conservation (\[Conserv calP\]) of the (matter) current is exactly the same equation as the local energy conservation (\[Local Conserv E-2\]) with the definitions (\[Def w Phi\]). Whence our use of the expression “matter energy current" to designate the four-vector field (\[Matter current-GR\]). Thus, according to the result [@Padmanabhan2015], the situation regarding the energy and momentum conservation would be nearly the same in GR as it is in the ether theory [@A35], namely: (i) there is a (scalar) local conservation equation of the type (\[Local Conserv E\]) for the total (material plus gravitational) energy; (ii) the energy density and its flux involved in that conservation equation depend on the reference frame; [^8] (iii) there is no equivalent local conservation equation for the spatial momentum \[by this we mean a (frame-dependent) space vector equation, thus three scalar equations, as Eq. (\[Local Conserv P\])\]. Anyway, as discussed in Ref. [@A35], the conservation of the total momentum, when it takes place, precludes a conservation of the momentum of [*matter*]{}.
Uniqueness and tensoriality of the Hilbert tensor {#Unique Hilbert tensor}
-------------------------------------------------
In addition to the difficulty described in the foregoing subsection, which did not seem solvable in the framework of the said theories (but may be solved by the work [@Padmanabhan2015]), there is a point that needs clarification. In a curved spacetime, the Hilbert tensor field is taken as the [*source of the gravitational field*]{} — in general relativity and in many other relativistic theories of gravity. Clearly, that source has to be [*locally*]{} defined: it is not the global value (the space integral) of that matters to determine the gravitational field, but indeed the distribution of its local value. However, could not the Hilbert tensor be subject to “relocalizations", due to the fact that the Lagrangian determining the equations of motion is not unique?\
Let us add to the Lagrangian a total divergence: with \[add div to L\] , where is a spacetime vector field. \[Of course, the partial derivatives in Eq. (\[add div to L\]) apply once the relevant fields have been substituted for the arguments of , see before Eq. (\[delta\_S\_coordinate-change-0\]).\] Then the Euler-Lagrange equations (\[E-L general\]) stay [*unchanged,*]{} see e.g. Ref. [@Olver2012]. Note that, of course, the modified Lagrangian is also an invariant scalar if is. But, [*a priori*]{}, should not the T-tensor generally change? This would indirectly contradict a statement of Forger & Römer [@ForgerRoemer2004], according to which the energy-momentum tensor field “ is the rank 2 tensor field on space-time depending on the fields of the theory which satisfies \[delta\_S\_metric-change-T\] for every compact subset of and for every variation of the metric tensor with support contained in ". \[Here, is the dimension of the spacetime, thus as far as we know. For us, is the compact closure of the bounded [*open domain*]{} , ; see Note \[K vs U\].\] Indeed, from this statement one easily concludes that a total divergence does not change {see [@ForgerRoemer2004], and see a detailed proof here around Eq. (\[S special-div\])}. However:\
([*a*]{}) It is not precised what is meant exactly by a “variation of the metric tensor" and what is meant exactly by (applied to the action integral) in the statement reproduced above, included in Theorem 4.2 of Ref. [@ForgerRoemer2004]. In the arguments (pp. 360–361) which lead the authors to state that theorem, the same situation is considered as in Subsect. \[Hilbert Tensor\] hereabove. I.e., the variation of the metric occurs due to an infinitesimal coordinate change (or equivalently due to an infinitesimal diffeomorphism), generated by a [*vector field*]{} (noted in Ref. [@ForgerRoemer2004]), with support contained in . (The variation of the metric is thus appropriately noted in these arguments, and the variation of the action is noted .) It is in this precise situation that one can at the same time derive the basic equation (\[delta\_S\_coordinate-change\]) and state that actually — which is needed to derive the “covariant conservation" (\[Semicolon\_Cons\]).\
([*b*]{}) In that situation \[i.e., the variation of the metric resulting thus from a coordinate change\], Eq. (\[delta\_S\_metric-change-T\]) is equivalent to our Eq. (\[delta\_S\_coordinate-change-T\]). However, we do not see how it could be the case that the validity of Eq. (\[delta\_S\_coordinate-change-T\]) “for every \[regular\] compact subset of and for every variation of the metric tensor with support contained in " would [*characterize*]{} (determine) some field object having components . Indeed, since the action is invariant under coordinate changes, it follows that the l.h.s. of (\[delta\_S\_coordinate-change-T\]) is automatically [*zero* ]{} (for every \[regular\] compact subset etc.). Hence, for example, [*is a solution of (\[delta\_S\_coordinate-change-T\]) (for every \[regular\] compact subset etc.), as well as is (\[Hilbert tensor\])*]{}.\
([*c*]{}) In that same situation, the following argument in Ref. [@ForgerRoemer2004] does not work. According to this argument, “ does not change when is modified by the addition of a total divergence (...), simply because the addition of such a term does not affect the l.h.s. of \[Eq. (\[delta\_S\_metric-change-T\])\]." The last statement is true, but since in that situation the l.h.s. of Eq. (\[delta\_S\_metric-change-T\]) is always zero, it can not prove that does not change.\
([*d*]{}) Therefore, it seems that, instead of the foregoing situation, the “variation of the metric tensor" alluded to in Theorem 4.2 of Ref. [@ForgerRoemer2004] be a variation of the metric itself (thus even in a fixed coordinate system): \[delta g per se\] , where the field , with components , is a given field of symmetric $(2\ 0)$ tensors defined on . (The matter fields are thus left unchanged.) However, if that is indeed the case, then Eq. (\[delta\_S\_metric-change-T\]) has a different meaning than Eq. (\[delta\_S\_coordinate-change-T\]): for example, the variation of the metric now depends on the ten independent parameters instead of merely the four parameters as is the case in the arguments which lead to the statement of Theorem 4.2 in Ref. [@ForgerRoemer2004], as well as in Subsect. \[Hilbert Tensor\] hereabove. Thus, the validity of Eq. (\[delta\_S\_metric-change-T\]) in that different situation has to be proved separately. This proof takes Points (i) and (ii) of the following theorem, whose conclusive part is its Point (iii).
#### Theorem 2. {#Theorem 2}
[*Let be a Lagrangian that is defined and smooth whenever the determinant of the matrix is negative, and that is invariant under general coordinate changes. Let be a bounded open domain of the spacetime manifold and let be its compact closure. Let be the vector space of the symmetric tensor fields which are defined and continuous on and which are on . Let be the subset of made of the tensor fields such that, for any chart defined in a neighborhood of* ]{} (assuming there does exist such charts), [*we have over the domain , where is the component matrix . The matter fields being given functions which are defined and continuous on and which are on , define an invariant functional on by*]{} \[S special\] .
\(i) [*Given any two tensor fields and , there is a number such that, for , the tensor field is in . We have:*]{} \[delta S special\] ,\
(ii) [*If , , and when , then we have, in any chart whose domain contains :*]{} \[delta S special-2\] ,\
(iii) [*Equation (\[delta S special-2\]) determines uniquely the continuous functions , . It follows that the “Hilbert tensor" with components (\[Hilbert tensor\]) is not modified by the addition of a four-divergence. Moreover, this is indeed a tensor field.* ]{}\
As a preliminary, recall that, if the matrix of some tensor field in one chart verifies over , then the corresponding matrix in any other chart in the atlas of , whose domain also contains , verifies over : indeed the Jacobian matrix is invertible for these two compatible charts, and we have . Also remind that the invariance of the functional (\[S special\]) under the change of the chart follows from the definition of a Lagrangian that is invariant under general coordinate changes, Eq. (\[L invariant\]), and from the invariance of the four-volume measure .\
(i) Let and , thus in particular these are two continuous functions defined over the compact set . The real function is defined and over the vector space of the real matrices. Choose a chart defined in a neighborhood of . For , we note the matrix . Let us note also . Since , we have for . Because is a continuous function over the compact , it is bounded and reaches its bounds. Hence, for some number , we have for . The set of matrices is compact, hence we have . Also, . Therefore, we have for any and any : . Hence, there is a number such that, for , we have for any . Thus, for , we have , as announced. Since that statement does not depend on the chart, the number does not depend on the chart, either, thus .\
Denoting henceforth for brevity, we define a function from into by setting \[Def f\] . Using the definition (\[delta g per se\]), we have: , . Therefore, we get \[df/d eps\] & = &\
& = & . From the definitions (\[S special\]) and (\[Def f\]), we have , hence \[d S/d eps\] . Since at we have , Eq. (\[delta S special\]) follows from Eqs. (\[df/d eps\]) and (\[d S/d eps\]).\
(ii) The second term in the integrand on the r.h.s. of (\[delta S special\]) can be written as \[surface term\] & = &\
& & .The first term on the r.h.s. of (\[surface term\]) is a divergence in and its integral on vanishes if on , thus if on . \[Remind: .\] Hence, in that case, (\[delta S special\]) rewrites as \[delta S special-3\] . In view of Eq. (\[Hilbert tensor\]), this is Eq. (\[delta S special-2\]).\
(iii) Consider a given tensor field and, in a given chart whose domain contains , let and be two sets of functions () defined and continuous over , each set being symmetric, such that both verify Eq. (\[delta S special-2\]) for any tensor field that vanishes on . \[We do [*not*]{} assume that either or is given by Eq. (\[Hilbert tensor\]).\] We claim that over . Denoting , we thus have for any such tensor field : \[integral T=0\] . Consider a given pair of indices. Let be any real function which is defined and continuous over , which is over , and that has compact support . Hence, implies . If , we have since is open, hence . Therefore, by setting in the chart , we define a tensor field such that for . We can thus apply (\[integral T=0\]) to get \[integral T=0 fixed indices\] = . (The second equality follows from the symmetry of and .) Since this is true for any such function , we deduce that is zero almost everywhere in . But since this is a continuous function, it is zero everywhere in the open set , and therefore it is zero also in its closure . Then, since over , we have over . This proves our precise statement about uniqueness at the beginning of this paragraph.\
Now suppose the Lagrangian is a four-divergence: with . Then the integral (\[S special\]) rewrites as: \[S special-div\] (setting , and with and ) — when this integral makes sense, which is true if , , and . If, moreover, on , we have for any , so that the integral (\[S special-div\]) does not depend on . Therefore, the l.h.s. of Eq. (\[delta S special-2\]) is zero. Since we have shown that this equation determines uniquely the functions , these functions are zero.\
Let us finally prove the actual tensoriality of the “Hilbert tensor", whose components are defined by Eq. (\[Hilbert tensor\]). Considering now any two charts and whose domain contains , Eq. (\[delta S special-2\]) is true for any tensor field that vanishes on , using either or on the r.h.s. (with primes for ). Since the l.h.s. of (\[delta S special-2\]) is invariant as is the action, so is the r.h.s.; i.e., we have for any such : \[invariance delta S\] . Composing with the reverse coordinate maps and , we may regard as functions defined over , and we have for any such : \[invariance delta S-U\] . For any given tensor field , set and , which are thus two continuous functions on . Consider any open domain with . Take any function which is defined and continuous over , which is over , and that has compact support . Define . This is a tensor field that belongs to and vanishes on . Therefore, the open domain being arbitrary in the already proved point (ii) and hence in (\[invariance delta S-U\]), we can apply (\[invariance delta S-U\]) with instead of . We thus get: \[invariance delta S-W\] . Because this is true for any such function , it follows that we have almost everywhere in . Since these are continuous functions, we have in . And since this is true for any open domain with , we have in , and hence also in . That is, is invariant under coordinate changes, for whatever tensor field . Considering a given point , we define a linear form on the vector space of the tensors at , by setting: \[Def Phi\] , which is thus independent of the chart. But the dual space of is known (and easily checked) to be the vector space of the tensors at . Hence, there is a unique tensor for which, in any chart, we have \[Phi=Phi\_S\] . From (\[Def Phi\]) and (\[Phi=Phi\_S\]), it follows that, in any chart, the numbers are the components of the unique tensor , Q.E.D. $\square$\
A uniqueness result for the energy balance {#Uniqueness E}
==========================================
Is the energy balance equation unique?
--------------------------------------
We begin with a discussion of this question for a system of isentropically deformable media in Newtonian gravity (NG). The energy balance (\[Newton Matter Energy Balance\]) established in Section \[NG\] for the [*matter field equations*]{} of NG has the form \[Matter Balance\] , with the four-components column vector being here the “matter current" made with the matter energy density and flux: \[V\^mu m\] . As we saw, Eq. (\[Matter Balance\]) \[i.e. Eq. (\[Newton Matter Energy Balance\])\] is verified as soon as the following three equations are verified among the matter field equations: Newton’s second law (\[Newton continuum\]), the isentropy equation (\[Isentropy\]), and the continuity equation (\[Continuity Eqn\]). For instance, we did not use the “constitutive equation" that relates the stress tensor to some deformation tensor, or (for a barotropic fluid) that relates the pressure with the density. We note that, in view of Eqs. (\[w\_m NG\]) and (\[Phi\_m NG\]), the matter current is polynomial in the local values of the fields that appear in those equations. (Thus, [*assigning in this section — contrary to Section \[T-tensor\] — a different number to different components of a given vector or tensor field:*]{} here. The gravitational potential plays the same role as does the metric tensor in a Lagrangian for the matter fields in a curved spacetime, as was the case in the foregoing section.) Now we ask if we can find a different expression for the matter current, say , for which the l.h.s. of (\[Matter Balance\]) would be [*always*]{} the same as with the current (\[V\^mu m\]), so that the same balance equation (\[Matter Balance\]) would be valid with , when it is with .Thus: Can we change the matter current for another one , also polynomial with respect to the local values of the fields at any spacetime point , , so that the l.h.s. of (\[Matter Balance\]) would be unchanged [*for whatever values of the fields?*]{} I.e., can we find a column four-vector which would be polynomial in the ’s, and such that we would have ?
A uniqueness result
-------------------
Thus, let be an order- polynomial in the field values , its coefficients being allowed to depend on the spacetime position : \[W mu\] (). Assume that its 4-divergence vanishes identically, : \[Div W mu\] & &\
& & (). I.e., at any spacetime point , Eq. (\[Div W mu\]) is valid for whatever [*possible*]{} values and of the fields and their derivatives at . But, for [*whatever*]{} values and of these variables, there exists smooth functions , defined in some neighborhood of , such that we have , . Thus our assumption means that on the r.h.s. of (\[Div W mu\]) the polynomial function in the real variables and is identically zero. Hence its coefficients are all zero. In particular: \[C=0\] . Thus all coefficients in (\[W mu\]) are zero — except perhaps , with .\
We thus got that we [*cannot*]{} alter the analytical expression of and on the l.h.s. of the matter energy balance (\[Newton Matter Energy Balance\]). \[Apart from arbitrarily adding a zero-divergence vector field that is [*independent*]{} of the matter fields — this is indeed obviously possible, but we can get rid of this by asking that the matter current be polynomial in the fields [*and have no zero-order term,*]{} as is indeed the case in all concrete examples.\] The gravitational energy balance (\[Newton Grav Energy Balance\]) has just the same form: \[Gravitational Balance\] , where is polynomial in the gravitational field . It is valid when the gravitational field equation is. Therefore, similarly as we found for the matter field energy balance, we cannot alter the analytical expression (\[Newton Grav Energy Balance\]) of the gravitational energy balance.
Generalization
--------------
These results are clearly general. Consider e.g. the Maxwell electromagnetic field instead of the Newtonian gravitational field. The energy balance of the e.m. field is: \[Maxwell Field Energy\] , with the [*volume energy density of the electromagnetic field,*]{} and the [*electromagnetic energy flux*]{}. The same uniqueness result says that we cannot find an alternative expression for and on the l.h.s., which would be valid for whatever values of the fields and and their first derivatives.
Conclusion
==========
The classical concept of energy emerges from an analysis of the power done, first in the case of a mass point and then for the case of a volume element in a continuous medium. We have argued that, in the case of a continuous medium or a system of fields, the meaning of the energy conservation is primarily local: it says that, in [*any*]{} bounded domain, the energy loss or gain is due only to a well-identified flux that goes through the boundary of that domain. Thus it expresses in a general way the Lavoisier principle: “Nothing is lost, nothing is created, everything transforms." While it is of course interesting also and even often important to have global energy conservation laws, this interest is limited by two facts: i) an exact global conservation law can be hoped, strictly speaking, only for the Universe as a whole, because there are energy exchanges at all scales — but physics can not be reduced to cosmology. ii) A global energy conservation law says merely that one number is a constant: the total energy; in the most favorable case with global conservation of the energy-momentum and angular momentum, ten numbers are constant. In the relevant case of a system of fields, however, there is an infinite number of degrees of freedom, so this is only a small part of the information needed.\
We have tried to precisely state and prove the main results regarding the derivation of the Hilbert tensor from the invariance of the action in generally-covariant theories. We hope to have proved these results in a convincing way, keeping the mathematical sophistication to the minimum needed. The Hilbert tensor’s theory is beautiful and is essential to general relativity. It is important also in relativistic quantum mechanics. One should note, however, that historically the main examples of the energy-momentum tensor have been derived from the corresponding local conservation equations for energy and momentum [@Provost2015]. Whence the interest in examining the uniqueness of the latter kind of equations.
Appendix: Regular domains {#Domains}
=========================
#### Definition 1 [@BergerGostiaux]. {#Definition 1}
[*Let be a differentiable manifold, with dimension . One will call*]{} closed domain
*of , any closed subset of such that, for any ,*
either
(i)
*there is an open subset of such that ,*
or
\(ii) [*there is a chart with and for , such that*]{} \[Case ii\] , .
#### Proposition 1 [@BergerGostiaux]. {#Proposition 1}
[*Let be a closed domain of a –dimensional differentiable manifold . In case*]{} (i), [*the point is in , the interior of (i.e. the largest open set of , such that ). In case* ]{} (ii), [*the point is in , the boundary of , which is a –dimensional submanifold of the differentiable manifold .*]{}\
Recall that the (topological) boundary of any subset of is defined to be , where the overbar means the adherence (or closure) in and means the complementary set of in . It is easy to prove (cf. [@DieudonneTome1]) that we have always \[int A inter dA = empty\] , \[int A U dA = Abar\] . If is an oriented manifold, then the Stokes theorem (and thus also the divergence theorem) applies to any differential –form (respectively to any continuously differentiable vector field), in any closed domain of with its boundary [@BergerGostiaux].
#### Proposition 2. {#Proposition 2}
[*Let be a*]{} closed domain [*(see ) of a –dimensional differentiable manifold and let be its interior. We have \[U bar = D\] , i.e., a closed domain is the closure of its interior. We thus call an*]{} open domain [*of . Moreover, we have*]{} \[C D bar = C U\] .
Since is a closed set such that , we have . Due to (\[int A U dA = Abar\]), in order to prove that , we just have to prove that . If , we may apply to it Case (ii) of . Let be any open neighborhood of . We will show that it intersects both and ; to show this, we may assume that , with the domain of the chart . Thus is an open neighborhood of in , hence it contains a ball . [*a*]{}) Take first such that and : then . But we get from (\[Case ii\]) that any point such that is in the interior of . Thus is in the interior of , or equivalently is in the interior of , hence in . So , hence (\[U bar = D\]) is proved. [*b*]{}) On the other hand, take now such that and : also , but we get from (\[Case ii\]) that is in . Thus we have also , whence . But, from (\[int A inter dA = empty\]) and (\[int A U dA = Abar\]), we have , or . Therefore, , which proves (\[C D bar = C U\]). $\square$
#### Corollary. {#Corollary}
[*Let be an open domain of and let be a continuous real function defined in a neighborhood of . In order that , it is necessary and sufficient that if .*]{}\
The support of , , is defined to be the smallest closed set containing the set of the points such that , or equivalently is the largest open set such that . Therefore, being an open set, . Since is continuous, is equivalent to — that is, from (\[C D bar = C U\]), to . $\square$\
I am grateful to T. Padmanabhan for pointing out Ref. [@Padmanabhan2015] to me.\
The author declares that there is no conflict of interest regarding the publication of this article.
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[^1]: Of course, is the spatial vector having components in any Cartesian coordinate system (“Cartesian components") — i.e., in any coordinate system for which we have the identity , where is the (Euclidean) spatial metric. Also, ${\textcolor{noir}{{\bf div}\, {{{\boldsymbol{\sigma }}}}}}$ is the spatial vector with Cartesian components ${\textcolor{noir}{({\bf div}\, {{{\boldsymbol{\sigma }}}})^i=\sigma ^{ij}_{\ \,,j}}}$. And is the mixed spatial tensor having Cartesian components . Mixed tensors are identified with linear mappings: is the spatial vector having components , which is the same vector independently of the coordinate system. We use the spatial metric to raise or lower the indices, e.g. . The equations in this section are valid in any spatial coordinates, whether Cartesian or curvilinear.
[^2]: is the vector transformed of by the linear mapping associated with the mixed tensor form of , thus . And is the scalar product of mixed tensors, defined by double contraction.
[^3]: If is a section of a vector bundle with base , say , to write its local expression needs that not only a chart on but also a frame field on is given. However, in the case of a tensor field, a relevant frame field is determined uniquely by the data of the chart with the associated natural basis and the dual basis . In a very general case, has the form with an usual tensor bundle on and a vector bundle of a different kind. Then it is natural to take a frame field of the form , with a frame field on , determined by the chart , and with a frame field on , which is left unchanged when changing the chart. Thus the transformation law on changing the chart is determined.
[^4]: \[K vs U\] Considering “every compact subset of " [@ForgerRoemer2004] is too general, because one needs to use the divergence theorem on to eliminate a surface term, as is done below to get Eqs. (\[d S\_epsilon/d epsilon-2\]) and (\[delta\_S\_coordinate-change\]), and as is done for the equation after Eq. (222) in Ref. [@ForgerRoemer2004]. One should assume that the compact set is a “closed domain". A closed domain is the closure of its interior (), hence it is equivalent to start from a bounded open domain and to define , which is a compact domain, or to start from and to define . Then, of course, the smooth boundary has measure zero with respect to the invariant four-dimensional measure . Hence, the action integral (\[S general\]), as well as its variation (\[delta\_S\_coordinate-change\]) or (\[delta\_S\_coordinate-change-T\]), are unaltered if instead of we consider its closure as the integration domain, as considered in Ref. [@ForgerRoemer2004]. Thus, when the compact considered in Ref. [@ForgerRoemer2004] is a closed domain, as it should be imposed, the definition of the action and the condition imposed on the vector field in Ref. [@ForgerRoemer2004] are equivalent to those considered here for and .
[^5]: In Ref. , it is “set at the integration limits" and, later on, it is stated that “the vanish at the limits of the integration", without any explanation nor any link between the two statements. Note that assuming merely that on the boundary does [*not*]{} imply that on .
[^6]: \[g\_mu nu independent\] All sixteen ’s () are considered as independent variables in for the calculation of , even though : see Note $\ddagger $ on p. 269 in Ref. . Thus, all sixty-four ’s () are also considered as independent variables for the calculation of .
[^7]: Reminding Note (\[g\_mu nu independent\]), the symmetry of the metric does not by itself imply the symmetry of the components (\[Hilbert tensor\]): check e.g. , which of course is not invariant.
[^8]: Actually, in the theory [@A35], the energy density and its flux have been defined only in the preferred reference frame assumed by the theory.
|
---
abstract: 'We present a class of asymptotically anti-de Sitter charged rotating black hole solutions in $f(T)$ gravity in $N$-dimensions, where $f(T)=T+\alpha T^{2}$. These solutions are nontrivial extensions of the solutions presented in [@Lemos:1994xp] and [@Awad:2002cz] in the context of general relativity. They are characterized by cylindrical, toroidal or flat horizons, depending on global identifications. The static charged black hole configurations obtained in [@Awad:2017tyz] are recovered as special cases when the rotation parameters vanish. Similar to [@Awad:2017tyz] the static black holes solutions have two different electric multipole terms in the potential with related moments. Furthermore, these solutions have milder singularities compared to their general relativity counterparts. Using the conserved charges expressions obtained in [@Ulhoa:2013gca] and [@Maluf:2008ug] we calculate the total mass/energy and the angular momentum of these solutions.'
author:
- 'A. M. Awad [^1]'
- 'G. G. L. Nashed,[^2]'
- 'W. El Hanafy [^3]'
title: 'Rotating charged AdS solutions in quadratic $f(T)$ gravity'
---
Introduction {#S1}
============
In the last two decades there has been a growing interest in gravitational solutions with cosmological constant in general relativity (GR) and its extensions. This interest has been generated by seminal observational and theoretical breakthroughs, namely, the discovery of cosmic acceleration [@Riess:1998cb; @Perlmutter:1998np] and the gauge/gravity dualities [@Maldacena:1997re]. Black hole solutions play a very important role in unraveling several classical and quantum mechanical aspects of the underlying gravitational theory. Therefore, it is viewed as an important tool to study various extensions of GR. Contrary to asymptotically flat black holes, asymptotically de Sitter (dS) and anti-de Sitter (AdS) black hole solutions possess more than one type of horizon topology. They could have spherical, hyperbolic, or flat horizons. dS and AdS black hole solutions have been obtained and studied in GR extensively, as well as teleparallel gravities, please see [@Lemos:1994xp; @Hawking:1998kw; @Chamblin:1999hg; @Awad:2002cz; @Nashed:2003ee; @Hanafy:2015yya; @Klemm:1997ea; @Iorio:2012cm; @2012ChPhL..29e0402G; @Xie:2013vua; @Awad:2005ff; @Awad:1999xx; @Awad:2000ac], for diverse black hole solutions.
Since the confirmation of the above cosmological observations there have been several proposed extensions of GR which are based on Riemannian as well as other types of geometries. Gravitational theories based on Riemannian geometry have been extended through $f(R)$ gravitational theory which was proposed in [@Nojiri:2007as; @Bamba:2008ut]. In such a theory, the Ricci scalar $R$ is replaced by an arbitrary function $f(R)$ in Einstein-Hilbert action. Other extensions consider a Lagrangian density on the form of $f(R,{\cal T})$ where ${\cal T}$ the trace of the energy-momentum tensor of the matter component [@Harko:2011kv], or some $f(R,G)$ where $G$ is Gauss-Bonnet scalar [@Cognola:2006eg; @Bamba:2011pz; @Bamba:2010wb; @Bamba:2012vg; @Myrzakulov:2010vz]. Different approach, however, has been developed within Weitzenböck geometry by introducing the teleparallel torsion scalar, $T$, as the Lagrangian density instead of the Ricci scalar, that is the teleparallel equivalent of general relativity (TEGR) theory. Motivated by the $f(R)$ gravity extension, TEGR has been generalized to $f(T)$ gravity by replacing $T$ by an arbitrary function $f(T)$ [@DeLaurentis:2015fea]. The $f(T)$ gravity is considered to be one of the simplest extensions of GR, since its field equations are still second order [@Bengochea:2008gz; @Linder:2010py; @Cai:2015emx] in spite of having arbitrary torsion scalar terms. Although there is an equivalence between GR and TEGR on the field equations level, their generalizations $f(R)$ and $f(T)$ are not equivalent.
In general, finding an exact nontrivial black hole solution in the above extensions, including $f(T)$ gravity, is not an easy task [@Li:2011wu; @Li:2010cg; @Nashed:2015pda; @Nashed:2013bfa; @Nashed:2016tbj; @Awad:2017tyz; @Capozziello:2012zj; @Nashed:uja]. In this work, we present a rotating black hole in all dimensions within Maxwell-$f(T)$ theory with a negative cosmological constant, where $f(T)=T+\alpha T^{2}$. These asymptotically AdS black holes are characterized by cylindrical, toroidal or flat horizons depending on the global identifications of some coordinates. These solutions can be constructed from coordinate transformations which are allowed locally on a manifold but not globally [@Stachel:1981fg]. They are the $f(T)$ analogue of the solutions found in GR by Lemos [@Lemos:1994xp] and their generalizations in higher dimensions that were introduced by one of us in [@Awad:2002cz]. The charged static configurations obtained in [@Awad:2017tyz] are recovered in the limit of vanishing rotation parameters. These interesting black hole solutions have two different electric multipole terms in the electric potential with related multipole moments. In addition, they have milder singularities at $r=0$, similar to that of the static solutions obtained in [@Awad:2017tyz], compared to Reissner Nordström solutions in GR. We calculate the energy and the angular momentum of the black hole using the conserved quantities in the framework of teleparallel gravity.
This work is arranged as follow: In Section \[S2\], a brief account of $f(T)$ gravitational theories are provided in addition to the previous solutions derived in [@Awad:2017tyz] within the framework of $f(T)$ gravitational theory. In Section \[S3\], charged rotating N dimensional exact solutions are derived. These solutions have monopoles and quadrupole moments which are not independent, in addition of being asymptotically AdS. In Section \[S4\], we calculate the energy and angular momentum of these solutions. In the final section we comment on some physical aspects of these black hole solutions.
Maxwell-$f(T)$ Gravity {#S2}
======================
Teleparallel geometry
---------------------
A Vielbein space can be defined as a pair ($M$, $e_a$), where $M$ is an $N$-dimensional differentiable manifold and the set $\{e_a\}$ contains $N$ independent vector fields defined globally on $M$, this set at point $p$ is the basis of its tangent space $T_{p} M$. Because of the independence of $e_{a}$, the determinant $e\equiv
\det (e_{a}{^{\mu}})$ is nonzero. The vielbein vector fields satisfy $e_{a}{^{\mu}}e^{a}{_{\nu}}=\delta^{\mu}_{\nu}\quad
\textmd{and}\quad e_{a}{^{\mu}}e^{b}{_{\mu}}=\delta^{b}_{a}$, where $\delta$ is the Kronecker tensor. Thus, we can construct an associated (pseudo-Riemannian) metric and its inverse, respectively, for any set of basis $g_{\mu \nu} \equiv
\eta_{ab}e^{a}{_{\mu}}e^{b}{_{\nu}},\quad g^{\mu \nu} =
\eta^{ab}e_{a}{^{\mu}}e_{b}{^{\nu}}$, where $\eta_{i j}=(-,+,+,+,
\cdots)$ is the metric of $N$-dimensions Minkowski spacetime. Also, it can be shown that $e=\sqrt{-g}$, where $g\equiv \det(g)$. Thus, we go further to define the symmetric Levi-Civita connection. In this sense, the vielbein space is a pseudo-Riemannian as well. However, if we decide not to use curvature as the basic description of gravity, we may begin with the vielbein vector fields as the fundamental field variables. Then, we define the nonsymmetric linear (Weitzenböck) connection [@Wr] $W^{\alpha}{_{\mu\nu}}\equiv
e_{a}{^{\alpha}}\partial_{\nu}e^{a}{_{\mu}}=-e^{a}{_{\mu}}\partial_{\nu}e_{a}{^{\alpha}}$. This connection is characterized by the property that $\nabla_{\nu}e_{a}{^{\mu}}\equiv\partial_{\nu}{e_a}^\mu+{W^\mu}_{\lambda
\nu} {e_a}^\lambda\equiv 0$, where the covariant derivative $\nabla_{\nu}$ is associated to the Weitzenböck connection. This nonsymmetric connection uniquely determines the teleparallel geometry, since the vielbein vector fields are parallel with respect to it. Indeed, the Weitzenböck connection is curvature free, but it has a non vanishing torsion ${T^\alpha}_{\mu \nu} =
{W^\alpha}_{\nu \mu}-{W^\alpha}_{\mu \nu} ={e_i}^\alpha
[\partial_\mu{e^i}_\nu-\partial_\nu{e^i}_\mu]$. Now we can go directly to construct the teleparallel torsion scalar $$\label{tor}
T ={T^\alpha}_{\mu \nu} {S_\alpha}^{\mu \nu},$$ where the superpotential tensor is defined as ${S_\alpha}^{\mu \nu} :=
\frac{1}{2}\left({K^{\mu\nu}}_\alpha+\delta^\mu_\alpha{T^{\beta
\nu}}_\beta-\delta^\nu_\alpha{T^{\beta \mu}}_\beta\right)$ and the Contortion tensor is $K_{\alpha \mu
\nu}=\frac{1}{2}\left(T_{\nu\alpha\mu}+T_{\alpha\mu\nu}-T_{\mu\alpha\nu}\right)$.
The theory
----------
We take the action of the $f(T)$-Maxwell theory in $N$-dimensional for asymptotically (Anti)-de-Sitter spacetimes as $$\label{q7}
\mathcal{S}_{g}+\mathcal{S}_{em}=\frac{1}{2\kappa}\int d^{N}x~ |e|\left(f(T)-2\Lambda\right)-\frac{1}{2\kappa}\int d^{N}x~|e| { F}\wedge ^{\star}{F},$$ where $\Lambda=-\frac{(N-1)(N-2)}{2l^2}$ is the $N$-dimensional cosmological constant in $N$ dimensions, $l$ is the length scale of AdS spacetime, $\kappa$ is a dimensional constant which can be related to the Newton constant $G_N$ by $\kappa
=2(N-3)\Omega_{N-2} G_N$, where $\Omega_{N-2} =
\frac{2\pi^{(N-1)/2}}{\Gamma([N-1]/2)}$ is the volume of $(N-2)$-dimensional unit sphere and $\Gamma$ function being the argument that depends on the dimension of the spacetime[^4]$^{,}$[^5]. Also, in the Maxwell action, $F = d{\cal A}$, with ${\cal A}={\cal A}_{\mu}dx^\mu$ being the gauge potential 1-form [@Awad:2017tyz; @Capozziello:2012zj].
Varying the action (\[q7\]) with respect to the vielbein and the vector potential ${\cal A}_\mu$, one gets, respectively, the field equations [@Bengochea:2008gz] $$\begin{aligned}
\label{q8}
& & \mathfrak{I{^\nu}{_\mu}}={S_\mu}^{\rho \nu} \partial_{\rho} T
f_{TT}+\left[e^{-1}{e^a}_\mu\partial_\rho\left(e{e_a}^\alpha
{S_\alpha}^{\rho \nu}\right)-{T^\alpha}_{\lambda \mu}{S_\alpha}^{\nu \lambda}\right]f_T-\frac{\delta^\nu_\mu}{4}\left(f+\frac{(N-1)(N-2)}{l^2}\right) +{\kappa \over 2} {{\mathop{\mathfrak{T}}\limits^{\tiny\textbf{em}}}}{^\nu}{_\mu},\nonumber\\
&&\partial_\nu \left( \sqrt{-g} F^{\mu \nu} \right)=0,\end{aligned}$$ where $f := f(T)$, $f_{T}:=\frac{\partial f(T)}{\partial T}$, $f_{TT}:=\frac{\partial^2 f(T)}{\partial T^2}$ and ${{\mathop{\mathfrak{T}}\limits^{\tiny\textbf{em}}}}{^\nu}{_\mu}$ is the energy momentum tensor of the electromagnetic field which is given by [@Capozziello:2012zj] $${{\mathop{\mathfrak{T}}\limits^{\tiny\textbf{em}}}}{^\nu}{_\mu}=F_{\mu \alpha}F^{\nu \alpha}-\frac{1}{4} \delta_\mu{}^\nu F_{\alpha \beta}F^{\alpha \beta}.$$
AdS charged black holes with flat horizons
------------------------------------------
In a previous work [@Awad:2017tyz] we have introduced the following diagonal vielbein which describes a static configuration in $N$-dimensions with the coordinates ($t$, $r$, $\phi_1$, $\phi_2$, $\cdots$, $\phi_{n}$, $z_1$, $z_2$ $\cdots$ $z_k$, $k=1,2
\cdots$ $N-n-2$) $$\label{tetrad}
\hspace{-0.3cm}\begin{tabular}{l}
$\left({e^{i}}_{\mu}\right)=\left( \sqrt{A(r)}, \; \frac{1}{\sqrt{B(r)}}, \; r, \; r, \; r\;\cdots \right)$,
\end{tabular}$$ where $0\leq r< \infty$, $-\infty < t < \infty$, $0\leq \phi_{n}< 2\pi$ and $-\infty < z_k < \infty$. The functions $A(r)$ and $B(r)$ are two unknown functions of the radial coordinate $r$. Thus, the spacetime which can be generated by (\[tetrad\]) is $$\label{m2}
ds^2=
-A(r)dt^2+\frac{1}{B(r)}dr^2+r^2\left(\sum_{i=1}^{n}d\phi^2_i+\sum_{k=1}^{N-n-2}{dz^2_k
\over l^2}\right).$$ Substituting from Eq. (\[tetrad\]) into Eq. (\[tor\]), we evaluate the torsion scalar as[^6] $$\label{tor1}
{\mathbf T=2(N-2)\frac{A'B}{rA}+(N-2)(N-3)\frac{B}{r^2}}.$$ Using the [*N*]{}-dimensional spacetime of Eq. (\[tetrad\]) with Eq. (\[tor1\]) and the vector potential ${\cal A} = \Phi(r) dt$, we obtain the following equations (\[q8\]): $$\begin{aligned}
& &{\mathbf \mathfrak{I}^t{}_t= \frac{2(N-2)Bf_{TT} T'}{r}+\frac{(N-2)f_T[2(N-3)AB+rBA'+rAB']}{r^2A}-f+2\Lambda+\frac{2\Phi'^2(r)B}{A}=0},\nonumber\\
& & {\mathbf \mathfrak{I}^r{}_r= 2Tf_T+2\Lambda-f+\frac{2\Phi'^2(r)B}{A}=0},\nonumber\\
& & {\mathbf\mathfrak{I}^{\phi_1}{}_{\phi_1}= \mathfrak{I}^{\phi_2}{}_{\phi_2}=\cdots \mathfrak{I}^{\phi_n}{}_{\phi_n}=\mathfrak{I}^{z_1}{}_{z_1}=\mathfrak{I}^{z_2}{}_{z_2}\cdots =\mathfrak{I}^{z_k}{}_{z_k}= \frac{f_{TT} [r^2T+(N-2)(N-3)B]T'}{(N-2)r}+\frac{f_T}{2r^2{A}^2}\Biggl\{2r^2ABA''}\nonumber\\
& & {\mathbf-r^2BA'^2+4(N-3)^2A^2B+2(2N-5)rABA'+r^2AA'B'+2(N-3)rA^2B'\Biggr\}-f+2\Lambda-\frac{2\Phi'^2(r)B}{A}=0}, \nonumber\\\end{aligned}$$ where $\Phi'=\frac{d\Phi}{dr}$. The general [*N*]{}-dimensional solutions with flat horizons of the Maxwell-$f(T)$ theory, where $f(T)=T+\alpha T^2$ of the above differential equations takes the form [@Awad:2017tyz] $$\begin{aligned}
\label{sol}
A(r)&=&r^2\Lambda_{eff}-\frac{m}{r^{N-3}}+\frac{3(N-3)q{}^2}{(N-2)r^{2(N-3)}}+\frac{2\sqrt{6\left|\alpha\right|}(N-3)^3q{}^3}{(2N-5)(N-2)r^{3N-8}},
\nonumber\\
B(r)&=&A(r)\left[1+\frac{(N-3)q\sqrt{6\left|\alpha\right|}}{r^{N-2}}\right]^{-2},\label{sol1}\nonumber\\
\Phi(r)&=&\frac{q}{r^{N-3}}+\frac{(N-3)^2q{}^2\sqrt{6\left|\alpha\right|}}{(2N-5)r^{2N-5}}, \end{aligned}$$ where $\Lambda_{eff}= \frac{1}{6(N-1)(N-2)\left|\alpha\right|}$, $m$ is the mass parameter, $q$ is the charge parameter and $\Phi(r)$ is the electric potential which defines the vector potential ${\cal A}=\Phi(r)dt$. As it is clear from Eq. (\[sol\]), that the potential $\Phi(r)$ depends on a monopole and quadrupole moments. By setting $q=0$ both momenta vanish and we get a non-charged solution. It is worth mentioning that the solution (\[sol\]) has been derived for the quadratic polynomial $f(T)$ theory in the presence of the constraint $\Lambda=\frac{1}{24\alpha}$. Consequently, one expects the model parameter to be $\alpha<0$, since the cosmological constant is negative. The reason of the constraint $\Lambda=\frac{1}{24\alpha}$ is as follows: If one chooses an ansatz for the charged solution in which the functions $A(r)$ and $B(r)$ are equal then one gets constant potential, i.e., a trivial potential for a charged solution! In order to avoid this trivial potential, we choose $A(r)= C(r) B(r)$. In this case the potential will not be trivial, but $A(r)$ and $B(r)$ are neither unique nor in closed form. For example, for the 5-dimensional uncharged solution, $\Phi(r)=0$, we have $C(r)=const.$ and $$A(r)=\frac{[1 \pm \sqrt{1-24 \alpha \Lambda}]r^2}{ 72 \alpha} + \frac{c_1}{r^2}$$ which shows that $A(r)$ is not unique. An extra complication is obtained when the potential is not constant, in this case, $A(r)$ and $B(r)$ can not be expressed in a closed form. Choosing $\Lambda=\frac{1}{24\alpha}$ leave the solution unique and in a closed form.
Before closing this section, we note that the black hole solution at hand cannot be considered as a special case of the cubic polynomial $f(T)$ gravity which has been studied in [@Nashed:2018cth]. In the later, the solution has been obtained under a specific constraint whereas the coefficient of cubic term cannot be made to vanish.
AS charged rotating black holes with flat horizons {#S3}
==================================================
One way to add an angular momentum for the above solution in four dimensions[^7] $$\label{m2}
ds^2= -A(r)dt^2+\frac{1}{B(r)}dr^2+r^2\left(d\phi^2+{dz^2 \over
l^2}\right).$$ We follow the procedure developed in [@Lemos:1994xp; @Awad:2002cz], applying the transformations $$\label{t1}
\bar{\phi} =-\Xi~ {\phi}+\frac{ \omega}{l^2}~t,\qquad \qquad \qquad
\bar{t}= \Xi~ t-\omega~ \phi.$$ We note that these transformations are allowed locally but not globally on a manifold as will be clarified below. Thus the spacetime (\[m2\]) reads $$\label{m1}
ds^2=-A(r)\left[\Xi d\bar{t} - \omega d\bar{\phi}
\right]^2+\frac{dr^2}{B(r)}+\frac{r^2}{l^4} \left[\omega
d\bar{t}-\Xi l^2 d\bar{\phi} \right]^2+ {r^2 \over l^2} dz^2,$$ where $$\Xi:=\sqrt{1+\frac{\omega^2}{l^2}}.$$ According to Stachel [@Stachel:1981fg] if the first Betti number of the manifold is non-vanishing, which is the case for the equivalent Riemannian manifold of these solutions, there are no global diffeomorphisms that can map one of these metrics to the other, leaving the new manifold with an additional parameter “$\omega$”. Since in $N$ dimensions we have more than one rotation parameter, the construction of the rotating tetrad or metric is not as obvious as the one rotation parameter case as was shown in [@Awad:2002cz]. It requires the addition of other terms which are not obtained by the above coordinate transformations. In the higher dimensional case the proposed form of the tetrad for more than one rotation parameter is given by
$$\begin{aligned}
\label{tetrad1}
\left({e^{i}}_{\mu}\right)=\left(
\begin{array}{cccccccccccccc}
\Xi\sqrt{A(r)} & 0 & -\omega_1\sqrt{A(r)}&-\omega_2\sqrt{A(r)}\cdots & -\omega_{n}\sqrt{A(r)}&0&0&\cdots&0 \\[5pt]
0&\frac{1}{\sqrt{B(r)}} &0 &0\cdots &0&0&0&\cdots & 0\\[5pt]
\frac{\omega_1r}{l^2} &0 &-\Xi r&0 \cdots &0&0&0&\cdots & 0\\[5pt]
\frac{ \omega_2r}{l^2} &0 &0 &-\Xi r\cdots & 0&0&0&\cdots & 0\\[5pt]
\vdots & \vdots &\vdots&\vdots&\vdots &\vdots&\vdots& \cdots & \vdots \\[5pt]
\frac{ \omega_nr}{l^2} & 0 &0&0 \cdots & -\Xi r&0&0&\cdots & 0 \\[5pt]
0 & 0 &0&0 \cdots &0&r&0&\cdots & 0\\[5pt]
0 & 0 &0&0 \cdots &0&0&r&\cdots & 0\\[5pt]
0 & 0 &0&0 \cdots &0&0&0&\cdots & r\\
\end{array}
\right),\end{aligned}$$
where $n= \lfloor(N - 1)/2\rfloor$ is the number of rotation parameters with $\lfloor y \rfloor$ is the integer part of $y$, $\omega_j$ are the rotation parameters and $\Xi$ is defined as $$\Xi:=\sqrt{1+\sum\limits_{j=1}^{{n}}\frac{ {\omega_j}^2}{l^2}}.$$ Also, the functions $A(r)$ and $B(r)$ are given by (\[sol\]). In addition, the gauge potential takes the form $$\label{Rotpot}
\bar{\Phi}(r)=-\Phi(r)\left[\omega_i~ d\bar{\phi}_i-\Xi~ d\bar{t}\,\right] .$$ We note that Eqs. (\[sol\]) and (\[Rotpot\]) are also solutions of the stationary configuration (\[tetrad1\]). Since transformation (\[t1\]) mixes compact and noncompact coordinates, it leaves the local properties of spacetime the same. However, it does change the spacetime properties globally, c.f. [@Lemos:1994xp]. On other words, the vielbein (\[tetrad\]) and (\[tetrad1\]) can be locally mapped into each other but not globally [@Lemos:1994xp; @Awad:2002cz]. One can show that the spacetime which is generated by the vielbein (\[tetrad1\]) takes the form $$\label{m1}
ds^2=-A(r)\left[\Xi d\bar{t} -\sum\limits_{i=1}^{n} \omega_{i}d\bar{\phi} \right]^2+\frac{dr^2}{B(r)}+\frac{r^2}{l^4}\sum\limits_{i=1 }^{n}\left[\omega_{i}d\bar{t}-\Xi l^2 d\bar{\phi}_i\right]^2+ {r^2 \over l^2} d\Sigma^2-\frac{r^2}{l^2}\sum\limits_{i<j }^{n}\left(\omega_{i}d\bar{\phi}_j-\omega_{j}d\bar{\phi}_i\right)^2,$$ where $0\leq r< \infty$, $-\infty < t < \infty$, $0 \leq \phi_{i}< 2\pi$, $i=1,2 \cdots n$ and $-\infty < z_k < \infty$, $d\Sigma^2=dz^kdz^k$ is the Euclidean metric on ($N-n-2$)-dimensions and $k = 1,2,\cdots, N-3$. We note that the static configuration (\[m2\]) can be recovered as a special case when the rotation parameters $\omega_j$ are chosen to be vanished. These charged rotating solutions do not correspond to any known solutions in GR or TEGR since by sending $\alpha \rightarrow 0$ we do not get a well defined tetrad or metric. Notice that upon setting the mass parameter $m=0$ and the charge $q=0$, the line-element (\[m1\]) reduces to the $N$-dimensional AdS metric in an unusual coordinate system. One can easily check that the resulting boundary metric is indeed Minkowski through checking the vanishing of its torsion components. Furthermore, this shows that the whole metric in this limit (i.e., line-element with $m=0$ and the charge $q=0$) is the AdS metric. In the next section, we are going to study the main feature of solution (\[tetrad1\]).
Conserved Charges {#S4}
=================
Four-momentum
-------------
Before we calculate the energy or total mass of these black holes, let us follow [@Ulhoa:2013gca] deriving the conserved four-momentum for $f(T)$ gravity in few lines. Variation of the action (\[q7\]) with respect to the vielbein gives the field equations in the form $$\label{q88}
{S_\mu}^{\rho \nu} \partial_{\rho} T
f_{TT}+\left[e^{-1}{e^a}_\mu\partial_\rho\left(e{e_a}^\alpha
{S_\alpha}^{\rho \nu}\right)-{T^\alpha}_{\lambda \mu}{S_\alpha}^{\nu
\lambda}\right]f_T
-\frac{\delta^\nu_\mu}{4}\left(f+\frac{(N-1)(N-2)}{l^2}\right) =-{\kappa \over 2}{\mathfrak{T}}{^\nu}{_\mu},$$ where ${\mathfrak{T}}{^\nu}{_\mu}$ is the energy-momentum of the matter. Equation (\[q88\]) can be rewritten as $$\begin{aligned}
\label{q888}
\partial_\rho\left(eS^{a \nu \rho }f_T \right)={\kappa \over 2} |e| \left(t^{a \nu}+T^{a \nu}\right), \end{aligned}$$ where $$t^{a \nu}={2 \over \kappa}\left[ f_TS^{b c \nu } T_{b c}{}^a-\frac{\delta^\nu_\mu}{4}\left(f+\frac{(N-1)(N-2)}{l^2}\right) \right].$$\
Taking the derivative of Eq. (\[q888\]) with respect to $x^\nu$, we get $$\begin{aligned}
\label{con}
\partial_\nu\partial_\rho\left(eS^{a \nu \rho }f_T \right)=0 \qquad \textrm {which leads to} \qquad \partial_\nu\left[ {\kappa \over 2} |e| \left(t^{a \nu}+T^{a \nu}\right)\right]=0. \end{aligned}$$ Equations (\[con\]) give the conserved $N$-momentum of $f(T)$ gravitational theory in the form $$\begin{aligned}
\label{energy}
P^a= \int_{V} d^{N-1}x |e| t^{0 a}.\end{aligned}$$ Equation (\[energy\]) which defines the $N$-momentum of $f(T)$ gravity was derived before in [@Ulhoa:2013gca]. This has been used, mostly, to calculate energy for asymptotically flat spacetime background. However, the solutions (\[tetrad1\]) are asymptotically AdS. Here we adopt the point of view of the authors in [@PhysRevD.15.2752; @Gibbons:1976pt; @Gibbons:1978ac; @Nashed:2011fg] to calculate conserved quantities of a gravitational solution in reference to a specific background spacetime. These backgrounds are naturally chosen as Minkowski spacetime for asymptotically flat solutions and AdS or dS for asymptotically AdS or dS solutions. Furthermore, infinities due to the asymptotic regions are canceled out in this subtraction prescription leaving the physical quantities finite. For example, the total energy of an AdS black hole, measured by a stationary observer at very large radial distance, is considered to be the difference in energy between the AdS black hole and the AdS space itself. Therefore, in calculating the conserved quantities, it is natural to subtract the contribution due to pure AdS spacetime from that of the solution. Therefore Eqs. (\[energy\]) and (\[an\]) take the form $$P^a=\int_{V} d^{N-1}x \left[|e|t^{0 a}\right]_{reg},$$ where the subscript “*reg*” stands for the regularized value of the physical quantity.
Let us now calculate the energy related to the rotating charged black holes given by Eqs. (\[tetrad1\]). Using Eq. (\[energy\]), it is possible to derive the components that are necessary for the calculations of energy in the form[^8]: $$\label{se} S^{(0)(0)1}=\frac{(N-2){B}}{2r}.$$$$\label{en1} P^0=E=\frac{(N-2)[m-\Lambda_{eff}\,r^{(N-1)}]\Xi}{3(N-3)G_N}+\left(\frac{1}{r}\right)+...,$$where $n\geq 1$. expression of Eq. (\[energy\]) takes the form of a surface integral $$\label{enr} P^a_{reg}:=\frac{2}{\kappa}\int_{\partial V} d^{N-2}x \left[e{S}^{a 0
\mu}\, n_{\mu} f_T\right]-\frac{2}{\kappa}\int_{\partial V} d^{N-2}x
\left[e{S}^{a 0 \mu}\, n_{\mu}\,f_T\right]_{AdS},$$where $n_{\mu}$ is the normal vector to the surface $\partial V$ and *AdS* means evaluating the second expression of Eq. (\[energy\]) for pure Anti-de-Sitter space. Using (\[enr\]) in solution (\[tetrad1\]), we get $$\label{en11}
E_{reg} = \frac{2(N-2)\,\Xi\,M}{3\,(N-3)},$$ where the mass parameter is taken to be $m=2\,G_N\,M$. As expected, the black hole energy is fully characterized by its mass.
Angular momentum
----------------
Although there is a hamiltonian formulation in teleparallel equivalent of general relativity (i.e., $f(T)=T$) which produces some known expressions for the conserved four-momenta and angular momentum [@Maluf:2008ug], there is no known expression for angular momentum in $f(T)$ gravity. But since the angular momentum is independent of the charge $q$, by sending $q\rightarrow 0$ we obtain a solution with constant torsion scale $T$, since the scalar torsion is given by, $$T = {-1 \over 6 \alpha}+{2q\sqrt{6} \over 3 \sqrt{\alpha}\,\, r^3}.$$ A solution in $f(T)$ gravity with constant torsion scalars, $T=T_c$, is equivalent to a solution in TEGR with constant torsion, where $T'=f(T_c)$. Therefore, one can use Maluf’s expression in [@Maluf:2008ug] to calculate the angular momentum of our solution in this limit. Following [@Maluf:2008ug], the angular momentum tensor can be written in terms of the superpotential $S^{a b c}$ in the following form $$\begin{aligned}
\label{sig}
M^{a \mu c}\equiv |e| e_b{}^\mu \left[S^{a b c}-S^{c a b}\right]=|e| \left[S^{a \mu c}-S^{c a \mu}\right] =-\frac{1}{2}\partial_\nu\{|e| \,[e^{a\nu}e^{c \mu}-e^{a\mu}e^{c \nu}] \, \}.\end{aligned}$$ From equation (\[sig\]) one can easily show that $$\begin{aligned}
\label{sig1}
\
\partial_\mu M^{a \mu c}=0.\end{aligned}$$ Using Eqs. (\[sig\]) and (\[sig1\]) the conserved angular momentum is given by $$\begin{aligned}
\label{an}
L^{a b}=\int_{V} d^{N-1} M^{a 0 b}&=&-\frac{1}{\kappa}\int_{V} d^{N-1}x \partial_\nu\{|e| \, [e^{a\nu}e^{b 0}-e^{a0}e^{b \nu}] \, \} \nonumber \\
&=&-\frac{1}{\kappa}\int_{\partial V} d^{N-2}x\; n_\rho |e| \, [e^{a\rho}e^{b 0}-e^{a0}e^{b \rho}],\end{aligned}$$ where $n_\rho$ is the outward unit normal vector.
Now we are going to calculate the angular momentum of solution (\[tetrad1\]) in the limit $q\rightarrow 0$. For this aim we are going to list the necessary components needed for these calculations. The non-vanishing components of the torsion tensor, $T^{a b c}=e^a{}_\mu e^b{}_\nu e^c{}_\rho T^{\mu \nu \rho}$, and the superpotential tensor, $S^{a b c}=e^a{}_\mu e^b{}_\nu e^c{}_\rho S^{\mu \nu \rho}$, are $$\begin{aligned}
\label{se1}
&&T_{(0) (1) (0) }=\frac{A'\sqrt{B}}{2A},\qquad\\[5pt]
&&T_{(N-i) (N-j) (1) }=\frac{\Bigg(\Bigg[l^2\Xi^2-\sum\limits_{a=1}^{n}\omega_a{}^2\Bigg]\delta_{i j}+\omega_i\omega_j\Bigg)\sqrt{B}}{l^2\Xi^2 r},\qquad \\[5pt]
&&T_{(N-n-\sum\limits_{k=1}^{(N-n-2)}\;\;k) (N-n-\sum\limits_{k=1}^{(N-n-2)}\;\;k) (1) }=\frac{\sqrt{B}}{r},\qquad \\[5pt]
&&S_{(0) (0) 1}=\frac{(N-2){B}}{2r}, \\[5pt]
&&S_{(N-i) (1) (N-j) }=\frac{\sqrt{B}\Bigg(\Bigg[l^2\Xi^2-\sum\limits_{a=1}^{n}\omega_a{}^2\Bigg]\delta_{i j}+\omega_i\omega_j\Bigg)\Bigg[2(N-3)A+rA'\Bigg]}{4Al^2\Xi^2 r},\qquad \\[5pt]
&&S_{(N-n-\sum\limits_{k=1}^{(N-n-2)}\;\;k) (1) (N-n-\sum\limits_{k=1}^{(N-n-2)}\;\;k) }=\frac{\sqrt{B}\Bigg[2(N-3)A+rA'\Bigg]}{4A r}.\qquad\end{aligned}$$ Similar to the energy calculations, we are going to use the background subtraction prescription to calculate the angular momentum of the black hole relative to the AdS space background.
$$L^{i j} =-\frac{1}{\kappa}\int_V d^{N-1}x e^i{}_\mu e^j{}_\nu |e|
\left[(S^{\mu 0\nu}-S^{\nu 0 \mu})\right]_{reg}.$$
Using the above equation one gets $$\label{an1}
{J_i|}_{reg}=\frac{ \omega_i M }{2\, (N-3)},$$ where $$\label{an12} {J_i}=\epsilon_{i j k}L^{j k}.$$ As clear from the above equations that the angular momentum vanishes when the rotation parameters $\omega_i$ vanish. In conclusion, under the constraint $q\to 0$, equations (\[en11\]) and (\[an1\]) show that the black holes are characterized by their masses and angular momenta.
Conclusions {#S5}
===========
In this work, we present a new class of charged rotating solutions in $f(T)$ theories in $N$ dimensions. These solutions are obtained for $f(T)=T+\alpha T^2$, where the parameter $\alpha <0$. It is worth to mention that these solutions cannot be considered as special cases of the solutions of the cubic polynomial $f(T)$ gravity which have been recently studied in [@Nashed:2018cth]. This is because the later are obtained whereas the cubic contribution is parameterized by an extra parameter which cannot be made to vanish. One of the attractive features of the solutions at hand is that their electric potential has related monopole and quadrupole moments. The relation between these moments is a result of demanding an asymptotically AdS solution. It is intriguing to note that all these black holes have a singularity at $r=0$, which is milder than that of their corresponding solutions in TEGR or GR. The asymptotic behavior of the Kretschmann invariant, the Ricci tensor squared, and the Ricci scalar have the same form of the charged ones presented in [@Awad:2017tyz], i.e. $K=R_{\mu \nu}R^{\mu \nu} \sim
r^{-2(N-2)}$, $R\sim r^{-(N-2)}$. This is in contrast with their corresponding known solutions in Einstein-Maxwell theory in both GR and TEGR. Also it is important to mention that, in spite that the charged rotating black hole has different components for $g_{tt}$ and $g^{rr}$, their Killing and event horizons coincide.
To understand these solutions more, we calculate their total energy and angular momentum. For this aim we have used the mass/energy expression in the framework of $f(T)$ obtained by [@Ulhoa:2013gca]. For the angular momentum we have used the expression in [@Maluf:2008ug]. We have used the form of the energy-momentum tensor to calculate the total energy of the rotating charged black holes and have shown that the resulting form depends on the mass of the black hole which is consistent with the derived form in [@Awad:2017tyz].
For calculating the angular momentum of the solutions one notices that, although there is a hamiltonian formulation for TEGR, which produces a known expression for the angular momentum, there is no known expression for angular momentum in $f(T)$ gravity. We argue that since the angular momentum in our solution is independent of the charge $q$, by sending $q\rightarrow 0$ we obtain a solution with constant torsion scaler $T$, therefore, one can use the angular momentum expression for TEGR following [@Maluf:2008ug]. As a results we have used the expressions obtained in [@Ulhoa:2013gca] and [@Maluf:2008ug] to calculate the mass and angular momentum of these solutions together with the subtraction technique used for asymptotically de-Sitter and Anti-de-Sitter solutions. One of the interesting features that we would like to check in future works is that if these milder curvature singularities are weak enough to make these singularities “Tipler weak” according to Tipler’s criteria [@Tipler:1977]. If it is weak enough, this might leads to possible extensions of the manifold as was shown in the same theory (i.e., $f(T)=T+\alpha T^2$) for some cosmological solutions in [@Awad:2017sau].
acknowledgments {#acknowledgments .unnumbered}
===============
This work is partially supported by the Egyptian Ministry of Scientific Research under project No. 24-2-12. \[2\][\#2]{}
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: For $N= 4$, one can recover $2(N-3)\Omega_{N-2}
= 8 \pi G_4$.
[^5]: The spacetime indices are given by $\mu,\,
\nu \cdots$ and the SO(3,1) indices are given by $a,\, b,\, \cdots$ in which all of them run from 0 to 3. The Latin indices $i, j,
\cdots$ are denote to the SO(3,1) spatial components.
[^6]: For abbreviation we will write $A(r)\equiv A$, $B(r)\equiv B$, $A'\equiv\frac{dA}{dr}$,$A''\equiv\frac{d^2A}{dr^2}$,$B''\equiv\frac{d^2B}{dr^2}$ and $B'\equiv\frac{dB}{dr}$ .
[^7]: It is well known, even in GR, that the addition of cosmological constant might produce different types of rotating black holes among them is the class under consideration here, please see [@Klemm:1997ea] for a discussion on these types of rotating black holes
[^8]: The square parentheses in the quantities $S^{(0)(0)1}$ refer to the tangent components, i.e., $S^{(0)(0)1}=e^0{}_\mu e^0{}_\nu S^{\mu \nu 1}$.
|
---
abstract: 'Scene understanding of full-scale 3D models of an urban area remains a challenging task. While advanced computer vision techniques offer cost-effective approaches to analyse 3D urban elements, a precise and densely labelled dataset is quintessential. The paper presents the first-ever labelled dataset for a highly dense Aerial Laser Scanning (ALS) point cloud at city-scale. This work introduces a novel benchmark dataset that includes a manually annotated point cloud for over 260 million laser scanning points into 100’000 (approx.) assets from Dublin LiDAR point cloud [@laefer20172015] in 2015. Objects are labelled into 13 classes using hierarchical levels of detail from large (building, vegetation and ground) to refined (window, door and tree) elements. To validate the performance of our dataset, two different applications are showcased. Firstly, the labelled point cloud is employed for training Convolutional Neural Networks (CNNs) to classify urban elements. The dataset is tested on the well-known state-of-the-art CNNs (PointNet, PointNet++ and So-Net). Secondly, the complete ALS dataset is applied as detailed ground truth for city-scale image-based 3D reconstruction.'
bibliography:
- 'ms.bib'
title: 'DublinCity: Annotated LiDAR Point Cloud and its Applications'
---
|
---
abstract: 'We determine Kelly criterion for a game with variable pay-off. The Kelly fraction satisfies a fundamental integral equation and is smaller than the classical Kelly fraction for the same game with the constant average pay-off.'
address: 'CNRS, LAGA UMR 7539, Université Paris XIII, 99, Avenue J.-B. Clément, 93430-Villetaneuse, France'
author:
- Ricardo Pérez Marco
date: 'November 4, 2014'
title: 'Kelly criterion for variable pay-off'
---
Introduction.
=============
Kelly criterion (see [@Ke]), also called “Fortune Formula”, is the fundamental tool to ensure an optimal positive return when playing with repetition a favorable game or strategy. It was first proposed by Edward Thorp for the money management of his card counting strategy to beat casino blackjack (see [@Th1]). Later he applied the same ideas to the financial markets (see [@Ka-Th]), and L. Breinman (see [@Bre]) proved that it was the optimal strategy for long run accumulation of capital.
One needs to modify the theory for common situation where the advantage is not known exactly but is a random variable (see [@MG-PM]). In this “fuzzy advantage” situation one needs to be far more conservative (see section 3 in [@MG-PM]). When we apply Kelly criterion to decision making in real situations, we are confronted with a fuzzy advantage but also a variable reward or pay-off. In this article we study how to modify Kelly criterion when the pay-off is a random variable with a known distribution. We obtain the same type of conclusion than in [@MG-PM]: The optimal Kelly fraction is more conservative than the one in the same game with the constant average pay-off.
The classical Kelly criterion.
==============================
We assume that we are playing a game with repetition. At each round we risk a fraction $0\leq f\leq 1$ of our capital $X$. With probability $0 < p <1$ we win and the pay-off is $b$-to-$1$, with $b\geq 0$. This means that if $X$ is our current capital, if we lose the bet (which happens with probability $q=1-p$) we substract $fX$ to our capital, and if we win, we add $bfX$ to our capital . Thus, the expected gain is $${{\mathbb E}}(\Delta X) = p b f X - q fX= (p b-q) fX = (p(1+b)-1) fX \ .$$ If we play a game with advantage, that is $p(1+b)>1$, then we expect an exponential growth of our initial bankroll $X_0$ if we follow a reasonable betting strategy. We assume that there is no minimal unit of bet. By homogeneity of the problem, a sharp strategy must consist in betting a proportion $f(p)$ of the total bankroll. In the classical Kelly criterion, $p$ and $b$ are assumed to be constant at each round. In this article we assume that $p$ is constant but the pay-off $b$ is a random variable. We first review the classical Kelly criterion with a constant pay-off that finds $f(p)$ in order to maximize the expected exponential growth. Our bankroll after having played $n$ rounds of the game have is $$X_n=X_0 \prod_{i=1}^n (1+b_i f(p))$$ where $b_i=b$ if we won the $i$-th round, and ${\epsilon}_i=-1$ if we lost the $i$-th round. The exponential rate of growth of the bankroll is $$G_n =\frac1n \log \frac{X_n}{X_0}=
\frac1n \sum_{i=1}^n \log (1+b_i f(p)).$$ The Kelly criterion maximizes the expected value of the exponential rate of growth:
**(Kelly criterion)** For a game with advantage, that is $p(b+1) >1$, the expected value of the exponential rate of growth $G_n$ is maximized for $$f^*(p, b)=p-q/b= \frac{p(1+b)-1}{b}\ .$$
The argument is straightforward. Observe that the expected value is $${{\mathbb E}}(G_n)= {{\mathbb E}}(G_1)=p\log (1+bf)+(1-p) \log (1-f)=g(f).$$ This function of the variable $f$ has a derivative, $$g'(f)=\frac{pb}{1+bf} - \frac{1-p}{1-f}=\frac{(p(1+b)-1)-fb}{(1+fb)(1-f)} \ ,$$ and $g(f)\to 0$ when $f\to 0^+$, and $g(f) \to - \infty$ when $f\to 1^-$. Also $g'(f) >0$ near $0$, and $g'$ is decreasing, so $g$ is concave. Thus $g(f)$ increases from $0$ to its maximum attained at $$f^*=f^*(p, b)=\frac{p(1+b)-1}{b}$$ and then decreases to $-\infty$.
Kelly criterion with variable pay-off.
======================================
This situation arises in a number of practical situations. The original problem that motivated the use of Kelly criterion was casino blackjack where the pay-off is constant $b=1$. But in other card games, like poker cash, the pay-off is variable. Also some trading strategies cannot set a predetermined pay-off, for example when speculating with a price rebound in a volatile market. The historic of trades of traders present a certain distribution of pay-offs for the successful trades. Therefore in these situations we cannot consider $b$ constant. We assume that the pay-off $b$ is a random variable with a known non-negative distribution $\rho: {{\mathbb R}}_+ \to {{\mathbb R}}_+$, with $\rho(x) dx$ giving the probability that the pay-off is in the infinitesimal interval $[x, x+dx]$. In practice $\rho$ has compact support and can be obtained empirically, although the model in [@PM] shows that the tail is of Pareto type. We do not need to assume that the distribution is absolutely continuous with respect to the Lebesgue measure (same proofs).
We can use a similar argument as in the previous section in order to determine the sharp fraction to bet $\hat f$.
To determine when the game is favorable we compute $${{\mathbb E}}(X_1-X_0)= (1-p) f X_0 + p f X_0 \int_0^{+\infty } b \ \rho (b) \ db \ ,$$ Thus the condition for a favorable game is $$\label{advantage}
p\left ( 1 + \int_0^{+\infty } b \ \rho (b) \ db \right ) >1 \ ,$$ which (naturally) is the same condition than that of a constant pay-off game where the pay-off is the average pay-off: $$p( 1 + \bar b ) >1 \ ,$$ and $$\bar b =\int_0^{+\infty} b \ \rho (b) \ db.$$
After $n$ rounds, if $b_i$ is the pay-off in the $i$-th round, the expected exponential growth is $$\begin{aligned}
g(f)={{\mathbb E}}(G_n) &=\frac1n \sum_{i=1}^n {{\mathbb E}}(\log (1+b_i f)) \\
&= q \log (1-f) + p \int_0^{+\infty} \log(1+b f) \ \rho(b) \ db \ ,\end{aligned}$$ under the integrable assumption that the density of the pay-off function makes the integral finite.
Again $g(f) \to 0$ when $f\to 0^+$, $g(f) \to -\infty$ when $f\to 1^-$, and $$g'(f)= - \frac{q}{1-f} + p \ \int_0^{+\infty} \frac{b \rho (b)}{1+bf} \ db \ .$$ We observe that when $f\to 0^+$, $$g'(f) \approx -(1-p) + p \int_0^{+\infty } b \ \rho (b) \ db$$ thus $g'(f) >0$ by the favorable game condition (\[advantage\]). Moreover $g'$ is strictly decreasing, so $g$ is strictly concave, and tends to $-\infty$ when $f\to 1^-$. Thus there is exactly one value $\hat f=\hat f(p)$ that maximizes this expression and annihilates $g'$. It is the unique solution $\hat f (p, \rho)$ to the fundamental integral equation $$\label{fundamental_equation}
p\int_0^{+\infty } \frac{b \ \rho (b)}{1+b \hat f } \ db-\frac{1-p}{1-\hat f} =0 \ .$$
**(Kelly criterion for variable pay-off)** A game with variable pay-off with a distribution $\rho$ is favorable if $$p\left ( 1 + \int_0^{+\infty } b \ \rho (b) \ db \right ) >1 \ .$$ The expected value of the exponential rate of growth is maximized for $0<\hat f=\hat f(p, \rho) <1$ satisfying the fundamental integral equation $$p\int_0^{+\infty } \frac{b \ \rho (b)}{1+b \hat f } \ db-\frac{1-p}{1-\hat f} =0 \ .$$
The optimal Kelly fraction $\hat f (p, \rho)$ for a favorable game with variable pay-off is smaller than the optimal Kelly fraction $f^*(p, \bar b)$ for the same game with constant pay-off equal to the average pay-off $\bar b$, $$\bar b =\int_0^{+\infty} b \ \rho (b) \ db \ .$$ We have $$\hat f (p, \rho)\leq f^*(p,\bar b) \ .$$ We only have equality when the pay-off is constant.
The function of $b$, $$h(b)= \frac{b}{1+b\hat f} \ ,$$ is strictly concave and therefore, by Jensen’s inequality, we have $$\int_0^{+\infty} \frac{b \ \rho(b)}{1+b\hat f} \ db \leq \frac{\bar b}{1+\bar b f} \ .$$ So, using the fundamental equation (\[fundamental\_equation\])we get $$p \frac{\bar b}{1+\bar b f}-\frac{1-p}{1-\hat f}= -\frac{1-p}{1-f^*}-\frac{1-p}{1-\hat f}\geq 0 \ ,$$ and the result follows. The case of equality follows from the case of equality of Jensen’s inequality and only occurs for a Dirac distribution.
**Conclusion:**
In a situation when the pay-off is variable one needs to adjust the Kelly fraction in a conservative way.
**Remark.**
The analysis generalizes to situations where the risk is larger than the fraction waged. As noted by Thorp, this happens in a leveraged investment in the financial markets.
[9]{}
BREIMAN, L.; [*Optimal gambling systems for favourable games*]{}, Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. **1**, Univ. Calif. Press, Berkeley, CA, 1961.
KASSOUF, S.T.; THORP, E.; [*Beat the market: A scientific stock market system*]{}, Random House, 1967.
KELLY, J.R.; [*A new interpretation of information rate*]{}, Bell System Technical Journal, **35**, (4), p.917–926.
MUÑOZ GARCÍA, E.; PÉREZ-MARCO, R.; [*The standard deviation effect*]{}, arXiv:math 0006017, 2000.
PÉREZ-MARCO, R.; [*A simple dynamical model leading to Pareto wealth distribution and stability*]{}, arXiv:1409.4857, 2014.
THORP, E.; [*Beat the dealer*]{}, Blaisdell Pub. Co, 1962.
|
---
abstract: 'Caustic rings of dark matter with *tricusp* cross section were predicted to lie in the galactic disk. Their radii increase on cosmological time scales at a rate of order $1$ kpc/Gyr. When a caustic ring passes through the orbit of a star, the orbit is strongly perturbed. We find that a star moving in a nearly circular orbit is first attracted towards the caustic ring, then moves with and oscillates about the caustic for approximately $1$ Gyr before returning to its original orbit. As a result, a stellar overdensity forms around the caustic ring. We predict such overdensities to be of order $120$% near the second caustic ring where the Monoceros Ring is observed and of order $45$, $30$ and $15$% near the third, fourth and fifth caustic rings, respectively. We show that the associated bulk velocities of the stars near the caustic rings are less than a few km/s. We also determine the density profile of interstellar gas near the fifth caustic ring assuming it is in thermal equilibrium in the gravitational potential of the caustic and of the gas itself.'
author:
- 'Sankha S. Chakrabarty and Pierre Sikivie'
bibliography:
- 'caustic\_bib.bib'
title: |
Effects of a caustic ring of dark matter on\
the distribution of stars and interstellar gas
---
\[I\]Introduction
=================
From various observations, it is known that $23\%$ of the energy density of the Universe is made of cold dark matter (CDM) [@Bertone; @*Weinberg; @*Kolb]. Axions are one of the widely accepted cold dark matter candidates [@Sikivie08; @*Marsh16]. The QCD axion was originally proposed as a solution to the strong *CP* problem in the standard model [@Peccei77; @*Peccei2; @*Weinberg78; @*Wilczek78; @Kim79; @*Shifman80; @*Zhitnitskii80; @*Dine81]. It became a cold dark matter candidate when it was shown that QCD axions are produced abundantly in the early universe with very small velocity dispersion [@Preskill83; @*Abbott83; @*Dine83; @*Ipser83].
Since dark matter particles are collisionless, they are described in six-dimensional phase space. Because they have very small velocity dispersion, CDM particles lie on a three-dimensional hypersurface embedded in that space. One inevitable consequence of this is the formation of *caustics* [@Zeldovich70; @*Arnold82; @*Shandarin89] [@Sikivie99] [@Natarajan05; @*Natarajan06]. Caustics are surfaces in physical space where the density is infinite in the limit of zero velocity dispersion. In a galactic halo, both outer and inner caustics are formed. As outer caustics appear in the outer regions of the galactic halo (e.g. at a distance of order $100 \; \text{kpc}$ from the Milky Way center), they hardly affect the stellar dynamics in the disk. In this paper, we only discuss the inner caustics which form when the particles are at their closest approach to the galactic center (see Fig. \[tricusp\_flows\]). The $n$th inner caustic forms in the flow of particles experiencing their $n$th infall in the galactic potential well. If the total angular momentum of dark matter particles is dominated by net overall rotation, each inner caustic is a closed tube whose cross section has the shape of a tricusp [@Sikivie98; @Sikivie99] (see Fig. \[tricusp\]). This structure is called a *caustic ring* of dark matter.
The caustic ring model [@Sikivie99; @Natarajan06; @Duffy08] is a proposal for the full phase space distribution of cold dark matter halos. It is axially symmetric, reflection symmetric and has self-similar time evolution. It predicts that caustic rings lie in the galactic plane and that their radii $a_n(t)$ increase on cosmological time scale as $a_n(t) \propto t^{4/3}$. There is observational evidence in support of the caustic ring model [@Sikivie03; @Banik17]. Furthermore, it was shown that net overall rotation, self-similarity and axial symmetry are the expected outcomes of the rethermalization of Bose-Einstein condensed axion dark matter [@Sikivie09; @Erken12] before it falls onto a galactic halo [@Sikivie11; @Banik13]. In the Milky Way, the present radius of the $n$th caustic ring is approximately $\frac{40 \; \text{kpc}}{n}$. Since the Solar System is about $8.5$ kpc away from the galactic center [@Binney], the $n$ = 1, 2, 3 and 4 caustic rings have passed the solar orbit while the $n = 5$ caustic ring is approaching.
We find that star orbits are strongly perturbed by a passing caustic ring (see Fig. \[Eplot\]). A star on a nearly circular orbit is first attracted toward the caustic ring, then moves with and oscillates about the caustic ring for approximately $1$ Gyr, and finally returns to its original orbit (see Fig. \[rtplot\]). This implies that a star overdensity forms near a caustic ring. The overdensity is determined by the depth $(\Phi_c)$ of gravitational potential well of the caustic and the velocity dispersion $(\sigma)$ of the stellar population near the caustic. If $\sigma^2$ is smaller than $\Phi_c$, the stellar distribution is heavily affected by the caustic. Due to larger infall rates [@Duffy08], caustic rings with smaller $n$, i.e. with larger radii, have stronger gravitational fields. Also, the stars near caustic rings of small $n$, i.e. those in the outer regions of the galactic disk, have relatively small velocity dispersions. As a result, large star overdensities form near the caustic rings of small $n$. We predict such overdensities around various caustic rings (see Figs. \[overdensityX\] and \[overdensityZ\]) by simulating the dynamics of half a million stars for each. We perform the simulations for the radial and vertical motions independently as it is computationally expensive to do so for the coupled two-dimensional motions of a large number of stars. We estimate the total overdensity near a caustic ring as the sum of overdensities formed due to the radial and vertical motions. Large star overdensities would attract more stars and interstellar gas, and are expected to be enhanced further. Such feedbacks are not considered here.
The Monoceros Ring [@Newberg02; @*Yanny03; @*Ibata03; @*Rocha03] has been observed at the location of the second caustic ring. We find that the star overdensity near the $n=2$ caustic ring is of order $120\%$. This reinforces the claim of Ref. [@Natarajan07] that the Monoceros Ring may be caused by the second caustic ring in our galaxy. We estimate the size of the tricusp of the $n=2$ ring to be $p \sim 2.5$ kpc based on the size of the Monoceros Ring. We find the star overdensity of order $45\%$ near the $n=3$ caustic ring, which may explain the existence of the Binney and Dehnen ring [@Binney97] at $13.6$ kpc. The overdensities for caustic rings with larger $n$ are smaller, e.g. approximately $30\%$ and $15 \%$ for $n=4$ and $5$ caustic rings, respectively. Such overdensities may be observed in upcoming astronomical data. Recently, three independent groups [@Widrow12; @*Carlin13; @*Williams13] have observed position dependent bulk velocities of order $10$ km/s for the stars in the extended solar neighborhood. Our work was originally motivated to investigate if the passing of a caustic ring through the solar neighborhood may explain such observations. A caustic ring passing the solar neighborhood moves with speed $\sim 1$ km/s $= 1.02$ kpc/Gyr. In Sec. \[IIIBa\], we show that the resultant bulk velocities of the stars are of order $\big( \frac{\Delta d}{d} \big)$ km/s, where $d$ is density of stars and $\frac{\Delta d}{d}$ their relative overdensity near the caustic ring. Even if the overdensity were of order $100 \%$, the bulk velocities are quite small compared to the observed ones. Hence, a passing caustic ring cannot explain the observed bulk velocities. More prominent astrophysical signatures of the caustic rings with large $n$ may be found in the distribution of interstellar gas. The interstellar gas has much smaller velocity dispersion than the stars and is strongly affected by caustic rings. We study the effects of the $n=5$ caustic ring on interstellar gas assuming the gas to be in thermal equilibrium in the gravitational potential of the caustic ring and of the gas itself. The caustic ring is taken to be static here because the dynamics of gas is fast compared to the time scale over which the radius of the caustic ring changes. We find that the density of the gas in a cross-sectional plane of the caustic ring has a triangular shape reminiscent of the tricusp. Triangular features in both tangent directions to the nearest caustic ring ($n=5$) have been observed in the IRAS [@Sikivie03; @Banik17] and GAIA[@Chakrabarty18] maps of the galactic plane. Interestingly, the observed features are sharper than those obtained under the above stated assumptions.
A brief outline of the paper is as follows. In Sec. \[II\], we describe the caustic ring model and determine the gravitational field and potential of a caustic ring. In Sec. \[III\], we study the dynamics of the stars in the vicinity of a caustic ring. In sec. \[IV\], we study its effect on the distribution of interstellar gas. Sec. \[V\] provides a summary.
\[II\]Caustic rings
===================
\[IIA\]Cold dark matter caustics
--------------------------------
In the limit of zero velocity dispersion, dark matter particles lie on a three-dimensional hypersurface in six-dimensional phase space. As the particles in a galactic halo are huge in number, they can be labeled by a set of three continuous parameters, $\vec{\alpha} = (\alpha_1, \alpha_2, \alpha_3)$. Let $\vec{x}(\vec{\alpha},t)$ be the position of particle $\vec{\alpha}$ at time $t$. For an arbitrary physical point $\vec{r}$, let the equation $\vec{x}(\vec{\alpha},t) = \vec{r}$ have solutions $\vec{\alpha}_j (\vec{r},t)$ with $j = 1, 2, ..., N_f(\vec{r},t)$. $N_f(\vec{r},t)$ is the number of flows through $\vec{r}$ at time $t$.
If $\frac{d^3 N}{d \alpha^3} (\vec{\alpha})$ is the number density of particles in the chosen parameter space, the mass density in physical space is given by [@Sikivie99] $$d(\vec{r},t) = m \sum_{j = 1}^{N_f(\vec{r},t)} \frac{d^3 N}{d \alpha^3} (\vec{\alpha}) \; \frac{1}{|D(\vec{\alpha},t)|}\Bigg|_{\vec{\alpha} = \vec{\alpha}_j (\vec{r},t)} \label{drt}$$ where $m$ is the mass of each particle and $|D(\vec{\alpha},t)| \equiv |\text{det} \Big( \frac{\partial \vec{x}(\vec{\alpha},t)}{\partial \vec{\alpha}} \Big)|$ is the Jacobian of the map from $\vec{\alpha}$ to $\vec{x}$. Caustics are locations in physical space where the density $d(\vec{r},t)$ diverges because the map is singular, i.e. $D(\vec{\alpha},t) = 0$.
\[IIB\]Flows near a caustic ring
--------------------------------
Each particle in an axially symmetric flow of cold dark matter is labeled by two parameters ($\alpha, \tau$). The third parameter labeling a particle is its azimuth which is irrelevant in the case of axial symmetry. $\alpha$ is the angle from the $z=0$ plane at the time of the particle’s most recent turnaround, i.e. $\alpha = \frac{\pi}{2} - \theta$ where $\theta$ is the polar angle in spherical coordinates. For each value of $\alpha$, $\tau = 0$ is defined as the time when the particle crosses the $z=0$ plane, i.e. $z(\alpha, \tau = 0) = 0$. The coordinates $(\rho, z)$ of a particle near the caustic ring are given by [@Sikivie99] $$\begin{aligned}
\rho && = a + \frac{1}{2} u (\tau - \tau_{0})^{2} - \frac{1}{2} s \alpha^{2} \label{rho} \\
z && = b \alpha \tau \label{z} \end{aligned}$$ where $a, \; u, \; \tau_{0}, \; s$ and $b$ are constants for a given caustic. The caustic occurs where the Jacobian $|D_{2} (\alpha, \tau)| \equiv |\text{det} \frac{\partial(\rho, z)}{\partial(\alpha, \tau)}|$ is $0$. Its location in the $\rho$-$z$ plane as a function of the parameter $\tau$ is given by $$\begin{aligned}
\rho &&= a + \frac{1}{2} u (\tau - \tau_{0})(2\tau - \tau_{0}) \label{rho_tau} \\
z &&= \pm b \sqrt{\frac{u}{s} \tau^{3} (\tau_{0} - \tau)} \;\; .\label{z_tau}\end{aligned}$$ Equations (\[rho\_tau\]) and (\[z\_tau\]) define the tricusp (see Figs. \[tricusp\_flows\] and \[tricusp\]). Its size is $p$ in the $\rho$-direction and $q$ in the $z$-direction: $p = \frac{1}{2} u \tau_{0}^{2}, \;\;\; q = \frac{\sqrt{27}}{4} \frac{b}{\sqrt{us}} p$. The density in physical space is [@Sikivie99] $$d(\rho, z) = \frac{1}{\rho} \sum_{j = 1}^{N_f(\vec{r},t)} \frac{dM}{d\Omega d\tau} (\alpha, \tau) \frac{cos \alpha}{|D_{2} (\alpha, \tau)|} \Bigg|_{\alpha_j (\rho, z), \tau_j (\rho, z)} \; , \label{drhoz}$$ where $\frac{dM}{d\Omega d\tau}$ is the mass of dark matter particles falling in per unit solid angle per unit time.
\[IIC\]Self-similarity
----------------------
The evolution of dark matter particles in a galactic halo is self-similar if there is no special time in its history [@Gunn77][@Fillmore84][@Bert85]. Self-similarity implies that the phase space distribution remains identical to itself except for an overall rescaling of its density, size and velocity. As pointed out in Ref. [@Duffy08], self-similarity does not require any symmetry. In the self-similar model, the radius of each caustic ring increases on cosmological time sale as $a(t) \propto t^{\beta}$ where $\beta = \frac{2}{3} + \frac{2}{9\epsilon}$. From the slope of the power spectrum of density perturbation on galactic scales, $\epsilon$ is determined to be in the range $0.25 < \epsilon < 0.35$ [@Sikivie97]. The value $\epsilon = \frac{1}{3}$ (hence $\beta = \frac{4}{3}$) is consistent with the observational evidence for caustic rings, the slope of the power spectrum on galactic scales and with the value from tidal torquing [@Sikivie11].
\[IID\]Gravitational field of a caustic ring
--------------------------------------------
The gravitational field inside and near a caustic ring has been calculated in Refs. [@Sikivie99; @Tam12]. Assuming the size ($p, q$) of the cross section of the ring to be much smaller than the radius $a$ of the ring, the gravitational field in terms of the rescaled variables $\big( X = \frac{\rho - a}{p}, Z = \frac{z}{p}\big)$ is given by $$\vec{g} = -8\pi G\; \frac{dM}{d\Omega d\tau} \; \frac{1}{b \rho} [I_{\rho} (X, Z) \hat{\rho} + I_{z} (X, Z) \hat{z}] \; , \label{gfield3}$$ where $$\begin{aligned}
I_{\rho} &&= \frac{1}{2\pi} \int_{-\infty}^{\infty} dA \int_{-\infty}^{\infty} dT \; \frac{X - (T - 1)^{2} + \xi A^{2}}{(X - (T - 1)^{2} + \xi A^{2})^{2} + (Z - 2AT)^{2}} \;\; ,\label{I_rho} \nonumber \\
I_{z} &&= \frac{1}{2\pi} \int_{-\infty}^{\infty} dA \int_{-\infty}^{\infty} dT \; \frac{(Z - 2AT)}{(X - (T - 1)^{2} + \xi A^{2})^{2} + (Z - 2AT)^{2}} \;\; \label{I_z} \nonumber\end{aligned}$$ and $\xi = \frac{su}{b^{2}}$. The variables $(\alpha, \tau)$ have been replaced by $\big( A = \frac{b \alpha}{u \tau_{0}}, T = \frac{\tau}{\tau_{0}} \big)$. In Ref. [@Tam12], $\xi$ was taken to be unity and the integrals were calculated both analytically, using residue theory, and numerically with consistent results. Equation (\[gfield3\]) gives the gravitational field of the whole flow forming the caustic. The gravitational field due to the flows without caustics is that of a smooth halo which results in a flat rotation curve. We are interested in the modified gravitational field only due to the formation of the caustic. In the caustic ring model, one cannot remove a flow without removing the caustic in the flow. To separate the gravitational field of a caustic ring from that of the flow of which it is part, we introduce a long distance damping factor, $$\vec{g}_{c} (\rho, z) = \exp \Big(- \frac{s^2}{R^2}\Big) \; \vec{g} (\rho, z) \label{g_c}$$ where $s^2 = (\rho - a - \frac{p}{4})^2 + z^2$ is the distance of the point $(\rho, z)$ from the center of the caustic and $R \sim 1.5 p$ is the distance scale over which the effects of the caustic ring are significant.
The parameters for various caustic rings have been listed in Ref. [@Duffy08]. The values for the infall rate $\frac{dM}{d\Omega d\tau}$ given in Ref. [@Duffy08] are based on the assumption of isotropic infall of dark matter particles. However, Ref. [@Banik13] argues that, because the vortices in the axion Bose-Einstein condensate attract each other, numerous smaller vortices join to form a huge vortex along the rotation axis of the galaxy. As a result, axions fall in preferentially along the galactic plane and caustic rings are enhanced. This explains why the bumps in the Milky Way rotation curve at the locations of the caustic rings are typically a factor of $4$ larger than that attributed only to the caustic rings with isotropic infall [@Sikivie03]. We therefore multiply the infall rates in Ref. [@Duffy08] by a factor $4$ to account for the formation of the ‘big vortex’, giving them the values $$\Big(\frac{dM}{d\Omega d\tau}\Big|_n : n = 1, 2, 3, 4, 5, ... \Big) \approx ( 210, 95, 60, 40, 32, ...)\; \frac{M_{\odot}}{\text{sterad-yr}}. \label{infallratecaustics}$$ In this paper, unless mentioned otherwise, we use the following parameters to describe a caustic ring passing through the solar neighborhood: $p = 0.5$ kpc, $b = 523$ km/s, $V = \frac{da}{dt} = 1$ kpc/Gyr and $\frac{dM}{d\Omega d\tau} = 32 \frac{M_{\odot}}{\text{sterad-yr}}$. The chosen infall rate is similar to that of the $n=5$ caustic ring, which is the one closest to us. In Fig. \[gplot\], we show the radial and vertical components ($g_{c\rho}$, $g_{cz}$) of the gravitational field at $Z = \frac{z}{p} = 0.25$ as a function of $X = \frac{\rho - a}{p}$.
\[IIE\]Gravitational potential of a caustic
-------------------------------------------
Assuming the size of the cross section of a caustic ring to be much smaller than its radius (i.e. $p, q \ll a$; see Fig. \[tricusp\]), the gravitational potential $\Phi(X, Z)$ near the caustic ring is given by $$\Phi(X, Z) - \Phi(X_0, Z_0) = 2 G \frac{dM}{d\Omega d\tau} \frac{p}{ba} J(X, Z; X_0, Z_0) \label{phixz}$$ where $$J(X, Z; X_0, Z_0) = \int_{-\infty}^{\infty} dA \int_{-\infty}^{\infty} dT \ln \frac{(X - (T-1)^2 + A^2)^2 + (Z-2AT)^2}{(X_0 - (T-1)^2 + A^2)^2 + (Z_0-2AT)^2} \;\; . \label{jxz}$$ We choose the center of the tricusp as the reference point ($X_0, Z_0$). Figure \[causticpot\] shows the two-dimensional plot of the caustic potential $$\Phi_c (X, Z) = \Phi(X, Z) - \Phi(X_0 = 0.25, Z_0 = 0) . \label{phicxz}$$ It looks smooth and continuous, although its second derivative $\nabla^2 \Phi_c$ diverges at the caustic.
\[III\]Effects on stars
=======================
\[IIIA\]A single star
---------------------
For the stars near the galactic disk, small radial and vertical oscillations can be treated independently [@Binney] in the absence of a caustic. The effective potential for radial motion consists of a logarithmic gravitational potential and the angular momentum barrier: $$\Phi_{\text{eff}} (\rho) = v_{\text{rot}}^2 \ln \rho + \frac{l^2}{2 \rho^2} \label{Phi_eff}$$ where $l = \rho^2 \dot{\phi} = \text{constant}$. For small radial oscillations about the minimum $\rho_0 = \frac{l}{v_{\text{rot}}}$ of the effective potential, the angular frequency is $\omega_{\rho} = \frac{\sqrt{2}v_{\text{rot}}}{\rho_0}$. For vertical motion, we choose the potential $\Phi_z (z)$ of an isothermal stellar disk with velocity dispersion $\sigma_{z}$ and scale height $z_0$ [@Binney] $$\Phi_z (z) = \sigma_{z}^2 \; \ln \Big[\cosh^2 \Big( \frac{z}{2z_0} \Big) \Big] . \label{Phi_z}$$ In the presence of a passing caustic ring, the radial and vertical accelerations of a star are given by $$\begin{aligned}
a_{\rho} (\rho, z) &&= - \frac{v_{\text{rot}}^2}{\rho} + \frac{l^2}{\rho^3} + g_{c\rho} (\rho, z, a(t)) \\ \label{arho_c}
a_z (\rho, z) &&= - \frac{\sigma_{z}^2}{z_0} \tanh \Big( \frac{z}{2z_0} \Big) + g_{cz}(\rho, z, a(t)) \label{az_c}\end{aligned}$$ where $g_{c\rho} (\rho, z, a(t))$ and $g_{cz}(\rho, z, a(t))$ are the contributions \[see Eqs. (\[gfield3\]) and (\[g\_c\])\] from the caustic ring with radius $a(t)$. To visualize the effects of the caustic ring on the star, we define radial and vertical energies per unit mass $$\begin{aligned}
E_{\rho} &&= \frac{1}{2} v_{\rho}^2 + \Phi_{\text{eff}} (\rho) - \Phi_{\text{eff}} (\rho_0) \;\; , \label{E_rho} \\
E_z &&= \frac{1}{2} v_z^2 + \Phi_{z} (z) \;\; , \label{E_z}\end{aligned}$$ where $\Phi_{\text{eff}} (\rho)$ and $\Phi_{z} (z)$ are given by Eqs. (\[Phi\_eff\]) and (\[Phi\_z\]) and $\rho_0$ is the minimum of $\Phi_{\text{eff}}(\rho)$. $E_{\rho}$ and $E_z$ remain constant in the absence of a caustic because radial and vertical motions are independent.
We numerically solve the equations of motion of a star in the $\rho$-$z$ plane with arbitrary initial conditions as the caustic ring passes through its orbit. The velocity dispersions of the stars in the solar neighborhood are $40$ km/s in the radial direction and $20$ km/s in the vertical direction, and the scale height $z_0 = 0.5$ kpc [@Binney]. As an example, we choose a star that orbits the galaxy at $\rho_0 = 8$ kpc with small radial and vertical oscillations such that $v_{\rho}^{\text{max}} = 10$ km/s and $v_z^{\text{max}} = 5$ km/s. In Fig. \[Eplot\], we show the energies, $E_{\rho}$ and $E_z$, of the star as the caustic ring passes through its orbit. The fluctuations in the energies are large and occur on a time scale of approximately $2$ Gyr. The fluctuations are smaller for larger initial values of $E_{\rho}$ and $E_z$.
The stars which are most affected by a passing caustic ring are those with nearly circular orbits. In Fig. \[rtplot\], we plot the radial coordinate $\rho$ of a star, and the locations of the rear ($a$) and front ($a+p$) of the tricusp of the caustic ring as a function of time. Initially, the star in Fig. \[rtplot\](a) has an exactly circular orbit with radius $\rho_0 = 8$ kpc, whereas the star in Fig. \[rtplot\](b) has an almost circular orbit with $\rho_0 = 8$ kpc and $v_{\rho}^{\text{max}} = v_z^{\text{max}} = 5$ km/s. The tricusp moves radially outward with a speed $\frac{da}{dt} = 1$ kpc/Gyr with $a(t=0) = 6$ kpc. In both cases, as the figures show, the star is first attracted towards the tricusp and then moves with and oscillates about the tricusp for approximately $1$ Gyr before returning to its initial orbit due to conservation of the angular momentum. For the chosen caustic ring parameters, we find that all the stars with $v_{\rho}^{\text{max}}, v_z^{\text{max}} \leq 10$ km/s exhibit such behavior. The intermediate phase of following the tricusp causes stellar overdensities around the caustic ring.
\[IIIB\]Distribution of stars
-----------------------------
When a caustic ring passes through a relaxed distribution of stars, it generates bulk velocities of the stars and perturbs the density profile of the stellar population.
### \[IIIBa\] Bulk velocities
From the continuity equation, $\frac{\partial d}{\partial t} + \vec{\nabla}. (d\; \vec{v}) = 0$, the bulk velocities of the stars are of order, $$v \sim \frac{\Delta d}{d} \; \frac{\Delta x}{\Delta t} \;\; , \label{v_bulk}$$ where $\frac{\Delta d}{d}$ is the relative overdensity caused by the caustic and, $\Delta x$ and $\Delta t$ are the length and time scales over which the stellar distribution changes. For a caustic ring with radius $a$ and tricusp of size $p$, $\Delta x \sim p$ and $\Delta t \sim \frac{p}{V}$ where $V \sim \frac{da}{dt}$ is the speed of the tricusp in the radial direction. All the caustic rings move slowly with speed $V \sim 1$ km/s. Hence, the bulk velocities are $v \sim 1\; \text{km/s} \; \big(\frac{\Delta d}{d}\big)$. Even if the overdensities are as large as $\sim 100$%, the bulk velocities induced by the caustic rings cannot be more than a few km/s. This is too small to explain the recently observed [@Widrow12; @*Carlin13; @*Williams13] bulk velocities of order $10$ km/s for the stars in the solar neighborhood.
### \[IIIBb\] Overdensities
A stellar population with smaller velocity dispersion is more susceptible to a passing caustic ring. The present locations of the caustic rings are given by $$(a_n : n = 1, 2, 3, 4, 5, ...) \approx (40, 20, 13.3, 10, 8, ...) \; \text{kpc}. \label{ancaustics}$$ The radial velocity dispersion $\sigma_{\rho}$ of the stars at a distance $\rho$ from the galactic center decays exponentially with $\rho$ [@Lewis89; @*Kubryk15]: $$\sigma_{\rho} (\rho) \approx (40\; \text{km/s})\;\text{exp}\Big[- \frac{(\rho - 8.5 \; \text{kpc})}{8 \; \text{kpc}}\Big]. \label{sigmavar}$$ Therefore, the radial velocity dispersions of the stars near the first five caustic rings are: $$(\sigma_{n\rho} : n = 1, 2, 3, 4, 5, ...) \approx (1, 10, 20, 30, 40, ...)\; \text{km/s}. \label{sigmacaustics}$$ The vertical velocity dispersions $\sigma_{z}$ are typically half of the radial ones. The infall rates for the caustic rings are given by Eq. (\[infallratecaustics\]). Caustic rings with smaller $n$ have stronger gravitational field \[see Eq. (\[gfield3\])\] and are surrounded by stars with smaller velocity dispersions.
In the presence of a caustic ring, the stellar dynamics in the radial and vertical directions are not independent. Simulating the dynamics of a large number of stars in the $\rho$-$z$ plane near the caustic ring is computationally expensive. For each caustic ring, we simulate one-dimensional motion of the stars in the radial and vertical directions independently. To determine how the star overdensities change with radial (vertical) coordinates, we simulate the radial (vertical) motions and suppress the vertical (radial) dynamics of the stars. The initial motion in the radial direction is assumed to be simple harmonic, i.e. a star oscillating about $\rho = \rho_0$ moves in a harmonic potential with $\omega_{\rho} = \frac{\sqrt{2} v_{\text{rot}}}{\rho_0}$ where $v_{\text{rot}} = 220$ km/s \[see Eq. (\[Phi\_eff\])\]. For the radial motion of the stars near each caustic ring, we generate a relaxed distribution of $500,000$ stars with phase space density $f(v_{\rho}, \rho) \sim \text{exp}\Big(- \frac{v_{\rho}^2 + \omega_{\rho}^2 (\rho - \rho_0)^2}{2\sigma_{\rho}^2}\Big)$. The initial vertical motion is determined by the potential $\Phi_z(z)$ in Eq. (\[Phi\_z\]) and the phase space density is given by $f(v_z, z) \sim \text{sech}^2 \Big( \frac{z}{2z_0} \Big) \text{exp}\Big(- \frac{v_{z}^2}{2\sigma_{z}^2}\Big)$. In the absence of the caustics, each stellar distribution remains in equilibrium; i.e. the number density profile $d_{\text{eq}}(\rho)$ or $d_{\text{eq}}(z)$ does not change with time.
For the $n$th caustic ring, the parameters for the radial and vertical dynamics of the stellar distributions are chosen as follows: $\sigma_{\rho} = 2\sigma_{z} = \sigma_{n\rho}$ \[see Eq. (\[sigmacaustics\])\], $\langle \rho \rangle = \rho_0 = a_n$ \[see Eq. (\[ancaustics\])\] and $z_0 = 0.5$ kpc. To minimize the error due to finite size of the stellar population, we choose the size $p$ of the tricusp of each caustic ring to be much smaller than the size of the corresponding stellar population. While simulating the dynamics of the stars as the caustic rings pass through them, we take several snapshots of each stellar distribution and determine the relative overdensity $\frac{\Delta d}{d} = \frac{d - d_{\text{eq}}}{d_{\text{eq}}}$. We find that, as long as the size $p$ of the tricusp is much smaller than the spread of the stellar population, the overdensity is independent of the size $p$. In Fig. \[overdensityX\], we plot the stellar overdensities $\frac{\Delta d}{d}$ due to the radial motion at the locations of different caustic rings as functions of the rescaled radial coordinate $X = \frac{\rho - a}{p}$. In Fig. \[overdensityZ\], we plot the same for the vertical motion as functions of the rescaled vertical coordinate $Z = \frac{z}{p}$. We did not consider the $n = 1$ caustic ring at $40$ kpc. In the linear approximation, the total overdensity near the caustic is the sum of the overdensities obtained for both directions. We find the maximum total overdensity to be of order $120\%, 45\%, 30\%$ and $15\%$ for the $n = 2, 3, 4$ and $5$ caustic rings. If the star overdensity near a caustic ring is large, it enhances the effects of the caustic by attracting more stars and interstellar gas. We did not include such backreactions in our simulation here. The large star overdensities near the $n = 2$ and $3$ caustic rings may explain the existence of the Monoceros Ring [@Newberg02; @*Yanny03; @*Ibata03; @*Rocha03] at $20$ kpc and the Binney and Dehnen Ring [@Binney97] at $13.6$ kpc. The Monoceros Ring has a vertical scale height of order $10$ kpc [@Newberg02]. According to Fig. \[overdensityZ\], the vertical scale height of the overdensity due to the $n=2$ caustic ring is of order $4p$. So, to form an overdensity of vertical size 10 kpc, the $n=2$ caustic ring is required to have a size $p \sim 2.5$ kpc. The sizes of the $n=1$ and $2$ caustic rings are not known from the bumps in the rotation curve while the size of the third caustic has been determined to be 1 kpc [@Duffy08]. Since the overdensities near the $n = 4$ and $5$ caustic rings occur within a distance of few kpc from the Sun, they may be observed in upcoming astronomical data such as from GAIA.
\[IV\]Effects on interstellar gas
=================================
The bulk properties of a distribution of stars in the solar neighborhood are affected by the caustic rings only at the $15\%$ level because the square of the velocity dispersion ($\sigma^2$) of the stars is larger than the depth of the potential ($\Phi_c$) of a caustic ring. Since gas and dust in the interstellar medium have smaller velocity dispersions [@Binney], their bulk properties, e.g. density, are expected to be affected more. Assuming the gas to be in thermal equilibrium in the gravitational potential of both the caustic and the gas itself, we study its density profile near the $n=5$ caustic ring. The caustic is taken to be static as the gas dynamics is fast compared to the time scale over which the radius of the caustic ring changes.
In the absence of the caustic ring, the interstellar gas is assumed to be in thermal equilibrium through self-gravitational interactions. Its potential and density are given by [@Binney]: $$\begin{aligned}
\Phi_g (\rho, z) &&= \sigma_g^2 \; \ln \Big[\cosh^2 \Big( \frac{z}{2z_g} \Big) \Big] \label{Phigas} \\
d_g (\rho, z) &&= d^{0}_{g} \; \text{sech}^2 \Big( \frac{z}{2z_g} \Big) \label{rhogas}\end{aligned}$$ where $z_g = \frac{\sigma_g}{\sqrt{8\pi G d^{0}_{g}}}$ is the scale height and $\sigma_g$ is the velocity dispersion. The parameters in the solar neighborhood are the following: $z_g \approx 65$ pc, $\sigma_g \approx 5$ km/s and $d^{0}_{g} \approx 0.05 \; \text{M}_{\odot}/ \text{pc}^{3}$ [@Binney]. Since the scale height of the gas is much smaller than that of disk stars ($300$ pc for thin and $500$ pc for thick disks), we ignore the gravity of the stars. The potential and density are taken to be independent of the radial coordinate $\rho$ since they do not change significantly with $\rho$ over the size of the tricusp.
We change our coordinate system from $(\rho, z)$ to $(X, Z) = (\frac{\rho - a}{p}, \frac{z}{p})$ as the latter is more convenient. In the presence of a caustic ring, the potential $\Phi (X, Z)$ due to gas is the solution of Poisson’s equation, $$\nabla^2 \Phi (X, Z) = 4\pi G d_g (X, Z) \label{Poisson}$$ where $$d_g (X, Z) = d_g (X_0, Z_0) \exp \Big(-\frac{\Phi(X, Z) + \Phi_c (X, Z)}{\sigma_g^2}\Big) \label{rhoxz}$$ assuming $\Phi(X_0, Z_0) = \Phi_c (X_0, Z_0) = 0$. We calculate the new potential $\Phi (X, Z)$ using a two-dimensional Green’s function. In the $X$-$Z$ plane, we choose a sufficiently large region around the tricusp such that, at the boundary of the region, $\Phi (X, Z)$ tends to be $\Phi_g (X, Z)$ \[see Eq. (\[Phigas\])\]. Choosing a Green’s function $G(X, Z; X', Z')$ that vanishes at the boundary, we have $$\Phi (X, Z) = \int dX' \int dZ' \; G(X, Z; X', Z') 4\pi G d_g (X', Z') + \oint dl' \; \Phi_g (X', Z') \frac{\partial G}{\partial n'}(X, Z; X', Z') . \label{phixzn}$$ We solve the above equation numerically. The solution for $\Phi(X, Z)$ converges after few iterations. In Fig. \[densityplot\], we plot the density of gas $d_g (X, Z)$ near the tricusp of the $n=5$ caustic ring with size $p = 150$ pc. As evident from the figure, the density profile has a triangular shape as does the caustic potential shown in Fig. \[causticpot\]. In the recent GAIA sky map of the Milky Way [@GAIADR1; @*GAIADR2], triangular features are observed [@Chakrabarty18] in both tangent directions to the fifth caustic ring. However, the observed features have sharper edges than the triangular shape of Fig. \[densityplot\].
\[V\]Summary
============
The radii $a_n$ of caustic rings increase slowly at the rate of approximately $1\; \text{km/s} \; \big(\frac{a_n}{8\; \text{kpc}}\big)$. We study the dynamics of stars and interstellar gas as the caustic rings pass through them. We find that stellar orbits are highly perturbed by a passing caustic ring (see Fig. \[Eplot\]). A star moving in a nearly circular orbit is first attracted towards the caustic ring, then moves with and oscillates about the caustic for approximately $1$ Gyr, and finally returns to its original orbit (Fig. \[rtplot\]) as a result of angular momentum conservation.
Next, we study how the bulk properties of a distribution of stars are affected by the passing of a caustic ring. We find that the induced bulk velocities of the stars cannot be more than a few km/s. The star overdensity around a caustic ring depends upon the velocity dispersion of the stars. Since a caustic ring with smaller $n$ has stronger gravitational field and is surrounded by stars with lower velocity dispersion, it causes a larger stellar overdensity. Figures \[overdensityX\]-\[overdensityZ\] show the overdensities of stars around different caustic rings due to radial and vertical dynamics. The maximum total overdensity is approximately $120\%$ around the $n=2$ caustic ring at $20$ kpc which supports the claim of Ref. [@Natarajan07] that the Monoceros Ring [@Newberg02; @*Yanny03; @*Ibata03; @*Rocha03] may be caused by the second caustic ring. The total overdensity of $45\%$ near the $n = 3$ caustic ring may explain the existence of the Binney and Dehnen ring at $13.6$ kpc [@Binney97]. The overdensities are smaller for larger $n$, e.g. approximately $30\%$ and $15\%$ for the $n= 4$ and $5$ caustic rings, respectively. These overdensities, within a few kpc from us, may be observable in upcoming astronomical data.
We also study the distribution of interstellar gas near the $n=5$ caustic ring which is the one closest to us. The gas dynamics is much faster than the time scale over which the radius of a caustic ring changes. We consider the caustic ring to be static and assume the gas to be in thermal equilibrium in the gravitational potential of the caustic (shown in Fig. \[causticpot\]) and of the gas itself. Using a Green’s function, we iteratively solve the Poisson’s equation to find the density of gas. The density profile of gas in $\rho$-$z$ plane shown in Fig. \[densityplot\] has a triangular shape as a result of the tricusp cross section of the caustic ring. In the recent GAIA sky map [@GAIADR1; @*GAIADR2], two triangular features in both tangent directions to the fifth caustic ring are observed [@Chakrabarty18]. However, the observed features have sharper edges than the features (Fig. \[densityplot\]) we obtain.
The authors are grateful to Heywood Tam for the code for calculating the gravitational field of a caustic ring. This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-97ER41209 and by the Heising-Simons Foundation under Grant No. 2015-109.
![\[causticpot\]Two-dimensional plot of the gravitational potential $\Phi_c (X, Z)$ of a caustic ring as a function of the rescaled coordinates $X = \frac{\rho - a}{p}$ and $Z = \frac{z}{p}$. The parameters of the caustic ring are the same as those in Fig. \[gplot\]. The center of the tricusp ($X_0 = 0.25$, $Z_0 = 0$) is chosen as the reference point. The potential has a triangular shape inherited from the tricusp.](plot_caustic_pot_2d.png)
![\[overdensityX\]Relative overdensities of the stars $\frac{\Delta d}{d} = \frac{d - d_{\text{eq}}}{d_{\text{eq}}}$ due to radial motion near the $n = 2, 3, 4$ and $5$ caustic rings as functions of the rescaled radial coordinate $X = \frac{\rho - a}{p}$. Stellar overdensities are higher near the caustic rings with smaller $n$ because they have stronger gravitational field and are surrounded by stars with lower velocity dispersions.](overdensity_X.png)
![\[overdensityZ\]Relative overdensities of the stars (same as Fig. \[overdensityX\]) due to vertical motion as functions of the rescaled vertical coordinate $Z = \frac{z}{p}$.](overdensity_Z.png)
![\[densityplot\]Two-dimensional plot of the density of gas $d_g (X, Z)$ (Eqs. \[rhoxz\], \[phixzn\]) when the gas is in thermal equilibrium in the gravitational potential of the caustic (shown in Fig. \[causticpot\]) and of the gas itself. The velocity dispersion $\sigma_g$ of the gas is chosen to be $5$ km/s. The density profile has a triangular shape reflecting that of the tricusp.](plot_density_2d_final.png)
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---
abstract: 'We present a study of the structural properties of (x)Na$_2$S-(1-x)GeS$_2$ glasses through DFT-based molecular dynamics simulations, at different sodium concentrations ($0<x<0.5$). We computed the radial pair correlation functions as well as the total and partial structure factors. We also analyzed the evolution of the corner- and edge-sharing intertetrahedral links with the sodium concentration and show that the sodium ions exclusively destroy the former. With the increase of the sodium concentration the “standard” FSDP disappears and a new pre-peak appears in the structure factor which can be traced back in the Na-Na partial structure factor. This self organization of the sodium ions is coherent with Na-rich zones that we find at high modifier concentration.'
address: |
Laboratoire de Physicochimie de la Matière Condensée, Université Montpellier 2,\
Place E. Bataillon, Case 03, 34095 Montpellier, France
author:
- Sébastien Blaineau and Philippe Jund
title: 'Structural properties of various sodium thiogermanate glasses through DFT-based molecular dynamics simulations'
---
Introduction
============
Amorphous (x)Na$_2$S-(1-x)GeS$_2$ systems are interesting glasses, presenting a high ionic conductivity at room temperature. These materials can be used as model systems for efficient solid electrolytes even though practical applications are more often based on glasses containing lithium or silver ions for example. Although the ionic conduction process has been clearly observed in sodium thiogermanate glasses, its microscopic origin is still not clearly understood. Like in many other glassy systems, the atomic mechanisms responsible for these transport properties have not yet been clearly determined which explains the large number of studies dedicated to this topic in the past few years [@jund]. In particular, it is interesting to determine if the modifier ions follow preferential pathways inside the amorphous system [@greaves], and if this process is dependent on the glass type. To that end, theoretical studies such as molecular dynamics (MD) simulations can provide detailed information concerning the microscopic properties of these chalcogenide glasses.\
In previous studies we have analyzed the physical and chemical properties of GeS$_2$ glasses via Density Functionnal Theory (DFT)-based molecular dynamics simulations [@blaineau1; @blaineau2; @blaineau3]. The results obtained from our model were in very good agreement with the available experimental data. Subsequently we studied the vibrational and electronic properties of sodium thiogermanate (Na-Ge-S) glasses using the same description [@blaineau4]. In the present work, we focus on the structural properties of (x)Na$_2$S-(1-x)GeS$_2$ amorphous systems using the same DFT-based model in order to determine the impact of the Na$^+$ ions on the structure of germanium disulfide glasses. It should be noted that there is a lack of experimental results in the literature concerning these doped chalcogenide glasses. Thus no extensive comparison with experiments is possible at this time, and our results should be considered as a first step in the study of the structural properties of (x)Na$_2$S-(1-x)GeS$_2$ glasses.\
This paper is organized as follows: section II is dedicated to the description of the theoretical foundations of our model. Subsequently, we present the results obtained in our simulations in section III. Finally, section IV contains the major conclusions of our work.
Model
=====
The mathematical model we have used was developed by Sankey and Niklewski [@sankey]. It is included in a first-principles type molecular-dynamics code, called , that is based on the DFT [@hohenberg-kohn], within the Local Density Approximation (LDA) [@ceperley; @perdew]. A tight-binding-like linear combination of pseudoatomic orbitals, satisfying the atomic self-consistent Hohenberg-Kohn-Sham equations [@kohn-sham], is used to determine the electronic eigenstates of the system. A basis set of one $s$ and three $p$ pseudo-orbitals, slightly excited since they vanish outside a cut-off radius of $5a_o$ (2.645 Å), is required.\
The pseudopotential approximation is used to replace the core electrons with an effective potential which acts on the valence electrons, and Hamman-Schluter-Chiang pseudopotentials were used in this work [@schluter]. The Hamiltonian is calculated through the use of the Harris functional [@harris], which avoids long self-consistent calculations by a zero$^{th}$ order approximation of the self-consistent density. Further details on the method can be found in the original paper [@fireball]. Periodic boundary conditions are applied in order to limit surface effects, and only the $\Gamma$ point is used to sample the Brillouin zone. This model has given excellent results in several different chalcogenide systems over the past few years [@blaineau1; @drabold; @junli].\
In the present work we computationally melted a crystalline $\alpha$-GeS$_2$ configuration containing 258 particles at 2000K during 60 ps (24000 timesteps) until we obtained an equilibrated liquid. Subsequently we randomly replaced GeS$_4$ tetrahedral units by Na$_2$S$_3$ units, in order to obtain a given concentration of modifier without affecting the global electrical neutrality of the system. This procedure has been used in a similar manner in SiO$_2$ glasses [@jund1], and keeps the total number of atoms, N, unchanged (258 particles in this work).\
Following this scheme, we generated eight (x)Na$_2$S-(1-x)GeS$_2$ samples with different sodium concentrations (x= 0, 0.015, 0.03, 0.06, 0.11, 0.2, 0.33 and 0.5). The bounding box was rescaled for each sample so that the density matched its experimental counterpart (with cell lengths ranging from 19.21 Å for x=0[@boolchand] to 18.3 Å for x=0.5[@ribes]). Since the experimental densities are not exactly the equilibrium densities of the model, the pressure inside the samples is not exactly equal to zero. Nevertheless it is extremely small ($<$ 0.4 GPa) demonstrating the quality of the model. The resulting system was subsequently melted at 2000K during 60 ps in order for the system to lose memory of the previous artificial configuration. When the liquid reached an equilibrium state, the systems were quenched at a quenching rate of 6.8$\times$10$^{14}$ K/s, decreasing the temperature to 300K, passing through the glass transition temperature T$_g$. The samples were then relaxed at 300K over a period of 100 ps. In order to improve the statistics of our results, five samples, starting from independent liquid configurations, were generated for each concentration. The results presented here have been averaged over these five samples.
Results
=======
In a previous work [@blaineau3], we determined that the atomic charges in GeS$_2$ glasses using a Löwdin description [@lowdin] are equal to +0.94 for Ge atoms and -0.47 for the S particles. Even though homopolar bonds exist in this glass [@blaineau2], no Ge-Na bonds were seen in our samples at the concentrations studied here ($0< x \le 0.5$). Therefore we can study the impact of the sodium ions on the short-range order of germanium disulfide glasses by analysis of the S-Na bonds exclusively.\
Among the radial pair correlation functions $g_{\alpha \beta}(r)$ defined as: $$g_{\alpha \beta}(r)=\frac{V}{4 \pi r^{2} N_{\alpha} dr}~dn_{\beta}$$ for the different ($\alpha$,$\beta$) pairs, we can hence concentrate on the $g_{SNa}(r)$ function. It can be seen in Fig.1 that the $g_{SNa}(r)$ for S-Na pairs does not change significantly with the concentration of the sodium modifier. The first-neighbor peak appears at 2.4 Å for all concentrations ($0< x\le 0.5$), which is slightly inferior to experimental data concerning crystalline sodium thiogermanate systems (2.71 Å [@foix]). However we tested several glassy samples with the program [@siesta] in a self-consistent $ab~initio$ description, using the most accurate basis set available in the code for the description of the atomic orbitals (Double Zeta Polarized + Generalized Gradient Approximation): the results were very similar (d$_{S-Na}=2.45$ Å) to those obtained with .\
The different coordination numbers $n_{\alpha \beta}$ can also be calculated. They correspond to the average number of $\beta$ neighbors of a given $\alpha$ atom within a sphere of radius r$_{min}$, r$_{min}$ being the first minimum of the corresponding $g_{\alpha \beta}(r)$. We show in Fig.2 the evolution of the coordination number of the sulfur atoms with the three elements present in our system. It can be seen that the value of $n_{S-Ge}$ decreases with the addition of sodium modifier from 1.95 for x=0 to 1.3 for x=0.5. This indicates that the Na$^+$ ions break several inter-tetrahedral connections in order to be connected to the sulfur atoms, which explains the increasing value of $n_{S-Na}$ (2.45 for x=0.5). The value of $n_{S-S}$ relative to homopolar bonds remains constant without any dependency with the sodium concentration. Finally the average coordination number of the sodium atoms relative to the sulfur atoms, $n_{Na-S}$, has been found close to 3.27 for x=0.5. In fact, the Na$^+$ particles in the system are connected to 3 or 4 sulfur atoms, which is consistent with the results obtained by Foix $et~al.$ in cluster simulations of Na$_2$GeS$_3$ glasses using a Hartree-Fock model. [@foix].\
We reported in a previous study [@blaineau2] that 14.53$\%$ of the sulfur atoms in glassy GeS$_2$ systems are non-bridging (i.e. connected to only one Ge atom). These bond defects have been found [@blaineau3] to create a negatively charged electronic environment around the terminal sulfur. Therefore it is interesting to study the evolution of these non-bridging sulfur particles with the introduction of Na$^+$ cations. We distinguish here between the terminal sulfur atoms connected to (a) zero, (b) one, (c) two or (d) three sodium ions, and represent in Fig.3 the evolution of these structural entities in (x)Na$_2$S-(1-x)GeS$_2$ for $0 \le x \le
0.5$.\
One can see that the proportion of terminal sulfur atoms that are not linked to a sodium ion decreases significantly as soon as x is non zero. This variation is counterbalanced by the increase of the proportion of terminal sulfur atoms connected to one sodium atom, which reaches approximately the same value of $\approx$15$\%$ of the S particles in the sample. We can therefore deduce that, as expected, the non-bridging sulfurs present in $g$-GeS$_2$ significantly attract the Na$^+$ ions. Between $x=0.11$ and $x=0.33$ these structural entities decrease (Fig.3(b)), and are counterbalanced by the increase of terminal sulfur connected to two Na ions (Fig.3(c)), that also reaches the value of $\approx$15 $\%$. Finally, for $x > 0.33$ the proportion of terminal sulfurs connected to three sodium ions reaches $\approx$15 $\%$ as in the aforementioned cases. This shows that the non-bridging sulfur atoms observed in $g$-GeS$_2$ attract $several$ Na$^+$ ions in sodium thiogermanate glasses.\
The study of the short-range charge deviation in $g$-GeS$_2$ has also revealed the existence of “positively” charged zones, which are mainly caused by 3-fold coordinated sulfur and homopolar bonds [@blaineau3]. We find that these positively charged zones remain basically unchanged in (x)Na$_2$S-(1-x)GeS$_2$ glasses, and that the Na$^+$ ions never connect to 3-fold coordinated sulfur atoms. Hence negatively and positively charged zones in $g$-GeS$_2$ appear to attract and repel the Na$^+$ cations respectively. This observation, which seems to be self-evident, is in fact relevant since all the different samples have been melted, quenched, and relaxed [*independently*]{}. Despite that, we find similarly charged zones in comparable proportions in all of these samples.\
The decrease of the $n_{S-Ge}$ coordination number with increasing sodium content observed in Fig. 2 shows that an increasing number of intertetrahedral bonds are destroyed by the addition of sodium ions. Since two different types of intertetrahedral links exist in glassy GeS$_2$ (edge-sharing and corner-sharing links), it is interesting to determine which type of S-Ge connections are destroyed by the introduction of the Na$^+$ ions. These two types of intertetrahedral connections can be distinguished in the radial pair correlation function of germanium-germanium pairs [@blaineau2], leading to Ge-Ge distances equal to 2.91 Å (edge-sharing) and 3.41 Å (corner-sharing). In Fig. 4 we present the [*total*]{} number of corner-sharing and edge-sharing links in the different samples as a function of sodium concentration. It can clearly be seen that exclusively corner-sharing links are destroyed by the introduction of sodium cations whereas the number of edge-sharing links remains constant for all sodium concentrations. This result in not predictable [*a priori*]{} since very different results may be obtained depending on the glass and the modifier. For example, experimental studies on chalcogenide SiS$_2$ glasses have shown that Li$^+$ cations destroy exclusively edge-sharing links [@micoulaut; @pradel], while the addition of Na$^+$ ions was found to break corner-sharing and edge-sharing links in equal proportions [@pradel2]. Here we find that sodium ions destroy exclusively corner-sharing links in GeS$_2$ glasses but this result still has to be confirmed (or refuted) experimentally.\
In order to analyze the global impact of the sodium ions on the structure of thiogermanate glasses, we studied next the total static structure factor $S(q)$ of our (x)Na$_2$S-(1-x)GeS$_2$ glasses. The total static structure factor was computed using the following formulae: $$S(q) = 1 + 4 \pi \frac{N}{V} \int_0^{r_{max}} ( g(r) - 1 ) \frac{\sin(qr)}{qr} r^2 dr$$ where $$g(r) = \frac{ \left[\displaystyle \sum_{\alpha,\beta} c_{\alpha} b_{\alpha} c_{\beta} b_{\beta} g_{\alpha \beta}(r) \right] }
{\left[ \displaystyle \sum_{\alpha} c_{\alpha} b_{\alpha} \right]^2 }$$ In the above equations, $r_{max}=L/2$ where $L$ is the edge of the box, the $g_{\alpha \beta}(r)$ are the radial pair correlation functions ($\alpha,\beta=$ Ge,S,Na), $c_{\alpha}$ and $c_{\beta}$ are the concentrations of the species $\alpha$ and $\beta$ and $b_{\alpha}$ and $b_{\beta}$ are their scattering lengths taken equal to 8.18 fm for Ge, 2.84 fm for S and to 3.63 fm for Na [@Sq]. The results for $S(q)$, calculated using the above definition, are shown in Fig.5 for the different values of $x$ (even though these results are directly comparable to neutron scattering experiments, no experimental data is available to our knowledge).\
The first sharp diffraction peak (FSDP), which is a signature of amorphous materials, appears at $q=$1 Å$^{-1}$ in $g$-GeS$_2$ [@blaineau1]. This peak reveals the existence of a structural order on a length scale $d$ corresponding to $d=2\pi /q\approx
6.3$ Å in germanium disulfide glasses. In sodium thiogermanate glasses, it can be seen that the intensity of the FSDP decreases significantly with $x $ for $x\leq 0.2$. This decrease of the FSDP has also been observed experimentally in silver thiogermanate glasses [@pradel]. This means that the introduction of ions disturbs the intermediate range order of the glassy matrix by breaking corner-sharing intertetrahedral connections. However for $x\geq 0.2$, a new feature appears in the structure factor at a lower value of $q$ ($q=0.5$ Å$^{-1}$). For x=0.5 this pre-peak reaches almost the same intensity than the original FSDP, which almost completely vanishes and shifts to a slightly higher $q$ value. In order to find the origin of this new peak we calculated the partial structure factors (computed from the radial pair correlation functions $g_{\alpha \beta}(r)$) in the different samples, and in particular the Na-Na partial structure factor (Fig.6). We find that the sodium-sodium pairs are mainly responsible for the pre-peak observed in Fig.5. It can indeed be seen that a peak emerges at $q=0.5$ Å$^{-1}$ for $x \geq 0.11$ and subsequently grows with increasing $x$. To analyze the structural arrangement of the sodium atoms, which are obviously not distributed homogeneously inside the simulation box, we present the density of sodium atoms in a (0.33)Na$_2$S-(0.66)GeS$_2$ averaged over the relaxation time in Fig.7. To that end, we divide the bounding box into 1000 (10$\times$10$\times$10) elementary volume elements. We subsequently calculated the number of Na atoms located in each specific volume element, and integrated this quantity over the relaxation time. Hence each sphere represented in Fig.7 is located at the center of a volume element, and its size increases (becomes more red in color) with the local Na density (the largest sphere and thus the highest density corresponds to 0.154 Å$^{-3}$). This figure shows that our sodium thiogermanate sample contains several zones which have a high concentration of Na$^+$ and other zones which have a poor concentration of Na$^+$ ions. When we analyzed the local structure around the sodium atoms we found that most of them are in a Na$_2$S-like environment, since for $x$=0.5, 23.5 $\%$ of all the sulfur atoms are [*exclusively*]{} connected to Na ions, i.e. without any germanium atom as their nearest-neighbors. These sodium rich zones could be the static “remainders” of conduction channels that could be observed at higher temperatures similarly to what has been reported in sodium enriched silicate glasses via [*classical*]{} molecular dynamics simulations and inelastic neutron scattering experiments [@jund1; @meyer]. If we accept this hypothesis, then, since the new pre-peak appears only for high sodium concentrations (see Fig.6), conduction channels would be absent at low sodium content and this could explain the experimentally observed increase in the sodium diffusion constant when the sodium concentration is greater than $x$=0.10 [@annie]. Unfortunately, the DFT-based model that we used to perform these simulations are too computationally expensive making it difficult to reach the simulation time required to study the diffusion of the Na$^+$ ions, and to analyze the evolution of these zones from a dynamical point of view. This is the reason why the sodium rich zones presented in Fig.7 cannot be directly assimilated to the diffusion channels.\
Most of theses results have no experimental counterparts and therefore need still to be confirmed or invalidated, but neutron diffraction studies on these systems have been performed and the results should be available in a not so distant future.\
To pursue this study further it would also be interesting to analyze the influence of the nature of the modifier on the results. Thus similar studies on silver thiogermanate glasses are currently in progress.
Conclusion
==========
We have studied the structural properties of sodium thiogermanate glasses via DFT-based molecular dynamics simulations for eight different modifier concentrations. We find that the structure of these chalcogenide glasses is significantly influenced by the extended positively and negatively charged zones observed in amorphous GeS$_2$ [@blaineau3]. As expected, the sodium cations are mainly attracted by the negative zones and avoid the positive regions. At high sodium concentrations, the Na$^+$ ions therefore become localized in space, and zones containing high and low density of sodium cations can be observed. These Na$_2$S-rich zones could be the low temperature analogs of the “conduction channels” observed in silicate glasses, where the diffusing ions follow preferential pathways. We have also determined that the addition of sodium cations inside the (x)Na$_2$S-(1-x)GeS$_2$ system destroys exclusively the corner-sharing links for $x\le 0.5$.\
The structure factor of the different samples shows that the FSDP that appears in amorphous GeS$_2$ at $q=$1 Å$^{-1}$, decreases significantly with the addition of sodium ions. At high sodium concentrations ($x \ge 0.11$), a new pre-peak appears at $q=0.5 \AA^{-1}$, which corresponds to a distance in real space of 12.6 Å. The Na-Na partial $S(q)$ shows that the Na-Na pairs are responsible of this pre-peak indicating a non homogeneous distribution of the Na$^+$ ions inside the glass which is consistent with the existence of the aforementioned sodium rich zones.\
Parts of the simulations have been performed on the computers of the “Centre Informatique National de l’Enseignement Supérieur” (CINES) in Montpellier.\
Dr. N.A. Ramsahye is acknowledged for proof reading the manuscript.\
------------------------------------------------------------------------
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abstract: 'In this paper, we focus on analyzing the period distribution of the inversive pseudorandom number generators (IPRNGs) over finite field $({\rm Z}_{N},+,\times)$, where $N>3$ is a prime. The sequences generated by the IPRNGs are transformed to $2$-dimensional linear feedback shift register (LFSR) sequences. By employing the generating function method and the finite field theory, the period distribution is obtained analytically. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods. The analysis results show that there are many small periods if $N$ is not chosen properly. The experimental examples show the effectiveness of the theoretical analysis.'
author:
- |
Bo Zhou$^{1,*}$, Qiankun Song$^{2}$\
[^1] [^2]\
title: Period Distribution of Inversive Pseudorandom Number Generators Over Finite Fields
---
*Keywords:* Inversive pseudorandom number generators (IPRNG); Linear feedback shift register (LFSR); Period distribution; Finite field.
Introduction
============
Pseudoramdom number generators (PRNGs) are deterministic algorithm that produces a long sequence of numbers that appear random and indistinguishable from a stream of random numbers [@s17], which are widely employed in science and engineering, such as Monte Carlo simulations, computer games and cryptography. In recent years, a variety of PRNGs based on nonlinear congruential method [@e8; @k13], chaotic maps [@j8; @a1; @a11] and linear feedback shift registers (LFSRs) [@r11; @k12] are proposed. These PRNGs are implemented on finite state machines, which lead to the fact that sequence generated by them are ultimately periodic. In cryptographic applications, a long period is often required. Once the period is not long enough, the encryption algorithms may be vulnerable to attacks, e.g., in [@k12], Kocarev *et al.* proposed a public key encryption algorithms based on Chebyshev polynomials over the finite field, but in [@c4; @c5], Chen *et al.* showed that if the period of the sequence generated by the Chebyshev polynomials is not sufficiently long, the public key encryption algorithm is easy to be decrypted. Therefore, it is worth to making clear that what are the possible periods of a PRNG and how to choose suitable control parameters and initial values such that the PRNG fits specific period, these knowledge helps in algorithm design and its related applications.
In [@c4; @c5], Chen *et al.* analyzed the period distribution of the sequence generated by the Chebyshev polynomials over finite fields and integer rings, respectively, by employing the generating function method. In [@c3], Chen *et al.* analyzed the period distribution of the generalized discrete Arnold cat map over Galois rings by employing the generating function method and the Hensel lifting method. In [@c6], Chen *et al.* summarized their works on the period distribution of the sequence generated by the linear maps.
In [@c9], Chou described all possible period lengths of IPRNG (1) and showed that these period lengths are related to the periods of some polynomials. However, the author did not give the full information on period distribution, this leads to the limitation of the applications of IPRNGs. In [@s7], Solé *et al.* proposed an open problem of arithmetic interest to study the period of the IPRNGs and to give conditions bearing on $a,b$ to achieve maximal period. Although their considered state space is a Galois ring, it is also significant to study this problem in finite field. Recent results on the distribution property in parts of the period of this generator over finite fields can be found in [@g16; @n17] and it would be interesting to generalize these results to arbitrary parts of the period. If the the full information on the period distribution is known, we could do such a work.
Motivated by the above discussions, we focus on analyzing the period distribution of the IPRNGs over the finite field $({\rm Z}_{N},+,\times)$, where $N>3$ is a prime. The analysis process is that, first, to make exact statistics on the periods of model (1), then count the number of IPRNGs for each specific period when $a$, $b$ and $x_{0}$ traverse all elements in ${\rm Z}_{N}$. The sequences generated by model (1) are transformed to $2$-dimensional LFSR sequences which is the foundation of the stream ciphers [@g21]. Then, the detailed period distribution of IPRNGs is obtained by employing the generating function method and the finite field theory. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods.
This paper is organized as follows. To make this paper self-contained, Section II presents some preliminaries that help to understand our analysis. In Section III, detailed analysis of the period distribution of the sequences generated by IPRNGs with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in{\rm Z}_{N}$. Then Section IV presents the detailed analysis of the period distribution of the sequences generated by IPRNGs with $a \in {\rm Z}^{\times }_{N}$, $b \in {\rm Z}^{\times }_{N}$ and $x_{0}\in{\rm Z}_{N}$. Finally, conclusion and some suggestions for future work are made in Section V.
Preliminaries
=============
In this section, we introduce relevant notation and definition to facilitate the presentation of main results in the ensuing sections. For the knowledge of finite fields, please refer to [@l2].
Recurring relation over the finite field
----------------------------------------
Let ${\rm Z}_{N}$ be the residue ring of integers modulo $N$. When $N$ is prime, $({\rm Z}_{N},+,\times)$ forms a finite field to which the modular operation is required in addition and multiplication.
[@l2]. A sequence $a_{0},a_{1},\ldots$ satisfying the relation over $({\rm Z}_{N},+,\times)$: $$\begin{aligned}
a_{n+k}=c_{1}a_{n+k-1}+c_{2}a_{n+k-2}+\ldots+c_{k}a_{n} \quad {\rm mod}N,\end{aligned}$$ where $c_{i}\in {\rm Z}_{N}$ for all $i=1,2,\ldots$, is called a linear recurring sequence in ${\rm Z}_{N}$.
The generation of the linear recurring sequences can be implemented on a linear feedback shift register which is a special kind of electronic switching circuit handling information in the form of elements in ${\rm Z}_{N}$.
[@l2]. $f(t)=t^{k}-c_{1}t^{k-1}-\cdots-c_{k}$ is called the characteristic polynomial of recurring relation (1). Also, the sequence $a_{0},a_{1},\ldots$ is called the sequence generated by $f(t)$ in ${\rm Z}_{N}$.
The characteristic polynomial $f(t)$ plays an important role in analyzing the period of the sequence generated by recurring relation (1). It follows from [@c4] that if all roots of $f(t)$ are with multiplicity $1$, then the period $T$ of $a_{0},a_{1},\ldots$ equals to ${\rm per}(f)$. ${\rm per}(f)$ is the smallest integer such that $f(t)\mid t^{{\rm per}(f)}-1$, which is called the period of $f(t)$. Then, we have the following proposition on ${\rm per}(f)$.
If $f(t)$ can be factorized as $f(t)=(t-\alpha_{1})(t-\alpha_{2})\ldots(t-\alpha_{m})$, where $\alpha_{i}\neq\alpha_{j}$ for all $1\leq i,j\leq m$ and $i\neq j$, then ${\rm per}(f)={\rm lcm}({\rm ord}(\alpha_{1}),{\rm ord}(\alpha_{2}),\ldots,{\rm ord}(\alpha_{m}))$, where ${\rm lcm}({\rm ord}(\alpha_{1}),{\rm ord}(\alpha_{2}),\ldots,{\rm ord}(\alpha_{m}))$ is the least common multiple of ${\rm ord}(\alpha_{1}),{\rm ord}(\alpha_{2}),\ldots,{\rm ord}(\alpha_{m})$.
Let $L={\rm lcm}({\rm ord}(\alpha_{1}),{\rm ord}(\alpha_{2}),\ldots,{\rm ord}(\alpha_{m}))$. Since $\alpha^{L}_{i}-1=0$ for all $i=1,2,\ldots,m$, it is valid that $$t-\alpha_{i}\mid t^{L}-1$$ for all $i=1,2,\ldots,m$. Since $\alpha_{i}\neq\alpha_{j}$ for all $1\leq i,j\leq m$ and $i\neq j$, it is valid that $t-\alpha_{i}$ and $t-\alpha_{j}$ are coprime for all $i,j$. Thus, $(t-\alpha_{1})(t-\alpha_{2})\ldots(t-\alpha_{m})\mid t^{L}-1$, which means that $f(t)\mid t^{L}-1$. By the property of the order, we have ${\rm per}(f)=L$. The proof is completed.
In [@c4; @c5], Proposition 1 is employed to analyze the period distributions of two linear maps: the Chebyshev map and the generalized discrete cat map, whose characteristic polynomials can be expressed as $f(t)=t^{2}+a t+1 \in {\rm Z}_{N}[t]$, where $N$ is an integer. If $\alpha$ and $\beta$ are roots of $f(t)$, then it must hold that $\alpha\beta=1$. Thus, ${\rm ord}(\alpha)={\rm ord}(\beta)$. By Proposition 1, we have ${\rm per}(f)={\rm ord}(\alpha)$, so $T={\rm ord}(\alpha)$. However, if the characteristic polynomial is $f(t)=t^{2}+a t+b \in {\rm Z}_{N}[t]$, whose roots are $\alpha$ and $\beta$, where $b\neq 1$, we can not conclude that $ {\rm ord}(\alpha)={\rm ord}(\beta)$. In order to analyze the period $T$, we should analyze $ {\rm ord}(\alpha)$ and ${\rm ord}(\beta)$, respectively. If $N$ is not chosen properly, i.e., both $N-1$ and $N+1$ has many divisors, the analysis process is rather complicated. This obstacle prompts us to adopt another approach which will be presented in Section IV.
IPRNGs over the finite field
----------------------------
In this paper, we consider the following IPRNG proposed in [@e8] over $({\rm Z}_{N},+,\times)$: $$\begin{aligned}
x_{n+1}=\left\{\begin{array}{cccc}
a x_{n}^{-1}+b &x_{n}\in{\rm Z}^{\times}_{N}\\
b&x_{n}=0
\end{array}\right.,\end{aligned}$$ for all $n\geq 1$, where $N>3$ is a prime, $a,b\in {\rm Z}_{N}$. The initial value associated with model (2) is given by $x_{0}\in{\rm Z}_{N}$.
Hereafter, we denote $S(x_{0};a,b)$ as the sequence generated by model (2) starts from $x_{0}$ for given $a$, $b$. Then, we have the following definition on the period of $S(x_{0};a,b)$.
For every initial value $x_{0}\in{\rm Z}_{N}$, the smallest integer $L(x_{0};a,b)$ such that $x_{n+L(x_{0};a,b)}=x_{n}$ for all $n\geq n_{0}\geq0$ is called the period of the IPRNGs correspond to $a$, $b$ and $x_{0}$, where $n_{0}$ is a nonnegative integer.
It is noteworthy that the sequence generated by the IPRNGs may not be purely periodic, i.e. every period start from $x_{0}$, which is different from the case for the Chebyshev map and the generalized discrete Arnold cat map. Its period depends on not only the control parameters $a,b$ but also the initial value $x_{0}$, this will be illustrated in Section III and Section IV.
Throughout this paper, ${\rm Z}_{N}$ denotes the residue ring of integers modulo $N$. ${\rm Z}^{\times}_{N}$ denotes the group of all units in ${\rm Z}_{N}$. $({\rm Z}_{N},+,\times)$ denotes the finite field where addition and multiplication are all modular operations. For $\alpha\in{\rm Z}_{N}$, denote ${\rm ord}(\alpha)$ as the order of $\alpha$ in ${\rm Z}_{N}$. ${\rm GF}(N^{2})$ denotes a finite field with $N^{2}$ elements. $\varphi(n)$, i.e., Euler¡¯s totient function, denotes the number of positive integers which are both less than or equal to the positive integer and coprime with $n$.
Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$
======================================================================================
When $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in{\rm Z}_{N}$, there are $2N^{2}-N$ IPRNGs. It would be better if we have an impression on what the period distribution with $ab=0$ in $ {\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$ looks like. Fig. 1 is a plot of the period distribution of IPRNGs (2) with $ab=0$ in ${\rm Z}_{31}$ and $x_{0}\in {\rm Z}_{31}$. It can be seen from Fig. 1 that the periods distribute very sparsely, some exist and some do not.
In [@c9], Chou has considered the periods of IPRNGs for $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$. The results are listed as follows
Suppose $a=0$, then $x_{n}=b$ for all $n\geq 1$ and $L(x_{0};0,b)=1$.
Suppose $a\neq 0$ and $b=0$.
(P1) If $x_{0}=0$, then $x_{n}=0$ for all $n\geq 1$ and $L(0;a,b)=1$.
(P2) If $a=x_{0}^{2}$ and $x_{0}\neq0$, then $x_{n}=x_{0}$ for all $n\geq1$ and $L(x_{0};x^{2}_{0},b)=1$.
(P3) If $a\neq x_{0}^{2}$ and $x_{0}\neq0$, then $x_{n+2}=x_{n}$ for all $n\geq1$ and $L(x_{0};a,b)=2$.
Now, all the possible periods for this case are revealed. In the following, we will count the number of IPRNGs for each specific period and present the period distribution.
For IPRNG (2) with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$ , the possible periods and the number of each special period are given in Table I.
[|c|c|]{} **Periods & **Number of IPRNGs\
****
-----
$1$
-----
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$.[]{data-label="table_example"}
& $N^{2}+2N-2$\
-----
$2$
-----
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$.[]{data-label="table_example"}
&
--------------
$(N-2)(N-1)$
--------------
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{N}$ and $x_{0}\in {\rm Z}_{N}$.[]{data-label="table_example"}
\
For $L(x_{0};a,b)=1$, there are three cases:
\(i) $a=0$. Here, the choice of $a$ is unique and there are $N$ choices of $b$ and $N$ choices of $x_{0}$. Thus, there are $N^{2}$ IPRNGs.
\(ii) $a\neq 0$, $b=0$ and $x_{0}=0$. Here, there are $N-1$ choices of $a$ and the choices of $b$ and $x_{0}$ are unique. Thus, there are $N-1$ IPRNGs.
\(iii) $a\neq 0$, $b=0$ and $a=x^{2}_{0}$. Here, there is a unique choice of $b$. Since $a\neq 0$ and $a=x^{2}_{0}$, it is valid that $x_{0}\neq0$. Thus, there are $N-1$ choices of $x_{0}$. Once $x_{0}$ is chosen, $a$ is uniquely determined. Thus, there are $N-1$ IPRNGs.
Combining (i), (ii) and (iii), we have there are $N^{2}+2N-2$ IPRNGs for $L(x_{0};a,b)=1$.
For $L(x_{0};a,b)=2$, since $x_{0}\neq0$, there are $N-1$ choices of $x_{0}$. Once $x_{0}$ is chosen, combining $a\neq0$, there are $N-2$ choices of $a$ and a unique choice of $b$. Thus, there are $(N-2)(N-1)$ IPRNGs. The proof is completed.
The following example is given to compare experimental and the theoretical results. A computer program has been written to exhaust all possible IPRNGs with $ab=0$ in ${\rm Z}_{31}$ and $x_{0}\in {\rm Z}_{31}$ to find the period by brute force, the results are shown in Fig. 1.
Table II lists the complete result we have obtained. It provides the period distribution of the IPRNGs. As it is shown in Fig. 1 and Table II, the theoretical and experimental results fit well. The maximal period is $2$ while the minimal period is $1$. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods.
[|c|c|]{} **Periods & **Number of IPRNGs\
****
-----
$1$
-----
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{31}$ and $x_{0}\in {\rm Z}_{31}$.[]{data-label="table_example"}
& $1021$\
-----
$2$
-----
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{31}$ and $x_{0}\in {\rm Z}_{31}$.[]{data-label="table_example"}
&
-------
$870$
-------
: Period distribution of IPRNGs with $ab=0$ in ${\rm Z}_{31}$ and $x_{0}\in {\rm Z}_{31}$.[]{data-label="table_example"}
\
Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$
=====================================================================================================================
In [@c9], Chou described all possible periods of the model (2) with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$ and showed that these periods were related to the periods of several polynomials, see Theorem 2 and Theorem 4 in [@c9]. However, the author did not provide a feasible way to evaluate these periods. In the following, we will characterize the full information on the period distribution of sequences generated by IPRNG (2) with $a,b$ traverse all elements in ${\rm Z}^{\times}_{N}$ and $x_{0}$ traverses all elements in ${\rm Z}_{N}$.
When $a$, $b$ traverse all elements in ${\rm Z}^{\times}_{N}$ and $x_{0}$ traverse all elements in ${\rm Z}_{N}$, there are $(N-1)^{2}N$ IPRNGs. It would be better if we have an impression on what the period distribution with $a\in{\rm Z}^{\times}_{N}$, $b\in {\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$ looks like. Fig. 2 is a plot of the period distribution of IPRNGs (2) with $a\in{\rm Z}^{\times}_{31}$, $b\in{\rm Z}^{\times}_{31}$ and $x_{0}\in {\rm Z}_{31}$. It can be seen from Fig. 2 that the periods distribute very sparsely, some exist and some do not. In the following, the period distribution rules for $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in{\rm Z}_{N}$ will be worked out analytically.
In order to get the main results in the rest of this paper, we provide an important lemma in [@c9] which transforms the sequence generated by IPRNGs to 2-dimensional LFSR sequences.
[@c9]. Let $a$, $b$ and $x_{0}$ are in ${\rm Z}_{N}$. Define the LFSR $$\begin{aligned}
y_{n+2}=by_{n+1}+ay_{n},\end{aligned}$$ for all $n\geq0$, where $y_{0}=1$, $y_{1}=x_{0}$. Then if $m\geq0$ is an integer such that $y_{n}\in {\rm Z}^{\times}_{p^{e}}$ for all $0\leq n\leq m$, then $x_{n}=y_{n+1}y^{-1}_{n}$ for all $0\leq n\leq m$. Moreover, $m$ is the smallest positive integer satisfying $x_{m}=$ if and only if $m+1$ is the smallest integer satisfying $y_{m+1}=0$.
Let $f(t)=t^{2}-bt-a$ be the characteristic polynomial of LFSR (3). If $f(t)$ has a root with multiplicity $2$, i.e., $f(t)=(t-\alpha)^{2}$, then $a=-\alpha^{2}$ and $b=2\alpha$. It follows from (3) that $$\begin{aligned}
y_{n+2}= 2\alpha y_{n+1}-\alpha^{2}y_{n}.\end{aligned}$$ By simple calculation, we can get the general term of (4) $$\begin{aligned}
y_{n}=\alpha^{n}(n(\alpha^{-1}x_{0}-1)+1).\end{aligned}$$
If $f(t)$ has two distinct roots with multiplicity $1$, i.e., $f(t)=(t-\alpha)(t-\beta)$ and $\alpha\neq\beta$, then $a=-\alpha\beta$ and $b=\alpha+\beta$. It follows from (3) that $$\begin{aligned}
y_{n+2}= (\alpha+\beta) y_{n+1}-\alpha\beta y_{n}.\end{aligned}$$ By simple calculation, we can get the general term of (6) $$\begin{aligned}
y_{n}=(\alpha-\beta)^{-1}((x_{0}-\beta)\alpha^{n}+(\alpha-x_{0})\beta^{n}).\end{aligned}$$
It can be observed from (5) and (7) that the general terms of (3) are different when $f(t)$ has a root with multiplicity $2$ and has two distinct roots with multiplicity $1$. Thus, we will discuss these two cases separately.
$f(t)$ has a root with multiplicity $2$
---------------------------------------
We suppose that $\alpha$ is a root of $f(t)$, i.e., $f(t)=(t-\alpha)^{2}$. In this case, it must holds that $\alpha\in{\rm Z}_{N}$. In fact, if $\alpha\notin {\rm Z}_{N}$, which means that $f(t)$ is irreducible in ${\rm Z}_{N}[t]$, then $f(t)$ must have two roots in ${\rm GF}(N^{2})$ and all roots of $f(t)$ are $\alpha$ and $\alpha^{N}$, where $\alpha$ and $\alpha^{N}$ are in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$. Since $f(t)$ has a root with multiplicity $2$, it must hold that $\alpha^{N}=\alpha$. Thus, $\alpha^{N-1}=1$, which means that ${\rm ord}(\alpha)\mid N-1$. Therefore, $\alpha\in {\rm Z}_{N}$, which is a contradiction.
It follows from (5) that if $x_{0}\neq\alpha$, then $y_{n}$ must contain $0$, which means that $S(x_{0};a,b)$ must contain some elements in $0$; Otherwise, $y_{n}$ dose not contain $0$, which means that $S(x_{0};a,b)$ does not contain $0$.
Suppose $f(t)$ has a root with multiplicity $2$ in ${\rm Z}_{N}$. If $x_{0}\neq\alpha$, then $L(x_{0};a,b)=N-1$ and there are $(N-1)^{2}$ IPRNGs of period $N-1$.
Period analysis.
Since $x_{0}\neq\alpha$, it is valid that $y_{n}$ must contain $0$. Thus, $L(x_{0};a,b)=L(b;a,b)$. When $x_{0}=2\alpha$, it follows from (5) that $y_{n}=(n+1)\alpha^{n}$. Thus, $n=N-1$ is the smallest integer such that $y_{n}=0$. By lemma 1, we have $N-2$ is the smallest integer such that $x_{N-2}=0$. Thus, $x_{N-1}=b$, which means that $L(b;a,b)=N-1$.
Counting.
When $\alpha$ traverses all elements in ${\rm Z}^{\times}_{N}$, there are $N-1$ choices of $\alpha$. Since $f(t)=(t-\alpha)^{2}$, it is valid that $a$ and $b$ are uniquely determined by a chosen $\alpha$. Also, it follows from $x_{0}\neq \alpha$ that there are $N-1$ choices of $x_{0}$. Thus, there are $(N-1)^{2}$ IPRNGs of period $N-1$. The proof is completed.
Suppose $f(t)$ has a root with multiplicity $2$ in ${\rm Z}_{N}[t]$. If $x_{0}=\alpha$, then $L(x_{0};a,b)=1$ and there are $N-1$ IPRNGs of period $1$.
Period analysis.
Since $x_{0}=\alpha$, it is valid that $y_{n}$ does not contain $0$. It follows from (5) that $y_{n}=\alpha^{n}$. By lemma 1, we can get that $x_{n}=\alpha$ for all $n=1,2,\ldots$. Thus, $L(x_{0};a,b)=1$.
Counting.
When $\alpha$ traverses all elements in ${\rm Z}^{\times}_{N}$, there are $N-1$ choices of $\alpha$. Since $f(t)=(t-\alpha)^{2}$, it is valid that $a$ and $b$ are uniquely determined by a chosen $\alpha$. Also, it follows from $x_{0}=\alpha$ that there is a unique choice of $x_{0}$. Thus, there are $N-1$ IPRNGs of period $1$. The proof is completed.
$f(t)$ has two distinct roots with multiplicity $1$
---------------------------------------------------
It follows from (7) that $y_{n}=0$ if and only if $$\begin{aligned}
(x_{0}-\alpha)(x_{0}-\beta)^{-1}=(\alpha\beta^{-1})^{n}.\end{aligned}$$
For presentation convenience, we denote set $\Omega=\{\alpha\beta^{-1},(\alpha\beta^{-1})^{2},\ldots,(\alpha\beta^{-1})^{{\rm ord}(\alpha\beta^{-1})-1}\}$.
If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$, there exists $1\leq n\leq p-1$ such that (8) holds, thus, $S(x_{0};a,b)$ must contains some elements in $0$; if $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, there does not exist any $n$ such that (10) holds, thus, $S(x_{0};a,b)$ does not contain any element in $0$.
On the other hand, if either $x_{0}-\alpha=0$ or $x_{0}-\beta=0$, then $y_{n}\neq 0$ for all $n=1,2,\ldots$, which means that $S(x_{0};a,b)$ does not contain any element in $0$.
In the following, we will provide three lemmas which are necessary for our analysis.
Suppose $a\in{\rm Z}_{N}^{\times}$, $b\in{\rm Z}_{N}^{\times}$. Then, if $\alpha,\beta$ are two distinct roots of $f(t)$, then ${\rm ord}(\alpha\beta^{-1})>2$.
Since $b\in{\rm Z}^{\times}_{N}$ and $b=\alpha+\beta$, it holds that $\alpha+\beta\neq0$. Combining $\alpha-\beta\neq0$, we have $\alpha\beta^{-1}-\alpha^{-1}\beta\neq0$, which means that $\alpha\beta^{-1}\neq \alpha^{-1}\beta$. If ${\rm ord}(\alpha\beta^{-1})=1$, then it must hold that $\alpha\beta^{-1}=1$ and $\alpha\beta^{-1}=1$, which contradicts to $\alpha\beta^{-1}\neq \alpha^{-1}\beta$. If ${\rm ord}(\alpha\beta^{-1})=2$, then it follows from $\varphi(2)=1$. Thus, $\alpha\beta^{-1}= \alpha^{-1}\beta$, which is a contradiction. The proof is completed.
Suppose $a\in{\rm Z}_{N}^{\times}$, $b\in{\rm Z}_{N}^{\times}$. If $\alpha,\beta$ are two distinct roots of $f(t)$, then $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are two roots of $g(t)=t^{2}+(a^{-1}b^{2}+2)t+1$.
Since $\alpha,\beta$ are two distinct roots of $f(t)$, it is valid that $a=-\alpha\beta$ and $b=\alpha+\beta$. Then, it is easy to verify that $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$. The proof is completed.
Suppose $a\in{\rm Z}_{N}^{\times}$, $b\in{\rm Z}_{N}^{\times}$. If $\alpha,\beta$ are two distinct roots of $f(t)$, then $a^{-1}b^{2}$ is uniquely determined by $\alpha\beta^{-1}$.
Since $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$, it holds that $a^{-1}b^{2}+2=\alpha\beta^{-1}+\alpha^{-1}\beta$.
If $a^{-1}b^{2}$ is not uniquely determined by $\alpha\beta^{-1}$ or $\alpha^{-1}\beta$, then there exist $\alpha_{1}\beta^{-1}_{1}$ and $\alpha_{2}\beta^{-1}_{2}$ with $\alpha_{1}\beta^{-1}_{1}\neq \alpha_{2}\beta^{-1}_{2}$ and $\alpha_{1}\beta^{-1}_{1}\neq (\alpha_{2}\beta^{-1}_{2})^{-1}$, such that $\alpha_{1}\beta^{-1}_{1}+\alpha^{-1}_{1}\beta_{1}=\alpha_{2}\beta^{-1}_{2}+\alpha^{-1}_{2}\beta_{2}$. Let $\gamma_{1}=\alpha_{1}\beta^{-1}_{1}$ and $\gamma_{2}=\alpha_{2}\beta^{-1}_{2}$, then we have $\gamma_{1}\neq\gamma^{-1}_{2}$ and $\gamma_{1}\neq\gamma_{2}$. However, by simple calculation, we have $\gamma_{1}+\gamma^{-1}_{1}=\gamma_{2}+\gamma^{-1}_{2}$ if and only if $ (\gamma_{1}\gamma_{2}-1)(\gamma_{1}-\gamma_{2})=0$, which means that either $\gamma_{1}\gamma_{2}=1$ or $\gamma_{1}=\gamma_{2}$. These are the contradictions. The proof is completed.
When $f(t)$ has a root with multiplicity $2$, its roots are in ${\rm Z}_{N}$. However, when $f(t)$ has two distinct roots with multiplicity $1$, its roots may be in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$. Therefore, it is nature to consider the the following two cases separetely: 1) $\alpha$ and $\beta$ are in ${\rm Z}_{N}$; 2) $\alpha$ and $\beta$ are in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$.
### $\alpha$ and $\beta$ are in ${\rm Z}_{N}$
Suppose $f(t)$ has two distinct roots with multiplicity $1$ in ${\rm Z}_{N}$. If $(x_{0}-\alpha)(x_{0}-\beta)\neq 0$ and $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$, then $L(x_{0};a,b)$ traverses the set $\{k-1:k>2,k\mid N-1\}$. For each $k$, there are $(k-1)(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k-1$.
Period analysis.
If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$, then $S(x_{0};a,b)$ must contain $0$. Thus, $L(x_{0};a,b)=L(b;a,b)$. Then, we consider the case that $x_{0}=b$, which means that $x_{0}=\alpha+\beta$. By (7), we have $y_{n}=0$ if and only if $(\alpha\beta^{-1})^{n+1}=1$. Thus, $n={\rm ord}(\alpha\beta^{-1})-1$ is the smallest integer such that $y_{n}=0$. By Lemma 1, we have $x_{n-1}=0$, thus, $x_{n}=b$, which means that $L(x_{0};a,b)={\rm ord}(\alpha\beta^{-1})-1$.
Since $\alpha\beta^{-1}\in {\rm Z}^{\times}_{N}$, it holds that ${\rm ord}(\alpha\beta^{-1})\mid N-1$ and ${\rm ord}(\alpha\beta^{-1})>2$. Hence, $L(x_{0};a,b)$ traverses the set $\{k-1:k>2,k\mid p-1\}$.
Counting.
For $L(x_{0};a,b)=k-1$, there are $k-1$ $x_{0}$’s such that $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$. Thus, there are $k-1$ choices of $x_{0}$.
Since $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$, it holds that $a^{-1}b^{2}+2=\alpha\beta^{-1}+\alpha^{-1}\beta$. Thus, $a=b^{2}(\alpha\beta^{-1}+\alpha^{-1}\beta-2)$. By Lemma 4, we have $a^{-1}b^{2}$ is uniquely determined by $\alpha\beta^{-1}$. Thus, when ${\rm ord}(\alpha\beta^{-1})=k$, there are $\frac{\varphi(k)}{2}$ different $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ ’s. Thus, there are $\frac{\varphi(k)}{2}$ choices of $\alpha\beta^{-1}+\alpha^{-1}\beta-2 $.
As a result of ${\rm ord}(\alpha\beta^{-1})>2$, we have $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ is a unit. The number of choices of $b$ is $N-1$. Once $b$ and $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ are chosen, $a$ is uniquely determined. Hence, for each $k$, there are $(k-1)(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k-1$. The proof is completed.
Suppose $f(t)$ has two distinct roots with multiplicity $1$ in ${\rm Z}_{N}$. If $(x_{0}-\alpha)(x_{0}-\beta)\neq 0$ and $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, then $L(x_{0};a,b)$ traverses the set $\{k:2<k<N-1,k\mid N-1\}$. For each $k$, there are $(N-(k-1))(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k-1$.
Period analysis.
If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, then $S(x_{0};a,b)$ does not contain $0$. It follows from Lemma 1 and (7) that $x_{n}=x_{0}$ if and only if $$\begin{aligned}
(x_{0}-\alpha)(x_{0}-\beta)\alpha^{n}=(x_{0}-\alpha)(x_{0}-\beta)\beta^{n}.\end{aligned}$$ Since $(x_{0}-\alpha)(x_{0}-\beta)\neq 0$, (9) is equivalent to $(\alpha\beta^{-1})^{n}=1$. Thus, $L(x_{0};a,b)={\rm ord}(\alpha\beta^{-1})$.
By lemma 2, we have ${\rm ord}(\alpha\beta^{-1})>2$. On the other hand, since $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, it must hold that $\alpha\beta^{-1}$ is not a primitive element in ${\rm Z}_{N}$, which means that ${\rm ord}(\alpha\beta^{-1})\neq N-1$ Hence, $L(x_{0};a,b)$ traverses the set $\{k:2<k<N-1,k\mid N-1\}$.
Counting.
For $L(x_{0};a,b)=k$, there are $N-(k-1)$ $x_{0}$’s such that $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$. Thus, there are $N-(k-1)$ choices of $x_{0}$.
Since $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$, it holds that $a^{-1}b^{2}+2=\alpha\beta^{-1}+\alpha^{-1}\beta$. Thus, $a=b^{2}(\alpha\beta^{-1}+\alpha^{-1}\beta-2)$. By Lemma 4, we have $a^{-1}b^{2}$ is uniquely determined by $\alpha\beta^{-1}$. Thus, when ${\rm ord}(\alpha\beta^{-1})=k$, there are $\frac{\varphi(k)}{2}$ different $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ ’s. Thus, there are $\frac{\varphi(k)}{2}$ choices of $\alpha\beta^{-1}+\alpha^{-1}\beta-2 $.
As a result of ${\rm ord}(\alpha\beta^{-1})>2$, we have $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ is a unit. The number of choices of $b$ is $N-1$. Once $b$ and $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ are chosen, $a$ is uniquely determined. Hence, for each $k$, there are $(N-(k-1))(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k$. The proof is completed.
Suppose $f(t)$ has two distinct roots with multiplicity $1$ in ${\rm Z}_{N}$. If $(x_{0}-\alpha)(x_{0}-\beta)= 0$, then $L(x_{0};a,b)=1$ and there are $(N-3)(N-1)$ IPRNGs of period $k$.
Period analysis.
If $(x_{0}-\alpha)(x_{0}-\beta)=0$, then $y_{n}=x^{n}_{0}$. Thus, $x_{n}=x_{0}$ for all $n=1,2,\ldots$, which means that $L(x_{0};a,b)=1$.
Counting.
For $L(x_{0};a,b)=1$, $\alpha,\beta$ traverses all suitable elements in ${\rm Z}^{\times}_{N}$, i.e. both $\alpha-\beta$ and $\alpha+\beta$ are units, there are $\frac{(N-3)(N-1)}{2}$ pairs of $\alpha,\beta$. Once $\alpha,\beta$ are chosen, there are $2$ choices of $x_{0}$. Thus, there are $(N-3)(N-1)$ IPRNGs of period $1$. The proof is completed.
### $\alpha$ and $\beta$ are in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$
In this case, it must hold that $(x_{0}-\alpha)(x_{0}-\beta)\neq0$. Then, we have the following results on the period distribution of IPRNGs for this case.
Suppose $f(t)$ has two distinct roots with multiplicity $1$ in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$. If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$, then $L(x_{0};a,b)$ traverses the set $\{k-1:k>2,k\mid N+1\}$. For each $k$, there are $(k-1)(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k-1$.
Period analysis.
If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$, then $S(x_{0};a,b)$ must contain $0$. Thus, $L(x_{0};a,b)=L(b;a,b)$. Then, we consider the case that $x_{0}=b$, which means that $x_{0}=\alpha+\beta$. By (7), we have $y_{n}=0$ if and only if $(\alpha\beta^{-1})^{n+1}=1$. Thus, $n={\rm ord}(\alpha\beta^{-1})-1$ is the smallest integer such that $y_{n}=0$. By Lemma 1, we have $x_{n-1}=0$, thus, $x_{n}=b$, which means that $L(x_{0};a,b)={\rm ord}(\alpha\beta^{-1})-1$.
By lemma 2, we have ${\rm ord}(\alpha\beta^{-1})>2$. Since $\alpha\beta^{-1}\in {\rm GF}(N^{2})$, it holds that ${\rm ord}(\alpha\beta^{-1})\mid N^{2}-1$. Notice that $\alpha$ and $\beta$ are not in ${\rm Z}_{N}$ and $\alpha\neq\beta$, it is valid that $\alpha\beta^{-1}\notin{\rm Z}_{N} $. Since ${\rm Z}_{N}\subseteq {\rm GF}(N^{2})$, it is valid that all units in ${\rm Z}_{N}$ are contained in ${\rm GF}(N^{2})$, which means that ${\rm ord}(\alpha\beta^{-1})\nmid N-1$. Thus, ${\rm ord}(\alpha\beta^{-1})\mid N+1$. Hence, $L(x_{0};a,b)$ traverses the set $\{k-1:k>2,k\mid N+1\}$.
Counting.
For $L(x_{0};a,b)=k-1$, there are $k-1$ $x_{0}$’s such that $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\in\Omega$. Thus, there are $k-1$ choices of $x_{0}$.
Since $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$, it holds that $a^{-1}b^{2}+2=\alpha\beta^{-1}+\alpha^{-1}\beta$. Thus, $a=b^{2}(\alpha\beta^{-1}+\alpha^{-1}\beta-2)$. By Lemma 4, we have $a^{-1}b^{2}$ is uniquely determined by $\alpha\beta^{-1}$. Thus, when ${\rm ord}(\alpha\beta^{-1})=k$, there are $\frac{\varphi(k)}{2}$ different $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ ’s. Hence, there are $\frac{\varphi(k)}{2}$ choices of $\alpha\beta^{-1}+\alpha^{-1}\beta-2 $.
As a result of ${\rm ord}(\alpha\beta^{-1})>2$, we have $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ is a unit. The number of choices of $b$ is $N-1$. Once $b$ and $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ are chosen, $a$ is uniquely determined. Hence, for each $k$, there are $(k-1)(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k-1$. The proof is completed.
Suppose $f(t)$ has two distinct roots with multiplicity $1$ in ${\rm GF}(N^{2})$ but not in ${\rm Z}_{N}$. If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, then $L(x_{0};a,b)$ traverses the set $\{k:2<k<N+1,k\mid N+1\}$. For each $k$, there are $(N-(k-1))(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k$.
Period analysis.
If $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, then $S(x_{0};a,b)$ does not contain $0$. It follows from Lemma 1 and (7) that $x_{n}=x_{0}$ if and only if $$\begin{aligned}
(x_{0}-\alpha)(x_{0}-\beta)\alpha^{n}=(x_{0}-\alpha)(x_{0}-\beta)\beta^{n}.\end{aligned}$$ Since $(x_{0}-\alpha)(x_{0}-\beta)\neq 0$, (10) is equivalent to $(\alpha\beta^{-1})^{n}=1$. Thus, $L(x_{0};a,b)={\rm ord}(\alpha\beta^{-1})$.
By lemma 2, we have ${\rm ord}(\alpha\beta^{-1})>2$. Since $\alpha\beta^{-1}\in {\rm GF}(N^{2})$, it holds that ${\rm ord}(\alpha\beta^{-1})\mid N^{2}-1$. Notice that $\alpha$ and $\beta$ are not in ${\rm Z}_{N}$ and $\alpha\neq\beta$, it is valid that $\alpha\beta^{-1}\notin{\rm Z}_{N} $. Since ${\rm Z}_{N}\subseteq {\rm GF}(N^{2})$, it is valid that all units in ${\rm Z}_{N}$ are contained in ${\rm GF}(N^{2})$, which means that ${\rm ord}(\alpha\beta^{-1})\nmid N-1$. Thus, ${\rm ord}(\alpha\beta^{-1})\mid N+1$.
On the other hand, since $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$, it must hold that $\alpha\beta^{-1}$ is not a primitive element in ${\rm GF}(N^{2})$, which means that ${\rm ord}(\alpha\beta^{-1})\neq N+1$ Hence, $L(x_{0};a,b)$ traverses the set $\{k:2<k<N+1,k\mid N+1\}$.
Counting.
For $L(x_{0};a,b)=k$, there are $N-(k-1)$ $x_{0}$’s such that $(x_{0}-\alpha)(x_{0}-\beta)^{-1}\notin\Omega$. Thus, there are $N-(k-1)$ choices of $x_{0}$.
Since $\alpha\beta^{-1}$ and $\alpha^{-1}\beta$ are roots of $g(t)$, it holds that $a^{-1}b^{2}+2=\alpha\beta^{-1}+\alpha^{-1}\beta$. Thus, $a=b^{2}(\alpha\beta^{-1}+\alpha^{-1}\beta-2)$. By Lemma 4, we have $a^{-1}b^{2}$ is uniquely determined by $\alpha\beta^{-1}$. Thus, when ${\rm ord}(\alpha\beta^{-1})=k$, there are $\frac{\varphi(k)}{2}$ different $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ ’s. Thus, there are $\frac{\varphi(k)}{2}$ choices of $\alpha\beta^{-1}+\alpha^{-1}\beta-2 $.
As a result of ${\rm ord}(\alpha\beta^{-1})>2$, we have $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ is a unit. The number of choices of $b$ is $N-1$. Once $b$ and $\alpha\beta^{-1}+\alpha^{-1}\beta-2$ are chosen, $a$ is uniquely determined. Hence, for each $k$, there are $(N-(k-1))(N-1)\frac{\varphi(k)}{2}$ IPRNGs of period $k$. The proof is completed.
Now, we summarize the results in the following theorem.
For IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in{\rm Z}_{N}$, the possible periods and the number of each special period are given in Table III.
[|c|c|]{} **Periods & **Number of IPRNGs\
****
-----
$1$
-----
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
& $(N-2)(N-1)$\
-------
$N-1$
-------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
&
-------------
$(N-1)^{2}$
-------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
\
-------------------------
$\{k-1:k>2,k\mid N-1\}$
-------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
&
----------------------------------
$(k-1)(N-1)\frac{\varphi(k)}{2}$
----------------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
\
-------------------------
$\{k-1:k>2,k\mid N+1\}$
-------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
&
----------------------------------
$(k-1)(N-1)\frac{\varphi(k)}{2}$
----------------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
\
---------------------------
$\{k:2<k<N-1,k\mid N-1\}$
---------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
&
--------------------------------------
$(N-(k-1))(N-1)\frac{\varphi(k)}{2}$
--------------------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
\
---------------------------
$\{k:2<k<N+1,k\mid N+1\}$
---------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
&
--------------------------------------
$(N-(k-1))(N-1)\frac{\varphi(k)}{2}$
--------------------------------------
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{N}$, $b\in{\rm Z}^{\times}_{N}$ and $x_{0}\in {\rm Z}_{N}$. []{data-label="table_example"}
\
It should be mentioned that $N>3$ is an important condition in Theorem 3, because of some periods require $k>2,k\mid N-1$, which implies that $N>3$.
The following example is given to compare experimental and the theoretical results. A computer program has been written to exhaust all possible IPRNGs with $a\in{\rm Z}^{\times}_{31}$ and $b\in{\rm Z}^{\times}_{31}$ and $x_{0}\in{\rm Z}_{31}$ to find the period by brute force, the results are shown in Fig. 2.
Table IV lists the complete result we have obtained. It provides the period distribution of the IPRNGs. As it is shown in Fig. 2 and Table IV, the theoretical and experimental results fit well. The maximal period is $31$ while the minimal period is $1$. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods.
Periods 1 2 3 4 5 6 7 8
------------------ ----- ------ ------ ------ ------ ------ ----- ------ --
Number of IPRNGs 870 60 900 1080 1650 720 420 1440
Periods 9 10 14 15 16 29 30 31
Number of IPRNGs 540 1200 1680 3600 1920 3480 900 7440
: Period distribution of IPRNGs with $a\in{\rm Z}^{\times}_{31}$, $b\in{\rm Z}^{\times}_{31}$ and $x_{0}\in {\rm Z}_{31}$[]{data-label="table_example"}
Conclusion
==========
The period distribution of the IPRNGs over $({\rm Z}_{N},+,\times)$ for prime $N>3$ has been analyzed. The period distribution of IPRNGs is obtained by the generating function method and the finite field theory. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods. The analysis results show that the period distribution is poor if $N$ is not chosen properly and there are many small periods.
A feasible way to resolve the open problem proposed by Solé *et al.* in [@s7] is to analyze the period distribution of the sequence generated by IPRNGs over Galois rings. However, the period distribution of IPRNG sequences varies substantially as $N$ changes, when $N$ is a prime, $({\rm Z}_{N},+,\times)$ is a finite field; when $N$ is a power of prime, i.e., $N=p^{e}$, $({\rm Z}_{N},+,\times)$ is a Galois ring. The structure of $({\rm Z}_{p^{e}},+,\times)$ is more complicated than that of $({\rm Z}_{N},+,\times)$, because of $({\rm Z}_{p^{e}},+,\times)$ contains many zero divisors but $({\rm Z}_{N},+,\times)$ does not, this difference makes the fact that the analysis in Galois rings is more complicated than that in finite fields, which is challenging and deserves intensive study. Another important problem is to characterize the security properties of the IPRNGs. These topics are interesting and need further research.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by the National Natural Science Foundation of China under Grant 60974132, the Natural Science Foundation Project of CQ CSTC2011BA6026 and the Scientific & Technological Research Projects of CQ KJ110424.
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[^1]: \* Corresponding author.
[^2]: E-mail addresses: [email protected] (B. Zhou), [email protected] (Q. Song)
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abstract: 'In an X-ray binary system, the companion star feeds the compact neutron star with plasma materials via accretions. The spinning neutron star is likely covered with a thin “magnetized ocean" and may support [*magnetohydrodynamic (MHD) tidal waves*]{}. While modulating the thermal properties of the ocean, MHD tidal waves periodically shake the base of the stellar magnetosphere that traps energetic particles, including radiating relativistic electrons. For a radio pulsar, MHD tidal waves in the stellar surface layer may modulate radio emission processes and leave indelible signatures on timescales different from the spin period. Accretion activities are capable of exciting these waves but may also obstruct or obscure their detections meanwhile. Under fortuitous conditions, MHD tidal waves might be detectable and offer valuable means to probe properties of the underlying neutron star. Similar situations may also occur for a cataclysmic variable – an accretion binary system that contains a rotating magnetic white dwarf. This Letter presents the theory for MHD tidal waves in the magnetized ocean of a rotating degenerate star and emphasizes their potential diagnostics in X-ray and radio emissions.'
author:
- 'Yu-Qing Lou'
title: MHD tidal waves on a spinning magnetic compact star
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Introduction
============
For a magnetized spinning degenerate compact star covered with a thin plasma layer referred to as an [*ocean*]{}, various magnetohydrodynamic (MHD) tidal waves, excited by relevant processes, may exist. This scenario is plausible for an accretion binary system in which the Roche-overflow from a companion star accretes onto a spinning magnetized compact star. This Letter describes the theory of MHD tidal waves that propagate in a thin magnetized ocean of a spinning compact star. When combined with pertinent dynamic processes and emission mechanisms, such MHD tidal waves may leave indelible signatures of electromagnetic emissions in a wide spectrum ranging from the radio to GeV gamma rays and could provide key diagnostics that might offer clues and hints of otherwise complex phenomena. In general, MHD tidal waves may be excited gravitationally, thermally, electromagnetically, or by impacts of accreting materials. While accretion activities may stir up and sustain these waves, wave signals might be undesirably entangled with or overwhelmed by accretion noises. Once observationally established, these waves may be utilized to probe the structures in the [*surface layers*]{} of neutron stars and white dwarfs by extending or adapting the helioseismological techniques (Papaloizou & Pringle 1978; van Horn 1980; Saio 1982; Lou 2000, 2001b). Furthermore, damping properties of these waves might provide information of the stellar constituent materials and diffusion processes (Lou 1995).
Theoretical Model Analysis and Results
======================================
MHD tidal waves may exist on spinning magnetic degenerate stars in either isolation or binaries (Chanmugam 1992; Phinney & Kulkarni 1994; Bildsten & Strohmayer 1999), although they might have a better chance to be detected in high-mass and low-mass x-ray binaries (HMXBs and LMXBs) or cataclysmic variables. By strong gravity, the magnetized plasma ocean on a degenerate star of radius $R$ is thin. As the ratio $eR^3\Omega B/(GMm_pc)$ of electric force to surface gravity can be large for either neutron stars or white dwarfs in rotation (Goldreich & Julian 1969), where $e$ is the electron charge, $\Omega$ is the stellar angular spin rate, $B$ is the poloidal magnetic field, $c$ is the speed of light, $G$ is the gravitational constant, $M$ is the stellar mass and $m_p$ is the proton mass, excess surface charges will be pulled upwards against gravity to form stellar magnetospheres. It is presumed that the ocean and an extended magnetosphere coexist on a compact star, that the magnetosphere maintains a pressure at the interface between the ocean and the magnetosphere much as the Earth’s atmosphere does on the ocean (Gill 1982), and that magnetic fields (idealized as a split monople) are in and out radially across the thin ocean of thickness $h$. For a zonal magnetic field, the Alfvén-Rossby waves have been studied earlier in the solar/stellar context (Gilman 1969; Lou 1987, 2000).
For $h/L\ll 1$ and a small Rossby number ${\cal R}\equiv U/(2\Omega L)$ where $U$ and $L$ are typical horizontal velocity and spatial scales, the shallow-magnetofluid approach is valid. With a quasi-static vertical balance, the MHD tidal equations in spherical polar coordinates $(r,\theta,\phi)$ are: $${\partial v_{\theta}\over\partial t}-f v_{\phi}
=-{1\over R}{\partial\Theta\over\partial\theta}
-{C_A^2\over c^2}{\partial v_{\theta}\over\partial t}\ ,\eqno(1)$$ $${\partial v_{\phi}\over\partial t}+f v_{\theta}
=-{1\over R\sin\theta}{\partial\Theta\over\partial\phi}
-{C_A^2\over c^2}{\partial v_{\phi}\over\partial t}\ ,\eqno(2)$$ $${\partial\eta\over\partial t}+{h\over R\sin\theta }
\bigg[{\partial\over\partial\theta }(\sin\theta v_{\theta})+
{\partial v_{\phi}\over\partial\phi}\bigg]=0\ ,\eqno(3)$$ $${\partial b_r\over\partial t}+{B\over R\sin\theta}
\bigg[{\partial\over\partial\theta }(\sin\theta v_{\theta})+
{\partial v_{\phi}\over\partial\phi}\bigg]=0\ ,\eqno(4)$$ where variables $v_{\theta}$, $v_{\phi}$, $\eta$ and $b_r$ are the perturbations of $\hat\theta-$component velocity, $\hat\phi-$component velocity, ocean surface elevation, and $\hat r-$component of magnetic field, respectively, $f\equiv 2\Omega\cos\theta$, $g\equiv GM/R^2$, $B$ and $\rho$ are the Coriolis parameter at colatitude $\theta$, the surface gravity, the radial magnetic field, and the mean mass density, respectively, $C_A\equiv B/(4\pi\rho)^{1/2}$ is the Alfvén speed, $\Theta\equiv g\eta +Bb_r/(4\pi\rho)$ is proportional to the total pressure perturbation, and the [*displacement current*]{} effect is retained in case of $C_A{\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}c$ (Lou 1995). The energy conservation follows from eqns. (1)-(4) with the energy flux density $\vec v_{\perp}[\rho g\eta+Bb_r/(4\pi)]$ where $\vec v_{\perp}\equiv (v_{\theta}, v_{\phi})$. By the $\exp[i(\omega t+m\phi)]$ dependence with angular frequency $\omega$ and integer $m$, the resulting equation of $\Theta(\mu)$ becomes $$\begin{aligned}
& & \!\!\!\!
{d\over d\mu}\bigg({1-\mu^2\over s^2-\mu^2}
{d\Theta\over d\mu}\bigg)-{1\over s^2-\mu^2}
\bigg({m\over s}{s^2+\mu^2\over s^2-\mu^2}
+{m^2\over 1-\mu^2}\bigg)\Theta \nonumber \\
& &\quad\qquad\qquad\qquad\quad
+{4R^2\Omega^2\Theta\over (gh+C_A^2)
(1+C_A^2/c^2)}=0 ,\qquad\ \nonumber (5)\end{aligned}$$ where $\mu\equiv\cos\theta$ and $s\equiv\omega (1+C_A^2/c^2)/(2\Omega)$. Variables $v_{\theta}$, $v_{\phi}$, $\eta$ and $b_r$ may be explicitly expressed in terms of $\Theta(\mu)$. The $\Theta (\mu)$ solutions, regular at the two poles, involve the [*Hough functions*]{} (Hough 1897, 1898; Flattery 1967; Longuet-Higgins 1968; Chapman & Lindzen 1970). Given relevant physical parameters, all properties of MHD tidal waves can be derived computationally.
Without magnetic field, eqn (5) is equivalent to eqns (7) and (8) of Bildsten et al. (1996) with their two parameters $q^2\equiv s^{-2}$ and $\lambda\equiv R^2\omega^2/(gh)$. In their “traditional approximation", the thin compressible layer is disturbed adiabatically, the radial component of the Coriolis force is ignored, the radial displacement is much less than the horizontal displacement, and $\omega$ is much lower than the Brunt-Väisälä frequency. They focus on internal gravity modes of a rotating ocean. While internal magneto-gravity modes (Lou 1996) are excluded by invoking the quasi-static vertical balance, our incompressible formulation contains various MHD tidal waves in different regimes associated with the distinguished names of Kelvin (1879), Alfvén (1942), Rossby (1939), and Poincaré (1910), leading to the acronym of KARP waves.
For analytical simplicity, we hereafter invoke the $\beta-$plane approximation (Rossby et al. 1939) that retains the essential physics in the spherical geometry. In the local $\beta-$plane approximation of eqns $(1)-(4)$ with $\beta\equiv 2\Omega\sin\theta/R$, a Cartesian coordinate system $(x,y,z)$ is set up so that $\hat x$, $\hat y$ and $\hat z$ point eastward (the sense of spin), northward and radially outward, respectively. One derives $$\bigg[
{\partial\over\partial t}\bigg({1\over c^{*2}}
{\partial^2\over\partial t^2}+{f^{*2}\over c^{*2}}
-{\partial^2\over\partial x^2}
-{\partial^2\over\partial y^2}\bigg)
-\beta^{*}{\partial\over\partial x}\bigg]
v_{\theta}=0\ ,\eqno(6)$$ where $f^{*}\equiv f/(1+C_A^2/c^2)$ is the [*modified $f-$parameter*]{}, $\beta^{*}\equiv\beta/(1+C_A^2/c^2)$ is the [*modified $\beta-$parameter*]{}, and $c^{*2}\equiv
(gh+C_A^2)/(1+C_A^2/c^2)$ defines a characteristic speed $c^{*}$. For free MHD tidal waves of [*small*]{} $y-$variation scales and with $v_{\theta}\propto\exp(i\omega t-ik_xx-ik_yy)$ where $k_x$ and $k_y$ are $x-$ and $y-$component wavenumbers, the cubic dispersion relation is $
\omega^3-[f^{*2}+c^{*2}(k_x^2+k_y^2)]
\omega-\beta^{*} c^{*2}k_x=0.
$ Global versions of KARP waves are contained in eqn (5).
Equatorially Trapped MHD Tidal Waves
====================================
By a latitudinal variation of $f=\beta y$, the [*equatorial $\beta-$plane*]{} becomes an effective [*waveguide*]{} for trapping, within the equatorial zone, MHD tidal waves that propagate in the $\hat\phi-$direction and this equatorial trapping effect becomes stronger in the regime of larger $4R^2\Omega^2/[(gh+C_A^2)(1+C_A^2/c^2)]$ in eqn (5) (Longuet-Higgins 1968). We examine main properties of such [*equatorially trapped MHD tidal waves*]{} below.
For [*equatorially trapped Alfvén-Kelvin waves*]{} (Thompson 1879; Alfvén 1942) around $\theta=\pi/2$ that propagate only [*eastward*]{} at speed $c^{*}$, one must directly solve eqns $(1)-(4)$ with $v_{\theta}=0$ to obtain $$\!
\{v_{\phi},\eta, b_r\}=\{c^{*}, h, B\}
\exp[-\beta^{*}y^2/(2c^{*})]
{\cal G}(x-c^{*}t) ,\eqno(7)$$ where ${\cal G}(\cdots)$ is a [*small arbitrary*]{} dimensionless waveform and $\beta^{*}$ appears in the Gaussian $y-$profile that decays at large $|y|$ (e.g., Gill 1982).
With $f=\beta y$ and a [*nonzero*]{} $v_{\theta}\propto\exp(ik_xx-i\omega t)$ in the equatorial zone around $\theta=\pi/2$, eqn (6) becomes the Schrödinger equation of a harmonic oscillator. The normalized real solution takes the form of $$v_{\theta}={\cal N}
\exp\bigg[-{\beta^{*}y^2\over 2c^{*}}\bigg]
H_n[(\beta^{*}/c^{*})^{1/2}y]
\cos(k_xx-\omega t)\ \eqno(8)$$ with $H_n(\xi)$ being a Hermite polynomial of order $n$ and the normalization constant ${\cal N}\equiv 2^{-n/2}(n!)^{-1/2}
[\beta^{*}/(\pi c^{*})]^{1/4}$. Variables $v_{\phi}$, $\eta$, and $b_r$ can be derived accordingly. The resulting [*dimensionless*]{} cubic dispersion relation for [*equatorially trapped KARP waves*]{} becomes $$\tilde\omega^3-(\tilde k_x^2+n+1/2)\tilde\omega
-\tilde k_x/2=0\ , \eqno(9)$$ where $\tilde\omega\equiv\omega/(2\beta^{*}c^{*})^{1/2}$, $\tilde k_x\equiv k_x/(2\beta^{*}/c^{*})^{1/2}$, and integer $n\geq 0$ gives the number of nodes along $y-$direction. Results of eqn (9) are displayed in Figure 1 with the straight dashed line for the Alfvén-Kelvin mode (7) [*added*]{}. For $n\geq 1$, there are two classes of MHD tidal waves. The high-frequency class (upper solid lines) corresponds to [*equatorially trapped Alfvén-Poincaré waves*]{} (Poincaré 1910; Alfvén 1942) that propagate either eastward or westward but with [*different*]{} phase speeds $\omega/k_x$ for a given $|k_x|$. The low-frequency class (lower solid lines) corresponds to [*equatorially trapped Alfvén-Rossby waves*]{} (Rossby et al. 1939; Alfvén 1942) with westward phase speeds; the group velocity $d\omega/dk_x$ is eastward and westward for short- and long-waves, respectively. The $n=0$ case (dash-dot line) of eqn (9) corresponds to $
\tilde\omega=\tilde k_x/2\pm
(\tilde k_x^2/4+1/2)^{1/2}
$ and is referred to as [*equatorially trapped mixed Alfvén-Rossby-Poincaré waves*]{} (Poincaré 1910; Rossby et al. 1939; Alfvén 1942) whose phase velocity can be to the east (plus sign) or west (minus sign) yet with a group velocity always to the east; for large $k_x>0$ it behaves more like Alfvén-Poincaré waves (plus sign), while for large $|k_x|$ with $k_x<0$ it behaves more like Alfvén-Rossby waves (minus sign). There are no standing modes (i.e., $\tilde k_x=0$) for equatorially trapped Alfvén-Kelvin and Alfvén-Rossby waves (Fig. 1). In contrast, for equatorially trapped Alfvén-Poincaré waves and mixed Alfvén-Rossby-Poincaré waves, [*axisymmetric standing modes*]{} exist with $\tilde k_x=0$ in eq. (9) and thus $\tilde\omega^2=n+1/2$ where $n=0,1,2,\cdots$.
Estimates and Diagnostics
=========================
The preceding analysis is succinct with potential applications to diverse stellar settings. The estimates here focus on typical neutron stars. One can extend the analysis to white dwarfs with different parameters. With $M\sim 1.4M_{\odot}$ and $R\sim 10^6$cm, the surface gravity of a neutron star is $g\sim 10^{14}\hbox{cm s}^{-2}$. The ocean surface gravity wave speed is $(gh)^{1/2}\sim 10^9h_4^{1/2}\hbox{cm s}^{-1}$ and the Alfvén speed is $C_A\sim 10^9B_{12}\rho_5^{-1/2}\hbox{cm s}^{-1}$ for HMXBs or $C_A\sim 10^6B_{9}\rho_5^{-1/2}\hbox{cm s}^{-1}$ for LMXBs, where $h_{\delta}$, $\rho_5$, and $B_{\alpha}$ stand for $h$, $\rho$, and $B$ in units of $10^{\delta}$cm, $10^5$g, and $10^{\alpha}$G, respectively. For an extreme magnetar of $C_A\sim 10^{12}B_{15}\rho_5^{-1/2}\hbox{cm s}^{-1}$, relativistic and quantum effects are involved. For spin periods ranging from $\sim 16$ms to $\sim 8.5$s, the rotation speed $\Omega R$ ranges from $\sim 10$ to $\sim 10^4\hbox{ km s}^{-1}$. It is clear that MHD tidal waves are important in the ocean dynamics of a spinning neutron star.
For a magnetar or neutron star in a HMXB, the situation of $C_A{\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}c$, $\Omega R$ and $(gh)^{1/2}$ may happen with a stronger $B$ and a less dense ocean. The inclusion of the displacement current term in eqns (1), (2), (5) and (6) is necessary (see the definitions of $f^*$, $\beta^*$, and $c^*$ following eq \[6\]) and the magnetic field is dynamically significant in MHD tidal waves besides potential diagnostic roles.
For a neutron star in a LMXB, $C_A\ll c$, $C_A<(gh)^{1/2}$, and $C_A\cong\Omega R$ unless $B_{9}\gg 1$, $\rho_{5}\ll 1$ and $h_2\ll 1$. Thus, $B$ affects only the rotational contribution to the speeds and frequencies of tidal waves significantly. Flow velocities of KARP waves are powerful to push magnetic fields around as $v_{\theta}^2+v_{\phi}^2>b_r^2/(4\pi\rho)$. Meanwhile, a $B$ of $\sim 10^{8}-10^9$ G would be strong enough to partially guide materials from an accretion disk onto the magnetic poles (White et al. 1983), leading to intense X-ray emissions as well as sporadic bursts (Bildsten & Strohmayer 1999).
KARP waves might be driven to excitation by radiative thermal or nuclear processes, by polar accretion impacts, by magnetic thrust and dragging, or by tidal interactions of the accretion disk, and in turn, are capable of periodically modulating thermal properties of the ocean and buffeting magnetic fields in polar regions to affect the height and cross section of polar accretions. As an example of illustration, we now discuss possible diagnostics for equatorial MHD tidal waves of a neutron star in a LMXB. In spherical geometry, $k_x=m/R$ for an integer $m$ and $\tilde k_x\sim 0.18m(M_{33}h_4)^{1/4}$ $(R_6\nu_{3})^{-1/2}$ for a [*small*]{} $C_A$, where $\nu_3$, $M_{33}$, and $R_6$ stand for $\nu\equiv\Omega/(2\pi)$, $M$, and $R$ in units of $10^3$Hz, $10^{33}$g, and $10^6$cm, respectively. The Rossby radius of deformation is $D_{rossby}\equiv ({2\beta^{*}/c^{*}})^{-1/2}$ $\sim 1.8\times 10^5(M_{33}h_{4})^{1/4}\nu_{3}^{-1/2}\hbox{ cm}$. For $R_6\sim 1$, $M_{33}\sim 2.8$ and $\nu_3\sim 0.5$, equatorially trapped KARP waves influence a latitude range ${\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}\pm 20^{\circ}$ across the equator. Take a plausible geometry favorable for a detection. The spin and magnetic axes are misaligned with the two magnetic poles at latitudes outside $\pm 45^{\circ}$. The spin axis, likely aligned with the binary orbital axis, is taken to be perpendicular to the line of sight. During the phase of moderate and steady polar accretions from the surrounding disk, the stellar surface gives off a largely isotropic X-ray flux sufficiently away from the two magnetic polar hot spots. In this scenario, one may not directly detect either X-ray hot spots. [*Instead, accretion-powered KARP waves trapped in the equatorial belt may give rise to relatively smooth large-scale ($L{\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}R$) periodical modulations of X-ray brightness*]{}. In principle, all these waves may coexist. Estimates are given below for $h_4\sim 1$, $R_6\sim 1$, and $M_{33}\sim 2.8$. For $m=0$ or $\tilde k_x=0$, dispersion relation (9) gives [*standing*]{} MHD tidal wave frequencies $f_{\hbox{\eightrm w}}$ as $$\begin{aligned}
& &\!\!\!
f_{\hbox{\eightrm w}}
=\pm{(2n+1)^{1/2}\over 2\pi}
\bigg[{2\Omega\over R}{(GMh/R^2+C_A^2)^{1/2}\over
(1+C_A^2/c^2)^{3/2}}\bigg]^{1/2}\nonumber \\
& &\ \ \cong\pm 5.1\times 10^2(2n+1)^{1/2}\, \hbox{ }
\nu_3^{1/2}(M_{33}h_4)^{1/4}R_6^{-1}\,\,\hbox{Hz}\ \ \nonumber\ (10)\end{aligned}$$ with $n=0,1,2,\cdots$. Given a detected $f_{\hbox{\eightrm w}}$, the stellar spin rate of $\nu$ can be inferred for $n=0$ – the special case of mixed Alfvén-Rossby-Poincaré standing waves (the dash-dot line in Fig. 1 with $k_x=0$). Or, for standing Alfvén-Poincaré waves of larger $n\geq 1$, $\nu$ would be smaller.
For nonaxisymmetric waves, the frequencies of equatorially trapped Alfvén-Kelvin waves (7) (the dashed straight line of Fig. 1) seen in an inertial reference frame are $$f_{\hbox{\eightrm w}}=10^3m\nu_3
+1.3\times 10^2m(M_{33}h_4)^{1/2}R_6^{-2}\quad\hbox{Hz}\ \eqno(11)$$ for a [*small*]{} $C_A$, where the corotation effect is included. With $m=1$ and a detected $f_{\hbox{\eightrm w}}$, one can infer $\nu$. For $m=2$ and 3, $f_{\hbox{\eightrm w}}$ would be [*precisely*]{} 2 and 3 times the fundamental frequency. For small $\tilde k_x\sim 0.18m(M_{33}h_4)^{1/4}(R_6\nu_{3})^{-1/2}$ with $m\neq 0$, the frequencies of equatorially trapped Alfvén-Rossby waves (lower solid lines of Fig. 1) as seen in an inertial reference frame are $$f_{\hbox{\eightrm w}}\cong 10^3m\nu_3-
1.3\times 10^2m(M_{33}h_4)^{1/2}R_6^{-2}/(2n+1)\ .\eqno(12)$$ With $m=1$, $n=1$, and a detected $f_{\hbox{\eightrm w}}$, one can infer $\nu$. For $m=2$ and 3, $f_{\hbox{\eightrm w}}$ would be [*approximately*]{} 2 and 3 times the fundamental frequency.
If only one apparent $f_{\hbox{\eightrm w}}$ were detected, it would be difficult to distinguish different KARP wave possibilities. A detection of two or more harmonics at 2 and 3 or more times the fundamental frequency may offer valuable hints. For nonaxisymmetric equatorially trapped Alfvén-Poincaré waves and mixed Alfvén-Rossby-Poincaré waves traveling around the globe, relation (10) must be modified to include the corotation effect $m\nu$ and other $m-$dependence (Fig. 1); the overall $m-$dependence of these frequencies is not linear. Similarly, the [*more accurate*]{} version of relation (12) for Alfvén-Rossby waves does not contain a [*strictly*]{} linear $m-$dependence either (Fig. 1). By detecting more harmonics, one can constrain the type of KARP waves involved according to the dispersion relations (Fig. 1). As $f_{\hbox{\eightrm w}}$ given by eqns (10)$-$(12) vary with $\nu$, the expected stellar $\nu$ spin-up in a LMXB (or unexpected variations, such as accreting X-ray pulsar Cen X-3) would lead to a $f_{\hbox{\eightrm w}}$ spin-up (or corresponding variations). There have been recent theoretical development on stellar $r-$mode (Papaloizou & Pringle 1978; Saio 1982) instabilities caused by gravitational radiations (Andersson et al. 1999) to limit spin rates of hot young neutron stars at birth. The prospect that fast spinning neutron stars in LMXBs might emit gravitational waves (Bildsten 1999) observable by ground-based detectors currently under construction (e.g., the enhanced LIGO) offers another potential test for the KARP wave scenario advanced here by independent determinations of neutron star spin rates.
In the case that the stellar surface of a radio pulsar is covered with an ocean, various MHD tidal waves may modulate or interfere with radio pulse emissions from the magnetic polar regions by (quasi-)periodically varying polar magnetic field strengths or changing magnetospheric conditions (Lou 2001a). This may lead to observable effects because magnetic and spin axes misalign for radio pulsars to shine. With $B{\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}10^{12}$ G for a pulsar, $C_A$ and $(gh)^{1/2}$ can be comparable and be both ${\;\lower4pt\hbox{${\buildrel\displaystyle >\over\sim}$}\;}\Omega R$. For equatorially trapped Alfvén-Poincaré waves, Alfvén-Kelvin waves, and the high-frequency branch of mixed Alfvén-Rossby-Poincaré waves, modulation timescales would be [*shorter than or comparable to*]{} radio pulse periods. For equatorially trapped Alfvén-Rossby waves and the low-frequency branch of mix Alfvén-Rossby-Poincaré waves, modulation timescales would be [*longer than*]{} radio pulse periods. These are intriguing possibilities to be explored further.
This research was supported in part by grants from US NSF (AST-9731623) to the University of Chicago, by the ASCI Center for Astrophysical Thermonuclear Flashes at the University of Chicago under Department of Energy contract B341495, by the Visiting Scientist Programs at the Institute of Astronomy and Astrophysics, Academia Sinica (NSC-88-2816-M-001-0010-6) and at the National Center of Theoretical Sciences (Physics Division), National Tsinghua University, and by the Collaborative Research Fund from the NSF of China for Young Overseas Chinese Scholars (NSFC 10028306) at the National Astronomical Observatory, Chinese Academy of Sciences.
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abstract: 'This paper deals with complex structures on Lie algebras $\ct_{\pi} \hh=\hh \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_{\pi} \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constrain $J\hh=V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non singular derivations of $\hh$. This fact derives different kind of applications. Finally an approach to the pseudo Kähler geometry was done.'
address:
- 'R. Campoamor-Stursberg: IMI y FCM, Universidad Complutense de Madrid'
- 'G. Ovando: CONICET y ECEN-FCEIA, Universidad Nacional de Rosario'
author:
- 'Rutwig Campoamor-Stursberg'
- 'Gabriela P. Ovando'
date: '17/2/2009'
title: complex structures on tangent and cotangent Lie algebras of dimension six
---
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[^2]
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Introduction
============
While the existence of complex structures on reductive Lie algebras of even dimension was solved in different steps (starting with [@Sm] and [@Wa]), the solvable case remains an open problem. For dimensions up to four, complex structures were studied in [@Sa1; @SJ; @O]; in dimension six the classification and induced complex geometry has been considered in the nilpotent case in [@CFGU1; @CFGU2; @CFU; @FPS; @GR; @Mg; @Sl; @Ug]. Since those works are mainly done on the basis of a case by case study, one of the principal obstructions in classifying complex (and more general) structures on solvable Lie groups of dimensions equal or greater than six relies in the high number of isomorphism classes. This implies that different criteria have to be developed in order to describe any kind of geometry on Lie groups. One of these is the notion of generalized complex structure, introduced by Hitchin in [@Hi] and treated by various authors (see for instance [@ABDF; @CG; @G] and references therein). On the other hand, in order to study the complex geometry, special types of complex structures were considered, the so called abelian [@BDM] and nilpotent [@CFGU1; @CFGU2], specific for nilpotent Lie algebras, and which have been shown to be of considerable interest, in particular in combination with other compatible geometric structures.
The aim of this work is the study of complex structures on tangent and cotangent Lie algebras, that is Lie algebras which are semidirect products $\ct_{\pi} \hh= \hh \ltimes_{\pi} V$, for $\dim
V=\hh$, originally for $\pi$ the adjoint or the coadjoint representation. We focus in the following existence questions:
i\) complex structures satisfying the condition $J\hh=V$,
ii\) complex structures on $\ct \hh$ and $\ct^*\hh$ where $\hh$ is a three dimensional real Lie algebra,
iii\) symplectic structures which are compatible for a complex structure in ii), therefore inducing pseudo Kähler geometries.
Complex structures appearing in i) are called [*totally real*]{}. They have become objects of importance in the construction of weak mirror pairs (see for instance [@CLP] and references therein).
Complex and symplectic geometry are extremal special cases of generalized complex geometry. Once a Lie algebra $\hh$ was fixed, the corresponding underlying geometric structure arises either as a complex structure on $\hh$ or as a totally real complex structure on $\ct^*\hh$, which is Hermitian for the canonical metric on $\ct^*\hh$.
For the adjoint representation we prove that a totally real complex structure corresponds to a non singular derivation of $\hh$. Therefore the existence of such a totally real complex structure on $\ct \hh$ imposes the condition on $\hh$ to be nilpotent (Theorem (\[le22\])). In dimension three only $\ct \hh_1$, where $\hh_1$ denotes the Heisenberg Lie algebra of dimension three, posseses a totally real complex structure. As application one proves the existence of a generalized complex structure of symplectic type on some kind of nilpotent Lie algebras $\hh$ and the existence of lagrangian symplectic structures on $\ct^*\hh$.
For the coadjoint representation, we give the general form of totally real complex structures $J$ on $\ct^*\hh$, proving the existence when $\hh$ is one of the following three dimensional Lie algebras: the Heisenberg Lie algebra, the Lie algebra of the group of rigid motions of the Minkowski 2-space $\rr_{3,-1}$, the Lie algebra of the group of rigid motions of the Euclidean 2-space $\rr_{3,0}'$ and the one dimensional trivial central extension of the Lie algebra of the group of affine motions, usually denoted as $\aff(\RR)$.
In addition to the Lie algebras obtained in i) the six dimensional tangent Lie algebras admitting complex structures correspond to a Lie algebra $\hh$ which is either simple: $\mathfrak{sl}(2)$, $\mathfrak{so}(3)$ or solvable and isomorphic to $\RR \times \aff(\RR)$. In the cotangent case we add $\mathfrak{sl}(2)$, $\mathfrak{so}(3)$ $\rr_{3,1}$ and $\rr_{3,\eta}'$ for $\eta >0$.
Concerning iii) we study the geometry that derives from the Kähler pairs. The only Lie algebras carrying such a structure are: the tangent and the cotangent of the Heisenberg Lie algebra and the tangent of $\RR\times \aff(\RR)$. The study in the nilpotent case says that there are flat and non flat pseudo Kähler metrics, result which extends those in [@CFU]. Again in this case totally real complex structures provide examples for Kähler pairs. The second author expresses her gratitude to the hospitality of the Departmento de Geometría y Topología of the Universidad Complutense de Madrid, where part of this work was written.
Generalities on Complex structures
==================================
An [*almost complex*]{} structure on a Lie algebra $\ggo$ is an endomorphism $J:\ggo \to \ggo$ satisfying $J^2=-{\rm I}$, where ${\rm I}$ is the identity map.
Let $\ggo^{\CC}=\ggo \otimes \CC$ denote the complexification of $\ggo$ whose elements have the form $v \otimes c$, with $v \in
\ggo,\quad c\in \CC$. An almost complex structure $J$ on $\ggo$ can be extended to a complex linear endomorphism of $\ggo^{\CC}$ that we also denote by $J$, by setting $J(v \otimes c)=Jv \otimes
c$.
As usual, we identify $v\in \ggo$, with $v \otimes 1 \in
\ggo^{\CC}$, and hence any element in $\ggo^{\CC}$ can be written as $x+iy$ where $x, y \in \ggo$. With this identification, the eigenspace corresponding to the imaginary eigenvalue $i$ of $J$ is the subspace $\mm$ of $\ggo^{\CC}$ given by $$\mm =\{ x - i Jx \, : \, x \in \ggo\}.$$ If we denote by $\sigma$ the conjugation map on $\ggo^{\CC}$, that is, $\sigma(x+iy)= x-iy$, the eigenspace corresponding to $-i$ is $\sigma \mm$, and we obtain the direct sum of vector spaces $$\label{des}
\ggo = \mm \oplus \sigma \mm.$$ Conversely any decomposition of type (\[des\]) induces an almost complex structure on $\ggo$. In fact if $x\in \ggo \subseteq \ggo^{\CC}$ and $x$ can be written uniquely as $x=u + v\in \mm \oplus \sigma \mm$ define an endomorphism $J$ by $J x =i u -i v$. Since $\sigma \circ J = J \circ \sigma$, the map $J$ gives rise to an almost complex structure on $\ggo$.
The integrability condition of an almost complex structure $J$ is expressed in terms of the Nijenhuis tensor $N_J$ $$\label{NJ}
N_J(x,y)=[Jx,Jy]-[x,y] - J[Jx,y] - J[x,Jy], \qquad\mbox{ for all }
x,y \in \ggo.$$
It is straightforward to verify that $N_J(Jx,Jy)=-N_J(x,y)$ for any $x,y\in \ggo$. Hence, if $\ggo= \uu \oplus J \uu$ is a direct sum as vector subspaces, then $N_J\equiv 0$ if and only if $N_J(u,v)=0=N_J(u,Jv)$ for all $u,v \in \uu$.
An almost complex structure $J$ on $\ggo$ is called [*integrable*]{} if $N_J\equiv 0$. In this case $J$ is called a [*complex structure*]{} on $\ggo$. Equivalently, $J$ is integrable if and only if $\mm$ (and hence $\sigma \mm$) satisfying (\[des\]) is a complex subalgebra of $\ggo^{\CC}$.
Special types of almost complex structures are determined by those endomorphisms $J :\ggo \to \ggo$ satisfying $J^2 = -{\rm I}$ and one of the following conditions for any $x, y\in \ggo$: $$\mbox{c1)}\,\,J[x,y]= [x,Jy] \qquad \qquad
\mbox{c2)}\,\,[Jx,Jy]= [x,y].$$ In any case they are integrable. Complex structures of type c1) determine a structure of complex Lie algebra on $\ggo$, they are sometimes called [*bi-invariant*]{}. The subalgebra corresponding to the eigenvalue $\pm i$ is actually an ideal of $\ggo^{\CC}$. Structures of type c2) are called [*abelian*]{}, and the corresponding eigenspaces for the eigenvalues $\pm i$ are complex abelian subalgebras of $\ggo^{\CC}$. It should be remarked that not any Lie algebra admitting a complex structure can be endowed with an abelian complex structure, as shows the following example.
\[exa1\] Let $\hh_1$ be the Heisenberg Lie algebra of dimension three and let $\ct^*\hh_1$ denote the cotangent Lie algebra of $\hh_1$. This is spanned by the vectors $e_1, e_2, e_3,
e_4, e_5, e_6$ with the non trivial Lie bracket relations $$[e_1,e_2]=e_3, \quad [e_1, e_6]=-e_5,\quad [e_2, e_6]=e_4.$$ Assume $J:\ct^*\hh_1 \to \ct^*\hh_1$ is an endomorphism satisfying $[Jx,Jy]=[x,y]$ or, in equivalent form, $[Jx,y]=-[x,Jy]$ for all $x,y \in \ggo$. If $y$ belongs to the center of $\ct^*\hh_1$, by c2) one has that $Jy\in \zz(\ct^*\hh_1)$, thus $J$ restricts to the center $\zz(\ct^*\hh_1)$, and therefore $J$ cannot be almost complex since the dimension of $\zz(\ct^*\hh_1)$ is odd. To see that $\ct^*\hh_1$ admits a complex structure see (\[coth1\]).
As proved in the previous example, if $\ggo$ carries an abelian complex structure, then the center of $\ggo$ must be $J$-invariant and therefore even dimensional. Another necessary condition to have abelian complex structures is that $\ggo$ is 2-step solvable, which means that the commutator subalgebra $C(\ggo)$ is abelian (see [@P] for instance). For the sake of completeness we include here a proof.
Let $\ggo$ be a Lie algebra with a decomposition as direct sum of vector subspaces $\ggo={\mathfrak{a} }\oplus \bb$, where ${\mathfrak{a} }$ and $\bb$ are abelian subalgebras. Denote $[a,b']=a^*+b''$ and $[a',b]=a''+b^*$, with $a, a', a'', a^*\in {\mathfrak{a} }$ and $b,b', b'', b^*\in \bb$. One has $$\begin{array}{rcl}
[[a,b], [a',b']] & = & [[[a,b], a'], b']-[[[a,b],b'],a']\\
& = & -[[[b,a'],a],b']+ [[[b',a],b],a']\\
& = & [[b^*,a],b'] -[[a^*,b],a']\\
& = & -[[a,b'], b^*]+ [[b,a'], a^*]\\
& = & -[a^*, b^*] -[b^*,a^*]=0,
\end{array}$$
which proves that the Lie bracket on $C(\ggo)$ is trivial.
Let $\ggo$ be a Lie algebra and let $J$ be a fixed almost complex structure on $\ggo$. For any $l\geq 0$ we define the set ${\mathfrak{a} }_l(J)$ inductively as: $${\mathfrak{a} }_0(J)=\{0\}, \qquad {\mathfrak{a} }_l(J)=\{ X
\in \ggo,\, /\, [X, \ggo] \subset {\mathfrak{a} }_{l-1} \mbox{ and } [JX,
\ggo] \subset {\mathfrak{a} }_{l-1}\} \quad l \geq 1.$$ It is easy to verify that $${\mathfrak{a} }_0(J) \quad \subseteq \quad {\mathfrak{a} }_1(J) \quad \subseteq \quad {\mathfrak{a} }_2(J) \quad \subseteq \quad \hdots$$
For a fixed $X\in {\mathfrak{a} }_{i+1}(J)$ we have that $[X,Y]\in
{\mathfrak{a} }_{i}(J)\subseteq {\mathfrak{a} }_{i+1}(J)$ for all $Y\in \ggo$, and clearly $[J[X,Y],Z]\in {\mathfrak{a} }_i(J) \subset {\mathfrak{a} }_{i+1}(J)$ for all $Y,Z\in
\ggo$. Therefore ${\mathfrak{a} }_i(J)$ is a $J$-invariant ideal of $\ggo$ for any $i\geq 0$.
The almost complex structure $J$ is called [*nilpotent*]{} if there exists a $t$ such that ${\mathfrak{a} }_t(J) =\ggo$. This implies that $\ggo$ must be nilpotent. For a nilpotent almost complex structure $J$ on a $s$-step nilpotent Lie algebra of even dimension we shall say that it is $r$-step nilpotent if $r$ is the first nonnegative integer such that ${\mathfrak{a} }_r(J)=\ggo$; this satisfies the inequality $s\leq r \leq n$ [@CFGU2]. In the following sections examples are given that show that these bounds are actually reached (see (\[coth1\]) and (\[tah1\])). Notice that if $J$ is a nilpotent almost complex structure on a nilpotent Lie algebra $\ggo$, then any term of the ascending series of $\ggo$ admits a two dimensional $J$-invariant subspace. Clearly, if $J$ is integrable, the condition of being nilpotent is stronger than asking the corresponding $\mm$ for $J$ to be nilpotent.
Notice $a_1(J)\subseteq \zz(\ggo)$. If $J$ is abelian then the equality holds $a_1(J)=\zz(\ggo)$.
The canonical complex structure of a nilpotent complex Lie algebra is nilpotent (see (\[cn\])).
An equivalence relation is defined among Lie algebras with complex structures. Lie algebras with complex structures $(\ggo_1,J_1)$ and $(\ggo_2,J_2)$ are called [*holomorphically equivalent*]{} if there exists an isomorphism of Lie algebras $\alpha:\ggo_1 \to \ggo_2$ such that $J_2 \circ \alpha = \alpha \circ J_1$. In particular when $\ggo_1=\ggo_2$ we simply say that $J_1$ and $J_2$ are [*equivalent*]{} and a classification of complex structures can be done.
\[le\] Let $\ggo$ be an even dimensional real Lie algebra.
i\) The class of an abelian complex structure, if non-empty, consists only of abelian complex structures.
ii\) Let $J, J'$ be complex structures on $\ggo$ such that $J'=\sigma J \sigma^{-1}$. Then $\sigma {\mathfrak{a} }_l(J)={\mathfrak{a} }_l(J')$ for any $l\geq 0$.
In particular the class of a nilpotent complex structure on a given nilpotent Lie algebra consists only of nilpotent complex structures, all of them being nilpotent of the same type.
iii\) The class of a bi-invariant complex structure has only bi-invariant complex structures.
i\) Let $J$ an abelian complex structure on $\ggo$. If $J'=\sigma J \sigma^{-1}$, the result follows using that $\sigma$ is an automorphism and $J$ is abelian.
ii\) For $i=0$ it is clear that $\sigma {\mathfrak{a} }_0(J)={\mathfrak{a} }_0(J')$. Assume that $\sigma {\mathfrak{a} }_l(J) ={\mathfrak{a} }_l(J')$ for all $l\leq i$. Let $X\in {\mathfrak{a} }_{i+1}(J)$. It follows at once that the following identity is satisfied $$[\sigma X, \ggo]=[\sigma X, \sigma \ggo]=\sigma [X, \ggo] \in \sigma {\mathfrak{a} }_i(J) ={\mathfrak{a} }_i(J').$$ On the other hand, $$[J'\sigma X, \ggo]=[\sigma J X, \ggo]=\sigma [J X, \ggo]\in \sigma {\mathfrak{a} }_i(J) ={\mathfrak{a} }_i(J').$$ Therefore $\sigma {\mathfrak{a} }_{i+1}(J) \subset {\mathfrak{a} }_{i+1}(J')$. A similar argument, interchanging the roles of $J$ and $J'$, proves that $\sigma {\mathfrak{a} }_i(J)={\mathfrak{a} }_i(J')$.
iii\) follows by direct application of the definitions.
From the definitions above it is not immediately clear which is the relationship between nilpotent complex structures and complex structures whose corresponding $i$-eigenspace $\mm$ is nilpotent. In (\[tah1\]) we show that the tangent Lie algebra of the Heisenberg Lie algebra $\hh_1$ carries only 2-step nilpotent complex structures, some of them being abelian, and others having $\pm i$-eigenspaces which are 2-step nilpotent subalgebras.
Totally real complex structures on tangent and cotangent Lie algebras
=====================================================================
The aim of this section is the study of totally real complex structures on tangent and cotangent Lie algebras, that is complex structures $J$ on $\ct_{\pi} \hh$ such that $J\hh=V$.
We briefly recall the construction. Let $\hh$ denote a real Lie algebra and let $(\pi,V)$ be a finite dimensional representation of $\hh$. By endowing $V$ with the trivial Lie bracket, consider the semidirect product of $\hh$ and $V$ relative to $\pi$, $\ct_{\pi} \hh:= \hh \ltimes_{\pi} V$, where the Lie bracket is: $$[(x,v), (x', v')]=([x,x'], \pi(x) v' - \pi(x') v)\qquad x,x'\in \hh, v,v'\in V.$$
In this work we focus on the the adjoint and the coadjoint representations. In both cases $V$ is a real vector space with the same dimension as that of $\hh$. The adjoint representation $\ad: \hh \to \mathfrak{gl}(\hh)$ is given by $\ad(x) y = [x,y]$, and it defines the [*tangent*]{} Lie algebra that we denote with $\ct \hh$.
For the coadjoint representation $\ad^*:\hh \to \mathfrak{gl}(\hh^*)$, that is $V=\hh^*$, and $$\label{coad}
\ad(x)^* \varphi (y) = - \varphi \circ \ad(x) y \qquad \mbox{ for } x, y
\in \hh, \, \varphi \in \hh^*;$$ we call the resulting Lie algebra [*cotangent Lie algebra*]{} and we denote it as $\ct^*\hh$.
A question concerning complex structures when we look at the algebraic structure of the Lie algebra $\ct_{\pi} \hh=\hh \ltimes_{\pi} V$ is whether there exists an almost complex structure $J$ such that $J \hh=V$. Such a $J$ induces a linear isomorphism $j:\hh \to V$, and conversely any such $j:\hh \to V$ determines an almost complex structure on $\ct_{\pi} \hh$ such that $J\hh=V$, by means of $$\label{j}
J(x,v)=(-j^{-1} v, j x) \qquad x\in \hh, v\in V, \,j: \hh \to V.$$
It follows that both $\hh$ and $V$ are totally real with respect to $J$. We adopt the next terminology, following [@CLP].
Let $\ct_{\pi} \hh:=\hh\ltimes_{\pi} V$ be the semidirect product of a Lie algebra $\hh$ with the real vector space $V$ such that $\dim V=\dim \hh$ and let $J$ denote an (almost) complex structure on $\ct_{\pi}\hh$. If $J\hh=V$ we say that $J$ is a [*totally real*]{} (almost) complex structure on $\ggo$.
Assume $(\pi, V)$ denotes a finite dimensional representation of $\hh$ and let $J$ be a totally real almost complex structure on $\ct_{\pi} \hh= \hh \ltimes_{\pi} V$ like in (\[j\]). In this case, the integrability condition for $J$ reduces to $$\label{e12}
0 = [x,y] - j^{-1} \pi(x) j y + j^{-1} \pi(y) j x\qquad \mbox{ for all } x, y\in \hh.$$
Fix a Lie algebra $\hh$, recall that the representations $(V, \pi)$ and $(V',\pi')$ are called [*equivalent*]{} if there is a linear isomorphism $T:V \to V'$ such that $T^{-1} \pi'(x) T = \pi(x)$ for all $x\in \hh$.
Actually for any $\psi\in Aut(\hh)$, the map $\varphi:\ct_{\pi} \hh \to \ct_{\pi'} \hh$ given by $\varphi=\psi+T$ is a Lie algebra isomorphism. In fact for all $x,y\in \hh$, $u,v\in V$ $$\begin{array}{rcl}
\varphi[x+u, y+v] & = & \varphi([x,y]+\pi(x) v - \pi(y)u)\\
& = & \psi [x,y]+ T\pi(x) v - T\pi(y)u\\
& = & [\psi x, \psi y] + \pi'(x) T v - \pi'(y) Tu\\
& = & [\varphi(x+u), \varphi(y+v)]
\end{array}$$ And therefore if $J$ denote a complex structure on $\ct_{\pi}\hh$ then $J':=\varphi \circ J \circ \varphi^{-1}$ denotes a complex structure on $\ct_{\pi'} \hh$ making of $(\ct_{\pi} \hh, J)$ and $(\ct_{\pi'} \hh, J')$ a pair of holomorphically equivalent Lie algebras.
In particular if $J$ is a totally real complex structure on $\ct_{\pi} \hh$, then $(\ct_{\pi}\hh, J)$ is holomorphically equivalent to $(\ct_{\pi'}\hh, \tilde{J})$ where $\tilde{J}_{|{_\hh}}:\hh \to V'$ is $\tilde{J}=T \circ J_{|_{\hh}}$ and extended as in (\[j\]). The proof of the following result follows by using these relations and the integrability condition (\[e12\]).
\[ceq\] Let $(V, \pi)$ and $(V', \pi')$ be equivalent representations of a Lie algebra $\hh$ such that $\dim V = \dim V'=\dim \hh$. Complex structures on $\ct_{\pi}\hh$ are in one to one correspondence with complex structures on $\ct_{\pi'} \hh$.
In particular, totally real complex structures on $\ct_{\pi} \hh$ are homomorphically equivalent to totally real complex structures on $\ct_{\pi'} \hh$.
A first consequence of (\[e12\]) concerns abelian complex structures.
\[ca\] Let $\hh$ be a Lie algebra and let $V$ denote the underlying vector space of $\hh$. Let $\ct_{\pi} \hh:=\hh \ltimes_{\pi} V$ denote the semidirect product and let $J$ be an abelian totally real complex structure on $\ct_{\pi} \hh$. Then $\hh$ is abelian and $\pi$ and $J$ are related by $\pi(x) Jy = \pi(y) Jx$ for all $x,y\in \hh$.
The converse of the previous corollary is true. Let $\hh$ denote an abelian Lie algebra and let $\pi$ be a representation of $\hh$ into $\hh$. Then if $j:\hh\to \hh$ is a non singular map such that $\pi(x)jy =\pi(y) jx$ for all $x,y\in \hh$, then the almost complex structure on $\hh \ltimes_{\pi} \hh$ given as in (\[j\]) is integrable and totally real with respect to $\hh$. See the final section for an explicit example.
Consider $\RR^n$ with the canonical basis $\{e_1, e_2, \hdots, e_n\}$ and let $A$ be a non singular $n\times n$ real matrix. Let $C_A$ denote the centralizer of $A$ in $gl(n,\RR)$ that is, the set of $n\times n$ matrices $B$ such that $BA=AB$. Let $B_1, B_2, \hdots ,B_n$ be $n$ matrices in $C_A$ such that they are pairwise in involution, $B_i B_j = B_j B_i$ for all $i,j$. Take $\pi$ the representation of $\RR^n$ which extends linearly the mapping $e_i \to B_i$ (notice that this could be trivial depending on $A$). The map $j$ represented by $A$ amounts to a totally real abelian complex structure on $\ct_{\pi} \RR^n$.
When $\pi$ is the adjoint representation, the solutions of (\[e12\]) have an algebraic interpretation as it will be seen next. Recall first that a [*derivation*]{} of a Lie algebra $\hh$ is a linear map $d:\hh \to \hh$ such that $$d[x,y]=[dx, y] + [x,dy]\qquad \qquad \mbox{ for all } x,y \in \hh.$$
Jacobson proved that if a Lie algebra $\hh$ admits a non singular derivation then it must be nilpotent [@Ja].
\[le22\] Let $\ct \hh$ denote the tangent Lie algebra of $\hh$. The set of totally real complex structures on $\ct \hh$ is in one to one correspondence with the set of non singular derivations of $\hh$. If one (and therefore both) of these sets is non empty, then $\hh$ is nilpotent.
Let $\ad$ denote the adjoint representation of $\hh$. The integrability condition (\[e12\]) becomes $$0 = j[x,y] - \ad(x) jy + \ad(y) jx \qquad \mbox{ for all } x,y \in \hh.$$ This shows, via identifications, that the complex structure $J$ determined by $j$ (\[j\]) corresponds to a non singular derivation of $\hh$. The proof is completed after the application of the result of Jacobson.
\[derh1\] Let $\hh_1$ denote the Heisenberg Lie algebra of dimension three (see \[lie3\]). By Lemma (\[le22\]) if $J$ is a totally real complex structure on $\ct \hh_1$, then it corresponds to a non singular derivation of $\hh_1$. Any non singular derivation of $\hh_1$ has a matrix representation in the basis of (\[lie3\]) given by: $$\left( \begin{matrix} A & 0 \\ * & tr(A) \end{matrix}
\right),\qquad \mbox{ with } A\in GL(2,\RR)\quad \mbox{ and } tr(A)\neq 0$$ where $tr$ denotes the trace of the matrix $A$.
More generally any non singular derivation $d$ of the Heisenberg Lie algebra $\hh_n$ of dimension $2n+1$, has a matrix representation as above with $A\in GL(2n, \RR)$ and $tr(A)\neq 0$. This induces a totally real complex structure $J$ on the tangent Lie algebra $\ct \hh_n$.
Totally real complex structures on six dimensional cotangent Lie algebras
-------------------------------------------------------------------------
We now proceed to analyze the existence of totally real complex structures on six dimensional cotangent Lie algebras $\ct^*\hh$. To this extend, recall the classification of three dimensional Lie algebras as given e.g. in [@GOV] or [@Mi].
\[lie3\] Let $\hh$ be a real Lie algebra of dimension three spanned by $e_1,e_2,e_3$. Then it is isomorphic to one and only one in the following list: $$\begin{array}{rll}
\hh_1 & [e_1,e_2]=e_3 \\
\rr_{3} & [e_1,e_2]=e_2,\, [e_1,e_3]= e_2 + e_3 \\
\rr_{3,\lambda} & [e_1,e_2]=e_2,\, [e_1,e_3]= \lambda e_3 & |\lambda| \leq 1 \\
\rr_{3,\eta}' & [e_1,e_2]=\eta e_2- e_3,\, [e_1,e_3]= e_2 + \eta e_3 & \eta \geq 0\\
\mathfrak{sl}(2) & [e_1,e_2]=e_3,\, [e_3,e_1]= 2e_1,\, [e_3,e_2]= -2e_2\\
\mathfrak{so}(3) & [e_1,e_2]=e_3,\, [e_3,e_1]= e_2,\, [e_3,e_2]= -e_1
\end{array}$$
A Lie algebra $\ggo$ which satisfies $tr(\ad(x))=0$ for all $x\in
\ggo$ is called unimodular. Among the Lie algebras above, the unimodular ones are: $\hh_1$, $\rr_{3,-1}$ and $\rr_{3,0}'$. The Lie algebra $\hh_1$ is known as the Heisenberg Lie algebra, while $\rr_{3,-1}$ is the Lie algebra of the group of rigid motions of the Minkowski 2-space and $\rr_{3,0}'$ corresponds to the Lie algebra of the group of rigid motions of the Euclidean 2-space. The Lie algebra $\rr_{3,0}$ denotes the central extension of the Lie algebra of the group of affine motions in $\RR$, usually denoted as $\aff(\RR)$.
\[trct\] Let $\ct^*\hh=\hh \ltimes \hh^*$ be a cotangent Lie algebra of a three dimensional Lie algebra $\hh$. Then totally real complex structures on $\ct^{\ast}\hh$ exist whenever $\hh$ is either unimodular or isomorphic to $\RR \times \aff(\RR)$. In those cases the map $j:\hh \to \hh^*$ admits a matrix representation as follows $$\ct^* \hh_1\quad \left( \begin{matrix} a_{41} & a_{42} & a_{43}
\\ a_{51} & a_{52} & a_{53} \\ -a_{43} & -a_{53} & 0 \end{matrix}
\right); \qquad \ct^*\rr_{3,-1} \quad \left( \begin{matrix} a_{41}
& a_{42} & a_{43} \\ -a_{42} & 0 & a_{53} \\ -a_{43} & -a_{53} & 0 \end{matrix} \right);$$ $$\ct^*\rr_{3,0} \quad \left( \begin{matrix} a_{41} & a_{42} & a_{43}
\\ -a_{42} & 0 & 0 \\ a_{61} & 0 & a_{63} \end{matrix} \right); \qquad
\ct^*\rr_{3,0}' \quad \left( \begin{matrix} a_{41} & a_{42} &
a_{43} \\ -a_{42} & 0 & a_{53} \\ -a_{43} & -a_{53} & 0
\end{matrix} \right);$$ where the matrix should be non singular.
The proof follows by direct computation of (\[e12\]) taking $\pi$ as the coadjoint representation. In the cases not listed above, the maps $j$ solving (\[e12\]) are singular, hence they cannot induce a complex structure on $\ct^* \hh$.
Let $\sg$ denote a semisimple Lie algebra. Since the Killing form is non degenerate this induces an ad-invariant metric on $\sg$. Therefore the adjoint and coadjoint representation are equivalent. By (\[ceq\]) the existence of totally real complex structures on $\ct^*\sg$ reduces to the existence of totally real complex structures on $\ct \sg$ and this cannot admit a totally real complex structure by (\[le22\]). Observe that in dimension three the simple Lie algebras are $\mathfrak{sl}(2)$ and $\mathfrak{so}(3)$ (see (\[lie3\])).
complex structures on tangent and cotangent Lie algebras of dimension six
=========================================================================
Examples of six dimensional real Lie algebras with complex structures arise from three dimensional complex Lie algebras. In fact let $\tilde{\ggo}$ denote a three dimensional complex Lie algebra, then the underlying real Lie algebra $\ggo:=\tilde{\ggo}_{\RR}$, is naturally equipped with a bi-invariant complex structure induced by the multiplication by $i$ on $\tilde{\ggo}$. In this way this complex structure on $\ggo$ is bi-invariant.
\[cn\] Let $\ggo$ denote a six dimensional two-step nilpotent Lie algebra equipped with bi-invariant complex structure $J$. We shall see that $\ggo$ is isomorphic to the real Lie algebra underlying $\hh_1\otimes \CC$, the complexification of the Heisenberg Lie algebra of dimension three.
Since $J \circ \ad(x) = \ad(x) \circ J$ for any $x\in \ggo$, we obtain the inclusions $J\zz(\ggo)\subseteq \zz(\ggo)$ and $J C(\ggo)\subset C(\ggo)$. Furthermore, there is a central ideal $\vv \subseteq \zz(\ggo)$ such that $$\zz(\ggo)=C(\ggo) \times \vv \qquad \mbox{ direct sum of $J$-invariant Lie algebras.}$$
A way to produce such a $\vv$ is the following. Take $\la \,, \, \ra$ an inner product on $\zz(\ggo)$ which is Hermitian for $J$ and let $\vv =C(\ggo)^{\perp}$.
Thus $\ggo=\vv \times \nn$ where $\nn$ is a two-step nilpotent Lie algebra such that $C(\ggo)=C(\nn)=\zz(\nn)$ and $\nn$ is equipped with a bi-invariant complex structure $J$, induced from that one on $\ggo$. Now if $\vv$ is non empty then it may be two or four dimensional. If it is two dimensional, then $\nn$ is four dimensional and is equipped with a bi-invariant complex structure, therefore it must be abelian (in dimension four a solvable Lie algebra endowed with a bi-invariant complex structure is either abelian or isomorphic to $\aff(\CC)$, see [@O2] for instance). A similar reasoning applies when $\vv$ has dimension four, and therefore $\zz(\ggo)=C(\ggo)$.
Now let $z\in \zz(\ggo)= C(\ggo)$. Then there exist $x, y\in \ggo$ such that $[x,y]=z$. The set $\{x, y, z\}$ is linearly independent and the set $\{x,y,z, Jx, Jy, Jz\}$ is a basis of $\ggo$. Due to the bi-invariance condition on $J$ one has the following Lie bracket relations $$[x,y]=z\quad [Jx,y]=Jz\quad [x,Jy]=Jz\quad [Jx,Jy]=-z$$ moreover $[x,Jx]= 0 =[y,Jy]$, and therefore the Lie algebra $\ggo$ is isomorphic to the real Lie algebra of dimension six underlying $\hh_1 \otimes \CC$.
In the previous section we gave examples of complex structures in tangent and cotangent Lie algebras of dimension six. Now we shall study the existence problem of complex structures on any tangent or cotangent Lie algebra corresponding to a three dimensional real Lie algebra as in (\[lie3\]). Some considerations about abelian and nilpotent complex structures are also be given.
[**The simple case**]{}. Among the Lie algebras listed in (\[lie3\]) the simple ones are $\mathfrak{sl}(2)$ and $\mathfrak{so}(3)$. Since the Killing form is non degenerate in both cases, the adjoint and the coadjoint representations are equivalent. After (\[ceq\]) for a semisimple Lie algebra $\mathfrak{s}$, the existence of complex structures on $\ct \mathfrak{s}$ determines one on $\ct^*\mathfrak{s}$ and viceversa. Recall that complex structures on compact semisimple and more generally on reductive Lie algebras were extended studied (see for instance [@Sm][@Wa] [@Sa1] [@Sa2] [@SD]). We perform below a construction of a complex structure on $\ct \mathfrak{sl}(2)$ and $\ct \mathfrak{so}(3)$. Explicitly the Lie brackets are given by: $$\begin{array}{llll}
\ct \mathfrak{sl}(2) & [e_1,e_2]=e_3 & [e_3,e_1]= 2e_1 & [e_3,e_2]= -2e_2\\
& [e_1,e_5]=e_6 & [e_1,e_6]= -2e_4 & [e_2,e_4]= -e_6\\
& [e_2,e_6]=2e_5 & [e_3,e_4]= 2e_4 & [e_3,e_5]= -2e_5\\
\ct \mathfrak{so}(3) & [e_1,e_2]=e_3 & [e_3,e_1]= e_2 & [e_3,e_2]= -e_1\\
& [e_1,e_5]=-e_6 & [e_1,e_6]= e_5 & [e_2,e_4]= -e_6\\
& [e_2,e_6]=e_4 & [e_3,e_4]= -e_5 & [e_3,e_5]= e_4
\end{array}$$ In any case an almost complex structure $J$ can be defined by $$J e_3= e_6\qquad Je_2=e_1 \qquad Je_4= e_5.$$ By calculating $N_J$ one verifies that $J$ is integrable. Hence
[*The tangent Lie algebras $\ct \mathfrak{so}(3)$ and $\ct \mathfrak{sl}(2)$ (and therefore $\ct^*\mathfrak{so}(3)$ and $\ct^* \mathfrak{sl}(2)$) carry complex structures*]{}
[**The solvable case.**]{} Suppose that $\ggo$ is a six dimensional tangent $\ct \hh$ or cotangent Lie algebra $\ct^* \hh$ being $\hh$ a solvable real Lie algebra of dimension three. It admits a complex structure if and only if $\ggo^{\CC}$ decomposes as a direct sum of vector subspaces $\ggo^{\CC}=\mm
\oplus \sigma \mm$, where $\mm$ (resp. $\sigma \mm$) is a complex subalgebra. Without lost of generality assume that $\mm$ is spanned by vectors $U, V, W$ as follows: $$\label{uv} U= e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6
\quad V= b_2 e_2 + b_3 e_3 + b_4 e_4 + b_5 e_5 + b_6 e_6,$$ $$W= c_2 e_2 + c_3 e_3 + c_4 e_4 + c_5 e_5 + c_6 e_6, \quad a_i, b_j, c_k\in \CC, \forall i, j, k=2, \hdots 6.$$ Let ${\mathfrak{a} }:=span\{V,W\}$. We claim that ${\mathfrak{a} }$ is a ideal in $\mm$. In fact, according to the Lie brackets in $\ggo$ (see (\[ta3\]) and (\[cot3\]) below), one verifies that $U\notin C(\ggo)$, hence for any $x,y\in \mm$, $[x,y]\in C(\mm) \subseteq {\mathfrak{a} }$. Thus $\mm =\CC U \ltimes {\mathfrak{a} }$, being ${\mathfrak{a} }$ a ideal of $\mm$ of dimension two and therefore isomorphic either to i) $\CC^2$ or to ii) $\aff(\RR)$, the two dimensional complex Lie algebra spanned by vector $X, Y$ with $[X,Y]=Y$. We may assume in the last situation that $V,W$ satisfy the Lie bracket relation $[V,W]=W$.
In case $\mm=\CC U \ltimes \CC^2$, the action of $U$ on ${\mathfrak{a} }$ admits a basis whose matrix is one of the following ones $$\label{typ1}
\begin{array}{ll}{\mbox{(1)}\quad \left( \begin{matrix}
\nu & 0\\ 0 & \mu \end{matrix} \right),\qquad \nu,\, \mu \in
\CC};& {\qquad \qquad \qquad \mbox{(2)} \quad \left( \begin{matrix}
\nu & 1\\ 0 & \nu \end{matrix} \right),\qquad \nu \in \CC}.\\
\end{array}$$
In case $\mm = \CC U \ltimes \aff(\RR)$ the action of $U$ on ${\mathfrak{a} }$ is a derivation of $\aff(\RR)$ thus over the basis $\{V,W\}$ we have a matrix $$\label{typ2}
\left( \begin{matrix}
0 & 0\\ a & b \end{matrix} \right)
\qquad \qquad a, b\in \CC$$
By making use of this we shall derive the existence or non existence of complex structures on any tangent or cotangent Lie algebra corresponding to a three dimensional solvable real Lie algebra.
Complex structures on six dimensional tangent Lie algebras
----------------------------------------------------------
If $H$ denotes a Lie group, its tangent bundle $T H$ is identified with $H\times \hh$, which inherits a natural Lie group structure as the semidirect product under the adjoint representation. Its Lie algebra, the tangent Lie algebra $\ct \hh$, is the semidirect product via the adjoint representation $\hh \ltimes_{\ad} V$, where $V$ is the underying vector space to $\hh$ equipped with the trivial Lie bracket.
\[ta3\] Let $\hh$ be a solvable real Lie algebra of dimension three and let $\ct \hh$ denote the tangent Lie algebra spanned by $e_1, e_2, e_3, e_4, e_5, e_6$. Then the non zero Lie brackets are presented in the following list: $$\begin{array}{ll}
\ct \hh_1: & [e_1,e_2]=e_3, \, [e_1, e_5]=e_6, [e_2, e_4]= -e_6\\
\ct \rr_{3}: & [e_1,e_2]=e_2,\, [e_1,e_3]= e_2 + e_3,\, \\
& [e_1, e_5]= e_5,\, [e_1, e_6]=e_5 + e_6,\, [e_2, e_4]=-e_5,\, [e_3, e_4]=-e_5-e_6 \\
\ct \rr_{3,\lambda}: & [e_1,e_2]=e_2,\, [e_1,e_3]= \lambda e_3 \\
|\lambda| \leq 1& [e_1, e_5]= e_5,\, [e_1, e_6]=\lambda e_6,\, [e_2, e_4]=-e_5,
\, [e_3,e_4]=-\lambda e_6\\ \ct \rr_{3,\eta}': & [e_1,e_2]=\eta
e_2- e_3,\, [e_1,e_3]= e_2 + \eta e_3,
[e_1, e_5]= \eta e_5-e_6,\, \\
\eta \geq 0 & [e_1, e_6]= e_5+\eta e_6,\, [e_2,e_4]=-\eta e_5+e_6,\, [e_3, e_4]= -e_5-\eta e_6\\
\end{array}$$
\[teot\] Let $\hh$ denote a three dimensional Lie algebra, then $\ct \hh$ admits a complex structure if and only if $\hh$ is either isomorphic to $\hh_1$ or $\RR \times \aff(\RR)$.
The proof can be derived from the next paragraphs.
If $\mm$ is a complex subalgebra of $\ct \hh$ being $\hh$ a three dimensional solvable real Lie algebra such that $\ct \hh^{\CC}=\mm \oplus \sigma \mm$ then $\mm \simeq \CC \ltimes \CC^2$.
According to the previous paragraphs it should hold $\mm\simeq \CC
\ltimes \CC^2$ or $\mm \simeq \CC \ltimes \aff(\RR)$. We shall prove that the last situation is not possible. In fact, from the Lie brackets above (\[ta3\]) we see that $[V,W]\in span\{e_5, e_6\}$ so that $c_2=0=c_3=c_4$. But by computing one has $[V,W]=0$ implying $W=0$ and therefore no complex structure can be derived from this situation.
With the previous Lemma it follows to analyze next the existence of complex structures attached to complex Lie subalgebras $\mm$ such that $\mm \simeq \CC \ltimes \CC^ 2$.
We already know that the tangent Lie algebra of the three dimensional Heisenberg Lie algebra $\ct \hh_1$, admits complex structures, moreover totally real ones. Recall that any totally real complex structure on $\ct \hh_1$ corresponds to a non singular derivation of $\hh_1$ (\[derh1\]). No one of these complex structures is abelian. However $\ct \hh_1$ can be equipped with abelian complex structures as we show below.
Let $\mm$ be a complex subalgebra of $\ct \hh_1$ spanned by vectors $U,V,W$ as in (\[uv\]). The subspace ${\mathfrak{a} }=span\{V,W\}$ is a ideal of $\mm$ and $\mm = \CC U \ltimes {\mathfrak{a} }$. Since $\ct \hh_1$ is nilpotent, ${\mathfrak{a} }$ is abelian and the action of $U$ on ${\mathfrak{a} }$ is of type (\[typ1\]) and moreover case 1) holds for $\mu=\nu=0$ while case 2) holds for $\nu =0$. Case 1) gives rise to abelian complex structures, while case 2) corresponds to non abelian ones.
Computing the Lie brackets $[V,W]$, $[U,V]$ and $[U,W]$, and imposing these brackets to be zero, we get $$U = e_1 + a_2 e_2 + a_3 e_3 +a_4 e_4 + a_5 e_5 + a_6 e_6,$$ $$V= b_3 e_3 + b_4 e_4 -a_2 b_4 e_5 + b_6 e_6,\qquad W= c_3 e_3+ c_4 e_4 - a_2 c_4 e_5 + c_6 e_6.$$ If the set $\{U,V,W, \sigma U, \sigma V, \sigma W\}$ spans a basis of $(\ct
\hh_1)^{\CC}$, the tangent algebra $\ct \hh_1$ carries an abelian complex structure $J$. For instance the linear homomorphism $J$ given by $$\label{h1abe}
J e_1 = e_2 \qquad J e_6 = e_3 \qquad J e_4 = e_5,$$ and such that $J^2=-{\rm I}$ defines a abelian complex structure on $\ct \hh_1$. After [@Mg] there is only one class among abelian complex structures.
Any abelian complex structure is 2-step nilpotent. In fact, since $J$ is abelian ${\mathfrak{a} }_1(J)=\zz(\ct \hh_1)$ and clearly the condition $C(\ct \hh_1) = \zz(\ct \hh_1)$ shows that ${\mathfrak{a} }_2(J)=\ct \hh_1$.
On the other hand the following set of vectors on $\ct \hh_1^{\CC}$ is a basis of the complex subalgebra $\mm$ corresponding to a totally real complex structure on $\ct \hh_1$: $$e_1 - i (a e_4 + b e_5 + e e_6); \quad e_2 -i (c e_4 + d e_5+ f e_6), \quad e_3 - i (a+d) e_6$$ with $a, b, c, d, e, f\in\RR$, $a+d\neq 0$ and $ad-bc \neq 0$. They induce non abelian complex structures, and furthermore there are more non abelian complex structures than the totally real ones. Let $\mm$ be a complex subalgebra of $(\ct \hh_1)^{\CC}$ spanned by $U,V,W$ as in (\[uv\]). Requiring that $[U,V]=0=[V,W]$ and $[U,W]=V$ we deduce that any complex subalgebra $\mm$ of $(\ct
\hh_1)^{\CC}$ spanned by $$U = e_1 + a_2 e_2 + a_3 e_3 +a_4 e_4 + a_5 e_5 + a_6 e_6,$$ $$V= c_2 e_3+ (c_5 - a_2 c_4 + a_4 c_2)e_6 \qquad W= c_2 e_2 + c_3 e_3 + c_4 e_4 + c_5 e_5 + c_6 e_6,\qquad$$ and such that $U,V, W, \sigma U, \sigma V, \sigma W$ is a basis of $(\ct
\hh_1)^{\CC}$, induces a non abelian complex structure on $\ct \hh_1$. The class of non abelian complex structures $J$ is 2-step nilpotent. Actually the vector $X:= W +\sigma W$ belongs to the center of $\ct \hh_1$ and also $JX\in \zz(\ct \hh_1)$. Since ${\mathfrak{a} }_1(J)=span\{X,JX\}=\zz(\ct
\hh_1)$ and $C(\ct \hh_1)=\zz(\ct \hh_1)$, we conclude that ${\mathfrak{a} }_2(J)=\ct \hh_1$.
After [@Mg] in the set of non abelian complex structures, one has the following non equivalent complex structures (the extension is such that $J^2=-{\rm I}$): $$\label{h1tr}
J_s e_1= e_4\qquad J_s e_2 = -s e_4+e_5 \qquad J_s e_3= 2e_6\qquad s=0,1,$$ which are totally real, and next $$\label{h1ntr}
J_{\nu} e_1= e_2 + (1-\nu) e_4+\frac{1-\nu}\nu e_5\quad J e_2=-\nu e_1 + (1-\nu) e_4\quad J_\nu e_3=e_6\quad \nu \in \RR-\{0\}$$ which are neither abelian nor totally real.
\[tah1\] The tangent Lie algebra $\ct \hh_1$ admits abelian and nonabelian complex structures, which are in every case 2-step nilpotent.
The Lie algebra $\ct \hh_1$ is isomorphic to $\mathcal G_{6,1}$ in [@Mg] and to $\mathfrak h_4$ in [@CFU].
Next we show a family of tangent Lie algebras which cannot be equipped with a complex structure.
\[noct\] The Lie algebras $\ct \rr_3$ and $\ct \rr_{3, \eta}'$ $\eta \geq 0$ do not admit complex structures.
We shall give the complete proof for the tangent Lie algebra $\ct \rr_3$. The proof for $\ct \rr_{3, \eta}'$ can be achieved with a similar reasonning.
Assume that $\mm$ is a complex subalgebra of $\ct \rr_3^{\CC}$ corresponding to a complex structure of $\ct \rr_3$. Let $U, V, W\in \mm$ be linearly independent vectors as in (\[uv\]) such that $\mm= \CC U \ltimes {\mathfrak{a} }$ with ${\mathfrak{a} }=span\{V,W\}$.
Let $\mm$ be a complex subalgebra of $\ct \rr_3^{\CC}$ spanned by vectors $U,V, W$ as in (\[uv\]) and assume that ${\mathfrak{a} }=span\{V,W\}$ is abelian. The Lie bracket relations on $\mm$ are: $$\begin{array}{rcl}
[V,W] & = & (b_4 c_2 - b_2 c_4 + b_4 c_3 - b_3 c_4) e_5 + (b_4 c_3 - b_3 c_4) e_6, \\
\, [U,V] & = & (b_2 + b_3) e_2 + b_3 e_3 + (b_5 + b_6+a_4 b_2 - a_2 b_4 + a_4 b_3 - a_3 b_4) e_5 + \\
& & + (b_6+a_4 b_3 - a_3 b_4) e_6,\\
\, [U,W] & = & (c_2 + c_3) e_2 + c_3 e_3 + (c_5 + c_6+a_4 c_2 - a_2 c_4 + a_4 c_3 - a_3 c_4) e_5 + \\
& & + (c_6+a_4 c_3 - a_3 c_4) e_6.
\end{array}$$ Assume $[V,W]=0$, $[U,V]=\nu V$ and $[U,W]= \mu W$.
We see that $\nu b_4=0$ implies either $\nu=0$ or $b_4=0$.
If $\nu=0$, from $[U,V]=0$ one has $b_2=0=b_3$ and $V$ has the form $V=b_4 (e_4 + a_2 e_5 + a_3 e_6)$.
We also have $\mu c_4 = 0$. If $\mu=0$ then $c_2 =0=c_3$ and so $U,V, \sigma V \sigma W\in span\{e_4, e_5, e_6\}$; therefore this set is not linearly independent. Hence $\mu \neq 0$ and $c_4=0$. But from $[V,W]=0$ it must hold $b_4 c_3=0=b_4 c_2$ and since $b_4\neq 0$ one has $c_2 = 0=c_3$ but then $V, W, \sigma V \sigma W\in span\{e_4, e_5, e_6\}$ which gives no basis of $\ct \rr_3^{\CC}$.
From this contradiction we get that $\nu\neq 0$ and so $b_4=0$.
From $[V,W]=0$ it may hold $b_2 c_4=0=c_4b_3$. If $c_4\neq 0$ then $b_2=0=b_3$. Moreover if $\nu\neq 1$ then $b_5=0=b_6$ which derives in the contradiction $V=0$. Therefore $\nu=1$ implies $b_6=0$ and so $V\in span\{e_5\}$ which does not allow a basis of $\ct \rr_3^{\CC}$. Hence $c_4=0$. If $\nu\neq 1$ then $b_2=0=b_3=b_5=b_6$ thus $V=0$ is a contradiction. Now $\nu=1$ implies $b_3=0$ and $b_6=a_4b_2$ so that $V\in span\{e_2, e_5, e_6\}$.
From $[U,W]=\mu W$ we have that if $\mu\neq 1$ then $c_2=0=c_3$ and in this way $V, W, \sigma V, \sigma W\in span\{e_2, e_5,
e_6\}$ which cannot build a basis of $\ct \rr_3^{\CC}$. Thus $\mu=1$ and so $c_3=0$, and therefore the set $V, W, \sigma V, \sigma
W\in span\{e_2, e_5, e_6\}$ does not induce a basis of $\ct \rr_3^ {\CC}$, implying the non existence of a complex subalgebra $\mm$ in this case.
Assume now $\mm=span\{U,V,W\}$ is a complex subalgebra such that $\ct \rr_3^{\CC}=\mm \oplus \sigma \mm$ and $[V,W]=0$ $[U,V]=\nu V$ $[U,W]=V+\nu W$ for some $\nu \in \CC$.
If $\nu\neq 1$ it follows that $b_2=0=b_3$ which also implies $c_2=0=c_3$. Therefore $V,W, \sigma V, \sigma W\in span\{e_4,e_5,e_6\}$ and we have no complex structure.
Hence $\nu=1$. Thus $b_3=0=b_4=c_4$ and $c_3=b_2$. From $[U,V]=V$ one gets $b_6+a_4b_2=0$ and from $[U,V]=V+W$ one has $b_6-a_4 b_2=0$; therefore $b_6=0=a_4 b_2$. Since $a_4\neq 0$ then $b_2=0$ but this implies $V=b_5 e_5$ and so $V, \sigma V$ are not linearly independent. Hence no complex structure arise in this case.
The following statements are equivalent:
i\) $\ct \rr_{3,\lambda}$ can be endowed with a complex structure;
ii\) $\ct \rr_{3, \lambda}$ carries an abelian complex structure;
iii\) $\lambda=0$.
One proves i) $\Longrightarrow$ iii) $\Longrightarrow$ ii) $\Longrightarrow$ i). It is easy to see ii) $\Longrightarrow$ iii).
We will give a general line for the proof.
Let $\mm$ denote a complex subalgebra of $\ct \rr_{3,\lambda}^{\CC}$ spanned by vectors $U,V, W$ as in (\[uv\]), with $[V,W]=0$.
The Lie brackets follow $$\begin{array}{rcl}
[V,W] & = & (b_4 c_2 - b_2 c_4) e_5 + \lambda (b_4 c_3 - b_3 c_4)e_6 \\
\,[U,V] & = & b_2 e_2 + \lambda b_3 e_3 + b_5 e_5 + \lambda b_6 e_6 +(a_4 b_2-a_2b_4) e_5
+ \lambda (a_4 b_3-a_3b_4) e_6\\
\,[U,W] & = & c_2 e_2 + \lambda c_3 e_3 + c_5 e_5 + \lambda c_6 e_6+ (a_4 c_2-a_2 c_4)
e_5 + \lambda (a_4c-3-a-3c_4) e_6
\end{array}$$
If the action of $U$ on $span\{V,W\}$ is of type (1) in (\[typ1\]), by solving the corresponding system, one gets a basis of $(\ct
\rr_{3, \lambda})^{\CC}$ only if $\lambda =0$, with the additional constraints $\nu=0=\mu$. Explicitly, the vectors adopt the form $$U=e_1 + a_2 e_2 + a_3 e_3+ a_4 e_4+ a_5 e_5 + a_6 e_6,$$ $$\label{uvwa}
V= b_3 e_3 + b_4 e_4 - a_2 b_4 e_5+ b_6 e_6, \qquad W= c_3 e_3 + c_4 e_4 - a_2 c_4 e_5+ c_6 e_6$$ whenever $U,V,W, \sigma U, \sigma V, \sigma W$ is a basis of $(\ct
\rr_{3,0})^{\CC}$. It follows at once that the induced complex structure on $\ct \rr_{3,0}$ is abelian.
If the action of $U$ on $span\{V,W\}\simeq \CC^2$ is of type (2) in (\[typ1\]), then we cannot find a complex structure for any value of $\lambda$. This argument shows i) $\Longrightarrow$ iii).
For ii) $\Longrightarrow$ iii) one works out the equations deriving from $[V, W]=0=[U,V]=[U,W]$ to obtain that a solution exists only for $\lambda=0$. In this case, one gets the vectors $U,V,W$ above (\[uvwa\]). For instance the following $J$ gives rise to a abelian complex structure on $\ct \rr_{3,0}$: $$\label{abaffr}
J e_1 = e_2 \qquad J e_3 = -e_6 \qquad J e_4 = e_5.$$ To prove iii) $\Longrightarrow$ ii) one must solve the equation $[V,W]=0$, $[U,V]=\nu
V$ and $[U,W]=\mu W$ for $\lambda =0$. It is possible to see that the only way to get solutions is for $\nu=\mu=0$, finishing the proof.
Complex structures on cotangent Lie algebras of dimension six
-------------------------------------------------------------
Recall the Lie group counterpart of the cotangent Lie algebra. The zero section in the cotangent bundle $T^*H$ of a Lie group $H$, can be identified with $H$, as well as the fibre over $(e,0)$ with $\hh^*$. As a Lie group, the cotangent bundle of $H$ is the semidirect product of $H$ with $\hh^*$ via the coadjoint representation. The tangent space of $T^*H$ at the identity is naturally identified with the [*cotangent*]{} Lie algebra $\ct^*\hh:= \hh \ltimes_{\coad} \hh^*$, the semidirect product of $\hh$ and its dual $\hh^*$ via the coadjoint action.
\[cot3\] Let $\hh$ be a solvable real Lie algebra of dimension three and let $\ct^* \hh$ denote the cotangent Lie algebra spanned by $e_1, e_2, e_3, e_4, e_5, e_6$. The non zero Lie brackets are listed below: $$\begin{array}{ll}
\ct^*\hh_1: & [e_1,e_2]=e_3, \, [e_1, e_6]=-e_5, [e_2, e_6]=e_4\\
\ct^*\rr_{3}: & [e_1,e_2]=e_2,\, [e_1,e_3]= e_2 + e_3,\, \\
& [e_1, e_5]= -e_5 -e_6,\, [e_1, e_6]=-e_6,\, [e_2, e_5]=e_4,\, [e_3, e_5]=e_4,\, [e_3,e_6]=e_4 \\
\ct^*\rr_{3,\lambda}: & [e_1,e_2]=e_2,\, [e_1,e_3]= \lambda e_3 \\
|\lambda| \leq 1& [e_1, e_5]= -e_5,\, [e_1, e_6]=-\lambda e_6,\, [e_2, e_5]=e_4,
\, [e_3,e_6]=\lambda e_4\\
\ct^* \rr_{3,\eta}': & [e_1,e_2]=\eta e_2- e_3,\, [e_1,e_3]= e_2 + \eta e_3,
[e_1, e_5]= -\eta e_5-e_6,\, \\
\eta \geq 0 & [e_1, e_6]= e_5-\eta e_6,\, [e_2,e_5]=\eta e_4,\, [e_2,e_6]= -e_4,
\, [e_3, e_5]= e_4,\, [e_3,e_6]=\eta e_4\\
\end{array}$$
\[teoct\] Let $\hh$ denote a three dimensional solvable real Lie algebra, if $\ct^*\hh$ admits a complex structure then $\hh$ is isomorphic to one of the following Lie algebras: $\hh_1$, $\RR \times \aff(\RR)$, $\rr_{3,1}$, $\rr_{3,-1}$, $\rr_{3, \eta}'$ for any $\eta \geq 0$.
The following propositions gives the proof of the theorem above.
Let $\hh$ denote a solvable real Lie algebra of dimension three. If $\mm$ is a complex subalgebra of $\ct^*\hh$ such that there exist $V,W\in \mm$ satisfying $[V,W]=W$ then $W=0$.
Assume $\mm$ is a complex subalgebra of $\ct^*\hh$ with $V,W\in \mm$ satisfying $[V,W]=W$. From the Lie bracket relations in (\[cot3\]) one has $W= c_4
e_4$ which implies $W=0$.
Let $\hh$ denote a solvable real Lie algebra of dimension three. If $(\ct^*\hh)^{\CC}$ splits as a direct sum $(\ct^*\hh)^{\CC}=\mm \oplus \sigma \mm$, where $\mm$ is a complex subalgebra and $\sigma$ is the conjugation with respect to $\ct^*\hh$, then $\mm\simeq \CC \ltimes \CC^2$.
For the existence problem of complex structures it remains the study of complex subalgebras $\mm$ such that $\mm
\simeq \CC\ltimes \CC^ 2$, we do below.
\[coth1\] Every complex structure on the Lie algebra $\ct^*\hh_1$ is three step nilpotent.
In (\[exa1\]) we proved that $\ct^*\hh_1$ cannot admit an abelian complex structure. For the sake of completeness we shall see first that $\ct^*\hh_1$ has a complex structure. Since $\ct^*\hh_1$ is nilpotent, if it admits a complex structure, then the corresponding complex subalgebra $\mm$ must be nilpotent, hence $\mm =span\{U,V,W\}$ with $[V,W]=0$, the action of $U$ of type (2) in (\[typ1\]) so that $[U,V]=0$ and $[U,W]=V$ . Explicitly
- $[V,W]=0$ implies $b_2 c_6 - b_6 c_2=0,$
- $[U,V]=0$ and $[U,W]=V$ imply the following systems of equations: $$\begin{array}{rclrcl}
b_2 & = & 0, & -c_6 & = & b_5,\\
a_2 b_6 - a_6 b_2 & = & 0, & c_2 & = & b_3,\\
a_2 c_6 - a_6 c_2 & = & b_4, &\quad b_6 & = & 0.
\end{array}$$
Since these systems have solutions, complex subalgebras of $(\ct^*\hh_1)^{\CC}$ spanned by $U,V,W$ of the form $$U=e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 +a_6 e_6,$$ $$\, V = c_2 e_3 + (a_2 c_6 - a_6 c_2) e_4 - c_6 e_5, \qquad W= c_2 e_2 + c_3 e_3 + c_4 e_4 + c_5 e_5 + c_6 e_6$$ induce complex structures if and only if the vectors $U,V,W, \sigma{U},
\sigma{V}, \sigma{W}$ span a basis of $(\ct^*\hh_1)^{\CC}$.
After [@Mg] there is only one class of complex structures, therefore anyone is equivalent to the complex structure $J$ given by $$\label{h1cs} J e_1 = e_4 \qquad J e_2 = e_6 \qquad Je_5 = e_3.$$
For this complex structure, notice that ${\mathfrak{a} }_0(J)=\{0\}$, ${\mathfrak{a} }_1(J)=span\{e_3, e_5\}$, ${\mathfrak{a} }_2(J)=span\{e_1, e_3, e_4, e_5\}$, ${\mathfrak{a} }_3(J)=\ggo$. Hence from lemma (\[le\]) we conclude that any complex structure on $\ct^*
\hh_1$ is nilpotent.
However notice that there are complex structures which are not totally real, for instance the next one $$\label{h1notr}
Je_1=e_2-e_4 \qquad J e_2 =e_6 \qquad Je_5= e_3+ e_4.$$
This Lie algebra is isomorphic to $\mathcal G_{6,3}$ in [@Mg] and to $\mathfrak h_7$ in [@CFU].
The Lie algebra $\ct^*\rr_3$ cannot be endowed with a complex structure.
Assume $\mm$ is a complex subalgebra of $(\ct^*\rr_3)^{\CC}$ spanned by vectors $U,V,W$ as in (\[uv\]) and such that $[V,W]=0$. Assume $U$ acts on ${\mathfrak{a} }= span\{V,W\}$.
Case 1). We assume the action of $U$ on is of type (1) in (\[typ1\]), then if $[U,V]= \nu V$ and $[U,W]= \mu W$ we have that $U= e_1 + a_2 e_2+a_3e_3+a_4 e_4 + a_5 e_5 +a_6
e_6$, $V= b_2 e_2 + b_4 e_4+ b_6 e_6$, $W= c_2 e_2 + c_4 e_4+ c_6
e_6$. It is not difficult to verify that $U,V,W, \sigma{U},
\sigma{V}, \sigma{W}$ cannot be a basis of $(\ct^*\rr_3)^{\CC}$.
Case 2) Suppose that the action of $U$ on ${\mathfrak{a} }= span\{V,W\}$ is of type (2). From the constraint $[U,V]= \nu V$ and $[U,W]=
V + \nu W$, we get that there is no possibility of choosing $\{U,V,W, \sigma
U, \sigma V, \sigma W\}$ as a linearly independent set in $(\ct^*
\rr_3)^{\CC}$, hence there is no complex structure associated to such $\mm$.
If the Lie algebra $\ct^*\rr_{3, \lambda}$ admits a complex structure then $\lambda =0, 1, -1$.
Let $\mm$ be a complex subalgebra of $(\ct^*\rr_{3,\lambda})^{\CC}$ spanned by the vectors $U,V,W$ as in (\[uv\]) and such that $[V,W]=0$. The following constraint must be satisfied: $$b_2 c_5 - b_5 c_2 + (b_3 c_6 - b_6 c_3) \lambda=0.$$
The Lie brackets on $\ct^*\rr_{3, \lambda}$ follow $$\begin{array}{rcl}
[U,V] & = & b_2 e_2 +\lambda b_3 e_3 + [a_2 b_5-a_5 b_2+\lambda(a_3
b_6-a_6 b_3)] e_4 - b_5 e_5 - \lambda b_6 e_6\\
\, [U,W] & = & c_2 e_2 +\lambda c_3 e_3 + [a_2 c_5 -a_5 c_2 +\lambda(a_3 c_6 -a_6 c_3)]e_4 -c_5
e_5-\lambda c_6 e_6\\
\, [V,W] & = & [b_2 c_5 - b_5 c_2 +\lambda(b_3 c_6- b_6 c_3)] e_4
\end{array}$$
Case 1) Assume the action of $U$ on ${\mathfrak{a} }= span\{V,W\}$ is of type (1) in (\[typ1\]). The conditions $[U,V]= \nu V$ and $[U,W]=\mu W$ shows that a subalgebra $\mm$ exists if $\lambda\in\left\{0,1,-1\right\}$. Moreover, such an $\mm$ is spanned by $U,V,W$ as given in the following table:
$$\begin{array}{|c|l|} \hline
& \quad U = e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6,\\ \quad \lambda=0 \quad
& \quad V =b_3 e_3 + b_4 e_4 + b_6 e_6, \, W=-a_2 c_5 e_4 + c_5 e_5 \quad \mbox{ or } \quad \\
& \quad U, \, V \,\mbox{as above and }\, W=c_2 e_2 -a_5 c_2 e_4 \\ \hline
& \quad U = e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6,\, \\
& \quad V =b_2 e_2 +b_3 e_3 - (a_5 b_2 + a_6 b_3) e_4, \, \\ \lambda=1 & \quad W=-\frac{c_5}{b_3}
(a_2 b_3-a_3b_2)+ c_5 e_5 - \frac{b_2c_5}{b_3} e_6\quad \mbox{ with }\quad b_3 \neq 0 \quad \mbox{ or } \\
& \quad U \quad \mbox{as above and }\, \\ & \quad V= -(a_2 b_5 + a_3 b_6) e_4 + b_5 e_5 + b_6 e_6,\,\\
& \quad W=c_2 e_2 -\frac{b_5 c_2}{b_6} e_3 -\frac{c_2}{b_6}(a_5 b_6 -a_6 b_5)\quad \mbox{ with }
\quad b_6 \neq 0 \\ \hline
& \quad U = e_1 + a_2 e_2 + a_3 e_3 + a_4 e_4 + a_5 e_5 + a_6 e_6,\, \\ & \quad V
=-\frac{c_3b_6}{c_5} e_2 -\frac{b_6}{c_5} (a_5 c_3 - a_6 c_5) e_4 + b_6 e_6 \, \\ \lambda=-1 &
\quad W=c_3 e_3 -(a_2 c_5+a_6c_3) \quad \mbox{ with }\quad c_5 \neq 0 \quad \mbox{ or } \\
& \quad U \quad \mbox{as above and }\, \\ & \quad V= b_3 e_3 -(a_2 b_5 + a_6 b_3) e_4 + b_5 e_5,\,\\
& \quad W=c_2 e_2 -\frac{c_2}{b_3}(a_3 b_5 -a_5 b_3)e_4-\frac{b_5 c_2}{b_3}e_6
\quad \mbox{ with } \quad b_3 \neq 0 \\ \hline
\end{array}$$
In all cases, $U,V, W, \sigma U, \sigma V, \sigma W$ turn out to be a basis of $(\ct^* \rr_{3,\lambda})^{\CC}$. We also observe that none of these complex structure is abelian.
For instance, the linear map on $\ct^*\rr_{3,0}$ given by $$\label{cl0tr}
Je_1=e_5\qquad Je_2=-e_4 \qquad Je_3=e_6$$ and such that $J^2=-{\rm I}$ defines a totally real complex structure on the cotangent Lie algebra $\ct^*\rr_{3,0}$, while the $J$ taken as $$\label{cl0notr}
Je_1=e_2\qquad Je_4=e_5 \qquad Je_3=e_6$$ gives rise to a complex structure which is not totally real.
For $\lambda=-1$ the linear homomorphism such that $J^2=-{\rm I}$ given by $$\label{cl-1tr}
Je_1=e_4 \qquad Je_2=e_6 \qquad Je_3=-e_5$$ gives a totally real complex structure on $\ct^*\rr_{3,-1}$. While the $J$ satisfying $J^2=-{\rm I}$ and $$\label{cl-1notr}
Je_3=-(e_1+e_6)\qquad Je_5=e_3-e_4 \qquad Je_6=-(e_2+e_4)$$ induces a non totally real complex structure on $\ct^*\rr_{3,-1}$.
Now for $\lambda=1$ no complex structure on $\ct^*\rr_{3,1}$ is totally real. For instance $$\label{cl1}
Je_1=e_4\qquad Je_2=e_3 \qquad Je_5=e_6$$ is a complex structure on $\ct^*\rr_{3,1}$.
The Lie algebra $\ct^*\rr_{3, \eta}'$ carries a complex structure for any $\eta \geq 0$.
The linear isomorphisms $J$ such that $J^2=-{\rm I}$ given by $$\label{cseta}
J e_1 = \pm e_4 \qquad Je_2 = e_3\qquad J e_5=e_6$$ define complex structures on $\ct^*\rr_{3, \eta}'$ for any $\eta\geq 0$.
Notice that on $\ct^*\rr_{3,0}'$ one has totally real complex structures (\[trct\]), for instance $$\label{csetatr}
J e_1 = \pm e_4 \qquad Je_2 = e_6\qquad J e_3=-e_5.$$
Complex structures and related geometric structures
===================================================
In these paragraphs we relate complex structures to some geometric structures. We are mainly interested on hermitian structures, symplectic and Kähler structures.
On Hermitian complex structures.
--------------------------------
A [*metric*]{} on a Lie algebra $\ggo$ is a non degenerate symmetric bilinear map, $\la \, , \, \ra:\ggo \times \ggo \to \RR$. It is called [*ad-invariant*]{} if $$\la [x,y],z\ra + \la y, [x,z]\ra=0 \qquad \qquad \mbox{for all }x,y \in \ggo.$$
\[mecot\] The canonical metric on a cotangent Lie algebra $\ct^*\hh$ is defined by $$\la (x,\varphi), (x', \varphi')\ra=\varphi'(x)+\varphi(x')\qquad \mbox{ for all }x,x'\in \hh,\, \varphi, \varphi'\in\hh^*.$$ It is neutral and ad-invariant.
A subspace $\ww \subseteq (\ggo, \la \,,\,\ra)$ is called [*isotropic*]{} if $\la x, y\ra=0$ for all $x,y \in \ww$, that is, if $\ww \subseteq \ww^{\perp}$, where $$\ww^{\perp}=\{ y\in \ggo \, \mbox{ such that }\, \la x,y \ra =0 \mbox{ for all } x\in \ww\},$$ furthermore $\ww$ is called [*totally isotropic*]{} whenever $\ww = \ww^{\perp}$.
On $\ct^*\hh$ equipped with its canonical metric, both subspaces $\hh$ and $\hh^*$ are totally isotropic.
Let $(\ggo, \la \, , \, \ra)$ denote a real Lie algebra equipped with a metric. An (almost) complex structure $J$ on $\ggo$ is called [*Hermitian*]{} if $$\label{her}
\la Jx, Jy\ra = \la x, y\ra \qquad \quad \mbox{ for all } x,y\in \ggo.$$
If the metric is positive definite a Hermitian complex structure is also called a orthogonal complex structure.
Notice that if $J$ is Hermitian, then $\la x, Jx\ra=0$ for all $x\in \ggo$. The non degeneracity property of $\la \, , \, \ra$ says that there is $y\in \ggo$ such that $\la x, y\ra \neq 0$. Therefore the subspace $\ww= span\{x, Jx, y, Jy\} \subseteq \ggo$ is non degenerate and $J$-invariant. Furthermore $$\ggo=\ww \oplus \ww^{\perp}$$ is a orthogonal direct sum of $J$-invariant non degenerate subspaces of $\ggo$. A similar argument can be done in the proof of the following lemma.
Let $\ggo$ denote a real Lie algebra endowed with a metric $\la \, ,\, \ra$ and let $J$ be an almost complex structure on $\ggo$. Assume $\vv$ is a totally real and totally isotropic subspace on $\ggo$, then
i\) $\ggo$ admits a decomposition into a direct sum of totally real and totally isotropic vector subspaces $$\ggo=\vv \oplus J\vv;$$
ii\) $\ggo$ splits into a orthogonal direct sum $$\ggo= \ww_1 \oplus \ww_2 \oplus \hdots \oplus \ww_n$$ of $J$-invariant non degenerate subspaces $\ww_1, \hdots, \ww_n$, where $\dim \ww_i\equiv 0$ (mod 4).
Let $\hh$ denote a real Lie algebra, and let $\la\,,\,\ra$ denote a metric on $\ct_{\pi}\hh$ for which $\hh$ is totally isotropic and assume $J$ is a totally real almost complex structure on $\ct_{\pi} \hh$. Then the dimension of $\hh$ is even.
A generalized complex structure on a Lie algebra $\hh$ is a Hermitian complex structure on $(\ct^*\hh, \la \, , \, \ra)$ where $\la \,,\,\ra$ denotes the canonical metric on $\ct^*\hh$.
A Hermitian complex structure $J$ on $\ct^*\hh$ which leaves $\hh$ invariant is called a generalized complex structure of [*complex type*]{} and it corresponds to a complex structure on $\hh$. A Hermitian complex structure $J$ on $\ct^*\hh$ which is totally real, that is $J \hh=\hh^*$, is said a generalized complex structure of [*symplectic type*]{}. It corresponds to a symplectic structure on $\hh$.
Assume $\hh$ is a Lie algebra which is equipped with an ad-invariant metric $(\,,\,)$. In this case it can easily be shown that the adjoint and the coadjoint representations are equivalent. In fact if $\ell:\hh \to \hh^{\ast}$ is the linear isomorphism given by $x \to \ell(x)$ such that $\ell(x)y=(x, y)$, it is straighforward to verify that $\ell^{-1} \coad(x) \ell = \ad(x)$ for all $x\in \hh$.
Results (\[ceq\]) and (\[le22\]) of previous sections imply that totally real complex structures $J$ on $\ct^*\hh$ correspond to non singular derivations of $\hh$. Explicitly, a non singular derivation $d$ on $\hh$ induces the map $\ell \circ d: \hh \to \hh^*$ giving rise to a complex structure on $\ct^*\hh$.
Consider now the canonical neutral metric $\la \, , \,\ra$ on $\ct^{\ast}\hh$ defined by $$\la (x, \ell_y),(x', \ell_{y'})\ra=(x, y') + (x', y).$$ Since $\hh$ and $\hh^*$ are isotropic subspaces for $\la \,, \, \ra$, a complex structure $J$ such that $J\hh=\hh^*$ is Hermitian if and only if $$\la y, Jx\ra =-\la x, J y \ra\qquad \quad \mbox{ for } x,y\in \hh.$$ Now, since $J$ is associated to a non singular map $j:\hh \to \hh^*$, the latter corresponding to a non singular derivation $d$ of $\hh$, we have the linear isomorphism $j:\hh \to \hh^{\ast}$ equals $$\label{jj}
j =\ell \circ d,$$ thus both (\[her\]) and (\[jj\]) imply $$(x, d y) = - (dx, y) \qquad \mbox{ for all } x,y \in \hh$$ this means that the Hermitian complex structures on $\ct^* \hh$, such that $J\hh=\hh^*$ correspond to non singular skew symmetric derivations of $(\hh, (\, , \,))$.
The previous explanations and [@Ja] derive the following.
Let $\hh$ denote an even dimensional Lie algebra endowed with an ad-invariant metric $(\,,\,)$. The following statements are equivalent
i\) $\hh$ admits a generalized complex structure of symplectic type;
ii\) $\hh$ admits a symplectic structure;
iii\) $\hh$ admits a non singular derivation which is skew symmetric for $( \,, \,)$.
Furthermore if any of these structures exists then $\hh$ is nilpotent.
Complex structures and symplectic structures
--------------------------------------------
A [*symplectic structure*]{} on a even dimensional Lie algebra $\ggo$ is a 2-form $\omega \in
\Lambda^2(\ggo^{\ast})$ such that $\omega$ has maximal rank, i.e., $\bigwedge^{\frac{1}{2}\dim \ggo}\omega\neq 0$ and it is closed: $$\label{close}
\omega([x,y], z)+\omega([y,z],x)+\omega(z,[x,y])=0 \qquad\mbox{for all } x,y, z\in \ggo.$$
Let ($\ct_{\pi} \hh=\hh \ltimes V, \omega)$ denote a semidirect product equipped with a symplectic structure. Following [@CLP] we say that $\ct_{\pi} \hh$ is [*lagrangian*]{}, if both $\hh$ and $V$ are lagrangian subspaces relative to $\omega$. We also say that $\omega$ is [*lagrangian symplectic*]{}.
Let $\ct_{\pi} \hh$ denote a generalized tangent Lie algebra, then its dual Lie algebra is the semidirect product $\ct_{\pi^*} \hh := \hh \ltimes_{\pi^*} V^*$, where $\pi^*$ is the dual representation $$(\pi^*(x)a)(u) := -a(\pi(x)(u)) \qquad \qquad x\in \hh, a\in V^*, u\in V.$$ Note that the cotangent Lie algebra $\ct^*\hh$ is the dual of the tangent Lie algebra $\ct\hh$.
Suppose $\ct_{\pi} \hh= \hh \ltimes_{\pi} V$ is a Lie algebra equipped with a totally real complex structure $J$ (that is $J\hh=V$). This enables us to define on $\ct_{\pi^*} \hh:= \hh \ltimes_{\pi^*}V^*$ a two-form $\omega_J$ by $$\omega_J(x + u, y + v) := v(Jx) - u(Jy),\qquad \quad
\mbox{where $x, y$ are in $\hh$ and $u, v$ are in ($J\hh=V)^*$}.$$ Then $\omega_J$ is non-degenerate and symplectic since $J$ is integrable (see [@BD] or [@CLP] for instance). Furthermore the converse is also true, lagrangian symplectic structures on $\ct_{\pi} \hh$ give rise to totally real complex structures on $\ct_{\pi^*} \hh$. Therefore
Totally real complex structures on $\ct_{\pi} \hh=\hh \ltimes_{\pi} V$ are in correspondence to lagrangian symplectic structures on $\ct_{\pi^*} \hh$.
i\) Let $\ct^*\hh$ denote a cotangent Lie algebra. If it admits a lagrangian symplectic structure then $\hh$ is nilpotent.
ii\) The tangent Lie algebra $\ct \hh$ admits a lagrangian symplectic structure for any $\hh$ isomorphic to $\hh_1$, $\rr_{3,-1}$, $\rr_{3,0}'$ or $\RR\times \aff(\RR)$.
If $\ggo$ is a Lie algebra which carries a symplectic structure and a complex structure, one searches for a compatible pair $(\omega, J)$, called a Kähler structure, $$\label{kp}
\omega(Jx,Jy)=\omega(x,y) \qquad \qquad \mbox{ for all } x,y \in \ggo.$$
Let $\psi\in Aut(\ggo)$ denote a automorphism of $\ggo$, since $\wedge^n \psi^* \omega=\psi^*\wedge^n\omega$ and $\psi^*d\omega=d(\psi^*\omega)$, one has that the existence of a compatible symplectic for a fixed complex structure $J$ is equivalent to the existence of a compatible symplectic structure for every complex structure in the orbit of $J$ under the action of $Aut(\ggo)$.
In fact if $J$ is compatible with $\omega$ and $J'=\psi^{-1}J\psi$ then $\omega'=\psi^*\omega$ is compatible with $J'$: $$\omega'(J'x,J'y)=\omega'(\psi^{-1}J\psi x, \psi^{-1}J\psi y)=\omega(J\psi x, J \psi
y)=\omega(\psi x, \psi y)=\omega'(x,y).$$
\[ek\] Let $\omega$ denote a two form on $\ggo$ which is compatible with the complex structure $J$. Let $\psi\in Aut(\ggo)$ be a automorphism such that $J'=\psi^{-1}J\psi$, then $\psi^*\omega$ is compatible with $J'$.
A [*Kähler Lie algebra*]{} is a triple $(\ggo,J,
\omega)$ consisting of a Lie algebra equipped with a Kähler structure. The Kähler pair $(J, \omega)$ origines a Hermitian structure on $\ggo$ by defining a metric $g$ as $$\label{km}
g(x,y)=\omega(Jx,y) \qquad \qquad \mbox{ for all } x,y \in \ggo.$$
This kind of Hermitian structures satisfies the parallel condition $$\nabla J\equiv 0$$ where $\nabla$ denotes the Levi Civita connection for $g$. The pair $(J,g)$ is called a [*pseudo-Kähler metric*]{} on $\ggo$.
A Lie algebra $\hh$ equipped with an ad-invariant metric $\la \, , \, \ra$ cannot carry a complex structure $J$ which is Hermitian and parallel with respect to the Levi Civita connection of $\la \, ,\, \ra$ (see [@ABO]). However $\hh$ can admit a pseudo Kähler metric if one relaxes the assumption on the metric to be ad-invariant (see (\[psch1\])).
It is our aim to investigate the existence of pseudo Kähler metrics on the Lie algebras $\ct \hh$ and $\ct^*\hh$ where $\hh$ denotes a three dimensional real Lie algebra.
We denote by $e^{ij\hdots}$ the exterior product $e^i\wedge e^j \wedge \hdots$, being $e^1, \hdots, e^6$ the dual basis of $e_1, \hdots ,e_6$.
The following Lie algebras do not carry a symplectic structure:
i\) $\ct^*\rr_{3, \lambda}$ for any $\lambda$.
ii\) $\ct^*\rr_{3,\eta}'$ for any $\eta\geq 0$.
The proof follows along the following lines. Let $\alpha_{ij} \in \RR$ be arbitrary constants and define the generic 2-form on $\ct^*\hh$ $$\label{2f}
\theta=\sum_{i<j} \alpha_{ij} e^{ij} \qquad \qquad i=1,\hdots, 5.$$
If one requires $\theta$ to be closed, the condition $d\theta=0$ generates a system depending on the parameters $\alpha_{ij}$.
We examplify here one case. The Maurer-Cartan equations on $\ct^*\rr_{3,\lambda}$ are given by $$\begin{array}{rclrclrcl}
de^1 & = & 0 & de^2 & = & e^{12} & de^3 & = & \lambda e^{13}\\
de^4 & = & e^{25}+\lambda e^{36} & de^5 & = & -e^{15} & de^6 & = & -\lambda e^{16}
\end{array}$$ By the expansion of this expression making use of $de^{ij}=de^i \wedge
e^j-e^i\wedge de^j$ one gets conditions on the parameters $\alpha_{ij}$. For instance in the case of the Lie algebras considered in i), one obtains that $\alpha_{1j}=0$ for all $j=1, \hdots
,6$, therefore a closed 2-form $\theta$ belongs to $\Lambda^2 \vv^*$ being $\vv=span\{e_1,
\hdots, e_5\}$, in this implies that $\theta$ cannot be symplectic.
A similar reasonning applies on $\ct^*\rr_{3,\eta}'$.
On $\ct \hh_1$ if a two form $\theta$ as in (\[2f\]) is closed, the constrains may satisfy $$\label{cc1}
0=\alpha_{36}=\alpha_{46}=\alpha_{56}\qquad 0=\alpha_{34}+\alpha_{16}=
\alpha_{35}+\alpha_{26}.$$ The compatibility condition with the abelian complex structure $J$ given by $Je_1=e_2$ $Je_3=-e_6$ $Je_4=e_5$ (\[h1abe\]) implies $$\label{cc2}
\alpha_{34}=\alpha_{56}\qquad \alpha_{35}=-\alpha_{46}.$$ Now (\[cc1\]) and (\[cc2\]) amounts to $\alpha_{i6}=0$ for i=1,2,3,4,5, therefore a closed two form cannot be symplectic. According to results in [@Mg] any abelian complex structure is equivalent to the previous one. Following a similar argument but now searching for the compatibility condition between $\theta$ and the totally real complex structure $J_s$ (\[h1tr\]), with s=0,1, one obtains that any such a two form cannot be symplectic.
Consider now the complex structure $J$ given by $$\label{h1ka}
Je_1= 2 e_4\qquad Je_2=-e_5\qquad Je_3=e_6$$ this is totally real and it is compatible with the following closed two forms $$\theta = a(e^{45}-2e^{12})+be^{14}+c(e^{24}-2e^{15})+d e^{25}+ e(e^{26}+e^{35})+fe^{36}.$$ For instance the following two forms give rise to Kähler pairs $$\label{ka1}
\omega= e^{45}-2e^{12}+ \mu e^{36}\qquad \qquad \mu \neq 0,$$ $$\label{ka2}
\omega=e^{14} + \nu(e^{26}-e^{35}) \qquad \qquad \nu \neq 0.$$
This together with (\[ek\]) prove the next result.
\[psth1\] The Lie algebra $\ct \hh_1$ carries several Kähler structures. However no abelian structure admits a compatible Kähler pair.
Next we see that the cotangent Lie algebra $\ct^*\hh_1$ possesses several Kähler pairs $(J, \theta)$. Let $\theta$ denote a closed two form on $\ct^*\hh_1$, thus $$\theta=\sum_{i<j} \alpha_{ij} e^{ij}\qquad \qquad
\alpha_{34}=0=\alpha_{35}=\alpha_{45},\quad \alpha_{36}=\alpha_{14}+\alpha_{25}.$$
Consider the complex structure on $\ct^*\hh_1$ given by $Je_1=e_4$ $Je_2=e_6$ $Je_3=-e_5$ (\[h1cs\]). The compatibility condition between $J$ and the closed two form $\theta$ implies the following conditions on the contrains $\alpha_{ij}$: $$\begin{array}{rclrclrcl}
\alpha_{12}& = & \alpha_{46} \quad & \alpha_{13} & = & -\alpha_{45} \quad & \alpha_{15} &= &-\alpha_{34} \\
\alpha_{16}& = & \alpha_{24} & \alpha_{23} & = & \alpha_{56} & \alpha_{25} &= &-\alpha_{36}
\end{array}$$ therefore any two form on $\ggo$ which is compatible with $J$ has the form $$\omega=a(e^{12}+e^{46}) + b(2e^{14}-e^{25}+e^{36})+c(e^{16}+e^{24})+d(e^{23}+e^{56})+e e^{26}+f e^{35}.$$
\[psch1\] The free 2-step nilpotent Lie algebra in three generators $\ct^*\hh_1$ admits several Kähler structures.
\[kaff\] The tangent Lie algebra $\ct (\RR\times\aff(\RR))$ carries several Kähler structures.
Let $J$ denote the complex structure on $\ct \rr_{3,0}$ given by $Je_1=e_2$, $Je_3=-e_6$, $Je_4=e_5$ (\[abaffr\]). Canonical computations show that this complex structure $J$ is compatible with the symplectic forms $$\label{oabg}
\omega_{\alpha,\beta,\gamma}=\alpha e^{12} +\beta(e^{15}-e^{24})+\gamma e^{36}\qquad
\qquad \beta\gamma\neq 0$$
therefore the pairs $(J, \omega_{\alpha,\beta,\gamma})$ amount to Kähler pairs on $\ct \rr_{3,0}$.
In view of explanations before, the proof of the theorem below is straighforward.
Let $\hh$ denote a real Lie algebra of dimension three.
i\) If $\hh$ is solvable and $\ct\hh$ admits a complex structure then it carries a Kähler structure.
ii\) If $\ct^*\hh$ carries a Kähler structure then $\hh$ is nilpotent.
On the geometry of some pseudo Kähler homogeneous manifolds
-----------------------------------------------------------
The goal is the study of some geometric features on the homogeneous manifolds arising in the previous paragraphs in Lemmas (\[psth1\]) (\[psch1\]) and (\[kaff\]).
In particular we shall see that in the nilpotent case there are flat and non flat metrics. It was already proved in [@FPS] that pseudo-Kähler metrics on nilmanifolds are Ricci flat.
Let $G_1$ denotes the simply connected Lie group whose Lie algebra is $\ct \hh_1$. Its underlying differentiable manifold is $\RR^6$ together with the multiplication $$\begin{array}{rcl}
(x_1,x_2,x_3,x_4,x_5,x_6) \cdot (y_1,y_2,y_3,y_4,y_5,y_6) & = & (x_1+y_1,x_2+y_2,\\
& & x_3+y_3+\frac12 (x_1y_2-x_2y_1),\\
& & x_4+y_4+\frac12 (x_2y_6-x_6y_2),\\
& & x_5+y_5 + \frac12 (x_6y_1-x_1y_6), x_6+y_6).
\end{array}$$ The left invariant vector fields at $Y=(y_1,y_2,y_3,y_4,y_5,y_6)\in G_1$ are $$e_1(Y) = \frac{\partial}{\partial x_1} - \frac12 y_2\frac{\partial}{\partial x_3} +\frac12 y_5 \frac{\partial}{\partial x_6} \quad
e_2(Y) = \frac{\partial}{\partial x_2} + \frac12 y_1\frac{\partial}{\partial x_3} + \frac12 y_4 \frac{\partial}{\partial x_6} \quad
e_3(Y) = \frac{\partial}{\partial x_3}$$ $$e_4(Y)=\frac{\partial}{\partial x_4}-\frac12 y_2 \frac{\partial}{\partial x_6} \qquad e_5(Y)=\frac{\partial}{\partial x_5}+\frac12 y_1 \frac{\partial}{\partial x_6}\qquad e_6(Y)= \frac{\partial}{\partial x_6}.$$
They satisfy the Lie bracket relations of $\ct \hh_1$. Let $e^i$ denote the dual basis of $e_i$ for i=1,2,3,4,5,6, and let $\cdot$ denote the symmetric product. The pseudo-Kähler metric for (\[ka1\]) is $$\label{pka1}
g_{\mu}= 2e^1\cdot e^5+e^2\cdot e^4 +\mu(e^3\cdot e^3+e^6\cdot e^6) \qquad \mu\neq 0$$ while for (\[ka2\]) is $$\label{pka2}
g_{\nu}= 2e^1\cdot e^1+ \frac12 e^4 \cdot e^4 +\nu(e^2\cdot e^3+e^5\cdot e^6)\qquad \nu\neq 0.$$
Let $\nabla^{\mu}$ and $\nabla^{\nu}$ denote the corresponding Levi Civita connections for $g_{\mu}$ and $g_{\nu}$ respectively. From the Koszul formula, for $X=x_i e_i$ one gets $$\nabla^{\mu}_X=\frac14 \left(\begin{matrix}
-\mu x_6 & 0 & 0 & 0 & 0 & -\mu x_1\\
0 & 2 \mu x_6 & 0 & 0 & 0 & 2\mu x_2\\
-2 x_2 & 2 x_1 & 0 & 0 & 0 & 0\\
-2\mu x_3 & 0 & -2\mu x_1 & -2\mu x_6 & 0 & -2\mu x_4\\
0 & \mu x_3 & \mu x_2 & 0 & \mu x_6 & \mu x_5\\
-2 x_5 & 2 x_4 & 0 & -2x_2 & 2 x_1 & 0
\end{matrix}
\right)$$ $$\nabla^{\nu}_X=\frac12 \left(\begin{matrix}
0 & x_2 & 0 & 0 & \nu x_5 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
- 2x_2 & 0 & 0 & -x_5 & -x_4 & 0\\
0 & 2\nu x_5 & 0 & 0 & 2\nu x_2 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
-2 x_5 & x_4 & 0 & -x_2 & 0 & 0
\end{matrix}
\right)$$ Clearly the Lie subgroup $H_1$ of $G_1$ with Lie algebra $\hh_1\subset \ct \hh_1$ is totally geodesic for $g_{\nu}$ for every $\nu$ but it is not totally geodesic for $g_{\mu}$ for any $\mu$.
Let $R(X,Y):=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ denote the curvature tensor, for $\nabla$ either the Levi Civita connection $\nabla^{\mu}$ or $\nabla^{\nu}$ . Notice that $\nabla^{\nu}_{[X,Y]}\equiv 0$ for all $X,Y$.
By computing them one can verify:
$\bullet$ The pseudo Kähler metrics $g_{\mu}$ are non flat.
$\bullet$ A pseudo Kähler metric $g_{\nu}$ is flat if $\nu=1$.
The Lie algebra $\ct \hh_1$ carry flat and non flat pseudo Kähler metrics.
The Lie algebra $\ct^*\hh_1$ is the free 2-step nilpotent Lie algebra in three generators. Its simply connected Lie group $G_2$, lies on $\RR^6$ with the multiplication given by $$\begin{array}{rcl}
(x_1,x_2,x_3,x_4,x_5,x_6) \cdot (y_1,y_2,y_3,y_4,y_5,y_6) & = & (x_1+y_1,x_2+y_2,\\
& & x_3+y_3+\frac12 (x_1y_2-x_2y_1),\\
& & x_4+y_4+\frac12 (x_2y_6-x_6y_2),\\
& & x_5+y_5 + \frac12 (x_6y_1-x_1y_6), x_6+y_6).
\end{array}$$
The left invariant vector fields at $Y=(y_1,y_2,y_3,y_4,y_5,y_6)$ are $$e_1(Y) = \frac{\partial}{\partial x_1} - \frac12 y_2\frac{\partial}{\partial x_3} +\frac12 y_6 \frac{\partial}{\partial x_5} \quad
e_2(Y) = \frac{\partial}{\partial x_2} + \frac12 y_1\frac{\partial}{\partial x_3} - \frac12 y_6 \frac{\partial}{\partial x_4} \quad
e_3(Y) = \frac{\partial}{\partial x_3}$$ $$e_4(Y)=\frac{\partial}{\partial x_4}\qquad e_5(Y)=\frac{\partial}{\partial x_5}\qquad e_6(Y)= \frac12 y_2 \frac{\partial}{\partial x_4}-\frac12 y_1 \frac{\partial}{\partial x_5} +\frac{\partial}{\partial x_6},$$ and let $e^i$ denote the dual left invariant 1-forms for i=1,2,3,4,5,6. Consider the metric on $G$ given by $$g= 2 e^1 \cdot e^1 + e^2\cdot e^3 + 2 e^4\cdot e^4 - e^5\cdot e^6$$ where $\cdot$ denotes the symmetric product. In particular $$\ct^*\hh_1=\hh_1 \oplus J\hh_1$$ denotes a orthogonal direct sum as vector spaces of totally real subalgebras.
The corresponding Levi Civita connection is given by $$\nabla_X=\left( \begin{matrix} 0 & \frac{x_2}2 & 0 & 0 & 0 & -\frac{x_6}2\\
0 & 0 & 0 & 0 & 0 & 0\\
-x_2 & 0 & 0 & 0 & 0 & -x_4\\
0 & 0 & 0 & 0& 0 & x_2\\
x_6 & -x_4 & 0 & x_2 & 0 & 0\\
0 & 0 & 0& 0& 0& 0
\end{matrix}
\right)
\qquad \mbox{ for } X=\sum x_i e_i.$$ One can verify that the Lie subgroup $H_1$ that corresponds to the Lie subalgebra $\hh_1 $, spanned by $e_1,e_2,e_3$, is totally geodesic.
The corresponding curvature tensor $R(X,Y)$ is given by $$R(X,Y)Z=(x_2y_6-x_6y_2)(\frac32 z_6 e_3+\frac12 z_2 e_5).$$
The Lie algebra $\ct^*\hh_1$ admits non flat but Ricci flat pseudo Kähler metrics.
The simply connected Lie group $G_3$ with Lie algebra $\ct \rr_{3,0}$ is, as a manifold, diffeomorphic to $\RR^6$. Let $(x_1, x_2, \hdots, , x_6)$ denote an arbitrary element in $\RR^6$, then the rule multiplication is given by $$\begin{array}{rcl}
(x_1,x_2,x_3,x_4,x_5,x_6) \cdot (y_1,y_2,y_3,y_4,y_5,y_6) & = & (x_1+y_1,x_2+e^{x_6}y_2,
x_3+e^{x_6}y_3+ \\ & & + \frac{e^{x_6}}2 (x_1y_2-x_2y_1), x_4+y_4, x_5+y_5, \\
& & x_6+y_6).
\end{array}$$
The left invariant vector fields at $Y=(y_1,y_2,y_3,y_4,y_5,y_6)\in G_3$ are $$e_1(Y)=\frac{\partial}{\partial x_1} - e^{y_6} y_2\frac{\partial}{\partial x_3}\qquad
e_2(Y)=e^{y_6}(\frac{\partial}{\partial x_2} + y_1\frac{\partial}{\partial x_3})$$ $$e_3(Y)=e^{y_6}\frac{\partial}{\partial x_3}\qquad e_4(Y)=\frac{\partial}{\partial x_4}\qquad e_5(Y)=\frac{\partial}{\partial x_5}\qquad e_6(Y)=\frac{\partial}{\partial x_6},$$
and let $e^i$ denote the dual 1-forms for i=1,2,3,4,5,6.
Consider the metric $\la \, , \, \ra$ on $G$ for which the vector fields above satisfy the non zero relations $$g = \alpha (e_1\cdot e_1+ e_2\cdot e_2) + \beta (e_1\cdot e_4+ e_2\cdot e_5)+\gamma( e_3\cdot e_3+ e_6\cdot e_6)\quad \beta \gamma \neq 0.$$
This metric is clearly non definite.
The complex structure on $G$ is defined as the linear map $J:T_Y G \to T_YG$ such that $J^2=-{\rm I}$ and $$J e_1=e_2 \qquad Je_3=e_6 \qquad Je_4=e_5.$$
This gives a complex structure on $\RR^6$ which is invariant under the action of the Lie group $G$, the action is induced from the multiplication on $G$. Moreover the complex structure is Hermitian for the metric above and it is parallel for the corresponding Levi Civita connection $\nabla$, which on the basis of left invariant vector fields is given by $$\nabla_X=\left( \begin{matrix}
0 & x_2 & 0 & 0 & 0 & 0\\
-x_2 & 0 & 0 & 0 & 0 & 0\\
0 &0 & 0 & 0 & 0 & 0 \\
0 & x_5 & 0 & 0 & x_2 & 0\\
-x_5 & 0 & 0 & -x_2 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right) \qquad \qquad \mbox{ for } X=\sum_{i=1}^6 x_i e_i.$$
Let $\hh$ denote the involutive distribution spanned by $e_2,e_6,e_5$, then it admits a complementary orthogonal distribution $J\hh$, therefore $T_YG=\hh\oplus J\hh$ as orthogonal direct sum. At the Lie algebra level, one has the following short exact sequence
$$0 \longrightarrow \hh \longrightarrow \ggo \longrightarrow J\hh \longrightarrow 0.$$
Notice that $\hh$ is a abelian ideal while $J\hh$ is a abelian subalgebra. Moreover, [*the complex structure $J$ is totally real*]{} with respect to this decomposition and the representation $\pi$ deriving from the adjoint action satisfies the conditions of Corollary (\[ca\]).
Let $H$ denote the Lie subgroup corresponding to the distribution $\hh $ and $JH$ the Lie subgroup corresponding to $J\hh$, which is totally geodesic. In fact, making use of the formula for $\nabla$, one verifies $$\nabla_X Y\subseteq J\hh, \qquad \nabla_{JX}{JY}\subseteq J\hh,\qquad \mbox{ for } X,Y\in \hh$$ and since $(J,g)$ is Kähler, $\nabla_X JY=J\nabla_X Y$ and $\nabla_{JX} Y=-J\nabla_{JX} JY$ for $X,Y\in \hh$.
The curvature tensor $R$ is given by $$R(x,y)=-\nabla_{[x,y]}$$ which implies that $J H$ is flat.
The Ricci tensor $r$ follows $r(X,Y)=2(x_1y_1 + x_2y_2)$ for $X=\sum x_i e_1$, $Y=\sum y_i e_i$, therefore $G_3$ is neither flat nor Ricci flat.
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[^1]: R.C.S. was partially supported by the research projects MTM2006-09152 of the M.E.C. and CCG07-UCM/ESP-2922 of the C.M.
[^2]: G.O. was partially supported by CONICET, ANPCyT, SECyT-UNC, SCyT-UNR
[^3]: Keywords: Complex structure, totally real, solvable Lie algebra, pseudo Kähler metric.
|
---
abstract: 'Conceptually different from the decoherence program, we present a novel theoretical approach to macroscopic realism and classical physics within quantum theory. It focuses on the limits of observability of quantum effects of macroscopic objects, i.e., on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. First, we demonstrate that for unrestricted measurement accuracy no classical description is possible for arbitrarily large systems. Then we show for a certain time evolution that under coarse-grained measurements not only macrorealism but even the classical Newtonian laws emerge out of the Schrödinger equation and the projection postulate.'
author:
- Johannes Kofler
- '[Č]{}aslav Brukner'
title: 'Classical world arising out of quantum physics under the restriction of coarse-grained measurements'
---
Quantum physics is in conflict with a classical world view both conceptually and mathematically. The assumptions of a genuine classical world—local realism and macroscopic realism—are at variance with quantum mechanical predictions as characterized by the violation of the Bell and Leggett–Garg inequality, respectively [@Bell1964; @Legg1985]. Does this mean that the classical world is substantially different from the quantum world? When and how do physical systems stop to behave quantumly and begin to behave classically? Although questions like these date back to Schrödinger’s famous cat paper [@Schr1935], the opinions in the physics community still differ dramatically. Various views range from the mere experimental difficulty of sufficiently isolating any system from its environment (decoherence) [@Zure1991] to the principal impossibility of superpositions of macroscopically distinct states due to the breakdown of quantum physical laws at some quantum-classical border (collapse models) [@Ghir1986].
Macrorealism is defined by the conjunction of two postulates [@Legg1985]: *Macrorealism per se*: A macroscopic object which has available to it two or more macroscopically distinct states is at any given time in a definite one of those states. *Non-invasive measurability*: It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics. These assumptions allow to derive the Leggett–Garg inequalities.
In this Letter—inspired by the thoughts of Peres on the classical limit [@Pere1995]—we present a novel theoretical approach to macroscopic realism and classical physics *within* quantum theory. We first show that, if consecutive eigenvalues $m$ of a spin component can sufficiently be experimentally resolved, a *Leggett–Garg inequality will be violated for arbitrary spin lengths* $j$ and the violation persists even in the limit of infinitely large spins. This contradicts the naive assumption that the predictions of quantum mechanics reduce to those of classical physics when a system becomes large and was demonstrated for local realism by Garg and Mermin [@Garg1982]. Note that due to the resolution of consecutive eigenvalues one cannot speak about violation of macrorealism. If, however, for a certain time evolution one goes into the limit of large spin lengths but can experimentally only resolve eigenvalues $m$ which are separated by much more than the square root of the spin length (the intrinsic quantum uncertainty), i.e., $\Delta m\gg\!\sqrt{j}%
$, the macroscopically distinct outcomes appear to obey classical (Newtonian) laws. This suggests that *macrorealism and classical laws emerge out of quantum physics under the restriction of coarse-grained measurements*.
While our approach is not at variance with the decoherence program, it differs conceptually from it. It is not dynamical and puts the stress on the limits of observability of quantum effects of macroscopic objects. The term macroscopic throughout the paper is used to denote a system with a high dimensionality rather than a low-dimensional system with a large parameter such as mass or size.
Consider a physical system and a quantity $Q$, which whenever measured is found to take one of the values $\pm1$ only. Further consider a series of runs starting from identical initial conditions such that on the first set of runs $Q$ is measured only at times $t_{1}$ and $t_{2}$, only at $t_{2}$ and $t_{3}$ on the second, at $t_{3}$ and $t_{4}$ on the third, and at $t_{1}$ and $t_{4}$ on the fourth $(0\leq t_{1}<t_{2}<t_{3}<t_{4})$. Introducing temporal correlations $C_{ij}\equiv\langle Q(t_{i})\,Q(t_{j})\rangle$, any macrorealistic theory predicts the Leggett–Garg inequality [@Legg1985]$$K\equiv C_{12}+C_{23}+C_{34}-C_{14}\leq2\,. \label{eq Leggett}%$$ This inequality is violated, e.g., by the precession of a spin-$\frac{1}{2}$ particle with the Hamiltonian $\hat{H}=\frac{1}{2}\,\omega\,\hat{\sigma}_{x}$ with $\omega$ the angular precession frequency and $\hat{\sigma}_{x}$ the Pauli $x$-matrix. (We use units in which the reduced Planck constant is $\hbar=1$). Measuring the spin along the $z$-direction, we obtain the correlations $C_{ij}=\langle\hat{\sigma}_{z}(t_{i})\,\hat{\sigma}_{z}%
(t_{j})\rangle=\cos[\omega(t_{j}\!-\!t_{i})]$. Choosing, e.g., equidistant measurement times with time difference $\Delta t=\pi/4\omega$, ineq. (\[eq Leggett\]) is violated as $K=2\sqrt{2}$, which is understandable since a spin-$\frac{1}{2}$ particle is a genuine quantum object. In contrast, any rotating classical spin vector always satisfies the inequality.
In the following, we show that the Leggett–Garg inequality (\[eq Leggett\]) is violated for arbitrarily large spin lengths $j$. As the first measurement will act as a preparation of the state for the subsequent measurement, the initial state is not decisive and it is sufficient to consider the maximally mixed state$$\hat{\rho}(0)\equiv\dfrac{1}{2j+1}\,%
%TCIMACRO{\dsum \nolimits_{m=-j}^{j}}%
%BeginExpansion
{\displaystyle\sum\nolimits_{m=-j}^{j}}
%EndExpansion
\left\vert m\right\rangle \!\left\langle m\right\vert =\dfrac{\openone}{2j+1}
\label{eq mixed state}%$$ with $\openone$ the identity operator and $\left\vert m\right\rangle $ the $\hat{J}_{z}$ (spin $z$-component) eigenstates. The Hamiltonian be$$\hat{H}=\hat{\mathbf{J}}^{2}/2I+\omega\,\hat{J}_{x}\,, \label{eq Hamiltonian}%$$ where $\hat{\mathbf{J}}$ is the rotor’s total spin vector operator, $\hat
{J}_{x}$ its $x$-component, $I$ the moment of inertia and $\omega$ the angular precession frequency. Here, $\hat{\mathbf{J}}^{2}/2I$ commutes with the individual spin components and does not contribute to the time evolution. The solution of the Schrödinger equation produces a rotation about the $x$-axis, represented by the time evolution operator $\hat{U}_{t}=\;$e$^{-\text{i}\omega
t\hat{J}_{x}}$ with $t$ the time. We define the parity measurement $\hat
{Q}\equiv%
%TCIMACRO{\tsum \nolimits_{m=-j}^{j}}%
%BeginExpansion
{\textstyle\sum\nolimits_{m=-j}^{j}}
%EndExpansion
(-1)^{j-m}\left\vert m\right\rangle \!\left\langle m\right\vert =$e$^{\text{i}\pi(j-\hat{J}_{z})}$ with possible dichotomic outcomes $\pm$ (identifying $\pm\equiv\pm1$). The correlation function between results of the parity measurement $\hat{Q}$ at different times $t_{1}$ and $t_{2}$ is $C_{12}\equiv p_{+}\,q_{+|+}+p_{-}\,q_{-|-}-p_{+}\,q_{-|+}-p_{-}\,q_{+|-}$, where $p_{+}$ ($p_{-}$) is the probability for measuring $+$ ($-$) at $t_{1}$ and $q_{l|k}$ is the probability for measuring $l$ at $t_{2}$ given that $k$ was measured at $t_{1}$ ($k,l=+,-$). Furthermore, $p_{+}=1-p_{-}=\frac{1}%
{2}\,(\langle\hat{Q}_{t_{1}}\rangle+1)$, $q_{+|\pm}=1-q_{-|\pm}=\frac{1}%
{2}\,(\langle\hat{Q}_{t_{2}}\rangle_{\pm}+1)$. Here $\langle\hat{Q}_{t_{1}%
}\rangle$ is the expectation value of $\hat{Q}$ at $t_{1}$ and $\langle\hat
{Q}_{t_{2}}\rangle_{\pm}$ is the expectation value of $\hat{Q}$ at $t_{2}$ given the outcome $\pm$ at $t_{1}$.
Using $\hat{\rho}(t_{1})=\hat{U}_{t_{1}}\,\hat{\rho}(0)\,\hat{U}_{t_{1}%
}^{\dagger}=\hat{\rho}(0)$, we find $\langle\hat{Q}_{t_{1}}\rangle\equiv
\;$Tr$[\hat{\rho}(t_{1})\,\hat{Q}]\approx0$. The approximate sign is accurate for half integer $j$ and in the macroscopic limit $j\gg1$, which is assumed from now on. Hence, as expected we have $p_{+}=\tfrac{1}{2}$. Depending on the measurement result at $t_{1}$, the state is reduced to $\hat{\rho}_{\pm}%
(t_{1})\equiv\hat{P}_{\pm}\,\hat{\rho}(t_{1})\,\hat{P}_{\pm}/$Tr$[\hat{P}%
_{\pm}\,\hat{\rho}(t_{1})\,\hat{P}_{\pm}]=(\openone\pm\hat{Q})/(2j+1)$ with $\hat{P}_{\pm}\equiv\tfrac{1}{2}(\openone\pm\hat{Q})$ the projection operator onto positive (negative) parity states. Denoting $\Delta t\equiv t_{2}-t_{1}$ and $\theta\equiv\omega\,\Delta t$, we obtain $\langle\hat{Q}_{t_{2}}%
\rangle_{\pm}\equiv\;$Tr$[\hat{U}_{\Delta t}\,\hat{\rho}_{\pm}(t_{1})\,\hat
{U}_{\Delta t}^{\dagger}\,\hat{Q}]=\pm$Tr$[$e$^{2\text{i}\theta\hat{J}_{x}%
}]/(2j+1)=\pm\sin[(2j+1)\,\omega\,\Delta t]/(2j+1)\sin[\omega\,\Delta t]$.
From $\langle\hat{Q}_{t_{2}}\rangle_{+}=-\langle\hat{Q}_{t_{2}}\rangle_{-}$ it follows $q_{+|+}+q_{+|-}=1$. Using this and $p_{+}=\tfrac{1}{2}$, the temporal correlation becomes $C_{12}=\left\langle \!\right. \hat{Q}_{t_{2}}\left.
\!\right\rangle _{+}$. With equidistant times, time distance $\Delta t$, and the abbreviation $x\equiv(2j+1)\,\omega\,\Delta t$ the Leggett–Garg inequality (\[eq Leggett\]) reads$$K\approx\dfrac{3\sin x}{x}-\dfrac{\sin3x}{3x}\leq2\,. \label{eq Leggett K(x)}%$$ The sine function in the denominator was approximated, assuming $\tfrac
{x}{2j+1}\ll1$. Inequality (\[eq Leggett K(x)\]) is violated for all positive $x\lesssim1.656$ and maximally violated for $x\approx1.054$ where $K\approx2.481$ (compare with Ref. [@Pere1995] for the violation of local realism). *We can conclude that a violation of the Leggett–Garg inequality is possible for arbitrarily high-dimensional systems and also for totally mixed states*, given that consecutive values of $m$ can be resolved.
In the second part of the paper we will show that inaccurate measurements not only lead to validity of macrorealism but even to the *emergence of classical physics*.
In quantum theory any two different eigenvalues $m_{1}$ and $m_{2}$ in a measurement of a spin’s $z$-component correspond to orthogonal states *without any concept of closeness or distance*. The terms close or distant only make sense in a classical context, where those eigenvalues are treated as close which correspond to *neighboring* outcomes in the real configuration space. For example, the eigenvalue labels $m$ and $m\!+\!1$ of a spin observable correspond to neighboring outcomes in a Stern-Gerlach experiment. (Such observables are sometimes called classical or reasonable [@Yaff1982; @Pere1995].) It is those neighboring eigenvalues which we conflate to coarse-grained observables in measurements of limited accuracy. It seems thus unavoidable that certain features of classicality have to be assumed beforehand.
In what follows we will first consider the special case of a single spin coherent state and then generalize the transition to classicality for arbitrary states. *Spin-*$j$ *coherent states* $\left\vert
\vartheta,\varphi\right\rangle $ [@Radc1971] are the eigenstates with maximal eigenvalue of a spin operator pointing into the ($\vartheta,\varphi
$)-direction, where $\vartheta$ and $\varphi$ are the polar and azimuthal angle, respectively: $\hat{\mathbf{J}}\left\vert \vartheta,\varphi
\right\rangle =j\left\vert \vartheta,\varphi\right\rangle $. At time $t=0$ let us consider $|\vartheta_{0},\varphi_{0}\rangle=%
%TCIMACRO{\tsum \nolimits_{m}}%
%BeginExpansion
{\textstyle\sum\nolimits_{m}}
%EndExpansion
\left( \!%
%TCIMACRO{\QTATOP{2j}{j+m}}%
%BeginExpansion
\genfrac{}{}{0pt}{1}{2j}{j+m}%
%EndExpansion
\!\right) \!^{1/2}\cos^{j+m}\!\tfrac{\vartheta_{0}}{2}\,\sin^{j-m}%
\!\tfrac{\vartheta_{0}}{2}\;$e$^{-\text{i}m\varphi_{0}}\,|m\rangle$. Under time evolution $\hat{U}_{t}=\;$e$^{-\text{i}\omega t\hat{J}_{x}}$ the probability that a $\hat{J}_{z}$ measurement at time $t$ has outcome $m$ is $p(m,t)=|\langle m|\vartheta,\varphi\rangle|^{2}$ with $\cos\vartheta
=\sin\omega t\,\sin\vartheta_{0}\,\sin\varphi_{0}+\cos\omega t\,\cos
\vartheta_{0}$, where $\vartheta$ and $\varphi$ are the polar and azimuthal angle of the (rotated) spin coherent state $\left\vert \vartheta
,\varphi\right\rangle $ at time $t$. In the macroscopic limit, $j\gg1$, the binomial can be well approximated by a Gaussian distribution$$p(m,t)\approx\dfrac{1}{\sqrt{2\pi}\,\sigma}\,\text{e}^{-(m-\mu)^{2}%
/2\sigma^{2}} \label{eq p(m1,t1)}%$$ with $\sigma\equiv\sqrt{j/2}\,\sin\vartheta$ the width and $\mu\equiv
j\cos\vartheta$ the mean.
Under the magnifying glass of sharp measurements we can see separate eigenvalues $m$ and resolve the Gaussian probability distribution $p(m,t)$, as shown in Fig. \[fig1\](a). Let us now assume that the *resolution of the measurement apparatus*, $\Delta m$, is finite and subdivides the $2j+1$ possible outcomes $m$ into a smaller number of $\frac{2j+1}{\Delta m}$ coarse-grained slots. If the slot size is much larger than the standard deviation $\sigma\sim\!\sqrt{j}$, i.e., $\Delta m\gg\!\sqrt{j}$, the sharply peaked Gaussian cannot be distinguished anymore from the discrete Kronecker delta,$$\Delta m\gg\!\sqrt{j}:\quad p(m,t)\rightarrow\delta_{\bar{m},\bar{\mu}}\,,
\label{eq Kronecker}%$$ where $\bar{m}$ is numbering the slots (from $-j+\frac{\Delta m}{2}$ to $j-\frac{\Delta m}{2}$ in steps $\Delta m$) and $\bar{\mu}$ is the number of the slot in which the center $\mu$ of the Gaussian lies, as indicated in Fig. \[fig1\](b). In the *limit of infinite dimensionality*, $j\rightarrow\infty$, one can distinguish two cases: (1) If the inaccuracy $\Delta m$ scales linearly with $j$, i.e, $\Delta m=O(j)$, the discreteness remains. (2) If $\Delta m$ scales slower than $j$, i.e., $\Delta m=o(j)$ but still $\Delta m\gg\!\sqrt{j}$, then the slots *seem* to become infinitely *narrow*. Pictorially, the *real space length* of the eigenvalue axis, representing the $2j+1$ possible outcomes $m$, is *limited* in any laboratory, e.g., by the size of the observation screen after a Stern–Gerlach magnet, whereas the number of slots grows with $j/\Delta m$. Then, in the limit $j\rightarrow\infty$, the Kronecker delta becomes the Dirac delta function,$$\Delta m\gg\!\sqrt{j}\;\;\&\;\;j\rightarrow\infty:\quad p(m,t)\rightarrow
\delta(\bar{m}\!-\!\bar{\mu})\,, \label{eq delta}%$$ which is shown in Fig. \[fig1\](c).
![An initial spin-$j$ coherent state $\left\vert
\vartheta_{0},\varphi_{0}\right\rangle $ precesses into the coherent state $\left\vert \vartheta,\varphi\right\rangle $ at time $t$ under a quantum time evolution. (a) The probability $p(m,t)$ for the outcome $m$ in a measurement of the spin’s $z$-component is given by a Gaussian distribution with width $\sigma$ and mean $\mu$, which can be seen under the magnifying glass of sharp measurements. (b) The measurement resolution $\Delta m$ is finite and subdivides the $2j+1$ possible outcomes into a smaller number of coarse-grained slots. If the measurement accuracy is much poorer than the width $\sigma$, i.e., $\Delta m\gg\!\sqrt{j}%
$, the sharply peaked Gaussian cannot be distinguished anymore from the discrete Kronecker delta $\delta_{\bar{m},\bar{\mu}}$ where $\bar{m}$ is numbering the slots and $\bar{\mu}$ is the slot in which the center $\mu$ of the Gaussian lies. (c) In the limit $j\rightarrow\infty$ the slots *seem* to become infinitely narrow and $\delta_{\bar{m},\bar{\mu}}$ becomes the Dirac delta function $\delta(\bar{m}\!-\!\bar{\mu})$.[]{data-label="fig1"}](figure1.eps)
Under a fuzzy measurement the reduced (projected) state is essentially the state before the measurement. If $\left\vert \vartheta,\varphi\right\rangle $ is centered well inside the slot, the disturbance is exponentially small. Only in the cases where it is close to the border between two slots, the measurement is invasive. Assuming that the measurement times and/or slot positions chosen by the observer are statistically independent of the (initial) position of the coherent state, a disturbance happens merely in the fraction $\sigma/\Delta m\ll1$ of all measurements. This is equivalent to the already assumed condition $\!\sqrt{j}\ll\Delta m$. Therefore, fuzzy measurements of a spin coherent state are largely non-invasive such as in any macrorealistic theory, in particular classical Newtonian physics. Small errors may accumulate over many measurements and eventually there might appear deviations from the classical time evolution. This, however, is unavoidable in any explanation of classicality *gradually* emerging out of quantum theory. To which extent this effect is relevant for our every-day experience is an open issue [^1].
Hence, at the coarse-grained level the physics of the (quantum) spin system can completely be described by a new formalism, utilizing a (classical) *spin vector* $\mathbf{J}$ at time $t=0$, pointing in the ($\vartheta_{0},\varphi_{0}$)-direction with length $J\equiv|\mathbf{J}|=\!\sqrt{j(j\!+\!1)}\approx j$, where $j\gg1$, and a (Hamilton) function$$H=\mathbf{J}^{2}/2I+\omega\,J_{x}\,. \label{eq Hamilton}%$$ At any time the probability that the spin vector’s $z$-component $J\cos\vartheta\approx j\cos\vartheta$ is in slot $\bar{m}$ is given by $\delta_{\bar{m},\bar{\mu}}$, eq. (\[eq Kronecker\]), *as if* the time evolution of the spin components $J_{i}$ ($i=x,y,z$) is given by the Poisson brackets, $\dot{J}_{i}=[J_{i},H]_{\text{PB}}$, and measurements are non-invasive. Only the term $\omega\,J_{x}$ in eq. (\[eq Hamilton\]) governs the time evolution and the solutions correspond to a rotation around the $x$-axis. In the proper continuum limit the spin vector at time $t$ points in the ($\vartheta,\varphi$)-direction where $\vartheta$ and $\varphi$ are the same as for the spin coherent state and the prediction is given by $\delta(\bar{m}\!-\!\bar{\mu})$, eq. (\[eq delta\]). *This is classical (Newtonian) mechanics*.
Finally, we show that the time evolution of *any* spin-$j$ quantum state becomes classical under the restriction of coarse-grained measurements. At all times *any* (pure or mixed) spin-$j$ density matrix can be written in the diagonal form [@Arec1972]$$\hat{\rho}=%
%TCIMACRO{\diint \nolimits_{\Omega}}%
%BeginExpansion
{\displaystyle\iint\nolimits_{\Omega}}
%EndExpansion
\,f(\vartheta,\varphi)\,|\vartheta,\varphi\rangle\langle\vartheta
,\varphi|\,\text{d}^{2}\Omega\label{eq rho}%$$ with d$^{2}\Omega\equiv\sin\vartheta\,$d$\vartheta\,$d$\varphi$ the infinitesimal solid angle element and $f(\vartheta,\varphi)$—usually known as $P$-function—a *not necessarily positive* real function (normalization $%
%TCIMACRO{\tiint \nolimits_{\Omega}}%
%BeginExpansion
{\textstyle\iint\nolimits_{\Omega}}
%EndExpansion
\,f(\vartheta,\varphi)\,$d$^{2}\Omega=1$).
The probability for an outcome $m$ in a $\hat{J}_{z}$ measurement in the state (\[eq rho\]) is $P(m)=%
%TCIMACRO{\tiint \nolimits_{\Omega}}%
%BeginExpansion
{\textstyle\iint\nolimits_{\Omega}}
%EndExpansion
\,f(\vartheta,\varphi)\,p(m)\,$d$^{2}\Omega$, where $p(m)$ is given by eq. (\[eq p(m1,t1)\]). At the coarse-grained level of classical physics only the probability for a slot outcome $\bar{m}$ can be measured, i.e., $\bar
{P}(\bar{m})\equiv%
%TCIMACRO{\tsum \nolimits_{m\in\{\bar{m}\}}}%
%BeginExpansion
{\textstyle\sum\nolimits_{m\in\{\bar{m}\}}}
%EndExpansion
P(m)$ with $\{\bar{m}\}$ the set of all $m$ belonging to $\bar{m}$. For $\Delta m\gg\!\sqrt{j}$ and large $j$ this can be well approximated by $$\bar{P}(\bar{m})\approx%
%TCIMACRO{\dint \nolimits_{0}^{2\pi}}%
%BeginExpansion
{\displaystyle\int\nolimits_{0}^{2\pi}}
%EndExpansion
\!%
%TCIMACRO{\dint \nolimits_{\vartheta_{1}(\bar{m})}^{\vartheta_{2}(\bar{m})}}%
%BeginExpansion
{\displaystyle\int\nolimits_{\vartheta_{1}(\bar{m})}^{\vartheta_{2}(\bar{m})}}
%EndExpansion
f(\vartheta,\varphi)\sin\vartheta\,\text{d}\vartheta\,\text{d}\varphi
\,,\label{eq P(m) f}%$$ where $\vartheta_{1}(\bar{m})$, $\vartheta_{2}(\bar{m})$ are the borders of the polar angle region corresponding to a projection onto $\bar{m}$. We will show that $\bar{P}(\bar{m})$ can be obtained from a positive probability distribution of classical spin vectors. Consider the function$$g(\vartheta,\varphi)\equiv\frac{2j+1}{4\,\pi}\,%
%TCIMACRO{\diint \nolimits_{\Omega^{\prime}}}%
%BeginExpansion
{\displaystyle\iint\nolimits_{\Omega^{\prime}}}
%EndExpansion
\,f(\vartheta^{\prime},\varphi^{\prime})\,\cos^{4j}\!\tfrac{\Theta}%
{2}\,\,\text{d}^{2}\Omega^{\prime}\label{eq g}%$$ with d$^{2}\Omega^{\prime}\equiv\sin\vartheta^{\prime}\,$d$\vartheta^{\prime
}\,$d$\varphi^{\prime}$ and $\Theta$ the angle between the directions $(\vartheta,\varphi)$ and $(\vartheta^{\prime},\varphi^{\prime})$. The distribution $g(\vartheta,\varphi)$ is *positive* (and normalized) because it is, up to a normalization factor, the expectation value Tr$[\hat{\rho}\,|\vartheta,\varphi\rangle\langle\vartheta,\varphi
|]=\langle\vartheta,\varphi|\,\hat{\rho}\,|\vartheta,\varphi\rangle$ of the state $\left\vert \vartheta,\varphi\right\rangle $. It is usually known as the $Q$-function [@Agar1981].
For fuzzy measurements with inaccuracy $\Delta\Theta\sim\vartheta_{2}(\bar
{m})-\vartheta_{1}(\bar{m})\gg1/\!\sqrt{j}$, which is equivalent to $\Delta
m\gg\!\sqrt{j}$, the probability for having an outcome $\bar{m}$ can now be expressed only in terms of the positive distribution $g$:$$\bar{P}(\bar{m})\approx%
%TCIMACRO{\dint \nolimits_{0}^{2\pi}}%
%BeginExpansion
{\displaystyle\int\nolimits_{0}^{2\pi}}
%EndExpansion
\!%
%TCIMACRO{\dint \nolimits_{\vartheta_{1}(\bar{m})}^{\vartheta_{2}(\bar{m})}}%
%BeginExpansion
{\displaystyle\int\nolimits_{\vartheta_{1}(\bar{m})}^{\vartheta_{2}(\bar{m})}}
%EndExpansion
g(\vartheta,\varphi)\sin\vartheta\,\text{d}\vartheta\,\text{d}\varphi\,.
\label{eq P(m) g}%$$ The approximate equivalence of eqs. (\[eq P(m) f\]) and (\[eq P(m) g\]) is shown by substituting eq. (\[eq g\]) into (\[eq P(m) g\]). Note, however, that $g$ is a mere mathematical tool and not experimentally accessible. Operationally, because of $\Delta m\gg\!\sqrt{j}$ an averaged version of $g$, denoted as $h$, is used by the experimenter to describe the system in the classical limit. Mathematically, this function $h$ is obtained by integrating $g$ over solid angle elements corresponding to the measurement inaccuracy. Without the magnifying glass the regions given by the experimenter’s resolution become points on the sphere where $h$ is defined. Thus, a *full description* is provided by an *ensemble of classical spins with the probability distribution* $h$.
The *time evolution* of the general state (\[eq rho\]) is determined by (\[eq Hamiltonian\]). In the classical limit it can be described by an ensemble of classical spins characterized by the initial distribution $g$ ($h$), where each spin is rotating according to the Hamilton function (\[eq Hamilton\]). From eq. (\[eq P(m) g\]) one can see that for the non-invasiveness at the classical level *it is the change of the* $g$ *(*$h$*) distribution which is important and not the change of the quantum state or equivalently f itself*. In fact, upon a fuzzy $\hat
{J}_{z}$ measurement the state $\hat{\rho}$ *is* reduced to one particular state, say to $\hat{\rho}_{\bar{m}}$, with the corresponding (normalized) functions $f_{\bar{m}}$, $g_{\bar{m}}$ and $h_{\bar{m}}$. The reduction to $\hat{\rho}_{\bar{m}}$ happens with probability $\bar{P}(\bar
{m})$, which is given by eq. (\[eq P(m) f\]) or (\[eq P(m) g\]). Whereas the $f$-function can change dramatically upon reduction, $g_{\bar{m}}$ is (up to normalization) approximately the same as $g$ in the region $\Omega_{\bar
{m}}$ between two circles of latitude corresponding to the slot $\bar{m}$ and zero outside. If $\hat{Q}_{\bar{m}}\equiv%
%TCIMACRO{\tsum \nolimits_{m\in\{\bar{m}\}}}%
%BeginExpansion
{\textstyle\sum\nolimits_{m\in\{\bar{m}\}}}
%EndExpansion
\left\vert m\right\rangle \!\left\langle m\right\vert $ denotes the projector onto the slot $\bar{m}$, then $\hat{Q}_{\bar{m}}\,|\vartheta,\varphi\rangle$ is *almost* zero ($|\vartheta,\varphi\rangle$) for all coherent states lying outside (inside) $\Omega_{\bar{m}}$. Thus, $g_{\bar{m}}\propto
\langle\vartheta,\varphi|\,\hat{\rho}_{\bar{m}}\,|\vartheta,\varphi
\rangle\propto\langle\vartheta,\varphi|\,\hat{Q}_{\bar{m}}\,\hat{\rho}%
\,\hat{Q}_{\bar{m}}\,|\vartheta,\varphi\rangle\approx\langle\vartheta
,\varphi|\,\hat{\rho}\,|\vartheta,\varphi\rangle\propto g$ inside and almost zero outside. Hence, at the coarse-grained level the distribution $g_{\bar{m}%
}$ ($h_{\bar{m}}$) of the reduced state after the measurement can always be understood approximately as a subensemble of the (classical) distribution $g$ before the measurement. Effectively, the measurement only reveals already existing properties in the mixture and does not alter the subsequent time evolution of the individual classical spins.
The disturbance at that level is quantified by how much $g_{\bar{m}}$ differs from a function which is (up to normalization) $g$ within $\Omega_{\bar{m}}$ and zero outside. One may think of dividing all $g$ distributions into two extreme classes, i.e., the ones which show narrow pronounced regions of size comparable to individual coherent states and the ones which change smoothly over regions larger or comparable to the slot size. The former is highly disturbed but in an extremely rare fraction of all measurements. The latter is disturbed in general in a single measurement but to very small extent, as the weight on the slot borders ($\propto\!\!\sqrt{j}$) is small compared to the weight well inside the slot ($\propto\!\Delta m$). (In the intermediate cases one has a trade-off between these two scenarios.) The typical fraction of these weights is $\!\sqrt{j}/\Delta m\ll1$. Thus, in any case classicality arises with overwhelming statistical weight.
*Conclusion*.—We showed that the time evolution of an arbitrarily large spin cannot be understood classically, as long as consecutive outcomes in a spin component measurement are resolved. For certain Hamiltonians, given the limitation of coarse-grained measurements, not only is macrorealism valid, but even the Newtonian time evolution of an ensemble of classical spins emerges out of a full quantum description of an arbitrary spin state—even for isolated systems. This suggests that classical physics can be seen as implied by quantum mechanics under the restriction of fuzzy measurements.
We thank M. Aspelmeyer, T. Paterek, M. Paternostro and A. Zeilinger for helpful remarks. This work was supported by the Austrian Science Foundation FWF, the European Commission, Project QAP (No. 015846), and the FWF Doctoral Program CoQuS. J. K. is recipient of a DOC fellowship of the Austrian Academy of Sciences.
[99]{}
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[^1]: For the general trade-off between measurement accuracy and state disturbance for more realistic smoothed positive-operator-valued measurements and for related approaches to classicality see Refs. [@Pere1995; @Poul2005].
|
---
abstract: 'Social participatory sensing is a newly proposed paradigm that tries to address the limitations of participatory sensing by leveraging online social networks as an infrastructure. A critical issue in the success of this paradigm is to assure the trustworthiness of contributions provided by participants. In this paper, we propose an application-agnostic reputation framework for social participatory sensing systems. Our framework considers both the quality of contribution and the trustworthiness level of participant within the social network. These two aspects are then combined via a fuzzy inference system to arrive at a final trust rating for a contribution. A reputation score is also calculated for each participant as a resultant of the trust ratings assigned to him. We adopt the utilization of PageRank algorithm as the building block for our reputation module. Extensive simulations demonstrate the efficacy of our framework in achieving high overall trust and assigning accurate reputation scores.'
author:
- Haleh Amintoosi
- |
Salil S. Kanhere\
University of New South Wales, Australia\
\
\
title: Providing Trustworthy Contributions via a Reputation Framework in Social Participatory Sensing Systems
---
Introduction {#intro}
============
Recent advances in mobile technologies have paved the way for a novel paradigm for achieving large-scale city-wide sensing known as Participatory Sensing [@Burke]. In Participatory sensing, the key idea is to recruit ordinary people to contribute in sensor data collection using their mobile phones. This revolutionary paradigm has been operationally used to crowdsource information ranging from personal health [@DietSense] and prices of consumer goods [@PetrolWatch] to environment monitoring [@EarPhone].\
As a crowdsensing platform, a key challenge in the success of participatory sensing is the recruitment of sufficient participants. Typically, participatory sensing campaigns rely on voluntary contributors without any explicit incentives for participation. The lack of adequate motivation may result in few participants which in turn, reduces the data reliability. Another challenge is the suitability of participants particularly for those tasks which require domain-specific knowledge or expertise [@reddyrecruitment].\
To address the aforementioned challenges, one proposed idea is to employ online social networks as the underlying substrate for recruiting well-suited contributors [@integrate; @integratemain]. This marriage of participatory sensing and online social networks, referred to as *social participatory sensing*, offers the following advantages. First, the identification of suitable participants can be done easily through the public profile information such as interests, expertise and education. Second, social ties can act as an effective motivation to contribute to tasks created by friends, since people normally like to be helpful to their friends. Third, it is possible to offer incentives in the form of e-coins [@ecash] or reputation points which can be published in participants’ profile and seen by others. A real-world instantiation of social participatory sensing was recently presented in [@twitter], wherein, Twitter was used as the underlying social network substrate. The proposed system was tested in the context of two applications: weather radar and noise mapping. Their experiment resulted in a considerable smartphone-based participation from Twitter members even without an incentive structure. This clearly demonstrated the suitability of online social networks as a publish-subscribe infrastructure for tasking/utilizing smartphones and pave the way for ubiquitous crowd-sourced sensing and social collaboration applications.
The open nature of participatory sensing which allows everyone to contribute, while valuable for encouraging participants, facilitates erroneous and untrusted data preparation. When combined with social network, new trust issues arise. For instance, following the devastation incurred due to Hurricane Sandy in the US in October 2012, social media was flooded with misinformation and fake photos [^1]. While some of these were easy to identify as fake data (e.g., photoshopped images of sharks swimming in New York streets), several other fake pictures and reports were initially thought to be true. In fact, the widespread use of social networks, along with fast and easy-to-use dissemination facilities such as re-sharing (a fake photo) or re-tweeting (a false event) make it difficult to identify the origin of the data and investigate its credibility. This clearly highlights the need for a trust system which is responsible for performing necessary validations both from the perspective of data trustworthiness and also the reliability of data contributors. In other words, it is important to know who and with what level of social trustworthiness produces the data and how much of the data can be trusted. While there exist works that address the issue of data trustworthiness in participatory sensing (see Section \[rel\]), they do not provide means to include social trust and as such cannot be readily adopted for social participatory sensing.\
In this paper, we present an application agnostic framework to evaluate trust in social participatory sensing systems. Our system independently assesses the quality of the data and the trustworthiness of the participants and combines these metrics using fuzzy logic to arrive at a comprehensive trust rating for each contribution. These trust ratings are then used to calculate and update the reputation score of participants. By adopting a fuzzy approach, our proposed system is able to concretely quantify uncertain and imprecise information, such as trust, which is normally expressed by linguistic terms rather than numerical values. We undertake extensive simulations to demonstrate the effectiveness of our reputation framework and benchmark against the state-of-the-art. The results demonstrate that considering social relations makes trust evaluation more realistic, as it resembles human behaviour in establishing trustful social communications. We also show that our framework is able to quickly adapt to rapid changes in the participant’s behaviour by prompt and correct detection and revocation of unreliable contributions and accurate update of participant’s reputation score. Moreover, we find that leveraging fuzzy logic provides considerable flexibility in combining the underlying components which leads to better assessment of the trustworthiness of contributions. Our framework results in a considerable increase in the overall trust over a method which solely associates trust based on the quality of contribution.
The rest of the paper is organised as follows. Related work is discussed in Section \[rel\]. We present the details of our framework in Section \[pro\]. Simulation results are discussed in Section \[sim\]. Finally, Section \[con\] concludes the paper.
![Reputation framework architecture[]{data-label="fig:frm"}](framework.eps){width="12cm"}
Related Work {#rel}
============
To the best of our knowledge, the issue of trust in social participatory sensing hasn’t been addressed in prior work. As such, we discuss about related research focussing on trust issues in participatory sensing.\
In a participatory sensing system, trustworthiness can be viewed as the quality of the sensed data. In order to ascertain the data trustworthiness, it is highly desirable to ascertain that the sensor data has been captured from the said location and at the said time. [@Lenders] has proposed a secure service which allows participants to tag their content with a spatial timestamp indicating its physical location, which is later used by a co-located infrastructure for verification. A similar approach has been proposed in [@LocProof], in the form of a small piece of metadata issued by a wireless infrastructure which offers a timestamped signed location proof. Since these works rely on external infrastructure, they have limited scalability. Moreover, neither approach will work in situations where the infrastructure is not installed. In our proposed framework, we assume that sensor data is tagged with GPS coordinates/system time before being stored in phone memory, which is then used by trust server for verification. Data trustworthiness has been investigated from another point of view which tries to confirm that uploaded data preserves the characteristics of the original sensed data and has not been changed unintentionally or maliciously. In particular, there are several works which make use of Trusted Platform Module (TPM)[@TPM], which is a micro-controller embedded in the mobile device and provides it with hardware-based cryptography as well as secure storage for sensitive credentials. In [@Dua], each device has a trusted hardware element that implements cryptographic algorithms for content protection. [@Imasensor] presents two TPM-based design alternatives: the first architecture relies on a piece of trusted code and the second design incorporates trusted computing primitives into sensors to enable them sign their readings. YouProve [@YouProve] is another TPM-based system that allows client applications to directly control the fidelity of data they upload and services to verify that the meaning of source data is preserved. However, TPM chips are yet to be widely adopted in mobile devices. There is also recent work that does not require TPM. [@RFSN] proposes a reputation-based framework which makes use of Beta reputation [@beta] to assign a reputation score to each sensor node in a wireless sensor network. Beta reputation has simple updating rules as well as facilitates easy integration of ageing. However, it is less aggressive in penalizing users with poor quality contributions. A reputation framework for participatory sensing was proposed in [@Brian]. A watchdog module computes a cooperative rating for each device according to its short-term behaviour which acts as input to the reputation module which utilizes Gompertz function [@gompertz] to build a long-term reputation score. Their results show an improvement over the non-trust aggregation based approaches and Beta reputation system. However, the parameters related to the participants’ social accountability have not been considered. As such, their system cannot be readily used in our context.
Fuzzy Trust Framework {#pro}
=====================
In this section, we explain the proposed framework for evaluating trust and reputation in social participatory sensing systems. An overview of the architecture is presented in Section \[arc\] followed by a detailed discussion of each component in Section \[comp\].
Framework Architecture {#arc}
----------------------
Since our framework attempts to mimic how human’s perceive trust, we first present a simple illustrative example. Suppose John is a member of an online social network (e.g., Facebook). He has made a profile and has friended several people. John is a vegetarian and is the member of several vegetarian social communities. He is also on a budget and is keen to spend the least possible amount for his weekly groceries. He decides to leverage his social circle to find out the cheapest stores where he can buy vegetarian products. Specifically, he asks his friends or community members to capture geotagged photos of price labels of vegetarian food items when they are out shopping and send these back to him. One of his friends, Alex decides to help out and provides him with several photos of price labels. In order to decide whether to rely on Alex’s contributions, John would naturally take into account two aspects: (i) his personal trust perception of Alex, which would depend on various aspects such as the nature of friendship (close vs. distant), Alex’s awareness of vegetarian foods, Alex’s location, etc and (ii) the quality of Alex’s data which would depend on the quality of the pictures, relevance of products, etc. In other words, John in his mind computes a trust rating for Alex’s contribution based on these two aspects. Our proposed trust framework provides a means to obtain such trust ratings by mimicking an approach similar to John’s perception of trustworthiness in a scalable and automated manner. This trust rating helps John to select trustable contributions and accordingly plan for his weekend shopping. Moreover, the trust server provides a reputation score for each of the participating friends, according to the trustworthiness of their successive contributions.
Fig. \[fig:frm\] illustrates the architecture of the proposed reputation framework. The social network serves as the underlying publish-subscribe substrate for recruiting friends as participants. In fact, the basic participatory sensing procedures (i.e., task distribution and uploading contributions) are performed by utilizing the social network communication primitives. A person wishing to start a participatory sensing campaign acts as a requester and disseminates the tasks to his friends via email, private message or by writing as a post on their profiles (e.g., Facebook wall). Friends transfer their contributions via email or in the form of a message. We can also benefit from group construction facilities in Facebook or community memberships in Google Plus. The contributions received in response to a campaign are transferred (e.g., by using Facebook Graph API[^2]) to a third party trust server, which incorporates the proposed fuzzy inference system and arrives at an objective trust rating for each contribution. This trust rating is used as a criterion to accept the contribution or revoke it, by comparing against a predefined threshold.
At the end of each campaign, a cumulative objective trust rating, referred to as $Trust_{RP}$ is automatically updated for each participant, which denotes the trustworthiness degree of Requester upon the Participant. $Trust_{RP}$ is dependent on the trustworthiness of the contribution that the participant has prepared for the requester.
For certain campaigns, depending on the nature of task, the requester may desire to add a subjective evaluation in order to indicate how much the contribution is compatible with his needs and expectations. In such a case, this subjective rating is combined with the system-computed rating to update $Trust_{RP}$.
At regular intervals, a reputation score is also calculated for each participant, which is a combination of the trust ratings that requesters have assigned to him. This reputation score is further used as a weight for participant’s evaluations, ratings or reviews. More details about trust update, subjective rating and reputation calculation are presented in Section \[repsec\].
Framework Components {#comp}
--------------------
This section provides a detailed explanation of the framework components. In particular we focus on the trust sever, fuzzy inference system and reputation module.
### Trust Server {#server}
The trust server is responsible for maintaining and evaluating a comprehensive trust rating for each contribution and calculating a reputation score for each participant. As discussed in Section \[intro\], there are two aspects that need to be considered: (1) Quality of Contribution (QoC) and (2) Trust of Participant (ToP). The server maintains a trust database, which contains the required information about participants and the history of their past contributions. When a contribution is received by the trust server, the effective parameters that contribute to the two aforementioned components are evaluated by the Evaluator and then combined to arrive at a single quantitative value for each. The two measures serve as inputs for the fuzzy inference system, which computes the trustworthiness of contribution. In the following, we present a brief discussion about the underlying parameters and the evaluation methods.\
\
In participatory sensing, contributions can be of any modality such as images or sounds. The quality of the data is affected not only by fidelity of the embedded sensor but also the sensing action initiated by the participant. The in-built sensors in mobile devices can vary significantly in precision. Moreover, they may not be correctly calibrated or even worse not functioning correctly, thus providing erroneous data. Participants may also use the sensors improperly while collecting data,(e.g., not focussing on the target when capturing images). Moreover, human-as-sensor applications such as weather radar in [@twitter] are exposed to variability in the data quality due to subjectivity. For example, what is hot for one person may be comfortable for another. In order to quantify QoC, a group of parameters must be evaluated such as: relevance to the campaign (e.g., groceries in the above example), ability in determining a particular feature (e.g., price tag), fulfilment of task requirements (e.g., specified diet restrictions), etc. There already exists research that has proposed methods for evaluating the quality of data in participatory sensing. Examples include image processing algorithms proposed in [@DietSense] and outlier detection [@outlier] for sound-based sensing tasks. Rather than reinventing the wheel, our system relies on these state-of-the-art methods for determining the QoC.\
\
![(a) Gompertz function for Friendship score (b) Inverse Gompertz function for time gap score[]{data-label="fig:gompertz"}](gompertz.eps){width="12cm"}
ToP is a combination of personal and social factors. Personal factors consist of the following parameters:\
*Expertise(E):* It is defined as the measure of a participant’s knowledge and is particularly important in tasks that require domain expertise. Greater credence is placed in contributions made by a participant who has expertise in the campaign. We propose to use expert finding systems for evaluating expertise. These systems employ social networks analysis and natural language processing (text mining, text classification, and semantic text similarity methods) to analyse explicit information such as public profile data and group memberships as well as implicit information such as textual posts to extract user interests and fields of expertise [@expert]. In particular, Dmoz[^3] open directory project is used for expertise classification. Expertise evaluation is done by incorporating text similarity analysis to find a match between the task keywords (e.g., vegetarian) and participant’s expertise. We assume that the set *TE* contains the Task’s required Expertise and *PE* is the set of Participant’s Expertises. In this case, the expertise score of each participant is defined as Eq. \[eq:eqE\]: $$\label{eq:eqE}
E= \frac{n(TE\cap PE)}{n(TE)}$$ where *n(A)* is the number of elements in set A.\
*Timeliness(T):* Timeliness measures how promptly a participant performs prescribed tasks. It depends on the contribution response time *(t)* and the task deadline *(d)*. To evaluate this parameter, inverse Gompertz function defined as $ T(t)= 1 - e^{-be^{-ct}} $ is used because of its compatibility with timeliness evolution: timeliness score is highest when the contribution is received immediately after the task release time. The score begins to decrease as the response time increases, reaching the minimum value when the response is received just before the deadline. In the original inverse Gompertz function, the lower asymptote is zero; it means that the curve approaches to zero in infinity. In our case, timeliness rate will only be zero if contribution is received after the deadline; otherwise, a value between *x* and 1 is assigned to it. It means that the lowest timeliness rating will be *x* if contribution is received before the deadline, and is zero if received after the deadline. So, we modify the function as Eq. \[eq:eq1\] to calculate the timeliness (T):
$$\label{eq:eq1}
T(t)= \begin{cases}
1 - [(1 - x)e^{-be^{-ct}}] & \text{ if } t<d \\
0& \text{otherwise}
\end{cases}$$
*Locality(L):* Another significant parameter is locality, which is a measure of the participant’s familiarity with the region where the task is to be performed. We argue that contributions received from people with high locality to the tasking region are more trustable than those received from participants who are not local, since the first group is more acquainted with and has better understanding of that region. According to the experimental results presented in [@shirazi], people tend to perform tasks that are near to their home or work place (places that they are considered ‘local’ to them). This implies that if we log the location of participants’ contributions, we can estimate their locality. A participant’s locality would be highest at locations from where they make maximum number of contributions. In order to evaluate locality, we assume that the sensing area has been divided to *n* regions, and a vector *V* with the length equal to *n* is defined for each participant, where, *V(i)* is number of samples collected in region *i*. In this case, locality of a participant to region *i* is calculated by Eq. \[eq:eq2\]: $$\label{eq:eq2}
L(i)= \frac{V(i)}{\sum_{i=0}^{n-1} V(i)}$$
-- -- -- -- -- -- -- --
-- -- -- -- -- -- -- --
Next, we explain the social factors that affect ToP:\
*Friendship duration(F):* In real as well as virtual communications, long lasting friendship relations normally translate to greater trust between two friends. So, friendship duration which is an estimation of friendship length is a prominent parameter in trust development. We use the Gompertz function depicted in Fig. \[fig:gompertz\](a) to quantify friendship duration, since its shape is a perfect match for how friendships evolve. Slow growth at start resembles the friendship gestation stage. This is followed by a period of accumulation where the relationship strengthens culminating in a steady stage. As such, the friendship duration is evaluated according to Eq. \[eq:eq3\], in which, *b* and *c* are system-defined constants and *t* is the time in years.
$$\label{eq:eq3}
F(t)= e^{-be^{-ct}}
\vspace{-1mm}$$
*Interaction time gap(I):* In every friendship relation, interactions happen in form of sending requests and receiving responses. Interaction time gap, measures the time between the consequent interactions and is a good indicator of the strength of friendship ties. If two individuals interact frequently, then it implies that they share a strong relationship, which translates to greater trust. We propose to use the inverse Gompertz function depicted in Fig. \[fig:gompertz\](b) to quantify the interaction time gap, since a smaller time gap indicates stronger relationship, which leads to high social trust and vice-versa. So, the interaction time gap is evaluated according to Eq. \[eq:eq4\], in which, *b* and *c* are system-defined constants and *t* is the gap (in days) between the current time and the Latest Interaction(LI) time.
$$\label{eq:eq4}
I(t)= 1-e^{-be^{-ct}}$$
The aforementioned parameters are combined by the Evaluator to arrive at a single value for ToP, as depicted in Eq. \[eq:top\], $$\label{eq:top}
ToP= w_1\times E + w_2 \times T + w_3\times L + w_4 \times F + w_5 \times I$$ where, *$w_{i}$* is the weight of each parameter, and $\sum_{i=1}^{5}(w_i)$ equals to 1. The adjustment of the weights depends on the nature of the task. For example, in location-based tasks, $w_3$ is set to be considerably high to give more impression to Locality parameter. Similarly, for tasks where real-time information is important, a higher weight may be associated with Timeliness ($w_2$).
### Fuzzy inference system {#fuzzy}
Our proposed framework employs fuzzy logic to calculate a comprehensive trust rating for each contribution, referred to as the Trust of Contribution (ToC). We cover all possible combinations of trust aspects and address them by leveraging fuzzy logic in mimicking the human decision-making process. The inputs to the fuzzy inference system are the crisp values of QoC and ToP. In the following, we describe the fuzzy inference system components.
*Fuzzifier:* The fuzzifier converts the crisp values of input parameters into a linguistic variable according to their membership functions. In other words, it determines the degree to which these inputs belong to each of the corresponding fuzzy sets. The fuzzy sets for QoC, ToP and ToC are defined as:\
T(QoC)=T(ToP)={Low, Med1, Med2, High}\
T(ToC)= { VL, L, M, H, VH}.\
For any set $X$, a membership function on $X$ is any function from $X$ to the real unit interval \[0,1\]. The membership function which represents a fuzzy set $A$ is usually denoted by $\mu_{A}$. The membership degree $\mu_{A}(x)$ quantifies the grade of membership of the element $x$ to the fuzzy set $A$. The value 0 means that $x$ is not a member of the fuzzy set; the value 1 means that $x$ is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially.\
Fig.\[fig:mfinput\] represents the membership function of QoC and ToP and Fig.\[fig:mfoutput\] depicts the ToC membership function. We used trapezoidal shaped membership functions since they provide adequate representation of the expert knowledge, and at the same time, significantly simplify the process of computation.
*Inference Engine:* The role of inference engine is to convert fuzzy inputs (QoC and ToP) to the fuzzy output (ToC) by leveraging If-Then type fuzzy rules. The combination of the above mentioned fuzzy sets create 4\*4=16 different states which have been addressed by 16 fuzzy rules as shown in Table \[tab:rule\]. Fuzzy rules help in describing how we balance the various trust aspects. The rule based design is based on the experience and beliefs on how the system should work. To define the output zone, we used *max-min* composition method as: $ \mu_{T(ToC)}(ToC)= max[\underset{\substack{X \in T(ToP),\\Y\in T(QoC)}}{min}(\mu_{X}(ToP), \mu_{Y}(QoC))] $. The result of the inference engine is the ToC which is a linguistic fuzzy value.
*Defuzzifier:* A defuzzifier converts the ToC fuzzy value to a crisp value in the range of \[0, 1\]. We employed the Centre of Gravity (COG) [@cog] defuzzification method, which computes the center of gravity of the area under ToC membership function. COG is perhaps the most commonly used and popular defuzzification technique with the advantage of quick and highly accurate computations.
### Reputation Module {#repsec}
Once the ToC is defined for a contribution, the corresponding requester-participant mutual trust is updated, which is then used to calculate/update the participant’s reputation score. In the following, we describe these steps in details:\
As mentioned before, for some tasks, it is desirable for the requester to assign a subjective rating to participants’ contributions. This is particularly relevant for campaigns where it is difficult for the requester to express his real needs, desires or restrictions via task definition. Subjective rating is also useful when the requester does not have enough knowledge about the task and needs an expert review to confirm the validity of the contributions. For example, assume a requester with a strict gluten-free diet who asks his friends to take photos from the price tag and ingredients of gluten-free products. The term gluten-free is generally used to indicate a supposedly harmless level of gluten rather than a complete absence. For those with serious celiac disease, the maximum safe level of gluten in a finished product is even lower than the amount that exists in normal gluten-free products. So, a double check with product ingredients is essential to be performed either by the requester himself or by a nutritionist to assure that it is safe to be consumed. To be brief, although the objective rating assigned to a contribution is perfect for many tasks, sometimes, a subjective rating is added to reassure the conformance of contribution to the specific needs of requester. In such a case, the need for such subjective evaluation is defined by the requester in the task definition step.
We denote the subjective rating as Requester Evaluation (RE) which implies the trustworthiness of contribution from the requester’s point of view. Although RE value can be in any range, in our simulation in Section \[sim\], we assume that RE has a value in the range of $(ToC-\mu , ToC+\mu)$, where $\mu$ = 1- $\rho_{Req}$ and $\rho_{Req}$ is the requester’s reputation score. For a requester with high reputation score, the value of $\mu$ is small, resulting in RE close to ToC. It means that a requester with high reputation score is likely to assign a rating, which is close to the system-computed rating.
In the absence of subjective ratings, the requester simply relies on the objective ratings assigned by the trust server. In this case, $\mu$ is simply set to zero, resulting in RE=ToC.
Based on the ToC assigned to each contribution, the trust of requester upon the corresponding participant ($Trust_{RP}$) is updated. In fact, we adopt a reward/penalty policy for this update. A participant with ToC values greater than a predefined threshold1$(Th_1)$ is rewarded, and the amount of $\left |ToC-\rho_{Req}*RE \right |$ is added to $Trust_{RP}$. Similarly, a participant with ToC less than a predefined threshold2$(Th_2)$ is penalized, and the amount of $\left |ToC-\rho_{Req}*RE \right |$ is reduced from $Trust_{RP}$. This can be summarized in Eq. \[eq:tr\]. In our simulations in Section \[sim\], we set $(Th_1)=0.7$ and $(Th_2)=0.3$.
$$\label{eq:tr}
\small{Trust_{RP} =\left\{\begin{matrix}
Trust_{RP}+ \left |ToC-\rho_{Req}*RE \right | & \; if\, ToC>Th_1\\
Trust_{RP}- \left |ToC-\rho_{Req}*RE \right | & \; if\, ToC<Th_2
\end{matrix}\right.}$$
Note that in the this equation, we use the requester’s reputation score $(\rho_{Req})$ as a weight for his evaluation($RE$), since we believe an evaluation from a requester with high reputation score is more trustworthy than an evaluation from a low reputable requester.
![A sample social graph of 4 members with mutual trust ratings[]{data-label="fig:graph"}](graph.eps){width="5cm"}
This process is repeated for all participants at the end of each sensing campaign, and $Trust_{RP}$ is updated for all of them.
After every $n$ campaigns, $Trust_{RP}$ values upon each active participant act as inputs for reputation module, which updates the participant’s reputation score accordingly.
While there are already different crowdsourcing applications of online reputation systems [@survey] such as eBay [^4], Epinions [^5] and Amazon [^6], we use Web Page ranking algorithms as the basis for computing reputation scores. We draw parallels between the rank of a page in a set of web pages and the reputation score of a member in a social network. Moreover, the weights of links from different pages to a specific page are considered to be equivalent to the trust ratings of one member as determined by the other members of the social network.
Having a set of objects, a ranking algorithm calculates a relative importance of all objects in the set and makes an ordered list according to the importance. Web page ranking algorithms such as PageRank [@PageRank] calculate and assign a rank to a web page by analysing the web graph. Roughly speaking, PageRank ranks a page according to how many other pages are pointing at it. This can be described as a reputation system, because the collection of hyperlinks to a given page can be seen as public information that can be combined to derive a reputation score. A single hyperlink to a given web page can be seen as a trust rating of that web page.
In PageRank, the rank of page $P$, denoted by $\rho(P)$ is defined as: $ \rho(P)=\frac{\sum\limits_{P_i\rightarrow P}(\rho(P_i))}{L(P_i)} $ in which, $P_i$ is the set of all pages which have an outgoing link to page $P$, and $L(P_i)$ is the number of outgoing links from page $P_i$.\
In the original PageRank algorithm, it is assumed that all the outgoing links have equal weights. This is not always true, since not all outgoing links from a web page are equally important. So, we adopted the extension offered in [@kaltix] which modifies the above equation as Eq. \[eq:pr\], $$\label{eq:pr}
\rho(P)=\sum\limits_{P_i\rightarrow P}{\frac{w_i}{\sum\limits_{P_i\rightarrow P_j}{w_j}} \rho(P_i)}$$ in which, $w_i$ is the weight of the outgoing link, and the sum of weights of outgoing links is equal to 1.\
We explain this further by presenting an illustrative example. Consider the graph in Fig. \[fig:graph\] in which, $P_1$, $P_2$, $P_3$ and $P_4$ are the social network members. Links represent friendship relations with weights equal to the mutual trust between the pairs. In this case, according to Eq. \[eq:pr\]:\
$ \rho(P_1)= T_{21}\times \rho(P_2)$\
$ \rho(P_2)= T_{32}\times \rho(P_3)$\
$ \rho(P_3)= T_{13}\times \rho(P_1)$\
$ \rho(P_4)= T_{14}\times \rho(P_1) + T_{24}\times \rho(P_2) + T_{34}\times \rho(P_3)$\
As can be seen in the above expressions, reputation calculation is an iterative process and continues until convergence is obtained. In our simulation in Section \[sim\], we assume that the convergence occurs when $\left |\rho_k(P_i)-\rho_{k-1}(P_i) \right| \leq 10^{-10}$ for all $P_i$.\
To summarize, once a campaign is launched, participants begin to send a series of contributions. For each contribution, the Evaluator computes a value for QoC and ToP. These values are fed to fuzzy inference engine which calculates ToC for that contribution. The trust of requester upon each participant($Trust_{RP}$) is updated according to his ToC. The server utilizes $Trust_{RP}$ and $\rho_{Req}$ to update the reputation score of each participant.
Experimental Evaluation {#sim}
=======================
This section presents simulation-based evaluation of the proposed trust system. The simulation setup is outlined in Section \[setup\] and the results are in Section \[res\].
Simulation Setup {#setup}
----------------
To undertake the preliminary evaluations outlined herein, we chose to conduct simulations, since real experiments in social participatory sensing are difficult to organise. Simulations afford a controlled environment where we can carefully vary certain parameters and observe the impact on the system performance. We developed a custom Java simulator for this purpose. We simulate an online social network where 100 members participate in 5000 campaigns, producing one contribution for each. We assume that each member is connected to all others, similar to a social group; So, there are totally 10000 friendship relations. All members can serve both as requesters who launch sensing campaigns and as participants who contribute data to these sensing campaigns.
In our previous work [@Mobi], we assumed of categorizing participants according to the trade-offs between ToP and QoC. We wanted to observe how accurate the system assigns trust ratings to contributions in case of different ToP and QoC levels. Moreover, we artificially created scenarios where participants begin producing contributions with low QoC, which results in a decrease in ToC. We wanted to see if the system is able to quickly detect this transition and revoke low trustable contributions in an accurate and robust manner.
In this paper, instead of categorizing the participants according to ToP and QoC, we designed the categories according to the trade-offs between personal factors and social factors inside ToP, and simply assumed that QoC has a value in the range of $(ToP-\mu , ToP+\mu)$. In fact, we are going to observe how the system reacts to behavioural changes of participants and how much it is successful to update the reputation scores in case of such fluctuations. As mentioned in Section \[comp\], ToP parameters can be divided into two groups: social factors which include Friendship duration and Interaction time gap, and personal factors which include Timeliness, Expertise and Locality. In the real-world, there are often situations where a friend with a high rating of social factors (i.e., a very close friend with whom one has repeated interactions) has a low rating for personal factors for a period of time (i.e., does not have related expertise or does not produce timely contributions). It other words, we may have participants who have high social trust, but low personal trust, and vice versa. We have thus 4 different states based on the combination of different levels of personal and social trusts.
Specifically, we assumed that 60 members (out of 100) belong to Category A whereas the remaining 40 belong to Category B, adding the assumption that category A members have high personal trust, while category B members have low personal trust. We also assume that for each member $P_A$ in category A, all other members score $P_A$ with high social trust, and for each member $P_B$ in category B, all other members score $P_B$ with low social trust.
When $P_A$ serves as requester, other members form two subcategories:\
A-1: which includes 59 members from category A, excluding $P_A$. They have high personal trust and score $P_A$ with high social trust.\
A-2: which includes 40 members from category B. They have low personal trust and score $P_A$ with high social trust.
Similarly, when $P_B$ serves as requester, other members form two subcategories:\
B-1: which includes 60 members from category A. They have high personal trust and score $P_B$ with low social trust.\
B-2: which includes 39 workers from category B, excluding $P_B$. They have low personal trust and score $P_B$ with low social trust.
It is but natural that not all friends in a social network would contribute data to sensing campaigns. As such, we assume that 10% of the members in category A and 50% of the members in category B do not upload any data. The rationale for assuming unequal percentages is that the first group are close friends and hence a higher percentage would be willing to contribute, whereas the second group are not so and have less willingness to contribute.
Whenever a task is launched, one of the participants is selected to be the requester. Without loss of generality we assume that tasks are launched in sequential order by the social network members, i.e., member 1 launches the first campaign, member 2 launches the second campaign and so on.\
In the following, we will discuss the initialisation of the various parameters introduced in Section \[comp\].
In order to set the Expertise value for a participant, we assume that there are a total of 6 expertise areas defined and that each task needs at most 3 expertise areas $(n(TE)=3)$. To calculate the Expertise score for each participant, we assign a value to $n(PE)$ based on his category, as shown in Table \[tab:setting\]. The expertise score $E$ is then calculated using Eq. \[eq:eqE\].
For Timeliness, we first set the response time *(rt)* for each participant. Considering the task deadline to be 7 days (d=7), *rt* is assigned a value according to the participant’s personal category, as depicted in Table \[tab:setting\]. For example, for a participant $P_A$ belonging to category A, with probability of 0.4, *rt* is at most one day, with the probability=0.65, *rt* is at most half of a week, and with the probability of 0.9, *rt* is at most one week (Note that the greatest probability is 0.9, since with the probability of 0.1 (10%), $P_A$ does not attend in sensing campaign). *rt* then acts as the input value for Eq. \[eq:eq1\] which results in Timeliness score $T$ for participant. Other input parameters for Eq. \[eq:eq1\] have been set as *x*=0.3, *b*= 6, *c*=0.6, and *d*=7 days.
For Locality, we assume that there are a total of 25 regions and that each participant is local to 3 regions (i.e., Locality score $L$ for these 3 regions is 1). We also assume that when a participant has the maximum Locality score to a region, he has a relatively high locality to its surrounding regions. So, Locality score $L$ is assigned to the surrounding regions up to 3 levels of neighborhood,i.e., $N_1$, $N_2$ and $N_3$, based on participant’s category, as shown in Table \[tab:setting\].
For Friendship duration, as mentioned in Section \[comp\], the input parameter $(t)$ is the time (in years) passed from the beginning of friendship establishment. The initial value of *t* is set according to the participant’s category, as shown in Table \[tab:setting\] and a constant value of 0.02 is added to *t* upon each participation. $t$ is then serves as the input value for Eq. \[eq:eq3\] which computes the Friendship duration score $F$ for the participant. Other input parameters for Eq. \[eq:eq3\] have been set as *b*=5 and *c*=1.
Finally, for the Interaction time gap, as mentioned in Section \[comp\], the input parameter $t$ is the gap (in days) between the current time and the Latest Interaction($LI$) time. We set $LI$ based on the category of each participant, as shown in Table \[tab:setting\], and calculate $t$ accordingly. $t$ is then fed to Eq. \[eq:eq4\] which calculates the Interaction time gap score $I$ for the participant. Other input parameters for Eq. \[eq:eq4\] have been set as *b*=10 and *c*=0.2.
Once all of the aforementioned parameters are computed, ToP is calculated by simply averaging them. In other words, we simply assume that $w_i=1/5$ in Eq. \[eq:top\]. QoC is then assigned a value in a range of $(ToP-\mu , ToP+\mu)$ with $\mu=0.1$.
ToC is then calculated and $Trust_{RP}$ is updated according to Eq. \[eq:tr\]. At intervals, reputation score is also updated for participants. We set the reputation interval to be after every 100 campaigns ($n$=100).
In the first scenario, we assume that ToPs follow the category settings throughout the entire simulation. In the second scenario, we assume that ToP parameters change for a group of participants which results in a transition from one category to another (details in Section \[res\]).
As mentioned in Section \[arc\], a ToC rating is calculated for each contribution and those with ToC lower than a predefined threshold are revoked from further calculations. The ToCs for the non-revoked contributions are then combined to form an overall trust for that campaign. In other words, $ Overall Trust = \frac{\sum_{i=1}^{n} ToC}{n} $ in which, *n* is the number of non-revoked contributions. The revocation threshold is set to 0.5. We consider the overall trust as the evaluation metric. The greater the overall trust the better the ability of the system to revoke untrusted contributions. Overall trust has a value in the range of \[0, 1\]. We also calculate the reputation scores for all participants to see whether they reflect the behaviour of participants in normal and transition settings. Reputation score value is a number in the range of \[0, 1\] with initial value of 0.5 for each participant.
We compare the performance of our framework against the following systems: (1) Baseline-Rep: which follows the approach in [@Brian] by calculating a reputation score for each participant according to the QoC of his successive contributions. This reputation score is used as a weight for QoC. In other words, $ ToC= \sqrt{Rep* QoC}$ (2) Average: which includes ToP but computes the ToC simply as an average of ToP and QoC (3) Fuzzy: our proposed framework.
Simulation Results {#res}
------------------
We first present the simulation results for the first scenario. Fig. \[fig:ch1\] depicts the evolution of the average overall trust as a function of the number of campaigns. As shown in the figure, our fuzzy trust method outperforms the other methods. This confirms its success in mimicking the human trust establishing process by correct settings of fuzzy rules. In particular, we have set the rules in a way that results in early detection and severe punishment of untrusted contributions and also put greater emphasis on highly trusted contributions. The former has been done by assigning a very low (VL) value to ToC in case of low ToP and QoC (i.e., Rule no. 1 in Table \[tab:rule\]), whereas the latter has been obtained through assigning very high(VH) value to ToC in case of high QoC and above average ToP (i.e., Rule no. 15 and 16 in Table \[tab:rule\]).
![Evolution of average overall trust for all methods, Scenario 1[]{data-label="fig:ch1"}](avgoverall.eps){width="8cm"}
![Evolution of overall trust, Fuzzy method, Scenario 1[]{data-label="fig:ch2"}](otrust1.eps){width="8cm"}
![Reputation score for all members, Fuzzy method, Scenario 1[]{data-label="fig:ch3"}](rep.eps){width="8cm"}
Figure \[fig:ch2\] depicts the evolution of overall trust for 1000 contributions with Fuzzy method. As can be seen in this figure, at each interval containing 100 contributions, two different levels of overall trust are achieved. Remembering the order of requesters which is equal to members’ order, higher level of overall trust is obtained when the requester is from category A. So, participants are located either in subcategory A-1 or A-2. This will result in either high ToC values (when participants are from category A-1) or medium ToC values (when participants are from category A-2), which in turn, results in high overall trust. Similarly, lower level of overall trust is obtained when the requester is from category B. So workers are located either in category B-1 or B-2. This will lead to either medium ToC values (when participants are from category B-1) or low ToC values (when participants are from category B-2), which results in low overall trust. This variation is repeated regularly at each interval of 100 contributions.
Fig. \[fig:ch3\] presents the reputation of 100 participants after attending in 5000 sensing campaigns. As mentioned before, the initial value of reputation score for all participants is 0.5. Category A participants who have high ToPs, produce contributions with high ToC and hence, they get rewarded. This reward results in $trust_{RP}$ increase upon them, which in turn, increases their reputation score. On the contrary, for category B participants with low ToPs, ToCs will also be low, and hence, they are penalized, which results in the reduction of their reputation score. To summarize, our system continually tracks the contributions made over a series of campaigns and detects participants’ behaviour, which is accurately reflected in the evolution of the reputation scores.\
Next, we present results for the second scenario, wherein, the behaviour of participants change for a period of time, which results in a transition from one category to another. This scenario allows us to observe the performance of the schemes in the presence of noise. For example, consider a participant $P_A$ who is in category A, changes his behaviour for a period of time and behaves in a different manner which results in decrease of his personal and (hence) social trust. For example $P_A$ no longer provides timely contributions or does not care enough about the requirements of the task. This behavioural change results in a decrease in his personal trust, and consequently, others score him low with social trust. In other words, a participant may encounter a transition from category A to category B. In this scenario, we assume that 10 from 60 participants of category A transition to category B (e.g., a reduction in their personal and social factor values is created) in the period between 1000^th^ and 4000^th^ campaigns.\
Fig. \[fig:ch4\] shows the reputation score of 100 participants at the end of transition period (i.e., after attending in 4000 campaigns). As can be seen in this figure, the reputation of first ten participants who encounter such transition has a considerable decrease in comparison with others not encountering such transition. This again demonstrate the ability of our reputation module to adjust the reputation scores as a reflection of behavioural changes of participants.\
Finally, Fig. \[fig:ch5\] shows the reputation score evolution of participant no. 9 encountering such transition between 10^th^ and 40^th^ reputation intervals (between 1000^th^ and 4000^th^ campaigns). As can be observed, our proposed method shows an explicit and considerable reaction to this behavioural change, as compared with other methods. There is a decrease in reputation score due to dishonest behaviour during the transition period. At the end of transition period, transition encountered participant resumes his normal behaviour which results in a considerable increase in his reputation score.
![Reputation score for all members at Campaign 4000th, Fuzzy method, Scenario 2[]{data-label="fig:ch4"}](repnoise.eps){width="8cm"}
![Evolution of Reputation score for participant no.9 in all methods, Scenario 2[]{data-label="fig:ch5"}](repuser.eps){width="8cm"}
Conclusions {#con}
===========
In this paper, we proposed an application agnostic reputation framework for social participatory sensing system. Our system independently assesses the quality of the data and the trustworthiness of the participants and combines these metrics using fuzzy inference engine to arrive at a comprehensive trust rating for each contribution. The system is then assigns a reputation score to participants by leveraging the concepts utilised in PageRank algorithm. Simulations demonstrated that our scheme increases the overall trust by over 15% as compared to other methods, and assigns reputation scores to participants in a robust and reliable manner.
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[^1]: http://news.yahoo.com/10-fake-photos-hurricane-sandy-075500934.html
[^2]: http://developers.facebook.com/docs/reference/api/
[^3]: http://www.dmoz.org
[^4]: http://www.ebay.com/
[^5]: http://www.epinions.com/
[^6]: http://www.amazon.com/
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abstract: 'It is known by the author that there exist 20 families of Dehn surgeries in the Poincaré homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeries in $\Sigma(2,3,6n\pm1)$ and $\Sigma(2,2s+1,2(2s+1)n\pm1)$ and so on. As other examples, we give lens space surgeries in graph homology spheres, which are obtained by splicing two Brieskorn homology spheres.'
address: 'Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan'
author:
- Motoo Tange
title: Homology spheres yielding lens spaces
---
¶
\[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Conjecture]{} \[thm\][Fact]{}
\[section\]
\[section\]
\[section\] \[rem\][**[Construction]{}**]{} \[rem\][**[Example]{}**]{}
[^1]
Introduction {#intro}
============
Lens space surgeries and double-primitive knots
-----------------------------------------------
We define lens space $L(p,q)$ to be the $+p/q$-surgery of the unknot in $S^3$. Let $Y$ be a homology sphere. If a knot $K\subset Y$ yields a lens space by some (positive) integral Dehn surgery, we say that $K\subset Y$ admits ([*positive*]{}) [*lens surgery*]{}. If in general the surgered manifold is e.g., Seifert manifold, we say the surgery [*Seifert surgery*]{} and so on. We write $p$-surgery of $K\subset Y$ as $Y_p(K)$. If $K$ admits lens space surgery or Seifert surgery, then such a knot $K$ is called [*lens space knot*]{} or [*Seifert space knot*]{} respectively. The number $p$ of the Dehn surgery $Y_p(K)$ is called [*slope*]{} and $[\tilde{K}]\in H_1(Y_p(K))$ is called [*dual class*]{} in this paper. [*Throughout this paper, any slope is a positive integer if there are no special statements.*]{} Here $\tilde{K}\subset Y_p(K)$ is dual knot (i.e., core circle of the attached solid torus) of the surgery. We call the pair of slope and dual class [*(e.g., lens surgery or Seifert surgery) parameter*]{}, depending on the surgered manifold.
As examples of lens space knots in $S^3$ it is well-known that there are the torus knots, some cable knots of torus knots and some hyperbolic knots and so on. Berge in [@B] defined double-primitive knot in $S^3$ which generalizes these examples. Here we define double-primitive knot in a general way.
Let $Y$ be a homology sphere with at most 2 Heegaard genus. Suppose $K\subset Y$ is a knot and $H_0\cup_{\Sigma_2} H_1$ is a genus 2 Heegaard decomposition of $Y$, where each $H_i$ is the genus 2 handlebody and $\Sigma_2$ is the Heegaard surface. If $K\subset Y$ is isotopic to a knot $K'$ satisfying the following conditions, then $K$ is called a [*double-primitive knot*]{}.
- $K'$ lies in $\Sigma_2$.
- For $i=0,1$ the induced elements $[K']\in \pi_1(H_i)\cong F_2$ (the rank 2 free group) are both primitive elements in $F_2$.
The main ideas by Berge in [@B] are the definition of double-primitive knots and the two basic facts which are written below.
Any double-primitive knot is a lens space knot.
The Dehn surgery by the surface slope for the genus two surface gives rise to a Heegaard genus one manifold, because of double-primitive condition. This is the easy proof for this fact. Note that even if a homology sphere which includes a double-primitive knot is not $S^3$, then the double-primitive knot becomes a lens space knot. Gordon conjectured that double-primitive knots in $S^3$ are all of lens space knots in $S^3$. This conjecture is known as [*Berge conjecture*]{} (Problem 1.78 in Kirby’s problem list [@Kirby]). This conjecture is open so far. Any simple (1,1)-knot $\tilde{K}_{p,k}$ in a lens space gives an integral ${\mathbb Z}$HS surgery uniquely (read Section \[simple11knot\]). We denote by $K_{p,k}$ the dual knot. ${\mathbb Z}$HS is the abbreviation of integral homology sphere. The following is a second basic fact:
For relatively prime positive integers $p,k$, the dual knot $K_{p,k}$ of simple (1,1)-knot in a lens space is a double-primitive knot in a homology sphere.
The dual means the core circle of the integral ${\mathbb Z}$HS surgery of $\tilde{K}_{p,k}$. By taking the union of the neighborhood of $K_{p,k}$ and one solid torus with respect to the Heegaard decomposition of $L(p,q)$, one can understand this fact. For example, see [@S]. These terminologies will be defined in Section \[simple11knot\] in this paper. The reason for us to restrict lens space knot to $K_{p,k}$ is to control the homology sphere containing a lens space knot. In fact in many cases, the homology sphere $Y_{p,k}$ containing $K_{p,k}$ can be Seifert homology sphere and the Heegaard genus is at most two. For example, if one considers more general homology sphere surgery of a lens space, the obtained homology sphere is complicated in general (possibly it may be a higher Heegaard genus ${\mathbb Z}$HS).
Berge in [@B], furthermore, listed $K_{p,k}$ in $S^3$. We call these knots [*Berge knots*]{} here. The examples are organized as in [Table]{} \[s3case\] (due to Rasmussen [@R]). The integer $p$ is the slope (positive integer) and $k$ is the dual class of lens space surgery.
Note that as long as we deal with a lens space surgery of a homology sphere, the dual class $k$ is regarded as an integer with $0<k<p/2$ canonically up to taking the inverse in the multiplicative group $({\mathbb Z}/p{\mathbb Z})^\times $ and multiplying it by $-1$.
Furthermore, if a pair of coprime integers $(p,k)$ is given, it uniquely determines a dual knot $K_{p,k}$ of simple (1,1)-knot in a lens space with the lens surgery parameter $(p,k)$. $K_{p,k}$ lies in a homology sphere $Y_{p,k}$. In other words, $K_{p,k}$ is a representative in lens space knots with the parameter $(p,k)$.
Greene in [@G] proved if a lens space is constructed by an integral Dehn surgery of a knot $K$ in $S^3$, then there exists a Berge knot $B$ such that the parameter of $B$ is the same as the one of $K$. This implies that Berge knots include all double-primitive knots in $S^3$. Namely [Table]{} \[s3case\] is the table of the parameters $(p,k)$ of double-primitive knots in $S^3$.
Type $p$ $k$ Condition
---------- ----------------------- ------------ ---------------------------------
I $ik\pm1$ $k$ $(i,k)=1$
II $ik\pm1$ $k$ $(i,k)=2$
III $\pm(2k\mp1)d\ (k^2)$ $k$ $d|k\pm 1,\frac{k\pm 1}{d}$ odd
IV $\pm(k\mp1)d\ (k^2)$ $k$ $d|2k\pm1$
V $\pm(k\mp 1)d\ (k^2)$ $k$ $d|k\pm1,d$ odd
VII,VIII $p$ $k$ $k^2\pm k\pm1=0\ (p)$
IX $22\ell^2+9\ell+1$ $11\ell+2$ $\ell\in {\mathbb Z}$
X $22\ell^2+13\ell+2$ $11\ell+3$ $\ell\in {\mathbb Z}$
: A list of Berge knots due to Rasmussen [@R].[]{data-label="s3case"}
The author in [@MT2] found a list of 20 families of $K_{p,k}$ with $Y_{p,k}=\Sigma(2,3,5)$ that the positive $p$-surgery of $K_{p,k}$ is a lens space $L(p,k^2)$. The families consist of 19 quadratic families and 1 isolated example as illustrated in [Table]{} \[po\]. The $\ell$ is a non-zero integer and $k_2$ is the integer with $kk_2\equiv \pm1\bmod p$ and $0<k_2<p/2$. The $g'$ stands for $2g(K)-p-1$. It is an open question whether the table lists all $K_{p,k}$’s with $Y_{p,k}=\Sigma(2,3,5)$ or not, which it would be Greene’s type statement if the question is yes.
$$\begin{array}{|c|c|c|c|c|c|}\hline
\text{Type} & p & k&k_2&g'\\\hline
\A_1 & 14\ell^2+7\ell+1 & 7\ell+2&14\ell+3&-|\ell|\\\hline
\A_2 & 20\ell^2+15\ell+3 & 5\ell+2&20\ell+7&-|\ell|\\\hline
\B & 30\ell^2+9\ell+1 & 6\ell+1&15\ell+2&-|\ell|\\\hline
\C_1 & 42\ell^2+23\ell+3 & 7\ell+2&42\ell+11&-|\ell|\\\hline
\C_2 & 42\ell^2+47\ell+13 & 7\ell+4&42\ell+23&-|\ell|\\\hline
{\Delta}_1 & 52\ell^2+15\ell+1 & 13\ell+2&52\ell+7&-|\ell|\\\hline
{\Delta}_2 & 52\ell^2+63\ell+19 & 13\ell+8&52\ell+31&-|\ell|\\\hline
\E_1 & 54\ell^2+15\ell+1 & 27\ell+4&54\ell+7&-|\ell|\\\hline
\E_2 & 54\ell^2+39\ell+7 & 27\ell+10&54\ell+19&-|\ell|\\\hline
\F_1 & 69\ell^2+17\ell+1 & 23\ell+3&69\ell+8&-2|\ell|\\\hline
\F_2 & 69\ell^2+29\ell+3 & 23\ell+5&69\ell+14&-2|\ell|\\\hline
{\Gamma}_1 & 85\ell^2+19\ell+1 & 17\ell+2&85\ell+9&-2|\ell|\\\hline
{\Gamma}_2 & 85\ell^2+49\ell+7 & 17\ell+5&85\ell+24&-2|\ell|\\\hline
\H_1 & 99\ell^2+35\ell+3 & 11\ell+2&99\ell+17&-2|\ell|\\\hline
\H_2 & 99\ell^2+53\ell+7 & 11\ell+3&99\ell+26&-2|\ell|\\\hline
\I_1 & 120\ell^2+16\ell+1 & 12\ell+1&20\ell+1&-2|\ell|\\\hline
\I_2 & 120\ell^2+20\ell+1 & 20\ell+2&30\ell+2&-2|\ell|\\\hline
\I_3 & 120\ell^2+36\ell+3 & 12\ell+2&30\ell+4&-2|\ell|\\\hline
\J & 120\ell^2+104\ell+22 & 12\ell+5&20\ell+9&-|2\ell+1|\\\hline
\K & 191 & 15&51&-2\\\hline
\end{array}$$
It is conjectured that [Table]{} \[po\] lists all $K_{p,k}$’s with $Y_{p,k}=\Sigma(2,3,5)$ with $2g(K)<p+1$. The similar conjecture is Conjecture 1 in [@R]. This genus restriction is needed because as mentioned in [@MT2], there is a counterexample. For a partial circumstantial evidence for this conjecture, for example see [@MT3]. On the other hand, in [@MT5] it is shown that $-\Sigma(2,3,5)$ never admit any positive lens space knot and $\Sigma(2,3,5)\#(-\Sigma(2,3,5))$ never have any lens space knots. Our main question is the following:
Which homology spheres other than $S^3$ or $\Sigma(2,3,5)$ include a double-primitive knot? Does such a homology sphere have a list analogous to [Table]{} \[s3case\] or \[po\]?
Tables of lens space surgeries in Brieskorn homology spheres.
-------------------------------------------------------------
In this paper we mainly focus on $K_{p,k}$ in some Brieskorn homology spheres and some graph homology spheres. We give a table of $K_{p,k}$ in $\Sigma(2,3,7)$.
\[237casethm\] [Table]{} \[lens237\] is a collection of examples of $K_{p',k}$ with $Y_{p',k}=\Sigma(2,3,7)$. The positive $p'$-surgery of $K_{p',k}$ gives the lens space $L(p',k^2)$ in the table. Here $\ell$ in the table is a non-zero integer.
We conjecture the following statement on the genus $g(K_{p',k})$.
The genus $g(K_{p',k})$ for $(p',k)$ which is any parameter in [Table]{} \[lens237\] is $(g'+p'+1)/2$, where $g'$ is the value in the table.
As one can see, [Table]{} \[lens237\] is parallel to [Table]{} \[po\]. The slope $p$ in [Table]{} \[po\] corresponding slope $p'$ is related by the formula by the last column in [Table]{} \[lens237\]. For example, on the type $\A_1$, between corresponding two lens space surgeries, the formula $(p+p')/2=(4\ell+1)^2$ is satisfied.
$p'$ $k$ $g'$ $\frac{p'+p}2$
------- ------------------------ ------------- ------------- ----------------
$A_1$ $18\ell^2+9\ell+1$ $9\ell+2$ $|\ell|$ $(4\ell+1)^2$
$A_2$ $30\ell^2+25\ell+5$ $5\ell+2$ $|\ell|$ $(5\ell+2)^2$
$B$ $42\ell^2+15\ell+1$ $ 6\ell+1$ $|\ell|$ $(6\ell+1)^2$
$C_1$ $56\ell^2+33\ell+5$ $7\ell+2$ $|\ell|$ $(7\ell+2)^2$
$C_2$ $56\ell^2+65\ell+19$ $7\ell+4$ $|\ell|$ $(7\ell+4)^2$
$D_1$ $76\ell^2+17\ell+1$ $19\ell+2$ $|\ell|$ $(8\ell+1)^2$
$D_2$ $76\ell^2+97\ell+31$ $19\ell+12$ $|\ell|$ $(8\ell+5)^2$
$E_1$ $74\ell^2+17\ell+1$ $37\ell+4$ $|\ell|$ $(8\ell+1)^2$
$E_2$ $74\ell^2+57\ell+11$ $37\ell+14$ $|\ell|$ $(8\ell+3)^2$
$F_1$ $93\ell^2+19\ell+1$ $31\ell+3$ $|2\ell|$ $(9\ell+1)^2$
$F_2$ $93\ell^2+43\ell+5$ $31\ell+7$ $|2\ell|$ $(9\ell+2)^2$
$G_1$ $115\ell^2+21\ell+1$ $23\ell+2$ $|2\ell|$ $(10\ell+1)^2$
$G_2$ $115\ell^2+71\ell+11$ $23\ell+7$ $|2\ell|$ $(10\ell+3)^2$
$H_1$ $143\ell^2+53\ell+5$ $11\ell+2$ $|2\ell|$ $(11\ell+2)^2$
$H_2$ $143\ell^2+79\ell+11$ $11\ell+3$ $|2\ell|$ $(11\ell+3)^2$
$I_1$ $168\ell^2+32\ell+1$ $12\ell+1$ $|2\ell|$ $(12\ell+1)^2$
$I_2$ $168\ell^2+60\ell+5$ $12\ell+2$ $|2\ell|$ $(12\ell+2)^2$
$I_3$ $168\ell^2+28\ell+1$ $28\ell+2$ $|2\ell|$ $(12\ell+1)^2$
$J$ $168\ell^2+136\ell+28$ $12\ell+5$ $|2\ell+1|$ $(12\ell+5)^2$
$K$ $259$ $15$ $2$ $15^2$
: $K_{p,k}$ for each $(p,k)$ is a lens space knot in $\Sigma(2,3,7)$ yielding $L(p,k^2)$.[]{data-label="lens237"}
Here we address the following conjectures combining the statement above.
[Table]{} \[po\] is a list of all $K_{p,k}$’s in $\Sigma(2,3,5)$ if $2g(K)< p+1$ and [Table]{} \[lens237\] is a list of all $K_{p',k}$’s in $\Sigma(2,3,7)$ if $2g(K)>p'+1$.
We recall estimate $p\le 2g(K)-1$ in the cases of lens space surgeries over the non-L-space homology sphere (see Theorem \[rasmussenthm\]). Rasmussen in [@R] proved the following:
\[rasmussenthm\] Let $K$ be a lens space knot in a homology sphere $Y$. If $2g(K)-1<p$ then, $Y$ is an L-space, while if $2g(K)-1>p$, then $Y$ is a non-L-space.
We give families of $K_{p,k}$ in $\Sigma(2,3,6n\pm1)$ as extension of lens space surgeries in [Table]{} \[po\] and \[lens237\].
\[sigma236npm1\] [Table]{} \[lens236n1\] is a collection of examples of $K_{p,k}$ in $\Sigma(2,3,6n\pm1)$ for a non-zero integer $\ell$. The positive $p$-surgery of $K_{p,k}$ gives $L(p,k^2)$.
In the same way we state the following conjecture:
The genus $g(K_{p,k})$ is $(g'+p+1)/2$, where $g'$ is the value in [Table]{} \[lens236n1\].
Type $p$ $k$ $g'$
-------- ------------------------------ ------------ ------------------------
$\A_2$ $nk^2\pm (5\ell^2+5\ell+1)$ $5\ell+2$ $|k(n-1)\pm\ell|$
B $nk^2\pm (6\ell^2+3\ell)$ $6\ell+1$ $|k(n-1)\pm\ell|$
$\C_1$ $nk^2\pm(7\ell^2+5\ell+1)$ $7\ell+2$ $|k(n-1)\pm\ell|$
$\C_2$ $nk^2\pm(7\ell^2+9\ell+3)$ $7\ell+4$ $|k(n-1)\pm\ell|$
$\H_1$ $nk^2\pm(22\ell^2+9\ell+1)$ $11\ell+2$ $|k(n-1)\pm2\ell|$
$\H_2$ $nk^2\pm(22\ell^2+13\ell+2)$ $11\ell+3$ $|k(n-1)\pm2\ell|$
$\I_1$ $nk^2\pm(24\ell^2+8\ell)$ $12\ell+1$ $|k(n-1)\pm2\ell|$
$\I_3$ $nk^2\pm(24\ell^2+12\ell+1)$ $12\ell+2$ $|k(n-1)\pm2\ell|$
J $nk^2\pm(24\ell^2+16\ell+3)$ $12\ell+5$ $|k(n-1)\pm(2\ell+1)|$
K $nk^2\pm 34$ $15$ $|k(n-1)\pm2|$
: $K_{p,k}$ in $\Sigma(2,3,6n\pm1)$ yielding $L(p,k^2)$.[]{data-label="lens236n1"}
\[2s+1theorem\] [Table]{} \[lens22s+11\] is a collection of $K_{p,k}$ in $\Sigma(2,2s+1,2(2s+1)\pm1)$. Here $\ell$ is a non-zero integer.
$p$ $k$
-------------- ---------------------------------- ---------------------------------
$\A_2$ $\frac{1}{4}(k^2\mp(k-2)\ell)$ $(4(s+1)\pm1)\ell+2$
$\B_1$ $k^2\pm(k+2)\ell$ $(4s+2)\ell+1$
$\B_2$ $k^2\pm((k+2s)\ell+s-1)$ $(4s+2)\ell+2s-1$
${\Delta}_1$ $\frac{1}{4}(k^2\mp(3k+2)\ell)$ $(4(3s+1)\pm3)\ell+2$
${\Delta}_2$ $\frac{1}{4}(k^2\mp(3l+2)(k-2))$ $(4(3s+1)\pm3)\ell+2(4s+1)\pm2$
${\Gamma}_1$ $\frac{1}{4}(k^2\mp(3k+2)\ell)$ $(4(3s+2)\pm3)\ell+2$
${\Gamma}_2$ $\frac{1}{4}(k^2\mp(3l+1)(k-2))$ $(4(3s+2)\pm3)\ell+2(2s+1)\pm1$
$\I_1$ $k^2\pm(2k\ell+(k-1)/2)$ $4(2s+1)\ell+2s-1$
$\I_2$ $\frac{1}{4}(k^2\mp4(k-2)\ell)$ $4(2(2s+1)\pm1)\ell+2$
J $k^2\pm (2k\ell+(k+1)/2)$ $4(2s+1)\ell+2s+3$
: $K_{p,k}$ in $\Sigma(2,2s+1,2(2s+1)\pm1)$ yielding $L(p,k^2)$.[]{data-label="lens22s+11"}
\[22s+1ntheorem\] [Table]{} \[lens22s+1n1\] is a collection of $K_{p,k}$ in $\Sigma(2,2s+1,2(2s+1)n\pm1)$. Here $\ell$ is a non-zero integer.
Type $p$ $k$
-------- ---------------------------- --------------------
$\B_1$ $nk^2\pm(k+2)\ell$ $(4s+2)\ell+1$
$\B_2$ $nk^2\pm((k+2s)\ell+s-1)$ $(4s+2)\ell+2s-1$
$\I_1$ $nk^2\pm(2k\ell+(k-1)/2)$ $4(2s+1)\ell+2s-1$
J $nk^2\pm (2k\ell+(k+1)/2)$ $4(2s+1)\ell+2s+3$
: $K_{p,k}$ in $\Sigma(2,2s+1,2(2s+1)n\pm1)$ yielding $L(p,k^2)$.[]{data-label="lens22s+1n1"}
To obtain these tables, we will look back $K_{p,k}$’s in [Table]{} \[po\] and give the knot diagrams of them (Appendix 1, 2). By using these diagrams, we give straight extensions of $K_
{p,k}$ to Brieskorn homology spheres, $\Sigma(2,3,6n\pm1)$, $\Sigma(2,2s+1,2(2s+1)\pm1)$ and $\Sigma(2,2s+1,2(2s+1)n\pm1)$. For lens surgeries of other types of Brieskorn homology spheres, see Section \[Ataipu\] to \[ktypelensspaceknotsection\]. For examples, Brieskorn homology spheres $$\Sigma(m,ms+m-1,m(ms+m-1)n\pm1),$$ for positive integers $s,n,m$ admit similar quadratic families of lens space surgeries.
Suppose on data of [Table]{} \[s\] we impose the restriction that $r_1,s_1,t_1$ are pairwise relatively prime in addition to the condition indicated in the table. The Seifert data in [Table]{} \[s\] present Brieskorn homology spheres $\Sigma(|r_1|,|s_1|,|t_1|)$ (up to orientation) containing $K_{p,k}$.
Type $S(e,(r_1,r_2),(s_1,s_2),(t_1,t_2))$ Condition
------ ---------------------------------------- ------------------------------------------------------------------- --
A $S(1,(2,1),(2s+1,1),\ast)$ $s\in{\mathbb Z}_{>0}$
B $S(1, (2a-1,a),\ast,\ast)$ $a\in {\mathbb Z}\setminus\{0\}$
CD $S(1,(m,m-1),(\beta,1),\ast)$ $m\in {\mathbb Z}\setminus\{1\},\beta\in {\mathbb Z}_{>0}$
E $S(1,(2m-1,m-1),(\alpha,1),(\beta,1))$ $m\in {\mathbb Z}\setminus\{1\},\alpha,\beta\in {\mathbb Z}_{>0}$
FGH $S(1,(a,1),(b,1),\ast)$ $a,b\in {\mathbb Z}_{>0}$
I $S(1,(c,1),\ast,\ast)$ $c\in {\mathbb Z}_{>0}$
J $S(1,(2,1),\ast,\ast)$
K $S(1,(3,1),(m,m-1),\ast)$ $m\in {\mathbb Z}\setminus\{1\}$
: The Seifert data (up to orientation) of Brieskorn homology spheres containing $K_{p,k}$, where $\ast$ means any multiplicity.[]{data-label="s"}
The data in [Table]{} \[s\] do not give all the Brieskorn homology spheres. For example, $\Sigma(7,11,15)$ has the Seifert data $S(1,(7,2),(11,2),(15,8))$. Hence, it is not determined whether $\Sigma(7,11,15)$ contains $K_{p,k}$ or generally any lens space knot or not. On the other hand, we would like to address the following conjecture.
Any Brieskorn homology sphere contains $K_{p,k}$.
As an analog of the result in [@MT5], we conjecture the following. This conjecture can be checked for our examples appeared in this paper.
Any Brieskorn homology sphere $-\Sigma(p,q,r)$ with the orientation opposite to the usual one does not contain $K_{p,k}$.
In the case of $(p,q,r)=(2,3,5)$, there is no positive lens space knots in $-\Sigma(2,3,5)$, as proven in [@MT5].
The final result of this paper is to give families of double-primitive knots in some graph homology spheres. Here we denote the graph manifold with the plumbing diagram as in [Figure]{} \[gra\] by $G(\{A,B\},\{C,D\};a,b)$ for four rationals $A,B,C,D$ and two integers $a,b$. We refer [@Sav] for the plumbing diagram for plumbed 3-manifold.
![The plumbing graph for $G(\{A,B\},\{C,D\};a,b)$[]{data-label="gra"}](graph-eps-converted-to.pdf)
\[graphhomspheres\] Suppose that graph homology spheres of the form above have the following condition:
1. $G(\{2,3\},\{P/Q,m/(m-1)\};0,0)$ where $m$ is any integer, $P,Q,m$ satisfy $-5P+5mQ-Pm=\pm1$.
2. $G(\{2,5\},\{P/Q,m/(m-1)\};0,0)$ where $m$ is any integer, $P,Q,m$ satisfy $-7P+7mQ-3mP=\pm1$.
Then the homology spheres contain $K_{p,k}$ and produce the lens space $L(p,q)$ with $$p/q=[\alpha_n,\cdots, \alpha_1,\ell+1,2,-m,3,-\ell],$$ $$p/q=[\alpha_n,\cdots, \alpha_1,\ell+1,2,-m,5,-\ell],$$ where $[\alpha_1,\alpha_2,\cdots, \alpha_n]=P/Q$ is satisfied.
There is a natural question here: Are there any hyperbolic homology sphere $Y_{p,k}$? In this article, we deal with non-hyperbolic $Y_{p,k}$ only. In the forthcoming article we will give examples of hyperbolic $Y_{p,k}$.
Application to the $b_2=1$ 4-manifold with lens space boundary. {#b2=1examplesection}
---------------------------------------------------------------
The computation of this article is definitely useful to construct simply connected 4-manifolds with small $b_2$ whose boundaries are lens spaces. ‘Small’ means $b_2=1$ here. If a lens space is constructed by integral Dehn surgery of $S^3$, then the lens space is the boundary of a 4-manifold with $b_2=1$. Our several lists in this article may give interesting small $b_2$ bounds of lens spaces even if the lens spaces are not Dehn surgeries of knots in $S^3$. Namely, the 4-manifolds are simply connected and homologically $S^2$ but never admit exact two Morse critical points. The strategy is to give lens space surgeries over homology spheres which bound contractible 4-manifolds. The reason for the lens space not to bound a 4-manifold with exact two Morse critical points is by the work of Greene in [@G]. For example, such Brieskorn homology spheres are $\Sigma(2,5,7)$, $\Sigma(2,3,13)$ or $\Sigma(3,4,5)$ etc. See [@AK].
As mentioned at Section 1.6 in [@G], $L(17,15)$ cannot bound a 4-manifold with index 0 and 2. However the lens space bounds a 4-manifold with $b_2=1$. Here, we give families of lens spaces (including $L(17,15)$) which bound 4-manifold with $b_2=1$. Lens space surgeries over $\Sigma(2,3,13)$ can be already viewed in Theorem \[sigma236npm1\] by taking $n=2$ and plus sign.
\[Joshua\] For any non-zero integer $\ell$, there are quadratic families of lens space surgeries on $\Sigma(2,5,7)$ as follows: $$L(35\ell^2+21\ell+3,(7\ell+2)^2)\ \ \ (\A \text{ type }surgeries)$$ $$L(117\ell^2+37\ell+3,(13\ell+2)^2)\ \ \ (\C_1{\Delta}_1 \text{ type }surgeries)$$ $$L(117\ell^2+145\ell+45,(13\ell+8)^2)\ \ \ (\C_2{\Delta}_2 \text{ type }surgeries)$$ For any non-zero integer $\ell$, there are quadratic families of lens space surgeries on $\Sigma(3,4,5)$ as follows: $$L(86\ell^2+37\ell+4,(43\ell+9)^2)\ \ \ (\E \text{ type }surgeries)$$ $$L(86\ell^2+49\ell+7,(43\ell+12)^2)\ \ \ (\E \text{ type }surgeries)$$
In particular, $L(17,15)$ is given by an A type $17$-surgery in $\Sigma(2,5,7)$ in [Table]{} \[s\]. In fact, this knot $K$ is isotopic to $K_{17,7}$. $\C_1D_1$ type or $\C_2{\Delta}_2$ type will be explained in Section \[CDEtaipu\].
This proposition implies the existence of sufficiently many families of lens spaces which bound simply connected 4-manifolds with $b_2=1$ but do not have exact two Morse critical points. On the other hand $L(17,2)$ (the orientation reversing $L(17,15)$) appears in $\A_2$ type ($n=2$, minus sign, and $\ell=-1$) in [Table]{} \[lens236n1\]. $L(17,2)$ is the 17-surgery of a lens space knot in $\Sigma(2,3,11)$. Read [@G].
In this way, easily the reader would be able to find lens spaces bounding 4-manifolds $b_2=1$.
Preliminary {#simple11knot}
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Simple (1,1)-knot. {#simple11knotsection}
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Let $p$ be a positive integer. Let $\tilde{K}$ be a knot in $L(p,q)$ that is a generator in $H_1(L(p,q))$. Let $m$ be the meridian of $\tilde{K}$. The $m$ lies on $\partial L(p,q)_0$, where $L(p,q)_0$ is the exterior of $\tilde{K}$. We choose a longitude $l$ of $\tilde{K}$. The pair $([m],[l])$ produces an oriented basis on $H_1(\partial L(p,q)_0)$. Since $H_1(L(p,q))\cong {\mathbb Z}/p{\mathbb Z}$, $a[m]+p[l]$ is null-homologous in $H_1(L(p,q)_0)$ for some integer $a$. Namely, $H_1(L(p,q)_0)= \langle [m],[l]\rangle/a[m]+p[l]\cong {\mathbb Z}$ holds. The number $a\bmod p$ is the invariant of $\tilde{K}$ because another choice of $l$ changes $a$ to $a+np$ for some integer $n$. The number $a$ modulo $p$ is called a [*self-linking number*]{}. The computation of the linking form $\ell k:H_1\times H_1\to {\mathbb Q}/{\mathbb Z}$ gives $\ell k([\tilde{K}], [\tilde{K}])=(a[m]+p[l])\cdot [l]/p=a/p\bmod 1$.
If $\tilde{K}$ is a generator is a generators in $H_1(L(p,q))$, $L(p,q)_0$ is homeomorphic to the exterior of a knot in a homology sphere. Suppose that the integral $b$-surgery produces a homology sphere $Y$ with the dual knot $K$. Then $a[m]+p[l]$ represents a longitude of $K$ and $b[m]+[l]$ is a meridian of $K$. For $\epsilon=\pm1$, if $(\epsilon(b[m]+[l]),a[m]+p[l])$ is the oriented basis with the same orientation as $([m],[l])$, then $\epsilon(bp-a)=1$ holds. The Dehn surgery of $K$ produces $L(p,q)$.
If the positive $p$-surgery $Y_p(K)$ is homeomorphic to $L(p,q)$, then $a\equiv 1\bmod p$.
Since $p\epsilon(b[m]+[l])+a[m]+p[l]$ must be homologous to $\pm [m]$, $\epsilon=-1$ holds. Thus $a=bp-\epsilon\equiv 1\bmod p$ holds.
As a result we have the following corollary:
\[1deter\] For a longitude $l$ of a homologically generating knot $\tilde{K}\subset L(p,q)$, there exists an integer $b$ uniquely such that $b$-surgery of $\tilde{K}$ is a homology sphere.
If the integral $b$-surgery of $\tilde{K}$ with respect to $l$ is a homology sphere, then $pb+1=a$, where the $a[m]+p[l]$ is null-homologous in $H_1(L(p,q)_0)$. Thus $b$ is uniquely determined.
Here we give a corollary.
\[positivesurgery\] Let $p$ be a positive integer. If $L(p,q)=Y_p(K)$ with dual class $k$ and with a homology sphere $Y$, then $k^2=q\bmod p$.
Suppose that $\tilde{K}$ is the dual knot of the lens space surgery with $[\tilde{K}]=k[c]$ for the core circle $c$ in $V_1$ of genus one Heegaard decomposition $V_0\cup V_1$ of $L(p,q)$, where $V_1$ is the neighborhood of the dual knot of $S^3_{p/q}(\text{unknot})$. Thus comparing the linking form $1/p=\ell k(k[c],k[c])=k^2\cdot \ell k([c],[c])=k^2q'/p\bmod p$, where the self-linking number of $[c]$ is $q'$, where $q'$ is the inverse of $q$ in $({\mathbb Z}/p{\mathbb Z})^\times$. Thus $q=k^2\bmod p$ holds.
Here we give definitions of (1,1)-knot and simple (1,1)-knot.
\[definitionofsimple11knot\] Let $L$ be a 3-manifold with Heegaard genus at most 1. Let $V_0\cup V_1$ be a genus 1 Heegaard splitting of $L$. We say that a knot $K'$ in $L$ is a (1,1)-knot if $K'$ is isotopic to $K$ satisfying the following conditions:
1. (transversality condition) $K$ transversely intersects with the Heegaard surface $\partial V_0=\partial V_1$.
2. ($\partial$-parallel condition) $K_i=V_i\cap K$ is a proper-embedded [*boundary-parallel*]{} arc in $V_i$, i.e., there exists an embedded disk $D_i$ in $V_i$ with $\partial D_i=K_i\cup {\alpha}_i$ where ${\alpha}_i$ is a simple arc in $\partial V_i$.
We call such a position of the knot $K$ in $L$ [*(1,1)-position*]{}. The (1,1) means that the manifold has a genus one Heegaard decomposition and the knot is 1-bridge.
Furthermore, we define a [*simple*]{} (1,1)-knot.
Let $K$ be a (1,1)-knot in the genus 1 Heegaard splitting $V_0\cup V_1$ of lens space $L(p,q)$. Let $D_i$ be the meridional disk of $V_i$. Let $\alpha,\beta$ be $\partial D_0$ and the image of $\partial D_1$ on $\partial V_0$ respectively ([Figure]{} \[simple\]). We assume that ${\alpha}$ and ${\beta}$ are minimally intersecting. Then if $K$ is isotopic to the union of the two arcs each of which is embedded in $D_i$, we call such a knot $K$ a [*simple (1,1)-knot*]{} in $L(p,q)$. The embedded arc $K_0$ in $V_0$ is described as in [Figure]{} \[simple\].
Note that non-simple (1,1)-knot cannot be [*simultaneously*]{} embedded in the meridional disks of the genus one Heegaard decomposition.
As in [Figure]{} \[simple\], the minimal intersection ${\alpha}\,\cap\, {\beta}$ consists of $p$ points. We name the points as $\{0,1,\cdots, p-1\}$ in order. Suppose that the arc $K_0$ of simple $(1,1)$-knot in $V_0$ is connecting the points $0$ and $k$ in the meridian disk. Then we denote such a simple (1,1)-knot by $\tilde{K}_{p,q,k}$.
![A simple (1,1)-knot $K_0$ in $V_0$.[]{data-label="simple"}](simple-eps-converted-to.pdf)
By the definition of simple (1,1)-knot, $\tilde{K}_{p,q,k}$ is isotopic to $\tilde{K}_{p,q,-k}$. We denote $\tilde{K}_{p,k^2,k}$ by $\tilde{K}_{p,k}$ and the integral ${\mathbb Z}$HS surgery of $\tilde{K}_{p,k}$ by $Y_{p,k}$. We denote the dual knot of the surgery by $K_{p,k}\subset Y_{p,k}$. Hence, we can write the surgery as $(Y_{p,k})_{p}(K_{p,k})=L(p,k^2)$.
By exchanging the role of the solid tori, the dual class $k$ is changed as $k\mapsto \pm k^{-1}$ in $({\mathbb Z}/p{\mathbb Z})^\times$. Hence, we have the ambiguity $\mathcal{K}:=\{\pm k^{\pm1}\}\subset \{0,1,\cdots, p-1\}$ as a set of the smallest positive remainders. We choose $k$ as the minimal element in $\mathcal{K}$ and $k_2$ is the second minimal one with $kk_2=\pm1\bmod p$ (possibly $k=k_2$). Throughout the paper we define $\frak{c}$ to be $(k-1)(k+1-p)/2$.
Brieskorn homology spheres.
---------------------------
Brieskorn sphere $\Sigma(a_1,a_2,a_3)$ is defined to be $\{(z_1,z_2,z_3)\in {\mathbb C}^3|z_1^{a_1}+z_2^{a_2}+z_3^{a_3}=0\}\cap S^5$. $\Sigma(a_1,a_2,a_3)$ is a Seifert 3-manifold with $S^2$ base space with three multiple fibers. The general properties of Seifert manifolds lie in [@Or]. The Seifert manifold is presented by [*multiplicity*]{} $(a,b)$ for the multiple fiber [*Euler number*]{} $e$, where the numbers $(a,b)$ are coprime integers. Hence, the Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ is presented as follows: $$S(e,(a_1,b_1),(a_2,b_2),(a_3,b_3)).$$ We call the presentation [*Seifert data*]{}. If the Brieskorn sphere $\Sigma(a_1,a_2,a_3)$ is a homology sphere, then integers $(a_1,a_2,a_3)$ are pairwise relatively prime. The presentation satisfies with $$e-\sum_{i=1}^3\frac{b_i}{a_i}=\frac{\pm1}{a_1a_2a_3}$$ and is described by Kirby diagram as in [Figure]{} \[236n-2\]. For example, $\Sigma(2,2s+1,2(2s+1)n\pm1)$ admits the following Seifert data:
$$\mp S(1,(2,1),(2s+1,s),(2(2s+1)n\pm1,n)).$$
![Kirby diagram of $\Sigma(a_1,a_2,a_3)$ with Seifert data $S(e,(a_1,b_1),(a_2,b_2), (a_3,b_3))$.[]{data-label="236n-2"}](brieskorn-eps-converted-to.pdf)
Knot diagrams of $\tilde{K}_{p,k}$ and $K_{p,k}$.
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The purpose of this section is to give a process to obtain knot diagrams of $\tilde{K}_{p,k}$ and $K_{p,k}$.
Involution on $L(p,k^2)$ and $\tilde{K}_{p,k}$.
-----------------------------------------------
To describe $\tilde{K}_{p,k}$ in a lens space $L(p,k^2)$, we use the involution $\iota$ on $L(p,q)$. It is a well-known that any lens space is a double branched cover along a 2-bridge knot or link in $S^3$. Consider the genus 1 Heegaard splitting of a lens space $V_0\cup V_1$. The Heegaard suface $T=\partial V_0=\partial V_1$ is an invariant set i.e., $\iota(T)=T$. The restriction of the involution $\iota$ to $T$ is the same thing as the $-1$-multiplication on ${\mathbb C}/({\mathbb Z}+i{\mathbb Z})$ (topologically $180^\circ$-rotation as in [Figure]{} \[pillow\]). We set $\bar{V}_i=V_i/\iota$. Each $\bar{V}_i$ is homeomorphic to a 3-ball. The set $\gamma_i$ is the image of the fixed points set $\text{Fix}(V_i,\iota)$ in $\bar{V}_i$, which are two properly embedded arcs (the red dashed lines in [Figure]{} \[pillow\]). We call such a 3-ball $\bar{V}_0$ [*pillowcase*]{}. The boundary points $\partial \gamma_i$ are four points and we call the points [*vertices*]{} of the pillowcase. As a result, the quotient $L(p,q)/\iota$ gives a genus 0 Heegaard splitting $\bar{V}_0\cup_{S^2}\bar{V}_1$ of $S^3$. We set the gluing map as $g_{p,q}:\partial \bar{V}_1\to \partial \bar{V}_0$. The map is isotopic to the identity map as a homeomorphism on $S^2$.
We consider $\bar{V}_0$. We denote the positions of the images of two parallel meridians $\alpha_0,\alpha_1$ and two parallel longitudes $l_0,l_1$ in $V_0$ by $\bar{\alpha}_0$(top), $\bar{\alpha}_1$(bottom), $\bar{l}_0$(left), $\bar{l}_1$(right), which are four arcs. The union of the four arcs makes a [*rectangle*]{} $R$ on $\partial \bar{V}_0$ as the first row in [Figure]{} \[pillow\]. The four vertices of $R$ are the vertices of the pillowcase. The $\bar{l}_i,\bar{\alpha}_i$ and $R$ are used as ‘coordinate’ of pillowcase. We can also see a similar picture in $\bar{V}_1$. We denote the images of two parallel meridians $\beta_0,\beta_1$ in $V_1$ as in [Figure]{} \[pillow\] by $\bar{\beta}_0,\bar{\beta}_1$.
![The involution of a lens space and a branched locus of $S^3$. The red curves are fixed point sets, an example of the measure of the pillowcase with slope $p/q=\pm5/4$[]{data-label="pillow"}](pillow-eps-converted-to.pdf)
Let $K$ be a simple (1,1)-knot $\tilde{K}_{p,q,k}$. Let $K_i$ be the same thing as that in Definition \[definitionofsimple11knot\]. The $K_i$ may be chosen to be an invariant arc as indicated in the left of above two pictures in [Figure]{} \[pillow\]. Then, each arc $K_i$ is an invariant set with respect to $\iota$. Let $K'_i$ be the image of $K_i$ in $\bar{V}_i$. Then $K'_0$ (or $K'_1$) is an arc connecting a point in $\gamma_0$ (or $\gamma_1$) and a point of $\bar{\alpha}_0$ (or $\bar{\beta}_0$). We determine the point $\bar{\alpha}_0\cap \partial K_0'$. The image of $\bar{\beta}_0\cup \bar{\beta}_1$ by the gluing map $g_{p,q}$ is two lines in $\partial \bar{V}_0$ with slope $p/q$ as in [Figure]{} \[not1\]. We call $g_{p,q}(\bar{\beta}_0\cup\bar{\beta}_1)\subset \partial \bar{V}_0$ [*pillowcase slope*]{}. If $p/q>0$ then the pillowcase slope is called [*positive*]{} and if $p/q<0$ then [*negative*]{}. The inclined lines in [Figure]{} \[not1\] stand for pillowcase slope in $\partial \bar{V}_0$. On the $\bar{\alpha}_i$ there are $p+1$ intersection points with $g_{p,q}(\bar{\beta}_0\cup \bar{\beta}_1)$. In the same way we put $q+1$ points on $\bar{l}_i\subset \partial V_0$. These $2(p+q)$ points on the rectangle are called [*integral points*]{}. The points are named $0,1,\cdots, p$ on $\bar{\alpha}_i$ from the left in order and $0,1,\cdots, q$ on $\bar{l}_i$ from the top in order and we call this scale of the rectangle [*measure*]{}. As an example of measure see the third row of [Figure]{} \[pillow\]. The boundary points of $K'_0$ are $k\in \bar{\alpha}_0$ and a point in $\gamma_0$.
![The notation of pillow case description[]{data-label="not1"}](notation1-eps-converted-to.pdf){width="\textwidth"}
Pillowcase method {#pme}
-----------------
It is the main purpose in this section that by deforming the gluing map $g_{p,q}$ of two pillowcases into the identity map, we keep track of the image of the simple (1,1)-knot in $\bar{V}_0$. We call the following process [*pillowcase method*]{} here. For coprime integers $P,Q$ we consider a triple $(P/Q,\gamma,K')$ in the pillowcase $\bar{V}$, where $\bar{V}$ is a pillowcase $\bar{V}_0$ with rectangle $R$. The rectangle $R$ is the same meaning as the previous section. The pillowcase slope is $P/Q$, and $\gamma$ is a 2-component tangle whose end points are the four vertices in $\partial\bar{V}$. $\text{Int}(K')\subset \bar{V}\setminus \gamma$ is an edge connecting a point in $R$ and a point in $\gamma$. Here $\text{Int}$ means the interior. If the slope is positive, negative, $0$ or $\infty$, we write $+$, $-$, $0$ or $\infty$ respectively on the left top of $R$, as in [Figure]{} \[not1\]. If the integral point of $K'\cap R$ is $h$, then we write the number under the pillowcase. When the $K$ does not intersect with $R$, we write no number any more there. The second row of [Figure]{} \[not1\] describes a part of $\gamma$ and $K'$. The intersection $K'\cap R$ is an integral point $h$ in the measure of $\bar{\alpha}_0\cup \bar{l}_0\subset R$. Such a triple is called [*pillowcase triple*]{}.
Now we suppose pillowcase triple $(P/Q,\gamma,K')$ satisfies $|P|>|Q|$ and the one end point $K'\cap R$ is an integral point in $\bar{\alpha}_0$, i.e., it is in $\{0,1,2,\cdots, p\}$. The picture is either of the top pictures in [Figure]{} \[pslope\] or \[nslope\] depending on the sign of $P/Q$. The black boxes stand for some tangles of $\gamma$ and $K'$. We divide $h$ by $Q$ as follows: $$h=bQ-h'.$$ Here $b,h'$ are some integers. Untwisting the end point $h\in \bar{\alpha}_0$ along the slope $P/Q$, we get the one pattern of four cases as in [Figure]{} \[pslope\] or \[nslope\]. Here if necessary, we turn the pillowcase by $180^\circ$ so that the end point $K'\cap R$ can be moved to $|h'|\in \bar{l}_0$.
We suppose the pillowcase triple satisfies $|P|<|Q|$. The one end point $K'\cap R$ is an integral point in $\bar{l}_0$ as in [Figure]{} \[pslopep\] or \[nslopep\]. In the same way as the case of $|P|>|Q|$, we divide $h$ by $P$ as follows: $$h=bP-h'.$$ Here $b,h'$ are some integers. We untwist the arc $K'$ along the slope on $\partial \bar{V}$ so that the boundary of $K'\cap R$ lies in $\bar{\alpha}_0\cup \bar{\alpha}_1$. Similarly, if necessary, we turn the pillowcase by $180^\circ$ so that the end point $K'\cap R$ is an integral point in $\bar{\alpha}_0$. The pattern is one of four in [Figure]{} \[pslopep\] or \[nslopep\]. These processes are called [*untwisting process of*]{} $K'$. As a result, by untwisting process of $K'$, for the pillowcase triple $(P/Q,\gamma,K')$ with $P/Q\neq 0$ and $\infty$, $K'\cap R\in \bar{l}_0$ if $|P/Q|>1$ or $K'\cap R\in \bar{\alpha}_0$ if $|P/Q|<1$.
![The positive slope pillowcase in the case of $|P/Q|>1$ and the untwisting by $h=bQ-h'$.[]{data-label="pslope"}](step-eps-converted-to.pdf)
![The negative slope pillowcase in the case of $|P/Q|>1$ and the untwisting by $h=bQ-h'$.[]{data-label="nslope"}](stepp-eps-converted-to.pdf)
![The positive slope pillowcase in the case of $|P/Q|<1$ and untwisting by $h=bP-h'$.[]{data-label="pslopep"}](steppp-eps-converted-to.pdf)
![The negative slope pillowcase in the case of $|P/Q|<1$ and untwisting by $h=bP-h'$.[]{data-label="nslopep"}](stepppp-eps-converted-to.pdf)
![Untwisting the pillowcase with $|P/Q|>1$.[]{data-label="undopi"}](undopi-eps-converted-to.pdf)
Next, we untwist the pillowcase. We suppose that $|P/Q|>1$ and $P/Q\neq\infty$. See [Figure]{} \[undopi\]. Divide $P$ by $Q$ and we get $P=aQ-P'$, where $a$ and $P'$ are some integers. Here we untwist the pillowcase in $\bar{V}$ by half $|a|$-times along the central horizontal line of $\bar{V}$ in the $\text{sgn}(-P/Q)$ direction. Hence, the new pillowcase slope becomes $-P'/Q$ (the bottom pictures in [Figure]{} \[undopi\]). The box with an integer stands for the half twist by the number.
For the assumption $|P/Q|<1$, we untwist the pillowcase similarly. Divide $Q$ by $P$ and we get $Q=aP-P'$, where $a,P'$ are some integers. Then we deform the pillowcase as the second row in [Figure]{} \[undopi2\] according to the sign of $-P/P'$. These pictures are the cases of $P/Q>0$. In the case of $P/Q<0$, we can also describe a similar picture.
![Untwisting the pillowcase with $|P/Q|<1$.[]{data-label="undopi2"}](undopi2-eps-converted-to.pdf)
We iterate the untwisting process of $\bar{V}$ or $K'$ alternatively, $$(p/q,\gamma,K')\to (p/q,\gamma,K'')\to (p/p',\gamma',K'')\to (p/p',\gamma',K''')\to \cdots.$$ After several untwisting processes, we obtain $P/Q=0$ or $P/Q=\infty$. Then via the gluing map, we can push the other 2-component arcs $K_1'$ in $\bar{V}$ and gluing $\bar{V}_1$, we get a 2-bridge knot or link attaching an arc in $S^3$.
Expand the continued fraction of $p/q$ where $p=p_1,q=p_2$ as follows: $$p_i=a_ip_{i+1}-p_{i+2}\ (i=1,\cdots, n-1),$$ and $$p_n=a_np_{n+1},|p_{n+1}|=1.$$ Such a sequence $p_1,p_2,\cdots, p_{n+1}$ is not uniquely determined from $p/q$, while the existence is guaranteed by the Euclidean algorithm. Thus, an iteration of untwisting processes can deform any triple $(p/q,\gamma,K')$ into a triple with slope $1/0$ or $0/1$.
Namely, we obtain the continued fraction $$p/p_2=[a_1,a_2,\cdots, a_n]=a_1-\frac{1}{a_2-\frac{1}{a_3-\cdots-\frac{1}{a_n}}}.$$ Here we denote $\epsilon_i=(-1)^{i-1}\text{sign}(a_i)$. Each $\epsilon_i$ corresponds to the signature of the pillowcase slope.
Here we consider the sequences $b_1,b_2,\cdots, b_n$ and $k=h_1,h_2,\cdots, h_n$ defined as follows: $$k=b_1p_2-h_2,h_i=b_{i}p_{i+1}-h_{i+1},h_{n}=b_{n}p_{n+1}.$$ Hence, we have $k=\sum_{i=1}^{n}b_i(-1)^{i-1}p_{i+1}$. For the lens surgery parameter $(p,k)$ we call $(a_1,\cdots,a_n)$ its [*$a$-sequence*]{}, $(b_1,\cdots, b_{n})$ its [*$b$-sequence*]{}, and $(h_1,\cdots, h_n)$ its $h$-sequence. In [Table]{} \[continuedpo\] and \[continuedpo2\], we give the $a$-sequences (of continued fractions of $p/q$), and $b$-sequences for lens surgeries in [Table]{} \[po\]: $$\Sigma(2,3,5)_p(K_{p,k})=L(p,q).$$ The $p,q$ in [Table]{} \[continuedpo\] correspond to $p_1$ and $p_2$ for the continued fraction of $p/q$. Note that the $b$-sequence for $\Sigma(2,3,7)$ is the one of $b$-sequence with the type multiplied by $-1$ as a vector.\
[**Example: An example $S^3_5(\text{trefoil})=L(5,2^2)$: pillowcase, untwisting of pillowcase or $K'$, Kirby calculus.**]{} Here we give an example $S_5^3(\text{trefoil})=L(5,2^2)$. This example is well-known as the first non-trivial lens space surgery in $S^3$. The continued fraction is $$5/4=1-1/(-4)=[1,-4].$$ The pillowcase method is (1)-(5) in [Figure]{} \[expi\]. We give the simple (1,1)-knot $\tilde{K}_{5,2}$ in (6)-(7). The diagram of $K_{5,2}\subset S^3$ is given in (8)-(9). We explain each process in detail.
First, we give a simple (1,1)-knot in $L(5,2^2)$ with dual class $2$ in the pillowcase: (1) in [Figure]{} \[expi\]. The red curves in [Figure]{} \[expi\] are $\gamma$. We untwist $K'$ as the deformation from (1) to (2) in [Figure]{} \[expi\]. We untwist the pillowcase (from the (2) to (3)). From (3) we obtain (5) by untwisting pillowcase and $K'$ in the similar way. Pushing $K'_1$ in $\bar{V}$, we obtain (6). The double branched cover along the 2-bridge knot is (7). The good reference from (6) to (7) is [@MW]. This diagram presents $Y_{5,2^2}=S^3$. The $-1$-framed component in (7) is $\tilde{K}_{5,2}$. The 0-framed meridian $\tilde{K}_{5,2}$ in (8) is the trefoil with $+5$-framing as in (9) by trivializing the diagram other than 0-framed meridian.
![Example: The pillowcase method for $(5/4,\gamma,K')$, the double covering along the 2-bridge knot or link(red curve), and the knot diagram of $K_{5,2}$ in $S^3$.[]{data-label="expi"}](examplepillow-eps-converted-to.pdf)
type $p$ $q$ $a$-sequence
-------------- ------------------------ --------------------- -------------------------------
$\A_1$ $14\ell^2+7\ell+1$ $7\ell^2+7\ell+1$ $[2,\ell+1,7,-\ell]$
$\A_2$ $20\ell^2+15\ell+3$ $5\ell^2+5\ell+1$ $[4,\ell+1,5,-\ell]$
B $30\ell^2+9\ell+1$ $6\ell^2+3\ell$ $[5,\ell+1,3,-\ell+1,2]$
$\C_1$ $42\ell^2+23\ell+3$ $7\ell^2+5\ell+1$ $[6,\ell,-3,2,-\ell]$
$\C_2$ $42\ell^2+47\ell+13$ $7\ell^2+9\ell+3$ $[6,\ell+1,2,-3,-\ell-1]$
${\Delta}_1$ $52\ell^2+15\ell+1$ $13\ell^2+7\ell+1$ $[4,\ell,-3,4,-\ell]$
${\Delta}_2$ $52\ell^2+63\ell+19$ $13\ell^2+19\ell+7$ $[4,\ell+1,4,-3,-\ell-1]$
$\E_1$ $54\ell^2+15\ell+1$ $27\ell^2+21\ell+3$ $[2,\ell+1,3,2,6,-\ell]$
$\E_2$ $54\ell^2+39\ell+7$ $27\ell^2+33\ell+9$ $[2,\ell+1,6,2,3,-\ell]$
$\F_1$ $69\ell^2+17\ell+1$ $46\ell^2+19\ell+2$ $[2,2,\ell+1,4,6,-\ell]$
$\F_2$ $69\ell^2+29\ell+3$ $46\ell^2+27\ell+4$ $[2,2,\ell+1,6,4,-\ell]$
${\Gamma}_1$ $85\ell^2+19\ell+1$ $34\ell^2+11\ell+1$ $[3,2,\ell+1,3,6,-\ell]$
${\Gamma}_2$ $85\ell^2+49\ell+7$ $34\ell^2+23\ell+4$ $[3,2,\ell+1,6,3,-\ell]$
$\H_1$ $99\ell^2+35\ell+3$ $22\ell^2+9\ell+1$ $[5,2,\ell+1,3,4,-\ell]$
$\H_2$ $99\ell^2+53\ell+7$ $22\ell^2+13\ell+2$ $[5,2,\ell+1,4,3,-\ell]$
$\I_1$ $120\ell^2+16\ell+1$ $24\ell^2+8\ell$ $[5,\ell+1,2,3,2,-\ell+1,3]$
$\I_2$ $120\ell^2+20\ell+1$ $40\ell^2+20\ell+1$ $[3,\ell+1,2,6,2,-\ell+1,2]$
$\I_3$ $120\ell^2+36\ell+3$ $24\ell^2+12\ell+1$ $[5,\ell+1,2,4,2,-\ell+1,2]$
J $120\ell^2+104\ell+22$ $24\ell^2+16\ell+3$ $[5,-\ell-1,-2,-3,-2,\ell,3]$
K $191$ $34$ $[6,2,-2,-3,-3]$
: Table of continued fractions of $p/q$ for $L(p,q)$ in [Figure]{} \[po\].[]{data-label="continuedpo"}
type $(p_3,\cdots, p_n,p_{n+1})$ $k$
-------------- ------------------------------------------------ -------------
$\A_1$ $(7\ell+1,\ell,-1)$ $7\ell+2$
$\A_2$ $(5\ell+1,\ell,-1)$ $5\ell+2$
B $(6\ell-1,2\ell-1,-2,-1)$ $6\ell+1$
$\C_1$ $(7\ell+3,5\ell+2,3\ell+1,-1)$ $7\ell+2$
$\C_2$ $(7\ell+5,3\ell+2,2\ell+1,\ell,-1)$ $7\ell+4$
${\Delta}_1$ $(13\ell+3,9\ell+2,5\ell+1,\ell,-1)$ $13\ell+2$
${\Delta}_2$ $(13\ell+9,3\ell+2,2\ell+1,\ell,-1)$ $13\ell+8$
$\E_1$ $(27\ell+5,11\ell+2,6\ell+1,\ell,-1)$ $27\ell+4$
$\E_2$ $(27\ell+11,5\ell+2,3\ell+1,\ell,-1)$ $27\ell+10$
$\F_1$ $(23\ell^2+21\ell+3,23\ell+4,6\ell+1,\ell,-1)$ $23\ell+3$
$\F_2$ $(23\ell^2+25\ell+5,23\ell+6,4\ell+1,\ell,-1)$ $23\ell+5$
${\Gamma}_1$ $(17\ell^2+14\ell+2,17\ell+3,6\ell+1,\ell,-1)$ $17\ell+2$
${\Gamma}_2$ $(17\ell^2+20\ell+5,17\ell+6,3\ell+1,\ell,-1)$ $17\ell+5$
$\H_1$ $(11\ell^2+10\ell+2,11\ell+3,4\ell+1,\ell,-1)$ $11\ell+2$
$\H_2$ $(11\ell^2+12\ell+3,11\ell+4,3\ell+1,\ell,-1)$ $11\ell+3$
$\I_1$ $(24\ell-1,15\ell-1,6\ell-1,3\ell-2,3,1)$ $12\ell+1$
$\I_2$ $(40\ell+2,22\ell+1,4\ell,2\ell-1,2,1)$ $20\ell+2$
$\I_3$ $(24\ell+2,14\ell+1,4\ell,2\ell-1,2,1)$ $12\ell+2$
J $(-24\ell-7,15\ell+4,-6\ell-1,3\ell-1,3,1)$ $12\ell+5$
K $(13,-8,3,-1)$ $15$
: Tables of the sequence $(p_3,p_4,\cdots p_{n+1})$ for the continued fractions of $p/q$ for $L(p,q)$ in [Figure]{} \[po\] and $b$-sequences.[]{data-label="continuedpo2"}
\
type $b$-sequence
------ --------------------
$(0,-1,0,1)$
$(0,-1,0,1,0)$
$(0,-1,0,0,1)$
$(0,0,1,0,0,-1)$
$(0,0,1,0,-1,0,0)$
$(0,-1,0,-1,1)$
: Tables of the sequence $(p_3,p_4,\cdots p_{n+1})$ for the continued fractions of $p/q$ for $L(p,q)$ in [Figure]{} \[po\] and $b$-sequences.[]{data-label="continuedpo2"}
$K_{p,k}$ in $\Sigma(2,3,5)$. {#235secion}
-----------------------------
In this section we concretely give knot diagrams of $\tilde{K}_{p,k}$ and $K_{p,k}$ for $(p,k)$ given in [Table]{} \[po\] according to the pillowcase method (untwisting pillowcase and $K'$) which is explained in the previous subsection. The simple but concrete example is already illustrated in [Figure]{} \[expi\]. Here we carry out the positive $\ell$ cases only. The descriptions of negative $\ell$ cases are similar. The difference is the direction of the untwisting, hence we omit them.
### To find $\tilde{K}_{p,k}$ in the pillowcase (Appendix 1).
In Appendix 1 we untwist the pillowcase and $K'$ for $K_{p,k}$ to end up the pillowcase with the slope $0$ or $\infty$ according to the $a$-sequence in [Table]{} \[continuedpo\]. For example, [Figure]{} \[2bribandA\] presents pillowcase method for $\A_1$ type knots. We omit the writing slope lines (as (1)-(4) in [Figure]{} \[expi\]) and describe $(P/Q,\gamma,K')$ and $h$-sequence only. The first picture is $((14\ell^2+7\ell+1)/(7\ell^2+7\ell+1),\gamma,K')$ and $7\ell+2$ and $\gamma$ is the 2-component arcs with the four end points attached at the vertices of pillowcase and $K'$ is the segment from one component of $\gamma$ to a point in $\bar{\alpha}_0$. We untwist $K'$ and the picture becomes the 2nd picture, and untwist the pillowcase (3rd picture). By iterating these methods alternatively, we obtain the last picture in [Figure]{} \[2bribandA\].
In the similar way to this case, we obtain each pillowcase with slope $0$ or $\infty$ according to the $a$-sequence of the type.
### To find $\tilde{K}_{p,k}$ in $L(p,k^2)$ (Appendix 2).
We move our procedure to Appendix 2. By using the Montesinos trick, we develop $K'$ from the last picture of Appendix 1 along the 2-bridge knot or link. Then we obtain a knot $\tilde{K}_{p,k}$ in the chain of unknots. The framing is uniquely determined so that the surgery can obtain a homology sphere.
### To find $K_{p,k}$ in $Y_{p,k}$ (to the first pictures in Appendix 3) {#0framedmeridian}
The last pictures in Appendix 2 are Kirby diagrams of homology sphere $Y_{p,k}$ for [Table]{} \[po\]. We give a knot diagram of $K_{p,k}$ by the usual way. By attaching 0-framed meridian to $\tilde{K}_{p,k}$ and trivializing the diagram other than the 0-framed meridian to deform into the diagram of $S(1,(2,1),(3,1),(5,1))$. Then we obtain a knot in the diagram. This is the knot diagram $K_{p,k}$ in the Poincaré homology sphere $\Sigma(2,3,5)$ ($K_{p,k}$ in Appendix 3). The knot diagrams are described to extend the underlying space to other homology spheres later on (Section \[extend\]).
### To check $\Sigma(2,3,5)_p(K_{p,k})=L(p,k^2)$.
To check that the first diagrams in Appendix 3 is $K_{p,k}$ in the case of $Y_{p,k}=\Sigma(2,3,5)$, we can do the Kirby calculus for the Dehn surgery descriptions along the processes in Appendix 3. Each last graph in the sequence of figures is a plumbing diagram. This diagram is defined in [@Sav]. Each of resulting linear plumbing diagrams gives the lens space $L(p,k^2)$. This means that the surgeries are all positive integral surgeries in $\Sigma(2,3,5)$ by using Lemma \[positivesurgery\] and $b$-sequence. This fact can be also checked by [@MT5].
[Table]{} \[processtable\] presents the index of the figure numbers per each type. As $b$-sequences in [Table]{} \[continuedpo2\] are indicated, we have only to do a sequence of processes, for six patterns: A, B, CDE, FGH, IJ, and K types. We call the plumbing diagram containing X type surgery [*X type plumbing diagram*]{}.
Type Pillowcase $\tilde{K}_{p,k}$ $K_{p,k}$ Extension
------ ----------------- ------------------- ------------- -----------
A \[2bribandA\] \[bandsumA\] \[DualA\] \[extA\]
B \[2bribandB\] \[bandsumB\] \[DualB\] \[extB\]
CDE \[2bribandCDE\] \[bandsumCDE\] \[DualCD\] \[extE\]
FGH \[2bribandFGH\] \[bandsumFGH\] \[DualFGH\] \[extE\]
I \[2bribandI\] \[bandsumI\] \[DualI\] \[extI\]
J \[2bribandJ\] \[bandsumJ\] \[DualJ\] \[extJ\]
K \[2bribandK\] \[bandsumK\] \[DualK\] \[extK\]
: The figure numbers for the processes to obtain $\tilde{K}_{p,k}$ and $K_{p,k}$.[]{data-label="processtable"}
$K_{p,k}$ in $\Sigma(2,3,7)$. {#237section}
-----------------------------
Here we prove Theorem \[237casethm\].
The Seifert data of $\Sigma(2,3,7)$ is $-S(1,(2,1),(3,1),(7,1))$. Thus we can give Kirby diagram of $\Sigma(2,3,7)$ by replacing the multiplicity $5$ in $\Sigma(2,3,5)$ with $7$ and reversing the orientation.
By doing the pillowcase method (untwisting pillowcase and $K'$ (in the sense of Section \[pme\])) for $\tilde{K}_{p,k}$ in the list of [Table]{} \[po\], we obtain the pillowcases corresponding to [Figure]{} \[2bribandA\] to \[2bribandK\]. These knots are realized as $K_{p,k}$ in $\Sigma(2,3,7)$, by seeing the $b$-sequence. Appendix 2 and 3 include these families.
Finally, we prove that the surgeries are all positive integral surgeries. For example, by substituting $(7,3)$ for $(a,b)$ in [Figure]{} \[bandsumA\], we get the following continued fraction $$[2,\ell+1,9,-\ell]=(18\ell^2+9\ell+1)/(9\ell^2+9\ell+1),$$ $$(p_2,p_3,p_4,p_5)=(9\ell^2+9\ell+1,9\ell+1,\ell,-1).$$ The lens space is $L(18\ell^2+9\ell+1,9\ell^2+9\ell+1)=L(18\ell^2+9\ell+1,-(9\ell+2)^2)$. Here recall that this type is A. The $b$-sequence is $(0,1,0,-1)$ for the case of $\Sigma(2,3,5)$. In the present case $(0,-1,0,1)$ should be the $b$-sequence because the orientation is the opposite direction. Thus the dual class is $k=9\ell+2$. Since this surgery is performed in $-\Sigma(2,3,7)$, Lemma \[positivesurgery\] tells us that this family gives positive integral surgeries. By doing the same procedure, the families in [Table]{} \[lens237\] are all positive surgeries.
Lens space surgeries in Brieskorn homology spheres. {#extend}
===================================================
In Appendix 3, we generalize the lens surgeries to plumbed 3-manifold surgeries in plumbed homology spheres other than $S^3$ and $\Sigma(2,3,6\pm1)$. Hence, it is easy to give some lens surgeries in Brieskorn homology spheres or several plumbed homology spheres. We give the graph deformation as in [Table]{} \[graphdeformation\]. Each deformation in the list means that some integral Dehn surgery of a knot in the left plumbing 3-manifold is deformed into the right plumbed 3-manifold. In this section we compute lens space surgeries over Brieskorn homology spheres.
Lens space surgeries in Brieskorn homology spheres derived from the list of [Table]{} \[po\]. {#brieskornsection}
---------------------------------------------------------------------------------------------
### A type lens space knots. {#Ataipu}
The pillowcase method of an A type dual knot $\tilde{K}_{p,k}$ is basically given in [Figure]{} \[2bribandA\]. By using double covering of 2-bridge knots or links (the Montesinos trick as in [@MW]), we get $\tilde{K}_{p,k}$ in lens spaces as in [Figure]{} \[bandsumA\]. Here the cases of $(a,b)=(3,5)$ and $(5,3)$ correspond to $\A_1$ and $\A_2$ respectively. By Montesinos trick for the 2-bridge knot (or link) with arc $K'$ (in the sense of Section \[pme\]) we obtain knot $\tilde{K}_{p,k}$ in the lens space [Figure]{} \[bandsumA\]. By Corollary \[1deter\], the $0$-framing in [Figure]{} \[bandsumA\] for integral ${\mathbb Z}$HS-surgery is uniquely determined. Thus, for the slope, we have only to compute the order of $H_1$. Hence, the knot $K_{p,k}$ is described as in the left picture in [Figure]{} \[DualA\]. Here we attach 0-framed meridian of $\tilde{K}_{p,k}$ in [Figure]{} \[bandsumA\] and move the Kirby diagram other than the meridian to the diagram as in [Figure]{} \[236n-2\]. This is already explained as the first example in Section \[0framedmeridian\].
These surgeries are generalized to knots in the Seifert manifolds as the left picture in [Figure]{} \[acde\]. If these diagrams present homology spheres, then the integer $a$ must be odd number because another multiplicity is $2$. These diagrams give Brieskorn homology spheres as in [Table]{} \[s\]. Note that to get the usual orientations of the Brieskorn homology spheres, we must change the orientation of the diagram, if necessary. Then there are examples of the following Brieskorn homology spheres with A type. Here $s,s',m$ are positive integers. As examples satisfying these conditions, we illustrate the below.
\[we\] There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres
- $\Sigma(2,2s+1,2(2s+1)\pm1)$\
($a=2(2s+1)\pm1$ and $b=(2s+1)/s$).
- $\Sigma(2,2s\pm1,2am-s)$\
($a=2s\pm1$. $b=(2am-s)/((a-2)m-s')$, and $s=2s'+1$)
- $\Sigma(2,2s\pm1,2am+s)$\
($a=2s\pm1$. $b=(2am+s)/((a-2)m+s')$, and $s=2s'+1$)
such that the resulting lens space $L(p,q)$ are $$[\beta_r,\cdots, \beta_1-1,\ell+1,a+2,-\ell],$$ where $b=[\beta_1,\cdots ,\beta_r]$.
That such a Brieskorn homology sphere includes $K_{p,k}$ is obtained by substituting the indicated values for the parameters $a,b$ in the extended Dehn surgery in [Figure]{} \[DualA\] in Appendix 3. Here note that one must change the orientation if necessary.
Here we illustrate the way to give the dual class for the example. The $b$-sequence of the lens surgery corresponds to the extension of the sequence of A type knot in [Table]{} \[continuedpo2\]. Therefore, such a knot is realized as $K_{p,k}$.
Here we compute the first example as above with the plus case only: $$p/q=[s,-1,-\ell-1,-2(2s+1)-3,\ell]=((4s^2+9s+5)\ell^2+(4s+5)\ell+1)/(4s+5)\ell^2$$ $$L(p,q)=L((4s^2+9s+5)\ell^2+(4s+5)\ell+1,((4s+5)\ell+2)^2)$$ $$(p_2,p_3,p_4,p_5,p_6)=((4s+5)\ell^2,-(4s+5)\ell^2-(4s+5)\ell-1,(4s+5)\ell+1,\ell,-1).$$ Thus the $b$-sequence is $$(b_1,b_2,b_3,b_4,b_5)=(0,0,1,0,-1).$$ Thus the dual class is $(4s+t)\ell+2$. This means that these $p$-surgeries over $\Sigma(2,2s+1,2(2s+1)+1)$ give lens spaces as above. Doing the pillowcase method along these data, we can obtain the straight extension of [Figure]{} \[2bribandA\].
The method of this proof of Proposition \[we\] is available for other propositions in this section. If we give no proof to them, it is regarded as omitting the proof.
![A and B type plumbing diagrams.[]{data-label="acde"}](AZS-eps-converted-to.pdf)
[**Proof of Proposition \[Joshua\] in the case of lens space knot in $\Sigma(2,5,7)$ with type A.**]{} The Seifert data of $\Sigma(2,5,7)$ is $-S(1,(2,1),(5,1),(7,2))$. We suppose that $B=[3,-2]$ and $a=5$ and $A=\emptyset$ on A type surgery in [Figure]{} \[graphdeformation\]. Then we can obtain a family of lens spaces with continued fraction: $$[-2,3,1,\ell+2,7,-\ell]=-\frac{35\ell^2+21\ell+3}{14\ell^2+7\ell+1}$$ $$L(35\ell^2+21\ell+3,14\ell^2+7\ell+1)=L(35\ell^2+21\ell+3,(7\ell+2)^2)$$ $$(p_2,p_3,p_4,p_5)=(14\ell^2+7\ell+1,7\ell^2,-7\ell-1,\ell,1)$$ and $b$-sequence is $(0,0,-1,0,1)$. Thus this is an A type surgery. From Corollary \[positivesurgery\] the surgeries are all positive integral surgeries on $\Sigma(2,5,7)$. $\Box$\
Specially, the case of $\ell=-1$ is the lens space surgery $\Sigma(2,5,7)_{17}(K)=L(17,15)$, which is appeared in Section \[b2=1examplesection\].
### B type lens space knots. {#Btaipu}
The pillowcase method and the double branched covering for B type dual knot $\tilde{K}_{p,k}$ are given in [Figure]{} \[2bribandB\] and \[bandsumB\]. Then B type plumbing diagram is the left in [Figure]{} \[acde\]. The B type Seifert data up to orientation is $$S(1,(2a-1,a),(b_1,b_2),(c_1,c_2)).$$ For example, the following Brieskorn homology spheres are included. Here $s,n$ are positive integers and $b=b_1/b_2$ and $c=c_1/c_2$ are rational numbers.
There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres
- $\Sigma(2,2s+1,2(2s+1)n\pm1)$ ($a=-s$, $b=2$, and $c=2(2s+1)\pm1/n$)
- $\Sigma(3,12s-8,18s-13)$ ($a=-9s+7$, $b=3$ and $c=(12s-8)/(2s-1)$)
- $\Sigma(3,12s-8,18s-11)$ ($a=-9s+6$, $b=3$ and $c=(12s-8)/(2s-1)$)
- $\Sigma(3,6s-1,18s-5)$ ($a=-9s+3$, $b=3$ and $c=(6s-1)/s$)
- $\Sigma(3,6s-1,18s-1)$ ($a=-9s+1$, $b=3$ and $c=(6s-1)/s$)
- $\Sigma(3,6n-1,3(6n-1)s\pm x)$\
($a=-3n+1$, $b=3$, and $c=(3(6n-1)s\pm x)/((3n+1)s\pm y)$, where $x,y$ are some integers satisfying $(3n+1)x-3(6n-1)y=\pm1$).
- $\Sigma(3,6n+1,3(6n+1)s\pm x)$\
($a=-3n$, $b=3$, and $c=(3(6n+1)s\pm x)/((3n+2)s\pm y)$, where $x,y$ are some integers satisfying $(3n+2)x-3(6n+1)y=\pm1$).
such that the resulting lens spaces $L(p,q)$ are of the form $$p/q=[\beta_r,\cdots,\beta_1,\ell,a,2,-\ell+1,\gamma_1,\cdots,\gamma_s],$$ where $b_1/b_2=[\beta_1,\cdots, \beta_r]$ and $c_1/c_2=[\gamma_1,\cdots, \gamma_s]$.
We list examples for a negative integer $a$ and a rational $b$ with $b\le 3$ above, while one can also find B type lens space knots with positive $a$ or higher $b$. The family of B type with $a=-1$ is symmetric if one exchanges the roles of $b$ and $c$. Thus we could find a pair of two families of type B. If $a$ is not $-1$, then we can find two families of type B lens space knots. For example, deal with the first example in the list above. Then we can find two families $\B_1,\B_2$ as below. Below, we describe the resulting lens spaces obtained by type $\B_1$ and $\B_2$.
![$\B_1$: $a=-s$, $b=2(2s+1)\pm1/n$, $c=2$; $\B_2$: $a=-s$, $b=2$ and $c=2(2s+1)\pm1/n$ in $\Sigma(2,2s+1,2(2s+1)n\pm1)$.[]{data-label="ntb"}](newtypeb-eps-converted-to.pdf){width=".9\textwidth"}
We state the result here.
There exist double-primitive knots $B_{p,k}$ in the Brieskorn homology spheres $\Sigma(2,2s+1,2(2s+1)n\pm1)$ with the following coefficients $a=-s$ and, $b,c$ in [Figure]{} \[acde\].
- ($\B_1$ type)
- $\Sigma(2,2s+1,2(2s+1)n-1)$ ($b=2(2s+1)-1/n$, and $c=2$)
- $\Sigma(2,2s+1,2(2s+1)n+1)$ ($b=2(2s+1)+1/n$, and $c=2$)
- ($\B_2$ type)
- $\Sigma(2,2s+1,2(2s+1)n-1)$ ($b=2$, and $c=2(2s+1)-1/n$)
- $\Sigma(2,2s+1,2(2s+1)n+1)$ ($b=2$, and $c=2(2s+1)+1/n$)
such that the resulting lens spaces $L(p,q)$ is reprsented in [Figure]{} \[ntb\].
### CDE type lens space knots. {#CDEtaipu}
Next, consider CDE type lens space knots in Brieskorn homology spheres. The pillowcase and double branched covering methods are in [Figure]{} \[2bribandCDE\] and \[bandsumCDE\] respectively. In the deformation of CDE type in [Figure]{} \[graphdeformation\], we assume that the $A,B,C,D$ are all single chains of unknots. Then homology spheres of these types are of form of the left of [Figure]{} \[tcde\] for and $a,b,d,s\in {\mathbb Q}$.
![CDE type and FGH type plumbing diagrams.[]{data-label="tcde"}](CDEZS-eps-converted-to.pdf)
For example, the plumbing diagrams of the cases of $\{a,b\}=\{1,2\}$ and $(c,d)=(5,2)$ in [Figure]{} \[graphdeformation\] are $\Sigma(2,3,5)$ and the deformations are C type lens space knots in [Table]{} \[po\].
We assume that $a=1$ or $b=1$ and $d=m/(m-1)$ for an integer $m$ with $m\neq 1$. Furthermore, the following cases produce Brieskorn homology spheres.
\[1\] $a=1$ \[2\] $b=1$ \[3\] $c=\pm1$
Consider the cases \[1\] or \[2\]. The Seifert data are $S(1,(m,m-1),(b+1,1),(c_1,c_2))$ and $S(1,(m,m-1),(a+1,1),(c_1,c_2))$ respectively where $c=c_1/c_2$. In the case of $m=2$, we obtain CD type in [Table]{} \[po\]. For example, $\Sigma(2,5,7)$ ($m=2$, $b+1=5$, and $c=7/2$) or $\Sigma(3,10,13)$ ($m=3$, $b+1=10$ and $c=13/3$). These surgeries are called CD type. Furthermore, $\C_1{\Delta}_1$ type means the case of $a=1$ and $\C_2{\Delta}_2$ type means the case of $b=1$.
Consider the case of \[3\]. If $c=-1$ holds, then the Seifert data is $S(-1,(a,1),(b,1),(1,m-1))$. The possible case is $S^3$ only. In the case of $c=1$, the Seifert data is $S(1,(2m-1,m-1),(b,1),(a,1))$. For example, $\Sigma(3,4,5)$ ($m=3$, $a=3$, $b=4$) $\Sigma(4,5,9)$ ($m=-4$, $a=4$, and $b=5$). These cases are called $\E$ type. As a result we obtain the following proposition:
The Brieskorn homology spheres with the following Seifert data
1. $S(1,(m,m-1),(b+1,1),(c_1,c_2))$ ($\C_1{\Delta}_1$ type)
2. $S(1,(m,m-1),(a+1,1),(c_1,c_2))$ ($\C_2{\Delta}_2$ type)
3. $S(1,(2m-1,m-1),(b,1),(a,1))$ (E type)
contain $K_{p,k}$ and the resulting lens spaces $L(p,q)$ satisfy $$p/q=[\gamma_n,\gamma_{n-1},\cdots, \gamma_1,\ell,-m-1,b,-\ell],$$ $$p/q=[\gamma_n,\gamma_{n-1},\cdots, \gamma_1,\ell+1,a,-m-1,-\ell-1].$$ $$p/q=[2,\ell+1,a,-m,b-\ell]$$ respectively. In the case of (i) or (ii) the continued fraction $[\gamma_1,\gamma_2,\cdots,\gamma_n]$ presents $(c_1+c_2)/c_2$.
Furthermore we can give other examples.
There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres.
- $\Sigma(2,3,6n\pm 1)$ ($m=2$, $\{a,b\}=\{1,3\}$, and $c_1/c_2=6\pm 1/n$.) (C type)
- $\Sigma(2,2s+1,2(2s+1)\pm1)$
- ($m=2$ $\{a,b\}=\{1,2(2s+1)\pm1-1\}$ and $c=(2s+1)/s$) (D type)
- ($m=s+1$, $\{a,b\}=\{2,2(2s+1)\pm1\}$ and $c=1$ (E type)
The cases where the homology spheres with CDE type plumbing diagram are graph manifolds are remained. These cases will be written in later section.
Here we prove the remaining part of Proposition \[Joshua\].\
[**Proof of Proposition \[Joshua\] in the cases of $\Sigma(2,5,7)$, $\Sigma(3,4,5)$ with CDE type.**]{} In the case of $(c_1+c_2)/c_2=7/2+1=9/2=[5,2]$, $m=2$ and $b=4$ or $a=4$, the lens spaces $L(p,q)$ are $$[2,5,\ell,-3,4-\ell]=\frac{117\ell^2+37\ell+3}{65\ell^2+22\ell+2}$$ $$[2,5,\ell+1,4,-3,-\ell-1]=\frac{117\ell^2+145\ell+45}{65\ell^2+82\ell+26}$$ respectively. The dual classes are $13\ell+2$ and $13\ell+8$ because the $b$-sequence of these examples corresponds to that of the CD type surgery.
This Seifert presentation of $\Sigma(3,4,5)$ is $-S(1,(3,1),(4,1),(5,2))$. In the case of $m=3$ and $(a,b)=(3,4)$ or $(4,3)$ for E type lens space surgery we obtain the following lens spaces $L(p,q)$. $$p/q=[2,\ell+1,3,-3,4-\ell]=\frac{86\ell^2+37\ell+4}{43\ell^2+40\ell+7}$$ $$p/q=[2,\ell+1,4,-3,3-\ell]=\frac{86\ell^2+49\ell+7}{43\ell^2+46\ell+10}$$ The dual class is $43\ell+9$ and $43\ell+12$, because the $b$-sequences of E type surgery corresponds to the one of E type surgery.$\Box$
### FGH type lens space knots. {#FGHtaipu}
The case that the parameter $m$ in the CDE type surgery is equal to $1$, equivalently $d=\infty$ is remaining. This case also should become some lens space surgeries. Actually, the pillowcase and double covering method are [Figure]{} \[2bribandFGH\] and \[bandsumFGH\]. In [Figure]{} \[graphdeformation\] we extend deformations of FGH type surgery to deformations of plumbing diagrams. We call such a surgery [*FGH type surgery*]{}.
We assume that the links in $A,B$ and $C$ in [Figure]{} \[graphdeformation\] are single linear chains. Then we obtain the right picture in [Figure]{} \[tcde\]. Here $a,b,c'$ are rationals. Furthermore, if the resulting manifolds are lens spaces, then we can find $a,b\in {\mathbb Z}$ and $c'\in{\mathbb Q}$. In the case of $d=2$, $c'=d-1/c=(2c-1)/c$ for integers $c$, and $\{a,b,c\}=\{2,3,5\}$ or $\{2,3,7\}$ we have FGH type of $\Sigma(2,3,5)$, or $\Sigma(2,3,7)$ respectively. Thus the Seifert data yielding lens spaces are $S(1,(a,1),(b,1),(c_1,c_2))$, where $c=c_1/c_2$.
For example, we obtain the following proposition.
There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres
1. $\Sigma(2,3,6n\pm1)$ ($\{a,b\}=\{2,3\}$ and $c_1/c_2=6\pm1/n$) (H type),
2. $\Sigma(2,2s+1,2(2s+1)\pm1)$ ($\{a,b\}=\{2,2(2s+1)\pm1$ $c=2+1/s$) (G type)
such that the resulting lens spaces are $$[\mp n,6,2,\ell+1,3,4,-\ell],[\mp n,6,2,\ell+1,4,3,-\ell]$$ and $$[-s,2,2,\ell+1,3,2(2s+1)\pm1+1,-\ell],[-s,2,2,\ell+1,2(2s+1)\pm1+1,3,-\ell]$$ respectively.
### IJ type lens space knot. {#IJtaipu}
First, consider I type lens space knots in Brieskorn homology spheres. Due to [Figure]{} \[graphdeformation\], if the $A$ and $B$ are two unknots with rational framings $a,b$, integral framing $c$ and empty $C$, then the resulting manifold is a lens space. Then, the Seifert data of the Brieskorn homology spheres are $S(1,(a_1,a_2),(b_1,b_2),(c,1))$, where $a=a_1/a_2$ and $b=b_1/b_2$.
There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres $\Sigma(2,2s+1,2(2s+1)n\pm1)$ such that
1. $\Sigma(2,2s+1,2(2s+1)n\pm1)$ $(a_1/a_2=2(2s+1)\pm1/n$, $b_1/b_2=(2s+1)/s$, and $c=2$) ($\I_1$ type)
2. $\Sigma(2,2s+1,2(2s+1)\pm1)$
1. $a_1/a_2=2(2s+1)\pm 1$, $b_1/b_2=(2s+1)/s$ and $c=2$ ($\I_1$ type)
2. $a_1/a_2=(2s+1)/s$, $b_1/b_2=2$ and $c=2(2s+1)\pm 1$ ($\I_2$ type)
3. $\Sigma(2,3,6n\pm1)$
1. $a_1/a_2=6\pm1/n$, $b_1/b_2=3$ and $c=2$ ($\I_1$ type)
2. $a_1/a_2=6\pm1/n$, $b_1/b_2=2$ and $c=3$ ($\I_3$ type)
such that the resulting lens spaces $L(p,q)$ satisfy with $$\label{Itypecontinued}
p/q=[\alpha_n,\cdots, \alpha_1,\ell+1,-2,c,-2,-\ell,\beta_1,\beta_2\cdots, \beta_m],$$ where $a_1/a_2=[\alpha_1,\cdots, \alpha_n]$ and $b_1/b_2=[\beta_1,\beta_2,\cdots, \beta_m]$ for $\alpha_i,\beta_j\in {\mathbb Z}$.
Further, there are many other examples.
There exist double-primitive knots $K_{p,k}$ in the following Brieskorn homology spheres
1. $\Sigma(3,3n+1,12n+5)$ ($a_1/a_2=[3,-n]$, $b_1/b_2=[3,n+1,4]$, $c=3$ $(n>0)$) (I type),
2. $\Sigma(3,9n+4,18n+5)$ ($a_1/a_2=[5,2,n+1]$, $b_1/b_2=[3,2,2,2,n,2]$, $c=3, (n>0)$) (I type),
such that the resulting lens spaces $L(p,q)$ satisfy (\[Itypecontinued\]), equivalently the continued fraction can be deformed as follows: $$p/q=[-n,3,\ell+1,2,4,2,-\ell+1,3,n+1,4],$$ $$p/q=[n+1,2,5,\ell+1,2,4,2,-\ell+1,3,2,2,2,n,2]$$ respectively.
Next, consider J type lens space surgery. If a Brieskorn homology sphere admits Seifert data $S(1,(2,1),(a_1,a_2),(b_1,b_2))$, then we can do the Dehn surgery of [Figure]{} \[DualJ\] of \[extJ\]. Then the resulting manifolds are lens spaces $L(p,q)$ with $$p/q=[\alpha_n,\cdots, \alpha_1,\ell+1,3,3,-\ell,\beta_1,\beta_2\cdots, \beta_m],$$ where $a_1/a_2=[\alpha_1,\cdots, \alpha_n]$ and $b_1/b_2=[\beta_1,\beta_2,\cdots, \beta_m]$ for $\alpha_i,\beta_j\in {\mathbb Z}$.
There exist double-primitive knots $K_{p,k}$ in Brieskorn homology spheres $\Sigma(2,2s+1,2(2s+1)n\pm1)$, $\Sigma(2,4n+1,12n+5)$ and $\Sigma(2,8n+3,12n+5)$ satisfying the following coefficients $A,B,c$ in [Figure]{} \[graphdeformation\]:
1. $\Sigma(2,2s+1,2(2s+1)n\pm1)$ ($A=[2(2s+1),\mp n]$, $B=[2,-s], s>0$),
2. $\Sigma(2,4n+1,12n+5)$ ($A=[4,-n]$, $B=[4,n+1,3], n>0)$,
3. $\Sigma(2,8n+3,12n+5)$ ($A=[8n+3]$, $B=[2,-2n,3], n>0)$,
such that the resulting lens spaces $L(p,q)$ of (vi), (vii), and (viii) are the following $$p/q=[\mp n,2(2s+1),-\ell,3,3,\ell+1,2,-s],$$ $$p/q=[-n,4,-\ell,3,3,\ell+1,4,n+1,3],$$ $$p/q=[8n+3,-\ell,3,3,\ell+1,2,-2n,3]$$ respectively.
According to [@AL] the homology spheres of (iv) (vii) bound rational 4-balls. Thus the resulting lens spaces bound 4-dimensional rational 2-spheres $\mathcal{S}$, namely $H_\ast(\mathcal{S},{\mathbb Q})=H_\ast(S^2,{\mathbb Q})$ and $\partial \mathcal{S}=L(p,q)$. These Brieskorn homology spheres are just examples of the J type lens space surgeries, hence, one can find double-primitive knots in other Brieskorn homology spheres of J type lens space surgery.
The resolution diagram of type II Brieskorn homology spheres with Seifert data $S(1,(2,1),(a_1,a_2),(b_1,b_2))$ are classified in [@Matsumoto].
![IJ type plumbing diagram.[]{data-label="tij"}](IJZS-eps-converted-to.pdf)
### K type lens space knot. {#ktypelensspaceknotsection}
We consider K type lens surgery. This case is exceptional in terms of quadratic family, because this family contains one example only in $\Sigma(2,3,5)$. The pillowcase method and double covering are [Figure]{} \[2bribandK\] and \[bandsumK\]. From the extension of Dehn surgery in [Figure]{} \[DualK\] and \[extK\], if the K type Seifert homology spheres yield lens spaces, then the Seifert data are $S(1,(3,1),(m,m-1),(a_1,a_2))$ is [Figure]{} \[tk\].
There exists a double-primitive knot $K_{p,k}$ in the Brieskorn homology sphere $\Sigma(2,3,6n\pm1)$ satisfying the following data:
1. $\Sigma(2,3,6n\pm1)$ ($m=2$ and $a_1/a_2=6\pm 1/n$)
such that the resulting lens space $L(p,q)$ satisfy with $$p/q=[\mp n,6,3,3,3,2].$$
![K type plumbing diagram.[]{data-label="tk"}](KZS-eps-converted-to.pdf)
![The deformations of plumbing diagrams obtained by extending lens space surgeries in $\Sigma(2,3,5)$.[]{data-label="graphdeformation"}](plumbingdeformation-eps-converted-to.pdf)
Proofs of Theorems
------------------
In this section we prove the theorem stated in Section \[intro\].\
[**Proofs of Theorem \[sigma236npm1\], \[2s+1theorem\], and \[22s+1ntheorem\]**]{}. Those proofs that those homology spheres contain these lens space knots immediately follows from propositions which are proven from Section \[Ataipu\] to Section \[ktypelensspaceknotsection\]. $\Box$\
[**Proof of Theorem \[graphhomspheres\].**]{} As described in [Figure]{} \[graphdeformation\], there exist graph homology spheres with not Brieskorn but CDE type plumbing diagram. We have only to check that the graph manifolds for the diagrams are homology spheres. $\Box$\
Dehn surgeries in $S^3$ as graph deformations in [Figure]{} \[graphdeformation\].
---------------------------------------------------------------------------------
Our method up to this point can be applied to construct some families of the lens space, connected sum of lens space and Seifert manifold surgeries in $S^3$. Does this viewpoint have somehow relation to networking theory of Seifert surgeries by Deruelle, Miyazaki and Motegi in [@DMT]? This may be clarified in future research.
In [Figure]{} \[graphdeformation\] we give graph deformations coming from lens surgeries of $\Sigma(2,3,5)$. We illustrate that some graph deformations in these figures give families of lens space surgeries in $S^3$. To clarify those examples one has only to check that the corresponding left diagrams in [Figure]{} \[graphdeformation\] present $S^3$.
- In A type surgery set $a=3$, $A=B=\emptyset$, then the resulting lens space is $L(5\ell^2+5\ell+1,5\ell^2)$ and $k=5\ell+2$. This family belongs to Berge’s type VII.
- In A type surgery, set $A=[0]$, $B=[3]$, then the resulting manifolds are $L(2\ell+1,\ell)\#L(\ell,-1)$. This family presents reducible surgeries on torus knots.
- In B type surgery in putting $A=[1]$, $B=[1,2]$ and $C=[-1,-3]$, then the resulting lens spaces are $L(2\ell^2-11\ell+14,2\ell^2-9\ell+11)$. This family is surgeries of $(2\ell-5,\ell-3)$-torus knot.
- In J type surgery, set $A=[-1,-2,\cdots, -2]$, $B=[1,2,\cdots,2]$, and $C=[-1,-2,\cdots,-2]$, $e=-a+b-c\pm1$, where the lengths of arms are $a,b$ and $c$. Then the resulting manifolds are Seifert manifold with Seifert data $S(-1,(2\ell-2a+1,a-\ell-1),(-a+b\pm1,1),(2\ell-2b+1,b-\ell))$.
Appendix 1 {#appendix-1 .unnumbered}
==========
In this appendix we give pillowcase methods ([Figure]{} \[2bribandA\], \[2bribandB\], \[2bribandCDE\], \[2bribandFGH\], \[2bribandI\], \[2bribandJ\] and \[2bribandK\]).
![The pillowcase method for an A type simple (1,1)-knot with the band (arc $K'$ in the sense of Section \[pme\]).[]{data-label="2bribandA"}](Amove1-eps-converted-to.pdf){width=".9\textwidth"}
![The pillowcase method for a B type simple (1,1)-knot with the band.[]{data-label="2bribandB"}](Bmove1-eps-converted-to.pdf){width=".9\textwidth"}
![The pillowcase method for a CDE type simple (1,1)-knot with the band.[]{data-label="2bribandCDE"}](CDEmove1-eps-converted-to.pdf){width=".8\textwidth"}
![The pillowcase method for an FGH type simple (1,1)-knot with the band.[]{data-label="2bribandFGH"}](FGHmove1-eps-converted-to.pdf){width=".8\textwidth"}
![The pillowcase method for an I type simple (1,1)-knot with the band.[]{data-label="2bribandI"}](Imove1-eps-converted-to.pdf){width=".9\textwidth"}
![The pillowcase method for a J type simple (1,1)-knot with the band.[]{data-label="2bribandJ"}](Jmove1-eps-converted-to.pdf){width=".85\textwidth"}
![The pillowcase method for a K type simple (1,1)-knot with the band.[]{data-label="2bribandK"}](Kmove1-eps-converted-to.pdf){width=".85\textwidth"}
Appendix 2 {#appendix-2 .unnumbered}
==========
In Appendix 2, we give double branched covering along the 2-bridge knot or link obtained in Appendix 1. In the covering space the attached arc in the 2-bridge knot or link is mapped to $\tilde{K}_{p,k}$. We describe the diagrams in generalized forms ([Figure]{} \[bandsumA\], [Figure]{} \[bandsumB\], [Figure]{} \[bandsumCDE\], [Figure]{} \[bandsumFGH\], [Figure]{} \[bandsumI\], [Figure]{} \[bandsumJ\], and [Figure]{} \[bandsumK\]).
![A type simple (1,1)-knot $\tilde{K}_{p,k}$ ($a\in {\mathbb Z}$, $b\in {\mathbb Q}$).[]{data-label="bandsumA"}](A11-eps-converted-to.pdf)
![B type simple (1,1)-knot $\tilde{K}_{p,k}$ ($b,c\in {\mathbb Q}$, $a\in {\mathbb Z}$).[]{data-label="bandsumB"}](B11-eps-converted-to.pdf)
![CDE type simple (1,1)-knot $\tilde{K}_{p,k}$ ($a,b\in {\mathbb Z}$, $c\in {\mathbb Q}$).[]{data-label="bandsumCDE"}](CDE11-eps-converted-to.pdf){width="\textwidth"}
![FGH type simple (1,1)-knot $\tilde{K}_{p,k}$ ($a,b,d\in {\mathbb Z}$ and $c\in {\mathbb Q}$).[]{data-label="bandsumFGH"}](FGH11-eps-converted-to.pdf)
![I type simple (1,1)-knot $\tilde{K}_{p,k}$ ($a,b\in {\mathbb Q}$ and $c\in {\mathbb Z}$).[]{data-label="bandsumI"}](I11-eps-converted-to.pdf)
![J type simple (1,1)-knot $\tilde{K}_{p,k}$ ($a,b\in {\mathbb Q}$).[]{data-label="bandsumJ"}](J11-eps-converted-to.pdf)
![K type simple (1,1)-knot $\tilde{K}_{p,k}$ ($m\in {\mathbb Z}$, $a\in {\mathbb Q}$).[]{data-label="bandsumK"}](K11-eps-converted-to.pdf)
Appendix 3 {#appendix-3 .unnumbered}
==========
In Appendix, we describe families of the knot diagrams $K_{p,k}$ in Seifert manifolds. Attaching 0-framed meridian for the last diagrams in figures in Appendix 2 and deforming the diagrams other than the 0-framed meridian into the form of the Seifert structure, we obtain the knot diagrams (the first picture in Appendix 3). Then we start Kirby calculus for the surgery diagrams and obtain lens spaces ([Figure]{} \[DualA\], \[DualB\], \[DualCD\], \[DualFGH\], \[DualI\], \[DualJ\] and \[DualK\]). These calculus easily extend to some Seifert surgeries (including lens surgeries) over Seifert manifolds (e.g., including Brieskorn homology spheres or graph homology spheres). See [Figure]{} \[extA\], \[extB\], \[extE\], \[extI\], \[extJ\] and \[extK\].
![A type $K_{p,k}$ (thickened line) for $a\in {\mathbb Z}$, $b\in {\mathbb Q}$ and the surgery calculus move to a lens space. The cases of $(a,b)=(3,6\pm1/n)$ and $(2(2s+1)\pm1,(2s+1)/2)$ correspond to lens surgeries with $\A_1$ and $\A_2$ types in $\Sigma(2,3,6n\pm1)$ and $\Sigma(2,2s+1,2(2s+1)\pm1)$.[]{data-label="DualA"}](A1-eps-converted-to.pdf){width=".9\textwidth"}
![The diagram of extended A type Dehn surgery along $K$.[]{data-label="extA"}](extendA-eps-converted-to.pdf)
![B type $K_{p,k}$ ($a\in {\mathbb Z}$ and $b,c\in {\mathbb Q}$) and the surgery calculus move to a lens space. The case of $(a,b,c)=(-1,5,2)$ or $(a,b,c)=(-1,2,5)$ is B type surgery in $\Sigma(2,3,5)$.[]{data-label="DualB"}](B1-eps-converted-to.pdf){width=".9\textwidth"}
![The diagram of extended B type Dehn surgery along $K$.[]{data-label="extB"}](extendB-eps-converted-to.pdf)
![CDE type $K_{p,k}$ ($a,b\in {\mathbb Z}$ and $c\in {\mathbb Q}$) and surgery calculus to a lens space. The cases of $(a,b,c)=(1,2,5), (2,1,5), (1,4,3),(4,1,3), (2,5,1),(5,2,1)$ are $\C_1$ $\C_2$, ${\Delta}_1$, ${\Delta}_2$, $\E_1$ and $\E_2$ type surgeries in $\Sigma(2,3,5)$.[]{data-label="DualCD"}](CD1-eps-converted-to.pdf){width=".9\textwidth"}
![FGH type $K_{p,k}$ and surgery calculus to a lens space. The cases of $d=2$ and $\{a,b,c\}=\{2,3,5\}$ are FGH type surgeries in $\Sigma(2,3,5)$. []{data-label="DualFGH"}](FGH1-eps-converted-to.pdf){width=".9\textwidth"}
![The diagram of extended CDE and FGH type Dehn surgery along $K$.[]{data-label="extE"}](extendE-eps-converted-to.pdf)
![I type $K_{p,k}$ ($a,b\in {\mathbb Q}$, $c\in {\mathbb Z}$) and surgery calculus to a lens space. The cases of $\{a,b,c\}=\{5,3,2\}$ are $\I_1,\I_2$ and $\I_3$ type knots in $\Sigma(2,3,5)$.[]{data-label="DualI"}](I1-eps-converted-to.pdf){width=".9\textwidth"}
![The diagram of extended I type Dehn surgery along $K$.[]{data-label="extI"}](extendI-eps-converted-to.pdf)
![J type $K_{p,k}$ ($a,b\in {\mathbb Q}$ $c=2$) and surgery calculus to a lens space. The cases of $\{a,b\}=\{3,5\}$ and $c=2$ are J type surgeries in $\Sigma(2,3,5)$.[]{data-label="DualJ"}](J1-eps-converted-to.pdf){width=".9\textwidth"}
![Extended J type Dehn surgery.[]{data-label="extJ"}](extendJ-eps-converted-to.pdf)
![K type $K_{p,k}$ and the surgery calculus to a lens space. The case of $(a,b,c)=(5,2,1)$ is K type surgery in $\Sigma(2,3,5)$.[]{data-label="DualK"}](K1-eps-converted-to.pdf){width="\textwidth"}
![The diagram of extended K type Dehn surgery along $K$.[]{data-label="extK"}](extendK-eps-converted-to.pdf)
[**Acknowledgements:**]{} This result is based on my talk presented by workshop “Topology of Knots X” at Tokyo Woman’s Christian University in 2007. The author is grateful for giving me the opportunity to talk in the conference. The motivation to write this paper is a private communication [@G1] with Joshua Greene at Gökova Geometry/Topology Conference 2011 as written as a private communication in [@G]. The author is grateful for him with respect to this point. We discuss the lens space surgery in $\Sigma(2,5,7)$ yielding $L(17,15)$ in Proposition \[Joshua\]. In 8 years after that, I received the similar questions about the private discussion from Daniel Ruberman and Kyungbae Park independently. It would be possibly somehow worthy that such computation and results are available. The author is so grateful that their communications give the opportunity to write this paper. In particular, K. Park gave the opportunity to talk in KIAS topology seminar in 2016 spring. The author thanks them for reminding me the computation. The author also thanks anonymous referees for correcting my first manuscript so detailed.
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[^1]: This research was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (Start-up) Grant Number 19840029 and (B) Grand Number 17K14180.
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abstract: 'The flux density from undetected sources in the observing beam of a telescope produces source confusion noise in the resulting maps of the sky, and thus limits sensitivity. In a recent paper [@BIS] we discussed this effect in the millimetre/submillimetre (mm/submm) waveband, using a simple model of galaxy evolution that could account for the gravitationally lensed images of distant dusty galaxies newly discovered in the fields of clusters of galaxies [@SIB]. New models explain all the available mm, submm and far-infrared data [@BSIK]. The associated predictions of source confusion noise are presented here. The new and old estimates agree to within a factor of two on angular scales between about 0.3 and 50arcmin, and are most similar on angular scales of about 1arcmin. The new estimates are greater/less than the old ones on larger/smaller angular scales.'
address:
- 'Cavendish Laboratory, Madingley Road, Cambridge, UK. '
- 'Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, UK. '
- 'Department of Physics, University of Durham, South Road, Durham, UK. '
- 'Observatoire Midi-Pyrenees, 14 Avenue E. Belin, 31400 Toulouse, France. '
author:
- 'A.W. Blain,$^{1}$ R.J. Ivison,$^{2}$ Ian Smail,$^{3}$ J.-P. Kneib$^{4}$'
---
Source confusion noise is a significant concern for both galaxy surveys and cosmic microwave background radiation (CMBR) anisotropy measurements in the mm/submm waveband. The dominant source of confusion noise is expected to be distant dusty star-forming galaxies and active galactic nuclei (AGN). The population of these objects was determined for the first time last year [@SIB], and used to make the first estimates of mm/submm-wave confusion noise [@BIS] based on direct observations.
By modeling the results of the observations in more detail, and taking into account the recently measured spectrum of extragalactic background radiation in the mm, submm and far-infrared wavebands, the form of evolution of distant dusty galaxies has now been determined more accurately. Families of galaxy evolution models that are consistent with all the observations are described elsewhere [@BSIK]. These new results allow us to derive more definitive estimates of source confusion noise, and to determine the model-dependent uncertainties in the estimates: see Fig.1. More details and references to the range of ground-based and space-borne telescopes and instruments included in Fig.1 can be found in [@BIS].
At longer wavelengths in the mm waveband the contribution of non-thermal radio sources to the population of confusing sources is expected to dominate that of dusty galaxies and AGN. The importance of confusing radio sources at frequencies from about 10 to 100GHz is rather uncertain at present, but is under active investigation because of its significance for CMBR anisotropy measurements.
-0.25cm
This paper provides the latest estimates of source confusion noise due to distant dusty galaxies and AGN in the mm, submm and far-infrared wavebands. The accuracy of the predictions as a function of the angular resolution and frequency of observations is assessed for the first time.
[99]{}[ Blain A.W., Ivison R.J., & Smail I., 1998, MNRAS 296, L29 (astro-ph/9710003). Blain A.W., Smail I., Ivison R.J., & Kneib J.-P., 1998, MNRAS, submitted (astro-ph/9806062). Smail I., Ivison R.J., & Blain A.W., 1997, ApJ, 490, L5 (astro-ph/9708135). ]{}
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---
abstract: 'A seminal result in operator theory is the Sz.-Nagy–Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$: in short, any completely nonunitary contraction operator $T$ is unitarily equivalent to its functional model $T(\Theta)$ on a certain functional Hilbert space (just a vectorial Hardy space over the unit disk in the simplest case) which can be constructed explicitly from the so-called characteristic function $\Theta = \Theta_T$ of $T$. The constructions also incorporate explicit geometric constructions for the minimal isometric and unitary dilation of the operator $T$. The goal of the present paper is to push this theory to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy–Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equivalence. In addition to the characteristic function $\Theta_T$, we identify additional invariants $({\mathbb G}, {\mathbb W})$ which can be used to construct a functional model for the commuting pair $(T_1, T_2)$ and which have good uniqueness properties: if two commutative contractive pairs $(T_1, T_2)$ and $(T''_1, T''_2)$ are unitarily equivalent, then their characteristic triples $(\Theta, {\mathbb G}, {\mathbb W})_T$ and $(\Theta, {\mathbb G}, {\mathbb W})_{T''}$ coincide in a natural sense. We illustrate the theory with several simple cases where the characteristic triples can be explicitly computed. This work extends earlier results of Berger-Coburn-Lebow [@B-C-L] for the case where $(T_1, T_2)$ is a pair of commuting isometries, and of Das-Sarkar [@D-S], Das-Sarkar-Sarkar [@D-S-S] and the second author [@sauAndo] for the case where $T = T_1T_2$ is pure (the operator sequence $T^{*n}$ tends strongly to $0$). Finally we use the model to study the structure of joint invariant subspaces for a commutative, contractive operator pair, extending results of Sz.-Nagy–Foias for the single-operator case.'
address:
- |
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA\
[email protected]
- |
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA\
[email protected], [email protected]
author:
- 'Joseph A. Ball'
- Haripada Sau
title: Functional Models and Invariant Subspaces for Pairs of Commuting Contractions
---
[^1]
Introduction {#S:intro}
============
The starting point for many future developments in nonselfadjoint operator theory was the Sz.-Nagy dilation theorem from 1953 [@sz-nagy]: [*if $T$ is a contraction operator on a Hilbert space ${{\mathcal H}}$, then there is a unitary operator ${{\mathcal U}}$ on a larger Hilbert space $\widetilde {{\mathcal K}}\supset {{\mathcal H}}$ such that $T^n = P_{{\mathcal H}}{{\mathcal U}}^n |_{{\mathcal H}}$ for all $n=0,1,2,\dots$.*]{} While the original proofs were more existential than constructive, there followed more concrete constructive proofs (e.g., the Schäffer-matrix construction to be discussed below) which evolved into a detailed geometric picture of the dilation space (see [@Nagy-Foias Chapter II]). Analysis of how the original Hilbert space ${{\mathcal H}}$ fit into the dilation space $\widetilde {{\mathcal K}}$ and the discovery of appropriate transforms to convert the abstract spaces to spaces of functions (holomorphic or measurable as the case may be) led to the discovery of the characteristic function $\Theta_T$ of any c.n.u. contraction operator $T$ and how the c.n.u. contraction operator $T$ can be represented (up to unitary equivalence) as a compressed multiplication operator on a functional-model Hilbert space constructed directly from $\Theta_T$.
The Andô dilation theorem [@ando], coming ten years later, provides a 2-variable analogue of the Sz.-Nagy dilation theorem: [*given a commuting pair of contraction operators $(T_1, T_2)$ on a Hilbert space ${{\mathcal H}}$, there is a commutative pair of unitary operators $({{\mathcal U}}_1, {{\mathcal U}}_2)$ on a larger Hilbert space $\widetilde {{\mathcal K}}\supset {{\mathcal H}}$ so that, for all $n,m \ge 0$, $T_1^n T_2^m = P_{{\mathcal H}}{{\mathcal U}}_1^n {{\mathcal U}}_2^m |_{{\mathcal H}}$.*]{} The proof was an ad hoc expanded extension of the Schäffer-matrix construction for the single-operator case which shed no light on the geometry of the dilation space (a consequence of the lack of uniqueness up to a notion of unitary equivalence for Andô dilations). Consequently there has been essentially no follow-up to the Andô result in the direction of a Sz.-Nagy–Foias-type model theory for a commuting pair of contraction operators as there was in the single-operator setting, although there have been some preliminary results (see [@A-M-Dist-Var; @BV]).
In an independent development, Berger-Coburn-Lebow [@B-C-L] obtained a model for a commutative-tuple of isometries $(V_1, \cdots, V_d)$ by considering the Wold decomposition for the product $V= V_1 \cdots V_d$ and understanding what form the factors $V_1, \dots, V_d$ must take in the shift-part of $V$ so as (i) to be themselves commuting isometries, and (ii) to have product equal to the shift operator $V$.
Much of this paper can be seen as an effort to extend the Berger-Coburn-Lebow results for the case $d=2$ to the setting where the commutative pair of isometries $(V_1, V_2)$ is replaced by a commutative pair of contractions $(T_1, T_2)$ such that the product $T = T_1 T_2$ is c.n.u. We can then place $T$ into its Sz.-Nagy–Foias functional model determined by the characteristic function $\Theta_T$ of $T$, and look for the form a pair of operators $(T_1, T_2)$ also defined on the functional model space must have so that (i) each of $T_1$ and $T_2$ is a contraction operator, and (ii) $T_1 T_2 = T_2 T_1 = T$. By using recent progress on construction of isometric Andô dilations on a non-minimal extended version of the Sz.-Nagy–Foias functional model space for the minimal isometric lift of $T$, we are able to identify the invariant (the characteristic triple $({\mathbb G}, {\mathbb W}, \Theta_T)$—an expanded version of the Sz.-Nagy–Foias invariant $\Theta_T$) which converts the study of the operator triple $(T_1, T_2, T= T_1 T_2)$ to function theory on a Sz.-Nagy-Foias model space $(H^2({{\mathcal D}}_{T^*}) \oplus \overline{\Delta_T L^2({{\mathcal D}}_T) })
\ominus \{ \Theta f \oplus \Delta_T f \colon f \in H^2({{\mathcal D}}_T) \} $. The ingredient ${\mathbb G}$ in the characteristic triple is closely connected to the [*fundamental operators*]{} coming up recently in the study of $\Gamma$-contractions (commutative operator pairs having the symmetrized bidisk as a spectral set [@B-P-SR]) as well as in the study of tetrablock contractions (commutative triples of operators having the tetrablock ${\mathbb E}$ as a spectral set [@sir's; @tetrablock; @paper]). Let us mention that earlier of Das-Sarkar-Sarkar [@D-S-S] found such a model for a pair of commuting contractions and earlier work of the second author [@sauAndo] worked out the invariant $({\mathbb G}, \Theta_T)$, but only for the case where $T = T_1 \cdot T_2$ is [*pure*]{} (i.e., SOT-$\lim_{n \to \infty} T^{*n} = 0$), in which case the second component ${\mathbb W}$ of the characteristic triple is vacuous.
The construction of the piece ${\mathbb G}$ for the characteristic triple $({\mathbb G}, {\mathbb W}, \Theta_T)$ for a given commutative contractive pair $(T_1, T_2)$ depends crucially on a collection of spaces and operators $({{\mathcal F}}, \Lambda, P, U)$ constructed from $(T_1, T_2)$ which we shall call an [*Andô tuple*]{} for $(T_1, T_2)$ (also playing a key role in [@sauAndo]). Let us note that this structure of [*Andô tuple*]{} appears at least implicitly already in the original construction by Andô of a joint isometric lift for a commutative pair of contractions [@ando]. Let us now discuss the notion of Andô tuple more precisely.
We first recall the notion of [*regular factorization*]{} for a general (not necessarily square or commutative) factorization $T = T_1 T_2$ where $T \colon {{\mathcal H}}\to {{\mathcal H}}^{\prime \prime}$, $T_1 \colon {{\mathcal H}}' \to {{\mathcal H}}^{\prime \prime}$ and $T_2 \colon {{\mathcal H}}\to {{\mathcal H}}'$ are all contraction operators with ${{\mathcal H}}$, ${{\mathcal H}}'$, ${{\mathcal H}}^{\prime \prime}$ all possibly different Hilbert spaces. We let $$D_T = (I - T^* T)^{\frac{1}{2}}, \quad D_{T_1} = (I - T_1^* T_1)^{\frac{1}{2}}, \quad
D_{T_2} = (I - T_2^* T_2)^{\frac{1}{2}}$$ denote the associated defect operators with ranges denoted as $${{\mathcal D}}_T = \overline{\operatorname{Ran}}\, D_T \subset {{\mathcal H}}, \quad
{{\mathcal D}}_{T_1} = \overline{\operatorname{Ran}}\, D_{T_1} \subset {{\mathcal H}}', \quad
{{\mathcal D}}_{T_2} = \overline{\operatorname{Ran}}\, D_{T_2} \subset {{\mathcal H}}.$$ Then the identity $$\label{id1}
D_T^2 = I - T_2^* T_2 +T_2^* T_2 - T_2^* T_1^* T_1 T_2 = D_{T_2}^2 + T_2^* D_{T_1}^2 T_2$$ shows that the operator $\Lambda \colon {{\mathcal D}}_T \to {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ defined densely by $$\label{defLambda}
\Lambda \colon D_T h = D_{T_1} T_2 h \oplus D_{T_2} h \text{ for all } h \in {{\mathcal H}}$$ is an isometry from ${{\mathcal D}}_T$ into ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$. In case this isometry is onto (so $\Lambda$ is in fact a unitary transformation from ${{\mathcal D}}_T$ onto ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$), we say that the contractive factorization $T = T_1 \cdot T_2$ is [*regular*]{}.
\[R:regfact\] [In addition to the context of Andô tuples, isometries of the form play a key role in characterizing invariant subspaces of a contraction operator $T$ in terms of its Sz.-Nagy–Foias model. Indeed, invariant subspaces correspond to factorizations of the characteristic function $\Theta = \Theta_1 \Theta_2$ (where $\Theta_1$ and $\Theta_2$ are also contractive analytic operator-valued functions) which are [*pointwise regular on the unit circle*]{}, i.e., the operator $Z$ defined by $$\label{Z}
Z \colon D_{\Theta}(\zeta)u \mapsto D_{\Theta_1}(\zeta) \Theta_2(\zeta) h \oplus D_{\Theta_2}(\zeta) h$$ which is always an isometry from ${{\mathcal D}}_{\Theta(\zeta)} $ into ${{\mathcal D}}_{\Theta_1(\zeta)} \oplus {{\mathcal D}}_{\Theta_2(\zeta)}$ is actually onto and hence unitary for almost all $\zeta \in{\mathbb T}$ (see [@Nagy-Foias Chapter VII]). Whenever this happens, following [@Nagy-Foias Chapter VII], we shall say that $\Theta =
\Theta_1 \cdot \Theta_2$ is a [*regular factorization*]{} of the contractive operator function $\Theta$. We shall present an extension of Sz.-Nagy–Foias characterization of invariant subspaces to the context of a commutative pair $(T_1, T_2)$ of contractions in Section \[S:invsub\] below. ]{}
We now restrict to the square case where ${{\mathcal H}}= {{\mathcal H}}' = {{\mathcal H}}^{\prime \prime}$ and $T$ has a commutative contractive factorization $T = T_1 \cdot T_2 = T_2 \cdot T_1$ with $T_1$ and $T_2$ also contraction operators on ${{\mathcal H}}$. Let us introduce the notation $$\begin{aligned}
& {{\mathcal D}}_{U_0} : =\operatorname{clos.} \{ D_{T_1} T_2 h \oplus D_{T_2} h \colon h \in {{\mathcal H}}\} \subset {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}, \notag \\
& {{\mathcal R}}_{U_0}:=\operatorname{clos.} \{ D_{T_1} h \oplus D_{T_2} T_1 h \colon h \in {{\mathcal H}}\} \subset {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}.
\label{U0dom-codom}\end{aligned}$$ Making use of the assumption that now $T_1$ and $T_2$ commute, in addition to the identity one can verify the following additional identity: $$\label{id2}
D_{T_2}^2 + T_2^* D_{T_1}^2 T_2 = D_{T_1}^2 + T_1^* D_{T_2}^2 T_1.$$ We conclude that the operator $U_0 \colon {{\mathcal D}}_{U_0} \to {{\mathcal R}}_{U_0}$ defined densely by $$\label{U0}
U_0 \colon D_{T_1} T_2 h \oplus D_{T_2} h \mapsto D_{T_1} h \oplus D_{T_2} T_1 h \text{ for all } h \in {{\mathcal H}}$$ is unitary. If $$\label{equal-codim}
\dim ({{\mathcal D}}_{U_0})^\perp = \dim ({{\mathcal R}}_{U_0})^\perp$$ (orthogonal complements with respect to the ambient space ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$), we can extend $U_0$ to a unitary operator $U$ on all of $ {{\mathcal F}}:= {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$. In case condition fails, we may enlarge the ambient space ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ to the space ${{\mathcal F}}: = {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2} \oplus \ell^2$ (here $\ell^2$ can be taken to be any separable infinite-dimensional Hilbert space) with respect to which condition does hold (with value $\infty$ equal to the common co-dimension), and thereby come up with a unitary extension $U$ of $U_0$ on the larger space ${{\mathcal F}}$. Finally, given ${{\mathcal F}}$, $\Lambda$, $U$ constructed in this way, we let $P$ be the orthogonal projection onto the first component as an operator on ${{\mathcal F}}$: $$\begin{aligned}
& P \colon f \oplus g \mapsto f \oplus 0 \text{ in case } {{\mathcal F}}= {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}, \\
& P \colon f \oplus g \oplus h \mapsto f \oplus 0 \oplus 0 \text{ in case } {{\mathcal F}}= {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2} \oplus \ell^2.
\end{aligned}$$
\[D:Ando-tuple\] [Given a commutative contractive factorization $T = T_1 \cdot T_2 = T_2\cdot T_1$ for a contraction operator $T$, we say that the collection of spaces and operators $({{\mathcal F}}, \Lambda, P , U)$ is an [*Andô tuple*]{} for the pair $(T_1, T_2)$ if it arises via the construction given in the preceding paragraph. ]{}
\[R:Ando-tuple-unique\] [Let us note that in general the Andô tuple depends nontrivially on the choice of unitary extension $U$ of the partially defined isometry $U_0$. An Andô tuple is uniquely determined from $(T_1, T_2)$ exactly when both ${{\mathcal D}}_{U_0}$ and ${{\mathcal R}}_{U_0}$ are equal to all of ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$. Note that ${{\mathcal D}}_{U_0} = {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ corresponds to $T = T_1 \cdot T_2$ being a regular factorization while the ${{\mathcal R}}_{U_0} = {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ corresponds to $T = T_2 \cdot T_1$ being a regular factorization. We conclude that [*an Andô tuple of the commutative pair of contractions $(T_1, T_2)$ is uniquely determined exactly when both $T = T_1 \cdot T_2$ and $T = T_2 \cdot T_1$ are regular factorizations*]{}. We shall see that in the finite-dimensional setting it is enough to assume that one of these factorizations is regular while in the infinite-dimensional setting this is not the case (see Theorem \[T:left-right\] below. ]{}
We shall actually need a couple of variations of the notion of Andô tuple ([*coordinate-free Andô tuple for $(T_1, T_2)$*]{} and [*coordinate-free Andô tuple for $(T_1^*, T_2^*)$*]{})—see Definitions \[D:Ando-c\] and \[D:Ando-c\*\] below.
The paper is organized as follows. Following this Introduction, in Section \[S:1D\] we review the geometric structure associated with a unitary dilation ${{\mathcal U}}$ or isometric lift $V$ of a contraction operator $T$ from [@Nagy-Foias Chapter II], with special additional attention to the case where ${{\mathcal U}}$ or $V$ may not be minimal. We then explain how particular choices of coordinates in this structure lead to three distinct functional models for the minimal unitary-dilation or isometric-lift space associated with the names of Schäffer, Douglas, and Sz.-Nagy–Foias; the term [*functional*]{} is somewhat loose and strictly applies only to the Sz.-Nagy–Foias case (see Section \[S:epilogue\]). In any case this analysis enables us to find an explicit identification between the Douglas isometric-lift space and the Sz.-Nagy-Foias isometric-lift space for a given c.n.u. contraction operator; this in turn is crucial for defining the second component ${\mathbb W}$ in a characteristic triple $({\mathbb G}, {\mathbb W}, \Theta_T)$ for a given pair of commuting contraction operators ($T_1, T_2)$ to come in Section \[S:NFmodel\].
Section \[S:c\] is an attempt to mimic Chapter II of [@Nagy-Foias] for the case of a contractive commutative operator pair $(T_1, T_2)$ in place of a single contraction operator $T$. We use the existence of a Andô isometric lift $(V_1, V_2)$ for a commutative contractive pair $(T_1, T_2)$ together with the ingredients from modified “coordinate-free” Andô tuples to arrive at three functional-model forms (of Schäffer, Douglas, and Sz.-Nagy–Foias type) for an Andô dilation. Unlike the single-operator case, the resulting Andô dilations need not have much to do with the original assumed (coordinate-free) dilation, nor with each other, as the construction depends on a choice of Andô tuple which in turns depends (except in the nongeneric case where both the factorization $T = T_1 \cdot T_2 = T_2 \cdot T_1$ are regular) on an arbitrary choice of unitary extension $U$ of the partially defined isometry $U_0$. While it appears that the Schäffer Andô isometric lift does not have much to do with the Douglas or Sz.-Nagy–Foias isometric lift (even when one tries to match the respective unitary operators $U$), it does appear that the Douglas and Sz.-Nagy–Foias isometric lifts can be arranged to be unitarily equivalent with appropriate matching choices of unitary extensions $U$ in the respective constructions of an Andô tuple. Strictly speaking, the construction here (based on so-called coordinate-free Andô tuple for $(T_1, T_2)$ and coordinate-free Andô tuple for $(T_1^*, T_2^*)$ and assumed coordinate-free Andô isometric lift $(V_1, V_2)$ of $(T_1, T_2)$) does not prove the existence of an Andô isometric lift; however, the proof can be rearranged, based on the “coordinate-dependent" definition of Andô tuple (Definition \[D:Ando-tuple\] above) to prove from scratch the existence of a Schäffer-type and Douglas-type Andô isometric lift; this is done in [@sauAndo].
In Section \[S:NFmodel\] we introduce the characteristic triple $({\mathbb G}, {\mathbb W}, \Theta_T)$ for a commutative, contractive pair ${\mathbf T} = (T_1, T_2)$ with $T = T_1 \cdot T_2 = T_2 \cdot T_1$, and show that this has all the invariance properties for the commutative, contractive pair $(T_1, T_2)$ as the Sz.-Nagy–Foias characteristic function $\Theta_T$ has for a single c.n.u. contraction operator $T$. In particular, there is a functional-model pair of commutative contractions $({\mathbf T}_1, {\mathbf T}_2)$ acting on the Sz.-Nagy–Foias functional model space ${{\mathcal H}}_{NF}$ such that the original abstract commutative, contractive pair $(T_1, T_2)$ in unitarily equivalent to the concrete functional-model commutative, contractive pair $({\mathbf T}_1, {\mathbf T}_2)$, and the characteristic triple for $T$ serves as a complete unitary invariant for $T$. We include some simple examples of characteristic triples to illustrate the ideas.
The final Section \[S:invsub\] extends the Sz.-Nagy–Foias analysis of invariant subspaces in the functional model to the commutative-contractive-pair setting.
Finally, let us mention that we are in the process of extending the framework of this paper to the setting of commutative $d$-tuples $(T_1, \dots, T_d)$ of contraction operators on a Hilbert space ${{\mathcal H}}$ [@BSprep].
Models for a unitary and isometric dilation of a contraction operator {#S:1D}
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The coordinate-free version
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Let $T$ be a contraction operator on a Hilbert space ${{\mathcal H}}$. We say that the pair $(\widetilde \Pi, {{\mathcal U}})$ is a [*unitary dilation*]{} of $T$ if (i) $\widetilde \Pi$ is an isometric embedding of ${{\mathcal H}}$ into a Hilbert space $\widetilde {{\mathcal K}}$, and (ii) ${{\mathcal U}}$ is a unitary operator on $\widetilde {{\mathcal K}}$ such that $$\widetilde \Pi^* {{\mathcal U}}^n \widetilde \Pi = \begin{cases} T^n &\text{if } n \ge 0, \\
T^{*n} &\text{if } n < 0, \end{cases}$$ or, equivalently in local form, for all $h, h' \in {{\mathcal H}}$ and integers $n,m$ we have $$\langle {{\mathcal U}}^n \widetilde \Pi h, \, {{\mathcal U}}^m \widetilde \Pi h' \rangle =
\begin{cases} \langle T^{n-m} h, \, h' \rangle &\text{if } n \ge m,\\
\langle h, T^{m-n} h' \rangle &\text{if } n < m. \end{cases}$$ We say that $(\widetilde \Pi, {{\mathcal U}})$ is a [*minimal unitary dilation*]{} of $T$ if in addition the smallest ${{\mathcal U}}$-reducing subspace of $\widetilde {{\mathcal K}}$ containing $\widetilde \Pi {{\mathcal H}}$ is all of $\widetilde {{\mathcal K}}$.
We say that the pair $(\Pi, V)$ is an [*isometric dilation*]{} of the contraction operator $T$ if (i) $\Pi$ is an isometric embedding of ${{\mathcal H}}$ into the Hilbert space ${{\mathcal K}}$, and (ii) $V$ is an isometric operator on ${{\mathcal K}}$ such that the dilation property holds: $$\Pi^* V^n \Pi = T^n \text{ for } n=0,1,2,\dots,$$ or, in local form, for all $h,h' \in {{\mathcal H}}$ and $n=1,2,3,\dots$ we have $$\langle V^n \Pi h, \Pi h' \rangle = \langle T^n h, h' \rangle.$$ We say that $(\Pi, V)$ is a [*minimal isometric dilation*]{} if the smallest invariant subspace if ${{\mathcal K}}$ containing ${{\mathcal H}}$ is all of ${{\mathcal K}}$
In case $(\Pi, V)$ is a minimal isometric dilation of $T$, then in fact $(\Pi, V)$ is a [*isometric lift*]{} (after isometric embedding) of $T$, meaning that $V^*$ is a [*coisometric extension*]{} (after isometric embedding) of $T^*$, i.e., $V^*\Pi = \Pi T^*$. When we speak about isometric dilations, we are usually interested in minimal isometric dilations, and therefore, as is usually done, we assume at the outset that the isometry is actually a lift. Thus we abuse the terminology slightly and say that $(\Pi, V)$ is an [*isometric dilation*]{} of $T$ if (i) $\Pi$ is an isometric embedding of ${{\mathcal H}}$ into a Hilbert space ${{\mathcal K}}$, and (ii) $V$ is an isometric operator on ${{\mathcal K}}$ such that $\Pi T^*= V^* \Pi$. We say that $(\Pi, V)$ is a [*minimal isometric dilation*]{} of $T$ if in addition ${{\mathcal K}}$ is the smallest invariant subspace of $V$ containing $\Pi {{\mathcal H}}$. We say that two isometric dilations $(\Pi, V)$ and $(\Pi', V')$ are [*unitarily equivalent*]{} if there is a unitary operator $U$ from ${{\mathcal K}}$ to ${{\mathcal K}}'$ so that $\Pi' = U \Pi$ and $U V = V' U$.
The classical formulation of unitary/isometric dilation is the case where one takes ${{\mathcal H}}\subset {{\mathcal K}}\subset \widetilde {{\mathcal K}}$ and $\Pi = \iota_{{{\mathcal H}}\to {{\mathcal K}}} \colon {{\mathcal H}}\to {{\mathcal K}}$ and $\widetilde \Pi = \iota_{{{\mathcal H}}\to \widetilde {{\mathcal K}}} \colon {{\mathcal H}}\to \widetilde {{\mathcal K}}$ are the inclusion maps. We can construct unitary/isometric dilations of $T$ via e.g. the Schäffer-matrix construction, so existence of unitary/isometric dilations is not an issue. Let us assume that ${{\mathcal U}}$ on $\widetilde {{\mathcal K}}$ and $V$ on ${{\mathcal K}}$ is a unitary (respectively, isometric) dilation of $T$ in the classical sense (${{\mathcal H}}\subset {{\mathcal K}}\subset \widetilde {{\mathcal K}}$ with $\Pi \colon {{\mathcal H}}\to {{\mathcal K}}$ and $\widetilde \Pi \colon {{\mathcal H}}\to \widetilde {{\mathcal K}}$ equal to the respective inclusion maps). For this case there is a nice coordinate-free description of the geometry behind any minimal unitary/isometric dilation as follows (see [@Nagy-Foias Chapter II]).
The space $\widetilde {{\mathcal K}}$ has two internal orthogonal direct-sum decompositions $$\begin{aligned}
\widetilde {{\mathcal K}}& = M({{\mathcal L}}_*) \oplus {{\mathcal R}}\label{decom1} \\
& = M_-({{\mathcal L}}_*) \oplus {{\mathcal H}}\oplus M_+({{\mathcal L}}) \oplus {{\mathcal R}}_0 \label{decom2}\end{aligned}$$ where $$\begin{aligned}
& {{\mathcal L}}_* = \overline{{{\mathcal U}}({{\mathcal U}}^* - T^*) {{\mathcal H}}} = \overline{(I - {{\mathcal U}}T^*) {{\mathcal H}}}, \quad {{\mathcal L}}= \overline{({{\mathcal U}}- T) {{\mathcal H}}}, \\
& M({{\mathcal L}}_*) = \bigoplus_{n=-\infty}^\infty {{\mathcal U}}^n {{\mathcal L}}=
M_-({{\mathcal L}}_*) \oplus M_+({{\mathcal L}}_*)\end{aligned}$$ where we set $$\begin{aligned}
& M_-({{\mathcal L}}_*) = \bigoplus_{n=-\infty}^{-1} {{\mathcal U}}^n {{\mathcal L}}_*, \quad M_+({{\mathcal L}}_*) = \bigoplus_{n=0}^\infty {{\mathcal U}}^n {{\mathcal L}}_*, \\
& M_-({{\mathcal L}}) = \bigoplus_{n=-\infty}^{-1} {{\mathcal U}}^n {{\mathcal L}}, \quad M_+({{\mathcal L}}) = \bigoplus_{n=0}^\infty {{\mathcal U}}^n {{\mathcal L}}\end{aligned}$$ and where $$\label{cR}
{{\mathcal R}}= \widetilde {{\mathcal K}}\ominus M({{\mathcal L}}_*), \quad {{\mathcal R}}_0 = [M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}] \ominus [ {{\mathcal H}}\oplus M_+({{\mathcal L}})].$$
We note that ${{\mathcal U}}|_{M({{\mathcal L}}_*)}$ and ${{\mathcal U}}|_{M({{\mathcal L}})}$ are bilateral shifts while ${{\mathcal U}}|_{M_+({{\mathcal L}}_*)}$ and ${{\mathcal U}}|_{M_+({{\mathcal L}})}$ are unilateral shift operators, with respective wandering subspaces equal to ${{\mathcal L}}_*$ and ${{\mathcal L}}$.
We next collect a few additional properties concerning the geometry of the minimal isometric lift of $T$ of the form $(\iota_{{{\mathcal H}}\to {{\mathcal K}}_+}, V)$.
\[P:dil\] Let $T$ be a contraction operator on a Hilbert space with unitary dilation ${{\mathcal U}}$ on $\widetilde {{\mathcal K}}$ and isometric dilation $V$ on ${{\mathcal K}}$ as described above. Then:
1. The maps $ \iota_* \colon {{\mathcal L}}_* \to {{\mathcal D}}_{T^*}$ and $\iota \colon {{\mathcal L}}\to {{\mathcal D}}_T$ given densely by $$\iota_* \colon (I - V T^*) h \mapsto D_{T^*} h, \quad
\iota \colon (V - T) h \mapsto D_T h$$ for $h \in {{\mathcal H}}$ extend to define unitary identification maps from ${{\mathcal L}}_*$ onto ${{\mathcal D}}_{T^*}$ and from ${{\mathcal L}}$ onto ${{\mathcal D}}_T$ respectively.
2. The projection $P_{{{\mathcal L}}_*}$ onto the wandering subspace ${{\mathcal L}}_*$ restricted to ${{\mathcal H}}$ is given by $$\label{ortho-proj}
P_{{{\mathcal L}}_*}|_{{\mathcal H}}= (I - V T^*)|_{{\mathcal H}}.$$
3. For $h \in {{\mathcal H}}$, the orthogonal projection $P_{M_+({{\mathcal L}}_*)} h$ of $h$ onto $M_+({{\mathcal L}}_*)$ is given by $$\label{PM+L}
P_{M_+({{\mathcal L}}_*)} h = \sum_{n=0}^\infty V^n (I - V T^* ) T^{*n} h.$$ with $$\label{proj-norm1}
\| P_{M_+({{\mathcal L}}_*)} h \|^2 = \sum_{n=0}^\infty \| D_{T^*} T^{*n} h \|^2.$$
4. For $h \in {{\mathcal H}}$ we have $$\label{proj-norm2}
\| P_{{{\mathcal R}}} h \|^2 = \langle Q^2 h, h \rangle$$ where we set $Q^2 = \operatorname{SOT-lim}_{n \to \infty} T^n T^{*n}$, i.e., for all $h \in {{\mathcal H}}$, $\| (T^n T^{*n} - Q^2)h \| \to 0$ as $n \to \infty$.
5. $\overline{P_{{\mathcal R}}{{\mathcal H}}}$ is invariant under $V^*$ and $V^*|_{\overline{P_{{\mathcal R}}{{\mathcal H}}}}$ is isometric.
6. Either of the following conditions is necessary and sufficient for ${{\mathcal U}}$ on $\widetilde {{\mathcal K}}$ to be a minimal unitary dilation of $T$, or equivalently for $V$ on ${{\mathcal K}}$ to be minimal isometric dilation of $T$:
1. $\widetilde {{\mathcal K}}= M_-({{\mathcal L}}_*) \oplus {{\mathcal H}}\oplus M_+({{\mathcal L}})$ (so the subspace ${{\mathcal R}}_0$ in is zero), or ${{\mathcal K}}= \overline{ \cup_{n=0}^\infty V^n {{\mathcal H}}} = {{\mathcal H}}\oplus M_+({{\mathcal L}})$;
2. the linear manifold $\bigcup_{n=0,1,2,\dots} V^n P_{{\mathcal R}}{{\mathcal H}}$ is dense in ${{\mathcal R}}$.
Furthermore the condition
1. $M({{\mathcal L}}_*) + M({{\mathcal L}})$ is dense in $\widetilde {{\mathcal K}}$
is sufficient for the minimality of $(\iota_{{{\mathcal H}}\to \widetilde {{\mathcal K}}}, {{\mathcal U}})$ as a unitary dilation of $T$ (respectively, minimality of $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ as an isometric dilation of $T$).
7. Suppose that $T$ is completely nonunitary. Then condition (c) above is both necessary and sufficient for minimality of $(\iota_{{{\mathcal H}}\to \widetilde {{\mathcal K}}}, {{\mathcal U}})$ as a unitary dilation of $T$ (respectively, the minimality of $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ as an isometric dilation of $T$).
In this case the decompositions – simplify to $$\label{decom3}
\widetilde {{\mathcal K}}= M({{\mathcal L}}_*) \oplus {{\mathcal R}}= M_-({{\mathcal L}}_*) \oplus {{\mathcal H}}\oplus M_+({{\mathcal L}})$$ with corresponding decompositions for the space ${{\mathcal K}}: = \widetilde {{\mathcal K}}\ominus M_-({{\mathcal L}}_*)$ $$\label{decom4}
{{\mathcal K}}= M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}= {{\mathcal H}}\oplus M_+({{\mathcal L}}),$$ and we have the following density condition: $$\label{cRDeltaTheta}
\overline{ (I - P_{M({{\mathcal L}}_*)}) M({{\mathcal L}})} = \overline{ P_{{\mathcal R}}M({{\mathcal L}}) }= {{\mathcal R}}.$$
\(1) and (2) follow from elementary computations as in [@Nagy-Foias Chapter II Section 1].
As ${{\mathcal L}}_*$ is the wandering subspace for $M_+({{\mathcal L}}_*)$, we have in general that $P_{M_+({{\mathcal L}}_*)} = \sum_{n=0}^\infty V^n P_{{\mathcal L}}V^{*n}$. Given $h \in {{\mathcal H}}$, by making use of the fact that $T^* = V^*|_{{\mathcal H}}$ together with the formula for $P_{{{\mathcal L}}_*} |{{\mathcal H}}$, the formula for $P_{M_+({{\mathcal L}}_*)} h$ follows immediately. It then follows that $$\begin{aligned}
\| P_{M_+({{\mathcal L}}_*)} h \|^2 & = \sum_{n=0}^\infty \| (I - V T^*) T^{*n} h \|^2 \\
& = \sum_{n=0}^\infty \| D_{T^*} T^{*n} h \|^2 \text{ by part (1) above}\end{aligned}$$ verifying and part (3) follows.
As for (4), note that $\operatorname{SOT-lim}_{n \to \infty} T^n T^{*n}$ exists since $T^n T^{*n}$ is a monotonically decreasing sequence of positive-semidefinite operators since $T^*$ is a contraction operator and hence we may define a positive-semidefinite operator $Q$ on ${{\mathcal H}}$ by $$\label{Q}
Q^2 = \operatorname{SOT-lim}_{n \to \infty} T^n T^{*n}.$$ From – we see that ${{\mathcal H}}\subset M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}$, and hence, for $h \in {{\mathcal H}}$, $$\| P_{{\mathcal R}}h \|^2 = \| h \|^2 - \| P_{M_+({{\mathcal L}}_*)} \|^2.$$ Thus $$\begin{aligned}
& \| P_{{\mathcal R}}h \|^2 = \| h \|^2 - \| P_{M_+({{\mathcal L}}_*)} h \|^2 = \| h \|^2 - \sum_{n=0}^\infty \| D_{T^*} T^{*n} h \|^2 \\
& \quad \quad = \| h \|^2 - \lim_{N\to \infty} \sum_{n=0}^N \langle T^n (I - T T^*) T^{*n} h, h \rangle \\
& \quad \quad = \| h \|^2 - \left( \| h \|^2 - \lim_{N \to \infty} \langle T^{N+1} T^{*(N+1)} h, h \rangle \right) \\
& \quad \quad = \lim_{N \to \infty} \langle T^{N+1} T^{*(N+1)} h, h \rangle = \langle Q^2 h, h \rangle\end{aligned}$$ (4) follows.
As for (5), note that $$V^* P_{{\mathcal R}}h = P_{{\mathcal R}}V^* h = P_{{\mathcal R}}T^* h,$$ we conclude that $\overline{P_{{\mathcal R}}{{\mathcal H}}}$ is invariant for $V^*$. Furthermore, since ${{\mathcal R}}$ is reducing for ${{\mathcal U}}$ we have $$\| V^* P_{{\mathcal R}}h \| = \| {{\mathcal U}}^* P_{{\mathcal R}}h \| = \| P_{{\mathcal R}}h \|$$ and we see that $V^*|_{\overline{P_{{\mathcal R}}{{\mathcal H}}}}$ is isometric, and (4) follows.
We now discuss the characterizations of minimality in (6). Note first that any ${{\mathcal U}}$-reducing subspace containing ${{\mathcal H}}$ must contain $M_-({{\mathcal L}}_*)$, since ${{\mathcal U}}^* {{\mathcal L}}_* =
\overline{({{\mathcal U}}^* - T^*) {{\mathcal H}}}$. Similarly, any ${{\mathcal U}}$-invariant subspace containing ${{\mathcal H}}$ (i.e., any $V$-invariant subspace containing ${{\mathcal H}}$) must contain $M_+({{\mathcal L}})$ since ${{\mathcal L}}= \overline{( {{\mathcal U}}- T) {{\mathcal H}}}$.
From the decompositions $$(I - {{\mathcal U}}T^*) h = (I - T T^*) h - ({{\mathcal U}}- T) T^* h, \quad {{\mathcal U}}h = Th + ({{\mathcal U}}- T) h$$ we see that $${{\mathcal L}}_* + {{\mathcal U}}{{\mathcal H}}\subset {{\mathcal H}}+ {{\mathcal L}}.$$ Similarly, from the decompositions $$h = (I - {{\mathcal U}}T^*) h + {{\mathcal U}}T^* h, \quad ({{\mathcal U}}- T) h = -(I - {{\mathcal U}}T^*)T h + {{\mathcal U}}(I - T^* T) h$$ we see that we have the reverse containment $${{\mathcal H}}+ {{\mathcal L}}\subset {{\mathcal L}}_* + {{\mathcal U}}{{\mathcal H}}.$$ From the decomposition we see that these decompositions are in fact orthogonal, so we have the identity of subspaces of ${{\mathcal K}}$: $${{\mathcal L}}_* \oplus {{\mathcal U}}{{\mathcal H}}= {{\mathcal H}}\oplus {{\mathcal L}}.$$ We may use this identity to read off that the subspace $$\widetilde {{\mathcal K}}_{\rm min}: = M_-({{\mathcal L}}_*) \oplus {{\mathcal H}}\oplus M_+({{\mathcal L}})$$ is reducing for ${{\mathcal U}}$, and that $${{\mathcal K}}_{\rm min}:= {{\mathcal H}}\oplus M_+({{\mathcal L}})$$ is invariant for $V$. From the preceding discussion we know that any reducing subspace for ${{\mathcal U}}$ containing ${{\mathcal H}}$ must contain $\widetilde {{\mathcal K}}_{\rm min}$ and that any invariant subspace for $V$ containing ${{\mathcal H}}$ must contain ${{\mathcal K}}_{\rm min}$. In this way we see that condition (a) is necessary and sufficient for minimality.
We have observed above that $\widetilde {{\mathcal K}}_{\rm min}: = M_-({{\mathcal L}}_*) \oplus {{\mathcal H}}\oplus M_+({{\mathcal L}})$ is ${{\mathcal U}}$-reducing, and hence ${{\mathcal R}}_0 = \widetilde {{\mathcal K}}\ominus \widetilde {{\mathcal K}}_{\rm min}$ is ${{\mathcal U}}$-reducing as well. From we see that ${{\mathcal R}}_0 \perp M_-({{\mathcal L}}_*)$. As ${{\mathcal R}}_0$ is ${{\mathcal U}}$-reducing this forces that in fact ${{\mathcal R}}_0 \perp M({{\mathcal L}}_*)$, i.e. (from ), ${{\mathcal R}}_0 \subset {{\mathcal R}}$. Thus the characterization of ${{\mathcal R}}_0$ in can be rewritten as $${{\mathcal R}}_0 = \{ k \in {{\mathcal R}}\colon k \perp {{\mathcal H}}\oplus {{\mathcal M}}_+({{\mathcal L}}) \}.$$ From the fact that ${{\mathcal L}}= (V - T) {{\mathcal H}}$, one can see that ${{\mathcal H}}\oplus M_+({{\mathcal L}}) = \overline{ \cup_{n=0}^\infty V^n {{\mathcal H}}}$. Thus we may rewrite our characterization of ${{\mathcal R}}_0$ as $$\label{cR0}
{{\mathcal R}}_0 = \{ k \in {{\mathcal R}}\colon k \perp \bigcup_{n=0}^\infty V^n {{\mathcal H}}\}.$$
Suppose now that $k \in {{\mathcal R}}$ is such that $k \perp \bigcup_{n=0}^\infty V^n P_{{\mathcal R}}{{\mathcal H}}$. Then $k \in {{\mathcal R}}$ is such that for all $h \in {{\mathcal H}}$ and $n=0,1,2,\dots$ we have $$0 = \langle k, V^n P_{{\mathcal R}}h \rangle = \langle k, {{\mathcal U}}^n P_{{\mathcal R}}h \rangle = \langle k, P_{{\mathcal R}}{{\mathcal U}}^n h \rangle =
\langle k, V^n h \rangle,$$ i.e., $k \in {{\mathcal R}}$ is such that $k \perp \bigcup_{n=0}^\infty V^n {{\mathcal H}}$, or $k \in {{\mathcal R}}_0$ by . The argument is reversible: if $k \in {{\mathcal R}}_0$, then $k$ is orthogonal to $\overline{\cup_{n=0}^\infty V^n P_{{\mathcal R}}{{\mathcal H}}}$. We have thus verified $${{\mathcal R}}_0 = \{k \in {{\mathcal R}}\colon k \perp \bigcup_{n=0}^\infty V^n P_{{\mathcal R}}{{\mathcal H}}\}.$$ In particular, ${{\mathcal R}}_0 = \{0\}$ (i.e., ${{\mathcal U}}$ is a minimal unitary dilation of $T$ or equivalently $V$ is a minimal isometric lift of $T$ by criterion (a) already proved) if and only if $\bigcup_{n=0}^\infty V^n P_{{\mathcal R}}{{\mathcal H}}$ is dense in ${{\mathcal R}}$, and it follows that (b) is also a criterion for minimality.
Recall that an argument in the penultimate paragraph above shows that ${{\mathcal R}}_0 \subset {{\mathcal R}}= \widetilde {{\mathcal K}}\ominus M({{\mathcal L}}_*)$, so we get ${{\mathcal R}}_0 \perp M({{\mathcal L}}_*)$. From the decomposition we see that ${{\mathcal R}}_0 \perp M_+({{\mathcal L}})$; as ${{\mathcal R}}_0$ is ${{\mathcal U}}$-reducing, this implies that furthermore ${{\mathcal R}}_0 \perp M({{\mathcal L}})$. Thus we always have $${{\mathcal R}}_0 \subset ( M({{\mathcal L}}_*) + M({{\mathcal L}}) )^\perp.$$ In particular, if $M({{\mathcal L}}_*) + M({{\mathcal L}})$ is dense in $\widetilde {{\mathcal K}}$, it follows that ${{\mathcal R}}_0 = \{0\}$ and ${{\mathcal U}}$ is a minimal unitary dilation (as well as $V$ is a minimal isometric lift) of the contraction operator $T$. Thus criterion (c) is sufficient for minimality. This completes the proof of part (6) of Proposition \[P:dil\].
Suppose next that $T$ is completely nonunitary contraction operator and that minimality holds, so ${{\mathcal R}}_0 = \{0 \}$. Suppose that $k \in \widetilde {{\mathcal K}}$ is orthogonal to $M({{\mathcal L}}_*) + M({{\mathcal L}})$. In particular, we see that $k \perp M_-({{\mathcal L}}_*)$ and $k \perp M_+({{\mathcal L}})$. From the decomposition with ${{\mathcal R}}_0 = \{0\}$, we see that $k \in {{\mathcal H}}$. As $k$ is also orthogonal to $M_-({{\mathcal L}})$, we also have, for all $h' \in {{\mathcal H}}$ and $n=1,2,3,\dots$, $$\begin{aligned}
0 & = \langle h, {{\mathcal U}}^{*n} ( {{\mathcal U}}- T) h' \rangle = \langle {{\mathcal U}}^{n-1} h, h' \rangle - \langle {{\mathcal U}}^n h, T h' \rangle \\
& = \langle T^{n-1} h, h' \rangle - \langle T^n h, Th \rangle = \langle (I - T^* T) T^{n-1} h, h' \rangle\end{aligned}$$ from which we conclude that $(I - T^* T)T^{n-1} h = 0$ for $n=1,2,3,\dots$. This in turn gives $$\| h \|^2 = \| T h \|^2 = \| T^2 h \|^2 = \dots = \| T^n h \|^2 = \dots,$$ or $h$ is in the isometric subspace for $T$ (the largest invariant subspace ${{\mathcal H}}_i$ for $T$ such that $T|_{{{\mathcal H}}_{\rm i}}$ is isometric. Similarly $k \in {{\mathcal H}}$ and $k \perp M_+({{\mathcal L}}_*)$ implies that $k \in {{\mathcal H}}_{\rm ci}$ where ${{\mathcal H}}_{\rm ci}$ is the largest $T^*$-invariant subspace such that $T^*|_{{{\mathcal H}}_{\rm ci}}$ is isometric. Putting all this together means that $h \in {{\mathcal H}}_{\rm i} \cap {{\mathcal H}}_{\rm ci} =: {{\mathcal H}}_{\rm u}$, where ${{\mathcal H}}_{\rm u}$ is the largest $T$-reducing subspace such that $T|_{{{\mathcal H}}_{\rm u}}$ is unitary. The hypothesis that $T$ is completely nonunitary amounts to saying that ${{\mathcal H}}_{\rm u} = \{0\}$. Thus finally $k = 0$ and we conclude $M({{\mathcal L}}) + M({{\mathcal L}}_*)$ is dense in $\widetilde {{\mathcal K}}$. It follows that criterion (c) is also necessary for minimality in case $T$ is completely nonunitary.
Finally the validity of for the case where $M({{\mathcal L}}_*) + M({{\mathcal L}})$ is dense in $\widetilde {{\mathcal K}}$ is part of Theorem II.2.1 in [@Nagy-Foias]. This concludes the proof of part (7) of Proposition \[P:dil\].
We close this section with a useful uniqueness result for minimal isometric lifts of a c.n.u. contraction operator $T$.
\[T:min-iso-lift\] Suppose that $T$ is a c.n.u. contraction operator on a Hilbert space ${{\mathcal H}}$ and that $({{\mathcal K}}, \Pi, V)$ and $({{\mathcal K}}', \Pi', V')$ are two minimal isometric lifts of $T$. Then $({{\mathcal K}}, \Pi, V)$ and $({{\mathcal K}}', \Pi', V')$ are unitarily equivalent as isometric lifts of $T$, i.e., there is a unitary operator $U \colon {{\mathcal K}}\to {{\mathcal K}}'$ such that $$\label{define-equiv}
U \Pi = \Pi', \quad U V = V' U,$$ and furthermore $U$ so specified is unique.
This is an adaptation of Theorem I.4.1 in [@Nagy-Foias] where the classical case is considered. The minimality of $({{\mathcal K}}, \Pi, V)$ and $({{\mathcal K}}', \Pi', V')$ as isometric lifts of $T$ forces the spaces ${{\mathcal K}}$ and ${{\mathcal K}}'$ to be given by $${{\mathcal K}}= \overline{\operatorname{span}} \{ V^n \Pi {{\mathcal H}}\colon n=0,1,2,\dots\}, \quad
{{\mathcal K}}' = \overline{\operatorname{span}} \{ V^{\prime n} \Pi' {{\mathcal H}}\colon n=0,1,2,\dots\}.$$ If $({{\mathcal K}}, \Pi, V)$ is any minimal isometric lift, one can check that $$\langle V^n \Pi h, V^m \Pi \widetilde h \rangle_{{\mathcal K}}=
\begin{cases} \langle V^{n-m} \Pi h, \Pi \widetilde h \rangle_{{\mathcal K}}= \langle T^{n-m} h, \widetilde h \rangle_{{\mathcal H}}& \text{if } n \ge m \ge 0, \\
\langle h, V^{m-n} \widetilde h \rangle_{{\mathcal K}}= \langle h, T^{m-n} \widetilde h \rangle_{{\mathcal H}}&\text{if }
m \ge n \ge 0, \end{cases}$$ and thus $\langle V^n h, V^m \widetilde h \rangle_{{\mathcal K}}$ does not depend on the choice of minimal isometric lift. Thus, if $({{\mathcal K}}, \Pi, V)$ and $({{\mathcal K}}', \Pi', V')$ are two minimal isometric lifts, the map defined densely by $$U \colon \sum_{n=0}^N V^n \Pi h \mapsto \sum_{n=0}^N V^{\prime n} \Pi' h$$ extends by continuity to a unitary map from ${{\mathcal K}}$ to ${{\mathcal K}}'$ implementing a unitary equivalence between the minimal isometric lifts $({{\mathcal K}}, \Pi, V)$ and $({{\mathcal K}}', \Pi', V')$. Furthermore from the defining intertwining conditions for unitary equivalence, we see that any such unitary equivalence must be of this form.
Function-theoretic models
-------------------------
We shall assume in the sequel that $T$ is a completely nonunitary contraction operator on ${{\mathcal H}}$.
### The Schäffer model for the minimal isometric dilation {#S:Schaffer}
The Schäffer model is based on the second of the decompositions for the minimal isometric-dilation space ${{\mathcal K}}$ for $T$. Note that we can extend the unitary identification $\iota \colon {{\mathcal L}}\to {{\mathcal D}}_{T}$ in part (1) of Proposition \[P:dil\] to a unitary identification ${{\boldsymbol \iota}}\colon M_+({{\mathcal L}}) \to H^2({{\mathcal D}}_{T^*})$ according to the formula $$\label{biota}
{{\boldsymbol \iota}}\colon \sum_{n=0}^\infty V^n \ell_{n} \mapsto \sum_{n=0}^\infty (\iota \ell_{n}) z^n.$$ Let us write ${{\mathcal K}}_{S}$ for the [*Schäffer isometric lift space*]{} $${{\mathcal K}}_{S} = \begin{bmatrix} {{\mathcal H}}\\ H^2({{\mathcal D}}_T) \end{bmatrix}$$ and $\Pi_S$ for the isometric embedding operator $$\Pi_S = {\left[\begin{smallmatrix} I_{{\mathcal H}}\\ 0
\end{smallmatrix}\right]} \colon {{\mathcal H}}\to {{\mathcal K}}_{S}$$ and let $V_S$ on ${{\mathcal K}}_{S}$ be the operator given by $$V_S \colon \begin{bmatrix} h \\ f \end{bmatrix} \mapsto \begin{bmatrix} T & 0 \\ D_T & M_z \end{bmatrix}
\begin{bmatrix} h \\ f \end{bmatrix} = \begin{bmatrix} T h \\ D_T h + z f(z) \end{bmatrix}$$ where $D_T$ in the lower left corner of the block matrix ${\left[\begin{smallmatrix} T & 0 \\ D_T & M_z
\end{smallmatrix}\right]}$ is to be interpreted as the operator $D_T \colon {{\mathcal H}}\to H^2({{\mathcal D}}_T)$ mapping the vector $h \in {{\mathcal H}}$ to the constant function $D_T h$ in $H^2({{\mathcal D}}_T)$. Then one can check that $V_S$ is isometric on ${{\mathcal K}}_{S}$ and that $\Pi_S T^* = V_S^* \Pi_S$ (due to the block lower-triangular form in the matrix representation of $V_S$). We conclude that $(\Pi_S, V_S)$ is an isometric dilation of $T$, called the [*Schäffer-matrix isometric dilation of $T$*]{}.
Let us next write $U_S \colon {{\mathcal K}}\to {{\mathcal K}}_{S}$ for the unitary identification map $$U_S = \begin{bmatrix} P_{{\mathcal H}}\\ \iota P_{M_+({{\mathcal L}})} \end{bmatrix}.$$ Then one can check the intertwinings $$\Pi_S = U_S\, \iota_{{{\mathcal H}}\to {{\mathcal K}}}.$$ Furthermore, the calculation $$\begin{aligned}
& U_S V (h + \sum_{n=0}^\infty V^n \ell_n ) = U_S \left( T h + (V-T) h + \sum_{n=0}^\infty V^{n+1} \ell_n \right) \\
& \quad \quad = \begin{bmatrix} T h \\ D_T h + M_z \cdot \sum_{n=0}^\infty (\iota \ell_n) z^n \end{bmatrix}
= \quad \quad \begin{bmatrix} T & 0 \\ D_T & M_z \end{bmatrix} U_S ( h + \sum_{n=0}^\infty V^n z^n )\end{aligned}$$ shows that $$\label{Schaffer-dil}
U_S V = V_S U_S.$$ We conclude that $U_S$ implements a unitary equivalence between the minimal coordinate-free isometric dilation $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ and the Schäffer-matrix isometric dilation $(\Pi_S, V_S)$. As the coordinate-free isometric dilation $(\iota_{{{\mathcal H}}\to {{\mathcal K}}_+}, V)$ is minimal, we conclude that the Schäffer-matrix isometric dilation $(\Pi_S, V_S)$ is also minimal.
### The Douglas model for the minimal isometric dilation {#S:Douglas}
We now show how to use the first decomposition in for the minimal isometric-dilation space of $T$ to arrive at the Douglas model for the minimal isometric dilation, as derived by Douglas from first principles in [@Doug-Dilation].
Let us now introduce the extension of the unitary identification $\iota_* \colon {{\mathcal L}}_* \to {{\mathcal D}}_{T^*}$ in part (1) of Proposition \[P:dil\] to the unitary identification ${{\boldsymbol \iota}}_* \colon M_+({{\mathcal L}}_*) \to H^2({{\mathcal L}}_*)$ given by $$\label{biota*}
{{\boldsymbol \iota}}_* \colon \sum V^n \ell_{*n} \mapsto \sum_{n=0}^\infty (\iota_* \ell_{*n}) z^n.$$ By part (3) of Proposition \[P:dil\] (see formula , we know that, for $h \in {{\mathcal H}}$, $$P_{M_+({{\mathcal L}}_*)} h = \sum_{n=0}^\infty V^n (I - V T^*) T^{*n} h$$ and hence $$\label{biota*M+L}
{{\boldsymbol \iota}}_* P_{M_+({{\mathcal L}}_*)} h = \sum_{n=0}^\infty \iota_* (I - V T^*) T^{*n} h =
\sum_{n=0}^\infty (D_{T^*} T^{*n} h) z^n = : \widehat {{\mathcal O}}_{D_{T^*}, T^*}(z) h$$ where $\widehat {{\mathcal O}}_{D_{T^*}, T^*}$ is the frequency-domain observability operator associated with the state/output linear system $$\left\{ \begin{array}{rcl} x(t+1) & = & T^* x(t) \\ y(t) & = & D_{T^*} x(t), \end{array} \right. t = 0, 1, 2, \dots.$$ From the construction we see that $$\widehat {{\mathcal O}}_{D_{T^*}, T^*} T^* h = (M_z)^* \widehat {{\mathcal O}}_{D_{T^*}, T^*} h.$$ and from formulas and we see that $$\label{isom-id}
\| \widehat {{\mathcal O}}_{D_{T^*}, T^*} h \|^2_{H^2 ({{\mathcal D}}_{T^*})} = \| P_{M_+({{\mathcal L}}_*)} h \|^2, \quad
\| Q h \|^2 = \langle Q^2 h, h \rangle = \| P_{{\mathcal R}}h \|^2$$ where $Q^2 = \operatorname{SOT-lim}_{n \to \infty} T^n T^{*n}$. Let us define the subspace ${{\mathcal R}}_0$ of ${{\mathcal R}}$ by $${{\mathcal R}}_0 = \overline{P_{{\mathcal R}}{{\mathcal H}}}.$$ Hence we can define an isometric map $\omega_D \colon {{\mathcal R}}_0 \to \overline{\operatorname{Ran}}\, Q$ by action on the dense subset $P_{{\mathcal R}}{{\mathcal H}}\subset {{\mathcal R}}_0$ given by $$\label{omega}
\omega_D \colon P_{{\mathcal R}}h \mapsto Q h.$$
We now note that $T Q^2 T^* = Q^2$; hence the formula $$\label{defX}
X^*Qh = Q T^* h$$ defines an isometry on $\overline{\operatorname{Ran}} \, Q$ which (if not already unitary) has a minimal unitary extension on a space ${{\mathcal R}}_D \supset \overline{\operatorname{Ran}} \, Q$ which we denote by $(W^{D})^*$. A dense subspace of ${{\mathcal R}}_D$ is $\bigcup_{n=0}^\infty W_D^{n} \operatorname{Ran} Q$ and then the extension $W^*$ is given densely by $$\begin{aligned}
\label{IntofQ}
W_D^* W_D^n Q h = W_D^{n-1} Q h \text{ for } n \ge 1, W_D^* Q h = X^* Q h = Q T^* h.\end{aligned}$$ As we are assuming that $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ is a minimal isometric lift of $T$, from part (6b) of Proposition \[P:dil\] we see that the linear manifold $\bigcup_{n=0}^\infty V^n P_{{\mathcal R}}{{\mathcal H}}$ is dense in ${{\mathcal R}}$. Hence we can extend the map $\omega_D$ given by to a unitary map, still denoted as $\omega_D$, from all of ${{\mathcal R}}$ onto all of ${{\mathcal R}}_D$ densely defined according to the formula $$\label{omega-ext}
\omega_D \colon V^n P_{{\mathcal R}}h \mapsto W_D^n Q h \text{ for } n = 0,1,2,\dots.$$ One can check that $\omega_D$ satisfies the intertwining relation $$\omega_D (V|_{{\mathcal R}}) = W_D \omega_D.$$ As $\omega_D$ is unitary, this relation can equivalently be written as $$\omega_D (V^*|_{{\mathcal R}}) = W_D^* \omega_D.$$
Let us introduce the Hilbert space ${{\mathcal K}}_D = {\left[\begin{smallmatrix} H^2({{\mathcal D}}_{T^*}) \\ {{\mathcal R}}_D
\end{smallmatrix}\right]}$ and define an isometric operator $V_D$ on ${{\mathcal K}}_{D}$ by $$\begin{aligned}
\label{Vd}
V_D = \begin{bmatrix} M_z & 0 \\ 0 & W_D \end{bmatrix}.\end{aligned}$$
There is a canonical isometric embedding operator $\Pi_D \colon {{\mathcal H}}\to {{\mathcal K}}_D$ given by $$\begin{aligned}
\label{PiD}
\Pi_D \colon h \mapsto \begin{bmatrix} {{\mathcal O}}_{D_{T^*}, T^*} h \\ Q h \end{bmatrix}.\end{aligned}$$ Furthermore we have the intertwining relation $$\Pi_D T^* = (V_D)^* \, \Pi_D.$$ Thus $(\Pi_D, V_D)$ is an isometric dilation of $T$.
We can now define a map $U_D \colon {{\mathcal K}}\to H^2({{\mathcal D}}_{T^*}) \oplus {{\mathcal R}}_D$ by $$U_D \colon k \mapsto {{\boldsymbol \iota}}_* P_{M_+({{\mathcal L}}_*)} k \oplus \omega_D P_{{\mathcal R}}k$$ or, in operator form with column notation, $U_D \colon {{\mathcal K}}\to {{\mathcal K}}_{D}: = \begin{bmatrix} H^2({{\mathcal D}}_{T^*}) \\ {{\mathcal R}}_D \end{bmatrix}$ is given by $$\begin{aligned}
\label{Ud}
U_D = \begin{bmatrix} {{\boldsymbol \iota}}_* P_{M_+({{\mathcal L}}_*)} \\ \omega_D P_{{\mathcal R}}\end{bmatrix}.\end{aligned}$$ For $k \in {{\mathcal K}}$, since ${{\mathcal K}}= M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}$ by the first decomposition in , we have $\| k \|^2 = \| P_{M_+({{\mathcal L}}_*)} k \|^2 + \| P_{{\mathcal R}}k \|^2$. Since ${{\boldsymbol \iota}}_* \colon M_+({{\mathcal L}}_*)\to H^2({{\mathcal D}}_{T^*})$ and $\omega_D \colon {{\mathcal R}}\to {{\mathcal R}}_D$ are unitary maps, we then see that $\Pi_D \colon {{\mathcal K}}\to {\left[\begin{smallmatrix} H^2({{\mathcal D}}_{T^*}) \\ {{\mathcal R}}_D
\end{smallmatrix}\right]}$ is unitary. Furthermore one can check the intertwinings: $$\begin{aligned}
\label{Dintwin}
\Pi_D = U_D \, \iota_{{{\mathcal H}}\to {{\mathcal K}}}, \quad U_D V = V_D U_D.\end{aligned}$$ Thus $(\Pi_D, V_D)$ and $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ are unitarily equivalent as isometric dilations of $T$. As $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ is minimal, it follows that $(U_D, V_D)$ is minimal as well.
### The Sz.-Nagy–Foias functional model for the minimal isometric dilation {#S:SNFmodel}
The Sz.-Nagy–Foias isometric dilation for a completely nonunitary contraction operator can be derived as follows. Let $(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$ be the coordinate-free minimal isometric dilation of the completely nonunitary contraction operator as in Proposition \[P:dil\]. Define the operator ${{\boldsymbol{\Theta}}}\colon M({{\mathcal L}}) \to M({{\mathcal L}}_*)$ as the restricted projection $${{\boldsymbol{\Theta}}}= P_{M({{\mathcal L}}_*)}|_{M({{\mathcal L}})}.$$ Let ${{\boldsymbol \iota}}_* \colon M({{\mathcal L}}_*) \to L^2({{\mathcal D}}_{T^*})$ and ${{\boldsymbol \iota}}\colon M({{\mathcal L}}) \to L^2({{\mathcal D}}_T)$ be the Fourier representation operators $$\label{biota-biota*-2sided}
{{\boldsymbol \iota}}_* \colon \sum_{n=-\infty}^\infty {{\mathcal U}}^n \ell_{*n} \mapsto \sum_{n=-\infty}^\infty \ell_{*n} e^{int}, \quad
{{\boldsymbol \iota}}\colon \sum_{n=-\infty}^\infty {{\mathcal U}}^n \ell_{n} \mapsto \sum_{n=-\infty}^\infty \ell_{n} e^{int},$$ i.e., the operator ${{\boldsymbol \iota}}_*$ in extended in the natural way to a unitary identification of the bilateral-shift space $M({{\mathcal L}})$ to the $L^2$-space $L^2({{\mathcal D}}_{T^*})$ and similarly for the operator ${{\boldsymbol \iota}}$ in ). Then it is easily checked that $${{\boldsymbol{\Theta}}}({{\mathcal U}}|_{M({{\mathcal L}})}) = ({{\mathcal U}}|_{M({{\mathcal L}}_*)}) {{\boldsymbol{\Theta}}}.$$ Let $\widehat {{\boldsymbol{\Theta}}}= {{\boldsymbol \iota}}_* {{\boldsymbol{\Theta}}}{{\boldsymbol \iota}}\colon L^2({{\mathcal D}}_T) \to L^2({{\mathcal D}}_{T^*})$. Then the previous intertwining relation becomes the function-space intertwining $$\widehat {{\boldsymbol{\Theta}}}M_{\zeta} = M_{\zeta} \widehat {{\boldsymbol{\Theta}}}.$$ By a standard result (see e.g. [@Nagy-Foias], it follows that $\widehat {{\boldsymbol{\Theta}}}$ is a multiplication operator $$\widehat {{\boldsymbol{\Theta}}}\colon h(\zeta) \mapsto \Theta(\zeta) \cdot h(\zeta)$$ for a measurable ${{\mathcal L}}({{\mathcal D}}_T, {{\mathcal D}}_{T^*})$-valued function $\zeta \mapsto \Theta(\zeta)$. As ${{\boldsymbol{\Theta}}}$ is a restricted projection, it follows that $\| {{\boldsymbol{\Theta}}}\| \le 1$, and also $\| M_\Theta \| \le 1$ as an operator from $L^2({{\mathcal D}}_T)$ to $L^2({{\mathcal D}}_{T^*})$, from which it follows that $\| \Theta(\zeta) \| \le 1$ for almost all $\zeta$ in the unit circle. Furthermore, from the second decomposition in , we see that $M_\Theta$ maps $H^2({{\mathcal D}}_T)$ into $H^2({{\mathcal D}}_{T^*})$; thus in fact $\Theta$ is a contractive $H^\infty$-function with values in ${{\mathcal L}}({{\mathcal D}}_T, {{\mathcal D}}_{T^*})$—known as the [*Sz.-Nagy-Foias characteristic function*]{} of $T$, i.e., $$\begin{aligned}
\label{charcfunction}
\Theta(z)=\Theta_T(z)=-T+zD_{T^*}(I_{{{\mathcal H}}}-zT^*)^{-1}D_{T}.
\end{aligned}$$
Suppose next that $k \in {{\mathcal R}}$ has the form $k = P_{{\mathcal R}}\ell$ for some $\ell \in M({{\mathcal L}})$. Then $$\| k \|^2 = \| P_{{\mathcal R}}\ell \|^2 = \| \ell \|^2 - \| {{\boldsymbol{\Theta}}}\ell \|^2 =
\| {{\boldsymbol \iota}}\ell \|^2 - \| \Theta_T \cdot {{\boldsymbol \iota}}\ell \|^2 = \| \Delta_{T} \cdot {{\boldsymbol \iota}}\ell \|^2$$ where we let $\Delta_{T}$ be the ${{\mathcal D}}_T$-valued operator function on the unit circle ${\mathbb T}$ given by $$\Delta_{T}(\zeta) := (I - \Theta_T(\zeta)^* \Theta_T(\zeta))^{1/2}.$$ By part (7) of Proposition \[P:dil\] we know that the space $(I - P_{M({{\mathcal L}}_*)}) M({{\mathcal L}}) = P_{{\mathcal R}}M({{\mathcal L}})$ is dense in ${{\mathcal R}}$. Hence we can define a unitary map $\omega_{NF}$ from ${{\mathcal R}}$ to $\overline{ \Delta_{T}L^2({{\mathcal D}}_T) }$ densely defined on $P_{{\mathcal R}}M({{\mathcal L}})$ by $$\label{omegaNF}
\omega_{NF} \colon P_{{\mathcal R}}\ell \mapsto \Delta_{T} \cdot {{\boldsymbol \iota}}\ell.$$ From this formula we can read off the validity of the intertwining relation $$\label{intertwine1}
\omega_{NF} (V|_{{\mathcal R}}) = M_{\zeta}\, \omega_{NF}.$$
We now can define a unitary identification map $U_{NF}$ from ${{\mathcal K}}= M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}$ to ${{\mathcal K}}_{NF} = {\left[\begin{smallmatrix} H^2({{\mathcal D}}_{T^*}) \\
\overline{ \Delta_{T}L^2({{\mathcal D}}_T) }
\end{smallmatrix}\right]}$ by $$\begin{aligned}
\label{Unf}
U_{NF} k = \begin{bmatrix} {{\boldsymbol \iota}}_* P_{M_+({{\mathcal L}}_*)} k \\ \omega_{NF} P_{{\mathcal R}}k \end{bmatrix}.\end{aligned}$$ Since ${{\boldsymbol \iota}}_*$ is unitary from $M_+({{\mathcal L}}_*)$ to $H^2({{\mathcal D}}_{T^*})$, $\omega_{NF}$ is unitary from ${{\mathcal R}}$ to $\overline{ \Delta_{T}L^2({{\mathcal D}}_T) }$, and ${{\mathcal K}}$ has the internal orthogonal decomposition ${{\mathcal K}}= M_+({{\mathcal L}}_*) \oplus {{\mathcal R}}$, we see that $U_{NF}$ so defined is unitary from ${{\mathcal K}}$ onto ${{\mathcal K}}_{NF}$. Observing the intertwining relation $M_z {{\boldsymbol \iota}}_*|_{M_+(({{\mathcal L}}_*)} = {{\boldsymbol \iota}}V|_{M_+({{\mathcal L}}_*)}$ and recalling , we arrive at the intertwining relation $$\label{NFintwin}
U_{NF} V = V_{NF} U_{NF}$$ where we set $V_{NF}$ equal to the isometric operator on ${{\mathcal K}}_{NF}$ given by $$\begin{aligned}
\label{Vnf}
V_{NF} = \begin{bmatrix} M_z & 0 \\ 0 & M_{\zeta} \end{bmatrix}.\end{aligned}$$ Define the isometric embedding $\Pi_{NF}$ of ${{\mathcal H}}$ into ${{\mathcal K}}_{NF}$ as $$\begin{aligned}
\label{Unf&Pinf}
\Pi_{NF} = U_{NF} \iota_{{{\mathcal H}}\to {{\mathcal K}}}.\end{aligned}$$ The space $U_{NF} M_+({{\mathcal L}})$ can be identified explicitly as follows: for $\ell \in M_+({{\mathcal L}})$, $$U_{NF} \ell = \begin{bmatrix} \Theta_T \cdot {{\boldsymbol \iota}}\ell \\ \omega_{NF} P_{{\mathcal R}}\ell \end{bmatrix} =
\begin{bmatrix} \Theta_T \cdot {{\boldsymbol \iota}}\ell \\ \Delta_{T} \cdot {{\boldsymbol \iota}}\ell \end{bmatrix}$$ and hence the space ${{\mathcal H}}_{NF} :=\Pi_{NF}{{\mathcal H}}= U_{NF} {{\mathcal H}}$ is given by $$\begin{aligned}
\label{HNF}
{{\mathcal H}}_{NF} = \begin{bmatrix} H^2({{\mathcal D}}_{T^*}) \\ \overline{ \Delta_{T}L^2({{\mathcal D}}_T)} \end{bmatrix}
\ominus \begin{bmatrix} \Theta_T \\ \Delta_{T} \end{bmatrix} \cdot H^2({{\mathcal D}}_T).\end{aligned}$$ Note that the subspace $U_{NF} M_+({{\mathcal L}}) = {\left[\begin{smallmatrix} \Theta_T \\ \Delta_{T}
\end{smallmatrix}\right]} \cdot H^2({{\mathcal D}}_T)$ is invariant for $V_{NF}$ and hence ${{\mathcal H}}_{NF}$ is invariant for $V_{NF}^*$. Rewrite in the form $$U_{NF} V^* = V_{NF}^* U_{NF}$$ and restrict this identity to ${{\mathcal H}}$ to arrive at $$\Pi_{NF} T^* = V_{NF}^* \Pi_{NF}.$$
We now have all the pieces needed to conclude that $(\Pi_{NF}, U_{NF})$ is an isometric lift of $T$ (the [*Sz.-Nagy–Foias functional-model isometric lift*]{} of $T$). Furthermore the operator $U_{NF} \colon {{\mathcal K}}\to {{\mathcal K}}_{NF}$ implements a unitary equivalence of the Sz.-Nagy–Foias isometric lift $(\Pi_{NF}, U_{NF})$ with the coordinate-free minimal isometric lift $( \iota_{{{\mathcal H}}\to {{\mathcal K}}}, V)$, and hence the Sz.-Nagy–Foias functional-model isometric lift $(\Pi_{NF}, V_{NF})$ is also minimal.
It is known that any two minimal isometric dilations of a contraction operator are unitarily equivalent. It also follows that the unitary operator involved in the equivalence of two minimal dilations is unique. We conclude this section by finding the explicit unitary involved in the equivalence of the minimal isometric dilations $(\Pi_{D},V_{D})$ and $(\Pi_{NF},V_{NF})$. Let $\omega_D$ and $\omega_{NF}$ be the unitaries defined in (\[omega-ext\]) and (\[omegaNF\]), respectively. Define the unitary $$\begin{aligned}
\label{DNFintwin}
U_{\text{min}}:=I_{H^2({{\mathcal D}}_{T^*})}\oplus\omega_{NF}\omega_D^*:H^2({{\mathcal D}}_{T^*})\oplus{{\mathcal R}}_D\to H^2({{\mathcal D}}_{T^*})\oplus \overline{\Delta_{T}L^2({{\mathcal D}}_T)}.\end{aligned}$$From equations (\[Dintwin\]) and (\[NFintwin\]), we see that the following diagram is commutative: $$\begin{tikzcd}
(\iota_{{{\mathcal H}}\to {{\mathcal K}}}, V, M_+({{\mathcal L}}_*)\oplus {{\mathcal R}})\arrow{r}{{{\boldsymbol \iota}}_*\oplus\omega_D } \arrow{rd}{{{\boldsymbol \iota}}_*\oplus\omega_{NF}}
& (\Pi_D, V_D, H^2({{\mathcal D}}_{T^*})\oplus{{\mathcal R}}_D) \arrow{d}{U_{\text{min}}} \\
& (\Pi_{NF},V_{NF}, H^2({{\mathcal D}}_{T^*})\oplus\overline{\Delta_{T}L^2({{\mathcal D}}_T)}).
\end{tikzcd}$$ Therefore the unitary $U_{\text{min}}$ intertwines the Douglas model and the Sz.-Nagy–Foias model of the minimal isometric dilation, i.e., $$\begin{aligned}
\label{DNFint}
U_{\text{min}} V_{D}=V_{NF}U_{\text{min}}\end{aligned}$$where $V_D$ and $V_{NF}$ are as defined in (\[Vd\]) and (\[Vnf\]), respectively. It also follows from the first equation in (\[Dintwin\]) and (\[Unf&Pinf\]) that $$\begin{aligned}
\label{uniqDNF}
U_{\text{min}}\Pi_D=\Pi_{NF}.\end{aligned}$$ This unitary identification will be used in what follows.
### Epilogue: the de Branges-Rovnyak functional model {#S:epilogue}
The term [*functional model*]{} for the Schäffer model (Section \[S:Schaffer\]) as well for the Douglas model (Section \[S:Douglas\]) is a bit loose since the Schäffer isometric-lift space ${{\mathcal K}}_S = {{\mathcal H}}\oplus H^2({{\mathcal D}}_T)$ has only its second component $H^2({{\mathcal D}}_T)$ as a space of functions while the first component remains an abstract Hilbert space, and similarly the Douglas isometric-lift space ${{\mathcal K}}_D = H^2({{\mathcal D}}_{T^*}) \oplus {{\mathcal R}}_D$ has only its first component $H^2({{\mathcal D}}_{T^*})$ equal to a space of functions while the second component ${{\mathcal R}}_D$ is an abstract Hilbert space. On the other hand the Sz.-Nagy–Foias isometric lift space ${{\mathcal K}}_{NF} = H^2({{\mathcal D}}_{T^*}) \oplus \overline{\Delta_T L^2({{\mathcal D}}_T)}$ in Section \[S:SNFmodel\] has both components equal to spaces of functions, the first component consisting of holomorphic functions with the second component consisting of measurable functions. Indeed, note that the transformation $\omega_{NF} \omega_D^{-1} \colon {{\mathcal R}}_D \to \overline{\Delta_T L^2({{\mathcal D}}_T)}$ can be viewed as a type of Fourier transform implementing a concrete spectral theory for the unitary operator $W_D$ in the following sense: $\omega_{NF} \omega_D^{-1}$ converts the abstract Hilbert space ${{\mathcal R}}_D$ to the space of measurable operator-valued functions $ \overline{\Delta_T L^2({{\mathcal D}}_T)}$ on the unit circle while at the same time converting the abstract unitary operator $W_D$ to the operator $M_{\zeta}$ of multiplication by the coordinate function on the function space $ \overline{\Delta_T L^2({{\mathcal D}}_T)}$. There is yet another functional model space for c.n.u. contraction operators, namely that of de Branges-Rovnyak (see [@dBR1; @dBR2] for the original work and [@BB; @BC; @BK; @NV1; @NV2] for later treatments) consisting of two (in general) coupled components, each of which is a holomorphic function on the unit disk ${\mathbb D}$, so the study of c.n.u. contraction operators is again reduced to pure holomorphic function theory rather than a hybrid of holomorphic- and measurable-function theory. We recommend the survey article [@BB] as an entrance to this rich topic.
Andô dilations: two coordinate-free constructions {#S:c}
=================================================
The goal of this section is to introduce coordinate-free and functional-model formulations for an Andô isometric lift for a pair of commuting contractions $(T_1, T_2)$. However we first add some additional information concerning Anô tuples.
More on Andô tuples
-------------------
A piece of unfinished business from the Introduction is to complete the discussion of uniqueness of the Andô tuple for a given commutative contractive pair $(T_1, T_2)$ (see Remark \[R:Ando-tuple-unique\]). We shall also need more coordinate-free versions of an An
There we saw that the Andô tuple is unique if and only if both $T_1 \cdot T_2$ and $T_2 \cdot T_1$ is a regular factorization. The question is whether it suffices to assume that one of these factorizations is regular. This issue is resolved by the following result.
\[T:left-right\] Let $(T_1, T_2)$ be a commutative, contractive pair of contraction operators on a Hilbert space ${{\mathcal H}}$.
1. Assume that all defect spaces ${{\mathcal D}}_{T}$, ${{\mathcal D}}_{T_1}$, ${{\mathcal D}}_{T_2}$ are finite-dimensional. Then $T_1 \cdot T_2$ is a regular factorization if and only if $T_2 \cdot T_1$ is a regular factorization.
2. In the infinite-dimensional setting, it is possible for one of the factorizations $T_1 \cdot T_2$ to be regular while the other $T_2 \cdot T_1$ is not regular.
Let us write $\Lambda_r$ for the counterpart of $\Lambda$ when the roles of $T_1$ and $T_2$ are interchanged: $$\Lambda_r \colon D_T h \mapsto D_{T_1} T_2 h \oplus D_{T_2} h.$$ Then by the same computation with the role of the indices $(1,2)$ interchanged, we see that $\Lambda_r$ is an isometry from ${{\mathcal D}}_T$ onto ${{\mathcal R}}(U_0)$. Moreover, Ii $T_1 \cdot T_2$ is a regular factorization, then $$\begin{aligned}
\dim {{\mathcal D}}_T & = \dim {{\mathcal D}}(U_0) \text{ (since $\Lambda$ is an isometry from ${{\mathcal D}}_T$ onto ${{\mathcal D}}_{U_0}$)} \\
& = \dim ({{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}) \text{ (by regularity of $T_1 \cdot T_2$)} \\
& \ge \dim {{\mathcal R}}_{U_0} \text{ (since ${{\mathcal R}}_{U_0} \subset {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$) }\\
& = \dim {{\mathcal D}}_T \text{ (since $\Lambda_r$ is an isometry from ${{\mathcal D}}_T$ onto ${{\mathcal R}}_{U_0}$).}
\end{aligned}$$ Thus we see that necessarily the inequality in line 3 must be an equality. If we are in the finite-dimensional setting (${{\mathcal D}}_T$, ${{\mathcal D}}_{T_1}$, ${{\mathcal D}}_{T_2}$ all finite-dimensional), we necessarily then have ${{\mathcal R}}_{U_0} = {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ which is the statement that the factorization $T_2 \cdot T_1$ is also regular.
[*Proof of (2):*]{} Let $(T_1, T_2)$ be the following pair of contractions on ${{\mathcal H}}: = H^2 \oplus H^2$: $$(T_1, T_2) =\left( \begin{bmatrix} 0 & 0 \\ I & 0 \end{bmatrix}, \begin{bmatrix}T_z & 0 \\ 0 & T_z \end{bmatrix} \right)$$ where $T_z$ is the Toeplitz operator $f(z) \mapsto z f(z)$ on $H^2$. Note that $$T_1 T_2 = \begin{bmatrix} 0 & 0 \\ T_z & 0 \end{bmatrix} = T_2 T_1 := T.$$ Then $$\begin{aligned}
& D_T^2 = \begin{bmatrix} I_{H^2} & 0 \\ 0 & I_{H^2} \end{bmatrix} -
\begin{bmatrix} 0 & T_z^* \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ T_z & 0 \end{bmatrix}
= \begin{bmatrix} 0 & 0 \\ 0 & I_{H^2} \end{bmatrix} = D_T, \quad {{\mathcal D}}_T = \begin{bmatrix} 0 \\ H^2 \end{bmatrix} \\
& D_{T_1}^2 = \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix} - \begin{bmatrix} 0 & I \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} 0 & 0 \\ I & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & I \end{bmatrix} = D_{T_1}, \quad
{{\mathcal D}}_{T_1} = \begin{bmatrix} 0 \\ H^2 \end{bmatrix}, \\
& D_{T_2}^2 = \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix} - \begin{bmatrix} T_z^* & 0 \\ 0 & T_z^* \end{bmatrix}
\begin{bmatrix} T_z & 0 \\ 0 & T_z \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \quad
{{\mathcal D}}_{T_2} = \{0 \},\end{aligned}$$ and ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2} = {\left[\begin{smallmatrix} 0 \\ H^2
\end{smallmatrix}\right]}$.
We let $\Lambda$ be the map associated with the factorization $T_1 \cdot T_2$ given by while $\Lambda_r$ is the same map associated with the factorization $T_2 \cdot T_1$ (i.e., but with the indices and then the components interchanged). For $h = {\left[\begin{smallmatrix} h_1 \\ h_2
\end{smallmatrix}\right]} \in {{\mathcal H}}= {\left[\begin{smallmatrix} H^2 \\ H^2
\end{smallmatrix}\right]}$ we compute $$\Lambda \colon D_T h = {\left[\begin{smallmatrix} 0 \\ h_2
\end{smallmatrix}\right]} \mapsto
D_{T_1} T_2 h \oplus D_{T_2} h = {\left[\begin{smallmatrix} 0 \\ T_z h
\end{smallmatrix}\right]}$$ and we conclude that $$\operatorname{Ran} \Lambda = {\left[\begin{smallmatrix} 0 \\ zH^2
\end{smallmatrix}\right]} \underset{\ne}\subset {\left[\begin{smallmatrix} 0 \\ H^2
\end{smallmatrix}\right]} = {{\mathcal D}}_{T_1} \oplus
{{\mathcal D}}_{T_2}$$ implying that $T_1 \cdot T_2$ is not a regular factorization. On the other hand, $$\Lambda_r \colon D_T h = {\left[\begin{smallmatrix} 0 \\ h_2
\end{smallmatrix}\right]} \mapsto
D_{T_1} h \oplus D_{T_2} T_1 h = {\left[\begin{smallmatrix} 0 \\ h_2
\end{smallmatrix}\right]} \oplus 0$$ from which we see that $$\operatorname{Ran} \Lambda_r = {\left[\begin{smallmatrix} 0 \\ H^2
\end{smallmatrix}\right]} = {{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$$ implying that the factorization $T_2 \cdot T_1$ is regular.
We shall also need the following extension of parts of Proposition \[P:dil\] to the setting of a commutative contractive operator-pairs.
\[P:LandL\*\] Let $(T_1,T_2)$ be a pair of commuting contractions on a Hilbert space $\mathcal{H}$ and let $(V_1,V_2)$ be an Andô commutative isometric dilation for $(T_1, T_2)$ acting on $\mathcal{K} \supset {{\mathcal H}}$. Define the following subspaces of ${{\mathcal K}}$: $$\begin{aligned}
\label{L}
&\mathcal{L}_1=\overline{(V_1-T_1)\mathcal{H}},\;\mathcal{L}_2=\overline{(V_2-T_2)\mathcal{H}},\\\label{L*}
&\mathcal{L}_{1*}=\overline{(I-V_1T_1^*)\mathcal{H}},\;\mathcal{L}_{2*}=\overline{(I-V_2T_2^*)\mathcal{H}}.\end{aligned}$$ Then for each $k=1,2$, the maps $$\iota_{k*} \colon {{\mathcal L}}_{k*} \to {{\mathcal D}}_{T_k^*}\text{ and }\iota_k \colon {{\mathcal L}}_k \to {{\mathcal D}}_{T_k}$$ given densely by $$\iota_{k*} \colon (I - V_k T_k^*) h \mapsto D_{T_k^*} h, \text{ and }
\iota_k \colon (V_k - T_k) h \mapsto D_{T_k} h$$ for $h \in {{\mathcal H}}$ extend to define unitary identification maps from ${{\mathcal L}}_{k*}$ onto ${{\mathcal D}}_{T_k^*}$ and from ${{\mathcal L}}_k$ onto ${{\mathcal D}}_{T_k}$ respectively.
This amounts to statement (1) of Proposition \[P:dil\] with the single contraction operator $T$ replaced by the pair of contraction operators $(T_1, T_2)$. The proof comes down to the simple inner product computation, for $k=1,2$ and $h\in \mathcal{H}$, $$\begin{aligned}
\|(I - V_k T_k^*) h\|^2 &=& \langle (I - V_k T_k^*) h, (I - V_k T_k^*) h \rangle \\
&=&\|h\|^2-\langle V_kT_k^*h,h \rangle -\langle h, V_kT_k^*h \rangle+\|T_k^*h\|^2\\
&=&\|h\|^2-\|T_k^*h\|^2=\|D_{T_k^*}h\|^2.\end{aligned}$$ A similar computation shows that $\|(V_k - T_k) h\|^2=\|D_{T_k}h\|^2$.
We next discuss the following “coordinate-free” adjustment of the notion of Andô tuple discussed in the Introduction (see Definition \[D:Ando-tuple\]). Using part (1) of Proposition \[P:dil\] and Proposition \[P:LandL\*\], we observe that for $h\in\mathcal{H}$, $$\begin{aligned}
\nonumber\|(V-T)h\|^2 =\|D_Th\|^2&=&\|D_{T_1}T_2h\|^2+\|D_{T_2}h\|^2=\|(V_1-T_1)T_2h\|^2+\|(V_2-T_2)h\|^2\\\label{IsoUniGen}
&=&\|D_{T_1}h\|^2+\|D_{T_2}T_1h\|^2=\|(V_1-T_1)h\|^2+\|(V_2-T_2)T_1h\|^2,\end{aligned}$$ which implies that $\Lambda_c:{{\mathcal L}}\to{{\mathcal L}}_1\oplus{{\mathcal L}}_2$ (here the subscript $c$ suggests [*coordinate-free*]{}) densely defined by $$\begin{aligned}
\label{Lambda-c}
\Lambda_c:(V-T)h\mapsto (V_1-T_1)T_2h\oplus (V_2-T_2)h\end{aligned}$$ is an isometry. Let us introduce the notation $$\begin{aligned}
& {{\mathcal D}}_{U_{c0}} = \operatorname{Ran} \Lambda_c = \operatorname{clos.}
\{ (V_1-T_1)T_2h\oplus (V_2-T_2)h \colon h \in {{\mathcal H}}\}, \\
& {{\mathcal R}}_{U_{c0}} = \operatorname{clos.} \{(V_1 - T_1) h \oplus (V_2 - T_2) T_1 h \colon
h \in {{\mathcal H}}\}.\end{aligned}$$ Then a consequence of the equality between the fourth and sixth terms in the chain of equalities is there is a unitary map $U_{c0} \colon {{\mathcal D}}_{U_{c0}} \to {{\mathcal R}}{U_{c0}}$ densely defined by $$\begin{aligned}
\label{U-c}
U_{c0} \colon (V_1-T_1)T_2h\oplus (V_2-T_2)h\mapsto (V_1-T_1)h\oplus (V_2-T_2)T_1 h.\end{aligned}$$ We add an infinite-dimensional summand $\ell^2$ (if necessary) to the external direct sum ${{\mathcal L}}_1\oplus{{\mathcal L}}_2$ to ensure that $U_{c0}$ can be extended to a unitary map $U_c$ on the spaces ${{\mathcal F}}_c: ={{\mathcal L}}_1 \oplus {{\mathcal L}}_2$ (in case the $\ell^2$-summand is not required) or ${{\mathcal F}}_c: =
{{\mathcal L}}_1 \oplus {{\mathcal L}}_2 \oplus \ell^2$ (otherwise). Let $P_c$ denote the orthogonal projection of ${{\mathcal F}}_c$ onto ${{\mathcal L}}_1$: $$\begin{aligned}
& P_c \colon f \oplus g \mapsto f \oplus 0 \text{ if } {{\mathcal F}}_c = {{\mathcal L}}_1 \oplus {{\mathcal L}}_2, \notag \\
& P_c \colon f \oplus g \oplus h \mapsto f \oplus 0 \oplus 0 \text{ if } {{\mathcal F}}_c = {{\mathcal L}}_1 \oplus {{\mathcal L}}_2 \oplus \ell^2.
\label{Pc}\end{aligned}$$
\[D:Ando-c\] [*[The tuple $({{\mathcal F}}_c,\Lambda_c,P_c,U_c)$ of Hilbert spaces and operators arising as above from a commutative contractive pair $(T_1, T_2)$ having commutative isometric dilation $(V_1, V_2)$ as above will be called a[ *[coordinate-free Andô tuple]{}*]{} for $(T_1,T_2)$.]{}*]{}
It will also be useful to spell out the notion of [*coordinate-free Andô tuple for $(T_1^*, T_2^*)$*]{}. Via a computation similar to what was done in we see that $$\begin{aligned}
\|(I-VT^*)h\|^2 &=\|(I-V_1T_1^*)T_2^*h\|^2+\|(I-V_2T_2^*)h\|^2 \notag \\
&=\|(I-V_1T_1^*)h\|^2+\|(I-V_2T_2^*)T_1^*h\|^2.
\label{IsoUniGen*}\end{aligned}$$ We conclude in particular that there is an isometry $\Lambda_{c*}:{{\mathcal L}}_*\to {{\mathcal L}}_{1*}\oplus{{\mathcal L}}_{2*}$ densely defined as $$\label{Gamma-c*}
\Lambda_{c*}:(I-VT^*)h\mapsto (I-V_1T_1^*)T_2^*h\oplus (I-V_2T_2^*) h.$$ Introduce the notation $$\begin{aligned}
& {{\mathcal D}}_{U_{c*0}} = \operatorname{Ran} \Lambda_{c*} = \operatorname{clos} \{ (I-V_1T_1^*)T_2^*h
\oplus (I-V_2T_2^*) h \colon h \in {{\mathcal H}}\}, \\
& {{\mathcal R}}_{U_{c*0}} = \operatorname{clos} \{ (I - V_1 T_1^*) h \oplus (I - V_2 T_2^*) T_1 h \colon h \in {{\mathcal H}}\}.\end{aligned}$$ Then the last equality in the chain of equalities implies that there is unitary $U_{c*0} \colon {{\mathcal D}}_{U_{c*0}} \to {{\mathcal R}}_{U_{c*0}}$ densely defined by $$U_{c*0} \colon (I-V_1T_1^*)T_2^*h \oplus (I-V_2T_2^*) h \mapsto
(I - V_1 T_1^*) h \oplus (I - V_2 T_2^*) T_1 h.$$ Add an infinite dimensional summand to ${{\mathcal L}}_{1*}\oplus{{\mathcal L}}_{2*}$ if necessary to guarantee that the isometry $U_{c0*}$ viewed as an operator from $\overline{\operatorname{Ran}}\, \Lambda_{c*}$ into ${{\mathcal F}}_{c*}$ can be extended to a unitary operator, denotes as $U_{c*}$, on ${{\mathcal F}}_{c*}$, where ${{\mathcal F}}_{c*}$ is either ${{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*}$, in case $\dim ({{\mathcal D}}_{U_{c*0}})^\perp =
\dim ({{\mathcal R}}_{U_{c*0}})^\perp$ (viewed as subspaces of the ambient space ${{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*}$, and ${{\mathcal F}}_{c*} =
{{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*} \oplus \ell_+^2$ otherwise. We fix a choice of such a unitary extension $U_{c*}$ of the isometry and denote it also as $U_{c*}$. We denote by $P_{c*}$ the orthogonal projection of ${{\mathcal F}}_{c*}$ onto its first component, given by $$\begin{aligned}
& P_{c*} \colon f \oplus g \mapsto f \oplus 0 \text{ in case } {{\mathcal F}}_{c*} = {{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*}, \\
& P_{c*} \colon f \oplus g \oplus h \mapsto f \oplus 0 \oplus 0 \text{ in case }
{{\mathcal F}}_{c*} = {{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*} \oplus \ell^2.\end{aligned}$$ We then make the following formal definition.
\[D:Ando-c\*\] [Let $(T_1, T_2)$ be a commutative pair of contractions on ${{\mathcal H}}$. Any collection of spaces $({{\mathcal F}}_{c*}, \Lambda_{c*}, P_{c*}, U_{c*} )$ arising via the construction as described in the previous paragraph will be called a [*coordinate-free Andô tuple for $(T_1^*, T_2^*)$*]{}.]{}
\[R:Ando-tuple\] [We point out here that Remark \[R:Ando-tuple-unique\] applies equally well to the coordinate-free Andô tuples in Definitions \[D:Ando-c\] and \[D:Ando-c\*\]. Each construction of an Andô tuple involves an arbitrary choice of unitary extension of an in general only partially defined isometry ($U_{0}$, $U_{c0}$,and $U_{c0*}$ respectively). Hence we cannot expect any result to the effect that Andô tuples are unique up to a unitary equivalence connecting the various operators and spaces except in the case when (or ${{\mathcal D}}_{U_0}$ and ${{\mathcal R}}_{U_0}$ is the whole space ${{\mathcal D}}_{T_1} \oplus {{\mathcal D}}_{T_2}$ (respectively, ${{\mathcal D}}_{U_{c0}}$ and ${{\mathcal R}}_{U_{c0}}$ is the whole space ${{\mathcal L}}_1 \oplus {{\mathcal L}}_2$, respectively ${{\mathcal D}}_{U_{c*0}}$ and ${{\mathcal R}}_{U_{c*0}}$ is the whole space ${{\mathcal L}}_{1*} \oplus {{\mathcal L}}_{2*}$). One can use the unitary identification maps $\iota_j$ and $\iota_{j*}$ ($j = 1, 2$) to see that this holds for one of the three cases (i.e., Definitions \[D:Ando-tuple\], \[D:Ando-c\], and \[D:Ando-c\*\]) if and only if holds for the all the cases. As observed in Remark \[R:Ando-tuple-unique\], this happens if and only if both $T = T_1 \cdot T_2$ and $T = T_2 \cdot T_1$ are regular factorizations. ]{}
The Schäffer coordinate-free model for an Andô dilation
-------------------------------------------------------
Let $(T_1,T_2)$ be a pair of commuting contractions on a Hilbert space $\mathcal{H}$ and $(V_1,V_2)$ be an isometric dilation of $(T_1,T_2)$ acting on $\mathcal{K}$. Note that it is highly unlikely for the lifting $V=V_1V_2$ of $T=T_1T_2$ to be minimal, because the space $\mathcal{K}$ contains $$\overline{\text {span}}\{V_1^mV_2^n h:h\in\mathcal{H}\text{ and }m,n\geq 0 \},$$which, in general, is bigger than the resulting space when we choose $m=n$ above. Therefore by part (6) of Proposition \[P:dil\], the dilation space $\mathcal{K}$ should, in general, be of the form $$\begin{aligned}
\label{CFdilspace}
{{\mathcal H}}\oplus M_+({{\mathcal L}}) \oplus {{\mathcal R}}_0,\end{aligned}$$ where ${{\mathcal L}}= \overline{(V- T) {{\mathcal H}}}$ and for some non-zero $\mathcal{R}_0$. Unlike the single-variable case, there is no canonical way of constructing a minimal isometric dilation of a pair $(T_1,T_2)$ of commuting contractions. As a consequence, construction of an Andô dilation requires a non-zero and noncanonical choice of subspace $\mathcal{R}_0$. We fix a choice of coordinate-free Andô tuple $({{\mathcal F}}_c, \Lambda_c, P_c, U_c)$ for $(T_1, T_2)$ (see Definition \[D:Ando-c\]) and then choose $$\begin{aligned}
\label{R0}
{{\mathcal R}}_0=\ell^2_+({{\mathcal F}}_c\ominus\Lambda_c{{\mathcal L}}).\end{aligned}$$ Consider the following identification of the space in (\[CFdilspace\]): $$\begin{aligned}
\label{CFLspace}
{{\mathcal K}}_c:={{\mathcal H}}\oplus \ell^2_+(\Lambda_c {{\mathcal L}}) \oplus {{\mathcal R}}_0={{\mathcal H}}\oplus \ell^2_+(\Lambda_c {{\mathcal L}}) \oplus \ell^2_+({{\mathcal F}}_c\ominus\Lambda_c{{\mathcal L}})={{\mathcal H}}\oplus \ell^2_+({{\mathcal F}}_c).\end{aligned}$$
Below we show that this choice of ${{\mathcal R}}_0$ works, i.e., it is possible to find an Andô dilation on a space of the form (\[CFdilspace\]) with ${{\mathcal R}}_0$ as in (\[R0\]). Let $S$ denote the forward shift on $\ell^2$. Then $S\otimes I_{{{\mathcal F}}_c}$ is the forward shift on $\ell^2_+({{\mathcal F}}_c)
\cong \ell^2_+ \otimes {{\mathcal F}}_{c}$. Define $\tilde{V}$ on ${{\mathcal K}}_c$ by the following $2\times2$ block operator matrix: $$\begin{aligned}
\label{tildeV}
\tilde{V}:= \begin{bmatrix}
T & 0 \\ \Lambda(V-T) & S\otimes I_{{{\mathcal F}}_c}\\ \end{bmatrix} \colon
\begin{bmatrix} {{\mathcal H}}\\ \ell^2_+({{\mathcal F}}_c) \end{bmatrix} \to \begin{bmatrix} {{\mathcal H}}\\ \ell^2_+({{\mathcal F}}_c) \end{bmatrix}.\end{aligned}$$ A matrix computation verifies that $\tilde{V}$ is an isometry.
We next find a commuting pair $(\tilde{V}_1,\tilde{V}_2)$ of isometries on ${{\mathcal K}}_c$ such that $(\tilde{V}_1, \tilde{V}_2)$ is a lift of $(T_1,T_2)$ and $\tilde{V}_1\tilde{V}_2=\tilde{V}$. The requirement that $(\tilde{V}_1,\tilde{V}_2)$ to be a lift of $(T_1,T_2)$ forces $(\tilde{V_1}, \tilde{V}_2)$ to have the form $$\tilde{V}_1=\begin{bmatrix}
T_1 & 0 \\
C_1 & D_1 \\
\end{bmatrix}
\text{ and }
\tilde{V}_2= \begin{bmatrix}
T_2 & 0 \\
C_2 & D_2 \\ \end{bmatrix}.$$ From the fact that the pair $(D_1,D_2)$ is the restriction of $(\tilde{V}_1,\tilde{V}_2)$ to ${\ell^2_+({{\mathcal F}}_c)}$, we see that $(D_1, D_2)$ is a commuting pair of isometries. Moreover, the condition that $\tilde{V}_1\tilde{V}_2=\tilde{V}$ forces $(D_1,D_2)$ to satisfy $D_1D_2=S\otimes I_{{{\mathcal F}}_c}$. Hence by the result of Berger-Coburn-Lebow [@B-C-L], there exists a projection $P$ and a unitary $U$ on ${{\mathcal F}}_c$ such that $$(D_1,D_2)=(I_{\ell^2_+}\otimes P^\perp U+S\otimes PU,I_{\ell^2_+}\otimes U^*P+S\otimes U^*P^\perp ).$$ Since $\tilde{V}_1\tilde{V}_2=\tilde{V}$, we have $\tilde{V}_1=\tilde{V}_2^*\tilde{V}$ and $\tilde{V}_2=\tilde{V}_1^*\tilde{V}$, which then give $$\begin{aligned}
\label{1*2}
\begin{cases}
\text{(a) } C_1=(I_{\ell^2_+}\otimes U^*P+S\otimes U^*P^\perp)^*\Lambda_c(V-T)=PU\Lambda_c(V-T), \\
\text{(b) } C_2=(I_{\ell^2_+}\otimes P^\perp U+S\otimes PU)^*\Lambda_c(V-T)=U^*P^\perp\Lambda_c(V-T), \\
\text{(c) } T_1-T_2^*T=(V-T)^*\Lambda_c^*P^\perp U\Lambda_c(V-T), \text{ and} \\
\text{(d) } T_2-T_1^*T=(V-T)^*\Lambda_c^*U^*P\Lambda_c(V-T).
\end{cases}\end{aligned}$$ We do not know if one can embed an arbitrary projection $P$ and unitary $U$ into an Andô dilation, but we show in the following theorem that if any $(P,U)$ of the form $(P_c, U_c)$ with $P_c$ and $U_c$ as in the coordinate-free Andô tuple can be embedded in an Andô dilation for $(T_1, T_2)$.
For a pair $(T_1,T_2)$ of commuting contractions, let $({{\mathcal F}}_c,\Lambda_c,P_c,U_c)$ be a coordinate-free Andô tuple for $(T_1,T_2)$. The the pair $(V_{1c},V_{2c})$ defined on ${{\mathcal H}}\oplus \ell^2_+({{\mathcal F}}_c)$ by $$\begin{aligned}
\label{CFSchaf}
\begin{cases}
V_{1c}&= \begin{bmatrix}
T_1 & 0 \\
P_cU_c\Lambda_c(V-T) & I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c \end{bmatrix} \text{ and}\\
V_{2c}&= \begin{bmatrix}
T_2 & 0 \\
U_c^*P_c^\perp\Lambda_c(V-T) &I_{\ell^2_+}\otimes U_c^*P_c+S\otimes U_c^*P_c^\perp \end{bmatrix}
\end{cases}\end{aligned}$$ is an isometric dilation of $(T_1,T_2)$.
The pair $(V_{1c},V_{2c})$ is clearly a dilation of $(T_1,T_2)$ because it is a co-extension. We first show that the $(2,1)$-entry in the matrix representation of both $V_{1c}V_{2c}$ and $V_{2c}V_{1c}$ is $\Lambda_c (V-T)$, i.e., $$\begin{aligned}
&P_cU_c\Lambda_c (V-T)T_2+(I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c)U_c^*
P_c^\perp\Lambda_c (V-T)\\
=&U_c^*P_c^\perp\Lambda_c (V-T)T_1+(I_{\ell^2_+}\otimes U_c^*P_c+S\otimes U_c^*P_c^\perp)
P_cU_c\Lambda_c (V-T)=\Lambda_c (V-T).\end{aligned}$$ This is true because for all $h\in \mathcal H$ we see that $$\begin{aligned}
&P_cU_c\Lambda_c (V-T)T_2+(I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c)U_c^*P_c^\perp\Lambda_c (V-T)h\\
& \quad =P_cU_c\Lambda_c (V-T)T_2+P_c^\perp \Lambda_c (V-T)h\\
& \quad = \Lambda_c (V-T)h\\
& \quad = U_c^*U_c((V_1-T_1)T_2h\oplus (V_2-T_2)h)\\
& \quad = U_c^*((V_1-T_1)h\oplus 0)+U_c^*(0\oplus (V_2-T_2)T_1h)\\
& \quad = U_c^*P_cU_c\Lambda_c (V-T)h+U_c^*P_c^\perp\Lambda_c (V-T)T_1h\\
& \quad = U_c^*P_c^\perp\Lambda_c (V-T)T_1h+(I_{\ell^2_+}\otimes U_c^*P_c+S\otimes U_c^*P_c^\perp) P_cU_c\Lambda_c (V-T)h.\end{aligned}$$ This proves the commutativity part because $$\begin{aligned}
&(I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c)(I_{\ell^2_+}\otimes U_c^*P_c+S\otimes U_c^*P_c^\perp)\\
=&(I_{\ell^2_+}\otimes U_c^*P_c+S\otimes U_c^*P_c^\perp)( I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c)=S\otimes I_{{{\mathcal F}}_c}.\end{aligned}$$
It remains to show that $V_{1c}$ and $V_{2c}$ are isometries. A simple matrix computation shows that $V_{1c}$ is an isometry if and only if the following equalities hold: $$\begin{aligned}
T_1^*T_1+(V-T)\Lambda_c^*(U_c^*P_c)(U_c^*P_c)^*\Lambda_c (V-T)=I_{\mathcal H}
\label{proof-2-1}\\
(V-T)\Lambda_c^*(U_c^*P_c)(I_{\ell^2_+}\otimes P_c^\perp U_c+S\otimes P_cU_c)=0.
\label{proof-2-2}\end{aligned}$$ The first equality is true because for every $h,h'\in \mathcal H$, $$\begin{aligned}
\langle P_cU_c\Lambda_c (V-T)h,P_cU_c\Lambda_c (V-T)h' \rangle=\langle (V_1-T_1)h\oplus 0,(V_1-T_1)h'\oplus 0\rangle =\langle D_{T_1}^2h,h' \rangle.\end{aligned}$$ As for the second equality, note that our conventions are such that the operator $U_c^*P_c$ appearing in the second equality is short hand for the operator $ P_{\{e_0\}} \otimes U_c^* P_c$ acting on the spaced $\ell^2_+({{\mathcal F}}_c) \cong \ell^2_+ \otimes {{\mathcal F}}_c$. Let us compute $$\begin{aligned}
(U_c^* P_c) (I_{\ell^2_+} \otimes P_c^\perp U_c + S \otimes P_c U_c) := &
(P_{\{ e_0\}} \otimes U_c^* P_c) (I_{\ell^2_+} \otimes P_c^\perp U_c + S \otimes P_c U_c) \\
=& P_{\{e_0\}} \otimes U_c^* P_c P_c^\perp U_c + ( P_{\{e_0\}} S \otimes U_c^* P_c U_c) \\
=& P_{\{e_0\}} \otimes 0 + 0 \otimes U_c^* P_c U_c = 0.\end{aligned}$$ Hence also $$(V-T) \Lambda_c^* \cdot (U_c^* P_c) (I_{\ell^2_+} \otimes P_c^\perp U_c + S \otimes P_c U_c) = 0.$$ Thus follows as well and we conclude that $V_{1c}$ is an isometry. Similarly one can show that $V_{2c}$ is an isometry too. This completes the proof.
The Douglas coordinate-free model for an Andô dilation
------------------------------------------------------
We now work with the first decomposition of the minimal isometric dilation space as in (\[decom4\]). As before we first must find a non-zero ${{\mathcal R}}_0$ such that the Andô dilation space is given by $$\begin{aligned}
\label{CFDspace*}
{{\mathcal K}}=M_+({{\mathcal L}}_*)\oplus {{\mathcal R}}\oplus {{\mathcal R}}_0,\end{aligned}$$where ${{\mathcal R}}$ is the minimal component, i.e., by part 6(b) of Proposition \[P:dil\], $${{\mathcal R}}=\overline{\operatorname{span}}\{V^n P_{{\mathcal R}}h: n\geq 0,\;h\in{{\mathcal H}}\}.$$ For this construction we make use of a coordinate-free Andô tuple $({{\mathcal F}}_{c*}, \Lambda_{c*}, P_{c*}, U_{c*})$ for $(T_1^*, T_2^*)$, (see Definition \[D:Ando-c\*\]). We then choose $${{\mathcal R}}_0 = \ell^2_+({{\mathcal F}}_{c*}\ominus\Lambda_{c*}{{\mathcal L}}_{*})$$ and consider the following identification of the space given in (\[CFDspace\*\]) $${{\mathcal K}}= \ell^2_+(\Lambda_{c*} {{\mathcal L}}_*) \oplus {{\mathcal R}}\oplus \ell^2_+({{\mathcal F}}_{c*}\ominus\Lambda_{c*}{{\mathcal L}}_{*})
=\ell^2_+({{\mathcal F}}_{c*}) \oplus {{\mathcal R}}.$$
Let the operators $Q$, $W_D$ be as in the discussion around –. Let $S_{{{\mathcal F}}_{c*}}$ denote the forward shift on $\ell^2_+({{\mathcal F}}_{c*})$; as was the case above where coefficient spaces other than ${{\mathcal F}}_{c*}$ came up, when we view $\ell^2_+({{\mathcal F}}_{c*})$ as $\ell^2_+ \otimes {{\mathcal F}}_{c*}$, then we may view $S_{{{\mathcal F}}_{c*}}$ as equal to $S \otimes I_{{{\mathcal F}}_{c*}}$. Define $V^D_c$ on $\ell^2_+({{\mathcal F}}_{c*})\oplus{{\mathcal R}}$ as $$\begin{aligned}
\label{VDc}
V^D_c=S_{{{\mathcal F}}_{c*}}\oplus W_D\end{aligned}$$and $\Pi_{D}^c:{{\mathcal H}}\to \ell^2_+({{\mathcal F}}_{c*})\oplus{{\mathcal R}}$ as $$\begin{aligned}
\label{PiDc}
\Pi_D^ch=\sum_{n=0}^{\infty}S_{{{\mathcal F}}_{c*}}^n E_0^* \Lambda_{c*}(I-VT^*)T^{* n}h \oplus Qh\end{aligned}$$ where $E_0^* \colon {{\mathcal F}}_{c*} \to \ell^2_+({{\mathcal F}}_{c*})$ is the embedding of ${{\mathcal F}}_{c*}$ into the $0$-component $\{f \oplus 0 \oplus 0 \oplus 0 \oplus \cdots \colon f \in {{\mathcal F}}_{c*}\}$ of $\ell^2_+({{\mathcal F}}_{c*})$.
The following computation shows that $V^D_c$ is a lift of $T$, where we use the second equation in (\[IntofQ\]): $$\begin{aligned}
\Pi_D^cT^*h & =\sum_{n=0}^{\infty} S_{{{\mathcal F}}_{c*}}^n E_0^* \Lambda_{c*} (I-VT^*)T^{* n+1}h \oplus QT^*h\\
& = S^*_{{{\mathcal F}}_{c*}} \left( \sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^{n+1} E_0^* \Lambda_{c*} (I - V T^*) T^{* n+1} h \right)
\oplus Q T^* h \\
&=S_{{{\mathcal F}}_{c*}}^* \left( \sum_{n=1}^{\infty} S_{{{\mathcal F}}_{c*}}^n E_0^* \Lambda_{c*} (I-VT^*)T^{* n}h \right)
\oplus W_D^*Qh \\
& = S_{{{\mathcal F}}_{c*}}^* \left( \sum_{n=0}^\infty
S_{{{\mathcal F}}_c*}^n E_0^* \Lambda_{c*} (I - V T^*) T^{*n} h \right) \oplus W_D^* Q h = (V^{D}_c)^*\Pi_D^c h.\end{aligned}$$
We shall find an Andô dilation $(V_{c1}^D,V_{c2}^D)$ of $(T_1,T_2)$ on $\ell^2_+({{\mathcal F}}_{c*})
\oplus{{\mathcal R}}$ such that $V_{c1}^DV_{c2}^D=V^D_c$. To that end, we do further analysis of the operator $Q$ defined as the positive square root of $Q^2 : = \operatorname{SOT-}
\lim_{n \to \infty} T^nT^{* n}$, where here $T=T_1T_2=T_2T_1$. We first note that $Q$ has the following additional properties $$\begin{aligned}
\label{prop1}
T_1Q^2T_1^* \preceq Q^2 \text{ and }T_2Q^2T_2^* \preceq Q^2,\end{aligned}$$the first one of which follows easily from the following inner product computation for every $h\in\mathcal{H}$ (the proof for the second one is similar): $$\begin{aligned}
\langle T_1Q^2T_1^*h,h\rangle = \lim_n\langle T^n(T_1T_1^*){T^*}^nh,h \rangle \leq \lim_n\langle T^n{T^*}^nh,h\rangle = \langle Q^2h,h\rangle.\end{aligned}$$We now recall the following simple but telling result of Douglas.
\[DougLem\] Let $A$ and $B$ be two bounded operators on a Hilbert space $\mathcal{H}$. Then there exists a contraction $C$ such that $A=BC$ if and only if $$AA^*\preceq BB^*.$$
See the paper [@Douglas] for a general version of the above lemma. Inequalities in (\[prop1\]) and Lemma \[DougLem\] imply that there exist two contractions say $X_1^*$ and $X_2^*$ such that $$\begin{aligned}
\label{X1X2}
X_1^*Q=QT_1^* \text{ and } X_2^*Q=QT_2^*.\end{aligned}$$ From the equalities in (\[X1X2\]) and (\[defX\]) it is clear that $X_1$ and $X_2$ commute and that $$\begin{aligned}
X^*=X_1^*X_2^*.\end{aligned}$$ Since $X^*$ is an isometry, both $X_1^*$ and $X_2^*$ are isometries, as a consequence of the general fact that, whenever $T$ is an isometry with factorization $T = T_1 T_2$ for some commuting contractions $T_1$ and $T_2$, then in fact $T_1$ and $T_2$ are also isometries; indeed consider the following norm equalities which we have already seen in the discussion of Andô tuples (Definition \[D:Ando-tuple\]): $$\|D_{T_1}T_2h\|^2+\|D_{T_2}h\|^2=\|D_Th\|^2=\|D_{T_1}h\|^2+\|D_{T_2}T_1h\|^2 \text{ for all }h \in \mathcal{H}.$$ Also, note that the same is true if the word ‘isometry’ is replaced by ‘unitary’ because the above equalities hold for every contraction, in particular, for $T_1^*$ and $T_2^*$ also. We now recall the following result from [@sauAndo].
\[special-ext\] Let $\underline{V}=(V_1,V_2,\dots, V_n,V)$ be a commuting tuple of isometries on a Hilbert space $\mathcal{H}$ such that $V=V_1V_2\cdots V_n$, then $\underline{V}$ has a commuting unitary extension $\underline{Y}=(Y_1,Y_2,\dots, Y_n,Y)$ such that $Y=Y_1Y_2\cdots Y_n$ is the minimal unitary extension of $V$.
By this lemma we get a commuting unitary extension $(W_{\partial1}^*,W_{\partial2}^*)$ on $\mathcal{R}_D$ of $(X_1^*,X_2^*)$ such that $W_D^*=W_{\partial1}^*W_{\partial2}^*$ is the minimal unitary extension of $X^*=X_1^*X_2^*$, equivalently, $W_D$ is the minimal isometric dilation of $X$, hence $$\begin{aligned}
\label{TheSpace-R}
\mathcal{R}_D=\overline{\text{span}}\{W_D^{ n}x: x\in \overline{\operatorname{Ran}}\,Q\text{ and }n\geq0\}.\end{aligned}$$ The lemma below shows that the pair $(W_1,W_2)$ depends uniquely on $(T_1,T_2)$.
Let $(T_1,T_2)$ on $\mathcal{H}$ and $(T_1',T_2')$ on $\mathcal{H'}$ be two pairs of commuting contractions and $(W_{\partial1},W_{\partial2})$ on $\mathcal{R}_D$ and $(W_{\partial1}',W_{\partial2}')$ on $\mathcal{R}_D'$ be the respective pairs of commuting unitaries obtained from them as above. If $(T_1,T_2)$ is unitarily equivalent to $(T_1',T_2')$ via the unitary similarity $\phi \colon \mathcal{H} \to \mathcal{H'}$, then $(W_{\partial1},W_{\partial2})$ and $(W_{\partial1}',W_{\partial2}')$ are unitarily equivalent via the induced unitary transformation $\tau_\phi \colon \mathcal{R}_D\to \mathcal{R}_D'$ determined by $\tau_\phi \colon W_D^nQh \to W_D'^n Q'\phi h$. In particular, if $(T_1, T_2) = (T_1', T_2')$, then $(W_{\partial1},W_{\partial2}) = (W_{\partial1}',W_{\partial2}')$.
Let $\phi:\mathcal{H}\to\mathcal{H'}$ be a unitary that intertwines $(T_1,T_2)$ and $(T_1',T_2')$. Let us denote $T=T_1T_2$ and $T'=T_1'T_2'$. Let $Q$ and $Q'$ be the limits of $T^nT^{* n}$ and $T'^nT'^{* n}$, respectively, in the strong operator topology. Clearly, $\phi$ intertwines $Q$ and $Q'$. Therefore $\phi$ takes $\mathcal{Q}\equiv\overline{\operatorname{Ran}}\,Q$ onto $\mathcal{Q'}\equiv\overline{\operatorname{Ran}}\,Q'$. We denote the restriction of $\phi$ to $\overline{\operatorname{Ran}}\,Q$ by $\phi$ itself. Let $(X_1,X_2)$ on $\overline{\operatorname{Ran}}\,Q$ and $(X_1',X_2')$ on $\overline{\operatorname{Ran}}\,Q'$ be the pairs of commuting co-isometries corresponding to the pairs $(T_1,T_2)$ and $(T_1',T_2')$ as in (\[X1X2\]), respectively. It is easy to see from the definition that $$\phi(X_1,X_2)=(X_1',X_2')\phi.$$Let $(W_{\partial1},W_{\partial2})$ on $\mathcal{R}_D$ and $(W_{\partial1}',W_{\partial2}')$ on $\mathcal{R}_D'$ be the pairs of commuting unitaries corresponding to $(T_1,T_2)$ and $(T_1',T_2')$, respectively. Remembering the formula (\[TheSpace-R\]) for the spaces $\mathcal{R}_D$ and $\mathcal{R}_D'$, we define $\tau_\phi:\mathcal{R}_D\to\mathcal{R}_D'$ by $$\tau_\phi: W_D^{ n}x\mapsto W_D'^{ n}\phi x, \text{ for every } x\in \mathcal{Q} \text{ and } n\geq 0$$and extend linearly and continuously. Trivially, $\tau_\phi$ is unitary and intertwines $W_D$ and $W_D'$. For a non-negative integer $n$ and $x$ in $\mathcal{Q}$, we have $$\begin{aligned}
\tau_\phi W_{\partial1}(W_D^{n}x)=\tau_\phi W_D^{n+1}(W_{\partial2}^*x)&=&W_D'^{ n+1}\phi(X_2^*x)\\
&=&W_D'^{n+1}W_{\partial2}'^*\phi x=W_{\partial1}'W_D'^{n}\phi x=W_{\partial1}'\tau_\phi(W_D^{ n}x).\end{aligned}$$A similar computation shows that $\tau_\phi$ intertwines $W_{\partial2}$ and $W_{\partial2}'$ too.
Define the following two operators on $\ell^2_+({{\mathcal F}}_{c*})\oplus{{\mathcal R}}$: $$\begin{aligned}
\label{CFDAndo}
\begin{cases}
V^D_{c1}=(I_{\ell^2_+} \otimes U_{c*}^* P_{c*}^\perp + S \otimes U_{c*}^* P_{c*}) \oplus W_{\partial1},\\
V^D_{c2} = (I_{\ell^2_+} \otimes P_{c*} U_{c*} + S \otimes P^\perp_{c*} U_{c*} ) \oplus W_{\partial2}.
\end{cases}\end{aligned}$$
\[Thm:CFDAndo\] With the operators $V_{c1}^D$ and $V_{c2}^D$ as defined in (\[CFDAndo\]), $(\Pi_D^c,V_{c1}^D,V_{c2}^D)$ is a commuting isometric lift of $(T_1,T_2)$.
Note that $(V_{c1}^D, V_{c2}^D)$ is a commutative pair of isometries since it has the form of a Berger-Coburn-Lebow model for a commuting pair of isometries. It remains to show that, for $j=1,2,$ $V_{cj}^D$ is a lift of $T_j$. We do in detail only the case $j=1$ as the other case is similar. All that remains to show is that $$\label{toshow}
V_{c1}^{D*} \Pi_D^c h = \Pi_D^c T_1^* h \text{ for all } h \in {{\mathcal H}}.$$ We use the definitions to compute $$\begin{aligned}
\label{step1}
& V_{c1}^{D*} \Pi_D^c h \\
=& \left( (I_{\ell^2_+} \oplus P^\perp_{c*} U_{c*}) + (S^* \otimes P_{c*} U_{c*}) \right)
\sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^n E_0^* \Lambda_{c*} (I - V T^*)T^{*n} h \oplus W_{\partial1}^* Q h \notag \\
=& \left( \sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^n E_0^* P^\perp_{c*} U_{c*} E_0^* \Lambda_{c*}
(I - VT^*) T^{*n} h + \sum_{n=0}^\infty S_{{{\mathcal F}}_c}^n E_0^* U_{c*} \Lambda_{c*} (I - VT^*) T^{*n+1} h \right) \oplus
QT_1^* h.
\notag\end{aligned}$$ As a side computation let us note that $$\begin{aligned}
\label{side1}
& P_{c*}^\perp U_{c*} \Lambda_{c*} (I - VT^*) T^{*n} h
= P_{c*}^\perp U_{c*} \left( (I - V_1 T_1^*) T_2^* T^{*n} h \oplus ( I - V_2 T_2^*) T^{*n} h \right) \\
& \quad = P_{c*}^\perp \left( (I - V_1 T_1^*) T^{*n} h \oplus (I - V_2 T_2^*) T_1^* T^{*n} h \right) \notag \\
& \quad = 0 \oplus (I - V_2 T_2^*) T_1^* T^{*n} h \notag\end{aligned}$$ while $$\begin{aligned}
\nonumber& P_{c*} U_{c*} \Lambda_{c*} (I - V T^*) T^{*n+1} h\\
&=P_{c*} U_{c*} \left( (I - V_1 T_1^*) T_2^* T^{* n+1} h \oplus (I - V_2 T_2^*) T^{*n+1} h \right) \label{side2} \\
&= P_{c*} \left( (I - V_1 T_1^*) T^{*n+1} h \oplus (I - V_2 T_2^*) T_1^* T^{*n+1} h \right) \notag \\
&= ( I - V_1 T_1^*) T^{*n+1} h \oplus 0. \notag\end{aligned}$$ Plugging and back into then leads us to $$\begin{aligned}
V_{c1}^{D*} \Pi_D^c h & = \sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^n E_0^* \left( (I - V_1 T_1^*) T^{*n+1} h
\oplus (I - V_2 T_2^*) T_1^* T^{*n} h \right) \\
& = \sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^n E_0^* \left( (I - V_1 T_1^*) T_2^* T^{*n} T_1^* h
\oplus (I - V_2 T_2^*) T^{*n} T_1^* h \right) \\
& = \sum_{n=0}^\infty S_{{{\mathcal F}}_{c*}}^n E_0^* \Lambda_{c*} T_1^* h = \Pi_D^c T_1^* h\end{aligned}$$ and now follows.
We can convert the preceding analysis to a functional-model form as follows. Define the $Z$-transform $Z_+ \colon \ell^2_+({{\mathcal F}}_{c*}) \to H^2({{\mathcal F}}_{c*})$ by $$Z_+ \colon \{ f_n \}_{n \ge 0} \mapsto \sum_{n=0}^\infty z^n f_n .$$ Then $Z_+$ is unitary from $\ell^2_+({{\mathcal F}}_{c*})$ onto $H^2({{\mathcal F}}_{c*})$ and intertwines the respective shift operators: $$M_z Z_+ = Z_+ S_{{{\mathcal F}}_{c*}}.$$ Let $\omega_D$ and $\omega_{NF}$ be the unitaries as defined in (\[omega-ext\]) and (\[omegaNF\]), respectively. We observed in §\[S:SNFmodel\] that the unitary $$u:=\omega_{NF}\omega_D^*:{{\mathcal R}}_D\to\overline{\Delta_TL^2({{\mathcal D}}_T)}$$intertwines $W_D$ and $M_{\zeta}|_{\overline{\Delta_TL^2({{\mathcal D}}_T)}}$. Consequently, the operator $$\begin{aligned}
{{\mathcal Z}}:=Z_+\oplus u:\ell^2_+({{\mathcal F}}_{c*})\oplus{{\mathcal R}}_D\to H^2({{\mathcal F}}_{c*})\oplus\overline{\Delta_TL^2({{\mathcal D}}_T)}\end{aligned}$$has the following intertwining property $$\begin{aligned}
{{\mathcal Z}}V_c^D=(M_z\oplus M_{\zeta}|_{\overline{\Delta_TL^2({{\mathcal D}}_T)}}){{\mathcal Z}}.\end{aligned}$$ We adopt the following notations $$\begin{aligned}
\label{WandVNFs}
\begin{cases}
(W_{\sharp1},W_{\sharp 2},M_{\zeta}|_{\overline{\Delta_TL^2({{\mathcal D}}_T)}}):=(uW_{\partial1}u^*,uW_{\partial2}u^*,uW_Du^*),\\
(V_{\flat1},V_{\flat2},V_{\flat}):=({{\mathcal Z}}V_{c1}^D{{\mathcal Z}}^*,{{\mathcal Z}}V_{c2}^D{{\mathcal Z}}^*,{{\mathcal Z}}V_c^D{{\mathcal Z}}^*)
\end{cases}\end{aligned}$$ and see that for every $f=\sum_{n=0}^{\infty}z^na_n$ in $H^2({{\mathcal F}}_{c*})$ and $g$ in $\overline{\Delta_TL^2({{\mathcal D}}_T)}$, $$\begin{aligned}
{{\mathcal Z}}V_{c1}^D{{\mathcal Z}}^*(f\oplus g)&={{\mathcal Z}}V_{c1}^D{{\mathcal Z}}^*\left(\sum_{n=0}^{\infty}z^na_n\oplus g\right)\\
&={{\mathcal Z}}(I_{\ell^2_+} \otimes U_{c*}^* P_{c*}^\perp + S \otimes U_{c*}^* P_{c*})\{a_n\}_{n\geq 0}\oplus W_{\sharp1}g\\
&={{\mathcal Z}}\{U_{c*}^* P_{c*}^\perp a_n+U_{c*}^* P_{c*}a_{n-1}\}_{n\geq0}\oplus W_{\sharp1}g\quad[a_{-1}:=0]\\
&=M_{U_{c*}^* P_{c*}^\perp + z U_{c*}^* P_{c*}} \oplus W_{\sharp1}(f\oplus g).\end{aligned}$$ Therefore (after a similar computation with ${{\mathcal Z}}V_{c2}^D{{\mathcal Z}}^*$) we obtain $$\begin{aligned}
\label{NFAndo}
\nonumber&(V_{\flat1},V_{\flat2},V_{\flat})=(V_{\flat1},V_{\flat2},V_{\flat1}V_{\flat2})\\
=&(M_{U_{c*}^* P_{c*}^\perp + z U_{c*}^* P_{c*}} \oplus W_{\sharp1},M_{P_{c*} U_{c*} + z P^\perp_{c*} U_{c*}} \oplus W_{\sharp2},M_z\oplus M_{\zeta}|_{\overline{\Delta_TL^2({{\mathcal D}}_T)}}).\end{aligned}$$ Finally define an isometry $\Pi_{\flat}:{{\mathcal H}}\to H^2({{\mathcal F}}_{c*})\oplus\overline{\Delta_TL^2({{\mathcal D}}_T)}$ as $$\begin{aligned}
\label{NF-iso}
\Pi_{\flat} h={{\mathcal Z}}\Pi^c_Dh=\sum_{n=0}^{\infty}z^n\Lambda_{c*} (I-VT^*)T^{* n}h\oplus u(Qh).\end{aligned}$$Therefore by Theorem \[Thm:CFDAndo\] we have proved the following theorem, which gives a Sz.-Nagy–Foias model for an Andô dilation for a pair of commuting contractions.
\[Thm:NFAndo\] Let $(T_1,T_2)$ be a pair of commuting contractions and let $({{\mathcal F}}_{c*},\Lambda_{c*},P_{c*},U_{c*})$ be a coordinate-free Andô tuple for $(T_1^*, T_2^*)$. Then with the notations as in (\[NFAndo\]) and (\[NF-iso\]) the triple $(\Pi_{\flat},V_{\flat1},V_{\flat2})$ is an isometric lift of $(T_1,T_2)$.
Characteristic/Admissible triples and functional model for a commuting pair of contractions {#S:NFmodel}
===========================================================================================
Characteristic triples and functional models
--------------------------------------------
Let $(T_1,T_2)$ be a pair of commuting contractions and $({{\mathcal F}}_{c*},\Lambda_{c*},P_{c*},U_{c*})$ be a coordinate-free Andô tuple for $(T_1^*,T_2^*)$. Define a pair $(G_{\sharp1},G_{\sharp2})$ of contractions on ${{\mathcal D}}_{T^*}$ as $$\begin{aligned}
\label{Gs}
(G_{\sharp1},G_{\sharp2}):=(\iota_*\Lambda_{c*}^*P_{c*}^\perp U_{c*}\Lambda_{c*}\iota_*^*,\iota_*\Lambda_{c*}^*U_{c*}^*P_{c*}\Lambda_{c*}\iota_*^*).\end{aligned}$$Let the pair $(W_{\sharp1},W_{\sharp2})$ of commuting unitaries be as defined in (\[WandVNFs\]). The objective of this section is to study the triple $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$, where $\Theta_T$ is the characteristic triple for the contraction $T=T_1T_2$. We start by giving the triple a name.
The triple $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ is called the characteristic triple for $(T_1,T_2)$.
Recall that a Andô tuple for $(T_1^*, T_2^*)$ depends on an arbitrary choice of unitary extension of a given partially defined isometric map, and hence is highly nonunique except in the case where $T = T_1 \cdot T_2$ is a regular factorization. Thus, as the definition of characteristic triple for a commutative contractive pair $(T_1, T_2)$ depends on a choice of Andô tuple for $(T_1^*, T_2^*)$, it would appear that $(T_1, T_2)$ does not uniquely determine its characteristic triple, specifically the component ${\mathbb G} =
(G_{\sharp 1}, G_{\sharp 2})$. This presumption however remarkably turns out to be wrong, as noted in the following result.
\[P:fundamental\] Let $(T_1, T_2)$ be a commutative contractive operator pair. Then the operators $G_{\sharp 1}, G_{\sharp 2}$ in a characteristic triple $((G_{\sharp 1}, G_{\sharp 2}), (W_{\sharp 1}, W_{\sharp 2}, \Theta_T)$ are uniquely determined from $(T_1, T_2)$ as solutions of the equations $$\label{fundamental}
T_i^* - T_j T^* = D_{T^*} G_{\sharp i } D_{T^*} \text{ where } (i,j) = (1,2) \text{ or } (2,1).$$
\[R:uniqueness\] [As the operators $G_{\sharp 1}$ and $G_{\sharp 2}$ are by definition operators on the space ${{\mathcal D}}_{T^*} = \overline{\operatorname{Ran}} \, D_{T^*}$, any solutions of the equations , assuming that such exist, must be unique. Operators $G_{\sharp 1}$ and $G_{\sharp 2}$ satisfying equations of this type appear in the theory of $\Gamma$-contractions (commutative operator pairs having the symmetrized bidisk as a complete spectral set) and of tetrablock contractions (commutative triples of operators having the tetrablock domain as a complete spectral set)—we refer to [@B-P-SR], [@sir's; @tetrablock; @paper] for further details. ]{}
We sketch the proof of only for the case $(i,j) = (1,2)$ as the case $(i,j) = (2,1)$ is similar.
Let $(V_1,V_2)$ be an arbitrary Andô dilation for $(T_1,T_2)$ and set $V=V_1V_2$. Then for $h, h' \in {{\mathcal H}}$ we compute $$\begin{aligned}
\langle D_{T^*}G_{\sharp1}D_{T^*}h,h'\rangle_{{\mathcal H}}&=
\langle P_{c*}^\perp U_{c*}\Lambda_{c*}\iota_*^*D_{T^*}h, \Lambda_{c*}\iota_*^*D_{T^*}h'\rangle_{{{\mathcal F}}_{c*}}
\text{ (by \eqref{Gs})}\\
&=\langle P_{c*}^\perp U_{c*}\Lambda_{c*}(I_{{{\mathcal H}}}-VT^*)h,\Lambda_{c*}(I_{{{\mathcal H}}}-VT^*)h'\rangle_{{{\mathcal F}}_{c*}}
\text{ (by Proposition \ref{P:dil})}\\
&=\langle (I_{{{\mathcal H}}}-V_2T_2^*)T_1^*h,(I_{{{\mathcal H}}}-V_2T_2^*)h' \rangle_{{{\mathcal L}}_{2*}} \\
& = \langle D_{T_2} T_1^* h, D_{T_2} h' \rangle_{{{\mathcal F}}_{c*}}
\text{ (by Proposition \ref{P:dil} with $T_2$ in place of $T$)} \\
& = \langle (I - T_2^* T_2) T_1^* h, h' \rangle_{{\mathcal H}}= \langle (T_1^* - T_2 T^*) h, h' \rangle_{{\mathcal H}}\end{aligned}$$ and the result follows.
The characteristic triple $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ for a commuting pair of contractions $(T_1, T_2)$ is the invariant leading to the construction of a Sz.-Nagy–Foias-type functional model for $(T_1, T_2)$ as follows.
\[Thm:SNFmodelPair\] Let $(T_1,T_2)$ be a pair of commuting contractions and let its characteristic triple be $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$. Then the Sz.-Nagy–Foias model space $$\label{HNF'}
{{\mathcal H}}_{NF}=\left(H^2({{\mathcal D}}_{T^*})\oplus\overline{\Delta_T(L^2({{\mathcal D}}_T))})\right)\ominus\{\Theta_Tf\oplus\Delta_Tf:f\in H^2({{\mathcal D}}_T)\}$$ is coinvariant under $$(M_{G_{\sharp1}^*+zG_{\sharp2}}\oplus W_{\sharp1},M_{G_{\sharp2}^*+zG_{\sharp1}}\oplus W_{\sharp2},M_z\oplus M_{\zeta}|_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}})$$and $(T_1,T_2,T_1T_2)$ is unitarily equivalent to $$\begin{aligned}
\label{NFmodelPair}
P_{{{\mathcal H}}_{NF}}(M_{G_{\sharp1}^*+zG_{\sharp2}}\oplus W_{\sharp1},M_{G_{\sharp2}^*+zG_{\sharp1}}\oplus W_{\sharp2},M_z\oplus M_{\zeta}|_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}})|_{{{\mathcal H}}_{NF}}.\end{aligned}$$
Let us denote $$\begin{aligned}
\label{Vs}
\nonumber\underline{V}_{NF}:=&(S^{NF}_1,S^{NF}_2,V_{NF})\\=&(M_{G_{\sharp1}^*+zG_{\sharp2}}\oplus W_{\sharp1},M_{G_{\sharp2}^*+zG_{\sharp1}}\oplus W_{\sharp2},M_z\oplus M_{\zeta}|_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}}).\end{aligned}$$ Note that with the triple $(\Pi_{\flat},V_{\flat1},V_{\flat2})$ as in Theorem \[Thm:NFAndo\] and with the isometry $L$ given by $$L:=(I_{H^2}\otimes\Lambda_{c*}\iota_*^*\oplus I_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}}):H^2({{\mathcal D}}_{T^*})\oplus\overline{\Delta_T(L^2({{\mathcal D}}_T))}\to H^2({{\mathcal F}}_{c*})\oplus\overline{\Delta_T(L^2({{\mathcal D}}_T))},$$ it is a direct check using the definitions that $$\label{check}
L^*(V_{\flat1},V_{\flat2},V_{\flat})L=(S^{NF}_1,S^{NF}_2,V_{NF})=\underline{V}_{NF}
\text{ and } L\Pi_{NF}=\Pi_{\flat}.$$ By Theorem \[Thm:NFAndo\] we also have $$\label{intertwine1}
(V_{\flat 1}, V_{\flat 2}, V_\flat)^* \Pi_\flat = \Pi_\flat (T_1^*, T_2^*, T^*).$$ Using the second equation in then leads to $$(V_{\flat 1}, V_{\flat 2}, V_\flat)^* L \Pi_{NF} = L \Pi_{NF} (T_1^*, T_2^*, T^*).$$ Multiplying by $L^*$ on the left and using that $L$ is an isometry then gives us $$L^* (V_{\flat 1}, V_{\flat 2}, V_\flat )^* L \Pi_{NF} = \Pi_{NF} (T_1^*, T_2^*, T^*).$$ Using the first equation in then gives us $$\label{Eg1}
(S^{NF}_1,S^{NF}_2,V_{NF})^*\Pi_{NF}=\Pi_{NF}(T_1,T_2,T_1T_2)^*.$$ We have seen in §\[S:SNFmodel\] that $\operatorname{Ran}\Pi_{NF}$ is ${{\mathcal H}}_{NF}$ as in . Hence from the intertwining (\[Eg1\]) we see that the unitary transformation $\Pi_{NF} \colon {{\mathcal H}}\to {{\mathcal H}}_{NF}$ establishes the unitary equivalence between the operator tuple on ${{\mathcal H}}_{NF}$ and $(T_1, T_2, T)$ on ${{\mathcal H}}$, and the theorem follows.
Characteristic triple as a complete unitary invariant
-----------------------------------------------------
It was proved by Sz.-Nagy–Foias (see [@Nagy-Foias Chapter VI] that the characteristic function $\Theta_T$ for a c.n.u. contraction $T$ is a complete unitary invariant. This means that two c.n.u. contractions $T$ and $T'$ are unitarily equivalent if and only if their characteristic functions [*[coincide]{}*]{} in the sense that there exist unitary operators $u: \mathcal{D}_T \to \mathcal{D}_{T'}$ and $u_{*}: \mathcal{D}_{T^*} \to \mathcal{D}_{{T'}^*}$ such that the following diagram commutes for every $z\in\mathbb D$: $$\begin{aligned}
\label{coindiagram}
\begin{CD}
\mathcal{D}_T @>\Theta_T(z)>> \mathcal{D}_{T^*}\\
@Vu VV @VVu_{*} V\\
\mathcal{D}_{T'} @>>\Theta_{T'}(z)> \mathcal{D}_{{T'}^*}
\end{CD}.\end{aligned}$$ Theorem \[UnitaryInv\] below shows that such a result holds for characteristic triples of pairs of commuting contractions also. First we define a notion of coincidence for such a triple.
A contractive analytic function $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$ is a ${{\mathcal B}}({{\mathcal D}},{{\mathcal D}}_*)$-valued analytic function on $\mathbb{D}$ such that $$\|\Theta(z)\|\leq 1 \text{ for all } z\in\mathbb{D}.$$ Such a function is called [*purely contractive*]{} if $\Theta(0)$ does not preserve the norm of any nonzero vector, i.e., $$\label{pureCAF}
\|\Theta(0)\xi\|_{{{\mathcal D}}_*}<\|\xi\|_{{{\mathcal D}}} \text{ for all nonzero }\xi\in{{\mathcal D}}.$$ We note that a Sz.-Nagy–Foias characteristic function $\Theta_T$ is always purely contractive (see [@Nagy-Foias Section VI.1]), and that it is always the case that a general contractive analytic function $({{\mathcal D}}, {{\mathcal D}}_*, \Theta)$ has a block diagonal decompostiton $\Theta = \Theta' \oplus \Theta^0$ where $({{\mathcal D}}', {{\mathcal D}}_*', \Theta')$ is a unitary constant function and $({{\mathcal D}}^0, {{\mathcal D}}_*^0, \Theta^0)$ is purely contractive. A key easily checked property of this decomposition is the following:
\[O:reduction\] The model space $${{\mathcal H}}_{NF} = \begin{bmatrix} H^2({{\mathcal D}}_*) \\ \overline{ \Delta_\Theta L^2({{\mathcal D}})} \end{bmatrix}
\ominus \begin{bmatrix} \Theta \\ \Delta_\Theta \end{bmatrix} H^2({{\mathcal D}})$$ and the associated model operator $$T_{NF} = P_{{{\mathcal H}}_{NF}} \begin{bmatrix} M_z & 0 \\ 0 & M_\zeta \end{bmatrix} \big|_{{{\mathcal H}}_{NF}}$$ remain exactly the same (after some natural identification of respective coefficient spaces) when $\Theta$ is replaced by $\Theta^0$.
Thus only completely contractive analytic functions are relevant when discussing Sz.-Nagy–Foias functional models. For the moment we consider only purely contractive analytic functions.
\[coincidence\] Let $(\mathcal{D},\mathcal{D}_*,\Theta)$, $(\mathcal{D'},\mathcal{D'_*},\Theta')$ be two purely contractive analytic functions, $\mathbb{G}=(G_1,G_2)$ on $\mathcal{D}_*$, $\mathbb{G}'=(G_1',G_2')$ on $\mathcal{D'_*}$ be two pairs of contractions and $\mathbb{W}=(W_1,W_2)$ on $\overline{\Delta_\Theta L^2(\mathcal{D})}$, $\mathbb{W}'=(W_1',W_2')$ on $\overline{\Delta_{\Theta'} L^2(\mathcal{D'})}$ be two pairs of commuting unitaries such that their product is $M_{\zeta}$ on the respective spaces. We say that the two triples $(\mathbb{G},\mathbb{W},\Theta)$ and $(\mathbb{G}',\mathbb{W}',\Theta')$ coincide, if
- $(\mathcal{D},\mathcal{D_*},\Theta)$ and $(\mathcal{D'},\mathcal{D'_*},\Theta')$ coincide, i.e., there exist unitary operators $u: \mathcal{D} \to \mathcal{D'}$ and $u_{*}: \mathcal{D}_{*} \to \mathcal{D'}_{*}$ such that the diagram (\[coindiagram\]) commutes with $\Theta_T$ and $\Theta_{T^*}$ in place of $\Theta$ and $\Theta'$, respectively;
- the unitary operators $u$, $u_*$ have the following intertwining properties: $$\begin{aligned}
\begin{cases}
\mathbb{G}'=(G_1',G_2')=u_*\mathbb{G}u_*^*=(u_*G_1u_*^*,u_*G_2u_*^*) \text{ and }\\
\mathbb{W}'=(W_1',W_2')=\omega_u\mathbb{W}\omega_u^*=(\omega_uW_1\omega_u^*,\omega_uW_2\omega_u^*),
\end{cases}\end{aligned}$$ where $\omega_u:\overline{\Delta_{\Theta} L^2(\mathcal{D})}\to\overline{\Delta_{\Theta'} L^2(\mathcal{D'})}$ is the following unitary map induced by $u$: $$\begin{aligned}
\label{omega-u}
\omega_u:=(I_{L^2}\otimes u)|_{\overline{\Delta_{\Theta} L^2(\mathcal{D})}}.\end{aligned}$$
We shall use the following uniqueness result from [@sauAndo] later in this section. For a pair $(T_1,T_2)$ of commuting contractions on $\mathcal{H}$ and $\underline{T}=(T_1,T_2,T_1T_2)$, let $$\begin{aligned}
\mathcal{U}_{\underline{T}}&:=&
\{(\Pi,\mathcal{K}, \underline{V}): \underline{V}=(S_1,S_2,V),\; \Pi:\mathcal{H}\to\mathcal{K} \text{ is an isometry such that }\\&& \underline{V}^*\Pi=\Pi\underline{T}^*, \;(\Pi, V)\text{ is the minimal isometric dilation of } T=T_1T_2,\\&&
(S_1,V),\; (S_2,V) \text{ are commuting and } S_1=S_2^*V.\}\end{aligned}$$ We next exhibit a concrete example of a member of $\mathcal{U}_{\underline{T}}$ for a given pair $(T_1,T_2)$ of commuting contractions.
\[TheEg\] [Let $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ be the characteristic triple for $(T_1,T_2)$. Let $\Pi_{NF}:{{\mathcal H}}\to H^2({{\mathcal D}}_{T^*})\oplus\overline{\Delta_T(L^2({{\mathcal D}}_T))}$ be the isometry as in (\[Unf&Pinf\]), i.e., $$\begin{aligned}
\Pi_{NF}h=\sum_{n=0}^{\infty}z^nD_{T^*}T^{*n}h\oplus u(Qh)=U_{\text{min}}\Pi_Dh.\end{aligned}$$ Consider the triple $(\Pi_{NF},{{\mathcal K}}_{NF},\underline{V}_{NF})$, where $\underline{V}_{NF}=(S^{NF}_1,S^{NF}_2,V_{NF})$ is as in (\[Vs\]). It can be checked easily from the definition that $(S_1^{NF},V_{NF})$ and $(S_2^{NF},V_{NF})$ are commuting and that $S_1^{NF}={S_2^{NF}}^*V_{NF}$. We have also seen in §\[S:SNFmodel\] that $(\Pi_{NF}, V_{NF})$ is the Sz.-Nagy–Foias minimal isometric dilation of $T=T_1T_2$. Therefore with ${{\mathcal K}}_{NF}=H^2({{\mathcal D}}_{T^*})\oplus\overline{\Delta_T(L^2({{\mathcal D}}_T))}$, we conclude that the triple $(\mathcal{K}_{NF}, \Pi_{NF}, \underline{V}_{NF})$ is in $\mathcal{U}_{\underline{T}}$ for $(T_1,T_2)$. ]{}
The theorem below, proved in [@sauAndo], asserts that any triple $({{\mathcal K}}, \Pi, V)$ in ${{\mathcal U}}_T$ is unitarily equivalent as an element of ${{\mathcal K}}_T$ to the model triple $(\mathcal{K}_{NF}, \Pi_{NF}, \underline{V}_{NF})$.
\[uniqueness\] For a pair $\underline{T}:=(T_1,T_2)$ of commuting contractions, the family $\mathcal{U}_{\underline{T}}$ is a singleton set under unitary equivalence, i.e., if $(\mathcal{K}, \Pi, \underline{V})$ and $(\mathcal{K}' , \Pi', \underline{V'})$ are in $\mathcal{U}_T$, then there exists a unitary $U:\mathcal{K}\to\mathcal{K'}$ such that $$\begin{aligned}
\label{IntertwiningU}
U\underline{V}=\underline{V'}U \text{ and }U(\Pi h)=\Pi'h, \text{ for all }h\in \mathcal{H}.
\end{aligned}$$
As we have seen in Theorem \[T:min-iso-lift\], if $(\Pi,V)$ and $(\Pi',V')$ are any two minimal isometric dilations of a contraction $T$, then there is a [*[unique]{}*]{} unitary $U$ that intertwines $V$ and $V'$ and $U\Pi=\Pi'$. This means that if $V=V'$, then $U$ has to be the identity operator. Let now $(\Pi,\mathcal{K}, \underline{V})$ and $(\Pi',\mathcal{K'}, \underline{V'})$ be two members in $\mathcal{U}_{\underline{T}}$ for a pair $(T_1,T_2)$ of commuting contractions such that the last entries of $\underline{V}$ and $\underline{V'}$ are the same. Then by (\[IntertwiningU\]) we see that this forces $\underline{V}=\underline{V'}$, and hence the following corollary is easily obtained.
\[Uniqueness-Cor\] For a pair of commuting contractions $(T_1,T_2)$, let the triples $(\mathcal{K}, \Pi, \underline{V})$ and $(\mathcal{K'}, \Pi', \underline{V'})$ be in $\mathcal{U}_{\underline{T}}$ such that the last entries of $\underline{V}$ and $\underline{V'}$ are the same. Then $\underline{V}=\underline{V'}.$
\[UnitaryInv\] Let $(T_1,T_2)$ and $(T_1',T_2')$ be two pairs of commuting contractions such that their products $T=T_1T_2$ and $T'=T_1'T_2'$ are c.n.u. contractions. Then $(T_1,T_2)$ and $(T_1',T_2')$ are unitarily equivalent if and only if their characteristic triples coincide.
Let $(T_1,T_2)$ on $\mathcal{H}$ and $(T_1',T_2')$ on $\mathcal{H}'$ be unitarily equivalent via the unitary operator $U:\mathcal{H}\to\mathcal{H}'$ and $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ and $((G'_{\sharp1},G'_{\sharp2}),(W_{\sharp1}',W_{\sharp2}'),\Theta_{T'})$ be their characteristic triples, respectively. It is easy to see that $UD_T=D_{T'}U$ and $UD_{T^*}=D_{T'^*}U$ and that the unitaries $$\begin{aligned}
u:=U|_{\mathcal{D}_T}: \mathcal{D}_{T} \to \mathcal{D}_{T'}\text{ and }u_{*}:=U|_{\mathcal{D}_{T^*}}: \mathcal{D}_{T^*} \to \mathcal{D}_{T'^{*}}\end{aligned}$$ have the following property: $$\begin{aligned}
\label{coincd}
u_*\Theta_T=\Theta_{T'}u.\end{aligned}$$ Hence $\Theta_T$ and $\Theta_{T'}$ coincide. We now show that the unitary $u_*$ above plays the role in unitary equivalence of $(G_{\sharp1},G_{\sharp2})$ and $(G_{\sharp1}',G_{\sharp2}')$.
To this end we note that by Proposition \[P:fundamental\] the following set of equations is satisfied: $$T_i^* - T_j T^* = D_{T^*} G_{\sharp i} D_{T^*}, \quad
T_i^{\prime*} - T'_j T^{\prime *} = D_{T^{\prime *}} G'_{\sharp i} D_{T^{\prime *}} \text{ for }
(i,j) = (1,2) \text{ or } (2,1).$$ It then follows that $$\begin{aligned}
\label{fundequiv}
u_*(G_{\sharp1},G_{\sharp2})=(G_{\sharp1}',G_{\sharp2}')u_*.\end{aligned}$$ We have noticed in Example \[TheEg\] that for a pair $(T_1,T_2)$ of commuting contractions, the triple $(\Pi_{NF},\mathcal{K}_{NF}, \underline{V}_{NF})$ is always in $\mathcal{U}_{\underline{T}}$. Let $(\Pi'_{NF},\mathcal{K}'_{NF}, \underline{V'}_{NF})$ be the corresponding triple for $(T_1',T_2')$. Let us denote by $\Pi''$ the following isometry $$\begin{aligned}
\label{Pi''}
\Pi'':=((I_{H^2}\otimes u_*^*)\oplus \omega_u^*)\Pi_{NF}'U:\mathcal{H}\to H^2(\mathcal{D}_{T^*})\oplus\overline{\Delta_{T}L^2(\mathcal{D}_{T})}\end{aligned}$$where $\omega_u:\overline{\Delta_{T}L^2(\mathcal{D}_T)}\to\overline{\Delta_{T'}L^2(\mathcal{D}_{T'})}$ is the unitary $\omega_u=(I_{L^2}\otimes u)|_{\overline{\Delta_{T}L^2(\mathcal{D}_T)}}$. We observe that the triple $(\Pi'',\mathcal{K}_{NF},\underline{V''})$ is also in $\mathcal{U}_{\underline{T}}$, where with $(W_1'',W_2'')=\omega_u^*(W_{\sharp1}',W_{\sharp2}')\omega_u$, $$\underline{V''}:=(M_{G_{\sharp1}^*+zG_{\sharp2}}\oplus W_1'',M_{G_{\sharp2}^*+zG_{\sharp1}}\oplus W_2'',M_z\oplus M_{\zeta}|_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}}),$$ because using (\[fundequiv\]) we have for $(i,j)=(1,2)$ or $(2,1)$ $$\begin{aligned}
\Pi''T_i^*&=&((I_{H^2}\otimes u_*^*)\oplus \omega_u^*)\Pi_{NF}'{T'_i}^*U \text{ (by \eqref{Pi''})} \\
&=&((I_{H^2}\otimes u_*^*)\oplus \omega_u^*)(M_{{G'_{\sharp i}}^*+z{G'_{\sharp j}}}^*\oplus {W'_{\sharp i}}^*)\Pi_{NF}'U\\
&=&(M_{G_{\sharp i}^*+zG_{\sharp j}}\oplus {W''_i}^*)((I_{H^2}\otimes u_*^*)\oplus \omega_u^*)\Pi_{NF}'U
\text{ (by \eqref{fundequiv}).}\end{aligned}$$ Now since the last entry of $\underline{V''}$ is the same as that of $\underline{V}_{NF}$, applying Corollary \[Uniqueness-Cor\], we get $$(W_1'',W_2'')=\omega_u^*(W_{\sharp1}',W_{\sharp2}')\omega_u=(W_{\sharp1},W_{\sharp2}),$$ which together with equations (\[coincd\]) and (\[fundequiv\]) establish the first part of the theorem.
Conversely, let $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ and $((G'_{\sharp1},G'_{\sharp2}),(W_{\sharp1}',W_{\sharp2}'),\Theta_{T'})$ be the characteristic triples of $(T_1,T_2)$ and $(T_1',T_2')$, respectively and suppose the respective characteristic triples coincide. Thus there exist unitaries $u:\mathcal{D}_T\to\mathcal{D}_{T'}$ and $u_*:\mathcal{D}_{T^*}\to\mathcal{D}_{T'^*}$ such that part $(i)$ and part $(ii)$ in Definition \[coincidence\] hold. Let $\omega_u$ be the unitary induced by $u$ as defined in $(\ref{omega-u})$. Then it is easy to see that the unitary $$\begin{aligned}
\label{unitary-coin}
(I_{H^2}\otimes u_*)\oplus \omega_u:H^2(\mathcal{D}_{T^*})\oplus \overline{\Delta_TL^2(\mathcal{D}_T)}\to H^2(\mathcal{D}_{T'^*})\oplus \overline{\Delta_{T'}L^2(\mathcal{D}_{T'})}\end{aligned}$$intertwines $$\begin{aligned}
&\underline{V}=(M_{G_{\sharp1}^*+zG_{\sharp2}}\oplus W_{\sharp1},M_{G_{\sharp2}^*+zG_{\sharp1}}\oplus W_{\sharp 2},M_z\oplus M_{\zeta}|_{\overline{\Delta_T(L^2({{\mathcal D}}_T))}}) \text{ and }\\
&\underline{V'}=(M_{G_{\sharp1}'^*+zG'_{\sharp2}}\oplus W_{\sharp1}',M_{G_{\sharp2}'^*+zG'_{\sharp1}}\oplus W_{\sharp2}',M_z\oplus M_{\zeta}|_{\overline{\Delta_{T'}(L^2({{\mathcal D}}_{T'}))}}).\end{aligned}$$Also, the unitary in $(\ref{unitary-coin})$ clearly takes the space $\{\Theta_Tf\oplus \Delta_T\tilde{f}: f\in H^2(\mathcal{D}_T)\}$ onto $\{\Theta_{T'}f\oplus \Delta_{T'}\tilde{f}: f\in H^2(\mathcal{D}_{T'})\}$ and hence $$\big(H^2(\mathcal{D}_{T^*})\oplus \overline{\Delta_T L^2(\mathcal{D}_T)}\big)\ominus \{\Theta_Tf\oplus \Delta_T\tilde{f}: f\in H^2(\mathcal{D}_T)\}$$onto $$\big(H^2(\mathcal{D}_{T'^*})\oplus \overline{\Delta_{T'} L^2(\mathcal{D}_{T'})}\big)\ominus \{\Theta_{T'}f\oplus \Delta_{T'}\tilde{f}: f\in H^2(\mathcal{D}_{T'})\}.$$This implies that the functional models for $(T_1,T_2)$ and $(T_1',T_2')$ as in (\[NFmodelPair\]) are unitarily equivalent and hence by (\[NFmodelPair\]) the pairs $(T_1,T_2)$ and $(T_1',T_2')$ are unitarily equivalent also.
Admissible triples
------------------
In this subsection, we consider general contractive analytic functions $({{\mathcal D}}, {{\mathcal D}}_*, \Theta)$ and do not insist that $\Theta$ be purely contractive. We start with a contractive analytic function $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$, a pair of commuting unitaries $(W_1,W_2)$ and a pair of contractions $(G_1,G_2)$ and investigate when the triple $((G_1,G_2),(W_1,W_2),\Theta)$ gives rise to a pair $(T_1,T_2)$ of commuting contractions such that $T=T_1T_2$ is completely non-unitary.
We note from equation (\[Eg1\]) that the characteristic triple $((G_{\sharp1},G_{\sharp2}),(W_{\sharp1},W_{\sharp2}),\Theta_T)$ for a pair $(T_1,T_2)$ of commuting contractions satisfies the following set of [*admissibility conditions*]{}, where $(i,j)=(1,2),(2,1)$ and we write $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$ in place of $({{\mathcal D}}_{T},{{\mathcal D}}_{T^*},\Theta_T)$ and $\Delta_\Theta$ in place of $\Delta_T$:
\[D:admis-cond\]
**Admissibility conditions:**
1. each of $M_{G_{\sharp i}^* + z G_{\sharp j}} \oplus W_{\sharp i}$ is a contraction,
2. $W_{\sharp1}W_{\sharp2}=W_{\sharp2}W_{\sharp1}=M_{\zeta}|_{\overline{\Delta_\Theta L^2({{\mathcal D}})}}$,
3. ${{\mathcal Q}}_{NF}=(\operatorname{Ran}\Pi_{NF})^\perp=\{\Theta f\oplus\Delta_\Theta f:f\in H^2({{\mathcal D}})\}$ is jointly invariant under $(M_{G_{\sharp1}^* + z G_{\sharp2}} \oplus W_{\sharp1},M_{G_{\sharp2}^* + z G_{\sharp1}} \oplus W_{\sharp2},M_z\oplus M_{\zeta}|_{\overline{\Delta_\Theta L^2({{\mathcal D}})}})$ and
4. with ${{\mathcal H}}_{NF}=H^2({{\mathcal D}}_{*})\oplus\overline{\Delta_\Theta L^2({{\mathcal D}})}\ominus{{\mathcal Q}}_{NF}$ we have $$(M_{G_{\sharp i}^* + z G_{\sharp j}}^*\oplus {W_{\sharp i}}^*)(M_{G_{\sharp j}^* + z G_{\sharp i}}^*\oplus {W_{\sharp j}}^*)|_{{{\mathcal H}}_{NF}}=(M_z^*\oplus M_{\zeta}^*|_{\overline{\Delta_\Theta L^2({{\mathcal D}})}})|_{{{\mathcal H}}_{NF}}.$$
This motivates us to define the following.
Let $(\mathcal{D},\mathcal{D}_*,\Theta)$ be a contractive analytic function and $(G_1,G_2)$ on $\mathcal{D}_*$ be a pair of contractions. Let $(W_1,W_2)$ be a pair of commuting unitaries on $\overline{\Delta_\Theta L^2(\mathcal{D})}$. We say that the triple $((G_1,G_2),(W_1,W_2),\Theta)$ is admissible if it satisfies the admissibility conditions $(1)$–$(4)$ in Definition \[D:admis-cond\]. We then say that the triple $$\begin{aligned}
\label{AdmisFuncModel}
\nonumber&({\bf T}_1,{\bf T}_2,{\bf T}_1{\bf T}_2)\\
&:=P_{\mathcal{H}_\Theta}(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2,M_z\oplus M_{\zeta}|_{\overline{\Delta_\Theta(L^2({{\mathcal D}}))}})|_{\mathcal{H}_\Theta}\end{aligned}$$ is the functional model associated with the admissible triple $((G_1,G_2),(W_1,W_2),\Theta)$.
Let us say that the admissible triple $({\mathbb G}, {\mathbb W}, \Theta)$ is [*pure*]{} if its last component $\Theta$ is a purely contractive analytic function. Then we have the following analogue of Observation \[O:reduction\] for the Sz.-Nagy–Foias model.
\[P:pure-adm-triple\] We suppose that $((G_1, G_2), (W_1, W_2), \Theta)$ is an admissible triple and that $\Theta$ has a (possibly nontrivial) decomposition $\Theta = \Theta' \oplus \Theta^0$ with $({{\mathcal D}}', {{\mathcal D}}'_*, \Theta')$ a unitary constant function and $\Theta^0$ a purely contractive analytic function. Then there is an admissible triple of the form $((G_1^0, G_2^0),
(W_1^0, W_2^0), \Theta^0)$ so that the functional model for $((G_1, G_2), (W_1, W_2), \Theta)$ is unitarily equivalent to the functional model for $((G_1^0, G_2^0), (W_1^0, W_2^0), \Theta^0)$.
We shall refer to $((G_1^0, G_2^0), (W_1^0, W_2^0), \Theta^0)$ as the [*pure part*]{} of $((G_1, G_2), (W_1, W_2), \Theta)$.
We suppose that $((G_1, G_2), (W_1, W_2), \Theta)$ is an admissible triple and that $\Theta$ has a (possibly nontrivial) decomposition $\Theta = \Theta' \oplus \Theta^0$ with $({{\mathcal D}}', {{\mathcal D}}'_*, \Theta')$ a unitary constant function and $({{\mathcal D}}^0, {{\mathcal D}}_*^0, \Theta^0)$ a purely contractive analytic function. Let $${{\mathcal H}}_\Theta = {{\mathcal K}}_\Theta \ominus \begin{bmatrix} \Theta \\ \Delta_\Theta \end{bmatrix} H^2({{\mathcal D}})$$ be the Sz.-Nagy–Foias functional model space associated with $\Theta$ (and hence also the functional model space associated with the admissible triple $((G_1, G_2), (W_1, W_2), \Theta)$), and let $$({\mathbf T}_1^*, {\mathbf T}_2^*, {\mathbf T}^*) =
\left((M_{G_1^* + z G_2} \oplus W_1)^*, (M_{G_2^* + z G_1} \oplus W_2)^*,
(M_z \oplus M_\zeta|_{\overline{\Delta_\Theta L^2({{\mathcal D}})}})^* \right) \big|_{{{\mathcal H}}(\Theta)}$$ be the associated functional-model triple of contraction operators. (with ${\mathbf T} = {\mathbf T}_1 {\mathbf T}_2$). As a result of [@Nagy-Foias Theorem VI.3.1], we know that ${\mathbf T}$ is c.n.u. with characteristic function $\Theta_{\mathbf T}$ coinciding with $\Theta^0$. Thus the characteristic triple for $({\mathbf T}_1^*, {\mathbf T}_2^*, {\mathbf T}^*)$ has the form $$\widetilde \Xi : = ((\widetilde G_1, \widetilde G_2), (\widetilde W_1, \widetilde W_2), \Theta_{\mathbf T})$$ and by Theorem \[Thm:SNFmodelPair\] it follows that $({\mathbf T}_1, {\mathbf T}_2, {\mathbf T})$ is unitarily equivalent to the model operators associated with $\widetilde \Xi$. As already noted, $\Theta_{\mathbf T}$ coincides with $\Theta^0$; hence there are unitary operators $u \colon {{\mathcal D}}_T \to {{\mathcal U}}^0$, $u_* \colon {{\mathcal D}}_{T^*} \to {{\mathcal U}}_*^0$ so that $$\Theta^0(z) u = u_* \Theta_T(z) \text{ for all } z \in {\mathbb D}.$$ Define operators $G_1^0$, $G_2^0$ on ${{\mathcal D}}_*^0$ and $W_1^0$, $W_2^0$ on $\overline{\Delta_{\Theta^0} L^2({{\mathcal D}}^0)}$ by $$G_i^0 = u_* \widetilde G_i u_*^*, \quad W_i^0 = (u \otimes I_{L^2} \widetilde W_i (u^* \otimes I_{L^2})$$ for $i = 1,2$. Then by construction the triple $$\Xi^0 = \left( (G_1^0, G_2^0), (W_1^0, W_2^0), \Theta^0 \right)$$ coincides with $\widetilde \Xi$ and hence is also admissible. Then by Theorem \[UnitaryInv\] the commutative contractive pair $({\mathbf T}_1, {\mathbf T}_2)$ is also unitarily equivalent to the functional-model commutative contractive pair associated with the admissible triple $\Xi^0$. This completes the proof of Proposition \[P:pure-adm-triple\].
For $\Theta$ a purely contractive analytic function, we have the following result.
\[AdmisCharc\] Let $(\mathcal{D},\mathcal{D}_*,\Theta)$ be a purely contractive analytic function, $(G_1,G_2)$ on $\mathcal{D}_*$ be a pair of contractions and $(W_1,W_2)$ on $\overline{\Delta_\Theta L^2(\mathcal{D})}$ be a pair of commuting unitaries such that their product is $M_{\zeta}|_{\overline{\Delta_\Theta L^2({{\mathcal D}})}}$. Then the triple $((G_1,G_2),(W_1,W_2),\Theta)$ is admissible if and only if it is the characteristic triple for some pair of commuting contractions with their product being a c.n.u. contraction. In fact, $((G_1,G_2),(W_1,W_2),\Theta)$ coincides with the characteristic triple of its functional model as defined in (\[AdmisFuncModel\]).
We have already observed that the characteristic triple of a pair $(T_1,T_2)$ of commuting contractions with $T=T_1T_2$ being a c.n.u. contraction is indeed a pure admissible triple (since characteristic functions $\Theta_T$ are necessarily purely contractive).
Conversely suppose that $((G_1,G_2),(W_1,W_2),\Theta)$ is a pure admissible triple. This means that the pair $({\bf T}_1,{\bf T}_2)$ defined on $$\mathcal{H}_\Theta:=\big(H^2(\mathcal{D_*})\oplus \overline{\Delta_\Theta L^2(\mathcal{D})}\big)\ominus \{\Theta f\oplus \Delta_\Theta f:f\in H^2(\mathcal{D})\}$$ by $$({\bf T}_1,{\bf T}_2):=P_{\mathcal{H}_\Theta}(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}
\oplus W_2)|_{\mathcal{H}_\Theta}$$ is a commuting pair of contractions and such that their product is given by $$\begin{aligned}
\label{producT}
{\bf T}:={\bf T}_1{\bf T}_2=
P_{\mathcal{H}_{\Theta}}(M_z\oplus M_{\zeta}|_{\overline{\Delta_\Theta(L^2({{\mathcal D}}))}})|_{\mathcal{H}_\Theta}.\end{aligned}$$ By the Sz.-Nagy–Foias model theory for a single contraction operator $T$ (see [@Nagy-Foias Theorem VI.3.1]), we conclude that ${\bf T}$ is a c.n.u. contraction. We claim that the triple $((G_1,G_2),(W_1,W_2),\Theta)$ coincides with the characteristic triple for $({\bf T}_1,{\bf T}_2)$, which we assume to be $((G'_1,G'_2),(W'_1,W'_2),\Theta_{{\bf T}})$. Since $\Theta$ is a purely contractive analytic function, by (\[producT\]) and Theorem VI.3.1 in [@Nagy-Foias], we conclude that $\Theta$ coincides with $\Theta_{{\bf T}}$. By definition this means that there exist unitaries $u:\mathcal{D}\to\mathcal{D}_{\bf T}$ and $u_*:\mathcal{D}_*\to\mathcal{D}_{{\bf T}^*}$ such that $\Theta_{{\bf T}}u=u_*\Theta$. Clearly the unitary operator $u_*\oplus\omega_u$ takes $H^2(\mathcal{D_*})\oplus\overline{\Delta_\Theta L^2(\mathcal{D})}$ onto $H^2(\mathcal{D}_{{\bf T}^*})\oplus\overline{\Delta_{{\bf T}} L^2(\mathcal{D}_{{\bf T}})}$. Denote by $\tau$ the restriction of $u_*\oplus\omega_u$ to $\mathcal{H}_\Theta$. The following commutative diagram, where $i$ and $i'$ are the inclusion maps and $K_\Theta:= H^2(\mathcal{D_*})\oplus \overline{\Delta_\Theta L^2(\mathcal{D})}$, $$\begin{CD}
\mathcal{H}_\Theta @> i>> \mathcal K_\Theta\\
@V\tau VV @VVu_*\oplus\omega_u V\\
\mathcal{H}_{NF}@>>i'> \mathcal{K}_{NF}
\end{CD}$$ shows that if we denote by $\underline{V}$ and $\underline{V'}$ the respective triples $$\begin{aligned}
&\underline{V}=(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2,M_z\oplus M_{\zeta}|_{\overline{\Delta_\Theta(L^2({{\mathcal D}}))}}), \\
&\underline{V'}=(M_{G_1'^*+zG_2'}\oplus W_1',M_{G_2'^*+zG_1'}\oplus W_2',M_z\oplus M_{\zeta}|_{\overline{\Delta_{{\bf T}}(L^2({{\mathcal D}}_{\bf T}))}}),\end{aligned}$$ then $(i,\mathcal{K}_\Theta, \underline{V})$ and $(i'\circ\tau,\mathcal{K}_{NF}, \underline{V'})$ both are in $\mathcal{U}_{\underline{T}}$. Applying the uniqueness result Theorem \[uniqueness\], we get a unitary from $\mathcal{K}_\Theta$ onto $\mathcal{K}_{NF}$ that intertwines $\underline{V}$ and $\underline{V'}$ and the restriction of which to $\mathcal{H}_\Theta$ is $\tau$. Since the last entries of $\underline{V}$ and $\underline{V'}$ are the minimal isometric dilations of $T=T_1T_2$, such a unitary is in fact unique as a consequence of Theorem \[T:min-iso-lift\]. Since $u_*\oplus\omega_u$ is one such unitary, we get $$(u_*\oplus\omega_u)\underline{V}=\underline{V'}(u_*\oplus\omega_u).$$ Consequently $((G_1,G_2),(W_1,W_2),\Theta)$ coincides with $((G'_1,G'_2),(W'_1,W'_2),\Theta_{{\bf T}})$ and the theorem follows.
The results of this and the previous subsection can be stated more succinctly in the language of Category Theory as follows.
\[P:catlan\] Define the following categories:
1. Let ${\mathfrak C}_1$ be the category of all commuting pairs of contraction operators ${\mathbf T} = (T_1, T_2)$ where we set $T = T_1 \cdot T_2 = T_2 \cdot T_1$ and we assume that $T$ is c.n.u.
2. Let ${\mathfrak C}_2$ be the category of all purely contractive admissible triples $({\mathbb G}, {\mathbb W}, \Theta)$.
Define functors ${\mathfrak f} \colon {\mathfrak C}_1 \to {\mathfrak C}_2$ and ${\mathfrak g} \colon {\mathfrak C}_2 \to {\mathfrak C}_1$ by $$\begin{aligned}
& {\mathfrak f} \colon {\mathbf T} \mapsto ({\mathbb G}, {\mathbb W}, \Theta)_{\mathbf T} =
\text{ characteristic triple for } {\mathbf T}, \\
& {\mathfrak g} \colon ({\mathbb G}, {\mathbb W}, \Theta) \mapsto {\mathbf T}_{{\mathbb G}, {\mathbb W}, \Theta}
= \text{functional-model commutative contractive pair} \\
& \quad \quad \quad \quad \text{ associated with } ({\mathbb G}, {\mathbb W}, \Theta)
\text{ as in } \eqref{AdmisFuncModel}.\end{aligned}$$ Then, for ${\mathbf T}, {\mathbf T}' \in {\mathfrak C}_1$ and $({\mathbb G}, {\mathbb W}, \Theta),
({\mathbb G}', {\mathbb W}', \Theta') \in {\mathfrak C}_2$, we have
1. ${\mathbf T} \underset{u}\cong {\mathbf T'} \Leftrightarrow
{\mathfrak f}({\mathbf T}) \underset{c}\cong {\mathfrak f}({\mathbf T}')$,
2. $({\mathbb G}, {\mathbb W}, \Theta) \underset{c} \cong ({\mathbb G}', {\mathbb W}', \Theta') \Leftrightarrow
{\mathfrak g}({\mathbb G}, {\mathbb W}, \Theta) \underset{u} \cong
{\mathfrak g}({\mathbb G}', {\mathbb W}', \Theta')$,
3. ${\mathfrak g} \circ {\mathfrak f}({\mathbf T}) \underset{u}\cong {\mathbf T}$,
4. ${\mathfrak f} \circ {\mathfrak g}(({\mathbb G}, {\mathbb W}, \Theta)) \underset{c}\cong
({\mathbb G}, {\mathbb W}, \Theta)$
where $\underset{u} \cong$ denotes [*unitary equivalence of operator tuples*]{} and $\underset{c} \cong$ denotes [*coincidence of admissible triples*]{}.
Concrete Examples
-----------------
In this subsection, we shall exhibit several examples of characteristic triples. In particular, in the following result we characterize when a triple of the form $(g_1,g_2,\vartheta)$ can be admissible, where $g_1,g_2$ are scalars and $\vartheta$ is a Blaschke function with simple non-zero zeros.
\[P:concrete\] Let $(g_1,g_2)\in\mathbb{C}$ and $\vartheta$ be a scalar-valued inner function. If
1. If $\vartheta(z)=z^N$ for some positive integer $N$, or
2. If $\vartheta(z)=z^N\prod_{j=1}^M\frac{z-a_j}{1-\overline{a_j}z}$ for some positive integers $N$ and $M$ and distinct nonzero $a_j \in {\mathbb D}$ for $j=1, \dots, M$,
then $(g_1,g_2,\vartheta)$ is admissible if and only if $$\label{criterion}
\text{either } (g_1, g_2) \text{ or } (g_2,g_1) \text{ is in } \overline{\mathbb{D}}\times \{0\}.$$
Let us define $\varphi_1(z)=\overline{g_1}+zg_2$ and $\varphi_2(z)=\overline{g_2}+zg_1$. For a function $\varphi$, let us denote by $\widetilde{\varphi}$ the function $\widetilde{\varphi}(z)=\overline{\varphi(\bar{z})}$. Then note that for non-zero $z$, $$z \widetilde{\varphi_1}(\frac{1}{z})=z(g_1+\overline{g_2}\frac{1}{z})
= z g_1 + \overline{g_2} =\varphi_2(z).$$ By definition, admissibility of $(g_1,g_2,\vartheta)$ means the following three conditions hold true:
1. the multiplication operators $M_{\varphi_1}$ and $M_{\varphi_2}$ are contractions on $H^2(\mathbb{D})$;
2. $\mathcal{M}:=\vartheta H^2(\mathbb{D})$ is joint $(M_{\varphi_1},M_{\varphi_2})$-invariant; and
3. $M_{\varphi_1}^*M_{\varphi_2}^*|_{\mathcal{M}^\perp}=M_{\varphi_2}^*M_{\varphi_1}^*|_{\mathcal{M}^\perp}=M_z^*|_{\mathcal{M}^\perp}.$
It is well known that condition (1) is equivalent to $\|\varphi_i\|_{\infty,\mathbb{D}}$ being at most $1$, for $i=1,2$. We see that condition (2) is automatic since we are in the scalar case. It remains to analyze condition (3). For this analysis we must handle the two cases $\vartheta(z) = z^N$ and $\vartheta(z) = z^N \prod_{j=1}^M
\frac{z-a_j}{1 - \overline{aJ} z}$ separately.
**Case 1: $\theta(z) = z^N$.** For simplicity we consider only the case $N=2$ as the case $N=1$ is trivial and $N\ge 3$ is similar but with a heavier notational burden. Then ${{\mathcal M}}^\perp$ has a orthonormal basis consisting of $\{1, z\}$. With respect to this orthonormal basis, the relevant operators have matrix representations $$\begin{aligned}
& M_{\varphi_1}^*|_{{{\mathcal M}}^\perp} = {\left[\begin{smallmatrix} \overline{\varphi_1(0)} & \overline{ \varphi'_1(0)} \\ 0 & \overline{\varphi_1(0)}
\end{smallmatrix}\right]}
= {\left[\begin{smallmatrix} g_1 & \overline{g_2} \\ 0 & g_1
\end{smallmatrix}\right]}, \quad
M_{\varphi_2}^*|_{{{\mathcal M}}^\perp} = {\left[\begin{smallmatrix} \overline{\varphi_2(0)} & \overline{ \varphi'_2(0)} \\ 0 & \overline{\varphi_2(0)}
\end{smallmatrix}\right]}
= {\left[\begin{smallmatrix} g_2 & \overline{g_1} \\ 0 & g_2
\end{smallmatrix}\right]} \\
& M_z^*|_{{{\mathcal M}}^\perp} = {\left[\begin{smallmatrix} 0 & 1 \\ 0 & 1
\end{smallmatrix}\right]}.\end{aligned}$$ Condition (3) then comes down to the requirement that $${\left[\begin{smallmatrix} g_1 & \overline{g_2} \\ 0 & g_1
\end{smallmatrix}\right]} {\left[\begin{smallmatrix} g_2 & \overline{g_1} \\ 0 & g_2
\end{smallmatrix}\right]} = {\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0
\end{smallmatrix}\right]}$$ which is to say $${\left[\begin{smallmatrix} g_1 g_2 & | g_1 |^2 + | g_2 |^2 \\ 0 & g_1 g_2
\end{smallmatrix}\right]} = {\left[\begin{smallmatrix} 0 & 1 \\ 0 & 0
\end{smallmatrix}\right]}$$ which in turn reduces to the condition that either $(g_1, g_2)$ or $(g_2, g_1)$ is in ${\mathbb T} \times \{0\}$. This completes the verification for Case 1.
**Case 2: $\vartheta(z) = z^N \prod_{j=1}^M \frac{ z - a_j}{1 - \overline{a_j} z}$.** In this case $${{\mathcal M}}^\perp=\operatorname{span}\{1,k_{a_j}:1\leq j \leq M\}.$$ Let us impose the convention that $a_0 = 0$. Then condition (3) is equivalent to the set of interpolation constraints: $$\begin{aligned}
\label{Interpol}
&\varphi_1(a_j)\varphi_2(a_j)=a_j, \text{ for } 0 \leq j \leq N \notag \\
& \Leftrightarrow (\overline{g_1}+a_jg_2)(\overline{g_2}+a_jg_1)=a_j \text{ for } 0\leq j \leq N \notag \\
& \Leftrightarrow (|g_1|^2+|g_2|^2)a_j+\overline{g_1g_2}+g_1g_2a_j^2=a_j\text{ for } 0\leq j \leq N.\end{aligned}$$ Putting $j=0$ in (\[Interpol\]), we get $g_1g_2=0$, which implies that either or both of $g_1$, $g_2$ are $0$. But $g_1=0=g_2$ violates the last equivalent statement in once $j > 0$. Therefore we assume that $g_1=0$ and $g_2$ is any non-zero number of modulus at most one. Then the final equivalent statement in (\[Interpol\]) collapses to $|g_2|^2 a_j = a_j$. As $a_j \ne 0$ for $j > 0$, we conclude that $|g_2|^2 = 1$, i.e., $g_2 \in {\mathbb T}$. This completes the verification of Case 2.
\[R:interpretation\] [We note the following interpretation of the criterion for admissibility of a triple of the form $(g_1, g_2, \vartheta(z))$ with $g_1, g_2$ complex numbers and $\vartheta$ a scalar inner function:]{} Suppose that $T$ is a c.n.u. contraction operator on a Hilbert space ${{\mathcal H}}$ with characteristic function $\vartheta$, and the criterion for complex numbers $g_1$, $g_2$ to be such that $(g_1, g_2, \vartheta)$ is an admissible triple is given by criterion . Suppose that $T_1$, $T_2$ is a commutative pair of contractions on ${{\mathcal H}}$ such that $T = T_1 \cdot T_2$. Then there is a number $\omega$ on the unit circle so that $T_1 = \omega I_{{\mathbb C}^N}$, $T_2 =\overline{\omega} T$, or the reverse.
We next give a perhaps somewhat less trivial example of a characteristic triple for a commutative contractive pair $(T_1, T_2)$ on ${\mathbb C}^2$.
[Let $a,b,x,y$ be any four complex numbers with moduli strictly less than one such that $ay=bx$. One can check that the triple $$\left(\begin{bmatrix}
0 & \frac{a(1-|y|^2)}{1-|ay|^2} \\
\frac{b(1-|x|^2)}{1-|bx|^2} & 0 \\
\end{bmatrix},\begin{bmatrix}
0 & \frac{x(1-|b|^2)}{1-|xb|^2} \\
\frac{y(1-|a|^2)}{1-|ay|^2} & 0 \\
\end{bmatrix},\frac{z-ay}{1-\overline{ay}z}I_{\mathbb{C}^2}\right)$$ is the characteristic triple for the commutative, contractive pair $$\begin{aligned}
\label{NRegExamp}
T_1=\begin{bmatrix}
0 & a \\
b & 0 \\
\end{bmatrix},\;
T_2=\begin{bmatrix}
0 & x \\
y & 0 \\
\end{bmatrix}.\end{aligned}$$ ]{}
Characterization of invariant subspaces for pairs of commuting contractions {#S:invsub}
===========================================================================
In this section we characterize invariant subspaces for pairs $(T_1,T_2)$ of commuting contractions such that $T=T_1T_2$ is a c.n.u. contraction. Sz.-Nagy and Foias characterized how invariant subspaces for c.n.u. contractions arise in the functional model. They showed that invariant subspaces of a c.n.u. contraction $T$ are in one-to-one correspondence with regular factorizations of the characteristic function of $T$. A minor complication in the theory is that the factors in a regular factorization of a purely contractive analytic function need not again be purely contractive. We now recall their result as we shall have use of it later in this section.
\[NFcnu\] Let $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$ be a purely contractive analytic function and $\bf T$ be the contraction on $$\begin{aligned}
\label{cnuMspace}
{{\mathbb{H}}}=H^2({{\mathcal D}}_*)\oplus\overline{\Delta_{\Theta} L^2({{\mathcal D}})}\ominus\{\Theta f\oplus\Delta_{\Theta} f: f\in H^2({{\mathcal D}})\}\end{aligned}$$defined by $$\begin{aligned}
\label{NFMop}
{\bf T}=P_{{{\mathbb{H}}}}\big(M_z\oplus M_{\zeta}\big)|_{{{\mathbb{H}}}}.\end{aligned}$$A subspace ${{\mathbb{H}}}'$ of ${{\mathbb{H}}}$ is invariant under ${\bf T}$ if and only if there exist contractive analytic functions $(\mathcal{D},\mathcal{F},\Theta')$, $(\mathcal{F},\mathcal{D}_{*},\Theta'')$ such that $$\begin{aligned}
\label{cnuRegFact}
\Theta=\Theta''\Theta'\end{aligned}$$is a regular factorization, and with the unitary $Z$ as in (\[Z\]) we have $$\begin{aligned}
\label{cnuH'}
{{\mathbb{H}}}'=\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g):\;f\in H^2(\mathcal{F}),g\in\overline{\Delta_{\Theta'}L^2(\mathcal{D})}\}\\\nonumber
\ominus\{\Theta h\oplus\Delta_{\Theta}h:h\in H^2(\mathcal{D})\}\end{aligned}$$ and $$\begin{aligned}
\label{cnuH''}
{{\mathbb{H}}}'' := {{\mathbb{H}}}\ominus{{\mathbb{H}}}'=H^2({{\mathcal D}}_*)\oplus &Z^{-1}(\overline{\Delta_{\Theta''}L^2(\mathcal{F})}\oplus\{0\})\\\nonumber
&\ominus\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus 0):f\in H^2(\mathcal{F})\}.\end{aligned}$$ Moreover, the characteristic function of ${\mathbf T}|_{{\mathbb H}'}$ coincides with the purely contractive part of $\Theta'$, and the characteristic function of $P_{{\mathbb H}''} {\mathbf T}|_{{\mathbb H}''}$ coincides with the purely contractive part of $\Theta''$.
Let $T$ be a c.n.u. contraction such that $T=T_1T_2$ for a pair $(T_1,T_2)$ of commuting contractions. It is natural that one would need more conditions than (\[cnuH’\]) and (\[cnuH”\]) for an invariant subspace of $T$ to be jointly invariant under $(T_1,T_2)$.
\[T:inv-cnu\] Let $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$ be a pure contractive analytic function and let the triple $((G_1,G_2),(W_1,W_2),\Theta)$ be admissible. Define the pair $({\bf T_1},{\bf T_2})$ of commuting contractions on $$\begin{aligned}
\label{cnuMspace}
{{\mathbb{H}}}=H^2({{\mathcal D}}_*)\oplus\overline{\Delta_{\Theta} L^2({{\mathcal D}})}\ominus\{\Theta f\oplus\Delta_{\Theta} f: f\in H^2({{\mathcal D}})\}\end{aligned}$$by $$\begin{aligned}
\label{Mop}
({\bf T_1},{\bf T_2})=P_{{{\mathbb{H}}}}\big(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2\big)|_{{{\mathbb{H}}}}.\end{aligned}$$ A subspace ${{\mathbb{H}}}'$ of ${{\mathbb{H}}}$ is jointly invariant under $({\bf T_1},{\bf T_2})$ if and only if there exist two contractions $G'_1$, $G'_2$ in $\mathcal{B}(\mathcal{F})$, a pair $(W'_1,W'_2)$ of unitary operators on $\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}$ with the property $$\begin{aligned}
\label{propW'}
W'_1W'_2=W'_2W'_1=M_{\zeta}|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}}\end{aligned}$$ along with two contractive analytic functions $(\mathcal{D},\mathcal{F},\Theta')$, $(\mathcal{F},\mathcal{D}_{*},\Theta'')$ such that $$\begin{aligned}
\Theta=\Theta'' \Theta'\end{aligned}$$ is a regular factorization and also, with $Z$ the pointwise unitary operator as in (\[Z\]), $$\begin{aligned}
\label{H'}
{{\mathbb{H}}}'=\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g):\;f\in H^2(\mathcal{F}),g\in\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}\}\\\nonumber
\ominus\{\Theta h\oplus\Delta_{\Theta}h:h\in H^2({{\mathcal D}})\},\end{aligned}$$ $$\begin{aligned}
\label{H''}
{{\mathbb{H}}}'':={{\mathbb{H}}}\ominus{{\mathbb{H}}}'=&\;H^2({{\mathcal D}}_*)\oplus Z^{-1}(\overline{\Delta_{\Theta''}L^2(\mathcal{F})}\oplus\{0\})\\&\nonumber\ominus\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus 0):f\in H^2(\mathcal{F})\},\end{aligned}$$ and for every $f\in H^2(\mathcal{F})$ and $g\in\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}$ $$\begin{aligned}
\label{ExtraCond}
\begin{bmatrix}
M_{G_i^*+zG_j} & 0 \\
0 & W_i \\
\end{bmatrix}
\begin{bmatrix}
\Theta''f\\Z^{-1}(\Delta_{\Theta''}f\oplus g)
\end{bmatrix}
=\begin{bmatrix}
\Theta''M_{G'_i+zG'_j}f\\Z^{-1}(\Delta_{\Theta''}M_{G'_i+zG'_j}f\oplus W'_i g)
\end{bmatrix},\end{aligned}$$where $(i,j)=(1,2),(2,1)$.
We first prove the easier part—the proof of sufficiency. Suppose that $$((G_1,G_2),(W_1,W_2),\Theta)$$ is a purely contractive admissible triple (i.e., $( {{\mathcal D}}, {{\mathcal D}}_*, \Theta)$ is a purely contractive analytic function) such that $\Theta$ has a regular factorization $\Theta = \Theta'' \Theta'$ with $({{\mathcal D}}, {{\mathcal F}}, \Theta')$ and $({{\mathcal F}}, {{\mathcal D}}_*, \Theta'')$ contractive analytic functions. We suppose also that $G_1'$ and $G_2'$ are contraction operators on ${{\mathcal F}}$, $W_1'$, $W_2'$ are unitary operators on $\overline{\Delta_{\Theta'} L^2({{\mathcal D}})}$ so that and hold. Then we have all the ingredients to define ${\mathbb H}'$ and ${\mathbb H}''$ as in and . Note next that ${\mathbb H}'$ is indeed a subspace of ${\mathbb H}$. We wish to show that the space ${{\mathbb{H}}}'$ given in (\[H’\]) is jointly invariant under the pair $({\bf T_1},\bf{T_2})$ defined in (\[Mop\]). Firstly, it is easy to see that ${{\mathbb{H}}}'$ is a subspace of ${{\mathbb{H}}}$. Since the operator $$\begin{aligned}
\label{iso-cnu}
{\mathcal I} \colon H^2(\mathcal{F})\oplus\overline{\Delta_{\Theta'}L^2({{\mathcal D}})} &\to
H^2({{\mathcal D}}_*)\oplus\overline{\Delta_{\Theta}L^2({{\mathcal D}})}\\\nonumber
f\oplus g &\mapsto \Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g)\end{aligned}$$ is an isometry, the space $$\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g):f\in H^2(\mathcal{F})
\text{ and } g \in \overline{\Delta_{\Theta'}L^2({{\mathcal D}})}\}$$ is closed and by (\[ExtraCond\]) we see that it is jointly invariant under $$(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2,M_z\oplus M_{\zeta}).$$ We also see that $$\operatorname{Ran}\, \mathcal{I}=\big(H^2({{\mathcal D}}_*)\oplus \overline{\Delta_{\Theta} L^2({{\mathcal D}})} \big)
\ominus({{\mathbb{H}}}\ominus\mathbb{H}').$$ Now the sufficiency follows from the definition of $({\bf T_1},\bf{T_2})$ and from the general fact that if $V$ is an operator on $\mathcal{K}$ containing $\mathcal{H}$, $V({{\mathcal K}}\ominus{{\mathcal H}})\subset{{\mathcal K}}\ominus{{\mathcal H}}$, and $V^*|_{\mathcal{H}}=T^*$, then for a subspace $\mathcal{H}'$ of $\mathcal{H}$, $$V(\mathcal{K}\ominus(\mathcal{H}\ominus\mathcal{H}'))\subseteq (\mathcal{K}\ominus(\mathcal{H}\ominus\mathcal{H}')) \text{ if and only if } T(\mathcal{H}')\subseteq\mathcal{H}'.$$
Now we show that the conditions are necessary. The first step of the proof is an application of Theorem \[NFcnu\]. Indeed, if ${{\mathbb{H}}}'\subset{{\mathbb{H}}}$ is jointly invariant under $({\bf T_1},\bf{T_2})$, then it is also invariant under the product ${\bf T_1T_2}$ and by definition of admissibility $${\bf T}={\bf T_1T_2}={\bf T_2T_1}=P_{{{\mathbb{H}}}}\big(M_z\oplus M_{\zeta}\big)|_{{{\mathbb{H}}}}.$$ Hence by Theorem \[NFcnu\], there exist two contractive analytic functions $$(\mathcal{D},\mathcal{F},\Theta'), \quad (\mathcal{F},\mathcal{D}_{*},\Theta'')$$ such that $\Theta=\Theta''\Theta'$ is a regular factorization and the spaces ${{\mathbb{H}}}'$ and ${{\mathbb{H}}}''$ are realized as in (\[H’\]) and (\[H”\]), respectively. It only remains to produce contraction operators $G_1'$, $G_2'$ on ${{\mathcal F}}$ and unitary operators $W_1'$, $W_2'$ on $\overline{\Delta_{\Theta'} L^2({{\mathcal D}})}$ so that conditions and hold. Note that, once we have found $G_1'$, $G_2'$, $W_1'$, $W_2'$, verification of breaks up into three linear pieces, where $(i,j) = (1,2)$ or $(2,1)$: $$\begin{aligned}
& M_{G_i^* + z G_j} \Theta'' f = \Theta'' M_{G_i' + z G'_j} f \text{ for all } f \in H^2({{\mathcal F}}), \label{verify1} \\
& W_i (Z^{-1} (\Delta_{\Theta''} f \oplus 0)= Z^{-1} (\Delta_{\Theta''} M_{G_i' + z G_j'} f \oplus 0)
\text{ for all } f \in H^2({{\mathcal F}}), \label{verify2} \\
& W_i Z^{-1} (0 \oplus g) = Z^{-1}(0 \oplus W_i' g) \text{ for all } g \in \overline{\Delta_{\Theta'} L^2({{\mathcal D}})}.
\label{verify3}\end{aligned}$$
As a first step, we define operators $X_i$ on $H^2(\mathcal{F})$ and $W_i'$ on $\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}$, for $i=1,2$, such that for every $f\in H^2(\mathcal{F})$ and $g\in\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}$, $$\begin{aligned}
\label{Intertwining2}
\nonumber\begin{bmatrix}
M_{G_i^*+zG_j} & 0 \\
0 & W_i \\
\end{bmatrix}\mathcal{I}
\begin{bmatrix}
f\\g
\end{bmatrix}&=
\begin{bmatrix}
M_{G_i^*+zG_j} & 0 \\
0 & W_i \\
\end{bmatrix}
\begin{bmatrix}
\Theta''f\\Z^{-1}(\Delta_{\Theta''}f\oplus g)
\end{bmatrix}\\
&=\begin{bmatrix}
\Theta''X_if\\Z^{-1}(\Delta_{\Theta''}X_if\oplus W_i'g)
\end{bmatrix}=
\mathcal{I}
\begin{bmatrix}
X_i & 0 \\
0 & W_i' \\
\end{bmatrix}
\begin{bmatrix}
f\\g
\end{bmatrix},\end{aligned}$$ where ${{\mathcal I}}$ is the isometry as defined in (\[iso-cnu\]). The operators $X_1,X_2$ and $W_1',W_2'$ are well-defined because the operator $\mathcal{I}$ is an isometry. Indeed, it follows that $X_1,X_2$ and $W_1',W_2'$ are contractions. Since the unitary $Z$ commutes with $M_{\zeta}$, it is easy to see from the definition of $\mathcal I$ that it has the following intertwining property $$\label{Intertwining1}
\mathcal{I} (M_z \oplus M_{\zeta}|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}}) =
(M_z\oplus M_{\zeta}|_{\overline{\Delta_{\Theta}L^2({{\mathcal D}})}})\mathcal{I}.$$ From the intertwining properties (\[Intertwining1\]) and (\[Intertwining2\]) of $\mathcal{I}$, we get for $i=1,2$ $$(X_i\oplus W_i')(M_z\oplus M_{\zeta}|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}})=
(M_z\oplus M_{\zeta}|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}})(X_i\oplus W_i'),$$ which implies that $(X_1,X_2)=(M_{\varphi_1},M_{\varphi_2})$, for some $\varphi_1$ and $\varphi_2$ in $L^\infty(\mathcal{B}(\mathcal{F}))$. We next show that $\varphi_1$ and $\varphi_2$ are actually linear pencils. Toward this end, notice from (\[Intertwining2\]) that $$\begin{aligned}
\label{subeqn1}
\begin{bmatrix}
M_{\varphi_1} & 0 \\
0 & W_1' \\
\end{bmatrix}&= \mathcal{I}^*
\begin{bmatrix}
M_{G_1^*+zG_2} & 0 \\
0 & W_1 \\
\end{bmatrix}\mathcal{I} \\\label{subeqn}
\begin{bmatrix}
M_{\varphi_2} & 0 \\
0 & W_2' \\
\end{bmatrix}&= \mathcal{I}^*
\begin{bmatrix}
M_{G_2^*+zG_1} & 0 \\
0 & W_2 \\
\end{bmatrix}\mathcal{I}\end{aligned}$$ Now multiplying (\[subeqn1\]) on the left by $M_z^*\oplus M_{\zeta}^*|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}}$, then using the intertwining property (\[Intertwining1\]) of ${{\mathcal I}}$ and then remembering that $(W_1,W_2)$ is a commuting pair of unitaries such that $W_1W_2=M_{\zeta}^*|_{\overline{\Delta_{\Theta}L^2({{\mathcal D}})}}$, we get $$\begin{bmatrix}
M_{z}^* & 0 \\
0 & M^*_{\zeta}|_{\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}} \\
\end{bmatrix}
\begin{bmatrix}
M_{\varphi_1} & 0 \\
0 & W_1' \\
\end{bmatrix}
=\mathcal{I}^*
\begin{bmatrix}
M^*_{G_2^*+zG_1} & 0 \\
0 & W_2^* \\
\end{bmatrix}\mathcal{I}=
\begin{bmatrix}
M_{\varphi_2}^* & 0 \\
0 & W'^*_2 \\
\end{bmatrix}.$$ Consequently, $M_{\varphi_2}=M_{\varphi_1}^*M_z$. A similar argument as above yields $M_{\varphi_1}=M_{\varphi_2}^*M_z$. Considering these two relations and the power series expansions of $\varphi_1$ and $\varphi_2$, we get $$\label{varphi}
\varphi_1(z)=G_1'^*+zG_2' \text{ and }\varphi_2(z)=G_2'^*+zG_1',$$ for some $G_1',G_2'\in\mathcal{B}(\mathcal{F})$. The fact that $M_{\varphi_1}$ (and $M_{\varphi_2}$) is a contraction implies that $G_1'$ and $G_2'$ are contractions too. Recalling and the substitution $(X_1, X_2) = (M_{\varphi_1}, M_{\varphi_2})$ where $\varphi_1$ and $\varphi_2$ are given by , we see that we have established with the choice of $G_1'$, $G_2'$ as in .
Next note that the bottom component of gives us $$\label{ConseqInt2}
W_iZ^{-1}(\Delta_{\Theta''}f\oplus g)=Z^{-1}(\Delta_{\Theta''}X_if\oplus W_i'g).$$ for all $f \in H^2({{\mathcal F}})$, $g \in \overline{\Delta_{\Theta'} L^2({{\mathcal D}})}$, and $i=1,2$. In particular, setting $g=0$ and recalling that $X_i = M_{\varphi_i} = M_{G_i^{\prime *} + z G_j'}$, we get $$\label{verify2'}
W_i Z^{-1} (\Delta_{\Theta''} f \oplus 0) = Z^{-1} (\Delta_{\Theta''} M_{G_i^{\prime *} + z G_j'} f \oplus 0),$$ thereby verifying . We next consider with $f=0$ and $g$ equal to a general element of $\overline{\Delta_{\Theta'} L^2({{\mathcal D}})}$ to get $$\label{verify3'}
W_i Z^{-1}(0\oplus g)=Z^{-1}(0\oplus W'_i g),$$ thereby verifying and hence also completing the proof of .
It remains to show that $(W_1', W_2')$ is a commuting pair of unitary operators satisfying condition . Toward this goal, let us rewrite in the form $$\label{verify3''}
Z W_i Z^{-1} (0 \oplus g) = 0 \oplus W_i' g.$$ which implies that, for $i=1,2$, $$ZW_iZ^{-1}(\{0\}\oplus\overline{\Delta_{\Theta'}L^2({{\mathcal D}})})\subseteq(\{0\}\oplus\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}).$$ On the other hand, using and noting that $M_{\zeta}|_{\overline{\Delta_{\Theta}L^2({{\mathcal D}})}}$ commutes with $Z,W_1,W_2$ and $\Delta_{\Theta''}$, we get for every $f\in H^2(\mathcal{F})$ and $n\geq 0$ $$ZW_iZ^{-1}(\Delta_{\Theta''}e^{-int}f\oplus 0)=(\Delta_{\Theta''}e^{-int}X_if\oplus 0),$$ which implies that $ZW_iZ^{-1}(\overline{\Delta_{\Theta}L^2({{\mathcal D}})}\oplus \{0\})\subseteq
(\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}\oplus\{0\})$, for $i=1,2$. We conclude that $Z^{-1} ( \{0\} \oplus
\overline{\Delta_{\Theta'} L^2({{\mathcal D}})})$ is a reducing subspace for the pair of unitaries $(W_1, W_2)$ and hence $(W_1, W_2) |_{Z^{-1} (\{0\} \oplus \overline{\Delta_{\Theta'} L^2({{\mathcal D}})}}$ is a pair of commuting unitary operators. The intertwining shows that the pair $(W_1', W_2')$ on $\overline{\Delta_{\Delta'} L^2({{\mathcal D}})}$ is jointly unitarily equivalent to the commutative unitary pair $(W_1, W_2)|_{Z^{-1}(\{0\} \oplus \overline{\Delta_\Theta'} L^2({{\mathcal D}})}$ and hence is itself a commutative unitary pair. Furthermore, since $W_1 W_2 = M_\zeta$ in particular on $Z^{-1} ( \{0\} \oplus \overline{\Delta_{\Theta'}
L^2({{\mathcal S}})})$ and $M_\zeta$ commutes past $Z$ and $Z^{-1}$, we conclude that condition holds as well. This completes the proof of the necessary part.
As we see from the last part of the statement of Theorem \[T:inv-cnu\], Sz.-Nagy and Foias went on to prove that, under the conditions of Theorem \[NFcnu\], the characteristic functions of $\bf T|_{{{\mathbb{H}}}'}$ and $P_{{{\mathbb{H}}}\ominus{{\mathbb{H}}}'} {\bf T}|_{{{\mathbb{H}}}\ominus{{\mathbb{H}}}'}$ coincide with the purely contractive parts of $\Theta'$ and $\Theta''$, respectively. Below we find an analogous result (at least for the first part of this statement) for pairs of commuting contractions. The strategy of the proof is the same as that of Sz.-Nagy–Foias, namely: application of model theory.
Under the conditions of Theorem \[T:inv-cnu\], let ${{\mathbb{H}}}'$ be a joint invariant subspace of ${{\mathbb{H}}}$ induced by the regular factorization $\Theta=\Theta''\Theta'$. Then with the notations as in Theorem \[T:inv-cnu\], the triple $((G_1',G_2'),(W_1',W_2'),\Theta')$ is admissible and its purely contractive part coincides with the characteristic triple for $({\bf T_1},{\bf T_2})|_{{{\mathbb{H}}}'}$.
With the isometry ${{\mathcal I}}$ as in (\[iso-cnu\]), define a unitary $U:={{\mathcal I}}^*|_{\operatorname{Ran }{{\mathcal I}}}$. Therefore $$\begin{aligned}
\label{iso-cnu^*}
\nonumber
U:\{\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g):f\in H^2(\mathcal{F}),\;g\in\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}\}
&\to H^2(\mathcal{F})\oplus\overline{\Delta_{\Theta'}L^2({{\mathcal D}})} \\
U \colon \Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g) \mapsto f\oplus g.\end{aligned}$$ For every $g\in H^2({{\mathcal D}})$, $$\begin{aligned}
\label{ortho}
U(\Theta g\oplus\Delta_{\Theta}g)=U(\Theta''\Theta'g\oplus Z^{-1}(\Delta_{\Theta''}\Theta'g\oplus\Delta_{\Theta'}g))
=\Theta'g\oplus\Delta_{\Theta'}g,\end{aligned}$$which implies that $U$ takes ${{\mathbb{H}}}'$ as given in (\[H’\]) onto the Hilbert space $$\begin{aligned}
\label{fracH'}
\mathfrak{H}':=H^2(\mathcal{F})\oplus\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}\ominus\{\Theta'g\oplus
\Delta_{\Theta'}g:g\in H^2({{\mathcal D}})\}\end{aligned}$$ The basis of the proof is the following unitary equivalences: $$\begin{aligned}
\label{goalcnu} U(M_{G_i^*+zG_j}\oplus W_i) U^*&=M_{G_i'^*+zG_j'}\oplus W_i' \text{ for } (i,j)=(1,2),(2,1), \\
\label{goalcnu1} U(M_z\oplus M_{\zeta})U^*&=(M_z\oplus M_{\zeta}).\end{aligned}$$ To verify –, proceed as follows. Since $M_{\zeta}$ commutes with $Z$ and $\Delta_{\Theta''}$, (\[goalcnu1\]) follows easily. We establish equation (\[goalcnu\]) only for $(i,j)=(1,2)$ and omit the proof for the other case because it is similar. For $f\in H^2(\mathcal{F})$ and $g\in\overline{\Delta_{\Theta'}L^2({{\mathcal D}})}$, $$\begin{aligned}
&U(M_{G_1^*+zG_2}\oplus W_1)U^*(f\oplus g)=U(M_{G_1^*+zG_2}\oplus W_1)(\Theta''f\oplus Z^{-1}(\Delta_{\Theta''}f\oplus g))\\
=&\; U(\Theta''M_{G_1'^*+zG_2'}f\oplus Z^{-1}(\Delta_{\Theta''}M_{G_1'^*+zG_2'}f\oplus W_1'g))\;[\text{by }(\ref{ExtraCond})]\\
=&\; M_{G_1'^*+zG_2'}f\oplus W_1'g\end{aligned}$$ and also follows.
We now show that the triple $((G_1',G_2'),(W_1',W_2'),\Theta')$ is admissible. Recall that in the course of the proof of Theorem \[T:inv-cnu\], we saw that both $G_1'$ and $G_2'$ are contractions and that $(W_1',W_2')$ is a pair of commuting unitaries satisfying (\[propW’\]). From (\[goalcnu\]) we see that for every $f\in H^2({{\mathcal F}})$ and $(i,j)=(1,2),(2,1)$, $$\begin{aligned}
(M_{G_i'^*+zG_j'}\oplus W_i') (\Theta'f\oplus\Delta_{\Theta'}f)
=&\;U(M_{G_i^*+zG_j}\oplus W_i) U^*(\Theta'f\oplus\Delta_{\Theta'}f)\\
=&\;U(M_{G_i^*+zG_j}\oplus W_i)\big(\Theta''\Theta'f\oplus Z^{-1}(\Delta_{\Theta''}\Theta'f\oplus \Delta_{\Theta'}f\big)\\
=&\;U(M_{G_i^*+zG_j}\oplus W_i)\big(\Theta f\oplus \Delta_{\Theta}f\big).\end{aligned}$$ From the admissibility of $((G_1,G_2),(W_1,W_2),\Theta)$, we know that each of the contraction operators $(M_{G_i^*+zG_j}\oplus W_i)$ takes the space $\{\Theta f\oplus \Delta_{\Theta}f:f\in H^2({{\mathcal D}})\}$ into itself. Therefore from the last term of the above computation and (\[ortho\]), we see that for each $(i,j)=(1,2),(2,1)$, $$(M_{G_i'^*+zG_j'}\oplus W_i')\big(\{\Theta' f\oplus \Delta_{\Theta'}f:f\in H^2({{\mathcal D}})\}\big)\subset\{\Theta' f\oplus \Delta_{\Theta'}f:f\in H^2({{\mathcal D}})\}.$$ From (\[goalcnu\]) it is also clear that for each $(i,j)=(1,2),(2,1)$, the operators $(M_{G_i'^*+zG_j'}\oplus W_i')$ are contractions and that with $\mathfrak{H}'$ as in (\[fracH’\]) $$\begin{aligned}
&(M_{G_i'^*+zG_j'}\oplus W_i')^*(M_{G_j'^*+zG_i'}\oplus W_j')^*|_{\mathfrak{H}'}=U(M_{G_i^*+zG_j}
\oplus W_i)^*(M_{G_j^*+zG_i}\oplus W_j)^*|_{{{\mathbb{H}}}}\\
=&\;U(M_z\oplus M_{\zeta})|_{{{\mathbb{H}}}}=U(M_z\oplus M_{\zeta})U^*|_{\mathfrak{H}'}=
(M_z\oplus M_{\zeta})|_{\mathfrak{H}'} \quad [\text{by } (\ref{goalcnu1})].\end{aligned}$$ This completes the proof of admissibility of $((G_1',G_2'),(W_1',W_2'),\Theta')$.
And finally to prove the last part we first observe that $$\begin{aligned}
({\bf T_1},{\bf T_2})=P_{{{\mathbb{H}}}'}(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2)|_{{{\mathbb{H}}}'}.\end{aligned}$$ Now from equations (\[goalcnu\]) and (\[goalcnu1\]) again and from the fact that $U({{\mathbb{H}}}')=\mathfrak{H}'$ (hence $UP_{{{\mathbb{H}}}'}=P_{\mathfrak{H}'}U$), we conclude that $$\begin{aligned}
({\bf T_1},{\bf T_2},{\bf T_1T_2})|_{{{\mathbb{H}}}'}=P_{{{\mathbb{H}}}'}(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2,M_z\oplus M_{\zeta})|_{{{\mathbb{H}}}'}\end{aligned}$$ is unitarily equivalent to the functional model associated to $((G_1',G_2'),(W_1',W_2'),\Theta')$, i.e., $$\begin{aligned}
P_{\mathfrak H'}(M_{G_1^*+zG_2}\oplus W_1,M_{G_2^*+zG_1}\oplus W_2,M_z\oplus M_{\zeta})|_{\mathfrak H'}\end{aligned}$$ via the unitary $U|_{{{\mathbb{H}}}'}:{{\mathbb{H}}}'\to\mathfrak{H}'$. Therefore appeal to Theorem \[UnitaryInv\], Theorem \[AdmisCharc\] and Proposition \[P:pure-adm-triple\] completes the proof.
In case the purely contractive analytic function $({{\mathcal D}},{{\mathcal D}}_{*},\Theta)$ is inner, the results above are much simpler, as in the following statement.
Let $({{\mathcal D}},{{\mathcal D}}_*,\Theta)$ be an inner function and $((G_1,G_2),\Theta)$ be an admissible pair. Define the pair $({\bf T_1},{\bf T_2})$ of commuting contractions on $$\begin{aligned}
\label{cnuMspace}
{{\mathbb{H}}}=H^2({{\mathcal D}}_*)\ominus\{\Theta f: f\in H^2({{\mathcal D}})\}\end{aligned}$$by $$\begin{aligned}
\label{Mop}
({\bf T_1},{\bf T_2})=P_{{{\mathbb{H}}}}\big(M_{G_1^*+zG_2},M_{G_2^*+zG_1}\big)|_{{{\mathbb{H}}}}.\end{aligned}$$ A subspace ${{\mathbb{H}}}'$ of ${{\mathbb{H}}}$ is jointly invariant under $({\bf T_1},{\bf T_2})$ if and only if there exist two inner functions $(\mathcal{D},\mathcal{F},\Theta')$, $(\mathcal{F},\mathcal{D}_{*},\Theta'')$ such that $$\begin{aligned}
\Theta=\Theta''\Theta'\end{aligned}$$ is a regular factorization, $$\begin{aligned}
\label{H'}
{{\mathbb{H}}}'=\{\Theta''f:f\in H^2(\mathcal{F})\}\ominus\{\Theta h:h\in H^2({{\mathcal D}})\},\end{aligned}$$ $$\begin{aligned}
\label{H''}
{{\mathbb{H}}}'':={{\mathbb{H}}}\ominus{{\mathbb{H}}}'=&\;H^2({{\mathcal D}}_*)\ominus\{\Theta''f:f\in H^2(\mathcal{F})\},\end{aligned}$$and two contractions $G'_1,G'_2$ in $\mathcal{B}(\mathcal{F})$ such that $$\begin{aligned}
\label{ExtraCond'}
M_{G_i^*+zG_j}M_{\Theta''}=M_{\Theta''}M_{G'_i+zG'_j}.\end{aligned}$$ Moreover, the pair $((G_1',G_2'),\Theta')$ coincides with the characteristic pair for $(T_1,T_2)|_{\mathcal{H}'}$.
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[^1]: The research of the second author is supported by SERB Indo-US Postdoctoral Research Fellowship, 2017.
|
---
abstract: 'Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class derivative-assets. The payoff of the derivative-assets may be path-dependent. Additionally, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative-assets: a vanilla option on a defaultable stock, a path-dependent option on a non-defaultable stock, and a bond in a short-rate model.'
author:
- |
Matthew J. Lorig[^1]\
*ORFE Department, Princeton University*
bibliography:
- 'Bibtex-Master-Multiscale.bib'
title: 'Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach'
---
[**Keywords:**]{} derivative pricing, stochastic volatility, local volatility, default, knock-out, barrier, spectral theory, eigenfunction, singular perturbation theory, regular perturbation theory.
Introduction
============
The spectral representation for the transition density of a general one-dimensional diffusion was obtained in a seminal paper by @McKean1956. Since that time, *spectral theory* – and more specifically, the study of eigenfunction expansions of linear operators – has become an essential tool for analysing diffusions. As a diffusion often serves as the underlying process on which financial models are built, it is not surprising that methods from spectral theory have made their way into mathematical finance as well.
In particular, many problems related to the pricing of derivative-assets have been solved analytically by using methods from spectral theory. An overview of the spectral method applied to derivative pricing is as follows. Using risk-neutral pricing, one expresses the value of a derivative-asset $u(t,x)$ as a risk-neutral expectation of some function of the future value of an underlying process $X$. Mathematically, this is expressed as $$\begin{aligned}
u(t,x) = \Et_x [ H(X_t) ] = \int H(y) \, p(t,x,y) \, dy . \label{eq:u}\end{aligned}$$ Here, $p(t,x,y)$ is the transition density of the $X$ under $\Pt$. If it turns out that the ininitesmal generator $\L$ of the underlying process is self-adjoint [^2] on a Hilbert space with weighting measure $m(x)dx$ and if the spectrum of $\L$ is purely discrete, then the transition density of $X$ has an eigenfunction expansion $$\begin{aligned}
p(t,x,y) = m(y) \sum_n e^{-\lam_n t} \psi_n(y) \psi_n(x) , \label{eq:p}\end{aligned}$$ where $\left\{ \lam_n \right\}$ are the eigenvalues of $(-\L)$ and $\left\{ \psi_n \right\}$ are the corresponding eigenfunctions $$\begin{aligned}
-\L \, \psi_n = \lam_n \, \psi_n .\end{aligned}$$ The value of a derivative-asset can then be expressed analytically by inserting into $$\begin{aligned}
u(t,x)
&= \sum c_n \, e^{-\lam_n t} \, \psi_n(x) , &
c_n
&= {\left(}\psi_n , H {\right)}:= \int H(y) \psi_n(y) m(y) dy .\end{aligned}$$
Under some basic assumptions, the infinitesimal generator of a general one-dimensional diffusion $$\begin{aligned}
\L &= \frac{1}{2} a^2(x) \d^2_{xx} + b(x) \d_x - k(x) , &
x &\in (e_1,e_2) , \label{eq:L}\end{aligned}$$ with domain $\text{dom}(\L)$ (described in appendix \[sec:BCs\]) is *always* self-adjoint on the Hilbert space $\H = L^2(I,\m)$, where $I \subset \R$ is an interval with endpoints $e_1$ and $e_2$ and $\m$ is the speed density of the diffusion $$\begin{aligned}
\m(x) &:= \frac{2}{a^2(x)}\exp {\left(}\int_{x_0}^x \frac{2 \, b(y)}{a^2(y)} dy {\right)}. & &(\text{speed density)} \label{eq:m}\end{aligned}$$ The lower limit of integration $x_0 \in I$ is arbitrary. Thus, when a one-dimensional diffusion is adequate for describing the dynamics of an underlying, the spectral method outlined above serves as a powerful tool for analytically pricing derivatives on that underlying. Among the topics that have been addressed by applying spectral methods to one-dimensional diffusions are option pricing (both vanilla and exotic), mortgages valuation, interest rate modeling, volatility modeling, and credit risk (see @linetsky2001 [@linetsky2002; @linetsky2003; @linetsky2004; @AlbaneseLawi2005; @albanese2004unifying; @albanese2001black; @lewis1998; @lipton2002universal; @Goldstein1997; @linetsky2004blackinterest; @linetsky2007intensity; @JDCEV; @linetsky2004lookback; @linetsky2004asian; @linetsky2006bankruptcy]). A useful reference on the topic of spectral methods for one-dimensional diffusions in finance is @linetskybook.
As widely applicable as one-dimensional diffusions are in finance, there are applications in which one-dimensional diffusions are not adequate for describing the dynamics of an underlying. This is the case, for example, in a stochastic volatility setting, where the volatility of the asset that underlies a derivative is controlled by (possibly multiple) nonlocal diffusions. Ideally, one would like to employ techniques from spectral theory to solve problems that relate to multidimensional diffusions. Unfortunately, whereas the infinitesimal generator of a one-dimensional diffusion is practically guaranteed to be self-adjoint, the infinitesimal generator of a multidimensional diffusion is only self-adjoint when the drift vector satisfies certain constraints imposed by the volatility matrix. The drift constraint is not satisfied by any of the most prominent stochastic volatility models – @heston, @hullwhite1987, @stein1991stock and the SABR model by @sabr – which complicates the use of spectral methods.
Recently, @lorig2, show one way to deal with this issue. By combining techniques from singular perturbation theory and spectral theory, the authors are able to express the approximate price of a (possibly path-dependent) option as an eigenfunction expansion, even though the infinitesimal generator of the two-dimensional diffusion they work with is not self-adjoint. As notable as their work is, the results of @lorig2 are valid only when the asset underlying the option is a Black-Scholes-like geometric Brownian motion (GBM) with fast mean-reverting stochastic volatility.
In this paper, we extend the work of @lorig2 in four important ways.
1. As a “base” model, we work with a general one-dimensional diffusion $dX_t = \nu(X_t) dt + a(X_t)dW_t$. This is in contrast to @lorig2, where the only base model considered is a GBM: $dX_t = \mu X_t dt + \sig X_t dW_t$.
2. The general diffusion we work with may exhibit killing (jump to default) at a rate $h(X_t) \geq 0$. In the GBM case considered in @lorig2, $X$ is always strictly positive.
3. To our general diffusion we add two factors of nonlocal volatility: $a(X_t) \rightarrow a(X_t) f(Y_t,Z_t)$. The first factor $Y$ is a fast-varying factor. The second factor $Z$ is slow-varying. Thus, our model is a *multiscale* stochastic volatility model. Again, this is in contrast to @lorig2, where the analysis is limited to a single fast mean-reverting factor of volatility $\sig X_t \rightarrow f(Y_t) X_t$.
4. In changing from the physical probability measure to the risk-neutral pricing measure, we consider a class of market prices of risk that is general enough to treat credit, equity, and interest rate derivatives in a single framework. In @lorig2 the form chosen for the market price of risk restricts the authors to equity derivatives only.
As in @lorig2, we will derive an eigenfunction expansion for the approximate price of a derivative-asset despite the fact that the infinitesimal generator we consider is not (in general) self-adjoint. Unlike @lorig2, because our multidimensional diffusion contains both a fast-varying and a slow-varying factor of volatility, we must combine techniques from both singular *and* regular perturbation theory to achieve our result. In @lorig2, only singular perturbation techniques are required, due to the presence of a single fast mean-reverting factor of volatility.
Of course, the idea of combining singular and regular perturbation techniques in a multiscale stochastic volatility setting is not particularly new or unique. The seminal paper on the subject, applied in a Black-Scholes-like GBM setting, is due to @fouque2004multiscale. Further application of the singular and regular perturbation methods developed in @fouque2004multiscale led to papers concerning bond-pricing, interest rate derivatives, credit derivatives, and option pricing in a CEV-like setting (see @foque2008bond [@cfps; @fouque2006stochastic; @fouque2008modeling; @fouqueCEV]). There is also a book by @fpss, which contains the many of the key results from the above mentioned publications. What this paper contributes to the existing literature on multiscale diffusions is flexibility and simplicity. From a flexibility standpoint, the methods developed in this paper are able to encapsulate, in a unified framework, many of the results contained in @fouqueCEV [@cfps; @foque2008bond; @fouque2006stochastic; @fouque2004multiscale; @fpss; @fouque2008modeling], as well as further results, which are not contained in these works (e.g., jump to default CEV with multiscale stochastic volatility, see section \[sec:JDCEV\]). With regards to simplicity, the spectral method we develop reduces the derivative pricing problem to that of solving a single, one-dimensional eigenvalue equation. Once this equation is solved, the approximate price of a derivative-asset can be calculated formulaically by computing a few simple inner products. This is in contrast to the methods developed in @fouqueCEV [@cfps; @foque2008bond; @fouque2006stochastic; @fouque2004multiscale; @fpss; @fouque2008modeling], where, in order to express the approximate price of a derivative-asset, an inhomogeneous partial differential equation (PDE) must be solved.
The rest of this paper proceeds as follows. In section \[sec:model\] we introduce a class of models described by multiscale diffusions. We also explain the kind of derivative-asset we wish to consider. In section \[sec:pricing\] we solve (approximately), the problem of pricing a derivative-asset. This is done in several steps. First, using risk-neutral pricing, we derive a Cauchy problem which, if solved, would yield the exact value of a derivative-asset. Next, we use techniques from singular and regular perturbation theory to formally derive three simpler Cauchy problems, which, if solved, would yield the approximate value of a derivative-asset. Finally, using eigenfunction expansion techniques, we solve these Cauchy problems explicitly. The solutions are given in Theorems \[thm:u00\], \[thm:u10\] and \[thm:u01\]. In section \[sec:examples\], we illustrate our method of pricing derivative-assets with three examples. We also provide an appendix, which contains some mathematical results that we use throughout this paper.
A Class of Multiscale Models {#sec:model}
============================
Let $(\Om,\F,\P)$ be a probability space supporting correlated Brownian motions $(W^x,W^y,W^z)$ and an exponential random variable $\mathscr{E} \sim \text{Exp}(1)$, which is independent of $(W^x,W^y,W^z)$. We shall consider a three-factor economy described by a time-homogenous, continuous-time Markov process $\Xc = (X,Y,Z)$, which takes values in some state space $E = I \times \R \times \R$. Here, $I$ is an interval in $\R$ with endpoints $e_1$ and $e_2$ such that $-\infty \leq e_1 < e_2 \leq \infty$. We assume that $\Xc$ starts in $E$ and is instantaneously killed (sent to an isolated cemetery state $\Delta$) as soon as $X$ leaves $I$. Specifically, the dynamics of $\Xc$ under the physical measure $\P$ are as follows: $$\begin{aligned}
\Xc_t &= \begin{cases}
{\left(}X_t, Y_t, Z_t {\right)}&\tau_I > t \\
\Delta &\tau_I \leq t
\end{cases} , &
\tau_I &= \inf \left\{ t > 0 : X_t \notin I \right\} ,\end{aligned}$$ where ${\left(}X,Y,Z{\right)}$ are given by $$\begin{aligned}
\left\{ \begin{aligned}
dX_t &= \nu(X_t) \, dt + a(X_t) f(Y_t,Z_t) \, dW^x_t , \\
dY_t &= \frac{1}{\eps} \alpha(Y_t) dt + \frac{1}{\sqrt{\eps}} \beta(Y_t) \, dW^y_t , \\
dZ_t &= \del c(Z_t) dt + \sqrt{\del} g(Z_t) \, dW^z_t , \\
d{\left\langle}W^x,W^y{\right\rangle}_t &= \rho_{xy} \, dt , \\
d{\left\langle}W^x,W^z{\right\rangle}_t &= \rho_{xz} \, dt , \\
d{\left\langle}W^y,W^z{\right\rangle}_t &= \rho_{yz} \, dt , \\
(X_0,Y_0,Z_0) &= (x,y,z) \in E .
\end{aligned} \right. \label{eq:Physical}\end{aligned}$$ Here, $(\rho_{xy},\rho_{xz} ,\rho_{yz})$ satisfy $|\rho_{xy}|, |\rho_{xz}| ,|\rho_{yz}| \leq 1$ and $1 + 2 \rho_{xy} \rho_{xz} \rho_{yz} - \rho_{xy}^2 - \rho_{xz}^2 - \rho_{yz}^2 \geq 0$ so that the correlation matrix of the Brownian motions is positive-semidefinite.
The process $X$ could represent a variety of things. For example, it could represent the price of a stock, the value of an index, the risk-free short-rate of interest, etc. More generally, $X$ could represent an exogenous factor that controls the value of any or all of the items mentioned above. Under the physical measure $\P$, the process $X$ has instantaneous drift $\nu(X_t)$ and stochastic volatility $a(X_t) f(Y_t,Z_t) > 0$, which contains both a local component $a(X_t)$ and nonlocal component $f(Y_t,Z_t)$. The nonlocal component of volatility $f(Y_t,Z_t)$ is controlled by two factors: $Y$ and $Z$. We note that the infinitesimal generators of $Y$ and $Z$ $$\begin{aligned}
\L_Y^\eps &= \frac{1}{\eps} {\left(}\frac{1}{2}\beta^2(y) \, \d^2_{yy}+ \alpha(y) \, \d_y {\right)}, \label{eq:Ly}\\
\L_Z^\del &= \del {\left(}\frac{1}{2}g^2(z) \, \d^2_{zz}+ c(z) \, \d_z {\right)}, \label{eq:Lz}\end{aligned}$$ are scaled by factors $1/\eps$ and $\del$ respectively. Thus, $Y$ and $Z$ have intrinsic time-scales $\eps>0$ and $1/\del>0$. We assume $\eps<<1$ and $\del << 1$ so that the intrinsic time-scale of $Y$ is small and the intrinsic time-scale of $Z$ is large. Hence, $Y$ represents a fast-varying factor of volatility and $Z$ represents a slow-varying factor. Note that $\L_Y^\eps$ and $\L_Z^\del$ have the form with $k(x)=0$ for all $x \in I$. Throughout this paper, we will assume that the domain of any operator of the form is given by equation of appendix \[sec:BCs\].
We are interested in pricing a (possibly defaultable) derivative-asset, whose payoff at time $t>0$ may depend on the path of $X$. Specifically, we shall consider payoffs of the form $$\begin{aligned}
\text{Payoff} &= H(X_t) \, \I_{\left\{ \tau > t \right\}} . \label{eq:Payoff}\end{aligned}$$ Here, $\tau$ is a random time, which represents the default time of the derivative-asset. Because we are interested in pricing derivatives, we must specify the dynamics of $(X,Y,Z)$ under the risk-neutral pricing measure, which we denote as $\Pt$. We have the following risk-neutral dynamics $$\begin{aligned}
\left\{ \begin{aligned}
dX_t &= \Big( b(X_t) - a(X_t) f(Y_t,Z_t) \Om(Y_t,Z_t) \Big) dt + a(X_t) f(Y_t,Z_t) \, d\Wt^x_t , \\
dY_t &= {\left(}\frac{1}{\eps} \alpha(Y_t) - \frac{1}{\sqrt{\eps}} \beta(Y_t) \Lam(Y_t,Z_t) {\right)}dt + \frac{1}{\sqrt{\eps}} \beta(Y_t) \, d\Wt^y_t , \\
dZ_t &= {\left(}\del c(Z_t) - \sqrt{\del} g(Z_t) \Gam(Y_t,Z_t) {\right)}dt + \sqrt{\del} g(Z_t) \, d\Wt^z_t , \\
d{\left\langle}\Wt^x,\Wt^y{\right\rangle}_t &= \rho_{xy} \, dt , \\
d{\left\langle}\Wt^x,\Wt^z{\right\rangle}_t &= \rho_{xz} \, dt , \\
d{\left\langle}\Wt^y,\Wt^z{\right\rangle}_t &= \rho_{yz} \, dt , \\
(X_0,Y_0,Z_0) &= (x,y,z) \in E ,
\end{aligned} \right. \label{eq:RiskNeutral}\end{aligned}$$ where $$\begin{aligned}
d\Wt_t^x &:= dW_t^x + {\left(}\frac{ \nu(X_t) - b(X_t) }{a(X_t) f(Y_t,Z_t) } + \Om{\left(}Y_t, Z_t {\right)}{\right)}\, dt, \\
d\Wt_t^y &:= dW_t^y + \Lam(Y_t,Z_t) \, dt, \\
d\Wt_t^z &:= dW_t^z + \Gam(Y_t,Z_t) \, dt,\end{aligned}$$ are driftless BM’s under $\Pt$. We assume has a unique strong solution.
As mentioned above, the random time $\tau$ represents the default time of the derivative-asset. In our framework, default can occur in one of two ways. Either default occurs when $X$ exits the interval $I$, or default occurs at a random time $\tau_h$, which is controlled by an instantaneous hazard rate $h(X_t) \geq 0$. Mathematically, we express the default time $\tau$ as follows $$\begin{aligned}
\left\{
\begin{aligned}
\tau &= \tau_I \wedge \tau_h , \\
\tau_I &= \inf \left\{ t \geq 0 : X_t \notin I \right\} , \\
\tau_h &= \inf \left\{ t \geq 0 : \int_0^t h(X_s) \, ds \geq \mathscr{E} \right\} , & \mathscr{E} &\sim \text{Exp}(1) , & &\mathscr{E} {\perp \! \! \! \perp}(X,Y,Z) .
\end{aligned}
\right. \label{eq:tau}\end{aligned}$$ Note that the exponentially distributed random variable $\mathscr{E}$ is independent of $(X,Y,Z)$.
Following @elliot2000, to keep track of $\tau_h$, we introduce the indicator process $D_t:=\I_{\{t \geq \tau_h \}}$. Denote by $\mathbb{D}=\{\D_t,t\geq0\}$ the filtration generated by $D$ and by $\mathbb{F}=\{\F_t,t\geq0\}$ the filtration generated by $(W^x,W^y,W^z)$. Define the enlarged filtration $\mathbb{G}=\{\G_t,t\geq0\}$ where $\G_t=\F_t \vee \D_t$. Note that $(X,Y,Z)$ is adapted to $\mathbb{G}$ and $\tau$ is a $\mathbb{G}$-stopping time (i.e., $\{\tau \leq t\}\in\G_t$ for every $t \geq 0$).
We shall assume our economy includes a risk-free asset, which grows instantaneously at short-rate $r(X_t) \geq 0$. Thus, if our economy includes, for example, a non-dividend-paying defaultable asset $S$, whose price process is described by $S_t = \I_{\{\tau>t\}}X_t$, where the state space of $X$ was $I=(0,\infty)$, then the discounted asset price $\{e^{-\int_0^t r(X_s)ds}S_t,t\geq0\}$ must be a $(\Pt,\mathbb{G})$-martingale. The martingale property can be achieved by setting $b(X_t)={\left[}r(X_t)+h(X_t){\right]}X_t$ and $\Om(Y_t,Z_t)=0$ in . The reason for adding the hazard rate $h(X_t)$ to the risk-free rate of interest $r(X_t)$ in the drift of $X$ is to compensate for the possibility of a default (see @JDCEV, Section 2).
On the other hand, if $X$ only describes the risk-free rate of interest through $r(X_t)$, then in changing from the physical measure $\P$ to the pricing measure $\Pt$, one may not have a reason to change the drift of $X$ from $\nu(X_t)$ to $b(X_t)$. However, one may still wish to consider the effect of including a market price of risk. In this case, one could set $b(X_t) = \nu(X_t)$ and keep $\Om(Y_t,Z_t) \neq 0$ in .
We have now described our economy under both the physical and risk-neutral pricing measures, and we have specified the kind of derivative-asset we wish to price. However, we have not been specific about certain technical assumptions, which we shall need in order to prove the accuracy of our pricing approximation. Specific model assumptions can be found in Appendix \[sec:assumptions\].
Derivative Pricing {#sec:pricing}
==================
We wish to price a derivative-asset whose payoff is of the form , where the default time $\tau$ is given by . Using risk-neutral pricing and the Markov property of $\Xc$, the value $u^{\eps,\del}(t,x,y,z)$ of such a derivative-asset at time zero is given by $$\begin{aligned}
u^{\eps,\del}(t,x,y,z)
&= \Et_{x,y,z} {\left[}\exp {\left(}-\int_0^t r(X_s) \, ds {\right)}H(X_t) \, \I_{\left\{ t > \tau \right\}} {\right]}, \label{eq:Expectation}\end{aligned}$$ where $(x,y,z) \in E$ represents the starting point of the process $(X,Y,Z)$. By conditioning on the path of $X$ (see p. 225 of @linetskybook) and by using the Feynman-Kac formula, one can show that $u^{\eps,\del}(t,x,y,z)$ satisfies the following Cauchy problem $$\begin{aligned}
{\left(}-\d_t + \L^{\eps,\del} {\right)}u^{\eps,\del} &= 0 , & & (x,y,z) \in E , \, t \in \R^{+} , \label{eq:uPDE} \\
u^{\eps,\del}(0,x,y,z) &= H(x) ,
\label{eq:uBC}\end{aligned}$$ where the operator $\L^{\eps,\del}$ is given by $$\begin{aligned}
\L^{\eps,\del} &= \frac{1}{\eps} \L_0 + \frac{1}{\sqrt{\eps}} \L_1 + \L_2 + \sqrt{\frac{\del}{\eps}} \M_3
+ \sqrt{\del} \M_1 + \del \M_2, \label{eq:L,eps,del} \\
\L_0 &= \frac{1}{2} \beta^2(y) \, \d^2_{yy}+ \alpha(y) \, \d_y, \\
\L_1 &= \beta(y) \Big( \rho_{xy} a(x) f(y,z) \d_x - \Lam(y,z) \Big) \d_y, \label{eq:L-10} \\
\L_2 &= \frac{1}{2} a^2(x) f^2(y,z) \d^2_{xx} + \Big( b(x) - a(x) \Om(y,z) f(y,z) \Big) \d_x - k(x) , \label{eq:L00} \\
\M_3 &= \rho_{xz} \beta(y) g(z) \d^2_{yz} , \\
\M_1 &= g(z) \Big( \rho_{xz} a(x) f(y,z) \d_x - \Gam(y,z) \Big) \d_z, \\
\M_2 &= \frac{1}{2}g^2(z) \d^2_{zz}+ c(z) \d_z , \\
k(x) &= r(x) + h(x) .\end{aligned}$$ Aside from the initial condition , the function $u^{\eps,\del}(t,x,y,z)$ must satisfy additional boundary conditions (BCs) at the endpoints $e_1$ and $e_2$ of the interval $I$. The BCs at $e_1$ and $e_2$ are understood to be contained in the domain of $\L^{\eps,\del}$ and will depend on the nature of the process $\Xc$ near the endpoints of $I$. Appropriate BCs are discussed in appendix \[sec:BCs\].
From equation we see that $\L_0=\L_Y^1$. We assume that a diffusion with generator $\L_Y^1$ has an invariant distribution $\Pi$ with density $\pi$. In section \[sec:asymptotics\], it will be important to note that the operator $\L_0$ with $\text{dom}(\L_0)=L^2(\R,\pi)$ is self-adjoint acting on the Hilbert space $L^2(\R,\pi)$.
Formal Asymptotic Analysis {#sec:asymptotics}
--------------------------
We wish to solve Cauchy problem -. For general $(f,\alpha,\beta,\Lam,c,g,\Gam)$, no analytic solution exists. However, we notice that, for fixed $\del$, the terms in containing $\eps$ are diverging in the small-$\eps$ limit, giving rise to a *singular* perturbation. Meanwhile, for fixed $\eps$, the terms containing $\delta$ are small in the small-$\del$ limit, giving rise to a *regular* perturbation. Thus, the small-$\eps$ and small-$\del$ regime gives rise to a combined singular-regular perturbation about the $\O(1)$ operator $\L_2$. This suggests that we seek an asymptotic solution to Cauchy problem -. To this end, we expand $u^{\eps,\del}$ in powers of $\sqrt{\eps}$ and $\sqrt{\del}$ as follows $$\begin{aligned}
u^{\eps,\del}
&= \sum_{j \geq 0} \sum_{i \geq 0} \sqrt{\eps}^{\,i} \sqrt{\del}^{\,j} u_{i,j} . \label{eq:uexpand}\end{aligned}$$ Our goal will be to find an approximation of the price $u^{\eps,\del} \approx u_{0,0} + \sqrt{\eps} u_{1,0} + \sqrt{\del} u_{0,1}$. The choice of expanding in half-integer powers of $\eps$ and $\del$ is natural given the form of $\L^{\eps,\del}$. We will justify this expansion when we prove the accuracy of our pricing approximation in Theorem \[thm:accuracy\].
Because we are performing a dual expansion in half-integer powers of $\eps$ and $\del$, we must decide which of these parameters we will expand in first. We choose to perform a regular perturbation expansion with respect to $\del$ first. Then, within each of the equations that result from the regular perturbation analysis, we will perform a singular perturbation expansion with respect to $\eps$. [^3]
### Regular Perturbation Analysis of Equation {#regular-perturbation-analysis-of-equation .unnumbered}
The regular perturbation expansion proceeds by separating terms in $\L^{\eps,\del}$ and $u^{\eps,\del}$ by powers of $\sqrt{\del}$ $$\begin{aligned}
\L^{\eps,\del} &= \L^\eps + \sqrt{\del} \, \M^\eps + \del \M_2 , &
u^{\eps,\del} &= \sum_{j \geq 0} {\left(}\sqrt{\del} {\right)}^j u_j^{\eps} \label{eq:u,del},\end{aligned}$$ where $$\begin{aligned}
\L^\eps &= \frac{1}{\eps} \L_0 + \frac{1}{\sqrt{\eps}} \L_1 + \L_2 , &
\M^\eps &= \frac{1}{\sqrt{\eps}} \M_3 + \M_1 , \label{eq:L,eps}\\
u_j^{\eps} &= \sum_{i \geq 0} {\left(}\sqrt{\eps} {\right)}^i u_{i,j} , \label{eq:u,eps}\end{aligned}$$ Inserting expansions into PDE and collecting terms of like-powers of $\sqrt{\del}$ we find that the lowest order equations of the regular perturbation expansion are $$\begin{aligned}
\O(1):&&
0
&= {\left(}-\d_t + \L^\eps {\right)}u_0^\eps , \label{eq:order1}\\
\O(\sqrt{\del}) :&&
0
&= {\left(}-\d_t + \L^\eps {\right)}u_1^\eps + \M^\eps u_0^\eps . \label{eq:orderhalf}\end{aligned}$$ Now, within equations and , we will perform a singular perturbation expansion with respect to the parameter $\eps$. We begin with , the $\O(1)$ equation.
### Singular Perturbation Analysis of Equation {#singular-perturbation-analysis-of-equation .unnumbered}
We insert expansions and into and collect terms of like-powers of $\sqrt{\eps}$. The resulting order $\O{\left(}1/ \eps {\right)}$ and $ \O{\left(}1/ \sqrt{ \eps} {\right)}$ equations are $$\begin{aligned}
\O{\left(}1/ \eps {\right)}:& &
0
&= \L_0 u_{0,0} , \label{eq:eps,-2} \\
\O{\left(}1/ \sqrt{ \eps} {\right)}:& &
0
&= \L_0 u_{1,0} + \L_1 u_{0,0} . \label{eq:eps,-1}\end{aligned}$$ We note that all terms in $\L_0$ and $\L_1$ take derivatives with respect to $y$. Therefore, if $ u_{0,0}$ and $ u_{1,0}$ are independent of $y$, equations and will be satisfied. Thus, we choose $u_{0,0}=u_{0,0}(t,x,z)$ and $u_{1,0}=u_{1,0}(t,x,z)$. Continuing the asymptotic analysis, the order $\O{\left(}1 {\right)}$ and $ \O{\left(}\sqrt{ \eps} {\right)}$ equations are $$\begin{aligned}
\O( 1 ) :& &
0
&= \L_0 u_{2,0} + {\left(}-\d_t + \L_2 {\right)}u_{0,0} , \label{eq:PoissonA} \\
\O( \sqrt{ \eps} ) :& &
0
&= \L_0 u_{3,0} + \L_1 u_{2,0} + {\left(}-\d_t + \L_2 {\right)}u_{1,0} . \label{eq:PoissonF}\end{aligned}$$ where we have used $ \L_1u_{1,0}=0$ in . Equations and are Poisson equations of the form $$\begin{aligned}
0
&= \L_0 u + \chi . \label{eq:PoissonGeneric}\end{aligned}$$ Recall that $\L_0$ is a self-adjoint operator acting on $L^2(\R,\pi)$. By the Fredholm alternative [^4], in order for equations of the form to admit solutions $u \in \text{dom}(\L_0)=L^2(\R,\pi)$, the following *centering condition* condition must be satisfied $$\begin{aligned}
{\left\langle}\chi {\right\rangle}&:= \int \chi(y) \, \pi(y) \, dy = 0 , \label{eq:center}\end{aligned}$$ where we have introduced the notation ${\left\langle}\cdot {\right\rangle}$ to indicate averaging over the invariant distribution $\Pi$. In equations and centering condition corresponds to $$\begin{aligned}
\O(1) :& &
0
&= {\left(}-\d_t +{\left\langle}\L_2 {\right\rangle}{\right)}u_{0,0}, \label{eq:CenterA} \\
\O( \sqrt{ \eps} ) :& &
0
&= {\left\langle}\L_1 u_{2,0}{\right\rangle}+ {\left(}-\d_t +{\left\langle}\L_2 {\right\rangle}{\right)}u_{1,0}. \label{eq:CenterB}\end{aligned}$$ The operator ${\left\langle}\L_2 {\right\rangle}$ is given by $$\begin{aligned}
{\left\langle}\L_2 {\right\rangle}&= \frac{1}{2} \sigb^2\, a^2(x) \d^2_{xx} + \Big( b(x) - \fOmb\,\, a(x) \Big) \d_x - k(x), &
x &\in (e_1,e_2) , \label{eq:<L00>}\end{aligned}$$ where we have defined $$\begin{aligned}
\sigb^2(z) &:= {\left\langle}f^2(\cdot,z) {\right\rangle}, &
\fOmb(z) &:= {\left\langle}f(\cdot,z) \Om(\cdot,z) {\right\rangle}, \label{eq:fOmbar}\end{aligned}$$ We assume $\sigb^2(z) < \infty $ and $\fOmb(z) < \infty$. Given appropriate BCs at $e_1$ and $e_2$, one can find a unique solution $u_{0,0}$ to PDE . However, in order to make use of we need an expression for ${\left\langle}\L_1 u_{2,0} {\right\rangle}$. To this end, we note from that $$\begin{aligned}
\L_0 u_{2,0}
&= - {\left(}-\d_t + \L_2 {\right)}u_{0,0} \\
&= - {\left(}-\d_t + \L_2 {\right)}u_{0,0} + {\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{0,0} \\
&= - {\left(}\frac{1}{2} a^2 \Big( f^2 - \sigb^2 \Big) \d^2_{xx} - a \Big( f\Om - \fOmb \Big) \d_x {\right)}u_{0,0} .
\label{eq:L-20psi20}\end{aligned}$$ Now, we introduce $\phi(y,z)$ and $\eta(y,z) $ as the solutions to the following Poisson equations $$\begin{aligned}
\L_0 \, \phi &= f^2 - \sigb^2 , &
\L_0 \, \eta &= f \Om - \fOmb . \label{eq:eta} \end{aligned}$$ Using , we can express $u_{2,0}$ as $$\begin{aligned}
u_{2,0} &= - {\left(}\frac{1}{2} a^2 \, \phi \, \d^2_{xx} - a \, \eta \, \d_x {\right)}u_{0,0} + C. \label{eq:psi20}\end{aligned}$$ Note that $C$ is a constant that is independent of $y$. Now, inserting and into ${\left\langle}\L_0 u_{2,0}{\right\rangle}$ we find $$\begin{aligned}
{\left\langle}\L_1 u_{2,0} {\right\rangle}&= - {\left\langle}{\left(}\beta \Big( \rho_{xy}\, a\, f \,\d_x - \Lam \Big) \d_y {\right)}{\left(}\frac{1}{2} \,a^2\, \phi\, \d^2_{xx} - a \eta\, \d_x {\right)}u_{0,0} {\right\rangle}= - \A \, u_{0,0} . \label{eq:<Lpsi>=A}\end{aligned}$$ The operator $\A$ is given by $$\begin{aligned}
\A
&= - \Vc_3 \, a(x) \d_x a^2(x) \d^2_{xx} - \Vc_2 \, a^2(x) \d^2_{xx} - \Uc_2 \, a(x) \d_x a(x) \d_x - \Uc_1 \, a(x) \d_x, \label{eq:A10}\end{aligned}$$ where we have defined four *group parameters* [^5] $$\begin{aligned}
\Vc_3
&= -\frac{\rho_{xy}}{2} \Big\langle \beta f \d_y \phi \Big\rangle, &
\Vc_2
&= \frac{1}{2} \Big\langle \beta \Lam \d_y \phi \Big\rangle , &
\Uc_2
&= \rho_{xy} \Big\langle \beta f \d_y \eta \Big\rangle , &
\Uc_1
&= - \Big\langle \beta \Lam \d_y \eta \Big\rangle .\end{aligned}$$ Inserting into we find $$\begin{aligned}
\A \, u_{0,0}
&= {\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{1,0} . \label{eq:eigentemp}\end{aligned}$$ Given an expression for $u_{0,0}$ and appropriate BCs, one can use PDE to find an expression for $u_{1,0}$. This is as far as we will take the analysis of equation . We now return to the $\O(\sqrt{\del})$ equation .
### Singular Perturbation Analysis of Equation {#singular-perturbation-analysis-of-equation-1 .unnumbered}
The singular perturbation analysis of proceeds by inserting expansions and into and collecting terms of like-powers of $\sqrt{\eps}$. The resulting order $\O( \sqrt{\del}/ \eps )$ and $ \O( \sqrt{\del}/ \sqrt{ \eps} )$ equations are $$\begin{aligned}
\O( \sqrt{\del}/ \eps ) :& &
0
&= \L_0 u_{0,1} , \label{eq:order-21} \\
\O( \sqrt{\del}/ \sqrt{ \eps} ) :& &
0
&= \L_0 u_{1,1} + \L_1 u_{0,1} , \label{eq:order-11}\end{aligned}$$ where we have used $\M_3 u_{0,0} = 0$. We note that if $ u_{0,1}$ and $ u_{1,1}$ are independent of $y$, equations and will automatically be satisfied. Thus, we choose $u_{0,1}=u_{0,1}(x,z)$ and $u_{1,1}=u_{1,1}(x,z)$. Continuing the asymptotic analysis, the order $\O( \sqrt{\del} )$ equation is $$\begin{aligned}
\O( \sqrt{\del} ) :& &
0
&= \L_0 u_{2,1} + {\left(}-\d_t+ \L_2 {\right)}u_{0,1} + \M_1 u_{0,0} , \label{eq:PoissonC}\end{aligned}$$ where we have used $\L_1 u_{1,1} = 0$ and $ \M_3 u_{1,0}=0$. We note that equation is a Poisson equation for $u_{2,1}$ of form . By the Fredholm alternative, in order for to admit a solution $u_{2,1} \in L^2(\R,\pi)$ centering condition must be satisfied. In centering condition corresponds to $$\begin{aligned}
0
&= {\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{0,1} + {\left\langle}\M_1 {\right\rangle}u_{0,0} . \label{eq:EigenTemp2}\end{aligned}$$ Note that $u_{0,0}(t,x,z)$ depends on $z$ only through $\sigb(z)$ and $\fOmb(z)$. Thus, in ${\left\langle}\M_1 {\right\rangle}$ can be written $$\begin{aligned}
{\left\langle}\M_1 {\right\rangle}&= - \B\d_z , \label{eq:<L01>} \\
\B
&= - \Vc_1 a(x) \d_x - \Vc_0 , &
\Vc_1
&:= g \rho_{xz}{\left\langle}f{\right\rangle}, &
\Vc_0
&:= - g {\left\langle}\Gam {\right\rangle}, \label{eq:B} \\
\d_z
&= \sigb' \d_{\sigb} + \fOmb' \d_{\fOmb} , &
\sigb'
&:= \d_z \sigb , &
\fOmb'
&:= \d_z \fOmb . \label{eq:dz} \end{aligned}$$ Note that we have introduced four more group parameters: $\Vc_1$, $\Vc_0$, $\sigb'$ and $\fOmb'$. This is as far as we will take the asymptotic analysis of equation . For convenience, we review the most important results of this section.
### Main Results of the Asymptotic Analysis {#main-results-of-the-asymptotic-analysis .unnumbered}
$$\begin{aligned}
\O ( 1 ):&&
{\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{0,0}
&= 0, &
u_{0,0}(0,x,z)
&= H(x) , \label{eq:EigenA} \\
\O ( \sqrt{ \eps } ):&&
{\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{1,0}
&= \A u_{0,0} , &
u_{1,0}(0,x,z)
&= 0 \label{eq:EigenB} \\
\O ( \sqrt{ \del } ) :&&
{\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{0,1}
&= \B\d_z u_{0,0} , &
u_{0,1}(0,x,z)
&= 0 . \label{eq:EigenC} \end{aligned}$$
The operators ${\left\langle}\L_2{\right\rangle}$, $\A$, $\B$ and $\d_z$ are defined in , , and respectively. Note that we have imposed BCs at $t=0$.
Explicit Solutions for $u_{0,0}$, $u_{1,0}$ and $u_{0,1}$ {#sec:u}
---------------------------------------------------------
In this section we shall explicitly solve equations , and in terms of the eigenfunctions $\{\psi_n \}$ and eigenvalues $\{\lam_n\}$ of the operator ${\left\langle}\L_2{\right\rangle}$. To begin, we note that ${\left\langle}\L_2{\right\rangle}$, given by , has the form of an infinitesimal generator of a one-dimensional diffusion with volatility $\sigb \, a(x)$, drift ${\left(}b(x) - \fOmb\,\, a(x) {\right)}$ and killing rate $k(x)$. The $\text{dom}({\left\langle}\L_2{\right\rangle})$ includes BCs, which must be imposed at the endpoints $e_1$ and $e_2$. Appendix \[sec:BCs\] describes the appropriate BCs to impose for a general one-dimensional diffusion with a generator of the form .
Throughout this section we assume ${\left\langle}\L_2{\right\rangle}$ has a purely discrete spectrum. We fix a Hilbert space $\H = L^2(I,\m)$ where $\m$ is the speed density corresponding to ${\left\langle}\L_2 {\right\rangle}$. The operator ${\left\langle}\L_2{\right\rangle}$ is self-adjoint in $\H$ and its domain is a dense subset of $\H$. Thus, the eigenfunctions $\{ \psi_n \}$ of ${\left\langle}\L_2{\right\rangle}$ form an orthonormal basis in $\H$. It is not necessarily true that either $\A : \H \to \H$, $\B : \H \to \H$ or $\d_z : \H \to \H$. As such, we define $$\begin{aligned}
\text{dom}(\A) &:= \left\{ \psi \in \H : \A \psi \in \H \right\} , &
\text{dom}(\B) &:= \left\{ \psi \in \H : \B \psi \in \H \right\} , \\
\text{dom}(\d_z) &:= \left\{ \psi \in \H : \d_z \psi \in \H \right\} .
\label{eq:domain,A}\end{aligned}$$
\[thm:u00\] Assume that we can solve the following eigenvalue equation $$\begin{aligned}
- {\left\langle}\L_2{\right\rangle}\psi_n
&= \lam_n \psi_n , &
\psi_n
&\in \text{\emph{dom}}{\left(}{\left\langle}\L_2{\right\rangle}{\right)}, \label{eq:Eigen00}\end{aligned}$$ and assume $H \in \H$. Then the solution $u_{0,0}$ to is given by $$\begin{aligned}
u_{0,0}
&= \sum_n c_n \psi_n T_n , &
c_n
&= {\left(}\psi_n , H {\right)}, &
T_n
&= e^{ -t \, \lam_n } .\end{aligned}$$
One can easily verify that $u_{0,0}$ satisfies PDE assuming holds. To see that the BC $u_{0,0}(0,x,z)=H(x)$ is satisfied, notice that $T_n(0,z)=1$ and apply equation to the payoff function $H$ $$\begin{aligned}
\text{Id} \, H
&= \sum_n {\left(}\psi_n , H {\right)}\psi_n = \sum_n c_n \psi_n .\end{aligned}$$
\[thm:u10\] Let $c_n$, $ \psi_n$ and $T_n$ be as described in Theorem \[thm:u00\] and define $$\begin{aligned}
\A_{k,n}
&:= {\left(}\psi_k , \A \psi_n {\right)}, &
U_{k,n}
&:= \frac{T_k-T_n}{\lam_k-\lam_n} .\end{aligned}$$ Then the solution $u_{1,0}$ to equation is $$\begin{aligned}
u_{1,0}
&= \sum_n \sum_{k \neq n} c_n \, \A_{k,n} \psi_k U_{k,n}
- \sum_n c_n \, \A_{n,n} \psi_n \, t \, T_n . \label{eq:u10,long}\end{aligned}$$
See appendix \[sec:u10proof\].
Note that $u_{1,0}$ is linear in the group parameters $(\Vc_3, \Vc_2, \Uc_2, \Uc_1)$.
\[thm:u01\] Let $c_n$, $ \psi_n$ and $T_n$ be as described in Theorem \[thm:u00\], let $U_{k,n}$ be as described in Theorem \[thm:u10\] and define $$\begin{aligned}
\Bt_{k,n}
&:= {\left(}\psi_k , \B\d_z \psi_n {\right)}, &
\B_{k,n}
&:= {\left(}\psi_k , \B \psi_n {\right)}, &
V_{k,n}
&:= \frac{T_k-T_n}{{\left(}\lam_k - \lam_n {\right)}^2}
+ \frac{t \, T_n}{\lam_k - \lam_n} .\end{aligned}$$ Then the solution $u_{0,1}$ to equation is $$\begin{aligned}
u_{0,1}
&= \sum_n \sum_{k \neq n} c_n \Bt_{k,n} \psi_k U_{k,n} -
\sum_n c_n \, \Bt_{n,n} \psi_n \, t \, T_n
\\ &\qquad +
\sum_n \sum_{k \neq n} (\d_z c_n )\B_{k,n} \psi_k U_{k,n} -
\sum_n ( \d_z c_n ) \B_{n,n} \psi_n \, t \, T_n
\\ &\qquad +
\sum_n \sum_{k \neq n} c_n \B_{k,n} \psi_k {\left(}\d_z \lam_n {\right)}V_{k,n} +
\sum_n c_n \B_{n,n} \psi_n {\left(}\d_z \lam_n {\right)}\tfrac{1}{2} t^2 \, T_n .
\label{eq:u01,long}\end{aligned}$$
See appendix \[sec:u01proof\].
Note that $u_{0,1}$ is linear in $(\Vc_1 \sigb', \Vc_1 \fOmb', \Vc_0 \sigb', \Vc_0 \fOmb')$.
### Accuracy of the Pricing Approximation {#accuracy-of-the-pricing-approximation .unnumbered}
We have now derived an approximation $u^{\eps,\del} \approx u_{0,0} + \sqrt{\eps} \, u_{1,0} + \sqrt{\del} \, u_{0,1}$ for the price of a derivative-asset. However, this derivation relied on formal singular and regular perturbation arguments. In what follows, we establish the accuracy of our approximation. For our accuracy result, in addition to the assumptions listed in section \[sec:assumptions\], we shall need one additional assumption
- The payoff function $H(x)$ and all of its derivatives are smooth and bounded.
Obviously, many common derivatives – e.g., call and put options – do not fit this assumption. To prove the accuracy of our pricing approximation for calls and puts would require regularizing the option payoff as is done in @fouque2003proof. The regularization procedure is beyond the scope of this paper. As such, we limit our analysis to options with smooth and bounded payoffs. Our accuracy result is as follows:
\[thm:accuracy\] For fixed $(t,x,y,z)$, there exists a constant $C$ such that for any $\eps \leq 1$, $\del \leq 1$ we have $$\begin{aligned}
\left| u^{\eps,\del} - {\left(}u_{0,0} + \sqrt{\eps} \, u_{0,0} + \sqrt{\del} \, u_{0,1} {\right)}\right| \leq C {\left(}\eps + \del {\right)}.
\label{eq:accuracy}\end{aligned}$$
See appendix \[sec:accuracy\].
Theorem \[thm:accuracy\] gives us information about how our pricing approximation behaves as $\eps \to 0$ and $\del \to 0$. In practice, both $\eps$ and $\del$ are small, but fixed (they do not go to zero). Without knowing what the constant $C$ is in theorem \[thm:accuracy\], it is difficult to gauge exactly how good our pricing approximation is. As such, in the examples provided in section \[sec:examples\], we will compare the approximate prices of derivative-assets (calculated using Theorems \[thm:u00\], \[thm:u10\] and \[thm:u01\]) to their exact prices (calculated via Monte Carlo simulation).
Examples {#sec:examples}
========
In this section we compute the approximate price of three derivative-assets: a double-barrier call option, a bond in a short-rate model, and a European call on a defaultable stock.
Double-Barrier Call Option with Multiscale Stochastic Volatility {#sec:BS}
----------------------------------------------------------------
In our first example, we let $X$ represent the value of a non-dividend paying asset (e.g., a stock, index, etc.). Often, $X$ is modeled as a GBM with constant volatility (e.g., Black-Scholes). Here, we model $X$ as a GBM with multiscale stochastic volatility. Specifically, the $\Pt$ dynamics of $X$ are given by $$\begin{aligned}
dX_t &= r X_t \, dt + f{\left(}Y_t, Z_t {\right)}X_t \, d\Wt^x_t , &
h(X_t) &= 0 , \label{eq:X,BS}\end{aligned}$$ where $r$ is the risk-free rate of interest and $Y$ and $Z$ are fast- and slow-varying factors of volatility, as described in . Note that, as it should be, the discounted price of the asset ${\left(}e^{-rt}X_t {\right)}$ is a martingale under $\Pt$. We will calculate the approximate price of a double-barrier call option written on $X$.
To start, we use equations and to write the operator ${\left\langle}\L_2{\right\rangle}$ and its associated speed density $\m(x)$ $$\begin{aligned}
{\left\langle}\L_2 {\right\rangle}&= \frac{1}{2} \sigb^2 x^2 \d^2_{xx} + r\, x \, \d_x - r , &
\m(x) &= \frac{2}{\sigb^2 x^2 } \exp {\left(}\frac{2 r }{ \sigb^2} \log x {\right)}.\label{eq:LBS}\end{aligned}$$ For a double-barrier call option with knock-out barriers at $L$ and $R$, the option payoff is $$\begin{aligned}
H(X_t) \, \I_{\{ \tau>t \}} &= {\left(}X_t - K {\right)}^{+} \, \I_{\{ \tau_I > t \}}, &
I &= (L,R) , &
0 < L < K < R , \label{eq:DBpayoff}\end{aligned}$$ To calculate the value of this option we must first solve eigenvalue equation with ${\left\langle}\L_2{\right\rangle}$ given by and with BCs $$\begin{aligned}
\lim_{x \searrow L} \psi_n(x) &= 0 , &
\lim_{x \nearrow R} \psi_n(x) &= 0 . \label{eq:BS,BC}\end{aligned}$$ Note that we have imposed the regular killing BC at the endpoints $L$ and $R$. The solution to with the above BCs can be found on page 262 of @linetskybook $$\begin{aligned}
\psi_n(x)
&= \frac{\sigb \sqrt{x}}{\sqrt{\log ( R / L )}} \exp {\left(}\frac{-r}{\sigb^2} \log x{\right)}\sin {\left(}\frac{n \pi \log ( x / L )}{\log ( R / L ) } {\right)}, &
n
&= 1, 2, 3, \cdots , \\
\lam_n
&= \frac{1}{2} {\left(}\frac{n \pi \sigb}{\log (R / L )} {\right)}^2 + {\left(}\frac{\nu^2}{2} + r{\right)}, &
\nu
&= \frac{r}{\sigb} - \frac{\sigb}{2} . \label{eq:QBS}\end{aligned}$$ Next, we use expressions and to write expressions for the operators $\A$ and $\B$ $$\begin{aligned}
\A &= - \Vc_3 x \, \d_x x^2 \d^2_{xx} - \Vc_2 x^2 \d^2_{xx} , &
\B &= - \Vc_1 x \, \d_x - \Vc_0 . \label{eq:A,BS}\end{aligned}$$ Using it is now straightforward to calculate inner products $\A_{k,n}$, $\B_{k,n}$ and $\Bt_{k,n}$. For $k \neq n$ we find $$\begin{aligned}
\A_{k,n}
&= - \Vc_3 {\left(}\frac{{\left(}-1+(-1)^{k+n}{\right)}k n {\left(}4 n^2 \pi^2 \sigb^4+{\left(}-12 r^2+4 r \sigb^2+\sigb^4{\right)}\log^2 (R/L) {\right)}}
{2 {\left(}k^2-n^2{\right)}\sigb^4 \log^3 (R/L)} {\right)}\\ &\qquad
- \Vc_2 {\left(}\frac{4 {\left(}-1+(-1)^{k+n}{\right)}k n r}{{\left(}k^2-n^2{\right)}\sigb^2 \log (R/L)} {\right)}, \\
\B_{k,n}
&= \Vc_1 \frac{2 {\left(}-1+(-1)^{k+n}{\right)}k n }{(k-n) (k+n) \log(R/L)}, \\
\Bt_{k,n}
&= - \Vc_1 \sigb' {\left(}\Upsilon_{k,n} {\right)}- \Vc_0 \sigb'
{\left(}\frac{8 {\left(}-1+(-1)^{k+n}{\right)}k n r \log(R/L)}{{\left(}k^2-n^2{\right)}^2 \pi^2 \sigb^3} {\right)}, \\
\Upsilon_{k,n}
&:= \frac{4 k n r {\left(}\log(L)-(-1)^{k+n} \log(R){\right)}}{{\left(}k^2-n^2{\right)}\sigb^3 \log (R/L)} \\ &\qquad
- \frac{2 {\left(}-1+(-1)^{k+n}{\right)}k n {\left(}(k-n) (k+n) \pi^2 \sigb^4-2 r {\left(}-2 r+\sigb^2{\right)}\log^2 (R/L){\right)}}
{{\left(}k^2-n^2{\right)}^2 \pi^2 \sigb^5 \log (R/L)} ,\end{aligned}$$ and for $k = n$ we find $$\begin{aligned}
\A_{n,n}
&= - \Vc_3 {\left(}\frac{1}{\sigb^3}{\left(}\frac{3 n^2 \pi ^2 \nu }{\log^2(R/L)}-\nu^3 {\right)}- \frac{1}{\sigb^2} {\left(}\nu^2-\frac{n^2 \pi ^2}{\log^2(R/L)} {\right)}{\right)}- \Vc_2 {\left(}\frac{1}{\sigb^2} {\left(}\nu^2 - \frac{n^2 \pi^2}{\log^2(R/L)} {\right)}+ \frac{\nu}{\sigb} {\right)},\\
\B_{n,n}
&= \Vc_1 {\left(}\frac{2 r - \sigb^2}{2 \sigb^2}{\right)}- \Vc_0 , \\
\Bt_{n,n}
&= - \Vc_1 \sigb' {\left(}\frac{1}{2 \sigb} - \frac{r \nu {\left(}\log^2(R)-\log^2(L){\right)}}{ \sigb^4 \log{\left(}R/L{\right)}} {\right)}- \Vc_0 \sigb' {\left(}\frac{1}{\sigb }+\frac{r {\left(}\log^2(R)-\log^2(L){\right)}}{\sigb^3 \log{\left(}R/L{\right)}} {\right)}.\end{aligned}$$ The calculation of $c_n$ can be found on page 262 of @linetskybook $$\begin{aligned}
c_n &= {\left(}\psi_n(\cdot) , ( \cdot - K )^{+} {\right)}= \frac{L^{\nu/\sigb}}{\log {\left(}R / L{\right)}} \Big( L \, \Phi_n(\nu+\sigb) - K \, \Phi_n(\nu) \Big) , \\
\Phi_n(z) &:= \frac{2}{\om_n^2 + z^2} \Big( \exp {\left(}\k z {\right)}\big( \om_n \cos {\left(}\om_n \k{\right)}- z \sin {\left(}\om_n \k {\right)}\big) - \exp {\left(}\u z {\right)}(-1)^n \om_n \Big) , \\
\om_n &:= \frac{n \pi}{\u} , \qquad
\k := \frac{1}{\sigb} \log {\left(}\frac{K}{L} {\right)}, \qquad
\u := \frac{1}{\sigb} \log {\left(}\frac{R}{L} {\right)}.\end{aligned}$$ Approximate option prices can now be computed using Theorems \[thm:u00\], \[thm:u10\] and \[thm:u01\].
On the left side of figure \[fig:DBslow\] we plot the approximate price $u_{0,0}+\sqrt{\eps}\,u_{1,0}$ of a double-barrier call option for a specific model that has only a fast-varying factor of volatility. We suppose the dynamics of $Y$ and the volatility function $f$ are given by $$\begin{aligned}
dY_t
&= {\left(}- \frac{1}{\eps} \, Y_t - \frac{1}{\sqrt{\eps}}\beta \, \text{Erf}(Y_t) {\right)}dt + \beta \, d\Wt_t^y , &
f(Y_t)
&= \frac{\sigma \, \exp {\left(}Y_t{\right)}}{ \exp{\left(}- \beta^2 / 2{\right)}} , \label{eq:Yexample} \\
\text{Erf}(y)
&:= \frac{2}{\sqrt{\pi}} \int_0^y e^{-t^2} dt . \end{aligned}$$ From comparison we also plot the full price $u^\eps$ (calculated by Monte Carlo simulation) and $u_{0,0}$, which corresponds to the Black-Scholes price with volatility $\sigb$. On the right side of figure \[fig:DBslow\] we plot the approximate price $u_{0,0}+\sqrt{\del}\,u_{0,1}$ of a double-barrier call option for a specific model that contains only a slow-varying factor of volatility. We suppose the dynamics of $Z$ and the volatility function $f$ are given by $$\begin{aligned}
dZ_t
&= {\left(}- \del \, Z_t - \sqrt{\del} \, g \, \text{Erf}(Z_t) {\right)}dt + g \, d\Wt_t^z , &
f(Z_t)
&= \frac{\sigma \, \exp {\left(}Z_t {\right)}}{\exp {\left(}z {\right)}} . \label{eq:Zexample}\end{aligned}$$ For comparison, we also plot the full price $u^\del$ (calculated by Monte Carlo simulation) and the Black-Scholes price $u_{0,0}$. As expected, as $\eps$ and $\del$ go to zero, the approximate price converges to the full price, which conveges to the Black-Scholes price.
Vasicek Short-Rate with Multiscale Stochastic Volatility
--------------------------------------------------------
In our second example, we let $X$ represent the short-rate of interest. One of the most widely known short-rate models is that of @vasicek, in which $X$ is modeled as an OU process. Here, we model $X$ as an OU with multiscale stochastic volatility. Specifically, the $\Pt$ dynamics of $X$ are given by $$\begin{aligned}
dX_t &= \Big( \kappa {\left(}\theta - X_t {\right)}- f(Y_t,Z_t) \Om(Y_t,Z_t) \Big) dt + f{\left(}Y_t, Z_t {\right)}\, d\Wt^x_t , &
r(X_t) &= X_t , &
h(X_t) &= 0 ,\end{aligned}$$ where $Y$ and $Z$ are fast- and slow-varying factors of volatility, as described in . We will calculate the approximate price of zero-coupon bond in this setting. [^6]
To start, we use equations and to write the operator ${\left\langle}\L_2{\right\rangle}$ and its associated speed density $\m(x)$ $$\begin{aligned}
{\left\langle}\L_2 {\right\rangle}&= \frac{1}{2} \sigb^2 \d^2_{xx} + \kappa {\left(}\thb - x {\right)}\d_x - x , &
\m(x) &= \frac{2}{\sigb^2} \exp {\left(}\frac{-\kappa}{\sigb^2} {\left(}\thb -x {\right)}^2 {\right)}, &
\thb &= \theta - \tfrac{1}{\kappa}\fOmb . \label{eq:Lvasicek}\end{aligned}$$ For a zero-coupon bond, the payoff at maturity is $$\begin{aligned}
H(X_t) \, \I_{\left\{ \tau>t \right\}} &= 1 . \label{eq:payoffBond}\end{aligned}$$
In order to price a bond with payoff , we must solve eigenvalue equation on the interval $I=(-\infty,\infty)$ with ${\left\langle}\L_2 {\right\rangle}$ given by . As both $-\infty$ and $\infty$ are natural boundaries, no BCs need to be specified. The solution to this eigenvalue problem can be found in equation (4.6) of @linetsky2004blackinterest $$\begin{aligned}
\psi_n &= \Ncal_n \exp {\left(}- A \, \xi - \frac{1}{2} A^2 {\right)}H_n{\left(}\xi + A {\right)}, &
\Ncal_n &= {\left(}\sqrt{\frac{\kappa}{\pi}} \frac{\sigb}{2^{n+1}n!}{\right)}^{1/2} , \\
A &= \frac{\sigb}{\kappa^{3/2}} , &
\xi &= \frac{\sqrt{\kappa}}{\sigb} {\left(}x - \thb {\right)}, \\
\lam_n &= \lam_n = \thb - \frac{\sigb^2}{2 \kappa^2} + \kappa \, n , &
n &= 0, 1, 2, \cdots .\end{aligned}$$ Here, $\{H_n\}$ are the (physicists’) Hermite polynomials. Next, we use and to write expressions for the operators $\A$ and $\B$ $$\begin{aligned}
\A
&= - \Vc_3 \, \d^3_{xxx} - {\left(}\Vc_2 + \Uc_2 {\right)}\d^2_{xx} - \Uc_1 \, \d_x, &
\B
&= - \Vc_1 \, \d_x - \Vc_0 .\end{aligned}$$ It is now straightforward to calculate inner products $\A_{k,n}$, $\B_{k,n}$ and $\Bt_{k,n}$. Using the recursion relations $$\begin{aligned}
\d_x H_n &= 2 \,n H_{n-1} , &
2 \,x \, H_n &= H_{n+1} + \d_x H_n ,\end{aligned}$$ we find $$\begin{aligned}
\A_{k,n}
&= - \Vc_3 \left\{
\sum_{m=0}^{3 \wedge n} {\left(}\begin{array}{c} 3 \\ m \end{array} {\right)}{\left(}\frac{-1}{\kappa} {\right)}^{3-m} {\left(}\frac{2 \sqrt{\kappa}}{\sigb} {\right)}^m
\frac{n! \, \Ncal_n}{(n-m)! \, \Ncal_{n-m}} \del_{k,n-m} \right\} \\ &\qquad
- {\left(}\Vc_2 + \Uc_2 {\right)}\left\{
\sum_{m=0}^{2 \wedge n} {\left(}\begin{array}{c} 2 \\ m \end{array} {\right)}{\left(}\frac{-1}{\kappa} {\right)}^{2-m}
{\left(}\frac{2 \sqrt{\kappa}}{\sigb} {\right)}^m
\frac{n! \, \Ncal_n}{(n-m)! \, \Ncal_{n-m}} \del_{k,n-m} \right\} \\ &\qquad
- \Uc_1 \left\{
{\left(}\frac{-1}{\kappa} {\right)}\delta_{k,n} + {\left(}\frac{2\sqrt{\kappa}}{\sigb} {\right)}\frac{ n! \, \Ncal_n}{(n-1)! \Ncal_{n-1}} \delta_{k,n-1} \right\} , \\
\B_{k,n}
&= - \Vc_1 \left\{
{\left(}\frac{-1}{\kappa} {\right)}\delta_{k,n} + {\left(}\frac{2\sqrt{\kappa}}{\sigb} {\right)}\frac{ n! \, \Ncal_n}{(n-1)! \Ncal_{n-1}} \delta_{k,n-1} \right\}
- \Vc_0 \, \delta_{k,n} , \\
\Bt_{k,n}
&= - \Vc_1 \sigb' \left\{
{\left[}{\left(}\frac{-1}{\kappa} {\right)}{\left(}\frac{1}{2 \sigb} - \frac{\sigb}{\kappa^3} - \frac{n}{\sigb} {\right)}{\right]}\del_{k,n} \right. \\ &\left. \qquad \qquad
+ {\left[}{\left(}\frac{-1}{\kappa} {\right)}{\left(}\frac{4}{\kappa^{3/2}} {\right)}+ {\left(}\frac{2 \sqrt{\kappa}}{\sigb} {\right)}{\left(}\frac{1}{2 \sigb} - \frac{\sigb}{\kappa^3} - \frac{n}{\sigb} {\right)}{\right]}\frac{n! \Ncal_n}{(n-1)!\Ncal_{n-1}} \del_{k,n-1} \right. \\ &\left. \qquad \qquad
+ {\left[}{\left(}\frac{-1}{\kappa} {\right)}{\left(}\frac{-2}{\sigb} {\right)}+ {\left(}\frac{2 \sqrt{\kappa}}{\sigb} {\right)}{\left(}\frac{4}{\kappa^{3/2}} {\right)}{\right]}\frac{n!\Ncal_n}{(n-2)! \Ncal_{n-2}}\del_{k,n-2} \right. \\ &\left. \qquad \qquad
+ {\left[}{\left(}\frac{2 \sqrt{\kappa}}{\sigb} {\right)}{\left(}\frac{-2}{\sigb} {\right)}{\right]}\frac{n! \Ncal_n}{(n-3)!\Ncal_{n-3}}\del_{k,n-3} \right\} \\ &\qquad - \Vc_0 \sigb' \left\{
{\left(}\frac{1}{2 \sigb} - \frac{\sigb}{\kappa^3} - \frac{n}{\sigb} {\right)}\del_{k,n}
+ {\left(}\frac{4}{\kappa^{3/2}} {\right)}\frac{ n! \, \Ncal_{n} }{ (n-1)!\Ncal_{n-1} } \del_{k,n-1} \right.
\\&\left. \qquad \qquad
+ {\left(}\frac{-2}{\sigb} {\right)}\frac{ n ! \Ncal_n }{ (n-2)! \Ncal_{n-2} }\del_{k,n-2} \right\} \\ &\qquad
- \Vc_1 \fOmb' \left\{
{\left(}\frac{1}{\kappa^3} {\right)}\delta_{k,n} +
{\left(}\frac{-4}{\sigb \kappa^{3/2}} {\right)}\frac{n! \Ncal_n}{(n-1)!\Ncal_{n-1}} \delta_{k,n-1} +
{\left(}\frac{4}{\sigb^2} {\right)}\frac{n! \Ncal_n}{(n-2)!\Ncal_{n-2}} \delta_{k,n-2}
\right\} \\ &\qquad
- \Vc_0 \fOmb' \left\{
{\left(}\frac{-1}{\kappa^2} {\right)}\delta_{k,n} + {\left(}\frac{2}{\sigb \sqrt{\kappa}} {\right)}\frac{n! \Ncal_n}{(n-1)!\Ncal_{n-1}} \delta_{k,n-1} \right\} .\end{aligned}$$ The computation of $c_n$ be found on page 63 of in @linetsky2004blackinterest $$\begin{aligned}
c_n &= {\left(}\psi_n , 1 {\right)}= \frac{2}{\sigb} \sqrt{ \frac{\pi}{\kappa} } \Ncal_n A^n e^{-A^2 /4} .\end{aligned}$$ The approximate price of a bond can now be calculated using Theorems \[thm:u00\], \[thm:u10\] and \[thm:u01\].\
### Yield Curve {#yield-curve .unnumbered}
For a zero-coupon bond, it is often the yield curve, rather than the bond price itself, that is of fundamental importance. The yield $R^{\eps,\del}$ of a zero-coupon bond that pays one dollar at time $t$ is defined via the relation $$\begin{aligned}
u^{\eps,\del} &= \exp {\left(}-R^{\eps,\del} t {\right)}.\end{aligned}$$ We can obtain an approximation for the yield of a zero-coupon bond by expanding both the bond price $u^{\eps,\del}$ and yield $R^{\eps,\del}$ in powers of $\sqrt{\eps}$ and $\sqrt{\del}$ as follows $$\begin{aligned}
u_{0,0} + \sqrt{\eps} \, u_{1,0} + \sqrt{\del} \, u_{0,1} + \cdots
&= e^{ - {\left(}R_{0,0} + \sqrt{\eps} \, R_{1,0} + \sqrt{\del} \, R_{0,1} + \cdots {\right)}\, t } \\
&= e^{ -R_{0,0} t } + \sqrt{\eps} {\left(}- R_{1,0} \, t {\right)}e^{ -R_{0,0} t } + \sqrt{\del} {\left(}- R_{0,1} \, t {\right)}e^{ -R_{0,0} t } + \cdots .\end{aligned}$$ Matching terms of like-powers of $\sqrt{\eps}$ and $\sqrt{\del}$ we obtain $$\begin{aligned}
R^{\eps,\del} &\approx R_{0,0} + \sqrt{\eps} \, R_{1,0} + \sqrt{\del} \, R_{0,1}, \\
R_{0,0} &= - \tfrac{1}{t}\log {\left(}u_{0,0} {\right)}, \qquad
R_{1,0} = \frac{ - u_{1,0} }{ t \, u_{0,0} } , \qquad
R_{0,1} = \frac{ - u_{0,1} }{ t \, u_{0,0} } ,\end{aligned}$$
On the left side of figure \[fig:Vasicekslow\] we plot the approximate yield $R_{0,0}+\sqrt{\eps}\,R_{1,0}$ of a zero coupon bond for a specific model that has only a fast-varying factor of volatility. We suppose the dynamics of $Y$ and the volatility function $f$ are given by . For comparison, we also plot the full yield $R^\eps$ (calculated by Monte Carlo simulation) and the Vasicek yield $R_{0,0}$. On the right side of figure \[fig:Vasicekslow\] we plot the approximate yield $R_{0,0}+\sqrt{\del}\,R_{0,1}$ of a zero coupon bond for a specific model that has only a slow-varying factor of volatility. We suppose the dynamics of $Z$ and the volatility function $f$ are given by . For comparison, we also plot the full yield $R^\del$ (calculated by Monte Carlo simulation) and the Vasicek yield $R_{0,0}$. As expected, as $\eps$ and $\del$ go to zero, the approximate yield converges to the full yield, which converges to the Vasicek yield.
Jump to Default CEV with Multiscale Stochastic Volatility {#sec:JDCEV}
---------------------------------------------------------
In our final example, we consider a non-dividend-paying, defaultable asset $S_t = \I_{\{\tau>t\}}X_t$. As $S$ must be non-negative, we let the state space of $X$ be $(e_1,e_2) = (0, \infty)$. We base our multiscale diffusion on the jump to default constant elastic variance model (JDCEV) of @JDCEV. Specifically, the $\Pt$ dynamics of $X$ prior to default are given by $$\begin{aligned}
dX_t &= {\left(}\mu + c \, X_t^{2\eta} {\right)}X_t \, dt + {\left(}f(Y_t,Z_t) \, X_t^\eta {\right)}X_t \, d\Wt_t^x , &
h(X_t) &= \mu + c \, X_t^{2 \eta} .\end{aligned}$$ For computational convenience we have set the risk-free interest rate to zero: $r=0$. The constants $\mu$ and $c$ are assumed to be strictly positive. As always, $Y$ and $Z$ are fast- and slow-varying factors of volatility, as described in . Note that the volatility of $X$ has both a local component $X_t^\eta$ and a nonlocal multiscale component $f(Y_t,Z_t)$. We assume $\eta<0$ so that the local component of volatility $X_t^\eta$ *increases* as $X_t$ decreases, reflecting the fact that price and volatility are negatively correlated. The stochastic hazard rate $h(X_t)$ also increases as $X$ decreases, capturing the idea that the probability of default increases as $X$ tends to zero. Note that $S$ is a $\Pt$-martingale, as it should be. We will calculate the approximate price of a European put option written on $S$. The price of a European call option can be obtained through put-call parity.
To begin, we use and to write the operator ${\left\langle}\L_2{\right\rangle}$ and its associated speed density $\m(x)$ $$\begin{aligned}
{\left\langle}\L_2 {\right\rangle}&= \frac{1}{2} \sigb^2 x^{2 \eta + 2} \d^2_{xx} + {\left(}\mu + c \, x^{2 \eta} {\right)}x \, \d_x - {\left(}\mu + c \, x^{2 \eta} {\right)}, \label{eq:LJDCEV} \\
\m(x) &= \frac{2}{\sigb^2} x^{2c/\sigb^2-2-2\eta} \exp {\left(}A \, x^{-2 \eta} {\right)}, &
A &= \frac{\mu}{\sigb^2 | \eta |} \end{aligned}$$ For the diffusion associated with infinitesimal generator the endpoint $e_2 = \infty$ is a natural boundary. However, the classification of endpoint $e_1=0$ depends on the values of $\eta$ and $c/\sigb^2$. The classification is as follows $$\begin{aligned}
c/\sigb^2 &\geq 1/2 &&\text{and} &\eta &< 0 , &&\text{$e_1=0$ is natural,} \\
c/\sigb^2 &\in (0,1/2) &&\text{and} &\eta &\in [c/\sigb^2-1/2,0) , &&\text{$e_1=0$ is exit,} \\
c/\sigb^2 &\in (0,1/2) &&\text{and} &\eta &< c/\sigb^2-1/2 , &&\text{$e_1=0$ is regular.}\\end{aligned}$$ If the parameters ($c$, $\sigb$, $\eta$) are chosen such that $e_1=0$ is regular, then we specify $e_1=0$ as a killing boundary. To calculate the approximate price of a European put we must solve the eigenvalue equation on the interval $(0,\infty)$ with ${\left\langle}\L_2 {\right\rangle}$ given by and with the BC $$\begin{aligned}
\lim_{x \searrow 0} \psi_n(x) &= 0 , &
&\text{if} &
c/\sigb^2 &\in (0,1/2) .\end{aligned}$$ The solution is given in equation (8.11) of Theorem 8.2 in @carr $$\begin{aligned}
\psi_n
&= A^{\nu/2} \sqrt{\frac{(n-1)! \,\mu}{\Gam(\nu + n)}}\, x\, \exp{\left(}-A \, x^{-2\eta}{\right)}L_{n-1}^{(\nu)}{\left(}A \, x^{-2 \eta} {\right)}, &
n
&= 1, 2, 3, \cdots ,\label{eq:psi,JDCEV} \\
\lam_n
&= 2 \mu | \eta | ( n + \nu ) , &
\nu
&= \frac{1 + 2 \, (c/\sigb^2)}{2 |\eta|} ,\end{aligned}$$ where $\{L_{n}^{(\nu)}\}$ are the generalized Laguerre polynomials. Next, we use and to write expressions for the operators $\A$ and $\B$ $$\begin{aligned}
\A
&= - \Vc_3 \, x^{ \eta+1} \d_x x^{2 \eta + 2} \d^2_{xx} - \Vc_2 \, x^{2 \eta + 2} \d^2_{xx}, &
\B
&= - \Vc_1 \, x^{ \eta + 1} \d_x - \Vc_0 .\end{aligned}$$ Analytic expressions for $\A_{k,n}$, $\B_{k,n}$ and $\Bt_{k,n}$ are easily derived by making the change of variables $A \, x^{-2\eta} \to y$, using $\d_y L_n^{\nu}(y) = - L_{n-1}^{(\nu+1)}(y)$ and $$\begin{aligned}
&\int_0^\infty y^\gamma e^{-y} L_n^{(\alpha)}(y) L_m^{(\beta)}(y) \, dy \\
&\qquad = \frac{\Gamma(\alpha-\gamma+n)\Gamma(\beta+1+m)\Gamma(\gamma+1)}{\Gamma(\alpha-\gamma)\Gamma(\beta+1)\,n!\,m!} \,
{}_3 F_2 {\left(}\begin{array}{ccc} \gamma+1, & -m, & \gamma+1-\alpha \\ \beta+1, & \gamma+1-\alpha-n , & \end{array} ; 1 {\right)},\end{aligned}$$ where where ${}_p F_q$ is a generalized hypergeometric function (the above formula is given in equation (14) of @laguerre). As the formulas for $\A_{k,n}$, $\B_{k,n}$ and $\Bt_{k,n}$ are quite long, for the sake of brevity, we do not provide them here.
The payoff of a European put option with strike price $K>0$ can be decomposed as follows $$\begin{aligned}
(K - S_t)^+
&= (K-X_t)^{+} \, \I_{ \left\{ \tau > t \right\}} + K {\left(}1 - \I_{ \left\{ \tau > t \right\}} {\right)}. \label{eq:PutDecomp}\end{aligned}$$ The first term on the RHS of represents the payoff of a put given no default prior to time $t$. The second term represents the payoff of a put option given a default occurs prior to time $t$. Thus, the value of a put option with strike price $K$ – denoted $u^{\eps,\del}(t,x;K)$ – can be expressed as the sum of two parts $$\begin{aligned}
u^{\eps,\del}(t,x;K)
&= u_0^{\eps,\del}(t,x;K) + u_D^{\eps,\del}(t,x;K) , \label{eq:Put,temp}\end{aligned}$$ where $$\begin{aligned}
u_0^{\eps,\del}(t,x;K) &= \Et_{x,y,z} {\left[}(K - X_t)^+ \I_{ \left\{ \tau > t \right\} }{\right]}, \label{eq:u0def} \\
u_D^{\eps,\del}(t,x;K) &= K - K \, \Et_{x,y,z} {\left[}\I_{ \left\{ \tau > t \right\} } {\right]}\\
&= K - K \int_0^\infty \Et_{x,y,z} {\left[}\delta_{x'}(X_t) \, \I_{ \left\{ \tau > t \right\} } {\right]}dx' \\
&= K - K \, \int_0^\infty u_1^{\eps,\del}(t,x;x') \, dx' , \label{eq:u1integral} \\
u_1^{\eps,\del}(t,x;x') &= \Et_{x,y,z} {\left[}\delta_{x'}(X_t) \, \I_{ \left\{ \tau > t \right\} } {\right]}. \label{eq:u1def}\end{aligned}$$ Note, because $1 \notin L^2(\R^+,\m)$, we have used the fact that $1 = \int_0^\infty \delta_{x'}(X_t) \, dx'$ on the set $\left\{ \tau > t \right\}$. This substitution comes at a cost; the integral in must be computed numerically. However, numerical evaluation of is not computationally intensive and does not pose any major difficulties.
Since the payoff functions $H_0(x) = (K-x)^+$ and $H_1(x) = \del_{x'}(x)$ belong to $L^2(\R^+,\m)$, we may calculate $$\begin{aligned}
c_{0,n} &= {\left(}\psi_n(\cdot) , (k-\cdot)^+ {\right)}, &
c_{1,n} &= {\left(}\psi_n , \del_{x'} {\right)}.\end{aligned}$$ The expression for $c_{0,n}$ can be found in equation (8.15) of Theorem 8.4 in @carr. The expression for $c_{1,n}$ is computed trivially. We have $$\begin{aligned}
c_{0,n}
&= \frac{ A^{\nu /2+1} K^{2c/\sigb^2+1-2\eta} \sqrt{ \Gam (\nu+n) } }{ \Gam(\nu +1) \sqrt{ \mu (n-1)! } } \, \times \\
& \qquad {\left[}\frac{|\eta|}{(c/\sigb^2)+|\eta|} \,{}_2 F_2 {\left(}\begin{array}{cc} 1-n, & \tfrac{c/\sigb^2}{|\eta|}+1 \\ \nu+1, & \tfrac{c/\sigb^2}{|\eta|}+2 \end{array} ; A \, K^{-2\eta }{\right)}- \frac{ \Gam(\nu+1)(n-1)! }{ \Gam(\nu + n + 1) } L_{n-1}^{(\nu+1)}(A \, K^{-2\eta})
{\right]}, \label{eq:c0n} \\
c_{1,n}
&= \psi_n(x') \, \m(x') .\end{aligned}$$ The approximate price of a European put option can now be computed using Theorems \[thm:u00\], \[thm:u10\] and \[thm:u01\].
For European options, it is often the implied volatility induced by an option price, rather than the option price itself that is of primary interest. Recall that the implied volatility $I^{\eps,\del}$ of a put option with price $u^{\eps,\del}(t,x;K)$ is defined implicitly through $$\begin{aligned}
u^{\eps,\del}(t,x;K)=u^{\text{BS}}(t,x,I^{\eps,\del};K)\end{aligned}$$ where $u^{\text{BS}}(t,x,I^{\eps,\del};K)$ is the Black-Scholes price of a put as calculated with volatility $I^{\eps,\del}$.
On the left side of figure \[fig:JDCEVslow\] we plot the implied volatility induced by the approximate price $u_{0,0}+\sqrt{\eps}\,u_{1,0}$ of a put option for a specific model that has only a fast-varying factor of volatility. We suppose the dynamics of $Y$ and the volatility function $f$ are given by . For comparison, we also plot the implied volatility induced by the full price $u^\eps$ (calculated by Monte Carlo simulation) and the implied volatility induced by the JDCEV price $u_{0,0}$. On the right side of figure \[fig:JDCEVslow\] we plot the implied volatility induced by the approximate price $u_{0,0}+\sqrt{\del}\,u_{0,1}$ of a put option for a specific model that has only a slow-varying factor of volatility. We suppose the dynamics of $Z$ and the volatility function $f$ are given by . For comparison, we also plot the implied volatility induced by the full price $u^\del$ (calculated by Monte Carlo simulation) and the implied volatility induced by the JDCEV price $u_{0,0}$. As expected, as $\eps$ and $\del$ go to zero, the implied volatility induced by the approximate price converges to the implied volatility induced by the full price, which converges to the implied volatility induced by the JDCEV price.
Review and Conclusions
======================
This paper develops a general method for obtaining the approximate price for a large class of derivative-assets. The payoff of the derivatives may be path-dependent and the process underlying the derivative-assets may exhibit jump to default as well as combined local/nonlocal stochastic volatility. The intensity of the jump to default event may be state-dependent and the nonlocal component of volatility may be multiscale, driven by one fast-varying and one slow-varying factor.
One key advantage of our pricing methodology is that, by combining techniques from spectral theory, singular perturbation theory and regular perturbation theory, we reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once this equation is solved, the approximate price of a derivative-asset may be calculated formulaically. We have illustrated the simplicity and flexibility of our method by calculating the approximate prices of thre derivative assets: a double-barrier option on a non-defaultable stock, a European option on a defaultable stock, and a non-defaultable bond in a short-rate model.
We believe that the flexibility of our framework, as well as the analytic tractability that our pricing methodology provides merit further research in this area. A logical next step, for example, would be to extend the results of this paper to include cases where the eigenvalue equation does not have a purely discrete spectrum.
Thanks {#thanks .unnumbered}
------
The authors of this paper would like to thank Stephan Sturm, Ronnie Sircar and Jean-Pierre Fouque for helpful conversations. Additionally, the authors would like to thank two anonymous referees, whose comments vastly improved both the quality and readability of this manuscript.
Appendix
========
Self-Adjoint Operators acting on a Hilbert Space {#sec:Hilbert}
------------------------------------------------
In this appendix we summarize some basic properties of self-adjoint operators acting on a Hilbert space. A detailed exposition on this topic (including proofs) can be found in @reedsimon. We shall closely follow @linetskybook, who provides a more streamlined review.
Let $\H$ be a real, separable [^7] Hilbert space with inner product $(\cdot,\cdot)$. A *linear operator* is a pair $(\text{dom}(\L),\L)$ where $\text{dom}(\L) $ is a linear subset of $\H$ and $\L$ is a linear map $\L:\text{dom}(\L) \to \H$. The *adjoint* of an operator $\L$ is an operator $\L^{*}$ such that $(\L f,g) = (f, \L^{*} g), \forall \, f \in \text{dom}(\L), g \in \text{dom}(\L^{*})$, where $$\begin{aligned}
\text{dom}(\L^{*}):=\{ g \in \H : \exists \, h \in \H \text{ such that } (\L f, g) = (f,h) \,\, \forall \, f \in \text{dom}(\L) \} .\end{aligned}$$ An operator $(\text{dom}(\L),\L)$ is said to be *self-adjoint* in $\H$ if $$\begin{aligned}
\text{dom}{(\L)} &= \text{dom}(\L^*) , &
(\L f,g) &= (f,\L g) & \forall \, f,g \in \text{dom}(\L).\end{aligned}$$ Throughout this appendix, for any self-adjoint operator $\L$, we will assume that $\text{dom}(\L)$ is a dense subset of $\H$.
Given a linear operator $\L$, the *resolvent set* $\rho(\L)$ is defined as the set of $\lam \in \mathbb{C}$ such that the mapping $(\L - \text{Id} \, \lam)$ is one-to-one and $R_\lam:=(\L - \text{Id} \, \lam)^{-1}$ is continuous with $\text{dom}( R_\lam ) = \H$. The operator $R_\lam:\H \to \H$ is called the *resolvent*. The *spectrum* $\sig(\L)$ of an operator $\L$ is defined as $\sig(\L):= \mathbb{C}\setminus\rho(\L)$. If $\L$ is self-adjoint, its spectrum is non-empty and real. We say that $\lam \in \sig(\L)$ is an *eigenvalue* of $\L$ if there exists $\psi \in \text{dom}(\L)$ such that the *eigenvalue equation* is satisfied $$\begin{aligned}
\L \, \psi &= \lam \, \psi . \label{eq:EigenvalueEquation}\end{aligned}$$ A function $\psi$ that solves is called an *eigenfunction* of $\L$ corresponding to $\lam$. The *multiplicity* of an eigenvalue $\lam$ is the number of linearly independent eigenfunctions for which equation is satisfied. The spectrum of an operator $\L$ can be decomposed into two disjoint sets called the *discrete* and *essential* spectrum $\sig(\L)=\sig_d(\L) \cup \sig_e(\L)$. For a self-adjoint operator $\L$, a number $\lam \in \R$ belongs to $\sig_d(\L)$ if and only if $\lam$ is an isolated point of $\sig(\L)$ and $\lam$ is an eigenvalue of finite multiplicity.
The *spectral representation Theorem* is an important tool for analysing self-adjoint operators acting on a Hilbert space. We state this theorem below in a form which is convenient for the computations in this paper.
\[thm:spectral\] Assume $\L$ is a self-adjoint operator in $\H$ and assume $\L$ has a purely discrete spectrum (i.e., $\sig_e(\L) = \{\emptyset\}$). The *Spectral Representation Theorem* states that $\L f$ has an eigenfunction expansion $$\begin{aligned}
\L f
&= \sum_n \lam_n \, ( \psi_n ,f ) \, \psi_n , &
\forall \, f
&\in \text{\emph{dom}}(\L) ,\end{aligned}$$ where the sum runs over all solutions $\{\lam_n ,\psi_n \}$ of the eigenvalue equation . Furthermore, for any real-valued Borel-measurable function on $\R$ one can define an operator $\phi(\L)$ using *functional calculus* $$\begin{aligned}
\phi(\L) f
&:= \sum_n \phi(\lam_n) \, ( \psi_n ,f ) \, \psi_n , &
\forall \, f
&\in \text{\emph{dom}}(\phi(\L)) , \label{eq:functional} \\
\text{\emph{dom}}(\phi(\L))
&:= \{ f \in \text{\emph{dom}}(\L) : \sum_n \phi^2(\lam_n) {\left(}\psi_n, f {\right)}^2 < \infty \} .\end{aligned}$$ The operator $\phi(\L)$ is self-adjoint in $\H$ and $\text{\emph{dom}}(\phi(\L)) \subseteq \text{\emph{dom}}(\L)$.
See [@reedsimon] Theorem VIII.6.
Note that setting $\phi(\lam)=\text{Id}$ yields $$\begin{aligned}
\text{Id} \, f
&= \sum_n ( \psi_n ,f ) \, \psi_n , &
\forall \, f
&\in \H , \label{eq:Identity}\end{aligned}$$ which is equivalent to saying that the eigenfunctions $\{ \psi_n \}$ of a densely defined self-adjoint operator in $\H$ form a Schauder basis. In fact, the basis can be chosen to be orthonormal $(\psi_n,\psi_m)=\delta_{n,m}$. Also note, setting $\phi(\lam) = R_\lam$ yields an eigenfunction representation of the resolvent operator $$\begin{aligned}
R_\lam \, f
&= \sum_n \frac{( \psi_n ,f )}{\lam_n -\lam} \, \psi_n , &
\forall \, f
&\in \H , \, \lam \in \rho(\L) .\end{aligned}$$
Boundary Conditions {#sec:BCs}
-------------------
According to @feller1954, the endpoints $e_1$ and $e_2$ of a one-dimensional diffusion in an interval $I$ can be classified as either *natural*, *exit*, *entrance* or *regular*. The classification, which can be found in @borodin [@linetskybook], is done as follows. For a general infinitesimal generator $\L$ of the form one can associate a *scale density* $$\begin{aligned}
\s(x) &:= \exp {\left(}- \int_{x_0}^x \frac{2 b(y)}{a^2(y)} dy {\right)}, & &(\text{scale density)} \label{eq:s}\end{aligned}$$ where the lower limit of integration $x_0 \in (e_1,e_2)$ may be chosen arbitrarily. From $\s$ one can define a *scale function* $\S$ $$\begin{aligned}
\S {\left(}{\left[}x, y {\right]}{\right)}&:= \int_x^y \s(z) \, dz , &
x,y &\in {\left(}e_1, e_2 {\right)}, \\
\S {\left(}{\left(}e_1, y {\right]}{\right)}&:= \lim_{x \searrow e_1} \S {\left(}{\left[}x, y {\right]}{\right)}, &
\S {\left(}{\left[}x, e_2 {\right)}{\right)}&:= \lim_{y \nearrow e_2} \S {\left(}{\left[}x, y {\right]}{\right)}.\end{aligned}$$ Note that the above limits may be infinite. For some arbitrary $y \in {\left(}e_1, e_2 {\right)}$ we define $$\begin{aligned}
I_1 &:= \int_{e_1}^y \S {\left(}{\left(}e_1, x {\right]}{\right)}{\left(}1 + k(x) {\right)}\m(x) \, dx , &
I_2 &:= \int_y^{e_2} \S {\left(}{\left[}x, e_2 {\right)}{\right)}{\left(}1 + k(x) {\right)}\m(x) \, dx , \\
J_1 &:= \int_{e_1}^y \S {\left(}{\left[}x, y {\right]}{\right)}{\left(}1 + k(x) {\right)}\m(x) \, dx , &
J_2 &:= \int_y^{e_2} \S {\left(}{\left[}y, x {\right]}{\right)}{\left(}1 + k(x) {\right)}\m(x) \, dx .\end{aligned}$$ An endpoint $e_i$ is classified as
- **Natural** if $I_i = \infty$ and $J_i = \infty$. No BC needs to be specified at a natural boundary. The interval $I$ is taken to be open at a natural boundary.
- **Exit** if $I_i < \infty$ and $J_i = \infty$. The appropriate BC at an exit boundary is $$\begin{aligned}
\lim_{x \rightarrow e_i} \psi(x) &= 0 .\end{aligned}$$ The interval $I$ is taken to be open at an exit boundary.
- **Entrance** if $I_i = \infty$ and $J_i < \infty$. The appropriate BC at an entrance boundary is $$\begin{aligned}
\lim_{x \rightarrow e_i} \frac{\d_x \psi(x)}{\s(x)} &= 0 .\end{aligned}$$ The interval $I$ is taken to be open at an entrance boundary.
- **Regular** if $I_i < \infty$ and $J_i < \infty$. We must specify the behavior of a diffusion at a regular boundary. Here, we consider only *killing* and *instantaneously reflecting* behavior, for which the appropriate BCs are $$\begin{aligned}
\lim_{x \rightarrow e_i} \psi(x) &= 0 \qquad \text{(killing BC)} , &
\lim_{x \rightarrow e_i} \frac{\d_x \psi(x)}{\s(x)} &= 0 \qquad \text{(instantaneously reflecting BC)}\end{aligned}$$ The interval $I$ is taken to be open at a regular boundary specified as a killing boundary and closed at a regular boundary specified as instantaneously reflecting.
The domain of $\L$ is then $$\begin{aligned}
\text{Dom}{\left(}\L {\right)}&= \left\{ f \in L^2{\left(}I, \m {\right)}: f, \d_x f \in AC_{\text{loc}}(I), \L f \in L^2{\left(}I, \m {\right)}, \text{BCs at $e_1$ and $e_2$} \right\} , \label{eq:domainL}\end{aligned}$$ where $AC_{\text{loc}}(I)$ is the space of functions that are absolutely continuous over each compact subinterval of $I$ (see @linetskybook, p. 242). The BCs at $e_1$ and $e_2$ correspond to the BCs specified above for natural, exit, entrance and regular boundaries.
Specific Model Assumptions {#sec:assumptions}
--------------------------
1. We assume existence and uniqueness of $(X,Y,Z)$ as the strong solution to .
2. We assume existence and uniqueness of $(X,Y,Z)$ as the strong solution to .
3. There exist positive constants $C_\Lam < \infty $ and $C_\Gam < \infty $ such that $|| \Lam ||_\infty <C_\Lam$ and $|| \Gam ||_\infty < C_\Gam$. \[item:GammaLambdaBound\]
4. \[item:Y\] Define the time-rescaled process $Y^{(1)}_t:=Y_{\eps \, t}$. Under $\P$, the process $Y^{(1)}$ has infinitesimal generator $\L_0$. Under $\P$ we assume:
1. The process $Y^{(1)}$ is ergodic and has a unique invariant distribution $\Pi$ with density $\pi$.
2. The operator $\L_0$ has a strictly positive spectral gap – meaning the smallest non-zero eigenvalue $\lam_{min}$ of ${\left(}- \L_0 {\right)}$ is strictly positive.
3. The process $Y^{(1)}$ is reversible – meaning $\L_0$ is self-adjoint acting on $L^2{\left(}\R, \pi {\right)}$.
These assumptions guarantee (see @fpss, p. 93) exponential convergence of $Y^{(1)}$ to its invariant distribution $$\begin{aligned}
\left| \E {\left[}g{\left(}Y_t^{(1)} {\right)}{\right]}- {\left\langle}g{\right\rangle}\right| &\leq C \, \exp {\left(}-\lam_{min} t {\right)}, &
\forall \, g &\in L^2{\left(}\R, \pi {\right)}. &\end{aligned}$$ The above assumptions also ensure (see @fpss, p. 139) that for all $k \in \mathbb{N}$ there exists $C(k)<\infty$ such that $$\begin{aligned}
\sup_t \E {\left[}\left| Y_t^{(1)} \right|^k {\right]}&\leq C(k) . \label{eq:Ybound}\end{aligned}$$
5. Define the time-rescaled process $Z^{(1)}_{t} := Z_{t/\del}$. Under $\P$, the process $Z^{(1)}$ has infinitesimal generator $\M_2$. Under $\P$ we assume the process $Z^{(1)}$ admits moments that are uniformly bounded in $s<t$. That is, for all $k \in \mathbb{N}$ there exists $C(t,k)<\infty$ such that $$\begin{aligned}
\sup_{s \leq t} \E {\left[}\left| Z_s^{(1)} \right|^k {\right]}&\leq C(t,k) .\label{eq:Zbound}\end{aligned}$$
6. \[item:sig,fOm\] We assume that the functions $f(y,z)$ and $\Om(y,z)$ satisfy $\sigb^2(z) <\infty$, $\fOmb(z) <\infty$ and the solutions $\phi(y,z)$ and $\eta(y,z)$ to Poisson equations are at most polynomially growing.
7. The functions $a(x)$ $b(x)$, $r(x)$ and $h(x)$ satisfy $a>0$, $a \in C^2( I )$, $b \in C^1( I )$, $r \geq 0$, $r \in C(I)$, $h \geq 0$, and $h \in C(I)$.
8. \[item:L\] The spectrum of the operator ${\left\langle}\L_2{\right\rangle}$, defined in , is simple and purely discrete.
We note that two of the processes that are most commonly used to model volatility – the Cox-Ingersoll-Ross (CIR) and Ornstein-Uhlenbeck (OU) processes – satisfy the assumptions placed on both $Y^{(1)}$ and $Z^{(1)}$.
Poisson Equations and the Fredholm Alternative {#sec:Poisson}
----------------------------------------------
In this appendix we review the existence and uniqueness of solutions to Poisson equations. Central to this discussion will be a statement of the Fredholm alternative. Our presentation follows page 93 of @fpss, as well as page 124 of @fouque2007wave.
Let $\L$ be a self-adjoint operator densely defined on a real separable Hilbert space $\H$, and let $\{\psi_n,\lam_n\}$ be the complete set of solutions to eigenvalue equation $\L \psi_n = \lam_n \psi_n$. Consider the following Poisson problem: find, $\psi \in \H$ such that $$\begin{aligned}
{\left(}\L - \lam {\right)}\psi &= \chi , \label{eq:PoissonB}\end{aligned}$$ where the function $\chi \in \H$ and the constant $\lam$ are given.
\[thm:fredholm\] The *Fredholm Alternative* states that one of the following is true:
1. Either $\lam$ is not an eigenvalue of $\L$, in which case equation has a unique solution $$\begin{aligned}
\psi
&= R_\lam \, \chi = \sum_n \frac{( \psi_n ,\chi )}{\lam_n -\lam} \, \psi_n .\end{aligned}$$
2. Or, $\lam$ is an eigenvalue of $\L$. Suppose this is the case. Let $\lam = \lam_1 = \lam_2 = \cdots = \lam_m$ (i.e., the eigenvalue $\lam$ has multiplicity $m$). Then has a solution if and only if ${\left(}\psi_n, \chi {\right)}= 0$ for all $n \leq m$. Assuming ${\left(}\psi_n, \chi {\right)}= 0$ for all $n \leq m$, a solution to has the form $$\begin{aligned}
\psi
&= \sum_{n > m} \frac{( \psi_n ,\chi )}{\lam_n -\lam_k}\, \psi_n
+ \sum_{n \leq m } c_n \psi_n, &
c_n
&\in \R .\end{aligned}$$
See @reedsimon, Theorem VI.14 and the ensuing corollary.
Classically, the Fredholm alternative Theorem holds for compact operators on a Hilbert space. However, the Theorem also holds true for differential operators $\L$ of the form , with domain acting on the Hilbert space $\H = L^2(I,\m)$, where $\m$ is the speed density corresponding to $\L$ (see section 9.4.2 of @haberman).
In particular, we note that $\lam=0$ is an eigenvalue of $\L_0$, which is a self-adjoint operator in $L^2(\R,\pi)$. The corresponding (normalized) eigenfunction is the constant $\psi_\lam=1$. Thus, in order for $\L_0 u = \chi$ to have a solution $u \in L^2(\R,\pi)$ we must have ${\left(}1,\chi{\right)}=\int \chi \pi dy =: {\left\langle}\chi {\right\rangle}= 0$, which is the centering condition .
Proof of Theorem \[thm:u10\] {#sec:u10proof}
----------------------------
We must show that $u_{1,0}$, given by satisfies PDE and BC . It is obvious that the BC $u_{1,0}(0,x,z)=0$ is satisfied. To show that $u_{1,0}$ satisfies PDE we note that $$\begin{aligned}
\A u_{0,0}
&= \sum_n c_n {\left(}\A \psi_n {\right)}T_n
= \sum_n \sum_k c_n \A_{k,n} \psi_k T_n, \label{eq:A10u00}\end{aligned}$$ where we have used in the second equality. Now, using and $$\begin{aligned}
{\left(}-\d_t - \lam_k {\right)}U_{k,n}
&= T_n , &
{\left(}-\d_t - \lam_n {\right)}t \, T_n
&= - T_n , \label{eq:dt}\end{aligned}$$ it is easy to show that $$\begin{aligned}
{\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{1,0} = \eqref{eq:A10u00}.\end{aligned}$$
Proof of Theorem \[thm:u01\] {#sec:u01proof}
----------------------------
We must show that $u_{0,1}$, given by satisfies PDE and BC . It is obvious that the BC $u_{0,1}(0,x,z)=0$ is satisfied. To show that $u_{0,1}$ satisfies PDE we note that $$\begin{aligned}
\B\d_z u_{0,0}
&= \sum_n c_n {\left(}\B\d_z \psi_n {\right)}T_n
+ \sum_n {\left(}\d_z c_n {\right)}{\left(}\B \psi_n {\right)}T_n
+ \sum_n c_n {\left(}\B \psi_n {\right)}{\left(}\d_z T_n {\right)}\\
&= \sum_n \sum_k c_n \Bt_{k,n} \psi_k T_n
+ \sum_n \sum_k ( \d_z c_n ) \B_{k,n} \psi_k T_n
- \sum_n \sum_k c_n \B_{k,n} \psi_k ( \d_z \lam_n ) t \, T_n ,
\label{eq:Mdzu00}\end{aligned}$$ where we have used in the second equality. Now, using , and $$\begin{aligned}
{\left(}-\d_t - \lam_k{\right)}V_{k,n}
&= - t \, T_n , &
{\left(}-\d_t - \lam_n{\right)}\tfrac{1}{2} t^2\, T_n
&= - t \, T_n\end{aligned}$$ it is easy to show that $$\begin{aligned}
{\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{0,1}
&= \eqref{eq:Mdzu00}.\end{aligned}$$
Proof of accuracy {#sec:accuracy}
-----------------
Before establishing our main accuracy result – Theorem \[thm:accuracy\] – we shall need the following lemma.
\[thm:polynomial\] Suppose $J(y,z)$ is at most polynomially growing. Then, for every $(y,z)$ and $s<t$, there exists a positive constant $C<\infty$ such that for any $\eps \leq 1$ and $\del \leq 1$, we have the following inequality $$\begin{aligned}
\Et_{y,z} {\left[}\, | J(Y_s, Z_s) | \, {\right]}&\leq C .\end{aligned}$$
It is enough to prove the result for $J(y,z)=y^k$ and $J(y,z)=z^k$ for any $k \in \mathbb{N}$. We begin with $J(y,z) = z^k$. Under the physical measure $\P$ we have $$\begin{aligned}
\E {\left[}| Z_s |^k {\right]}&= \E {\left[}| Z_{\del s}^{(1)} |^k {\right]}\leq \sup_{\del \leq 1} \E {\left[}| Z_{\del s}^{(1)} |^k {\right]}\leq C(s,k) \leq C(t,k) ,\end{aligned}$$ by . Now define an exponential martingale $M_t^{(\Gam)}$, which relates the dynamics of $Z$ under the risk-neutral measure $\Pt$ to its dynamics under the physical measure $\P$. We have $$\begin{aligned}
M_t^{(\Gam)} &:= \exp {\left(}-\int_0^t \Gam(Y_s,Z_s) \, dW_s^z - \frac{1}{2} \int_0^t \Gam^2(Y_s,Z_s) \, ds {\right)}= \left. \frac{d\Pt}{d\P} \right|_{\F_t}.\end{aligned}$$ The $\Pt$-expectation of $\left| Z_s \right|^k$ can be found as follows: $$\begin{aligned}
\Et {\left[}\left| Z_s \right|^k {\right]}&= \E {\left[}| Z_s |^k M_s^{(\Gam)} {\right]}\\
&= \E {\left[}| Z_s |^k \exp {\left(}\frac{1}{2} \int_0^s \Gam^2(Y_u,Z_u) \, du {\right)}{\left(}M_s^{(2 \Gam)} {\right)}^{1/2} {\right]}\\
&\leq {\left(}\E {\left[}| Z_s |^{2k} \exp {\left(}\int_0^s \Gam^2(Y_u,Z_u) \, du {\right)}{\right]}{\right)}^{1/2} {\left(}\E {\left[}M_s^{(2 \Gam)} {\right]}{\right)}^{1/2} & &\text{(by Cuachy-Schwarz)} \\
&= {\left(}\E {\left[}| Z_s |^{2k} \exp {\left(}\int_0^s \Gam^2(Y_u,Z_u) \, du {\right)}{\right]}{\right)}^{1/2} & &\text{($M^{(2\Gam)}$ is a $\P$-martingale)} \\
&\leq {\left(}\E {\left[}| Z_{\del s}^{(1)} |^{2k} {\right]}\exp {\left(}s \, || \Gam ||_\infty^2) {\right)}{\right)}^{1/2} \leq C ,\end{aligned}$$ where we have used assumption \[item:GammaLambdaBound\] of section \[sec:assumptions\] in the last line. We now examine the case $J(y,z)=y^k$. We have $$\begin{aligned}
\E {\left[}| Y_s |^k {\right]}&= \E {\left[}| Y_{s / \eps}^{(1)} |^k {\right]}\leq \sup_{\eps \leq 1} \E {\left[}| Y_{s / \eps}^{(1)} |^k {\right]}\leq C(k) ,\end{aligned}$$ by . Using the same argument as above, one can easily show $$\begin{aligned}
\Et {\left[}\left| Y_s \right|^k {\right]}&= \E {\left[}| Y_s |^k M_s^{(\Lam)} {\right]}\leq {\left(}\E {\left[}| Y_{s / \eps}^{(1)} |^{2k} {\right]}\exp {\left(}s \, || \Lam ||_\infty^2) {\right)}{\right)}^{1/2} \leq C ,\end{aligned}$$ which proves the lemma.
We are now in a position to prove Theorem \[thm:accuracy\]. We begin by defining a remainder term $R^{\eps,\del}$ by $$\begin{aligned}
u^{\eps,\del}
&= u_{0,0} + \sqrt{\eps} \, u_{0,0} + \sqrt{\del} \, u_{0,1} + \eps {\left(}u_{2,0} + \sqrt{\eps} \, u_{3,0} {\right)}+ \sqrt{\del} {\left(}\sqrt{\eps} \, u_{1,1} + \eps \, u_{2,1} {\right)}+ R^{\eps,\del} .\end{aligned}$$ The functions $u_{0,0}$, $u_{1,0}$ and $u_{0,1}$ are the unique solutions to , and respectively. The function $u_{2,0}$ is given by . And $u_{2,0}$ is the solution to Poisson equation . To characterize $u_{1,1}$ and $u_{2,1}$ we must continue the singular perturbation analysis of equation a bit further. The $\O(\sqrt{\eps \, \del})$ equation that results from continuing the asymptotic analysis is $$\begin{aligned}
\O( \sqrt{\eps \, \del} ) :& &
0
&= \L_0 u_{3,1} + \L_1 u_{2,1} + {\left(}-\d_t + \L_2{\right)}u_{1,1} + \M_3 u_{2,0} + \M_1 u_{1,0} \label{eq:PoissonD}\end{aligned}$$ Equation is a Poisson equation of the form . In order for to admit a solution $u_{3,1}$ in $L^2(\R,\pi)$, centering condition must in satisfied. In the centering condition corresponds to $$\begin{aligned}
0
&= {\left\langle}\L_1 u_{2,1} {\right\rangle}+ {\left(}-\d_t + {\left\langle}\L_2 {\right\rangle}{\right)}u_{1,1} + {\left\langle}\M_3 u_{2,0} {\right\rangle}+ {\left\langle}\M_1 {\right\rangle}u_{1,0} . \label{eq:RandomName}\end{aligned}$$ Now, by introducing $\xi(y,z)$ and $\zeta(y,z)$ as solutions to $$\begin{aligned}
\L_0 \xi &= f - {\left\langle}f {\right\rangle}, &
\L_0 \zeta &= \Gam - {\left\langle}\Gam {\right\rangle}. \label{eq:zeta} \end{aligned}$$ and by subtracting from , we can express $u_{2,1}$ as $$\begin{aligned}
u_{2,1}
&= - {\left(}\frac{1}{2} a^2\, \phi\, \d^2_{xx} - a\, \eta \, \d_x {\right)}u_{0,1}
- g \, \Big( \rho_{xz}\, a \, \xi \, \d_x - \zeta \, \Big) \d_z u_{0,0} + D , \label{eq:Psi21}\end{aligned}$$ where $D(x,z)$ is a constant which is independent of $y$. Substituting into characterizes $u_{1,1}$ in terms of $u_{0,0}$, $u_{1,0}$, $u_{1,0}$ and $u_{0,1}$. We choose $u_{1,1}$ as the solution to with BC $u(0,x,z)=0$.
Now, we compute $$\begin{aligned}
0 &= {\left(}-\d_t + \L^{\eps,\del} {\right)}u^{\eps,\del} \\
&= {\left(}-\d_t + \L^{\eps,\del} {\right)}R^{\eps,\del}
+ \frac{1}{\eps} F_0 + \frac{1}{\sqrt{\eps}} F_1 + F_2
+ \sqrt{\del} {\left(}\frac{1}{\eps} F_3 + \frac{1}{\sqrt{\eps}} F_4 + F_5 {\right)}\\
&\qquad + \eps \, R_1^\eps + \sqrt{\eps \, \del} \, R_2^\eps + \del R_3^\eps , \label{eq:RandF}\end{aligned}$$ where $$\begin{aligned}
F_0 &= \L_0 u_{0,0} , \\
F_1 &= \L_0 u_{1,0} + \L_1 u_{0,0} , \\
F_2 &= \L_0 u_{2,0} + \L_1 u_{1,0} + {\left(}-\d_t + \L_2 {\right)}u_{0,0} , \\
F_3 &= \L_0 u_{0,1} , \\
F_4 &= \L_0 u_{1,1} + \L_1 u_{0,1} + \M_3 u_{0,0}, \\
F_5 &= \L_0 u_{2,1} + \L_1 u_{1,1} + \M_3 u_{1,0} + \M_1 u_{0,0} + {\left(}-\d_t + \L_2 {\right)}u_{0,1} ,\end{aligned}$$ and $$\begin{aligned}
R_1^\eps &= {\left(}-\d_t + \L_2 {\right)}u_{2,0} + \L_1 u_{3,0} + \sqrt{\eps} {\left(}-\d_t + \L_2 {\right)}u_{3,0}, \\
R_2^\eps &= {\left(}-\d_t + \L_2 {\right)}u_{1,1} + \L_1 u_{2,1} + \M_1 u_{1,0} + \M_3 u_{2,0} \\
&\qquad + \, \sqrt{\eps} {\left(}{\left(}-\d_t + \L_2 {\right)}u_{2,1} + \M_1 u_{2,0} + \M_3 u_{3,0} {\right)}+ \eps \, \M_1 u_{3,0}, \\
R_3^\eps &= \M_1 u_{0,1} + \M_2 u_{0,0} + \M_3 u_{1,1} + \sqrt{\eps} {\left(}\M_1 u_{1,1} + \M_2 u_{1,0} + \M_3 u_{2,1} {\right)}\\
&\qquad + \, \eps {\left(}\M_1 u_{2,1} + \M_2 u_{2,0} {\right)}.\end{aligned}$$ From the choices made in section \[sec:asymptotics\], it is straightforward to show $F_0 = F_1 = F_2 = F_3 = F_4 = F_5 = 0$. Hence, from we have $$\begin{aligned}
0
&= {\left(}-\d_t + \L^{\eps,\del} {\right)}R^{\eps,\del} + \eps \, R_1^\eps + \sqrt{\eps \, \del} \, R_2^\eps
+ \del R_3^\eps , \label{eq:RPDE} \\
R(0,x,y,z)
&= \eps \, G_1^\eps(x,y,z) + \sqrt{\eps \, \del} \, G_2^\eps(x,y,z) , \label{eq:RBC}\end{aligned}$$ where $$\begin{aligned}
G_1^\eps(x,y,z)
&:= - u_{2,0}(0,x,y,z) - \sqrt{\eps} \, u_{3,0}(0,x,y,z) , \\
G_2^\eps(x,y,z)
&:= - u_{1,1}(0,x,y,z) - \sqrt{\eps} \, u_{2,1}(0,x,y,z) .\end{aligned}$$ Using the Feynman-Kac formula, we can express $R^{\eps,\del}(t,x,y,z)$, which is the solution to PDE with BC , as an expectation $$\begin{aligned}
R^{\eps,\del}(t,x,y,z)
&= \eps \, \Et_{x,y,z} {\left[}e^{-\int_0^t k(X_s) ds} G_1^\eps(X_t,Y_t,Z_t) + \int_0^t e^{-\int_0^s k(X_u) du} R_1^\eps(s, X_s,Y_s,Z_s) \, ds {\right]}\\
&\qquad + \, \sqrt{\eps\,\del} \, \Et_{x,y,z} {\left[}e^{-\int_0^t k(X_s) ds} G_2^\eps(X_t,Y_t,Z_t) + \int_0^t e^{-\int_0^s k(X_u) du} R_2^\eps(s, X_s,Y_s,Z_s) \, ds {\right]}\\
&\qquad + \, \del \, \Et_{x,y,z} {\left[}\int_0^t e^{-\int_0^s k(X_u) du} R_3^\eps(s, X_s,Y_s,Z_s) \, ds {\right]}.\end{aligned}$$ From the assumptions of section \[sec:assumptions\] one can deduce that the functions ${\left(}R_1^\eps, R_2^\eps, R_3^\eps, G_1^\eps, G_2^\eps {\right)}$ are bounded in $x$ and at most polynomially growing in $(y,z)$ (see @fpss). Hence, by Lemma \[thm:polynomial\] we have $$\begin{aligned}
\left| R^{\eps,\del} \right| &\leq \eps \, C_1 + \sqrt{\eps\,\del} \, C_2 + \del \, C_3 \leq {\left(}\eps + \del {\right)}\, C_4 .\end{aligned}$$ Finally $$\begin{aligned}
&\left| u^{\eps,\del} - {\left(}u_{0,0} + \sqrt{\eps} \, u_{1,0}+ \sqrt{\del} \, u_{0,1}{\right)}\right| \\
&\qquad \leq \left| R^{\eps,\del} \right| + \left| \eps \, u_{2,0} + \eps^{3/2} u_{3,0} + \sqrt{\eps \, \del} \, u_{1,1} + \eps \sqrt{\del} \, u_{2,1} \right| \\
&\qquad \leq {\left(}\eps + \del {\right)}C_4 + \eps \, \left| u_{2,0} + \sqrt{\eps} \, u_{3,0} \right| + \sqrt{\eps \, \del} \, \left| u_{1,1} + \sqrt{\eps} \, u_{2,1} \right| \\
&\qquad \leq {\left(}\eps + \del {\right)}C ,\end{aligned}$$ which is the claimed accuracy result.
![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBeps1new.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBdel0.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
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![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBeps01new.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBdel1.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBeps001new.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The price of a double-barrier call option is plotted as a function of the underlying $x$. On the left we consider the Black-Scholes model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Black-Scholes model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full price of the option, the dashed line corresponds to our approximation, and the dotted line corresponds to the Black-Scholes price. For the plots on the left we use parameters $t=1/12$, $y=0$, $r=0.05$, $\sigma=0.34$, $\rho_{xy}=-0.5$, $\beta=1$, $L=1.5$, $K=2.0$, $R=2.5$. For the plots on the right we use parameters $t=1/12$, $z=2$, $r=0.05$, $\sigma=0.34$, $\rho_{xz}=-0.5$, $g=2$, $L=1.5$, $K=2.0$, $R=2.5$.[]{data-label="fig:DBslow"}](EPS/DBdel01.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicek0.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicekdel01.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
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![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicek1.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicekdel001.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicek01.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The yield of a zero coupon bond is plotted as a function of the time to maturity $t$. On the left we consider the Vasicek model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the Vasicek model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full yield of the bond, the dashed line corresponds to our approximation, and the dotted line corresponds to the Vasicek yield. For the plots on the left we use parameters $x=0.03$, $y=0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xy}=-0.5$, $\beta=1$, $\Om = 0.1\,e^{\beta^2/4}$. For the plots on the right we use parameters $x=0.03$, $z=1.0$, $\theta=0.05$, $\sigma=0.02$, $\rho_{xz}=-0.5$, $g=1$, $\Om = 0.1$.[]{data-label="fig:Vasicekslow"}](EPS/Vasicekdel0001.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVeps01.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVdel0.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
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![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVeps001.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVdel1.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVeps0001.eps "fig:"){width=".5\textwidth" height=".25\textheight"} ![The implied volatility of a European put option is plotted as a function of the strike price $K$. On the left we consider a JDCEV model with only a fast-varying factor of volatility $Y$ whose dynamics are given by \[eq:Yexample\]. On the right, we consider the JDCEV model with only a slow-varying factor of volatility $Z$ whose dynamics a given by \[eq:Zexample\]. In each plot, the solid black line corresponds to the full implied volatility, the dashed line corresponds to our approximation, and the dotted line corresponds to the JDCEV implied volatility. For the plots on the left we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xy}=-0.5$, $y=0$ and $\beta=2$. For the plots on the right we use parameters $t=1$, $x=50$, $\mu=0.05$, $\sig=10$, $\eta=-1$, $c=0.5$, $\rho_{xz}=-0.5$, $z=2$ and $g=2$.[]{data-label="fig:JDCEVslow"}](EPS/JDCEVdel01.eps "fig:"){width=".5\textwidth" height=".25\textheight"}
[^1]: Work partially supported by NSF grant DMS-0739195.
[^2]: An operator $\L$ is *self-adjoint* on a Hilbert space $\H$ with inner product $(\cdot,\cdot)$ if $\text{dom}{(\L)}=\text{dom}(\L^*)$ and $(\L f,g)=(f,\L g)$ for all $f,g \in \text{dom}(\L)$. Please see appendix \[sec:Hilbert\] for a brief review of self-adjoint operators in Hilbert Spaces.
[^3]: Note that we do not take a limit as $\eps$ and $\del$ go to zero simultaneously.
[^4]: Please refer to Appendix \[sec:Poisson\] for an discussion of the Fredholm alternative
[^5]: The phrase *group parameter* refers to any $z$-dependent parameter which can be calculated as a moment of model-specific functions. As we shall see, the effect that the functions ($\beta$, $\Lam$, $g$, $\Gamma$) have on the approximate price of a derivative asset is felt only through eight group parameters.
[^6]: We note that $r(X_t)=X_t$ may become negative when $X$ is described by an OU process. As such, one may wish to impose a reflecting boundary condition at $x=0$, as carried out in @linetsky2004blackinterest. However, as an OU *without* a reflecting boundary is most prevalent in literature, this is the case we treat here.
[^7]: A Hilbert space is separable if and only if it admits a countable orthonormal basis (i.e., Schauder basis).
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abstract: 'The statistical shapes of the highest pulse have been studied by aligned method. A wavelet package analysis technique and a developed pulse–finding algorithm have been applied to select the highest pulse from burst profiles observed by BATSE on board CGRO from 1991 April 21 to 1999 January 26. The results of this work show that the statistical shapes of the highest pulses are related to energy: the higher the energy, the narrower the pulse. However, the characteristic structures of the pulses have nothing to do with energy, which strongly supports the previous conclusion that the temporal profiles in different channels are self–similar. The characteristic structures of the pulses can be well described by a model proposed by Norris et al. (1996). The fitting parameters are: $t_r$=0.12, $t_d$=0.16, $\upsilon =1.09$, the ratio of $t_r$ to $t_d$ for the pulse is 0.75. The result leads to our conjecture that the mechanisms of bursts in different gamma-ray bands might be the same. The shock, either an internal or an external one, producing the pulse, might emit photons over the four energy channels in the same way.'
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Introduction
============
Gamma–ray bursts(GRBs), which are still mysterious, have very complex temporal structure. Their temporal profiles are enormously varied — no two bursts have ever been found to have exactly the same temporal and spectral development. The temporal activity is suggestive of a stochastic process (Nemiroff et al. 1993). The diversity of the bursts seems to be due to random realization of the same process that is self-similar over the whole range of timescale. Attempts to quantify these structures have not been successful (e.g., Fishman 1999).
Most of the observed profiles of GRBs are composed of pulses, each comprising a fast rise and an exponential decay (a FRED; e. g., Desai 1981, Fishman et al. 1994). Many methods for pulse analysis have been developed, e. g., the parametric analysis in model fitting (Nemiroff et al. 1993, Norris et al. 1996), the auto-correlation method (Fenimore et al. 1995), the nonparametric method (Li & Fenimore 1996), the peak alignment and normalized flux averaging method (Mitrofanov et al. 1996, Mitrofanov et al. 1998, Ramirez–Ruiz & Fenimore 1999, Ramirez–Ruiz & Fenimore 2000), and the pulse decomposition analysis method (Lee et al. 2000), etc. These statistical studies have revealed many observed temporal signatures of pulses. The pulses are hypothesized to have the same shape at all energies, differing only by scale factors in time and amplitude (“pulse scale conjecture”). And, the pulses are hypothesized to start at the same time, independent of energy (“pulse start conjecture”). The two conjectures were confirmed by Nemiroff (2000). In general, higher energy channels show shorter temporal scale factors (e.g., Norris et al. 1996., Nemiroff 2000). It is found that the temporal scale factors between a pulse measured at different energies are related to that energy by a power law, possibly indicating a simple relativistic mechanism is at work (Fenimore et al. 1995, Norris et al. 1996., Nemiroff 2000).
The statistical pulse shape has been well studied by the peak alignment and normalized flux averaging method. The peak aligned averaging pulse is spiky. A succinct pulse model, which well describes many pulse shapes, was proposed by Norris et al. (1996): $$I(t)=I_0e^{-(|t-t_{\max }|/{t_{r,d}})^\upsilon }$$ where $t_{max}$ is the time of the pulse’s maximum intensity ($I_0$); $t_r$ and $t_d$ are the rise and decay time constants, respectively; and $\upsilon $ is a measure of the pulse sharpness, which was referred to “peakness” by Norris et al. (1996).
However, both the duration and total count of pulses vary significantly. Statistical properties of the pulses revealed by the peak alignment and flux–normalized averaging method are limited. In this paper, this method is developed to study the shape of pulses in a more detailed manner.
To reach a result of high quality, we concern in this paper only the highest pulse of bursts, where one finds the highest level of signal–to–noise. We make the noise decomposition for the time profile of bursts by performing the wavelet analysis (which is described in section 2), then modify the pulse–finding algorithm proposed by Li & Fenimore (Li & Fenimore 1996) to identify the highest pulse in a burst profile (see section 3). In sections 4 and 5, we employ and develop the pulse aligned method to study the flux–normalized aligned averaging pulse shape and the count–and–duration–normalized aligned averaging pulse shape, respectively. Conclusions and discussion are presented in section 6.
Data Analysis
=============
The data used for analyzing is the 64 ms temporal resolution and four–channel spectral resolution GRB data observed by BATSE from 1991 April 21 to 1999 January 26. There are 1738 bursts included. It is a concatenation of three standard BATSE data types, DISCLA, PREB, and DISCSC. All these data types are derived from the on-board data stream of BATSE’s eight Large Area Detectors (LADs). There are four energy channels observed, with the following approximate channel boundaries: 25-55 keV, 55-110 keV, 110-320 keV, and $>$320 keV. The DISCLA data are a continuous stream with 1.024 second resolution. They are independent of burst occurrence and taken as the background. The PREB data cover the interval 2.048 second just prior to a burst trigger.
We make the noise decomposition for the time profiles by the wavelet package analysis technique. The technique is suitable to treat those signals which cannot be analyzed by the traditional Fourier method. It was successful in de–noising the original signal and identifying the structure within a burst (e.g., Hurley et al. 1997, Quilligan et al. 1999, Lee et al. 2000). We use DB3 wavelet to make the first–class decomposition with the MATLAB software. The profile is decomposed into the signal component and the noise component. Figure 1 illustrates an example of the decomposition.
The method of the background treatment used here is similar to that in Li & Fenimore (1996). Since the DISCLA data are a continuous stream prior to and independent of the burst occurrence, they are always taken as the background of bursts. The data of the background is obtained by a linear fitting to the DISCLA data.
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Pulse–Finding Algorithm and Sample Selection
============================================
Many burst time profiles appear to be composed of a series of overlapping pulses, mingling with noises. It is not easy to determine their actual light curves and to distinguish a pulse from the time profile. The result of pulse analysis strongly relies on the algorithm of pulse–finding and sample selection. Several pulse–finding algorithms have been proposed (e.g., [@LF96], Norris et al. 1996,Mitrofanov et al. 1998). Li & Fenimore (1996) suggested an efficacious algorithm to identify “true peaks” in a profile. A “true peak” is not necessarily to be regarded as a pulse. If a profile is composed of only one “true peak”, the “true peak” can be regarded as a pulse. However, most of the profiles are composed of many overlapping “true peaks”. To identify a pulse in such situation is not easy. Norris et al. (1996) introduced a definition of “inseparable pulse”. We adopt this concept and regard an “inseparable pulse” as a true pulse. Our pulse–finding algorithm is described in the following.
\(1) The peak–finding criterion proposed by Li & Fenimore is $%
C_p-C_{1,2}\geq N_{var}\sqrt{C_p}$ , where $C_i$ ($i=1,2$) is the photon count at time bin $t_i$, $C_p$ (at $t_p$) is the maximum count of a candidate peak, and $N_{var}$ is an adjustable parameter, typically $3\leq N_{var}\leq 5$. This criterion strongly relies on absolute photon count of the candidate peak. We follow Norris et al. (1996) to employ the concept of “inseparable pulse”, and then the pulse–finding criterion becomes $%
1-C_{1,2}/C_p\geq 0.5$. It means that a candidate peak is a true peak only when $C_1$ (at $t_1$) and $C_2$ (at $t_2$) are lower than the half of the $C%
_p$ on both sides of $t_p$. With this method, one might find more than one true peaks within a burst. For the reason mentioned in section 1, we select only the highest one.
\(2) In order to maintain a high level of signal–to–noise, we adopt the intensity criterion as $C_{max}>10\sigma $, where $\sigma $ is the standard deviation of the background.
\(3) Only those pulses with at least 10 bins of time are selected. Those with less bins do not provide enough structure information and thus are ignored.
We apply the above pulse–finding algorithm to select highest pulses in the profiles of bursts. There are 760, 885, 885, and 334 bursts, for which the highest pulses can be identified, in channels 1 to 4, respectively. The number of bursts for which all of the highest pulses in four channels can be identified is 275. The flux in this sample ranges widely, from 0.513 photons.cm$%
^{-1}$.s$^{-1}$ to 183.370 photons.cm$^{-1}$.s$^{-1}$. We select the sample to study the statistical properties of the pulse morphology.
The Flux–Normalized aligned Averaging Pulse
===========================================
It was found that the peak–aligned averaging pulse well illustrates how the average pulse evolves with energy. To do that, one averages the time profile of individual events by the normalized peak–alignment technique, where each time profile is normalized by the peak number of counts C$_{max}$, aligned at the peak time bin t$_{max}$, and then averaged for all bins along the timescale(e. g., Norris et al. 1996, Mitrofanov et al. 1996, Ramirez–Ruiz & Fenimore 2000). Though the pulse–finding method and the sample adopted in this paper are somewhat different from the previous ones, the peak–aligned averaging pulses we obtain are quite similar to that in Norris et al. (1996) and Ramirez-Ruiz & Fenimore (2000) (see Fig. 2). Figure 2 shows the same result that the higher the energy, the narrower the pulse. One can also find this from the flux–normalized–and–beginning–aligned averaging pulses shown in Fig. 3.
We find that the shapes of pulses in Fig. 2 are quite different from that in Fig. 3. Though both figures come from the sum of the same normalized pulses, but the ways of the alignment are different. The difference between Figs. 2 and 3 must come from the diversity of the duration and the asymmetry of the shape of the normalized pulses. Obviously, the peak–aligned method would lead to a spiky shape. The statistical result must conceal most of the diversity of the duration and the asymmetry of the pulses. This explains why the averaging pulses in Fig. 2 are very different from observation. Also, the flux–normalized–beginning–aligned method does not take into account the diversity of the duration of pulses. Thus, neither Fig. 2 or 3 truly embodies the temporal structure of pulses.
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The Count-and-Duration-Normalized aligned Averaging Pulse
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We observe that, the above peak–aligned method allows those pulses with longer durations to possess larger total counts and then to contribute more significantly to the averaging pulse. To investigate the averaged shape of pulses, one needs to get rid of the effects from both the total count and the duration of the selected pulses. This leads to the count-and-duration-normalized aligned averaging method employed bellow.
First, both the total count and the duration for each selected pulse are normalized. Then the averaging pulses for the four channels are obtained with the same way used in last section. The results are shown in Figs. 4 and 5. They are different respectively from that in Figs. 2 and 3.
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Figure 4 shows that the count–and–duration–normalized–peak–aligned averaging pulses in the four channels are almost the same. One can not tell any difference between the count–and–duration–normalized–beginning–aligned averaging pulses of the four channels. It shows that the shapes of these pulses are independent of energy bands. The pulses of the four channels have the same temporal structure.
The pulse model in Norris et al. (1996) is employed to fit them, and it shows that these pulses can be well described by the model. The fitting parameters are: $t_r$=0.28, $t_d$=0.38, $\upsilon
=2.29$ for the count–and–duration–normalized–peak–aligned averaging pulse (in channel 3); $t_r$=0.12, $t_d$=0.16, $\upsilon
=1.09$ for the count–and–duration–normalized–beginning–aligned averaging pulse (in channel 3). The ratios of $t_r$ to $t_d$ for both pulses are about 0.75.
Conclusions and Discussion
==========================
In this paper we study the flux–normalized–peak(beginning)–aligned averaging pulses and the count-and-duration-normalized–peak(beginning)–aligned averaging pulses with the highest pulse in the profiles of GRBs. We apply the wavelet package analysis technique and a developed pulse–finding algorithm to select the highest pulse. The sample we get includes 275 bursts which fluxes range from 0.513 erg.cm$^{-1}$.s$^{-1}$ to 183.370 photons.cm$^{-1}$.s$^{-1}$.
The wavelet package analysis technique is suitable to treat those signals which cannot be analyzed by the traditional Fourier method. It is successful in de–noising the original signal and identifying the structure within a burst. The pulse–finding algorithm used in Li & Fenimore (1996) is developed in this paper so that the selection of pulses depends on the relative value of counts rather than the absolute value.
The number of bursts concerned in this paper is the largest one of that used for pulse analysis so far, and the sample adopted here covers the biggest flux range. Many samples of very bright bursts have been employed to study statistical properties of pulses (e.g., Norris et al. 1996). Though the numbers of bursts concerned are much smaller, the authors were able to get more pulses by selecting not only the highest pulse but also other pulses in a burst (e.g., Norris et al. 1996). Their results refer only to bright bursts, and the pulses so selected might include possible evolutionary effects of pulses.
Figures 2 and 3 support the well-known conclusion that the higher the energy, the narrower the pulse.
Different from the previous works, we make in this paper not only the normalization of the total count but also the normalization of the duration. In this way, all the pulses (strong or weak) contribute equally to the averaging pulses. And, in this way, the averaging pulses obtained stand only for the statistical shape of pulses. The effects from both the duration and the total count are removed. The results so obtained are quite different from that got by the previous method. One can find this by comparing Figs. 2 and 4.
For the pulses shown in Figs. 4 and 5, we prefer those in the latter. Since the pulses in Fig. 4 come from aligning the normalized pulses at the moment of maximum count of the pulses, those asymmetric normalized pulses would contribute differently to different sides of the averaging pulses. Thus the distribution of the peak count position in the shape of selected pulses must be at work. It would lead to a spiky shape of the averaging pulses. Fig. 4 must conceal most of the diversities of the duration and the asymmetry of pulses. Differently, the pulses shown in Fig. 5 stand only for the average shape of the original pulses. These pulses are well described by the model in Norris et al. (1996). The fitting parameters for the pulse in channel 3 in Fig. 5 are: $t%
_r$=0.12, $t_d$=0.16, $\upsilon =1.09$. The ratios of $t_r$ to $t_d$ for the pulse is 0.75.
We find that the count-and-duration-normalized–peak(beginning)–aligned averaging pulses are the same for different channels. Our results strongly support the previous conclusion that the temporal profiles in different channels are self–similar. The averaging pulse shape is independent of energy bands. Due to these results, we believe that the mechanisms of bursts in different gamma-ray bands must be the same.
The mechanism generating the bursts is still unknown. Many models for interpreting the origin and emission of the event have been proposed (e. g., Rees & M$\acute{e}$sz$\acute{a}$ros 1992, Vietri et al. 1998, Fuller et al. 1998, Dai & Lu 1998, Daigne et al. 1998, [@Pa00], etc.), mostly in the context of two major scenarios involving relativistic shells. An approach frequently used in these models is to identify each pulse in the light curve with a single event. Depending on the model chosen, this event could be the collision between inhomogeneities in a relativistic wind in the internal models or the “activation” of a region on a single external shell. Our study shows that, the shock, either an internal or an external one, producing the pulse, might produce photons over the four energy channels in the same way.
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abstract: 'In this paper we consider a correspondence between the holographic dark energy density and interacting generalized Chaplygin gas energy density in FRW universe. Then we reconstruct the potential of the scalar field which describe the generalized Chaplygin cosmology.'
author:
- |
M.R. Setare [^1]\
[Department of Science, Payame Noor University. Bijar, Iran]{}
title: '**[Interacting holographic generalized Chaplygin gas model ]{}**'
---
c ł Ł ø Ø ¶
Introduction
============
The accelerated expansion that based on recent astrophysical data [@exp], our universe is experiencing is today’s most important problem of cosmology. Missing energy density - with negative pressure - responsible for this expansion has been dubbed Dark Energy (DE). Wide range of scenarios have been proposed to explain this acceleration while most of them can not explain all the features of universe or they have so many parameters that makes them difficult to fit. The models which have been discussed widely in literature are those which consider vacuum energy (cosmological constant) [@cosmo] as DE, introduce fifth elements and dub it quintessence [@quint] or scenarios named phantom [@phant] with $w<-1$ , where $w$ is parameter of state.
An approach to the problem of DE arises from holographic principle that states that the number of degrees of freedom related directly to entropy scales with the enclosing area of the system. It was shown by ’tHooft and Susskind [@hologram] that effective local quantum field theories greatly overcount degrees of freedom because the entropy scales extensively for an effective quantum field theory in a box of size $L$ with UV cut-off $ \Lambda$. As pointed out by [@myung], attempting to solve this problem, Cohen [*et al.*]{} showed [@cohen] that in quantum field theory, short distance cut-off $\Lambda$ is related to long distance cut-off $L$ due to the limit set by forming a black hole. In other words the total energy of the system with size $L$ should not exceed the mass of the same size black hole i.e. $L^3 \rho_{\Lambda}\leq LM_p^2$ where $\rho_{\Lambda}$ is the quantum zero-point energy density caused by UV cutoff $\Lambda$ and $M_P$ denotes Planck mass ( $M_p^2=1/{G})$. The largest $L$ is required to saturate this inequality. Then its holographic energy density is given by $\rho_{\Lambda}= 3c^2M_p^2/
L^2$ in which $c$ is free dimensionless parameter and coefficient 3 is for convenience. Based on cosmological state of holographic principle, proposed by Fischler and Susskind [@fischler], the Holographic model of Dark Energy (HDE) has been proposed and studied widely in the literature [@miao; @HDE].\
Some experimental data has implied that our universe is not a perfectly flat universe and recent papers have favored the universe with spatial curvature . As a matter of fact, we want to remark that although it is believed that our universe is flat, a contribution to the Friedmann equation from spatial curvature is still possible if the number of e-foldings is not very large [@miao2]. Defining the appropriate distance, for the case of non-flat universe has another story. Some aspects of the problem has been discussed in [@miao2; @guberina]. In this case, the event horizon can not be considered as the system’s IR cut-off, because for instance, when the dark energy is dominated and $c=1$, where $c$ is a positive constant, $\Omega_\Lambda=1+ \Omega_k$, we find $\dot
R_h<0$, while we know that in this situation we must be in de Sitter space with constant EoS. To solve this problem, another distance is considered- radial size of the event horizon measured on the sphere of the horizon, denoted by $L$- and the evolution of holographic model of dark energy in non-flat universe is investigated.\
It is fair to claim that simplicity and reasonability of HDE provides more reliable frame to investigate the problem of DE rather than other models proposed in the literature[@cosmo; @quint; @phant]. For instance the coincidence or “why now” problem is easily solved in some models of HDE based on this fundamental assumption that matter and holographic dark energy do not conserve separately, but the matter energy density decays into the holographic energy density [@intde]. In fact a suitable evolution of the Universe is obtained when, in addition to the holographic dark energy, an interaction (decay of dark energy to matter) is assumed.\
In a very interesting paper Kamenshchik, Moschella, and Pasquier [@kmp]have studied a homogeneous model based on a single fluid obeying the Chaplygin gas equation of state \[chp\] P= where $P$ and $\rho$ are respectively pressure and energy density in comoving reference frame, with $\rho> 0$; $A$ is a positive constant. This equation of state has raised a certain interest [@jac] because of its many interesting and, in some sense, intriguingly unique features. Some possible motivations for this model from the field theory points of view are investigated in [@a]. The Chaplygin gas emerges as an effective fluid associated with d-branes [@b] and can also be obtained from the Born-Infeld action [@c].\
Inserting the equation of state (\[chp\]) into the relativistic energy conservation equation, leads to a density evolving as \[enerd\]\_= where $B$ is an integration constant.\
In present paper, using the generalized Chaplygin gas model of dark energy, we obtain equation of state for interacting Chaplygin gas energy density in non-flat universe. The current available observational data imply that the dark energy behaves as phantom-type dark energy, i.e. the equation-of-state of dark energy crosses the cosmological-constant boundary $w=-1$ during the evolution history. We show this phantomic description of the interacting generalized Chaplygin gas dark energy in non-flat universe, and reconstruct the potential of the phantom scalar field. Finally we introduce the squared speeds of sound for generalized Chaplygin gas and interacting holographic fluid. We find that the squared speed for generalized Chaplygin gas is negative when, $\alpha<0$, $A> 0$ or $\alpha >0$, $A <0$. Also we show that the squared speed for interacting holographic fluid is negative when choosing $c = 0.84$ , and taking $ \Omega_{\Lambda}= 0.73$, $
\Omega_{k} = 0.01$ for the present time.
Interacting generalized Chaplygin gas model
============================================
In this section we obtain the equation of state for the generalized Chaplygin gas when there is an interaction between generalized Chaplygin gas energy density $\rho_{\Lambda}$ and a Cold Dark Matter(CDM) with $w_{m}=0$. The continuity equations for dark energy and CDM are $$\begin{aligned}
\label{2eq1}&& \dot{\rho}_{\rm \Lambda}+3H(1+w_{\rm \Lambda})\rho_{\rm \Lambda} =-Q, \\
\label{2eq2}&& \dot{\rho}_{\rm m}+3H\rho_{\rm m}=Q.\end{aligned}$$ The interaction is given by the quantity $Q=\Gamma \rho_{\Lambda}$. This is a decaying of the generalized Chaplygin gas component into CDM with the decay rate $\Gamma$. Taking a ratio of two energy densities as $r=\rho_{\rm m}/\rho_{\rm \Lambda}$, the above equations lead to $$\label{2eq3} \dot{r}=3Hr\Big[w_{\rm \Lambda}+
\frac{1+r}{r}\frac{\Gamma}{3H}\Big]$$ Following Ref.[@Kim:2005at], if we define $$\begin{aligned}
\label{eff}
w_\Lambda ^{\rm eff}=w_\Lambda+{{\Gamma}\over {3H}}\;, \qquad w_m
^{\rm eff}=-{1\over r}{{\Gamma}\over {3H}}\;.\end{aligned}$$ Then, the continuity equations can be written in their standard form $$\dot{\rho}_\Lambda + 3H(1+w_\Lambda^{\rm eff})\rho_\Lambda =
0\;,\label{definew1}$$ $$\dot{\rho}_m + 3H(1+w_m^{\rm eff})\rho_m = 0\; \label{definew2}$$ We consider the non-flat Friedmann-Robertson-Walker universe with line element \[metr\] ds\^[2]{}=-dt\^[2]{}+a\^[2]{}(t)(+r\^2d\^[2]{}). where $k$ denotes the curvature of space k=0,1,-1 for flat, closed and open universe respectively. A closed universe with a small positive curvature ($\Omega_k\sim 0.01$) is compatible with observations . We use the Friedmann equation to relate the curvature of the universe to the energy density. The first Friedmann equation is given by $$\label{2eq7} H^2+\frac{k}{a^2}=\frac{1}{3M^2_p}\Big[
\rho_{\rm \Lambda}+\rho_{\rm m}\Big].$$ Define as usual $$\label{2eq9} \Omega_{\rm
m}=\frac{\rho_{m}}{\rho_{cr}}=\frac{ \rho_{\rm
m}}{3M_p^2H^2},\hspace{1cm}\Omega_{\rm
\Lambda}=\frac{\rho_{\Lambda}}{\rho_{cr}}=\frac{ \rho_{\rm
\Lambda}}{3M^2_pH^2},\hspace{1cm}\Omega_{k}=\frac{k}{a^2H^2}$$ Now we can rewrite the first Friedmann equation as $$\label{2eq10} \Omega_{\rm m}+\Omega_{\rm
\Lambda}=1+\Omega_{k}.$$ Using Eqs.(\[2eq9\],\[2eq10\]) we obtain following relation for ratio of energy densities $r$ as $$\label{ratio}
r=\frac{1+\Omega_{k}-\Omega_{\Lambda}}{\Omega_{\Lambda}}$$ In the generalized Chaplygin gas approach [@c], the equation of state to (\[chp\]) is generalized to \[chpge\] P\_= The above equation of state leads to a density evolution as \[denge\] \_=\[A+\]\^ By considering the above equations, one can find $$\label{stateq}
w_{\rm
\Lambda}=\frac{P_{\Lambda}}{\rho_{\Lambda}}=\frac{-A}{a^{3(1+\alpha)}[A+B
a^{-3(1+\alpha)}]}.$$ From Eqs.(\[eff\], \[stateq\]), we have the effective equation of state as $$\label{3eq401}
w_{\rm \Lambda}^{eff}=\frac{-A}{a^{3(1+\alpha)}[A+B
a^{-3(1+\alpha)}]}+\frac{\Gamma}{3H}.$$ Here as in Ref.[@WGA], we choose the following relation for decay rate $$\label{decayeq}
\Gamma=3b^2(1+r)H$$ with the coupling constant $b^2$. Now using the definition of generalized Chaplygin gas energy density $\rho_{\rm \Lambda}$, and using $\Omega_{\Lambda}$, we can rewrite Eq.(\[3eq401\]) as $$\label{stateq2}
w_{\rm \Lambda}^{eff}=\frac{-A}{(3M_{p}^{2} H^2 \Omega_{\rm
\Lambda})^{1+\alpha}}+\frac{b^2(1+\Omega_{k})}{\Omega_{\rm \Lambda}}$$ Now we suggest a correspondence between the holographic dark energy scenario and the generalized Chaplygin gas dark energy model.\
In non-flat universe, our choice for holographic dark energy density is \[holoda\] \_=3c\^2M\_[p]{}\^[2]{}L\^[-2]{}. $L$ is defined as the following form[@miao2]: $$\label{leq}
L=ar(t),$$ here, $a$, is scale factor and $r(t)$ is relevant to the future event horizon of the universe. Given the fact that $$\begin{aligned}
\int_0^{r_1}{dr\over \sqrt{1-kr^2}}&=&\frac{1}{\sqrt{|k|}}{\rm
sinn}^{-1}(\sqrt{|k|}\,r_1)\nonumber\\
&=&\left\{\begin{array}{ll}
\sin^{-1}(\sqrt{|k|}\,r_1)/\sqrt{|k|},\ \ \ \ \ \ &k=1,\\
r_1,&k=0,\\
\sinh^{-1}(\sqrt{|k|}\,r_1)/\sqrt{|k|},&k=-1,
\end{array}\right.\end{aligned}$$ one can easily derive \[leh\] L=,where $R_h$ is the future event horizon given by R\_h= a\_t\^=a\_a\^ By considering the definition of holographic energy density $\rho_{\rm \Lambda}$, one can find : $$\label{stateq4}
w_{\rm \Lambda}=-[\frac{1}{3}+\frac{2\sqrt{\Omega_{\rm
\Lambda}}}{3c}\frac{1}{\sqrt{|k|}}\rm
cosn(\sqrt{|k|}\,R_{h}/a)+\frac{\Gamma}{3H}].$$ where $$\frac{1}{\sqrt{|k|}}{\rm cosn}(\sqrt{|k|}x)
=\left\{\begin{array}{ll}
\cos(x),\ \ \ \ \ \ &k=1,\\
1,&k=0,\\
\cosh(x),&k=-1.
\end{array}\right.$$ Substituting Eq.(\[decayeq\]) into Eq.(\[stateq\]), and using Eq.(\[eff\]) one can find $$\label{3eq402}
w_{\rm \Lambda}^{eff}=-\frac{1}{3}-\frac{2\sqrt{\Omega_{\rm
\Lambda}-c^2\Omega_{k}}}{3c}.$$ If we establish the correspondence between the holographic dark energy and generalized Chaplygin gas energy density, then using Eqs.(\[denge\], \[holoda\])we have \[aco\]A=(3c\^2M\_[p]{}\^[2]{}L\^[-2]{})\^[1+]{}- Using definitions $\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{cr}}$ and $\rho_{cr}=3M_{p}^{2}H^2$, we get
$$\label{hl}
HL=\frac{c}{\sqrt{\Omega_{\Lambda}}}$$
Now, by comparing the effective equation of states (\[stateq2\], \[3eq402\]) we obtain [^2] \[bco\]A=(3M\_[p]{}\^[2]{} H\^2 \_)\^[1+]{}(++) Substituting the above relation into Eq.(\[aco\]) we have \[aco1\]B=(3M\_[p]{}\^[2]{} H\^2 \_a\^3)\^[1+]{}\[1-(++)\]
The correspondence between interacting generalized Chaplygin gas and holographic phantom
========================================================================================
For the non-flat universe, the authors of [@obsnonflat] used the data coming from the SN and CMB to constrain the holographic dark energy model, and got the 1 $\sigma$ fit results: $c=0.84^{+0.16}_{-0.03}$. If we take $c=0.84$, and taking $\Omega_{\Lambda}=0.73$, $\Omega_{k}=0.01$ for the present time, using Eq.(\[3eq402\]) we obtain $w_{\rm \Lambda}^{eff}=-1.007$. Also for the flat case, the X-ray gas mass fraction of rich clusters, as a function of redshift, has also been used to constrain the holographic dark energy model. The main results, i.e. the 1 $\sigma$ fit values for $c$ is: $c=0.61^{+0.45}_{-0.21}$, in this case also we obtain $w_{\rm
\Lambda}^{eff}<-1$. This implies that one can generate phantom-like equation of state from an interacting holographic dark energy model in flat and non-flat universe only if $c\leq 0.84$. It must be pointed out that the choice of $c\leq 0.84$, on theoretical level, will bring some troubles. The Gibbons-Hawking entropy will thus decrease since the event horizon shrinks, which violates the second law of thermodynamics as well. However, the current observational data indicate that the parameter $c$ in the holographic model seems smaller than 1. Now we reconstruct the phantom potential and the dynamics of the scalar field in light of the holographic dark energy with $c\leq 0.84$. According to the following forms of phantom energy density and pressure \[roph1\] \_=-\^[2]{}+V() \[roph2\] P\_=-\^[2]{}-V() One can easily derive the scalar potential and kinetic energy term as \[v\] V()=(1-w\_)\_ \[phi\]\^[2]{} =-(1+w\_)\_ Differenating Eq.(\[2eq7\]) with respect to the cosmic time $t$, one find \[hdot\]=+ where $\rho=\rho_{m}+\rho_{\Lambda}$ is the total energy density, now using Eqs.(\[2eq1\], \[2eq2\]) \[doro\] =-3H(1+w)where \[weq\]w== Substituting $\dot{\rho}$ into Eq.(\[hdot\]), we obtain \[weq2\] w=-1 Using Eqs.(\[weq\], \[weq2\]), one can rewrite the holographic energy equation of state as \[eqes1\] w\_=(2+3H\^2+) Substituting the above $w_{\Lambda}$ into Eqs.(\[v\], \[phi\]), we obtain \[v1\] V()= \[2+3H\^2(1+\_)+\]\[phi2\]\^[2]{} =M\_[p]{}\^[2]{}\[2+3H\^2(1-\_)+\]In similar to the , we can define $\dot{\phi}^{2}$ and $V(\phi)$ in terms of single function $f(\phi)$ as \[v2\] V()= \[2f’()+3f\^[2]{}()(1+\_)+\]\[phi3\]1=M\_[p]{}\^[2]{} \[2f’()+3f\^[2]{}()(1-\_)+\]In the spatially flat case the Eqs.(\[v2\], \[phi3\]) solved only in case of presence of two scalar potentials $V(\phi)$, and $\omega(\phi)$. Here we have claimed that in the presence of curvature term $\frac{k}{a^2}$, Eqs.(\[v2\], \[phi3\]) may be solved with potential $V(\phi)$. Hence, the following solution are obtained \[sol\] =t, H=f(t)\
From Eq.(\[phi3\]) we get \[keq\]=3f\^[2]{}()(\_-1)-2f’()+ Substituting the above $\frac{k}{a^2}$ into Eq.(\[v2\]), we obtain the scalar potential as \[pottac111\]V()=3M\_[p]{}\^[2]{}\_ f\^2()+ One can check that the solution (\[sol\]) satisfies the following scalar field equation \[phieq\]--3H+V’()=0 Therefore by the above condition, $f(\phi)$ in our model must satisfy following relation \[coneq\] 3f()=V’() In the other hand, using Eqs.(\[holoda\], \[stateq\]) we have \[v4\]V()=(++) \[phi1\] =\[-1++\]\^[1/2]{} Using Eq.(\[phi1\]), we can rewrite Eq.(\[v4\]) as \[v5\]V()=3 M\_[p]{}\^[2]{}\_ H\^2(1+),or in another form as following \[v5\]V()=3 M\_[p]{}\^[2]{}\_ \[f\^[2]{}()+\]=3M\_[p]{}\^[2]{}\_ f\^2()+which is exactly the result (\[pottac111\]).\
From Eq.(\[phi3\]) for the flat case we have \[flat\] 2f’()=3f\^[2]{}()(\_-1)+By derivative of the above equation respect to $\phi$ we obtain \[flat1\] 2f”()=6ff’()(\_-1)+3f\^2\_’then \[flat2\] \_’=+(1-\_)Now using Eqs. (\[coneq\]), (\[v5\]) we have \[flat3\] 2f’\_+f\_’=Substituting $\Omega_{\Lambda}'$ from Eq.(\[flat2\]) into the above equation we obtain \[flat4\] 2f”+6f’f-=0Therefore, $f(\phi)$ must satisfy the above equation in flat case. Using Maple software one can obtain following relation \[w\]\^[f()]{}=+c\_2 where $W$ is the Lambert $W$-function.[^3]
Squared speed for generalized Chaplygin gas and interacting holographic dark energy
===================================================================================
Here we introduce the squared speed of generalized Chaplygin gas as \[sp\] v\_[g]{}\^[2]{}= Using Eq.(14), we have \[sp1\] v\_[g]{}\^[2]{}= For $\alpha<0$, $A> 0$ or $\alpha >0$, $A <0$ (see recent paper [@lo]) ; $v_{g}^{2}<0$ , in this cases the generalized Chaplygin gas model is instable (see also [@my]). The squared speed of interacting holographic dark energy fluid is as \[sp2\] v\_\^[2]{}== where \[sp3\]\_= \_\^[eff]{}\_+w\_\^[eff]{}\_ with \[sp4\] \_\^[eff]{}= H where $x=Ln a$. Using Eq.(27) and following equation \[evol\] ==3\_(1+\_[k]{}-\_) \[+cosn(R\_[h]{}/a)\] We can write \[evol1\] \_\^[eff]{}= \_(1+\_[k]{}-\_) \[+cosn(R\_[h]{}/a)\] Where we have assumed $\frac{d\Omega_{k}}{dx}=0$. Substituting the above equation and Eq.(38) into Eq.(65) we obtain (to see the non-interacting case relation refer to [@my] ) \[sp5\] v\_\^[2]{}=w\_\^[eff]{}- From Eq.(69) one can see that , $\dot{w}_{\rm \Lambda}^{eff}<0$, also as we have mentioned in section 3 if we take $c = 0.84$ , and taking $ \Omega_{\Lambda}=
0.73$, $ \Omega_{k} = 0.01$ for the present time, we obtain $w_{\rm
\Lambda}^{eff}=-1.007 $. One can see from Eq.(40)that in the phantom phase where $\dot{H}>0$ ,$w+1 <0$ , hence we obtain a negative value for squared speed of interacting holographic fluid. Due to this the interacting holographic fluid similare to generalized Chaplygin gas is instable. In a recent paper Myung [@my] has shown that the perfect fluid for holographic dark energy is classically unstable, our result show interacting fluid of holographic dark energy is also unstable. However, in contrast to the Chaplygin gas fluid where the squared speed is always non-negative, for the generalized Chaplygin gas may be one can obtain negative value for the squared speed. Hence the holographic interpretation for generalized Chaplygin gas in contrast with Chaplygin gas is not problematic.
Conclusions
===========
In order to solve cosmological problems and because the lack of our knowledge, for instance to determine what could be the best candidate for DE to explain the accelerated expansion of universe, the cosmologists try to approach to best results as precise as they can by considering all the possibilities they have. Within the different candidates to play the role of the dark energy, the Chaplygin gas, has emerged as a possible unification of dark matter and dark energy, since its cosmological evolution is similar to an initial dust like matter and a cosmological constant for late times. Inspired by the fact that the Chaplygin gas possesses a negative pressure, people [@mas] have undertaken the simple task of studying a FRW cosmology of a universe filled with this type of fluid.\
In this paper, by considering an interaction between generalized Chaplygin gas energy density and CDM, we have obtained the equation of state for the interacting generalized Chaplygin gas energy density in the non-flat universe. Then we have considered a correspondence between the holographic dark energy density and interacting generalized Chaplygin gas energy density in FRW universe. Then we have reconstructed the potential of the scalar field which describe the generalized Chaplygin cosmology. Also we calculated the squared speeds of sound for generalized Chaplygin gas and interacting holographic fluid, then we have shown that interacting holographic fluid similar to the generalized Chaplygin gas is instable.
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[^1]: E-mail: [email protected]
[^2]: As one can see in this case the $A$ and $B$ can change with time. Similar situation can arise when the cosmological constant has dynamic, see for example eq.(12) of [@kmp], (see also [@set4]), according to this equation A=(+\_[m]{}) therefore, if $\Lambda$ vary with time [@shap], $A$ does not remain constant.
[^3]: Consideration of Lambert $W$ function can be traced back to J. Lambert around 1758, and later, it was considered by L. Euler but it was recently established as a special function of mathematics on its own[@31].\
The Lambert $W$ function is defined to be the function satisfying \[w2\]W\[z\]e\^[W\[z\]]{}=z It is a multivalued function defined in general for z complex and assuming values $W[z]$ complex. If $z$ is real and $z < -1/e$, then $W[z]$ is multivalued complex. If $z$ is real and $-1/e \leq z \leq 0$, there are two possible real values of $W[z]$. The one real value of $W[z]$ is the branch satisfying $\leq-1 W[z]$, denoted by $W_{0}[z]$, and it is called the principal branch of the W function. The other branch is $W[z]
\leq -1$ and is denoted by $W_{-1}[z]$. If $z$ is real and $z \geq
0$, there is a single real value for $W[z]$ which also belongs to the principal branch $W_{0}[z]$. Special values of the principal branch of the Lambert $W$ function are $W_{0}[0] = 0 $and $W_{0}[-1/e] = -1$. The Taylor series of $W_{0}[z]$ about $z = 0$ can be found using the Lagrange inversion theorem and is given by [@32] \[w2\]W\[z\] = \_[1]{}\^= z\^[n]{} = z - z\^[2]{} + z\^[3]{} - z\^[4]{} + z\^[5]{} - z\^[6]{} + . . . . The ratio test establishes that this series converges if $|z| < 1/e$.
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---
abstract: 'It is generally accepted that the entropy of an asymptotically de Sitter universe is bounded by the area, in Planck units, of the de Sitter horizon. Based on an analysis of the entropy associated to the vacuum quantum fluctuations, we suggest that the existence of such a holographic bound constitutes a possible explanation for the observed value of the cosmological constant, theoretically justifying a relation proposed $35$ years ago by Zel’dovich.'
author:
- Saulo Carneiro
title: 'On the vacuum entropy and the cosmological constant[^1]'
---
As extensively discussed in the literature [@9], there is a fundamental problem related to the existence of a positive cosmological constant. If $\Lambda$ originates from vacuum energy, its expected value, on the basis of quantum field calculations with a cutoff given by the Planck energy, has the order of $l_{Planck}^{-2} \approx 10^{70}$m$^{-2}$. This is $122$ orders of magnitude greater than the observed value $\Lambda \approx
10^{-52}$m$^{-2}$ [@8]. Even considering smaller cutoffs, as the energy scales of electroweak or QCD phase transitions, the expected value is still over 40 orders of magnitude too high. This huge discrepancy is known as the cosmological constant problem.
We have shown elsewhere [@GS; @preprint] that this problem can be related to other open issues in cosmology, as the large numbers coincidence and the cosmic coincidence problem, with the help of the holographic principle [@11]. In this essay, we will try to show that the application of this principle to an asymptotically de Sitter universe leads to a value for $\Lambda$ in accordance with observation.
In a simplified form, the holographic principle can be described as the extension, to any gravitating system, of the Bekenstein-Hawking formula for black-hole entropy. More precisely, it establishes that the number of degrees of freedom of the system is not bounded by its volume in Planck units, as expected from quantum theories of space-time, but by the area, in Planck units, of its delimiting surface.
The use of this principle to the universe as a whole depends on the definition of such a surface at a cosmic scale. The existence of a positive cosmological constant naturally introduces a characteristic surface of radius $\Lambda^{-1/2}$ [@18]. Another possibility is to use the Hubble horizon, with radius $H^{-1}$, for it defines the scale of causal connections for any observer [@17]. It is clear that in a homogeneous and isotropic, infinite universe, filled with dust and a positive cosmological constant, this last version implies the former, for the Hubble radius tends asymptotically to $\sqrt{3/\Lambda}$. Therefore, in the context of an asymptotically de Sitter universe, the holographic bound establishes that the maximum number of available degrees of freedom is given by $$\label{Nmax} N_{max} \approx \Lambda^{-1}.$$
How much entropy has our universe? We know that the number of baryons is of the order of $10^{80}$, and that the cosmic background radiation contains about $10^{8}$ photons per baryon. It is reasonable to believe that dark matter contributes with a similar figure. But the major contribution to the entropy of matter seems to come from massive black-holes present in galactic nuclei, which represent an entropy of the order of $10^{101}$ [@Penrose].
But what about the entropy associated to the vacuum fluctuations? To put this question in a proper way and to clarify its role in solving the cosmological constant problem, let us inquire more carefully on the origin and meaning of this problem.
The difficulties appear when we calculate the vacuum energy density with the help of quantum field theories in flat space-time. The vacuum energy comes from two kinds of contributions. The first one is the energy associated to the vacuum expectation value of self-interacting fields, as the Higgs field, the quark and gluon condensates of QCD, or any other field associated to vacuum phase transitions. The second kind comes from the zero-point fluctuations of the fields, which lead to infinite results. To regulate them, it is used to impose some energy cutoffs, but, as said before, this leads to results still many orders of magnitude too high compared to observation.
Nevertheless, as discussed by some authors (see for example [@Ralf]), any contribution, infinite or not, to the vacuum energy density predicted by quantum field theories in flat space-time must be exactly canceled by a bare cosmological constant in the Einstein equations, because in the flat space-time the right-hand side of those equations is identically zero. Therefore, to properly pose the problem, we have to calculate the vacuum energy density in a curved background. As before, we find a divergent result as well. But now a physically meaningful (renormalized) value for $\Lambda$ should be obtained by subtracting the Minkowskian result. Since the space-time of our universe is not strongly curved at cosmic scale, we expect to obtain a small value for $\Lambda$, in accordance with observations.
The situation is analog to what occurs in the Casimir effect. There, the zero-point fluctuations of the electromagnetic field give rise to an infinite contribution to the vacuum energy density, inside and outside the region between the Casimir plates. But what is physically meaningful, leading to observable effects, is the difference between the values in the two regions, which is shown to be finite. In our case, the role of Casimir plates is played by gravity.
Calculating the vacuum energy density in a curved background is a difficult task. An example of a rough estimation in the line of the above reasoning was recently given by Schützhold [@Ralf]. He estimates the contribution for $\Lambda$ from the chiral anomaly of QCD in a curved, expanding space-time, obtaining $\Lambda \approx H \Lambda_{QCD}^3$, where $\Lambda_{QCD}$ is the energy scale of the chiral phase transition. In the limiting case of a de Sitter universe, $H \approx \sqrt{\Lambda}$, and his result leads to $$\label{Lambda} \Lambda \approx m^6$$ (where we have made $m = \Lambda_{QCD}$).
This expression was derived $35$ years ago by Zel’dovich, from empirical arguments [@Zel'dovich]. Although it is sensible to the parameter $m$, (\[Lambda\]) leads to the correct order of magnitude: using $\Lambda_{QCD} \approx 150$ MeV, we obtain $\Lambda \approx 10^{-51}$ m$^{-2}$, in good agreement with observation, considering that it was not taken into account numerical factors.
An alternative to circumvent the difficulties involved in quantum field calculations in a curved background is to use a thermodynamic approach, which does not depend on the details of the field dynamics. The idea is to obtain a superior limit for the vacuum entropy, instead of its energy density, and to compare the result with the holographic bound (\[Nmax\]). The reader may argue that it is not trivial to define the number of virtual particles in curved backgrounds. Let us remind, however, that our universe has a quasi-flat space-time. Therefore, the estimation given below can be considered a good approximation.
It is clear that non-trivial vacuum configurations of classical fields (as the vacuum expectation value of the Higgs field or the QCD condensates) do not contribute to the vacuum entropy. In what concerns the zero-point fluctuations, they have, properly speaking, an infinite entropy density, because (if we do not impose any energy cutoff) the number of modes is infinite. But if we regulate their energy, by introducing an ultraviolet cutoff $m$, we also regulate their entropy. A simple estimation of the resulting entropy bound can be derived as follows.
Limiting the energy-momentum space associated to the zero-point fluctuations leads to the quantization of their configuration space, with a minimum size given by $l \approx m^{-1}$. This results in a superior bound to the number of available degrees of freedom in a given volume, say, the volume inside the Hubble horizon. The maximum number $N$ of observable degrees of freedom will be of the order of $V/l^3$, where $V$ is the Hubble volume. That is, $$\label{N} N \approx \left(\frac{m}{H}\right)^3.$$
Now, if we take for $H$ the de Sitter asymptotic value $H \approx
\sqrt{\Lambda}$, we can identify (\[N\]) with the holographic bound (\[Nmax\]). It is easy to verify that this leads to Zel’dovich’s relation (\[Lambda\]).
But why $m$ coincides to the energy scale of the QCD phase transition? The common belief is that a natural cutoff should be given by the Planck energy, for at the Planck scale the classical picture of space-time breaks down. Nevertheless, it is not difficult to see that, equating (\[Nmax\]) to (\[N\]), with $H\approx\sqrt{\Lambda}$ and $m=m_{Planck}$, one obtains a Hubble radius of the order of $l_{Planck}$, which is not consistent with our universe.
One can also argue that the zero-point fluctuations of other fields than quarks and gluons contribute to the entropy as well. It is then intriguing that just $\Lambda_{QCD}$ enters in Zel’dovich’s relation. Note, however, that in a curved space-time the different sectors of the standard model of particles interactions are coupled by gravity. On the other hand, the de Sitter universe is a stationary space-time, and, therefore, all the (interacting) vacuum fields should tend to a state of thermodynamic equilibrium, at the temperature of the last vacuum phase transition. But the last of such transitions was the chiral transition of QCD, at a temperature given by $\Lambda_{QCD}$.
Finally, let us note that the vacuum entropy already dominates the universe entropy. Indeed, taking for $H$ the value observed nowadays, $H \approx 65$ km/(sMpc) [@8], we obtain from (\[N\]) (with $m \approx \Lambda_{QCD}$) $N \approx 10^{122}$, a figure that predominates over the matter entropy referred above, of order $10^{101}$.
${}$
I am thankful to F.S. Navarra, M.R. Robilotta, A. Saa, A.E. Santana, and R. Schützhold, for useful discussions. I am also grateful to Prof. P.F. González-Díaz, G.A. Mena Marugán and L. Garay, from CSIC (Madrid), where I have started to work with this subject.
S. Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989); V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D [**9**]{}, 373 (2000); S.E. Rugh and H. Zinkernagel, Stud. Hist. Philos. Mod. Phys. [**33**]{}, 663 (2002); N. Straumann, astro-ph/0203330; A.D. Dolgov, hep-ph/0203245; T. Padmanabhan, Phys. Rept. [**380**]{}, 235 (2003).
M. Tegmark, M. Zaldarriaga, and A.J.S. Hamilton, Phys. Rev. D [**63**]{}, 043007 (2001); X. Wang, M. Tegmark, and M. Zaldarriaga, Phys. Rev. D [**65**]{}, 123001 (2002).
G.A. Mena Marugán and S. Carneiro, Phys. Rev. D [**65**]{}, 087303 (2002).
S. Carneiro, gr-qc/0206064.
G. ’t Hooft, gr-qc/9310026; L. Susskind, J. Math. Phys. [**36**]{}, 6377 (1995). See also T. Padmanabhan, Gen. Rel. Grav. [**34**]{}, 2029 (2002); Ted Jacobson and R. Parentani, Found. Phys. [**33**]{}, 323 (2003).
R. Bousso, JHEP [**0011**]{}, 038 (2000); Rev. Mod. Phys. [**74**]{}, 825 (2002).
G. Veneziano, Phys. Lett. B [**454**]{}, 22 (1999); R. Easther and D. Lowe, Phys. Rev. Lett. [**82**]{}, 4967 (1999); N. Kaloper and A. Linde, Phys. Rev. D [**60**]{}, 103509 (1999).
R. Penrose, [*The Emperor’s New Mind*]{} (Oxford University Press, 1989), chapter $7$.
R. Schützhold, Phys. Rev. Lett. [**89**]{}, 081302 (2002); Int. J. Mod. Phys. A [**17**]{}, 4359 (2002). The relation between chiral anomaly and vacuum energy density is also discussed in M. Novello, J. Barcelos-Neto and J.M. Salim, Class. Quant. Grav. [**18**]{}, 1261 (2001); [**19**]{}, 3107 (2002).
Ya.B. Zel’dovich, JETP Lett. [**6**]{}, 316 (1967); Sov. Phys. Uspekhi [**11**]{}, 381 (1968). Zel’dovich used for $m$ the proton mass. The idea associating $m$ to the energy scale of the QCD phase transition can also be found in R.A.J. Matthews, Astron. & Geophys. [**39**]{}, 19 (1998).
[^1]: This essay received an “honorable mention" in the 2003 Essay Competition of the Gravity Research Foundation.
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abstract: 'Let $(M,\omega)$ be a connected symplectic manifold on which a connected Lie group $G$ acts properly and in a Hamiltonian fashion with moment map $\mu:M \lra {\mathfrak{g}}^*$. Our purpose is investigate multiplicity-free actions, giving criteria to decide a multiplicity freenes of the action. As an application we give the complete classification of multiplicity-free actions of compact Lie groups acting isometrically and in a Hamiltonian fashion on Hermitian symmetric spaces of noncompact type. Successively we make a connection between multiplicity-free actions on $M$ and multiplicity-free actions on the symplectic reduction and on the symplectic cut, which allow us to give new examples of multiplicity-free actions.'
address: 'Dipartimento di Matematica, Università Politecnica delle Marche, Via Brecce Bianche, 60131, Ancona Italy'
author:
- Leonardo Biliotti
title: 'Some results on Multiplcity-free spaces'
---
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Definition]{} \[thm\][Notation]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Problem]{} \[thm\][Remark]{}
Introduction
============
Let $(M, \omega)$ be a connected symplectic manifold with a proper and Hamiltonian action of a connected Lie group $G$ and let $\mu:
M \lra \gd$ be a corresponding moment map. In 1984 Guillemin and Sternberg [@GS], motivated by geometric quantization, introduced the notion of multiplicity-free space when the ring of the $G-$invariant functions on $M$ is commutative with respect to the Poisson-bracket. The manifold $M$ is called $G$ multiplicity-free space and the $G-$action is called multiplicity-free. The term multiplicity-free comes from the representation theory of Lie groups.
A unitary representation of a Lie group $G$ on a Hilbert space $\H$ is said to be multiplicity-free if each irreducible representation of $G$ occurs with multiplicity zero or one in $\H$. The relationships between the two definitions comes via the theory of geometric quantization. The condition that a unitary representation of $G$ on $\H$ be multiplicity-free is equivalent to the condition that the ring of bounded operators on $\H$ be commutative.
In this paper we investigate multiplicity-free actions, which we also may call coisotropic actions, on a symplectic manifold $M$, imposing that $G$ acts properly and in a Hamiltonian fashion on $M$ and a technical condition, which is needed for applying the symplectic slice, see [@lb] and [@sl], and the symplectic stratification of the reduced space given in [@lb], [@sl], which is the following.
For every $\alpha \in {\mathfrak{g}}^*$, ${\mathfrak{g}}^*$ is the dual of the Lie algebra of $G$, $G\alpha$ is a locally closed coadjoint orbit of $G$. Observe that the condition of a coadjoint orbit being locally closed is automatic for reductive groups and for their semidirect products with vector spaces. There exists an example of a solvable group due to Mautner [@mau] p.$512$, with non-locally closed coadjoint orbits.
One of our purpose is to give Equivalence theorem for multiplicity-free action, which shall allow us to prove that the complete classification of compact Lie groups acting multiplicity-free on irreducible Hermitian symmetric spaces of noncompact type follows from one given in a compact case.
We also prove a reduction principle for multiplicity-free actions and we make a connection between multiplicity-free actions on $M$ and multiplicity-free actions on the symplectic reduction and on the symplectic cut, mainly in a Kähler geometry. As an application we give new examples of multiplicity-free actions on compact Kähler manifolds which are not Hermitian symmetric spaces.
Hamiltonian viewpoint
=====================
Let $M$ be a connected differential manifold equipped with a non-degenerate closed $2-$form $\omega$. The pair $(M,\omega)$ is called symplectic manifold. Here we consider a finite-dimensional connected Lie group acting smoothly and properly on $M$ so that $g^* \omega=\omega$ for all $g \in G,$ i.e. $G$ acts as a group of canonical or symplectic diffeomorphism. For $f,g \in
C^{\infty}(M),$ define $\{ f,g \}= \omega (X_f , X_g),$ where $X_f$ and $X_g$ are the vector fields which is uniquely defined by $df=i_{X_f} \omega$ and $dg= i_{X_g} \omega.$ It follows that $(C^{\infty}(M), \{,\})$ is a Poisson algebra.
The $G$-action is called [*Hamiltonian*]{}, and we said that $G$ acts in a Hamiltonian fashion, if there exists a map $$\mu: M \lra {\mathfrak{g}}^*,$$ which is called moment map, satisfying:
1. for each $X\in {\mathfrak{g}}$ let
- $\mu^{X}: M \lra \R,\ \mu^{X} (p)= \mu(p) (X),$ the component of $\mu$ along $X,$
- $X^\#$ be the vector field on $M$ generated by the one parameter subgroup $\{ \exp (tX):
t \in \R \} \subseteq G$.
Then $${\rm d} \mu^{X}= {\rm i}_{X^\#} \omega$$ i.e. $\mu^{X}$ is a Hamiltonian function for the vector field $X^\#.$
2. $\mu$ is $G-$equivariant, i.e. $\mu (gp)=Ad^* (g)
(\mu(p)),$ where $Ad^*$ is the coadjoint representation on ${\mathfrak{g}}^*$.
Let $x \in M$ and ${\rm d} \mu_x: T_x M \lra T_{\mu(x)} {\mathfrak{g}}^*$ be the differential of $\mu$ at $x$. Then $$\label{nucleo}
Ker {\rm d} \mu_x = (T_x G(x))^{\perp_{\omega}}:= \{ v \in T_x M:
\omega (v,w)=0,\ \forall w \in T_x G(x)\}.$$ If we restrict $\mu$ to a $G-$orbit $Gx$, then we have the homogeneous fibration $\mu: Gx \lra Ad^*(G)\mu(x)$ and the restriction of the ambient symplectic form $\omega$ on the orbit $Gx$ is the pullback by the moment map $\mu$ of the symplectic form on the coadjoint orbit through $\mu(x)$ $$\label{KKS}
\omega_{|_{Gx}}= \mu^* (\omega_{ Ad^*(G) \mu(x)})_{|_{Gx}}$$ see [@lb] p. 211, where $\omega_{G \mu(x)}$ is the Kirillov-Konstant-Souriau (KKS) symplectic form on the coadjoint orbit of $\mu(x)$ in ${\mathfrak{g}}^*$. This implies the following result.
\[c1\] A $G$-orbit $Gx$ is a symplectic submanifold of $M$ if and only if the moment map restricted to $Gx$ into $G\mu(x)$, $\mu_{|Gx}:Gx \longrightarrow G\mu(x)$, is a covering map. In particular if $G$ is a compact Lie group then $G_x=G_{\mu(x)}$ and $\mu_{|_{Gx}}:Gx \longrightarrow G\mu(x)$ is a diffeomorphism onto.
The first affirmation follows immediately from (\[KKS\]). If $G$ is compact, coadjoint orbits are of the form $G/C(T)$, where $C(T)$ is the centralizer of the torus $T$. In particular such orbits are simply connected, form which one may deduce $G_x=G_{\mu(x)}$.
Multiplicity-free spaces
========================
Let $(M,\omega)$ be a connected symplectic manifold and let $G$ be a connected Lie group acting properly and as a group of symplectic diffeomorphism on $M$.
The $G$-action is called *multiplicity-free*, $M$ is called a $G$ multiplicity-free space, if the space of invariant function $C^{\infty}(M)^G$ is a commutative Lie algebra with respect the Poisson bracket.
By the definition follows that if $K \subset G$ and $M$ is a $K$ multiplicity-free space then $M$ is a $G$ multiplicity-free space as well.
The multiplicity-free actions are also called *coisotropic actions*. This is justified by the following discussion.
A submanifold $N$ of a symplectic manifold $(M,\omega)$ is said to be *coisotropic* if and only if for every $x\in N$, $(T_x
N)^{\perp_{\omega}} \subset T_x N$. In particular a $G$-action on $M$ is called coisotropic if and only if there exists an open dense subset $U\subset M$ with $Gx$ coisotropic for every $x\in
U$.
$M$ is a $G$ multiplicity-free space if and only the $G$-action on $M$ is coisotropic.
First of all we note the following easy fact: if $f\in
C^{\infty}(M)^G$ then for every $\xi \in {\mathfrak{g}}$ we have $\{f,
f_{\xi} \}=0,$ where $f_{\xi}$ is defined by $f_{\xi}(x)=\mu(x)(\xi).$
Assume that a generic orbit $Gx$ is coisotropic.
Let $f,g \in C^{\infty}(M)^G.$ Since $\{ f, f_{\xi} \}=\{g, f_\xi \}=0$ we have $X_f, X_g \in (T_x Gx)^{\perp_{\omega}}
\subset T_x Gx,$ since $Gx$ is coisotropic, for every $x\in U.$ Hence $$\{f,g \}(x) =\omega(X_f,X_g)=0, \ \forall x \in U,$$ which implies $\{f,g\}=0$, since $U$ is an open dense subset.
Vice-versa, let $x\in M$ be a regular point. By the slice-theorem there are functions $f_1,\ldots,f_k \in C^{\infty}(M)^G$ with $df_1 \wedge \ldots \wedge df_k \neq 0$ in some neighborhood $W$ of $Gx$ and $$Gx=\{ y \in W : f_1(y)=\ldots=f_k(y)=0 \}.$$ From $\{f_i,f_j \}=0$ one may deduce that $X_{f_i} \in T_x Gx.$ Therefore, since $X_{f_i} \in (T_x Gx)^{\perp_{\omega}},$ $i=1,
\ldots, k$ and $X_{f_1},\ldots ,X_{f_k}$ are independent in $W$, we have that $Gx$ is a coisotropic submanifold.
Our proof is essentially one given in [@HW]. However in [@HW] the authors assumed that $G$ is a compact Lie group, their proof works also when the $G$-action is a proper action.
It is standard that given a $G$-orbit $Gx=G/G_x$, study the slice representation, i.e. the linear representation of $G_x$ induced from the $G$-action on $T_x M / T_x Gx$. In [@HW] p. 274, as a consequence of the arguments used in the Restriction Lemma, it was proved that given a complex orbit $Gp=G/G_p$ then $G$ acts coisotropically on $M$ if and only if $G_p$ acts coisotropically on the slice, whenever $M$ is a compact Kähler manifold and $G$ is a subgroup of its full isometric group. Here, we give the same result in symplectic context.
Let $(M,\omega)$ be a symplectic manifold and let $G$ be a Lie group which acts properly and in a Hamiltonian fashion on $M$ with moment map $\mu: M \lra {\mathfrak{g}}^*$. Let $Gx$ be a symplectic orbit. If $M$ is a $G$ multiplicity-free space then the slice representation is a multiplicity-free representation. Moreover, if $G$ is compact the vice-versa holds as well.
It follows from symplectic slice, see [@lb], [@or] [@sl].
There exists a neighborhood of the orbit $Gx$ which is $G$-equivariantly symplectomorphic to a neighborhood of the zero section of the symplectic manifold $ (Y=G \times_{G_x}( ({\mathfrak{g}}_{\beta} / {\mathfrak{g}}_x )^* \oplus V), \tau)$, see [@lb], [@sl] for an explicit description of the symplectic form $\tau$, with a $G$-moment map $\mu$ given by $$\mu([g,m,v])=Ad(g)(\beta+j(m)+ i(\mu_{V}(v))),$$ where $\beta=\mu(x)$, $i: {\mathfrak{g}}^*_x \hookrightarrow {\mathfrak{g}}^*$ is the transpose of the projection $p:{\mathfrak{g}} \lra {\mathfrak{g}}_x$, $j:( {\mathfrak{g}}_{\beta} / {\mathfrak{g}}_x )^* \hookrightarrow {\mathfrak{g}}^o_x$ (${\mathfrak{g}}^o_x$ is the annihilator of ${\mathfrak{g}}_x$ in ${\mathfrak{g}}^*$) is defined by a choice of a $G_x$-equivariant splitting and finally $\mu_V$ is the moment map of the $G_x$-action on the symplectic subspace $V$ of $(T_x Gx, \omega(x))$. Note that $V$ is isomorphic to the quotient $((T_x Gx)^{\perp_{\omega}} / ((T_x Gx)^{\perp_{\omega}}
\cap T_x Gx))$. In the sequel we denote by $\omega_V=
\omega(x)_{|_{V}}$.
Since $Gx$ is symplectic, $ (Y=G \times_{G_x} V, \tau)$ and the moment map $\mu$ becomes $$\mu([g,m])=Ad^*(g)(\beta+ i(\mu_{V}(m))),$$ Assume that $M$ is a $G$-multiplicity-free space and let $y=[e,m]\in Y$ be such that $Gy$ is coisotropic.
Let $Y \in Ker {\rm d} (\mu_{V})_m$. Noting ${\rm
d}\mu_{[e,m]}(0,Y)={\rm d}( \mu_V)_m (Y)=0$, which implies, from (\[nucleo\]), $Y\in (T_y Gy)^{\perp_{\omega}} \subset T_y Gy$. This claims $Y\in T_y Gy \cap V= T_m G_x m$, i.e the slice representation is multiplicity-free.
Assume that $G$ is a compact Lie group. It is well known $$\rm{cohom}(G,M)=\rm{cohom}(G_x ,V),$$ which follows from the classical slice theorem for proper actions [@Path], and rk($G$)=rk($G_x$), since $Gx$ is a symplectic manifold, where cohom($G$,$M$) denotes the codimension of a principal orbit and for every compact group $K$, rk($K$) denotes the rank, namely the dimension of the maximal torus. If $G_x$ acts multiplicity-free on $V$ then cohom($G_x,V)$=rk($G_x$)-rk($G_{{\rm princ}}),$ see [@HW], where $G_{{\rm princ}}$ is the principal isotropy subgroup of the action, which implies that $${\rm cohom}(G,M)=rk(G)-rk(G_{{\rm princ}}).$$ Therefore, from Theorem $3$ [@HW] p. $269$, we get $G$ acts multiplicity-free on $M$.
\[hnc\] Let $M$ be an irreducible Hermitian irreducible symmetric space of non compact type. Let $G$ be a compact group which acts in a Hamiltonian fashion on $M$. Then $G$ acts multiplicity-free on $M$ if and only if it acts multiplicity-free on $M^*$, the corresponding irreducible Hermitian symmetric space of compact type.
Since $G$ is compact it has a fixed point $x\in M$, from a Theorem of Cartan, see [@he]. Hence $G$ acts multiplicity-free on $M$ if and only if $G$ acts multiplicity-free on the tangent space at $x$ if and only if it acts multiplicity free on $M^*$.
Corollary \[hnc\] classifies completely the compact Lie groups acting in a Hamiltonian fashion and coisotropically on the irreducible Hermitian symmetric spaces of noncompact type, due the results proved in [@BG], [@Bi], [@PTh].
Equivalence Theorems for multiplicity-free action
=================================================
From now on we assume that $(M,\omega)$ is a connected symplectic manifold acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. We denote by $\mu:M \lra {\mathfrak{g}}^*$ the corresponding moment map for the $G$-action on $M$.
Let $\alpha \in \gd$. We define the corresponding reduced space $${M_{\alpha}}=\mu^{-1}(G\alpha)/ G,$$ to be the topological quotient of the subset $\mu^{-1}(G \alpha)$ of $M$ by the action of $G.$ It is well known, see [@acg], [@lb], [@sl], that the reduced space ${M_{\alpha}}$ is a union of a symplectic manifolds and it can be endowed with a Poisson structure which arise from Poisson structure on the orbits space. Hence ${M_{\alpha}}$ is a symplectic stratified space and the manifolds which decompose ${M_{\alpha}}$ are called pieces.
Here we analyze the case when $G=G_1 \times G_2,$ where $G_i, \ i=1,2$ are closed connected subgroup of $G$. We assume also that the coadjoint orbits of $G, G_1$ and $G_2$ are locally closed spaces.
Obviously ${\mathfrak{g}}^*= {\mathfrak{g}}_1^* \oplus {\mathfrak{g}}_2^*$ and the moment map $\mu=\mu_1 + \mu_2$, where $\mu_i$ is the corresponding moment map for the $G_i$-action on $M,$ $i=1,2. $ Since $\mu$ is $G$-equivariant, we have that $\mu_1$ is invariant under $G_2$ and $\mu_2$ is invariant under $G_1$.
Let $\alpha=\alpha_1 + \alpha_2.$ The $G_1$-action on the pieces of the reduced space ${M_{\alpha_2}}=\mu_2^{-1}(G_2
\alpha_2)/G_2$ is symplectic. These moment maps on the pieces fit together to form an application $$\mu_{1,2}: {M_{\alpha_{2}}} \lra
{\mathfrak{g}}^*_1 ,$$ such that $\mu_{1,2} =\mu_1 \circ \pi_2,$ where $\pi_2$ is the projection on ${M_{\alpha_2}}.$ Clearly $G_2$ acts on the reduced space ${M_{\alpha_1}}=\mu_1^{-1}(G_1 \alpha_1) /G_1$ with moment map $$\mu_{2,1}: {M_{\alpha_{1}}} \lra {\mathfrak{g}}^*_2$$ such that $\mu_{2,1}=\mu_2 \circ \pi_1,$ where $\pi_1$ is the projection on ${M_{\alpha_1}}.$
We introduce the notion of multiplicity-free space for the reduced space. Indeed, we say that the $G_1$-action on ${M_{\alpha_2}}$ is multiplicity-free if the ring of $G_1$-invariant functions of ${M_{\alpha_{2}}}$, [@acg], is a commutative Poisson algebra.
We may also define the reduced space with respect the $G_1$-action on ${M_{\alpha_2}}$ to be $$(M_{\alpha_2})_{\alpha_1}= \mu_{1,2}^{-1}(G_1 \alpha_1 )/G_1.$$ The same definition holds for the $G_2$-action on ${M_{\alpha_1}}.$
Now, we shall give a criterion for a $G$-action to be a multiplicity-free action. We begin with the following lemma.
Let $(M,\omega)$ be a symplectic manifold and let $G=G_1 \times G_2$ be a connected Lie group which acts in a Hamiltonian fashion on $M$. Let $\alpha_1 \in {\mathfrak{g}}_1^*.$ Then the $G_2$-action on ${M_{\alpha_1}}$ is multiplicity-free if and only if for every $\alpha_2 \in {\mathfrak{g}}_2^*$ the reduced space $({M_{\alpha_1}})_{\alpha_2}$ are points
In the sequel we always refer to [@acg] and [@lb].
Let $\mu_2 : M \lra {\mathfrak{g}}_1^*$ be the moment map of the $G_1$-action and let $\mu_{2,1}: {M_{\alpha_1}} \lra {\mathfrak{g}}_2^*$ be the corresponding moment map of the $G_2$-action on the reduced space. We recall that the smooth function on the reduced spaces are defined by $$C^{\infty}({M_{\alpha_1}})= C^{\infty}(M)^{G_1} |_{\mu_1^{-1} (G_1
\alpha_1)}$$ and the reduced space is a locally finite union of symplectic manifolds (symplectic pieces) which are the following manifolds.
Let $H$ be a subgroup of $G.$ The set $M^{(H)}$ of orbit of type $H,$ i.e. the set of points which orbits are isomorphic to $G/H,$ is a submanifold of $M$ [@Path]. The set $(\mu_1^{-1}
(G_1 \alpha_1 ) \cap M^{(H)})
$ is a submanifold of constant rank and the quotient $$(
{M_{\alpha_1}})^{(H)}=(\mu_1^{-1} (G_1 \alpha_1 ) \cap
M^{(H)})/G_1,$$ is a symplectic manifold which inclusion $(
{M_{\alpha_1}})^{(H)} \hookrightarrow {M_{\alpha_1}}$ is a Poisson map [@lb].
The $G_2$-action preserves $({M_{\alpha_1}})^{(H)},$ and the following topological space $$( ( {M_{\alpha_1}})^{(H)} \cap \mu_{2,1}^{-1}(G_2 \alpha_2))
/G_2=\cup_{i \in I} S_i$$ is a stratified symplectic manifold which restrictions map $$r^{H}_{\alpha_2}:C^{\infty}(( {M_{\alpha_1}})^{(H)} \cap
\mu_{2,1}^{-1}(G_1 \alpha_1) )^{G_2} \lra C^{\infty}(S_i)$$ are Poisson and surjective. Therefore, if $C^{\infty}({M_{\alpha_1}})^{G_2}$ is abelian, the algebra $C^{\infty}(S_i),\ i\in I$ must be abelian, and $S_i$ must be discrete and therefore a points.
Vice-versa, if all reduced spaces are points then $r^{H}_{\alpha_2}(\{f,g \})=0$ for all $\alpha_2 \in g_2^*,$ and every $H$ subgroup of $G_1,$ so that $\{ f,g \}=0.$
\[ghigi\] Let $(M, \omega)$ be a symplectic manifold with a proper and Hamiltonian action of a connected Lie group $G=G_1 \times G_2.$ Assume also that $G_i,\ i=1,2$ are closed connected Lie group and the coadjoint orbits of $G, G_1$ and $G_2$ are locally closed. Hence $M$ is a $G$ multiplicity-free space if and only if for every $\alpha=\alpha_1+ \alpha_2 \in \gd$ ${M_{\alpha_1}}$ is a $G_1$ multiplicity-free space if and only if ${M_{\alpha_2}}$ is a $G_2$ multiplicity-free space.
It follows immediately from the above result. Indeed, it is easy to check that ${M_{\alpha}}=\mu^{-1}(G\alpha) /G$ is homemorphic to $(M_{\alpha_1})_{\alpha_2}$ or equivalently is homemorphic to $(M_{\alpha_2})_{\alpha_1};$ the homeomorphism is given by the natural application $$( M_{\alpha_1} )_{\alpha_2} \longrightarrow {M_{\alpha}}, \ \ \
[[x]] \longrightarrow [x]$$ which preserves the symplectic pieces, concluding the prove.
Theorem \[ghigi\] is not difficult to prove. However, from Theorem \[ghigi\], we may deduce some interesting facts.
\[restriction\] Let $N$ be closed $G-$invariant symplectic submanifold of $M.$ If $M$ is a $G$ multiplicity-free space then so is $N$.
$G$ acts on $N$ in a Hamiltonian fashion with moment map $\overline{\mu}: N \lra {\mathfrak{g}}^* ,$ $\overline{\mu}(x) =\mu(x),$ which is the restriction of $\mu$ on $N$. In particular, for every $\alpha \in {\mathfrak{g}}^*$ the reduced space $N_{\alpha} \subset
{M_{\alpha}},$ which implies that the topological space $N_{\alpha}$ are points.
If $G$ is a compact Lie group acting multiplicity-free on $M$ then $$M^G:=\{x \in M: Gx=x \},$$ must be a finite set.
Another interesting application of Theorem \[ghigi\] is the following result.
\[po\] Let $(M,\omega)$ be a symplectic manifold with a Hamiltonian circle action. Let $K$ be a connected Lie group which acts properly and in a Hamiltonian fashion on $M.$ Assume also that the $K$-action commutes with the circle action. If $M$ is a $K$ multiplicity-free space then so is the $K$-action induced on any symplectic cut.
We briefly describe the symplectic cut, see [@lerman] and [@gbl] for more details.
Let $(M,\omega)$ be a symplectic manifold with a Hamiltonian action of a one-dimensional torus $\To^1$ with moment map $\phi: M \lra \R.$ We consider the symplectic manifold $N=M \times \C$, equipped with the symplectic form $\omega - \frac{i}{2} dz \wedge d \overline{z}.$ $\To^1$ acts on $N$ with its product action and this action is a Hamiltonian action with moment map $\psi(p,z)=\phi(p)+ |z|^2.$ The reduced space $\mk= \psi^{-1}(\lambda)/ S^1,$ $\lambda \in \R$ is the symplectic cut.
The $K-$action on $M \times \C$ is given by $k(m,z)=(km,z).$ Since the $K-$action commutes with the $\To^1$-action, it induces a Hamiltonian action on the symplectic cut with moment map $\overline{\mu}([x,z])=\mu(x)$ where $\mu$ is the moment map of the $K-$action on $M.$ Note that $K \times \To^1$-action is multiplicity-free on $M\times \C$ if and only if the $K$-action is on $M$. Therefore, if $K$ acts multiplicity-free on $M$, from Theorem \[ghigi\], $K$ acts multiplicity-free on the symplectic cut.
Let $H$ be a compact subgroup of $G$ and let $N(H)$ be its normalizer in $G$. It is well-known that the Lie group $L=N(H)/H$ acts freely and properly on the submanifold $M^{H}:=\{ x\in M: G_x=H \}$ [@Path]. Moreover, since $T_x M^H = (T_x M )^H$, $M^H$ is symplectic.
In [@lb] it was proved that $L$ acts in a Hamiltonian fashion on $M^{H}$, the dual of the Lie algebra of $L$ is naturally isomorphic to the subspace $({\mathfrak{h}}^o)^H$ of the $H$-fixed vectors in the annihilator of ${\mathfrak{h}}$=Lie($H$) in ${\mathfrak{g}}^*$. Furthermore, given $\alpha=\mu(x)$, where $x\in M^{H}$, then $$G\mu^{-1}(\alpha) \cap M^{H}/ G \cong (M^{H})_{\alpha_o },$$ $\cong$ means symplectically diffeomorphic, where $\alpha_o=\pi(\alpha)$ and $\pi: ({\mathfrak{h}}^o)^{H} \lra l^*$ be the natural projection. This proves that if $M$ is a $G$ multiplicity-free space then, from Theorem \[ghigi\], $M^H$ is a $L$ multiplicity-free space. Hence, we have the following result.
\[oba\] Let $H$ be a compact subgroup of $G$. If $G$ acts coisotropically on $M$ then $L$ acts coisotropically on $M^{H}$.
Next, we claim the reduction principle for a multiplicity-free action.
(**Reduction principle**) Let $G$ be a connected Lie group acting properly on a connected symplectic manifold $M$. Let $H$ be a principal isotropy for the $G$-action. Then $G$ acts coisotropically on $M$ if and only if $L=N(H)/H$ acts coisotropically on $M^{H}$.
Since the $G$-action is proper and preserves $\omega$, there exists a $G$-invariant almost complex structure $J$, i.e. $J:TM \lra TM$ be such that $J^2 =-Id$, adapted to $\omega$, i.e. $\omega(J\cdot, J\cdot)=\omega(\cdot, \cdot )$ and $g=\omega(\cdot, J \cdot)$ is a Riemannian metric, see [@lb].
Now let $H$ be a principal isotropy and let $L=N(H)/H$. It is well-known that $$M^{H} \cong N(H)/H \times (T_{x}Gx)^{\perp_g },$$ see [@Path], which implies $T_x Lx= (T_x Gx )^H$.
Since $(T_x Gx )^{\perp_{\omega}}=J(( T_x Gx )^{\perp_g})$ and $(T_x Gx)^{\perp_g}\subset (T_x M )^H$, recall that $H$ acts trivially on the slice due the fact that $Gx$ is a principal orbit, we get that $$J( (T_x Gx )^{\perp_g }) \subset T_x Gx \Leftrightarrow
J( (T_x Gx )^{\perp_g }) \subset (T_x Gx )^H .$$ Therefore, recall that $(T_x Lx )^{{\perp_g }_{|_{T_x M^H }}}=(T_x Gx )^{\perp_g }$ since $Gx$ is principal, we have that $Gx$ is coisotropic in $M$ if and only if $Lx$ is in $M^H$.
We conclude this section giving the Equivariant mapping lemma, see [@HW], in a symplectic context.
\[equi\] Let $(M,\omega)$ and $(N, \omega_o)$ be connected symplectic manifolds and $G$ be a connected Lie group acting on both manifolds properly, and in a Hamiltonian fashion. Let $\phi: M
\lra N$ be a smooth surjective $G-$equivariant map with ${\rm rank} \phi=\dim N.$ Assume that for every $p \in M,$ $Ker d
\phi_p$ is a symplectic subspace and $$d\phi_p : ((Ker
\phi)^{\perp_{\omega}}, \omega) \lra (T_p N, \omega_o)$$ is a symplectomorphism. If $M$ is a $G$ multiplicity-free space then so is $N$.
Let $f \in C^{\infty}(N)^G .$ The function $\tilde{f}= f \circ
\phi$ is a $G$-invariant function of $M.$ Take the vector field $X_f$ such that $df= i_{X_f } \omega_o.$ By assumption the vector field $\tilde X \in (Ker \phi)^{\perp_{\o}}$ such that $d\phi(
\tilde X )= X_f$ is the symplectic gradient of $\tilde f.$ Hence, given $f,g \in C^{\infty}(N)^G$ there exist $\tilde f,$ $\tilde g
\in C^{\infty}(M)^G$ such that $\{ f, g \}(\phi(x) )= \{\tilde f ,
\tilde g \}(x)$ which conclude our proof.
multiplicity-free spaces in Kähler geometry {#pussavia}
===========================================
In the sequel we shall assume that $M$ is a compact Kähler manifold and $G$ is a closed subgroup of its full isometry group acting in a Hamiltonian fashion on $M$. Note that this action is automatically holomorphic by a Theorem of Konstant (see [@KN] p.242).
In [@Pu] it was introduced the [*homogeneity rank*]{} of $(G,M)$ as the following integer $${\rm homrk}(G,M)= {\rm rk}(G)- {\rm rk} (G_{{\rm princ}})- {\rm
cohom}(G,M),$$ where $G_{{\rm princ}}$ is the principal isotropy subgroup of the action, the integer cohom$(G,M)$ is the codimension of the principal orbit and, for a compact Lie group $H,$ [rk]{}$(H)$ denotes the rank, namely the dimension of the maximal torus.
In [@HW] it was proved that a group $G$ acts multiplicity-free on $M$ if and only if the homogeneity rank vanishes.
Our purpose is to make a connection between homogeneity rank of $(G,M)$ and homogeneity rank of $(G, \mr)$, where $\mr$ is the reduced space obtained from a torus action.
Let $K$ be a semisimple compact Lie subgroup of $G$ and let $\To^k$ be a $k$-dimensional connected torus which centralizes $K$ in $G$, i.e. $\To^k
\subset C_G (K)$. In the sequel we denote by $$\phi:M \lra {\mathfrak{k}} \oplus {\mathfrak{t}}_k,$$ where ${\mathfrak{t}}_k=$Lie$(\To^k)$, the moment map of the $\To^k \cdot K$-action on $M$ and with $\mu$, respectively with $\psi$, the moment map of the $K$-action on $M$, respectively a moment map of the $\To^k$-action on $M$.
Let $\lambda \in {\mathfrak{t}}_k$ be such that $\To^k$ acts freely on $\psi^{-1}(\lambda)$. The symplectic reduction $$(M_{\lambda}= \psi^{-1}(\lambda) / \To^k, \omega_{\lambda}),$$ is a symplectic manifold and $\omega_{\lambda}$ satisfies $$\pi^*(\omega_{\lambda})= i^*(\omega),$$ where $\pi$ is the natural projection $\psi^{-1}(\lambda)
\stackrel{\pi}{\lra} \mr$ and $i$ is the inclusion $\mk \hookrightarrow M$, [@Ca], [@smd]. Since $K$ commutes with $\To^k$, $K$ acts on $M_{\lambda}$ in a Hamiltonian fashion with moment map $$\overline{\mu}: M_{\lambda} \lra {\mathfrak{k}} , \ \overline\mu ([x])= \mu(x).$$ Indeed, It is easy to see that $\overline{\mu}$ is $K-$equivariant. Hence the problem is then restricted to verify that for every $Z \in {\mathfrak{k}}$ we have $d \overline{\psi}^{Z} = i_{ \tilde {Z}^\# } \omr,$ where $\tilde{Z}^\#$ is the vector field on $\mr$ generated by the one parameter subgroup $\exp(tZ).$
Let $X \in T_{[x]} \mr$ and let $\tilde{X} \in T_{x} \mu^{-1}(\lambda)$ such that $\pi_* (\tilde{X}) = X.$ Since $\pi_* (Z^\#)=\tilde{Z}^\#$, where $Z^\#$ is the Killing field induced from $Z$ in $M$, it follows $$d \overline{\psi}^Z (X) = d \psi^Z (\tilde{X})=
i_{Z^\#} \omega (\tilde{X})= \pi^* \omr (Z^\#, \tilde{X})
= i_{\tilde{Z}^\#} \omr (X),$$ thus $K$ acts in a Hamiltonian fashion on $\mk.$
Let $[p]\in M_{\lambda}$. It is easy to see that $k[p]=[p]$ if and only if there exists $r(k)\in \To^k$ such that $kp=r(k)p$, which is unique since $\To^k$ acts freely on $\psi^{-1}(\lambda)$. This means that the following application $$\label{pq}
K_{[p]} \stackrel{F}{\lra} (\To^k \cdot K)_p,\ F(k)=k r(k)^{-1},$$ is a covering map, due the fact that $K$ is semisimple. Hence $$\label{pw}
\dim K[p]= \dim (\To^k \cdot K)p - \dim \To^k.$$ Since $M$ is a compact manifold, we may extend the $\To^k$-action to a holomorphic action of $(\C^*)^n$ which commutes with the $K-$action. This can be easily deduced from the following easy fact: let $X,Y$ be holomorphic fields. If $[X,Y]=0$ then $[X,J(Y)]=0$, since $[X,J(Y)]=J([X,Y])=0$, due the fact that $M$ is Kähler. In particular the infinitesimal generatores of the $K-$action commute with ones of the $(\C^* )^n -$action, proving that the two action commute as well.
The set $(\C^* )^n \cdot
\psi^{-1}(\lambda)$ is an open subset. Indeed, for every $p \in
\psi^{-1}(\lambda)$, denoting with ${\mathfrak{z}}_p$ the vector subspace of $T_p M$ spanned by the infinitesimal generator of the $\To^k$-action on $M$, we have $T_p \psi^{-1}(\lambda) \oplus
J({\mathfrak{z}}_p)=T_p M$, since $\lambda$ is a regular value, which implies our affirmation. In particular the open subset $(\C^*)^n \cdot \psi^{-1}(\lambda)$ contains regular points. Hence there exists an element $$q=\rho_1 \cdots \rho_n \exp(i \theta_1) \cdots \exp(i \theta_n)p
=\rho \exp(i \Theta)p \in (\C^*)^n \cdot \psi^{-1}(\lambda),$$ such that the orbit $( \To^k\cdot K) q$ is a principal orbit. Since $K$ commutes with $(\C^* )^n$, we get that $(\To^k \cdot K)_p = (\To^k \cdot K )_q$ which means that $p$, which lies in $\psi^{-1}(\lambda)$, is a regular point. Therefore, from (\[pq\]) we deduce that $K[p]$ is a principal orbit and from (\[pw\]), we get $${\rm homrk} (K, \mr)= {\rm homrk}(\To^k \cdot K, M),$$ which proves the following result.
\[red\] Let $G$ be a connected compact Lie group acting isometrically and in a Hamiltonian fashion on a compact Kähler manifold $M$. Let $K$ be a compact semisimple Lie group of $G$ which centralizer in $G$ contains a $k$-dimensional connected torus $\To^k$. Let $\lambda \in {\mathfrak{t}}_k$ be such that $\To^k$ acts freely on $\psi^{-1}(\lambda)$, where $\psi$ is a moment map of $\To^k$-action on $M$. Then the $(\To^k \cdot K)$-action is coisotropic on $M$ if and only if the $K$-action is on $M_{\lambda}=\psi^{-1}(\lambda)/ \To^k$.
If we consider a one-dimensional torus $\To^1$ we may investigate the $K$-action on the Kähler cut $\mk$ obtained from the $\To^1$-action. Here we only assume that the $K$-action commutes with the $\To^1$-action. It is easy to check that $K_{[v,z]}=K_v$ when $z \neq 0$. Since $\{[(v,z]\in \mk : \ z\neq 0 \}$ is an open subset, one may deduce that $$\label{k2} {\rm homrk}(K,M)={\rm
homrk}(K,\mk).$$ Hence $K$ acts coisotropically on $M$ if and only if $K$ acts on $\mk$ which proves the following result
\[cut\] Let $G$ be a connected compact Lie group acting isometrically and in a Hamiltonian fashion on a compact Kähler manifold $M$. Let $K$ be a compact Lie group of $G$ whose centralizer in $G$ contains a one-dimensional torus $\To^1$. Let $\lambda \in {\mathfrak{t}}_1$ be such that $\To^1$ acts freely on $\psi^{-1}(\lambda)$, where $\psi$ is a moment map of $\To^1$-action on $M$. Then $K$-action is coisotropic on $M$ if and only if the $K$-action on the Kähler cut $\mr$ is.
Let $\omega= \sqrt{-1} \sum_{i=1}^{n+1} d z_i \wedge d
\overline{z}_i$ be a Kähler form on $\C^{n+1}.$ Consider the following $S^1 -$action on $(\C^{n+1}, \omega):$ $$t \in S^1 \mapsto \theta_t={\rm multiplication\ by\ }e^{it}.$$ The action is Hamiltonian with moment map $\mu(z)= | z |^2+$ constant. If we choose the constant to be $-1,$ then $\mu^{-1}(0)=S^{2n+1}$ is the unit sphere on which $S^1$ acts freely and the Kähler reduction $\mu^{-1}(0)/S^1$ is just $\P^n
( \C)={{\rm SU}}(n+1) / {{{\rm S}({\rm U}({1}) \times {\rm U}({n}))}}.$ Therefore, by Proposition \[red\], a compact connected Lie subgroup $K$ of ${{\rm SU}}(n+1)$ acts multiplicity-free on $\P^n(\C)$ if and only if $S^1 \cdot K$ acts multiplicity-free on $\C^n.$ Kac [@Kac] and Benson and Ratclif [@BR] have given the complete classification of linear coisotropic actions, from which one has the full classification of coisotropic actions on $\P^n (\C).$ If we consider the cut of $\C^{n+1}$ at $\lambda>0$, with respect the above ${\rm S}^1$-action, we obtain, see [@gbl], $\P^{n+1}(\C)$, with $\lambda$ times the Fubini-Study metric. Hence $G\subset {{\rm SU}}(n+1)$ acts coisotropically on $\P^{n+1}(\C)$ if and only if it acts coisotropically on $\C^{n+1}$.
Multiplicity-free actions on compact non Hermitian symmetric spaces {#quo}
===================================================================
Let $\To^1$ acting on $\P^n (\C),$ as $$(t, [z_o, \ldots, z_n]) \longrightarrow [z_o,\cdots ,tz_n] .$$ This is a Hamiltonian action with moment map $$\phi ([z_o, \ldots, z_n] )= \frac{1}{2} \frac{\pr z_n \pr^2}{\pr
z_0 \pr^2 + \cdots + \pr z_n \pr^2} \ .$$ Note that $\phi([0, \ldots , 1])$ is the maximum value of $\phi$ and $\phi^{-1} (\frac{1}{2})=[0, \ldots ,
1].$ Hence (see [@gbl] page 5) if $\lambda=\frac{1}{2}-
\epsilon,$ $\epsilon \cong 0,$ then the Kähler cut $\P^n (\C)^{\lambda}$ is the blow up of $\P^n (\C)$ at $[0, \ldots , 1].$
Let $\To^n$ be a torus acting on $\P^n (\C)$ as follows $$(t_1,\ldots,t_n)([z_o, \ldots,z_n])=
(t_1 z_o, t_2 z_1, \ldots, t_n z_{n-1}, z_n]) .$$ This action is Hamiltonian and the principal orbits are Lagrangian; therefore $\To^n$ acts coisotropically on $\P^n (\C)$. Since $\To^n $-action commutes with the above $\To^1$-action, from Proposition \[cut\], we get $\To^n$ acts coisotropically on the blow-up at one point of $\P^n (\C)$.
We may generalize the above procedure as follows.
Let $G$ be a connected compact Lie group acting coistropically on a compact Kähler manifold. It is well-known that, see [@he], ${\mathfrak{g}}= {\mathfrak{z(g)}}\oplus [{\mathfrak{g}}, {\mathfrak{g}}]$, and if we denote by $G_{{\rm ss}}$ the semisimple connected compact Lie group whose Lie algebra is $[{\mathfrak{g}} , {\mathfrak{g}}]$, then $$G= Z(G) \cdot G_{{\rm ss}}.$$ From Proposition \[red\], if $G_{ss}$ acts coisotropically on $M$, then so is the $G_{ss}$-action induced on $\mk$, the reduced space obtained from $\To^k \subset Z(G)$.
Let $\To^1$ be a one-dimensional torus which lies on $Z(G)$. If $K\subset G$ is a compact Lie group acting coisotropically on $M$ then from Proposition \[cut\] $K$ acts coisotropically on the Kähler cut, obtained from the $\To^1$-action on $M$. In particular, see [@gbl], if $\lambda_o$ is a maximum for the moment map of the $\To^1$-action then $M^{\lambda}$, $\lambda=\lambda_o - \epsilon$, $\epsilon \cong 0$, is the blow-up of $M$ along the complex submanifold $\psi^{-1}( \lambda_o)$, where $\psi$ is the corresponding moment map for the $\To^1$-action on $M$.
In [@BG],[@Bi], [@PTh], the complete classification of coisotropic actions on irreducible Hermitian symmetric spaces of compact type is given. Therefore, it is easy to construct examples using the above strategy. For example, the ${{\rm SU}}(n)$-action on ${{\rm SO}}(2n)/{{\rm U}}(n)$ induces a coisotropic action on Kähler cut given by $Z({{\rm U}}(n))$. More generally, if $M=L/P$ is an irreducible Hermitian symmetric space of compact type, then $Z(P)$ is a one-dimensional torus. Since the $P$-action on $M$ is coisotropic, see [@BG], [@Bi], [@PTh], $P$ acts coisotropically on the Kähler cut with respect the $Z(P)$-action on $M$.
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abstract: |
The intrinsic idea of superdense coding is to find as many gates as possible such that they can be perfectly discriminated. In this paper, we consider a new scheme of discrimination of quantum gates, called ancilla-assisted discrimination, in which a set of quantum gates on a $d-$dimensional system are perfectly discriminated with assistance from an $r-$dimensional ancilla system. The main contribution of the present paper is two-fold: (1) The number of quantum gates that can be discriminated in this scheme is evaluated. We prove that any $rd+1$ quantum gates cannot be perfectly discriminated with assistance from the ancilla, and there exist $rd$ quantum gates which can be perfectly discriminated with assistance from the ancilla. (2) The dimensionality of the minimal ancilla system is estimated. We prove that there exists a constant positive number $c$ such that for any $k\leq cr$ quantum gates, if they are $d$-assisted discriminable, then they are also $r$-assisted discriminable, and there are $c^{\prime}r\textrm{
}(c^{\prime}>c)$ different quantum gates which can be discriminated with a $d-$dimensional ancilla, but they cannot be discriminated if the ancilla is reduced to an $r-$dimensional system. Thus, the order $O(r)$ of the number of quantum gates that can be discriminated with assistance from an $r-$dimensional ancilla is optimal. The results reported in this paper represent a preliminary step toward understanding the role ancilla system plays in discrimination of quantum gates as well as the power and limit of superdense coding.
author:
- 'Jianxin Chen and Mingsheng Ying [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'power.bib'
title: 'Ancilla-Assisted Discrimination of Quantum Gates'
---
superdense coding, discrimination, quantum gates, ancilla system, algebraic geometry.
Introduction {#section:intro}
============
to determine the oracle from many known candidates if the investigation into internal structures of oracle is not allowed? This is one of the central questions in theoretical computer science named as oracle identification problem or sometimes circuit verification problem.
In quantum information theory, a special case of the oracle identification problems is the discrimination of quantum states [@Yuen75]. This is a fundamental task in quantum information theory since information is encoded in states and after accomplishing some quantum information tasks, information encoded in the states has to be read out, that means the final state of the system has to be determined. It is well known that perfect discrimination is possible only for orthogonal quantum states [@Bennett92]. Thus, so-called unambiguous discrimination strategy is introduced for quantum states which are not necessarily mutually orthogonal [@Ivanovic87; @Dieks88; @Peres88; @Jaeger95; @Chefles98; @Chefles01; @Eldar03A; @Zhang01; @Chen01; @Feng02; @Feng04; @Jafarizadeh08]. In contrast to perfect discrimination where one can always identify the state, unambiguous discrimination guarantees that except for an inconclusive probability, one can always get the correct state with zero error probability. It was shown that a set of quantum pure states are unambiguously discriminable if and only if they are linearly independent [@Chefles98]. Furthermore, many variations of state discrimination problem such as local discrimination, discrimination of mixed states, discrimination of multipartite quantum states, minimum-error discrimination and asymptotic discrimination were proposed and have attracted much attention [@Walgate00; @Duan0702; @Xin08; @Rudolph03; @Takeoka03; @Raynal03; @Eldar01; @Eldar03B; @Eldar04; @Schmid2008; @Markham2008]. For a comprehensive survey of state discrimination, we refer to [@Chefles00; @Paris04].
Another class of the oracle identification problems arisen in quantum information theory is the discrimination of quantum gates which is indeed closely related to the discrimination of quantum states. The problem of discriminating oracle quantum gates can be formally stated as follows. Suppose we are given an unknown quantum gate, namely a unitary operator, which is secretely chosen from a set $\{U_1,...,U_k\}$ of quantum gates acting on a quantum system whose state space is a $d-$dimensional Hilbert space $\mathcal{H}_d$. The aim is to decide which one is the given gate. The discrimination of $U_1,...,U_k$ can be realized by transforming it to a corresponding problem of discriminating quantum states, more precisely, by choosing a suitable input state $|\psi\rangle\in\mathcal{H}_d$ and then discriminating $U_1|\psi\rangle,...,U_k|\psi\rangle$.
Several variants of the problem of discriminating quantum operations have been considered in the literature. For example, unambiguous discrimination is introduced to the discrimination of quantum operations [@Chefles07; @Wang06]. If we consider quantum gates acting on many particles, discrimination using local operations and classical communication is then proposed and has been extensively studied [@Zhou07; @Duan07; @Duan08].
It should be remarked that, for perfect discrimination of quantum gates, it was shown that any two different quantum gates can always be perfectly discriminated if multicopy is allowed, which is quite different from perfect discrimination of quantum states, where infinite copies are required in order to attain perfect discrimination [@Acin01]. Thus the multicopy strategy would greatly extends the ability to discriminate quantum gates.
In this paper, we consider another possibility of improving ability of discriminating quantum gates, namely discrimination with an ancilla system. If ancilla system is allowed, then the ability of discriminating quantum gates acting on $d$-dimensional Hilbert space should be improved. However, any reasonable ancilla system in practice generally cannot possibly extend to large scale. Thus we assume the dimension of ancilla system is bounded by some $r\leq d$. In this situation, we say these quantum gates can be $r$-assisted discriminated.
The purpose of the present paper is to address the following two questions and to explore the role ancilla system plays in discriminating among quantum oracle gates.
\[Problem:1\]\[**Maximal Discrimination Problem**\] How many quantum gates acting on $d$-dimensional Hilbert space at most can be perfectly discriminated with assistance from an $r$-dimensional ancilla?
\[Problem:2\]\[**r-Reduced Discrimination Problem**\] What is the maximum number $N$ such that for any $N$ quantum gates acting on $d$-dimensional Hilbert spaces, if they can be perfectly discriminated with some large ancilla system, then they can still be discriminated when the ancilla system is reduced to $r$-dimensional.
Ancilla-assisted discrimination of quantum gates is a fundamental problem in quantum information theory. Its importance is witnessed by the simple observation that the intrinsic idea of superdense coding is to find as many gates as possible such that they can be perfectly discriminated by some state [@Bennett92; @mozes05]. To be more precise, let’s recall the protocol for standard superdense coding. Suppose there are two parties, known as Alice and Bob, who are far away from one another. Alice possesses a $p$-qubit system $\mathcal{A}$ and Bob holds a $q$-qubit system $\mathcal{B}$. Their goal is to transmit as much information as possible. Then Alice can send $k$ bits of classical information to Bob using only $p$ qubits by following procedure if there exist ${2^k}$ quantum gates $\{U_i\}_{i=1}^{2^k}$ acting on $\mathcal{A}$ which can be $2^q$-assisted discriminable. Let Alice and Bob initially share a state ${\ensuremath{\left|{\psi}\right\rangle}}_{\mathcal{A}\mathcal{B}}$ which satisfies $${\ensuremath{\left\langle{\psi}\right |}}_{\mathcal{A}\mathcal{B}}{(U_i\otimes I_{\mathcal{B}})}^{\dagger}(U_j\otimes I_{\mathcal{B}}){\ensuremath{\left|{\psi}\right\rangle}}_{\mathcal{A}\mathcal{B}}=0$$ for all $1\leq i\neq j\leq 2^k$. The existence of such ${\ensuremath{\left|{\psi}\right\rangle}}$ is equivalent to $2^q$-assisted discriminability of these quantum gates. If Alice wishes to send the bit string ’$i_1i_2\cdots i_k$’ to Bob, she applies the quantum gate $U_{i_1i_2\cdots i_k}$ to her qubits, and then sends her qubits to Bob, giving Bob possession of the whole system. Notice that $${\ensuremath{\left\langle{\psi}\right |}}_{\mathcal{A}\mathcal{B}}{(U_i\otimes I_{\mathcal{B}})}^{\dagger}(U_j\otimes I_{\mathcal{B}}){\ensuremath{\left|{\psi}\right\rangle}}_{\mathcal{A}\mathcal{B}}=0$$ for all $1\leq i\neq j\leq 2^k$ is equivalent to $\{(U_i\otimes I_{\mathcal{B}}){\ensuremath{\left|{\psi}\right\rangle}}_{\mathcal{A}\mathcal{B}}\}_{i=1}^{2^k}$ form a subset of an orthogonal basis. Thus Bob can do a measurement in this basis, and then determine which of the bit strings Alice sent. Thus the natural strategy of optimal superdense coding scheme is to find out as many unitary gates as possible, which can be perfectly discriminated with assistance from $\mathcal{B}$.
The paper is structured as follows. After introducing our notations and giving some background on discrimination of quantum gates in section \[section:intro\], we give a straightforward and complete solution to Problem \[Problem:1\] in section \[section:1\]. Section \[section:r\] is devoted to answer Problem \[Problem:2\]. This section is divided into two subsections. In the first subsection, we present an upper bound for Problem \[Problem:2\], and a lower bound for this problem is given in the second subsection. To prove the theorem concerning the lower bound, some basic ideas from algebraic geometry are required. For convenience of the reader, the necessary concepts and theorems in algebraic geometry are included in the second subsection. Finally, we conclude and present a number of open problems.
Through this paper, we focus on perfectly discrimination strategy. And for simplicity, sometimes we may use the word “discrimination” directly instead of “perfectly discrimination” without explicit explanation.
Maximal Discrimination Problem {#section:1}
==============================
In this section, we will answer Problem \[Problem:1\]. We first give a formal definition of discriminability of quantum gates.
Suppose a $d$-dimensional Hilbert space $\mathcal{H}_d$ is given, consider a set $\mathcal{U}=\{U_i\}_{i=1}^k$ of quantum gates acting on $\mathcal{H}_d$.
1. We say $\mathcal{U}$ is (or $U_1,...,U_k$ are) discriminable if it (or they) will output a set of orthogonal states for some given initial input state; that is, there exists ${\ensuremath{\left|{\psi}\right\rangle}}\in \mathcal{H}_d$ such that ${\ensuremath{\left\langle{\psi}\right |}}{U_i}^{\dagger}{U_j}{\ensuremath{\left|{\psi}\right\rangle}}=0$ for all $1\leq i\neq
j\leq k$.
2. We say $\mathcal{U}$ is (or $U_1,...,U_k$ are) $r$-assisted discriminable if it is (or they are) discriminable with an $r$-dimensional ancilla system $\mathcal{H}_R$; that is, there exists a pure state ${\ensuremath{\left|{\psi}\right\rangle}}\in \mathcal{H}_d\otimes
\mathcal{H}_R$ such that ${\ensuremath{\left\langle{\psi}\right |}}{({U_i}\otimes
I_R)}^{\dagger}({U_j}\otimes I_R){\ensuremath{\left|{\psi}\right\rangle}}=0$ for all $1\leq i\neq
j\leq k$.
The following two technical lemmas will be needed in the proof of our main theorem in this section.
\[lemma:density\] $k$ quantum gates $\{U_i\}_{i=1}^k$ are $r$-assisted discriminable if and only if there exists a density operator $\rho$ with rank $\leq r$ such that $$tr(U_i^{\dagger}U_j\rho)=0\textrm{ }for \textrm{ }all \textrm{
}1\leq i\neq j\leq k.$$
Recall that a set $\{M_i\}_{i=1}^k$ of matrices are said to be mutually orthogonal if $tr(M_i^{\dagger}M_j)=0$ for all $1\leq i\neq
j\leq k$.
\[lemma:orthonal\] $k$ quantum gates $\{U_i\}_{i=1}^k$ acting on a $d$-dimensional Hilbert space are $r$-assisted discriminable if and only if there exists a matrix $S$ with rank $\leq r$ such that $\{U_iS\}_{i=1}^k$ are mutually orthogonal; in this case we say that $\{U_i\}_{i=1}^k$ are discriminated by $S$.
Now we are able to present the main result of this section which give a complete solution to Problem 1.
\[theorem:upper\]If $k$ quantum gates $\{U_i\}_{i=1}^k$ acting on $d$-dimensional Hilbert space are $r$-assisted discriminable, then $k\leq rd$, and the upper bound $rd$ can be achieved provided $r\leq d$.
As a simple application of the above theorem, we return back to the superdense coding scenario considered in the Introduction. With Theorem \[theorem:upper\], we have $$2^k\leq dim(\mathcal{A}) dim(\mathcal{B})=2^p2^q=2^{p+q}.$$ which follows $k\leq p+q$. Thus if Alice possesses $p$ qubits and Bob possess $q$ qubits, then Alice can transmit at most $p+q$ bits of classical information to Bob using her $p$ qubits of quantum information, this bound can be achieved provided $q\leq p$.
r-Reduced Discrimination Problem {#section:r}
================================
As pointed in the Introduction, this section is entirely devoted to address Problem 2. Suppose a set $\mathcal{U}$ of quantum gates acting on $d$-dimensional Hilbert space can be perfectly discriminated with assistance from an ancilla system. In the real world, sometimes the large ancilla system is not possible in practice and what we can deal with is a small ancilla system. In this section, we assume some quantum gates can be perfectly discriminated with assistance from a large ancilla system, and our goal is to reduce the ancilla such that these quantum gates still can be perfectly discriminated with assistance from the reduced ancilla system.
A simple application of the Schmidt decomposition indicates that the quantum gates acting on $d$-dimensional Hilbert space can always be perfectly discriminated with assistance from a $d$-dimensional ancilla system; in other words, they are always $d$-assisted discriminable. So, it is reasonable to rephrase Problem 2 as the following more specific question:
\[Problem:3\]\[**r-Reduced Discrimination Problem**\] Given a positive integer $r<d$. What is the maximum number $N$ satisfying that any $N$ $d$-assisted discriminable quantum gates acting on $d$-dimensional Hilbert space will preserve their discriminability when the ancilla system is reduced to dimension $r$.
We will answer this question by showing that, there exist $O(r)$ $d$-assisted discriminable quantum gates acting on $d$-dimensional Hilbert space which cannot be $r$-assisted discriminated. On the other hand, there exists some constant positive number $c$ such that for any $cr$ quantum gates, if they are perfectly discriminable with assistance from some large ancilla system, then the discriminability remains when the ancilla system is reduced to $r$-dimensional.
Upper Bound of $r$-Reduced Discrimination problem
-------------------------------------------------
In this subsection, we will show the upper bound for Problem 3 in the following three steps.
1. Starting from a simple observation at the ancilla-free discrimination.
2. We will derive an upper bound $2\sqrt{rd}$ for general situation.
3. If $r\leq \frac{d}{2}-1$, then our upper bound can be improved to $2r$.
From above arguments it follows immediately that there always exist $O(r)$ $d$-assisted discriminable quantum gates acting on $d$-dimensional Hilbert space which cannot be $r$-assisted discriminated.
Let’s label the diagonals of a $d\times d$ matrix by integers $k$, with $k$ increasing from lower left to upper right. So there are $2d-1$ diagonals, and the $i$-th diagonal has $d-|d-x|$ entries. It is easy to see that entries of $X^k$ are all 1’s in the $(d-k)$-th diagonal or $(2d-k)$-th diagonal, and all 0’s in other diagonals.
\[lemma:free\] There does not exist a state ${\ensuremath{\left|{\psi}\right\rangle}}\in \mathcal{H}_d$ such that ${\ensuremath{\left\langle{\psi}\right |}}XZ^k{\ensuremath{\left|{\psi}\right\rangle}}=0$ for all $0\leq k\leq d-1$ and ${\ensuremath{\left\langle{\psi}\right |}}Z^k{\ensuremath{\left|{\psi}\right\rangle}}=0$ for all $1\leq k\leq d-1$.
There exist $O(\sqrt{d})$ quantum gates acting on a $d$-dimensional Hilbert space which are $d$-assisted discriminable, but cannot be perfectly discriminated if the ancilla system is not allowed.
Next, we will generalize our idea to $r$-dimensional ancilla-assisted case and show that there exist $O(\sqrt{rd})$ $d$-assisted discriminable quantum gates which cannot be perfectly discriminated with assistance from an $r$-dimensional ancilla system.
\[lemma:r1\] There does not exist any density operator $\rho$ with rank $\leq r$ such that $$tr(X^iZ^j\rho)=0$$ for all $1\leq i\leq r, 0\leq j\leq
d-1$, and $$tr(Z^j\rho)=0$$ for all $1\leq j\leq d-1$.
\[lemma:r\] There exist $O(\sqrt{rd})$ quantum gates which are $d$-assisted discriminable but are not $r$-assisted discriminable.
The upper bound given in the above lemma can be improved in some special cases. Suppose $r\ll d$, or more specifically, $r\leq
\frac{d}{2}-1$. Then we have the following lemma which tightens our previous upper bound significantly.
\[lemma:rr\] Let $r\leq \frac{d}{2}-1$. Then there are $2r+2$ quantum gates which can be $d$-assisted discriminated but cannot be $r$-assisted discriminated.
Combining Lemmas \[lemma:r\] and \[lemma:rr\], we achieve the main result of this subsection:
\[theorem:upper2\] For any $r < d$, we can find $O(r)$ quantum gates which can be $d$-assisted discriminated but cannot be $r$-assisted discriminated.
Lower Bound of $r$-Reduced Problem
----------------------------------
We first state the main result of this subsection which presents a lower bound for Problem 3.
\[theorem:lower2\] Any $$k\leq \lfloor\frac{\sqrt{3(r-1)^2-19}+1}{2}\rfloor$$ quantum gates which are $d$-assisted discriminable must also be $r$-assisted discriminable.
The proof of the above requires certain mathematical tools from algebraic geometry. For convenience of the reader, we first recall some definitions and results in algebraic geometry. For details, we refer to [@artin91; @hartshorne77; @sha95].
Let $\mathbb{A}^n$ be an *affine* $n$-space, the set of all $n$-tuples of complex numbers. We write $\mathbb{C}[x_1,x_2,\cdots,x_n]$ for the polynomial ring in $n$ variables. A subset of $A^n$ is an *algebraic set* or *algebraic variety* if it is the common zeros of a finite set of polynomials $f_1,f_2,\cdots,f_r$ with $f_i\in \mathbb{C}[x_1,x_2,\cdots,x_n]$ for all $1\leq i\leq r$. Such an algebraic set is usually denoted by $Z(f_1,f_2,\cdots,f_r)$. It is not hard to check that the union of a finite number of algebraic sets is an algebraic set, and the intersection of any family of algebraic sets is again an algebraic set. Thus by taking the open subsets to be the complements of algebraic sets, we can define a topology, called the Zariski topology on $\mathbb{A}^n$. Notice that in some references an algebraic variety should also be irreducible in the sense that it cannot be expressed as the union of two proper algebraic sets.
Suppose we can write $\mathcal{X}=\bigcup X_i$, as a finite union of irreducible closed sets. If $X_i\nsubseteq X_j$ for all $i\neq j$, we say such a representation is irredundant, and the $X_i$ are irreducible components of $X$.
We define *projective n-space*, denoted by $\mathbb{P}^n$, to be the set of equivalence classes of $(n+1)-$tuples $(a_0,\cdots,a_n)$ of complex numbers, not all zero, under the equivalence relation given by $(a_0,\cdots,a_n)\sim(\lambda a_0,\cdots,\lambda a_n)$ for all $\lambda \in \mathbb{C}$, $\lambda\neq 0$.
A notion of algebraic variety may also be introduced in projective spaces, called projective algebraic variety: a subset $Y$ of $\mathbb{P}^n$ is an *algebraic set* if it is the common zeros of a finite set of homogeneous polynomials $f_1,f_2,\cdots,f_r$ with $f_i\in
\mathbb{C}[x_0,x_1,\cdots,x_n]$ for $1\leq i\leq r$. We call open subsets of irreducible projective varieties as quasi-projective varieties.
The variety of common zeros of $r$ polynomials $f_1$, $\cdots$, $f_r$ on an $n$-dimensional projective variety has dimension $\geq
n-r$.
Let $X, Y\subset \mathbb{P}^N$ be irreducible quasiprojective varieties with $dim(X)=n$ and $dim(Y)=m$. Then any (nonempty) component $Z$ of $X\bigcap Y$ has $dim(Z)\geq n+m-N$. Moreover, if $X$ and $Y$ are projective and $n+m\geq N$ then $X\bigcap Y\neq \emptyset$.
One may notice that a matrix $M$ has rank no more than $r$ if and only if all the determinants of its $(r+1)\times (r+1)$ submatrices are zero. Thus all $d_A\times d_B$ matrices with rank $\leq r$ can be considered as a variety, namely the determinantal variety, whose projective dimension is $d_Ad_B-(d_A-r)(d_B-r)-1$. Some applications of the determinantal variety in quantum information theory has been found recently [@Chen06; @Cubitt08]. For more details, we refer to [@Harris92].
It should be noted that above argument fails if we reduce the dimension $r$ from 2 to 1 since $$\mathcal{B}=\{Y \in M_r|rank(Y)\leq\lfloor\frac{t-1}{2}\rfloor=0\}$$ only contains zero matrix. But this doesn’t affect our argument very much because we can choose a proper constant coefficient $c$ to make the lower bound hold for all $r$.
Theorem \[theorem:lower2\] can be restated as the following:
There exists a constant positive number $c$ such that for any $k\leq cr$ quantum gates, if they are $d$-assisted discriminable, then they are also $r$-assisted discriminable.
By comparing the above corollary with Theorem \[theorem:upper2\], it can be seen that there is indeed no gap between the lower bound and the upper bound. Thus, this corollary together with Theorem \[theorem:upper2\] completely answers Problem 3.
Conclusions
===========
In this paper, we addressed the problem of ancilla-assisted perfectly discrimination of quantum gates. We evaluated the number of quantum gates which can be perfectly discriminated with an $r$-dimensional ancilla system. More specifically, we answered following two questions.
- How many quantum gates at most can be perfectly discriminated?
- Suppose a set of quantum gates can be perfectly discriminated with some ancilla system. Whether their discriminability remains if the ancilla system is reduced to a small dimension?
For the first question, it is quite simple to observe that any $rd+1$ quantum gates on a $d-$dimensional system cannot be perfectly discriminated with assistance from an $r$-dimensional ancilla. And we showed that there does exist $rd$ quantum gates on a $d-$dimensional system which can be perfectly discriminated with assistance from an $r-$dimensional ancilla.
Our results concerning the second question can be summarized as follows: (1) There exists a constant positive number $c$ such that for any $k\leq cr$ quantum gates on a $d-$dimensional system, if they are $d$-assisted discriminable, then they are also $r$-assisted discriminable. (2) $O(r)$ is optimal. More precisely, there are $c^{\prime}r\textrm{ }(c^{\prime}>c)$ different quantum gates on a $d-$dimensional system which can be $d$-assisted discriminated, but they cannot be perfectly discriminated if the ancilla is reduced to $r$-dimensional.
Discrimination problem is of widely interest because it has many surprising applications. This paper represents a preliminary step toward understanding the role ancilla system plays in discrimination of quantum gates. There are many open questions from this approach deserving further investigation. For example, we may combine our model with the multi-shot discrimination or generalize the input state to several input states as candidates.
Acknowledgment {#acknowledgment .unnumbered}
==============
We are thankful to the colleagues in the Quantum Computation and Information Research Group of Tsinghua University for helpful discussions. And J. Chen thanks N. Yu for bringing the original question to our attention. J. Chen also thanks J. Yu, Z. Ji for helpful discussion. This work was partly supported by the Natural Science Foundation of China (Grant No. 60736011, 60621062, 60503001) and the Hi-Tech Research and Development Program of China(863 project)(Grant No. 2006AA01Z102).
[^1]: J. Chen and M. Ying are with the State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China. e-mail: [email protected] (J. Chen), [email protected] (M. Ying)
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abstract: 'Conditional domain generation is a good way to interactively control sample generation process of deep generative models. However, once a conditional generative model has been created, it is often expensive to allow it to adapt to new conditional controls, especially the network structure is relatively deep. We propose a conditioned latent domain transfer framework across latent spaces of unconditional variational autoencoders(VAE). With this framework, we can allow unconditionally trained VAEs to generate images in its domain with conditionals provided by a latent representation of another domain. This framework does not assume commonalities between two domains. We demonstrate effectiveness and robustness of our model under widely used image datasets.'
author:
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Yingjing Lu\
Carnegie Mellon University\
`[email protected]`\
bibliography:
- 'biba.bib'
title: Cross Domain Image Generation through Latent Space Exploration with Adversarial Loss
---
Introduction
============
Humans are can easily learn to transfer knowledge of one domain to another. They can flexibly learn to connect knowledge they already learned in different domains together so that under conditionals within one domain they can recall or activate knowledge they learned from another. Deep generative models are well know for encoding implicit knowledge within one domain through mapping them to latent space. They can be controlled to generate specific samples within learned domain through conditionals. However, compared to humans, deep generative models are less flexible to make new connections from one domain to another. In another word, once it has learned to generate samples from one set of domain conditionals, making it to adopt to generate samples conditioned on another set of control is often hard and may require to retrain the model, which is often expensive.
There are many works proposed to address this issue by proposing different approaches to allow deep generative models to transfer knowledge from one to another more flexible. In particular, Engel et al[@engel2017latent] recently propose a solution to map conditional encoding to an unconditionally trained VAE to allow it to generate samples conditionally with user defined domain and has achieved excellent results. One limitation is that those conditionals are specifically defined through an one-hot vector. Doing this requires feature engineering and is less effective when we want to condition on some features that are implicit such as using image from one scene as conditional to generate related images in learned domain.
Another solution is provided by Domain transfer network[@taigman2016unsupervised] that trains an end to end model with the assumption of the two domains are somewhat related. In this way, the embedding produced by the autoencoder that encodes images from one domain can be used along with the generated sample to identify whether the transfer learned is effective. To improve on that we tend to develop a framework that make less assumptions between the two domains.
{width="0.8\columnwidth"}
Methodology
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We will use foot index such as $\mathcal{D}_1$ to define variables from one corresponding domain and head notes such as $x^{(i)}_1$ to define $i$th $x$ within domain 1. We also use bold alphabets to express random variables such as $\xv$. Our proposed framework lies on conditionally generating images from two different domains $\mathcal{D}_1$ with probability distribution $p_1(\xv_1)$, $\mathcal{D}_2$ with probability distribution $p_2(\xv_2)$ for images from domain 1 and domain 2 respectively. $\xv_1$ and $\xv_2$ here being i.i.d within each domain. We define $\gamma_1^{i} \rightarrow \gamma_2^{j}$ to be the user defined arbitrary condition to generate images in class $j$ from domain 2 under conditionals of images of class $i$ from domain 1. One example would be to generate image of ’1’ from learned domain 2 conditioned on a image of ’2’ in domain 1. We make no assumption of any commonalities between the two domains, nor do we assume that the conditional between the two domains involves explicit matching features (such as generating ’1’ from domain 2 conditioned on an image of ’1’ from domain 1). In order to map image generation process from two domain with a newly defined domain condition, our framework explores the potential transformation law to map defined class of image from one domain to the corresponding class of image in another domain through latent space.
Without loss of generality, we define two VAEs $V_1$ and $V_2$ with encoding function $z_1 = f_1(x_1)$, $z_2 = f_2(x_2)$ as encoder functions for each VAE and $g_1(z_1)$ and $g_2(z_2)$ as the corresponding decoder network that are trained unconditionally on the two image domains respectively. We use the word unconditionally to indicate that the two VAEs are both trained with standard unsupervised fashion without any additional conditionals involved in training.The objective function of the two VAEs are formulated by maximizing the estimated lower bound(ELBO)[@kingma2013auto]: $$\log p(x)
\geq KL[q_{\phiv}(\zv \mid \xv) \| p_{\thetav}(\zv \mid \xv)] - E_{z\sim q_{\phiv}(\zv \mid \xv)}[\log q_{\phiv}(\zv \mid \xv) - \log p_{\theta}(\xv, \zv)] = ELBO$$ Here $p(x)$ represent true distribution of the domain dataset. and encoder and decoder functions are parametrized by $\phi$ and $\theta$ respectively.Thus the total loss function we used in training our VAEs is: $$\mathcal{L}(\theta, \phi, \xv^{(i)}) = \frac{1}{N} \sum_{i=1}^{N} \lambda_1 \mathcal{C}(\xv^{(i)}, g(f(\xv^{(i)}))) + \lambda_2 ELBO^{(i)}$$ Here $\mathcal{C}(\xv^{(i)}, g(f(\xv^{(i)})))$ denotes the pixel-wise reconstruction cost and the $ELBO$ is described above. we here abused the notation of $\{\lambda_1, \lambda_2\}$ to indicate the two hyperparameters that are used to balance the reconstruction cost and the ELBO. The two hyperparameters are shared among the two VAEs throughout our experiments.
With the given cost functions for training we further assume that the two VAEs are well trained in that they can unconditionally reconstruct images with high fidelity given the images within their respective domains. Our training and implementation details are provided in the experiment section.
To learn a given domain conditional $\gamma_1^{i} \rightarrow \gamma_2^{j}$ we will sample images $x_1^{(i)}$ for arbitrary sample of class $i$ in $\mathcal{D}_1$ and $x_2^{j}$ for arbitrary sample of class $j$ from $\mathcal{D}_2$. We adopt the notion of generative adversarial network(GANs) and use a generator $G(\epsilon, z^{(i)})$ to transform embedding learning from one VAE to the corresponding embedding of another. The generator takes in a noise sampled from a simple prior distribution and the embedding $z^{(i)}$ serves as the conditional. We will follow our presentation structure and use the setting of transforming from domain of $VAE_1$ to domain of $VAE_2$ form class $i$ to class $j$. Thus the generator on domain 1 side generates encoding $z_2^{j}~fake = G_1(\epsilon, z_1^{(i)})$. During training, the result from $G_1$ is passed to the discriminator on the domain 2 side $D_2(z_2^{(j)}, z_2~arbitrary)$. The discriminator from domain 2 takes true encoding from $f_2(x_2^{j})$ as conditional and to determine whether $z_2~arbitrary$ is in the true embedded subspace of class $j$ in domain 2. To make the discriminators stronger within this model we expand our loss function from the ’traditional’ GAN loss. For clarity, we introduce $z^{'}$ for the arbitrary embedding z input to the discriminator, and $z$ for the true embedding generated by the encoder network. We shorthand notation $\mathcal{L}_{c=1}(z,z) \equiv -\log(D(z,z))$ for the loss of the true vector under true class, $\mathcal{L}_{c=1}(z,z^{'}) \equiv -(1-\log(D(z, z^{'})))$ as the discriminator loss for classifying a false embedding under true conditional, and finally an additional term $\mathcal{L}_{c=0}(z, \epsilon) \equiv -(1-\log(D(z, \epsilon))$ as the false classification for entering a random noise from a simple distribution. The third term here aims to strengthen the discriminator that under the conditional, the embedding generated from $G$ should not be a random noise vector that seems to comply with the pattern of the distribution. The full loss function of the discriminator is then: $$\mathcal{L}_D = \Eb_{z\sim q(z|x)}\big[ \mathcal{L}_{c=1}(z,z) \big] + \Eb_{z\sim G(z,\epsilon)}\big[ \mathcal{L}_{c=0}(z,z^{'}) \big] + \Eb_{\epsilon \sim p(\epsilon)}\big[ \mathcal{L}_{c=0}(z,\epsilon) \big]$$
Opposite to discriminator which also restrict from random noise, we introduce a regularization term that is inspired by the log regularization term proposed by [@engel2017latent] as $\frac{1}{n}|| \epsilon - G(\epsilon, z)||^2_2$ here $z$ represents the input embedding to the generator to be transformed. Intuitively, as the generator shift the simple noise to the mapped distribution, it would usually maximize the distance between the embedding generated and the original noise. With this term added in minimizing the loss function will create a force that “pulls back” $G$ from moving from $\epsilon$ too far, thus encourage variety of the embedding generated. Adding the regularization term resulting in out generator loss: $$\mathcal{L}_G = \Eb_{z\sim p(z), \epsilon \sim p(\epsilon)}\big[ \mathcal{L}_{c=1}G(z,\epsilon) + \frac{\lambda_{reg}}{n}|| \epsilon - G(\epsilon, z)||^2_2 \big]$$
During training, we first train two VAEs until convergence, then we train two pairs of generator and discriminator alternatively until convergence. During sampling, we sample from desired class $i$ in domain 1 and feed in $VAE_1$ then VAE will encode the sample to embedding $z_1^{(i)}$ and transformed by generator with a noise $G_1(\epsilon, z_1^{(i)})$, this new transformed embedding will be passe to decoder of $VAE_2$ to generate actual sample $g_2(G_1(\epsilon, z_1^{(i)}))$.
Experiments
===========
We performed our experiments mainly on MNIST[@lecun-mnisthandwrittendigit-2010] and SVHN[@netzer2011reading] datasets. MNIST dataset contains 28X28 hand written digits from 0 to 9 in grey scale with approximately 60000 training samples and 20000 testing samples. SVHN dataset contains digit photos captured in street cropped to 32X32 in RGB. Training set contains more than 73000 images and testing set contains more than 26000 images for digits from 1 to 9.
For **MNIST to MNIST** mapping, we split the training set and testing set with number as their classes. To adhere to our assumption and avoid feature matching, we assign domain 1 contain digits $\{0,1,2,3,4\}$ and domain 2 contains digits $\{5,6,7,8,9\}$. The conditional generation law is defined to generate a certain class of digit given a class of digit. For conditional generation from domain 1 to domain 2 we define $\{0 \rightarrow 5, 1 \rightarrow 6, 2 \rightarrow 7, 3 \rightarrow 8, 4 \rightarrow 9\}$ and conditional generation from domain 2 to domain 2 is the opposite direction.
To evaluate the accuracy quantitatively, we train a convolutional neural network on the full train set of MNIST that serve as a classifier for evaluating whether our image generated belongs to the right class. Our convolutional neural network achieve 99.2% classification accuracy on MNIST test set so that we are confident that our classifier would be a good fit to evaluate whether our model can generate sample in the right class. Aside from the classifier we also consider the recognizablility of a digit. If a digit is generated too vague to be identified or mixed between digits, we as humans regard those samples as false positives as they are not valid generation and wrongly projected in latent space. We ask three different volunteers to eyeball a subset of the image generated and record their classification result as a comparison. To avoid unknown effect of one digit can perform extremely well to conditionally generate another digit, we shuffle the conditional pair 3 times and report our results based on the average performance recorded from the three conditional pairs.
For **SVHN to MNIST** mapping we use similar scheme to regard digits as different classes. Since SVHN does not contain digit 0, we drop the pair 0-5 and only uses $\{ 1 \rightarrow 6, 2 \rightarrow 7, 3 \rightarrow 8, 4 \rightarrow 9\}$ as our major experiment target and shuffle pairs among those. Since we cannot train a classifier that can perform classification of generated SVHN images with high accuracy(the best classification accuracy we had was 66.8%), we use MNIST digits generated conditioned on SVHN images to report the performance. Similarly, we use both trained convolutional neural network and human as classifier to evaluate the results quantitatively.
For all of our experiments, we train our models on training set and report results from images generated from testing set.
{width="\columnwidth"}
{width="\columnwidth"}
{width="0.8\columnwidth"}
Quantitative results on image generation
----------------------------------------
From the quantitative perspective, we see that our framework generally perform well on MNIST dataset transformation. Probably due to MNIST’s simple distribution. The average accuracy of the CNN classifier reports higher than the average accuracy reported by human volunteers. We inspected and found that the classifier tend to be lenient when the sample is vague and tend not to classify to the right class where as human volunteers tend not to guess which class it belongs and simply report it wrong. Especially with small training size the samples have a lot of jitters and volunteers tend not to guess what the digit actually is. SVHN to MNIST yields less accuracy. From Figure 3 we see that the SVHN contains many disruptive samples that make the conditional less inclined to a particular class. Combining the results from the two datasets together we see that degradation of accuracy resulting in decreased training set is relatively small, suggesting that our model is relatively robust in the datasets we examined.
(r)[1-4]{} Dataset Size of training set Accuracy(CNN classifier) Accuracy(Human)
--------------------------- ---------------------- -------------------------- -----------------
MNIST $\rightarrow$ MNIST 500 0.63 0.58
MNIST $\rightarrow$ MNIST 1000 0.68 0.65
MNIST $\rightarrow$ MNIST 2000 0.66 0.67
MNIST $\rightarrow$ MNIST Full set 0.77 0.75
SVHN $\rightarrow$ MNIST 500 0.43 0.48
SVHN $\rightarrow$ MNIST 1000 0.48 0.44
SVHN $\rightarrow$ MNIST 2000 0.46 0.43
SVHN $\rightarrow$ MNIST Full set 0.61 0.58
: Accuracy of generated images[]{data-label="sample-table"}
Implementation details
----------------------
For the we uses symmetric architecture for the two variational autoencoders. we use three layer convolutional neural networks followed by a linear layer of size 256. Z embedding is set to 100 for all experiments. For the decoder we use 1 linear layer of size 2048 followed by 4 layers of deconvolution layers. We apply RELU on all layers except the output uses tanh and batch normalization on convolutional and deconvolutional layers. We use a hyperparameter $\alpha = 0.1$ as the coefficient applied to the $\sigma$ outputed from the encoder. This hyperparameter is used by Engel et.al.[@engel2017latent] as the authors in the this paper show that imposing this hyperparameter allow the distribution of the embedded space to be tighter.
For the generator and discriminator pair, we use a paired 4-layer fully connected network of size 512. The conditional and the noise $\epsilon$ was simply concatenated and feed to the first layer. The generator outputs a transformed embedding as well as a gating factor resulting from sigmoid of the transformed embedding. The final output transformed embedding is an interpolation between the input random noise $\epsilon$ and the transformed embedding with gateing factor as the interpolation coefficient. This structure is also introduced by [@engel2017latent]. We adopt this architecture and believe that this interpolation can introduce more variety of output.
Related works
=============
Generative adverserial networks or GAN for short, is well known for generating realistic samples through mapping latent manifolds with noise and conditionals inputed into generator(s).Many prior works has illustrates successful attempts to disentangle how GAN encode features into latent space and conditional GANs(cGAN) is a particular class of GANs that are proven to control output contents through concatenating tractable conditionals with selected noise. In the conditional setting, a lot of previous work has been done to explore possibilities to control samples generated from GANs through encoded conditionals[@mirza2014conditional; @reed2016generative; @isola2017image].Here in particular, Zhu et. al.[@NIPS2017_6650] proposed a way to transform style of image from one domain to another through a cycle encoding and verification structure and has achieved decent results. We view the BiGAN proposed by Donahue ei.al.[@donahue2016adversarial] as a concurrent work of ours in that it also explores the concept of conditional noise mapping. It would be interesting to see if the model can be adaptable to the setting not being trained end-to-end. The notion of approximating an implicit distribution is not restricted to GAN alone and many works have focused on incorporating adversarial loss to achieve more flexible latent representation approximation[@makhzani2015adversarial; @dumoulin2016adversarially; @mescheder2017adversarial].
Autoencoder(AE) is another class of deep generative model that compiles inputs and reconstruct samples through encoding features in latent space. Recent work done by Liu et.al.[@liu2017unsupervised] has shown that VAE can perform well on matching the latent space between images from two domains. The added adversarial loss facilitates the training of the VAE by preventing it from mode collapsing.
![MNIST to FASHION-MNIST[@xiao2017/online] generation. Labeling correlation is 0 - T-shirt, 1-Trouser, 2-Pullover, 3-Dress, 4-Coat. Bottom is the original MNIST reconstruction, top is the corresponding image generated using bottom as conditional mapping.](fashion-1){width="\columnwidth"}
![MNIST to FASHION-MNIST[@xiao2017/online] generation. Labeling correlation is 5-Sandal, 6- Shirt, 7-Sneaker, 8-Bag, 9-Ankleboot. Bottom is the original MNIST reconstruction, top is the corresponding image generated using bottom as conditional mapping.](fashion-2){width="\columnwidth"}
Discussion and future work
==========================
In this work we present a framework that can decouple the latent space of the variational encoder and match the arbitrary domains through adversarial training. To shift the conditional generation from one domain to another, the only thing to do is to train a separate pair of generator and discriminator that can transfer latent features from one domain to another. No need to retrain the entire VAE. Both qualitative and quantitative results have shown that this approach produces promising results and is easy to implement. In the future we will explore this framework on more complex datasets such as CIFAR-10[@krizhevsky2009learning] or Oxford Flower[@Nilsback08]. Those datasets either contain smaller images or with more complex distributions. Under those conditions a variational autoencoder may not be the best option to provide a clear domain boundary for discriminators in our framework to distinguish. But our framework is not limited to variational autoencoders and can be applied to other models with latent space embeddings. We will explore more generative models that can incorporate our framework in the future.
|
---
author:
- 'Karin de Langis$^{1}$ and Junaed Sattar$^{2}$ [^1]'
bibliography:
- 'citations.bib'
title: |
**Real-Time Multi-Diver Tracking and Re-identification\
for Underwater Human-Robot Collaboration**
---
[^1]: The authors are with the Department of Computer Science and Engineering, Minnesota Robotics Institute, University of Minnesota Twin Cities, Minneapolis, MN, USA. [{$^{1}$dento019, $^{2}$junaed} at umn.edu.]{}
|
---
abstract: 'A quantum deformation of three-dimensional de Sitter space was proposed in hep-th/0407188. We use this to calculate the entropy of Kerr-de Sitter space, using a canonical ensemble, and find agreement with the semiclassical result.'
author:
- 'Michael C. Abbott and David A. Lowe'
bibliography:
- 'de-sitter.bib'
title: 'Statistical entropy of three-dimensional q-deformed Kerr-de Sitter space'
---
introduction
============
Black holes are known to carry entropy proportional to their horizon area [@Bekenstein:1973ur; @Hawking:1974sw]. One of the main successes of string theory has been to provide a microscopic interpretation of this entropy [@Strominger:1996sh], at least in certain cases. Since the semiclassical arguments for this horizon area are not specific to horizons surrounding black holes, they should also apply to cosmological horizons [@Figari:1975km; @Unruh:1976db; @Gibbons:1977mu]. It would be of great interest to have microscopic state-counting arguments for such situations.
The natural place to investigate cosmological horizons is de Sitter space, the maximally symmetric spacetime with constant positive cosmological constant $\Lambda$ [@Parikh:2002py; @Strominger:2001pn; @Banks:2003cg; @Banks:2006rx; @Banks:2002wr]. Current observations suggest that our universe is now $\Lambda$-dominated , and thus asymptotically de Sitter in the future [@wmap3-cosmology; @Carroll:2000fy; @Fischler:2001yj].
Following the great success of the AdS/CFT correspondence [@Maldacena:1997re], there have been suggestions of a dual conformal field theory, living on the conformal boundary of de Sitter [@Park:1998qk; @Strominger:2001pn]. Unlike the anti-de Sitter case, this boundary is spacelike, and time-translation in the bulk corresponding to scale transformation on the boundary [@Balasubramanian:1999jd; @deBoer:1999xf; @Strominger:2001gp]. It also has two disconnected components, and it is not clear whether the boundary theory should live on one or both [@Parikh:2002py].
Since the area of an observer’s cosmological horizon is finite, de Sitter has finite Bekenstein-Hawking entropy. This immediately causes problems with finding a state counting interpretation, since the isometry group is non-compact [@Goheer:2002vf] and hence only has infinite dimensional unitary representations. This apparent contradiction is even stronger if the dimension of the Hilbert space is also finite [@Banks:2000fe; @Balasubramanian:2001rb; @Banks:2005bm]. It has been suggested that the correct inner product is not the naive local one, which changes the notion of unitarity [@Witten:2001kn; @Bousso:2001mw]. (Another approach is given in [@Parikh:2004ux; @Parikh:2004wh].)
Some of these difficulties might be tamed by noncommutative geometry [@connes94; @Madore:1999bi]. The approach used in [@Guijosa:2003ze] is to deform the group of isometries into a quantum group. This was further studied in [@Lowe:2004nw; @Guijosa:2005qi; @Lowe:2005de] and is the approach followed here. Quantum deformations of the Lorentz group in various dimensions are studied in [@Woronowicz-94; @Buffenoir:1997ih] and in [@Krishnan:2005ct; @Krishnan:2006bq].
The plan of the paper is as follows. Section \[sec:q-Deformed-de-Sitter\] of this paper is mostly a recap of [@Guijosa:2003ze; @Lowe:2004nw]; section \[sec:Entropy-of-Kerr–de\] contains a novel calculation of the entropy. The interpretation of the result is discussed and concluding remarks made in section \[sec:Conclusions\].
q-Deformed de Sitter space\[sec:q-Deformed-de-Sitter\]
======================================================
Three-dimensional de Sitter space can be defined as the hyperboloid $-(x^{0})^{2}+\sum_{i=1}^{3}(x^{i})^{2}=\ell^{2}$ in Minkowski space. This is a spacetime of constant curvature, representing a vacuum with positive cosmological constant $\Lambda=1/\ell^{2}$.
The isometries of this hyperboloid are just the rotations and boosts of the embedding space, which generate the Lorentz group $SO(3,1)$. We will focus on the Lie algebra, rather than the global properties of the group. The complex combinations of generators $X_{i}=J_{i}+iK_{i}$ (left) and $\overline{X}_{i}=J_{i}-iK_{i}$ (right) each obey the $su(2)$ commutation relation $[X_{i},X_{j}]=i\epsilon_{ijk}X_{k}$, and commute with each other.
In the complex algebra, the fact that $J$ and $K$ are Hermitian is encoded in the star operation $J^{\star}=J$, $K^{\star}=K$, and so $X_{i}^{\star}=\overline{X}_{i}$. The use of this star-structure specifies that we are dealing with the non-compact real form $so(3,1)$. We will also use the basis given by: $$\begin{aligned}
L_{0} & =X_{1} & \overline{L}_{0} & =-\overline{X}_{1}\\
L_{1} & =X_{2}-iX_{3} & \overline{L}_{1} & =-\overline{X}_{2}-i\overline{X}_{3}\\
L_{-1} & =-X_{2}-iX_{3} & \overline{L}_{-1} & =\overline{X}_{2}-i\overline{X}_{3}.\end{aligned}$$ These generators form the $n=0\pm1$ part of the Virasoro algebra $[L_{m},L_{n}]=(m-n)L_{m+n}$, but with real form $$L_{n}^{\star}=-\overline{L}_{n}.\label{eq:so31}$$
A field in de Sitter space will transform under isometries in some representation of this algebra. Since the group is non-compact, there are no finite-dimensional unitary representations, thus any field has infinitely many modes. For a field of mass $m>\ell$ the representation is in the principal series [@naimark64; @vilenkin91; @Joung:2006gj].
It was proposed in [@Guijosa:2003ze] that the Lie algebra of isometries should be deformed to a quantum group (Hopf algebra) [@Biedenharn:1996vv; @Klimyk:1997eb]. Taking the deformation parameter to be a root of unity $$q=e^{2\pi i/N}$$ limits the dimension of an irreducible representation to at most $N$. In particular, the deformed versions of non-compact algebras can have finite dimensional unitary representations, which become infinite in the classical limit $q\to1$ [@Dobrev:1993gi; @Steinacker:1999fj]. This was done explicitly for dS$_{2}$’s $so(2,1)$ principal series in [@Guijosa:2003ze]. The relation between $N$ and gravity quantities will be fixed momentarily.
In dS$_{3}$ however there is a complication which does not arise in dS$_{2}$: even the deformed algebra cannot have non-trivial unitary representations [@Lowe:2004nw]. Suppose $\left|\psi\right\rangle $ is an eigenstate of $L_{0}$ and $\overline{L}_{0}$ in a unitary representation. Then the state $L_{\pm1}\left|\psi\right\rangle $ has zero norm, since $L_{1}^{\star}$ does not lower the eigenvalue $L_{1}$ raised. So the representation must be trivial. (In the infinite-dimensional principal series representation, such a $\left|\psi\right\rangle $ lies outside the Hilbert space.)
Similar problems with unitarity are found in [@Krishnan:2006bq] in attempting to deform this and higher Lorentz groups, and multi-parameter families of deformations were studied in [@Krishnan:2005ct].
These algebraic problems are related to the problem of defining an inner product for fields on de Sitter space, which in turn induces a particular adjoint. The standard local Klein-Gordon one, which induces . Witten proposed to use the path integral from asymptotic past to future, with an extra insertion of $CPT$ [@Witten:2001kn]. Choosing the parity operation to be $P\, x^{3}=-x^{3}$, [@Lowe:2004nw] showed that this induces $$L_{n}^{\dagger}=-L_{n},\qquad\overline{L}_{n}^{\dagger}=-\overline{L}_{n}\label{eq:split-form}$$ or $X_{1,2}^{\dagger}=-X_{1,2}$, $X_{3}^{\dagger}=X_{3}$ and the same on the right. This amounts to using the the (non-compact) split real form $su(1,1)\oplus su(1,1)$, instead of $so(3,1)$.[^1]
With this real form, the natural deformation of the algebra to use is $$U_{q}\left(su(1,1)\right)\oplus U_{q}\left(su(1,1)\right).$$ The quantum group $U_{q}\left(su(1,1)\right)$ has unitary representations of dimension $N$. These are representations without highest weight, having $(X_{\pm})^{N}\neq0$, and are called cyclic representations ($\mathcal{B}$ in [@Biedenharn:1996vv]). It was shown in [@Guijosa:2003ze] that the parameters of a cyclic representation can be chosen so as to give the same Casimirs as the classical $su(1,1)=so(2,1)$ principal series, and in [@Lowe:2004nw] that a left-right product of two cyclic representations has the correct Casimirs to match the $so(3,1)$ principal series.
The geodesics lying in the embedding space’s 0-1 plane are the north and south poles of de Sitter space. The south pole is $r=0$ in the static coordinate patch, whose metric is$$ds^{2}=-\left(1-\frac{r^{2}}{\ell^{2}}\right)dt^{2}+\frac{dr^{2}}{1-r^{2}/\ell^{2}}+r^{2}d\phi^{2}.\label{eq:static-metric}$$ The generator of time translations here is$$-i\partial_{t}=K_{1}=-i\left(L_{0}+\overline{L}_{0}\right).$$ At the antipodal point $-x^{\mu}$ this generates instead reverse time translation. (This is the standard situation for a thermofield double, the canonical example of which is Rindler space.)
In these coordinates the horizon is at $r=\ell$. It has Hawking temperature $T=1/2\pi$ which can be derived most transparently for our purposes by tracing over modes living behind the horizon (which have negative frequency) to produce southern density matrix [@Bousso:2001mw] $$\rho^{\mathrm{south}}\propto e^{-\beta K_{1}}.$$
In the classical (principal series) case this operator has a continuous spectrum, while a single irreducible cyclic representation of the quantum group it has eigenvalues spaced approximately $1/\ell$ apart. So it was proposed in [@Lowe:2004nw] that the appropriate quantum representations are not the cyclic representations $\mathcal{B}$, of dimension $N$, but rather reducible representations $\bigoplus_{i=1}^{N}\mathcal{B}_{i}$ of dimension $N^{2}$. There is one phase parameter of the cyclic representation not fixed by matching the principal series’s Casimirs, and the sum is over different choices of this phase. In the resulting twisted representation, $-i(L_{o}+\overline{L}_{0})$ has eigenvalues spaced $\sim1/N\ell$, thus tending to a continuum in the classical limit.
The natural choice for $N$ is the de Sitter radius in Planck units: we set $$N=\frac{\ell}{G}.$$ The maximum eigenvalue of $J_{1}$, the generator of rotations about the south pole, is of order $N$, so this can be viewed as allowing only those rotations which move the most distant points by at least one Planck length. (The same $N$ would be obtained by the argument used in [@Lowe:2005de]. There, the semiclassical entropy is used to obtain an estimate of the mass gap, [@Preskill:1991tb] which is then matched to the spacing of the Hamiltonian’s eigenvalues.)
Note that the dimension of the twisted representation, $N^{2}=\ell^{2}/G^{2}$, is equal to the ratio of the Planck density $1/G^{3}$ to the vacuum energy density $\Lambda/G$. Thus the dimension of the Hilbert space associated with a single twisted representation is essentially one bit per unit Planck volume. In the next section, the horizon entropy (one bit per unit Planck area) is identified as the entropy of a thermal ensemble inside the full Hilbert space built out of tensor products of these representations.
Entropy of Kerr-de Sitter space\[sec:Entropy-of-Kerr–de\]
=========================================================
Kerr-de Sitter is obtained by placing a spinning point mass at the origin of the static patch, changing the metric to$$ds^{2}=-\mathcal{N}dt^{2}+\frac{dr^{2}}{\mathcal{N}}+r^{2}\left(d\phi-\frac{4GJ}{r^{2}}dt\right)^{2}\label{eq:Kerr-dS metric}$$ where$$\mathcal{N}=M-\frac{r^{2}}{\ell^{2}}+\frac{16G^{2}J^{2}}{r^{2}}.$$ The point has mass $E=(1-M)/8G$ and angular momentum $J$. There is still only one horizon, at radius $$r=r_{+}=\tfrac{1}{2}\left(\sqrt{\tau}+\sqrt{\overline{\tau}}\right)\ell$$ where $\tau=M+i8GJ/\ell$. It carries entropy$$S=\frac{A}{4G}=\frac{\pi\ell}{4G}\left(\sqrt{\tau}+\sqrt{\overline{\tau}}\right).\label{eq:BH-Entropy}$$
This space is a quotient of pure dS$_{3}$ by a discrete group, so is locally the same. In particular, it has the same $\Lambda$ and the same Lie algebra of isometries. We therefore use the same quantum deformation of this algebra, including the same $q$.
As for rotating black holes in flat space [@Birrell:1982ix], the rotation creates an angular potential $\Omega$ conjugate to $J$, in addition to the temperature $T$. The Boltzmann factor becomes [@Bousso:2001mw] $$e^{-\frac{\mathcal{M}+\Omega\mathcal{J}}{T}}=e^{\beta iL_{0}+\overline{\beta}i\overline{L}_{0}}$$ where the complex inverse temperature is given by $$\beta=\frac{1+i\Omega}{T}=\frac{2\pi\ell}{\sqrt{\overline{\tau}}}.$$
Now consider a field living in the above twisted representation. We propose that the microscopic CFT is formulated in terms of elementary degrees of freedom in the twisted representation discussed above. We will also assume that we are in a regime where free-field calculations in the CFT suffice to give a good approximation to the entropy. In this case, the multiparticle thermodynamic averages are as follows [^2]:$$\begin{aligned}
\left\langle -iL_{0}\right\rangle & =\sum_{-iL_{0}>0}e^{\beta iL_{0}}(-iL_{0})\\
& \approx N\ell\int_{0}^{\infty}dE\: e^{-\beta E}E\sim\frac{N\ell}{\beta^{2}}.\end{aligned}$$ The states $-iL_{0}<0$ are the ones traced over to produce this thermal behaviour. ($\sim$ here means equal up to numerical factors of order 1.) Similarly $\left\langle -i\overline{L}_{0}\right\rangle \sim N\ell/\overline{\beta}$. Notice that unitarity is restored at the level of thermal expectation values: $\left\langle i\overline{L}_{0}\right\rangle $ is the complex conjugate of $\left\langle iL_{0}\right\rangle $, matching $\left(iL_{0}\right)^{\star}=+i\,\overline{L}_{0}$.
Using the relationships $\left\langle -iL_{0}\right\rangle =-\frac{\partial}{\partial\beta}\log Z$ and $\left\langle -i\overline{L}_{0}\right\rangle =-\frac{\partial}{\partial\overline{\beta}}\log Z$, where $Z$ is the partition function, we can then calculate the entropy as follows:$$S=\left(1-\beta\frac{\partial}{\partial\beta}-\overline{\beta}\frac{\partial}{\partial\overline{\beta}}\right)\log Z\sim N\ell\left(\frac{1}{\beta}+\frac{1}{\,\overline{\beta}}\right)$$ matching the semiclassical result up to an overall numerical factor [^3].
Conclusions \[sec:Conclusions\]
===============================
In this paper, a quantum deformed CFT has been proposed as the holographic dual to a theory of gravity in a de Sitter background. This accounts for the Bekenstein-Hawking horizon entropy of de Sitter spacetime. The functional dependence on three independent parameters: cosmological constant, mass, and angular momentum, was reproduced precisely. This should be regarded as a very interesting success of this approach, as other approaches to quantizing gravity in a de Sitter background lead to divergent horizon entropy. One of the interesting features of this construction is that the entropy is really to be thought of as a thermal entropy, or more precisely the entropy in a canonical ensemble with fixed temperature and angular potential. On the other hand, the microcanonical entropy, with fixed total mass and angular momentum, will not agree with the canonical ensemble. Instead, the fundamental degrees of freedom are non-unitary with respect to the standard inner product of quantum fields in de Sitter space, which leads to imaginary angular momenta. Only when they are combined as an ensemble with fixed angular potential does the average total angular momentum agree with the macroscopic value of the Kerr de Sitter space.
These facts point to the instability of de Sitter spacetime. The analog of heat baths are needed for the fixed temperature and angular potential ensemble to make sense. The fact that we cannot ignore the presence of these heat baths at large $N$ and obtain agreement with the microcanonical ensemble suggests that the CFT is not a complete self-contained description of quantum gravity in a de Sitter background. Including other degrees of freedom becomes a necessity, which then opens the door to the complete theory describing more than just asymptotically de Sitter spacetime.
It would be interesting to understand whether these facts relate to the metastability of de Sitter backgrounds in string theory [@Kachru:2003aw]. The hope is that the type of formulation of dS/CFT described in the present work will provide an effective description of physics around such backgrounds for timescales smaller than the lifetime of the de Sitter phase.
This research is supported in part by DOE grant DE-FG02-91ER40688-Task A.
[^1]: Throughout this paper we use $\star$ for the so(3,1) involution , and $\dagger$ for this one.
[^2]: Here we assume Boltzmann statistics for simplicity, Fermi-Dirac or Bose-Einstein changes overall numerical factors only. Note also that for convenience the trivial tensor product is used rather than the usual coproduct of the quantum group [@Klimyk:1997eb]. We have checked via numerical calculation that this does not change the expression for the entropy for large $N$.
[^3]: In AdS/CFT there is a similar discrepancy in overall numerical coefficient in comparing the weakly coupled CFT calculation at finite temperature with the Bekenstein-Hawking entropy of the black hole.
|
---
abstract: 'A circle packing is a collection of disks with disjoint interiors in the plane. It naturally defines a graph by tangency. It is shown that there exists $p>0$ such that the following holds for every circle packing: If each disk is retained with probability $p$ independently, then the probability that there is a path of retained disks connecting the origin to infinity is zero. The following conclusions are derived using results on circle packings of planar graphs: (i) Site percolation with parameter $p$ has no infinite connected component on recurrent simple plane triangulations, or on Benjamini–Schramm limits of finite simple planar graphs. (ii) Site percolation with parameter $1-p$ has an infinite connected component on transient simple plane triangulations with bounded degree. These results lend support to recent conjectures of Benjamini. Extensions to graphs formed from the packing of shapes other than disks, in the plane and in higher dimensions, are presented. Several conjectures and open questions are discussed.'
address: |
Ron Peled Tel Aviv University\
School of Mathematical Sciences\
Tel Aviv, 69978, Israel.
author:
- Ron Peled
title: On the site percolation threshold of circle packings and planar graphs
---
Introduction {#sec:introduction}
============
*Site percolation with parameter $p\in[0,1]$ on a graph $G$* is the process of independently retaining each vertex in $G$ with probability $p$ and deleting it with probability $1-p$; the (random) induced subgraph on the retained vertices is denoted $G^p$. Percolation theory is concerned with the structure of the connected components of $G^p$. It forms a huge body of research in both the physics and mathematics communities; see [@G99; @BR06; @DC17] for two books and a recent review. Possibly the most basic question in the theory is whether $G^p$ has a connected component with infinitely many vertices. Kolmogorov’s zero-one law implies that the probability of this event is either zero or one for each $p$ (on locally finite graphs), and a standard coupling shows that the probability is non-decreasing in $p$. This leads to the definition of the *critical probability $p_c(G)$*, defined as the infimum over all $p$ such that $G^p$ has an infinite connected component almost surely.
The critical probability of the triangular lattice is exactly $1/2$, but for most lattices (and more general graphs) the critical probability is not predicted to have an explicit expression; simulations suggest that $p_c({\mathbb{Z}}^2)\approx0.59$. The critical probability of general graphs, or even planar graphs, may be arbitrarily small, as evidenced by the fact that the critical probability of the $d$-regular tree is $\frac{1}{d-1}$. The lower bound $p_c(G)\ge \frac{1}{\Delta(G)-1}$ holds for graphs $G$ with finite maximal degree $\Delta(G)$, as follows from a simple union bound or by coupling the percolation on the graph with a percolation on the regular tree. For which general classes of graphs can this lower bound be improved? Motivated by ideas of coarse conformal uniformization, Benjamini [@B2018] recently made two conjectures on the behavior of site percolation with parameter $p=1/2$ on large classes of planar triangulations (Conjecture \[conj:transient triangulations\] and Conjecture \[conj:square tiling\] below). The first conjecture relates the behavior to the recurrence properties of the triangulation (a connected graph is called *recurrent* if simple random walk on it returns to its starting vertex infinitely often, almost surely) while the second discusses connectivity probabilities in specific embeddings. The present work is motivated by Benjamini’s conjectures, as well as by applications to the study of planar loop models [@CGHP20]. We prove a (positive) uniform lower bound on the critical probability of locally finite planar graphs which can be represented as the tangency graph of a circle packing with at most countably many accumulation points (Corollary \[cor:circle packing accumulation points\]). This class of graphs includes *all recurrent simple plane triangulations* as well as *all Benjamini–Schramm limits of finite planar graphs*. A uniform *upper bound* on the critical probability is further obtained for *transient* simple plane triangulations with bounded degrees. In each of these cases, the fact that the critical probability cannot be arbitrarily close to $0$, or arbitrarily close to $1$ in the latter case, was not known before. The results lend support to Benjamini’s conjectures, verifying their analogues when the percolation probability 1/2 is replaced by a different universal constant (close to $0$ or close to $1$, according to context).
Our results are based on the following statement (Theorem \[thm:percolation for circle packing\]): There exists $p>0$ such that for *all* circle packings in the plane, after retaining each disk with probability $p$ and deleting it with probability $1-p$, there is no path of retained disks connecting the origin to infinity, almost surely. In the spirit of Benjamini’s conjectures, we conjecture that one may in fact take $p=1/2$ in this statement (see Section \[sec:discussion and open questions\]). Similar statements are obtained for packings of general shapes, in dimension two or higher, satisfying a regularity assumption.
Results
=======
Circle packings {#sec:circle packings}
---------------
A *circle packing* is a collection of closed (geometric) disks in ${\mathbb{R}}^2$ having positive radii (possibly changing from disk to disk) and disjoint interiors. A circle packing ${\mathcal{S}}$ naturally defines a graph $G_{\mathcal{S}}$ with vertex set ${\mathcal{S}}$ by declaring disks *adjacent* when they are tangent. We also write that a graph *$G$ is represented by ${\mathcal{S}}$* if $G = G_{\mathcal{S}}$. We note that a circle packing may have *accumulation points* - points in ${\mathbb{R}}^2$ with infinitely many disks of the packing intersecting each of their ${\mathbb{R}}^2$-neighborhoods. In addition, a disk may be tangent to infinitely many other disks.
As before, we denote by $G_{\mathcal{S}}^p$ the (random) induced subgraph on retained disks in a site percolation process on $G_{\mathcal{S}}$ with parameter $p$. The retained disks are termed *open* and the non-retained ones, *closed*. Given $s_0, s_1\in{\mathcal{S}}$ write $s_0\xleftrightarrow{{\mathcal{S}}, p} s_1$ for the event that $s_0$ and $s_1$ are connected in $G_{\mathcal{S}}^p$, i.e., that there is a finite path of open disks between them (in particular, $s_0$ and $s_1$ need to be open). Define the distance between $s_0$ and $s_1$ by $$\label{eq:distance between sets}
d(s_0, s_1):=\min\{\|x-y\|_\infty\colon x\in s_0, y\in s_1\}$$ where $\|\cdot\|_\infty$ denotes $\ell_\infty$ distance; we emphasize that this distance is measured *in the ambient space ${\mathbb{R}}^2$* rather than in the graph $G_{\mathcal{S}}$. Given $r>0$ and $s_0\in {\mathcal{S}}$ define the event that $s_0$ is connected by open disks to some $s\in{\mathcal{S}}$ at distance at least $r$ from it, $$\label{eq:connectivity event}
E_{{\mathcal{S}}, p}(s_0, r) := \{\exists s\in{\mathcal{S}}\text{ satisfying }d(s_0, s)\ge r\text{ and }s_0\xleftrightarrow{{\mathcal{S}}, p} s\}.$$ We also define the event that $s_0$ is connected to infinity; precisely, let $E_{{\mathcal{S}}, p}(s_0, \infty)$ be the event that $s_0$ is open and there is a sequence of open $s_1,s_2,\ldots$ in ${\mathcal{S}}$ with $s_n$ adjacent to $s_{n+1}$ in $G_{\mathcal{S}}$ for $n\ge 0$ and with $d(s_0, s_n)\to\infty$ as $n\to\infty$ (see Section \[sec:other connectivity notions\] for other connectivity notions). Lastly, let $\operatorname{diam}(s)$ be the diameter of a disk $s$. The following is our main result.
\[thm:percolation for circle packing\] There exists $p>0$ such that the following holds: Let ${\mathcal{S}}$ be a circle packing and $s_0\in {\mathcal{S}}$. Then $$\label{eq:no open path to infinity circle packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, \infty))=0.$$ Moreover, if $D:=\sup_{s\in{\mathcal{S}}}\operatorname{diam}(s)<\infty$ then for each $r>0$, $$\label{eq:bounded above squares circle packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le e^{-\frac{r}{D}}.$$
We emphasize that the theorem excludes only the existence of an open path to *infinity* in the percolation process on ${\mathcal{S}}$. If ${\mathcal{S}}$ has accumulation points, it may still happen that $G_{{\mathcal{S}}}^p$ has an infinite connected component. Indeed, for each $p>0$ there are circle packings for which this is the case, with probability one — e.g., circle packings of regular trees of sufficiently high degree. Another example is when ${\mathcal{S}}$ has a disk $s$ which is tangent to infinitely many other disks, in which case $G_{{\mathcal{S}}}^p$, for any $p>0$, has an infinite connected component with positive probability (with probability one on the event that $s$ is open). Infinite connected components of $G_{\mathcal{S}}^p$ are excluded, however, when ${\mathcal{S}}$ has at most countably many accumulation points and ${\mathcal{S}}$ is *locally finite* in the sense that each disk of ${\mathcal{S}}$ is tangent to only finitely many other disks.
\[cor:circle packing accumulation points\] Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. Let ${\mathcal{S}}$ be a locally finite circle packing with at most countably many accumulation points in ${\mathbb{R}}^2$. Then the probability that $G_{\mathcal{S}}^p$ has an infinite connected component is zero.
The corollary follows from Theorem \[thm:percolation for circle packing\] by applying Möbius transformations to ${\mathcal{S}}$, as detailed in Section \[sec:proof of corollary no accumulation points\].
Recurrent and transient triangulations {#sec:recurrent and transient triangulations}
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In this section we deduce consequences of Theorem \[thm:percolation for circle packing\] for plane triangulations. A main tool is a result of He and Schramm [@HS95] which connects the recurrence/transience properties of plane triangulations with their representing circle packings. The obtained results further shed light on two conjectures and a question of Benjamini [@B2018].
We start by describing required notation, following [@N18 Chapters 3,4]. A graph is *locally finite* if every vertex has finite degree, it is *simple* if it has no multiple edges or self loops and it is of *bounded degree* if the supremum of its degrees is finite. A *proper drawing* of a planar graph $G$ is a map sending the vertices of $G$ to distinct points in ${\mathbb{R}}^2$ and the edges of $G$ to continuous curves between the corresponding vertices so that no two curves intersect, except at the vertices shared by their edges. A *planar map* is a locally finite planar graph endowed with a cyclic permutation of the edges incident to each vertex, such that there exists a proper drawing of the graph for which the clockwise order of the curves touching the image of each vertex follows the cyclic permutation associated to that vertex. We often use the same notation for the planar map and its underlying graph. A planar map is *simple* (*connected, bounded degree*) if its underlying graph is simple (connected, bounded degree). The structure of a planar map allows to define its *faces*. To this end direct each of the edges of $G$ in both ways and say that a directed edge $\vec{e}$ of the map *precedes* the directed edge $\vec{f}$ if $\vec{e}=(x,v), \vec{f}=(v,y)$ and $y$ is the successor of $x$ in the cyclic permutation of $v$ (if $x$ is the only neighbor of $v$ we mean that $y=x$). Now define an equivalence relation on directed edges by saying that $\vec{e}$ and $\vec{f}$ are in the same face if there exists a directed path $\vec{e}_1,\ldots, \vec{e}_m$ in the graph with $\{\vec{e},\vec{f}\}=\{\vec{e}_1,\vec{e}_m\}$ and $\vec{e}_i$ preceding $\vec{e}_{i+1}$ for each $i$. Faces are defined as the equivalence classes of this relation. The planar map is a *triangulation* if it is connected and each of its faces has exactly 3 edges. A graph (or planar map) is *infinite* it has infinitely many vertices. An infinite graph is *one-ended* if removing any finite subset of its vertices (and their incident edges) leaves exactly one infinite connected component. Infinite one-ended triangulations are also called *plane triangulations* or *disk triangulations* as they have proper drawings which ‘cover’ the plane, or disk, in a suitable sense (see [@N18 Chapter 4.1]).
Itai Benjamini [@B2018] (see also [@BK18]) made several conjectures regarding percolation on planar triangulations which are suggested by assuming quasi-invariance under coarse conformal uniformization. The following conjecture and question serve as part of the motivation for our next results.
[@B2018 Conjecture 2.1]\[conj:transient triangulations\] Let $G$ be a bounded degree plane triangulation. If $G$ is transient then $G^{1/2}$ has an infinite connected component almost surely.
It is further pointed out in [@B2018] that the conjecture remains open even if $1/2$ is replaced by any other fixed $1/2<p<1$.
[@B2018 Section 2]\[ques:recurrent triangulations\] Let $G$ be a bounded degree plane triangulation. Does recurrence of $G$ imply that $G^{1/2}$ does not have an infinite connected component almost surely?
We say that a planar map $G$ is *represented* by a circle packing $\mathcal{S}$ if $G = G_{{\mathcal{S}}}$ and the cyclic order on the edges incident to each vertex $v$ equals the clockwise order in which the disks corresponding to the neighbors of $v$ appear around the disk corresponding to $v$. The *carrier* of a circle packing ${\mathcal{S}}$ representing a triangulation is the union of the closed disks of ${\mathcal{S}}$ together with the space between any three disks corresponding to a face of the triangulation. The definitions imply that all accumulation points of ${\mathcal{S}}$ lie outside of its carrier.
The following theorem is part of the main result of [@HS95] (see also [@N18 Chapter 4]). Write $\mathbb{D}:=\{z\in{\mathbb{R}}^2\colon \|z\|<1\}$ for the open unit disk.
(He–Schramm [@HS95])\[thm:He–Schramm\] Let $G$ be a simple plane triangulation.
1. If $G$ is recurrent then it may be represented by a circle packing whose carrier is ${\mathbb{R}}^2$.
2. If $G$ is transient and of bounded degree then it may be represented by a circle packing whose carrier is $\mathbb{D}$.\[item:He–Schramm transience\]
The first part of Theorem \[thm:percolation for circle packing\], given in , is not helpful for circle packings whose carrier is the unit disk. However, the second part, given in , is applicable and allows to derive the following lemma, which shows the existence of an infinite cluster for sufficiently large values of the percolation parameter.
\[lem:circle packings of unit disk\] Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. Let ${\mathcal{S}}$ be a circle packing representing a triangulation whose carrier is $\mathbb{D}$. Then the probability that $G_{\mathcal{S}}^{1-p}$ has an infinite connected component is one.
The method of proof of the lemma can yield the existence of infinite connected components on other triangulations. We formulate one more result of this type.
\[lem:circle packings with slowly growing radii\] Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. There exists $c>0$ such that the following holds. Let ${\mathcal{S}}$ be a circle packing representing a triangulation whose carrier is $\mathbb{{\mathbb{R}}}^2$. Let $f(r)$ be the maximal radius of a disk in ${\mathcal{S}}$ intersecting $r\mathbb{D}$. Suppose that $$\label{eq:radius growth}
\limsup_{r\to\infty} f(r)\cdot\frac{\log\log r}{r}\le c.$$ Then the probability that $G_{\mathcal{S}}^{1-p}$ has an infinite connected component is one.
We remark that the assertion of the lemma may fail if $f(r)$ grows linearly; see Section \[sec:value of p\_c\].
Corollary \[cor:circle packing accumulation points\], Theorem \[thm:He–Schramm\] and Lemma \[lem:circle packings of unit disk\] yield the following corollary, which resolves the versions of Conjecture \[conj:transient triangulations\] and Question \[ques:recurrent triangulations\] in which the parameter $1/2$ is replaced by sufficiently large and sufficiently small probabilities, respectively.
\[cor:percolation on plane triangulations\] Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. Let $G$ be a simple plane triangulation.
1. If $G$ is recurrent then the probability that $G^p$ has an infinite connected component is zero.\[item:recurrent plane triangulations\]
2. If $G$ is transient and has bounded degrees then the probability that $G^{1-p}$ has an infinite connected component is one.\[item:transient plane triangulations\]
Benjamini made a second conjecture regarding left-right crossings in percolation on square tilings.
[@B2018 Conjecture 2.2]\[conj:square tiling\] There exists $c>0$ so that the following holds. Tile the unit square with (possibly infinitely many) squares of varying sizes so that at most three squares meet at corners. Color each square black or white with equal probability independently. Then the probability of a black left-right crossing is at least $c$.
It is pointed out in [@B2018] that the conjecture is open even when the probability to color a square black is $2/3$, and to the author’s knowledge the conjecture is open for all fixed probabilities in $[1/2,1)$. Our results suffice to verify the version of the conjecture in which the probability of coloring a square black is an absolute constant close to $1$ and the tiling involves only finitely many squares (otherwise, the notion of left-right crossing may need to be made more precise; see Section \[sec:other connectivity notions\]). We prove a slightly stronger statement. Note that as the squares in Conjecture \[conj:square tiling\] form a triangulation, the event of a black left-right crossing equals the event that there is no white top-bottom crossing. We consider a collection of squares with disjoint interiors in the unit square (not necessarily a tiling, with four squares allowed to share a corner) and bound the probability of a white top-bottom crossing where the crossing is allowed to use diagonal connectivity when squares share a corner diagonally. The statement (and proof) involve Theorem \[thm:percolation for general packings\] of Section \[sec:general packings\].
\[cor:square tiling of square\] There exists $c>0$ so that the following holds. Let $p>0$ be the constant from Theorem \[thm:percolation for general packings\] for a packing of squares in ${\mathbb{R}}^2$. Pack finitely many squares in the unit square. Color each square white with probability $p$ independently. Then the probability of a white top-bottom crossing of the unit square (with diagonal connectivity allowed when four squares share a corner) is at most $1-c$.
Benjamini [@B2018] remarks that if conjecture \[conj:square tiling\] is true then the same should hold for a tiling, or a packing of a triangulation, with a set of shapes that are of bounded Hausdorff distance to circles. We note that a version of Corollary \[cor:square tiling of square\] with other shapes may be proved in a similar manner using Theorem \[thm:percolation for general packings\].
Benjamini–Schramm limits
------------------------
In this section we use Corollary \[cor:circle packing accumulation points\] to study percolation on Benjamini–Schramm limits of finite planar graphs. We conclude that on *all* such graphs, percolation with the parameter $p$ of Theorem \[thm:percolation for circle packing\] has no infinite connected component, almost surely.
We start by defining the necessary concepts (see also [@N18 Chapter 5]). A *rooted graph* $(G,\rho)$ is a graph with a distinguished vertex $\rho$. For a graph $G$, vertex $v$ of $G$ and integer $r\ge 0$, let $B_G(v,r)$ be the graph ball of radius $r$ in $G$ centered around $v$, that is, the induced subgraph on the set of vertices at graph distance at most $r$ from $v$, rooted at $v$. Suppose $(G_n)_{n\ge 1}$ is a sequence of, possibly random, finite graphs. Let $\rho_n$ be a uniformly sampled vertex of $G_n$ (if $G_n$ is random, one first samples $G_n$ and then samples $\rho_n$ uniformly in $G_n$). Let $(G,\rho)$ be a random rooted graph, with $G$ connected almost surely. Then $(G,\rho)$ is called the *Benjamini–Schramm limit* (or local limit) of $(G_n)$ if for each $r\ge 0$, the distribution of $B_{G_n}(\rho_n, r)$ converges as $n$ tends to infinity to the distribution of $B_G(\rho, r)$ (in the sense that for each $r\ge 0$, ${\mathbb{P}}(B_{G_n}(\rho_n,r)=(H,\sigma))\to{\mathbb{P}}(B_G(\rho,r)=(H,\sigma))$ for every rooted graph $(H,\sigma)$, where the equality sign denotes the existence of a root-preserving graph isomorphism). Note that if $(G,\rho)$ is the Benjamini–Schramm limit of the finite graphs $(G_n)$ then $G$ is locally finite, and that if the $(G_n)$ are simple then also $G$ is simple.
Benjamini and Schramm [@BS01] proved that every Benjamini–Schramm limit of finite simple *planar* graphs with uniformly bounded degrees may be represented by a circle packing with at most one accumulation point, almost surely. Asaf Nachmias explained to the author that the restriction on the degrees is not required for this conclusion (Benjamini and Schramm further proved that the limiting graph is recurrent, and for this conclusion the restriction is relevant) and suggested the following lemma and the idea of its proof.
\[lem:circle packing with at most one accumulation point\] Let $(G_n)$ be a sequence of, possibly random, finite simple planar graphs with Benjamini–Schramm limit $(G,\rho)$. Then, almost surely, there is a circle packing ${\mathcal{S}}$ with at most one accumulation point in ${\mathbb{R}}^2$ such that $G=G_{\mathcal{S}}$.
Combining Corollary \[cor:circle packing accumulation points\] and Lemma \[lem:circle packing with at most one accumulation point\] yields the following conclusion.
\[cor:percolation on Benjamini Schramm limits\] Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. Let $(G,\rho)$ be a Benjamini–Schramm limit of, possibly random, finite simple planar graphs. Then the probability that $G^p$ has an infinite connected component is zero.
This corollary is an important ingredient for the proof in [@CGHP20] that the loop $O(n)$ model exhibits macroscopic loops in a subset of positive measure of its phase diagram. This application makes use of the fact that the corollary holds without a bounded degree assumption.
General packings {#sec:general packings}
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The method of proof of Theorem \[thm:percolation for circle packing\] generalizes to higher dimensions and to a general class of packings. We describe a theorem to this effect following required notation.
A *packing* in ${\mathbb{R}}^d$ is a finite or countable collection of non-empty compact sets in ${\mathbb{R}}^d$ with disjoint interiors. A packing ${\mathcal{S}}$ defines a graph $G_{\mathcal{S}}$ with vertex set ${\mathcal{S}}$ by declaring two (distinct) sets *adjacent* when their intersection is non-empty. It is noted that these general definitions allow for sets in ${\mathcal{S}}$ to be disconnected, that even when $d=2$ the graph $G_{\mathcal{S}}$ need not be planar and that a set may be adjacent to infinitely many other sets in $G_{\mathcal{S}}$.
Again, denote by $G_{\mathcal{S}}^p$ the (random) induced subgraph on retained sets in a site percolation process on $G_{\mathcal{S}}$ with parameter $p$. Given $s_0, s_1\in{\mathcal{S}}$ write $s_0\xleftrightarrow{{\mathcal{S}}, p} s_1$ for the event that $s_0$ and $s_1$ are connected in $G_{\mathcal{S}}^p$, i.e., that there is a finite path of open sets between them (in particular, $s_0$ and $s_1$ need to be open). The distance $d(s_0, s_1)$ between sets is again given by where now $\|\cdot\|_\infty$ denotes the $\ell_\infty$ distance in ${\mathbb{R}}^d$. The event $E_{{\mathcal{S}},p}(s_0, r)$ that $s_0$ is connected by open sets to some $s\in{\mathcal{S}}$ at distance at least $r$ from it is again defined by . The event that $s_0$ is connected to infinity, $E_{{\mathcal{S}}, p}(s_0, \infty)$, is again the event that $s_0$ is open and there is a sequence of open $s_1,s_2,\ldots$ in ${\mathcal{S}}$ with $s_n$ adjacent to $s_{n+1}$ in $G_{\mathcal{S}}$ for $n\ge 0$ and with $d(s_0, s_n)\to\infty$ as $n\to\infty$. For a set $s$ in ${\mathbb{R}}^d$ define its diameter by $$\operatorname{diam}(s):=\sup\{\|x-y\|_\infty\colon x,y\in s\}.$$
For a measurable set $s$ in ${\mathbb{R}}^d$ let $\operatorname{vol}(s)$ be its Lebesgue measure and $\partial s$ stand for its boundary. A packing ${\mathcal{S}}$ in ${\mathbb{R}}^d$ is called *${\varepsilon}$-regular* if $\operatorname{vol}(\partial s)=0$ and $\operatorname{vol}(s)\ge {\varepsilon}\operatorname{diam}(s)^d$ for each $s\in{\mathcal{S}}$.
\[thm:percolation for general packings\] Let $d\ge 2$ and ${\varepsilon}>0$. There exists $p = p(d,{\varepsilon})>0$ such that the following holds: Let ${\mathcal{S}}$ be an ${\varepsilon}$-regular packing in ${\mathbb{R}}^d$ and $s_0\in {\mathcal{S}}$. Then $$\label{eq:no open path to infinity general packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, \infty))=0.$$ Moreover, if $D:=\sup_{s\in{\mathcal{S}}}\operatorname{diam}(s)<\infty$ then for each $r>0$, $$\label{eq:bounded above diameters general packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le e^{-\frac{r}{D}}.$$
We remark that the exponential decay bound has no leading constant in the exponent. However, examination of the proof shows that the bound ${\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le e^{-C\frac{r}{D}}$ with $C\ge 1$ is also valid upon replacing $p(d,{\varepsilon})$ with a suitable $p(d,{\varepsilon},C)>0$.
Our techniques can lead to a more general result, in which the sets are arbitrary subsets of some metric space (without the packing assumption) and restrictions are placed on the number of sets having diameter at least $d_1$ which can be at distance at most $k\cdot d_1$ from a set of diameter $d_2$. We do not elaborate on this extension.
Acknowledgements
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The author is grateful to Asaf Nachmias for enlightening discussions on the topic of circle packings and for suggesting the statement and idea of proof of Lemma \[lem:circle packing with at most one accumulation point\]. Further thanks are due to Itai Benjamini for enthusiasm and support and to Alexander Glazman, Ori Gurel-Gurevich and Matan Harel for very helpful conversations on the presented results.
The author is supported by the Israel Science Foundation grants 861/15 and 1971/19 and by the European Research Council starting grant 678520 (LocalOrder).
Percolation on square packings {#sec:percolation on square packings}
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A *square packing* ${\mathcal{S}}$ is a collection of closed squares in ${\mathbb{R}}^2$ with sides parallel to the coordinate axes and with disjoint interiors. It is a special case of the ${\varepsilon}$-regular packings discussed in Section \[sec:general packings\]. To explain the argument in its simplest form first, we start by explaining the proof of Theorem \[thm:percolation for general packings\] for square packings. The full theorem, of which the case of circle packings is a special case, is proved in Section \[sec:general packings proof\].
We prove the theorem with $$\label{eq:choice of p}
p := e^{-26}$$ (though fine-tuning the proof can give somewhat larger values).
As the first and main step of the proof, we show (a variant of) the bound under the additional assumption that the side lengths of the squares in the packing are bounded below: For integer $k\ge 0$ let $\Sigma^k$ be the collection of all pairs $({\mathcal{S}},s_0)$ with ${\mathcal{S}}$ a square packing with side lengths in $[1,2^k]$ and $s_0\in{\mathcal{S}}$. Define $$\label{eq:alpha def}
\alpha(k,r):=\sup_{({\mathcal{S}}, s_0)\in\Sigma^k}{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r)).$$ We shall prove that for integer $k\ge 0$ and real $r>0$, $$\label{eq:induction statement}
\alpha(k,r)\le e^{-\frac{r}{2^{k-1}}}.$$
We prove by a double induction on $k$ and $r$. We start with the base case $k=0$. Let $r>0$ and let $({\mathcal{S}}, s_0)\in \Sigma^0$. As all squares in ${\mathcal{S}}$ have side length $1$, if $s\in{\mathcal{S}}$ satisfies $d(s_0, s)\ge r$ then the graph distance of $s_0$ and $s$ in $G_{\mathcal{S}}$ is at least $\lceil r\rceil+1$. Simple geometric considerations show that each square in ${\mathcal{S}}$ can be adjacent to at most $8$ other squares in $G_{\mathcal{S}}$ (recalling that squares touching at a corner are adjacent) and thus the number of paths of length $L$ in $G_{\mathcal{S}}$ which start at $s_0$ is at most $8^L$. We conclude that $${\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0,r))\le {\mathbb{P}}(\text{there is a simple path in $G_{\mathcal{S}}^p$ of length $\lceil r\rceil+1$ from $s_0$})\le p\cdot(8p)^{\lceil r\rceil+1}\le e^{-2r}$$ for our choice of $p$. Thus is established for $k=0$ and all $r>0$.
Fix an integer $k\ge 1$. We assume by induction that is established for all $r>0$ when the $k$ of is replaced by $k-1$, and proceed to establish for all $r>0$ with our fixed $k$. This is achieved via a second induction on $r$. Suppose first, as a base case, that $0<r\le 2^k$. Then for each $({\mathcal{S}}, s_0)\in \Sigma^k$, recalling the choice of $p$, $$\label{eq:small r}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0,r))\le {\mathbb{P}}(s_0\in G_{\mathcal{S}}^p) = p <e^{-2}\le e^{-\frac{r}{2^{k-1}}}$$ as required. Now fix $r>2^k$ and assume, as the second induction hypothesis, that is established with our fixed $k$ when the $r$ of is restricted to $[0,r - 2^k]$.
Let $({\mathcal{S}}, s_0)\in \Sigma^k$. We first reduce to the case in which $\operatorname{diam}(s_0)\in[1, 2^{k-1}]$, if this is not already the case, by the following geometric construction (this is the place where it is convenient to work with squares instead of disks). If $\operatorname{diam}(s_0)\in(2^{k-1}, 2^k]$ we cut $s_0$ in half along both axes, dividing it into four squares $s_0^1, \ldots, s_0^4$ with half the side length of $s_0$ and with disjoint interiors. These give rise to pairs $({\mathcal{S}}^i, s_0^i)$, $1\le i\le 4$, where ${\mathcal{S}}^i := ({\mathcal{S}}\setminus\{s_0\})\cup\{s_0^i\}$. We use the natural coupling of $G_{{\mathcal{S}}}^p$ with $G_{{\mathcal{S}}^i}^p$ in which the vertex sets of these (random) graphs are the same except that $s_0\in V(G_{{\mathcal{S}}}^p)$ if and only if $s_0^i\in V(G_{{\mathcal{S}}^i}^p)$. The construction yields that $E_{{\mathcal{S}},p}(s_0,r)\subseteq\cup_{i=1}^4 E_{{\mathcal{S}}^i,p}(s_0^i, r)$ whence $${\mathbb{P}}(E_{{\mathcal{S}},p}(s_0,r))\le \sum_{i=1}^4 {\mathbb{P}}(E_{{\mathcal{S}}^i,p}(s_0^i, r)).$$ It thus suffices to prove that for each $({\mathcal{S}}, s_0)\in \Sigma^k$ with $\operatorname{diam}(s_0)\in[1,2^{k-1}]$ we have $$\label{eq:goal after dividing s 0}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0,r))\le \frac{1}{4}e^{-\frac{r}{2^{k-1}}}.$$ Fix such an $({\mathcal{S}}, s_0)$. We next aim to decompose the event $E_{{\mathcal{S}}, p}(s_0, r)$ into events to which our induction hypotheses apply. To this end, decompose ${\mathcal{S}}$ into $$\begin{split}
{\mathcal{S}}^{k-1} &:= \{s\in {\mathcal{S}}\colon \operatorname{diam}(s)\in[1,2^{k-1}]\},\\
{\mathcal{S}}_{k-1}^k &:= \{s\in {\mathcal{S}}\colon \operatorname{diam}(s)\in(2^{k-1},2^k]\}.
\end{split}$$ Note that $s_0\in{\mathcal{S}}^{k-1}$ by assumption. We couple $G_{{\mathcal{S}}^{k-1}}^p$ and $G_{{\mathcal{S}}}^p$ in the natural way, by setting $G_{{\mathcal{S}}^{k-1}}^p$ to be the induced subgraph of $G_{\mathcal{S}}^p$ on ${\mathcal{S}}^{k-1}$. Let $$E^{k-1} := E_{{\mathcal{S}}^{k-1},p}(s_0, r)$$ be the event that $s_0$ is connected to $\ell^\infty$ distance $r$ by a path in $G_{{\mathcal{S}}^{k-1}}^p$. In addition, define three events for each $s\in {\mathcal{S}}_{k-1}^k$: Let $E_s^1$ be the event that there is a path in $G_{{\mathcal{S}}^{k-1}}^p$ from $s_0$ to a square adjacent (in $G_{\mathcal{S}}$) to $s$. Let $E_s^2$ be the event that there is a path in $G_{{\mathcal{S}}}^p$ from $s$ to some $s_1$ with $d(s_0, s_1)\ge r$ (allowing the possibility that $s_1=s$), and let $E_s$ be the event that $E_s^1$ and $E_s^2$ occur disjointly, that is, that there exist two disjoint paths of open squares, one implying that $E_s^1$ occurs and the other implying that $E_s^2$ occurs (the notation $E_s := E_s^1\circ E_s^2$ is sometimes used for this operation). Our definitions imply the following decomposition $$E_{{\mathcal{S}},p}(s_0, r) = E^{k-1}\cup\bigcup_{s\in {\mathcal{S}}_{k-1}^k} E_s$$ so that $$\label{eq:E S p union bound}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0, r))\le {\mathbb{P}}(E^{k-1}) + \sum_{s\in {\mathcal{S}}_{k-1}^k} {\mathbb{P}}(E_s).$$ We proceed to estimate each of the probabilities on the right-hand side. First, $$\label{eq:E k-1 probability bound}
{\mathbb{P}}(E^{k-1}) \le \alpha(k-1,r)\le e^{-\frac{r}{2^{k-2}}}$$ by the (first) induction hypothesis. Second, as the two paths involved in the definition of $E_s$ are disjoint, we may invoke the van-den-Berg–Kesten inequality [@BK85] to obtain $${\mathbb{P}}(E_s) \le {\mathbb{P}}(E_s^1)\cdot{\mathbb{P}}(E_s^2)\le \min\{p, \alpha(k-1, d(s_0, s)-2^{k-1})\}\cdot\alpha(k, r - d(s_0, s) - 2^k).$$ where we define $\alpha(k,r')$ to be $1$ if $r'\le 0$ and where we have used that ${\mathbb{P}}(E_s^1)\le p$ as the square $s_0$ itself needs to be open for $E_s^1$ to occur. Our induction hypotheses thus show that $$\label{eq:E s bound}
{\mathbb{P}}(E_s)\le \min\{p, e^{\frac{-d(s_0, s)+2^{k-1}}{2^{k-2}}}\}\cdot e^{\frac{-r +d(s_0, s) + 2^k}{2^{k-1}}}.$$ We further decompose ${\mathcal{S}}_{k-1}^k$ to $$\begin{split}
{\mathcal{S}}_{k-1}^k(0) &:= \{s\in {\mathcal{S}}_{k-1}^k\colon d(s_0, s)\le 8\cdot2^k\},\\
{\mathcal{S}}_{k-1}^k(m) &:= \{s\in {\mathcal{S}}_{k-1}^k\colon m 2^k<d(s_0, s)\le (m+1)2^k\}, \quad m\ge 8
\end{split}$$ and simplify the expression in each case, $$\label{eq:E s probability estimates}
\begin{split}
{\mathbb{P}}(E_s)&\le p\cdot e^{-\frac{r}{2^{k-1}}+18},\ \ \quad\qquad\qquad\qquad s\in {\mathcal{S}}_{k-1}^k(0),\\
{\mathbb{P}}(E_s)&\le e^{-\frac{r + d(s_0, s)}{2^{k-1}} + 4}\le e^{-\frac{r}{2^{k-1}} - 2m + 4},\quad s\in {\mathcal{S}}_{k-1}^k(m),\, m\ge 8.
\end{split}$$ We proceed to upper bound the size of ${\mathcal{S}}_{k-1}^k(m)$. Each $s\in{\mathcal{S}}_{k-1}^k$ has area at least $2^{2k-2}$ and is fully contained in the $\ell^\infty$ annulus $A_s$ of in-radius $d(s_0, s)$ and out-radius $2^{k-2}+d(s_0, s)+2^k$ around the center of $s_0$. As $$\text{area}(A_s) = (2^{k-2}+d(s_0, s)+2^k)^2 - d(s_0, s)^2 \le 2^{2k+1}+2^{k+2}d(s_0, s)$$ we conclude that $$\label{eq:area estimates}
\begin{split}
|{\mathcal{S}}_{k-1}^k(0)|&\le \frac{2^{2k+1}+8\cdot 2^{2k+2}}{2^{2k-2}} = 136,\\
|{\mathcal{S}}_{k-1}^k(m)|&\le \frac{2^{2k+1}+(m+1)2^{2k+2}}{2^{2k-2}} = 16m + 24\le 20m,\quad m\ge 8.
\end{split}$$ Finally, plugging the bounds , and into implies that $$\label{eq:final bound}
\begin{split}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0, r)) &\le e^{-\frac{r}{2^{k-2}}} + 136 p\cdot e^{-\frac{r}{2^{k-1}}+18} + \sum_{m=8}^\infty 20m e^{-\frac{r}{2^{k-1}} - 2m + 4}\\
&= \frac{1}{4}e^{-\frac{r}{2^{k-1}}}(4 e^{-\frac{r}{2^{k-1}}} + 544 p e^{18} + 80 e^4\sum_{m=8}^\infty m e^{-2m}).
\end{split}$$ Recalling that $r>2^k$ (the complimentary case having been discussed in ) one checks that the expression inside the parenthesis in is at most $1$ for our choice of $p$, thus verifying and finishing the proof of .
As the second step of the proof, we verify the bound .
Suppose ${\mathcal{S}}$ is a square packing with side lengths in $[1,D]$ for some $D<\infty$ and let $s_0\in {\mathcal{S}}$. Then implies as for each $r>0$, $$\label{eq:bounded above squares with general R}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le e^{-\frac{r}{2^{k-1}}} \le e^{-\frac{r}{D}}.$$ where $k$ is an integer such that $2^{k-1}<D\le 2^k$. Moreover, if holds for a square packing ${\mathcal{S}}$ and square $s_0\in {\mathcal{S}}$ then, for any $\rho>0$, it holds also for the dilated square packing $\rho{\mathcal{S}}$ (having upper bound $\rho D$ on its side lengths) and square $\rho s_0$. Thus follows whenever the side lengths of ${\mathcal{S}}$ are bounded above and below by arbitrary finite positive numbers. Finally suppose ${\mathcal{S}}$ is a square packing with side lengths in $(0,D]$ (possibly with $\inf _{s\in {\mathcal{S}}} \operatorname{diam}(s) = 0$) and $s_0\in {\mathcal{S}}$. By definition, paths between squares in ${\mathcal{S}}$ are finite and thus in any such path there is a *positive* minimal side length for the squares involved. It follows that $E_{{\mathcal{S}}, p}(s_0, r) = \cup_{n=n_0}^\infty E_{{\mathcal{S}}_n, p}(s_0, r)$ with ${\mathcal{S}}_n = \{s\in {\mathcal{S}}\colon \operatorname{diam}(s)\ge \frac{1}{n}\}$ and $n_0=\lceil 1/\operatorname{diam}(s_0)\rceil$. As the union is increasing we have that ${\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r)) = \lim_{n\to\infty} {\mathbb{P}}(E_{{\mathcal{S}}_n, p}(s_0, r))$, implying in all cases.
As the final step of the proof we proceed to show that holds. Define $$\alpha := \sup_{({\mathcal{S}}, s_0)}{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, \infty))$$ where the supremum is over all square packings ${\mathcal{S}}$ and $s_0\in {\mathcal{S}}$.
Let ${\mathcal{S}}$ be a square packing and $s_0 \in {\mathcal{S}}$. By definition, the event $E_{{\mathcal{S}}, p}(s_0, \infty)$ entails the existence of a sequence of open squares $s_0, s_1,\ldots$ in ${\mathcal{S}}$ with $s_n$ adjacent to $s_{n+1}$ in $G_{\mathcal{S}}$ and $d(s_0, s_n)\to\infty$ as $n\to\infty$. Set $\ell_0 := \operatorname{diam}(s_0)$. Let $E^0_{{\mathcal{S}}, p}(s_0, \infty)$ be the sub-event of $E_{{\mathcal{S}}, p}(s_0, \infty)$ in which there exist $(s_n)$ as above with $\sup_n \operatorname{diam}(s_n)\le\ell_0$ and let $E^1_{{\mathcal{S}}, p}(s_0, \infty)$ be the complimentary sub-event. Write ${\mathcal{S}}^0:=\{s\in{\mathcal{S}}\colon \operatorname{diam}(s)\le \ell_0\}$. Clearly $E^0_{{\mathcal{S}}, p}(s_0, \infty) = E_{{\mathcal{S}}^0, p}(s_0, \infty)$ (with the natural coupling of the percolation processes) whence the bound shows that $$\label{eq:E 0 prob bound}
{\mathbb{P}}(E^0_{{\mathcal{S}}, p}(s_0, \infty))=0.$$
It remains to show that also $E^1_{{\mathcal{S}}, p}(s_0, \infty)$ is of zero probability. Let ${\mathcal{S}}^{\text{big}}:=\{s\in{\mathcal{S}}\colon \operatorname{diam}(s)>\ell_0\}$. For $s\in {\mathcal{S}}^{\text{big}}$ let $F_s^1$ be the event that there is a path of open squares of side lengths at most $\ell_0$ from $s_0$ to a neighbor of $s$. Let $F_s := F_s^1\circ E_{{\mathcal{S}}, p}(s, \infty)$ be the event that $F_s^1$ occurs disjointly from $E_{{\mathcal{S}}, p}(s, \infty)$, that is, that the two events occur due to disjoint open paths. By definition, $$E^1_{{\mathcal{S}}, p}(s_0, \infty) = \cup_{s\in {\mathcal{S}}^{\text{big}}} F_s.$$ A second use of the van-den-Berg–Kesten inequality [@BK85] implies that $$\label{eq:E 1 prob bound}
{\mathbb{P}}(E^1_{{\mathcal{S}}, p}(s_0, \infty))\le \sum_{s\in {\mathcal{S}}^{\text{big}}} {\mathbb{P}}(F_s^1){\mathbb{P}}(E_{{\mathcal{S}}, p}(s, \infty))\le \alpha\sum_{s\in {\mathcal{S}}^{\text{big}}} {\mathbb{P}}(F_s^1).$$ We proceed to bound the sum occurring in the last expression, arguing similarly to before. Write $$\begin{split}
{\mathcal{S}}(0) &:= \{s\in {\mathcal{S}}^{\text{big}}\colon d(s_0, s)\le 8\ell_0\},\\
{\mathcal{S}}(m) &:= \{s\in {\mathcal{S}}^{\text{big}}\colon m \ell_0<d(s_0, s)\le (m+1)\ell_0\}, \quad m\ge 8.
\end{split}$$ Each square $s\in{\mathcal{S}}(0)$ has area at least $\ell_0^2$ in the $\ell^\infty$ ball of radius $9\frac{1}{2}\ell_0$ around the center of $s_0$. Similarly, each square $s\in{\mathcal{S}}(m)$ has area at least $\ell_0^2$ in the annulus of in-radius $m\ell_0$ and out-radius $(m+\frac{5}{2})\ell_0$ around the center of $s_0$. Thus $|{\mathcal{S}}(0)|\le 19^2$ and $|{\mathcal{S}}(m)|\le (2m+5)^2-(2m)^2 \le 24m$. Applying the bound we conclude that $$\sum_{s\in {\mathcal{S}}^{\text{big}}} {\mathbb{P}}(F_s^1) \le |{\mathcal{S}}(0)|p + \sum_{m=8}^\infty 24m e^{-m} < 1-\delta$$ for some $\delta>0$, due to our choice of $p$. Combining this result with and shows that $${\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, \infty))\le (1-\delta)\alpha.$$ However, taking supremum over the square packing ${\mathcal{S}}$ and $s_0\in {\mathcal{S}}$ implies that $$\alpha \le (1-\delta)\alpha$$ which is only possible if $\alpha=0$, as required.
Percolation on general packings {#sec:general packings proof}
===============================
In this section we prove Theorem \[thm:percolation for general packings\], detailing the necessary changes from the proof for square packings in Section \[sec:percolation on square packings\].
Fix $d\ge 2$ and ${\varepsilon}>0$. The value $p = p(d,{\varepsilon})$ with which the theorem is proved is chosen small enough for the following arguments.
The main step is again to prove a variant of the bound under the additional assumption that the diameters of the shapes are bounded below. For integer $k\ge 0$ let $\Sigma^k$ be the collection of all pairs $({\mathcal{S}},s_0)$ with ${\mathcal{S}}$ a packing in ${\mathbb{R}}^d$ satisfying $\operatorname{diam}(s)\in[1,2^k]$ for $s\in{\mathcal{S}}$, $s_0\in{\mathcal{S}}$ and where we require ${\mathcal{S}}\setminus\{s_0\}$ to be ${\varepsilon}$-regular (i.e., ${\varepsilon}$-regularity is not required of $s_0$). As before, set $$\label{eq:alpha def general packing}
\alpha(k,r):=\sup_{({\mathcal{S}}, s_0)\in\Sigma^k}{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r)).$$ and we shall prove that for integer $k\ge 0$ and real $r>0$, $$\label{eq:induction statement general packing}
\alpha(k,r)\le e^{-\frac{r}{2^{k-1}}}.$$ We again use double induction on $k$ and $r$ to prove . For the case $k=0$ observe that as $\operatorname{diam}(s)=1$ for $s\in{\mathcal{S}}$ and $\operatorname{vol}(\partial s)=0$, $\operatorname{vol}(s)\ge {\varepsilon}$ for $s\in{\mathcal{S}}\setminus\{s_0\}$ it follows that each $s\in{\mathcal{S}}$ can be adjacent in $G_{\mathcal{S}}$ to at most $3^d/{\varepsilon}$ sets in ${\mathcal{S}}$. The case $k=0$ thus follows in the proof for square packings by taking $p<{\varepsilon}3^{-d}$.
Fix an integer $k\ge 1$. It is again assumed by induction that is established for all $r>0$ when the $k$ of is replaced by $k-1$. The case of (with our fixed $k$ and) with $0<r\le 2^{k-1} d$ follows as before by taking $p<e^{-d}$ and noting that $s_0$ itself needs to be open for the event $E_{{\mathcal{S}}, p}(s_0, r)$ to occur. Thus we fix $r>2^{k-1}d$ and assume, as the second induction hypothesis, that is established with our fixed $k$ when the $r$ of is restricted to $[0,r - 2^k]$.
Let $({\mathcal{S}}, s_0)\in \Sigma^k$. We again reduce to the case in which $\operatorname{diam}(s_0)\in[1, 2^{k-1}]$, if this is not already the case, by a geometric construction. If $\operatorname{diam}(s_0)\in(2^{k-1}, 2^k]$ we let $C_0$ be a cube of side length $2^k$ which contains $s_0$. We partition $C_0$ into $2^d$ sub-cubes $(C_0^i)$ of side length $2^{k-1}$ and set $s_0^i := C_0^i \cap s_0$, where it is noted that some of the $s_0^i$ may be empty and that even if $s_0$ is ${\varepsilon}$-regular the (non-empty) $s_0^i$ need not be such. Let $I = \{1\le i\le 2^d\colon s_0^i\neq\emptyset\}$. For each $i\in I$ let $({\mathcal{S}}^i, s_0^i)\in\Sigma^k$ where ${\mathcal{S}}^i := ({\mathcal{S}}\setminus\{s_0\})\cup\{s_0^i\}$. The natural coupling of $G_{{\mathcal{S}}}^p$ with $G_{{\mathcal{S}}^i}^p$ is used, where the vertex sets of these (random) graphs are the same except that $s_0\in V(G_{{\mathcal{S}}}^p)$ if and only if $s_0^i\in V(G_{{\mathcal{S}}^i}^p)$. Again, $E_{{\mathcal{S}},p}(s_0,r)\subseteq\cup_{i\in I} E_{{\mathcal{S}}^i,p}(s_0^i, r)$ whence $${\mathbb{P}}(E_{{\mathcal{S}},p}(s_0,r))\le \sum_{i\in I} {\mathbb{P}}(E_{{\mathcal{S}}^i,p}(s_0^i, r)).$$ It thus suffices to prove that for each $({\mathcal{S}}, s_0)\in \Sigma^k$ with $\operatorname{diam}(s_0)\in[1,2^{k-1}]$ we have $$\label{eq:goal after dividing s 0 general packing}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0,r))\le \frac{1}{2^d}e^{-\frac{r}{2^{k-1}}}.$$
Fix such an $({\mathcal{S}}, s_0)$. The proof of begins exactly as in the proof for square packings, as we briefly recall now. The set ${\mathcal{S}}$ is partitioned to ${\mathcal{S}}^{k-1}$ and ${\mathcal{S}}_{k-1}^k$ which hold, respectively, the sets of diameter in $[1,2^{k-1}]$ and in $(2^{k-1},2^k]$. Correspondingly, we have $E_{{\mathcal{S}},p}(s_0, r) = E^{k-1}\cup\bigcup_{s\in {\mathcal{S}}_{k-1}^k} E_s$ where $E^{k-1}$ is the event that there is an open path in ${\mathcal{S}}^{k-1}$ from $s_0$ to distance $r$ and, for each $s\in{\mathcal{S}}_{k-1}^k$, $E_s$ is the event that there is an open path in ${\mathcal{S}}^{k-1}$ from $s_0$ to a neighbor of $s$ and a disjoint open path in ${\mathcal{S}}$ from $s$ to distance $r$ from $s_0$. For the purpose of applying the induction hypotheses in the next step we note here that as the two paths are disjoint, the path from $s$ to distance $r$ from $s_0$ cannot contain $s_0$ and thus may be thought of as a path in the *${\varepsilon}$-regular packing* ${\mathcal{S}}\setminus\{s_0\}$. Applying the induction hypotheses and the van-den-Berg–Kesten inequality [@BK85], $$\label{eq:E S p union bound general packing}
\begin{split}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0, r))&\le {\mathbb{P}}(E^{k-1}) + \sum_{s\in {\mathcal{S}}_{k-1}^k} {\mathbb{P}}(E_s)\\
&\le e^{-\frac{r}{2^{k-2}}} + \sum_{s\in{\mathcal{S}}_{k-1}^k} \min\{p, \alpha(k-1, d(s_0, s)-2^{k-1})\}\cdot\alpha(k, r - d(s_0, s) - 2^k)\\
&\le e^{-\frac{r}{2^{k-2}}} + \sum_{s\in{\mathcal{S}}_{k-1}^k} \min\{p, e^{-\frac{d(s_0, s)-2^{k-1}}{2^{k-2}}}\}\cdot e^{-\frac{r - d(s_0, s) - 2^k}{2^{k-1}}},
\end{split}$$ where $\alpha(k,r'):=1$ when $r'\le 0$.
We again decompose ${\mathcal{S}}_{k-1}^k$ to $$\begin{split}
{\mathcal{S}}_{k-1}^k(0) &:= \{s\in {\mathcal{S}}_{k-1}^k\colon d(s_0, s)\le m_0\cdot2^k\},\\
{\mathcal{S}}_{k-1}^k(m) &:= \{s\in {\mathcal{S}}_{k-1}^k\colon m 2^k<d(s_0, s)\le (m+1)2^k\}, \quad m\ge m_0
\end{split}$$ with the integer $m_0 = m_0(d,{\varepsilon})$ sufficiently large for the following calculations, and proceed to upper bound the size of ${\mathcal{S}}_{k-1}^k(m)$. By the ${\varepsilon}$-regularity assumption (and the fact that $s_0\notin{\mathcal{S}}_{k-1}^k$), each $s\in{\mathcal{S}}_{k-1}^k$ has $\operatorname{vol}(\partial s)=0$, $\operatorname{vol}(s)\ge{\varepsilon}2^{(k-1)d}$ and is fully contained in the $\ell^\infty$ ball $B_s$ of radius $2^{k-2}+d(s_0, s)+2^k$ around the center of $s_0$. As $$\operatorname{vol}(B_s) = (2^{k-2}+d(s_0, s)+2^k)^d$$ we conclude that $$\label{eq:vol estimates general packing}
\begin{split}
|{\mathcal{S}}_{k-1}^k(0)|&\le \frac{((m_0+2)2^k)^d}{{\varepsilon}2^{(k-1)d}} = \frac{1}{{\varepsilon}}(2m_0+4)^d,\\
|{\mathcal{S}}_{k-1}^k(m)|&\le \frac{((m+3)2^k)^d}{{\varepsilon}2^{(k-1)d}} = \frac{1}{{\varepsilon}}(2m+6)^d,\quad m\ge m_0.
\end{split}$$ These bounds may be used in to obtain $$\label{eq:final bound general packing}
\begin{split}
{\mathbb{P}}(E_{{\mathcal{S}},p}(s_0, r)) &\le e^{-\frac{r}{2^{k-2}}} + \frac{1}{{\varepsilon}}(2m_0+4)^d\cdot p\cdot e^{-\frac{r}{2^{k-1}}+2m_0+2} + \sum_{m=m_0}^\infty \frac{1}{{\varepsilon}}(2m+6)^d e^{-\frac{r}{2^{k-1}} - 2m + 4}\\
&= \frac{1}{2^d}e^{-\frac{r}{2^{k-1}}}(2^d e^{-\frac{r}{2^{k-1}}} + \frac{1}{{\varepsilon}}(4m_0+8)^d\cdot e^{2m_0+2}\cdot p + \frac{e^4}{{\varepsilon}}\sum_{m=m_0}^\infty (4m+12)^d e^{-2m}).
\end{split}$$ Recalling that $r>2^{k-1}d$ one checks that the expression inside the parenthesis in can be made smaller than $1$ by choosing first $m_0 = m_0(d,{\varepsilon})$ sufficiently large and then $p = p(d,{\varepsilon})$ sufficiently small. This verifies and finishes the proof of .
The deduction of and from now works in the same way as in the proof for square packings, where, in deducing , one relies on volume estimates analogous to .
No accumulation points {#sec:proof of corollary no accumulation points}
======================
In this section we prove Corollary \[cor:circle packing accumulation points\].
Let ${\mathcal{S}}$ be a locally finite circle packing with at most countably many accumulation points in ${\mathbb{R}}^2$. First, suppose that $G_{\mathcal{S}}$ has an infinite connected component. By K[ö]{}nig’s lemma [@K27] there is an infinite sequence of distinct disks $s_0, s_1,\ldots$ in ${\mathcal{S}}$ such that $s_n$ is tangent to $s_{n+1}$ for $n\ge 0$. Write $o_j$ for the center of $s_j$. It follows that either
1. \[item:converges to infinity\] $(o_n)$ converges to infinity, implying that $d(s_0, s_n)\to\infty$, or
2. \[item:converges to accumulation point\] $(o_n)$ converges to a point of ${\mathbb{R}}^2$, which must then be an accumulation point of ${\mathcal{S}}$.
Let us prove that these are indeed the only alternatives. If $(o_n)$ does not converge to a point of ${\mathbb{R}}^2$ or to infinity then it has two distinct subsequential limit points $x,y$, one of which may be infinity. Suppose $x$ is not infinity. As ${\mathcal{S}}$ has at most countably many accumulation points, there exists a radius $r<\|x-y\|_2$ such that the circle $S(x,r):=\{z\in{\mathbb{R}}^2\colon \|x-z\|_2=r\}$ contains no accumulation point of ${\mathcal{S}}$. Thus $S(x,r)$ can intersect at most finitely many disks of ${\mathcal{S}}$. In particular, since the disks $(s_n)$ are distinct, we must have that the $(s_n)$ are completely inside or completely outside $S(x,r)$ starting from some $n$, but this contradicts the fact that both $x$ and $y$ are subsequential limit points for $(o_n)$.
Second, let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. By way of contradiction, suppose that $G^p_{{\mathcal{S}}}$ has positive probability to contain an infinite connected component. As $G^p_{{\mathcal{S}}}$ is locally finite with at most countably many accumulation points we may apply the above arguments to it. Thus there exists, with positive probability, a sequence of distinct open disks $s_0,s_1,\ldots$ with $s_n$ tangent to $s_{n+1}$ such that one of the alternatives above holds. Alternative is ruled out by Theorem \[thm:percolation for circle packing\], applying it to each of the countably many possible starting disks $s_0$. Thus alternative must hold for a random accumulation point $q$. As there are at most countably many accumulation points, alternative holds with positive probability for some *deterministic* accumulation point $q_0$. Note that $q_0$ may be on the boundary of at most two disks of ${\mathcal{S}}$ and we denote by ${\mathcal{S}}'$ the circle packing ${\mathcal{S}}$ with these disks removed (and naturally couple $G^p_{\mathcal{S}}$ with $G^p_{{\mathcal{S}}'}$). We may assume, without loss of generality, that $(s_n)$ does not contain any of the removed disks. Lastly, we apply a Möbius transformation $T$ to ${\mathcal{S}}'$ which sends $q_0$ to infinity. The transformation maps ${\mathcal{S}}'$ to a new circle packing (as $q_0$ is not on the boundary of any disk) and defines a coupling of $G^p_{{\mathcal{S}}'}$ with $G^p_{T({\mathcal{S}}')}$. Under this coupling, on the event that the centers of the disks $(s_n)$ converge to $q_0$, we have $d(T(s_0), T(s_n))\to \infty$. This possibility, however, has probability zero as explained when discussing alternative , yielding the required contradiction.
Existence of infinite connected components and crossings {#sec:existence of crossings}
========================================================
In this section we prove Lemma \[lem:circle packings of unit disk\], Lemma \[lem:circle packings with slowly growing radii\] and Corollary \[cor:square tiling of square\].
Proof of Lemma \[lem:circle packings of unit disk\]
---------------------------------------------------
Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\], let ${\mathcal{S}}$ be a circle packing representing a triangulation and having carrier $\mathbb{D}$ and let $E$ be the event that $G_{\mathcal{S}}^{1-p}$ does not have an infinite connected component. As $E$ is a tail event it suffices, by Kolmogorov’s zero-one law, to show that ${\mathbb{P}}(E)<1$. Let $\mathcal{P}$ be the collection of infinite paths in $G_{\mathcal{S}}$ which contain a disk intersecting $\frac{1}{2}\mathbb{D}=\{z\in{\mathbb{R}}^2\colon \|z\|<\frac{1}{2}\}$. We will prove that the probability that there exists a path in $\mathcal{P}$ consisting of open disks in $G_{\mathcal{S}}^{1-p}$ is positive.
Let $\mathcal{C}$ be the collection of cycles in $G_{\mathcal{S}}$ which intersect all paths in $\mathcal{P}$ (cycles which surround $\frac{1}{2}\mathbb{D}$). As ${\mathcal{S}}$ represents a triangulation and has carrier $\mathbb{D}$ it follows that there exists a path in $\mathcal{P}$ whose disks are open in $G_{\mathcal{S}}^{1-p}$ if and only if there does not exist a cycle in $\mathcal{C}$ whose disks are closed in $G_{\mathcal{S}}^{1-p}$. For a disk $s\in{\mathcal{S}}$, define the event $$\label{eq:E s def}
E_s:=\left\{\textstyle
\begin{array}{c}
\text{there exists $C\in\mathcal{C}$ whose disks are closed in $G_{\mathcal{S}}^{1-p}$}\\
\text{such that $s\in C$ and $s$ has maximal radius among the disks of $C$}
\end{array}\right\}.$$ Our reasoning so far shows that $$\label{eq:E contained in union of E s}
E \subset \cup_{s\in {\mathcal{S}}} E_s.$$ Let ${\mathcal{S}}_k$ be the set of disks in ${\mathcal{S}}$ with radius in $(2^{-(k+1)}, 2^{-k}]$. As the disks of ${\mathcal{S}}$ are contained in $\mathbb{D}$ it follows that ${\mathcal{S}}= \cup_{k=0}^\infty {\mathcal{S}}_k$. Moreover, by area considerations, $$\label{eq:size of S k}
|{\mathcal{S}}_k| \le 2^{2(k+1)},\quad k\ge 0.$$ Let $F$ be the event that all disks in the collection $\cup_{k=0}^{5}{\mathcal{S}}_k$ are open in $G_{\mathcal{S}}^{1-p}$; as there is a finite number of such disks, the probability of $F$ is positive. Thus, to prove that ${\mathbb{P}}(E)<1$ it suffices to show that ${\mathbb{P}}(E\,|\,F)<1$. Taking into account we have $$\label{eq:probability of E given F}
{\mathbb{P}}(E\,|\,F) \le \sum_{s\in {\mathcal{S}}} {\mathbb{P}}(E_s\,|\,F).$$ Our definitions imply that ${\mathbb{P}}(E_s\,|\,F) = 0$ when $s\in\cup_{k=0}^{5}{\mathcal{S}}_k$ and that $E_s$ is independent of $F$ when $s\in\cup_{k=6}^{\infty}{\mathcal{S}}_k$. Hence, $$\label{eq:probability of E s given F}
\sum_{s\in {\mathcal{S}}} {\mathbb{P}}(E_s\,|\,F)\le \sum_{k=6}^{\infty}\sum_{s\in{\mathcal{S}}_k}{\mathbb{P}}(E_s).$$ Fix $k\ge 6$ and $s\in {\mathcal{S}}_k$. Observe that any cycle $C\in\mathcal{C}$ containing $s$ as a disk of maximal radius must also contain a disk $s'$ with $d(s,s')\ge \frac{1}{2}$. Note also that the probability for a disk to be closed is the parameter $p$ of Theorem \[thm:percolation for circle packing\]. Applying that theorem to the circle packing $\cup_{m=k}^{\infty}{\mathcal{S}}_m$ with $s_0 = s$ thus implies that $${\mathbb{P}}(E_s)\le \exp(-2^{k-1}).$$ Combining this estimate with , and yields $${\mathbb{P}}(E\,|\,F)\le \sum_{s\in {\mathcal{S}}} {\mathbb{P}}(E_s\,|\,F)\le \sum_{k=6}^{\infty}2^{2(k+1)}\exp(-2^{k-1})<1$$ and thus also ${\mathbb{P}}(E)<1$, as we wanted to prove.
Proof of Lemma \[lem:circle packings with slowly growing radii\] {#sec:circle packings carried by R2}
----------------------------------------------------------------
The lemma is proved in a similar manner to Lemma \[lem:circle packings of unit disk\] so we will be brief on some of the details.
Let $p>0$ be the constant from Theorem \[thm:percolation for circle packing\]. Let ${\mathcal{S}}$ be a circle packing representing a triangulation whose carrier is ${\mathbb{R}}^2$ and satisfying the assumption for some $c>0$ sufficiently small for the following arguments. Let $r_0\ge 3$ be large enough so that $f(r)\le 2c\, r / \log\log r$ for all $r\ge r_0$. Let $E$ be the event that $G_{\mathcal{S}}^{1-p}$ does not have an infinite connected component, so that our goal is to show that ${\mathbb{P}}(E)<1$.
Let $\mathcal{P}$ be the collection of infinite paths in $G_{\mathcal{S}}$ which contain a disk intersecting $r_0\mathbb{D}$. Let $\mathcal{C}$ be the collection of cycles in $G_{\mathcal{S}}$ which intersect all paths in $\mathcal{P}$ (cycles which surround $r_0\mathbb{D}$). As ${\mathcal{S}}$ represents a triangulation carried by ${\mathbb{R}}^2$ it follows that there exists a path in $\mathcal{P}$ whose disks are open in $G_{\mathcal{S}}^{1-p}$ if and only if there does not exist a cycle in $\mathcal{C}$ whose disks are closed in $G_{\mathcal{S}}^{1-p}$. For $s\in {\mathcal{S}}$ define the event $E_s$ by , so that the relation again holds.
Let ${\mathcal{S}}_0$ be the set of disks intersecting $r_0\mathbb{D}$. Let ${\mathcal{S}}_{m,k}$ be the set of disks in ${\mathcal{S}}$ whose radius is in $(2^{m-k-1}r_0/\log(m+1),\, 2^{m-k} r_0/\log(m+1)]$ and which intersect the annulus $2^m r_0\mathbb{D}\setminus 2^{m-1} r_0\mathbb{D}$. For each $m\ge 1$, the definition of $r_0$ implies that ${\mathcal{S}}_{m,k}$ is empty for $k< C$ for some large integer $C$ depending only on $c$ and tending to infinity as $c$ tends to zero. Thus ${\mathcal{S}}$ is the union of ${\mathcal{S}}_0$ and the sets ${\mathcal{S}}_{m,k}$ with $m\ge 1$ and $k\ge C$. In addition, area considerations imply that $$\label{eq:S k m bound}
|{\mathcal{S}}_{m,k}| \le 2^{2(k+2)}\log(m+1)^2.$$
Let $F$ be the event that all disks in ${\mathcal{S}}_0$ are open. As ${\mathcal{S}}_0$ is finite, the probability of $F$ is positive whence it is enough to show that ${\mathbb{P}}(E\,|\,F)<1$. We now have $$\label{eq:E F big sum}
{\mathbb{P}}(E\,|\,F) \le \sum_{m=1}^\infty\sum_{k=C}^\infty \sum_{s\in {\mathcal{S}}_{m,k}}{\mathbb{P}}(E_s \,|\, F).$$ Let $m\ge 1, k\ge C$ and $s\in{\mathcal{S}}_{m,k}$. Observe that, as $C$ is large, any cycle $C\in\mathcal{C}$ containing $s$ as a disk of maximal radius must also contain a disk $s'$ with $d(s,s')\ge 2^{m-2}r_0$. Thus, applying Theorem \[thm:percolation for circle packing\], with $s_0 = s$ and the circle packing consisting of the disks in ${\mathcal{S}}$ whose radius is at most that of $s$ and which are not in ${\mathcal{S}}_0$, implies that $$\label{eq:E s F bound}
{\mathbb{P}}(E_s\,|\,F)\le \exp(-2^{k-2}\log(m+1)).$$ Combining , and we conclude that $${\mathbb{P}}(E \,|\, F) \le \sum_{m=1}^\infty\sum_{k=C}^\infty 2^{2(k+2)}\log(m+1)^2 \exp(-2^{k-2} \log(m+1))<1$$ when $C$ is a sufficiently large absolute constant, as we wanted to prove.
Proof of Corollary \[cor:square tiling of square\]
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The proof is again a variation on the proof of Lemma \[lem:circle packings of unit disk\].
Let $p>0$ be the constant from Theorem \[thm:percolation for general packings\] for a packing of squares in ${\mathbb{R}}^2$. Let ${\mathcal{S}}$ be a packing of finitely many squares in the unit square. Let $E$ be the event of a top-bottom crossing of the unit square in $G_{\mathcal{S}}^p$. We need to bound the probability of $E$ away from one uniformly in ${\mathcal{S}}$.
Let $\mathcal{C}$ be the collection of paths in $G_{\mathcal{S}}$ which connect the top of the unit square to its bottom. For a square $s\in{\mathcal{S}}$, define the event $$E_s:=\left\{\textstyle
\begin{array}{c}
\text{there exists $C\in\mathcal{C}$ whose squares are open in $G_{\mathcal{S}}^p$}\\
\text{such that $s\in C$ and $s$ has maximal diameter among the squares of $C$}
\end{array}\right\}.$$ Let ${\mathcal{S}}_k$ be the set of squares in ${\mathcal{S}}$ with diameter in $(2^{-(k+1)}, 2^{-k}]$. By area considerations, $$\label{eq:bound on number of squares}
|{\mathcal{S}}_k| \le 2^{2(k+1)}.$$ Let $F$ be the event that all squares in $\cup_{k=0}^5{\mathcal{S}}_k$ are closed. As the number of squares in this collection is at most an absolute constant by , the probability of $F$ is at least an absolute constant. Thus it suffices to bound ${\mathbb{P}}(E\,|\,F)$ from one, uniformly in ${\mathcal{S}}$. We use that $$\label{eq:P E F squares}
{\mathbb{P}}(E\,|\,F)\le \sum_{k=0}^\infty \sum_{s\in{\mathcal{S}}_k} {\mathbb{P}}(E_s\,|\, F) = \sum_{k=6}^\infty \sum_{s\in{\mathcal{S}}_k} {\mathbb{P}}(E_s).$$ Lastly, if $s\in{\mathcal{S}}_k$ for some $k\ge 6$ then any cycle $C\in\mathcal{C}$ containing $s$ as a square of maximal diameter must contain another square $s'\in{\mathcal{S}}$ with $d(s,s')\ge \frac{1}{4}$. Thus, combining Theorem \[thm:percolation for general packings\] (for each $s\in{\mathcal{S}}_k$ it is applied with $s = s_0$ and the square packing $\cup_{m=k}^\infty{\mathcal{S}}_m$) and the bounds and leads to $${\mathbb{P}}(E\,|\,F)\le \sum_{k=6}^\infty 2^{2(k+1)}\exp(-2^{k-2})$$ which is bounded away from one uniformly in ${\mathcal{S}}$, as we wanted to prove.
Circle packings of Benjamini–Schramm limits of finite planar graphs
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In this section we prove Lemma \[lem:circle packing with at most one accumulation point\]. The proof is encumbered by technical details so for the reader’s convenience we briefly describe its steps here: (i) Embed each $G_n$ inside a finite *triangulation* $\bar{G}_n$ in a way that the degrees in $\bar{G}_n$ of the vertices of $G_n$ are controlled. (ii) Use the circle packing theorem to obtain a circle packing of each $\bar{G}_n$. Translate and dilate the obtained circle packing so that a uniformly chosen root disk becomes the unit disk. (iii) For each $r$, use the assumption of Benjamini–Schramm convergence and the ring lemma (as $\bar{G}_n$ is a triangulation) to obtain tightness for the centers and radii of the disks of the circle packing which correspond to vertices of $G_n$ and are at graph distance at most $r$ from the root disk. (iv) Use the obtained tightness to extract, along a subsequence, a limiting circle packing of the disks corresponding to vertices of $G_n$. (v) The magic lemma of Benjamini–Schramm implies that the limiting circle packing has at most one accumulation point. (vi) Argue that the limiting circle packing represents the graph $G$ (for this last step we place additional restrictions on the triangulations $(\bar{G}_n)$).
Ingredients
-----------
We start by describing several tools which are used in the proof.
### Convergence of circle packings {#sec:convergence of circle packings}
Call a circle packing ${\mathcal{S}}$ *connected* (*locally finite*) if its underlying graph $G_{\mathcal{S}}$ is connected (locally finite). A rooted circle packing is a pair $({\mathcal{S}}, s)$ with ${\mathcal{S}}$ a circle packing and $s$ one of the disks in ${\mathcal{S}}$. For a locally finite rooted circle packing $({\mathcal{S}},s)$ and integer $r\ge 0$, write $B_{\mathcal{S}}(s,r)$ for the induced subgraph of $G_{\mathcal{S}}$ on the disks whose graph distance to $s$ is at most $r$, rooted at $s$. We proceed to define a notion of convergence for a sequence of rooted circle packings.
Let $({\mathcal{S}}_n, s_n)$ be a sequence of finite, connected, rooted circle packings. Say that $({\mathcal{S}}_n, s_n)$ *converge locally* to the triple $((G,\rho), {\mathcal{S}}, \tau)$, where $(G,\rho)$ is a locally finite connected rooted graph, ${\mathcal{S}}$ is a circle packing and $\tau$ is a bijection of the vertices of $G$ and the disks of ${\mathcal{S}}$, if
1. (local graph convergence) There is a non-decreasing sequence of integers $(r_n)$ tending to infinity such that for each $n$ there exists an isomorphism $I_n$ between the rooted graphs $B_{{\mathcal{S}}_n}(s_n, r_n)$ and $B_G(\rho, r_n)$.
2. (convergence of disks) For each vertex $v$ in $G$, the disk $I_n^{-1}(v)$ (well defined for large $n$) converges as $n$ tends to infinity to a non-trivial disk (i.e., its center converges to a point in ${\mathbb{R}}^2$ and its radius converges to a number in $(0,\infty)$). The mapping $\tau$ takes $v$ to the limiting disk and ${\mathcal{S}}$ is the set of all $\tau(v)$, $v\in G$ (it is straightforward that ${\mathcal{S}}$ is a circle packing).
It is pointed out that this convergence does not force the graph $G_{\mathcal{S}}$ of the circle packing ${\mathcal{S}}$ to coincide with $G$. Indeed, adjacency of $v$ to $w$ in $G$ implies that $\tau(v)$ is tangent to $\tau(w)$, but the converse implication may fail in general. Moreover, the graph $G_{\mathcal{S}}$ may even fail to be locally finite (e.g., it is possible that $G_{{\mathcal{S}}_n}$ is a path of length $n$ and $G_{\mathcal{S}}$ is a star graph with infinitely many ‘arms’ of length $1$). In general, we thus have only that $(G_{\mathcal{S}},\tau(\rho))$ contains $(G,\rho)$ as a rooted subgraph.
We now extend the above convergence notion to *random* rooted circle packings. A sequence of random finite, connected, rooted circle packings $({\mathcal{S}}_n, s_n)$ is said to *converge locally in distribution* to the random triple $((G,\rho), {\mathcal{S}}, \tau)$ with $(G,\rho), {\mathcal{S}}, \tau$ random objects of the above types if there exists a coupling of the $({\mathcal{S}}_n, s_n)$ and $((G,\rho), {\mathcal{S}}, \tau))$ so that, almost surely under this coupling, $({\mathcal{S}}_n, s_n)$ converges locally to $((G,\rho), {\mathcal{S}}, \tau)$.
The following *tightness condition* will be of use. Let $({\mathcal{S}}_n, s_n)$ be a sequence of random, finite, connected rooted circle packings. Let $(G,\rho)$ be a random locally finite connected rooted graph $(G,\rho)$. Suppose that (i) for each integer $r\ge 0$ and each rooted graph $(H,\sigma)$ it holds that ${\mathbb{P}}(B_{{\mathcal{S}}_n}(s_n, r)=(H,\sigma))\to {\mathbb{P}}(B_G(\rho, r)=(H,\sigma))$, (ii) for each $r\ge 0$, the centers of the disks in all $B_{{\mathcal{S}}_n}(s_n, r)$, as $n$ varies, are tight, when considered as random variables in ${\mathbb{R}}^2$, and (iii) for each $r\ge 0$, the radii of the disks in all $B_{{\mathcal{S}}_n}(s_n, r)$, as $n$ varies, are tight, when considered as random variables in $(0,\infty)$. Then, via compactness arguments (and the Skorohod representation theorem), there exists a subsequence $(n_k)$, random circle packing ${\mathcal{S}}$ and random bijection $\tau$ so that $({\mathcal{S}}_{n_k}, s_{n_k})$ converges locally in distribution to $((G,\rho), {\mathcal{S}}, \tau)$.
### Applying the magic lemma
Benjamini and Schramm proved that every Benjamini–Schramm limit of finite simple planar graphs with uniformly bounded degrees may almost surely be realized as the tangency graph of a circle packing with at most one accumulation point. The main tool in their proof is the so-called *magic lemma* [@BS01 Lemma 2.3]. We now apply the magic lemma to obtain a similar conclusion in our context.
Denote the closed unit disk by $\bar{\mathbb{D}}=\{z\in{\mathbb{R}}^2\colon \|z\|\le 1\}$.
(Benjamini–Schramm limit of finite circle packings)\[lem:magic lemma consequence\] Let $({\mathcal{S}}_n)$ be a sequence of random finite and connected circle packings. For each $n$, let $s_n$ be a uniformly sampled disk in ${\mathcal{S}}_n$ (again, first ${\mathcal{S}}_n$ is sampled and then $s_n$ is sampled uniformly from it). Let $T_n$ be the (unique) mapping $z\mapsto az+b$ with $a>0$ and $b\in{\mathbb{R}}^2$ for which $T_n(s_n) = \bar{\mathbb{D}}$. If $(T_n({\mathcal{S}}_n), \bar{\mathbb{D}})$ converges locally in distribution to some $((G,\rho), {\mathcal{S}}, \tau)$ then the circle packing ${\mathcal{S}}$ has at most one accumulation point in ${\mathbb{R}}^2$, almost surely.
The proof follows that of [@BS01 Proposition 2.2].
### The circle packing theorem
To make use of Lemma \[lem:magic lemma consequence\] we need a way to generate appropriate circle packings. This is provided by the following celebrated theorem of Koebe.
(The circle packing theorem [@K36], [@N18 Theorem 3.5])\[thm:circle packing\] For any finite planar map $G$ there exists a circle packing ${\mathcal{S}}$ which represents $G$.
### The ring lemma
The following lemma of Rodin and Sullivan will be used to check the tightness condition discussed in Section \[sec:convergence of circle packings\].
In a circle packing, we say that the disks $s_1,\ldots, s_M$ *completely surround* the disk $s_0$ if each $s_i$ is tangent to $s_0$ and $s_i$ is tangent to $s_{i+1}$ for $1\le i\le M$, where we set $s_{M+1}:=s_1$.
(The ring lemma [@RS87], [@N18 Lemma 4.2])\[lem:ring lemma\] For each integer $M>0$ there exists $c(M)>0$ for which the following holds. Suppose that $s_0, \ldots, s_M$ are disks in a circle packing and $s_1,\ldots, s_M$ completely surround $s_0$. Let $r_i$ be the radius of $s_i$. Then $r_i / r_0 \ge c(M)$ for all $i$.
### Extension to a triangulation
To verify the assumption of the ring lemma we will need to work with triangulations. The following lemma provides a way to extend a finite planar graph to a triangulation with control on the degrees of the vertices of the original graph as well as their new neighbors.
\[lem:extension to triangulation\] Let $H$ be a finite simple connected planar graph. There exists a finite simple *triangulation* $\bar{H}$ such that $H$ is contained in the graph of $\bar{H}$, $\deg_{\bar{H}}(v)=3\deg_H(v)$ for all vertices $v$ of $H$ (where $\deg_G(v)$ is the degree of $v$ in the graph $G$) and $\deg_{\bar{H}}(v)=5$ for all vertices in $\bar{H}\setminus H$ which are neighbors of the vertices of $H$.
Draw $H$ in the plane with straight lines for edges (e.g., with a circle packing) and consider $H$ with the resulting map structure. Each face $f$ of $H$ may be represented by a directed path $\vec{e}_1^f, \ldots, \vec{e}_{k(f)}^f$, with $k(f)\ge 3$ as $H$ is simple. Write $\vec{e}_j^f = (v_j^f, v_{j+1}^f)$ so that $v_1^f, \ldots, v_{k(f)}^f, v_{k(f)+1}^f=v_1^f$ is a cycle in which the vertices need not be distinct. For each face $f$: First, draw a new cycle $w_1^f, \ldots, w_{k(f)}^f, w_{k(f)+1}^f = w_1^f$ having distinct vertices in the region surrounded by $(v_j^f)$ in the drawing (the bounded region, unless $f$ is the ‘outer face’), and draw edges from $w_j^f$ to both $v_j^f$ and $v_{j+1}^f$ for $1\le j\le k$. Second, add a new vertex $u^f$ in the region surrounded by the cycle $(w_j^f)$ in the drawing and draw edges from $u^f$ to each of the $(w_j^f)$. The resulting proper drawing describes a simple triangulation which contains $H$ as a subgraph, and it is straightforward to check that the degree of each vertex of $H$ is exactly tripled in this construction. In addition, among the added vertices, the only ones neighboring those of $H$ are the $(w_j^f)$ and each of these has degree $5$ by construction.
Proof of Lemma \[lem:circle packing with at most one accumulation point\]
-------------------------------------------------------------------------
Let $(G_n)$ be a sequence of, possibly random, finite simple planar graphs with Benjamini–Schramm limit $(G,\rho)$. Let $\rho_n$ be uniformly sampled in $G_n$.
### First observations
We begin with some observations. First we may, and will, assume without loss of generality that each $G_n$ is connected. Indeed, otherwise we may replace $G_n$ with $\hat{G}_n$ where the distribution of $\hat{G}_n$ is obtained by first sampling $G_n$ and then sampling any one of the connected components of $G_n$ with probability proportional to the number of vertices in the component. It is simple to check that for any $r\ge 0$, the distribution of $B_{G_n}(\rho_n, r)$ is equal to that of $B_{\hat{G}_n}(\hat{\rho}_n, r)$, where $\hat{\rho}_n$ is uniformly sampled in $\hat{G}_n$. Thus $(G,\rho)$ is also the Benjamini–Schramm limit of $(\hat{G}_n)$.
Second, write $|H|$ for the number of vertices of a graph $H$. We assume without loss of generality that $|G|=\infty$, almost surely. Let us show that this is indeed without loss of generality. The assumption that $(G_n)$ converges to $(G,\rho)$ in the Benjamini–Schramm sense implies that there exists a coupling of $((G_n, \rho_n))_n$ and $(G_,\rho)$ so that for each integer $r\ge 0$, $B_{G_n}(\rho_n, r)$ converges to $B_G(\rho, r)$ almost surely (by the Skorohod representation theorem). Under the coupling, the event $\{|G|=\infty\}$ equals the event $\{\lim_n |G_n|=\infty\}$ (as $(G_n)$ are connected). On the event $\{|G|<\infty\}$ it is clear that $G$ can be represented by a circle packing with no accumulation points, by Theorem \[thm:circle packing\]. Assume that the probability of $\{|G|=\infty\}$ is positive, as otherwise the lemma follows trivially. Condition (under the coupling) on $\{|G|=\infty\}$ and note that it still holds under the conditioning that $\rho_n$ is uniformly distributed in $G_n$, as $\{|G|=\infty\}$ is independent of the choice of the $(\rho_n)$ in $(G_n)$ (it depends only on $|G_n|$, as explained above). We may now replace the distribution of each $G_n$ and the distribution of $(G,\rho)$ by their distribution conditioned on $\{|G|=\infty\}$ and preserve the property that $(G_n)$ converges in the Benjamini–Schramm sense to $(G,\rho)$, while adding the property that $|G|=\infty$ almost surely.
Third, the assumption that $|G|=\infty$ almost surely implies that $|G_n|$ converges to infinity in probability, i.e., that $$\label{eq:size of G_n tends to infinity}
\text{For each $M>0$,\quad ${\mathbb{P}}(|G_n|\le M)\to 0$\quad as $n\to\infty$}.$$ In addition, the Benjamini–Schramm convergence of $(G_n)$ to $(G,\rho)$ implies that for each integer $r\ge 0$, the number of vertices of $G_n$ at distance at most $r$ from $\rho_n$ is tight as $n$ tends to infinity. In other words, $$\label{eq:tightness of size of r neighborhoods}
\text{For each integer $r\ge 0$,\quad $\lim_{M\to\infty}\sup_{n\to\infty}{\mathbb{P}}(|B_{G_n}(\rho_n, r)|\ge M) = 0$}.$$ Combining and we conclude that for each random sequence of vertices $v_n\in G_n$ with $v_n$ independent of $\rho_n$, the distance between $v_n$ and $\rho_n$ tends to infinity in probability, i.e., $$\label{eq:v n out of neighborhood}
\text{For each integer $r\ge 0$,\quad ${\mathbb{P}}(v_n\in B_{G_n}(\rho_n, r))\to 0$ as $n\to\infty$}.$$ To see this, observe that the uniformity of $\rho_n$ (and its independence from $v_n$) implies that, for each integer $r\ge 0, M\ge 1$, $$\begin{split}
{\mathbb{P}}(v_n\in B_{G_n}(\rho_n, r)\, |\, G_n, v_n) &= {\mathbb{P}}(\{v_n\in B_{G_n}(\rho_n, r)\}\cap\{|B_{G_n}(v_n, r)|\ge M\}\, |\, G_n, v_n)\\
&+ {\mathbb{P}}(\{v_n\in B_{G_n}(\rho_n, r)\}\cap\{|B_{G_n}(v_n, r)|< M\}\, |\, G_n, v_n)\\
&\le {\mathbb{P}}(|B_{G_n}(\rho_n, 2r)|\ge M\, |\, G_n, v_n) + \min\left\{\frac{M}{|G_n|}, 1\right\}.
\end{split}$$ Averaging over $G_n$ and $v_n$ we see that both terms can be made as small as we like by choosing first $M$ large and then $n$ large, by and .
### Circle packings {#circle-packings}
We proceed to make use of the circle packing theorem. For each $n$, let $\bar{G}_n$ be the extension of $G_n$ to a (random) finite simple triangulation given by Lemma \[lem:extension to triangulation\]. Apply the circle packing theorem, Theorem \[thm:circle packing\], to $\bar{G}_n$ to obtain a (random) circle packing $\bar{S}_n$. Let ${\mathcal{S}}_n$ be the subset of $\bar{S}_n$ of the disks corresponding to $G_n$ (chosen arbitrarily if there is more than one correspondence). Let $s_n$ be uniformly sampled from the disks of ${\mathcal{S}}_n$. Let $T_n$ be the unique mapping $z\mapsto az+b$, with $a>0$ and $b\in{\mathbb{R}}^2$, for which $T_n(s_n) = \bar{\mathbb{D}}$. We will establish that
1. There is a subsequence $(n_k)$ for which $(T_{n_k}({\mathcal{S}}_{n_k}), \bar{\mathbb{D}})$ converges locally in distribution to $((G,\rho), {\mathcal{S}}, \tau)$ with $(G,\rho)$ the Benjamini–Schramm limit of $(G_n)$, ${\mathcal{S}}$ a (random) circle packing and $\tau$ a bijection of the vertices of $G$ and the disks of ${\mathcal{S}}$.
2. Almost surely, $G = G_{\mathcal{S}}$.
The two statements suffice to finish the proof as Lemma \[lem:magic lemma consequence\] implies that ${\mathcal{S}}$ has at most one accumulation point in ${\mathbb{R}}^2$.
### Tightness
We first prove the existence of $(n_k)$ for which $(T_{n_k}({\mathcal{S}}_{n_k}), \bar{\mathbb{D}})$ converges locally in distribution. We employ the tightness condition discussed in Section \[sec:convergence of circle packings\]. For brevity, write $B_{n,r}:=B_{T_n({\mathcal{S}}_n)}(\bar{\mathbb{D}}, r)$.
The fact that for each integer $r\ge 0$ and each rooted graph $(H,\sigma)$ the convergence ${\mathbb{P}}(B_{n,r}=(H,\sigma))\to {\mathbb{P}}(B_G(\rho, r)=(H,\sigma))$ holds is equivalent to our assumption that $(G,\rho)$ is the Benjamini–Schramm limit of $(G_n)$.
To see that for each $r$ the centers and radii of the disks in $(B_{n,r})$, as $n$ varies, are tight we make use of the ring lemma, Lemma \[lem:ring lemma\], applied to the larger circle packing $\bar{{\mathcal{S}}}_n$. Fix an integer $r\ge 0$. As $\bar{G}_n$ is a triangulation, every disk in $T_n(\bar{{\mathcal{S}}}_n)$, besides three disks $b_n^1, b_n^2, b_n^3$ (which border the ‘outer face’), is completely surrounded by other disks in $T_n(\bar{{\mathcal{S}}}_n)$. The three disks, or a subset of them, may belong to the smaller circle packing ${\mathcal{S}}_n$ but in any case they are unlikely to be in the neighborhood $B_{n,r}$. Precisely, by , $$\label{eq:boundary is far}
{\mathbb{P}}(\{b_n^1, b_n^2, b_n^3\}\cap B_{n,r}\neq \emptyset) \to 0\quad\text{as $n\to\infty$}.$$ On the event in , the ring lemma may be used for each of the disks in $B_{n,r}$, when considered inside the larger circle packing $T_n(\bar{{\mathcal{S}}}_n)$, by considering a shortest path in $B_{n,r}$ from the disk to $\bar{\mathbb{D}}$. Write $\bar{\Delta}_{n,r}$ for the maximal degree of the disks of $B_{n,r}$ when considered in the graph $G_{T_n(\bar{{\mathcal{S}}}_n)}$ and $\Delta_{n,r}$ for the maximal degree of the same disks when considered in the graph $G_{T_n({\mathcal{S}}_n)}$. Lemma \[lem:extension to triangulation\] shows that $\bar{\Delta}_{n,r} = 3\Delta_{n,r}$. Thus the ring lemma, together with , implies that the centers and radii of the disks in $B_{n,r}$ are tight if $\Delta_{n,r}$ is tight. This latter statement now follows from .
### Graph of the limiting circle packing
It remains to prove that $G = G_{\mathcal{S}}$ almost surely. The discussion in Section \[sec:convergence of circle packings\] already shows that $G$ is a subgraph of $G_{\mathcal{S}}$, so we need only show that if $v,w\in G$ are non-neighbors, then their disks $\tau(v), \tau(w)\in{\mathcal{S}}$ are non-tangent. This in turn is implied by the following claim: For each integer $r\ge 0$, the minimal distance between two non-tangent disks in $B_{n,r}$ is a tight random variable in $(0,\infty)$, as $n$ tends to infinity. To see the claim, introduce the intermediate circle packing $\hat{{\mathcal{S}}}_n$, consisting of the disks of ${\mathcal{S}}_n$ and the disks of $\bar{{\mathcal{S}}}_n$ which are tangent to them. As explained in the previous section, the disks in $B_{n,r}$ are completely surrounded by disks in $T_n(\hat{{\mathcal{S}}}_n)$, with probability tending to one as $n$ tends to infinity. These surrounding disks can act as a ‘barrier’, preventing two non-tangent disks of $B_{n,r}$ from coming too near to each other. With this idea, the above claim will follow once we show that the minimal radius of a disk in $B_{T_n(\hat{{\mathcal{S}}}_n)}(\bar{\mathbb{D}}, r+1)$ is tight in $(0,\infty)$ as $n$ tends to infinity. The proof is similar to that of the previous section, making use of two facts: (i) Each disk in $\hat{{\mathcal{S}}}_n\setminus {\mathcal{S}}_n$ is tangent to exactly 5 other disks in $\bar{{\mathcal{S}}}_n$ by Lemma \[lem:extension to triangulation\], and (ii) if a boundary disk $b_n^1, b_n^2$ or $b_n^3$ belongs to $T_n(\hat{{\mathcal{S}}}_n\setminus {\mathcal{S}}_n)$ and is also in $B_{T_n(\hat{{\mathcal{S}}}_n)}(\bar{\mathbb{D}}, r+1)$ then one of its 5 neighbors belongs to $B_{n,r+2}$, which is unlikely by .
Discussion, open questions and conjectures {#sec:discussion and open questions}
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Percolation on circle packings
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Theorem \[thm:percolation for circle packing\] shows that there exists $p>0$ such that for any circle packing, there is zero probability for the origin to be connected to infinity by open disks after site percolation with parameter $p$. What is the largest value of $p$ for which this statement holds? The example of the triangular lattice shows that $p$ cannot exceed $1/2$. The following conjecture states, with the notation of Section \[sec:circle packings\], that this bound is tight.
\[conj:p=1/2\] Let ${\mathcal{S}}$ be a circle packing and $s_0\in {\mathcal{S}}$. Then $${\mathbb{P}}(E_{{\mathcal{S}}, 1/2}(s_0, \infty))=0.$$
The conjecture is similar in spirit to Conjecture \[conj:square tiling\] of Benjamini. We emphasize that Conjecture \[conj:p=1/2\] does not make any assumptions on the circle packing but point out that it is open and interesting also when the circle packing represents a triangulation, has carrier ${\mathbb{R}}^2$ and uses disks whose radii are uniformly bounded from zero and infinity.
If the conjecture is verified then it would follow that site percolation with $p=1/2$ on the following classes of graphs has no infinite connected component almost surely (using the same proofs as the corresponding statements here): (i) Graphs represented by locally finite circle packings with at most countably many accumulation points (as in Corollary \[cor:circle packing accumulation points\]). (ii) Recurrent simple plane triangulations (as in part of Corollary \[cor:percolation on plane triangulations\]), giving a positive answer to Question \[ques:recurrent triangulations\] of Benjamini. (iii) Benjamini–Schramm limits of, possibly random, finite simple planar graphs (as in Corollary \[cor:percolation on Benjamini Schramm limits\]).
Exponential decay
-----------------
The second part of Theorem \[thm:percolation for circle packing\] states that if the radii of the disks in the circle packing are uniformly bounded above then the probability in the site percolation that the origin is connected to distance $r$ decays exponentially in $r$, at a rate which is uniform in the circle packing. Such a statement cannot hold at $p=1/2$, again due to the example of the triangular lattice for which this value of $p$ is critical. On transitive graphs, exponential decay of connection probabilities has been shown to hold in the sub-critical regime of percolation [@M86; @AB87; @DCT16; @DCRT19]. If Conjecture \[conj:p=1/2\] is verified, then it is natural to conjecture also that exponential decay holds for all $p<1/2$, uniformly in circle packings whose disks have bounded above radii.
\[conj:uniform exponential decay\] There exists $f:(0,1/2)\to(0,\infty)$ such that the following holds. Let $0<p<1/2$. Let ${\mathcal{S}}$ be a circle packing and $s_0\in {\mathcal{S}}$. Assume that $D:=\sup_{s\in{\mathcal{S}}}\operatorname{diam}(s)<\infty$. Then for each $r>0$, $$\label{eq:conj bounded above circle packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le \exp\left(-f(p)\frac{r}{D}\right).$$
The fact that $D$ enters via the ratio $\frac{r}{D}$ follows by a simple scaling consideration. Theorem \[thm:percolation for circle packing\] verifies Conjecture \[conj:uniform exponential decay\] for sufficiently small $p$. Again, the conjecture is open and interesting also when the circle packing represents a triangulation, has carrier ${\mathbb{R}}^2$ and uses disks whose radii are uniformly bounded from zero.
If Conjecture \[conj:uniform exponential decay\] is verified then it would follow that site percolation with any $p>1/2$ on transient simple bounded degree plane triangulations has an infinite connected component almost surely (as in part of Corollary \[cor:percolation on plane triangulations\]). Proving the existence of an infinite connected component at $p=1/2$ and thus verifying Conjecture \[conj:transient triangulations\] of Benjamini requires additional tools. One may similarly deduce an analogue of Conjecture \[conj:square tiling\] of Benjamini in which the squares are replaced by circles and the probability to color a circle black is strictly greater than $1/2$.
Value of $p_c$ and scaling limit {#sec:value of p_c}
--------------------------------
For well-behaved circle packings, it may be that the critical probability for site percolation is exactly $1/2$. We expect this for circle packings representing a triangulation, having carrier ${\mathbb{R}}^2$ and using disks whose radii are uniformly bounded from infinity. Indeed, this will follow, together with the fact that there is no infinite connected component at $p=1/2$, from Conjecture \[conj:p=1/2\] and Conjecture \[conj:uniform exponential decay\] (as in Lemma \[lem:circle packings with slowly growing radii\]). For such circle packings, it may even be that the scaling limit of the $p=1/2$ percolation is the Conformal Loop Ensemble [@S09; @SW12] with parameter $\kappa=6$ as for the triangular lattice [@S01; @CN04; @CN05; @CN06]. Beffara [@B08] discusses the question of finding embeddings of triangulations on which percolation has a conformal invariant scaling limit and makes a related conjecture [@B08 Conjecture 12].
![Part of the circle packing of the graph obtained by triangulating the product of a triangle with the natural numbers.[]{data-label="fig:circle packing triangle natural numbers"}](CP3.png)
We point out that the restriction that the radii are uniformly bounded above cannot be waived without a replacement. Indeed, consider the product of a triangle with the natural numbers, endowed with a natural map structure and triangulated with a diagonal edge added in every face with 4 edges. The circle packing of one such triangulation is depicted in Figure \[fig:circle packing triangle natural numbers\]. It is straightforward that is carried by ${\mathbb{R}}^2$ but uses disks whose radii are unbounded above, growing linearly with their distance to the origin. It is simple to see that the critical probability for site percolation on this graph is one.
General shapes and the dependence on the aspect ratio
-----------------------------------------------------
Theorem \[thm:percolation for general packings\] extends Theorem \[thm:percolation for circle packing\] from circle packings to ${\varepsilon}$-regular packings, showing that percolation on the latter class has no infinite connected component at fixed $p$ depending only on ${\varepsilon}$ and the dimension of the space. Possibly, Conjecture \[conj:p=1/2\] and Conjecture \[conj:uniform exponential decay\], apply also for ${\varepsilon}$-regular packings in ${\mathbb{R}}^2$, as long as they represent a planar graph (planarity may fail, for instance, for packings of squares in which four squares are allowed to share a corner). In Conjecture \[conj:uniform exponential decay\] the rate function $f(p)$ should then be allowed to depend on ${\varepsilon}$. We state this explicitly for ellipse packings. Define the *aspect ratio* of an elliptical disk as the ratio of lengths of the major axis and the minor axis of its bounding ellipse.
\[conj:ellipses\] There exists $f:(0,1/2)\times[1,\infty)\to(0,\infty)$ such that the following holds. Let $M\ge 1$. Let ${\mathcal{S}}$ be a packing of closed elliptical disks of aspect ratio at most $M$. Let $s_0\in {\mathcal{S}}$. Then $${\mathbb{P}}(E_{{\mathcal{S}}, 1/2}(s_0, \infty))=0.$$ If, in addition, $D:=\sup_{s\in{\mathcal{S}}}\operatorname{diam}(s)<\infty$, then for each $0<p<1/2$ and each $r>0$, $$\label{eq:conj bounded above ellipse packing}
{\mathbb{P}}(E_{{\mathcal{S}}, p}(s_0, r))\le \exp\left(-f(p,M)\frac{r}{D}\right).$$
The dependence of $f$ on the aspect ratio is unclear, even for small $p$. Consider, for instance, a packing of elliptical disks of fixed diameter $D$ and fixed aspect ratio $M$ formed as follows: Let one ellipse $s_0$ have its major axis on the $y$-axis and at approximately $M$ locations on the top part of $s_0$ start lines of infinitely many ellipses with their major axis parallel to the $x$-axis (see Figure \[fig:ellipse packing\]). In this case, it is straightforward that after site percolation with any $p>0$, there is chance approximately $e^{-1} p$ (for $M$ large) that $s_0$ is open and in one of the lines of ellipses above $s_0$, the first $\lfloor\log_{1/p}(M)\rfloor$ ellipses are open. On this event, $E_{{\mathcal{S}},p}(s_0, r)$ occurs for $r\approx\lfloor\log_{1/p}(M)\rfloor D$. Thus $f(p,M)\le \frac{\tilde{f}(p)}{\log M}$ for some $\tilde{f}(p)$ and large $M$. Matan Harel [@H19] has found a construction, for rectangle packings of aspect ratio at most $M$, showing that the analogous $f(p,M)$ must satisfy $f(p,M)\le \frac{\tilde{f}(p)}{M^{{\varepsilon}(p)}}$ for some functions $\tilde{f},{\varepsilon}$ of $p$ and large $M$. It is interesting to determine whether indeed the $f(p,M)$ depends on $M$ only through an inverse power as in Harel’s construction. If proved for more general packings, such a result would be useful in the study of the loop $O(n)$ model [@CGHP20], leading to quantitative estimates in *finite* volume.
When the ellipse packing represents a recurrent plane triangulation, one may use the He–Schramm theorem, Theorem \[thm:He–Schramm\], to pass to a circle packing representing the same graph and then apply Theorem \[thm:percolation for circle packing\] to deduce the absence of infinite connected components after site percolation with the parameter $p$ of that theorem. The obtained value of $p$ is better than what one would obtain by applying Theorem \[thm:percolation for general packings\] directly to the ellipse packing, as the latter value depends on the aspect ratio of the elliptical disks. However, even when this route is available, one still requires additional arguments in order to derive an exponential decay estimate such as (even at the value of $p$ given by Theorem \[thm:percolation for circle packing\]), since the mapping of the ellipse packing to the circle packing introduces non-trivial distortions to the underlying metric.
Recurrent planar graphs
-----------------------
In part of corollary \[cor:percolation on plane triangulations\] we prove the absence of an infinite connected component for site percolation on simple *recurrent* plane triangulations, at the value of $p$ given by Theorem \[thm:percolation for circle packing\]. In this result, unlike its counterpart for transient plane triangulations in part of the corollary, the assumption that the map is a triangulation plays a technical role, allowing us to rely on the results of He–Schramm, Theorem \[thm:He–Schramm\]. The role of the assumption that the graph is one-ended is also unclear. Is it in fact the case that there is no infinite connected component for site percolation, at some fixed $p>0$, or even at $p=1/2$ along the lines of Conjecture \[conj:p=1/2\] and Benjamini’s Question \[ques:recurrent triangulations\], for *all planar recurrent graphs*? This follows, with the $p$ of Theorem \[thm:percolation for circle packing\], for subgraphs of simple recurrent *plane triangulations*, by part of Corollary \[cor:percolation on plane triangulations\]. For multiply-ended planar recurrent graphs the following result of Gurel-Gurevich, Nachmias and Souto [@GNS17] is possibly of relevance: Let ${\mathcal{S}}$ be a circle packing of a bounded degree triangulation $G$. Then $G$ is recurrent if and only if the set of accumulation points of ${\mathcal{S}}$ is polar (where a set is polar if it is avoided by two-dimensional Brownian motion, almost surely).
Other connectivity notions {#sec:other connectivity notions}
--------------------------
For a circle packing with accumulation points, one may define notions of connectivity extending the one used here. Specifically, given a circle packing ${\mathcal{S}}$, parameter $0<p<1$ and points $x,y$ in ${\mathbb{R}}^2$ one may say that $x$ is connected to $y$ in $G^p_{\mathcal{S}}$ if $x$ and $y$ belong to the same connected component of the *closure* (in ${\mathbb{R}}^2$) of the union of open disks in $G^p$. A more restricted possibility is to say that $x$ and $y$ are connected if there is a *path* (in ${\mathbb{R}}^2$) connecting them in the closure of the union of open disks in $G^p$. Such definitions allow connections between $x$ and $y$ to ‘pass through’ accumulation points of ${\mathcal{S}}$. Similar definitions may be used to define a connection between $x$ and infinity. The proof of Theorem \[thm:percolation for circle packing\] does not apply to these extended notions of connectivity.
Bond percolation
----------------
The analogue of Theorem \[thm:percolation for circle packing\] fails for *bond* percolation. Precisely, given a circle packing ${\mathcal{S}}$ and $p\in[0,1]$, the $p$-bond-percolation process is the graph $G_{\mathcal{S}}^{p,\text{bond}}$ obtained from $G_{\mathcal{S}}$ by independently retaining each *edge* of the graph with probability $p$ and discarding it with probability $1-p$. For each $p>0$ there exist circle packings ${\mathcal{S}}$ without accumulation points such that $G_{\mathcal{S}}^{p,\text{bond}}$ contains an infinite connected component almost surely. Indeed, one may start with a convenient circle packing ${\mathcal{S}}_0$, such as the periodic circle packing of the square lattice ${\mathbb{Z}}^2$, and for each integer $M>0$ create a new circle packing ${\mathcal{S}}_M$ by adding for each pair of tangent circles $s_0, s_1$ in ${\mathcal{S}}_0$ additional $M$ circles tangent to both $s_0$ and $s_1$. For each $p>0$ one may take $M$ large enough so that the graph $G_{{\mathcal{S}}_M}^{p,\text{bond}}$ will contain an infinite connected component almost surely.
Related conjectures and results
-------------------------------
We mention several additional conjectures and results relating to site percolation on planar graphs and the special value $p=1/2$.
### Isoperimetric assumptions {#sec:isoperimetric}
Benjamini and Schramm [@BS96 Conjecture 3] conjectured that $p_c(G)\le 1/2$ for (bounded degree) plane triangulations $G$ for which $|\partial A|\ge f(|A|)\log|A|$ for some $f$ growing to infinity and all finite sets of vertices $A$ (see also [@B2018 Section 2.1]). Moreover, they conjecture that $p_c(G)<1/2$ if $G$ has positive Cheeger constant.
Georgakopoulos and Panagiotis [@GP2018 Section 11] make progress on this and other conjectures by proving that the *bond* percolation critical probability is at most $1/2$ for the following classes of graphs: (i) locally finite plane triangulations satisfying the above isoperimetric assumption, (ii) bounded degree transient plane triangulations (making progress towards Benjamini’s Conjecture \[conj:transient triangulations\]), (iii) bounded degree recurrent plane triangulations which can be represented by a circle packing whose disks have radii which are uniformly bounded above. They further improve the upper bound to $1/2-{\varepsilon}(\Delta(G))$ where $\Delta(G)$ is the maximal degree in $G$ and ${\varepsilon}(\Delta)\to0$ as $\Delta\to\infty$ (see [@GP2018 end of Section 11.2]). For the above graphs, they also explain how to prove the bound $p_c(G)\le 1 - \frac{1}{\Delta(G)-1}$ on the *site* percolation critical probability (this bound is related to the fact that $p_c(G)\ge \frac{1}{\Delta(G)-1}$ as mentioned in Section \[sec:introduction\]).
### Degree assumptions
Benjamini and Schramm [@BS96 Conjecture 7] also conjectured that $p_c(G)< 1/2$ for every planar graph $G$ with minimal degree at least $7$, and that there are infinitely many open connected components when $p\in(p_c(G), 1-p_c(G))$.
Angel, Benjamini and Horesh [@ABH18 Problem 4.2] asked whether $p_c(G)\le 1/2$ for plane triangulations with degrees at least $6$ (and posed related questions on bond percolation and the connective constant).
Haslegrave and Panagiotis [@HP19] resolve the first part of the conjecture of Benjamini and Schramm, proving that $p_c(G)<1/2$ when the minimal degree is at least $7$ (for planar graphs that have proper drawings without accumulation points in a suitable sense). They also make progress towards the conjecture of Angel, Benjamini and Horesh, proving that $p_c(G)\le 2/3$ when the minimal degree is at least $6$. In a third result they prove an upper bound for $p_c(G)$ when the minimal degree is at least $5$ and the minimal face degree is at least $4$.
### Volume growth and non-amenability
Benjamini [@B2018 Section 2.1] conjectures that $p_c(G)\ge 1/2$ for plane triangulations with polynomial volume growth. It is also conjectured there that $p_C(G)<1/2$ for nonamenable plane triangulations.
### Number of infinite connected components
A conjecture of a different nature of Benjamini and Schramm [@BS96 Conjecture 8] is that if site percolation with $p=1/2$ on a planar graph has an infinite connected component almost surely, then it has infinitely many infinite connected components almost surely. This is proved [@BS96 Theorem 5] when the planar graph admits an embedding in ${\mathbb{R}}^2$ in which the $x$-axis avoids all edges and vertices and every compact set intersects finitely many vertices and edges.
### Translation-invariant and unimodular graphs
Call a planar graph a *planar lattice* if it has an embedding in ${\mathbb{R}}^2$ which is invariant under a full-rank lattice of translations and satisfying that every compact set intersects only finitely many vertices and edges. On planar lattices there is no infinite connected component for site percolation with $p=1/2$. This follows from the classical results of Aizenman–Kesten–Newman [@AKN87] and Burton–Keane [@BK89] which rule out the existence of more than one infinite connected component and theorems showing that there is no coexistence of unique open and closed infinite connected components [@HJ06 Theorem 14.3],[@S05 Corollary 9.4.6], [@DCRT19 Theorem 1.5].
The notion of *unimodularity* [@H97; @BLPS99; @BS01; @AL07] of a random rooted graph can sometimes serve as a replacement for invariance properties. We do not provide the definition here (see [@AL07]) but mention that Benjamini–Schramm limits (sometimes called *sofic* graphs) are unimodular and it is a major open problem to determine if these are the only examples [@AL07 Section 10]. A theory of unimodular *planar* maps is developed by Angel, Hutchcroft, Nachmias and Ray [@AHNR16; @AHNR18] and extended by Timár [@T19] and Benjamini and Timár [@BT19]. In [@BT19 Corollary 3.2], ergodic unimodular plane triangulations, with finite expected degree of the root, are considered. Following Benjamini and Schramm [@BS01-2], it is shown that if such a map $G$ is nonamenable then $p_c(G)<1/2$ (and additional results regarding the number of infinite connected components).
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---
abstract: 'A Lorentz-noninvariant modification of quantum electrodynamics (QED) is considered, which has photons described by the nonbirefringent sector of modified Maxwell theory and electrons described by the standard Dirac theory. These photons and electrons are taken to propagate and interact in a Schwarzschild spacetime background. For appropriate Lorentz-violating parameters, the photons have an effective horizon lying outside the Schwarzschild horizon. A particular type of Compton scattering event, taking place between these two horizons (in the photonic ergoregion) and ultimately decreasing the mass of the black hole, is found to have a nonzero probability. These events perhaps allow for a violation of the generalized second law of thermodynamics in the Lorentz-noninvariant theory considered.'
address: |
Institute for Theoretical Physics, University of Karlsruhe,\
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
author:
- 'E. Kant'
- 'F.R. Klinkhamer'
- 'M. Schreck'
title: |
Lorentz violation and black-hole thermodynamics:\
Compton scattering process
---
Lorentz violation , ,Compton scattering 11.30.Cp ,04.70.Dy ,12.20.Ds
Introduction
============
Lorentz-violating theories coupled to gravity can have interesting black-hole solutions. Particles that obey Lorentz-violating dispersion relations may perceive an effective horizon different from the event horizon for standard Lorentz-invariant matter [@DubovskySibiryakov2006; @Eling-etal2007; @Betschart-etal2009]. It has been argued [@DubovskySibiryakov2006; @Eling-etal2007] that such multiple-horizon structures allow for the construction of a perpetuum mobile of the second kind (involving heat transfer from a cold body to a hot body, without other change).
This Letter considers modified Maxwell theory [@KosteleckyMewes2002] as a concrete realization of a Lorentz-violating theory. With an appropriate choice for the Lorentz-violating parameters, the nonstandard photons have an effective horizon lying outside the Schwarzschild event horizon for standard matter. Of interest, now, are Compton scattering events $\gamma e^{-} \to \gamma e^{-}$, which take place between these two horizons, that is, in the accessible part of the photonic ergosphere region. After the collision, the photon may carry negative Killing energy as it propagates inside the photonic ergosphere, so that the final electron carries away more Killing energy than the sum of the Killing energies of the ingoing particles. As shown in Sec. IV–B of Ref. [@Eling-etal2007], such a scattering event ultimately *reduces* the black-hole mass. In the following, it will be demonstrated that this particular Compton scattering event is kinematically allowed and has a nonvanishing probability to occur.
The purpose of this Letter is to give a concrete example of a Compton scattering event that can be used to reduce the black-hole mass. This requires a detailed discussion of the theory in Sec. \[sec:Setup\], which can, however, be skipped in a first reading. The main result is presented in Sec. \[sec:Compton\] and discussed in Sec. \[sec:Discussion\], both of which sections are reasonably self-contained.
Setup {#sec:Setup}
=====
Units and conventions
---------------------
Natural units are used with $c=G_\text{N}=\hbar =1$. Spacetime indices are denoted by Greek letters and correspond to $t,r,\theta, \phi$ for standard spherical Schwarzschild coordinates or to $\tau,R,\theta, \phi$ for Lemaître coordinates. Local Lorentz indices are denoted by Latin letters and run from $0$ to $3$. The flat-spacetime Minkowski metric is $\eta_{a b}$ and the curved-spacetime Einstein metric $g_{\mu\nu}$, both with signature $\left(+,-,-,-\right)$. The determinant of the metric is denoted by $g\equiv \text{det}\,g_{\mu\nu}$. The vierbeins are introduced in the standard way by writing $g_{\mu\nu}={e_{\mu}^{\;\;a}}{e_{\nu}^{\;\;b}}\,\eta_{ab}$ and obey the relations ${e^{\mu}_{\;\;a}}{e_{\mu}^{\;\;b}} = \delta_a^{~b}$ and ${e^{\mu}_{\;\;a}}{e_{\nu}^{\;\;a}} = \delta_{~\nu}^{\mu}$.
Modified QED in curved spacetime
--------------------------------
Modified Maxwell theory is an Abelian $U(1)$ gauge theory with a Lagrange density that consists of the standard Maxwell term and an additional Lorentz-violating bilinear term [@KosteleckyMewes2002; @BaileyKostelecky2004; @KlinkhamerRisse2008; @KlinkhamerSchreck2008]. The vierbein formalism is particularly well-suited for describing Lorentz-violating theories in curved spacetime, since it allows to distinguish between local Lorentz and general coordinate transformations [@Kostelecky2004] and to set the torsion identically to zero.
A minimal coupling procedure then yields the following Lagrange density for the photonic part of the action:
\[eq:LagMMgrav\] $$\begin{aligned}
\mathcal{L}_\text{modM}
&=&
-\frac{1}{4}\, g^{\mu\rho}g^{\nu\sigma}\, F_{\mu\nu}F_{\rho\sigma}
-\frac{1}{4}\, \kappa^{\mu\nu\rho\sigma}\, F_{\mu\nu}F_{\rho\sigma} \ ,
\label{eq:LagMMgrav-L}
\\[2mm]
\kappa^{\mu\nu\rho\sigma}
&\equiv&
\kappa^{abcd}\,e^\mu_{\;\;a}\,
e^\nu_{\;\;b}\,e^\rho_{\;\;c}\,e^\sigma_{\;\;d}\,,
\label{eq:LagMMgrav-kappa-tensor}\end{aligned}$$
in terms of the standard Maxwell field strength tensor $F_{\mu\nu}\equiv \partial_\mu A_\nu -\partial_\nu A_\mu$. The “tensor” $\kappa^{abcd}$ has the same symmetries as the Riemann curvature tensor, as well as a double-trace condition. The numbers $\kappa^{abcd}(x)$ are considered to be fixed parameters, with no field equations of their own.
In the following, we explicitly choose this background tensor field to be of the form [@BaileyKostelecky2004] $$\label{eq:ansatznonbire}
\kappa^{abcd}(x)=\frac{1}{2} \left(\eta^{ac}\,
\widetilde{\kappa}^{bd}(x)-\eta^{ad}\, \widetilde{\kappa}^{bc}(x)
+\eta^{bd}\, \widetilde{\kappa}^{ac}(x)-\eta^{bc}\, \widetilde{\kappa}^{ad}(x)\right),$$ in terms of a symmetric and traceless background field $\widetilde{\kappa}^{ab}(x)$. Physically, implies the restriction to the nonbirefringent sector of modified Maxwell theory. Moreover, we employ the following decomposition of $\widetilde\kappa^{ab}(x)$: $$\label{eq:ansatzxi}
\widetilde{\kappa}^{ab}(x)=\kappa\Big(\xi^a(x)\,\xi^b(x)-\eta^{ab}/4 \Big),$$ relative to a normalized parameter four-vector $\xi^a$ with $\xi_a \xi^a=1$. For our purpose, we will choose the parameter $\kappa$ in to be spacetime independent.
The breaking of Lorentz invariance in the electromagnetic theory is indicated by the fact that the flat-spacetime theory allows for maximal photon velocities different from $c=1$ (operationally defined by the maximum attainable velocity of standard Lorentz-invariant particles to be discussed shortly). See, e.g., Refs. [@KosteleckyMewes2002; @BaileyKostelecky2004; @KlinkhamerRisse2008; @KlinkhamerSchreck2008] for further details of the simplest version of modified Maxwell theory with constant $\kappa^{abcd}$ over Minkowski spacetime and physical bounds on its 19 parameters.
The charged particles (electrons) are described by the standard Dirac Lagrangian over curved spacetime [@Birrell] and gravity itself by the standard Einstein–Hilbert Lagrangian [@Wald1984]. All in all, this particular modification of quantum electrodynamics (QED) has action
\[eq:fullaction-LagEH-LagD\] $$\begin{aligned}
S &=& \int_{\mathbb{R}^4} d^4x\;\sqrt{-g}\;
\big(\mathcal{L}_\text{EH}+\mathcal{L}_\text{D}+\mathcal{L}_\text{modM}\big),
\label{eq:fullaction}\\[1mm]
\mathcal{L}_\text{EH} &=& R/(16\pi)\,,
\label{eq:LagEH} \\[1mm]
\mathcal{L}_\text{D} &=& \overline{\psi} \left( \frac{1}{2}\, \gamma^a
e^\mu_{\;\;a}
\; \text{i} \overset{\leftrightarrow}{\nabla}_\mu-m\right) \psi\,,
\label{eq:LagD}\end{aligned}$$
with Ricci curvature scalar $R$ from the metric $g_{\mu\nu}$, the usual Dirac matrices $\gamma^a$, and the gauge- and Lorentz-covariant derivative of a spinor [@Birrell],
$$\nabla_\mu\psi \equiv \partial_\mu\, \psi +\Gamma_\mu \, \psi-e A_\mu\, \psi\;,$$
with spin connection $$\begin{aligned}
\Gamma_\mu &=& \frac{1}{2}\, \Sigma^{ab}\, e_a^{\;\;\nu}\partial_\mu (e_{b\;
\nu}) \,,\quad \Sigma_{ab} \equiv \frac{1}{4}\left(\gamma_a\gamma_b
-\gamma_b\gamma_a\right)\,.\end{aligned}$$
Effective background for the photons
------------------------------------
As demonstrated in Sec. 3 of Ref. [@Betschart-etal2009], photons described by the Lagrange density with the Lorentz-violating parameters – propagate on null-geodesics of an effective metric. This effective metric is given by: $$\begin{aligned}
\widetilde{g}_{\mu\nu}(x)&=&
g_{\mu\nu}(x)-\frac{\kappa}{1+\kappa/2}\;
\xi_{\mu}(x)\xi_{\nu}(x)\,,
\label{eq:effectivemetric}\end{aligned}$$ with an inverse following from $\widetilde{g}^{\mu\nu}\widetilde{g}_{\nu\rho}=\delta^{\mu}_{\;\;\rho}$. All lowering or raising of indices is, however, understood to be performed by contraction with the original background metric $g_{\mu\nu}$ or its inverse $g^{\mu\nu}$, unless stated otherwise.
In order to avoid obvious difficulties with causality, we restrict our considerations to a subset of theories without space-like photon trajectories (with respect to the original metric). This is ensured by the choice $0\leq\kappa< 2$.
Schwarzschild spacetime metric
------------------------------
In the following, we consider a standard Schwarzschild geometry as given by the following line element:
\[eq:SS-lineelementstandard\] $$\begin{aligned}
ds^2 &=&\left(1-2 M/r\right) dt^2-\left(1-2 M/r\right)^{-1}
dr^2 -r^2 d\Omega^2\,,\\
d\Omega^2 &\equiv& d\theta^2+\sin^2\theta\, d\phi^2\,.\end{aligned}$$
It will be convenient to work with Lemaître coordinates, $$\label{eq:SS-lineelementlemaitre} ds^2 = d\tau^2-\left(\frac{3(R-\tau)}{4 M}\right)^{-2/3}dR^2
-\Big(3/2\, (R-\tau)\Big)^{4/3}(2 M)^{2/3}\,d\Omega^2\,,$$ as Lemaître coordinates describe the standard Schwarzschild solution in coordinates which are nonsingular at the horizon (corresponding to the reference frame of a free-falling observer).
The transformation to standard Schwarzschild coordinates reads
$$d \tau=dt+\frac{\sqrt{2M/r}}{1-2M/r}\;dr\;,$$
$$dR=dt+\frac{1}{\left(1-2M/r\right)\sqrt{2M/r}} \;dr\;,$$
and the horizon is described by $\left(R-\tau\right)=(4/3)\, M$. A suitable choice of the vierbein $e_\mu^{\;a}$ is given by $$\begin{aligned}
\label{eq:vierbein} e_{\tau}^{\;0} &=& 1\,,\quad
e_R^{\;1} = \sqrt{\left|g_{RR}\right|}\,,\quad
e_\theta^{\;2} = \sqrt{\left|g_{\theta\theta}\right|}\,,\quad
e_{\phi}^{\;3} = \sqrt{\left|g_{\phi\phi}\right|}\,,\end{aligned}$$ with all other components vanishing.
Effective Schwarzschild metric for the photons
----------------------------------------------
For the vector field $\xi^\mu(x)=e^\mu_{\;\;a}(x)\,\xi^a(x)$ entering the nonstandard part of the photonic action – and the effective Lorentz-violating parameter, we take
\[eq:xiexplicit-epsilon\] $$\begin{aligned}
\xi^\mu(x) &=& \left(1,0,0,0\right)\,,
\label{eq:xiexplicit}\\
\epsilon &\equiv& \frac{\kappa}{1-\kappa/2}\;,
\label{eq:epsilon}\end{aligned}$$
where the first expression (in Lemaître coordinates) makes clear that the photonic Lorentz violation is isotropic and the last expression introduces a convenient Lorentz-violating parameter for the theory considered. The particular parameter choices correspond to Case 1 in Ref. [@Betschart-etal2009]. Asymptotically ($R\to\infty$ for fixed $\tau$), the parameter $\kappa$ corresponds to $2\,\widetilde{\kappa}_\text{tr}$, in terms of the parameter $\widetilde{\kappa}_\text{tr}$ introduced by Ref. [@KosteleckyMewes2002] and bounded in Ref. [@KlinkhamerSchreck2008].
As shown in Sec. 4.1 of Ref. [@Betschart-etal2009], the effective background for the photons is again a Schwarzschild background, $$\label{eq:SS-lineelementlemaitremodmass} d\widetilde{s}^2=d\widetilde{\tau}^2-\left(\frac{3(\widetilde{R}-\widetilde{\tau})}
{4\widetilde{M}}\right)^{-2/3}d\widetilde{R}^2
-\Big(3/2\, (\widetilde{R}-\widetilde{\tau})\Big)^{4/3}(2 \widetilde{M})^{2/3}\,d\Omega^2\,,$$ with a rescaled mass $\widetilde{M} \equiv M{\left(1+\epsilon\right)}$ and modified horizon coordinate $r_\text{hor}=2M(1+\epsilon)$. The nonstandard photons perceive a horizon outside the standard Schwarzschild event horizon at $r=r_\text{Schw}\equiv 2M$.[^1] The space lying between these horizons, $2M<r<2M(1+\epsilon)$, will be referred to as the photonic ergoregion or ergoregion, for short.
Compton scattering {#sec:Compton}
==================
Generalities {#subsec:comptongeneralities}
------------
In this section, we present a concrete realization of the process proposed by Eling *et al.* [@Eling-etal2007], which, in an appropriate Lorentz-violating theory, corresponds to a type of Penrose-mechanism [@Penrose1969; @Piran-etal1975] to extract energy from the photonic ergosphere of a nonrotating Schwarzschild black hole.
In fact, we consider a Compton scattering event [@Compton1923; @Heitler1954; @JauchRohrlich1976; @Peskin] from modified Maxwell theory as defined in Sec. \[sec:Setup\]. Specifically, the theory is given by the total action in terms of the Lagrange densities , , and , with Lorentz-violating parameters given by , , and .
The scattering event is assumed to take place at $$\begin{aligned}
\label{eq:scattering-point} r_\text{scatter} &=&2M(1+\epsilon\, \rho)\,,\quad
\theta_\text{scatter} =\pi/2\,,\quad
\phi_\text{scatter} =0\,,\end{aligned}$$ with the Schwarzschild mass $M$ from the metric , the effective Lorentz-violating parameter $\epsilon$ defined by , and a free parameter $\rho$ taking values between $0$ and $1$. Using Lemaître coordinates , the transformation to a local inertial frame is given by $$\begin{aligned}
\left({e_{\tau}^{\;0}}\right)_\text{\,scatter}&=&1\,,\quad
\left({e_R^{\;1}}\right)_\text{\,scatter} =1/\sqrt{1+\epsilon\, \rho}\,,
\nonumber\\
\left({e_\theta^{\;2}}\right)_\text{\,scatter}&=&
\left({e_{\phi}^{\;3}}\right)_\text{\,scatter}= 2 M(1+\epsilon\, \rho)\,,
\label{eq:vierbein_LICS}\end{aligned}$$ with all other components vanishing. The asymptotically time-like Killing field in local coordinates at the scattering point reads $$\label{eq:Killing-local-coord}
\sigma^a_\text{\,scatter} \equiv e_\mu^{\; a}\,\sigma^\mu\;\Big|_\text{\,scatter}
= \left(1,\, \frac{1}{\sqrt{1+\epsilon\, \rho}},\, 0,\, 0\right)\,.$$
As explained in the Introduction, we are interested in a Compton scattering event (Fig. \[fig:Comptonscat\]) where the final scattered photon carries negative Killing energy: $$\label{eq:negativeKillingE}
E_{\text{Killing},\gamma,\text{out}}=\sigma^\mu\,
k^\nu_{\gamma,\text{out}}\; \widetilde{g}_{\mu\nu}\equiv \sigma^\mu\,
k_\mu^{\gamma,\text{out}}< 0\,,$$ with $k^\nu_{\gamma,\text{out}}$ the tangent vector to the path of the final photon. \[Here, and in the following, the label ‘in’ or ‘out’ on a particle momentum refers only to the scattering point and the label ‘out,’ in particular, does not foretell the ultimate destiny of the particle.\] Such processes are allowed, since the asymptotically time-like Killing field for the photon becomes space-like for $r< 2M(1+\epsilon)$. The final electron should, however, be able to leave to infinity, carrying more Killing energy than the sum of the initial Killing energies. \[The physical interpretation is that energy is extracted from the black hole. Thus, it is clear that the complete process is not just an isolated 2–2 scattering, but that the black hole itself should be considered as a participant, making this essentially a 3–3 scattering process. However, the treatment as a 2–2 scattering process in a fixed spacetime background is justified for a black-hole mass $M$ very much larger than all Killing energies involved.\] Moreover, we demand that such a *Gedankenexperiment* can be prepared in the asymptotically flat region of spacetime, i.e., that the two initial particles come in from spatial infinity.
These conditions impose several constraints on the initial and final four-vectors of the particles. For the sake of brevity, these constraints are omitted, but it has been checked that the example of Sec. \[subsec:example\] fulfills all requirements.
Parametrization {#subsec:Parametrization}
---------------
For our purpose, a useful parametrization of the Compton-scattering wave vectors (in the local inertial frame with Cartesian coordinates) is given by
\[eq:Ansatz\] $$\begin{aligned}
\big(k_a\big)^{\gamma,\text{out}} &=& E_{\gamma,\text{out}}\,
\left(1,\, -\zeta \omega_1,\, 0,\, \zeta\, \sqrt{1-\omega_1^2}\right), \\
\big(k_a\big)^{e,\text{out}} &=&
{p}_{e,\text{out}}\,\left(\sqrt{m^2/({p}_{e,\text{out}})^2+1},\,\;\widehat{p}_{e,\text{out}}\right)\,,\\
\big(k_a\big)^{\gamma,\text{in}} &=&
\widetilde{E}_{\gamma,\text{in}}\,
\left(1,\, -\zeta \beta_1,\, -\zeta \beta_2 ,\, s_1\zeta\, \sqrt{1-\beta_1^2-\beta_2^2} \right)\,,\\
k_a^{e,\text{in}} &=& k_a^{e,\text{out}}+k_a^{\gamma,\text{out}}
-k_a^{\gamma,\text{in}}\;,\end{aligned}$$
with arbitrary photon energy $E_{\gamma,\text{out}}>0$, electron three-momentum $\vec{p}\equiv\left(p_1,p_2,p_3\right)$ $\equiv$ $p_{e,\text{out}}\;\widehat{p}_{e,\text{out}}$ for modulus $p_{e,\text{out}}\equiv |\vec{p}_{e,\text{out}}| >0$, Lorentz-violating parameter $\zeta\equiv\sqrt{1+\epsilon}> 1$, and energy $\widetilde{E}_{\gamma,\text{in}}>0$ to be determined from the dispersion relation of the incoming electron. The parameters $\omega_1$, and $\beta_{1,2}$ vary between $-1$ and $1$, with the additional constraint $\beta_1^2+\beta_2^2\leq 1$. The parameter $s_{1}$ takes the value $+1$ or $-1$.
The *Ansatz* ensures that the dispersion relations for massless Lorentz-violating photons and massive electrons are fulfilled.
Concrete example {#subsec:example}
----------------
Since the experimental bounds on isotropic Lorentz violation are tight [@KlinkhamerSchreck2008], very small Lorentz violation ($0<\kappa\ll 1$) would be physically more interesting than large Lorentz violation ($\kappa \sim 1$). However, the Compton scattering process with negative Killing energy of the final photon appears to be kinematically forbidden for a small Lorentz-violating parameter $\kappa$ (see Sec. \[subsec:Small-LV\]).
The following example of allowed kinematics is, therefore, of purely theoretical interest. Specifically, the parameters are chosen to be
\[eq:example\] $$\begin{aligned}
\epsilon&=1/2\,,\quad
&\rho&=99/100\,,
\\
E_{\gamma,\text{out}} &=5\,m\,,\quad
&\omega_1&=9984/10000\,,
\\
p_{e,\text{out}}&=20\, E_{\gamma,\text{out}}\,, \quad
&\widehat{p}_{e,\text{out}}&=
\big(-41,\, 0,\, 3 \sqrt{91} \; \big)\big/50\,,
\\
\beta_1&=74/100,\;\;\;
&\beta_2&=0,\quad s_{1}=1\,,\end{aligned}$$
with corresponding Lorentz-violating parameter $\kappa=\epsilon/\big(1+\epsilon/2\big) = 2/5$. It has taken considerable effort to find this single example. Apparently, the allowed domain of the multi-dimensional parameter space is very small, which is confirmed by preliminary numerical calculations.
The above parameters allow for a Compton scattering event that ultimately reduces the black-hole mass, because the Killing energy of the final photon is negative: $\sigma^a\,k_a^{\gamma,\text{out}} <0$ using and the above numbers \[the actual value of this energy will be given in Sec. \[subsec:Gedankenexperiment\]\].
Squared matrix element
----------------------
To ensure that the Compton scattering event discussed above has a nonvanishing probability to occur, the corresponding matrix element must be nonzero. The squared matrix element for the Compton scattering process at tree level (calculated with flat-spacetime electron propagators) reads $$\begin{aligned}
\label{eq:matrix-element}
\hspace*{-10mm}&&
\frac{1}{4}\;\;\sum_{s_1,s_2=\pm 1/2}\;\;\sum_{\lambda_1,\lambda_2=\pm 1}\;\;
|\mathcal{M}|^2=\notag\\
\hspace*{-10mm}&&
\Pi_{ac}\,\Pi_{bd}\;
\frac{e^4}{4}\;\mathrm{tr}\left\{(\cancel{k}_{e,\text{out}}+m)
\left[\frac{\gamma^{a}\cancel{k}_{\gamma,\text{in}}\gamma^{b}
+2\gamma^{a}k_{e,\text{in}}^{b}}{2k_{e,\text{in}}\cdot k_{\gamma,\text{in}}
+k_{\gamma,\text{in}}^2}
+\frac{\gamma^{b}\cancel{k}_{\gamma,\text{out}}\gamma^{a}-2\gamma^{b}k_{e,\text{in}}^{a}}
{2k_{e,\text{in}}\cdot k_{\gamma,\text{out}}
-k_{\gamma,\text{out}}^2}\right] \right.
\notag \\
\hspace*{-10mm}&&
\left. \times (\cancel{k}_{e,\text{in}}+m)
\left[\frac{\gamma^{d}\cancel{k}_{\gamma,\text{in}}\gamma^{c}
+2\gamma^{c}k_{e,\text{in}}^{d}}{2k_{e,\text{in}} \cdot k_{\gamma,\text{in}}
+k_{\gamma,\text{in}}^2}+\frac{\gamma^{c} \cancel{k}_{\gamma,\text{out}}\gamma^{d}-2\gamma^{d}k_{e,\text{in}}^{c}}
{2k_{e,\text{in}}\cdot k_{\gamma,\text{out}}
-k_{\gamma,\text{out}}^2}\right]\right\}\,,\end{aligned}$$ with Feynman slash $\cancel{k} \equiv k_a\,\gamma^a$ and photon polarization sum $$\Pi_{ab}\equiv \sum_{\lambda=\pm 1}
\overline{(\varepsilon^{(\lambda)})}_{a}\,(\varepsilon^{(\lambda)})_{b}\,.$$ The Ward identities ensure that, in gauge-invariant expressions like the one leading up to , the polarization sum can be replaced by the following expression $$\Pi_{ab} \mapsto
\frac{1}{1+\kappa/2}\left(-\eta_{ab}+\frac{\kappa}{1+\kappa/2}\,\xi_{a}\xi_{b}\right)\,.$$ For $k_{\gamma,\text{in}}^2=k_{\gamma,\text{out}}^2=0$ and standard photon polarization sums, reproduces the standard squared matrix element of Compton scattering; see, for example, Eq. (5.81) in Ref. [@Peskin].
It has now been checked by explicit calculation that the average squared amplitude is nonzero for the large Lorentz-violating parameter and kinematics defined by . This particular Compton scattering event has, therefore, a nonvanishing probability to occur \[it has also been verified that the same holds for final photon energies $E_{\gamma,\text{out}}\geq m$, while keeping the other values in unchanged\].
Gedankenexperiment {#subsec:Gedankenexperiment}
------------------
At this moment, it may be instructive to give the numerical values of the four-vectors of the Compton scattering event –:
\[eq:example-4vectors\] $$\begin{aligned}
\big(k_a\big)^{e,\text{in\phantom{n}}}
&\approx&
m\, \left(17.0968,\, -8.44173,\, 0,\, -14.833628\right)\,,
\label{eq:example-4vector-e-in}
\\
\big(k_a\big)^{\gamma,\text{in\phantom{n}}}
&\approx&
m\,\left(87.9082,\, -79.6722,\, 0,\, +72.4163\right)\,,
\label{eq:example-4vector-gamma-in}
\\
\big(k_a\big)^{e,\text{out}}
&\approx&
m\,\left(100.005,\, -82.0000,\, 0,\, +57.2364\right)\,,
\label{eq:example-4vector-e-out}
\\
\big(k_a\big)^{\gamma,\text{out}}
&\approx&
m\, \left(5.00000,\, -6.11393,\, 0,\, +0.346272\right)\,,
\label{eq:example-4vector-gamma-out}\end{aligned}$$
where the three-momenta are seen to lie in a plane ($k_2=0$). The resulting (conserved) Killing energies of the particles are
\[eq:example-Killing-energies\] $$\begin{aligned}
\big(E_{\text{Killing}}\big)^{e,\text{in\phantom{n}}}
&\approx&\phantom{+}
10.19264 \, m\,,
\\\label{eq:example-Killing-e-in}
\big(E_{\text{Killing}}\big)^{\gamma,\text{in\phantom{n}}}
&\approx&\phantom{+}
22.74743\, m\,,
\\\label{eq:example-Killing-gamma-in}
\big(E_{\text{Killing}}\big)^{e,\text{out}}
&\approx&\phantom{+}
32.94041\,m\,,
\label{eq:example-Killing-e-out}
\\
\big(E_{\text{Killing}}\big)^{\gamma,\text{out}}
&\approx&
- \phantom{0} 0.00034\, m\,,
\label{eq:example-Killing-gamma-out}\end{aligned}$$
where the energy of the escaping electron is seen to be larger than the total energy of the two incoming particles, $E_{\text{Killing}}^{\text{in}} \approx 32.94007\,m$.
A possible *Gedankenexperiment* (in the *Gedankenwelt* of this Letter) consists of three steps. First, prepare electron and photon beams to give momenta – at the scattering point . Second, count the number of electrons scattered in the direction corresponding to and measure their energy. Third, determine the change of black-hole mass (for example, by measuring the change in the orbit of a test particle encircling the black hole).
Small Lorentz violation {#subsec:Small-LV}
-----------------------
A straightforward but tedious analysis for the case of vanishing electron mass, $m=0$, shows that the above Compton scattering process is not allowed for small (but finite) Lorentz-violating parameter $\epsilon$. Very briefly, the argument consists of two steps. First, the dispersion relation for the initial electron can be solved in terms of the energy of the initial photon. Second, this initial photon energy can be expanded in $\epsilon$. For any configuration of the parameters discussed in Sec. \[subsec:Parametrization\], it can be shown that this photon energy becomes negative or imaginary for sufficiently small $\epsilon$, if the constraints mentioned in the last paragraph of Sec. \[subsec:comptongeneralities\] are taken into account. It is not easy to get the explicit analytic bound, but a conservative bound can be found and is given by $\epsilon<1/10$. That is, it can be shown rigorously that the Compton scattering process with negative Killing energy of the final photon is kinematically forbidden for $\epsilon<1/10$.
For the case of nonvanishing electron mass, $m>0$, numerical investigations show that the process is, once more, kinematically forbidden for small enough $\epsilon$. A conservative bound is, again, given by $\epsilon<1/10$ (corresponding to $\kappa<2/21$).
The surprising result, then, is that the reduction of the black-hole mass by the specific Compton scattering process appears to be separated from the standard situation of non-decreasing black-hole mass [@Hawking1971] by a *finite gap* of the Lorentz-violating parameter $\kappa$. At the moment, it is not clear if this is just an artefact of the specific process considered (to be overcome by a more complicated setup) or if it indicates the existence of a mechanism that protects the Hawking area theorem [@Hawking1971] for the case of “small enough Lorentz violation.” This interesting question deserves further study.
Discussion {#sec:Discussion}
==========
This Letter investigated the kinematics of Compton scattering in the accessible part of the photonic ergoregion of a Schwarzschild black hole for nonbirefringent modified Maxwell theory –. More specifically, a Compton scattering event (Fig. \[fig:Comptonscat\]) was considered, for which the scattered photon carries away negative Killing energy and ultimately reduces the mass of the black hole. By giving a concrete example, it has been shown that such an event is kinematically allowed and has a nonzero matrix element.
This particular type of Compton scattering event has, therefore, a nonvanishing probability to occur, at least, for a relatively large Lorentz-violating parameter $\kappa$. In a *Gedankenexperiment* starting with a large number $N_\text{BH}$ of Schwarzschild black holes of identical mass $M$ and having a large number $N_\text{scatt}$ of repeated Compton scattering events on each of these black holes, it is then possible to find certain black holes for which the initial mass $M$ has been reduced by a macroscopic amount. In this way, the Hawking area theorem [@Hawking1971] is circumvented by the presence of negative-energy states outside the Schwarzschild radius, whose existence is due to the Lorentz violation of the photonic theory considered.
These area-reducing events are believed [@Eling-etal2007] to contradict the generalized second law of thermodynamics [@Bekenstein1974], since they may allow for a construction of a perpetuum mobile of the second kind. The basic idea is that such events decrease the mass of the black hole and with it the associated entropy. If the scattering were classical [@Eling-etal2007], the outgoing electron would not carry entropy.[^2] The whole process would, then, globally *decrease* entropy, in contradiction with the generalized second law [@Bekenstein1974].
However, the Compton scattering process discussed above *is* a quantum process. Certainly, a particular type of Compton scattering event has been shown to have a nonvanishing probability to decrease the black-hole mass and reduce the black-hole entropy. But there is also the possibility that both particle trajectories after the scattering head towards the black hole and that the black-hole entropy increases.
An analogous classical process with reduced black-hole entropy would surely be able to violate the generalized second law, since it would be possible to conceive of a deterministic experiment that would result in a decrease of entropy. But the possibility of an entropy-decreasing quantum process need not imply, by itself, the violation of the generalized second law. For example, already in a system with two types of molecules, there is a nonvanishing probability that a slow-moving (“cold”) molecule transfers energy to a fast-moving (“hot”) molecule. In fact, it is only the application of statistical mechanics to a system with a large number of molecules that recovers the second law of thermodynamics [@LandauLifshitzStatMech1].
A quantitative analysis would be needed to see whether or not the Compton scattering process of Sec. \[sec:Compton\] would be able to violate the generalized second law. This would require phase-space integrations with nontrivial cuts to determine the probabilities for the interesting Compton scattering events to occur. Perhaps one might, then, be able to show a violation of the fluctuation theorem [@Fluctuation1993; @Fluctuation2002], which might, in turn, imply the breakdown of the generalized second law.
A more speculative idea expands on the *Gedankenexperiment* discussed in the last paragraph of Sec. \[subsec:Gedankenexperiment\]. Perhaps it is possible to arrange for a cloud of electrically charged particles and a pulse of light coming in from infinity to scatter elastically at point with average momenta – and to have a final cloud and pulse taking off with average momenta –. If that arrangement were possible (admittedly a big ‘if’), the discussion of Sec. IV–B of Ref. [@Eling-etal2007] could be taken over literally, with the consequent violation of the generalized second law (the incoming and outgoing charged clouds would have the same velocity dispersion and other characteristics, the scattering being elastic by assumption).
Whether or not a violation of the generalized second law of thermodynamics occurs in Lorentz-violating theories remains, therefore, an open question.[^3] The present Letter tried to find a concrete realization of the promising idea [@Eling-etal2007] of exploiting a Penrose-mechanism-type process. However, as discussed above, we did not succeed in obtaining an entirely convincing and totally explicit *Gedankenexperiment* that is able to violate the generalized second law. Still, the presented Compton scattering events, being able to reduce the black-hole mass, may provide a step towards demonstrating the violation of the generalized second law in the Lorentz-noninvariant theory considered, if at all possible.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank H. Sahlmann and S.M. Sibiryakov for helpful comments on the manuscript.
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[^1]: The effective background agrees with the effective metric obtained in Ref. [@DubovskySibiryakov2006] for a minimally coupled scalar field interacting with the ghost condensate [@ArkaniHamed-etal2004; @Mukohyama2005; @Feldstein2009]. In the present article, the background field is introduced by hand. But it is also possible, as shown in [@Betschart-etal2009], to obtain this background field $\xi^\mu$ by spontaneous symmetry breaking from the ghost-condensate. For our purpose, though, it is more convenient to consider the background as coming from *explicit* Lorentz violation, avoiding discussion of the stability of the solution and the related flow of energy or entropy.
[^2]: This would precisely be the difference with the mining technique of Ref. [@UnruhWald1982], for which the black-hole mass is also reduced but the outgoing box (with the mined energy) does carry entropy, namely, that of the trapped “acceleration radiation.”
[^3]: A different mechanism to violate the generalized second law was suggested in Ref. [@Feldstein2009]. A quasi-stationary solution was constructed from the ghost condensate to describe the flow of negative energy into a black hole. Just as the process described in the present article, this flow of negative energy appears to be able to reduce the black-hole mass. Potential problems with the stability of this solution have been discussed and seem to be under control. But, whether or not this reduction of the black-hole mass results in a violation of the generalized second law remains, in our opinion, an open question, since the entropy of the effective ghost-condensate fluid has not been considered and the issue of turning the flow on and off requires further analysis. Similar and other reservations regarding the claim of generalized-second-law violation by Ref. [@Feldstein2009] have been presented in Ref. [@Mukohyama2009].
|
---
abstract: |
We define and study a class of entwined modules (stable anti-Yetter-Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter-Drinfeld modules, we find Hopf-Galois extensions with a flipped version of the Miyashita-Ulbrich action.
[*To cite this article: P. M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).*]{}
0.5
[**Résumé**]{} 0.5[**Modules anti-Yetter-Drinfeld stables**]{} Nous définissons et étudions une classe de modules enlacés (modules anti-Yetter-Drinfeld stables) qui servent de coefficients pour l’homologie et la cohomologie Hopf-cyclique. En particulier, nous expliquons leurs liens avec les modules de Yetter-Drinfeld et les doublets de Drinfeld. Parmi les sources d’exemples de modules anti-Yetter-Drinfeld stables, nous trouvons des extensions de Hopf-Galois munies d’une version transposée de l’action de Miyashita-Ulbrich.
[*Pour citer cet article : P. M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).*]{}
address:
- 'Instytut Matematyczny, Polska Akademia Nauk, ul. Śniadeckich 8, Warszawa, 00-956 Poland'
- 'Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoża 74, Warszawa, 00-682 Poland'
- 'Department of Mathematics, University of Western Ontario, London ON, Canada'
- 'Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany'
author:
- 'Piotr M. Hajac'
- Masoud Khalkhali
- Bahram Rangipour
- Yorck Sommerhäuser
title: 'Stable anti-Yetter-Drinfeld modules'
---
Introduction
============
The aim of this paper is to define and provide sources of examples of stable anti-Yetter-Drinfeld modules. They play the role of coefficients for Hopf-cyclic theory [@hkrsb]. In particular, we claim that modular pairs in involution of Connes and Moscovici are precisely 1-dimensional stable anti-Yetter-Drinfeld modules.
Throughout the paper we assume that $H$ is a Hopf algebra with a bijective antipode. On the one hand, the bijectivity of the antipode is implied by the existence of a modular pair in involution, so that then it need not be assumed. On the other hand, some parts of arguments might work even if the antipode is not bijective. We avoid such discussions. The coproduct, counit and antipode of $H$ are denoted by $\Delta$, ${\varepsilon}$ and $S$, respectively. For the coproduct we use the notation $\Delta(h)=h^{(1)}\otimes h^{(2)}$, for a left coaction on $M$ we write $_M\Delta(m)=m^{(-1)}\otimes m^{(0)}$, and for a right coaction $\Delta_M(m)=m^{(0)}\otimes m^{(1)}$. The summation symbol is suppressed everywhere. We assume all algebras to be associative, unital and over the same ground field $k$. The symbol ${\mathcal O}(X)$ stands for the algebra of polynomial functions on $X$.
P.M.H. acknowledges the Marie Curie Fellowship HPMF-CT-2000-00523 and KBN grant 2 P03A 013 24. All authors are grateful to T. Brzeziński and M. Furuta for their help and comments, and to D. Perrot for the French translation.
The transformation of Yetter-Drinfeld modules
=============================================
It turns out that, in order to incorporate coefficients into cyclic theory, we need to alter the concept of a Yetter-Drinfeld module by replacing the antipode by its inverse in the Yetter-Drinfeld compatibility condition between actions and coactions. We call the modules-comodules satisfying the thus modified Yetter-Drinfeld compatibility condition [*anti-Yetter-Drinfeld modules*]{}[^1]. Just as Yetter-Drinfeld modules come in 4 different versions depending on the side of actions and coactions (see [@cmz02 p.181] for a general formulation), so do the anti-Yetter-Drinfeld modules. All versions are completely equivalent and can be derived from one another by replacing a Hopf algebra $H$ by $H^{cop}$, $H^{op}$, or $H^{op,cop}$, respectively.\
\[ayd\] Let $H$ be a Hopf algebra with a bijective antipode $S$, and $M$ a module and comodule over $H$. We call $M$ an anti-Yetter-Drinfeld module iff the action and coaction are compatible in the following sense: $$\begin{aligned}
&\label{ayd1}
_M\Delta(hm)=h^{(1)}m^{(-1)}S^{-1}(h^{(3)})\otimes h^{(2)}m^{(0)}&
&\mbox{\em if $M$ is a left module and a left comodule},&
\\ &
\Delta_M(hm)=h^{(2)}m^{(0)}\otimes h^{(3)}m^{(1)}S(h^{(1)})&
&\mbox{\em if $M$ is a left module and a right comodule},&
\\ &
_M\Delta(mh)=S(h^{(3)})m^{(-1)}h^{(1)}\otimes m^{(0)}h^{(2)}&
&\mbox{\em if $M$ is a right module and a left comodule},&
\\ &\label{ayd4}
\Delta_M(mh)= m^{(0)}h^{(2)}\otimes S^{-1}(h^{(1)})m^{(1)}h^{(3)}&
&\mbox{\em if $M$ is a right module and a right comodule}.&\end{aligned}$$
To make cyclic theory work, we also need to assume that the action splits coaction, i.e., for all $m\in M$, $m^{(-1)}m^{(0)}=m$, $m^{(1)}m^{(0)}=m$, $m^{(0)}m^{(-1)}=m$, $m^{(0)}m^{(1)}=m$, for the left-left, left-right, right-left, and right-right versions, respectively. We call modules satisfying this condition [*stable*]{}. Let us emphasize that it is the anti-Yetter-Drinfeld condition rather than the Yetter-Drinfeld condition that makes the homomorphism $action\circ coaction$ $H$-linear and $H$-colinear. Therefore the stability condition $action\circ coaction= {\rm id}$ suits the former and not the latter. The first class of examples of stable anti-Yetter-Drinfeld modules is provided by modular pairs in involution [@cm00 p.8]. Since such pairs occur naturally in different contexts, Lemma \[1dim\] and Lemma \[affine\] guarantee ample amount of examples of anti-Yetter-Drinfeld modules.\
\[1dim\] Let the ground field $k$ be a right module over $H$ via a character $\delta$ and a left comodule over $H$ via a group-like $\sigma$. Then $k=^\sigma\!\!\!k_\delta$ is a stable anti-Yetter-Drinfeld module [*if and only if*]{} $(\delta,\sigma)$ is a modular pair in involution.
The anti-Yetter-Drinfeld modules do not form a monoidal category themselves, but rather a so-called $\mathcal C$-category over the category of Yetter-Drinfeld modules (see [@p-b77 p.351] for details). More precisely:\
\[affine\] Let $N$ be a Yetter-Drinfeld module and $M$ an anti-Yetter-Drinfeld module. Then $N\otimes M$ is an anti-Yetter-Drinfeld module via $h(n\otimes m)=h^{(1)}n\otimes h^{(2)}m$, $_{N\otimes M}\Delta(n\otimes m)=n^{(-1)}m^{(-1)}\otimes n^{(0)}\otimes m^{(0)}$, for the left-left case, and via $h(n\otimes m)=h^{(2)}n\otimes h^{(1)}m$, $\Delta_{N\otimes M}(n\otimes m)=n^{(0)}\otimes m^{(0)}\otimes n^{(1)}m^{(1)}$, for the left-right case. Similarly, $M\otimes N$ is an anti-Yetter-Drinfeld module via $(m\otimes n)h=mh^{(2)}\otimes nh^{(1)}$, $_{M\otimes N}\Delta(n\otimes m)=m^{(-1)}n^{(-1)}\otimes m^{(0)}\otimes n^{(0)}$, for the right-left case, and via $(m\otimes n)h=mh^{(1)}\otimes nh^{(2)}$, $\Delta_{M\otimes N}(m\otimes n)=m^{(0)}\otimes n^{(0)}\otimes m^{(1)}n^{(1)}$, for the left-right case.
Note that, just as the right-right Yetter-Drinfeld modules are entwined modules [@b-t99] for the entwining $\psi(h'\otimes h)=h^{(2)}\otimes S(h^{(1)})h'h^{(3)}$, the right-right anti-Yetter-Drinfeld modules are entwined with respect to $\psi(h'\otimes h)=h^{(2)}\otimes S^{-1}(h^{(1)})h'h^{(3)}$. (Other cases are completely analogous.)
An intermediate step between modular pairs in involution and stable anti-Yetter-Drinfeld modules is given by matched and comatched pairs of [@kr03]. Whenever the antipode is equal to its inverse, the difference between the Yetter-Drinfeld and anti-Yetter-Drinfeld conditions disappears. For a group ring Hopf algebra $kG$, a left $H$-comodule is simply a $G$-graded vector space $M={\bigoplus}_{g\in G}M_g$, where the coaction is defined by $M_g\ni m\mapsto g\otimes m$. An action of $G$ on $M$ defines an (anti-)Yetter-Drinfeld module if and only if for all $g,h\in G$ and $m\in M_g$ we have $hm\in M_{hgh^{-1}}$. The stability condition means simply that $gm=m$ for all $g\in G$, $m\in M_g$. A very concrete classical example of a stable (anti-)Yetter-Drinfeld module is provided by the Hopf fibration. Then $H={\mathcal O}(SU(2))$ and $M={\mathcal O}(S^2)$. Since $S^2\cong SU(2)/U(1)$, we have a natural left action of $SU(2)$ on $S^2$. Its pull-back makes $M$ a left $H$-comodule. On the other hand, one can view $S^2$ as the set of all traceless matrices of $SU(2)$. The pull-back of this embedding $j:S^2{\hookrightarrow}SU(2)$ together with the multiplication in ${\mathcal O}(S^2)$ defines a left action of $H$ on $M$. It turns out that the equivariance property $j(gx)=gj(x)g^{-1}$ guarantees the anti-Yetter-Drinfeld condition, and this combined with the injectivity of $j$ ensures the stability of $M$. This stability mechanism can be generalized in the following way.\
Let $M$ be an algebra and a left $H$-comodule. Assume that $\pi: H\rightarrow M$ is an epimorphism of algebras and the action $hm=\pi(h)m$ makes $M$ an anti-Yetter-Drinfeld module. Assume also that $\pi(1^{(-1)})1^{(0)}=1$. Then $M$ is a stable module.
Hopf-Galois extensions and the opposite Miyashita-Ulbrich action
================================================================
Another source of examples is provided by Hopf-Galois theory. These examples are purely quantum in the sense that the employed actions are automatically trivial for commutative algebras. To fix the notation and terminology, recall that an algebra and an $H$-comodule is called a comodule algebra if the coaction is an algebra homomorphism. An $H$-extension $B:=\{p\in P\;|\;\Delta_P(p)=p\otimes 1\}\subseteq P$ is called Hopf-Galois iff the canonical map $can: P\otimes_BP\rightarrow P\otimes H$, $can(p\otimes p')=
p\Delta(p')$, is bijective. The bijectivity assumption allows us to define the translation map $T:H\rightarrow P\otimes_BP$, $T(h):=can^{-1}(1\otimes h)
=:h^{[1]}\otimes_Bh^{[2]}$ (summation suppressed). It can be shown that when everything is over a field (our standing assumption), the centralizer $Z_B(P):=\{p\in P\;|\;bp=pb, \forall\, b\in B\}$ of $B$ in $P$ is a subcomodule of $P$. On the other hand, the formula $ph=h^{[1]}ph^{[2]}$ defines a right action on $Z_B(P)$ called the Miyashita-Ulbrich action. This action and coaction satisfy the Yetter-Drinfeld compatibility condition [@dt89 (3.11)]. The following proposition modifies the Miyashita-Ulbrich action so as to obtain stable anti-Yetter-Drinfeld modules.\
\[mu\] Let $B\subseteq P$ be a Hopf-Galois $H$-extension such that $B$ is central in $P$. Then $P$ is a right-right stable anti-Yetter-Drinfeld module via the action $
ph=(S^{-1}(h))^{[2]}p(S^{-1}(h))^{[1]}
$ and the right coaction on $P$.
The simplest examples are obtained for $P=H$. A broader class is given by the so-called Galois objects [@c-s98]. Then quantum-group coverings at roots of unity provide examples with central coinvariants bigger than the ground field (see [@dhs99] and examples therein). Finally, one can generalize Proposition \[mu\] to arbitrary Hopf-Galois extensions by replacing $P$ by $P/[B,P]$ [@js Remark 4.2].
The Drinfeld double comodule algebra
====================================
For finite-dimensional Hopf algebras, the Yetter-Drinfeld modules can be understood as modules over the Drinfeld double [@k-c95 p.220]. Much in the same way, the anti-Yetter-Drinfeld modules can also be understood as modules over a certain algebra. This makes the usual notions and operations for modules, like projectivity or induction, directly available for anti-Yetter-Drinfeld modules. To this end, the comodule structure of an anti-Yetter-Drinfeld module has to be converted into a module structure over the dual Hopf algebra $H^*$, so that from now on we assume that the Hopf algebra $H$ is finite-dimensional.\
Let $H$ be a finite-dimensional Hopf algebra. The formula $$(\varphi \otimes h)(\varphi' \otimes h') =
\varphi'{}^{(1)}(S^{-1}(h^{(3)})) \varphi'{}^{(3)}(S^2(h^{(1)})) \;
\varphi \varphi'{}^{(2)} \otimes h^{(2)} h'$$ turns the vector space $A(H):= H^* \otimes H$ into an associative algebra with the unit $\varepsilon \otimes 1$.
Note that the above product differs from the product in the Drinfeld double of $H$ [@k-c95 p.214] only by the additional squared antipode in the second factor. To relate the modules over $A(H)$ with anti-Yetter-Drinfeld modules, recall first that every right $H$-comodule $M$ becomes a left $H^*$-module via $\varphi m := \varphi(m^{(1)}) m^{(0)}$. Conversely, any left $H^*$-module yields a right $H$-comodule via $\Delta_M(m) = \sum_{i=1}^n h_i^* m \otimes h_i$. Here $\{h_1,\ldots,h_n\}$ is a basis of $H$ and $\{h_1^*,\ldots,h_n^*\}$ is the dual basis. (Of course, this comodule structure does not depend on the choice of a basis.) Using this, we get the following connection between the modules over $A(H)$ and anti-Yetter-Drinfeld modules:\
Let $H$ be a finite-dimensional Hopf algebra. If $M$ is a left-right anti-Yetter-Drinfeld module, it becomes a left $A(H)$-module by $(\varphi \otimes h)m := \varphi((hm)^{(1)}) \, (hm)^{(0)}$. Conversely, if $M$ is a left $A(H)$-module, it becomes a left-right anti-Yetter-Drinfeld module by $hm := (\varepsilon \otimes h)m$, $\Delta_M(m) := \linebreak\sum_{i=1}^n (h_i^* \otimes 1) m \otimes h_i$. Here $\{h_1,\ldots,h_n\}$ is a basis of $H$ and $\{h_1^*,\ldots,h_n^*\}$ its dual basis.
The claim of Lemma \[affine\] is reflected in the fact that although $A(H)$ is not a Hopf algebra itself, it can be shown that the formula $(\varphi \otimes h)
\mapsto (\varphi^{(2)} \otimes h^{(1)}) \otimes
(\varphi^{(1)} \otimes h^{(2)})$ makes $A(H)$ a right comodule algebra over the Drinfeld double $D(H)$.
[00]{}
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Caenepeel, S.; Militaru, G.; Zhu, S.: Frobenius and separable functors for generalized module categories and nonlinear equations. Lecture Notes in Mathematics, 1787, Springer-Verlag, Berlin, 2002.
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Dabrowski, L.; Hajac, P.M.; Siniscalco, P.: Explicit Hopf-Galois description of $SL_{e^{2i\pi\slash 3}}(2)$-induced Frobenius homomorphisms. Enlarged proceedings of the ISI GUCCIA workshop on quantum groups, non commutative geometry and fundamental physical interactions, D. Kastler, M. Rosso, T. Schucker (eds.), Commack – New York, Nova Science Pub, Inc., pp.279-298, 1999.
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[^1]: This concept was devised independently by Ch. Voigt and, also independently, by P. Jara and D. Ştefan.
|
---
abstract: 'We provide a method to describe quantum nonlocality for $n$-qubit systems. By treating the correlation function as an $n$-index tensor, we derive a generalized Bell inequality. Taking generalized Greenberger-Horne-Zeilinger (GHZ) state for example, we calculate quantum prediction under a series of measurement settings involving various angle parameters. We reveal the exact relationship between quantum prediction and the angle parameters. We show that there exists a set of optimal measurement settings and find the corresponding maximal quantum prediction for $n$-qubit generalized GHZ states. As an example, we consider an interesting situation involving only two angle parameters. Finally, we obtain a criterion for the violation of the generalized Bell inequality.'
author:
- 'Dong Ding$^{1}$, Yingqiu He$^{2}$'
- Fengli Yan$^3$
- Ting Gao$^4$
title: Violation of generalized Bell inequality and its optimal measurement settings
---
Introduction
============
To demonstrate the nonlocal quantum correlation of quantum system, in 1964, Bell [@Bell1964] proved that quantum predictions are incompatible with the local hidden variable (LHV) model by a simple logical contradiction. Inspired by this seminal paper, Clauser *et al.* [@CHSH1969] derived a correlation inequality, namely Clauser-Horne-Shimony-Holt (CHSH) inequality, which provides a way of experimentally testing the LHV theory. Then, a series of multipartite Bell-type inequalities have been proposed [@Mermin1990; @Ardehali1992; @BK1993; @WW2001; @ZB2002; @WYKO2007PRA75-032332; @GSDRS2009; @LF2012; @WZCG2013; @DHYG2015CPB; @HDYG2015EPL], where Werner-Wolf-Żukowski-Brukner (WWZB) inequalities [@WW2001; @ZB2002] are the most important because of the properties of investigating the possible connections between quantum nonlocality and entanglement for $n$-qubit systems. For two-qubit systems, the theorem of Gisin [@G91; @GP92] states that all pure entangled states violate the CHSH inequality. Then, Chen *et al.* [@CWKO2004] generalized Gisin’s theorem to three-qubit system and showed that all three-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states violate a Bell inequality for probabilities. On the other hand, given a set of standard Bell experiment settings involving two dichotomic observables for each position, there exist pure $n$-qubit entangled states that do not violate any Bell inequality [@ZBLW2002]. By now it is an open question whether Gisin’s theorem can be generalized to an arbitrary $n$-qubit system [@BCPSW2014]. In any event, the key to investigating Bell inequalities with multipartite correlation functions is to find a set of optimal measurement settings.
Optical quantum systems [@DDI2006; @Kok2007; @Pan2012; @LDJW2007; @AGACVMC2012; @CM2014; @LDQ2015; @KNM2017] are prominent candidates for testing nonlocal quantum correlations, since multiphoton entanglement, interferometry and measurement are relatively easy to perform in experiments, as long as the number of photons is not very large. In 2001, Weinfurter and Żukowski [@WZ2001] proposed a scheme to produce a superposition of four-photon GHZ state with a product state of two Bell states and investigated the features of this state by constructing a Bell-type inequality. By introducing a set of polarization correlation measurements, it has been shown that with a given set of measurement settings the maximal violation of the Bell-type inequality is $4\sqrt{2}/3$. Later, with this method, Li and Kobayashi [@LK2004] investigated another four-photon superposition state, and more recently, Ding *et al.* [@DHYG2017-4-photon] discussed a class of generic superposition of four-photon entangled states with a tunable angle parameter. Of course, a natural question is whether or not the given settings [@WZ2001; @LK2004; @DHYG2017-4-photon] are optimal or unique.
In this paper, we investigate quantum nonlocality for $n$-qubit entangled states. We first describe a method of treating correlation function as an $n$-index tensor and then derive a generalized Bell inequality. Under a set of measurement settings involving various angle parameters, we calculate quantum prediction of generalized GHZ states and show the exact relationship between quantum prediction and the angle parameters. We demonstrate that there is a set of optimal measurement settings and obtain the corresponding maximal quantum prediction for $n$-qubit generalized GHZ states. We analyze the interesting situation involving only two angle parameters in details. Finally, as an important application, a criterion for the violation of the generalized Bell inequality is provided.
A generalized Bell inequality for $n$-qubit system
==================================================
In LHV theory [@BCPSW2014], a correlation function represents an average over many runs of experiment. An $n$-partite correlation function for two alternative dichotomic measurements is generally given by $$\begin{aligned}
\label{E-LHV}
E_{\textrm{LHV}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n})
= \int d\lambda \rho(\lambda) I_{1}(\phi_1^{k_1}, \lambda) I_{2}(\phi_2^{k_2}, \lambda) \cdots I_{n}(\phi_n^{k_n}, \lambda),\end{aligned}$$ where $\lambda$ is a hidden variable and $\rho(\lambda)$ is the probability distribution function, $\phi_i^{k_i}$, $(k_i=0,1)$ indicates the local phase angle at site $i$, and $I_{i}(\phi_i^{k_i}, \lambda)=\pm 1$, $i=1,2,\cdots,n$, represents the predetermined binary outcomes of the measurements. As described by Weinfurter and Żukowski [@WZ2001], one can treat the four-photon correlation function as a four-index tensor. Here we consider an $n$-index tensor which is obtained by taking the form of a tensor product of $n$ two-dimensional real vectors $\textbf{v}_{i}^{\lambda}=(I_{i}(\phi_{i}^{0}, \lambda),I_{i}(\phi_{i}^{1}, \lambda))$, $i=1,2,\cdots,n$. That is, we define a tensor for $n$-partite system as $$\begin{aligned}
\label{E-LHV-lambda}
\hat{E}_{\textrm{LHV}}
= \int d\lambda \rho(\lambda) \textbf{v}_{1}^{\lambda} \otimes \textbf{v}_{2}^{\lambda} \otimes \cdots \otimes \textbf{v}_{n}^{\lambda}.\end{aligned}$$ Choose two orthogonal unit vectors $$\begin{aligned}
\label{}
\textbf{v}_{i}^{0}=(1,0),~~~~ \textbf{v}_{i}^{1}=(0,1).\end{aligned}$$ Since each of the vectors $\textbf{v}_{i}^{\lambda}$ can be written as $\sum_{k_{i}=0,1}I_{i}(\phi_{i}^{k_{i}}, \lambda)\textbf{v}_{i}^{k_{i}}$, one can simplify the correlation function as $$\begin{aligned}
\label{}
\hat{E}_{\textrm{LHV}} &=&
\sum_{k_1,k_2,\cdots,k_n = 0,1} E_{\textrm{LHV}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}) \textbf{v}_{1}^{k_1} \otimes \textbf{v}_{2}^{k_2} \otimes \cdots \otimes \textbf{v}_{n}^{k_n}.\end{aligned}$$
Let $\textbf{A}_{i}^{0}=(1,1)$ and $\textbf{A}_{i}^{1}=(1,-1)$. Obviously, $$\begin{aligned}
\label{}
\textbf{v}_{i}^{k_i} = \frac{1}{2}[\textbf{A}_{i}^{0}+(-1)^{k_i}\textbf{A}_{i}^{1}], ~~~ k_{i}=0,1, ~~~ i=1,2,\cdots,n.\end{aligned}$$ The correlation function can be expressed as $$\begin{aligned}
\label{}
\hat{E}_{\textrm{LHV}} &=&
\frac{1}{2^{n}} \sum_{j_1,j_2,\cdots,j_n = 0,1} \sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k} \centerdot \textbf{j}} \nonumber \\
& & \times E_{\textrm{LHV}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n})
\textbf{A}_{1}^{j_{1}} \otimes \textbf{A}_{2}^{j_{2}} \otimes \cdots \otimes \textbf{A}_{n}^{j_{n}},\end{aligned}$$ where $\textbf{k}=(k_1,k_2,\cdots,k_n)$ and $\textbf{j}=(j_1,j_2,\cdots,j_n)$. Let $$\begin{aligned}
\label{}
c_{j_1,j_2,\cdots,j_n} = \frac{1}{2^{n}} \sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k} \centerdot \textbf{j}} E_{\textrm{LHV}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n})\end{aligned}$$ be the LHV correlation coefficients. Then we have $$\begin{aligned}
\label{}
\hat{E}_{\textrm{LHV}} &=&
\sum_{j_1,j_2,\cdots,j_n = 0,1} c_{j_1,j_2,\cdots,j_n}
\textbf{A}_{1}^{j_{1}} \otimes \textbf{A}_{2}^{j_{2}} \otimes \cdots \otimes \textbf{A}_{n}^{j_{n}}.\end{aligned}$$
Let $\textbf{A}_{i}^{2}=(-1,-1)$ and $\textbf{A}_{i}^{3}=(-1,1)$, then we can use the probability to describe $n$-partite correlation function (2) as $$\label{}
\hat{E}_{\textrm{LHV}} =
\sum_{j_1,j_2,\cdots,j_n = 0,1,2,3} p_{j_1,j_2,\cdots,j_n}
\textbf{A}_{1}^{j_{1}} \otimes \textbf{A}_{2}^{j_{2}} \otimes \cdots \otimes \textbf{A}_{n}^{j_{n}},$$ where $p_{j_1,j_2,\cdots,j_n}$ is the probability of $\textbf{v}_{i}^{\lambda}$ being $\textbf {A}_i^{j_i}, i=1,2, \cdots, n, j_i=0,1,2,3$, i.e., $p_{j_1,j_2,\cdots,j_n}$ is the probability of obtaining measurement outcomes $\textbf {A}_1^{j_1}, \textbf {A}_2^{j_2}, \cdots, \textbf {A}_n^{j_n}$. Obviously, $$\sum_{j_1,j_2,\cdots,j_n = 0,1,2,3}p_{j_1,j_2,\cdots,j_n}=1.$$ Note that $\textbf A_i^{j_i+2}=-\textbf A_i^{j_i}$ with $j_i=0,1$, we obtain $$\begin{aligned}
\label{}
\hat{E}_{\textrm{LHV}} &=&
\sum_{j_1,j_2,\cdots,j_n = 0,1} (p_{j_1,j_2,\cdots,j_n}-p_{j_1+2,j_2,\cdots,j_n} - \cdots - p_{j_1,j_2,\cdots,j_{n-1},j_n+2}\nonumber\\
& &+p_{j_1+2,j_2+2,j_3,\cdots,j_n} + \cdots - p_{j_1+2,j_2+2,j_3+2,j_4,\cdots,j_n} - \cdots) \textbf{A}_{1}^{j_{1}}\otimes \textbf{A}_{2}^{j_{2}} \otimes \cdots \otimes \textbf{A}_{n}^{j_{n}}.\end{aligned}$$ Compared with Eq.(8), one gets $$\begin{aligned}
\label{}
&c_{j_1,j_2,\cdots,j_n}=&p_{j_1,j_2,\cdots,j_n}-p_{j_1+2,j_2,\cdots,j_n} - \cdots - p_{j_1,j_2,\cdots,j_{n-1},j_n+2}
+p_{j_1+2,j_2+2,j_3,\cdots,j_n} + \cdots \nonumber\\
&&- p_{j_1+2,j_2+2,j_3+2,j_4,\cdots,j_n} - \cdots.\end{aligned}$$ Then $$\begin{aligned}
\label{BI-LHV}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |c_{j_1,j_2,\cdots,j_n}| \leq 1.\end{aligned}$$ This inequality is derived from the natural generalization of the four-qubit correlation inequality [@WZ2001] to $n$-qubit systems. It is conventionally referred to as generalized Bell inequality, and can be used to test the LHV theory. It may be equivalent to WWZB inequality [@WW2001; @ZB2002] and limits the total amount of correlation allowed for the LHV theory.
On the other hand, quantum mechanically, suppose a measurement, described by measurement operator $$\label{}
\{M_{x} = |m_{x},\phi_{x}\rangle \langle m_{x},\phi_{x}|, m_x=\pm 1\}$$ is performed upon $x$-port with a detector placed at the corresponding output station, where $$\label{}
|m_{x},\phi_{x}\rangle = \frac{1}{\sqrt{2}}(|0\rangle_{x} + m_{x} \text{e}^{-\text{i}\phi_{x}} |1\rangle_{x}), ~~~ x=1,2,\cdots, n,$$ $\phi_x$ is a local phase setting chosen by each of the observers and $m_x$ represents the possible measurement result.
Consider a standard quantum correlation test, in which each observer chooses between two dichotomic measurements; that is, for each site one can label phase angle $\phi_x^{k_x}$, $k_x=0,1$ and take $m_x=\pm 1$. For an $n$-qubit entangled state $|\psi_n \rangle$, the probability of outcomes $m_{1},m_{2}, \cdots, m_{n}$ with the phase settings $\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}$ labels $p(m_1,m_2,\cdots,m_n|\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n})$ and then the correlation function can be represented by $$\begin{aligned}
\label{n-QM-correlation}
E_{\textrm{QM}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}) &=& \sum_{m_1,m_2,\cdots,m_n = \pm 1} p(m_1,m_2,\cdots,m_n|\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}) m_1m_2\cdots m_n,\end{aligned}$$ where the sum is over all possible runs of experiment. This set of operators is sufficient to describe the quantum correlation in contrast to the LHV case.
Similarly, the quantum correlation function can also be described by the $n$-fold tensor $$\begin{aligned}
\label{}
\hat{E}_{\textrm{QM}} &=&
\sum_{j_1,j_2,\cdots,j_n = 0,1} q_{j_1,j_2,\cdots,j_n}
\textbf{A}_{1}^{j_{1}} \otimes \textbf{A}_{2}^{j_{2}} \otimes \cdots \otimes \textbf{A}_{n}^{j_{n}},\end{aligned}$$ where $$\begin{aligned}
\label{}
q_{j_1,j_2,\cdots,j_n} = \frac{1}{2^{n}} \sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k} \centerdot \textbf{j}} E_{\textrm{QM}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n})\end{aligned}$$ are quantum correlation coefficients. Compared with the inequality (\[BI-LHV\]) derived from the LHV correlation, once quantum prediction $$\begin{aligned}
\label{quantum-prediction}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|\end{aligned}$$ is greater than the classical limit value 1, which means that Bell inequality is violated and quantum nonlocal correlations of quantum system occurs.
Quantum prediction for $n$-qubit generalized GHZ states
=======================================================
In a finite-dimensional Hilbert space, consider an $n$-qubit generalized GHZ state $$\begin{aligned}
\label{GGHZ}
|\psi_{n}\rangle &=& \alpha |00 \cdots 0\rangle_{12 \cdots n} + \beta |11 \cdots 1\rangle_{12 \cdots n},\end{aligned}$$ where $\alpha$ and $\beta$ are respectively the complex parameters satisfying the normalization condition $|\alpha|^{2}+|\beta|^{2}=1$. A computation reveals that the quantum correlation function determined by the dichotomic measurement parameter settings $\{\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}\}$ for the generalized GHZ state (19) is $$\begin{aligned}
\label{}
E_{\textrm{QM}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}) &=& 2 \text{Re}[\alpha\beta^{*}\text{e}^{-\text{i}(\phi_1^{k_1}+\phi_2^{k_2}+\cdots+\phi_n^{k_n})}].\end{aligned}$$ Without loss of generality we suppose that $\alpha$ and $\beta$ are real. Then the quantum prediction is $$\begin{aligned}
\label{sum-QC-1}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}| =
\frac{|\alpha\beta^{}|}{2^{n-1}} \sum_{j_1,j_2,\cdots,j_n = 0,1} |\sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k} \centerdot \textbf{j}}
\text{Re}[\text{e}^{-\text{i}(\phi_1^{k_1}+\phi_2^{k_2}+\cdots+\phi_n^{k_n})}]|.\end{aligned}$$
Let $$\alpha_l=\frac {(\phi_l^1+\phi_l^0)}{2}, ~~~~~ \beta_l=\frac {(\phi_l^1-\phi_l^0)}{2}, ~~~~~ l=1,2,\cdots,n.$$ We obtain the the quantum prediction $$\begin{aligned}
\label{sum-QC-1*}
&&\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|\nonumber\\
&=&|\alpha\beta|\{[|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|+|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|]\prod_{l=1,2,\cdots,n}(|\cos\beta_l|+|\sin\beta_l|)\nonumber\\
& &+[|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|-|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|]\prod_{l=1,2,\cdots,n}(|\cos\beta_l|-|\sin\beta_l|)\}.\end{aligned}$$ The details of derivation can be found in Appendix A. Eq. (\[sum-QC-1\*\]) is the exact quantum prediction under a series of measurement settings involving various angle parameters.
Let $\sum_{l=1,2,\cdots,n}\alpha_l=\beta_{0}$. In Appendix B, we prove that the maximum value of the quantum prediction for $n$-qubit generalized GHZ state is $$\textrm{max}\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|=|\alpha\beta|
2^{\frac {n+1}{2}},$$ which occurs at $\beta_{l}=(2k+1)\pi/4$, $k=0, \pm1,\pm2,\cdots,$ $l=0,1,2,\cdots, n$.
Up to now£¬ we have described the method to investigate the generalized Bell inequality. One of the attractive aspects of this result is that it provides a convenient way to choose the optimal measurement settings for testing Bell-type inequalities. In fact, according to this result it turns out that the previous settings [@WZ2001; @LK2004; @DHYG2017-4-photon] are optimal, but not unique.
An example
===========
As an example, we here discuss an interesting situation involving only two angle parameters. Choose a set of measurement settings satisfying $$\begin{aligned}
\label{}
&\phi_{i}^{0}=0, ~~~ \phi_{i}^{1}=\theta_{1}, ~~~ i=1,2,\cdots,l,\\
&\phi_{i}^{0}=\theta_{2}, ~~~ \phi_{i}^{1}= -\theta_{2}, ~~~ i=l+1,l+2,\cdots,n.\end{aligned}$$ In this architecture, obviously $$\begin{aligned}
\label{setting-2-angle}
\sum_{i=1,2,\cdots,n}\alpha_i=\frac{l\theta_{1}}{2}, ~~~ \beta_{i=1,2,\cdots,l}=\frac{\theta_{1}}{2}, ~~~ \beta_{i=l+1,l+2,\cdots,n}= -\theta_{2}.\end{aligned}$$ It is easily seen that for the generalized GHZ state, the quantum prediction $$\begin{aligned}
\label{Q-n-l}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}| &=& |\alpha\beta|
[
(|\text{cos}{\frac{l\theta_{1}}{2}}|+|\text{sin}{\frac{l\theta_{1}}{2}}|)(|\text{cos}{\frac{\theta_{1}}{2}}|+|\text{sin}{\frac{\theta_{1}}{2}}|)^{l} (|\text{cos}{\theta_{2}}|+|\text{sin}{\theta_{2}}|)^{n-l} \nonumber \\ & &
+(|\text{cos}{\frac{l\theta_{1}}{2}}|-|\text{sin}{\frac{l\theta_{1}}{2}}|)(|\text{cos}{\frac{\theta_{1}}{2}}|-|\text{sin}{\frac{\theta_{1}}{2}}|)^{l}
(|\text{cos}{\theta_{2}}|-|\text{sin}{\theta_{2}}|)^{n-l}
].\end{aligned}$$ This implies that the quantum prediction of the given $n$-qubit system varies as two angle parameters $\theta_{1}$, $\theta_{2}$ and the value $l$.
Here we analyze the quantum prediction for the expression (\[Q-n-l\]). For the sake of simplicity, we take $\alpha=\beta=1/\sqrt{2}$. As shown in Fig. 1, taking $n=4$ and $l=1$ for example, it is straightforward to show that in the range $0$ to $\pi/2$ the value of quantum prediction varies continuously as angle parameters $\theta_{1}$ and $\theta_{2}$, and the peak $2\sqrt{2}$ occurs at $\theta_1=\pi/2$ and $\theta_2=\pi/4$. Indeed, similar results can be found for any value of $n$ and $l$. This allows a further simplification in which one of two angle parameters is fixed, and then one can plot quantum prediction as a function of the remaining phase angle. In this way, one may, therefore, optimize the measurement settings. We also take $n=4$ for example. Setting $\theta_{1}=\pi/2$ and taking $\theta=\theta_{2}$, as shown in Fig. 2, this plot shows the quantum prediction varies as the angle parameter $\theta$ over a range from $0$ to $\pi/2$. By comparing the curves with different values of $l$, one sees immediately that the maximum value $2\sqrt{2}$ occurs at $\theta_{2}=\pi/4$ for $l=1$ and $l=3$, respectively.
To sum up, there are three experimentally significant points in our architecture. () The optimal measurement settings are $\theta_1=\pi/2$ and $\theta_2=\pi/4$ with all odd $l$, and the maximum of quantum prediction is $2^{(n-1)/2}$. () For $l=0$, it is not the optimal measurement settings and its maximal violation is $2^{(n-2)/2}$ with $\theta_{2}=\pi/4$. () For $l=n$, with odd $l$ the maximum of quantum prediction occurs at the optimal measurement settings $\theta_{1}=\pi/2$, while for even $l$ it is not the optimal settings. In fact, this result is easy to check directly by inserting $\theta_1=\pi/2$ and $\theta_2=\pi/4$ into measurement settings (\[setting-2-angle\]). Obviously, with an odd $l$, we have $\sum_{i=1,2,\cdots,n}\alpha_i={(2k+1)\pi}/{4}$, $\beta_{i=1,2,\cdots,l}={\pi}/{4}$ and $\beta_{i=l+1,l+2,\cdots,n}= -{\pi}/{4}$, then the maximum of quantum prediction occurs.
Generalization and application
==============================
We now consider the generalized GHZ states with complex coefficients $\alpha$ and $\beta$. Let $\alpha\beta^{*}=|\alpha\beta^{*}| \text{e}^{-\text{i}\phi}$. Then, quantum correlation function can be rewritten as $$\begin{aligned}
\label{}
E_{\textrm{QM}}(\phi_1^{k_1},\phi_2^{k_2},\cdots,\phi_n^{k_n}) &=& 2 |\alpha\beta^{*}|\text{cos} (\phi + \phi_1^{k_1}+\phi_2^{k_2}+\cdots+\phi_n^{k_n}).\end{aligned}$$ A similar calculation yields the quantum prediction $$\begin{aligned}
\label{QP-complex}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|
&=& |\alpha\beta^{*}| [(|\cos(\phi+\sum_{l=1,2,\cdots,n}\alpha_l)|+|\sin(\phi+\sum_{l=1,2,\cdots,n}\alpha_l)|)\prod_{l=1,2,\cdots,n}(|\cos\beta_l|+|\sin\beta_l|)\nonumber\\
& & +(|\cos(\phi+\sum_{l=1,2,\cdots,n}\alpha_l)|-|\sin(\phi+\sum_{l=1,2,\cdots,n}\alpha_l)|)\prod_{l=1,2,\cdots,n}(|\cos\beta_l|-|\sin\beta_l|)].\end{aligned}$$ When $\phi+\sum_{l=1,2,\cdots,n}\alpha_l$ and $\beta_l$ are $(2k+1)\pi/4$ $(k=0,\pm1,\pm2,\cdots)$, the maximum value $|\alpha\beta^{*}| 2^{{(n+1)}/{2}}$ occurs. According to this result, the generalized Bell inequality will be violated conditioned on $$\begin{aligned}
\label{}
|\alpha\beta^{*}| > 2^{-\frac{n+1}{2}}.\end{aligned}$$
As an important application, this provides a criterion for the violation of the generalized Bell inequality derived from the LHV theory; that is, under the present optimal measurement settings, for $|\alpha\beta^{*}| \leq 2^{-(n+1)/2}$ this inequality can not be violated. Especially, for real $\alpha$ and $\beta$, if we assume that $\alpha=\text{cos}\xi$ and $\beta=\text{sin}\xi$, then, consequently, for $\text{sin}2\xi \leq 2^{-(n-1)/2}$ the inequality can not be violated, which is consistent with the result in [@ZBLW2002] with an arbitrary odd $n$.
Discussion and summary
======================
Having investigated the generalized Bell inequality and its maximal violation, we now discuss the possible experimental realization using optical quantum technologies [@DDI2006; @Kok2007; @Pan2012]. One aspect of this framework is the preparation of multiphoton entangled states. With linear optics and multiphoton interferometry, more recently, Pan *et al.* [@Pan-8-photon-GHZ-2012; @Pan-10-photon-GHZ-2016] successively reported two schemes of observing eight-photon and ten-photon entanglement in experiment. By combining pairs of photons emitting from the parametric down-conversion processes [@SPDC1970; @PDC1995], eight-photon or ten-photon GHZ state can be engineered step by step. So, with currently available techniques, here one may produce the generalized GHZ state by considering a tunable angle parameter, which is settled by the orientation of a wave plate [@WSKPGW2008; @LL-NJP2009; @DHYG2017-4-photon]. Another way of preparing multiphoton entangled states is to utilize cross-Kerr nonlinearities [@Imoto1985; @SI1996; @LI2000; @NM2004; @RV2005; @Barrett2005; @MNBS2005; @Kok2008; @HDYG2015OE; @HDYG2016SR]. For example, we may consider an entangler of multiphoton GHZ states proposed by Ding *et al.* [@DYG2014]. By resetting an input state $(\alpha |H\rangle_1+\beta|V\rangle_1)\otimes(|H\rangle_2+|V\rangle_2)\otimes \cdots\otimes(|H\rangle_n+|V\rangle_n)/2^{(n-1)/2}$, after the homodyne measurement on the probe beam the entangler is capable of preparing an $n$-photon generalized GHZ state. The other aspect of the present architecture is concerned with performing polarization analysis. Similar to the four-photon entanglement experiment [@EGBKZW2003; @GBEKW2003; @BEKGWGHBLS2004; @XLG2006], polarization analysis in various bases can be performed in each of the $n$ outputs via quarter- and half-wave plates in front of polarizing beam splitters. Taking the settings (\[setting-2-angle\]) (with $l=1$, $\theta_1=\pi/2$ and $\theta_2=\pi/4$) for example, the observer at site 1 switches analysis angle between 0 and $\pi/2$, and the other observers at sites $2,3,\cdots,n$ switch analysis angles between $\pm \pi/4$. When the $n$ photons are detected by single photon avalanche detectors, an $n$-fold coincidence detection can be registered. With these registrations one can investigate the violation of the generalized Bell inequality.
In summary, we have shown a method to deal with quantum nonlocality for $n$-qubit systems. Calculating the correlation function as an $n$-index tensor leads to a generalized Bell inequality. In this architecture, for an arbitrary $n$-qubit generalized GHZ state, under a set of experimental settings with various angle parameters, we have obtained the exact relationship between the amount of violation of the generalized Bell inequality and the variable angle parameters. By calculating the value of quantum prediction, as a result, we find a set of optimal measurement settings. Furthermore, as an example, we have shown a simplified description of $n$-qubit system involving two angle parameters. The main result is that when $l$ is odd there exists a set of optimal measurement settings, $\theta_1=\pi/2$ and $\theta_2=\pi/4$, and otherwise it does not exist. Finally, we calculate the quantum prediction for the generalized GHZ state with complex coefficients $\alpha$ and $\beta$. With the modified optimal measurement settings, an important criterion for the violation of the generalized Bell inequality have been demonstrated. Indeed, it is an interesting and useful fact in experimental tests of multipartite Bell-type inequalities.
This work was supported by the National Natural Science Foundation of China under Grant Nos: 11475054, 11547169, the Hebei Natural Science Foundation of China under Grant Nos: A2016205145, A2018205125, the Fundamental Research Funds for the Central Universities of Ministry of Education of China under Grant Nos: 3142017069, 3142015044, the Foundation for High-Level Talents of Chengde Medical University under Grant No: 201701, the Research Project of Science and Technology in Higher Education of Hebei Province of China under Grant No: Z2015188.
In order to compute the quantum prediction $$\begin{aligned}
\label{sum-QC-1}
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}| =
\frac{|\alpha\beta^{}|}{2^{n-1}} \sum_{j_1,j_2,\cdots,j_n = 0,1} |\sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k} \centerdot \textbf{j}}
\text{Re}[\text{e}^{-\text{i}(\phi_1^{k_1}+\phi_2^{k_2}+\cdots+\phi_n^{k_n})}]|,\end{aligned}$$ we first let $$\begin{aligned}
\label{A-j}
A(\textbf{j}) = \sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{\textbf{k}{} \centerdot \textbf{j}} \text{e}^{ -\text{i} (\phi_1^{k_1}+\phi_2^{k_2}+\cdots+\phi_n^{k_n})}.\end{aligned}$$
A simple calculation shows that $$\begin{aligned}
A(\textbf{j})
&=& \sum_{k_1,k_2,\cdots,k_n = 0,1} (-1)^{k_1j_1} \text{e}^{ -\text{i} \phi_1^{k_1}}(-1)^{k_2j_2} \text{e}^{ -\text{i} \phi_2^{k_2}}\cdots(-1)^{k_nj_n} \text{e}^{ -\text{i} \phi_n^{k_n}}\nonumber\\
&=& \prod_{l=1,2,\cdots,n}[\text{e}^{-\text {i}\phi_l^0} + (-1)^{j_l}\text{e}^{-\text{i}\phi_l^1}]\nonumber\\
&=& \prod_{l=1,2,\cdots,n}\text{e}^{-\text{i}\frac {(\phi_l^1+\phi_l^0)}{2}}[\text{e}^{\text{i}\frac {(\phi_l^1-\phi_l^0)}{2}} + (-1)^{j_l}\text{e}^{-\text{i}\frac {(\phi_l^1-\phi_l^0)}{2}}]\nonumber\\
&=& \prod_{l=1,2,\cdots,n}\text{e}^{-\text{i}\frac {(\phi_l^1+\phi_l^0)}{2}}2[\text{cos}\frac {(\phi_l^1-\phi_l^0)}{2}]^{1-j_l}[\text{i}\text{sin}\frac {(\phi_l^1-\phi_l^0)}{2}]^{j_l}.\end{aligned}$$ It follows immediately the definition of $\alpha_l$, $\beta_l$ in Eq.(22) that $$A(\textbf{j}) = 2^n\text{e}^{\text{i}\sum_{l=1,2,\cdots,n}(-\alpha_l+{\frac{\pi}{2}}j_l)}\prod_{l=1,2,\cdots,n}[\text{cos}\beta_l]^{1-j_l}[\text{sin}\beta_l]^{j_l}.$$ Taking the real part of $A(\textbf{j})$, there is $$\text{Re} [A(\textbf{j})] = 2^n\text{cos}[\sum_{l=1,2,\cdots,n}(-\alpha_l+{\frac{\pi}{2}}j_l)]\prod_{l=1,2,\cdots,n}[\text{cos}\beta_l]^{1-j_l}[\text{sin}\beta_l]^{j_l}.$$
By calculation, one derive $$\begin{aligned}
& & \sum_{\textbf{j}}|\text{Re} [A(\textbf{j})]|\nonumber\\
& =& 2^n\sum_{\textbf{j}}|\text{cos}[\sum_{l=1,2,\cdots,n}(-\alpha_l+{\frac{\pi}{2}}j_l)]|\prod_{l=1,2,\cdots,n}|\text{cos}\beta_l|^{1-j_l}|\text{sin}\beta_l|^{j_l}\nonumber\\
& =& 2^n\sum_{\textbf{j}}|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)\cos(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)+
\sin(\sum_{l=1,2,\cdots,n}\alpha_l)\sin(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)|\prod_{l=1,2,\cdots,n}|\text{cos}\beta_l|^{1-j_l}|\text{sin}\beta_l|^{j_l} \nonumber\\
&=&2^n|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|\sum_{\textbf{j}, \sum_{l=1,2,\cdots,n}j_l=\text{odd}}\prod_{l=1,2,\cdots,n}|\text{cos}\beta_l|^{1-j_l}|\text{sin}\beta_l|^{j_l}
\nonumber\\ & &
+2^n|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|\sum_{\textbf{j}, \sum_{l=1,2,\cdots,n}j_l=\text{even}}\prod_{l=1,2,\cdots,n}|\text{cos}\beta_l|^{1-j_l}|\text{sin}\beta_l|^{j_l},\end{aligned}$$ where the following identities $$\sum_{l=1,2,\cdots,n}j_l=\text{odd}, ~~~~~ \sin(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)=\pm 1, ~~~~~~ \cos(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)=0,\\$$ $$\sum_{l=1,2,\cdots,n}j_l=\text{even}, ~~~~~ \sin(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)=0, ~~~~~~ \cos(\frac{\pi}{2}\sum_{l=1,2,\cdots,n}j_l)=\pm 1\\$$ are used in the last equality.
Then we obtain $$\begin{aligned}
\label{A-Re-A}
& & \sum_{\textbf{j}}|\text{Re} [A(\textbf{j})]|\nonumber\\
&=&2^{n-1}(|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|+|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|)\prod_{l=1,2,\cdots,n}(|\cos\beta_l|+|\sin\beta_l|)\nonumber\\
& &+2^{n-1}(|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|-|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|)\prod_{l=1,2,\cdots,n}(|\cos\beta_l|-|\sin\beta_l|).\end{aligned}$$ Therefore, the quantum prediction is given by $$\begin{aligned}
\nonumber
&&\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|\nonumber\\
&=&|\alpha\beta|\{[|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|+|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|]\prod_{l=1,2,\cdots,n}(|\cos\beta_l|+|\sin\beta_l|)\nonumber\\
& &+[|\cos(\sum_{l=1,2,\cdots,n}\alpha_l)|-|\sin(\sum_{l=1,2,\cdots,n}\alpha_l)|]\prod_{l=1,2,\cdots,n}(|\cos\beta_l|-|\sin\beta_l|)\},\end{aligned}$$ as desired.
By the definition $\sum_{l=1,2,\cdots,n}\alpha_l=\beta_{0}$, the quantum prediction $$\begin{aligned}
\nonumber
\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|=|\alpha\beta|\{\prod_{l=0,1,2,\cdots,n}(|\cos\beta_l|+|\sin\beta_l|)+\prod_{l=0,1,2,\cdots,n}(|\cos\beta_l|-|\sin\beta_l|)\}.\end{aligned}$$ Obviously, the quantum prediction is the function of $n+1$ independent variables $\beta_{0}, \beta_{1},\cdots, \beta_{n} \in [0,\pi]$ by virtue of the periodic nature of the absolute values of sine and cosine functions. Furthermore, one can divide each of variables $\beta_{0}, \beta_{1},\cdots, \beta_{n} \in [0,\pi]$ into two sections $[0,{\pi}/{2}]$ and $[{\pi}/{2}, \pi]$.
Note that for $\beta_{l} \in [0, {\pi}/{2}]$, $$|\cos \beta_{l} |+|\sin \beta_{l} |= \sqrt 2 \sin(\beta_{l} +\frac {\pi}{4}),$$ $$|\cos \beta_{l} |-|\sin \beta_{l} |= \sqrt 2 \cos(\beta_{l} +\frac {\pi}{4}).$$
While for $\beta_{l} \in [{\pi}/{2},\pi]$, $$|\cos \beta_{l}|+|\sin \beta_{l}|= \sqrt 2 \sin(\beta_{l}-\frac {\pi}{4})= \sqrt 2 \sin[(\beta_{l}-\pi/2)+\frac {\pi}{4}],$$ $$|\cos \beta_{l}|-|\sin \beta_{l}|= -\sqrt 2 \cos(\beta_{l}-\frac {\pi}{4})= -\sqrt 2 \cos[(\beta_{l}-\pi/2)+\frac {\pi}{4}].$$
We use $\beta_{l}^0$, $\beta_{l}^1$ to denote $\beta_{l} \in [0, {\pi}/{2}]$, $\beta_{l} \in [{\pi}/{2},\pi]$, respectively. Thus, the $n+1$ intervals $\beta_l$ in $[0, {\pi}]$, can be divided into $2^{n+1}$ sections with $\beta_l^0$ in $[0, {\pi}/2]$ and $\beta_l^1$ in $[{\pi}/2, \pi]$. Therefore, in the section where the number of $\beta_l^0$ in $[0, {\pi}/2]$ is $(n+1-m)$, while the number of $\beta_l^1$ in $[0, {\pi}/2]$ is $m$, the quantum prediction should be $$\begin{aligned}
\label{aa}
&&\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|\nonumber\\
&=&|\alpha\beta|
2^{\frac {n+1}{2}}\{\prod_{l=0}^{m-1}\sin[(\beta_{l}^1-\pi/2)+\frac {\pi}{4}]\prod_{l=m}^{n}\sin(\beta_l^0+\frac {\pi}{4})\nonumber\\
&&+(-1)^{m}\prod_{l=0}^{m-1}\cos[(\beta_{l}^1-\pi/2)+\frac {\pi}{4}]\prod_{l=m}^{n}\cos(\beta_l^0+\frac {\pi}{4})\}.\end{aligned}$$ Obviously, in this section Eq. (\[aa\]) is equivalent to $$\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|=|\alpha\beta|
2^{\frac {n+1}{2}}[
\prod_{l=0}^{n}\sin(\delta_l+\frac {\pi}{4})+(-1)^{m}\prod_{l=0}^{n}\cos(\delta_l+\frac {\pi}{4})], ~~~ \delta_l\in [0, {\pi}/2].$$ Let $$F=|\alpha\beta|2^{\frac {n+1}{2}}[
\prod_{l=0}^{n}\sin(\delta_l+\frac {\pi}{4})+\prod_{l=0}^{n}\cos(\delta_l+\frac {\pi}{4})], ~~~ \delta_l\in [0, {\pi}/2];$$ $$G=|\alpha\beta|2^{\frac {n+1}{2}}[
\prod_{l=0}^{n}\sin(\delta_l+\frac {\pi}{4})-\prod_{l=0}^{n}\cos(\delta_l+\frac {\pi}{4})], ~~~ \delta_l\in [0, {\pi}/2].$$
Then, maximum value of the quantum prediction $$\textrm{max}\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|=\textrm{max} \{\textrm{max} F, \textrm{max} G\}.$$
Solving sets of differential equations $\partial F/\partial \delta_{l} =0$ produces $\delta_{l}=\pi/4$, $l=0,1,2,\cdots, n$. With these values it is obvious that the $(n+1)\times (n+1)$ matrix $(\partial^{2} F/{\partial {\delta_{i}}{\partial\delta_{j}}})$ is negative. So we have $$\textrm{max} F=|\alpha\beta|2^{\frac {n+1}{2}}.$$ Similarly, there is $$\textrm{max} G=|\alpha\beta|2^{\frac {n+1}{2}},$$ while also occurs at $\delta_{l}=\pi/4$, $l=0,1,2,\cdots, n$, that means $\beta_{l}^0=\pi/4, \beta_{l}^1=\pi/4+\pi/2$.
Together with the periodicity condition, we arrive at the conclusion that the maximum value of the quantum prediction is $$\textrm{max}\sum_{j_1,j_2,\cdots,j_n = 0,1} |q_{j_1,j_2,\cdots,j_n}|=|\alpha\beta|
2^{\frac {n+1}{2}},$$ which occurs at $\beta_{l}=(2k+1)\pi/4$, $k=0, \pm1,\pm2,\cdots,$ $l=0,1,2,\cdots, n$.
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abstract: 'After discussing the key idea underlying the Maxwell’s Demon ensemble, we employ this idea for calculating fluctuations of ideal Bose gas condensates in traps with power-law single-particle energy spectra. Two essentially different cases have to be distinguished. If the heat capacity remains continuous at the condensation point in the large-$N$-limit, the fluctuations of the number of condensate particles vanish linearly with temperature, independent of the trap characteristics. If the heat capacity becomes discontinuous, the fluctuations vanish algebraically with temperature, with an exponent determined by the trap. Our results are based on an integral representation that yields the solution to both the canonical and the microcanonical fluctuation problem in a singularly transparent manner.'
address: 'Fachbereich Physik der Philipps-Universität, Renthof 6, D-35032 Marburg, Germany'
author:
- Siegfried Grossmann and Martin Holthaus
title: |
Maxwell’s Demon at work:\
Two types of Bose condensate fluctuations in power-law traps
---
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According to grand canonical statistics, the root-mean-square fluctuations $\delta N_\nu$ of the occupation numbers $N_\nu$ of an ideal Bose gases’s $\nu$-th single-particle state are given by \[\[LandauLifshitz59\],\[Pathria85\]\] $$\left(\delta N_\nu\right)^2 = N_\nu(N_\nu + 1) \; .
\label{FLUC}$$ This expression follows without any approximation from the grand canonical approach, but it faces a severe problem when applied to the fluctuations $\delta N_0$ of the ground state occupation number $N_0$ of [*isolated*]{} Bose gases: if the temperature approaches zero, all $N$ particles of an isolated Bose gas occupy the ground state, so that the actual fluctuations vanish, whereas Eq. (\[FLUC\]) predicts fluctuations $\delta N_0$ of the order $N$. This seems to be one of the most important examples where the different statistical ensembles can not be regarded as equivalent. When computing low-temperature fluctuations of the ground state occupation number for isolated Bose gases, one therefore has to give up the convenient grand canonical point of view, and to resort to a microcanonical treatment.
Although this problem had been well recognized and discussed some time ago \[\[FujiwaraEtAl70\],\[ZiffEtAl77\]\], tools for computing the microcanonical fluctuations $\delta N_0$ have been developed only recently \[\[GajdaRzazewski97\]\], spurred by the progress in preparing Bose–Einstein condensates of alkali atoms in magnetic traps \[\[AndersonEtAl95\],\[DavisEtAl95\],\[BradleyEtAl97\]\]. A particularly instructive model system for illustrating the microcanonical approach to fluctuations $\delta N_0$ is provided by $N$ ideal Bosons trapped in a one-dimensional harmonic potential. Since quasi one-dimensional harmonic trapping potentials can be realized as limiting cases of strongly anisotropic three-dimensional traps \[\[KetterleDruten96\],\[DrutenKetterle97\]\], this system is not merely of academic interest. The value of the model lies in the fact that it allows one to map the problem of evaluating the microcanonical statistics to problems also arising in analytic number theory, because the number of microstates accessible at some excitation energy $E$ equals the number of partitions of the integer $n = E/(\hbar\omega)$ into no more than $N$ summands, with $\omega$ being the oscillator frequency. Using the appropriate asymptotic formulae from partition theory, one finds that the microcanonical fluctuations $\delta N_0$ for this model system vanish linearly with temperature $T$ \[\[GH96\],\[GH97a\]\]: $$\delta N_0 \; \approx \; \frac{\pi}{\sqrt 6} \frac{k_B T}{\hbar\omega}
\qquad {\mbox{for}} \qquad
T \ll T_0^{(1)} \equiv \frac{\hbar\omega}{k_B} \frac{N}{\ln N} \; ,
\label{ODOF}$$ where $k_B$ is the Boltzmann constant, and $T_0^{(1)}$ denotes the temperature below which the ground state occupation becomes significant. As illustrated in Fig. 1, which compares the relative microcanonical fluctuations $\delta N_0/N$ to the corresponding grand canonical fluctuations and to the approximation (\[ODOF\]), for $N = 10^6$ particles, this approximation is quite good indeed. The very same result (\[ODOF\]) has also been obtained by Wilkens \[\[Wilkens96\]\] within a [*canonical*]{} approach, that is, for a trap in contact with a heat bath.
[Fig. 1. Full red line: microcanonical fluctuations $\delta N_0/N$ for a system of $N = 10^6$ ideal Bose particles trapped by a one-dimensional harmonic potential \[\[GH96\]\]. The temperature $T_0 \equiv T_0^{(1)}$ denotes the characteristic temperature below which the ground state occupation becomes significant, see Eq. (\[ODOF\]). Black short-dashed line: grand canonical fluctuations for the same system. Blue dashed line: low-temperature approximation provided by Eq. (\[ODOF\]).]{}
How can one generalize this finding to other trap types? A rather interesting suggestion has been made by Navez [*et al.*]{} \[\[NavezEtAl97\]\]. Denoting, for an ideal $N$-particle Bose gas in some arbitrary trap, the number of microstates with [*exactly*]{} $N_{\rm ex}$ excited particles as $\Phi(N_{\rm ex}|E)$, so that the total number of microstates accessible at the given energy $E$ reads $\Omega(E|N) = \sum_{N_{\rm ex}=0}^{N} \Phi(N_{\rm ex}|E)$, these authors consider the generating function $$\Upsilon(z,E) \; = \sum_{N_{\rm ex}=0}^{\infty} \!
z^{N_{\rm ex}} \, \Phi(N_{\rm ex}|E) \; .
\label{MWDF}$$ This function involves $\Phi(N_{\rm ex}|E)$ even for $N_{\rm ex} > N$, which appears to be unphysical: after all, the excitation energy $E$ can not be distributed over more than the $N$ particles. However, [*provided*]{} the microcanonical distributions for finding $N_{\rm ex}$ out of $N$ particles in an excited trap state, $$p_{\rm ex}(N_{\rm ex}|E) \; = \;
\frac{\Phi(N_{\rm ex}|E)}{\Omega(E|N)} \; ,
\qquad N_{\rm ex} = 1,2,\ldots,N \; ,
\label{DIST}$$ are strongly peaked around some value $\overline{N}_{\rm ex} \ll N$, which should be the case for temperatures well below the onset of Bose–Einstein condensation, we will have $\Phi(N_{\rm ex}|E)/\Omega(E|N) \approx 0$ for $N_{\rm ex} > N$. In that case the generating function (\[MWDF\]) would be quite useful, since one could obtain the microcanonical expectation value $\langle N_0 \rangle$ for the ground state occupation number, and its fluctuation, from $$N - \langle N_0 \rangle \; = \;
\left. z\frac{\partial}{\partial z}
\ln \Upsilon(z,E) \right|_{z=1}
\qquad {\mbox{and}} \qquad
\left(\delta N_0\right)^2 \; = \;
\left. \left(z\frac{\partial}{\partial z}\right)^2
\ln \Upsilon(z,E) \right|_{z=1} \; ,$$ respectively \[\[NavezEtAl97\]\]. The proviso can be formulated in more intuitive terms: the required well-peakedness of the distributions (\[DIST\]) means that those microstates where the energy $E$ is actually spread out over all $N$ particles carry only negligible statistical weight, so that the overwhelming majority of all microstates leaves a fraction of the particles in the ground state, forming the Bose condensate. Then the restriction on the number of microstates caused by the fact that there is only a finite number $N$ of particles becomes meaningless, so that, loosely speaking, “the system has no chance to know how many particles the condensate consists of”. But if this is the case, i.e., if the system’s properties become insensitive to the actual number of particles contained in the condensate, then one can act as if the condensate particles constituted an [*infinite*]{} reservoir. Thus, the generating function (\[MWDF\]) may be regarded as the partition function of a rather unusual ensemble, consisting of the excited-states subsystems of Bose gases that exchange particles with the ground state “reservoirs” without exchanging energy. Since such an exchange process, if performed by hand, requires a genius who is able to separate the hot, excited particles from the cold ones in the ground state, this new ensemble has been called the “Maxwell’s Demon ensemble” \[\[NavezEtAl97\]\].
But can we rely on Maxwell’s Demon, that is, does the proviso hold? This question needs to be answered first. A strong argument in favour of the Maxwell’s Demon ensemble has already been provided by the approximation (\[ODOF\]) to the low-temperature fluctuations for a Bose gas in a one-dimensional oscillator potential: these fluctuations are [*independent of the total particle number*]{} $N$, as they should be if the system really has no knowledge of the number of condensate particles, and thus of $N$. The awe-inspiring agreement with the actual microcanonical fluctuations depicted in Fig. 1 leaves no doubt that this approximation [*is*]{} reliable. To further substantiate the new ensemble, we also consider the microcanonical fluctuations $\delta N_0$ for an ideal Bose gas trapped by a three-dimensional isotropic harmonic oscillator potential \[\[GH97b\]\]. The numbers $\Omega(E|N)$ of microstates for some given excitation energy $E = n\hbar\omega$ can then be obtained from the canonical $N$-particle partition function $$Z_N(\beta) \; = \; \sum_{n=0}^{\infty} e^{-n\beta\hbar\omega} \,
\Omega(n\hbar\omega|N) \; ,
\label{CNPF}$$ which, in turn, can be calculated numerically with the help of the recursion relation \[\[BorrmannFranke93\],\[Eckhardt97\],\[WilkensWeiss97\]\] $$Z_N(\beta) \; = \; \frac{1}{N}\sum_{k=1}^{N}
Z_1(k\beta) Z_{N-k}(\beta) \; .$$ As usual, $\beta = 1/(k_BT)$ denotes the inverse temperature. By means of numerical saddle-point inversions of Eq. (\[CNPF\]), we compute the desired numbers $\Omega(E|N_{\rm ex})$ for $N_{\rm ex}$ ranging from $1$ to $N$ \[\[GH97b\]\], and get the differences $$\Phi(N_{\rm ex}|E) \; = \; \Omega(E|N_{\rm ex}) - \Omega(E|N_{\rm ex}-1)
\label{MDIF}$$ that determine the microcanonical distributions (\[DIST\]). Some of these distributions are displayed in Fig. 2, for $N = 1000$ and several “low” temperatures. What we find is exactly what is needed for Maxwell’s Demon: the distributions are well peaked for temperatures below the onset of condensation, and remarkably close to Gaussians \[\[GH97a\]\]. It is then no surprise that the corresponding microcanonical low-temperature fluctuations $\delta N_0$, obtained from the widths of these distributions, are — as long as a condensate exists! — once again independent of $N$, as exemplified in Fig. 3 for $N = 200$, $500$, and $1000$. As discussed above, it is precisely this $N$-independence, expressed mathematically by the appearance of the upper summation bound “$\infty$” rather than “$N$” in Eq. (\[MWDF\]), that lies at the bottom of the Maxwell’s Demon ensemble. But whereas this $N$-independence is, by construction, [*put into*]{} this ensemble, it has [*come out*]{} here as the result of a truly microcanonical calculation \[\[GH97a\],\[GH97b\]\] that works with the actual $N$, not with $\infty$.
[Fig. 2. Microcanonical probability distributions $p_{\rm ex}(N_{\rm ex}|n)$ for finding $N_{\rm ex}$ out of $N=1000$ ideal Bose particles, trapped by a three-dimensional isotropic harmonic potential, excited when the total excitation energy $E$ is $n \cdot \hbar\omega$, with $\omega$ denoting the oscillator frequency. The number $n$ determines the temperature $T$. The normalized temperatures $T/T_0$ corresponding to the blue, Gaussian-like distributions range from $0.3$ to $0.9$ (left to right, in steps of $0.1$); $T_0 = (\hbar\omega/k_B)(N/\zeta(3))^{1/3}$. Due to finite-$N$-effects, the condensation temperature is lowered from $T_0$ to about $0.93 \, T_0$. The temperature corresponding to the rightmost, red distribution is $T = 0.95 \, T_0$, lying slightly above the condensation point.]{}
[Fig. 3. Microcanonical fluctuations $\delta N_0$ for $N = 200$, $500$, and $1000$ ideal Bose particles trapped by a three-dimensional, isotropic harmonic potential. The fluctuations are maximal close to the respective condensation points. These maximal fluctuations scale approximately as $\sqrt{N}$, cf. Eqs. (\[CONT\]) and (\[MFL1\]). Note that the low-temperature fluctuations for all three systems agree perfectly, thus demonstrating the $N$-independence of $\delta N_0$ below the condensation point.]{}
Having thus gained confidence in the abilities of Maxwell’s Demon, we now set it to work in order to compute condensate fluctuations $\delta N_0$. To this end, we consider ideal Bose gases in $d$-dimensional traps with arbitrary single-particle energies ${\varepsilon_{\nu}}$; we stipulate $\varepsilon_0 = 0$. Denoting the grand canonical partition function by $\Xi(z,\beta)$, we base our analysis on its “excited” part $\Xi_{\rm ex}(z,\beta) \equiv (1 - z)\Xi(z,\beta)$. Since, by virtue of Eq. (\[MDIF\]), $$\Xi_{\rm ex}(z,\beta) \; = \sum_{N_{\rm ex}=0}^{\infty} \! z^{N_{\rm ex}} \!
\sum_{E} \, \Phi(N_{\rm ex}|E) \, e^{-\beta E} \; ,$$ this function has the decisive property $$\left.
\left(z\frac{\partial}{\partial z}\right)^{\!k} \!
\Xi_{\rm ex}(z,\beta) \right|_{z=1}
= \; \sum_E \left(
\sum_{N_{\rm ex}=0}^{\infty} N_{\rm ex}^k \, \Phi(N_{\rm ex}|E)
\right) e^{-\beta E} \; \equiv \; M_k(\beta) \; ,
\label{RICH}$$ i.e., it yields directly the non-normalized [*canonical*]{} moments $M_k(\beta)$, and generates the [*microcanonical*]{} moments $$\mu_k(E) \; \equiv \;
\sum_{N_{\rm ex}=0}^{\infty} \, N_{\rm ex}^k \, \Phi(N_{\rm ex}|E)
\qquad {\mbox{with}} \qquad k = 0,1,2,\ldots \; .
\label{MIMO}$$ We then employ the Maxwell’s Demon approximation: as long as there is a condensate, these moments (with $N_{\rm ex}$ ranging from $0$ to $\infty$) approximate the true moments of the physical set $\{ \, \Phi(N_{\rm ex}|E) \, \}$ (where the number $N_{\rm ex}$ of excited particles can not exceed the total particle number $N$); in this approximation one has the identity $\mu_0(E) = \Omega(E|N)$ \[\[GH97b\]\]. Now the calculations within the canonical ensemble become remarkably simple. The canonical expectation value $\langle N_{\rm ex} \rangle = N - \langle N_0 \rangle$ of the number of excited particles is given by $$\langle N_{\rm ex} \rangle \; = \; \frac{M_1(\beta)}{M_0(\beta)} \; ;$$ the canonical condensate fluctuations $\left(\delta N_0\right)^2_{cn} \; = \; \left(\delta N_{\rm ex}\right)^2_{cn}$ follow from $$\left(\delta N_0\right)^2_{cn} \; = \; \frac{M_2(\beta)}{M_0(\beta)}
- \left( \frac{M_1(\beta)}{M_0(\beta)} \right)^2 \; .$$ Without any further approximation, these expressions can be rewritten as complex integrals: $$\langle N_{\rm ex} \rangle \; = \;
\frac{1}{2\pi i} \int_{\tau-i\infty}^{\tau+i\infty} \!\! {\mbox d}t \,
\Gamma(t) \, Z(\beta,t) \, \zeta(t)
\label{FUN1}$$ and $$\left(\delta N_0\right)^2_{cn} \; = \;
\frac{1}{2\pi i} \int_{\tau-i\infty}^{\tau+i\infty} \!\! {\mbox d}t \,
\Gamma(t) \, Z(\beta,t) \, \zeta(t-1) \; ,
\label{FUN2}$$ where $\Gamma(t)$ and $\zeta(t)$ denote the Gamma function and Riemann’s Zeta function, respectively. All the information about the specific trap under consideration is embodied in its spectral Zeta function $$Z(\beta,t) \; = \;
\sum_{\nu=1}^{\infty} \frac{1}{(\beta\varepsilon_\nu)^t} \;$$ where the sum runs over the trap spectrum, excluding the ground state energy $\varepsilon_0$ $=$ $0$. The real number $\tau$ in Eqs. (\[FUN1\]) and (\[FUN2\]) has to be chosen such that the path of integration up the complex $t$-plane sees all poles to its left.
So far, the analysis is quite general. We now specialize the further deliberations to ideal Bose gases in $d$-dimensional traps with power-law single-particle spectra $$\varepsilon_{\{\nu_i\}} = \Delta \sum_{i=1}^{d} c_i \nu_i^\sigma \; ,
\qquad \nu_i = 0,1,2,\ldots \; , \qquad \sigma > 0 \; ,
\label{SPSP}$$ where the dimensionless coefficients $c_i$ characterize the trap’s anisotropy, normalized such that the lowest $c_i$ is unity; the characteristic energy $\Delta$ measures the gap between the ground state and the first excited state, and the exponent $\sigma$ is determined by the potential’s shape. Such systems have been studied first by de Groot [*et al.*]{} \[\[deGrootEtAl50\]\]; we have adopted here the notation also employed by Wilkens and Weiss \[\[WilkensWeiss97\]\].
If we consider $N$-asymptotically large systems and disregard finite-$N$-effects, that is, if we focus on gases consisting of at least some $10^5$ particles, say, then a good approximation to the density of states is provided by $$\rho(E) \; = \; \frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)}
\left(\frac{E}{\Delta}\right)^{\!d/\sigma-1} \!\frac{1}{\Delta}
\qquad {\mbox{with}} \qquad
A \; \equiv \; \frac{\Gamma\!\left(\frac{1}{\sigma}+1\right)^d}
{\left(\prod_{i=1}^d c_i\right)^{\!1/\sigma}} \; .$$ Using this density, and assuming that the anisotropy coefficients $c_i$ are not too different from each other \[\[DrutenKetterle97\]\], the usual line of reasoning shows that for $d/\sigma>1$ there is a sharp onset of Bose–Einstein condensation at the temperature $T_0$ given by \[\[deGrootEtAl50\]\] $$\frac{k_B T_0}{\Delta} \; = \; \frac{1}{A^{\sigma/d}}
\left(\frac{N}{\zeta\!\left(\frac{d}{\sigma}\right)}\right)^{\!\sigma/d}
\; .
\label{CONT}$$ Moreover, the spectral Zeta functions can now be well approximated by $$Z(\beta,t) \; \approx \; \frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)}
\left(\beta\Delta\right)^{-t}
\zeta(t+1-d/\sigma) \; ,$$ so Eqs. (\[FUN1\]) and (\[FUN2\]) adopt the transparent forms $$\langle N_{\rm ex} \rangle \; \approx \;
\frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)}
\frac{1}{2\pi i} \int_{\tau-i\infty}^{\tau+i\infty} \!\! {\mbox d}t \,
\left(\beta\Delta\right)^{-t} \Gamma(t) \,
\zeta(t + 1 - d/\sigma) \, \zeta(t)
\label{GREX}$$ and $$\left(\delta N_0\right)^2_{cn} \; \approx \;
\frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)}
\frac{1}{2\pi i} \int_{\tau-i\infty}^{\tau+i\infty} \!\! {\mbox d}t \,
\left(\beta\Delta\right)^{-t} \Gamma(t) \,
\zeta(t + 1 - d/\sigma) \, \zeta(t-1) \; .
\label{GRFL}$$
For $\Delta \ll k_BT$, the behavior of either integral is determined by the pole of its integrand farthest to the right in the complex plane. Keeping in mind that $\zeta(z)$ has merely one single pole at $z = 1$, with residue $1$, while the poles of $\Gamma(z)$ are located at $z = 0$, $-1$, $-2$, …, the decisive pole is provided [*either*]{} by the system’s Zeta function $\zeta(t + 1 - d/\sigma)$, [*or*]{} by the other Zeta function that is determined by the order of the cumulant one is asking for: by $\zeta(t)$, if one asks for the first cumulant $\langle N_{\rm ex} \rangle$, or by $\zeta(t-1)$, if one asks for the second cumulant $\left(\delta N_0\right)^2$. To see what the argument boils down to, let us first consider the evaluation of Eq. (\[GREX\]), where the system’s pole at $t = d/\sigma$ competes with the cumulant-order pole at $t=1$:
- If $d/\sigma > 1$, the low-temperature behavior of $\langle N_{\rm ex} \rangle$ is governed by the pole of $\zeta(t + 1 - d/\sigma)$ at $t = d/\sigma$. Hence the residue theorem yields $$\langle N_{\rm ex} \rangle \; \approx \; A \, \zeta(d/\sigma)
\left(\frac{k_B T}{\Delta}\right)^{\!d/\sigma} \; .
\label{EXOC}$$ This canonical result, valid for $T < T_0$, coincides precisely with the result of the customary grand canonical analysis \[\[deGrootEtAl50\]\]. For example, in the case of the three-dimensional isotropic harmonic oscillator potential (i.e., for $d = 3$, $\sigma = 1$, $A = 1$ and $\Delta = \hbar\omega$) Eq. (\[EXOC\]) yields the familiar formula $$\langle N_0 \rangle \; = \; N - \langle N_{\rm ex} \rangle \; = \;
N\!\left[1 - \left(\frac{T}{T_0}\right)^3\right]
\quad {\mbox{for}} \quad T < T_0
= \frac{\hbar\omega}{k_B}\left(\frac{N}{\zeta(3)}\right)^{1/3} \; .$$
- If $d/\sigma = 1$, both Zeta functions in Eq. (\[GREX\]) coincide. We then encounter a double pole at $t = 1$, and find $$\langle N_{\rm ex} \rangle \; \approx \; A \, \frac{k_B T}{\Delta}
\left[ \ln\!\left(\frac{k_B T}{\Delta}\right) + \gamma \right] \; ,
\label{EXO2}$$ where $\gamma = 0.5772\ldots$ is Euler’s constant. This corresponds to a result obtained already in 1950 by Nanda \[\[Nanda53\]\] with the help of the Euler-Maclaurin summation formula.
- If $0 < d/\sigma < 1$, the pole of $\zeta(t)$ at $t = 1$ takes over: $$\langle N_{\rm ex} \rangle \; \approx \;
\frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)} \,
\zeta(2-d/\sigma) \, \frac{k_B T}{\Delta} \; ,
\label{EXO3}$$ so that for sufficiently low temperatures $\langle N_{\rm ex} \rangle$ now depends linearly on $T$, regardless of the value of $d/\sigma$ that characterizes the trap.
A mere glance at Eq. (\[GRFL\]) then suffices to reveal that [*the very same scenario*]{} — a first pole at $t = d/\sigma$ that endows the temperature dependence with a trap-specific exponent as long as it lies to the right of a second one, which yields universal behavior when it becomes dominant — also governs the canonical condensate fluctuations, with the only difference that the second pole now is located at $t = 2$:
- If $d/\sigma > 2$, the pole of $\zeta(t+1-d/\sigma)$ at $t = d/\sigma$ wins, giving $$\left(\delta N_0\right)^2_{cn} \; \approx \; A \, \zeta(d/\sigma-1)
\left(\frac{k_B T}{\Delta}\right)^{\!d/\sigma} \; .
\label{MFL1}$$
- If $d/\sigma = 2$, we find at $t = 2$ the already familiar double pole, resulting in $$\left(\delta N_0\right)^2_{cn} \; \approx \; A \,
\left(\frac{k_B T}{\Delta}\right)^2
\left[ \ln\!\left(\frac{k_B T}{\Delta}\right) + \gamma + 1 \right] \; .
\label{MFL2}$$
- If $0 < d/\sigma < 2$, the pole of $\zeta(t-1)$ at $t = 2$ lies to the right of its rival, yielding $$\left(\delta N_0\right)^2_{cn} \; \approx \;
\frac{A}{\Gamma\!\left(\frac{d}{\sigma}\right)} \,
\zeta(3-d/\sigma) \left(\frac{k_B T}{\Delta}\right)^2 \; .
\label{MFL3}$$ In particular, for the one-dimensional harmonic oscillator we have $d = 1$, $\sigma = 1$, $A = 1$ and $\Delta = \hbar\omega$, so that we recover our previous microcanonical result (\[ODOF\]) within the canonical ensemble, recalling that $\zeta(2) = \pi^2/6$.
The above canonical fluctuations, derived from the integral representation (\[FUN2\]), reduce to the expression obtained by Politzer \[\[Politzer96\]\] in the case of the three-dimensional isotropic trap, and match the results obtained by Wilkens and Weiss \[\[WilkensWeiss97\]\].
The calculation of the corresponding [*microcanonical*]{} quantities now requires saddle-point inversions of Eq. (\[RICH\]) in order to obtain the microcanonical moments $\mu_k(E)$ from the canonical moments $M_k(\beta)$. Performing these inversions, and reexpressing energy in terms of temperature, we find that the integral (\[FUN1\]) — and, hence, the results (\[EXOC\]), (\[EXO2\]), and (\[EXO3\]) for the number of excited particles — remains valid within the microcanonical ensemble. The fluctuations require more care: Whereas canonical and microcanonical fluctuations coincide in the large-$N$-limit for $d/\sigma < 2$, the microcanonical mean-square fluctuations $\left(\delta N_0\right)_{mc}^2$ are distinctly lower than their canonical counterparts for $d/\sigma > 2$: $$\left(\delta N_0\right)^2_{cn} - \left(\delta N_0\right)^2_{mc}
\; \approx \; \frac{Ad}{d+\sigma}
\frac{\zeta^2\!\left(\frac{d}{\sigma}\right)}
{\zeta\!\left(\frac{d}{\sigma}+1\right)}
\left(\frac{k_B T}{\Delta}\right)^{\!d/\sigma}
\quad {\mbox{for}} \;\; d/\sigma > 2 \;\;
{\mbox{and}} \;\; T < T_0 \; .
\label{DIFF}$$ Thus, the exponent of $T$ is the same for both $\left(\delta N_0\right)^2_{cn}$ and $\left(\delta N_0\right)^2_{mc}$, but the prefactors can differ substantially. This Eq. (\[DIFF\]) contains as a special case the result obtained for the three-dimensional isotropic trap by Navez [*et al.*]{} \[\[NavezEtAl97\]\].
Before summarizing these findings, it is useful to also consider the heat capacities for trapped ideal Bose gases with the single-particle spectra (\[SPSP\]): for $d/\sigma > 1$, and temperatures below the condensation temperature $T_0$, the heat capacity per particle is given by $$\frac{C_<}{Nk_B} \; = \; \frac{d}{\sigma}\!\left(\frac{d}{\sigma}+1\right)
\frac{\zeta\!\left(\frac{d}{\sigma}+1\right)}
{\zeta\!\left(\frac{d}{\sigma}\right)}
\left(\frac{T}{T_0}\right)^{\!d/\sigma} \; ,$$ above $T_0$ by $$\frac{C_>}{Nk_B} \; = \; \frac{d}{\sigma}\!\left(\frac{d}{\sigma}+1\right)
\frac{g_{d/\sigma+1}(z)}{g_{d/\sigma}(z)}
\, - \, \frac{d^2}{\sigma^2}\frac{g_{d/\sigma}(z)}{g_{d/\sigma-1}(z)} \; .$$ Since the fugacity $z$ approaches unity from below when $T$ approaches $T_0$ from above, so that the Bose function $g_\alpha(z)$ approaches $\zeta(\alpha)$, we see that the heat capacity remains continuous at $T_0$ for $0 < d/\sigma \leq 2$, but exhibits a jump of size $$\left.\frac{C_< - C_>}{Nk_B}\right|_{T_0} \; = \; \frac{d^2}{\sigma^2}
\frac{\zeta\!\left(\frac{d}{\sigma}\right)}
{\zeta\!\left(\frac{d}{\sigma}-1\right)}$$ for $d/\sigma > 2$.
We thus arrive at the following picture: For any dimension $d$ and trap exponent $\sigma >0$, the fluctuation of the number of condensate particles is independent of the total particle number $N$. For isolated traps, this insensitivity of the system with respect to $N$ reflects the well-peakedness of the microcanonical distributions (\[DIST\]), see Fig. 2: if there is a condensate, the behavior of the ideal Bose gas does not depend on how many particles the condensate consists of. If $d/\sigma < 2$, so that the heat capacity remains continuous in the large-$N$-limit, canonical and microcanonical fluctuations $\delta N_0$ vanish linearly with temperature, see Eq. (\[MFL3\]). If $d/\sigma = 2$, there appears a logarithmic correction to the linear $T$-dependence, as quantified by Eq. (\[MFL2\]). But if $d/\sigma > 2$, so that the heat capacity becomes discontinuous, then the fluctuations $\delta N_0$ vanish proportionally to $T^{d/2\sigma}$, so that now the properties of the trap determine the way the fluctuations depend on temperature. In addition, in this case the microcanonical fluctuations are markedly lower than the fluctuations in a trap that exchanges energy with a heat bath.
Intuitively, one might have expected some sort of square root law for the fluctuations. Because of the $N$-independence of the condensate fluctuations, there is, of course, no “$\sqrt{N}$-dependence” of $\delta N_0$. The square root is hidden elsewhere: Since $\delta N_0 = \delta N_{\rm ex}$ for ensembles with fixed particle number $N$, we find
- $\delta N_{\rm ex} \propto \langle N_{\rm ex} \rangle^{1/2} \quad \, $ for $ \quad 2 < d/\sigma$;
- $\delta N_{\rm ex} \propto \langle N_{\rm ex} \rangle^{\sigma/d} \quad $ for $ \quad 1 < d/\sigma < 2$;
- $\delta N_{\rm ex} \propto \langle N_{\rm ex} \rangle \qquad \; $ for $ \quad 0 < d/\sigma < 1$,
with proportionality constants that are independent of temperature, both canonically and microcanonically. The first of these relations is just what one might have guessed, but the crossover from normal fluctuations for $d/\sigma > 2$ to much stronger fluctuations for $0 < d/\sigma < 1$ appears noteworthy.
It remains to be seen how much of this ideal structure survives in the case of weakly interacting Bose gases. It should also be recognized that Maxwell’s Demon, though it has provided the microcanonical low-temperature fluctuations, can not solve all problems of the ideal gas. When considering $d$-dimensional isotropic harmonic traps, the Maxwell’s Demon approximation (i.e., the replacement of the true upper summation bound “$N$” in Eq. (\[MIMO\]) by “$\infty$”) is [*exact*]{} below the “restriction temperature” (i.e., that temperature where the number $n = E/(\hbar\omega)$ of energy quanta equals the number $N$ of particles \[\[GH97b\]\]), but the description of the Bose–Einstein transition itself is beyond the capabilities of Maxwell’s Demon. Namely, that description requires the computation of the numbers $\Omega(n\hbar\omega|N)$ of microstates also under conditions where the restriction due to the finite $N$ becomes decisive. Incidentially, one meets the task of computing such restricted partitions of integers also in other problems of statistical mechanics, for example in the theory of the so-called compact lattice animals, or of the infinite-state Potts model \[\[WuEtAl96\]\].
Nonetheless, the results obtained with the help of the Maxwell’s Demon approximation have some interesting number-theoretical implications. Going once more back to Eq. (\[ODOF\]) for the one-dimensional oscillator, and inserting the energy–temperature relation $n = E/(\hbar\omega) \approx \zeta(2)(k_B T/\hbar\omega)^2$, we find the truly remarkable formula $$\delta N_0 \approx \sqrt{n} \; .
\label{AMAZ}$$ This has a twofold interpretation. The physicist, puzzeled by the loss of the square root fluctuation law at the level of $\langle N_{\rm ex} \rangle $, finds a substitute:
- [*For ideal Bose particles trapped at low temperatures by a one-dimensional harmonic potential, the root-mean-square fluctuation of the number of ground state particles is given by the square root of the number of energy quanta.*]{}
The mathematician, who approaches Eq. (\[AMAZ\]) from the viewpoint of partition theory, sees the solution to another problem:
- [*If one considers all unrestricted partitions of the integer $n$ into positive, integer summands, and asks for the root-mean-square fluctuation of the number of summands, then the answer is (asymptotically) just $\sqrt{n}$*]{}
— certainly one of the most amazing examples for the occurrence of square root fluctuations! The ease with which the solution to a seemingly difficult number-theoretical question has been obtained here is even aesthetically appealing. It is pleasing to conclude that ongoing developments in statistical mechanics, themselves being motivated by recent experimental achievements \[\[AndersonEtAl95\],\[DavisEtAl95\],\[BradleyEtAl97\]\], have a high potential for further fertilization across subfield boundaries.
|
---
abstract: 'In this paper, we propose a distributive queue-aware intra-cell user scheduling and inter-cell interference (ICI) management control design for a delay-optimal celluar downlink system with $M$ base stations (BSs), and $K$ users in each cell. Each BS has $K$ downlink queues for $K$ users respectively with heterogeneous arrivals and delay requirements. The ICI management control is adaptive to joint queue state information (QSI) over a slow time scale, while the user scheduling control is adaptive to both the joint QSI and the joint channel state information (CSI) over a faster time scale. We show that the problem can be modeled as an infinite horizon average cost Partially Observed Markov Decision Problem (POMDP), which is NP-hard in general. By exploiting the special structure of the problem, we shall derive an equivalent Bellman equation to solve the POMDP problem. To address the distributive requirement and the issue of dimensionality and computation complexity, we derive a distributive online stochastic learning algorithm, which only requires local QSI and local CSI at each of the $M$ BSs. We show that the proposed learning algorithm converges almost-surely (with probability 1) and has significant gain compared with various baselines. The proposed solution only has linear complexity order $O(MK)$.'
author:
- '[^1]'
bibliography:
- 'IEEEabrv.bib'
- 'multicell.bib'
title: 'Delay-Optimal User Scheduling and Inter-Cell Interference Management in Cellular Network via Distributive Stochastic Learning'
---
multi-cell systems, delay optimal control, partially observed Markov decision problem (POMDP), interference management, stochastic learning.
Introduction {#sec:intro}
============
It is well-known that cellular systems are [*interference limited*]{} and there are a lot of works to handle the [*inter-cell interference*]{} (ICI) in cellular systems. Specifically, the optimal binary power control (BPC) for the sum rate maximization has been studied in [@BPC:2008]. They showed that BPC could provide reasonable performance compared with the multi-level power control in the multi-link system. In [@pattern:2009], the authors studied a joint adaptive multi-pattern reuse and intra-cell user scheduling scheme, to maximize the long-term network-wide utility. The ICI management runs at a slower scale than the user selection strategy to reduce the communication overhead. In [@multicell:cooperation:2008] and the reference therein, cooperation or coordination is also shown to be a useful tool to manage ICI and improve the performance of the celluar network.
However, all of these works have assumed that there are infinite backlogs at the transmitter, and the control policy is only a function of channel state information (CSI). In practice, applications are delay sensitive, and it is critical to optimize the delay performance in the cellular network. A systematic approach in dealing with delay-optimal resource control in general delay regime is via Markov Decision Process (MDP) technique. In [@Delay_IT:2006; @Vincent:MIMO], the authors applied it to obtain the delay-optimal cross-layer control policy for broadcast channel and point-to-point link respectively. However, there are very limited works that studied the delay optimal control problem in the cellular network. Most existing works simply proposed heuristic control schemes with partial consideration of the queuing delay[@multicell:multiuser:2009]. As we shall illustrate, there are various technical challenges involved regarding delay-optimal cellular network.
- [**Curse of Dimensionality:**]{} Although MDP technique is the systematic approach to solve the delay-optimal control problem, a primal difficulty is the curse of dimensionality[@Bertsekas:2007]. For example, a huge state space (exponential in the number of users and number of cells) will be involved in the MDP and brute force value or policy iterations cannot lead to any implementable solution[^2] [@RL:survey; @Powell:2007]. Furthermore, brute force solutions require explicit knowledge of transition probability of system states, which is difficult to obtain in the complex systems.
- [**Complexity of the Interference Management:**]{} Jointly optimal ICI management and user scheduling requires heavy computation overhead even for the throughput optimization problem [@pattern:2009]. Although grouping clusters of cells [@BPC:2008] and considering only neighboring BSs [@BSs:neighbour] were proposed to reduce the complexity, complex operations on a slot by slot basis are still required, which is not suitable for the practical implementation.
- [**Decentralized Solution:**]{} For delay-optimal multi-cell control, the entire system state is characterized by the global CSI (CSI from any BS to any MS) and the global QSI (queue length of all users). Such system state information are distributed locally at each BS and centralized solution (which requires global knowledge of the CSI and QSI) will induce substantial signaling overhead between the BSs and the Base Station Controller (BSC).
In this paper, we consider the delay-optimal inter-cell ICI management control and intra-cell user scheduling for the cellular system. For implementation consideration, the ICI management control is computed at the BSC at a longer time scale and it is adaptive to the QSI only. On the other hand, the intra-cell user scheduling control is computed distributively at the BS at a smaller time scale and hence, it is adaptive to both the CSI and QSI. Due to the [*two time-scale*]{} control structure, the delay optimal control is formulated as an infinite-horizon average cost Partially Observed Markov Decision Process (POMDP). Exploiting the special structure, we propose an [*equivalent Bellman Equation*]{} to solve the POMDP. Based on the equivalent Bellman equation, we propose a distributive online learning algorithm to estimate a per-user value function as well as a per-user $\mathbb{Q}$-factor[^3]. Only the local CSI and QSI information is required in the learning process at each BS. We also establish the technical proof for the almost-sure convergence of the proposed distributive learning algorithm. The proposed algorithm is quite different from the iterative update algorithm for solving the deterministic NUM [@Palomar:NUM:2006], where the CSI is always assumed to be quasi-static during the iterative updates. However, the delay-optimal problem we considered is stochastic in nature, and during the iterative updates, the system state will not be quasi-static anymore. In addition, the proposed learning algorithm is also quite different from conventional stochastic learning[@Cao:2007; @Q-learning:2007]. For instance, conventional stochastic learning requires centralized update and global system state knowledge and the convergence proof follows from standard [*contraction mapping*]{} arguments[@Bertsekas:2007]. However, due to the distributive learning requirement and simultaneous learning of the per-user value function and $\mathbb{Q}$-factor, it is not trivial to establish the contraction mapping property and the associated convergence proof. We also illustrate the performance gain of the proposed solution against various baselines via numerical simulations. Furthermore, the solution has linear complexity order $O(MK)$ and it is quite suitable for the practical implementation.
System Model {#sec:model}
============
In this section, we shall elaborate the system model, as well as the control policies. We consider the downlink of a wireless celluar network consisting of $M$ BSs, and there are $K$ mobile users in each cell served by one BS. Specifically, let $\mathcal{M}=\{1,...,M\}$ and $\mathcal{K}_m=\{1,...,K\}$ denote the set of BSs and the set of users served by the BS $m$ respectively. $k\in\mathcal{K}_m$ denotes the $k$-th user served by BS $m$. The time dimension is partitioned into [*scheduling slots*]{} (every slot lasts for $\tau$ seconds). The system model is illustrated in Fig.\[fig:sys\_model\].
Source Model
------------
In each BS, there are $K$ independent application streams dedicated to the $K$ users respectively. Let $\mathbf{A}(t)=\{\mathbf{A}_m(t)\}_{m=1}^{M}$ and $\mathbf{A}_m(t)=\{A_{(m,k)}(t)\}_{k=1}^{K}$, where $A_{(m,k)}(t)$ represents the new arrivals (number of bits) for the user $k\in\mathcal{K}_m$ at the end of the slot $t$.
\[ass:source\_model\] We assume that the arrival process $A_{(m,k)}(t)$ is i.i.d over the scheduling slot $t$ according to a general distribution $\Pr\{A_{(m,k)}\}$ with average arrival rate $\lambda_{(m,k)}=\mathbb{E}[A_{(m,k)}]$, and the arrival processes for all the users are independent with each other, i.e., $\Pr\{A_{(m,k)}A_{(n,l)}\}=\Pr\{A_{(m,k)}\}\Pr\{A_{(n,l)}\}$ if $m\neq n$ or $k\neq l$.
Let $\mathbf{Q}(t)=\{\mathbf{Q}_m(t)\}_{m=1}^{M}\in\mathcal{Q}$ denote the global QSI in the system, where $\mathcal{Q}$ is the state space for the global QSI. $\mathbf{Q}_m(t)=\{Q_{(m,k)}(t)\}_{k=1}^K$ denotes the QSI in the BS $m$, where $Q_{(m,k)}(t)$ represents the number of bits for user $k\in\mathcal{K}_m$ at the beginning of the slot $t$, and $N_{Q}$ denotes the maximal buffer size (bits). When the buffer is full, i.e, $Q_{(m,k)}=N_Q$, new bits arrivals will be dropped. The cardinality of the global QSI is $I_Q=(1+N_Q)^{MK}$.
Channel Model and Physical Layer Model
--------------------------------------
Let $H_{(m,k)}^n(t)$ and $L_{(m,k)}^n$ denote the small scale channel fading gain and the path loss from the $n$-th BS to the user $k\in\mathcal{K}_m$ respectively, and $\mathbf{H}_{(m,k)}(t)=\{H_{(m,k)}^n(t)\}_{n=1}^M$ is the local CSI states for user $k$. $\mathbf{H}_m(t)=\{\mathbf{H}_{(m,k)}(t)\}_{k=1}^K$ denotes the local CSI states for BS $m$, and the global CSI is denoted as $\mathbf{H}(t)=\{\mathbf{H}_{m}(t)\}_{m=1}^M\in\mathcal{H}$, where $\mathcal{H}$ is the state space for the global CSI.
\[ass:csi\_model\] We assume that the global $\mathbf{H}$ is quasi-static in each slot. Furthermore, $H_{(m,k)}^n(t)$ is i.i.d over the scheduling slot $t$ according to a general distribution $\Pr\{H_{(m,k)}^n\}$ and the small scale channel fading gains for all users are independent with each other. The path loss $L_{(m,k)}^n$ remains constant for the duration of the communication session.
The cellular system shares a single common channel with bandwidth $W$Hz (all the BSs use the same channel). At the beginning of each slot, the BS is either turned on (with transmit power $P^m_{\max}$) or off (with transmit power 0)[^4], according to a [*ICI management control policy*]{}, which is defined later. At each slot, a BS can select only one user for its data transmission. Specifically, let $\mathbf{p}=\{p_{m}\}_{m=1}^M\in\mathcal{P}$ denotes an ICI management control pattern, where $p_{m}=1$ denotes BS $m$ is active, $p_{m}=0$ otherwise, and $\mathcal{P}$ denotes the set of all possible control patterns. Furthermore, let $\mathcal{M}_\mathbf{p}\in\mathcal{M}$ be the set of BSs activated by the pattern $\mathbf{p}$ and $\mathcal{P}_m\in\mathcal{P}$ be the set of patterns that activate the BS $m$. The signal received by the user $k\in\mathcal{K}_m$ at slot $t$, when pattern $\mathbf{p}\in\mathcal{P}_m$ is selected, is given by $$\label{eq:sys_model} \begin{array}{l} y_{(m,k)}[t]=
\sqrt{H^m_{(m,k)}L^m_{(m,k)}}x_{m}[t]+\\
\quad\quad\quad\quad\sum\nolimits_{n\neq
m,n\in\mathcal{M}_\mathbf{p}}\sqrt{H^n_{(m,k)}L_{(m,k)}^n}x_{n}[t]+z[t]
\end{array}$$ where $x_{m}[t]$ is the transmit signal from the $m$-th BS to the $k$-th user at slot $t$, and $\{z[t]\}_{t=1}^{\infty}$ is the i.i.d $\mathcal{N}(0,N_0)$ noise. The achievable data rate of user $k$ can be expressed by $$\label{eq:rate}\begin{array}{l} R_{(m,k)}=\\
\left\{\begin{array}{ll}
W\log_2\left(1+\frac{\xi
P^m_{\max}H^m_{(m,k)}L_{(m,k)}^m}{I_{(m,k)}+N_0W}\right)s_{(m,k)}
& \text{if $\mathbf{p}\in\mathcal{P}_m$}\\
0 & \text{otherwise}
\end{array}
\right.
\end{array}$$ where $I_{(m,k)}=\sum\limits_{n\neq m,n\in\mathcal{M}_\mathbf{p}}
P^n_{\max}H^n_{(m,k)}L_{(m,k)}^n$, $s_{(m,k)}\in\{0,1\}$ is an indicator variable with $s_{(m,k)} = 1$ when the user $k$ is scheduled. $\xi\in(0,1]$ is a constant can be used to model both the coded and uncoded systems[@Vincent:MIMO].
ICI Management and User Scheduling Control Policy {#sec:con_pol}
-------------------------------------------------
At the beginning of the slot, the BSC will decide which BSs are allowed to transmit according to a stationary ICI management control policy defined below.
A stationary ICI management control policy $\Omega_{\mathbf{p}}:
\mathcal{Q}\rightarrow\mathcal{P}$ is defined as the mapping from current global QSI to an ICI management pattern $\Omega_{\mathbf{p}}(\mathbf{Q})=\mathbf{p}$.
Let $\boldsymbol{\chi}(t)=\{\mathbf{H}(t),\mathbf{Q}(t)\}$ to be the global system state at the beginning of slot $t$. The active user at each cell is selected according to a user scheduling policy defined below.
A stationary user scheduling policy $\Omega_{\mathbf{s}}:\{\mathcal{Q},\mathcal{H}\}\rightarrow\mathcal{S}$ is defined as the mapping from current global system state $\boldsymbol{\chi}$ to current user scheduling action $\Omega_{\mathbf{s}}(\boldsymbol{\chi})=\mathbf{s}\in\mathcal{S}$. The scheduling action $\mathbf{s}$ is a set of all the users’ scheduling indicator variable, i.e., $\mathbf{s}=\{s_{(m,k)},\forall
k\in\mathcal{K}_m,\forall m\}$. It represents which users are scheduled and which users are not in any given slot. $\mathcal{S}$ is the set of all user scheduling actions.
For notation convenience, let $\Omega=\{\Omega_{\mathbf{p}},\Omega_{\mathbf{s}}\}$ to be the joint control policy, and $\Omega(\boldsymbol{\chi})=\{\mathbf{p},\mathbf{s}\}$ be the control action under state $\boldsymbol{\chi}$.
Problem Formulation {#sec:problem}
===================
In this section, we will first elaborate the dynamics of system state under a control policy $\Omega$. Based on that, we shall formally formulate the delay-optimal control problem.
Dynamics of System State
------------------------
Given the new arrival $A_{(m,k)}(t)$ at the end of the slot $t$, the current system state $\boldsymbol{\chi}(t)$ and the control action $\Omega(\boldsymbol{\chi}(t))$, The queue evolution for user $k\in\mathcal{K}_m$ is given by: $$Q_{(m,k)}(t+1)=\big[\big(Q_{(m,k)}(t)-U_{(m,k)}(t)\big)^++A_{(m,k)}(t)\big]_{\bigwedge
N_Q}$$ where $U_{(m,k)}(t)=\lfloor
R_{(m,k)}(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))\tau\rfloor$ is the number of bits delivered to user $k$ at slot $t$, and $R_{(m,k)}(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))$, given by (\[eq:rate\]), is the achievable data rate under the control action $\Omega(\boldsymbol{\chi}(t))$. $\lfloor x\rfloor$ denotes the floor of $x$, $(x)^+=\max(x,0)$, and $(x)_{\bigwedge
N_Q}=\min(x,N_Q)$. Let $\mathbf{U}(t)=\{\mathbf{U}_m(t)\}_{m=1}^M$, and $\mathbf{U}_m(t)=\{U_{(m,k)}(t)\}_{k=1}^K$, $U_{(m,k)}(t)=
R_{(m,k)}(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))\tau$ for the user $k\in\mathcal{K}_m$, and $\mathbf{\hat{Q}}(t+1)=\big[\big(\mathbf{Q}(t)-\mathbf{U}(t)\big)^++\mathbf{A}(t)\big]_{\bigwedge
N_Q}$. Therefore, given a control policy $\Omega$, the random process $\{\mathbf{H}(t), \mathbf{Q}(t)\}$ is a controlled Markov chain with transition probability $$\label{eq:sys_tran}\begin{array}{l}
\Pr\{\boldsymbol{\chi}(t+1)|\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t))\}=\\
\left\{\begin{array}{ll} \Pr\{\mathbf{H}(t+1)\}\Pr\{\mathbf{A}(t)\}
& \text{if
$\mathbf{Q}(t+1)=\mathbf{\hat{Q}}(t+1)$}\\
0 & \text{otherwise}
\end{array}
\right.
\end{array}$$
Delay Optimal Control Problem Formulation
-----------------------------------------
Given a stationary control policy $\Omega$, the average cost of the user $k\in\mathcal{K}_m$ is given by: $$\label{eq:T_single}
\overline{T}_{(m,k)}(\Omega)=\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}[f(Q_{(m,k)}(t))]$$ where $f(Q_{(m,k)})$ is a monotonic increasing cost function of $Q_{(m,k)}$. For example, when $f(Q_{(m,k)})=Q_{(m,k)}/\lambda_{(m,k)}$, using Little’s Law [@Delay_IT:2006; @Ross:2003], $\overline{T}_{(m,k)}(\Omega)$ is an approximation[^5] of the average delay of user $k$. When $f(Q_{(m,k)})=1_{\{Q_{(m,k)}\geq N_Q\}}$ and $A_{(m,k)}$ follows the bernoulli process, $\overline{T}_{(m,k)}(\Omega)$ is the [*bit dropping probability*]{} (conditioned on bit arrival). Note that, the $MK$ queues in the celluar system are coupled together via the control policy $\Omega$. In this paper, we seek to find an optimal stationary control policy $\Omega$ to minimize the average cost in (\[eq:T\_single\]). Specifically, we have:
\[prob:delay\] [^6] For some positive constants $\boldsymbol{\beta}=\{\beta_{(m,k)},,\forall
k\in\mathcal{K}_m,\forall m\}$, finding a stationary control policy $\Omega$ that minimizes: $$\begin{aligned}
\label{eq:problem}
\min_{\Omega}J_{\beta}^{\Omega}&=&\sum\nolimits_{m,k}\beta_{(m,k)}\overline{T}_{(m,k)}(\Omega)\\
&=&\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}^{\Omega}[g(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))]\nonumber\end{aligned}$$ where $g(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t))=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})$ is the per-slot cost, and $\mathbb{E}^{\Omega}$ denotes the expectation w.r.t. the induced measure (induced by the control policy $\Omega$ and the transition kernel in (\[eq:sys\_tran\])). The positive constants $\boldsymbol{\beta}$ indicate the relative importance of the users and for a given $\boldsymbol{\beta}$, the solution to (\[eq:problem\]) corresponds to a Pareto optimal point of the multi-objective optimization problem given by $\min_{\Omega}
\overline{T}_{(m,k)}(\Omega), \forall m,k$. Moreover, a control policy $\Omega^*$ is called Pareto optimal if for any control policy $\Omega^{\prime}\neq\Omega^*$ such that $\overline{T}_{(m,k)}(\Omega^{\prime})\leq\overline{T}_{(m,k)}(\Omega^*),
\forall m,k$, it implies that $\overline{T}_{(m,k)}(\Omega^{\prime})=\overline{T}_{(m,k)}(\Omega^*),
\forall m,k$. In other words, we cannot reduce $\overline{T}_{(m,k1)}$ without increasing other component (say $\overline{T}_{(m,k2)}$) at Pareto optimal control $\Omega^*$[@Boyd:2004].
General Solution to the Delay Optimal Problem {#sec:opt_solution}
=============================================
In this section, we will show that the delay optimal problem \[prob:delay\] can be modeled as an infinite horizon average cost POMDP, which is a very difficult problem. By exploiting the special structure, we shall derive an [*equivalent Bellman equation*]{} to solve the POMDP problem.
Preliminary on MDP and POMDP
----------------------------
An infinite horizon average cost MDP can be characterized by a tuple of four objects: $\{\mathbb{S},\mathbb{A},\Pr\{s^{\prime}|s,a\},g(s,a)\}$, where $\mathbb{S}$ is a finite set of states and $\mathbb{A}$ is the action space. $\Pr\{s^{\prime}|s,a\}$ is the transition probability from state $s$ to $s^{\prime}$, given that the action $a\in\mathbb{A}$ is taken. $g(s,a)$ is the per-slot cost function. The objective is to find the optimal policy $\mathbf{a}=\{a(s)\}$ so as to minimize the average per-slot cost $\theta$ as: $$\label{eq:per-theta} \theta=\min_{\mathbf{a}}\lim_{T\rightarrow
\infty}\sup\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}^{\mathbf{a}}[g(s(t),a(s(t)))]$$
If the policy space consists of [*unichain policies*]{} and the associated induced Markov chain is irreducible, it is well known that there exist a unique $\theta$ for each starting state[@Cao:2007; @Bertsekas:2007]. Furthermore, the optimal control policy $\mathbf{a}$ can be obtained by the following Bellman equation. $$\label{eq:bellman}
V(s)+\theta=\min_{a(s)}\left\{g(s,a(s))+\sum\nolimits_{s^{\prime}}\Pr\{s^{\prime}|s,a(s)\}V(s^{\prime}))\right\}$$ where $V(s)$ is called the value function. General offline solutions, [*value*]{} or [*policy iteration*]{}, can be used to find the value function $V(s)$ iteratively, as well as the optimal policy[@Bertsekas:2007].
POMDP is an extension of MDP when the control agent does not have direct observation of the entire system state (and hence it is called “partially observed MDP”). Specifically, an infinite horizon average cost POMDP can be characterized by a tuple [@Meuleau:1999; @POMDP:1998]: $\{\mathbb{S},\mathbb{A},\Pr\{s^{\prime}|s,a\},g(s,a),\mathbb{O},O(z,s,a)\}$, where $\{\mathbb{S},\mathbb{A},P(s^{\prime}|s,a),g(s,a)\}$ characterize a MDP and $\mathbb{O}$ is a finite set of observations. $O(z,s,a)$ is the observation function, which gives the probability (or stochastic relationship) between the partial observation $z$, the actual system state $s$ and the control action $a$. Specifically, $O(z,s,a)$ is the probability of getting a partial observation “$z$” given that the current system state is $s$ and the action $a$ was taken in the previous slot. A PODMP is a MDP where current system state and the actions are based on the observation $z$. The objective is to find the optimal policy $\mathbf{a}=\{a(z)\}$ so as to minimize the average per-slot cost $\theta$ in (\[eq:per-theta\]). However, in general, it is a [*NP-hard*]{} problem and there are various approximation solutions proposed based on the special structure of the studied problems[@POMDP:survey].
Equivalent Bellman Equation and Optimal Control Policy
------------------------------------------------------
In this subsection, we shall first illustrate that the optimization problem \[prob:delay\] is an infinite horizon average cost POMDP. We shall then exploit some special problem structure to simplify the complexity and derive an [*equivalent Bellman equation*]{} to solve the problem. For instance, in the delay optimal problem \[prob:delay\], the ICI management control policy $\Omega_{\mathbf{p}}$ is adaptive to the QSI $\mathbf{Q}$, while the user scheduling policy $\Omega_{\mathbf{s}}$ is adaptive to the complete system state $\{\mathbf{Q},\mathbf{H}\}$. Therefore, the optimal control policy $ \Omega^*$ cannot be obtained by solving a standard Bellman equation from conventional MDP[^7]. In fact, problem \[prob:delay\] is a POMDP with the following specification.
- [**State Space:**]{} The system state is the global QSI and CSI $\boldsymbol{\chi}=\{\mathbf{Q},\mathbf{H}\}\in\{\mathcal{Q},\mathcal{H}\}$.
- [**Action Space:**]{} The action is ICI management pattern and user scheduling $\{\mathbf{p},\mathbf{s}\}\in\{\mathcal{P},\mathcal{S}\}$.
- [**Transition Kernel:**]{} The transition probability $\Pr\{\boldsymbol{\chi}^{\prime}|\boldsymbol{\chi},\mathbf{p},\mathbf{s}\}$ is given in (\[eq:sys\_tran\]).
- [**Per-Slot Cost Function:**]{} The per-slot cost function is $g(\boldsymbol{\chi},\mathbf{p},\mathbf{s})=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})$.
- [**Observation:**]{} The observation for ICI management control policy is global QSI, i.e., $z_{\mathbf{p}}=\mathbf{Q}$, while the observation for User scheduling policy is the complete system state, i.e., $z_{\mathbf{s}}=\boldsymbol{\chi}$.
- [**Observation Function:**]{} The observation function for ICI management control policy is $O_{\mathbf{p}}(z_{\mathbf{p}},\boldsymbol{\chi},\mathbf{p},\mathbf{s})=1$, if $z_{\mathbf{p}}=\mathbf{Q}$, otherwise 0. Furthermore the observation function for user scheduling policy is $O_{\mathbf{s}}(z_{\mathbf{s}},\boldsymbol{\chi},\mathbf{p},\mathbf{s})=1$, if $z_{\mathbf{s}}=\boldsymbol{\chi}$, otherwise 0.
While POMDP is a very difficult problem in general, we shall utilize the notion of [*action partitioning*]{} in our problem to substantially simplify the problem. We first define [*partitioned actions*]{} below.
\[def:partitioned action\] Given a control policy $\Omega$, we define $\Omega(\mathbf{Q})=\{(\mathbf{p},\mathbf{s})=\Omega(\boldsymbol{\chi}):\boldsymbol{\chi}=(\mathbf{Q},\mathbf{H})\forall\mathbf{H}\in\mathcal{H}\}$ as the collection of actions under a given $\mathbf{Q}$ for all possible $\mathbf{H}\in\mathcal{H}$. The complete policy $\Omega$ is therefore equal to the union of all partitioned actions, i.e., $\Omega=\bigcup_{\mathbf{Q}}\Omega(\mathbf{Q})$.
Based on the action partitioning, we can transform the POMDP problem into a regular infinite-horizon average cost MDP. Furthermore, the optimal control policy $\Omega^*$ can be obtained by solving an [*equivalent Bellman equation*]{} which is summarized in the theorem below.
\[Thm:MDP\_cond\] The optimal control policy $\Omega^*=
(\Omega_{\mathbf{p}}^*, \Omega_{\mathbf{s}}^*)$ in problem \[prob:delay\] can be obtained by solving the [*equivalent Bellman equation*]{} given by: $$\label{eq:bellman_cond} V(\mathbf{Q})+\theta =
\min_{\Omega(\mathbf{Q})}\Big[\hat{g}(\mathbf{Q},\Omega(\mathbf{Q}))+\sum\limits_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}V(\mathbf{Q}^{\prime})\Big]$$ where $\hat{g}(\mathbf{Q},\Omega(\mathbf{Q}))=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})$ is the per-slot cost function, and the transition kernel is given by $\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}=\mathbb{E}_{\mathbf{H}}\left[\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi})\}\right]$, where $\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi})\}$ is given by $$\begin{array}{l}
\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi})\}=\\
\quad\quad\left\{\begin{array}{ll} \Pr\{\mathbf{A}\} & \text{if
$\mathbf{Q}^{\prime}=\Big[\big(\mathbf{Q}-\mathbf{U}\big)^++\mathbf{A}\Big]_{\bigwedge N_Q}$}\\
0 & \text{otherwise}
\end{array}
\right.
\end{array}$$ where $\mathbf{U}=\{\mathbf{U}_m\}_{m=1}^M$, and $\mathbf{U}_m=\{U_{(m,k)}\}_{k=1}^K$, and $U_{(m,k)}=
R_{(m,k)}(\boldsymbol{\chi},\Omega(\boldsymbol{\chi}))\tau$ for $k\in\mathcal{K}_m$. Suppose $\Omega^*(\mathbf{Q})=\{\mathbf{p}^*({\mathbf{Q}}),\bigcup_{\mathbf{H}}\mathbf{s}^*(\mathbf{Q},\mathbf{H})\}$ is a solution that solves the Bellman equation in (\[eq:bellman\_cond\]), the optimal control policy for the original Problem \[prob:delay\] is given by: $\Omega_{\mathbf{p}}^*=\bigcup_{\mathbf{Q}}\{\mathbf{p}^*(\mathbf{Q})\}$ and $\Omega_{\mathbf{s}}^*=\bigcup_{\mathbf{Q},\mathbf{H}}\{\mathbf{s}^*(\mathbf{Q},\mathbf{H})\}$. The value function $V(\mathbf{Q})$ that solves (\[eq:bellman\_cond\]) is a component-wise monotonic increasing function.
Please refer to Appendix A.
Note that solving (\[eq:bellman\_cond\]) will obtain an ICI management policy $\Omega_{\mathbf{p}}^*$ that is a function of QSI $\mathbf{Q}$ and a user scheduling policy $\Omega_{\mathbf{s}}^*$ that is a function of the QSI and CSI $\{\mathbf{Q},\mathbf{H}\}$. We shall illustrate this with a simple example below.
Suppose there are two BSs with equal transmitting power ($P^m_{\max}=P,\forall m$), and there are three ICI management control patterns in $\mathcal{P}$, given by $\mathbf{p}_1=\{p_1=1,p_2=0\}$ (BS 1 is active), $\mathbf{p}_2=\{p_1=0,p_2=1\}$ (BS 2 is active) and $\mathbf{p}_3=\{p_1=1,p_2=1\}$ (both BSs are active). Assume deterministic arrival where one bit will always arrive at each slot, i.e., $\Pr\{A_{(m,k)}=1\}=1$. The number of users served by each BS is $K=2$. The path loss $L_{(m,k)}^n=1$ for all $\{k,n,m\}$, and the small scale fading gain is chosen from two values $\{H_{g},H_{b}\}$ with equal probability. As a result, the global CSI state space[^8] is $\mathcal{H}=\{H_{g},H_{b}\}^{M^2K}$. Note that the cardinality of CSI state space $\mathcal{H}$ is $|\mathcal{H}|=2^{M^2K}=256$. Given a realization of the global QSI $\mathbf{Q}$, the [*partitioned actions*]{} (following Definition \[def:partitioned action\]) is given by: $$\Omega(\mathbf{Q})=\{\mathbf{p}(\mathbf{Q}),
\mathbf{s}(\mathbf{Q},\mathbf{H}^{(1)}),\cdots,
\mathbf{s}(\mathbf{Q},\mathbf{H}^{(256)})\}$$ Using Theorem \[Thm:MDP\_cond\], the optimal partitioned action $\Omega^*(\mathbf{Q})$ is given by solving the right hand side (RHS) of (\[eq:bellman\_cond\]): $$\begin{array}{l}
\Omega^*(\mathbf{Q})=\operatorname*{\arg\min}\limits_{\{\mathbf{p}(\mathbf{Q}),
\{\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}_{i=1}^{256}\}}\sum_{\mathbf{Q}^{\prime}}\sum_{\mathbf{H}^{(i)}
\in \mathcal{H} }\\
\quad\Big[\Pr\{\mathbf{H}^{(i)}\}
\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H}^{(i)},\mathbf{p}(\mathbf{Q}),
\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}V(\mathbf{Q}^{\prime})\Big]
\end{array}$$ where $$\begin{array}{l}
\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H}^{(i)},\mathbf{p}(\mathbf{Q}),\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}=\\
\quad\quad\quad\quad\left\{\begin{array}{ll} 1 & \text{if }
\mathbf{Q}^{\prime} =
\Big[\big(\mathbf{Q}-\mathbf{U}\big)^++\mathbf{1}\Big]_{\bigwedge N_Q}\\
0 & \text{otherwise}
\end{array}\right.
\end{array}$$ and $\mathbf{U}=\{U_{(1,1)},U_{(1,2)};U_{(2,1)},U_{(2,2)}\}$ is the number of departure bits. For a given ICI management control $\mathbf{p}(\mathbf{Q})=\mathbf{p}$, the optimal user scheduling policy $\{\mathbf{s}^*(\mathbf{Q},\mathbf{H}^{(i)})\}$ is $$\label{eq:tmp}\begin{array}{l}
\{\mathbf{s}^*(\mathbf{Q},\mathbf{H}^{(i)})\}=
\operatorname*{\arg\min}\limits_{\{\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}_{i=1}^{256}}\sum_{\mathbf{Q}^{\prime}}
\sum_{\mathbf{H}^{(i)} \in
\mathcal{H}}\\
\quad\Big[\Pr\{\mathbf{H}^{(i)}\}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H}^{(i)},\mathbf{p},
\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}V(\mathbf{Q}^{\prime})\Big]
\end{array}$$ Observe that the RHS of (\[eq:tmp\]) is a decoupled objective function w.r.t. the variables $\{\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}_{i=1}^{256}$ and hence, applying standard decomposition theory, $$\label{eq:opt_s}\begin{array}{l}
\mathbf{s}^*(\mathbf{Q},\mathbf{H}^{(i)})=\\
\quad\operatorname*{\arg\min}\limits_{\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})}\sum\limits_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H}^{(i)},\mathbf{p},\mathbf{s}(\mathbf{Q},\mathbf{H}^{(i)})\}V(\mathbf{Q}^{\prime})
\end{array}$$ As a result, the optimal ICI management control policy $\mathbf{p}^*(\mathbf{Q})$ is given by: $$\label{eq:opt_p}\begin{array}{l}
\mathbf{p}^*(\mathbf{Q})=\operatorname*{\arg\min}_{\mathbf{p}(\mathbf{Q})}\sum\nolimits_{\mathbf{Q}^{\prime}}
\sum\nolimits_{\mathbf{H}^{(i)}\in\mathcal{H}}\\
\quad\Big[\Pr\{\mathbf{H}^{(i)}\}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H}^{(i)},\mathbf{p}(\mathbf{Q}),
\mathbf{s}^*(\mathbf{Q},\mathbf{H}^{(i)})\}V(\mathbf{Q}^{\prime})\Big]
\end{array}$$ where $\mathbf{s}^*(\mathbf{Q},\mathbf{H}^{(i)})$ given in (\[eq:opt\_s\]) is the optimal user scheduling policy under the ICI management control policy $\mathbf{p}(\mathbf{Q})$. Using Theorem \[Thm:MDP\_cond\], the optimal ICI management control and user selection control of the original Problem \[prob:delay\] for a CSI realization $\mathbf{H}^{(i)}$ and QSI realization $\mathbf{Q}$ are given by $\mathbf{p}^*(\mathbf{Q})$ and $\mathbf{s}^*(\mathbf{Q},
\mathbf{H}^{(i)})$ respectively.
Distributive Value Function and $\mathbb{Q}$-factor Online Learning {#sec:learning}
===================================================================
The solution in Theorem \[Thm:MDP\_cond\] requires the knowledge of the value function $V(\mathbf{Q})$. However, obtaining the value function is not trivial as solving the Bellman equation (\[eq:bellman\_cond\]) involves solving a very large system of the nonlinear fixed point equations (corresponding to each realization of $\mathbf{Q}$ in (\[eq:bellman\_cond\])). Brute-force solution of $V(\mathbf{Q})$ require huge complexity, centralized implementation and knowledge of global CSI and QSI at the BSC. This will also induce huge signaling overhead because the QSI of all the users are maintained locally at the $M$ BSs. In this section, we shall propose a decentralized solution via distributive stochastic learning following the structure as illustrated in Fig. \[fig:learning\_structure\]. Moreover, we shall prove that the proposed distributive stochastic learning algorithm will converge almost-surely.
Post-Decision State Framework
-----------------------------
In this section, we first introduce the post-decision state also used framework, also used in [@Thesis:Salodkar] and the references therein, to lay ground for developing the online learning algorithm. The post-decision state is defined to be the virtual system state immediately after making an action but before the new bits arrive. For example, $\boldsymbol{\chi}=\{\mathbf{Q},\mathbf{H}\}$ is the state at the beginning of some time slot (also called the [*pre-decision state*]{}), and making an action $\Omega(\boldsymbol{\chi})=\{\mathbf{p},\mathbf{s}\}$, the post-decision state immediately after the action is $\widetilde{\boldsymbol{\chi}}=\{\widetilde{\mathbf{Q}},\mathbf{H}\}$, where the transition to $\widetilde{\mathbf{Q}}$ is given by $\widetilde{\mathbf{Q}}=\big(\mathbf{Q}-\mathbf{U}\big)^+$. If new arrivals $\mathbf{A}$ occur in the post-decision state, and the CSI changes to $\mathbf{H}^{\prime}$, then the system reaches the next actual state, i.e., pre-decision state, $\boldsymbol{\chi}^{\prime}=\{\big[\widetilde{\mathbf{Q}}+\mathbf{A}\big]_{\bigwedge
N_Q},\mathbf{H}^{\prime}\}$.
Using the action partitioning and defining the value function $\widetilde{\mathbf{V}}$ on post-decision state $\widetilde{\mathbf{Q}}$ (where pre-decision state is $\{\mathbf{Q}=\big[\widetilde{\mathbf{Q}}+\mathbf{A}\big]_{\bigwedge
N_Q},\mathbf{H}\}$), $\widetilde{\mathbf{V}}$ will satisfy the post-decision state Bellman equation[@Thesis:Salodkar] $$\label{eq:bellman_post}\begin{array}{l}
\widetilde{V}(\widetilde{\mathbf{Q}})+\theta=\sum_{\mathbf{A}}\Pr\{\mathbf{A}\}\bigg\{
\min_{\Omega(\mathbf{Q})}\Big[\widetilde{g}(\mathbf{Q},\Omega(\mathbf{Q}))\\
\quad\quad\quad\quad+\sum\nolimits_{\widetilde{\mathbf{Q}}^{\prime}}\Pr\{\widetilde{\mathbf{Q}}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}\widetilde{V}(\widetilde{\mathbf{Q}}^{\prime})
\Big]\bigg\}
\end{array}$$ where $\widetilde{g}(\mathbf{Q},\Omega(\mathbf{Q}))=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})$, $\Pr\{\widetilde{\mathbf{Q}}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}=\mathbb{E}_{\mathbf{H}}[\Pr\{\widetilde{\mathbf{Q}}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\mathbf{Q})\}]$, and $\widetilde{\mathbf{Q}}^{\prime}$ is the next post-decision state transited from $\mathbf{Q}$. As Theorem \[Thm:MDP\_cond\], $\widetilde{V}(\widetilde{\mathbf{Q}})$ is also a component-wise monotonic increasing function. The optimal policy is obtained by solving the RHS of Bellman equation (\[eq:bellman\_post\]).
Distributive User Scheduling Policy on the CSI Time Scale
---------------------------------------------------------
To reduce the size of the state space and to decentralize the user scheduling, we approximate $\widetilde{V}(\widetilde{\mathbf{Q}})$ in (\[eq:bellman\_post\]) by the sum of per-user post-decision state value function[^9] $\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})$, i.e., $$\label{eq:linear_value}
\widetilde{V}(\widetilde{\mathbf{Q}})\approx\sum\nolimits_{m,k}\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})$$ where $\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})$ is defined as the [*fixed point*]{} of the following per-user fixed point equation: $$\label{eq:user_selection}\begin{array}{l}
\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})+\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)}^I)=\\
\sum\nolimits_{A_{(m,k)}}\Pr\{A_{(m,k)}\}\Big[ \beta_{(m,k)}f(Q_{(m,k)})+\\
\sum\limits_{\widetilde{Q}_{(m,k)}^{\prime}}\Pr\{\widetilde{Q}_{(m,k)}^{\prime}|Q_{(m,k)},s_{(m,k)}=1,\widetilde{\mathbf{p}}_m^I\}\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)}^{\prime})
\Big]
\end{array}$$ where $Q_{(m,k)}=\widetilde{Q}_{(m,k)}+A_{(m,k)}$ is the pre-decision state, $s_{(m,k)}=1$ means that the user $k$ is scheduled to transmit at BS $m$, $\widetilde{Q}_{(m,k)}^I\in\{0,\cdots,N_Q\}$ is a reference state and $\widetilde{\mathbf{p}}_m^I\in\mathcal{P}_m$ is a reference ICI management pattern (with the BS $m$ active). The per-user value function $\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})$ is obtained by the proposed distributive online learning algorithm (explained in section \[subsec:learning\_algorithm\]). Note that the state space for the value function of $\widetilde{V}(\widetilde{\mathbf{Q}})$ is substantially reduced from $(N_Q+1)^{MK}$ (exponential growth w.r.t the number of all mobile users $MK$) to $MK(N_Q+1)$ (linear growth w.r.t the number of all mobile users).
\[cor:dstr\_policy\] Using the linear approximation in (\[eq:linear\_value\]), the user scheduling action of BS $m\in\mathcal{M}_{\mathbf{p}}$ under any given ICI management pattern $\mathbf{p}$ (obtained by solving the RHS of Bellman equation (\[eq:bellman\_post\])) is given by: $$\label{eq:sm} \mathbf{s}_{m}=\{s_{k^*}=1, s_{(m,k)}=0, \forall k\neq
k^*\text{ and } k,k^*\in\mathcal{K}_m\}$$ where $k^*=\arg\max_{k\in\mathcal{K}_m}\widetilde{\delta}_{(m,k)}(Q_{(m,k)})$, and $\widetilde{\delta}_{(m,k)}(Q_{(m,k)})=\widetilde{V}_{(m,k)}(Q_{(m,k)})-\widetilde{V}_{(m,k)}((Q_{(m,k)}-U_{(m,k)})^+)$[^10]. $U_{(m,k)}=\log_2\left(1+\frac{\xi
\phi_{(m,k)}}{\varphi_{(m,k)}}\right)\tau$, where $\varphi_{(m,k)}=\sum\limits_{n\neq m,n\in\mathcal{M}_{\mathbf{p}}}
P^n_{\max}H^n_{(m,k)}L_{(m,k)}^n+N_0W$ is the power sum of interference and noise, and $\phi_{(m,k)}=P^m_{\max}H^m_{(m,k)}L_{(m,k)}^m$ is the signal power.
Please refer to Appendix B.
The user scheduling action in (\[eq:sm\]) is both function of local CSI and QSI. Specifically, the number of bits to be delivered $U_{(m,k)}$ is controlled by the local CSI $\mathbf{H}_{(m,k)}$, and local QSI $Q_{(m,k)}$ will determine $\widetilde{\delta}_{(m,k)}(Q_{(m,k)})$. Each user estimates $\varphi_{(m,k)}$ and $\phi_{(m,k)}$ in the preamble phase, and sends $U_{(m,k)}$ to the associated BS $m$ according to the process as indicated in Fig.\[fig:learning\_structure\].
ICI Management Control Policy on the QSI Time Scale
---------------------------------------------------
To determine the ICI management control policy, we define the $\mathbb{Q}$-factor as follows [@Cao:2007]: $$\label{eq:q_factor_orgn}\begin{array}{l}
\mathbb{Q}(\mathbf{Q},\mathbf{p})=\sum\nolimits_{m,k}\beta_{(m,k)}f(Q_{(m,k)})+\\
\quad\quad\quad\sum\nolimits_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{p}\}
\min_{\mathbf{p}^{\prime}}\mathbb{Q}(\mathbf{Q}^{\prime},\mathbf{p}^{\prime})-\theta
\end{array}$$ where $\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{p}\}$ is the transition probability from current QSI $\mathbf{Q}$ to $\mathbf{Q}^{\prime}$, given current action $\mathbf{p}$, and $\theta$ is a constant. Note that the $\mathbb{Q}$-factor $\mathbb{Q}(\mathbf{Q},\mathbf{p})$ represents the potential cost of applying a control action $\mathbf{p}$ at the current QSI $\mathbf{Q}$ and applying the action $\arg\min_{\mathbf{p}^{\prime}}\mathbb{Q}(\mathbf{Q}^{\prime},\mathbf{p}^{\prime})$ for any system state $\mathbf{Q}^{\prime}$ in the future. Similar to (\[eq:linear\_value\]), we approximate the $\mathbb{Q}$-factor in (\[eq:q\_factor\_orgn\]) with a sum of per-user $\mathbb{Q}$-factor, i.e, $$\label{eq:q_approx}
\mathbb{Q}(\mathbf{Q},\mathbf{p})\approx\sum\nolimits_{m,k}\mathbb{Q}_{(m,k)}(Q_{(m,k)},\mathbf{p})$$ where $\mathbb{Q}_{(m,k)}$ is defined as the [*fixed point*]{} of the following per-user fixed point equation: $$\label{eq:Q_linear}\begin{array}{l}
\mathbb{Q}_{(m,k)}(Q_{(m,k)},\mathbf{p})=\\
\beta_{(m,k)}f(Q_{(m,k)})-
\mathbb{Q}_{(m,k)}(Q_{(m,k)}^I,\mathbf{p}^I_m)+\sum\limits_{Q_{(m,k)}^{\prime}}\\
\Big[\Pr\{Q_{(m,k)}^{\prime}|Q_{(m,k)},
s_{(m,k)}=1,\mathbf{p}\}\min\limits_{\mathbf{p}^{\prime}}\mathbb{Q}_{(m,k)}(Q_{(m,k)}^{\prime},
\mathbf{p}^{\prime})\Big]
\end{array}$$ where $\Pr\{Q_{(m,k)}^{\prime}|Q_{(m,k)},s_{(m,k)}=1,\mathbf{p}\}=\mathbb{E}_{\mathbf{H}_{(m,k)}}[\Pr\{Q_{(m,k)}^{\prime}|Q_{(m,k)},s_{(m,k)}=1,\mathbf{H}_{(m,k)},\mathbf{p}\}]$. $Q_{(m,k)}^I\in\{0,\cdots,N_Q\}$ is a reference state and $\mathbf{p}^I_m\in\mathcal{P}$ is a reference ICI management control pattern. The per-user $\mathbb{Q}$-factor $\mathbb{Q}_{(m,k)}$ is obtained by the proposed distributive online learning algorithm (explained in section \[subsec:learning\_algorithm\]). The BSC collects the per-BS $\mathbb{Q}$-information $\mathbb{Q}_m^t(\mathbf{p})=\sum_{(m,k)}\mathbb{Q}_{(m,k)}^t(Q_{(m,k)}^t,\mathbf{p})$ at the beginning of slot $t$, and the ICI management control policy is given by: $$\label{eq:learn_pattern_pol}
\mathbf{p}^t=\operatorname*{\arg\min}\nolimits_{\mathbf{p}}\sum\nolimits_m\mathbb{Q}_m^t(\mathbf{p})$$
In order to reduce the communication overhead between the $M$ BSs and the BSC, we could further partition the local QSI space into $N$ regions[^11] ($\mathcal{Q}_m=\bigcup_{n=1}^N \mathcal{R}_n$) as illustrated in Fig. \[fig:pkt\_region\]. At the beginning of the $t$-th slot, the $m$-th BS will update the BSC of the per-BS $\mathbb{Q}$-information if its QSI state belongs to a new region. Hence, the per-BS $\mathbb{Q}$-information at the BSC is updated according to the following dynamics: $$\label{eq:BSC_q_update}\begin{array}{l}
\mathbb{Q}_m^t(\mathbf{p})=\\
\left\{\begin{array}{ll}
\sum\limits_{m,k}\mathbb{Q}_{(m,k)}^t(Q_{(m,k)}^t,\mathbf{p}) &
\text{if $\mathbf{Q}_m^t\in\mathcal{R}_n,\mathbf{Q}_m^{t-1}\not\in
\mathcal{R}_n$}\\
\mathbb{Q}_m^{t-1}(\mathbf{p}) & \text{otherwise}
\end{array}
\right.
\end{array}$$
The communication overhead between the $M$ BS and the BSC is reduced from $O((N_Q+1)^{MK}+(N_H)^{M^2K})$ (exponential growth w.r.t the number of users $K$) to $O(M(\alpha)^{|\mathcal{P}|})$ for some constant $\alpha$ (O(1) w.r.t. $K$), where $N_H$ is the cardinality of the CSI state space for one link.
Online Per-User Value Function and Per-User $\mathbb{Q}$-factor Learning Algorithm {#subsec:learning_algorithm}
----------------------------------------------------------------------------------
The system procedure for distributive online learning is given below:
- [**Initialization**]{}: Each BS initiates the per-user value function and $\mathbb{Q}$-factor for its $K$ users, denoted as $\{\widetilde{V}_{(m,k)}^0\}$ and $\{\mathbb{Q}_{(m,k)}^0\}$, where $\widetilde{V}_{(m,k)}^0(Q^{\prime}_{(m,k)})>\widetilde{V}_{(m,k)}^0(Q_{(m,k)}),
\forall Q^{\prime}_{(m,k)}>Q_{(m,k)}$.
- [**ICI Management Control**]{}: At the beginning of the $t$-th slot, the BSC updates the $\mathbb{Q}$-information $\mathbb{Q}_m^t(\mathbf{p})$ as (\[eq:BSC\_q\_update\]) and determines the ICI management pattern as (\[eq:learn\_pattern\_pol\]).
- [**User Scheduling**]{}: If $m\in\mathcal{M}_{\mathbf{p}^t}$, BS $m$ is selected to transmit. The user scheduling policy is determined according to (\[eq:sm\]).
- [**Local Per-user Value Function and Per-user $\mathbb{Q}$-factor Update**]{}: Based on the current observations, each of the $M$ BSs updates the per-user value function $\widetilde{V}_{(m,k)}$ and the per-user $\mathbb{Q}$-factor $\mathbb{Q}_{(m,k)}$ according to Algorithm \[alg:learning\].
Fig. \[fig:learning\_structure\] illustrates the above procedure by a flowchart. The algorithm for the per-user value function and per-user $\mathbb{Q}$-factor update is given below:
\[alg:learning\] Let $\widetilde{\mathbf{Q}}_m$ and $\mathbf{Q}_m$ be the current observation of post-decision and pre-decision states respectively, $\mathbf{A}_m$ be the current observation of new arrival, $\{\mathbf{H}_{(m,k)}\}_{k=1}^K$ be the current observation of the local CSI, and $\mathbf{p}$ is the realization of the ICI management control pattern. The online learning algorithm for user $k\in\mathcal{K}_m$ is given by $$\label{eq:learn_value_f}\begin{array}{l}
\widetilde{V}_{(m,k)}^{t+1}(\widetilde{Q}_{(m,k)})=\\
\left\{\begin{array}{ll}
\widetilde{V}_{(m,k)}^t(\widetilde{Q}_{(m,k)})+\gamma(t)
\Bigl[\beta_{(m,k)}f(\widetilde{Q}_{(m,k)}+& \\
A_{(m,k)})+\widetilde{V}_{(m,k)}^{t}(\widetilde{Q}_{(m,k)}+A_{(m,k)}-U_{(m,k)})
&\text{if } \mathbf{p} =
\widetilde{\mathbf{p}}_m^I \\
\quad\quad-\widetilde{V}_{(m,k)}^{t}(\widetilde{Q}_{(m,k)}^I)-\widetilde{V}_{(m,k)}^{t}(\widetilde{Q}_{(m,k)})\Bigr]
&\\
\widetilde{V}_{(m,k)}^t(\widetilde{Q}_{(m,k)}) &
\text{otherwise}
\end{array}\right.\end{array}$$ $$\label{eq:learn_q_factor}\begin{array}{l}
\mathbb{Q}_{(m,k)}^{t+1}(Q_{(m,k)},\mathbf{p})=\mathbb{Q}_{(m,k)}^{t}(Q_{(m,k)},\mathbf{p})+
\gamma(t)\Bigl[\beta_{(m,k)}\\
\cdot f(Q_{(m,k)})-\mathbb{Q}_{(m,k)}^t(Q_{(m,k)}^I,\mathbf{p}_m^I)-\mathbb{Q}_{(m,k)}^{t}(Q_{(m,k)},\mathbf{p})\\
+\min_{\mathbf{p}^{\prime}}\mathbb{Q}_{(m,k)}^t(Q_{(m,k)}-U_{(m,k)}+A_{(m,k)},\mathbf{p}^{\prime})\Bigr]
\end{array}$$ where $U_{(m,k)}$ is the number of bits to be delivered for user $k$ (given in Corollary \[cor:dstr\_policy\] and depends indirectly on the local CSI observations $\mathbf{H}_{(m,k)}$), $\{\widetilde{Q}_{(m,k)}^I,\widetilde{\mathbf{p}}_m^I\}$ and $\{Q_{(m,k)}^I,\mathbf{p}_m^I\}$ are the reference state and reference ICI management pattern for the value function $\widetilde{V}_{(m,k)}$ in (\[eq:user\_selection\]) and $\mathbb{Q}$-factor $\mathbb{Q}_{(m,k)}$ in (\[eq:Q\_linear\]) respectively. $\gamma(n)$ is diminishing positive step size sequence satisfying $\sum_n\gamma(n)=\infty,\sum_n\gamma^2(n)<\infty$.
The proposed learning scheme only requires the observations of the local QSI $\widetilde{\mathbf{Q}}_m$ and $\mathbf{Q}_m$. Furthermore, each users only need to feedback $U_{(m,k)}$ instead of the local CSI $\mathbf{H}_m$, which is of similar feedback loading compared with HSDPA systems.
Convergence Analysis
--------------------
In this section we will establish the convergence proof of the proposed per-user learning algorithm \[alg:learning\]. We first define a mapping on the post-decision state $\widetilde{Q}_{(m,k)}$ as $$\label{eq:T_{(m,k)}}\begin{array}{l}
T_{(m,k)}(\widetilde{\mathbf{V}}_{(m,k)},\widetilde{Q}_{(m,k)})=\widetilde{g}_{(m,k)}(\widetilde{Q}_{(m,k)})+\\
\sum\limits_{\widetilde{Q}_{(m,k)}^{\prime}}\Pr\{\widetilde{Q}_{(m,k)}^{\prime}|\widetilde{Q}_{(m,k)},s_{(m,k)}=1,\widetilde{\mathbf{p}}_m^I\}\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)}^{\prime})
\end{array}$$ where $Q_{(m,k)}=\widetilde{Q}_{(m,k)}+A_{(m,k)}$ is the pre-decision state, $\widetilde{g}_{(m,k)}(\widetilde{Q}_{(m,k)})=
\mathbb{E}_{A_{(m,k)}}\big[\beta_{(m,k)}f(\widetilde{Q}_{(m,k)}+A_{(m,k)})\big]$, and $\Pr\{\widetilde{Q}_{(m,k)}^{\prime}|\widetilde{Q}_{(m,k)},s_{(m,k)}=1,\widetilde{\mathbf{p}}_m^I\}=
\mathbb{E}_{\mathbf{H}_{(m,k)},A_{(m,k)}}[\Pr\{\widetilde{Q}_{(m,k)}^{\prime}|Q_{(m,k)},\mathbf{H}_{(m,k)},
s_{(m,k)}=1,\widetilde{\mathbf{p}}_m^I\}]$. The vector form of the mapping is given by: $$\label{eq:Tk_vector}
\mathbf{T}_{(m,k)}(\widetilde{\mathbf{V}}_{(m,k)})=\widetilde{\mathbf{g}}_{(m,k)}+\mathbf{P}_{(m,k)}\widetilde{\mathbf{V}}_m$$ where $\mathbf{P}_{(m,k)}$ is $(N_Q+1)\times(N_Q+1)$ transition matrix for the post-decision state queue of the user $k$. and $\widetilde{\mathbf{V}}_{(m,k)}$ are $(N_Q+1)\times1$ vectors. Specifically, we have the following lemma for the per-user value function learning in (\[eq:learn\_value\_f\]).
\[lem:converge\_value\] The update of the per-user value function $\widetilde{\mathbf{V}}_{(m,k)}^t$ will converge almost-surely in the proposed learning algorithm \[alg:learning\], i.e., $\lim_{t\rightarrow\infty}\widetilde{\mathbf{V}}_{(m,k)}^t=\widetilde{\mathbf{V}}_{(m,k)}^{\infty},\forall
k,m$, and $\widetilde{V}_{(m,k)}^{\infty}(\widetilde{Q}_{(m,k)})$ is a monotonic increasing function satisfying: $$\label{eq:con_value}
\widetilde{\mathbf{V}}_{(m,k)}^{\infty}+\widetilde{V}_{(m,k)}^{\infty}(\widetilde{Q}_{(m,k)}^I)\mathbf{e}=\mathbf{T}_{(m,k)}(\widetilde{\mathbf{V}}_{(m,k)}^{\infty})$$
Please refer to Appendix C.
Note that (\[eq:con\_value\]) is equivalent to the per-user fixed point equation in (\[eq:user\_selection\]). This result illustrates that the proposed online distributive learning in (\[eq:learn\_value\_f\]) can converge to the target per-user fixed point solution in (\[eq:user\_selection\]). We define a mapping for the per-user $\mathbb{Q}$-factor $\mathbb{Q}_{(m,k)}$ as $$\label{eq:Q_map}\begin{array}{l}
T_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)},Q_{(m,k)},\mathbf{p})=
\beta_{(m,k)}f(Q_{(m,k)})+\sum\limits_{Q_{(m,k)}^{\prime}}\\
\Big[\Pr\{Q_{(m,k)}^{\prime}|Q_{(m,k)},s_{(m,k)}=1,
\mathbf{p}\}\min\limits_{\mathbf{p}^{\prime}}\mathbb{Q}_{(m,k)}(Q_{(m,k)}^{\prime},\mathbf{p}^{\prime})\Big]
\end{array}$$ Specifically, we have following lemma for the $\mathbb{Q}$-factor online learning in (\[eq:learn\_q\_factor\]).
\[lem:q\_converge\] The update of per-user $\mathbb{Q}$-factor $\mathbb{Q}_{(m,k)}$ will converge almost-surely in the proposed learning algorithm \[alg:learning\], i.e., $\lim_{t\rightarrow\infty}\mathbb{Q}_{(m,k)}^t=\mathbb{Q}_{(m,k)}^{\infty},\forall
k,m$, where the steady state $\mathbb{Q}$-factor $\{\mathbb{Q}_{(m,k)}^{\infty}\}$ satisfy: $$\label{eq:q_infty}\begin{array}{l}
\mathbb{Q}_{(m,k)}^{\infty}(Q_{(m,k)},\mathbf{p})=\\
\quad\quad
T_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)}^{\infty},Q_{(m,k)},\mathbf{p})-\mathbb{Q}_{(m,k)}^{\infty}(Q_{(m,k)}^I,\mathbf{p}^I_m)
\end{array}$$
Please refer to Appendix D.
Note that (\[eq:q\_infty\]) is equivalent to the per-user fixed point equation for $\mathbb{Q}_{(m,k)}$ in (\[eq:Q\_linear\]). This result illustrates that the proposed online distributive learning in (\[eq:learn\_q\_factor\]) can converge to the target per user fixed point solution in (\[eq:Q\_linear\]).
Lemma \[lem:converge\_value\] and \[lem:q\_converge\] only established the convergence of the proposed online learning algorithm. Strictly speaking, the converged result is not optimal due to the linear approximation of the value function $\widetilde{V}(\widetilde{\mathbf{Q}})$ and the $\mathbb{Q}$-factor $\mathbb{Q}(\mathbf{Q},\mathbf{p})$ in (\[eq:linear\_value\]) and (\[eq:q\_approx\]) respectively. The linear approximation is needed for distributive implementation. As illustrated in Fig. \[fig:opt\], the proposed distributive solution has close-to-optimal performance compared with brute-force centralized solution of the Bellman equation in (\[eq:bellman\_cond\]).
Simulation and Discussion {#sec:simulation}
=========================
In this section, we shall compare the proposed distributive queue-aware intra-cell user scheduling and ICI management control scheme with three baselines. Baseline 1 refers to the [*CSIT only*]{} scheme, where the user scheduling are adaptive to the CSIT only so as to optimize the achievable data rate. Baseline 2 refers to a throughput optimal policy (in stability sense) for the user scheduling, namely the [*Dynamic Backpressure*]{} scheme [@Georgiadis-Neely-Tassiulas:2006]. In both baseline 1 and 2, the traditional frequency reuse scheme (frequency reuse factor equals 3) is used for inter-cell interference management. Baseline 3 refers to the [*time-scale decomposition*]{} scheme proposed in [@pattern:2009], where the sets of possible ICI management patterns $\mathcal{P}$ is the same as the proposed scheme. In the simulation, we consider a two-tier celluar network composed of 19 BSs as in [@pattern:2009], each has a coverage of 500m. Channel models are implemented according to the Urban Macrocell Model in 3GPP and Jakes’ Rayleigh fading model. Specifically, the path loss model is given by $PL = 34.5 + 35\log_{10}(r)$, where $r$ (in m) is the distance from the transmitter to the receiver. The total BW is 10MHz. We consider Poisson packet arrival with average arrival rate $\mathbb{E}[A_{(m,k)}]=\lambda_{(m,k)}$ (packets/slot) and exponentially distributed random packet size $\overline{N}_{(m,k)}$ with $\mathbb{E}[\overline{N}_{(m,k)}]=5$Mbits. The scheduling slot duration $\tau$ is 5ms. The maximum buffer size $N_Q$ is 9 (in packets), where each user’s QSI is partitioned into 4 regions, given by $\big\{\{0,1,2\};\{3,4,5\};\{6,7\};\{8,9\}\big\}$. The cost function is given by $f(Q_{(m,k)})=\frac{Q_{(m,k)}}{\lambda_{(m,k)}}$ for all the users in the simulations.
Performance w.r.t. Transmit Power
---------------------------------
Fig.\[fig:delay\_tx\_pwr\] and Fig.\[fig:delay\_full\_buffer\] illustrate the performance of average delay and packet dropping probability (conditioned on packet arrival) per user versus transmit power $P_{\max}^m$ respectively. The number of users per BS $K=3$, and the average arrival rate $\lambda_{(m,k)}=1$. Note that the average delay and packet dropping probability of all the schemes decreases as the transmit power increases, and there is significant performance gain of the proposed scheme compared to all baselines. This gain is contributed by the QSI-aware user scheduling as well as ICI management control.
Performance w.r.t. Loading
--------------------------
Fig.\[fig:delay\_loading\] illustrates the average delay versus per user loading (average arrival rate $\lambda_{(m,k)}$) at transmit power of $P_{\max}^m=30$dBm and the number of users per BS $K=3$. It can also be observed that the proposed scheme achieved significant gain over all the baselines across a wide range of input loading.
Cumulative Distribution Function (CDF) of the Queue Length
----------------------------------------------------------
Fig.\[fig:delay\_cdf\] illustrates the Cumulative Distribution Function (CDF) of the queue length per user with transmit power $P_{\max}^m=25$dBm. The number of users per BS is $K=3$ and the average arrival rate $\lambda_{(m,k)}=1$. It can be also be verified that the proposed scheme achieves not only a smaller average delay but also a smaller delay percentile compared with the other baselines.
Convergence Performance
-----------------------
Fig.\[fig:delay\_learning\_vq\] illustrates the average delay per user versus the scheduling slot index with transmit power $P_{\max}^m=35$dBm. The number of users per BS is $K=3$ and the average arrival rate $\lambda_{(m,k)}=1.5$. It can be observed that the convergence rate of the online algorithm is quite fast. For example, the delay performance of the proposed scheme already out-performs all the baselines at the $400$-th slot. Furthermore, the delay performance at $400$-th slot is already quite close to the converged average delay. Finally, unlike the conventional iterative NUM approach where the iterations are done offline within the coherence time of the CSI, the proposed iterative algorithm is updated over the same time scale of the CSI and QSI updates. Moreover, the iterative algorithm is online, meaning that useful payload are transmitted during the iterations.
Summary {#sec:summary}
=======
In this paper, we study the design of a distributive queue-aware intra-cell user scheduling and inter-cell interference management control design for a delay-optimal celluar downlink system. We first model the problem as an infinite horizon average reward POMDP, which is NP-hard in general. By exploiting special problem structure, we derive an equivalent Bellman equation to solve the POMDP problem. To address the distributive requirement and the issue of dimensionality and computation complexity, we derive a distributive online stochastic learning algorithm, which only requires local QSI and local CSI at each of the $M$ BSs. We show that the proposed learning algorithm converges almost-surely and has significant gain compared with various baselines. The proposed algorithm only has linear complexity order $O(MK)$.
Appendix A: Proof of Theorem \[Thm:MDP\_cond\] {#app:V_increase .unnumbered}
==============================================
Based on the action partitioning, we can associate the MDP formulation in our delay-optimal control problem as follows:
- [**State Space:**]{} The system state of the MDP is global QSI $\mathbf{Q}\in\mathcal{Q}$.
- [**Action Space:**]{} The action on the system state $\mathbf{Q}$ is the partitioned action $\Omega(\mathbf{Q})$ given in Definition \[def:partitioned action\], and the action space is $\{\mathcal{P},\mathcal{S}\}$.
- [**Transition Kernel:**]{} The transition kernel is $\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}=\mathbb{E}_{\mathbf{H}}\left[\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi})\}\right]$, where $\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi})\}$ is given by (\[eq:sys\_tran\]).
- [**Per-Slot Cost:**]{} The per-slot cost function is $\hat{g}(\mathbf{Q},\Omega(\mathbf{Q}))=\mathbb{E}_{\mathbf{H}}[g(\mathbf{Q},\mathbf{H},\Omega(\boldsymbol{\chi}))]=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})$.
Therefore, the optimal partitioned action $\Omega^*(\mathbf{Q})$ can be determined from the equivalent Bellman equation in (\[eq:bellman\_cond\]).
Next, we shall prove that $V(\mathbf{Q})$ is a monotonic increasing function w.r.t. its component. Given the $V^l(\mathbf{Q})$ is the result of $l$-th iteration, $V^{l+1}(\mathbf{Q})$ is given by: $$V^{l+1}(\mathbf{Q})=T_{\Omega}(\mathbf{V}^{l},\mathbf{Q})-T_{\Omega}(\mathbf{V}^{l},\mathbf{Q}^I)$$ where $T_{\Omega}(\mathbf{V}^{l},\mathbf{Q})=\min\limits_{\Omega(\mathbf{Q})}\big[\widetilde{g}(\mathbf{Q},\Omega(\mathbf{Q}))+
\sum_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q},\Omega(\mathbf{Q})\}V^l(\mathbf{Q}^{\prime})\big]$, and $\mathbf{Q}^I$ is a reference state. Because $\lim_{l\rightarrow\infty}V^l(\mathbf{Q})=V(\mathbf{Q})$ [@Bertsekas:2007], it is sufficient to prove $V^l(\mathbf{Q}),\forall l$ is component-wise monotonic increasing. Using the induction method, we start from $V^0(\mathbf{Q})=0,
\forall \mathbf{Q}$. In the induction step, we assume that $\forall
\mathbf{Q}^1\succ\mathbf{Q}^2,V^l(\mathbf{Q}^1)>V^l(\mathbf{Q}^2)$, we get $$\begin{array}{ll}
&V^{l+1}(\mathbf{Q}^1)+T_{\Omega}(\mathbf{V}^{l},\mathbf{Q}^I)\\
=&\min\limits_{\Omega(\mathbf{Q}^1)}\left[\widetilde{g}(\mathbf{Q}^1,\Omega(\mathbf{Q}^1))+
\sum\limits_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q}^1,\Omega(\mathbf{Q}^1)\}V^l(\mathbf{Q}^{\prime})\right]\\
>&\sum\limits_{m,k}\beta_{(m,k)}f(Q_{(m,k)}^2)+\sum\limits_{\mathbf{A}}\Pr\{\mathbf{A}\}\mathbb{E}_{\mathbf{H}}
[V(\mathbf{Q}^2-\mathbf{U}^*+\mathbf{A})]\\
\geq&\min\limits_{\Omega(\mathbf{Q}^2)}\left[\widetilde{g}(\mathbf{Q}^2,\Omega(\mathbf{Q}^2))+
\sum\limits_{\mathbf{Q}^{\prime}}\Pr\{\mathbf{Q}^{\prime}|\mathbf{Q}^2,\Omega(\mathbf{Q}^2)\}V^l(\mathbf{Q}^{\prime})\right]\\
=&V^{l+1}(\mathbf{Q}^2)+T_{\Omega}(\mathbf{V}^{l},\mathbf{Q}^I)
\end{array}$$ where $\mathbf{U}^*$ is the delivered bits under the conditional action $\Omega^*(\mathbf{Q}^1)=\{\mathbf{p}^*,\mathbf{s}^*\}$ for all users. Specifically, $U_{(m,k)}(t)=
R_{(m,k)}(\mathbf{H},\mathbf{p}^*,\mathbf{s}^*)\tau$.
Appendix B: Proof of Corollary \[cor:dstr\_policy\] {#app:dstr_policy .unnumbered}
===================================================
Using the linear approximation in (\[eq:linear\_value\]), and the given ICI management pattern $\mathbf{p}$, the optimal user scheduling action $\mathbf{s}$ (obtained by solving the RHS of Bellman equation (\[eq:bellman\_post\])) is: $$\begin{array}{ll}
&\min_{\mathbf{s}(\mathbf{Q},\mathbf{H})\in\mathcal{S}}\Big[\widetilde{g}(\mathbf{Q},\mathbf{P},\mathbf{s}(\mathbf{Q},\mathbf{H}))+\\
&\quad\quad\sum_{\widetilde{\mathbf{Q}}^{\prime}}\Pr\{\widetilde{\mathbf{Q}}^{\prime}|\mathbf{Q},\mathbf{p},\mathbf{s}(\mathbf{Q},\mathbf{H})\}\widetilde{V}(\widetilde{\mathbf{Q}}^{\prime})
\Big]\\
\Rightarrow&\min_{\mathbf{s}(\mathbf{Q},\mathbf{H})\in\mathcal{S}}\bigg[
\sum_{m,k}\Big(\widetilde{V}_{(m,k)}(Q_{(m,k)})(1-s_{(m,k)})+\\
&\quad\quad\widetilde{V}_{(m,k)}((Q_{(m,k)}-U_{(m,k)})^+)s_{(m,k)}\Big)\bigg]\\
\Rightarrow&\max_{\mathbf{s}_m\in\mathcal{S}_m}\sum_{k\in\mathcal{K}_m}\Big(\widetilde{V}_{(m,k)}(Q_{(m,k)})-\\
&\quad\quad\widetilde{V}_{(m,k)}((Q_{(m,k)}-U_{(m,k)})^+)\Big)s_{(m,k)},
\forall m\in\mathcal{M}_{\mathbf{p}}
\end{array}$$ where $\mathcal{S}_m=\{\mathbf{s}_m:\sum_{k\in\mathcal{K}_m}s_{(m,k)}=1,s_{(m,k)}\in\{0,1\}\}$ is the set of all the possible user scheduling policy for BS $m$. As a result, Corollary \[cor:dstr\_policy\] is obvious from the above equation.
Appendix C: Proof of Lemma \[lem:converge\_value\] {#app:value_converge .unnumbered}
==================================================
From the definition of mapping $T_{(m,k)}$ in (\[eq:T\_[(m,k)]{}\]), the convergence property of the per-user value function update algorithm in (\[eq:learn\_value\_f\]) is equivalent to the following update equation[@Borkar:2008]: $$\label{eq:syn_learning}\begin{array}{l}
\widetilde{V}_{(m,k)}^{t+1}(\widetilde{Q}_{(m,k)})=\\
\quad\widetilde{V}_{(m,k)}^t(\widetilde{Q}_{(m,k)})+
\gamma(t)\Bigl[T_{(m,k)}(\widetilde{V}_{(m,k)}^{t},\widetilde{Q}_{(m,k)})-\\
\quad\widetilde{V}_{(m,k)}^{t}(\widetilde{Q}_{(m,k)}^I)
-\widetilde{V}_{(m,k)}^{t}(\widetilde{Q}_{(m,k)})+
Z^{t+1}_{(m,k)}(\widetilde{Q}_{(m,k)})\Bigr]
\end{array}$$ where $\widetilde{Z}^{t+1}_{(m,k)}(\widetilde{Q}_{(m,k)})=\beta_{(m,k)}f(\widetilde{Q}_{(m,k)}+A_{(m,k)})
+\widetilde{V}_{(m,k)}^{t}(Q_{(m,k)}^{\prime})-
T_{(m,k)}(\widetilde{V}_{(m,k)}^{t},\widetilde{Q}_{(m,k)})$, and $Q_{(m,k)}^{\prime}=\widetilde{Q}_{(m,k)}+A_{(m,k)}-U_{(m,k)}$. $U_{(m,k)}$ is determined by the ICI management control pattern $\mathbf{\widetilde{p}}_m^I$ and local CSI $\mathbf{H}_{(m,k)}$. Let $\mathbb{F}_t=\sigma(\mathbf{\widetilde{V}}_{(m,k)}^l,\mathbf{\widetilde{Z}}_{(m,k)}^l,l\leq
t)$ be the $\sigma$-algebra generated by $\{\mathbf{\widetilde{V}}_{(m,k)}^l,\mathbf{\widetilde{Z}}_{(m,k)}^l,l\leq
t\}$, It can be verified that $\mathbb{E}_{\{\mathbf{H}_{(m,k)},A_{(m,k)}\}}[
\mathbf{\widetilde{Z}}^{t+1}_{(m,k)}|\mathbb{F}_t]=0$, and $\mathbb{E}_{\{\mathbf{H}_{(m,k)},A_{(m,k)}\}}[
||\mathbf{\widetilde{Z}}^{t+1}_{(m,k)}||^2|\mathbb{F}_t]\leq
C_1(1+||\widetilde{\mathbf{V}}_{(m,k)}^t||^2)$ for a suitable constant $C_1$. Therefore, the learning algorithm in (\[eq:syn\_learning\]) is a standard stochastic learning algorithm with the Martingale difference noise $\mathbf{\widetilde{Z}}^{t+1}_{(m,k)}$. We use the ordinary differential equation (ODE) to analyze the convergence probability. Specifically, the limiting ODE associated for (\[eq:syn\_learning\]) to track asymptotically is given by: $$\label{eq:ode_syn}\begin{array}{l}
\dot{\widetilde{\mathbf{V}}}_{(m,k)}(t)=\mathbf{T}_{(m,k)}(\widetilde{\mathbf{V}}_{(m,k)}(t))-\widetilde{\mathbf{V}}_{(m,k)}(t)-\\
\quad\quad\quad\quad\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)}^I,t)\mathbf{e}=h(\widetilde{\mathbf{V}}_{(m,k)}(t))
\end{array}$$ Note that there is a unique fixed point $\widetilde{\mathbf{V}}^*_{(m,k)}$ that satisfies the Bellman equation $$\mathbf{T}_{(m,k)}(\widetilde{\mathbf{V}}^*_{(m,k)})-\widetilde{\mathbf{V}}^*_{(m,k)}-\widetilde{V}_{(m,k)}^{*}(\widetilde{Q}_{(m,k)}^I)\mathbf{e}=0$$ and it is proved in [@SA:ODE:globally] that $\widetilde{\mathbf{V}}^*_{(m,k)}$ is the globally asymptotically stable equilibrium for (\[eq:ode\_syn\]). Furthermore, define $h_r(\widetilde{\mathbf{V}}_{(m,k)})=h(r\widetilde{\mathbf{V}}_{(m,k)})/r,
\forall r>0$ and $h_{\infty}(\widetilde{\mathbf{V}}_{(m,k)})=\lim_{r\rightarrow\infty}h_r(\widetilde{\mathbf{V}}_{(m,k)})=\mathbf{P}_{(m,k)}\widetilde{\mathbf{V}}_{(m,k)}-\widetilde{\mathbf{V}}_{(m,k)}-\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)}^I)\mathbf{e}$. The origin is the globally asymptotically stable equilibrium point of the ODE $\dot{\widetilde{\mathbf{V}}}_{(m,k)}(t)=h_{\infty}(\widetilde{\mathbf{V}}_{(m,k)}(t))$ (This is merely a special case by setting $\widetilde{\mathbf{g}}_{(m,k)}=\mathbf{0}$ in the $\mathbf{T}_{(m,k)}(\widetilde{\mathbf{V}}_{(m,k)})$). By theorem 2.2 of [@SA:ODE:Bound], the iterates $\widetilde{\mathbf{V}}^t_{(m,k)}$ remains bounded almost-surely. By the ODE approach[@Borkar:2008 Chap.2], we can conclude that the iterates of the update $\widetilde{\mathbf{V}}_{(m,k)}^t\rightarrow\widetilde{\mathbf{V}}_{(m,k)}^*$ almost-surely, i.e., converging to the globally asymptotically stable equilibrium of the associated ODE.
Finally the proof of $\widetilde{V}_{(m,k)}^{\infty}(\widetilde{Q}_{(m,k)})=\widetilde{V}_{(m,k)}^{*}(\widetilde{Q}_{(m,k)})$ being a monotonic increasing function can be derived in the same way as Theorem \[Thm:MDP\_cond\].
Appendix D: Proof of Lemma \[lem:q\_converge\] {#app:q_converge .unnumbered}
==============================================
From the definition of mapping $T_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)},Q_{(m,k)},\mathbf{p})$ in (\[eq:Q\_map\]), defining the vector form mapping $\mathbf{T}_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)}):\mathbb{R}^{(1+N_Q)\times|\mathcal{P}|}\rightarrow
\mathbb{R}^{(1+N_Q)\times|\mathcal{P}|}$ where each elements is given by $T_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)},Q_{(m,k)},\mathbf{p})$. The convergence property of the per-user $\mathbb{Q}$-factor update algorithm in (\[eq:learn\_q\_factor\]) is equivalent to the following update equation[@Borkar:2008]: $$\begin{aligned}
\label{eq:Q_sync}\begin{array}{l} \mathbb{Q}_{(m,k)}^{t+1}=
\mathbb{Q}_{(m,k)}^{t}+
\gamma(t)\Bigl[\mathbf{T}_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)}^{t})-\\
\quad\quad\mathbb{Q}_{(m,k)}^{t}(Q_{(m,k)}^I,\mathbf{p}_m^I)\mathbf{e}
-\mathbb{Q}_{(m,k)}^{t}+ \mathbf{Z}^{t+1}_{(m,k)}\Bigr]
\end{array}\end{aligned}$$ where $\mathbf{Z}^{t+1}_{(m,k)}$ is the vector form of $Z^{t+1}_{(m,k)}(Q_{(m,k)},\mathbf{p})$, $Z^{t+1}_{(m,k)}(Q_{(m,k)},\mathbf{p})=\beta_{(m,k)}f(Q_{(m,k)})
+\mathbb{Q}_{(m,k)}^{t}(Q_{(m,k)}^{\prime})-
T_{(m,k)}^{\mathbb{Q}}(\mathbb{Q}_{(m,k)}^{t},Q_{(m,k)},\mathbf{p})$, and $Q_{(m,k)}^{\prime}=Q_{(m,k)}-U_{(m,k)}+A_{(m,k)}$. $U_{(m,k)}$ is determined by the ICI management control pattern $\mathbf{p}$ and local CSI $\mathbf{H}_{(m,k)}$. Let $\mathbb{F}_t=\sigma(\mathbb{Q}_{(m,k)}^l,\mathbf{Z}_{(m,k)}^l,l\leq
t)$ be the $\sigma$-algebra generated by $\{\mathbb{Q}_{(m,k)}^l,\mathbf{Z}_{(m,k)}^l,l\leq t\}$, It can be verified that $\mathbb{E}_{\{\mathbf{H}_{(m,k)},A_{(m,k)}\}}[
\mathbf{Z}^{t+1}_{(m,k)}|\mathbb{F}_t]=0$, and $\mathbb{E}_{\{\mathbf{H}_{(m,k)},A_{(m,k)}\}}[
||\mathbf{Z}^{t+1}_{(m,k)}||^2|\mathbb{F}_t]\leq
C_1(1+||\mathbb{Q}_{(m,k)}^t||^2)$ for a suitable constant $C_1$. Therefore, the learning algorithm in (\[eq:Q\_sync\]) is also a standard stochastic learning algorithm with the Martingale difference noise $\mathbf{Z}^{t+1}_{(m,k)}$. The limiting ODE associated to track asymptotically is given by: $$\label{eq:ode_syn_Q}\begin{array}{l}
\dot{\mathbb{Q}}_{(m,k)}(t)=\mathbf{T}_{(m,k)}^{
\mathbb{Q}}(\mathbb{Q}_{(m,k)}(t))-\\
\quad\quad\quad\mathbb{Q}_{(m,k)}(t)-\mathbb{Q}_{(m,k)}(Q_{(m,k)}^I,\mathbf{p}_m^I,t)\mathbf{e}
\end{array}$$ Furthermore, there is a unique fixed point $\mathbb{Q}_{(m,k)}^*$ satisfying the following equation [@SA:Q_learning]: $$\mathbf{T}_{(m,k)}^{
\mathbb{Q}}(\mathbb{Q}_{(m,k)}^*)-\mathbb{Q}_{(m,k)}^*-\mathbb{Q}_{(m,k)}^*(Q_{(m,k)}^I,\mathbf{p}_m^I)\mathbf{e}=0$$ and it is proved in [@SA:Q_learning] that $\mathbb{Q}^*_{(m,k)}$ is the globally asymptotically stable equilibrium for (\[eq:ode\_syn\_Q\]). As a result, following the same argument in the convergence proof of per-user value function in Lemma \[lem:converge\_value\], we can conclude that the iterates of the update $\mathbb{Q}_{(m,k)}^t\rightarrow\mathbb{Q}_{(m,k)}^*$ almost-surely.
[^1]: The authors are with the Department of Electronic and Computer Engineering (ECE), The Hong Kong University of Science and Technology (HKUST), Hong Kong. (email: [email protected], [email protected]).
[^2]: For a celluar system with 5 BSs, 5 users served by each BS, a buffer size of 5 per user and 5 CSI states for each link between one user and one BS, the system state space contains $(5+1)^{5\times5}\times5^{5\times5\times5}$ states, which is already unmanageable.
[^3]: The $\mathbb{Q}$-factor $\mathbb{Q}(s,a)$ is a function of the system state $s$ and the control action $a$, which represents the [*potential cost*]{} of applying a control action $a$ at the current state $s$ and applying the action $a^{\prime}=\arg\min_{a}\mathbb{Q}(s^{\prime},a)$ for any system state $s^{\prime}$ in the future[@Cao:2007].
[^4]: Note that the on-off BS control is shown to be close to optimal in[@BPC:2008; @pattern:2009]. Moreover, the solution framework can be easily extended to deal with discrete BS power control.
[^5]: Strictly speaking, the average delay is given by $\overline{T}_{(m,k)}(\Omega)=\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}[\frac{Q_{(m,k)}}{\lambda_{(m,k)}(1-\text{PBD}_{(m,k)})}]$, where $\text{PBD}_{(m,k)}$ is the bit dropping probability conditioned on bit arrival. Since our target bit dropping probability $\text{PBD}_{(m,k)}\ll 1$, $\overline{T}_{(m,k)}(\Omega)=\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}[\frac{Q_{(m,k)}}{\lambda_{(m,k)}}]\approx\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}[\frac{Q_{(m,k)}}{\lambda_{(m,k)}(1-\text{PBD}_{(m,k)})}]$.
[^6]: In fact, the proposed solution framework can be easily extended to deal with a more general QoS based optimization. For example, say we minimize the average delay subject to the constraints on average data rate: $\overline{R}_{(m,k)}(\Omega)=\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}[R_{(m,k)}(t)]\geq
R_T^k$. The Lagrangian of such constrained optimization is: $\min_{\Omega}J_{\beta}^{\Omega}=\sum\nolimits_{m,k}\left[
\beta_{(m,k)}\overline{T}_{(m,k)}(\Omega)+\mu_{(m,k)}\overline{R}_{(m,k)}(\Omega)\right]=\lim\sup_{T\rightarrow
\infty}\frac{1}{T}\sum\nolimits_{t=1}^T\mathbb{E}^{\Omega}
[g_{\mu}(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))]$, where $g_{\mu}(\boldsymbol{\chi}(t),\Omega(\boldsymbol{\chi}(t)))=\sum_{m,k}\beta_{(m,k)}f(Q_{(m,k)})+\mu_{(m,k)}R_{(m,k)}$, and $\mu_{(m,k)}$ is the Lagrange multiplier corresponding to the QoS constraint $\overline{R}_{(m,k)}(\Omega)\geq R_T^k$. Note that it has the same form as (\[eq:problem\]) and the proposed solution framework can be applied to the QoS constrained problem as well.
[^7]: The policy will be a function of the complete system state by solving a standard bellman equation.
[^8]: For the sake of easy discussion, we consider discrete state space in this example. Yet, the proposed algorithms and convergence results in the paper work for general continuous state space as well.
[^9]: Using the linear approximation in (\[eq:linear\_value\]), we can address the curse of dimensionality (complexity) as well as facilitate distributive implementation where each BS could solve for $\widetilde{V}_{(m,k)}(\widetilde{Q}_{(m,k)})$ based on local CSI and QSI only.
[^10]: Note that $\widetilde{\delta}_{(m,k)}(0)=0,\forall k$, and hence the users with empty buffer will not be scheduled and the activated BS $m$ will serve the users with non-empty buffer (the chance for the buffer of all $K$ users being empty at a given slot is very small).
[^11]: For example, one possible criteria is to partition the local QSI space so that the probability of $\mathbf{Q}_{m}$ belonging to any region is the same (uniform probability partitioning).
|
---
abstract: 'A semi-discrete fourth-order finite volume method is introduced to solve multi-dimensional magnetohydrodynamic (MHD) problems. This method employs a dimension by dimension approach followed by one dimensional fourth-order centrally weighted essentially non-oscillatory reconstruction (CWENO) to compute higher order point values of the physical quantities. Density, momentum and energy are discretized as volume averages. However, magnetic fields are discretized as area averages on the grid cell interfaces. The conservative semi-discrete finite volume scheme is combined with the constrained transport (CT) technique to enforce the divergence-free constraint on the magnetic field. Various 1D, 2D and 3D tests are performed to confirm fourth order accuracy, shock capturing nature and divergence-free property of the scheme.'
author:
- 'Prabal Singh Verma[^1], Wolf-Christian Müller'
title: Fourth Order Finite Volume CWENO Scheme For Multidimensional MHD
---
****
Introduction
============
Many astrophysical phenomena that exhibit discontinuous solutions can be approximately described by the ideal magnetohydrodynamic (MHD) equations. Various shock capturing numerical methods have been proposed to solve astrophysical MHD problems [@Evans; @brio-wu; @divb1; @divb2; @divb3; @divb4; @divb5; @divb6; @divb7; @divb8; @divb9; @cs13; @divb10; @athena; @zanna; @stone1; @stone2; @cpc2].
In this paper we introduce a semi-discrete finite volume scheme based on the computation of higher order local Lax-Friedrichs (LLF) fluxes (see for example [@shu1; @shu2]) at the grid cell interfaces. The LLF flux is an approximation to the Harten-Lax-van Leer (HLL) flux [@riemann1; @riemann6] as it only requires the maximum propagation speed of information at the discontinuity which exists at the grid cell interface. The LLF fluxes are first computed at the centers of grid cell interfaces using 1D fourth order CWENO reconstruction [@lpr7] following a dimension by dimension approach. Later an averaging is performed to obtain higher-order averaged LLF fluxes at the grid cell interfaces [@weno-2014]. The CWENO approach is the heart of the present scheme as it straightforwardly allows us to compute a higher-order point value of the conserved quantities at the center of each grid cell interface.
In this work, the constrained transport (CT) method [@Evans] is used to evolve the magnetic field. This technique automatically maintains the solenoidality of the magnetic field up to the machine precision. In this approach the magnetic fields are usually discretized as face averages and electric fields as edge averages in the numerical grid cells. It is not straight forward to merge the CT technique into conservative hyperbolic equations due to different discretization of the physical variables. So far, several methods have been suggested on how to combine these two consistently [@divb1; @divb2; @divb3; @divb4; @divb5; @divb6; @divb7; @divb8; @divb9; @divb10]. In this paper, we use a very simple 1D CWENO reconstruction [@lpr7] to compute higher order edge averaged electric field components from higher order LLF electric fluxes. These edge averaged electric field components are used to evolve the magnetic field in the CT. At the end, a fourth order Runge-Kutta method [@runge] is used to evolve the semi-discrete scheme in time. We have performed several multi-dimensional tests to confirm the fourth order accuracy and the divergence free property (${\bf\nabla. b} = 0$) of our scheme.
The manuscript is organized as follows. Section II introduces the ideal MHD equations, the finite volume semi-discrete scheme for conservation laws, computation of averaged fluxes at the grid cell interfaces and the CT technique. Section III has been devoted to the 1D fourth order CWENO reconstruction of all the physical quantities. In Section IV, we present the convergence of errors for 1D, 2D and 3D Alfven wave tests at large amplitude to show fourth order accuracy in the nonlinear regime. We also present the convergence of errors for the 2D Orszag-Tang vortex problem as long as the solution is smooth. Moreover, the 1D Shock tube, the Orszag-Tang vortex, the 2D blast wave and 2D Rotor problems are solved to demonstrate the shock capturing nature and divergence free property of the scheme. Section V contains a brief summary of the work presented in this manuscript.
Ideal MHD equations and solver
==============================
The ideal MHD equations, in conservative form, can be expressed as follows:
$$\label{con}
{\partial_t \rho} + {\bf \nabla .} (\rho {\bf v}) = 0,$$
$$\label{mom}
\partial_t (\rho {\bf v}) + {\bf \nabla .} \big [ \rho {\bf v v} + (p + \frac{1}{8 \pi}|{\bf b}|^2)I - \frac{1}{4 \pi}{\bf b b} \big ] = 0,$$
$$\label{energy}
\partial_t e + {\bf \nabla .} \big [(e + p + \frac{1}{8 \pi}|{\bf b}|^2){\bf v} - \frac{1}{4 \pi }{\bf (v.b)b} \big ] = 0,$$
$$\label{current}
{\partial_t {\bf b}} + {\bf \nabla} \times {\bf E} = 0,$$
with the constraint
$$\label{divb}
{\bf \nabla } . {\bf b} = 0.$$
Here ${\bf b}$ is the magnetic field, $\rho$ is the density, $e$ is the total energy density, ${\bf v}$ is the velocity, ${\bf E = - v \times b}$ is the electric field. The thermal pressure $p$ is computed from the ideal gas equation of state,
$$\label{press}
p = (\gamma -1) \big [e - \frac{1}{2}\rho |{\bf v}|^2 - \frac{1}{8 \pi}|{\bf b}|^2 \big ].$$
where $\gamma$ is the ratio of specific heats. The set of Eqs.- in compact form can be written as, $$\label{con1}
\frac{\partial {\bf U}}{\partial t}+\frac{\partial {\bf F}}{\partial x}+\frac{\partial {\bf G}}{\partial y}+\frac{\partial {\bf H}}{\partial z}=0$$
where,
$$\begin{aligned}
\label{UFG}
\nonumber {\bf U} = \left( \begin{array}{c} \rho\\ \rho v_x\\\rho v_y\\\rho v_z\\ e\\ b_x\\ b_y\\ b_z\end{array}\right), \hspace{0.2cm}
{\bf F} = \left( \begin{array}{c} \rho v_x\\ \rho v_x^2+p+ {\bf b}^2/{8 \pi}- { b_x}^2/4\pi\\\rho v_x v_y-b_x b_y/4\pi\\\rho v_x v_z - b_x b_z/4\pi\\
v_x(e+p+ {\bf b}^2/{8 \pi}) - b_x({\bf v.b})/{8 \pi}) \\ 0 \\ (v_x b_y - v_y b_x) \\ -(v_z b_x - v_x b_z) \end{array}\right), \hspace{0.2cm}
{\bf G} = \left( \begin{array}{c} \rho v_y\\\rho v_x v_y-b_x b_y/4\pi\\ \rho v_y^2+p+ {\bf b}^2/{8 \pi}- { b_y}^2/4\pi\\\rho v_y^2+p\\\rho v_y v_z- b_y b_z/4\pi
\\ v_y(e+p+ {\bf b}^2/{8 \pi}) - b_y({\bf v.b})/{8 \pi})\\ -(v_x b_y - v_y b_x)\\0\\(v_y b_z - v_z b_y)\end{array}\right), %\hspace{0.2cm}
\end{aligned}$$
$$\begin{aligned}
\label{H}
{\bf H} = \left( \begin{array}{c} \rho v_z\\ \rho v_x v_z- b_x b_z/4\pi\\\rho v_y v_z- b_y b_z/4\pi\\ \rho v_z^2+p
\\v_z(e+p+ {\bf b}^2/{8 \pi}) - b_z({\bf v.b})/{8 \pi}) \\ (v_z b_x - v_x b_z)\\-(v_y b_z - v_z b_y)\\0\end{array}\right).
\end{aligned}$$
All the symbols have their usual meanings.
Semi-discrete scheme for conservative quantities
------------------------------------------------
Since density, energy and momentum components are discretized as volume averages, Eqs.- are solved by a semi-discrete method (see for example [@divb5]) as follows:
$$\label{semi-dis}
\frac{d\bar u_{i,j,k}}{d t}+\frac{\mathcal{F}_{i+1/2,j,k}^x-\mathcal{F}_{i-1/2,j,k}^x}{\Delta x}+ \\
\frac{\mathcal{G}_{i,j+1/2,k}^y-\mathcal{G}_{i,j-1/2,k}^y}{\Delta y} + \\
\frac{\mathcal{H}_{i,j,k+1/2}^z-\mathcal{H}_{i,j,k-1/2}^z}{\Delta z} =0\,.$$
Here $\bar u_{i,j,k}$ is the volume average which is defined as,
$$\label{u_avg}
\bar u_{i,j,k} = \frac{1}{\Delta x\Delta y\Delta z}\int_{z_{i,j,k-1/2}}^{z_{i,j,k+1/2}} \\
\int_{y_{i,j-1/2,k}}^{y_{i,j+1/2,k}} \int_{x_{i-1/2,j,k}}^{x_{i+1/2,j,k}} \! \\
\mathcal{R}(x,y,z) \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z\,.% \int\limits_E\limits^F$$
The function $\mathcal{R}(x,y,z)$ is the higher order centrally weighted essentially non-oscillatory polynomial. The functions [$\mathcal{F}^x$]{}, [$\mathcal{G}^y$]{} and [$\mathcal{H}^z$]{} are the higher order averaged local Lax-Friedrichs (LLF) fluxes along $x$-, $y$-and $z$-directions respectively.
Computation of Averaged LLF Fluxes
----------------------------------
This subsection is devoted to the computation of the fluxes along the $x$-direction as an example, the generalization to other directions is straightforward. We first reconstruct the 1D CWENO polynomial along the $x$-direction from the given volume averages. Later this polynomial is used to compute the non-oscillatory point values centered at the cell interfaces along the $x$-direction. These reconstructed point values carry an area average information in the $yz$-plane. Considering them as cell averages, the 1D reconstruction is performed along the $y$-direction to obtain a point value in the cell center which still has an average information along the $z$-direction. Again considering the reconstructed values as cell averages we perform the same 1D reconstruction along the $z$-direction to get the point value in the cell center. This is how we obtain the point values of the conserved quantities at the center of the grid cell interfaces along the $x$-direction. We can then compute corresponding point value fluxes $f^x$ and hence point value local Lax-Friedrichs (LLF) fluxes $F^x$ as follows :
$$\label{fx}
F_{i+1/2,j,k}^x = \frac{f^x(u_{i+1/2,j,k}^W)+f^x(u_{i+1/2,j,k}^E)}{2} - \\
\frac{a^x_{i+1/2,j,k}}{2} (u^{W}_{i+1/2,j,k}-u^{E}_{i+1/2,j,k})\,.$$
The quantities $u^E$ and $u^W$ denote right and left reconstructed point values of the conserved quantities at the center of the grid cell interfaces. These point values are obtained by 1D CWENO reconstruction reconstruction [@lpr7] following dimension by dimension approach. The quantities $(f^x)^E$, $(f^x)^W$ are corresponding point value fluxes of the conserved quantities at ($x_{i+1/2},y_j,z_k$). Here $a^x$ is the local maximum speeds of propagation which is estimated as,
$$\label{ax}
a^{x}_{i+1/2,j,k} = \text{max}\{(|v_x| + c_f)^E_{i,j,k},(|v_x| + c_f)^W_{i+1,j,k}, 0\},$$
where $c_f$ is the fast magneto-sonic speed,
$$\label{cf}
c_f = \bigg[\frac{1}{2}\big((c_s^2 + c_A^2) + \big((c_s^2 + c_A^2)^2 -4 c_s^2 \frac{b_x^2}{4\pi \rho}\big)^{1/2}\big)\bigg]^{1/2}.$$
Here $c_s = (\gamma p / \rho)^{1/2}$ is the sound speed and $c_A = (|{\bf b}|^2 / {4 \pi \rho})^{1/2}$ is the Alfven speed [@divb5].
After we have computed the higher order point value LLF fluxes $F^x$, the higher order averaged LLF fluxes [$\mathcal{F}^x$]{} can be easily computed using simple averaging along the $y$ and $z$-directions [@weno-2014] as,
$$\begin{aligned}
\label{Fx}
\nonumber \mathcal{F}_{i+1/2,j,k}^x = F_{i+1/2,j,k}^x +\frac{1}{24} ( F_{i+1/2,j-1,k}^x
\nonumber - 2 F_{i+1/2,j,k}^x + F_{i+1/2,j+1,k}^x) \\
\nonumber +\frac{1}{24} ( F_{i+1/2,j,k-1}^x
\nonumber - 2 F_{i+1/2,j,k}^x + F_{i+1/2,j,k+1}^x).
\end{aligned}$$
The method, described above, requires all physical quantities to be volume averages. However, magnetic field components $b_x$, $b_y$ and $b_z$ are discretized as area averages on the $yz$, $xz$ and $xy$-planes respectively. We therefore compute volume averaged magnetic field components ($\bar{B_x}$, $\bar{B_y}$, $\bar{B_z}$) from their area averages ($\bar{b_x}$, $\bar{b_y}$, $\bar{b_z}$) as follows,
1. Lagrange interpolation is used to obtain a fourth order accurate area average value at the center of the cell. It is to be noted that interpolation is only allowed along the direction where the field remains smooth and the direction is determined by the solenoidality property of the magnetic field [@divb6; @zanna].
2. Now since each grid cell has three face average values, one at the center and other two on the opposite faces, we are able to apply Simpson’s $1/3$ rule to compute a fourth order accurate volume average.
Thus, fourth order accurate volume average approximations for the magnetic field components are expressed as,
$$\begin{aligned}
\label{Bx_avg}
\nonumber {\bar B}_{x i,j,k} = \frac{1}{6}\Bigg[
\frac{1}{32} \Big(90 {\bar b}_{x i+1/2,j,k}+60 {\bar b}_{x i-1/2,j,k}-20
{\bar b}_{x i+3/2,j,k}+3 {\bar b}_{x i+5/2,j,k}-5 {\bar b}_{x i-3/2,j,k}\Big) \\
+ {\bar b}_{x i-1/2,j,k}+{\bar b}_{x i+1/2,j,k} \Bigg].
\end{aligned}$$
$$\begin{aligned}
\label{By_avg}
\nonumber {\bar B}_{y i,j,k} = \frac{1}{6}\Bigg[
\frac{1}{32} \Big(90 {\bar b}_{y i,j+1/2,k}+60 {\bar b}_{y i,j-1/2,k}-20
{\bar b}_{y i,j+3/2,k}+3 {\bar b}_{y i,j+5/2,k}-5 {\bar b}_{y i,j-3/2,k}\Big) \\
+ {\bar b}_{y i,j-1/2,k}+{\bar b}_{y i,j+1/2,k} \Bigg].
\end{aligned}$$
$$\begin{aligned}
\label{Bz_avg}
\nonumber {\bar B}_{z i,j,k} = \frac{1}{6}\Bigg[
\frac{1}{32} \Big(90 {\bar b}_{z i,j,k+1/2}+60 {\bar b}_{z i,j,k-1/2}-20
{\bar b}_{z i,j,k+3/2}+3 {\bar b}_{z i,j,k+5/2}-5 {\bar b}_{z i,j,k-3/2}\Big) \\
+ {\bar b}_{z i,j,k-1/2}+{\bar b}_{z i,j,k+1/2} \Bigg].
\end{aligned}$$
For the computation of fluxes along the $x$-direction, $\bar{B_y}$ and $\bar{B_z}$ are being treated like other volume averaged variables (density, momentum etc.). However, $x$-component of the magnetic field is special as it does not require any reconstruction along the $x$-direction, but it does require the reconstructions along the $y$ and $z$-directions as it also has an average information in the $yz$-plane.
Computation of averaged fluxes in other directions is trivial due to the dimension by dimension approach.
Constrained transport
---------------------
Since magnetic fields are discretized as area averages, one may, therefore, introduce finite volume discretization on Eq. after applying Stoke’s theorem as follows [@Evans] :
$$\label{bx}
\frac{d }{d t} {\bar b}_{xi-1/2,j,k}= -\frac{{\bar E}_{zi-1/2,j+1/2,k}-{\bar E}_{zi-1/2,j-1/2,k}}{\Delta y} + \\
\frac{{\bar E}_{yi-1/2,j,k+1/2}-{\bar E}_{yi-1/2,j,k-1/2}}{\Delta z}.$$
$$\label{by}
\frac{d }{d t} {\bar b}_{yi,j-1/2,k}= \frac{{\bar E}_{zi+1/2,j-1/2,k}-{\bar E}_{zi-1/2,j-1/2,k}}{\Delta x} - \\
\frac{{\bar E}_{xi,j-1/2,k+1/2}-{\bar E}_{xi,j-1/2,k-1/2}}{\Delta z}.$$
$$\label{bz}
\frac{d }{d t} {\bar b}_{zi,j,k-1/2}= -\frac{{\bar E}_{yi+1/2,j,k-1/2}-{\bar E}_{yi-1/2,j,k-1/2}}{\Delta x} + \\
\frac{{\bar E}_{xi,j+1/2,k-1/2}-{\bar E}_{xi,j-1/2,k-1/2}}{\Delta y}.$$
The variables $\bar b_x$, $\bar b_y$, $\bar b_z$ are the area averaged magnetic field components and $\bar E_x$, $\bar E_y$, $\bar E_z$ are the edge averaged approximations of the electric field components.
From Eqs.-, one may easily show that $d ({\bf \nabla . \bar b})_{i,j,k}/dt = 0$ if the differentiation is carried out by using central differences along the cell edges. Thus, the evolved magnetic field remains solenoidal if so initially. The edge averaged electric field components are obtained from the higher order LLF electric fluxes. These fluxes exist because Eq. can also be written in divergence form as follows,
$$\label{current1}
{\partial_t {\bf b}} + {\bf \nabla} . \left( \begin{array}{ccc} 0 & E_z & -E_y\\ -E_z & 0 & E_x\\ E_y & -E_x & 0 \end{array}\right)= 0,$$
Thus LLF electric flux vector along the $x$-direction reads as,
$$\begin{aligned}
\label{Ex}
{\bf E}^x_{i+\frac{1}{2},j,k} = \frac{1}{2} \left[ \left( \begin{array}{c} 0 \\ -E_z \\ E_y \end{array}\right)_{i+1,j,k}^{W} +
\left( \begin{array}{c} 0 \\ -E_z \\ E_y \end{array}\right)_{i,j,k}^{E} \right] -
\frac{a^x_{i+1/2,j,k}}{2} \left[ \left( \begin{array}{c} b_x \\ b_y\\ b_z \end{array}\right)_{i+1,j,k}^{W} -
\left( \begin{array}{c} b_x \\ b_y\\ b_z \end{array}\right)_{i,j,k}^{E} \right].\end{aligned}$$
Note here that the $x$-component of the electric flux ${ E}^{xx}$ is zero because ${b_x}_{i,j,k}^{E}$ is equal to ${b_x}_{i+1,j,k}^{W}$. The vectors ${\bf E }^W$, ${\bf E }^E$ are the left and right reconstructed point values which can be computed by using the definition of the electric field
$$\begin{aligned}
\label{E_west}
{\bf E } = - {\bf v} \times {\bf b},\end{aligned}$$
and $a^x$ is already defined in Eq.. Once we have computed point value LLF electric fluxes along the $x$-direction, we apply the same averaging formula as in Eq. to obtain the face (area) averaged LLF electric fluxes as follows,
$$\begin{aligned}
\label{Ex}
\nonumber {\bf \mathcal{ E}}_{i+1/2,j,k}^x = {\bf E}_{i+1/2,j,k}^x +\frac{1}{24} ( {\bf E}_{i+1/2,j-1,k}^x
\nonumber - 2 {\bf E}_{i+1/2,j,k}^x + {\bf E}_{i+1/2,j+1,k}^x) \\
\nonumber +\frac{1}{24} ( {\bf E}_{i+1/2,j,k-1}^x
\nonumber - 2 {\bf E}_{i+1/2,j,k}^x + {\bf E}_{i+1/2,j,k+1}^x).
\end{aligned}$$
Now, since all the components of the electric fluxes ${\bf \mathcal{ E}}^x$ carry an average information along the $y$ and $z$-directions, 1D reconstruction along any of those directions will return an edge average value along the other direction. Thus, considering $- \mathcal{ E}^{xy}$ as cell average, if we reconstruct the point value at the cell edges along the $y$-direction, we get a contribution to the edge averaged electric field along the $z$-direction $\bar{E_z}$. Similarly, reconstruction of $ \mathcal{ E}^{xz}$ at the cell edges along the $z$-direction contributes to the edge averaged electric field along the $y$-direction $\bar{E_y}$.
Similarly, one can compute averaged LLF electric fluxes along $y$ and $z$-directions and there contributions to the edge averaged electric field components. At the end we take a mean of all the contributions to compute fourth order edge averaged electric field components.
Finally, Eqs. and - are evolved in time using a fourth order Runge-Kutta method [@runge] , once the L.H.S. of these equations are computed.
The spatial accuracy of this scheme is now completely determined by the order of spatial reconstruction of the physical quantities. We use 1D fourth order CWENO reconstruction [@lpr7] for the same.
Although, the fourth order CWENO reconstruction for 1D hyperbolic problems is well described by Levy et al. [@lpr7], we provide it here for the sake of completeness.
1D Fourth order CWENO reconstruction
====================================
In each cell $I_j$, one has to reconstruct a quadratic polynomial $R_j(x)$ which is a convex combination of three quadratic polynomials $P_{j-1}(x)$, $P_{j}(x)$ and $P_{j+1}(x)$ such that,
$$\label{R_J_4th}
R_j(x) = \sum_{k = j-1}^{j+1} w_{k} P_k(x),\hspace{0.2cm} \text{where} \hspace{0.2cm} \sum_{k = j-1}^{j+1} w_{k} = 1
,\hspace{0.2cm} w_{k} \geq 0, \hspace{0.2cm} \forall \hspace{0.1cm}k \in (j-1, j, j+1).$$
$R_j(x)$ is reconstructed so as to satisfy all the three constraints accuracy, conservation and non-oscillatory. The coefficients of the polynomial $P_k(x)$ are obtained uniquely by requiring it to conserve the cell averages $\bar u_{k-1}$, $\bar u_{k}$ and $\bar u_{k+1}$, where $k \in (j-1,j,j+1)$. Thus, each polynomial, $P_k(x)$, can be written as,
$$\begin{aligned}
\label{P_k}
\nonumber P_{k}(x) = \bar u_k - \frac{1}{24}(\bar u_{k+1}-2\bar u_k+\bar u_{k-1})
+ \frac{\bar u_{k+1} -\bar u_{k-1}}{2 \Delta x}(x-x_k) \\
+ \frac{(\bar u_{k+1}-2\bar u_k+\bar u_{k-1})} {2\Delta x^2} (x-x_k)^2,\hspace{0.2cm} k = j-1,j,j+1.\end{aligned}$$
The nonlinear weights $w_k$ are obtained as before, [*i.e.*]{}
$$\label{w_k}
w_k = \frac{\alpha_k}{\alpha_{j-1}+\alpha_{j}+\alpha_{j+1}}, \hspace{0.2cm}
\text{where} \hspace{0.2cm}\alpha_k = \frac{c_k}{(\epsilon + IS_k)^p}, \hspace{0.2cm} \forall \hspace{0.1cm}k \in (j-1,j,j+1).$$
Here $\epsilon$, $p$ are chosen to be $10^{-6}$ and $2$ respectively and $c_{j-1} = c_{j+1} = 1/6$, $c_j = 2/3$ [@lpr7]. $IS_k$ are the smoothness indicators which are defined as,
$$\label{ISk}
IS_k = \sum_{l=1}^2 \int_{x_{j-1/2}}^{x_{j+1/2}} \! (\Delta x)^{2l-1} (P_k^{(l)}(x))^2 \, \mathrm{d}x ,
\hspace{0.2cm} \forall \hspace{0.1cm}k \in (j-1,j,j+1).$$
Once we have reconstructed all the the polynomials ($P_{j-1}$, $P_{j}$, $P_{j+1}$), smoothness indicators can easily be computed using Eq. and hence nonlinear weights using Eq.. These weights are finally used to reconstruct non-oscillatory the polynomial $R_j(x)$ which provides fourth order accurate point values at the cell interfaces in smooth regions.
Results from the simulations
============================
The fourth order accuracy of the proposed method has been verified by computing the convergence of errors in 1D, 2D and 3D circularly polarized Alfven wave (CPAW) tests at large amplitude ($\Delta = 0.9$) so as to retain the nonlinear behavior. We further confirm the fourth order accuracy in another 2D test — Orszag-Tang vortex problem when the solution is still smooth but highly nonlinear.
In order to show the shock capturing behavior of the method we present the results for 1D Brio-Wu test, the evolution of the Orszag-Tang vortex problem, the 2D MHD blast wave and 2D Rotor problem. The ${\bf \nabla . b}$ errors in all the test are at the round-off level and do not grow in time.
Accuracy
--------
In this subsection, we present convergence of the truncation error for the 1D, 2D and 3D circularly polarized Alfven wave (CPAW) tests and for 2D Orszag-Tang vortex problem to validate the fourth order accuracy of the scheme.
The norm of the error for all the convergence studies is computed as follows, $$\delta E = \frac{1}{NG}\sum_{i = 1}^N |E_i^f - E_i^0|,$$ where $E_i^0$, $E_i^f$ are the reference and the numerical solutions as a function of grid resolution and $NG$ is the number of grid points.
After computing the norms of the errors, we obtain the experimental order of convergence $(EOC)$ using the formula, $$EOC(j) = \frac{|log(\delta E(NG(j)))|-|log(\delta E(NG(j-1)))|}{|log(NG(j))|-|log(NG(j-1))|},$$
here $j$ runs over the indices of the column vectors in the TABLES shown in the later subsections.
— The initial conditions for 1D, 2D and 3D CPAW tests are selected as follows:
$$\begin{aligned}
\label{alf1d}
\nonumber \rho = 1, \hspace{0.2cm} v_x = 0, \hspace{0.2cm}
v_y = \Delta \sin [2\pi x ],
\hspace{0.2cm} v_z = \Delta \cos [2\pi x ], \\
\nonumber p = 0.1, \hspace{0.2cm} b_x = 1, \hspace{0.2cm} b_y = \Delta \sin [2\pi x ],
\hspace{0.2cm} b_z = \Delta \cos [2\pi x ], \\
\hspace{0.2cm} [0,l_x] \times [0,l_y] \times [0,l_z]= [0,1]\times [0,1]\times [0,1]. %, \\\end{aligned}$$
$$\begin{aligned}
\label{alf2d}
\nonumber \rho = 1, \hspace{0.2cm} v_x = - \Delta \cos [2\pi(y + z)],
\hspace{0.2cm}
v_y = -\frac{\Delta}{\sqrt{2}} \sin [2\pi(y + z)], \\
\nonumber v_z = \frac{\Delta}{\sqrt{2}} \sin [2\pi(y + z)],
\hspace{0.2cm} p = 0.1, \hspace{0.2cm} b_x = \Delta \cos [2\pi(y + z)], \\
\nonumber b_y = 1 + \frac{\Delta}{\sqrt{2}} \sin [2\pi(y + z)],
\hspace{0.2cm} b_z = 1 - \frac{\Delta}{\sqrt{2}} \sin [2\pi(y + z)], \\
\hspace{0.2cm} [0,l_x] \times [0,l_y] \times [0,l_z]= [0,1]\times [0,1]\times [0,1].
\end{aligned}$$
$$\begin{aligned}
\label{alf3d}
\nonumber \rho = 1, \hspace{0.2cm} v_x = - \frac{\Delta}{l_y} \cos [2\pi(x/l_x + y/l_y+ z/l_z)]
- \frac{\Delta}{l_x \sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)], \\
\nonumber v_y = \frac{\Delta}{l_x} \cos [2\pi(x/l_x + y/l_y+ z/l_z)]
- \frac{\Delta}{l_y \sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)], \\
\nonumber v_z = \frac{\Delta}{\sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)], \hspace{0.2cm} p = 0.1,
\\ \nonumber b_x = \frac{1}{l_x\sqrt{2}}- \frac{\Delta}{l_y} \cos [2\pi(x/l_x + y/l_y+ z/l_z)]
- \frac{\Delta}{l_x \sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)], \\
\nonumber b_y = \frac{1}{l_y\sqrt{2}}+
\frac{\Delta}{l_x} \cos [2\pi(x/l_x + y/l_y+ z/l_z)]
- \frac{\Delta}{l_y \sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)],
\\ b_z = \frac{1}{\sqrt{2}} + \frac{\Delta}{\sqrt{2}} \sin [2\pi(x/l_x + y/l_y+ z/l_z)], % \\
\hspace{0.2cm} [0,l_x] \times [0,l_y] \times [0,l_z]= [0,2/\sqrt 3]\times [0,2]\times [0,1].
\end{aligned}$$
It is clear from the initial conditions that for the 1D case, CPAWs are propagating along the $x$-direction, for the 2D case, CPAW are propagating in the $yz$-plane at an angle $\pi/4$ to the $x$-axis and for the 3D case, CPAW propagate along the diagonal plane at an angle $\pi/4$ to the $z$-axis. $\gamma$ and $\Delta$ are chosen to be $5/3$ and $0.9$ respectively for all tests. Boundary conditions are periodic and the numerical solutions are obtained after one wave period.
TABLE \[tab1\], TABLE \[tab2\] and TABLE \[tab3\] contain the convergence of errors for the 1D, 2D and 3D CPAW tests respectively.
\[tab1\]
[ |p[1cm]{}|p[1cm]{}|p[5cm]{}|p[1cm]{}| ]{}\
j & NG & $\delta E$ & EOC\
1 & $16$ & 3.4341404679648566E-003 & -\
2 & $32$ & 1.4793985928679421E-004 &4.54\
3 & $64$ & 6.7699572887503336E-006 & 4.45\
4 & $128$ &3.5186704647354231E-007 &4.26\
5 & $256$& 2.0612437922231232E-008 &4.09\
\[tab2\]
[ |p[1cm]{}|p[1cm]{}|p[5cm]{}|p[1cm]{}| ]{}\
j &NG & $\delta E$ & EOC\
1 & $16^2$ & 6.7880543545184406E-004 & -\
2 & $32^2$ & 3.1515227458324933E-005 & 4.43\
3 & $64^2$ & 1.6701819661858828E-006 & 4.24\
4 & $128^2$ & 9.5043935626870201E-008 & 4.14\
5 & $256^2$& 5.6911207021282479E-009 & 4.06\
\[tab3\]
[ |p[1cm]{}|p[1cm]{}|p[5cm]{}|p[1cm]{}| ]{}\
j &NG & $\delta E$ & EOC\
1 & $16^3$ &7.2081207206123044E-003 & -\
2 & $32^3$ & 2.6369454325840709E-004 & 4.77\
3 & $64^3$ & 9.2593960801853198E-006 & 4.83\
4 & $128^3$ & 3.5135732619242231E-007 & 4.71\
5 & $256^3$ & 1.7011058257482867E-008 & 4.36\
— The initial condition for the Orszag-Tang Vortex test [@divb5] are expressed as, $$\begin{aligned}
\label{ot-vortex}
\nonumber \rho = \gamma^2, \hspace{0.2cm} v_x = -\sin (y), \hspace{0.2cm}
v_y = \sin (x),
\hspace{0.2cm} v_z = 0, \\
\nonumber p = \gamma, \hspace{0.2cm} b_x = -\sin (y), \hspace{0.2cm}
b_y = \sin (2x),
\hspace{0.2cm} b_z = 0, \\\nonumber
\hspace{0.2cm} [0,l_x] \times [0,l_y] \times [0,l_z]= [0,2\pi]\times [0,2\pi]\times [0,2\pi]. %, \\\end{aligned}$$
The ratio of specific heats $\gamma$ is chosen to be $5/3$. TABLE \[tab4\] contains the convergence of errors at $t = 0.5$, where the reference solution is obtained at higher resolution ($1056^2$).
\[tab4\]
[ |p[1cm]{}|p[1cm]{}|p[5cm]{}|p[1cm]{}| ]{}\
j & NG & $\delta E$ & EOC\
1 & $32^2$ & 8.3441012034921325E-003 & -\
2 & $96^2$ & 6.8376844996276276E-005 & 4.37\
3 & $160^2$ & 5.8164163709434336E-006 & 4.82\
4 & $288^2$ & 4.0090947783788522E-007 & 4.55\
TABLES \[tab1\]-\[tab4\] confirm the fourth order accuracy of the scheme.
Shock capturing behavior
------------------------
— This test is performed to demonstrate the shock capturing nature of the proposed method. The initial conditions for Brio-Wu Shock Tube test [@brio-wu] are :
$$\begin{aligned}
\label{brio-wu}
\nonumber (\rho, \hspace{0.1cm} v_x, \hspace{0.1cm}v_y, \hspace{0.1cm}v_z, \hspace{0.1cm} p,
\hspace{0.1cm} b_x, \hspace{0.1cm}b_y, \hspace{0.1cm}b_z) = (1, 0, 0, 0, 1, 0.75, 1, 0),
\hspace{0.2cm}\text{if} \hspace{0.1cm} x < l_x/2
\\ \nonumber (\rho, \hspace{0.1cm} v_x, \hspace{0.1cm}v_y, \hspace{0.1cm}v_z, \hspace{0.1cm} p,
\hspace{0.1cm} b_x, \hspace{0.1cm}b_y, \hspace{0.1cm}b_z) = (0.125, 0, 0, 0, 0.1, 0.75, -1, 0), \hspace{0.2cm}\text{if} \hspace{0.1cm} x \ge l_x/2.
\end{aligned}$$
Boundary conditions are open along the $x$-direction. The system length ($l_x$), number of grid points ($n_x$) and $\gamma$ are chosen to be $2$, $1024$ and $5/3$ respectively. Figures \[fig:fig1\]-\[fig:fig4\] contain the spatial profiles of $\rho$, $p$, $v_y$ and $b_y$ respectively at $t = 0.25$.
— The initial conditions for this test are already described in the previous subsection. Here we provide the space-time evolution of the density ($\rho$) and pressure at a resolution $256^2$ to demonstrate the shock capturing nature of our scheme. Figures \[fig:fig5\]-\[fig:fig6\] show the contour plots of the density ($\rho$) and pressure respectively at $t = 0.5$ when the solution is smooth, however, Figures \[fig:fig7\]-\[fig:fig8\] contain the snap shots of the same at $t = 3.0$ when the system exhibits discontinuities.
— The initial conditions for the 2D MHD Blast wave test [@athena] are chosen as follows:
$$\begin{aligned}
\label{mhdblast}
\nonumber (\rho, \hspace{0.1cm} v_x, \hspace{0.1cm}v_y, \hspace{0.1cm}v_z,\hspace{0.1cm} p) = (1, 0, 0, , 0, 10),
\hspace{0.2cm}\text{if} \hspace{0.1cm}\sqrt{(x-l_x/2)^2+(y-l_y/2)^2} \le 0.1
\\ \nonumber (\rho, \hspace{0.1cm} v_x, \hspace{0.1cm}v_y, \hspace{0.1cm}v_z, \hspace{0.1cm} p) = (1, 0, 0, 0, 0.1),
\hspace{0.2cm}\text{if} \hspace{0.1cm}\sqrt{(x-l_x/2)^2+(y-l_y/2)^2} \ge 0.1, \\
\nonumber b_x = 1/\sqrt{2}, \hspace{0.2cm}
b_y = 1/\sqrt{2},
\hspace{0.2cm} b_z = 0, \\\nonumber
\text{where} \hspace{0.1cm} [0,l_x] \times [0,l_y] = [0,1] \times [0,1] .
\end{aligned}$$
The ratio of specific heats $\gamma$ is chosen to be $5/3$ and grid resolution as $256^2$. Figures \[fig:fig9\]-\[fig:fig10\] contain the snap shots of the density ($\rho$) and pressure at $t = 0.15$ which clearly exhibit strong shock capturing nature of the present scheme.
— MHD Rotor problem is well described in ref.[@divb2; @divb3; @divb6; @athena]. The computational domain is chosen to be unit square $[0,l_x] \times [0,l_y] = [0,1] \times [0,1]$ and the resolution is $256^2$. The ratio of specific heats $\gamma$ is $1.4$. The density $\rho = 10$ and the velocity components $v_x = -v_0 (y-0.5)/r_0$, $v_y = v_0 (x-0.5)/r_0$ for $r \le r_0$, where $r = \sqrt{(x-l_x/2)^2+(y-l_y/2)^2}$, $r_0 = 0.1$ and $v_0 = 1$. For $r > r_1$, $\rho = 1$, $v_x = 0$ and $v_y = 0$, where $r_1 = 0.115$. However, for $r_0 < r < r_1$, $\rho = 1 + 9 f$, $v_x = -f v_0 (y-0.5)/r$, $v_y = f v_0 (x-0.5)/r$, where $f = (r_1 -r)/(r_1 - r_0)$. The pressure $p$ and the magnetic field components are uniform; $i.e.$, $p = 1$, $b_x = 5/\sqrt{4 \pi}$, $b_y = 0$. Figures \[fig:fig11\]-\[fig:fig12\] contain the snap shots of the density ($\rho$) and pressure at $t = 0.2$.
In all the tests ${\bf\nabla. b}$ remains conserved and does not grow in time.
Conclusion
==========
We have successfully developed a very simple fourth order finite volume scheme to study astrophysical MHD problems. Accuracy, shock capturing nature and divergence free property are confirmed through various multi-dimensional tests. The method, presented in this paper, opens a possibility for implementing even higher order schemes simply by using higher order 1D reconstruction along with higher order averaging of fluxes and higher order time integrator.
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![\[fig:fig1\]A snap shot of the density ($\rho$) at $t = 0.25$ in the Brio-Wu test.](fig1.pdf){height="3.5in" width="6in"}
![\[fig:fig2\]A snap shot of the pressure ($p$) at $t = 0.25$ in the Brio-Wu test.](fig2.pdf){height="3.5in" width="6in"}
![\[fig:fig3\]A snap shot of the $y$-component of the velocity ($v_y$) at $t = 0.25$ in the Brio-Wu test.](fig3.pdf){height="3.5in" width="6in"}
![\[fig:fig4\]A snap shot of the $y$-component of the magnetic field ($b_y$) at $t = 0.25$ in the Brio-Wu test.](fig4.pdf){height="3.5in" width="6in"}
![\[fig:fig5\]Contour plot of the density at $t = 0.5$ for the Orszag-Tang Vortex test](fig5.pdf){height="5.5in" width="5.5in"}
![\[fig:fig6\]Contour plot of the pressure at $t = 0.5$ for the Orszag-Tang Vortex test](fig6.pdf){height="5.1in" width="5.1in"}
![ \[fig:fig7\]Contour plot of the density at $t = 3.0$ for the Orszag-Tang Vortex test ](fig7.pdf){height="5.8in" width="5.5in"}
![\[fig:fig8\]Contour plot of the pressure at $t = 3.0$ for the Orszag-Tang Vortex test](fig8.pdf){height="5in" width="5in"}
![ \[fig:fig9\] Contour plot of the density at t = 0.15 for the 2D MHD blast wave ](fig9.pdf){height="5.5in" width="5.5in"}
![ \[fig:fig10\] Contour plot of the pressure at t = 0.15 for the 2D MHD blast wave ](fig10.pdf){height="5.3in" width="5.1in"}
![ \[fig:fig11\]Contour plot of the density at $t = 0.2$ for the 2D Rotor problem ](fig11.pdf){height="5.5in" width="5.4in"}
![ \[fig:fig12\]Contour plot of the pressure at $t = 0.2$ for the 2D Rotor problem ](fig12.pdf){height="5.5in" width="5.4in"}
[^1]: [email protected]
|
---
address: 'Artem A. Lopatin Omsk Branch of Federal State Budgetary Scientific Establishment Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (OB IM SBRAS). Pevtsova street, 13, 644043, Omsk, Russia. http://www.iitam.omsk.net.ru/\~lopatin'
author:
- '[Artem A. Lopatin]{}'
title: Identities for matrix invariants of the symplectic group
---
Introduction {#section_intro}
============
We work over an infinite field ${{\mathbb{F}}}$ of arbitrary characteristic $p={\mathop{\rm char}}{{{\mathbb{F}}}}$. All vector spaces, algebras and modules are over ${{\mathbb{F}}}$ and all algebras are associative with unity unless otherwise stated.
Consider a group $G$ from the list $GL(n)$, $O(n)=\{A\in {{\mathbb{F}}}^{n\times n}\,|\,A A^T =E\}$, $SO(n)=\{A\in O(n)\,|\,\det(A)=1\}$, ${S\!p}(n)=\{A\in {{\mathbb{F}}}^{n\times n}\,|\,AA^{\ast}=E\}$, where we assume that $p\neq2$ in case $G\in\{O(n),SO(n)\}$ and $n$ is even in case $G={S\!p}(n)$. Here ${{\mathbb{F}}}^{n\times n}$ is the space of $n\times n$ matrices over ${{\mathbb{F}}}$ and $A^{\ast}=-J A^T J$ is the [*symplectic*]{} transpose of $A$, where $J=\left(
\begin{array}{cc}
0& E \\
-E& 0\\
\end{array}
\right)$ is the matrix of the skew-symmetric bilinear form. The group $G$ acts on the space $V=({{\mathbb{F}}}^{n\times n})^{\oplus d}$ by the diagonal conjugation: $$g\cdot (A_1,\ldots,A_d)=(gA_1g^{-1},\ldots,gA_dg^{-1})$$ for $g\in G$ and $A_1,\ldots,A_d$ in ${{\mathbb{F}}}^{n\times n}$. The coordinate ring of $V$, i.e. the algebra of all polynomial maps $V\to {{\mathbb{F}}}$, is the polynomial ring $$R={{\mathbb{F}}}[x_{ij}(k)\,|\,1\leq i,j\leq n,\, 1\leq k\leq d]$$ in $n^2 d$ variables, where $x_{ij}(k)$ sends $(A_1,\ldots,A_d)$ to the $(i,j)^{\rm th}$ entry of $A_k$. The algebra of [*matrix $G$-invariants*]{} is the set of all polynomial maps $f\in R$ that are constants on $G$-orbits of $V$, i.e., $f(g\cdot v)=f(v)$ for all $g\in G$ and $v\in V$. We denote this algebra by $R^G$.
For $f\in R$ denote by $\deg{f}$ its [*degree*]{} and by ${\mathop{\rm mdeg}}{f}$ its [*multidegree*]{}, i.e., ${\mathop{\rm mdeg}}{f}=(t_1,\ldots,t_d)$, where $t_k$ is the total degree of the polynomial $f$ in $x_{ij}(k)$, $1\leq i,j\leq n$, and $\deg{f}=t_1+\cdots+t_d$. By the Hilbert–Nagata Theorem on invariants, each of the considered algebras of invariants $I$ is finitely generated algebras. The algebra $I$ also have ${{\mathbb{N}}}_0$-gradings by degrees and ${{\mathbb{N}}}_0^d$-grading by multidegrees, where ${{\mathbb{N}}}_0$ stands for non-negative integers. Denote by $D(I)$ the maximal degree of elements of a minimal (by inclusion) ${{\mathbb{N}}}^d$-homogeneous set of generators (m.h.s.g.) for $I$. Generating sets for the considered algebras of invariants are known (see Section \[section\_generators\]) as well as relations between generators for $R^{GL(n)}$ and $R^{O(n)}$ (see Section \[section\_identities\]). In case $p=0$ relations between generators for $R^{{S\!p}(n)}$ are also known (see [@Procesi76]). The key difference between the case of zero and positive characteristic is the following property obtained in [@DKZ02]:
\[eq\_weyl\] D(R\^[GL(n)]{})d0<pn.
In this paper we describe the ideal of relations for $R^{Sp(n)}$ over a field of odd characteristic (see Theorem \[theo\_relationsSp\] below).
Generators {#section_generators}
==========
To formulate the result describing generators of the algebra $R^G$, we introduce the following notations. The ring $R$ is generated by the entries of $n\times n$ [*generic*]{} matrices $X_k=(x_{ij}(k))_{1\leq i,j\leq n}$ ($1\leq k\leq d$).
Consider an arbitrary $n\times n$ matrix $A=(a_{ij})$ over some commutative ring. Denote coefficients in the characteristic polynomial of $A$ by $\sigma_t(A)$, i.e., $$\det(\lambda E - A)=\sum_{t=0}^{n} (-1)^t\lambda^{n-t}\sigma_t(A).$$ So, $\sigma_0(A)=1$, $\sigma_1(A)={\mathop{\rm tr}}(A)$ and $\sigma_n(A)=\det(A)$.
Part (a) of the following theorem was proven by Donkin [@Donkin92a], parts (b), (c) by Zubkov [@Zubkov99].
\[theo\_intro1\] The algebra of matrix $G$-invariants $R^G$ is generated by the following elements:
1. ${\sigma}_t(A)$ ($1\leq t\leq t$ and $A$ ranges over all monomials in $X_1,\ldots,X_d$), if $G=GL(n)$;
2. ${\sigma}_t(B)$ ($1\leq t\leq t$ and $B$ ranges over all monomials in $X_1,\ldots,X_d$, $X_1^T,\ldots,X_d^T)$, if $G=O(n)$;
3. ${\sigma}_t(C)$ ($1\leq t\leq t$ and $C$ ranges over all monomials in $X_1,\ldots,X_d$, $X_1^{\ast},\ldots,X_d^{\ast}$), if $G={S\!p}(n)$.
We can assume that in the formulation of Theorem \[theo\_intro1\] each of monomials $A,B,C$ is [*primitive*]{}, i.e., is not equal to a power of a shorter monomial. If $p=0$ or $p>n$, then in Theorem \[theo\_intro1\] it is enough to take traces ${\mathop{\rm tr}}(U)$, where $U$ can be non-primitive, instead of ${\sigma}_t(U)$ in order to generate the algebra $R^G$. The corresponding results were obtained earlier than Theorem \[theo\_intro1\] by Sibirskii [@Sibirskii68] and Procesi [@Procesi76].
It is not difficult to see that elements from Theorem \[theo\_intro1\] are in fact invariants. Namely, the action of $G$ on $V$ induces the action on $R$ as follows: $g\cdot x_{ij}(k)$ is the $(i,j)$-th entry of $g^{-1} X_k g$. Given $f\in R$, we have $f\in R^G$ if and only if $g\cdot f=f$ for all $g\in G$. Then the following properties give us the required: ${\sigma}_t(g A g^{-1})={\sigma}_t(A)$.
Identities {#section_identities}
==========
The following definitions were given in [@Lopatin12ART]. Let ${{\langle}X{\rangle}}$ be the semigroup (without unity) freely generated by [*letters*]{}
1. $x_1,\ldots,x_d$, if $G=GL(n)$;
2. $x_1,\ldots,x_d,x_1^T,\ldots,x_d^T$, otherwise.
Denote ${{\langle}X{\rangle}}^{\#}={{\langle}X{\rangle}}\sqcup\{1\}$, i.e., we endow ${{\langle}X{\rangle}}$ with the unity. Let ${{\mathbb{F}}}{{\langle}X{\rangle}}$ and ${{\mathbb{F}}}{{\langle}X{\rangle}}^{\#}$ be the free associative algebras (without and with unity, respectively) with the ${{\mathbb{F}}}$-bases ${{\langle}X{\rangle}}$ and ${{\langle}X{\rangle}}^{\#}$, respectively. Note that elements of ${{\mathbb{F}}}{{\langle}X{\rangle}}$ and ${{\mathbb{F}}}{{\langle}X{\rangle}}^{\#}$ are [*finite*]{} linear combinations of monomials from ${{\langle}X{\rangle}}$ and ${{\langle}X{\rangle}}^{\#}$, respectively. Assume that $a=a_1\cdots a_r$ and $b$ are elements of ${{\langle}X{\rangle}}$, where $a_1,\ldots,a_r$ are letters.
1. Introduce the involution ${}^T$ on ${{\langle}X{\rangle}}$ as follows. If $G=GL(n)$, then $a^T=a$. Otherwise, we set $b^{TT}=b$ for a letter $b$ and $a^T=a_r^T\cdots a_1^T\in{{\langle}X{\rangle}}$.
2. We say that $a$ and $b$ are [*cyclic equivalent*]{} and write $a\stackrel{c}{\sim} b$ if $a=a_1a_2b$ and $b=a_2a_1$ for some $a_1,a_2\in{{\langle}X{\rangle}}^{\#}$. If $a\stackrel{c}{\sim} b$ or $a\stackrel{c}{\sim} b^T$, then we say that $a$ and $b$ are [*equivalent*]{} and write $a\sim b$.
An element from ${{\langle}X{\rangle}}$ is called [*primitive*]{} if it is not equal to a power of a shorter monomial.
1. Let ${{\langle}\widetilde{X}{\rangle}}\subset{{\langle}X{\rangle}}$ be the subset of primitive elements. Note that if $a\sim b$ for $a\in{{\langle}\widetilde{X}{\rangle}}$, then $b\in{{\langle}\widetilde{X}{\rangle}}$.
2. Let ${\sigma}{{\langle}\widetilde{X}{\rangle}}$ (${\sigma}{{\langle}X{\rangle}}$, respectively) be the ring with unity of commutative polynomials over ${{\mathbb{F}}}$ freely generated by “symbolic” elements ${\sigma}_t(a)$, where $t>0$ and $a\in{{\langle}\widetilde{X}{\rangle}}$ ranges over $\sim$-equivalence classes ($a\in{{\mathbb{F}}}{{\langle}X{\rangle}}$, respectively).
We will use the following conventions: ${\sigma}_0(a)=1$ and ${\sigma}_1(a)={\mathop{\rm tr}}(a)$, where $a\in{{\langle}\widetilde{X}{\rangle}}$. For a letter $b\in{{\langle}X{\rangle}}$ define $$X_{b}=
\left\{
\begin{array}{rl}
X_{k},&\text{if } b=x_k\\
X_{k}^T,&\text{if } b=x_k^T \text{ and }G=O(n)\\
X_{k}^{\ast},&\text{if } b=x_k^T \text{ and }G={S\!p}(n)\\
\end{array}
\right..$$ Given $a=a_1\cdots a_r\in{{\langle}X{\rangle}}$, where $a_i$ is a letter, we set $X_{a}=X_{a_1}\cdots X_{a_r}$.
Consider the surjective homomorphism $$\phi_n:{\sigma}{{\langle}\widetilde{X}{\rangle}}\to R^G$$ defined by ${\sigma}_t(a) \to {\sigma}_t(X_a)$, if $t\leq n$, and ${\sigma}_t(a) \to 0$ otherwise. Note that for all $n\times n$ matrices $A,B$ over $R$ and $1\leq t\leq n$ we have ${\sigma}_t(A^{{\delta}})={\sigma}_t(A)$, $(A^{{\delta}})^{{\delta}}=A$, and $(AB)^{{\delta}}=B^{{\delta}}A^{{\delta}}$, where ${\delta}$ stands for the transposition or symplectic transposition. Hence the map $\phi_n$ is well defined. Its kernel $K_{n}$ is the ideal of [*relations*]{} for $R^G$.
In [@Lopatin13JPAA] it was shown that ${\sigma}{{\langle}\widetilde{X}{\rangle}}\simeq {\sigma}{{\langle}X{\rangle}}/L$ and the generators of the ideal $L$ were given. Therefore, any element of ${\sigma}{{\langle}X{\rangle}}$ can be considered as an element of ${\sigma}{{\langle}\widetilde{X}{\rangle}}$.
Assume that $G=O(n)$. Let us recall the definition of element ${\sigma}_{t,r}(a,b,c)$ of ${\sigma}{{\langle}X{\rangle}}$, where $t,r\geq0$ and $a,b,c\in{{\mathbb{F}}}{{\langle}X{\rangle}}$. For short, we set $x=x_1$, $y=x_2$, and $z=x_3$. Consider the quiver (i.e., the oriented graph) ${\mathcal{Q}}$: $${\begin{picture}(20,0)(0,0)
\put(-2,1){\llap{$\scriptstyle x$}} \put(11,3){\circle{20}} \put(20,6){\vector(1,-4){1}}
\end{picture}} \xymatrix@C=1cm@R=1cm{ {*+[o][F-]{\scriptscriptstyle 1}}\ar@1@/^/@{<-}[rr]^{y,y^T} &&{*+[o][F-]{\scriptscriptstyle 2}}\ar@1@/^/@{<-}[ll]^{z,z^T}\\
}{\begin{picture}(20,0)(0,0)
\put(22,1){$\scriptstyle x^T$} \put(9,3){\circle{20}} \put(0,6){\vector(-1,-4){1}}
\end{picture}}\qquad\qquad,$$ where there are two arrows from vertex $2$ to vertex $1$ as well as from $1$ to $2$. By abuse of notation arrows of ${\mathcal{Q}}$ are denoted by letters from ${{\langle}X{\rangle}}$. For an arrow $a$ denote by $a'$ its head and by $a''$ its tail. A sequence of arrows $a_1\cdots a_s$ of ${\mathcal{Q}}$ is a [*path*]{} of ${\mathcal{Q}}$ if $a_i''=a_{i+1}'$ for all $1\leq i< s$. The head of the path $a$ is $a'=a_1'$ and the tail is $a''=a_s''$. A path $a$ is [*closed*]{} if $a'=a''$. Denote the multidegree of a monomial $a$ in arrows of ${\mathcal{Q}}$ by ${\mathop{\rm mdeg}}(a)=(\deg_{x}(a) + \deg_{x^T}(a),\deg_{y}(a) + \deg_{y^T}(a),\deg_{z}(a) + \deg_{z^T}(a))$. We set $${\sigma}_{t,r}(x,y,z)=\sum (-1)^{\xi} {\sigma}_{k_1}(e_1)\cdots {\sigma}_{k_q}(e_q),$$ where the sum ranges over all closed paths $e_1,\ldots,e_q$ in ${\mathcal{Q}}$ that are pairwise different with respect to $\sim$-equivalence and $k_1,\ldots,k_q>0$ ($q>0$) satisfing $k_1{\mathop{\rm mdeg}}(e_1)+\cdots+k_q{\mathop{\rm mdeg}}(e_q)=(t,r,r)$. Here $\xi=t+\sum_{i=1}^q k_i(\deg_{y}{e_i}+\deg_{z}{e_i}+1)$. Given $a,b,c\in{{\mathbb{F}}}{{\langle}X{\rangle}}$ we define ${\sigma}_{t,r}(a,b,c)$ as the result of the substitutions $x\to a$, $y\to b$, $z\to c$ in ${\sigma}_{t,r}(x,y,z)$.
Part (a) of the following theorem was proven by Zubkov [@Zubkov96] and part (b) by Lopatin [@Lopatin12ART], [@Lopatin12JPAA]:
\[theo\_relationsGLO\] The ideal of relations $K_{n}$ for $R^G\simeq {\sigma}{{\langle}\widetilde{X}{\rangle}}/K_{n}$ is generated by
1. ${\sigma}_t(a)$ for $t>n$, if $G=GL(n)$;
2. $\sigma_{t,r}(a,b,c)$ for $t+2r>n$ ($t,r\geq0$), if $G=O(n)$ and $p\neq2$.
Here $a,b,c$ ranges over ${{\mathbb{F}}}{{\langle}X{\rangle}}$.
Assume $G={S\!p}(n)$. Define the element $\varrho_{t,r}(x,y,z)$ of ${\sigma}{{\langle}X{\rangle}}$ by $$\varrho_{t,r}(x,y,z)=\sum (-1)^{t+k_1+\cdots+k_q} {\sigma}_{k_1}(e_1)\cdots {\sigma}_{k_q}(e_q),$$ where $e_1,\ldots,e_q$, $k_1,\ldots,k_q$ are the same as in the definition of ${\sigma}_{t,r}$.
\[theo\_relationsSp\] Assume $p\neq2$. Then the ideal of relations $K_{n}$ for $R^{{S\!p}(n)}\simeq {\sigma}{{\langle}\widetilde{X}{\rangle}}/K_{n}$ is generated by $\varrho_{t,r}(a,b,c)$ for $t+2r>n$ ($t,r\geq0$), where $a,b\in{{\mathbb{F}}}{{\langle}X{\rangle}}$ and $c\in{{\mathbb{F}}}{{\langle}X{\rangle}}^{\#}$.
Key difference of the relations in case $G=O(n)$ and in case $G={S\!p}(n)$ is that in the first case the degree of any relation is greater than $n$, but in the second case there are non-trivial relations of degree $\frac{n}{2}+1$. The proof of the theorem is given at the end of Section \[section\_quivers\].
\[remark1\] Denote by ${{\mathbb{F}}}_p\subset {{\mathbb{F}}}$ the field of characteristic $p$, generated by $1$. Note that generators of $R^{{S\!p}(n)}$ as well as elements from the formulation of Theorem \[theo\_relationsSp\] are defined over ${{\mathbb{F}}}_p$. Hence the standard linear algebra arguments imply that without loss of generality we can assume that ${{\mathbb{F}}}$ is algebraically closed.
Isomorphism of algebras {#section_isomorphism}
=======================
In this section we assume that ${{\mathbb{F}}}$ is an algebraically closed field of characteristic different from two (see Remark \[remark1\]). To define a subalgebra $I_n$ of $R\otimes {{\mathbb{F}}}[y_{ij}\,|\,1\leq i<j\leq n]$ we denote by $Y$ the $n\times n$ skew-symmetric matrix with $(i,j)^{\rm th}$ entry equal to $y_{ij}$ for $i<j$. The algebra $J_n$ is generated by all ${\sigma}_t(A_1 Y\cdots A_r Y)$ for $1\leq t\leq n$, $r>0$, where $A_i\in\{X_1,\ldots,X_d,X_1^T,\ldots,X_d^T\}$ for all $i$. Consider the homomorphism $$\Psi_n:I_n\to R^{{S\!p}(n)}$$ defined by $X_k\to X_kJ$ and $Y\to -J$. (This notation means that $\Psi_n(x_{ij}(k))$ is equal to $(i,j)^{\rm th}$ entry of $X_kJ$ and similarly for $Y$).
\[lemma\_iso\] The map $\Psi_n$ is an isomorphism of algebras $I_n$ and $R^{{S\!p}(n)}$.
This lemma is proven at the end of this section. Let $I'_n$ be the subalgebra of $R\otimes {{\mathbb{F}}}[z_{ij}\,|\,1\leq i,j\leq n]$ generated by elements ${\sigma}_t(A_1 Z J Z^T\cdots A_r Z J Z^T)$ for $1\leq t\leq n$, $r>0$, where $A_i\in\{X_1,\ldots,X_d,X_1^T,\ldots,X_d^T\}$ for all $i$ and $Z=(z_{ij})_{1\leq i,j\leq n}$.
\[lemma\_hom\] The exists a unique homomorphism $\theta_n: I'_n\to I_n$ that sends ${\sigma}_t(A_1 Z J Z^T\cdots A_r Z J Z^T)$ to ${\sigma}_t(A_1 Y\cdots A_r Y)$.
Given a monomial $a=A_1 Z J Z^T\cdots A_r Z J Z^T$, we write $\theta_n(a)$ for the monomial $A_1 Y\cdots A_r Y$. Let $f=\sum_i{\alpha}_i {\sigma}_{t_{i1}}(a_{i1})\cdots{\sigma}_{t_{ir_i}}(a_{ir_i})$ be an element of $I'_n$, where ${\alpha}_i\in{{\mathbb{F}}}$. Denote by $h$ the element $\sum_i{\alpha}_i {\sigma}_{t_{i1}}(\theta_n(a_{i1}))\cdots{\sigma}_{t_{ir_i}}(\theta_n(a_{ir_i}))$ of $I_n$. To prove the lemma, it is enough to show that if $f=0$, then $h=0$.
Assume that $f=0$. Then the result of substitution $Z\to B$ in $f$ is zero for every $B\in{{\mathbb{F}}}^{n\times n}$. It is well-known that for any skew-symmetric $n\times n$ matrix $C$ over an algebraically closed field ${{\mathbb{F}}}$ with $p\neq 2$ there is $B\in {{\mathbb{F}}}^{n\times n}$ such that $BJB^T=C$. Therefore, the result of substitution $Y\to C$ in $h$ is zero for every skew-symmetric matrix $C\in{{\mathbb{F}}}^{n\times n}$. Since ${{\mathbb{F}}}$ is infinite, the last condition implies that $h=0$.
We define the homomorphism $\mu_n:R^{{S\!p}(n)}\to I'_n$ of algebras by $X_k\to Z^T X_k Z J$. Straightforward calculations imply that
\[eq\_id1\] \_n\_n \_n R\^[[Sp]{}(n)]{}
\[eq\_id2\] \_n\_n \_n R\^[[Sp]{}(n)]{},
where $\theta_n \circ \mu_n (x)$ stands for $\theta_n(\mu_n(x))$ and so on. Thus $\Psi_n^{-1}=\theta_n \circ \mu_n$ and Lemma \[lemma\_iso\] is proven.
Invariants of quivers {#section_quivers}
=====================
Consider the following quivers:
$$\xymatrix@C=1cm@R=1cm{ {*+[o][F-]{\scriptscriptstyle 1}}\ar@1@/^/@{<-}[rr]^{x_1,\ldots,x_d,x_1^T,\ldots,x_d^T} &&{*+[o][F-]{\scriptscriptstyle 2}}\ar@1@/^/@{<-}[ll]^{y}\\
}\qquad\text{ and } \qquad
\xymatrix@C=1cm@R=1cm{ {*+[o][F-]{\scriptscriptstyle 1}}\ar@1@/^/@{<-}[rr]^{x_1,\ldots,x_d,x_1^T,\ldots,x_d^T} &&{*+[o][F-]{\scriptscriptstyle 2}}\ar@1@/^/@{<-}[ll]^{z,z^T}\\
}.$$ Denote the left hand side quiver by ${\mathcal{G}}_y$ and the right hand side quiver by ${\mathcal{G}}_z$. As in Section \[section\_identities\], by abuse of notation some arrows of this quivers are denoted by letters from ${{\langle}X{\rangle}}$. Let ${\langle}{\mathcal{G}}_y{\rangle}$ (${\langle}{\mathcal{G}}_z{\rangle}$, respectively) be the set of all closed paths in ${\mathcal{G}}_y$ (${\mathcal{G}}_z$, respectively). By definition, we set $y^T=-y$. Then we define $\stackrel{T}{}$-involution and $\sim$-equivalence on ${\langle}{\mathcal{G}}_y{\rangle}$ and ${\langle}{\mathcal{G}}_z{\rangle}$ in the natural way. Considering ${\langle}{\mathcal{G}}_y{\rangle}$ instead of ${{\langle}X{\rangle}}$, we define ${\langle}\widetilde{{\mathcal{G}}_y}{\rangle}$, ${{\mathbb{F}}}{\langle}{\mathcal{G}}_y{\rangle}$, ${\sigma}{\langle}{\mathcal{G}}_y{\rangle}$ and ${\sigma}{\langle}\widetilde{G_y}{\rangle}$ similarly to ${{\langle}\widetilde{X}{\rangle}}$, ${{\mathbb{F}}}{{\langle}X{\rangle}}$, ${\sigma}{{\langle}X{\rangle}}$ and ${\sigma}{{\langle}\widetilde{X}{\rangle}}$, respectively (see Section \[section\_identities\]). The same notions we also introduce for ${\mathcal{G}}_z$.
We write $I({\mathcal{Q}}_y,n)$ for the algebra of polynomial invariants of dimension vector $(n,n)$ of the quiver ${\mathcal{Q}}_y$ with involution. By [@Lopatin09JAlg], $I({\mathcal{Q}}_y,n)=I_n$. We can also see that the algebra $I({\mathcal{Q}}_z,n)$ is generated by elements ${\sigma}_t(A_1B_1\cdots A_rB_r)$ for $1\leq t\leq n$, $r>0$, where $A_i\in\{X_1,\ldots,X_d,X_1^T,\ldots,X_d^T\}$ and $B_i\in\{Z,Z^T\}$ for all $i$ (see [@Zubkov05I]).
We set $X_y=Y$, $X_z=Z$, and $X_{z^T}=Z^T$, where matrices $Y$ and $Z$ were defined in Section \[section\_isomorphism\]. Hence $X_a$ is determined for every $a\in{\langle}{\mathcal{G}}_y{\rangle}$ as well as for $a\in{\langle}{\mathcal{G}}_z{\rangle}$ (see also Section \[section\_identities\]). Define the surjective homomorphisms $$\pi_{y,n}:{\sigma}{\langle}\widetilde{{\mathcal{G}}_y}{\rangle}\to I_n \text{ and } \pi_{z,n}:{\sigma}{\langle}\widetilde{{\mathcal{G}}_z}{\rangle}\to I({\mathcal{G}}_z,n)$$ defined by ${\sigma}_t(a) \to {\sigma}_t(X_a)$, if $t\leq n$, and ${\sigma}_t(a) \to 0$ otherwise. Kernels $T_{y,n}$ and $T_{z,n}$, respectively, of these maps are ideals of relations for $I_n$ and $I({\mathcal{G}}_z,n)$, respectively.
Assume that $v$ is a vertex of ${\mathcal{G}}_y$. Given an $a\in{{\mathbb{F}}}{\langle}{\mathcal{G}}_y{\rangle}$, we have $a=\sum_i{\alpha}_ia_i$ for some ${\alpha}_i\in{{\mathbb{F}}}$ and $a_i\in{\langle}{\mathcal{G}}_y{\rangle}$. If $a_i=v$ for all $i$, then we write $a'=v$. Similarly we define $a''$.
A triple $(a,b,c)$ of elements from ${{\mathbb{F}}}{\langle}{\mathcal{G}}_y{\rangle}$ is called [*admissible*]{} if $a'=a''=b'=c''=1$ and $b''=c'=2$.
\[lemma\_relIn\] Assume $p\neq2$. Then the ideal of relations $T_{y,n}$ for $I_n$ is generated by ${\sigma}_{t,r}(a,b,c)$ for $t+2r>n$ ($t,r\geq0$), where $(a,b,c)$ is an admissible triple of ${\mathcal{G}}_y$.
Consider a relation $f\in{\sigma}{\langle}\widetilde{{\mathcal{G}}_y}{\rangle}$ for $I_n$. Since ${{\mathbb{F}}}$ is infinite, without loss of generality we can assume that $f$ is multihomogeneous. In particular, each monomial of $f$ has one and the same degree $k\geq0$ in letter $y$. Denote by $h\in{\sigma}{\langle}\widetilde{{\mathcal{G}}_z}{\rangle}$ the result of substitution $y\to z-z^T$ in $f$. Obvioulsly, $h$ is a relation for $I({\mathcal{G}}_z,n)$. The general result by Zubkov [@Zubkov05II] implies that the ideal $T({\mathcal{G}}_z,n)$ of relations for ${\mathcal{G}}_z$ is generated by ${\sigma}_{t,r}(a,b,c)$ for $t+2r>n$ and admissible triples $(a,b,c)$ of ${\mathcal{G}}_z$. Denote by $l\in{\sigma}{\langle}\widetilde{{\mathcal{G}}_y}{\rangle}$ the result of substitution $z\to y$, $z^T\to y^T=-y$ in $h$. Then $l$ belongs to the ideal of ${\sigma}{\langle}\widetilde{{\mathcal{G}}_y}{\rangle}$ generated by ${\sigma}_{t,r}(a,b,c)$ for $t+2r>n$ and admissible triples $(a,b,c)$ of ${\mathcal{G}}_y$. On the other hand, $l=2^k \!f$ and the proof is completed.
Now we can prove Theorem \[theo\_relationsSp\]
By Lemma \[lemma\_iso\], relations for $R^{{S\!p}(n)}$ are images of relations for $I_n$ with respect to $\Psi_n$. Relations for $I_n$ are describes in Lemma \[lemma\_relIn\]. The starightforward calculations complete the proof.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper was supported by FAPESP No. 2011/51047-1. The author is grateful for this support.
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---
abstract: 'In this letter we explore beyond the Standard Model top-Higgs Yukawa couplings as a function of a CP-mixing parameter $\xi_t$ at the 14 TeV HL-LHC in the process $pp\to thj$. We observe that angular variables of the decay products of the top are non-trivially sensitive to $\xi_t$. This fact is exploited in a full detector level analysis that employs machine learning techniques to optimize signal sensitivity on a suite of variables, including lepton azimuthal angle. The key result of this study is an improved projected exclusion limit on $\xi_t$ even when including the realistic effects of detector smearing and a conservative estimate of systematic error.'
address: 'ARC Center of Excellence for Particle Physics at the Terascale, Department of Physics, University of Adelaide, 5005 Adelaide, South Australia'
author:
- Riley Patrick
- Andre Scaffidi
- Pankaj Sharma
bibliography:
- 'tqh.bib'
title: 'Top polarisation as a probe of CP-mixing top-Higgs coupling in $tjh$ signals'
---
top-Higgs Yukawa coupling, top polarisation, machine learning, boosted decision trees
Introduction
============
Since the discovery of a scalar particle at 125 GeV as predicted by the Standard Model (SM) at the Large Hadron Collider (LHC) in 2012 [@201230; @20121] attention has turned to narrowing down it’s properties. It is of great interest as to whether this particle behaves exactly as the SM predicts or if it is perhaps a beyond the SM (BSM) scalar that exists within an expanded Higgs sector.
The measurement of the Higgs couplings to fermions is an on-going area of research as many of these couplings can be small and require large luminosities to probe, which motivates the application of intelligent analysis techniques to probe the signal as optimally as possible. The Higgs coupling to the top is of particular interest as it is the largest, and so far the $t\bar{t}h$ coupling is probed via the loop-induced processes $hgg$ and $h\gamma\gamma$ which rely on decay rate measurements of the Higgs. It is well known that the production cross section of both $hgg$ and $h\gamma\gamma$ will be sensitive to a phase $\xi_t$ of the top Yukawa coupling [@Ren:2019xhp]. This phase mixes the CP-properties of the top-Higgs coupling and thus is CP violating. Assuming only the CP phase and keeping SM value for magnitude, strong constraints could be placed on the phase. However, those processes can only occur at one-loop level and beyond. Noting that new physics dynamics can contribute in the loop and thus may affect the accurate determination of top-Yukawa phase, the Higgs production in association with a top pair (dominant) and Higgs produced with a single top (sub-dominant) are the only means to directly probe this coupling. These processes have been established at above 5$\sigma$ in Refs [@Aaboud:2018urx; @Sirunyan:2018hoz]. It is found that increasing $|\xi_t|$ leads to a suppression of the $t\bar{t}h$ cross section and an enhancement of the $tjh$ cross section [@Chang:2014rfa; @Kobakhidze2014]. A study of this process has also been undertaken utilizing the Matrix-Element-Method (MEM) which shows that, with signal detection efficiencies on the order of a few percent, discovery could be made in the high luminosity phase of the LHC [@Kraus:2019myc].
The top quarks lifetime is so short that it decays before hadronization, meaning that its polarization information is preserved in the distribution of its decay products which can be measured directly by the detector, especially in its lepton angular distributions. In many new physics scenarios, it has been shown extensively in literature, Refs. [@Godbole2010; @Godbole_2006; @doi10106313327703; @PhysRevD88074018; @Huitu2011; @Godbole2012; @Rindani2011; @RINDANI2012413; @Rindani2013; @Rindani2015], that charged-lepton azimuthal distribution is a powerful probe of top quark polarisation in the lab frame. There are two advantages of studying the charged lepton azimuthal distribution: first it does not require reconstruction of top-rest frame which would need full information of top-quark momentum and second, it is unaffected by any new physics in the top-quark decay and thus making it an uncontaminated probe of top quark polarization. This variable is constructed by taking the azimuthal angle of the lepton decaying from the top with respect to the x-z plane, where the top quarks x-component is positive.
Top quark polarisation can be written in terms of $\xi_t$ [@Ellis2014; @Yue:2014tya] and thus the decay products differential distributions will be effected by $\xi_t$ allowing for the improvement of analysis of this process. This has been exploited in a range of studies [@Ellis2014; @Yue:2014tya; @Rindani:2016scj], however these have not been undertaken at detector level to provide more accurate reflections of achievable sensitivity. Furthermore, these studies have employed traditional cutflow methods rather than more modern and advanced machine learning (ML) techniques to optimize signal sensitivity. In this letter we for the first time employ a full detector level analysis to calculate the angular variables of the decay products of the process $pp\rightarrow thj$ with the optimization of signal sensitivity through ML algorithms.
This paper is structured as follows: section \[sec:tophiggscoup\] will cover the parametrization of the top-Higgs coupling and it’s implementation, section \[sec:sigback\] will outline signal and backgrounds, section \[sec:simreco\] will outline event simulation and reconstruction, section \[sec:results\] will present the results of this analysis and finally we will conclude in section \[sec:conc\].
CP-mixed Top-Higgs Coupling {#sec:tophiggscoup}
===========================
In this study, a CP-mixing parameter $\xi_t$ is introduced in the mass basis of the top-Higgs sector via the Lagrangian $$\begin{aligned}
\label{eqn:Lag}
\mathcal{L}_{tth} = -\frac{y_t}{\sqrt{2}} \bar{t}\left( \cos\xi_t + i\gamma_5\sin\xi_t\right){t}h\;.\end{aligned}$$ The SM limit corresponds to when the mixing angle $\xi_t = 0$ and the Yukawa coupling adopts its SM value $y_t\rightarrow y_t^{SM} = \sqrt{2} m_{t} / v$, where $m_t$ is the mass of the top and $v\simeq 246$ GeV is the standard model Higgs vacuum expectation. We adopt a model independent approach in where the interaction Lagrangian in Eqn. \[eqn:Lag\] arises from an effective field theory (EFT) such as the dimention-6 opperators discussed in Refs. [@AguilarSaavedra:2009mx; @AguilarSaavedra:2008zc; @Zhang:2012cd; @Belusca-Maito:2015lna; @He:2013tia]. We assume that the new physics scale $\Lambda$ of such an EFT is $\gtrsim$ TeV such that the mixing angle $\xi_t\in (-\pi, \pi]$ [@Harnik:2013aja].
Constraints on $y_t$ and $\xi_t$ from the $hgg$ and $h\gamma\gamma$ loop processes can be found in Refs [@Ellis2014; @Kobakhidze2014; @Mileo2016; @Nishiwaki2014; @PhysRevD92015019; @Cheung2013; @PhysRevD90095009]. Further constraints including unitary violation in W and Z scattering with the top have been defined in Refs [@PhysRevD.88.013014; @PhysRevD.87.011702]. The strongest constraints come from precision electron dipole-moment (EDM) measurements [@Brod2013; @Cirigliano:2016njn; @Chien2016], however these are done under assumptions which when relaxed allow for much looser constraints. Assuming a standard model value for $y_t$, collider constraints have rendered $\xi_t \in [0,2\pi/3]$ at 2$\sigma$ [@Mildner2016_1000065207]. For this study we also assume that $y_t$ and $WWh$ coupling adopt standard model values. Furthermore, for the sake of completeness, we perform the study on the entire region $\xi_t \in [0,\pi]$.
Signal and Background {#sec:sigback}
=====================
The process studied is Higgs production with an associated top quark and jet, $p p \to tjh$, at the 14 TeV LHC. Fig \[fig:prod-diag\] displays the dominant Feynman diagrams contributing to signal production.
![\[fig:prod-diag\] Feynman diagrams for the dominant production process $bq\to thj$. ](./figs/prod1.pdf "fig:") ![\[fig:prod-diag\] Feynman diagrams for the dominant production process $bq\to thj$. ](./figs/prod2.pdf "fig:")
Due to the extremely clean signature it provides, the decay mode of $h\to\gamma\gamma$ has comparable signal sensitivity to the $h\to b\bar{b}$ decay despite a much smaller branching ratio. We hence choose this decay mode of the Higgs for our analysis. In Fig \[fig:xsections\] one can see the effect of the CP-mixing parameter $\xi_t$ on the production cross section of $tjh$. This effect is in agreement with previous results seen in Ref [@Rindani:2016scj] and contains a maximum enhancement at $\xi_t = \pi$ of approximately 1200%.
![\[fig:xsections\] The production cross section (red) and production cross section times $h\to\gamma\gamma$ branching ratio (blue) normalized with respect to standard model values for $\xi_t\in [0,\pi]$.](./figs/xsections.pdf)
We demand a final state containing 1 or more $b$-jets, exactly 1 lepton (but not $\tau$) and at least 2 photons. The irreducible background for this process is $tj\gamma\gamma$ continuum. The sub-leading background is $t\bar{t}\gamma\gamma$ where a $b$-jet is mistagged as a light jet. Finally the background $Wjj\gamma\gamma$ where a light jet is mistagged as a $b$-jet exists, however it is found to be at least an order of magnitude lower than the previous two backgrounds [@Wu:2014dba; @Chang:2014rfa] and is ignored in this study.
Event Simulation and Reconstruction {#sec:simreco}
===================================
The parton level events are produced in `MG5_aMC_v2_6_0` [@Frederix:2018nkq] then passed to `PYTHIA8` [@Sjostrand:2007gs] for hadronization/fragmentation and finally `Delphes` [@deFavereau:2013fsa] for detector effects.
We employ anti-kt jet clustering and take a $b$-tag efficiency of 77%, a mistagging efficiency of 1% and a lepton selection efficiency of 100%. We also employ the following detector acceptance cuts: $$\begin{aligned}
p_T^{b,\ell} > 20 \text{GeV},~|\eta_{b,\ell}| < 2.5,~p^j_T > 25 \text{GeV},~|\eta_j| > 2.5\end{aligned}$$ The cut on $|\eta_j|$ is employed to take advantage of the forwardness of the light jet which is characteristic of the $tjh$ signal.
As we are selecting exactly 1 lepton we are able to calculate the longitudinal momentum of the neutrino decaying from the top. We do this using the following quadratic equation: $$p_\nu^z = \frac{1}{2p_{\ell T}^2}\left(A_Wp^z_{\ell}\pm E_\ell\sqrt{A_W^2\pm 4p^2_{\ell T}E^2_{\nu T}}\right),$$ where, $A_W = M^2_{W^{\pm}} + 2p_{\ell T}\cdot E_{\nu T}$. We choose the solution for $p_\nu^z$ that is real and that when combined with the remainder of the neutrino and lepton 4-vector components produces an invariant mass closest to the $W$ boson mass. After this is done the top quark is reconstructed from the neutrino, the lepton and the $b$-jet which best reproduces the top quark invariant mass.
In Fig \[fig:variables\] we present the variables selected for the numerical analysis to come. In the $\ell^\phi_0$ plot (upper right of each subfigure) the lepton azimuthal distribution generated from the prescription above can be seen for the hardest lepton in each event. It is clear that the value of $\xi_t$ is impacting this distribution significantly as the SM value of $\xi_t = 0$ presents a distribution identical to the background while the fully CP-odd value of $\xi_t = \frac{\pi}{2}$ presents a far more skewed distribution. In addition, the variables show that the reconstruction is faithfully producing invariant mass distributions for the top quark and Higgs.
[0.5]{} ![\[fig:variables\] Examples of variables employed in the analysis for the $\xi = 0$ (a) and $\xi = \frac{\pi}{2}$ (b) benchmarks.](./figs/xi-0-variables-custom.pdf "fig:")
\
[0.5]{} ![\[fig:variables\] Examples of variables employed in the analysis for the $\xi = 0$ (a) and $\xi = \frac{\pi}{2}$ (b) benchmarks.](./figs/xi-pi-2-variables-custom.pdf "fig:")
Results {#sec:results}
=======
As seen in Ref [@Rindani:2016scj], we can construct the lab frame left-right asymmetry of the charged lepton using: $$\begin{aligned}
A_{\phi}^{\ell} = \frac{ \sigma\left(\cos\phi > 0\right) - \sigma\left(\cos\phi < 0\right) }
{ \sigma\left(\cos\phi > 0\right) + \sigma\left(\cos\phi < 0\right)}\end{aligned}$$ Fig \[fig:asymmetry\_phi\_lep\] displays this asymmetry as a function of $\xi_t$ which takes a maximimum at $\xi_t = \frac{\pi}{2}$. However when compared to the parton level calculation of this asymmetry found in Ref [@Rindani:2016scj] it can be seen that the detector effects lead to a flattening of this curve.
![\[fig:asymmetry\_phi\_lep\] The $A^{\ell}_{\phi}$ asymmetry as a function of $\xi_t$ for the signal and background.](./figs/asymmetry_phi_lep-haa-allbackgrounds.pdf)
Standard cut flows for this signal have been performed in the past such as in Refs [@Yue:2014tya][@Rindani:2016scj] and thus we do not perform one in this work. Instead we employ a boosted decision tree analysis on the variables found in Fig \[fig:variables\] using the `Toolkit for Multivariate Data Analysis` (TMVA) [@Harnik:2013aja].
$\xi_t$ $N^b_s$ Cut $N^a_s$ $N^a_b$ $Z$ $Z_{0.2}$
------------------ --------- -------- --------- --------- ------- -----------
$0$ 8 0.1146 7 21 1.28 1.04
$\frac{\pi}{4}$ 23 0.0910 20 24 3.05 2.44
$\frac{\pi}{2}$ 89 0.0819 83 27 7.92 7.04
$\frac{3\pi}{4}$ 198 0.0317 191 35 12.70 11.52
$\pi$ 255 0.0503 244 34 14.64 13.55
: \[tab:significance\]\
Table of signal sensitivity defined as: $Z = S/\sqrt{S+B+\left(\Delta B\right)^2}$ after applying the optimized cut generated by BDT analysis. A luminosity of 3000fb$^{-1}$ is chosen and the number of background events before cuts is $N^b_b = 1076$.
Tab \[tab:significance\] presents the BDT results for each value of $\xi$. The column labels are as follows: $N^b_s$ is the number of signal events before cuts while $N^a_s$ and $N^a_b$ are the number of signal and background events after cuts respectively. The number of background events before cuts was $N^b_b = 1076$. The column labelled “Cut” is the position of the optimal cut on the BDT classifier distribution. $\sigma$ is the signal sensitivity considering no systematic error, while $\sigma_{0.2}$ is the signal sensitivity with a flat 20% systematic error. The systematic error of 20% was taken as an estimate of the overall level of systematic error in typical 1-lepton plus jets final state experiments [@Aaboud:2017aeu]. The results presented in this table show, as expected, high values of $\xi_t \geq \frac{\pi}{2}$ are strongly inconsistent with background only. The 95$\%$ C.L. exclusion of approximately $\xi_t = 0.54$ without systematic and $\xi_t = 0.68$ with systematic is expected for 3ab$^{-1}$ of data at the HL-LHC, a significant improvement on the $\xi_t = 0.79$ exclusion placed in Ref [@Rindani:2016scj]. This is due to a combination of the BDT analysis and the additional top polarisation variable included. It is likely that had this analysis been undertaken at parton level like previous analyses of this process then improvements on the constraints would be even larger, that is to say that detector effects have likely reduced the overall positive impact of the BDT analysis and top polarisation variable.
Conclusion {#sec:conc}
==========
The direct detection of top-Higgs coupling has now been achieved via the process $pp \to t\bar{t}h$, however the properties of this coupling still require further study. The process $pp \to tjh$ provides a good window into the charge-parity properties of the coupling as increasing values of the top-Higgs coupling phase $\xi_t$ lead to increased cross sections, while the $t\bar{t}h$ process experiences decreases.
In this work we have introduced a CP-mixing parameter $\xi_t$ to the SM top-Higgs coupling via an effective operator. We have explored the well studied effects of this variable on top and jet associated Higgs production. Previous studies were then expanded on by performing a full detector level analysis of this process including the variable defined from the azimuthal distribution of the lepton decaying from the top which provides a powerful insight into top polarisation. Results were then further improved via the application of a ML algorithm, namely boosted decision tree analysis, to optimize signal sensitivity.
The key result of this study is a projected 95$\%$ median exclusion of $\xi_t \leq 0.54$ when not considering systematic errors and $\xi_t \leq 0.68$ when considering a conservative level of systematic error with 3ab$^{-1}$ of luminosity, a significant improvement over previous analyses of this process. It is reasonable that the HL-LHC can provide very strong limits on $\xi_t$ even in pessimistic scenarios.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the University of Adelaide and the Australian Research Council through the ARC Center of Excellence for Particle Physics (CoEPP) at the Terascale (grant no. CE110001004).
|
---
abstract: 'Let $\cal M$ be a semi-finite von Neumann algebra equipped with a distinguished faithful, normal, semi-finite trace $\tau$. We introduce the notion of equi-integrability in non-commutative spaces and show that if a rearrangement invariant quasi-Banach function space $E$ on the positive semi-axis is $\alpha$-convex with constant $1$ and satisfies a non-trivial lower $q$-estimate with constant $1$, then the corresponding non-commutative space of measurable operators $E({\cal M}, \tau)$ has the following property: every bounded sequence in $E({\cal M}, \tau)$ has a subsequence that splits into a $E$-equi-integrable sequence and a sequence with pairwise disjoint projection supports. This result extends the well known Kadec-Pełczyński subsequence decomposition for Banach lattices to non-commutative spaces. As applications, we prove that for $1\leq p <\infty$, every subspace of $L^p(\cal M, \tau)$ either contains almost isometric copies of $\ell^p$ or is strongly embedded in $L^p(\cal M, \tau)$.'
address: 'Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056'
author:
- Narcisse Randrianantoanina
title: ' Sequences In non-commutative $L^p$-spaces'
---
\
[^1]
Introduction
============
In [@KP], Kadec and Pełczyński proved that if $1\leq p<\infty$ then every bounded sequence $\{f_n\}_{n=1}^\infty$ in $L^p[0,1]$ has a subsequence that can be decomposed into two extreme sequences $\{g_k\}_{k=1}^\infty$ and $\{h_k\}_{k=1}^\infty$, where the $h_k$’s are pairwise disjoint and the $g_k$’s are $L_p$-equi-integrable that is $\lim\limits_{m(A) \to 0}\sup\limits_{k}\Vert \chi_{A} g_{k}
\Vert_p \rightarrow 0$ and $h_k \perp g_k$ for every $k \geq 1$. This result was used to study different structures of subspaces of $L^p[0,1]$. Later, the same decomposition property was proved for larger classes of Banach function spaces (see [@FGJ] for Orlicz spaces with $\Delta_{2}$-condition and $q$-concave lattices, [@JMST] for some symmetric spaces). There are however Banach lattices with sequences for which the above decomposition is not possible. Examples of reflexive, $p$-convex Banach lattices without the subsequence decomposition can be found in a paper of Figiel [*et al*]{} [@FGJ]. Subsequently, Weis [@WE] characterized, in terms of uniform order continuity conditions and ultrapowers, all Banach lattices where such property is possible. For the case of rearrangement invariant function spaces, the order-continuous spaces, in which the above decomposition works, were fully characterized as those that have the Fatou property (equivalently, those that contains no subspace isomorphic to $c_{0}$). In [@D3LRS], a version of Kadec-Pełczynski decomposition was considered for preduals of semi-finite von Neumann algebras.
It is the intention of the present paper to give an extension of the Kadec-Pełczyński decomposition stated above to the case of general non-commutative symmetric spaces of measurable operators. Let $\cal M$ be a von Neumann algebra, equipped with a distinguished faithful, normal, semi-finite trace $\tau$ and $E$ be a rearrangement invariant Banach function space on $[0,1]$ or the half line $(0,\infty)$. We define equi-integrability in the non-commutative setting as generalizations of Akemann’s characterization of weak compactness on preduals of von Neumann algebras. Using such notion, we provide an analogue of the Kadec-Pełczyński decomposition for non-commutative spaces. Namely, we proved that if $E$ is order continuous and satisifies the Fatou property then the corresponding symmetric space of measurable operators $E(\cal M,\tau)$ has the subsequence splitting property. Our approach allows ones to consider more general spaces such as quasi-Banach rearrangement invariant spaces that are $\alpha$-convex with constant $1$ and satisfy non trivial $q$-lower estimate with constant $1$. In particular splitting of bounded sequences is valid in non-commutative $L^p$-spaces for $0<p<\infty$. It should be noted that Sukochev [@SU] obtain a similar result for the case of finite von Neumann algebras. We also remark that since $c_{0}$ fails the subsequence splitting property, our result for the case where $E$ is a Banach space case is the best possible.
As application of the main result, we study the structure of subspaces of $L^p(\M,\T)$ for $1<p<\infty$ which generalizes the case $p=1$ treated in [@D3LRS].
We refer to [@KR] and [@TAK] for general information concerning von Neumann algebras as well as non-commutative integration, to [@SA] and [@LT] for Banach lattice theory.
Definitions and preliminary results
===================================
Throughout, $H$ is a given Hilbert space and ${\cal M}\subset \cal{B}(H) $ denotes a semi-finite von Neumann algebra with a distinguished normal, faithful semi-finite trace $\tau$. The identity in $\cal M$ will be denoted by $\bf{1}$ and $\M_p$ will stand for the set of all (self adjoint) projections in $\M$. A closed and densely defined operator $a$ on $H$ is said to be affiliated with $\cal M$ if $ua = au$ for all unitary operator $u$ in the commutant $\cal{M}'$ of $\cal M$.
A closed and densely defined operator $x$, affiliated with $\cal M$, is called $\tau$-measurable if for every $\epsilon > 0$, there exists an orthogonal projection $p \in \M$ such that $p(H)\subseteq
\text{dom}(x)$, $\tau ({\bf 1} -p)<\epsilon$ and $xp \in \cal{M}$. The set of all $\tau$-measurable operators will be denoted by $\widetilde{\cal M}$. The set $\widetilde{\cal M}$ is a ${*}$-algebra with respect to the strong sum, the strong product and the adjoint operation. Given a self-adjoint operator $x$ in $\widetilde{\cal M}$, we denote by $e^{x}(\cdot)$ its spectral measure. Recall that $e^{\vert x \vert}(B) \in \cal{\cal M}$ for all Borel sets $B
\subseteq \R$ and $x \in \widetilde{\cal M}$. For fixed $x \in \widetilde{\cal M}$ and $t \geq 0$, we define $$\mu_{t}(x) = \inf \left\{s\geq 0 : \tau(e^{\vert x \vert}(s,
\infty))\leq t \right\}.$$ The function $\mu_{(.)}(x):[0, \infty) \rightarrow [0,
\infty]$ is called the generalized singular value function (or decreasing rearrangement) of $x$. We note that $\mu_t(x) < \infty$ for every $t > 0$. For a complete study of $\mu_{(.)}$, we refer the reader to [@FK]. The topology defined by the metric on $\widetilde{\M}$ obtained by setting $$d(x,y)=\inf\left\{t\geq 0:\ \mu_t(x-y)\leq t \right\},
\ \ \ \text{for}\
x, y \in \widetilde{\M},$$ is called the [*measure topology*]{}. It is well-known that a net $(x_\alpha)_{\alpha \in I}$ in $\widetilde{\M}$ converge to $x \in \widetilde{\M}$ in measure topology if and only if for every $\epsilon >0$, $\delta >0$, there exists $\alpha_0 \in I$ such that whenever $\alpha \geq \alpha_0$, there exists a projection $p \in \M_p$ such that $$\left\Vert (x_\alpha - x)p \right\Vert_{\M} <\epsilon \ \
\text{and}\ \ \T({\bf 1}-p)<\delta.$$ It was shown in [@N] that $(\widetilde{\M}, d)$ is a complete metric space.
Recall that if we consider $\cal{M} = L^{\infty}(\R^{+},m)$, where $m$ is the Lebesgue measure on $\R^{+}$ then $\cal{M}$ is an abelian von Neumann algebra acting on $L^2(\R^{+},m)$ via multiplication, with the trace being the integration with respect to $m$, then $\widetilde{\cal M} = L^{0}(\R^{+},m)$ (the usual space of all measurable functions on $\R^{+}$) and the generalized singular value $\mu(f)$ is precisely the decreasing rearrangement of the function $\vert f\vert$ (usually denoted by $f^*$ in Banach lattice theory).
A symmetric quasi-Banach function space on $\R^+$ is a quasi-Banach lattice $E$ of measurable functions with the following properties:
- $E$ is an order ideal in $L^{0}(\R^{+},m)$;
- $E$ is rearrangemant invariant in the sense of [@LT] (p .114);
- $E$ contains all finitely supported simple functions.
Unless stated otherwise, $E$ will always denote a symmetric quasi-Banach function space on $\R^+$. We define the symmetric space of measurable operators $E(\cal{ M}, \tau)$ by setting $$E(\cal{M},\tau) := \left\{x \in \widetilde{\cal M} : \mu(x)\in E
\right\}$$ and $$\left\Vert x \right\Vert_{E(\cal{M}, \tau)} =
\left\Vert \mu(x) \right\Vert_{E} \ \ \text{for \ all} \ x \in
E(\cal{M},\tau).$$
It is shown in [@X] (Lemma 4.1) that if $E$ is $\alpha$-convex (for some $0 < \alpha \leq 1$) with constant $1$, then $\Vert \cdot
\Vert_{E(\cal{M}, \tau)}$ is an $\alpha$-norm that is for every $x, y \in E(\cal{M},\tau)$, $$\left\Vert x +
y \right\Vert_{E(\cal{M}, \tau)}^\alpha \leq
\left\Vert x \right\Vert_{E(\cal{M}, \tau)}^\alpha + \left\Vert y
\right\Vert_{E(\cal{M}, \tau)}^\alpha.$$ Equipped with $\Vert \cdot \Vert_{E(\cal{M}, \tau)}$, the space $E(\cal{M}, \tau)$ is a $\alpha$-Banach space. The space $E(\cal{M}, \tau)$ is often referred to as the non-commutative analogue of the function space $E$. We remark that if $0<p<\infty$ and $E = L^p
(\R^{+}, m)$ then $E(\cal{M}, \tau)$ coincides with the usual non-commutative $L^p$-space associated to the semi-finite von Neumann algebra $\cal M$. Also if $E=L^\infty(\R^{+}, m)$, then $L^\infty(\M,\tau)$ is the von Neumann algebra $\M$. We refer to [@DDP1], [@DDP3] and [@X] for some background on the space $E(\cal{M},\tau)$.
A quasi-Banach function space $E$ is said to satisfy a lower $q$-estimate if there exists a positive constant $C>0$ such that for all finite sequences $\{x_{n}\}$ of mututally disjoint elements in $E$, $$\left(\sum \left\Vert x_n \right\Vert^q \right)^{\frac1{q}} \leq
C \left\Vert \sum x_n \right\Vert.$$
The least such constant $C$ is called the constant of the lower $q$-estimate. Recall that if $E$ is a quasi-Banach function space and $1<p<\infty$, $$E^{(p)} = \left\{x \in L^0(\R^{+}, m); \vert x \vert^{p} \in E \right\}
\ with \ \left\Vert x
\right\Vert_{E^{(p)}} = \left\Vert \ \vert x \vert^{p} \right\Vert_{E}^{\frac1{p}}.$$
We will need the following known result. A proof can be found in [@D3LS].
\[ordercontinuity\] Assume that $E$ is order-continuous and $\alpha$-convex with constant $1$ for some $0 < \alpha \leq 1$.
- If $x \in E(\cal{M}, \tau)$ and $e \leq f$ are projections in $\cal M$ then $\Vert exe \Vert_{E(\cal{M}, \tau)} \leq \Vert fxf \Vert_{E(\cal{M}, \tau)}$;
- If $x \in E(\cal{M}, \tau)$ and $e_{\beta}\downarrow_\beta 0$ is a net of projections in $\cal M$ then $\Vert x e_\beta\Vert_{E(\cal{M}, \tau)} \downarrow_\beta 0$.
The following definition isolates the main topic of this paper.
\[equi-integrability\] Let $E$ be a quasi-Banach function space on $\R^{+}$ and $K$ be a bounded subset of $E(\cal{M}, \tau)$. We will say that $K$ is [**$E$-equi-integrable**]{} if $\lim\limits_{n \to \infty} \sup\limits_{x \in K}
\left\Vert e_n x e_n \right\Vert_{E(\cal{M},\tau)} = 0$ for every decreasing sequence $\{e_n\}_{n=1}^\infty$ of projections with $e_n \downarrow_n 0$.
We remark, from Proposition \[ordercontinuity\], that since $\{e_n\}_{n=1}^\infty$ is decreasing, so is the sequence $\left\{\sup\limits_{x \in K}
\left\Vert e_n x e_n \right\Vert_{E(\cal{M},\tau)}\right\}_{n=1}^\infty$ and therefore the limit in the definition above always exists. This definition was motivated by the commutative case on one hand and the characterization of weakly compact subsets of $L^{1}(\cal{M}, \tau)$ by Akemann [@AK] (see also [@TAK] p.150) on the other. Using this terminology, Akemann’s characterization can be stated as in the commutative case: relatively weakly compact subsets of $L^1(\cal{M}, \tau)$ are exactly the equi-integrable sets. Such characterization is not valid in general. For $1<p<\infty$, any set of normalized disjoint sequence cannot be $L^p$-integrable but since $L^p$ is reflexive, such set is relatively weakly compact.
On the next proposition, we will show that for the general case, one implication always holds.
\[compact\] Assume that $E$ is an order-continuous symmetric Banach function space and $K$ is a $E$-equi-integrable set in $E(\cal{M},\tau)$ then $K$ is relatively weakly compact.
The proposition will be proved in several steps. Recall that if $E$ is a symmetric Banach function space, then $E(\cal{M},\tau)$ is a subset of $L^1(\cal{M} , \tau) + \cal{M}$ and therefore if $p$ is a projection in $L^1(\cal{M}, \tau ) \cap \cal{M}$ and $K$ is a subset of $E(\cal{M},\tau)$, then $pK$ and $Kp$ are subsets of $L^1(\cal{M},\tau)$.
\[finitecompact1\] Let $p$ be a projection in $L^1(\cal{M},\tau) \cap \cal{M}$ and $K$ be a $E$-equi-integrable subset of $E(\cal{M},\tau)$. The sets $pKp$ and $pK(1-p)$ are relatively weakly compact in $L^1(\cal{M},\tau)$.
To see this lemma, it is enough to check that these sets are $L^1$-equi-integrable. Let $T: E(\cal{M},\tau) \longrightarrow L^1(\cal{M},\tau)$ be the linear map defined by $x \to Tx= pxp$. This map is well-defined and one can deduce from the closed graph theorem that it is bounded. Let $\{e_n\}_{n=1}^\infty$ be a sequence of projections with $e_n \downarrow_n 0$. For each $n\geq 1$, set $f_n$ to be the right support projection of $e_np$. By the definition of support projections, $f_n \leq p$. So $\{f_n\}_{n=1}^\infty$ is a sequence of finite projections. We also note that (see for instance the proof of [@TAK Proposition 1.6 p.292]), $$f_n = e_n \vee ({\bf 1}-p) - ({\bf 1}-p)$$ and by Kaplansky formula (see for instance [@KR Theorem 6.1.6 p.403]), $$f_n \sim e_n - e_n \wedge ({\bf 1}-p).$$ Since $\tau(f_n) =\tau(e_n - e_n \wedge({\bf 1}-p)) \leq \tau(p)$ and $\{e_n - e_n \wedge({\bf 1}-p)\}_{n=1}^\infty$ converges to zero, $\{f_n\}_{n=1}^\infty$ converges to zero. Now since the $f_n$’s are finite projections, we conclude that if $g_n= \wedge_{k \geq n}f_k $, then $\{g_n\}_{n=1}^\infty$ converges to zero. Therefore, for every $x \in K$, $$\begin{aligned}
\left\Vert e_n pxp e_n \right\Vert_1
&=\left\Vert e_n p (g_n x g_n) pe_n \right\Vert_1 \cr
&\leq \left\Vert p(g_n x g_n)p \right\Vert_1 \cr
&\leq ||T|| \cdot \left\Vert g_n x g_n \right\Vert_{E(\cal{M}, \tau)}.\end{aligned}$$ Since $K$ is $E$-equi-integrable, one obtain that $$\lim_{n\to \infty}\sup_{y \in pKp} \left\Vert e_n y e_n \right\Vert_1
\leq ||T|| \cdot \lim_{n \to \infty}\sup_{x\in K}\left\Vert g_n x g_n
\right\Vert_{E(\cal{M},\tau)}=0$$ which concludes that $pKp$ is relatively weakly compact in $L^1(\cal{M},\tau)$.
For $pK({\bf 1}-p)$, set $S: E({\cal{M}},\tau) \to L^1(\M,\T)$ be the map defined by $x \to Sx=px({\bf 1}-p)$. As above, $S$ is bounded. Let $\{e_n\}_{n=1}^\infty$ and $\{g_n\}_{n=1}^\infty$ be sequences of projections as discribed above. For each $ n\geq 1$, let $s_n$ be the left support projection of $({\bf 1}-p)e_n$. Then $s_n =e_n \vee p - p $ for every $n \geq 1$ and the sequence $\{s_n\}_{n=1}^\infty$ is decreasing. It is claimed that $s_n \downarrow_n 0$.
For this, it is enough to check that $e_n \vee p \downarrow_n p$. In fact, $e_n \vee p - e_n \sim p - e_n \wedge p$ and the sequence defined by the right hand side of the equivalence converges to $p$ which implies that $$\lim_{n\to \infty} \tau(e_n \vee p - e_n)= \lim_{n \to \infty}
\tau(p- e_n\wedge p) =\tau(p)$$ which gives $$\lim_{n \to \infty} \tau(e_n \vee p - p - e_n)=0.$$ But since $(e_n \vee p - p - e_n)^2 =(e_n \vee p - p - e_n) + e_np + pe_n$, we can conclude that $$\lim_{n \to \infty} \Vert e_n \vee p - p - e_n \Vert_2 =0.$$ From this, we get (by passing to a subsequence if necessary) that $\{e_n \vee p - p - e_n\}_{n=1}^\infty$ converges to zero in measure. Similarly, $\{e_n \vee p - p -pe_n\}_{n=1}^\infty$ converges to zero in measure so $e_n \vee p \downarrow_n p$ hence $s_n \downarrow_n 0$.
To conclude the proof of Lemma \[finitecompact1\], note that $g_n \perp s_n$ so $g_n \vee s_n = g_n + s_n $. In particular, $g_n \vee s_n \downarrow_n 0$ and we get that $$\begin{split}
\lim_{n\to \infty}\sup_{y \in pK({\bf 1}-p)}
\left\Vert e_n y e_n
\right\Vert_1
&=\lim_{n\to \infty}\sup_{x \in K} \left\Vert e_n p x
({\bf 1}-p)
e_n \right\Vert_1\\
&=\lim_{n\to \infty}\sup_{x \in K} \left\Vert e_n p
(g_n \vee s_n) x (g_n \vee s_n)({\bf 1}-p) e_n \right\Vert_1 \\
&\leq\lim_{n\to \infty}\sup_{x \in K} \left\Vert p
(g_n \vee s_n) x (g_n \vee s_n)({\bf 1}-p) \right\Vert_1 \\
&\leq \|S\|\cdot \lim_{n\to \infty}\sup_{x \in K} \left\Vert
(g_n \vee s_n) x (g_n \vee s_n) \right\Vert_{E(\M,\tau)} =0.
\end{split}$$
The proof is complete.
\[finitecompact2\] Let $p$ and $K$ be as in Lemma \[finitecompact1\]. Then $pK$ is relatively weakly compact in $E(\M,\T)$.
Note first that $pK$ is $E$-equi-integrable. This can be seen by applying the series of argument used in Lemma \[finitecompact1\], considering the operators $T$ and $S$ as maps from $E(\M,\T)$ into $E(\M,\T)$.
Let $\{px_n\}_{n=1}^\infty$ be a bounded sequence in $pK$. From Lemma ref[finitecompact1]{}, we can assume that $\{px_n\}_{n=1}^\infty$ is weakly convergent in $L^1(\M,\T)$. Fix $\varphi \in
E^*(\M, \T)_{+}$ and let $\varphi=\int_{0}^\infty t \ de_t$ be its spectral decomposition. For each $k \geq 1$, set $q_k:=e^{\varphi}( (0,k))$. We remark that $\varphi q_k = q_k \varphi \in \M$. For $m, \ n \in \N$, $$\begin{aligned}
\langle \varphi, px_n -px_m \rangle
&=\langle \varphi -\varphi q_k, px_n -px_m \rangle +
\langle \varphi q_k, px_n -px_m \rangle \\
&=\langle \varphi ({\bf 1}-q_k), px_n -px_m \rangle +
\langle \varphi q_k, px_n -px_m \rangle \\
&=\langle ({\bf 1}-q_k)\varphi ({\bf 1}-q_k), px_n -px_m \rangle +
\langle \varphi q_k, px_n -px_m \rangle.\end{aligned}$$ This gives $$\left|\langle \varphi, px_n -px_m \rangle\right| \leq
\left|\T\left( \varphi (1-q_k)(px_n -px_m)(1-q_k) \right) \right| +
\left|\langle \varphi q_k, px_n -px_m \rangle \right|.$$ Since $\varphi q_k$ belongs to $\M$, $$\limsup\limits_{n,m \to \infty}\left|\langle \varphi q_k, px_n -px_m
\rangle\right|
\leq 2 \left\Vert\varphi \right\Vert_{E^*(\M,\T)}
\cdot \sup_{a \in K}\left\Vert ({\bf 1}-q_k)pa({\bf 1}-q_k)
\right\Vert_{E(\M,\T)}.$$ Now since ${\bf 1}-q_k \downarrow_k 0$ and $pK$ is $E$-equi-integrable, we obtain that $$\lim_{n,m \to \infty}\langle \varphi q_k, px_n -px_m
\rangle=0.$$ The lemma is proved.
To complete the proof of Proposition \[compact\], let $\{p_k\}_{k=1}^\infty$ be a sequence of projections that increases to ${\bf 1}$ and $\T(p_k)< \infty$ and fix $\epsilon > 0$. Choose $k_0 \geq 1$ such that $$\sup_{a \in K} \left\Vert ({\bf 1}-p_{k_0})a({\bf 1}-p_{k_0})\right\Vert_{E(\M,\T)} \leq
\epsilon.$$ We have $K=p_{k_0}K + ({\bf 1}-p_{k_0})Kp_{k_0} +
({\bf 1}-p_{k_0})K({\bf 1}-p_{k_0})$ which implies that $$K \subset p_{k_0}K + ({\bf 1}-p_{k_0})Kp_{k_0} + \epsilon B_{E(\M,\T)}$$ where $B_{E(\M,\T)}$ denotes the closed unit ball of $E(\M,\T)$. From Lemma \[finitecompact2\], the sets $p_{k_0}K$ and $({\bf 1}-p_{k_0})Kp_{k_0}$ are relatively weakly compact which concludes that $K$ is relatively weakly compact. The proof is complete.
If $\tau({\bf 1}) <\infty$, the proof above can be considerably shortened. In this case, $E(\M,\T) \subset L^1(\M,\T)$ so if $K$ is $E$-equi-integrable, then it is relatively weakly compact in $L^1(\M,\T)$ and on can argue as in the last part of Lemma \[finitecompact2\] to conclude that $K$ is relatively weakly compact in $E(\M,\T)$.
The following proposition should be compared with [@CS2](Theorem 5.1 and Theorem 5.2). It generalizes well known property of equi-integrable sets in function spaces to the non-commutative settings.
\[measure\] Let $E$ be a symmetric quasi-Banach space function space and $K$ be a $E$-equi-integrable subset of $E(\M,\T)$. For each sequence $\{x_n\}_{n=1}^\infty$ in $K$ and $x \in \overline{K}$, the following are equivalent:
- $\lim\limits_{n \to \infty} \left\Vert x_n -x
\right\Vert_{E(\M,\T)} =0$;
- $\{x_n\}_{n=1}^\infty$ converges to $x$ in measure (as $n \to \infty$).
The implication $(a) \Rightarrow (b)$ is trivial. For $(b) \Rightarrow (a)$, we will assume that $x=0$. Recall that there exists $0 < \alpha \leq 1$, such that $\M \cap L^\alpha(\M,\T) \subset E(\M,\T) \subset
\M + L^\alpha(\M,\T)$ with $\Vert x \Vert_{\M +L^\alpha(\M,\T)}
\leq \Vert x \Vert_{E(\M,\T)} \leq
2\Vert x \Vert_{\M \cap L^\alpha(\M,\T)}$ for every $x \in \M \cap L^\alpha(\M,\T)$. The proposition will be proved by showing the following lemma:
For every $p \in \M_p $ with $\T(p)<\infty$, $\lim\limits_{n \to \infty}
\left\Vert x_np \right\Vert_{E(\M,\T)} =
\lim\limits_{n \to \infty}
\left\Vert px_n \right\Vert_{E(\M,\T)} =0$.
To see this lemma, fix $\epsilon >0$ and let $C=\max\{1,\tau(p)\}$. Since $K$ is equi-integrable, there exists $\delta>0$ such that whenever $q \in \M_p$ satisfies $\tau(q)<\delta$, then for every $n \in \N$, $\left\Vert qx_nq \right\Vert_{E(\M,\T)} \leq \epsilon/(2)^{1/\alpha}$. Since both $\{x_n\}_{n=1}^\infty$ and $\{x_n^*\}_{n=1}^\infty$ converge to zero in measure, one can choose $n_0 \geq 1$ such that for each $n \geq n_0$, there exists a projection $p_n \in \M_p$ with $\T({\bf 1}-p_n) <\delta$, $$\left\Vert x_n p_n \right\Vert_{\M} <
\frac{\epsilon}{2[4C]^\frac{1}{\alpha}}$$ and $$\left\Vert x_n^*p_n \right\Vert_{\M} <
\frac{\epsilon}{2[4C]^\frac{1}{\alpha}}.$$ For $n \geq n_0$, $$\begin{aligned}
\left\Vert x_np \right\Vert_{E(\M,\T)}^\alpha
&\leq \left\Vert x_n p_n p \right\Vert_{E(\M,\T)}^\alpha
+ \left\Vert p_n x_n ({\bf 1}-p_n) p
\right\Vert_{E(\M,\T)}^\alpha
+ \left\Vert ({\bf 1}-p_n) x_n ({\bf 1}-p_n) p
\right\Vert_{E(\M,\T)}^\alpha \\
&\leq 2^\alpha \max\left\{\left\Vert x_n p_n
\right\Vert_{\M }^\alpha,
\left\Vert x_n p_n p \right\Vert_{L^\alpha(\M,\T)}^\alpha
\right\} \\
&\ +
2^\alpha
\max\left\{\left\Vert x_n^*p_n \right\Vert_{\M }^\alpha,
\left\Vert p({\bf 1}-p_n)x_n^*p_n \right\Vert_{L^\alpha(\M,\T)}^\alpha
\right\}
+
\left\Vert ({\bf 1}-p_n) x_n ({\bf 1}-p_n)
\right\Vert_{E(\M,\T)}^\alpha \\
&\leq 2. 2^\alpha \max\left\{ \epsilon^\alpha/ 2^\alpha 4C,
(\epsilon^\alpha/2^\alpha 4C) \tau(p) \right\} +
\epsilon^{\alpha}/2 \\
&\leq \epsilon^\alpha.\end{aligned}$$ A similar estimate works for $\{x_n^*p\}_{n=1}^\infty$. The lemma is verified.
To complete the proof of Proposition \[measure\], choose a mutually disjoint family $\{e_i\}_{i\in I}$ of projections in $\M$ with $\sum_{i \in I} e_i={\bf 1}$ for the strong operator topology and $\tau(e_i)<\infty$ for all $i \in I$. Using a similar argument as in [@X], one can get an at most countable subset $\{e_k\}_{k=1}^\infty$ of $\{e_i\}_{i \in I}$ such that for each $e_i$ outside of $\{e_k\}_{k=1}^\infty$, $e_ix_n =x_ne_i=0$ for every $n \in \N$. Let $e=\sum_{k \in \N} e_k$. Replacing $\M$ by $e{\M}e$ and $\tau$ by its restriction on $e{\M}e$, we may assume that $e={\bf 1}$. Let $p_n=\sum_{k\geq n}e_k$. It is clear that $p_n \downarrow_n 0$ and $\tau({\bf 1}-p_n) <\infty$ for every $n \in \N$. Fix $\epsilon>0$ and choose $n_0 \geq 1$ such that $$\sup_{n \in \N}\left\Vert p_{n_0} x_n p_{n_0}
\right\Vert_{E(\M,\T)} \leq \epsilon.$$ We get that $$\limsup_{n\to \infty}
\left\Vert x_n \right\Vert_{E(\M,\T)}^\alpha
\leq \lim_{n\to \infty}
\left\Vert x_n({\bf 1}-p_{n_0}) \right\Vert_{E(\M,\T)}^\alpha +
\lim_{n\to \infty}
\left\Vert x_n^*({\bf 1}-p_{n_0} \right\Vert_{E(\M,\T)}^\alpha +
\epsilon =\epsilon$$ and since $\epsilon$ is arbitrary, the proof is complete.
For the rest of this section, we collect some results that will be useful in the later sections of the paper.
The inequality given below can be viewed as the analogue of the well-known fact on normal functional on von Neumann algebra, $|\varphi(a)|^2 \leq \Vert \varphi \Vert \cdot \vert \varphi
\vert(aa^*)$ whenever $a \in \M$ and $\varphi \in \M_*$ [@TAK Proposition 4.6 p. 146], to the general case of symmetric spaces of measurable operators.
\[inequality\] Let $x \in E(\cal{M},\tau)$ and $y \in \cal{M}$ then $$\left\Vert xy \right\Vert_{E(\cal{M},\tau)} \leq \left\Vert \
\vert x \vert y \right\Vert_{E(\cal{M}, \tau)}
\leq \left\Vert x \right\Vert_{E(\cal{M},\tau)}^{\frac1{2}}
\cdot \left\Vert y^{*}\vert x \vert y
\right\Vert_{E(\cal{M},\tau)}^{\frac1{2}}.$$
Let $x = u \vert x \vert$ be the polar decomposition of $x$. Then $\left\Vert xy
\right\Vert_{E(\cal{M},\T)} = \left\Vert u \vert x \vert y
\right\Vert_{E(\cal{M},\T)}
\leq \left\Vert u
\right\Vert_{\infty} \cdot \left\Vert \ \vert x \vert y
\right\Vert_{E(\cal{M},\T)}$. Also $\left\Vert \ \vert
x \vert y \right\Vert_{E(\cal{M},\T)} = \left\Vert \ \vert x \vert^{\frac1{2}} \vert x
\vert^{\frac1{2}} y \right\Vert_{E(\cal{M},\T)}$ and using Hölder’s inequality, $$\begin{aligned}
\left\Vert \ \vert x \vert y \right\Vert_{E(\cal{M},\T)}
&\leq \left\Vert \ \vert x \vert^{\frac1{2}}
\right\Vert_{E^{(2)}{(\cal{M},\T)}} \cdot \left\Vert \
\vert x \vert^{\frac1{2}} y
\right\Vert_{E^{(2)}{(\cal{M},\T)}}\cr
&= \left\Vert x \right\Vert_{E(\cal{M},\T)}^{\frac{1}{2}}
\cdot \left\Vert y^* \vert x \vert y
\right\Vert_{E(\cal{M},\T)}^{\frac{1}{2}}.\end{aligned}$$
Let $K$ be a bounded subset of $E(\cal{M},\tau)$. If we set $|K|:=\left\{ |a|:\ a \in K \right\}$, then it is clear from Proposition \[inequality\] that if $\vert K \vert$ is $E$-equi-integrable then for every decreasing projections $e_n{\downarrow}_n 0$, $\lim\limits_{n \to \infty} \sup\limits_{x \in K}
\left\Vert x e_n
\right\Vert_{E(\M,\tau)}
= \lim\limits_{n \to \infty} \sup\limits_{x \in K}
\left\Vert e_n x \right\Vert_{E(\M,\tau)}= 0$. In particular if $|K|$ is $E$-equi-integrable then so is $K$.
\[equi-integrability2\] Assume that $E$ is $\alpha$-convex with constant 1 for some $0 < \alpha \leq 1$. Let $\{p_n\}_{n=1}^\infty$ be a sequence of decreasing projections in $\cal M$ and $K$ be a bounded subset of $E(\cal{M},\tau)$ such that:
- $p_n \downarrow_n 0$;
- For each $n \geq 1$, the sets $({\bf 1} - p_n)K$ and $\vert K({\bf 1}-p_n) \vert$ are $E$-equi-integrable.
Then $K$ is $E$-equi-integrable if and only if $\lim\limits_{n \to \infty}
\sup\limits_{a \in K} \left\Vert p_n a p_n \right\Vert_{E(\cal{M},\tau)} = 0.$
We will show the non trivial implication. Fix $f_k \downarrow_k 0$, a sequence in $\cal{M}_p$. We need to show that $\lim\limits_{k \to \infty} \sup\limits_{a \in K} \left\Vert f_k a f_k
\right\Vert_{E(\cal{M},\tau)} = 0$.
We will assume without loss of generality that $K$ is a subset of the unit ball of $E(\M,\T)$. For every $a \in K$, $$\begin{aligned}
f_k a f_k &= f_k({\bf 1}-p_n)af_k + f_k p_n a f_k \cr
&=f_k ({\bf 1}-p_n)a f_k + f_k p_n a ({\bf 1}-p_n)f_k + f_kp_n a p_n f_k.\end{aligned}$$ Since $E(\cal{M},\tau)$ is $\alpha$-convex, we get: $$\begin{aligned}
\left\Vert f_k a f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)}
&\leq \left\Vert f_k ({\bf 1}-p_n)a f_k
\right\Vert^{\alpha}_{E(\cal{M},\tau)}
+ \left\Vert f_k p_n a({\bf 1}-p_n)f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)}
+ \left\Vert f_k p_n a p_n f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)} \cr
&\leq \left\Vert f_k ({\bf 1}-p_n) a f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)} +
\left\Vert a ({\bf 1}-p_n) f_k
\right\Vert^{\alpha}_{E(\cal{M},\tau)}
+ \left\Vert p_n a p_n \right\Vert^{\alpha}_{E(\cal{M},\tau)}.\end{aligned}$$ Using Proposition \[inequality\] on the second term, we have $$\begin{aligned}
\left\Vert f_k a f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)}
&\leq \left\Vert f_k ({\bf 1}-p_n)a f_k
\right\Vert^{\alpha}_{E(\cal{M},\tau)} +
\left\Vert a ({\bf 1}-p_n) \right\Vert^{\frac\alpha{2}}_{E(\cal{M},\tau)}
\cdot \left\Vert
f_k \vert a({\bf 1}-p_n) \right\vert f_k \Vert^{\frac\alpha{2}}_{E(\cal{M},\tau)} \cr
&+ \left\Vert p_n a p_n \right\Vert^{\alpha}_{E(\cal{M},\tau)}.\end{aligned}$$ Let $\epsilon > 0$, choose $n_0$ large enough so that $\sup\limits_{a \in
K} \left\Vert p_{n_{0}}a p_{n_{0}} \right\Vert_{E(\cal{M},\tau)}
<\epsilon$. We conclude that $$\begin{aligned}
\lim\limits_{k \to \infty} \sup\limits_{a \in K}
\left\Vert f_k a f_k
\right\Vert^{\alpha}_{E(\cal{M},\tau)}
&\leq \lim\limits_{k \to \infty} \sup\limits_{a \in
K} \left\Vert f_k ({\bf 1}-p_{n_0})a f_k \right\Vert^{\alpha}_{E(\cal{M},\tau)}\cr
&+ \lim\limits_{k \to \infty} \sup\limits_{a \in K} \left\Vert f_k \vert
a({\bf 1}-p_{n_0}) \vert f_k
\right\Vert^{\frac\alpha{2}}_{E(\cal{M},\tau)}
+ \epsilon^\alpha.\end{aligned}$$ By (ii), the first two terms converge to zero so $\lim\limits_{k \to \infty}
\sup\limits_{a \in K} \Vert f_k a f_k \Vert_{E(\cal{M},\tau)} \leq \epsilon$ and since $\epsilon$ is arbitrary, the proof is complete.
The next proposition can be found in [@D3LS] (Proposition 2.5).
\[lower-estimate\] Assume that $E$ is $\alpha$-convex with constant $1$ for some $0 < \alpha \leq
1$ and satisfies a lower $q$-estimate with constant $1$ for some finite $q \geq \alpha$. If $k =2q/\alpha$, then for all $y \in E(\cal{M}, \tau)$, for all projections $e, f \in \cal{M}$ with $e + f = 1$ and $\tau(e) <
\infty$, it follows that $$\Vert e y e \Vert^k_{E(\cal{M},\tau)} +
\Vert e y f \Vert^k_{E(\cal{M},\tau)} +
\Vert f y e\Vert^k_{E(\cal{M},\tau)} +
\Vert f y f \Vert^k_{E(\cal{M},\tau)}
\leq \Vert y \Vert^k_{E(\cal{M}, \tau)}.$$
The proof of the next lemma is just a notational adjustment of the proof of a lemma from the $L^1$-case [@D3LRS] so we will leave the details to the interested readers.
\[subsequence\] Let $\{x_k\}_{k=1}^\infty$ be a bounded sequence in $E(\cal{M},\tau)$ and $\{e_n\}_{n=1}^\infty$ be a decreasing sequence of projections in $ \M$ such that $e_n\downarrow_n 0$. Assume that $\lim\limits_{n \to \infty} \sup\limits_{k \in \N} \Vert e_n
x_k e_n \Vert_{E(\cal{M},\tau)} =\gamma > 0$ then there exists a subsequence $\{x_{k_n}\}_{n=1}^\infty$ so that $\lim\limits_{n \to \infty} \Vert e_n x_{k_n} e_n \Vert_{E(\cal{M},\tau)} = \gamma$.
Kadec-Pełczyński theorem for symmetric spaces
=============================================
The main result of the present article is the following theorem.
\[main\] Let $E$ be an order continuous symmetric quasi-Banach function space in $\R^+$ that is $\alpha$-convex with constant $1$ for some $0 < \alpha \leq 1$ and suppose that $E$ satisfies a lower $q$-estimate with constant $1$ for some $q \geq \alpha$.
Let $\{x_n\}_{n=1}^\infty$ be a bounded sequence in $E(\cal{M},\tau)$ then there exists a subsequence $\{x_{n_k}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$, bounded sequences $\{y_k\}_{k=1}^\infty$ and $\{z_k\}_{k=1}^\infty$ in $E(\cal{M},\tau)$ and a decreasing sequence of projections $p_k \downarrow_n 0$ in $\cal M$ such that:
- $x_{n_k} = y_k + z_k$ for all $k \geq 1$;
- $\{y_k:\ k \geq 1\}$ is $E$-equi-integrable and $p_ky_kp_k = 0$ for all $k \geq 1$;
- $\{z_k\}_{k=1}^\infty$ is such that $p_kz_kp_k = z_k$ for all $k \geq
1$.
The proof will be divided into several steps. Without loss of generality, we will assume that the sequence $\{x_k\}_{k=1}^\infty$ is a subset of the unit ball of $E(\M,\T)$. Since we are dealing with sequences, we can and do assume without loss of generality that $\cal M$ is countably decomposable (see [@X] and the proof of Proposition \[measure\] above for the details of such reduction).
Set $\script{D}_1 : = \left\{\{e_n\}_{n=1}^\infty \subset \cal{M}_p:\ e_n \downarrow_n 0
\ \text{and}\ \tau(e_1)< \infty \right\}$ and consider $$\delta: = \sup\left\{\lim\limits_{n \to \infty} \sup\limits_{k \in \N}
\left\Vert e_n \vert x_k \vert e_n \right\Vert_{E(\cal{M},\tau)}:\
\{e_n\}_{n=1}^\infty \in \script{D}_1 \right\}.$$
As in [@D3LRS], one can show that $\delta$ is attained and using Lemma \[subsequence\], one can choose a subsequence of $\{x_{n}\}_{n=1}^\infty$ (which we will denote again by $\{x_n\}_{n=1}^\infty$) and $\{e_n\}_{n=1}^\infty \in \script{D}_1$ such that $$\label{delta}
\delta =\lim\limits_{n \to \infty} \left\Vert e_n \vert x_n \vert e_n
\right\Vert_{E(\cal{M},\tau)}$$ and $$\label{delta*}
\delta^*=\lim\limits_{n \to \infty} \left\Vert e_n \vert x^{*}_{n} \vert e_n
\right\Vert_{E(\cal{M},\tau)}
= \sup \left\{\lim\limits_{n \to \infty} \sup\limits_{k}
\left\Vert q_n \vert
x^{*}_{k} \vert q_n \right\Vert_{E(\cal{M},\tau)};
\{q_{n}\}_{n=1}^\infty \in \script{D}_1 \right\}.$$
For each $n \geq 1$, set $v_n: = x_n - e_n x_n e_n$ and let $V: = \left\{v_n:\ n \geq 1 \right\}$.
There exists a sequence of projections $\{g_{n}\}_{n=1}^\infty$ in $\cal{M}$ with:
- For every $n \geq 1$, $g_n \leq {\bf 1} - e_n$;
- $\tau(g_n)<\infty$, in particular $g_n$ is a finite projection;
- $g_n \uparrow^n \bf{1}$.
The lemma can be obtained inductively. Since $\cal M$ is countably decomposable, there exists $\varphi_0$ a faithful normal state in $\cal{M}_*$. Since ${\bf 1} - e_n$ is a semifinite projection, there exists a sequence of projections $\{g_j^{(n)}\}_{j=1}^\infty$ with $\tau (g_j^{(n)}) < \infty$ for every $j \geq
1$ and $g_j^{(n)} \uparrow^j {{\bf 1}-e_n}$. One can choose $j_n \geq 1$ such that $\varphi_0 (1-e_n) - \varphi_0(g^{(n)}_{j_n})< 1/n$. Set $$\begin{cases}
g_n: = g^{(1)}_{j_1} &\text{for } n=1 \cr
g_n = g^{(n)}_{j_n} \vee g_{n-1} &\text{for } n>1.
\end{cases}$$ It is easy to verify that $\{g_n\}_{n=1}^\infty$ satisfies the requirements of the lemma.
For each $n \geq 1$, let $p_n = {\bf 1} - g_n$. Clearly $p_n \downarrow_{n} 0$, ${\bf 1}-p_{n}$ is a finite projection and $p_n \geq e_n$ for each $n \geq 1$.
\[finite-equi-integrable\] For each $n \geq 1$, the sets $\left\vert V ({\bf 1}-p_n) \right\vert$ and $\left\vert({\bf 1}-p_n)V\right\vert$ are $E$-equi-integrable.
Let us show that for every $n \geq 1$, $\left\vert V (1-p_{n}) \right\vert$ is an $E$-equi-integrable set. Assume that there exists $ k_0 \geq 1$ such that $\vert V
({\bf 1}-p_{k_0}) \vert$ is not $E$-equi-integrable. By definition, there exists a decreasing sequence of projections $q_n \downarrow_n 0$ such that $\lim\limits_{n \to \infty} \sup\limits_{a \in \vert V ({\bf 1}-p_{k_0})\vert}
\left\Vert q_n\ a\ q_n \right\Vert_{E(\cal{M},\tau)} > 0$, that is $$\lim_{n \to \infty} \sup_{m \in \N}
\left\Vert q_n \vert v_m
({\bf 1}-p_{k_0}) \vert q_n \right\Vert_{E(\cal{M},\tau)} > 0.$$ Choose a strictly increasing sequence $\{m_n\}_{n=1}^\infty$ of $\N$ such that $$\lim_{n \to \infty} \left\Vert q_n \vert v_{m_n}(1-p_{k_0})
\vert q_n \right\Vert_{E(\cal{M},\tau)} > 0.$$ Let $u_{n,k_0}$ be a bounded operator such that $\vert v_{m_n} ({\bf 1}-p_{k_0})
\vert = u_{n,k_0} v_{m_n} ({\bf 1}-p_{k_0})$. We get that $$\begin{aligned}
\left\Vert q_n \vert v_{m_n} ({\bf 1}-p_{k_0}) \vert q_n
\right\Vert_{E(\cal{M},\tau)} &=
\left\Vert q_n u_{n,k_0} v_{m_n} ({\bf 1}-p_{k_0}) q_n
\right\Vert_{E(\cal{M},\tau)} \cr
&= \left\Vert q_n u_{n,k_0}[x_{m_n} - e_{m_n} x_{m_n} e_{m_n}]
( {\bf 1}-p_{k_0})q_n \right\Vert_{E(\cal{M},\tau)}.\end{aligned}$$ We recall that $e_{k_0} \leq p_{k_0}$ and since $\{e_n\}_{n=1}^\infty$ is decreasing, for $m_n
\geq k_0$, $e_{m_n} \leq p_{k_0}$ and therefore $e_{m_n} ({\bf 1}-p_{k_0}) = 0$ and since $\left\Vert q_n u_{n,k_0} \right\Vert_{\infty} \leq 1$, we obtain that for $n$ large enough, $$\begin{aligned}
\left\Vert q_n \vert v_{m_n} ({\bf 1}-p_{k_0}) \vert q_n
\right\Vert_{E(\cal{M},\tau)} &= \left\Vert q_n
u_{n,k_0} (x_{m_n}) ({\bf 1}-p_{k_0}) q_n \right\Vert_{E(\cal{M},\tau)} \cr
&\leq \left\Vert x_{m_n} ({\bf 1}-p_{k_0}) q_n \right\Vert_{E(\cal{M},\tau)}.\end{aligned}$$ Using Proposition \[inequality\], with $x = x_{m_n}$ and $y = ({\bf 1}-p_{k_0})q_n$, we get $$\begin{aligned}
\left\Vert q_n \vert v_{m_n}({\bf 1}-p_{k_0}) \vert p_n
\right\Vert_{E(\cal{M},\tau)} &\leq \left\Vert
x_{m_n} \right\Vert^{\frac1{2}}_{E(\cal{M},\tau)}\
\cdot \ \left\Vert q_n (1-p_{k_0}) \vert x_{m_n} \vert
({\bf 1}-p_{k_0}) q_n \right\Vert^{\frac1{2}}_{E(\cal{M},\tau)} \cr
&\leq \left\Vert q_n ({\bf 1}-p_{k_0}) \vert x_{m_n} \vert ({\bf 1}-p_{k_0}) q_n
\right\Vert^{\frac1{2}}_{E(\cal{M},\tau)}.\end{aligned}$$ This implies that $$\limsup_{n \to \infty}
\left\Vert q_n ({\bf 1}-p_{k_0}) \vert x_{m_n} \vert
({\bf 1}-p_{k_0}) q_n \right\Vert_{E(\cal{M},\tau)} > 0.$$ Let $s_n$ be the left support projection of $({\bf 1}-p_{k_0})q_n$ (this is equal to the right support projection of $q_n ({\bf 1}-p_{k_0})$). We have $$\begin{aligned}
\left\Vert q_n ({\bf 1}-p_{k_0}) \vert x_{m_n} \vert
({\bf 1}-p_{k_0})q_n \right\Vert_{E(\cal{M},\tau)} &=
\left\Vert q_n ({\bf 1}-p_{k_0}) s_n \vert x_{m_n} \vert s_n
({\bf 1}-p_{k_0}) q_n
\right\Vert_{E(\cal{M},\tau)} \cr
&\leq \left\Vert s_n \vert x_{m_n} \vert s_n \right\Vert_{E(\cal{M},\tau)}.\end{aligned}$$ By the definition of support projection, $s_n \leq ({\bf 1}-p_{k_0})$ for every $n
\geq 1$, so $\{s_n\}_{n=1}^\infty$ is a sequence of finite projections. As in proof of Lemma \[finitecompact1\], we note that $s_n = q_n \vee p_{k_0} - p_{k_0}$ and as before, $s_n \sim q_n -
q_n \wedge p_{k_0}$. Now since $q_n \downarrow_n 0$, $q_n - q_n \wedge
p_{k_0}\downarrow_n 0$ hence $\tau(s_n) = \tau(q_n - q_n \wedge p_{k_0})$ converges to zero which implies that $s_n \downarrow_n 0$. Therefore, $\{s_n\}_{n=1}^\infty \in \script{D}_1$.
In summary, we get $\{s_n\}_{n=1}^\infty \in \script{D}_1$ with $s_n
\leq 1-p_{k_0}$ for each $n\geq 1$ and for some $\gamma >0$, $$\label{gamma}
\limsup_{n \to\infty}
\left\Vert s_n \vert x_{m_n} \vert
s_n \right\Vert = \gamma.$$ Let $f_n: = s_n \vee e_{m_n}$.
For each $m_n \geq k_0$, $s_n \leq {\bf 1} - p_{k_0} \leq
{\bf 1} - e_{k_0}$ so $s_n
\perp e_{m_n}$ hence $f_n = s_n + e_{m_n}$. In particular $\{f_n\}_{n=1}^\infty \in \script{D}_1$.
Using Proposition \[lower-estimate\] (it applies since $\tau(f_n) < \infty)$, $$\begin{aligned}
\left\Vert f_n \vert x_{m_n} \vert f_n
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} &\geq
\left\Vert s_n \vert x_{m_n} \vert s_n
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert e_{m_n}
\vert x_{m_n} \vert s_n
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} \cr
&+ \left\Vert s_n \vert x_{m_n} \vert e_{m_n}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert e_{m_n} \vert x_{m_n} \vert e_{m_n}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} \cr
&\geq \left\Vert s_n \vert x_{m_n} \vert s_n
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert e_{m_n} \vert x_{m_n} \vert e_{m_n}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}.\end{aligned}$$
Taking the limit as $n$ tends to $\infty$, one gets from (\[delta\]) and (\[gamma\]) that $\delta^{\frac{2q}{\alpha}} \geq \gamma^{\frac{2q}{\alpha}} +
\delta^{\frac{2q}{\alpha}}$. This is a contradiction since $\gamma > 0$.
We conclude that for every $n \geq 1$, the set $\vert V({\bf 1}-p_n) \vert$ is an $E$-equi-integrable set.
For the case of $\vert ({\bf 1}-p_n)V \vert$, it is enough to repeat the argument above for $V^*({\bf 1}-p_n)$ using the definition of $\delta^*$ (instead of $\delta$). Details are left to the reader. This ends the proof of the lemma.
We will proceed to the proof of Theorem \[main\]. Consider two cases.
[*Case 1: Assume that $V$ is $E$-equi-integrable.*]{}
It is enough to set $y_n = x_{m_n} - e_{m_n} x_{m_n} e_{m_n}$ and $z_n = e_{m_n} x_{m_n} e_{m_n}$.
[*Case 2: Assume that $V$ is not $E$-equi-integrable.*]{}
Proposition \[equi-integrability2\] and Lemma \[finite-equi-integrable\] imply that there exists $\nu>0$ such that $$\lim\limits_{n
\to \infty} \sup_{v \in V}
\left\Vert p_n v p_n \right\Vert_{E(\cal{M},\tau)} = \nu > 0.$$ Choose a subsequence $\{v_{n_k}\}_{k=1}^\infty$ such that $$\label{nu}
\lim\limits_{k \to \infty}
\left\Vert p_k v_{n_k} p_k \right\Vert_{E(\cal{M},\tau)}
= \nu > 0.$$
For each $k \geq 1$, let $w_k:
= v_{n_k} - p_k v_{n_k} p_k$ and set $$W := \left\{w_k: \ k \geq 1 \right\}.$$
The set $W$ is $E$-equi-integrable.
We note first that if $k \geq n$, then $({\bf 1}-p_n)w_k = ({\bf 1}-p_n)v_k$ and $w_k(1-p_n) = v_k (1-p_n)$ so for fixed $n\geq 1$, $({\bf 1}-p_n)W = \left\{({\bf 1}-p_n)w_k:\ k <n \right\} \cup
\left\{({\bf 1}-p_n)v_k:\ k \geq n \right\}$. Similarly, $W({\bf 1}-p_n) = \left\{w_k({\bf 1}-p_n):\ k<n \right\}
\cup \left\{v_k({\bf 1}-p_n):\ k \geq n \right\}.$
Lemma \[finite-equi-integrable\] implies that for every $n \geq 1$, both $\vert W({\bf 1}-p_n) \vert$ and $({\bf 1}-p_n)W$ are $E$-equi-integrable sets. Therefore, if $W$ is not $E$-equi-integrable, there would be a subsequence $\{w_{k(j)}\}_{j=1}^\infty$ of $\{w_k\}_{k=1}^\infty$ and $\epsilon > 0$ such that $$\label{epsilon}
\lim_{j \to \infty} \left\Vert p_j w_{k(j)}p_j \right\Vert_{E(\cal{M},\tau
)} = \epsilon.$$
Using Proposition \[inequality\] on $v_{n_{k(j)}}$ and $p_j = (p_j - p_{k(j)}) +
p_{k(j)}$, we obtain: $$\begin{aligned}
\left\Vert p_j v_{n_{k(j)}} p_j \right\Vert^{\frac{2p}{\alpha}}_{E(\cal{M},\tau)}
\geq &\left\Vert
(p_j - p_{k(j)}) v_{n_{k(j)}} (p_j -p_{k(j)})
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert p_{k(j)} v_{n_{k(j)}} (p_j - p_{k(j)})
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} + \cr
&\left\Vert (p_j - p_{k(j)}) v_{n_{k(j)}} p_{k(j)}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert (p_j - p_{k(j)}) v_{n_{k(j)}} (p_j - p_{k(j)})
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}.\end{aligned}$$
Taking into account the identities, $(p_j - p_{k(j)}) v_{n_{k(j)}} (p_j - p_{k(j)}) = (p_j -
p_{k(j)}) w_{k(j)} (p_j - p_{k(j)})$, $(p_j - p_{k(j)}) v_{n_{k(j)}} p_{k(j)} = p_j w_{k(j)}
p_{k(j)}$ and $p_{k(j)} v_{n_{k(j)}} (p_j - p_{k(j)}) = p_{k(j)} w_{k(j)}
p_j$, one can deduce that, $$\begin{aligned}
\left\Vert p_j v_{n_{k(j)}} p_j
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}
\geq
&\left\Vert(p_j - p_{k(j)}) w_{k(j)} (p_j -p_{k(j)})
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert p_j w_{k(j)} p_{k(j)}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} + \cr
&\left\Vert p_{k(j)} w_{k(j)} p_j
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)} +
\left\Vert p_{k(j)} v_{n_{k(j)}} p_{k(j)}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}.\end{aligned}$$
Let $C(q,\alpha)$ be the norm of the identity map from $\ell^{\frac{2q}{\alpha}}_3$ onto $\ell^{\alpha}_3$, where $\ell^{\frac{2q}{\alpha}}_3$ (resp. $\ell^{\alpha}_3$) denotes the $3$-dimensional $\ell^{\frac{2q}{\alpha}}$-space (resp. $\ell^{\alpha}$-space). We have
$$\begin{aligned}
\left\Vert p_j v_{n_{k(j)}} p_j \right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}
\geq &C(q,
\alpha)^{\frac{2q}{\alpha}} \bigg[ \bigg( \left\Vert (p_j - p_{k(j)}) w_{k(j)}
(p_j - p_{k(j)}) \right\Vert^{\alpha}_{E(\cal{M},\tau)} + \cr
&\left\Vert p_j w_{k(j)} p_{k(j)} \right\Vert^{\alpha}_{E(\cal{M},\tau)}
+ \left\Vert p_{k(j)}
w_{k(j)} p_j \right\Vert^{\alpha}_{E(\cal{M},\tau)} \bigg)^{\frac1{\alpha}}
\bigg]^{\frac{2q}{\alpha}} + \cr
&\left\Vert p_{k(j)} v_{n_{k(j)}} p_{k(j)}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}.\end{aligned}$$
We remark that $p_j w_{k(j)} p_j = (p_j - p_{k(j)}) w_{k(j)} (p_j
- p_{k(j)}) + p_j w_{k(j)} p_{k(j)} + p_{k(j)} w_{k(j)} p_j$ and since $E(\cal{M},\tau)$ is $\alpha$-convex (with constant $1$), the above inequality implies $$\left\Vert p_j v_{n_{k(j)}} p_j \right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}
\geq
C(q,\alpha)^{\frac{2q}{\alpha}} \left\Vert p_j w_{k(j)} p_j
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)}
+ \left\Vert p_{k(j)} v_{n_{k(j)}} p_{
k(j)}
\right\Vert^{\frac{2q}{\alpha}}_{E(\cal{M},\tau)},$$ and taking the limit as $j \to \infty$, we get from (\[nu\]) and (\[epsilon\]) that $$\nu^{\frac{2q}{\alpha}} \geq C(q, \alpha)^{\frac{2q}{\alpha}}
\epsilon^{\frac{2q}{\alpha}} + \nu^{\frac{2q}{\alpha}}.$$ This is a contradiction since $\epsilon >0$, so $W$ is a $E$-equi-integrable set. The lemma is proved.
To complete the proof of Theorem \[main\], we note that $W = \{x_{n_{k}} - p_k x_{n_{k}} p_k: \ k \geq 1\}$ so if we set $y_k =
x_{n_{k}} - p_k x_{n_{k}} p_k$ and $z_k = p_k x_{n_{k}} p_k$. The proof of is complete.
\(1) If $\M =\cal{B}(\ell^2)$ with the usual trace, then every projection of finite trace is a finite rank projection so in the proof above, $\delta=\delta^*=0$. In the particular case of unitary matrix space $C_E$ where $E$ is a symmetric sequence space, one proceed directly to Case 2 by setting $W:=\left\{ x_n - p_n x_n p_n;\ n \geq 1 \right\}$ where $\{p_n\}_{n=1}^\infty$ is an arbitrary sequence of projections satisfying: $p_n \downarrow_n 0$ and for every $n \geq 1$, $1-p_n$ is a finite projection.
\(2) If $\M$ is a finite von Neumann algebra with a normalized finite trace $\tau$ and $E$ is a symmetric space on $[0,1]$ satisfying the assumptions of Theorem \[main\], it is enough to take $p_n =e_n$ (i.e $g_n ={\bf 1}-e_n$ on Lemma 3.2) and conclude immediately as in Lemma \[finite-equi-integrable\] that $V$ is $E$-equi-integrable.
\(3) In the proof above, it is clear that the projections $\{p_k\}_{k=1}^\infty$ are such that either $\T(p_1)< \infty$ or $\tau({\bf 1}-p_k)< \infty$ for all $k
\geq 1$. In fact, the argument above shows that if $\{e_n\}_{n=1}^\infty$ is a sequence in $\cal{D}_1$ that attained the quantities $\delta$ and $\delta^*$, then any sequence of projections satisfying $p_n \downarrow_n 0$, $e_n \leq p_n$ for each $n\geq 1$ and $\T({\bf 1}-p_n)<\infty$ for each $n\geq 1$, would satisfy the conclusion of Theorem \[main\].
The following extension shows that if one considers finitely many bounded sequences in $E(\M,\T)$, one can choose a single sequence of projections that works for each sequence.
If $\M$ and $E$ are as in Theorem \[main\] and $\left\{x^{(1)}\right\}_{n=1}^\infty,\left\{x^{(2)}\right\}_{n=1}^\infty,
\dots,
\left\{x^{(j_0)}\right\}_{n=1}^\infty$ be finitely many bounded sequences in $E(\M, \tau)$. Then there exist a strictly increasing sequence $\{n_k\}_{k=1}^\infty$ of $\N$ and a sequence of decreasing projections $p_k \downarrow_k 0$ in $\cal M$ such that for each $1 \leq j \leq j_0$, the set $\left\{x^{(j)}_{n_k}- p_k x^{(j)}_{n_k}p_k :\ k \geq 1 \right\}$ is $E$-equi-integrable.
For $1 \leq j \leq j_0$, we set, as in the proof of Theorem \[main\], $$\delta_j: = \sup\left\{\lim\limits_{n \to \infty}
\sup\limits_{k \in \N}
\left\Vert e_n \vert x^{(j)}_k \vert e_n \right\Vert_{E(\cal{M},\tau)}:\
\{e_n\}_{n=1}^\infty \in \script{D}_1 \right\}.$$
One can choose a strictly increasing sequence $\{n_k\}_{k=1}^\infty$ in $\N$ such that for each $1\leq j \leq j_0$, there exists a sequence $\{e^{(j)}_k\}_{k=1}^\infty \in \script{D}_1$ with $$\delta_j =\lim_{k \to \infty} \left\Vert e^{(j)}_k
\vert x^{(j)}_{n_k} \vert e^{(j)}_k
\right\Vert_{E(\cal{M},\tau)}$$ and $$\delta^*_j=\lim\limits_{k \to \infty} \left\Vert e^{(j)}_k
\vert {x^{(j)}_{n_k}}* \vert e^{(j)}_k
\right\Vert_{E(\cal{M},\tau)}
= \sup \left\{\lim\limits_{n \to \infty} \sup\limits_{k}
\left\Vert q_n \vert
{x^{(j)}_{n_k}}^* \vert q_n \right\Vert_{E(\cal{M},\tau)};
\{q_{n}\}_{n=1}^\infty \in \script{D}_1 \right\}.$$ For every $k \geq 1$, set $e_k := \vee_{1\leq j \leq j_0} e^{(j)}_k$. Since $\T(e_k) \leq \sum_{j=1}^{j_0} \T(e^{(j)}_k)$, it is clear that the sequence $\{e_k\}_{k=1}^\infty$ belongs to $\script{D}_1$ and each of the $\delta_j$’s and $\delta^{*}_j$’s are attained at $\{e_k\}_{k=1}^\infty$. One can complete the proof by procceding as in the proof of Theorem \[main\], simmultaneously on the finite set of sequences and the fixed $\{e_k\}_{k=1}^\infty$.
Our next result shows that the decreasing projections in the decomposition can be replaced by mutually disjoint projections.
\[main2\] Let $E$ be an order continuous quasi-Banach function space as in Theorem \[main\]. Let $\{x_n\}_{n=1}^\infty$ be a bounded sequence in $E(\cal{M},\tau)$ then there exists a subsequence $\{x_{n_{k}}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$, bounded sequences $\{\varphi_k\}_{k=1}^\infty$ and $\{\zeta_k\}_{k=1}^\infty$ in $E(\cal{M,\tau})$ and mutually disjoint sequence of projections $\{e_k\}_{k=1}^\infty$ such that:
- $x_{n_{k}} = \varphi_k + \zeta_k$ for all $k \geq 1$;
- $\{\varphi_k:\ k\geq 1\}$ is $E$-equi-integrable and $e_k \varphi_k e_k = 0 $ for all $\ k \geq 1$;
- $\{\zeta_k\}_{k=1}^\infty$ is such that $e_k \zeta_k e_k = \zeta_k$ for all $k \geq1$.
Let $\{x_n\}_{n=1}^\infty$ be a bounded sequence in $E(\cal{M},\tau)$ and suppose (by taking a subsequence if necessary), $x_n = y_n + z_n$ with $p_n y_n p_n = 0$, the set $\{y_n:\ n \geq 1\}$ is $E$-equi-integrable and $p_n z_n p_n = z_n$ for all $n \geq 1$, be the decomposition of $\{x_n\}_{n=1}^\infty$ as in Theorem \[main\].
Let $n_1 = 1$. Since $p_n \downarrow_n 0$ and $$p_1 z_1 p_1 - (p_1 - p_n) z_1 (p_1 - p_n) = p_n z_1 p_1 + p_1
z_1 p_n - p_n z_1 p_n,$$ Proposition ref[order-continuity]{}(part(ii)) shows that $$\lim_{n \to \infty} \Vert p_1 z_1 p_1 - (p_1 - p_n) z_1 (p_1 -
p_n) \Vert_{E(\cal{M},\tau)} = 0.$$ Choose $n_2 > n_1 = 1$ such that $$\Vert p_1 z_1 p_1 - (p_1 - p_{n_{2}}) z_1 (p_1 - p_{n_{2}})
\Vert_{E(\cal{M},\tau)} < \frac1{2}.$$ Inductively, one can construct $n_1 < n_2 < \dots < n_k < \dots$ such that $$\Vert p_{n_{k}} z_{n_{k}} p_{n_{k}} - (p_{n_{k}} - p_{n_{k + 1}})
z_{n_{k}} (p_{n_{k}} - p_{n_{k + 1}}) \Vert_{E(\cal{M},\tau)}
< {\frac1{2^k}}.$$ Since $z_n=p_n z_n p_n$ for every $n\geq 1$, one gets $$\Vert z_{n_{k}} - (p_{n_{k}} - p_{n_{k + 1}}) z_{n_{k}} (p_{n_{k}} - p_{n_{k +
1}}) \Vert_{E(\cal{M},\tau)} < {\frac1{2^k}}.$$ For every $ k \geq 1$, set $$\begin{aligned}
e_k : & = p_{n_{k}} - p_{n_{k + 1}} \cr
\zeta_k : & = (p_{n_{k}} - p_{n_{k + 1}}) z_{n_{k}} (p_{n_{k}} - p_{n_{k +
1}}) \cr
\varphi_k : & = y_{n_{k}} + [z_{n_{k}} - e_k z_{n_{k}} e_k] .\end{aligned}$$ Since $\{y_{n_{k}}:\ k \geq 1\}$ is a $E$-equi-integrable set and $\lim\limits_{k \to \infty} \Vert z_{n_{k}} - e_k z_{n_{k}} e_k
\Vert_{E(\cal{M},\tau)} = 0$, it is clear that $\{\varphi_k:\ k \geq 1\}$ is $E$-equi-integrable. Also $\{e_k\}_{k=1}^\infty$ is mutually disjoint. The proof is complete.
Let $E$ be an order-continuous symmetric Banach function space on $\R^{+}$ with the Fatou propery. Let $\{x_n\}_{n=1}$ be a bounded sequence in $E(\cal{M},\tau)$ then there exists a subsequence $\{x_{n_{k}}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$, bounded sequences $\{\varphi_k\}_{k=1}^\infty$ and $\{\zeta_k\}_{k=1}^\infty$ in $E(\cal{M,\tau})$ and mutually disjoint sequence of projections $\{e_k\}_{k=1}^\infty$ such that:
- $x_{n_{k}} = \varphi_k + \zeta_k$ for all $k \geq 1$;
- $\{\varphi_k:\ k\geq 1\}$ is $E$-equi-integrable and $e_k \varphi_k e_k = 0 $ for all $\ k \geq 1$;
- $\{\zeta_k\}_{k=1}^\infty$ is such that $e_k \zeta_k e_k = \zeta_k$ for all $k \geq1$.
Assume that $E$ has the Fatou property (equivalenty $E$ does not contain $c_0$). Since $E$ is symmetric, $E \not \supset c_0$ is equivalent to $E$ not containing $\ell^n_\infty$ uniformly, and therefore $E$ satisfies the $q$-lower estimate for some $q$ and one can renorm $E$ so that it satisfies the lower $q$-estimate of constant $1$. All of these facts can be found in [@LT].
The proof of Theorem \[main\] can be adjusted to obtain decompositions where the projections are taken only on one side, that is, the following result follows:
Let $E$ be an order continuous quasi-Banach function space in $\R^+$ that is $\alpha$-convex with constant $1$ for some $0 < \alpha \leq 1$ and suppose that $E$ satisfies a lower $q$-estimate with constant $1$ for some $q \geq \alpha$. Let $\{x_n\}_{n=1}^\infty$ be a bounded sequence in $E(\cal{M},\tau)$ then there exist a subsequence $\{x_{n_k}\}_{k=1}^\infty$ of $\{x_n\}_{n=1}^\infty$, bounded sequences $\{y_k\}_{k=1}^\infty$ and $\{z_k\}_{k=1}^\infty$ in $E(\cal{M},\tau)$ and decreasing projections $e_k \downarrow_n 0$ in $\cal M$ such that:
- $x_{n_k} = y_k + z_k$ for all $k \geq 1$;
- $e_ky_k = 0$ for all $k \geq 1$ and $\lim\limits_{n\to \infty}\sup_{k\geq 1}||f_n y_k||_{E(\M,\T)}=0$ for every $f_n \downarrow_n 0$.
- $\{z_k\}_{k=1}^\infty$ is such that $e_kz_k = z_k$ for all $k \geq
1$.
A subspace $X$ of $L^p(\M,\T)$ is called strongly embedded into $L^p(\M,\T)$ if the $L^p$ and the measure topologies on $X$ coincide.
The following result is a direct application of Proposition \[measure\] and Theorem \[main2\].
Let $1\leq p <\infty$. Every subspace of $L^p(\M,\T)$ either contains almost isometric copies of $\ell^p$ or is strongly embedded in $L^p(\M,\T)$.
The next corollary should be compared with [@SU Theorem 2.4].
Assume that $\M$ is finite and $p>2$. Every subspace of $L^p(\M,\T)$ either contains almost isometric copies of $\ell^p$ or is isomorphic to a Hilbert space.
For the commutative case, the space $\ell^p$ can not be strongly embedded in $L^p[0,1]$ for $0<p<2$. This is due to Kalton [@KA3] for $0<p<1$ and Rosenthal [@R6] for the case $1\leq p <2$ (see also [@CADI] for another approach). A non-commutative analogue should be of interest.
[**Problem:**]{} [*Let $\M$ be a semifinite von Neumann algebra and $0<p<2$. Does $\ell^p$ strongly embed into $L^p(\M,\T)$?*]{}
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[^1]: Supported in part by NSF grant DMS-9703789
|
---
abstract: |
We present the X-ray data and the optical identifications for a deep ROSAT PSPC observation in the “Marano field”. In the inner region of the ROSAT field (15 radius) we detected 50 X-ray sources with $S_x \ge 3.7 \times 10^{-15}$ . When corrected for the different sensitivity over the field, the estimated observed surface density at $S_x \ge 4 \times 10^{-15}$ is $272 \pm 40$ sources/sq.deg. Four X-ray sources, corresponding to 8% of the total sample, have been detected in radio images with a flux limit of about 0.2 mJy.
Careful statistical analysis of multi–colour CCD data in the error boxes of the 50 X-ray sources has led to the identification of 42 sources, corresponding to 84% of the X-ray sample. These 42 reliable identifications are 33 AGNs (including two radio galaxies and one BL Lac candidate; 79% of the identified sources), 2 galaxies, 3 groups or clusters of galaxies and 4 stars. If we divide our sample into two equally populated sub–samples as a function of flux, at $S_x = 6.5 \times 10^{-15}$ , we find that the percentage of identifications remains approximately constant (88% and 80% in the high and low flux sub–samples, respectively). AGNs are the dominant class of objects in both sub–samples (90% of the optical identifications in the high flux sub–sample and 65% in the low flux sub–sample), while the few identifications with clusters and galaxies are all in the low flux sub–sample.
We also show that it is likely that a few of the 8 unidentified sources are such because the derived X-ray positions may be offset with respect to the real ones due to confusion effects. The unidentified sources have a large ratio of X-ray to optical fluxes and most of them have harder than average X-ray spectra. Since most of the identified objects with these characteristics in our field and in the Lockman field are AGNs, we conclude that also most of these sources are likely to be AGNs.
Finally, comparing the optically and X-ray selected samples of AGNs in this field, we estimate that the “efficiency” of AGN selection with X-ray exposures reaching about $4 \times 10^{-15}$ is $\sim$65% and $\sim$20% in the magnitude ranges $m_B < 22.5$ and $22.5 < m_B < 23.5$, respectively. On the other hand, a not negligible fraction of the X-ray selected AGNs would have not been easily selected as AGN candidates on the basis of purely optical criteria, either because of colours similar to those of normal stars or because of morphological classification not consistent with that of point–like sources.
author:
- 'G.Zamorani'
- 'M.Mignoli'
- 'G. Hasinger'
- 'R.Burg'
- 'R.Giacconi'
- 'M.Schmidt'
- 'J.Trümper'
- 'P.Ciliegi'
- 'C.Gruppioni'
- 'B.Marano'
date: 'Received 21 January 1999 / Accepted 26 March 1999'
subtitle: ' V. X-ray Sources and Optical Identifications in the Marano Field'
title: 'The ROSAT deep survey [^1]'
---
Introduction
============
Complete or, at least, statistically well defined samples of optical identifications of faint X-ray sources, are important for a number of scientific goals. For example, combined optical and X-ray data allow to obtain information about the luminosity functions of various types of X-ray sources as well as their evolution with redshift. In turn, these informations can be used to further constrain models for the production of the X-ray background, discovered more than thirty years ago (Giacconi et al. 1962). There is a general consensus that the majority of the optically identified X-ray sources, at least for fluxes $ S_{0.5 - 2 keV} \ge 5 \times 10^{-15}$ , are active galactic nuclei (AGNs), i.e. quasars and Seyfert galaxies with broad emission lines (Shanks et al. 1991, Georgantopoulos et al. 1996, McHardy et al. 1998, Hasinger et al. 1998 (Paper I) and Schmidt et al. 1998 (Paper II)).
In the first two papers of this series we have presented the X-ray and optical data for a complete catalogue of 50 X-ray sources with PSPC fluxes (0.5-2 keV) above $ 5.5 \times 10^{-15}$ in the Lockman Field. Optical and X-ray results at a much fainter flux limit in this field, which has been observed with the deepest ROSAT PSPC and HRI observations, will be presented elsewhere.
In this paper we present and discuss the X-ray data (Section 2) and the optical data (Section 3) for 50 sources detected with the PSPC at a flux limit $ S_x \ge 3.7 \times 10^{-15}$ in the Marano Field. A discussion of the main results (percentages of identifications with different classes of objects, hardness ratio as a function of X-ray flux, comparison between X-ray and optically selected AGNs) is given in Section 4. Throughout the paper we use $H_0$ = 50 km s$^{-1}$ Mpc$^{-1}$ and $q_0$ = 0.5.
The X-ray Data
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The sample
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The ROSAT–PSPC pointed observations of the Marano field, centered at , (epoch 2000.0), have been carried out in the time interval December 1992 – July 1993, for a total of 56 ksec of observing time. The observations were performed in the “wobble mode”.
The X-ray analysis (e.g. source detection, position and flux determination of the detected sources, determination of sensitivity as a function of distance from the center, etc.) has been done applying a maximum likelihood method. For details about the various steps in this analysis we refer the reader to Hasinger et al. (1993 and 1998). Although detection of the sources has been run in various ROSAT bands, we report here results only for the hard band (0.5–2 keV), which has been shown (Hasinger et al. 1993) to be the most efficient for point source detection.
[ccccrrccr]{}
[\#]{} & RA (2000) & DEC (2000) & Err & Off-axis & ML & Net Counts & S$_x$ & HR \
& & &\[arcsec\]&\[arcmin\]& & &\[$\times 10^{-14}$ cgs\]&\
X013–01 & 3 13 47.1 & $-$55 11 48 & 1.6 & 12.1 & 448.2 & 210.1 & 5.00 $\pm$ 0.42 & $-$0.03 $\pm$ 0.06\
X012–02 & 3 13 29.2 & $-$55 10 20 & 2.7 & 14.8 & 203.3 & 155.7 & 3.82 $\pm$ 0.37 & 0.41 $\pm$ 0.12\
X046–03 & 3 14 32.4 & $-$55 14 43 & 1.6 & 5.5 & 303.1 & 130.1 & 2.99 $\pm$ 0.32 & 0.20 $\pm$ 0.10\
X036–04 & 3 16 50.3 & $-$55 11 10 & 2.4 & 14.5 & 175.4 & 118.9 & 2.91 $\pm$ 0.30 & $-$0.10 $\pm$ 0.08\
X021–05 & 3 14 56.4 & $-$55 20 08 & 1.8 & 6.7 & 190.4 & 102.3 & 2.36 $\pm$ 0.25 & 0.88 $\pm$ 0.20\
X025–06 & 3 15 49.5 & $-$55 18 11 & 1.8 & 7.1 & 184.7 & 99.8 & 2.31 $\pm$ 0.30 & 0.28 $\pm$ 0.13\
X027–07 & 3 16 05.5 & $-$55 15 44 & 2.2 & 8.1 & 124.6 & 71.8 & 1.66 $\pm$ 0.28 & $-$0.21 $\pm$ 0.10\
X041–08 & 3 15 28.7 & $-$55 10 32 & 2.6 & 4.1 & 99.7 & 62.0 & 1.42 $\pm$ 0.25 & 0.58 $\pm$ 0.15\
X240–09 & 3 16 38.2 & $-$55 06 41 & 3.9 & 14.4 & 56.1 & 50.9 & 1.25 $\pm$ 0.24 & 0.45 $\pm$ 0.25\
X033–10 & 3 15 58.3 & $-$55 26 40 & 4.2 & 14.6 & 46.7 & 50.9 & 1.25 $\pm$ 0.23 & $-$0.48 $\pm$ 0.11\
X042–11 & 3 15 40.4 & $-$55 12 25 & 2.3 & 4.5 & 77.1 & 52.4 & 1.20 $\pm$ 0.18 & 0.32 $\pm$ 0.32\
X043–12 & 3 15 09.8 & $-$55 13 18 & 2.3 & 0.5 & 88.2 & 48.5 & 1.10 $\pm$ 0.25 & 0.37 $\pm$ 0.32\
X023–13 & 3 15 25.2 & $-$55 18 28 & 2.7 & 5.2 & 59.1 & 43.5 & 1.00 $\pm$ 0.17 & $-$0.27 $\pm$ 0.19\
X108–14 & 3 15 37.9 & $-$55 01 42 & 4.4 & 12.7 & 30.8 & 41.0 & 0.98 $\pm$ 0.20 & 0.63 $\pm$ 0.62\
X030–15 & 3 16 26.1 & $-$55 23 00 & 4.3 & 14.2 & 28.3 & 39.6 & 0.97 $\pm$ 0.18 & 0.03 $\pm$ 0.20\
X304–16 & 3 15 11.4 & $-$55 09 30 & 3.1 & 4.3 & 56.5 & 40.8 & 0.94 $\pm$ 0.18 & 0.08 $\pm$ 0.27\
X019–17 & 3 14 21.8 & $-$55 23 55 & 3.6 & 12.3 & 43.0 & 38.8 & 0.93 $\pm$ 0.27 & 0.24 $\pm$ 0.31\
X029–18 & 3 16 29.9 & $-$55 19 06 & 3.8 & 12.5 & 30.1 & 37.0 & 0.89 $\pm$ 0.17 & 0.14 $\pm$ 0.30\
X039–19 & 3 15 52.9 & $-$55 08 20 & 3.2 & 8.1 & 47.5 & 38.0 & 0.88 $\pm$ 0.18 & 1.00 \
X001–20 & 3 15 20.7 & $-$55 02 33 & 3.6 & 11.3 & 46.4 & 37.4 & 0.88 $\pm$ 0.26 & $-$0.07 $\pm$ 0.24\
X049–21 & 3 15 06.0 & $-$55 09 42 & 3.0 & 4.1 & 49.4 & 35.9 & 0.82 $\pm$ 0.21 & $-$0.19 $\pm$ 0.20\
X211–22 & 3 13 45.5 & $-$55 19 25 & 4.4 & 13.4 & 21.5 & 33.8 & 0.81 $\pm$ 0.19 & $-$0.08 $\pm$ 0.32\
X404–23 & 3 16 48.9 & $-$55 12 40 & 9.3 & 14.1 & 10.2 & 29.7 & 0.73 $\pm$ 0.21 & 1.00 \
X031–24 & 3 15 48.7 & $-$55 22 50 & 4.4 & 10.6 & 17.4 & 30.2 & 0.71 $\pm$ 0.18 & $-$0.21 $\pm$ 0.29\
X050–25 & 3 15 07.6 & $-$55 04 58 & 3.8 & 8.8 & 10.0 & 30.1 & 0.70 $\pm$ 0.16 & $-$0.26 $\pm$ 0.23\
X235–26 & 3 16 31.7 & $-$55 12 27 & 5.2 & 11.7 & 14.3 & 26.7 & 0.63 $\pm$ 0.18 & 0.21 $\pm$ 0.45\
X045–27 & 3 15 10.7 & $-$55 15 23 & 3.5 & 1.6 & 30.8 & 26.9 & 0.61 $\pm$ 0.15 & 0.24 $\pm$ 0.33\
X409–28 & 3 14 26.3 & $-$55 17 41 & 5.2 & 7.4 & 22.0 & 26.2 & 0.61 $\pm$ 0.14 & 0.84 $\pm$ 0.85\
X301–29 & 3 14 36.3 & $-$55 14 04 & 4.1 & 4.9 & 20.2 & 26.0 & 0.60 $\pm$ 0.15 & $-$0.21 $\pm$ 0.27\
X408–30 & 3 14 50.3 & $-$55 19 39 & 4.1 & 6.6 & 17.1 & 25.6 & 0.59 $\pm$ 0.16 & 0.65 $\pm$ 0.87\
X207–31 & 3 13 50.3 & $-$55 13 00 & 4.9 & 11.5 & 16.1 & 24.7 & 0.59 $\pm$ 0.15 & $-$0.12 $\pm$ 0.42\
X040–32 & 3 15 43.1 & $-$55 07 46 & 3.5 & 7.6 & 27.5 & 24.9 & 0.58 $\pm$ 0.14 & 0.34 $\pm$ 0.41\
X028–33 & 3 16 21.8 & $-$55 18 00 & 4.8 & 11.0 & 16.5 & 23.8 & 0.56 $\pm$ 0.14 & 0.24 $\pm$ 0.45\
X250–34 & 3 15 23.3 & $-$55 04 03 & 4.4 & 9.9 & 17.4 & 24.0 & 0.56 $\pm$ 0.14 & 0.65 $\pm$ 0.31\
X011–35 & 3 13 39.9 & $-$55 07 21 & 6.1 & 14.4 & 11.2 & 22.7 & 0.56 $\pm$ 0.15 & 0.04 $\pm$ 0.37\
X032–36 & 3 15 38.7 & $-$55 22 33 & 5.4 & 9.7 & 13.4 & 23.2 & 0.54 $\pm$ 0.14 & $-$0.01 $\pm$ 0.43\
X251–37 & 3 15 31.0 & $-$55 04 43 & 6.0 & 9.5 & 16.8 & 22.5 & 0.52 $\pm$ 0.14 & 0.07 $\pm$ 0.67\
X024–38 & 3 15 34.7 & $-$55 19 27 & 5.4 & 6.7 & 12.4 & 22.7 & 0.52 $\pm$ 0.16 & $-$0.11 $\pm$ 0.23\
X015–39 & 3 13 51.6 & $-$55 18 33 & 6.9 & 12.2 & 11.8 & 21.4 & 0.51 $\pm$ 0.14 & $-$0.43 $\pm$ 0.22\
X236–40 & 3 16 24.0 & $-$55 11 44 & 4.3 & 10.7 & 15.9 & 21.3 & 0.50 $\pm$ 0.13 & 0.02 $\pm$ 0.32\
X234–41 & 3 16 23.7 & $-$55 15 17 & 5.3 & 10.6 & 10.7 & 20.7 & 0.48 $\pm$ 0.15 & $-$0.15 $\pm$ 0.41\
X306–42 & 3 15 50.1 & $-$55 09 15 & 4.5 & 7.2 & 17.0 & 20.1 & 0.47 $\pm$ 0.14 & $-$0.22 $\pm$ 0.38\
X051–43 & 3 15 01.6 & $-$55 03 40 & 6.6 & 10.2 & 11.1 & 19.9 & 0.46 $\pm$ 0.14 & 0.93 $\pm$ 1.00\
X407–44 & 3 14 12.4 & $-$55 25 52 & 7.3 & 14.7 & 9.8 & 18.2 & 0.45 $\pm$ 0.14 & $-$0.33 $\pm$ 0.25\
X233–45 & 3 16 18.9 & $-$55 14 29 & 7.2 & 9.8 & 9.8 & 19.3 & 0.45 $\pm$ 0.14 & 0.26 $\pm$ 0.41\
X109–46 & 3 16 08.1 & $-$55 17 25 & 4.8 & 9.0 & 11.9 & 18.3 & 0.43 $\pm$ 0.12 & 0.05 $\pm$ 0.30\
X213–47 & 3 14 12.1 & $-$55 18 24 & 5.4 & 9.5 & 11.7 & 17.9 & 0.42 $\pm$ 0.14 & 1.00 \
X022–48 & 3 15 03.4 & $-$55 19 06 & 4.4 & 5.4 & 12.3 & 18.3 & 0.42 $\pm$ 0.12 & 1.00 \
X215–49 & 3 14 29.6 & $-$55 16 44 & 5.3 & 6.5 & 12.3 & 17.7 & 0.41 $\pm$ 0.12 & 1.00 \
X264–50 & 3 14 49.1 & $-$55 22 24 & 5.2 & 9.2 & 12.0 & 15.8 & 0.37 $\pm$ 0.14 & 1.00 \
Table 1 contains the X-ray data for the complete sample of sources in a circular area with radius of 15. Within this area we detected 50 sources with a maximum likelihood value ML $\ge$ 9.8. With this adopted threshold in ML we expect less than one spurious source over the entire area. The first column gives an identification code for the X-ray sources: the first number identifies the sources in our working lists, while the second one (from 1 to 50) represents the source rank in order of decreasing X-ray flux. Columns 2 to 5 give the $\alpha$ (2000) and $\delta$ (2000) positions, with the one sigma error on each coordinate (in arcsec) derived from the maximum likelihood fitting, and the off–axis distance from the center of the X-ray field (in arcmin). The next three columns give the maximum likelihood value in the hard band, the net counts and the corresponding 0.5 – 2.0 keV flux and error. The flux has been computed assuming the same intrinsic spectrum for all the sources, i.e. a power law spectrum with a slope $\alpha$ = 1.0, with no intrinsic absorption and a galactic absorption corresponding to $N_H = 2.5 \times 10^{20}$. These assumed parameters correspond to the conversion factor 1 ct/s = 2.26 $\times 10^{-12} $ at the center of the field. The last column gives the hardness ratio with its statistical error.
Since most of the detected X-ray sources are too faint to obtain a full resolution spectrum, we have characterized their spectra using the hardness ratio technique. The hardness ratio is defined as HR=(H-S)/(H+S), where S and H are the net counts in the PSPC energy channels 11–41 and 52–201, respectively, corresponding approximately to the energy ranges 0.1–0.4 and 0.5–2.0 keV. The source counts in each band have been obtained summing all the counts from a circle centered at the source position. The radius of the extraction circle was 60. However, when the extraction circles of two sources overlapped, this radius was reduced to 30in order to avoid contamination from the nearby source. The resulting net counts for the instrumental radial vignetting and for the energy dependence of the ROSAT/PSPC Point Spread Function (Hasinger et al. 1994). The background counts were estimated in a circular area near the source position, after excluding the contribution from near–by sources. Five sources, corresponding to 10% of the total sample, have a formal hardness ratio equal to or greater 1.00 (i.e. zero or negative net counts detected in the soft band). Four of them are at the faint flux limit of the entire sample.
To eliminate a possible systematic error of a few arcsec in the X-ray positions, we have cross–correlated the positions of the 50 X-ray sources within 15 from the center with those of 29 previously known optically selected AGNs with m$_B \le$ 22.5 in the same area (Zitelli et al. 1992; Mignoli et al. in preparation). We have found 19 positional coincidences, with distances smaller than 12between X-ray and optical positions, while less than 0.3 random coincidences are expected on a statistical basis. The average offset between X-ray and optical positions is about 4in right ascension and less than 1in declination, with no additional trend as a function of position in the field. The one sigma uncertainty for the offset in each coordinate is of the order of 0.9. We have therefore applied this average offset to all the original X-ray positions and corrected the ML positional errors by adding in quadrature the uncertainty on the offset. Both the positions and the positional errors given in Table 1 are the “corrected” values.
Figure 1 shows a gray scale representation of the hard ROSAT image. The image has been slightly smoothed with a gaussian with $\sigma$ = 10. All the sources detected within a radius of 15 are labelled. The figure shows the capability of the maximum likelihood algorithm in retrieving pairs of sources relatively close to each other (see, for example, the two pairs 49–304 and 46–301, for which the distances between the X-ray positions are of the order of 50 arcsec). For distances between two X-ray centroids smaller than $\sim$40, however, our detection algorithm is unable to separate the two X-ray sources and will find a single source (Hasinger et al. 1993), with a position intermediate between the two true positions. In these cases the resulting X-ray position and its associated error would not be reliable and this would lead to problems in the optical identification process.
In order to estimate how many such cases we may have in our list of sources we have simulated 1000 random samples with the same number of detected sources and with the same radial surface densities of sources as the real sample (i.e. higher density in the inner region of the field and lower density in the outer region). For each sample we have then computed the number of sources which have a nearby companion at a distance smaller than the smallest observed distance in the real sample (i.e. 48). Figure 2 shows the normalized histogram of the number of such sources in the 1000 random samples. The figure shows that in only 2% of the cases there is no pair of sources with a distance smaller than 48 arcsec, while in the central 75% of the cases (hatched area) the number of such sources ranges from 4 to 10.
Since each pair is counted twice in this histogram, this means that we may expect that in about 2–5 cases a single source in our list may be produced by two different close–by sources, so that its position may be significantly wrong and we may not be able to find any reliable optical counterpart within the error box. Note that this estimate is based on purely geometric considerations; moreover, since in computing the expected number of close–by sources we have adopted the observed surface density of sources, we are not considering in this order of magnitude estimate the cases in which one or both the sources in a close pair are below the detection limit. Results from more detailed simulations, which take into account all the possible effects of source confusion, are discussed in Hasinger et al. (1998).
Figure 3 shows the hard X-ray flux versus the off–axis angle for all the X-ray sources listed in Table 1. Even if we have considered only sources within 15 from the center, the limiting flux is not constant over the adopted field of view. Its increase at distances greater than about 10 is mainly due to the increase of the width of the point spread function with the off-axis angle. The curve drawn in the figure shows our estimated limiting flux as a function of the off–axis angle for the complete sample. When corrected for the different sensitivity over the field, and using only the 48 sources with flux greater than the adopted limiting flux, the estimated observed surface density at $S_x \ge 4 \times 10^{-15}$ is $272 \pm 40$ sources/sq.deg. This value is consistent with the surface density (248 sources/sq.deg.) derived by Hasinger et al. (1993) from the composite observed log N – log S relationship using the deeper ROSAT data in the Lockman Hole together with a number of shallower fields.
The Identification of the X-ray Sources
=======================================
The Radio Data
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We observed the Marano field at 1.4 and 2.4 GHz with the Australia Telescope Compact Array (ATCA). The radio observations were carried out on 1994 January 4, 5, 6 and 7. Details about the radio observation and data reduction are given in Gruppioni et al. (1997), where catalogs of 5$\sigma_{local}$ radio sources are given. The radio limits are about 0.2 mJy at both frequencies. Cross–correlation of the entire radio and X-ray catalogs has produced four positional coincidences with a maximum difference of $\sim$10between radio and X-ray positions. All these distances are between 1.1 and 2.4 times the combined X-ray and radio positional error ($\epsilon$). Fifteen more radio – X-ray pairs have distances smaller than 90, but none has a distance smaller than 25. For all these pairs the distances between radio and X-ray positions are larger than five times the combined positional error and can therefore be considered random coincidences. >From the number of these random pairs we estimate that the expected number of similarly random pairs within 10is 0.20 $\pm$ 0.05. If, instead of using a radius of 10, which can be considered an “a posteriori” choice, we use for each X-ray source a radius corresponding to the 95% error circle, the estimated number of random pairs inside the 95% area becomes 0.40. On this basis we conclude that probably none and at most one of the observed radio – X-ray coincidences is not real. The four radio – X-ray pairs correspond to a percentage of radio detection of 8$\pm$4% for our X-ray sources at a radio limit of $\sim$ 0.2 mJy. Of the same order (10$\pm$5%) is the percentage of X-ray detections for our sample of radio sources. Both these percentages are in good agreement with what has been found by De Ruiter et al. (1996) from VLA observations in the region of the Lockman Hole. Table 2 lists these four radio – X-ray pairs, where the columns are X-ray number and flux from Table 1, radio number and flux from the 20 cm catalog in Gruppioni et al. 1997, difference in position ($\Delta$) in arcsec and normalized to the combined X-ray and radio error ($\Delta$/$\varepsilon$). Three of these four radio – X-ray pairs have ben optically identified (see Section 3.2). For all of them the offset between the radio and the optical position of the suggested identification is less than 1.8, consistent with the combined radio and optical positional errors.
[rc|rc|cc]{} & &\
\# & flux & \# & flux &$\Delta$&$\Delta$/$\varepsilon$\
&& \[mJy\] & \[arcsec\] &\
X013–01 & 5.00 & 15 & 158 & 3.8 & 2.0\
X021–05 & 2.36 & 38 & 1.25 & 2.3 & 1.1\
X409–28 & 0.61 & 30 & 6.32 & 8.2 & 1.5\
X408–30 & 0.59 & 35 & 0.41 & 10.3 & 2.4\
The Optical Data
-----------------
In addition to the ESO 3.6m plates (U, J, F bands) which were already available and have been used in the past to obtain a complete sample of optically selected AGNs with $m_B \le 22.0$ (Zitelli et al. 1992), in the years 1992–1994 we have obtained a set of U, B, V, R CCD images at the ESO NTT. The V and R images cover $\sim$ 90% of the circle with 15 radius and contain all but one of the X-ray sources within this radius; the U and B images, which have a smaller field of view, cover a smaller fraction of the 15 field, but still contain 45 out of 50 X-ray sources. Details about the observations, data reduction and optical catalogs obtained from these CCD observations will be given elsewhere (Mignoli et al. in preparation; see also Mignoli 1997).
The CCD limiting magnitudes vary from field to field, but typically are of the order of 23.5 in U, 25.0 in B, 24.0 in V and R. The typical surface density of objects at these limits is $\sim$ 60,000 per square degree. Since the total area covered by the 2$\sigma$ (3$\sigma$) error circles of the 50 X-ray sources corresponds to 5.4 (10.4) sq.arcmin., this implies a total expected number of $\sim$ 90 (175) catalogued objects inside the X-ray error boxes. In Table 3 we report the optical data for all the objects in our optical catalogs within a 3$\sigma$ error box, plus a few interesting objects at slightly larger distance. (We recall that, under the assumption that the distribution function of the positional errors follows a circular normal distribution, the 1$\sigma$, 2$\sigma$ and 3$\sigma$ error boxes correspond to radii equal to 1.51, 2.45 and 3.4 times the error in each coordinate, respectively.) The first ten columns in the Table give the X-ray number, the distance in right ascension and declination between the X-ray source and the optical objects (arcsec), the total distance both in arcsec and normalized to the X-ray positional error () as defined in Section 2.1, and R magnitudes and , , colours and a morphological classification (p for point-like source, e for extended, p/e or e/p for sources classified differently in the blue and red bands). The morphological classification is reliable only for objects which are more than $\sim$ 1.5 magnitudes brighter than the limiting magnitude (Flynn, Gould and Bahcall 1996). An asterisk in the magnitude columns means that the magnitudes have been measured from the plates and then converted to the Johnson system (see Gruppioni, Mignoli and Zamorani 1999).
The next two columns give the likelihood ratio (LR) computed from the B and R data. These LR values have been computed following the procedure described by Sutherland and Saunders (1992), which, differently from previous formulations, has been shown to be valid also in the case of multiple candidates in the error boxes. For each optical object its likelihood ratio is defined as:
$$LR = \frac{q(\rm{m,c})\; \em{f}(\rm{x,y})}{n(\rm{m,c})}$$
where $f$(x,y) is the probability distribution function of the positional errors, $n$(m,c) is the surface density of “background” objects with magnitude m and type c and $q$(m,c) is the probability distribution function in magnitude and type of the optical counterparts. With this definition LR is the ratio between the probability of finding the true optical counterpart with the observed offset (x,y) from the X-ray position and the observed magnitude and the probability of finding a similar chance background object (see Eq. 1 and related discussion in Sutherland and Saunders 1992). Assuming that the distribution function of the positional errors is gaussian,
$$f(x,y) = \frac { e^{- \frac {(x^2 + y^2)}{2 \sigma^2} } } {2 \pi \sigma^2}
%LR = \frac{q(\rm{m,c})\; \em{f}(\rm{x,y})}{n(\rm{m,c})}$$
where x and y are the offsets in right ascension and declination between the optical and X-ray sources and $\sigma$ is the positional error in each coordinate (see column 4 in Table 1). We considered two different types of optical objects, i.e. point–like and extended. For each of them the observed $n$(m) has been obtained from our own CCD data, while $q$(m) has been estimated from the magnitude distribution of the excess of objects in the 50 X-ray error boxes with respect to the expected number of “background” objects.
The last two columns give the redshift and the classification of the objects based on our spectroscopic data. Objects which we consider to be the correct identification are written in bold face in the last column. When the spectroscopic data suggest identification with a group or cluster of galaxies, all the galaxies at the redshift of the cluster are shown in bold face.
Figure 4 shows the finding charts for all the X-ray sources within 15, ordered by decreasing X-ray flux, and, when available, the spectrum of the most likely identification. For completeness we show here also the spectra of the previously known, optically selected AGNs already published in Zitelli et al. (1992). For all sources the optical images (1$\times$1) are taken from R CCDs, except for sources X046–03, X240–09 and X045–27, for which the images from the F plate are shown. The two circles drawn on the figures correspond to the 68% and 95% error boxes as defined above. The total number of objects in the 95% error boxes is $\sim$ 140. Spectroscopy for such a large number of faint objects was obviously not possible, and therefore we had to decide a strategy for the follow–up spectroscopic observations of the most promising candidates. Our adopted strategy was the following:
i\. First we have cross–correlated the positions of the 50 X-ray sources with those of the 29 optically selected AGNs previously known in the same area, finding 19 positional coincidences (see Section 2.1). With one exception (X019–17), all these AGNs have at least one of the two LR values higher than 1.6. We have considered them as likely identifications (these objects are indicated as [**AGN$^{(1)}$**]{} in the last column of Table 3) and we decided not to observe spectroscopically other objects in these error boxes, including the X-ray source X019–17 in which the AGN has a relatively low likelihood ratio. We note here that our LR values are based only on optical magnitudes and morphology and do not take into account the spectroscopic information. At the typical magnitudes of the optical counterparts identified with AGNs (21 $\le m_B \le$ 23) the ratio between the number of objects classified as point–like objects in our CCD data and that of broad–line AGNs (Mignoli and Zamorani 1998) is in the range 10–15. Therefore, the “a posteriori” LR, when an AGN is found spectroscopically in any given error box, would be about 10–15 times higher than that listed in Table 3.
ii\. Since AGNs are well known to be the dominant optical counterpart of faint X-ray sources, at least for $S_x \ge 5 \times 10^{-15} $ , we have then searched our optical catalogs for all the stellar or slightly fuzzy objects within the remaining 31 X-ray error boxes. We have then taken spectra for these objects helping us in making a priority list in each error box with additional information from the colours, the magnitude and the distance between the objects and the X–centroid (i.e. the LR value). In addition to the objects that on the basis of these data were considered to be the most likely AGN candidates, we gave high priority also to objects which coincide with a radio source (see Table 2) and to objects with colours typical of M stars, which are the dominant spectral type of stars found in faint X-ray surveys. M stars with the highest ratio between X-ray and optical fluxes and with an X-ray flux equal to our limiting flux are expected to have at most an optical magnitude of the order of $m_V \sim 19.75
\pm 0.25$ (see Maccacaro et al. 1988), corresponding to $m_B \sim 21.25 \pm 0.25$. From our own colour – colour diagrams (see also Marano et al. 1988) we estimate that the surface density of M stars with this limiting magnitude is $\sim$ 380 per square degree, in good agreement (within 15%) with the predictions of the Bahcall and Soneira model for the structure of the Galaxy (see, for example, Ratnatunga and Bahcall 1985). Following this recipe, we found spectroscopically four stars (two M stars with $m_b \sim 19.5$ (X030–15 and X028–33), and two very bright F stars with $m_b \sim 10$ (X046–03 and X045–27)), eight broad-line AGNs (X012–02, X042–11, X040–32, X032–36, X251–37, X024–38, X015–39, X306–42) and two radio galaxies (X021–05 and X409–28), one of which with broad MgII line. The blue magnitudes of the AGNs and the two radio galaxies are in the range $21.7 - 23.7$.
We also took a spectrum for the relatively bright object ($m_B$ = 21.44), with LR values greater than 5.5, which is right at the center of the error box of source X051–43. Its spectrum does not show any clear evidence for convincing emission or absorption features, so that, even if we do consider it as the likely counterpart of the X-ray source, we have not been able to spectroscopically classify it. The absence of features in the spectrum suggests that it might be a BL Lac object. If so, it would be a somewhat anomalous BL Lac object. In fact, being not detected in the radio at a flux limit of about 0.2 mJy, it would be classified as a radio quiet BL Lac, with a radio to optical spectral index $\alpha_{ro} <$0.26. None of the X-ray selected BL Lac objects in the Einstein Medium Sensitivity Survey (Stocke et al. 1991), and just a few in the much larger samples of ROSAT selected BL Lacs (see, for example, Perlman et al. 1998) have such a low value of $\alpha_{ro}$.
[rrrrrrrrrcrrll]{}\
ID. & $\Delta\alpha$ & $\Delta\delta$ & $\Delta$/$\displaystyle{\Delta\over\varepsilon_{\rm{x}}}$ & & & U-B & B-V & V-R & & & & & object notes\
& &\
X013–01 & 1.2 & -0.0 & 1.2/0.8 & $>$25.00 & 23.17 & & & $>$ 1.13 & e & & 4.73 & &\
& -4.0 & -1.7 & 4.3/2.8 & 19.96 & 19.61 & -0.55 & -0.09 & 0.44 & p & 1.25 & 1.63 & 1.663 & [**AGN$^{(1)}$**]{}\
X012–02 & -7.6 & 0.2 & 7.6/2.8 & 21.94 & 20.58 & -0.70 & 0.63 & 0.73 & p & 1.16 & 0.65 & 1.378 & [**AGN**]{}\
X046–03 & -1.4 & 0.9 & 1.6/1.0 & $\sim$9.3 & & & $\sim$0.5& & & & & 0.00 & [**F6IV star**]{}\
X036–04 & -0.4 & -0.3 & 0.5/0.2 & 18.04\* & 17.07 & -0.78 & 0.55 & 0.42 & p & 17.86 & 28.60 & 2.531 & [**AGN$^{(1)}$**]{}\
& 0.8 & 5.1 & 5.1/2.1 & $>$23.50\* & 23.44 & & $>$-0.67 & 0.73 & e & & 0.29 & &\
X021–05 & -2.7 & 0.0 & 2.7/1.5 & 21.79 & 19.28 & 0.66 & 1.48 & 1.03 & e & 6.83 & 3.58 & 0.387 & [**RadioGal**]{}\
X025–06 & 1.0 & -1.6 & 1.9/1.0 & 21.43 & 20.97 & -0.52 & 0.36 & 0.10 & p & 71.44 & 45.19 & 0.808 & [**AGN$^{(1)}$**]{}\
X027–07 & 3.0 & 3.3 & 4.4/2.0 & 21.17 & 20.66 & -0.55 & 0.19 & 0.32 & p & 11.89 & 7.52 & 0.636 & [**AGN$^{(1)}$**]{}\
& -3.4 & -5.6 & 6.6/2.9 & 23.34 & 22.87 & -0.86 & 0.29 & 0.18 & e & 0.12 & 0.09 & &\
X041–08 & 1.8 & 3.6 & 4.0/1.5 & 21.62 & 21.06 & -0.67 & 0.35 & 0.21 & p & 20.74 & 10.78 & 2.161 & [**AGN$^{(1)}$**]{}\
& -3.5 & -5.7 & 6.7/2.6 & 24.04 & 23.46 & -0.66 & 0.41 & 0.17 & e & 0.08 & 0.08 & &\
& 0.3 & -7.5 & 7.5/2.9 & 24.20 & $>$23.50 &$>$-0.70 & 0.48 & $<$ 0.22 & e & 0.04 & & &\
& -8.5 & 2.3 & 8.8/3.4 & 24.15 & $>$23.50 &$>$-0.65 & -0.21 & $<$ 0.86 & e & 0.01 & & &\
X240–09 & -0.3 & 4.0 & 4.0/1.0 & 21.81\* & 20.87\* & -0.71 & 0.52 & 0.42 & p & 17.75 & 9.99 & 0.854 & [**AGN$^{(1)}$**]{}\
& 12.4 & -1.9 & 12.5/3.2 & 22.54\* & 21.91\* & -0.74 & 0.34 & 0.29 & e & 0.01 & 0.02 & &\
X033–10 & 7.2 & 1.4 & 7.3/1.7 & 21.90\* & 20.91 & -0.90 & 0.65 & 0.34 & p & 5.70 & 3.21 & 0.983 & [**AGN$^{(1)}$**]{}\
& 1.0 & 12.6 & 12.6/3.0 & $>$23.50\* & 21.86 & & $>$ 0.72 & 0.92 & p & & 0.09 & &\
X042–11 & -0.8 & 2.4 & 2.5/1.1 & 22.99 & 22.04 & -0.33 & 0.58 & 0.37 &p/e& 19.50 & 5.43 & 1.062 & [**AGN**]{}\
& -6.8 & -3.8 & 7.8/3.3 & $-out-$ & 22.10 & & & 1.00 & e & & 0.04 & &\
X043–12 & 1.6 & 3.1 & 3.5/1.5 & 23.80 & 22.61 & -0.32 & 0.17 & 1.02 &e/p& 1.66 & 0.87 & 2.80: & [**AGN2**]{} (?)\
& 5.3 & -5.8 & 7.9/3.4 & 24.15 & 22.93 & -0.47 & 0.95 & 0.27 & e & 0.01 & 0.02 & &\
X023–13 & 0.3 & -0.7 & 0.8/0.3 & 22.24 & 21.99 & -0.45 & -0.10 & 0.35 & p & 36.47 & 19.86 & 1.573 & [**AGN$^{(1)}$**]{}\
& 4.5 & -8.1 & 9.3/3.4 & 23.94 & 21.32 &$>$-0.44 & 1.42 & 1.20 & e & 0.01 & 0.02 & &\
X108–14 & 3.6 & 0.6 & 3.6/0.8 & 22.42 & 21.63 & -1.55 & 0.23 & 0.56 & p & 10.04 & 5.47 & 1.374 & [**AGN$^{(1)}$**]{}\
& -4.8 & 3.1 & 5.7/1.3 & 21.30 & 20.36 & 0.05 & 0.52 & 0.42 & e & 2.90 & 1.71 & &\
X030–15 & 0.5 & 3.5 & 3.6/0.8 & 24.07 & $>$23.50 & -0.35 & $<$-0.23 & & p & 0.60 & & &\
& 0.5 & 7.6 & 7.6/1.8 & 19.28 & 16.86 & 1.20 & 1.59 & 0.83 & p & 1.95 & 2.59 & 0.00 & [**M star**]{}\
& -8.1 & 10.1 & 13.0/3.0 & 23.29 & 22.20 & -0.11 & 0.87 & 0.22 & e & 0.02 & 0.03 & &\
& -13.0 & -5.7 & 14.2/3.3 & 22.99 & 22.24 & -0.86 & 0.33 & 0.42 & e & 0.01 & 0.01 & &\
X304–16 & -1.0 & 1.6 & 1.9/0.6 & 21.53 & 20.73 & -0.74 & 0.39 & 0.41 & p & 39.05 & 21.97 & 1.192 & [**AGN$^{(1)}$**]{}\
& -2.6 & 5.7 & 6.3/2.0 & 24.04 & 23.05 & -0.84 & 0.40 & 0.59 & e & 0.21 & 0.21 & &\
& 9.8 & -2.2 & 10.1/3.2 & 23.59 & 22.27 & -0.70 & 0.46 & 0.86 & e & 0.02 & 0.03 & &\
X019–17 & 0.9 & -2.8 & 2.9/0.8 & 24.34 & 23.47 & -0.90 & 0.78 & 0.09 & e & 0.85 & 0.85 & &\
& -2.8 & 8.7 & 9.2/2.5 & 23.45 & 21.20 &$>$ 0.05 & 1.35 & 0.90 & p & 0.30 & 0.74 & &\
& -5.5 & -10.4 & 11.7/3.2 & 22.40\* & 21.69 & -0.70 & 0.49 & 0.22 &p/e& 0.12 & 0.03 & 0.614 & [**AGN$^{(1)}$**]{}\
X029–18 & 3.5 & 4.8 & 6.0/1.6 & 23.67 & 21.94 & -0.10 & 1.11 & 0.62 & e & 0.56 & 1.20 & &\
& 0.9 & -6.2 & 6.2/1.6 & 21.66 & 21.15 & -1.01 & 0.23 & 0.28 & p & 8.42 & 4.37 & 1.254 & [**AGN$^{(1)}$**]{}\
X039–19 & -10.5 & 2.2 & 10.7/3.3 & 23.63 & 22.41 & -0.20 & 0.99 & 0.23 & e & 0.01 & 0.02 & ?.??? & low S/N(A)\
X001–20 & -5.4 & -1.8 & 5.7/1.6 & 20.39 & 19.71 & -0.83 & 0.26 & 0.42 & p & 2.31 & 3.92 & 1.353 & [**AGN$^{(1)}$**]{}\
& -8.4 & 7.0 & 10.9/3.0 & 22.72 & 22.73 & -1.00 & -0.09 & 0.08 & p & 0.16 & 0.01 & &\
X049–21 & -3.2 & -1.7 & 3.6/1.2 & 23.86 & 22.36 & -0.61 & 0.68 & 0.82 & e & 1.50 & 2.82 & ?.??? & low S/N\
X211–22 & -0.2 & -6.9 & 6.9/1.6 & 24.27 & 23.31 & -0.51 & $<$-0.03 & $>$ 0.99 & e & 0.23 & 0.23 & &\
& 9.3 & 7.5 & 12.0/2.8 & 20.97 & 19.35 & 0.11 & 1.06 & 0.56 & e & 0.07 & 0.04 & 0.180 & early gal\
& -12.5 & 0.5 & 12.5/2.9 & 22.20 & 20.26 & -0.45 & 0.86 & 1.08 &p/e& 0.23 & 0.07 & 0.281 & [**AGN (Sy 1)**]{}\
& -12.3 & -4.0 & 13.0/3.0 & 24.37 & 22.99 & -0.16 & 0.43 & 0.95 & e & 0.01 & 0.02 & &\
& -13.5 & 0.1 & 13.5/3.1 & 22.12 & 21.36 & -0.04 & 0.41 & 0.35 & p & 0.12 & 0.10 & 0.00 & B star\
& 7.0 & 12.9 & 14.7/3.4 & 24.60 & 23.49 &$>$-1.10 & 0.64 & 0.47 & e & 0.00 & 0.00 & &\
[rrrrrrrrrcrrll]{}\
ID. & $\Delta\alpha$ & $\Delta\delta$ & $\Delta$/$\displaystyle{\Delta\over\varepsilon_{\rm{x}}}$ & & & U-B & B-V & V-R & & & & & object notes\
& &\
X404–23 & 11.8 & -1.4 & 11.9/1.3 & 24.03 & 22.74 &$>$-0.53 & 0.70 & 0.59 & e & 0.08 & 0.17 & &\
& 6.0 & 10.9 & 12.4/1.3 & $>$25.00 & 23.25 & & & $>$ 1.05 & e & & 0.07 & &\
& -9.8 & -10.1 & 14.1/1.5 & 23.67 & 22.54 & -0.38 & 0.87 & 0.26 & e & 0.10 & 0.12 & &\
& -16.1 & -1.6 & 16.2/1.7 & 24.25 & 22.67 &$>$-0.75 & 0.76 & 0.82 & e & 0.04 & 0.09 & &\
& 6.2 & -16.5 & 17.6/1.9 & 24.47 & 23.41 &$>$-0.97 & 0.26 & 0.80 & e & 0.03 & 0.03 & &\
& 8.9 & -16.5 & 18.8/2.0 & 24.89 & 23.18 &$>$-1.39 & 0.70 & 1.01 & e & 0.01 & 0.02 & &\
& -12.9 & -15.5 & 20.2/2.2 & 23.40 & 22.34 & -0.77 & 0.57 & 0.49 & e & 0.05 & 0.06 & 0.204 & NELG(G)\
& -18.0 & -9.3 & 20.3/2.2 & 24.23 & 23.60 & -0.17 & 0.16 & 0.47 & e & 0.02 & 0.01 & &\
& -3.7 & 21.3 & 21.6/2.3 & 24.32 & 23.47 & -0.66 & 0.41 & 0.44 & e & 0.01 & 0.01 & &\
& 11.9 & -19.3 & 22.6/2.4 & 23.55 & 21.01 &$>$-0.05 & 1.31 & 1.23 & p & 0.03 & 0.14 & 0.00 & M star(J)\
& 23.3 & 14.2 & 27.3/2.9 & 23.65 & 21.81 &$>$-0.15 & 0.85 & 0.99 & e & 0.00 & 0.01 & &\
& 24.6 & -12.6 & 27.7/3.0 & 23.80 & 22.20 & -1.18 & 0.75 & 0.85 & e & 0.00 & 0.01 & &\
& 24.1 & 13.7 & 27.7/3.0 & 23.82 & $>$23.50 &$>$-0.32 & -0.24 & $<$ 0.56 & e & 0.00 & & &\
& 13.6 & -24.3 & 27.8/3.0 & 23.67 & 23.04 & -0.67 & -0.16 & 0.79 & e & 0.00 & 0.00 & &\
& 27.8 & -5.3 & 28.3/3.0 & 22.51 & 21.70 & -0.36 & 0.52 & 0.29 & e & 0.00 & 0.01 & 0.157 & Sy2 (?) (N)\
& -27.1 & 8.9 & 28.5/3.1 & 23.76 & 21.43 &$>$-0.26 & 1.28 & 1.05 & p & 0.01 & 0.02 & &\
& -15.7 & 24.5 & 29.1/3.1 & $>$25.00 & 23.35 & & $>$ 0.77 & 0.88 & e & & 0.00 & &\
& 8.3 & 28.2 & 29.4/3.2 & 24.45 & 21.94 &$>$-0.95 & 1.51 & 1.00 & p & 0.00 & 0.01 & &\
X031–24 & 0.0 & 2.8 & 2.8/0.6 & 21.37\* & 20.71 & -0.72 & 0.32 & 0.34 & p & 17.50 & 11.07 & 0.409 & [**AGN$^{(1)}$**]{}\
& -1.3 & -3.7 & 3.9/0.9 & $>$23.50\* & 21.23 & & $>$ 1.19 & 1.08 & e & & 1.90 & &\
& 5.8 & 1.0 & 5.9/1.4 & $>$23.50\* & 23.18 & & $>$-0.77 & 1.09 & e & & 0.33 & &\
& -0.9 & -7.8 & 7.8/1.8 & $>$23.50\* & 21.71 & & $>$ 1.37 & 0.42 & e & & 0.64 & &\
& 0.1 & -13.6 & 13.6/3.1 & $>$23.50\* & 22.82 & & $>$ 0.21 & 0.47 & e & & 0.01 & &\
X050–25 & -0.4 & 0.7 & 0.8/0.2 & 20.77 & 20.39 & -0.72 & 0.04 & 0.34 & p & 18.47 & 13.93 & 1.315 & [**AGN$^{(1)}$**]{}\
& 4.0 & -5.8 & 7.1/1.9 & 24.17 & 23.08 &$>$-0.67 & 0.31 & 0.78 & e & 0.19 & 0.19 & &\
X235–26 & -0.9 & -1.7 & 1.9/0.4 & 21.40 & 20.96 & -0.00 & 0.26 & 0.18 & p & 13.94 & 8.82 & 2.536 & [**AGN$^{(1)}$**]{}\
& 2.5 & -1.0 & 2.7/0.5 & 24.92 & 23.42 &$>$-1.42 & $<$ 0.62 & $>$ 0.88 & e & 0.23 & 0.50 & &\
& -0.5 & -5.2 & 5.2/1.0 & 22.28 & 20.88 & -0.42 & 0.69 & 0.71 & e & 1.54 & 1.24 & &\
& 5.5 & -9.5 & 11.0/2.1 & 23.37 & 22.14 & -0.62 & 0.57 & 0.66 & e & 0.19 & 0.21 & &\
& 2.7 & 10.9 & 11.2/2.1 & 24.33 & 23.37 &$>$-0.83 & 0.40 & 0.56 & e & 0.06 & 0.06 & &\
& -10.5 & -9.7 & 14.3/2.7 & 24.46 & 23.73 & -0.64 & 0.03 & 0.70 & e & 0.01 & 0.01 & &\
& 7.6 & 14.9 & 16.8/3.2 & $>$25.00 & 22.55 & & $>$ 0.88 & 1.57 & e & & 0.01 & &\
& 11.6 & -13.4 & 17.7/3.4 & 23.45 & 22.10 & 0.08 & 0.49 & 0.86 & e & 0.01 & 0.01 & &\
X045–27 & 2.6 & -9.8 & 10.1/2.9 &$\sim$10.0 & & & $\sim$0.3& & & & & 0.00 & [**F5V star**]{}\
X409–28 & 0.1 & -7.5 & 7.5/1.4 & 23.70 & 21.81 & -0.65 & 0.97 & 0.92 & e & 0.36 & 0.79 & 0.957 & [**AGN (BLRG)**]{}\
& -6.5 & 5.0 & 8.2/1.6 & 24.67 & $>$23.50 &$>$-1.17 & $<$ 0.37 & & e & 0.10 & & &\
& -9.0 & -3.1 & 9.5/1.8 & 24.46 & 23.21 & -0.78 & 0.24 & 1.01 & e & 0.11 & 0.11 & &\
& 3.5 & -9.3 & 9.9/1.9 & 24.27 & 22.47 &$>$-0.77 & 0.87 & 0.93 & e & 0.09 & 0.32 & &\
& -6.5 & -8.6 & 10.7/2.0 & $>$25.00 & 22.76 & & & $>$ 1.54 & e & & 0.15 & &\
& -10.1 & -6.6 & 12.1/2.3 & 23.65 & 22.68 & -0.66 & 0.41 & 0.56 & e & 0.07 & 0.08 & &\
& 13.5 & 5.6 & 14.6/2.8 & 24.41 & 23.50 & -0.56 & 0.29 & 0.62 & e & 0.01 & 0.00 & &\
X301–29 & -2.5 & 0.4 & 2.6/0.6 & 21.13\* & 20.41\* & -0.88 & 0.40 & 0.32 & p & 19.92 & 10.18 & 1.709 & [**AGN$^{(1)}$**]{}\
& -0.9 & 5.9 & 6.0/1.5 & 23.86 & 22.67 & -0.42 & 0.56 & 0.63 & e & 0.57 & 0.68 & &\
& -2.7 & -10.4 & 10.8/2.6 & 19.13\* & 16.30\* & 1.09 & 1.59 & 1.24 & p & 0.32 & 0.25 & &\
& 4.7 & 11.3 & 12.2/3.0 & 22.79 & 21.13 & -1.38 & 0.96 & 0.70 & e & 0.03 & 0.04 & &\
X408–30 & -9.6 & 3.5 & 10.2/2.5 & 24.76 & $>$23.50 &$>$-1.26 & 0.40 & $<$ 0.86 & e & 0.02 & & &\
& 12.8 & 5.9 & 14.1/3.4 & 24.56 & 23.00 &$>$-1.06 & 0.70 & 0.86 & e & 0.00 & 0.00 & &\
& 4.6 & 13.8 & 14.6/3.6 & 23.49 & 21.53 & -0.30 & 1.04 & 0.92 & e & 0.00 & 0.01 & 0.814 & NELG(C)\
X207–31 & -5.7 & -0.7 & 5.7/1.2 & 23.37 & 22.19 & -0.38 & 0.64 & 0.54 & e & 0.98 & 1.11 & &\
& -1.6 & -8.0 & 8.1/1.7 & 23.78 & 22.43 & -0.44 & 0.84 & 0.51 & e & 0.29 & 0.56 & 0.58: & [**cluster gal**]{}\
& -10.0 & 4.8 & 11.1/2.3 & 24.05 & 22.97 &$>$-0.55 & 0.44 & 0.64 & e & 0.05 & 0.11 & &\
& -6.7 & 10.0 & 12.0/2.5 & 24.58 & 23.55 &$>$-1.08 & 0.45 & 0.58 & e & 0.02 & 0.02 & &\
& -12.3 & 1.0 & 12.3/2.5 & $>$25.00 & 23.22 & & & $>$ 1.08 & e & & 0.03 & &\
& -14.0 & -5.8 & 15.2/3.1 & 24.58 & $>$23.50 &$>$-1.08 & $<$ 0.28 & & e & 0.00 & & &\
& -15.7 & 6.4 & 16.9/3.5 & 23.94 & 22.38 & -0.43 & 0.57 & 0.99 & e & 0.00 & 0.01 & &\
& -5.8 & 17.3 & 18.2/3.7 & 24.34 & 23.69 & -0.84 & 0.11 & 0.54 & e & 0.00 & 0.00 & &\
& -17.3 & -6.3 & 18.5/3.8 & 22.68 & 19.65 &$>$ 0.82 & 1.72 & 1.31 & e & 0.00 & 0.00 & 0.584 & [**cD gal**]{}\
[rrrrrrrrrcrrll]{}\
ID. & $\Delta\alpha$ & $\Delta\delta$ & $\Delta$/$\displaystyle{\Delta\over\varepsilon_{\rm{x}}}$ & & & U-B & B-V & V-R & & & & & object notes\
& &\
X040–32 & 6.0 & 2.6 & 6.6/1.9 & 23.37 & 22.31 & -0.68 & 0.57 & 0.49 & p & 1.30 & 0.99 & 1.204 & [**AGN**]{}\
& -6.1 & 4.8 & 7.8/2.2 & 24.53 & $>$23.50 & -0.82 & -0.30 & $<$ 1.33 & e & 0.08 & & &\
& 8.3 & 1.5 & 8.4/2.4 & 23.26 & 21.99 & -0.74 & 0.52 & 0.75 & e & 0.21 & 0.27 & 0.804 & NELG\
& 9.0 & 4.1 & 9.8/2.8 & 23.61 & 22.27 & 0.41 & 0.52 & 0.82 & e & 0.04 & 0.08 & &\
X028–33 & -3.1 & -0.0 & 3.1/0.6 & $>$25.00 & $>$23.50 &$<$-1.77 & & & p & & & &\
& 4.3 & -0.3 & 4.3/0.9 & 23.20 & 22.20 & -0.78 & 0.40 & 0.60 & e & 1.35 & 1.53 & &\
& -5.5 & -1.8 & 5.8/1.2 & 19.16 & 17.22 & 1.06 & 1.29 & 0.65 & p & 3.64 & 3.59 & 0.00 & [**M star**]{}\
& -1.8 & -10.7 & 10.8/2.3 & 21.95 & 20.98 & -0.61 & 0.36 & 0.61 & e & 0.23 & 0.19 & 0.657 & NELG\
& 11.1 & -4.4 & 11.9/2.5 & 23.88 & 23.32 & -0.79 & 0.28 & 0.28 & e & 0.06 & 0.03 & &\
& 12.2 & 8.0 & 14.6/3.0 & 23.89 & 23.01 & -0.43 & 0.10 & 0.78 & p & 0.02 & 0.01 & &\
& 16.0 & 1.8 & 16.1/3.4 & 24.25 & 21.71 &$>$-0.75 & 1.26 & 1.28 & e & 0.00 & 0.01 & &\
X250–34 & 7.1 & 0.6 & 7.1/1.6 & 24.30 & 23.59 &$>$-0.80 & $<$ 0.00 & $>$ 0.71 & e & 0.22 & 0.16 & &\
& -8.9 & 5.6 & 10.5/2.4 & 23.29 & 22.94 &$>$ 0.21 & 0.08 & 0.27 & e & 0.13 & 0.10 & &\
& -9.1 & -6.0 & 10.9/2.5 & 24.32 & 23.68 &$>$-0.82 & $<$ 0.02 & $>$ 0.62 & e & 0.04 & 0.02 & &\
& 6.8 & -8.9 & 11.2/2.6 & $>$25.00 & 23.18 & & & $>$ 1.12 & e & & 0.03 & &\
X011–35 & 0.6 & -0.8 & 1.0/0.2 & 21.94 & 20.33 & -0.20 & 1.14 & 0.47 & e & 1.80 & 2.03 & 0.189 & abs.gal(A)\
& 1.7 & 2.5 & 3.1/0.5 & 24.49 & 23.75 &$>$-0.99 & $<$ 0.19 & $>$ 0.55 & e & 0.37 & 0.23 & &\
& 0.8 & 6.3 & 6.4/1.0 & 23.61 & 22.40 & -0.32 & 0.63 & 0.58 & e & 0.43 & 0.81 & &\
& -7.8 & 5.5 & 9.5/1.6 & 21.95 & 20.63 & -0.67 & 0.76 & 0.56 & e & 0.54 & 0.44 & 0.391 & [**NELG**]{}(D)\
& -7.6 & -5.8 & 9.6/1.6 & 23.99 & $>$23.50 &$>$-0.49 & -0.19 & $<$ 0.68 & e & 0.22 & & &\
& -1.0 & -10.2 & 10.2/1.7 & 24.03 & 21.34 &$>$-0.53 & 1.34 & 1.35 & e & 0.10 & 0.36 & 0.586 & early gal(F)\
& 7.3 & -7.4 & 10.4/1.7 & 24.02 & 23.42 & -0.97 & -0.05 & 0.65 & p & 0.10 & 0.10 & &\
& -11.9 & 0.2 & 11.9/1.9 & 23.63 & 22.94 &$>$-0.13 & 0.39 & 0.30 & e & 0.11 & 0.14 & &\
& -6.0 & -12.0 & 13.4/2.2 & 22.52 & 21.29 & -0.39 & 0.64 & 0.59 & e & 0.09 & 0.13 & ?.??? & low S/N(I)\
& -10.0 & -10.9 & 14.8/2.4 & 23.45 & 23.63 &$>$ 0.05 & 0.05 & -0.23 & e & 0.07 & 0.02 & &\
& 7.4 & -16.9 & 18.5/3.0 & 24.72 & 22.17 &$>$-1.22 & 1.41 & 1.14 & p & 0.00 & 0.02 & &\
& -15.0 & 13.4 & 20.1/3.3 & 22.00 & 20.15 &$>$ 1.00 & 1.14 & 0.71 & e & 0.01 & 0.01 & 0.390 & [**early+\[OII\]**]{}(L)\
& -4.8 & -21.2 & 21.7/3.5 & 24.02 & 22.66 &$>$-0.52 & 0.90 & 0.46 & e & 0.00 & 0.00 & &\
& -19.5 & 10.9 & 22.3/3.6 & $-out-$ & 22.86 & & & 0.93 & e & & 0.00 & &\
& 8.9 & -22.3 & 24.0/3.9 & 22.90 & 20.57 &$>$ 0.60 & 1.31 & 1.02 & e & 0.00 & 0.00 & 0.390 & [**early gal**]{}(O)\
X032–36 & 5.3 & -3.1 & 6.1/1.1 & 24.43 & 23.19 &$>$-0.93 & 0.51 & 0.73 & e & 0.28 & 0.28 & &\
& -3.8 & 9.4 & 10.1/1.9 & 24.24 & 22.81 &$>$-0.74 & 0.87 & 0.56 & e & 0.09 & 0.20 & &\
& 4.4 & 9.2 & 10.3/1.9 & 21.26 & 20.61 & -0.15 & 0.34 & 0.31 & p & 2.27 & 1.44 & 0.00 & B star\
& 0.5 & 12.4 & 12.4/2.3 & 23.36 & 21.98 & -0.56 & 0.79 & 0.59 & p & 0.54 & 0.37 & 1.190 & [**AGN**]{}\
& 10.6 & 9.0 & 13.9/2.6 & $>$25.00 & 22.88 & & $>$ 1.09 & 1.03 & e & & 0.04 & &\
& -0.2 & -14.3 & 14.3/2.6 & 24.21 & 23.46 &$>$-0.71 & 0.66 & 0.09 & p & 0.02 & 0.02 & &\
& -8.9 & -13.4 & 16.1/3.0 & 24.21 & 23.43 &$>$-0.71 & 0.22 & 0.56 & e & 0.01 & 0.01 & &\
& 16.3 & 1.6 & 16.4/3.0 & $>$25.00 & 23.27 & & & $>$ 1.03 & e & & 0.01 & &\
X251–37 & -6.1 & 4.5 & 7.6/1.3 & 21.72 & 20.68 & -0.20 & 0.70 & 0.34 & p & 5.70 & 3.20 & 2.710 & [**AGN**]{}\
& 4.3 & 6.9 & 8.1/1.3 & 24.34 & 23.40 &$>$-0.84 & 0.57 & 0.37 & e & 0.17 & 0.17 & &\
& 3.5 & 12.4 & 12.9/2.1 & 24.61 & $>$23.50 &$>$-1.11 & $<$ 0.31 & & e & 0.03 & & &\
& -13.7 & 0.8 & 13.7/2.3 & 20.03 & 18.40 & 0.28 & 1.07 & 0.56 & e & 0.23 & 0.14 & 0.091 & early gal\
& -12.3 & 8.0 & 14.7/2.4 & 23.95 & 22.80 &$>$-0.45 & 0.81 & 0.34 & e & 0.04 & 0.05 & &\
X024–38 & 0.8 & -0.0 & 0.8/0.1 & 22.19 & 20.76 & -0.37 & 0.77 & 0.66 & p & 9.29 & 8.72 & 1.430 & [**AGN**]{}\
& 3.6 & -3.6 & 5.1/0.9 & 17.94 & 16.82 & 0.02 & 0.45 & 0.67 & p & 3.39 & 5.06 & 0.00 & B star\
& 0.4 & 9.3 & 9.3/1.7 & 23.21 & 22.23 & -0.24 & 0.73 & 0.25 & p & 0.75 & 0.57 & 0.276 & NELG\
& 8.5 & 10.1 & 13.2/2.4 & 22.76 & 21.26 & 0.24 & 0.88 & 0.62 & e & 0.06 & 0.09 & &\
& 2.4 & 15.8 & 16.0/3.0 & 23.82 & 23.54 &$>$-0.32 & 0.50 & -0.22 & e & 0.01 & 0.01 & &\
& -17.4 & -1.8 & 17.5/3.2 & 24.35 & 22.65 &$>$-0.85 & 0.95 & 0.75 & e & 0.00 & 0.01 & &\
& 17.8 & -2.1 & 17.9/3.3 & 14.59 & 13.36 & -0.54 & 0.23 & 1.00 & p & 0.15 & 0.06 & 0.00 & A star\
& -19.1 & -1.1 & 19.1/3.5 & 23.95 & $>$23.50 &$>$-0.45 & -0.34 & $<$ 0.79 & e & 0.00 & & &\
& -10.1 & 16.9 & 19.7/3.6 & 16.99 & 15.68 & 0.29 & 0.42 & 0.89 & p & 0.00 & 0.00 & 0.00 & G star\
X015–39 & -5.4 & -5.8 & 7.9/1.1 & 23.27 & 22.01 & -0.46 & 0.69 & 0.57 & p & 1.04 & 0.79 & 0.500 & [**AGN (Sy 1)**]{}\
& 13.1 & -8.1 & 15.4/2.2 & $>$25.00 & 23.22 & & & $>$ 1.08 & e & & 0.03 & &\
& -17.4 & -2.8 & 17.6/2.5 & 23.48 & 22.73 & -0.68 & 0.15 & 0.60 & e & 0.04 & 0.03 & &\
& 7.2 & -17.2 & 18.7/2.7 & 24.64 & 23.43 &$>$-1.14 & $<$ 0.34 & $>$ 0.87 & e & 0.01 & 0.01 & &\
& -17.3 & -7.5 & 18.8/2.7 & 23.85 & 22.62 & -0.79 & 0.21 & 1.02 & e & 0.01 & 0.02 & ?.??? & low S/N\
[rrrrrrrrrcrrll]{}\
ID. & $\Delta\alpha$ & $\Delta\delta$ & $\Delta$/$\displaystyle{\Delta\over\varepsilon_{\rm{x}}}$ & & & U-B & B-V & V-R & & & & & object notes\
& &\
X236–40 & 5.6 & -0.3 & 5.6/1.3 & 22.23\* & 21.64 & -1.00 & 0.24 & 0.35 & p & 6.38 & 3.47 & 1.140 & [**AGN$^{(1)}$**]{}\
& -7.7 & 0.5 & 7.7/1.8 & $>$23.50\* & 22.00 & & $>$ 0.11 & 1.39 & e & & 0.57 & &\
& -8.7 & -2.1 & 8.9/2.1 & 24.55 & 23.73 &$>$-1.05 & 0.19 & 0.63 & e & 0.07 & 0.06 & &\
& -12.0 & -3.0 & 12.4/2.9 & $>$25.00 & 23.19 & & $>$ 0.39 & 1.42 & e & & 0.01 & &\
& -6.2 & 12.3 & 13.8/3.2 & 24.30 & 22.64 & -0.60 & 0.65 & 1.01 & e & 0.00 & 0.01 & &\
X234–41 & 0.9 & 1.2 & 1.5/0.3 & 23.38 & 22.18 & 0.17 & 0.77 & 0.43 & e & 1.58 & 1.78 & &\
& -0.5 & 4.1 & 4.1/0.8 & 23.83 & 23.29 & -0.54 & -0.03 & 0.57 & e & 0.73 & 0.41 & &\
& 1.5 & -4.1 & 4.3/0.8 & 23.72 & 22.58 & 0.41 & 0.75 & 0.39 & e & 0.71 & 0.86 & &\
& 2.3 & -5.6 & 6.1/1.1 & 23.83 & 22.51 & -0.69 & 0.50 & 0.82 & e & 0.51 & 0.62 & &\
& -1.1 & -6.4 & 6.5/1.2 & 23.67 & 22.08 & -0.24 & 0.39 & 1.20 & e & 0.47 & 0.88 & &\
& -8.5 & 2.8 & 9.0/1.7 & 23.18 & 22.84 & -0.65 & 0.03 & 0.31 & e & 0.39 & 0.28 & &\
& 11.4 & -6.3 & 13.1/2.5 & 23.56 & 22.48 & -0.78 & 0.37 & 0.71 & e & 0.05 & 0.09 & &\
& 0.2 & -14.7 & 14.7/2.8 & $>$25.00 & 23.22 & & & $>$ 1.08 & e & & 0.01 & &\
& 0.0 & 16.2 & 16.2/3.0 & 23.51 & 22.60 & -0.80 & 0.24 & 0.67 & e & 0.01 & 0.01 & &\
& -4.8 & -17.6 & 18.2/3.4 & 23.10 & 21.46 & -0.34 & 0.77 & 0.87 & e & 0.00 & 0.01 & &\
& -6.3 & 18.0 & 19.0/3.6 & 24.22 & $>$23.50 & -0.72 & 0.14 & $<$ 0.58 & e & 0.00 & & &\
& 17.3 & 8.8 & 19.4/3.6 & 18.93 & $<$18.02 & -0.15 & 0.55 & $>$ 0.36 & p & 0.01 & 0.00 & 0.00 & G star\
X306–42 & 3.1 & -3.6 & 4.7/1.0 & 24.08 & $>$23.50 & -0.74 & 0.11 & $<$ 0.47 & e & 0.44 & & &\
& -3.6 & 5.6 & 6.6/1.5 & 22.62 & 22.04 & -0.60 & 0.34 & 0.24 & p & 3.18 & 1.23 & 1.065 & [**AGN**]{}\
& -8.0 & 2.1 & 8.2/1.8 & 24.03 & 23.29 & -0.25 & 0.02 & 0.72 & e & 0.15 & 0.15 & &\
& 10.4 & -5.4 & 11.7/2.6 & 23.37 & 22.47 & -0.60 & 0.36 & 0.54 & e & 0.08 & 0.09 & &\
& -2.0 & 12.2 & 12.3/2.7 & $>$25.00 & 23.40 & & $>$ 0.34 & 1.26 & p & & 0.02 & &\
& 10.4 & 11.8 & 15.7/3.4 & 22.03 & 21.19 & -0.10 & 0.54 & 0.30 & e & 0.01 & 0.01 & 0.078 & NELG\
X051–43 & 1.1 & -1.1 & 1.5/0.2 & 21.44 & 20.60 & -0.12 & 0.48 & 0.36 & p & 9.00 & 5.70 & ?.??? & [**Bl Lac**]{} (?)\
& -6.3 & 6.9 & 9.4/1.4 & 24.45 & 23.55 &$>$-0.95 & $<$ 0.15 & $>$ 0.75 & e & 0.13 & 0.10 & &\
& -9.7 & 5.2 & 11.0/1.7 & 23.39 & $>$23.50 & -0.50 & -0.05 & $<$-0.06 & e & 0.27 & & &\
& -0.6 & 13.9 & 13.9/2.1 & 24.28 & 22.68 &$>$-0.78 & 0.78 & 0.82 & e & 0.04 & 0.09 & &\
& -13.1 & 4.8 & 14.0/2.1 & 23.71 & 23.59 &$>$-0.21 & -0.41 & 0.53 & e & 0.07 & 0.03 & &\
& 3.8 & -17.4 & 17.8/2.7 & 24.22 & $>$23.50 &$>$-0.72 & $<$-0.08 & & e & 0.01 & & &\
& -12.2 & -14.5 & 19.0/2.9 & 20.23 & 19.20 & -0.13 & 0.60 & 0.43 & e & 0.04 & 0.01 & 0.096 & Starburst\
& -11.7 & 16.6 & 20.3/3.1 & 23.86 & 23.51 &$>$-0.36 & -0.30 & 0.65 & e & 0.01 & 0.00 & &\
& -6.1 & 21.2 & 22.0/3.3 & 23.00 & 21.52 &$>$ 0.50 & 0.81 & 0.67 & p & 0.01 & 0.01 & &\
X407–44 & -3.3 & -4.7 & 5.7/0.8 & 21.41\* & 20.65 & -0.94 & 0.19 & 0.57 & p & 5.68 & 3.59 & 1.821 & [**AGN$^{(1)}$**]{}\
& 14.1 & 0.9 & 14.2/2.0 & $>$23.50\* & 21.08 & & & $>$ 1.42 & p & & 0.67 & &\
& -18.2 & -16.0 & 24.2/3.3 & 22.73\* & 20.99 & -0.92 & 0.93 & 0.81 & e & 0.00 & 0.00 & &\
X233-45 & 4.9 & -5.8 & 7.6/1.1 & 24.86 & 23.72 & -1.30 & $<$ 0.56 & $>$ 0.58 & e & 0.09 & 0.11 & &\
& -8.4 & -2.5 & 8.8/1.2 & 24.83 & 23.33 & -1.29 & 0.71 & 0.79 & e & 0.07 & 0.14 & &\
& -4.0 & -8.8 & 9.6/1.3 & 22.55 & 21.86 & -0.82 & 0.19 & 0.50 & e & 0.29 & 0.48 & ?.??? & low S/N\
& -9.2 & 5.9 & 10.9/1.5 & 24.69 & 21.76 &$>$-1.19 & 1.64 & 1.29 & e & 0.06 & 0.37 & &\
& -11.0 & 3.2 & 11.5/1.6 & 24.86 & 23.10 &$>$-1.36 & 0.57 & 1.19 & e & 0.04 & 0.08 & &\
& -13.3 & -1.8 & 13.4/1.9 & 22.51 & 21.70 & -0.66 & 0.25 & 0.56 & e & 0.12 & 0.20 & 1.180 & [**AGN**]{}\
& -12.0 & -11.9 & 16.9/2.4 & 21.29 & 18.62 & 1.19 & 1.51 & 1.16 & e & 0.49 & 0.21 & &\
& -16.1 & 10.4 & 19.2/2.7 & 24.49 & $>$23.50 & -0.49 & $<$ 0.19 & & e & 0.01 & & &\
& 6.3 & 20.7 & 21.6/3.0 & 22.41 & 21.12 & -0.54 & 0.58 & 0.71 & e & 0.01 & 0.01 & &\
& -17.6 & -17.0 & 24.5/3.4 & 24.50 & 23.34 & -0.69 & $<$ 0.20 & $>$ 0.96 & e & 0.00 & 0.00 & &\
X109-46 & -1.9 & -0.2 & 1.9/0.4 & 23.56 & 22.59 & -0.52 & 0.42 & 0.55 & p & 2.25 & 0.58 & &\
& 2.6 & 3.6 & 4.4/0.9 & 23.63 & 23.10 & -0.52 & 0.29 & 0.24 & e & 0.80 & 0.44 & &\
& -6.2 & 5.1 & 8.0/1.7 & 21.70 & 20.56 & -0.54 & 0.75 & 0.39 & e & 0.74 & 0.60 & 0.269 & Starburst(C)\
& 2.5 & 7.6 & 8.0/1.7 & 24.60 & 22.96 &$>$-1.10 & 1.04 & 0.60 & e & 0.11 & 0.36 & &\
& 2.1 & -9.2 & 9.4/2.0 & 24.30 & 21.36 &$>$-0.80 & 1.44 & 1.50 & p & 0.10 & 1.51 & 0.00 & M star(E)\
& 4.0 & -9.8 & 10.6/2.2 & 22.79 & 20.06 &$>$ 0.71 & 1.57 & 1.16 & p & 0.71 & 0.79 & 0.00 & M star(F)\
& 7.5 & 9.1 & 11.8/2.5 & 24.78 & $>$23.50 &$>$-1.28 & $<$ 0.48 & & e & 0.02 & & &\
& 0.5 & -12.7 & 12.7/2.7 & 24.75 & 23.44 &$>$-1.25 & 0.83 & 0.48 & e & 0.01 & 0.02 & &\
& -4.9 & 14.7 & 15.5/3.2 & 22.78 & 21.83 & -0.83 & 0.25 & 0.70 & e & 0.01 & 0.01 & 0.626 & NELG(I)\
X213–47 & -8.5 & -13.4 & 15.8/2.9 & 24.70 & $>$23.50 &$>$-1.20 & $<$ 0.40 & & e & 0.00 & & &\
& -13.8 & -13.8 & 19.5/3.5 & 24.51 & 23.54 &$>$-1.01 & 0.25 & 0.72 & e & 0.00 & 0.00 & &\
[rrrrrrrrrcrrll]{}\
ID. & $\Delta\alpha$ & $\Delta\delta$ & $\Delta$/$\displaystyle{\Delta\over\varepsilon_{\rm{x}}}$ & & & U-B & B-V & V-R & & & & & object notes\
& &\
X022–48 & -1.7 & -1.3 & 2.1/0.5 & 22.21 & 19.86 & 0.72 & 1.40 & 0.95 & e & 3.25 & 3.36 & 0.32: & [**Interactive gal**]{}\
& -1.6 & -4.6 & 4.9/1.1 & 23.31 & 22.06 & -0.20 & 0.74 & 0.51 & e & 1.30 & 1.47 & & companion\
& 5.9 & -6.0 & 8.5/2.0 & 24.04 & $>$23.50 &$>$-0.54 & -0.03 & $<$ 0.57 & e & 0.12 & & &\
& 8.9 & -7.9 & 11.9/2.7 & 22.33 & 21.00 & -0.38 & 0.66 & 0.67 & e & 0.09 & 0.07 & 0.474 & NELG\
& -3.3 & 11.5 & 12.0/2.8 & 21.90 & 20.70 & -0.38 & 0.64 & 0.56 & e & 0.08 & 0.07 & 0.389 & early+\[OII\]\
& -10.0 & -9.6 & 13.9/3.2 & 23.76 & 22.55 & -0.25 & 0.71 & 0.50 & e & 0.01 & 0.01 & &\
& -15.4 & -0.6 & 15.4/3.5 & 24.51 & $>$23.50 &$>$-1.01 & $<$ 0.21 & & e & 0.00 & & &\
& -6.5 & -14.9 & 16.3/3.7 & 21.03 & 20.01 & 0.01 & 0.64 & 0.38 & p & 0.02 & 0.01 & 0.00 & F/G star\
X215–49 & 5.4 & -7.4 & 9.2/1.7 & 23.12 & 22.44 & -0.69 & 0.18 & 0.50 & e & 0.37 & 0.42 & &\
& -0.1 & -14.3 & 14.3/2.7 & $>$25.00 & 23.32 & & & $>$ 0.98 & e & & 0.01 & &\
& 3.7 & 14.9 & 15.4/2.9 & 22.58 & 22.19 & -0.78 & 0.19 & 0.20 & e & 0.02 & 0.03 & 1.053 & stronggal (C)\
& -3.1 & 16.4 & 16.6/3.1 & 23.62 & 22.47 & -0.60 & 0.59 & 0.56 & e & 0.01 & 0.01 & &\
& 17.6 & 0.0 & 17.6/3.3 & 24.35 & 22.93 &$>$-0.85 & 0.77 & 0.65 & e & 0.00 & 0.00 & &\
X264–50 & 0.4 & 3.5 & 3.5/0.7 & 24.44 & $>$23.50 &$>$-0.94 & $<$ 0.14 & & e & 0.45 & & &\
& -2.1 & -7.2 & 7.5/1.4 & 21.84 & 20.77 & -0.42 & 0.49 & 0.58 & e & 0.89 & 0.73 & 0.568 & [**Starburst**]{}\
& 3.1 & -9.1 & 9.6/1.8 & 23.46 & 22.20 &$>$ 0.04 & 0.66 & 0.60 & e & 0.31 & 0.36 & &\
& -7.6 & -7.5 & 10.7/2.0 & 24.60 & $>$23.50 &$>$-1.10 & 0.10 & $<$ 1.00 & e & 0.05 & & &\
& -9.9 & -8.7 & 13.2/2.5 & 23.24 & 22.96 & -0.41 & -0.03 & 0.31 & p & 0.14 & 0.02 & &\
& 17.8 & 2.1 & 17.9/3.4 & 23.53 & 22.06 & -0.17 & 0.64 & 0.83 & e & 0.00 & 0.01 & &\
The LR values for most of these 15 suggested identifications are greater than 1 and are the highest in their error boxes. The only exceptions are the broad–line radio–galaxy identified with the source X409–28 and the AGN identified with the source X032–36, whose LR values are of the order of 0.5. In the latter error box there is also a brighter stellar object with higher LR values. However, its spectrum shows that it is a B type star and its identification with the X-ray source can therefore be excluded on the basis of the $f_x/f_v$ ratio (Maccacaro et al. 1988). A few other objects have been observed spectroscopically in these 15 error boxes (see last column in Table 3) and most of them have been found to be narrow emission line galaxies (NELG) or starburst galaxies. In all cases their LR value is significantly smaller than that of the best candidate and therefore we consider them unlikely to be associated to the X-ray sources. The estimated surface densities of AGNs with $ m_B \le$ 23.5 ($\sim$ 325 per square degree, Mignoli and Zamorani 1998) is similar to the surface density of M stars with $m_V < $ 20. From the sum of the two surface densities we estimate that only $\sim$ 0.7 random coincidences in the 95% error circles of the 31 X-ray sources are expected for AGNs and M stars in the magnitude ranges covered by these suggested identifications.
iii\. For the 16 error boxes which at this stage were still without a reliable optical identification, we have taken spectra also of objects classified as extended. At the typical magnitudes of the optical counterparts ($m_B >$ 22.0) the surface density of objects classified as extended in our optical images is higher than that of stellar objects. The ratio between the numbers of extended and stellar objects in our CCD catalogue is $\sim$ 2 at $m_B \sim$ 22.25 and $\sim$ 5 at $m_B \sim$ 23.25. Therefore, the large number of extended objects makes more difficult to deal with the problem of random coincidences and to find convincing optical identifications. Each of these sixteen error boxes is now discussed in some detail, in order of decreasing X-ray flux.
[**X043–12**]{}: the 95% error box contains only one object, classified as extended in the blue band and as point-like in the V and R bands. Its noisy spectrum shows two narrow lines well coincident with Ly$\alpha$ and CIV at z = 2.80. It appears to be a high redshift analogue of the low–z type–2 AGNs, similar to the QSO 2 at z = 2.35 found by Almaini et al. (1995) in the optical follow–up of other deep ROSAT fields. Although a higher quality spectrum would be needed to confirm the nature of this object, given its relatively large LR values, we consider it as the correct identification.
[**X039–19**]{} and [**X049–21**]{}: these are the brightest X-ray sources in our sample without any suggested spectroscopic identification. The 95% error box of X039–19 does not contain any object in our CCD data, while a faint ($m_B \sim 24$), extended object is present near the center of the error box of X049–21. Its relatively large LR values suggest that it might be the correct optical identification. However, both the spectrum of this object (shown in figure 4) and the spectrum of a similarly faint object just outside the error box of X039–19 (object A in figure 4) have extremely low S/N and do not allow any spectroscopic classification. Moreover, both X-ray sources have a close–by source at less than one arcmin (see figure 1), so that it is possible that their X-ray positions are not as well determined as those of the other sources with similar X-ray flux.
[**X211–22**]{}: we took spectra for three objects, all of them just outside the 95% radius. The brightest one is an early type galaxy at z = 0.180, while the other two, separated by about 2 arcsec from each other, are a Sey 1 galaxy (classified as point–like in the B band and as extended in the R band) at z = 0.281 and a B star. We consider the Sey 1 galaxy as the most likely identification.
[**X404–23**]{}: its large error box makes the identification difficult. The 3$\sigma$ error box contains 18 sources and all of them have low LR values. We took spectra for three objects, finding a NELG galaxy (object G) and a Sey 2 galaxy (object N) at different redshifts, while the third one (object J) is a faint ($m_B = 23.74$) M star. None of them is a convincing identification. Visual inspection of the X-ray image suggests the presence of two different sources, at about 40from each other, which our detection algorithm has not been able to separate. This is also supported by the large positional error resulting from the maximum likelihood fit. Since the two sources appear to have approximately the same number of counts, it is likely that both of them are very close to or just below our X-ray detection threshold.
=24.5cm
=24.5cm
=24.5cm
=24.5cm
=24.5cm
=24.5cm
[**X408–30**]{}: the radio source coinciding with X408-30 and indicated with a cross in figure 4 is not associated to any optical object (see Gruppioni, Mignoli and Zamorani 1999). If the radio and X-ray sources are really associated, then the optical counterpart of the X-ray source is fainter than our optical limit. The brightest object in the field (object C), but well outside the 95% error box, is a NELG at z = 0.814.
[**X207–31**]{}: all the objects in the error box are extended. We took spectra for two galaxies, one ($m_R$ = 22.43) just outside the 1$\sigma$ error box and the second one ($m_R$ = 19.65) just outside the 3$\sigma$ error box. Both of them have z $\sim$ 0.58; the second galaxy, which is the brightest and reddest of all the surrounding galaxies appears to be a cD in a cluster. (Figure 4 shows the spectrum of this galaxy). Just below our maximum likelihood threshold for X-ray detection, at about two arcmin from X207–31, there is an other X-ray source in a position where an overdensity of faint galaxies is clearly seen. Three of these galaxies have the same redshift as those in X207–31. We therefore identify X207–31 with a cluster, probably interacting with a second cluster at a distance of about 1.5 Mpc.
[**X250–34**]{}: the error box contains only faint ($m_B \ge$ 23.2, $m_R \ge$ 22.9) extended objects, none of which with a large LR. No spectrum was taken for any object.
[**X011–35**]{}: the density of objects in the error box is twice as high as the average density in our catalog. We took spectra for the six brightest objects ($m_R <$ 21.5) finding three galaxies at z $\sim$ 0.39 (objects D, L and O in figure 4), one absorption line galaxy at z = 0.189 (object A; this is the object closest to the X-ray position), one early–type galaxy at z = 0.586 (object F). The redshift of this galaxy is about the same as that of the clusters discussed in connection with X207–31. The angular distance between X011–35 and X207–31 ($\sim$5 arcmin) corresponds to about 4 Mpc, suggesting the existence of a large scale structure at this redshift. The sixth spectrum (object I) has a very low S/N and no redshift was derived. The spectrum in figure 4 is the spectrum of the galaxy closest to the X–position among those at z $\sim$ 0.39 (D). On the basis of these results, although with some possible ambiguity, we identify the X-ray source with a group of galaxies.
[**X234–41**]{}: also in this error box the density of faint extended objects is twice as high as the average density. In particular, in the inner 10there are six extended objects with $ 23 \le m_B \le 24 $, while about one would be expected. Although no spectrum was taken, because of the faintness of the objects, we tentatively identify this source with a group or cluster of galaxies. The bright object just outside the 3$\sigma$ error box is a G star with a too large X-ray to optical ratio to be associated with the X-ray source.
[**X233–45**]{}: we took spectra of the two brightest objects ($m_B \sim$ 22.5) within 15from the X-ray position. Both of them are classified as extended. One spectrum is very blue, with low S/N and no redshift was determined; the second object is an AGN at z = 1.180, which we consider to be the identification. No spectrum is available for the more distant and brighter extended object, whose colours suggest a low redshift elliptical galaxy.
[**X109–46**]{}: the error box contains three relatively bright objects ($m_R <$ 21.5), two of which are point–like and one extended. The two stellar objects are two M stars (objects E and F in figure 4), while the extended one is a starburst galaxy at z = 0.269 (object C). The two M stars appear to be too faint for being the counterpart of the X-ray source. The starburst galaxy might be considered a likely candidate. We note, however, that very close to the X-ray position, there are two faint, blue objects with LR values larger than that of the starburst galaxy. One of these two objects, classified as point–like, is a good candidate for being an AGN. Waiting for spectra for these objects, we consider this source not identified yet.
[**X213–47**]{}: no object appears in our optical catalogs within the 95% error box, although a very faint object ($m_R \ge$ 24.0) is barely detected in two different exposures of the R CCDs at $\sim$ 2 arcsec from the X-ray position.
[**X022–48**]{}: the noisy spectrum of the galaxy close to the center of the X-ray error box shows a narrow emission line which, if interpreted as \[OII\], corresponds to z = 0.32. Both the blue and the red CCDs show clear signs of interaction with a fainter, blue extended object located about 3south. Also on the basis of the LR values, we consider this complex of interacting galaxies as a likely identification. The two galaxies just outside the 2$\sigma$ error box have different redshifts, while the brighter, more distant point–like object is an F/G star.
[**X215–49**]{}: no obvious candidate is contained within the 95% error box. The spectrum of the galaxy just outside of it (object C) shows a strong \[OII\] emission line and MgII2800 absorption line at z = 1.053. Its LR is small and we do not consider this object as the correct identification.
[**X264–50**]{}: there is one object in the error box with LR $\sim$ 1. It is a relatively bright ($m_R$ = 20.77) starburst galaxy at z = 0.568. Despite the presence of a much fainter ($m_R \sim $ 23) blue, point-like source which may be an AGN candidate, we consider the galaxy as a possible identification.
Discussion
============
Summary of identifications and not identified sources
-----------------------------------------------------
On the basis of the criteria discussed above, we have found reliable spectroscopic identifications for 41 sources (82% of the total), indicated in bold face in the last column of Table 3. For one more source (X234–41) the identification is very likely to be with a faint cluster of galaxy, although no redshift is available. These 42 reliable identifications are 33 AGNs (including the two radio galaxies and the BL Lac candidate; 79% of the identified sources), 2 galaxies, 3 groups or clusters of galaxies and 4 stars. Except for the higher fraction of unidentified sources (see discussion below), the identification content of this sample is in excellent agreement with what has been found, at a similar flux limit, in the much deeper PSPC and HRI surveys in the Lockman field (Hasinger et al. 1998, Schmidt et al. 1998).
Figure 5 shows the expected and observed cumulative distributions of the distances normalized to the error on each coordinate between the optical counterparts and the X-ray sources. In the observed distribution we have excluded the three sources identified with groups or clusters of galaxies, because for these objects it is more difficult to unambiguously define the optical position. The excellent agreement between the two distributions shows the goodness of our derived ML estimates for the positional uncertainties, at least when the X-ray position is not affected by confusion problems (see below).
If we divide our sample into two equally populated sub–samples as a function of flux (S$> 6.5 \times 10^{-15}$ and S$< 6.5 \times 10^{-15}$ ), we find that the percentage of identifications remains approximately constant (88% and 80% in the high and low flux sub–samples, respectively). AGNs are the dominant class of objects in both sub–samples (90% of the optical identifications in the high flux sub–sample and 65% in the low flux sub–sample), while the few identifications with clusters and galaxies are all in the low flux sub–sample. With the two extreme assumptions that none or all of the still unidentified objects are AGNs we derive that the percentage of AGNs among [**all**]{} the X-ray sources in our sample is comprised between 66% and 82%.
Figure 6 shows the X-ray flux versus $m_R$ magnitudes for the sources in our sample and for the AGNs identified in the Lockman Field (Schmidt et al. 1998). The two bright F stars ($m_B \sim$ 10) are not shown here. The two straight dashed lines, corresponding to constant X-ray to optical ratios, show the approximate locus of this plane which contains most of the identified AGNs, both in our sample and in the Lockman Field. This range in X-ray to optical fluxes correspond to -0.6 $< log~f_x/f_v < $ 0.6, where $ log~f_x/f_v $ is defined as in Maccacaro et al. (1988). For any given X-ray flux the range in magnitude is about $\pm$ 1.5, corresponding to about $\pm$ a factor of 4 with respect to the average X-ray to optical ratio. Despite the much higher X-ray flux limit and, correspondingly, much brighter limiting magnitudes, also most ($\sim$ 90%) of the EMSS AGNs lie in the same band. For the three sources identified with groups or clusters of galaxies we have plotted the magnitude of the brightest galaxies. The two X-ray sources identified with galaxies have an X-ray to optical ratio close to the lower limit of those of AGNs.
The limits for the eight unidentified sources are plotted at the magnitude of the brightest object within the 95% error box which is not excluded from being the optical counterpart on the basis of the available spectra. Three of these upper limits lie outside the band shown in the figure and three more are very close to the upper bound of this band. If associated with AGNs, they would have an X-ray to optical ratio significantly higher than the average. As shown in the figure, 6 such AGNs have been identified in the Lockman field; four of them are classical broad line objects, while the other two, showing only narrow lines, have been classified as AGNs on the basis of the presence of \[NeV\]$\lambda$3426 emission (Schmidt et al. 1998). The HRI arcsec positions in the Lockman field has made these identifications with such faint optical objects possible. In our case, with the 95% error radius for these sources ranging from 8to 24, similar identifications are significantly more difficult. Alternatively, we can not exclude that some of these sources may have a “wrong” position because of confusion. In Section 2 we have estimated that in about 2–5 cases a single source in our list may be produced by two different close–by sources, so that its position may be significantly wrong and we may not be able to find any reliable optical counterpart within the error box. The much more detailed simulations performed by Hasinger et al. (1998) show that in a PSPC survey like ours up to almost 20% of the detected sources with $S_x < 1 \times 10^{-14}$ (corresponding to about 7 sources) may appear at a detected position more than 15away from the true position because of confusion. From these considerations it follows that only higher resolution, deep X-ray data (e.g. with AXAF and XMM) can fully clarify the situation for these sources.
Some additional clues on the nature of the unidentified sources can be obtained from the analysis of the hardness ratio. Figure 7 shows the hardness ratio versus X-ray flux for the sources in the Marano field (panel a) and in the Lockman field (panel b).
Analysis of this figure shows the following results:
i\. Most of the sources in both fields occupy well defined bands in HR. On average, the observed hardness ratio values in the Lockman field are smaller than those in the Marano field. The reason for this shift in HR is essentially due to the lower $N_H$ value in the Lockman field, so that the observed spectra appear softer. When converted to energy spectral indices, under the assumption of a single power law with the galactic $N_H$, the HR band in the Marano field corresponds to the range 0.5 $\le \alpha_x \le $ 2.0. Approximately the same range in $\alpha$ is derived for the HR band in the Lockman field assuming $N_H \sim 1 \times
10^{20}$, i.e. approximately twice the value of atomic hydrogen derived from measurements in the 21 cm line (Lockman et al. 1986). A similar $N_H$ value was derived by Hasinger et al. (1993) when fitting the average spectrum of all the X-ray sources in the Lockman field. This excess of X-ray absorption is consistent with the column density of ionized gas in the galactic disk derived from analyses of the dispersion measures of pulsars (Reynolds 1989). The presence of this additional column of ionised gas should be taken into account when analyzing ROSAT data, particularly in directions of low $N_H$ as, for example, in the Lockman field.
ii\. In both samples there are about (15–20)% of the sources which appear to have hard or absorbed spectra, with HR values close to 1. These sources appear to be reasonably well separated from the others, especially in the Lockman field where, because of the higher S/N ratio, the errors in HR are smaller. The fraction of sources without optical identification is significantly higher among these hard sources than among sources with “normal” X-ray spectrum. For example, in the Lockman field the unidentified sources are 3 out of 8 for sources with HR $>$ 0.6, to be compared with zero out of 42 for sources with HR $<$ 0.6; in the Marano field the corresponding fractions of unidentified sources in the same HR ranges are 6/12 and 2/38. The eleven identifications (Lockman and Marano fields together) among these hard sources comprise 2 galaxies (X264–50 and X022–48), and 9 AGNs (4 broad line AGNs, 2 radio galaxies, 1 possible BL Lac, and 2 narrow line objects classified as AGNs on the basis of the presence of \[NeV\]$\lambda$3426). This shows that AGNs are the dominant population also in this range of HR. However, contrary to what happens for objects with “normal” X-ray spectra, a relatively large fraction of them does not show the prominent broad lines typical of classical quasars. The absence of broad lines coupled with the presence of X-ray absorption are consistent with the standard unified models for AGNs. It is interesting to note that the two narrow line objects with \[NeV\]$\lambda$3426 have two of the highest X-ray to optical ratios in the Lockman field sample. Since the current optical limits for the unidentified sources in the Marano field correspond to X-ray to optical ratios similar to those of these sources, it is possible that at least some of our hard, unidentified sources belong to the same class of objects.
iii\. In both samples, but especially in the Marano field, the hard sources tend to be more numerous at low flux levels. While this is likely to be, at least in part, a real effect, as shown by the identification content of these sources discussed above, there is also an obvious selection effect which favours hard spectra near the detection limit. At these fluxes, because of the combined effect of confusion and statistical fluctuations on small number of counts, a not–negligible fraction of the detected sources has measured fluxes significantly higher than the true fluxes (see Hasinger et al. 1998). Since the sample is defined in the ROSAT hard band, the measured hardness ratio for some of these sources would be biased toward large values and therefore the observed fraction of hard sources is probably higher than the real one.
Optically and X-ray selected AGNs
---------------------------------
As mentioned in Section 1, in the same area covered by the X-ray survey we have conducted in the past years a search for optically selected AGNs, using multi-colour data from plates taken at the ESO 3.6m telescope. (Zitelli et al. 1992). This survey has later been extended to fainter magnitudes using CCD data (Mignoli et al. in preparation) and has produced, so far, spectroscopic data for 29 optically selected broad–line AGNs inside the ROSAT area. Ten of these have not been detected in the X-ray data and their corresponding X-ray upper limits are shown in Figure 6. Three of these AGNs have X-ray upper limits outside the band shown in Figure 6, in the area of very low X-ray to optical ratio.
On the basis of the available CCD and spectroscopic data, which do not cover the entire ROSAT area of about 0.2 sq.deg., Mignoli and Zamorani (1998) estimate surface densities of $\sim$ 185 and 140 AGNs per sq.deg. with $m_B < 22.5$ and $22.5 < m_B < 23.5$, respectively. These estimates are in good agreement with the predictions obtained by Zamorani (1995) on the basis of reasonable extrapolations from counts at slightly brighter magnitudes. Limiting ourselves at the classical broad–line AGNs and merging together the X-ray and the optically selected samples, we have a total of 35 such objects with $m_B \le$ 22.6 corresponding to a surface density of 178 $\pm$ 30 broad–line AGNs per sq.deg. This is the highest reported surface density for these objects so far at this magnitude. Taking into account that, as mentioned above, the optically selected sample is not complete over the entire ROSAT area, this density is significantly higher than the estimate of about 125 AGNs per sq.deg. with $m_B \le 22.6$ recently obtained in the Deep Multicolor Survey, on the basis of 53 spectroscopically confirmed AGNs (Kennefick et al. 1997). Our estimated surface densities for $m_B < 22.5$ and $22.5 < m_B < 23.5$, correspond to $36 \pm 6$ and $27 \pm 5$ AGNs inside the ROSAT area. Since in the same magnitude ranges ROSAT has detected 23 and 6 broad–line AGNs respectively, we conclude that the “efficiency” of AGN selection with X-ray exposures reaching about $4 \times 10^{-15}$ is $\sim$ 65% and $\sim$ 20% in the two magnitude ranges.
Having colours for all the X-ray selected AGNs, we can also estimate how many of them would have been missed by a purely optical selection. Figure 8 shows, with different symbols (see figure caption) all known AGNs in our field. The small dots represent all the point–like objects in our CCD data with $m_B \le$ 22.5. The dotted curve shows the approximate locus occupied by stars in this plane. This figure shows that two X-ray selected AGNs (X251–37 at z = 2.71 and X042–11 at z = 1.062) are inside the star locus and therefore would have not been easily selected as AGN candidates on the basis of this diagram. Moreover, two X-ray selected AGNs (X233–45 at z = 1.180, and X409–28, the broad line radio galaxy at z = 0.958) are classified as extended by our morphological algorithm and two more (X043–12, the AGN2 at z = 2.80, and X211–22 at z = 0.281) have uncertain classification (i.e. p/e in Table 3). Since the colour-colour area occupied by faint AGNs contains a number of extended objects which is about ten times higher than that of the point–like ones, also these four objects would have not been easily selected as AGN candidates by purely optical data. Moreover, the location of these objects in the magnitude – redshift plane is not the same as that of AGNs which are more easily selected optically; five have $m_B \ge$ 22.2 and only one has $m_B <$ 22.2, while the corresponding numbers for all the other AGNs with redshift in this field are 11 and 25. The same difference is also visible as a function of the absolute magnitude: five of the objects which would have been more difficult to detect from the optical data have $M_B \ge$ -22.4 and only one has $M_B <$ -22.4, while the corresponding numbers for all the other AGNs are 14 and 22. This clearly shows that not only colour–colour selection of AGN candidates among point–like objects can be significantly incomplete at faint apparent magnitudes, but also the incompleteness increases at faint absolute magnitudes. As a consequence, if these not so stellar AGNs were really missed in faint optical surveys, this would introduce a bias in the derived optical luminosity function, which would appear flatter than the real one. Finally, we note also that a not negligible number of X-ray selected AGNs are significantly redder in B-V than the bulk of AGNs (see figure 8). Also this has to be taken into account in devising the most efficient optical selection criteria for faint AGN candidates.
Conclusion
===========
We have presented the X-ray data and the optical identifications for a deep ROSAT PSPC observation in the Marano field. Careful statistical analysis of multi–colour CCD data in the error boxes of the 50 X-ray sources detected in the inner region of the ROSAT field (15 radius) has led to the identification of 42 sources, corresponding to 84% of the X-ray sample. These 42 reliable identifications are 33 AGNs (including two radio galaxies and one BL Lac candidate; 79% of the identified sources), 2 galaxies, 3 groups or clusters of galaxies and 4 stars. Except for the higher fraction of unidentified sources, the identification content of this sample is in excellent agreement with what has been found, at a similar flux limit, in the much deeper PSPC and HRI surveys in the Lockman field (Hasinger et al. 1998, Schmidt et al. 1998).
With simple simulations we have shown that in a few cases the reason for not having found an optical identification can be due to the fact that a single X-ray source in our list may be produced by two different close–by sources, so that its detected position may be significantly wrong. Most of the unidentified sources have a large ratio of X-ray to optical fluxes and harder than average X-ray spectra. Since most of the identified objects with these characteristics in our field and in the Lockman field are AGNs, we conclude that most of the sources with good position determination but without identification are likely to be AGNs.
Most of the sources in the Marano and Lockman fields occupy well defined bands in the plane hardness ratio versus X-ray flux. These bands correspond to the same energy spectral index range 0.5 $\le \alpha_x \le $ 2.0, only if the effective X-ray absorbing column in the Lockman field is about twice the value of atomic hydrogen derived from measurements in the 21 cm line. In both samples there are about (15–20)% of the sources which appear to have hard or absorbed spectra, with HR values close to 1 and, especially in the Marano field, they tend to be more numerous at low flux levels. However, since there is an obvious selection effect which favours hard spectra near the detection limit, it is difficult to quantify the reality of this effect. The fraction of sources without optical identification is significantly higher among these hard sources than among sources with “normal” X-ray spectrum. The eleven identifications (Lockman and Marano fields together) among these hard sources (9 AGNs and 2 galaxies) show that AGNs are the dominant population also in this range of HR. However, contrary to what happens for objects with “normal” X-ray spectrum, a relatively large fraction of them does not show the prominent broad lines typical of classical quasars.
Finally, comparing the optically and X-ray selected samples of AGNs in the same area, we estimate that the “efficiency” of AGN selection with X-ray exposures reaching about $4 \times 10^{-15}$ is $\sim$ 65% and $\sim$ 20% in the magnitude ranges $m_B < 22.5$ and $22.5 < m_B < 23.5$, respectively. On the other hand, a not negligible fraction of the X-ray selected AGNs would have not been easily selected as AGN candidates on the basis of purely optical criteria, either because of colours similar to those of normal stars or because of morphological classification not consistent with point–like sources.
Moreover, the location of these objects in the magnitude – redshift plane is not the same as that of AGNs which are more easily selected optically. They tend to be fainter in terms of both apparent and absolute magnitudes. As a consequence, if these not so stellar AGNs were really missed in faint optical surveys, this would introduce a bias in the derived optical luminosity function, which would appear flatter than the real one. Finally, we note also that a not negligible number of X-ray selected AGNs are significantly redder in B-V than the bulk of AGNS. Also this has to be taken into account in devising the most efficient optical selection criteria for faint AGN candidates.
The ROSAT project is supported by the Bunderministerium für Forschung und Technologie (BMFT), by the Science and Engineering Research Council (SERC) and by the National Aeronautics and Space Administration (NASA). This work was supported in part by NASA grants NAG5-1531 (M.S.), NAG8-794, NAG5-1649 and NAGW-2508 (R.B. and R.G). G.H. acknowledges the DARA grant FKZ 50 OR 9403 5; G.Z. acknowledges partial support by the Italian Space Agency (ASI) under ASI contract ARS-96-70 and by the Italian Ministry for University and Research (MURST) under grant Cofin98-02-32.
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[^1]: based on observations collected at the European Southern Observatory, La Silla, Chile.
|
---
author:
- |
Victor Fragoso$^{2,3}$ Chunhui Liu$^1$ Aayush Bansal$^1$ Deva Ramanan$^1$\
$^{1}$ Carnegie Mellon University $^2$ Microsoft C + AI $^3$ Western Virginia University\
bibliography:
- 'compnn.bib'
title: 'Patch Correspondences for Interpreting Pixel-level CNNs'
---
|
---
abstract: '*Objective*. Consistently changing physiological properties in developing children’s brains challenges new data heavy technologies, like brain-computer interfaces (BCI). Advancing signal processing methods in such technologies to be more sensitive to developmental changes could help improve their function and usability in paediatric populations. Taking advantage of the multi-dimensional structure of EEG data through tensor analysis offers a framework to extract relevant developmental features present in paediatric resting-state EEG datasets. *Methods*. Three paediatric datasets from varying developmental states and populations were analyzed using a developed two-step constrained Parallel Factor (PARAFAC) tensor decomposition. The datasets included the Muir Maxwell Epilepsy Centre, Children’s Hospital Boston-MIT and the Child Mind Institute, outlining two impaired and one healthy population, respectively. Within dataset cross-validation used support vector machines (SVM) for classification of out-of-fold data predicting subject age as a proxy measure of development. t-distributed Stochastic Neighbour Embedding (t-SNE) maps complemented classification analysis through visualization of the high-dimensional feature structures. *Main Results* Development-sensitive features were successfully identified for the developmental conditions of each dataset. SVM classification accuracy and misclassification costs were improved significantly for both healthy and impaired paediatric populations. t-SNE maps revealed suitable tensor factorization was key in extracting developmental features. *Significance* The described methods are a promising tool for incorporating the unique developmental features present throughout childhood EEG into new technologies like BCI and its applications.'
address:
- '$^1$School of Engineering, Institute for Digital Communications, The University of Edinburgh, Edinburgh EH9 3FB, United Kingdom'
- '$^2$The Muir Maxwell Epilepsy Centre, The University of Edinburgh, Edinburgh EH8 9XD, United Kingdom'
- '$^3$Royal Hospital for Sick Children, Edinburgh EH9 1LF, United Kingdom'
- '$^*$Corresponding Author'
author:
- 'E. Kinney-Lang$^{1,2,*}$, L. Spyrou$^1$, A. Ebied$^1$, R. FM Chin $^{2,3}$, and J. Escudero$^{1,2}$'
bibliography:
- 'mylibrary.bib'
title: 'Tensor-driven extraction of developmental features from varying paediatric EEG datasets'
---
[ ]{}
Introduction {#introduction .unnumbered}
============
Diseases and injuries sustained in childhood are a major public health issue worldwide. The resultant acute and/or chronic motor disability affect millions, with early-life sustained motor insult potentially leading to learned non-use of afflicted regions and potential complications later in life [@Manning2015a; @Johnston2004b]. While traditional therapeutic options for improving motor disability often involve exercise-based techniques [@Fluck2006a; @Damiano2006] like constraint-induced movement therapy (CIM) [@Manning2015a; @Taub2002; @Damiano2010], these exercise-based techniques have a trade-off in requiring residual movement and control in the patient’s afflicted appendage [@Manning2015a; @Millan2010b]. Brain-computer interfaces (BCI) and other emerging technologies are strong candidates for non-muscular neurorehabilitation options in clinical settings [@Ang2014; @Mrachacz-Kersting2015; @Wolpaw2007a; @VanDokkum2015a], with promising early results in adults [@Young2014; @Soekadar2015].
Accounting for varying electrophysiological properties in developing children, however, poses a hurdle for many of these data driven technologies, including BCI. Blending together engineering and medicine, BCIs provide direct communication channels between the brain and an output device, i.e. computer [@Wolpaw2002], through advanced signal processing. Many popular BCI applications measure and decode electric potentials created in the brain, such as specific thought patterns used to invoke motor imagery (MI) [@Nicolas-Alonso2012; @Ahn2014; @Vuckovic2014]. Hardware like electroencephalography (EEG) record the electric potentials over the scalp [@Wolpaw2002; @Nicolas-Alonso2012], with user intent then determined through signal analysis, feature identification, extraction and classification. These BCI signal processing chains often target relatively static electrophysiological signal features common in adult EEG recordings, relying on their a priori predictability in spatial/temporal/spectral feature selection and determining data outliers. However, assuming the adult features may not be appropriate for analyzing populations with more variable signal features, like children [@Kinney-Lang2016].
Signal properties and profiles of the brain are continually changing from birth through adulthood [@Kinney-Lang2016; @Marshall2002; @Gasser1988]. For example, the location and frequency of the well established EEG alpha rhythm in adults is thought to migrate throughout childhood from approximately 6-9 Hz until reaching 8-13 in adulthood [@Gasser1988; @Matsuura1985; @Miskovic2015]. EEG signal recordings from young children are further confounded by high background noise alongside the shifting EEG signal bands [@Marshall2002; @Matsuura1985; @Miskovic2015], resulting in obfuscated and weaker signals of interest. Therefore a means to identify and extract EEG features sensitive to the changing developmental profiles of children would be a critical tool in constructing paediatric BCI rehabilitation paradigms.
Tensor (or multi-way) analysis [@Cichocki2015; @Acar2009] provides a potential framework to capture the dynamic developmental profiles in paediatric EEG, through investigating the relationships present in the multi-dimensional EEG data [@Phan2010; @Cichocki2008]. Tensor analysis is a higher-order (i.e. multi-dimensional/multi-way) extension of standard matrix analysis techniques, which retains informative structural relationships between dimensions (domains or ways) in the data [@Cichocki2008]. Tensor analysis has already been adapted to adult BCI paradigms [@Cichocki2008; @Cichocki2015; @Liu2014], and thus offers a structure to build tools towards effective paediatric BCI paradigms.
This paper is an extension to our previous conference submission [@Kinney-Lang2017] which provided a proof-of-concept for characterizing developmental feature profiles of children using tensor analysis on an EEG dataset. Here an extended, robust feature selection paradigm is presented, introducing tensor component selection and model validation. Furthermore, the improved paradigm is demonstrated on several resting-state paediatric EEG datasets which span 1.) a rapidly developing preschool population with potential developmental impairments; 2.) a population spanning childhood to adulthood with potential developmental impairments; 3.) a healthy population during a stable developmental period of childhood. Successful characterization of key age-specific features within each dataset supports this approach as a potential adaptive tool for development-sensitive feature selection in paediatric EEG for applications like BCI.
Materials and Methods {#materials-and-methods .unnumbered}
=====================
Datasets {#datasets .unnumbered}
--------
### Muir Maxwell Epilepsy Centre {#muir-maxwell-epilepsy-centre .unnumbered}
A retrospective analysis of an epileptic preschool cohort ($<5$ years) from the Muir Maxwell Epilepsy Centre was included in this study, henceforth referred to as the MMEC dataset. The original cohort was prospectively recruited from National Health Service (NHS) hospitals in Fife and Lothian as part of the NEUROPROFILES study [@M.B.2015]. A 32-channel, unipolar montage captured routine EEG in the standard 10-20 system for each child. Of 64 children available, 14 were excluded from this study due to corrupted EEG data, inconsistent or incompatible EEG acquisition parameters and irregular recordings, resulting in a dataset of routine EEG from $n=50$ preschool children. If multiple resting-state EEG recordings existed, only the first recording was selected for each child to avoid weighting results toward children with more recordings and to select from the same awake resting-state data across all children.
### Children’s Hospital Boston-MIT {#childrens-hospital-boston-mit .unnumbered}
Publicly available data from a study at the Children’s Hospital Boston-MIT of epileptic patients from infancy to early adulthood [@Shoeb2009a] was used in this study, downloaded through Physiobank.org [@Goldberger2000] and henceforth referred to as the CHB-MIT dataset. A 28-channel, bipolar montage captured EEG recordings continuously over two days of monitoring. Of the 23 patients available, 6 subjects were determined to have inconsistent EEG acquisition parameters for this study, due to discrepancies in the montages and unsuitable recordings. This resulted in $n=17$ subjects (age 2-19 y.o.) for analysis. The 48-hour continuous recordings were separated into 4-hour time blocks, with an equal number of trials at each time selected for processing. Results were averaged across all time bins to render a holistic representation of the resting-state data for each subject.
### Child Mind Institute {#child-mind-institute .unnumbered}
Resting-state EEG data for healthy control participants was taken from the open science resource provided by [@Langer2017] and the Child Mind Institute. Data captured from high-density 129-channel resting-state paradigms of pre-adolescent subjects (one age 6, the rest age 8-11 y.o.) was used. Of 45 subjects available, one subject was excluded (age 11) due to abnormalities in EEG processing, resulting in $n=44$ subjects for analysis. The single 6-year-old in the dataset was grouped with the 8-year-old class to allow for cross-validated classification.
A summary of subject distribution per age for each dataset is included in Table \[DatasetAgedist\] of the supplementary data.
Pre-processing {#pre-processing .unnumbered}
--------------
Raw EEG data was processed using the Fieldtrip toolbox [@Oostenveld2011] in Matlab 2015a. A two-pass (zero-phase forward and reverse) bandpass filter between \[0.5-31\] Hz was applied to EEG time-series signals. The filter was detrended, and signals were separated into 10-second (5-second) long trials for the MMEC/CHB-MIT (CMI) datasets. EEG channels were re-referenced to a common average reference and auxiliary/reference specific channels were removed. The EEG channel montages in the MMEC and CHB-MIT were matched through adapting the bipolar EEG electrode information from the CHB-MIT to the unipolar MMEC setup. The high-density set-up of the CMI data was not adapted to this same convention to avoid potential information loss. Channels in the CMI data with NAN values were removed. Trials with any seizure activity were immediately excluded from processing. A multi-pass artifact rejection system removed muscle, jump and ocular artifacts automatically, followed by manual inspection of data to verify and remove any remaining artifacts. Automatic rejection was based on recommended thresholding values given in Fieldtrip.
Tensor Construction {#tensor-construction .unnumbered}
-------------------
Three-way tensors consisting of $[Spatial] \times [Spectral] \times [Subject]$ dimensions were created using the EEG channel, power spectra and subject age data for each dataset, resulting in $(19) \times (301) \times (50)$, $(19) \times (301) \times (17)$ and $(105) \times (61) \times (44)$ elements for the MMEC, CHB-MIT and CMI datasets respectively. Figure \[Figure1TensorConstruct\](a) provides a general illustration of the tensor construction.
Time-frequency analysis of clean trials using Fieldtrip’s multitaper method with a 0.5s Hanning window provided power spectra for each subject in the MMEC and CHB-MIT at 0.1 Hz resolution, and 0.5 Hz resolution for the CMI with data normalized. The power spectra in each subject was averaged across all trials providing a general spectral profile of the resting-state EEG for the $[Spectral]$ domain.
The $[Subject]$ domain in each dataset was specifically ordered so all subjects were strictly increasing in age from youngest to oldest. The structure of this domain is critical as a developmental proxy, through which features can be identified as age-specific or pan-developmental as indicated by their general association to age in years.
Tensor Factorization: Parallel Factor Analysis {#tensor-factorization-parallel-factor-analysis .unnumbered}
----------------------------------------------
Tensor factorization (decomposition) is a multiway extension of standard matrix analysis techniques (e.g. principal/independent component analysis) which describes a model of the structural relationship between tensor modes [@Cichocki2015; @Phan2010; @Kolda2008]. A common tensor factorization model for EEG data is Parallel Factor Analysis (PARAFAC, also known as Canonical Polyadic Decomposition CANDECOMP) [@Harshman1970; @Klaas2003; @Cong2015], which decomposes a tensor **** into a linear combination of rank-1 tensors coupled with a super-diagonal core [@Harshman1970; @Klaas2003]. EEG data readily supports a PARAFAC model due to its inherent higher-order structure, e.g. relationship between the time-series, channels and power spectra [@Cong2015]. Equation (\[parafac\_eq\]) illustrates the general *R*-component PARAFAC model of a 3-dimensional array $\textbf{\underline{X}}(I \times J \times K)$: $$x_{ijk} = \sum_{r=1}^{R} a_{ir}b_{jr}c_{kr} + e_{ijk}
\label{parafac_eq}$$
with $x_{ijk}, a_{ir}, b_{jr}, c_{kr}$ $(i=1...I; j=1...J,k=1...K)$ and $e_{ijk}$ as elements of $\textbf{\underline{X}}$, domains $\textbf{A}(I \times R)$, $\textbf{B}(J \times R)$, $\textbf{C}(K \times R)$ and residual $\textbf{\underline{E}}(I \times J \times K)$, respectively. Figure \[Figure1TensorConstruct\] B) illustrates tensor decomposition for varying components $R$ for a 3-dimensional tensor.
Tensor datasets were factored using an adapted PARAFAC function from the NWAY-toolbox for Matlab [@Andersson2000]. Several domain constraints were used in analysis to improve interpretation of results, and account for domain-specific properties like strictly non-negative components in the power spectra. Non-negativity constraints were applied to the $[Spatial]$ and $[Spectral]$ domains, with unimodality applied to the $[Subject]$ domain. Unimodality was imposed in order to extract components that are bound to specific age groups within a dataset. The $[Subject]$ domain structure permitted unimodal constraints since no subjects had repeated ages (data analyzed at ‘months-old’, grouped into ‘years-old’ for classification).
The PARAFAC model decomposition guarantees a 1:1 interaction between extracted factors across domains due to its super-diagonal core [@Cichocki2008; @Klaas2003]. Through imposing strict zero-values in the core tensor on all but the diagonal components (i.e. making the core super-diagonal), any given component in a domain can only interact with the corresponding component in other domains. In example, in Figure \[fullpageMMECtensor\] the first factor (*black*) in the $[Spatial]$ domain corresponds directly with only the first factor (*black*) in the $[Subject]$ and $[Spectral]$ domains as a result. Therefore, examining component interactions across domains in PARAFAC provides direct insight into the structural relationships within the data. In the presented work, this amounts to information on the extracted developmental features present throughout childhood.
Exploiting the mild conditions required for uniqueness of the PARAFAC model guarantees that the low-rank factor matrices of the PARAFAC decomposition retain their meaning [@Cichocki2015; @Kolda2008]. In this proposal, retaining decomposition uniqueness allows interpretation of how the developmental $[Subject]$ domain influences $[Spatial]/[Spectral]$ factors. Even if the underlying relationships are not obvious in the original EEG data, they are assured to be viable through the uniqueness condition. Generic uniqueness of PARAFAC holds under the sufficient condition [@Kolda2008]: $$\sum_{n=1}^{N} k_{\textbf{A}^{(n)}} \ge 2R+(N-1)
\label{parafac_uniq}$$ for an *N*-way tensor with *k* elements for each domain matrix $\textbf{A}^{(\textit{n})}$ (i.e. mode-$n$ rank of A) and $R$ factor components [@Kolda2008].
Tensor Factorization: Component Selection {#tensor-factorization-component-selection .unnumbered}
-----------------------------------------
Component selection is a critical step in tensor factorization. Choosing the optimal number of component factors for PARAFAC decomposition balances model suitability with proper representation of latent structural information. The unknown underlying developmental profiles in the data tensors are best captured by an unknown number of components, e.g. some non-minimal rank $R$ decomposition. In example, assume that there is a known number of EEG ‘sources’, $s$ underlying the power spectra of the $[Spectral]$ domain. Each source $s$ could be described by exactly one rank-1 tensor by selecting $r = s$ components for tensor decomposition. However, if there are more sources $s$ than components $r$ (i.e. too few components chosen) then the model may obscure the less obvious, but still important, structural relationships. Similarly, if too many components are chosen, this may lead to over-fitting the model. To help gauge viable component selection of a PARAFAC model, the core consistency diagnostic (corcondia) is used [@Bro2003]. Corcondia provides a direct measure of the suitability of a specific PARAFAC model in an easy to interpret fashion by describing the degree to which the approximate tensor model deviates from its ‘true’ super-diagonal core [@Bro2003]. The validity of the model can then be determined, with models under 40% corcondia considered as generally non-viable [@Bro2003].
Corcondia may also provide insight into the point in which latent structural relationships have been matched uniquely to underlying component ‘sources’. Corcondia has a property of permanently and sharply decreasing after some maximum number of components is chosen [@Bro2003]. This decrease is not necessarily monotonic, but rather assured to never be better than the corcondia given at said maximum number of components. This was considered to be the point in which the maximum number of possible underlying relationships have been taken into account, i.e. there is likely maximum matching between components and underlying sources. Therefore, the number of components immediately prior to the downfall can serve as an inflection point for determining viable versus non-viable models, as done in this study.
A two-step analysis chain over a varying set of components ($r = 1:20$) was constructed for this study to identify PARAFAC models which accounted for the maximum number of pan-developmental structures in the tensor while maintaining acceptable model viability. First, for each number of components $r$, five PARAFAC decompositions were run simultaneously with their corcondia and explained variance recorded. Using multiple simultaneous runs helped account for potential model convergence to local minimums. Any model with corcondia below 70% was considered to be non-viable and was removed. The best fit model was considered to be the maximum rank decomposition model still above the 70% corcondia. Then, a thresholding method based on our proof-of-concept work [@Kinney-Lang2017] was applied to select only the subset of factors which spanned the unimodal $[Subject]$ domain. Reduction of the factors to a development-specific subset helped remove any features reflecting properties of a single child, which could occur due to the unimodal constraint [@Kinney-Lang2017]. The reduced PARAFAC model then had its new corcondia evaluated, and the reduced model with the largest $R$-components maintaining corcondia above 70% in both steps was selected for analysis and training. Figure \[Figure1TensorConstruct\](b) and Figure \[Figure1TensorConstruct\](c) outline the grid search, corcondia evaluation and optimal model selection process. Supplemental Figure \[corcondiaReducedCompare\] shows full and reduced tensor model viability based on corcondia calculations for multiple components $r$ for one dataset.
Classification {#classification .unnumbered}
--------------
A modified direct projection of the extracted training factors onto the testing components was used for prediction validation, stemming from [@Escudero2015]. Direct projection traditionally includes a pseudo-inverse step, which inherently introduces negative testing component values [@Escudero2015]. Given the non-negative constraints on the training tensor decomposition, to retain meaningful prediction the projected test components must also be non-negative. Therefore, the non-negative least square (NNLS) solution of the Khatri-Rao product ($\odot$) between the non-$[Subject]$ dimensions (\[proj\_eq1\]) from the training tensor was introduced as an alternative to the pseudo-inverse step. Using the NNLS maintains the $[Spatial]$ and $[Spectral]$ non-negative domain constraints while still fulfilling the same approximate function as the pseudo-inverse. The NNLS solution was then multiplied by the $[Subject]$-domain matricized test tensor (\[proj\_eq2\]), resulting in a predicted $[Subject \times Factor]$ matrix for validation (\[proj\_eq3\]). Results from (\[proj\_eq1\]) were then multiplied by the $[Subject]$-domain matricized test tensor (\[proj\_eq2\]), resulting in a new predicted $[Subject \times Factor]$ non-negative matrix for validation classification. $$ProjectedFactor_{train} = NNLS([Train_{spectral}]^{T}\odot[Train_{spatial}])
\label{proj_eq1}$$ $$Test_{matrix} = [Test_{subject}]\times[Test_{spatial \cdot spectral}]
\label{proj_eq2}$$ $$Predicted_{test} = ProjectedFactor_{train} \times Test_{matrix}
\label{proj_eq3}$$
To maintain stringent integrity for classification, data was split into training and testing cross-validation folds prior to tensor decomposition. A multi-class, ordinal classification scheme was devised to evaluate the tensor extracted factor’s ability to predict subject age using the Weka toolbox [@Hall2009; @Frank2016]. Subject age (in years) was used for within dataset class labels. An ordinal cost-matrix was used to account for the multi-class, ordinal nature of each data constructed tensor. The ordinal cost-matrix penalized misclassification through linearly weighted differences based on class age, thereby increasing classification penalties for predicting subjects as drastically older/younger compared to their actual age. The cost-matrix was matched for each dataset to the unique subject ages in that dataset.
Using the ordinal cost-matrix, a non-linear radial basis function (RBF) support vector machine (SVM) was trained using the decomposed PARAFAC factors for each cross-validation fold in each dataset. The RBF-SVM was optimized using a grid search in Weka to find $C$ and $\gamma$ which provided the highest classification accuracy. Results were evaluated on their overall classification accuracy and total penalty costs (e.g. the sum of all misclassification penalties based on the ordinal cost-matrix). Random classification and naive classification (e.g. choosing a single class for all subjects) is included for comparison. Results are reported as averages across all training folds with standard deviation and a two-tailed Student’s *t*-test to infer differences from random and naive classification.
The distribution of subjects per age in the MMEC and CMI datasets allowed for 4-fold and 5-fold stratified cross-validation respectively. The CMI classification included the single six-year-old as a member of the ‘Age 8’ class to retain stratified cross-validation. Comparative classification within the CHB-MIT data was not possible due to the limited subjects per age.
Visualization {#visualization .unnumbered}
-------------
To complement classification of the extracted high-dimensional tensor features across the $[Spatial],[Spectral],[Subject]$ domains, results from factorized training folds for the MMEC and CMI datasets are displayed using t-distributed Stochastic Neighbour Embedding (t-SNE) [@VanDerMaaten2008]. Using t-SNE, high-dimensional data can be visualized to capture both the local and global structure of the data through presence of clusters at several scales. Demonstrating t-SNE maps on individual training folds in the data offers a visual companion to the classification analysis, showing the local and global structure underlying a single fold used in analysis.
Simulations {#simulations .unnumbered}
-----------
A simulation of pseudo-EEG data accompanies the real-world datasets. The Berlin Brain Connectivity Benchmark (BBCB) simulation code [@Haufe2016] was modified to include a shifting spectral frequency band of interest, similar to the alpha frequency seen in development [@Marshall2002; @Matsuura1985; @Gasser1988; @Miskovic2015]. The band of interest lower bound was set with mean $\pm1$ standard deviation as the simulated age, up to 8 years-old. Afterwards, each additional simulated year increased the sampling mean thereby gating the lower bound towards 8-Hz. The upper bound was set at 3-Hz $\pm2$ standard deviations higher than the lower bound. The upper bound variation was at least 1-Hz above the lower bound. Ten children were simulated per age using the modified code, from 5 to 11 years-old. Simulated EEG was converted to Fieldtrip for processing, with a simulated tensor constructed in an identical fashion to the real-world datasets.
Results {#results .unnumbered}
=======
The proposed tensor analysis successfully identified latent developmental features across subjects independently for each dataset. A detailed visual breakdown of the PARAFAC model decomposition and its resulting ‘developmental profile’ snapshots is given in Figure \[fullpageMMECtensor\] as an example, using one entire dataset (MMEC). Qualitative developmental feature profiles are illustrated in each tensor domain, where actual weighted values in the testing folds were used for classification purposes. Individual factor contributions are shown in the extended profiles to help clarify the latent developmental relationships in each domain. Component profiles in the $[Subject]$ domain reflect in which ages the extracted factor (feature) is most dominant and influential. Features of the $[Subject]$ domain are ordered to match highest to lowest explained variance from the $[Spatial]$ domain. The normalized topographic map of the $[Spatial]$ domain shows relative regional contributions of EEG channels for each ‘developmental feature profile’ (note this is not a topographic map of EEG activity). The $[Spectral]$ domain is shown up to 15 Hz, as higher frequencies for preschool children in resting-state data has little activity of interest and remains fairly flat.
![Detailed visualization of PARAFAC extracted developmental feature profiles of epileptic children from 0-5 years old from the MMEC dataset. Normalized topographic maps of the $[Spatial]$ domain in column 1 show EEG channel regions with higher/lower relative contribution for each individual feature. The $[Subject]$ domain x-axis is the child’s number (e.g. Child 1, Child 2), organized by increasing age. The combined and separated feature profiles for the $[Spatial] \times [Subject] \times [Spectral]$ domains are shown in columns 2-4. Full resolution figure available upon request.[]{data-label="fullpageMMECtensor"}](figure2){width="\textwidth"}
Preschool children with epilepsy: The Muir Maxwell Epilepsy Centre data {#preschool-children-with-epilepsy-the-muir-maxwell-epilepsy-centre-data .unnumbered}
-----------------------------------------------------------------------
Classification was significantly improved using the identified developmental factor profiles as features in the MMEC dataset. Table \[MMEC\_classif\_table\] contains the average classification results for the MMEC extracted features across all folds. The PARAFAC extracted features improved upon random classification by approximately 50%, and reduced total penalty cost by 9.5 points or 37% compared to naive classification (Student’s *t*-test, $p<0.05$). To put the reduced penalty cost in another context, if every test subject was misclassified by approximately 2 years using naive classification, PARAFAC reduced misclassification to approximately only 1.3 year for each test subject. Naive classification accuracy was improved upon using SVM and the extracted features by approximately 37% as well (Student’s *t*-test, $p<0.01$). Average corcondia across training folds was $85.74\pm4.86$.
Classification Penalty Cost Accuracy (%)
---------------- ----------------------- ----------------------------
SVM 16.0 $\pm4.1$ 30.0 $\pm3.5$
Random – 20.1 $\pm0.7^{\star}$
Naive 25.5 $\pm1.7^{\star}$ 22.0 $\pm3.6^{\star\star}$
: Average classification results across all MMEC cross-validation folds for SVM, Random and Naive classification for epileptic children 0-5 years of age. No penalty costs available for random classification due to its random nature. $^{\star}$ Indicates significant difference from SVM using student’s *t*-test at $p<0.05$. $^{\star\star}$ Indicates significant difference from SVM using student’s *t*-test at $p<0.01$.[]{data-label="MMEC_classif_table"}
While only a third of subjects were on average correctly classified in the MMEC dataset, the significantly reduced cost penalties indicate a move toward reductions in gross age misclassification (e.g. classifying a subject age 0 as age 3, 4, or 5). This is critical, as improved misclassification penalties indicate mistakes trended more towards closely related ages (i.e. $\pm1$ or 2 years) more often. These improvements along with the factor profiles imply success in identifying developmentally important features of preschool children’s EEG.
Child-to-Adult epilepsy spectrum: The CHB-MIT data {#child-to-adult-epilepsy-spectrum-the-chb-mit-data .unnumbered}
--------------------------------------------------
Results of the CHB-MIT dataset were used to demonstrate the scalable nature of the proposed analysis across a broad age range, for subjects under otherwise similar conditions (e.g. epilepsy afflicted subjects with matched EEG montages). Due to the limited number of subjects for each age (only one to two - see Supplemental Table \[DatasetAgedist\]), meaningful classification was not possible for the CHB-MIT dataset. Instead a qualitative illustration of the general developmental profile trends is presented in Figure \[halfpageCHBtensor\]. The extracted factor profiles reflecting dominating influences at specific ages which match expected developmental patterns [@Gasser1988; @Matsuura1985] indicate successful characterization of development sensitive features. These results reinforce the likely generalizable nature of the tensor analysis.
### Profiles {#profiles .unnumbered}
Figure \[halfpageCHBtensor\] demonstrates the PARAFAC model decomposition of the CHB-MIT dataset in a condensed format. Key developmental feature profiles are emphasized across the extracted feature domains, with probable ‘background’ profiles unaccented. The key feature profiles have been organized by the $[Subject]$ domain, with influential features prominent in early childhood to early adulthood ordered from top to bottom. The exact age of each subject is present on the $[Subject]$ domain axis. Sharp peaks in the $[Spectral]$ domain at 16, 19, 28 Hz are potentially residual artifacts from the time-frequency analysis and NAN-averaging across time-bins. Corcondia was 73.49 for the extracted factors, with 85% explained variance.
![A compact visualization of the key feature profiles from the CHB-MIT dataset for epileptic subjects from age 2 to 19. A normalized topographic map reflecting relative contributions of each EEG channel region to the extracted factor is shown in the first column. Combined and separated feature profiles for the $[Spatial] \times [Subject] \times [Spectral]$ domains are shown in columns 2-4. Full resolution figure available upon request.[]{data-label="halfpageCHBtensor"}](figure3){width="\textwidth"}
School-age controls: The Child Mind Institute Data {#school-age-controls-the-child-mind-institute-data .unnumbered}
--------------------------------------------------
Developmental features were successfully identified for the healthy control CMI dataset via PARAFAC. While the available control data could not be age-matched to the MMEC for direct comparison, factor profiles found in the CHB-MIT data (seen in Figure \[halfpageCHBtensor\]) at ages similar to CMI subjects supported analyzing the CMI dataset for feasibility in identifying developmentally sensitive features in normally developed children. Table \[CMI\_classif\_table\] contains the average classification results across the training/testing folds for CMI subjects. PARAFAC extracted features improved significantly upon random classification by approximately 25%, and reduced total penalty cost by 2 points or 19% compared to naive classification. Average corcondia across training folds was $78.25\pm6.57$.
Although classification results had smaller improvements in classifier performance and penalty reduction compared to the epileptic population of the MMEC dataset, significant improvements were still present using the PARAFAC model for SVM classifier training. Extra difficulty in discerning developmental differences in the CMI data was to be expected, given the smaller developmental window and higher homogeneity of subjects in the CMI dataset. Therefore, despite only marginal boons in accuracy and penalty costs, the classification improvements in the CMI dataset under PARAFAC are important. The positive results serve as evidence that the proposed tensor analysis is accessible to ‘normal’ developing children.
The CMI dataset results also further illustrates the scalable nature of the tensor analysis, demonstrating successful feature extraction on subjects more similar developmentally (i.e. healthy, between the ages of 8-11 y.o.) compared to the other datasets. Physiological changes across this age span are significantly less drastic compared to both the MMEC development window from infancy to early childhood (age 0-5 y.o.), and the CHB-MIT development window from infancy to early adult hood (age 2-19 y.o.).
Classification Penalty Cost Accuracy (%)
---------------- ----------------------- ----------------------------
SVM 9.6 $\pm0.9$ 34.2 $\pm1.9$
Random – 27.4 $\pm1.3^{\star\star}$
Naive 11.8 $\pm1.1^{\star}$ 29.8 $\pm3.6$
: Average classification results across all CMI cross-validation folds for SVM, Random and Naive classification between healthy children 8-11 years of age. No penalty costs are available for random classification due to its random nature. $^{\star}$ Indicates significant difference from SVM using student’s *t*-test at $p<0.05$. $^{\star\star}$ Indicates significant difference from SVM using student’s *t*-test at $p<0.01$. []{data-label="CMI_classif_table"}
t-SNE Visualization {#t-sne-visualization .unnumbered}
-------------------
Features from the first training fold tensor decomposition of the MMEC and CMI datasets are displayed as t-SNE maps in Figure \[tsneFig\](b) and Figure \[tsneFig\](d) respectively. Both t-SNE maps demonstrate strong local grouping of different age groups. Maps of EEG data prior to tensor factorization (Figure \[tsneFig\](a)) and when the decomposition has a randomly ordered $[Subject]$ domain in the MMEC data (Figure \[tsneFig\](c)) are included for comparison. The results demonstrate significantly improved feature grouping and clusters using the properly ordered tensor decomposition methodology compared to using the raw EEG or random ordered $[Subject]$ domain. The t-SNE maps help illustrate how SVM classification from the raw EEG time-series data using all frequency and spatial features do not perform better than random classification in the MMEC. Similar patterned results were found for CMI data.
Simulation {#simulation .unnumbered}
----------
A 3-component PARAFAC model revealed the underlying ‘ground truth’ developmental profiles built into the simulation tensor data. Resulting component factors are presented in Supplemental Figure \[SimulatedEEG\]. Corcondia was 99% for the model, with approximately 23% explained variance. These values are understandable for the model, as the BBCB pseudo-EEG simulation was designed to retain trilinearity in the data while introducing multiple levels of noise at the frequency band, brain background and sensor layers. Replicating developmental profile extraction results in the simulated data grants further support to our conclusions in real-world datasets.
Discussion {#discussion .unnumbered}
==========
The tensor analysis outlined in this paper lays a foundational framework capable of extracting latent structures and features associated with development in paediatric EEG. The unsupervised nature of this framework opens the door to encourage better personalized paradigms for data-driven technologies aimed at paediatric populations. Capitalizing on these developments could help translate new technologies to children, which are sensitive to developmental features in EEG.
Tensor analysis derives informative ‘development feature profiles’ of paediatric subjects {#tensor-analysis-derives-informative-development-feature-profiles-of-paediatric-subjects .unnumbered}
-----------------------------------------------------------------------------------------
The factor profiles derived in this study confirm the proposed PARAFAC decomposition simultaneously accounts for background EEG noise and shifting frequency bands across subject age, often explaining more than 85% of the data variance. Low-frequency, high power spectral activity typically associated with background EEG noise can be seen in Figure \[fullpageMMECtensor\], with factors 1-4 reflecting decaying power curves in the $[Spectral]$ domain, coupled with relatively strong contributions across all channels in the $[Spatial]$ domain. These factors are likely characterizing general ‘background’ EEG noise independent to potential signals of interest (e.g. factors 5-8). Critically, the $[Subject]$ domain demonstrates the tensor analysis has sensitivity to subtle developmental differences, since the features uniquely correspond to distinct age groupings, even within the potential background noise.
Shifts in spectral power and frequency due to development [@Marshall2002; @Matsuura1985; @Miskovic2015] are seen in the qualitative factor profiles of both the MMEC and CHB-MIT datasets. In the MMEC dataset, for example, the very low frequency, high power dominated spectral profile associated with infant and early-life EEG recordings seem to be reflected by Factor 7 in the $[Subject]$ and $[Spectral]$ domains. Meanwhile, factor 5 (light blue) is centrally located in the $[Subject]$ domain (covering approximately ages 1-3) and spans the 3-7 Hz range of the $[Spectral]$ domain. Factor 5 therefore highlights the likely dominant frequency range for those ages, while also reflecting a shift in power towards higher frequencies, which is expected with growing. Factor 8 illustrates a further shift towards higher frequencies contributing more to the spectral profile, as it steadily increases for subjects 31-50 (approximately ages 3-5 y.o.) in the $[Subject]$ domain. Also, factors 6 and 8 may represent the beginning separation between classical EEG bands of interest, i.e. the delta/theta bands and the alpha/beta bands respectively.
Similar developmental shifts are also seen in the CHB qualitative feature profiles. The key component factors show significantly reduced spectral power contributing more in higher $[Subject]$ domain ages, alongside a shift in the $[Spectral]$ dimension towards higher frequencies. These extracted profiles are reflective of the traditional movement and prevalence of the classical EEG bands, like alpha, throughout child development into adulthood [@Matsuura1985; @Miskovic2015].
Improved classification results verify tensor extracted features’ sensitivity to development {#improved-classification-results-verify-tensor-extracted-features-sensitivity-to-development .unnumbered}
--------------------------------------------------------------------------------------------
Improvements in classification coupled with the obtained tensor profiles of both impaired (epileptic) and healthy children indicate age-specific factors uncovered from EEG via PARAFAC contain structural information on latent developmental relationships. The scalable nature of the proposed analysis showed promise in identifying relevant features to development across varying developmental conditions, including both afflicted/healthy populations, and at slow/rapid developmental windows.
With better characterization of such features for the paediatric subjects comes a stronger case for translating signal processing and machine learning applications to children. Clear support for this is seen in the comparison between t-SNE maps for the full feature raw EEG time-series data and the PARAFAC processed data. Using the feature-full raw EEG time-series without processing led to complete failure in identifying developmental features. The resulting t-SNE map has no discernible structure with random clusters and groupings. Characterization of the underlying developmental profile was rendered completely imperceptible, which is likely reflected in the classification being no better than chance. On the other hand, the highly-structured t-SNE maps of the tensor extracted features for both the MMEC and CMI dataset reflect well characterized developmental profiles, which can be utilized in machine learning applications.
Importantly, the randomly ordered $[Subject]$ domain t-SNE map suggests that successful identification of key developmental features is not inherent to tensor factorization itself. Although the overarching global structure is capable of being identified, as seen in the similarity of global shapes of Figure \[tsneFig\](b) and Figure\[tsneFig\](c)), the local grouping is completely lost. Rather, the viability of determining developmentally sensitive features relies on proper construction of the $[Subject]$ domain e.g. having an inherent proxy to growing like strictly increasing subject age. Future work could investigate the effects of altering the inherent structure of the $[Subject]$ domain to reflect other developmental markers, such as cognitive or behavioural scores.
Exploiting the structural information from higher-order tensors constructed with careful construction of the $[Subject]$ domain as a proxy measure for child development provides a methodological framework designed to enhance sensitivity to physiological changes common throughout childhood. Higher sensitivity to these unique developmental profiles could be useful in applications like BCI. Through using a framework built to determine a current child’s developmental state at an electrophysiological level, the BCI could be automatically tuned and weighted appropriately for subjects at different points along development. Additionally, the results could provide ‘healthy development curves’ in studies for comparison to potentially developmentally impaired children.
Together these findings expand upon our previous results [@Kinney-Lang2017]. Our improved methodology is verified using multiple datasets. Results indicate the new proposed methods can account for developmental differences in background EEG and shifting spectral signals for children under a variety of different developmental conditions. The classification results illustrate a means for developmental feature extraction sensitive to progressive changes, while the profiles provide informative context regarding the relationship between $[Spatial]$ and $[Spectral]$ structures relative to subject age and development.
Limitations {#limitations .unnumbered}
-----------
Limitations in this study included restricted access to age/acquisition-matched paediatric datasets and the heterogeneity associated with epilepsy in subjects. Due to limited resources no direct age/acquisition-matched healthy control data was available for analysis to compare directly. To mitigate these drawbacks, however, multiple publicly available datasets were used to demonstrate the proposed methodology in multiple settings. The CHB-MIT dataset built upon our analysis from the MMEC across a wider age range using a similar disease condition, while the CMI dataset represented healthy control within the bounds of childhood (but not age-matched). Future work using a more homogeneous population with age-matched controls could help further validate the results, along with data from both resting-state and event-related EEG.
Conclusion {#conclusion .unnumbered}
==========
Advanced signal processing, like PARAFAC, combined with machine learning can help distinguish non-obvious developmental patterns from child EEG data. This study proposes tensor analysis can provide an intuitive sense of the latent developmental relationships in paediatric EEG data, and provide a way for development-sensitive feature extraction. The results indicate successful identification of factor profiles and benefits to classification analysis for a wide variety of developmental conditions, including both afflicted and healthy paediatric populations. This study lays a methodological framework which could improve applications for children reliant on EEG processing and analysis, like BCI. Further development on this framework could help improve BCI application sensitivity to developmental changes by setting the groundwork for a ‘developmental domain’ for tensor-based EEG classification in BCIs (e.g. [@Zink2016]). These advances could help immensely in translating the BCI technology to paediatric populations and pave the way for development of more readily accessible, effective rehabilitation strategies internationally.
Acknowledgement {#acknowledgement .unnumbered}
===============
Funding support for this project was provided by the RS McDonald Trust, Thomas Theodore Scott Ingram Memorial Fund and the Muir Maxwell Trust. The authors would like to thank the children, parents, clinicians and researchers who volunteered and comprised each of the datasets, and the Muir Maxwell Epilepsy Centre for providing access the MMEC dataset. The authors also thank Ephrem Zewdie for his comments and suggestions on figures.
References {#references .unnumbered}
==========
Supplementary Information {#supplementary-information .unnumbered}
=========================
[|cc|]{}\
\# of Subj. & Age (Months)\
11 & 0-12\
14 & 12-24\
4 & 24-36\
8 & 36-48\
10 & 48-60\
3 & 60-72\
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\# of Subj. & Age (Years)\
1 & 2\
2 & 3\
1 & 6\
2 & 7\
2 & 9\
2 & 11\
2 & 12\
1 & 13\
1 & 14\
1 & 16\
1 & 18\
1 & 19\
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\# of Subj. & Age (Years)\
1$^*$ & 6\
11 & 8\
12 & 9\
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7 & 11\
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---
abstract: 'We propose non-stationary spectral kernels for Gaussian process regression. We propose to model the spectral density of a non-stationary kernel function as a mixture of input-dependent Gaussian process frequency density surfaces. We solve the generalised Fourier transform with such a model, and present a family of non-stationary and non-monotonic kernels that can learn input-dependent and potentially long-range, non-monotonic covariances between inputs. We derive efficient inference using model whitening and marginalized posterior, and show with case studies that these kernels are necessary when modelling even rather simple time series, image or geospatial data with non-stationary characteristics.'
author:
- |
Sami Remes\
Markus Heinonen\
Samuel Kaski\
Helsinki Institute for Information Technology\
Department of Computer Science, Aalto University
bibliography:
- 'refs.bib'
title: 'Non-Stationary Spectral Kernels'
---
Introduction
============
Gaussian processes are a flexible method for non-linear regression [@rasmussen2006]. They define a distribution over functions, and their performance depends heavily on the covariance function that constrains the function values. Gaussian processes interpolate function values by considering the value of functions at other similar points, as defined by the kernel function. Standard kernels, such as the Gaussian kernel, lead to smooth neighborhood-dominated interpolation that is oblivious of any periodic or long-range connections within the input space, and can not adapt the similarity metric to different parts of the input space.
Two key properties of covariance functions are *stationarity* and *monotony*. A stationary kernel $K(x,x') = K(x+a,x'+a)$ is a function only of the distance $x-x'$ and not directly the value of $x$. Hence it encodes an identical similarity notion across the input space, while a monotonic kernel decreases over distance. Kernels that are both stationary and monotonic, such as the Gaussian and Matérn kernels, can encode neither input-dependent function dynamics nor long-range correlations within the input space. Non-monotonic and non-stationary functions are commonly encountered in realistic signal processing [@rioul1991], time series analysis [@huang1998], bioinformatics [@Grzegorczyk2008; @robinson2009], and in geostatistics applications [@higdon1999; @huang2008].
Recently, several authors have explored kernels that are either non-monotonic or non-stationary. A non-monotonic kernel can reveal informative manifolds over the input space by coupling distant points due to periodic or other effects. Non-monotonic kernels have been derived from the Fourier decomposition of kernels [@lazaro2010sparse; @sinha2016; @wilson2013], which renders them inherently stationary. Non-stationary kernels, on the other hand, are based on generalising monotonic base kernels, such as the Matérn family of kernels [@heinonen2016; @paciorek2004], by partitioning the input space [@gramacy2008], or by input transformations [@snoek2014]. We propose an expressive and efficient kernel family that is – in contrast to earlier methods – both non-stationary and non-monotonic, and hence can infer long-range or periodic relations in an input-dependent manner. We derive the kernel from first principles by solving the more expressive *generalised* Fourier decomposition of non-stationary functions, than the more limited standard Fourier decomposition exploited by earlier works. We propose and solve the generalised spectral density as a mixture of Gaussian process density surfaces that model flexible input-dependent frequency patterns. The kernel reduces to a stationary kernel with appropriate parameterisation. We show the expressivity of the kernel with experiments on time series data, image-based pattern recognition and extrapolation, and on climate data modelling.
Non-stationary spectral kernels {#sec:model}
===============================
This section introduces the main contributions. We employ the generalised spectral decomposition of non-stationary functions and derive a practical and efficient family of kernels based on non-stationary spectral components. Our approach relies on associating input-dependent frequencies for data inputs, and solving a kernel through the generalised spectral transform.
The most general family of kernels is the non-stationary kernels, which include stationary kernels as special cases [@genton2001]. A non-stationary kernel $k(x,x') \in \R$ for scalar inputs $x,x' \in \R$ can be characterized by its spectral density $S(s,s')$ over frequencies $s,s' \in \R$, and the two are related via a generalised Fourier transform[^1] $$\begin{aligned}
k(x,x') = \int_{\R} \int_{\R} e^{2\pi i (xs - x' s')} \mu_S(ds, ds') \; , \label{eq:fourier}\end{aligned}$$ where $\mu_S$ is a Lebesgue-Stieltjes measure associated to some positive semi-definite (PSD) spectral density function $S(s,s')$ with bounded variations [@genton2001; @loeve1978probability; @yaglom1987correlation], which we denote as the *spectral surface* since it considers the amplitude of frequency pairs (See Figure \[fig:bsm\]a).
The generalised Fourier transform specifies that a spectral surface $S(s,s')$ generates a PSD kernel $K(x,x')$ that is non-stationary unless the spectral measure mass is concentrated only on the diagonal $s=s'$. We design a practical, efficient and flexible parameterisation of spectral surfaces that, in turn, specifies novel non-stationary kernels with input-dependent characteristics and potentially long-range non-monotonic correlation structures.
Bivariate Spectral Mixture kernel {#sec:bivariate}
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[0.49]{} ![**(a)**: Spectral density surface of a single component bivariate spectral mixture kernel with $8$ permuted peaks. **(b)**: The corresponding kernel on inputs $x \in [-1,1]$.[]{data-label="fig:bsm"}](s_bsm-crop "fig:"){width="\textwidth"}
[0.49]{} ![**(a)**: Spectral density surface of a single component bivariate spectral mixture kernel with $8$ permuted peaks. **(b)**: The corresponding kernel on inputs $x \in [-1,1]$.[]{data-label="fig:bsm"}](k_bsm-crop "fig:"){width="\textwidth"}
Next, we introduce spectral kernels that remove the restriction of stationarity of earlier works. We start by modeling the spectral density as a mixture of $Q$ bivariate Gaussian components $$\begin{aligned}
S_i(s,s') = \sum_{ \bmu_i \in \pm \{ \mu_i, \mu_i'\}^2 } \N \left( \begin{pmatrix} s \\ s' \end{pmatrix} | \bmu_i, \Sigma_i\right), \qquad \Sigma_i = \begin{bmatrix} \sigma_i^2 & \rho_i \sigma_i \sigma_i' \\ \rho_i \sigma_i \sigma_i' & {\sigma_i'}^2 \end{bmatrix} \label{eq:spect} \; ,\end{aligned}$$ with parameterization using the correlation $\rho_i$, means $\mu_i,\mu_i'$ and variances $\sigma_i^2,{\sigma_i'}^2$. To produce a PSD spectral density $S_i$ as required by equation we need to include symmetries $S_i(s,s') = S_i(s',s)$ and sufficient diagonal components $S_i(s,s)$, $S_i(s',s')$. To additionally result in a real-valued kernel, symmetry is required with respect to the negative frequencies as well, i.e., $S_i(s,s') = S_i(-s,-s')$. The sum $\sum_{ \bmu_i \in \pm \{ \mu_i, \mu_i'\}^2 }$ satisfies all three requirements by iterating over the four permutations of $\{\mu_i,\mu_i'\}^2$ and the opposite signs $(-\mu_i, -\mu_i')$, resulting in eight components (see Figure \[fig:bsm\]a).
The generalised Fourier transform can be solved in closed form for a weighted spectral surface mixture $S(s,s') = \sum_{i=1}^Q w_i^2 S_i(s,s')$ using Gaussian integral identities (see the appendix): $$\begin{aligned}
k(x,x') %&= \iint S(s,s') e^{2\pi i (xs - x' s')} ds ds' \notag \\
&= \sum_{i=1}^Q w_i^2 \exp(- 2 \pi^2 \tilde{\x}^T \Sigma_i \tilde{\x}) \Psi_{\mu_i,\mu'_i}(x)^T\Psi_{\mu_i,\mu'_i}(x') \label{eq:wm}
\end{aligned}$$ where $$\begin{aligned}
\Psi_{\mu_i,\mu'_i}(x) &= \begin{pmatrix} \cos 2\pi\mu x + \cos 2\pi\mu' x \\ \sin 2\pi\mu x + \sin 2\pi\mu' x \end{pmatrix},\end{aligned}$$ and where we define $\tilde{\x} = ( x, -x')^T$ and introduce mixture weights $w_i$ for each component. We denote the proposed kernel as the *bivariate spectral mixture* (BSM) kernel (see Figure \[fig:bsm\]b). The positive definiteness of the kernel is guaranteed by the spectral transform, and is also easily verified since the sinusoidal components form an inner product and the exponential component resembles an unscaled Gaussian density.
We immediately notice that the BSM kernel vanishes rapidly outside the origin $(x,x') = (0,0)$. We would require a huge number of components centered at different points $x_i$ to cover a reasonably-sized input space.
Generalised Spectral Mixture (GSM) kernel
-----------------------------------------
To overcome the deficiencies of the kernel derived in Section \[sec:bivariate\], we extend it further by parameterizing the frequencies, length-scales and mixture weights as a Gaussian processes[^2], that form a smooth spectrogram (See Figure \[fig:surfaces\]l): $$\begin{aligned}
\log w_i(x) \sim \GP(0,k_w(x,x')), \\
\log \ell_i(x) \sim \GP(0,k_\ell(x,x')), \\
\logit \mu_i(x) \sim \GP(0,k_\mu(x,x')).\end{aligned}$$ Here the log transform is used to ensure the weights $w(x)$ and lengthscales $\ell(x)$ are non-negative, and the logit transform $\logit \mu(x) = \log\frac{\mu}{F_N-\mu}$ limits the learned frequencies between zero and the Nyquist frequency $F_N$, which is defined as half of the sampling rate of the signal.
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](k_gauss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](k_ss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](k_sm-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](k_gsm-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](s_gauss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](s_ss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](s_sm-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](s_gsm-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](g_gauss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](g_ss-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](g_sm-crop "fig:"){width="\textwidth"}
[0.245]{} ![**(a)-(d)**: Examples of kernel matrices on inputs $x \in [-1,1]$ for a Gaussian kernel (a), sparse spectrum kernel [@lazaro2010sparse] (b), spectral mixture kernel [@wilson2013] (c), and for the GSM kernel (d). **(e)-(h)**: The corresponding generalised spectral density surfaces of the four kernels. **(i)-(l)**: The corresponding spectrograms, that is, input-dependent frequency amplitudes. The GSM kernel is highlighted with a spectrogram mixture of $Q=2$ Gaussian process surface functions.[]{data-label="fig:surfaces"}](g_gsm-crop "fig:"){width="\textwidth"}
A GP prior $f(x) \sim \GP(0, k(x,x'))$ defines a distribution over zero-mean functions, and denotes the covariance between function values $\cov[ f(x), f(x')] = k(x,x')$ as the prior kernel For any collection of inputs, $x_1, \ldots, x_N$, the function values follow a multivariate normal distribution $(f(x_1), \ldots, f(x_N))^T \sim \N(\0, K)$, where $K_{ij} = k(x_i, x_j)$. The key property of Gaussian processes is that they can encode smooth functions by correlating function values of input points that are similar according to the kernel $k(x,x')$. We use standard Gaussian kernels $k_w$, $k_\ell$ and $k_\mu$.
We accommodate the input-dependent lengthscale by replacing the exponential part of by the Gibbs kernel $$\begin{aligned}
k_{\text{Gibbs},i}(x,x') = \sqrt{\frac{2\ell_i(x) \ell_i(x')}{\ell_i(x)^2+\ell_i(x')^2}}\exp\left(-\frac{(x-x')^2}{\ell_i(x)^2+\ell_i(x')^2}\right) \; ,\end{aligned}$$ which is a non-stationary generalisation of the Gaussian kernel [@gibbs1997; @heinonen2016; @paciorek2004]. We propose a non-stationary *generalised spectral mixture* (GSM) kernel with a simple closed form (see the appendix): $$\begin{aligned}
k_{\text{GSM}}(x,x') = \sum_{i=1}^Q w_i(x) w_i(x') k_{gibbs,i}(x,x') \cos(2 \pi (\mu_i(x) x - \mu_i(x') x')) \; . \label{eq:gsm}\end{aligned}$$ The kernel is a product of three PSD terms. The GSM kernel encodes the similarity between two data points based on their combined signal variance $w(x)w(x')$, and the frequency surface based on the frequencies $\mu(x),\mu(x')$ and frequency lengthscales $\ell(x),\ell(x')$ associated with both inputs. The GSM kernel encodes the spectrogram surface mixture into a relatively simple kernel. The kernel reduces to the stationary Spectral Mixture (SM) kernel [@wilson2013] with constant functions $w_i(x) = w_i$, $\mu_i(x) = \mu_i$ and $\ell_i(x) = 1/(2\pi \sigma_i)$ (see the appendix).
We have presented the proposed kernel for univariate inputs for simplicity. The kernel can be extended to multivariate inputs in a straightforward manner using the generalised Fourier transform with vector-valued inputs [@genton2001; @kakihara1985]. However, since in many applications multivariate inputs have a grid-like structure, for instance in geostatistics, image analysis and temporal models. We exploit this assumption and propose a multivariate extension that assumes the inputs to decompose across input dimensions [@flaxman2015; @wilson2013]: $$\begin{aligned}
k_{\text{GSM}}(\x,\x' | \boldsymbol\theta) = \prod_{p=1}^P k_{\text{GSM}}(x_p, x_p' | \bt_p) \; .\end{aligned}$$ Here $\x,\x' \in \R^P$, $\bt = (\bt_1, \ldots, \bt_P)$ collects the dimension-wise kernel parameters $\bt_p = (\w_{ip}, \bl_{ip}, \bmu_{ip})_{i=1}^Q$ of the $N$-dimensional realisations $\w_{ip},\bl_{ip},\bmu_{ip} \in \R^N$ per dimension $p$. Then, the kernel matrix can be expressed using Kronecker products as $\K_\bt = K_{\bt_1} \otimes \cdots \otimes K_{\bt_P}$, while missing values and data not on a regular grid can be handled with standard techniques [@flaxman2015; @saatcci2011scalable; @wilson2015kissgp].
Inference
=========
We use the Gaussian process regression framework and assume a Gaussian likelihood over $N^P$ data points $(\x_j,y_j)_{j=1}^{N^P}$ with all outputs collected into a vector $\y \in \R^{N^P}$, $$\begin{aligned}
y_j &= f(\x_j) + \varepsilon_j, \qquad \varepsilon_j \sim \N(0, \sigma_n^2) \notag \\
f(\x) &\sim \GP(0, k_{\text{GSM}}(\x,\x' | \bt)),\end{aligned}$$ with a standard predictive GP posterior $f(\x_\star | \y)$ for a new input point $\x_\star$ [@rasmussen2006]. The posterior can be efficiently computed using Kronecker identities [@saatcci2011scalable] (see the appendix).
We aim to infer the noise variance $\sigma_n^2$ and the kernel parameters $\boldsymbol\theta = (\w_{ip}, \bl_{ip}, \bmu_{ip})_{i=1,p=1}^{Q,P}$ that reveal the input-dependent frequency-based correlation structures in the data, while regularising the learned kernel to penalise overfitting. We perform MAP inference over the log marginalized posterior $\log p(\bt | \y) \propto \log p(\y | \bt) p(\bt) = \mathcal{L}(\bt)$, where the functions $f(x)$ have been marginalised out, $$\begin{aligned}
\mathcal{L}(\bt) &= \log \left( \N(\y | \0, \K_{\bt} + \sigma_n^2\I) \prod_{i,p=1}^{Q,P} \N(\w_{ip} | \0, K_{w_p}) \N(\bmu_{ip} | \0, K_{\mu_p}) \N(\bl_{ip} | \0, K_{\ell_p}) \right), \label{eq:gpmll}\end{aligned}$$ where $K_{w_p},K_{\mu_p},K_{\ell_p}$ are $N \times N$ prior matrices per dimensions $p$. The marginalized posterior automatically balances between parameters $\bt$ that fit the data and a model that is not overly complex [@rasmussen2006]. We can efficiently evaluate both the marginalized posterior and its gradients in $\mathcal{O}( P N^{\frac{P+1}{P}} )$ instead of the usual $\mathcal{O}( {N^P}^3 )$ complexity [@saatcci2011scalable] (see the appendix).
Gradient-based optimisation of is likely to converge very slowly due to parameters $\w_{ip}, \bmu_{ip}, \bl_{ip}$ being highly self-correlated. We remove the correlations by whitening the variables as $\tilde\bt = \LL^{-1}\bt$ where $\LL$ is the Cholesky decomposition of the prior covariances. We maximize $\L(\bt)$ using gradient ascent with respect to the whitened variables $\tbt$ by evaluating $\L( \LL \tbt)$ and the gradient as [@heinonen2016; @kuss2005] $$\begin{aligned}
\pdd{\L(\bt)}{\tbt} = \pdd{\L(\bt)}{\bt}\pdd{\bt}{\tbt} = \LL^T \pdd{\L(\bt)}{\bt}. \label{eq:grad}\end{aligned}$$
Related Work {#sec:related}
============
Bochner’s theorem for stationary signals, whose covariance can be written as $k(\tau) = k(x-x') = k(x,x')$, implies a Fourier dual [@wilson2013] $$\begin{aligned}
k(\tau) &= \int S(s) e^{2\pi i s \tau} ds \\
S(s) &= \int k(\tau) e^{-2\pi i s \tau} d\tau.\end{aligned}$$ The dual is a special case of the more general Fourier transform , and has been exploited to design rich, yet stationary kernel representations [@sinha2016; @yang2015] and used for large-scale inference [@rahimi2008]. Lazaro-Gredilla et al. proposed to directly learn the spectral density as a mixture of Dirac delta functions leading to a sparse spectrum (SS) kernel $k_{\text{SS}}(\tau) = \frac{1}{Q} \sum_{i=1}^Q \cos(2 \pi s_i^T \tau)$ [@lazaro2010sparse]. Wilson et al. derived a stationary spectral mixture (SM) kernel by modelling the univariate spectral density using a mixture of normals $S_{\text{SM}}(s) = \sum_i w_i [\N(s|\mu_i,\sigma_i^2) + \N(s| -\mu_i,\sigma_i^2)] / 2$ [@wilson2013], corresponding to the kernel function $k_\text{SM}(\tau) = \sum_i w_i\exp(-2\pi^2\sigma_i^2\tau)\cos(2\pi\mu_i\tau)$, which we generalized to the non-stationary case. Kernels derived from the spectral representation are particularly well suited to encoding long-range, non-monotonic or periodic kernels; however, they have so far been unable to handle non-stationarity.
Non-stationary kernels, on the other hand, have been constructed by non-stationary extensions of Matérn and Gaussian kernels with input-dependent lengthscales [@gibbs1997; @heinonen2016; @paciorek2004; @paciorek2006], input space warpings [@sampson1992; @snoek2014], and with local stationarity with products of stationary and non-stationary kernels [@genton2001; @silverman1957]. The simplest non-stationary kernel is arguably the dot product kernel [@rasmussen2006], which has been used as a way to assign input-dependent signal variances [@tolvanen2014]. Non-stationary kernels are a good match for functions with transitions in their dynamics, yet are unsuitable for modelling non-monotonic properties.
Our work can also be seen as a generalisation of wavelets, or time-dependent frequency components, into general and smooth input-dependent components. In signal processing, Hilbert-Huang transforms and Hilbert spectral analysis explore input-dependent frequencies, but with deterministic transform functions on the inputs [@huang2008; @huang1998].
Experiments {#sec:experiments}
===========
We apply our proposed kernel first on simple simulated time series, then on texture images and lastly on a land surface temperature dataset. With the image data, we compare our method to two stationary mixture kernels, specifically the spectral mixture (SM) [@wilson2013] and sparse spectrum (SS) kernels [@lazaro2010sparse], and the standard squared exponential (SE) kernel. We employ the GPML Matlab toolbox, which directly implements the SM and SE kernels, and the SS kernel as a meta kernel combining simple cosine kernels. The GPML toolbox also implements Kronecker inference automatically for these kernels. We implemented the proposed GSM kernel and inference in Matlab.
For optimizing the log posterior we employ the L-BFGS algorithm. For both our method and the comparisons, we restart the optimization from 10 different initialisations, each of which is chosen as the best among 100 randomly sampled hyperparameter values as evaluating the log posterior is cheap compared to evaluating gradients or running the full optimisation.
Simulated time series with a decreasing frequency component
-----------------------------------------------------------
First we experiment whether the GSM kernel can find a simulated time-varying frequency pattern. We simulated a dataset where the frequency of the signal changes deterministically as $\mu(x) = 1+(1-x)^2$ on the interval $x\in[-1,1]$. We built a single-component GSM kernel $K$ using the specified functions $\mu(x)$, $\ell(x) = \ell = \exp(-1)$ and $w(x) = w = 1$. We sampled a noisy function $\y \sim \N(\vec0,K+\sigma_n^2 I)$ with a noise variance $\sigma_n^2 = 0.1$. The example in Figure \[fig:incfreq\] shows the learned GSM kernel, as well as the data and the function posterior $f(x)$. For this 1D case, we also employed the empirical spectrogram for initializing the hyperparameter values. The kernel correctly captures the increasing frequency towards negative values (towards left in Figure \[fig:incfreq\]a).
[0.48]{} ![**(a)** A simulated time series with a single decreasing frequency component and a GP fitted using a GSM kernel. **(b)** The learned kernel shows that close to $x=-1$ the signal is highly correlated and anti-correlated with close time points, while these longer-range dependencies vanish when moving towards $x=1$.[]{data-label="fig:incfreq"}](inputdep_posterior-crop "fig:"){height="1.6in"}
[0.48]{} ![**(a)** A simulated time series with a single decreasing frequency component and a GP fitted using a GSM kernel. **(b)** The learned kernel shows that close to $x=-1$ the signal is highly correlated and anti-correlated with close time points, while these longer-range dependencies vanish when moving towards $x=1$.[]{data-label="fig:incfreq"}](inputdep_kernels-crop "fig:"){height="1.7in"}
Image data
----------
We applied our kernel to two texture images. The first image of a sheet of metal represents a mostly stationary periodic pattern. The second, a wood texture, represents an example of a very non-stationary pattern, especially on the horizontal axis. We use majority of the image as training data (the non-masked regions of Figure \[fig:incfreq\]a and \[fig:incfreq\]f) , and use the compared kernels to predict a missing cross-section in the middle, and also to extrapolate outside the borders of the original image.
Figure \[fig:metal\] shows the two texture images, and extrapolation predictions given by the proposed GSM kernel, with a comparison to the spectral mixture (SM), sparse spectrum (SS) and standard squared exponential (SE) kernels. For GSM, SM and SS we used $Q=5$ mixture components for the metal texture, and $Q=10$ components for the more complex wood texture.
The GSM kernel gives the most pleasing result visually, and fills in both patterns well with consistent external extrapolation as well. The stationary SM kernel does capture the cross-section, but has trouble extrapolation outside the borders. The SS kernel fails to represent even the training data, it lacks any smoothness in the frequency space. The gaussian kernel extrapolates poorly.
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](metal_ground "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](metal_GSM "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](metal_SM "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](metal_SS "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](metal_SE "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](wood_ground "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](wood_GSM "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](wood_SM "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](wood_SS "fig:"){width="\textwidth"}
[0.195]{} ![A metal texture data with $Q=5$ components used for GSM, SM and SS kernels shown in **(a)**-**(e)** and a wood texture in **(f)**-**(j)** (with $Q=10$ components). The GSM kernel performs the best, making the most believable extrapolation outside image borders in **(b)** and **(g)**. The SM kernel fills in the missing cross pattern in **(c)** but does not extrapolate well. In **(h)** the SM kernel fills in the vertical middle block only with the mean value while GSM in **(g)** is able to fill in a wood-like pattern. SS is not able discover enough structure in either texture **(d)** or **(i)**, while the SE kernel overfits by using a too short length-scale in **(e)** and **(j)**.[]{data-label="fig:metal"}](wood_SE "fig:"){width="\textwidth"}
Spatio-Temporal Analysis of Land Surface Temperatures
-----------------------------------------------------
NASA[^3] provides a land surface temperature dataset that we used to demonstrate our kernel in analysis of spatio-temporal data. Our primary objective is to demonstrate the capability of the kernel in inferring long-range, non-stationary spatial and temporal covariances.
We took a subset of four years (February 2000 to February 2004) of North American land temperatures for training data. In total we get 407,232 data points, constituting 48 monthly temperature measurements on a $84 \times 101$ map grid. The grid also contains water regions, which we imputed with the mean temperature of each month. We experimented with the data by learning a generalized spectral mixture kernel using $Q=5$ components.
Figure \[fig:spatial\] presents our results. Figure \[fig:spatial\]b highlights the training data and model fits for a winter and summer month, respectively. Figure \[fig:spatial\]a shows the non-stationary kernel slices at two locations across both latitude and longitude, as well as indicating that the spatial covariances are remarkably non-symmetric. Figure \[fig:spatial\]c indicates five months of successive training data followed by three months of test data predictions.
[0.49]{}
![**(a)** Demonstrates the non-stationary spatial covariances in the land surface data. The vertical black lines denote the point $x_0$ at which the kernel function $k(\cdot,x_0)$ is centered. **(b)** Sample reconstructions. In all plots, only the land area temperatures are shown. **(c)** Posterior for five last training months (until Jan 2004) and prediction for the three next months (February 2004 to April 2004), which the model is able to to construct reasonably accurately.[]{data-label="fig:spatial"}](spatial_covs_white){width="\textwidth"}
[0.49]{}
![**(a)** Demonstrates the non-stationary spatial covariances in the land surface data. The vertical black lines denote the point $x_0$ at which the kernel function $k(\cdot,x_0)$ is centered. **(b)** Sample reconstructions. In all plots, only the land area temperatures are shown. **(c)** Posterior for five last training months (until Jan 2004) and prediction for the three next months (February 2004 to April 2004), which the model is able to to construct reasonably accurately.[]{data-label="fig:spatial"}](spatial_feb2000_white "fig:"){width="\textwidth"}\
![**(a)** Demonstrates the non-stationary spatial covariances in the land surface data. The vertical black lines denote the point $x_0$ at which the kernel function $k(\cdot,x_0)$ is centered. **(b)** Sample reconstructions. In all plots, only the land area temperatures are shown. **(c)** Posterior for five last training months (until Jan 2004) and prediction for the three next months (February 2004 to April 2004), which the model is able to to construct reasonably accurately.[]{data-label="fig:spatial"}](spatial_jul2000_white "fig:"){width="\textwidth"}
![**(a)** Demonstrates the non-stationary spatial covariances in the land surface data. The vertical black lines denote the point $x_0$ at which the kernel function $k(\cdot,x_0)$ is centered. **(b)** Sample reconstructions. In all plots, only the land area temperatures are shown. **(c)** Posterior for five last training months (until Jan 2004) and prediction for the three next months (February 2004 to April 2004), which the model is able to to construct reasonably accurately.[]{data-label="fig:spatial"}](spatial_predict_future_white){width="\textwidth"}
Discussion {#sec:discussion}
==========
In this paper we have introduced non-stationary spectral mixture kernels, with treatment based on the generalised Fourier transform of non-stationary functions. We first derived the bivariate spectral mixture (BSM) kernel as a mixture of non-stationary spectral components. However, we argue it has only limited practical use due to requiring an impractical amount of components to cover any sufficiently sized input space. The main contribution of the paper is the generalised spectral mixture (GSM) kernel with input-dependent Gaussian process frequency surfaces. The Gaussian process components can cover non-trivial input spaces with just a few interpretable components. The GSM kernel is a flexible, practical and efficient kernel that can learn both local and global correlations across the input domains in an input-dependent manner. We highlighted the capability of the kernel to find interesting patterns in the data by applying it on climate data where it is highly unrealistic to assume the same (stationary) covariance pattern for every spatial location irrespective of spatial structures.
Even though the proposed kernel is motivated by the generalised Fourier transform, the solution to its spectral surface $$\begin{aligned}
S_{\text{GSM}}(s,s') = \iint k_{\text{GSM}}(x,x') e^{-2 \pi i (xs - x's')} dx dx' \label{eq:gsmsurface}\end{aligned}$$ remains unknown due to having multiple GP functions inside the integral. Figure \[fig:surfaces\]h highlights a numerical integration of the surface equation on an example GP frequency surface. Furthermore, the theoretical work of Kom Samo and Roberts [@komsamo2015] on generalised spectral transforms suggests that the GSM kernel may also be dense in the family of non-stationary kernels, that is, to reproduce arbitrary non-stationary kernels.
[^1]: We focus on scalar inputs and frequencies for simplicity. An extension based on vector-valued inputs and frequencies [@genton2001; @kakihara1985] is straightforward.
[^2]: See the appendix for a tutorial on Gaussian processes.
[^3]: <https://neo.sci.gsfc.nasa.gov/view.php?datasetId=MOD11C1_M_LSTDA>
|
---
abstract: 'We announce a new type of “Jacobi identity” for vertex operator algebras, incorporating values of the Riemann zeta function at negative integers. Using this we “explain” and generalize some recent work of S. Bloch’s relating values of the zeta function with the commutators of certain operators and Lie algebras of differential operators.'
author:
- 'J. Lepowsky'
date: '*Dedicated to Howard Garland on the occasion of his sixtieth birthday*'
title: '[**Vertex operator algebras and the zeta function**]{}'
---
amssym.def amssym
\[section\] \[rema\][Proposition]{} \[rema\][Theorem]{} \[rema\][Definition]{} \[rema\][Lemma]{} \[rema\][Corollary]{} \[rema\][Example]{}
Introduction
============
Consider the famous classical “formula” $$\label{1+}
1 + 2 + 3 + \cdots = - \frac{1}{12},$$ which has the rigorous meaning $$\label{zeta-1}
\zeta(-1) = - \frac{1}{12}.$$ Here $\zeta$ is of course the Riemann zeta function $$\label{zeta}
\zeta(s) = \sum_{n > 0} n^{-s}$$ (analytically continued), and (\[1+\]) is classically generalized by the formal equality $$\label{zeta-s}
\sum_{n>0}n^s = \zeta(-s)$$ for $s = 0,1,2,\dots$. The classical number theory behind this analytic continuation is well known to be related to the widely-pervasive issue of regularizing certain infinities in quantum field theory, in particular, in conformal field theory. Here we shall announce some general principles of vertex operator algebra theory that elucidate the passage from the unrigorous but suggestive formula (\[1+\]) to formula (\[zeta-1\]), and the generalization (\[zeta-s\]). In the process, we shall “explain” some recent work of S. Bloch’s involving zeta-values and differential operators. The work [@L2] contains details and related results. The material that we shall present involves foundational notions of vertex operator algebra theory, and we shall try to make this writeup accessible to nonspecialists by reviewing elementary matters.
We were motivated by a desire to understand some very interesting phenomena found by Bloch [@Bl] relating the values $\zeta (-n)$, $n = 1, 3, 5, \dots$, of the zeta function at negative odd integers to the commuatators of certain operators on an infinite-dimensional space. We shall begin with some elementary background and a brief description of this work, then we shall explain how to recover and somewhat generalize these results using vertex operator algebra theory, and finally, we shall place these ideas and results into a very general context and present some new general results in vertex operator algebra theory. These methods serve incidentally to enhance the many already-existing motivations for vertex operator algebra theory (see [@Bo], [@FLM]) and its underlying formal calculus (as developed in [@FLM] and [@FHL]).
One of our main themes is to “always” use generating functions—to introduce new formal variables and generating functions in order to try to make complicated things easier and more natural and at the same time, much more general, as in the corresponding parts of [@FLM]. We use commuting formal variables rather than complex variables because they provide the most natural way to handle the doubly-infinite series such as $\delta (x) = \sum_{n \in {\Bbb Z}}x^n$ that pervade the natural formulations and proofs. Other central themes are to exploit the formal exponential of the differential operator $x{\frac{d}{dx}}$ as a formal change-of-variables automorphism (again as in [@FLM]); to formulate Euler’s interpretation of the divergent series (\[zeta-s\]) by means of the operator product expansion in conformal field theory; and to place considerations about Lie algebras of differential operators into the very general context of what we termed the “Jacobi identity” [@FLM] for vertex operator algebras. There are some interesting points of contact between the present work and [@KR], [@M] and [@FKRW].
I am very pleased to dedicate this paper to Howard Garland on the occasion of his sixtieth birthday. This work was presented in a talk at Yale University in fall, 1997 at a seminar in his honor. I would like to mention here that it was from Howard Garland that Robert Wilson and I, in 1980, first learned about the idea of using formal delta-function calculus, which was also used in [@DKM]; cf. [@FLM]. This was one among many of Howard’s insights that have influenced us.
This work was also presented in a talk at the May, 1998 Conference on Representations of Affine and Quantum Affine Algebras and Related Topics at North Carolina State University. I would like to thank Naihuan Jing and Kailash Misra, the organizers, for a stimulating conference.
I am very grateful to Spencer Bloch for informing me about his work and for many valuable discussions.
This work was partially supported by NSF grants DMS-9401851 and DMS-9701150.
Background
==========
Consider the commutative associative algebra $\Bbb C [t,t^{-1}]$ of Laurent polynomials in an indeterminate $t$, and consider its Lie algebra $\frak {d}$ of derivations: $${\frak {d}} = \mbox{\rm Der}\ \Bbb C [t,t^{-1}],$$ the Lie algebra of formal vector fields on the circle, with basis $\{t^n D | n \in \Bbb Z\}$, where $$\label{D}
D = D_t = t \frac{d}{dt}.$$ (A preview of one of our main themes: The “homogeneous” differential operator $D_t$, rather than $\frac{d}{dt}$, will be the appropriate form of differentiation for our considerations, and we shall be using it for various variables as well as $t$.)
Consider also the Virasoro algebra ${\frak {v}}$, the well-known central extension $$\label{Vir}
0 \rightarrow \Bbb C c \rightarrow {\frak {v}} \rightarrow {\frak {d}}
\rightarrow 0,$$ where ${\frak {v}}$ has basis $\{L(n) | n \in \Bbb Z \}$ together with a central element $c$; the bracket relations among the $L(n)$ are given by $$\label{Virbrackets}
[L(m),L(n)] = (m-n)L(m+n) + \frac{1}{12} (m^3 - m) \delta_{m+n,0} c$$ and $L(n)$ maps to $-t^n D$ in (\[Vir\]). The number $\frac{1}{12}
(m^3 - m)$, being one-half the binomial coefficient ${m+1} \choose 3$, is closely related to a third derivative, which becomes visible when we use generating functions to write the bracket relations (\[Virbrackets\]), as we review below. The Virasoro algebra is naturally $\Bbb Z$-graded, with ${\rm deg}\ L(n) = n$ and ${\rm deg}\
c = 0$.
The following classical realization of the Lie algebra ${\frak {v}}$ is well known: We start with the Heisenberg Lie algebra with basis consisting of the symbols $h(n)$ for $n \in \Bbb Z$, $n \neq 0$ and a central element $1$, with the bracket relations $$\label{Heisbrackets}
[h(m),h(n)] = m \delta_{m+n,0} 1.$$ For convenience we adjoin an additional central basis element $h(0)$, so that the relations (\[Heisbrackets\]) hold for all $m,n \in
\Bbb Z$. This Lie algebra acts irreducibly on the polynomial algebra $$\label{S}
S = \Bbb C [h(-1),h(-2),h(-3),\dots]$$ as follows: For $n<0$, $h(n)$ acts as the multiplication operator; for $n>0$, $h(n)$ acts as the operator $n \frac{\partial}{\partial
h(-n)}$; $h(0)$ acts as 0; and $1$ acts as the identity operator. Then ${\frak {v}}$ acts on $S$ by means of the following operators: $$\label{c}
c \mapsto 1,$$ $$\label{Ln}
L(n) \mapsto \frac{1}{2} \sum_{j \in \Bbb Z} h(j)h(n-j) \ \ \mbox{for}
\ n \neq 0,$$ $$\label{L0}
L(0) \mapsto \frac{1}{2} \sum_{j \in \Bbb Z} h(-|j|)h(|j|).$$
It is important to observe that in the case of $L(0)$, the absolute values make the formal quadratic operator well defined, while for $n
\neq 0$ the operator is well defined as it stands, since $[h(j),h(n-j)] = 0$. Thus the operators (\[Ln\]) and (\[L0\]) are in “normal-ordered form,” that is, the “annihilation operators” $h(n)$ for $n>0$ act to the right of the “creation operators” $h(n)$ for $n<0$. Using colons to denote normal ordering (the appropriate reordering of the factors in the product if necessary), we thus have $$\label{Lnnormal}
L(n) \mapsto \frac{1}{2} {\mbox{\tiny ${\bullet\atop\bullet}$}}\sum_{j \in \Bbb Z}
h(j)h(n-j){\mbox{\tiny ${\bullet\atop\bullet}$}}$$ for all $n \in \Bbb Z$.
It is an instructive and not-so-trivial (classical) exercise to verify by direct computation that the operators (\[Lnnormal\]) indeed satisfy the bracket relations (\[Virbrackets\]). (This exercise and the related constructions are presented in [@FLM], for example, where the standard generalization of this construction of ${\frak
{v}}$ using a Heisenberg algebra based on a finite-dimensional space of operators $h(n)$ for each $n$ is also carried out.)
Vertex operator algebra theory and conformal field theory place this exercise into a very general, natural setting (among many other things), with conceptual approaches and techniques (cf. [@FLM]). It is standard procedure to embed operators such as $h(n)$ and $L(n)$ into generating functions and to compute with these generating functions, using a formal calculus, and to systematically avoid computing with the individual operators. Doing this vastly simplifies computations that would otherwise be complicated or sometimes almost impossible. In fact, we shall be using a number of generating-function ideas below.
The space $S$ carries a natural $\Bbb Z$-grading, determined by the rule ${\rm deg}\ h(j) = j$ for $j < 0$. Then $S$ is in fact graded by the nonpositive integers, and the ${\frak {v}}$-module $S$ is a graded module. It turns out to be appropriate to use the negative of this grading, that is, to define a new grading (by “conformal weights”) on the space $S$ by the rule ${\rm wt}\ h(-j) = j$ for $j > 0$. One reason why this is natural is that for each $n \geq 0$, the homogeneous subspace of $S$ of weight $n$ coincides with the eigenspace of the operator $L(0)$ with eigenvalue $n$, as is easy to see. For $n \in {\Bbb Z}$ (or $n \geq 0$) we define $S_n$ to be the homogeneous subspace of $S$ of weight $n$, and we consider the formal power series in the formal variable $q$ given by $$\mbox{dim}_* S = \sum_{n \geq 0} (\mbox{dim}\;S_n) q^n$$ (the “graded dimension” of the graded space $S$). Clearly, from the definitions, $$\mbox{dim}_* S = \prod_{n > 0} (1 - q^n)^{-1}.$$
Here are the main points about these classical considerations that we want to emphasize: As is well known in conformal field theory, removing the normal ordering in the definition of the operator $L(0)$ introduces an infinity which formally equals $\frac{1}{2} \zeta(-1)$: The unrigorous expression $$\label{Lbar(0)}
{\bar L}(0) = \frac{1}{2} \sum_{j \in \Bbb Z} h(-j)h(j)$$ formally equals (by the bracket relations (\[Heisbrackets\])) $$\label{Lbar(0)2}
L(0) + \frac{1}{2} (1 + 2 + 3 + \cdots),$$ which itself formally equals $$\label{Lbar(0)3}
L(0) + \frac{1}{2} \zeta(-1) = L(0) - \frac{1}{24}.$$ We rigorize ${\bar L}(0)$ by defining it as: $$\label{Lbar(0)rig}
{\bar L}(0) = L(0) + \frac{1}{2} \zeta(-1),$$ and we define $$\label{Lbar(n)}
{\bar L}(n) = L(n) \ \ \mbox{for} \ n \neq 0,$$ to get a new basis of the Lie algebra ${\frak {v}}$. (We are identifying the elements of ${\frak {v}}$ with operators on the space $S$.) The brackets become: $$\label{newVirbrackets}
[{\bar L}(m),{\bar L}(n)] = (m-n){\bar L}(m+n) + \frac{1}{12} m^3
\delta_{m+n,0};$$ that is, $m^3 - m$ in (\[Virbrackets\]) has become the pure monomial $m^3$.
Perhaps the most important (and also well-known) thing accomplished by this formal removal of the normal ordering is the following: We define a new grading of the space $S$ by using the eigenvalues of the modified operator ${\bar L}(0)$ in place of $L(0)$, so that the grading of $S$ is “shifted” from the previous grading by conformal weights by the subtraction of $\frac{1}{24}$ from the weights. We let $\chi (S)$ be the corresponding graded dimension, so that $$\label{chiS}
\chi (S) = \frac{1}{\eta (q)},$$ where $$\eta (q) = q^{\frac{1}{24}} \prod_{n > 0} (1 - q^n).$$ The point is that $\eta (q)$ has important (classical) modular transformation properties, unlike $\prod_{n > 0} (1 - q^n)$, when viewed as a function of $\tau$ in the upper half-plane via the substitution $q = e^{2 \pi i \tau}$; $\eta (q)$ is Dedekind’s eta-function.
Bloch [@Bl] extended this classical story in various ways, in particular, the following: Instead of the Lie algebra $\frak {d}$, we consider the larger Lie algbebra of formal differential operators, spanned by $$\{ t^n D^m | n \in \Bbb Z, \ m \geq 0 \}$$ or more precisely, we restrict to $m > 0$ and further, to the Lie subalgebra ${\cal D}^+$ spanned by the differential operators of the form $D^r (t^nD) D^r$ for $r \geq 0, \ n \in \Bbb Z$. Then we can construct a central extension of ${\cal D}^+$ using generalizations of the normally-ordered quadratic operators (\[Lnnormal\]) above: $$\label{Lrn}
L^{(r)}(n) = \frac{1}{2} \sum_{j \in \Bbb Z} j^r h(j) (n-j)^r h(n-j) \
\ \mbox{for} \ n \neq 0,$$ $$\label{Lr0}
L^{(r)}(0) = \frac{1}{2} \sum_{j \in \Bbb Z} (-j)^r h(-|j|) j^r
h(|j|),$$ that is, $$\label{Lrnnormal}
L^{(r)}(n) = {\mbox{\tiny ${\bullet\atop\bullet}$}}\frac{1}{2} \sum_{j \in \Bbb Z} j^r h(j) (n-j)^r
h(n-j){\mbox{\tiny ${\bullet\atop\bullet}$}}$$ for $n \in \Bbb Z$. Direct computation of the commutators among these operators [@Bl] shows that they provide a central extension of ${\cal D}^+$ such that $$L^{(r)}(n) \mapsto (-1)^{r+1} D^r (t^nD) D^r$$ (cf. [@KP]). (It is not surprising in retrospect that these operators $L^{(r)}(n)$ are related to differential operators, because the generating function of these operators as $n$ ranges through $\Bbb
Z$ is based on $D^r$, as we discuss below.)
A central point of [@Bl] is that the formal removal of the normal-ordering procedure in the definition (\[Lr0\]) of $L^{(r)}(0)$ adds the infinity $(-1)^r \frac{1}{2} \zeta(-2r-1) =$ “ $\sum_{n>0} n^{2r+1}$” (generalizing (\[Lbar(0)\])–(\[Lbar(0)3\])), and if we correspondingly define $$\label{lbarr(0)}
{\bar L}^{(r)} (0) = L^{(r)} (0) + (-1)^r \frac{1}{2} \zeta(-2r-1)$$ and ${\bar L}^{(r)} (n) = L^{(r)} (n)$ for $n \neq 0$ (generalizing (\[Lbar(0)rig\]) and (\[Lbar(n)\])), the commutators simplify in a remarkable way: As direct computation [@Bl] shows, the complicated polynomial in the scalar term of $[{\bar L}^{(r)} (m),{\bar L}^{(s)}
(-m)]$ reduces to a pure monomial in $m$, by analogy with, and generalizing, the passage from $m^3 - m$ to $m^3$ in (\[newVirbrackets\]). The precise formulas can be found in [@Bl], along with further results; for instance, in [@Bl], these considerations and results are generalized to Dirichlet $L$-series in place of the zeta function.
First “explanation” and generalization
======================================
Our goal is to present two layers of “explanation” and generalization of the results of [@Bl] sketched above. First we need some elementary formal background:
What does $\zeta(-2r-1)$ “mean,” for a nonnegative integer $r$?
It is a well-known classical fact that for $k > 1$, $$\label{zetaB}
\zeta(-k+1) = - \frac{B_{k}}{k},$$ where the $B_{k}$ are the Bernoulli numbers, defined by the generating function $$\label{Ber}
\frac{x}{e^x - 1} = \sum_{k \geq 0} \frac{B_{k}}{k!} x^k,$$ where $x$ is a formal variable. This formal power series in $x$ is understood to be computed (on the left-hand side) by expanding $e^x -
1$ as the formal series $x + \frac{x^2}{2!} + \cdots$ and performing the division of formal power series to obtain a formal power series with constant term $1$; this of course corresponds to expanding a complex function in a certain domain, but we are operating purely formally.
Why does $B_k$ defined in this way have anything to do with the formal series $\sum_{n>0} n^{k-1}$? We recall Euler’s heuristic interpretation of such formal sums as “$1 + 2 + \cdots$” (cf. the Preface of [@Hi]); actually, we give a variant of Euler’s interpretation adapted to the main theme that we shall introduce:
Consider the expansion (from (\[Ber\])) $$\label{fromBer}
\frac{1}{1 - e^x} = - \sum_{k \geq 0} \frac{B_{k}}{k!} x^{k-1}.$$ Expand the left-hand side [*unrigorously*]{} as the formal geometric series $$\label{unrig}
1 + e^x + e^{2x} + \cdots = 1 + \sum_{k \geq 0} \frac{1^k}{k!} x^k +
\sum_{k \geq 0} \frac{2^k}{k!} x^k + \cdots.$$ For $k>1$, the coefficient of $x^{k-1}$ in this formal expression is $$\frac{1}{(k-1)!} (1^{k-1} + 2^{k-1} + \cdots),$$ which looks like $\frac{1}{(k-1)!} \zeta (-k+1)$. Also, the coefficient of $x^0$ in (\[unrig\]) is formally $1 + \frac{1}{0!}
(1^0 + 2^0 + \cdots)$, which we formally view as $1 + \zeta (0)$ (and not as $\zeta (0)$). Thus, formally equating the coefficients of $x^l$ for $l \geq 0$ in (\[Ber\]) “explains” (\[zetaB\]) and the fact that $\zeta(0) = - B_1 - 1$ ($= - \frac{1}{2}$); now we know what (\[zeta-s\]) says.
The key point here is the interplay between the formal geometric series expansion (in powers of $e^x$) and the expansion in powers of $x$.
Now, how do we interpret all of this via vertex operator algebra theory?
First note that the expressions (\[Lrn\])–(\[Lrnnormal\]) above for $L^{(r)}(n)$ suggest $r^{\rm th}$ derivatives. We have already mentioned that a basic theme in vertex operator algebra theory is to always use appropriate generating functions (as we just did, incidentally, in the heuristic discussion above). First we put our individual operators into generating functions. Using a formal variable $x$, we define $$\label{h(x)}
h(x) = \sum_{n \in \Bbb Z} h(n)x^{-n}$$ and $$L^{(r)}(x) = \sum_{n \in \Bbb Z} L^{(r)}(n)x^{-n},$$ and using $D_x$ to denote the operator $x \frac{d}{dx}$ (recall the comment after (\[D\])), we observe that $$\label{Lrx}
L^{(r)}(x) = {\frac{1}{2}}{\mbox{\tiny ${\bullet\atop\bullet}$}}(D_x^r h(x))^2{\mbox{\tiny ${\bullet\atop\bullet}$}},$$ where the colons, as always, denote normal ordering (recall (\[Lnnormal\])). (For other purposes, other versions of these generating functions are used, in particular, $h(x) = \sum_{n \in \Bbb
Z} h(n)x^{-n-1}$, as in (\[Yh(-1)\]) below, in place of (\[h(x)\]), but we have chosen the appropriate generating functions for our purposes.)
Using standard elementary techniques, we could directly compute the brackets $[L^{(r)}(x_1),L^{(s)}(x_2)]$ of these generating functions, for $r,s \ge 0$, where $x_1$ and $x_2$ are independent commuting formal variables. (As always in vertex operator algebra theory or conformal field theory, when we consider such operations as brackets of generating functions, we need independent commuting formal variables; the expression $[L^{(r)}(x),L^{(s)}(x)]$, with the variable $x$ repeated, would be meaningless.) But this computation, which might be carried out as a more complicated variant of the argument on pp. 224–226 of [@FLM], for example, would not be simple. It would of course recover the information of the brackets $[L^{(r)}(m),L^{(s)}(n)]$ computed in [@Bl].
The best use of generating functions in this context is instead to also introduce suitable generating functions over the number of [*derivatives*]{}. Consider the elementary formal Taylor theorem equating the application of a formal exponential of a formal multiple of $\frac{d}{dx}$ with a formal substitution operation: $$\label{Taylor}
e^{y \frac{d}{dx}} f(x) = f(x+y),$$ where $f(x)$ is an arbitrary formal series of the form $\sum_n a_n
x^n$, and where it is understood that each binomial expression $(x+y)^n$ is to be expanded in nonnegative integral powers of $y$. Here $n$ is allowed to range over something very general, like $\Bbb
Z$ or even $\Bbb C$, say, and the $a_n$ lie in a fixed vector space; $f(x)$ is very definitely not necessarily the expansion of an analytic function. Formula (\[Taylor\]) is proved by direct formal expansion of both sides (cf. [@FLM], Proposition 8.3.1; Taylor’s theorem in this kind of generality is heavily exploited in Chapter 8 of [@FLM], for instance). Now $\frac{d}{dx}$ is of course a formal infinitesimal translation (as (\[Taylor\]) states), but for our present purposes we want the following formal multiplicative analogue of (\[Taylor\]): $$\label{infinitdil}
e^{y D_x} f(x) = f(e^y x),$$ with $f(x)$ as above (again cf. [@FLM], Proposition 8.3.1), which expresses the fact that $D_x$ is a formal infinitesimal dilation.
Now ${\mbox{\tiny ${\bullet\atop\bullet}$}}(D_x^r h(x))^2{\mbox{\tiny ${\bullet\atop\bullet}$}}$ (recall (\[Lrx\])) is hard to put into a “good” generating function over $r$, but we can make the problem easier by making it more general: Consider independently many derivatives on each of the two factors $h(x)$ in ${\mbox{\tiny ${\bullet\atop\bullet}$}}h(x)^2{\mbox{\tiny ${\bullet\atop\bullet}$}}$, use two new independent formal variables $y_1$ and $y_2$, and form the generating function $$\label{Ly1y2}
L^{(y_1,y_2)}(x) = {\frac{1}{2}}
{\mbox{\tiny ${\bullet\atop\bullet}$}}(e^{y_1 D_x} h(x))(e^{y_2 D_x}
h(x)){\mbox{\tiny ${\bullet\atop\bullet}$}}= {\frac{1}{2}}
{\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_1} x)h(e^{y_2} x){\mbox{\tiny ${\bullet\atop\bullet}$}}$$ (where we use (\[infinitdil\])), so that $L^{(r)}(x)$ is a “diagonal piece” of this generating function in the sense that it is $(r!)^2$ times the coefficient of $y_1^r y_2^r$ in $L^{(y_1,y_2)}(x)$. Using formal vertex operator calculus techniques (generalizing the argument on pp. 224-226 of [@FLM], for example), we can calculate $$\label{bracketofquadratics}
[{\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_1} x_1)h(e^{y_2} x_1){\mbox{\tiny ${\bullet\atop\bullet}$}},{\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_3} x_2)h(e^{y_4} x_2){\mbox{\tiny ${\bullet\atop\bullet}$}}].$$
Then, a nontrivial, and in fact quite tricky, vertex operator computation recovers Bloch’s formulas, in somewhat generalized form, as we explain next. Here are the main points:
The expression $h(x)^2$ is not rigorous (as we observe for instance by trying to compute the coefficient of any fixed power of $x$ in this expression), while the normal-ordered expression ${\mbox{\tiny ${\bullet\atop\bullet}$}}h(x)^2{\mbox{\tiny ${\bullet\atop\bullet}$}}$ is of course rigorous. The expression $h(e^{y_1}
x)h(e^{y_2} x)$ is still not rigorous (even though the expressions $e^{y_1} x$ and $e^{y_2} x$ are distinct), as we see by (for example) trying to compute the constant term in the variables $y_1$ and $y_2$ in this expression. The lack of rigor in this expression in fact corresponds exactly to the occurrence of formal sums like $\sum_{n>0}
n^r$ with $r>0$, as we have been discussing above.
However, we have $$\label{hx1hx2}
h(x_1)h(x_2) = {\mbox{\tiny ${\bullet\atop\bullet}$}}h(x_1)h(x_2){\mbox{\tiny ${\bullet\atop\bullet}$}}+ x_2
\frac{\partial}{\partial x_2}
\frac{1}{1 - x_2 / x_1}$$ (an exercise using elementary vertex operator techniques), and it follows that $$\label{hex1hex2}
h(e^{y_1} x_1)h(e^{y_2} x_2) = {\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_1} x_1)h(e^{y_2}
x_2){\mbox{\tiny ${\bullet\atop\bullet}$}}+ x_2
\frac{\partial}{\partial x_2} \frac{1}{1 - e^{y_2} x_2 / e^{y_1}
x_1}.$$ Note that $x_2\frac{\partial}{\partial x_2}$ can be replaced by $-
\frac{\partial}{\partial y_1}$ in the last expression (and this is one illustration of the naturalness of our emphasis on the “homogeneous” differential operator $D_x = x\frac{\partial}{\partial x}$ rather than $\frac{\partial}{\partial x}$). The expression $\frac{1}{1 - e^{y_2}
x_2 / e^{y_1} x_1}$ came from, and is, a geometric series expansion (recall (\[hx1hx2\])).
Now we try to set $x_1 = x_2 \ (= x)$ in (\[hex1hex2\]). The result of this procedure is unrigorous on the left-hand side, as we have pointed out above, [*but the result has rigorous meaning on the right-hand side*]{}, because the normal-ordered product ${\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_1} x)h(e^{y_2} x){\mbox{\tiny ${\bullet\atop\bullet}$}}$ is certainly well defined, and the expression $-
\frac{\partial}{\partial y_1} \frac{1}{1 - e^{-y_1 + y_2}}$ can be interpreted rigorously as in (\[Ber\]) and (\[fromBer\]); more precisely (the role of $x$ in (\[fromBer\]) being played here by $-y_1 + y_2$), we take $\frac{1}{1 - e^{-y_1 + y_2}}$ to mean the formal (Laurent) series in $y_1$ and $y_2$ of the shape $$\label{defof1/1-e}
\frac{1}{1 - e^{-y_1 + y_2}} = (y_1 - y_2)^{-1}F(y_1,y_2),$$ where $(y_1 - y_2)^{-1}$ is understood as the binomial expansion (geometric series) in nonnegative powers of $y_2$ and $F(y_1,y_2)$ is an (obvious) formal power series in (nonnegative powers of) $y_1$ and $y_2$. This motivates us to define a new “normal-ordering” procedure $$\label{hexhex}
{\mbox{\scriptsize ${+ \atop +}$}}h(e^{y_1} x)h(e^{y_2} x){\mbox{\scriptsize ${+ \atop +}$}}=
{\mbox{\tiny ${\bullet\atop\bullet}$}}h(e^{y_1} x)h(e^{y_2} x){\mbox{\tiny ${\bullet\atop\bullet}$}}-
\frac{\partial}{\partial y_1} \frac{1}{1 - e^{-y_1 + y_2}},$$ with the last part of the right-hand side being understood as we just indicated. Again compare this with the heuristic discussion above; this expression came from a geometric series, but it becomes rigorous only when we expand in the new way (actually, we might alternatively replace the binomial expansion $(y_1 - y_2)^{-1}$ by the different expansion of the same formal expression in nonnegative powers of $y_1$ rather than of $y_2$, but it is more natural to make the choice that we did).
Formula (\[hexhex\]) and its indicated interpretation give a natural “explanation” of the zeta-function-modified operators defined in (\[lbarr(0)\]): We use (\[hexhex\]) to define the following analogues of the operators (\[Ly1y2\]): $$\label{Lbary1y2}
{\bar L}^{(y_1,y_2)}(x) =
{\frac{1}{2}}
{\mbox{\scriptsize ${+ \atop +}$}}h(e^{y_1} x)h(e^{y_2} x){\mbox{\scriptsize ${+ \atop +}$}},$$ and it is easy to check that the operator ${\bar L}^{(r)} (n)$ is exactly $(r!)^2$ times the coefficient of $y_1^r y_2^r x_0^{-n}$ in (\[Lbary1y2\]); the significant case is the case $n = 0$.
We are now ready to formulate the result mentioned just after (\[bracketofquadratics\]) above. With the new normal ordering (\[hexhex\]) replacing the old one, remarkable cancellation occurs in the commutator (\[bracketofquadratics\]), and the result is:
\[theoremforLbar\] With the formal delta-function Laurent series $\delta(x)$ defined as $$\label{delta}
\delta(x) = \sum_{n \in {\Bbb Z}} x^n,$$ and with independent commuting formal variables as indicated, we have: $$\begin{aligned}
\label{Lbarbrackets}
\lefteqn{[{\bar L}^{(y_1,y_2)}(x_1),{\bar L}^{(y_3,y_4)}(x_2)]}\nonumber\\
&&= - {\frac{1}{2}} \frac{\partial}{\partial y_1}
\biggl({\bar L}^{(-y_1+y_2+y_3,y_4)}(x_2)
\delta \left({\frac{e^{y_1}x_1}{e^{y_3}x_2}}\right)\nonumber\\
&&\hspace{2em} + {\bar L}^{(-y_1+y_2+y_4,y_3)}(x_2)
\delta \left({\frac{e^{y_1}x_1}{e^{y_4}x_2}}\right)\biggr)\nonumber\\
&&\hspace{2em} - {\frac{1}{2}} \frac{\partial}{\partial y_2}
\biggl({\bar L}^{(y_1-y_2+y_3,y_4)}(x_2)
\delta \left({\frac{e^{y_2}x_1}{e^{y_3}x_2}}\right)\nonumber\\
&&\hspace{2em} + {\bar L}^{(y_1-y_2+y_4,y_3)}(x_2)
\delta \left({\frac{e^{y_2}x_1}{e^{y_4}x_2}}\right)\biggr).\end{aligned}$$
Actually, hidden in the right-hand side of (\[Lbarbrackets\]) are formal expressions (suitably expanded) of the type $(y_1 - y_2 - y_3 +
y_4)^{-3}$ because of the formal pole $y_1 = y_2$ in (\[hexhex\]), and these expressions, multiplied by the formal delta-function expressions, are the source of the pure monomials in $m$ that we set out to explain (recall the end of Section 2). Indeed, the delta-function expression $\delta
\left(e^{y_1}x_1 / e^{y_3}x_2 \right)$, for instance, can be written as $e^{y_1 D_{x_1}}e^{y_3 D_{x_2}}\delta
\left(x_1 / x_2 \right)$, and when we extract and equate the coefficients of the monomials in the variables $y_1^r y_2^r y_3^s
y_4^s$ on the two sides of (\[Lbarbrackets\]), we get expressions like $(D^j \delta)\left(x_1 / x_2 \right)$, whose expansion, in turn, in powers of $x_1$ and $x_2$ clearly yields a pure monomial analogous to and generalizing the expression $m^3$ in (\[newVirbrackets\]). In this way, we have an explicit generalization and explanation of Bloch’s formula for $[{\bar L}^{(r)}
(m),{\bar L}^{(s)} (n)]$ in terms of the operators ${\bar L}^{(j)}
(m+n)$ and a monomial in $m$.
We have been working all along with a Heisenberg algebra based on a one-dimensional space—that is, a Heisenberg algebra with only one dimension of operators, spanned by the element $h(n)$, for each $n$, but all of these considerations hold equally well in the more general situation where we start with a Heisenberg algebra based on a finite-dimensional space.
All of this has been an interesting (and nontrivial) special case of vertex operator algebra theory, but what is [*really*]{} happening?
Second “explanation” and generalization
=======================================
It turns out that Theorem \[theoremforLbar\] is an extremely special case of a something that can be formulated and proved for an arbitrary vertex operator algebra (and indeed this gives another motivation for the general theory). We recall the definition of the notion of vertex (operator) algebra from [@Bo], [@FLM] and [@FHL]; the principles that we have found are based heavily on the “Jacobi identity” as formulated in [@FLM] and [@FHL]:
\[VOA\] [A [*vertex operator algebra*]{} $(V, Y, {\bf 1}, \omega)$, or simply $V$ (over ${\Bbb C}$), is a ${\Bbb
Z}$-graded vector space (graded by [*weights*]{}) $$V=\coprod_{n\in {\Bbb Z}}V_{(n)}; \ \mbox{\rm for}\ v\in V_{(n)},\;n=\mbox{\rm wt}\ v;$$ such that $$\mbox{\rm dim }V_{(n)}<\infty\;\;\mbox{\rm for}\; n \in {\Bbb Z},$$ $$V_{(n)}=0\;\;\mbox{\rm for} \;n\; \mbox{\rm sufficiently small},$$ equipped with a linear map $V\otimes V\to V[[x, x^{-1}]]$, or equivalently, $$\begin{aligned}
V&\to&(\mbox{\rm End}\; V)[[x, x^{-1}]]\nonumber \\
v&\mapsto& Y(v, x)={\displaystyle \sum_{n\in{\Bbb Z}}}v_{n}x^{-n-1}
\;\;(\mbox{\rm where}\; v_{n}\in
\mbox{\rm End} \;V),\end{aligned}$$ $Y(v, x)$ denoting the [*vertex operator associated with*]{} $v$, and equipped also with two distinguished homogeneous vectors ${\bf 1}\in
V_{(0)}$ (the [*vacuum*]{}) and $\omega \in V_{(2)}$. The following conditions are assumed for $u, v \in V$: the [*lower truncation condition*]{} holds: $$u_{n}v=0\;\;\mbox{\rm for}\;n\; \mbox{\rm sufficiently large}$$ (or equivalently, $Y(u,x)v$ involves only finitely many negative powers of $x$); $$Y({\bf 1}, x)=1\;\; (1\;\mbox{\rm on the right being the identity
operator});$$ the [*creation property*]{} holds: $$Y(v, x){\bf 1} \in V[[x]]\;\;\mbox{\rm and}\;\;\lim_{x\rightarrow
0}Y(v, x){\bf 1}=v$$ (that is, $Y(v, x){\bf 1}$ involves only nonnegative integral powers of $x$ and the constant term is $v$); with $\delta(x)$ as in (\[delta\]) and with binomial expressions understood (as above) to be expanded in nonnegative powers of the second variable, the [*Jacobi identity*]{} (the main axiom) holds: $$\begin{aligned}
\label{jacobi}
&x_{0}^{-1}\delta
\left({\displaystyle\frac{x_{1}-x_{2}}{x_{0}}}\right)Y(u, x_{1})Y(v,
x_{2})-x_{0}^{-1} \delta
\left({\displaystyle\frac{x_{2}-x_{1}}{-x_{0}}}\right)Y(v, x_{2})Y(u,
x_{1})&\nonumber \\ &=x_{2}^{-1} \delta
\left({\displaystyle\frac{x_{1}-x_{0}}{x_{2}}}\right)Y(Y(u, x_{0})v,
x_{2})&\end{aligned}$$ (note that when each expression in (\[jacobi\]) is applied to any element of $V$, the coefficient of each monomial in the formal variables is a finite sum; on the right-hand side, the notation $Y(\cdot, x_{2})$ is understood to be extended in the obvious way to $V[[x_{0}, x^{-1}_{0}]]$); the Virasoro algebra relations hold (acting on $V$): $$[L(m), L(n)]=(m-n)L(m+n)+{\displaystyle\frac{1}{12}}
(m^{3}-m)\delta_{n+m,0}({\rm rank}\;V)1$$ for $m, n \in {\Bbb Z}$, where $$L(n)=\omega _{n+1}\;\; \mbox{\rm for} \;n\in{\Bbb Z}, \;\;{\rm
i.e.},\;\;Y(\omega, x)=\sum_{n\in{\Bbb Z}}L(n)x^{-n-2}$$ and $$\begin{aligned}
&{\rm rank}\;V\in {\Bbb C};&\\
&L(0)v=nv=(\mbox{\rm wt}\ v)v\;\;\mbox{\rm for}\;n \in {\Bbb
Z}\;\mbox{\rm and}\;v\in V_{(n)};&\\
&{\displaystyle \frac{d}{dx}}Y(v,x)=Y(L(-1)v, x)&\end{aligned}$$ (the [*$L(-1)$-derivative property*]{}).]{}
Note that as in Theorem \[theoremforLbar\], the Jacobi identity encapsulates higher derivatives of delta-function expressions, since the expression $\delta ((x_1 - x_2)/x_0)$, say, can be expanded by means of (\[Taylor\]). The use of the three formal variables, rather than complex variables (which would require changes in the formulas), allows the full symmetry of the Jacobi identity to reveal itself (see [@FLM] and [@FHL]).
The polynomial algebra $S$ (recall (\[S\])) carries a canonical vertex operator algebra structure of rank 1 with vacuum vector ${\bf
1}$ equal to $1 \in S$ and with the operators $L(n)$ agreeing with the operators defined in Section 2 (cf. [@FLM]). We will not describe the vertex operators $Y(v,x)$ here, except to say that $$\label{Yh(-1)}
Y(h(-1),x) = x^{-1}h(x) = \sum_{n \in \Bbb Z} h(n)x^{-n-1}$$ (recall (\[h(x)\])) and that the element $\omega$ is $\frac{1}{2}
(h(-1))^{2} \in S$.
If we omit the grading and the Virasoro algebra from Definition \[VOA\] and adjust the axioms appropriately, we have the notion of “vertex algebra” as introduced in [@Bo], except that Borcherds used certain special cases of the Jacobi identity instead of (\[jacobi\]). The identity (\[jacobi\]) is the canonical “maximal” axiom: It contains the “full” necessary information in compact form; it is analogous to the classical Jacobi identity in the definition of the notion of Lie algebra; and it is invariant in a natural sense under the symmetric group on three letters (see [@FLM] and [@FHL]).
There are also “minimal” axioms, stemming from the fact that the (suitably formulated) “commutativity” of the operators $Y(u,x_1)$ and $Y(v,x_2)$ implies “associativity” (again suitably formulated) and hence the Jacobi identity (see [@FLM] and [@FHL]; cf. [@BPZ] and [@G]). The simplest “minimal” axiom, as found in [@DL] (actually, in the greater generality of “abelian intertwining algebras”) states that for $u,v
\in V$, there exists $n \ge 0$ such that $$\label{weakcomm}
(x_1 - x_2)^n [Y(u,x_1),Y(v,x_2)] = 0$$ (see [@DL], formula (1.4)). However, it is still a nontrivial matter to construct examples, even relatively simple ones like $S$, of vertex operator algebras, partly because one has to extend the condition (\[weakcomm\]) from generators to arbitrary elements of $V$. A general and systematic approach and solution to this and related problems was first found by Li (see [@Li1], [@Li2]).
The commutativity condition asserts that for $u,v \in V$, $$Y(u,x_1)Y(v,x_2) \sim Y(v,x_2)Y(u,x_1),$$ where “$\sim$” denotes equality up to a suitable kind of generalized analytic continuation, and the associativity condition asserts that $$\label{assoc}
Y(u,x_1)Y(v,x_2) \sim Y(Y(u,x_1 - x_2)v,x_2),$$ where the right-hand side and the generalized analytic continuation have to be understood in suitable ways (again see [@FLM] and [@FHL] and cf. [@BPZ] and [@G]); the right-hand side of (\[assoc\]) is [*not*]{} a well-defined formal series in $x_1$ and $x_2$.
On the level of these basic principles, for any vertex operator algebra $V$ we shall now conceptually formulate and considerably generalize the normal-ordering procedure (\[hexhex\]) and we shall formulate a new general “Jacobi identity” which implies Theorem \[theoremforLbar\] in the very particular case of the vertex operator algebra $S$ and very special vertex operators. Ideas in Zhu’s work [@Z1], [@Z2] enter into our considerations.
Formally replacing $x_1$ by $e^{y}x_2$ in (\[assoc\]), we find (formally and unrigorously) that $$\label{assoc2}
Y(u,e^{y}x_2)Y(v,x_2) \sim Y(Y(u,(e^{y}-1)x_2)v,x_2).$$ Now we observe that while the left-hand side of (\[assoc2\]) is not a well-defined formal series in the formal variables $y$ and $x_2$, the right-hand side of (\[assoc2\]) [*is*]{} in fact a well-defined formal series in these formal variables. By replacing $x_1$ by $e^{y}x_2$ we have made the right-hand side of (\[assoc2\]) rigorous (and the left-hand side unrigorous). This situation should be compared with our motivation for introducing the normal-ordering procedure (\[hexhex\]) above.
Next, instead of the vertex operators $Y(v,x)$, we want the modified vertex operators defined for homogeneous elements $v
\in V$ by: $$X(v,x) = x^{{\rm wt}\;v}Y(v,x) = Y(x^{L(0)}v,x),$$ as in [@FLM], formula (8.5.27) (recall that $L(0)$-eigenvalues define the grading of $V$); the formula $X(v,x) = Y(x^{L(0)}v,x)$ works for [*all*]{} $v \in V$ (not necessarily homogeneous). For instance, the operator $h(x)$ (\[h(x)\]) is exactly $X(h(-1),x)$ (cf. (\[Yh(-1)\])).
What we really want is a rigorous expression that “equals” the unrigorous expression $X(u,e^{y}x_2)X(v,x_2)$; this will considerably generalize our interpretation of the unrigorous expression $h(e^{y}
x_2)h(x_2)$ (cf. (\[hex1hex2\]) and (\[hexhex\])). So we replace $u$ by $(e^{y}x_2)^{L(0)}u$ and $v$ by $x_2^{L(0)}v$ in (\[assoc2\]) and using basic techniques we get: $$\label{assoc3}
X(u,e^{y}x_2)X(v,x_2) \sim X(Y(e^{yL(0)}u,e^{y}-1)v,x_2),$$ and this right-hand side is still rigorous (and the left-hand side still unrigorous). But $Y(e^{yL(0)}u,e^{y}-1)$ is exactly Zhu’s operator $Y[u,y]$ in [@Z1], [@Z2], so that $$\label{assoc4}
X(u,e^{y}x_2)X(v,x_2) \sim X(Y[u,y]v,x_2).$$ By Zhu’s change-of-variables theorem, $x \mapsto Y[u,x]$ defines a new vertex operator algebra structure on the same vector space $V$ under suitable conditions; this theorem was a step in Zhu’s vertex-operator-algebraic proof of the modular-invariance properties of “characters” (cf. the comments surrounding (\[chiS\]) above). There have been two subsequent treatments of this change-of-variables theorem, in [@L1] and in [@H1], [@H2]; in the latter works, Huang considerably generalized this result (and removed a hypothesis of Zhu’s) using his geometric analysis of the Virasoro algebra structure in a vertex operator algebra. The formal relation (\[assoc4\]) generalizes to products of several operators.
So we want to bracket the (rigorous) expressions $X(Y[u,y]v,x)$, which are the correct natural generalization of the expression (\[hexhex\]) above (at least with $y_2 = 1$ in (\[hexhex\]), but this restriction is a minor issue since $y_2$ can easily be restored). Keep in mind that the formal relation (\[assoc4\]) naturally implements the formal relation (\[zeta-s\]) in a foundational way from the viewpoint of vertex operator algebras.
But just as in [@FLM], where the Jacobi identity for vertex operator algebras was the most natural general formula, here we find that the most natural thing to do is to formulate and prove a new Jacobi identity, based on the operators $X(Y[u,y]v,x)$, in the general setting of an arbitrary vertex (operator) algebra, rather than just to bracket the operators. It turns out that delta-function expressions of the type appearing on the right-hand side of (\[Lbarbrackets\]), and that in turn “explained” the phenomenon of pure monomials in $m$ (as discussed above), arise naturally in complete generality, and when we ask for a Jacobi identity rather than just a commutator formula in general, we find that delta-function expressions of this same type appear on the [*left-hand side*]{} as well as the right-hand side. This is another instance of the naturalness of “Jacobi identities,” which have symmetries that commutator formulas do not have. We state our result for the operators $X(v,x)$ rather than $X(Y[u,y]v,x)$ (i.e., the case where $u$ is the vacuum vector):
\[newjacobithm\] In any vertex (operator) algebra $V$, for $u,v \in V$ we have: $$\begin{aligned}
\label{newjacobiiden}
&x_{0}^{-1}\delta
\left(e^{y_{21}}{\displaystyle\frac{x_{1}}{x_{0}}}\right)X(u, x_{1})X(v,
x_{2})-x_{0}^{-1} \delta
\left(-e^{y_{12}}{\displaystyle\frac{x_{2}}{x_{0}}}\right)X(v, x_{2})X(u,
x_{1})&\nonumber \\ &=x_{2}^{-1} \delta
\left(e^{y_{01}}{\displaystyle\frac{x_{1}}{x_{2}}}\right)X(Y[u, -y_{01}]v,
x_{2}),&\end{aligned}$$ where $$y_{21} = \log \left(1-{\displaystyle\frac{x_{2}}{x_{1}}}\right),\;\;
y_{12} = \log \left(1-{\displaystyle\frac{x_{1}}{x_{2}}}\right),\;\;
y_{01} = \log \left(1-{\displaystyle\frac{x_{0}}{x_{1}}}\right).\nonumber$$
If we want the [*commutator*]{} $[X(u, x_{1}),X(v, x_{2})]$, we simply extract the coefficient of $x_0^{-1}$ (the formal residue in the variable $x_0$) on both sides, and it turns out that the resulting right-hand side can be put into an elegant form. If we replace $u$ and $v$ by expressions of the shape $Y[u,y]v$, we obtain naturally a formula that generalizes formula (\[Lbarbrackets\]) (Theorem \[theoremforLbar\]) to arbitrary elements of arbitrary vertex (operator) algebras. That is, interesting as they are, the phenomena that we have been discussing concerning central extensions of Lie algebras of differential operators form extremely special cases of general vertex-operator-algebraic phenomena. The detailed formulations and proofs, and generalizations, are found in [@L2]. Also, my student Antun Milas has generalized some of these results in a number of directions.
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[Department of Mathematics, Rutgers University, Piscataway, NJ 08854]{}
[*E-mail address*]{}: [email protected]
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abstract: 'The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. In the case of abelian four-folds, recent work in gauge theory suggests an approach using Spin(7) instantons. I advertise a class of examples due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplication to exhibit Cayley cycles representing the given Hodge classes. I find alternate complex structures with respect to which the given bundles are holomorphic, and close with a suggestion (due to G. Tian) as to how this may possibly be put to use.'
address: 'Abdus Salam I.C.T.P., 11 Strada Costiera, Trieste 34014 Italy '
author:
- 'T. R. Ramadas'
title: 'Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal'
---
[^1]
Introduction
============
Let $X$ be a smooth complex projective variety of dimension $n$, and $c$ a rational $(p,p)$ cohomology class ($0<p<n$). The Hodge Conjecture is that
H
: there exist finitely many (reduced, irreducible) $(n-p)$-dimensional subvarieties $Y_i$ and rational numbers $a_i$ such that $c=\sum_i
a_i [Y_i]$, where $[Y_i]$ is the (rational) cohomology class dual to $Y_i$ . That is, $c$ is dual to a rational algebraic cycle.
This is equivalent to
V
: there exists a holomorphic vector bundle $E$ such that its Chern character $ch(E)$ is equal to a rational multiple of $c$ modulo (classes of) rational algebraic cycles.
The second statement implies the first because the Chern character of a holomorphic (and therefore algebraic) bundle factors through the Chow ring of algebraic varieties. The converse also holds. In fact, as Narasimhan pointed out to me, it is known ([@M]) that the rational Chow ring is generated by stable vector bundles.
Let $X$, $c$ be as above. By a theorem of Atiyah-Hirzebruch ([@A-H], page 19), the Chern character map $ch:K^0(X) \otimes
\mathbb{Q} \to H^{even} (X,\mathbb{Q})$ is a bijection, where $K^0(X)$ is the Grothendieck group of (topological/smooth) vector bundles on $X$. Thus we are assured of the existence of a smooth bundle $E$ and and an integer $n > 0$ such that $ch(E )=rank
(E)+nc$. A possible strategy to show that a given class $c$ is algebraic suggests itself – find a suitable such bundle $E$ and then exhibit a holomorphic structure on it. This note is written to argue that recent progress in mathematical gauge theory, and in particular the work of G. Tian and C. Lewis, makes this worth pursuing, at least in the case of certain abelian four-folds. Such an approach to the Hodge Conjecture for the case of Calabi-Yau four-folds is surely known to the experts (and this has been confirmed to me), but I have only been able to locate some coy references. Claire Voisin ([@V]), following a similar approach, has much more definitive *negative* results in the case of *non-algebraic* tori.
Before proceeding, let us note that the known “easy" cases of the Hodge conjecture are proved essentially by the above method. First, given an integral class $c \in H^2(X, \mathbf{Z})$, a smooth hermitian line bundle $L$ exists with (first) Chern class equal to $c$. Given any real 2-form $\Omega$ representing $c$ there exists an unitary connection on $L$ with curvature $-2\pi i \Omega$. If $c$ is a $(1,1)$ class, it can be represented by an $\Omega$ which is $(1,1)$. The corresponding connection defines a holomorphic structure on $L$. If $c$ is an integral $(n-1,n-1)$ class, the strong Lefschetz theorem exhibits the dual class as a rational linear combination of complete intersections.
What follows is the result of much trial and error and computations - which I either only sketch or omit altogether - using [*Mathematica*]{}; the notebooks are available on request. (I used an exterior algebra package of Sotirios Bonanos, available from `http://www.inp.demokritos.gr/~sbonano/.` )
Mumford’s examples
==================
We consider Hodge classes on certain abelian four-folds. These examples are due to Mumford ([@P]).
It is best to start with some preliminary algebraic number theory. If $F$ is an algebraic number field, with $degree\ F =d$, the ring of algebraic integers $\Lambda \equiv {\mathfrak o}_F$ is a free $\mathbb{Z}$-module of rank $d$ which generates $F$ as a $\mathbb{Q}$-vector space. If $V$ denotes the real vector space $\mathbb{R} \otimes_{\mathbb{Q}} F$, then $\Lambda \subset V$ is a lattice and $X_r=V/\Lambda$ is a real $d$-torus.
Let $L$ denote the Galois saturation of $F$ in $\bar{\mathbb{Q}}
\subset \mathbb{C}$. (That is, $L$ is the smallest subfield Galois over $\mathbb{Q}$ and containing any (and therefore all) embeddings of $F$.) Then $G=Gal(L/\mathbb{Q})$ acts transitively on the set $E$ of embeddings $\iota:L \hookrightarrow \mathbb{Q}$ by $(g,\iota)
\mapsto g(\iota) = g \circ \iota$ ($g \in G,\ \iota \in E$), and the image by $\iota$ is the fixed field of the stabiliser of $\iota$. Further, the map $$\bar{\mathbb{Q}} \otimes_{\mathbb{Q}} F \to \bar{\mathbb{Q}}^E$$ given by $1 \otimes x \mapsto (\iota(x))_E$ is an isomorphism of $\bar{\mathbb{Q}}$ vector spaces.
Turning to the real torus $X_r$:
1. we have natural isomorphisms $H_1(X_r,\mathbb{Z})=\Lambda$ and $H_1(X_r,\bar{\mathbb{Q}})=\bar{\mathbb{Q}}^E$;
2. $H^1(X_r,\bar{\mathbb{Q}})$ has basis $\{dt_\iota\}_E$, where $dt_\iota$ is induced by the projection to the $\iota^{th}$ factor from $\bar{\mathbb{Q}}^E$.
In what follows we will identify the real or complex cohomolgy of $X_r$ with the corresponding spaces of translation-invariant forms on $X_r$.
We will need the following result, whose proof is straightforward.
A one-form $\omega=\sum_\iota \omega_\iota dt_\iota$ represents a rational class iff the coefficients $\omega_\iota$ belong to $L$ and satisfy the equivariance $$\omega_{g(\iota)}=g(\omega_\iota),\ g \in G$$ Similarly, a two-form $\phi=\sum_{\iota,\kappa} \phi_{\iota,\kappa}
dt_{\iota} \wedge dt_{\kappa}$ (with the coefficients antisymmetric functions of the two indices) represents a rational class iff $$\phi_{g(\iota),g(\kappa)}=g(\phi_{\iota,\kappa}),\ g \in G$$
Suppose now that the embeddings $E$ occur in complex conjugate pairs - $E=E' \sqcup E''$, with each $\iota \in E'$ corresponding to $\bar{\iota} \in E''$. Then the map $$V=\mathbb{R} \otimes_{\mathbb{Q}} F (\hookrightarrow \mathbb{C}
\otimes_{\mathbb{Q}} F \sim \mathbb{C}^E) \to \mathbb{C}^{E'}$$ is an isomorphism of real vector spaces and induces a (translation-invariant) complex structure on $X_r$, which becomes a complex torus, which we will denote simply $X$.
We turn now to specifics. Let $P=ax^4+bx^2+cx+d$ be an irreducible polynomial with rational coefficients and all roots $x_1,x_2,x_3,x_4$ real. We will suppose that the roots are numbered such that $x_1>x_2>x_3>x_4$. Let $L_1/\mathbb{Q}$ be the splitting field $L_1=\mathbb{Q}[x_1,x_2,x_3,x_4] \subset \mathbb{R}$. We suppose that $P$ is chosen such that the Galois group is $S_4$. This is equivalent to demanding that $[L_1:\mathbb{Q}]=24$. We set $L\equiv L_1[i]$. This is a Galois extension of $\mathbb{Q}$, with Galois group $S_4 \times \{e,\rho\}$, where $\rho$ is complex conjugation.
Consider a cube, with vertices labeled as in the figure:
{width="2in"} \[cube\]
Let $G$ denote the group of symmetries of the cube. We have the exact sequence: $$1 \to \{e,\rho\} \to G \to S_4 \to 1$$ where now $\rho$ denotes inversion, and $S_4$ is the group of permutations of the four diagonals. Splitting this, identifying $S_4$ with (special orthogonal) rotations implementing the corresponding permutation of diagonals. we get an identification $$G \sim S_4 \times \{e,\rho\} = Gal(L/\mathbb{Q})$$ Let $H$ denote the stabiliser of the vertex 1, $F$ the corresponding fixed field, and $\varphi_1:F \to L \to \mathbb{C}$ the corresponding embedding. The left cosets of $H$ can be identified with the vertices of the cube, as well as embeddings of $F$ in $\mathbb{C}$. We label the latter $\varphi_j, \varphi_{\bar{j}}\ \
(j=1,2,3,4)$.
Note that the field $F$ is invariant under complex conjugation, which therefore acts on it with fixed field $F_1$. Clearly, $F_1=\mathbb{Q}[x_1]$ . We set $$\mathbb{D}=(x_1-x_2)(x_1-x_3)(x_1-x_4)(x_2-x_3)(x_2-x_4)(x_3-x_4)$$ Given our ordering of the roots, $\mathbb{D}>0$. Note that $i\mathbb{D}\in F$, $F=F_1[i\mathbb{D}]$, and $\Delta \equiv
\mathbb{D}^2$ is a rational number. We will assume that (after multiplying all the $x_i$ by a common natural number if necessary) $\Delta$ is an integer (and so $\mathbb{D}$ is an algebraic integer). We will repeatedly use the fact that the Galois conjugates of $i\mathbb{D}\in F$ are given by $$\begin{split}
\phi_j(i\mathbb{D})&=-(-1)^j i\mathbb{D}\\
\phi_{\bar{j}}(i\mathbb{D})&=(-1)^j i\mathbb{D}\\
\end{split}$$
In our case $X_r$ is a real 8-torus. The embeddings $\varphi_i:F \to
\mathbb{C}$ induce $\mathbb{R}$-linear maps $z_i:V \to \mathbb{C}$, such that $\mathbf{z}=(z_1,z_2,z_3,z_4)$ is an isomorphism of $\mathbb{R}$-vector spaces $V \to \mathbb{C}^4$. We let $X$ denote the complex manifold $V/\Lambda$ obtained thus. Note that if $a \in
{\mathfrak o}_F$, multiplication by $a$ is a $\mathbb{Q}$-linear map $F \to F$ which induces a $\mathbb{R}$-linear map $V \to V$ taking the lattice $\Lambda$ to itself. If $\mathbf{z}(a)=(a_1,a_2,a_3,a_4)$, and $u\in V$ with $\mathbf{z}(u)=(z_1,z_2,z_3,z_4)$ we also have $\mathbf{z}(au)=(a_1z_1,a_2z_2,a_3z_3,a_4z_4)$, so that we see that this induces an analytic map (in fact an isogeny) $X \to X$. In other words, ${\mathfrak o}_F$ acts on $X$ by “complex multiplication".
As a complex torus, $X$ is certainly Kähler, and we shall see below that it is algebraic. What is relevant for our purposes is that it is possible to describe explicitly the Hodge decomposition as well as the rational structure of the complex cohomology of $X$. Let $T$ (for “top") denote the set of indices $\{1,2,3,4\}$ and $B$ (for “bottom") the indices $\{\bar{1},\bar{2},\bar{3},\bar{4}\}$. (The corresponding vertices are denoted $1b$, etc. in the figure.)
A basis of $H^{p,q}$ is labeled by subsets $P \subset T$, $Q \subset B$, with $|P|=p$, and $|Q|=q$, and given by the translation-invariant forms $dz^Pd{\bar z}^Q$, where for example, if $P=\{i,j\}$, with $i<j$ we set $dz^P=dz_i \wedge dz_j$, and if $Q=\{\bar{i},\bar{j}\}$ (again with $i<j$), we set $d{\bar z}^Q= d{\bar z}_i \wedge d{\bar z}_j$. A basis of the rational cohomology $H^r_{\mathbb{Q}}$ is labelled by pairs $(R,
\chi)$ where
- $R$ is an orbit of $G$ in the set of sequences $\mu \equiv (\mu_1,\dots,\mu_r)$ of distinct elements in $T \cup B$, and
- $\chi$ runs over a $\mathbb{Q}$-basis of $H_R$, the space of $G$-equivariant maps $R \to L$, satisfying $$\chi(\mu_{\sigma(1)},\dots,\mu_{\sigma(r)})=sign(\sigma)
\chi(\mu_1,\dots,\mu_r)$$ for any permutation $\sigma$ such that $\mu, \mu_{\sigma} \in R$.
The corresponding classes are given by the forms $$\sum_{\mu \in R} \chi(\mu) dz^\mu$$ We use the notation $dz^\mu=dz_{\mu_1}\wedge\dots\wedge dz_{\mu_r}$, with the convention that $dz_{\bar{1}}=d{\bar z}_1, etc.$
It is useful to note the following
Given $R$, the $\mathbb{Q}$-dimension of $H_R$ is $|R|/r!$.
Note that if $r=2p$, a rational class as above is of type $(p,p)$ iff the orbit consists of sequences with elements equally divided between the top and bottom faces of the cube. In particular, the rational $(1,1)$ classes correspond to the $G$-orbit of the sequence $(1,\bar{1})$. Since in this case $H_{R}$ has dimension 4, we see that the Neron-Severi group has rank 4.
Consider now the orbit of the sequence $(1,3,\bar{2},\bar{4})$. This corresponds to a two-dimensional space ${{\mathcal{M}}}$ of rational $(2,2)$ classes, which have the property that *these are not products of rational $(1,1)$ classes.* It is easy to check that but for (the $\mathbb{Q}$-span of) these, rational $(2,2)$ classes are generated by products of rational $(1,1)$ classes.
A $\mathbb{Q}$-basis of ${{\mathcal{M}}}$ is given by the classes
- $M= \mathbb{D}(dz_1 d{\bar z}_2 dz_3 d{\bar z}_4 + d{\bar z}_1 dz_2 d{\bar z}_3 dz_4)$
- $M'= i(dz_1 d{\bar z}_2 dz_3 d{\bar z}_4 - d{\bar z}_1 dz_2 d{\bar z}_3 dz_4)$
So the Hodge conjecture in this case would be that : [**the classes $M$ and $M'$ are algebraic.**]{}
We will use complex multiplication in an essential way later; here I illustrate its use by showing how it can be used to halve our work. Consider multiplication by the algebraic integer $a=1+i\mathbb{D} \in
{\mathfrak o}_F$. This induces a (covering) map $\pi_a:X \to X$ and one easily computes: $$\label{isogeny}
\begin{split}
\pi_a^* M &= ((1-\Delta)^2-4\Delta)M+4(1-\Delta)\Delta M' \\
\pi_a^* M' &= ((1-\Delta)^2-4\Delta)M'-4(1-\Delta)M\\
\end{split}$$ This proves
Algebraicity of either one of $M$ or $M'$ implies that of the other.
Before moving on, we find a positive rational $(1,1)$ form $\omega$ on $X$, which will show that it is projective. Let $\mu_1\in F_1$ (to be chosen in a moment) and consider the form $$\omega=\frac{i\mathbb{D}}{\Delta}(\mu_1 dz_1 d{\bar z}_1-\mu_2 dz_2
d{\bar z}_2+\mu_3 dz_3 d{\bar z}_3-\mu_4 dz_4 d{\bar z}_4)$$ where $\mu_i$ are Galois conjugates. Clearly this is a rational $(1,1)$ form, and it will be positive provided $(-1)^{j+1} \mu_j>0$. For example, we can take $\mu_1=(x_1-x_2)(x_1-x_3)(x_1-x_4)$, and we will do so. With this choice the holomorphic four-form $\theta
\equiv (1/\mathbb{D})dz_1 dz_2 dz_3 dz_4$ satisfies $$\label{omegatheta} \frac{\omega^4}{4!}=\theta \wedge \bar{\theta}$$
Expressing $M,\ M'$ in terms of Chern characters
================================================
Consider the $G$-orbit of $(1,3)$. The corresponding subspace of $H^2_{\mathbb{Q}}$ is spanned by the classes of the form $$A_1=a_{13}(x_1-x_3)dz_1dz_3+....$$ where $a_{13}$ belongs to the fixed field of the subgroup of $G$ that leaves the set of vertices $\{1,3\}$ invariant, and this coefficient determines the others in the sum by Galois covariance. We introduce the notation $$T_a=a_{13}a_{\bar{2}\bar{4}}(x_1-x_3)(x_2-x_4)-a_{1\bar{2}}a_{3\bar{4}}(x_1-x_2)(x_3-x_4)
+a_{1\bar{4}}a_{3\bar{2}}(x_1-x_4)(x_3-x_2)$$ Squaring $A_1$, we get $$\begin{split}
A_1^2=&2a_{13}a_{24}(x_1-x_3)(x_2-x_4) dz_1dz_3dz_2dz_4+..\\
+&2a_{\bar{1}2}a_{13}(x_1-x_2)(x_1-x_3)d{\bar z}_1dz_2dz_1dz_3+...\\
+&2a_{1\bar{2}}a_{2\bar{1}}(x_1-x_2)(x_2-x_1)dz_1d{\bar z}_2dz_2d{\bar z}_1+..\\
+&2T_a dz_1dz_3d{\bar z}_2d{\bar z}_4+..
\end{split}$$ If we make the replacement $a_{13} \rightsquigarrow
ic\mathbb{D}a_{13}$ ($c$ an integer introduced for later use in §7), we get a class $A_2$, such that $$\begin{split}
A_2^2/(c^2\Delta)=&2a_{13}a_{24}(x_1-x_3)(x_2-x_4) dz_1dz_3dz_2dz_4+..\\
+&2a_{\bar{1}2}a_{13}(x_1-x_2)(x_1-x_3)d{\bar z}_1dz_2dz_1dz_3+...\\
+&2a_{1\bar{2}}a_{2\bar{1}}(x_1-x_2)(x_2-x_1)dz_1d{\bar z}_2dz_2d{\bar z}_1\\
-&2T_a dz_1dz_3d{\bar z}_2d{\bar z}_4-..
\end{split}$$
Suppose now that the classes $A_i$ are integral. (This is easily arranged by clearing denominators.) Let $L_i$ ($i=1,2$) be the line bundle with Chern class $A_i$.
Let ${{\mathcal{V}}}_i=L_i \oplus L_i^{-1} ,\ i=1,2$. Then $$ch({{\mathcal{V}}}_1^{c^2\Delta} \ominus {{\mathcal{V}}}_2)=4c^2\Delta(T_a dz_1dz_3d{\bar z}_2d{\bar z}_4+..)\\$$ where the equality is modulo (rational) 0- and 8-forms.
We have the freedom to choose the coefficient $a_{13}$, which by Galois covariance determines the other coefficients, and hence the above classes. We now make the choice $$a_{13}=h_3$$ where for later use we introduce the notation $$\begin{split}
h_2=(x_1 x_2+x_3 x_4)\\
h_3=(x_1 x_3+x_2 x_4)\\
h_4=(x_1 x_4+x_2 x_3)
\end{split}$$ Then $T_a=-\mathbb{D}$, and we get
\[key\] With the above choice, $$ch({{\mathcal{V}}}_1^{c^2\Delta} \ominus {{\mathcal{V}}}_2)=4c^2\Delta M$$ where the equality is modulo (rational) 0- and 8-forms.
The virtual bundle ${{\mathcal{V}}}_1^{c^2\Delta} \ominus {{\mathcal{V}}}_2$ has the properties: $c_1=0$, and $c_2 \wedge \omega^2=0$, where $\omega$ is the rational Kähler class defined at the end of §2. (This is because the $M_i$, as can be easily seen, are orthogonal to $\omega$.) This will not do for reasons to do with the Bogomolov inequality, but this can be fixed because of a minor miracle:
\[miracle1\] With the above choices, $$\begin{split}
A_1^2 \wedge \omega = -2i\Delta \frac{1}{\mu_4}
dz_1d{\bar z}_1dz_2d{\bar z}_2dz_3d{\bar z}_3+...
\end{split}$$ In particular, $A_1^2 \wedge \omega $ is a (rational) (3,3) form.
For later use, we also record
\[miracle2\] With the above choices, $$\begin{split}
A_1 \wedge \omega^3 = 0\\
A_2 \wedge \omega^3 = 0\\
\end{split}$$
We will suppose that $k\omega$ (for some positive integer $k$) is an integral class, and let $L_{k\omega}$ denote a (holomorphic, in fact ample) line bundle with this Chern class. The following is a easy consequence of \[miracle1\].
\[bundles\] Let ${\hat{\mathcal{V}}}_1=L_1 \otimes L_{k\omega} \oplus L_1^{-1} \otimes L_{k\omega} $, and ${\hat{\mathcal{V}}}_2= L_2 \oplus L_2^{-1}$ and set ${{\mathcal{E}}}={\hat{\mathcal{V}}}_1^{c^2\Delta}
\ominus {\hat{\mathcal{V}}}_2$. Then $$ch({{\mathcal{E}}})= 2c^2\Delta k \omega+4c^2\Delta M+k^2c^2\Delta \omega^2$$ where the equality is modulo (rational) 0-, (3,3)- and 8-forms.
In particular, this (difference) bundle ${{\mathcal{E}}}$ satisfies the “Bogomolov inequality": $$\begin{split}
<c_2\omega^2>-\frac{2\Delta-3}{4(\Delta-1)}<c_1^2 \omega^2>&=\frac{c^2\Delta}{c^2\Delta-1}k^2<\omega^4>\\
&> 0
\end{split}$$
The symbol $<..>$ stands for integration against the fundamental class. We use the quote marks since we are not (yet!) talking of a holomorphic bundle ${{\mathcal{E}}}$. Since the virtual bundle has positive rank, we are justified, up to some non-canonical choices, in dropping the qualifiers “virtual”/“difference”.
We have concentrated on the Hodge class $M$ in this section; it is possible, with slight modifications to the above expressions, to find a smooth bundle ${{\mathcal{E}}}'$ whose Chern character similarly contains the Hodge class $M'$.
$Spin(7)$ instantons
====================
In this section we recall the definition of $Spin(7)$ instantons ([@B-K-S], [@T]), specialised to the case of a Kähler four-fold $X$ with trivial canonical bundle $K_X$. We fix a Ricci-flat Kähler form $\omega$, and let $\theta$ denote a trivialisation of $K_X$ satisfying (\[omegatheta\]). We define a (complex antilinear) endomorphism $\star:\Omega^{(0,2)} \to
\Omega^{(0,2)}$, by $$|\alpha|^2 \theta=\alpha \wedge \star \alpha$$ We have $\star^2=1$, so we can decompose the bundle into a self-dual and anti-self-dual part: $$\Omega^{(0,2)}=\Omega^{(0,2)}_+ \oplus \Omega^{(0,2)}_-$$
Let $E$ be a hermitian ($C^{\infty}$) vector bundle on $X$. A Spin(7) instanton is a hermitian connection $A$ on $E$, whose curvature $F$ satisfies $$F_+^{(0,2)}=0,\ \ \Lambda F=0$$ Here $\Lambda$ denotes as usual contraction with the Kähler form. A crucial point is the following ([@T],[@L]):
The $L^2$-norm of the curvature of a Spin(7) instanton satisfies $||F_-^{(0,2)}||_2^2=\int Tr (F \wedge F) \wedge \bar{\theta}$
In particular, if the invariant on the right vanishes, a Spin(7) instanton is equivalent to a holomorphic structure on $E$ together with a Hermite-Einstein connection. Clearly, such a bundle would be poly-stable, and hence (or directly from the Hermite-Einstein condition) satisfy the Bogomolov inequality: $$\label{Bogomolov} c_2(E).\omega^2 \ge \frac{r - 1}{2r}
c_1(E)^2.\omega^2$$ where $r$ denotes the rank of $E$.
*Now that we have embedded the problem of construction a holomorphic structure on ${{\mathcal{E}}}$ in a broader context – that of constructing an instanton connection – one can envisage deforming the complex structure in such a way $$\int c_2({{\mathcal{E}}}) \wedge \bar{\theta} \ne 0$$ and still hope to have the moduli space of semi-stable holomorphic structures on ${{\mathcal{E}}}$ deform as the moduli space of instanton connections.*
There are several possible approaches to the construction of such a connection.
1. Exhibit an instanton by glueing.
2. The fact that the bundles are exhibited as a difference of two vector bundles, each of which is in turn a sum of explicit line bundles, suggests the use of monads, possibly combined with a twistor construction. This would involve a matrix of sections of line bundles.
A third idea, suggested to me by G. Tian, is pursued in the last section of this paper.
Calibrations; Cayley submanifolds
=================================
In his thesis, C. Lewis [@L] shows how (in one particular case) one can construct an instanton by glueing around a suitable Cayley submanifold. (See also [@B].) We define these terms below, and then exhibit some relevant Cayley cycles that arise in our context. (References are [@H-L], and [@J]; but we follow the conventions of [@T].)
Let $M$ be a Riemannian manifold. A closed $l$-form $\phi$ is said to be a [*calibration*]{} if for every oriented tangent $l$-plane $\xi$, we have $$\phi|_{\xi} \le vol_{\xi}$$ where $vol_{\xi}$ is the (Riemannian) volume form. Given a calibration $\phi$, an oriented submanifold $N$ is said to be [*calibrated*]{} if $\phi$ restricts to $N$ as the Riemannian volume form.
It is easy to see that a calibrated submanifold is minimal. Two examples are relevant. First, if $M$ is Kähler, with Kähler form $\omega$, for any integer $p \ge 1$, the form $\frac{\omega^p}{p!}$ is a calibration, and the calibrated submanifolds are precisely the complex submanifolds.
The case that concerns us is that of a four-fold $X$ with trivial canonical bundle $K_X$. We fix an integral Ricci-flat Kähler form $\omega$, and let $\theta$ denote a trivialisation of $K_X$ with normalisation as in (\[omegatheta\]). Then $4 Re(\theta)$ is a second calibration, and the calibrated submanifolds are called [*Special Lagrangian submanifolds.*]{} There is a “linear combination" of the two, defined by the form $$\Omega=\frac{w^2}{2}+4Re(\theta)$$ which defines the [*Cayley calibration*]{}. The corresponding calibrated manifolds are called [*Cayley manifolds*]{}. Any smooth complex surface (on which the second term will restrict to zero) or any Special Lagrangian submanifold (on which the first term will vanish) furnish examples. In fact, the Cayley cycles we deal with will be of the latter kind.
Cayley manifolds are not easy to find. We will use the following result (Proposition 8.4.8 of [@J]):
Let $X$ be as above, and $\sigma:X \to X$ an anti-holomorphic isometric involution such that $\sigma^*\theta =
\bar{\theta}$. Then the fixed point set is a Special Lagrangian submanifold.
We return to the constructions of our paper. Recall that the field $F$ is invariant under complex conjugation, which therefore acts on it with fixed field $F_1$. This induces an involution $\hat{\sigma_1}:V
\to V$ such that $\mathbf{z}(\hat{\sigma_1} (u))=
\bar{\mathbf{z}}(u)$, where, if $\mathbf{z}=(z_1,z_2,z_3,z_4)$, we set $\bar{\mathbf{z}}=({\bar z}_1, {\bar z}_2,{\bar z}_3,{\bar z}_4)$. The induced involution $\sigma_1:X \to X$ has fixed locus which we will denote $Y$. Note that $\sigma$ satisfies the conditions of the previous Proposition and therefore $Y$ is Special Lagrangian.
There exist (rational) Cayley cycles representing the Hodge classes $M_i$.
Recall the isogeny $\pi_a:X \to X$, given by multiplication by the algebraic integer $a=1+i\mathbb{D}$. It is easy to check $$\begin{split}
\pi_a^* \omega = (1+\Delta) \omega\\
\pi_a^* \theta = (1+\Delta)^2 \theta\\
\end{split}$$ We will also need a second isogeny $\pi_b$, where $b=i\mathbb{D}$, which satisfies $$\begin{split}
\pi_b^* \omega = \Delta \omega\\
\pi_b^* \theta = \Delta^2 \theta\\
\end{split}$$ These equations guarantee the maps $\pi_a, \pi_b$ take Cayley cycles to Cayley cycles (possibly introducing singularities.)
We have the following table giving the action of the above isogenies on four-forms of various types (all the forms in the list are eigenvectors):
------ ------------------------- ------------------------- ----------------------------------------------------------------------------------------------------
Form eigenvalue of $\pi_a^*$ eigenvalue of $\pi_b^*$ “multiplicity"\
$dz_1 dz_2 dz_3 dz_4$ & $(1+\Delta)^2$ & $\Delta^2$ & $2 \times 1$\
$dz_1 d{\bar z}_1 dz_2 dz_3$ & $(1+\Delta)^2$ & $\Delta^2$ & $2 \times 8$\
$dz_1 d{\bar z}_1 dz_2 dz_4$ & $(1+\Delta)(1-i\mathbb{D})^2$ & $-\Delta^2$ & $2 \times 4$\
$d{\bar z}_1 dz_2 dz_3 dz_4$ & $(1+\Delta)(1-i\mathbb{D})^2$ & $-\Delta^2$ & $2 \times 4$\
$dz_1 d{\bar z}_1 dz_2 d{\bar z}_2$ & $(1+\Delta)^2$ & $\Delta^2$ & 6\
$dz_1 d{\bar z}_1 dz_2 d{\bar z}_3$ & $(1+\Delta)(1-i\mathbb{D})^2$ & $-\Delta^2$ & $2 \times 12$\
$dz_1 dz_2 d{\bar z}_3 d{\bar z}_4$ & $(1+\Delta)^2$ & $\Delta^2$ & 4\
$dz_1 d{\bar z}_2 dz_3 d{\bar z}_4$ & $(1+i\mathbb{D})^4$ & $\Delta^2$ & 1\
$d{\bar z}_1 dz_2 d{\bar z}_3 dz_4$ & $(1-i\mathbb{D})^4$ & $\Delta^2$ & 1\
------ ------------------------- ------------------------- ----------------------------------------------------------------------------------------------------
(We list only forms of type (4,0), (3,1) and (2,2), omitting types that are related to the ones in the list by conjugation. The term “multiplicity” refers to the number of forms of a given type, not the multiplicity of eigenvalues.)
Consider the operator $$\Phi_a=(\pi_a^*-(1+\Delta)^2)(\pi_b^*+\Delta^2)$$ From the list it follows that the space ${{\mathcal{M}}}\otimes_{\mathbb{Q}} \mathbb{C}$ (spanned by the $M_i$) is the sum of the eigenspaces of $\Phi_a$ corresponding to the non-zero eigenvalues. We have (using (\[isogeny\])) $$\begin{split}
\Phi_a^* M' &= -8 \Delta^2[2 \Delta M'+ (1-\Delta) M]\\
\Phi_a^* M &= - 8 \Delta^2[-(1-\Delta)\Delta M'+2 \Delta M]\\
\end{split}$$
Next, note that the Cayley cycle $Y$ defined above satisfies $$\begin{split}
<Y, M> &= 2\mathbb{D} \delta\\
<Y, M'> &= 0 \\
\end{split}$$ Here $<,>$ denotes the integration pairing of cycles and forms, and $\delta$ denotes the co-volume of the lattice $\mathfrak{o}_{F_1} \subset F_1 \otimes_{\mathbb{Q}} \mathbb{R} $. By standard facts in algebraic number theory, $\delta$ is a rational multiple of $\mathbb{D}$; so the above pairings are rational, as they had better be.
We now consider the Cayley cycle $$C_a= (\pi_a-(1+\Delta)^2)(\pi_b+\Delta^2) Y$$ By construction $C_a$ is orthogonal to all the forms in the above list except the $M_i$. Its pairings with these are as follows: $$\begin{split}
<C_a, M> &= -32 \Delta^3 \mathbb{D} \delta\\
<C_a, M'> & = -16 \Delta^2(1-\Delta)\mathbb{D} \delta \\
\end{split}$$
Let now ${\bar a}=(1-i\mathbb{D})$, and repeat the above construction with operators $\Phi_{{\bar a}}$, etc. $$\begin{split}
\Phi_{{\bar a}}^* M' &= -8 \Delta^2[2 \Delta M'-(1-\Delta) M]\\
\Phi_{{\bar a}}^* M &= - 8 \Delta^2[(1-\Delta)\Delta M'+2 \Delta M]\\
\end{split}$$ This gives a cycle $C_{{\bar a}}$ satisfying $$\begin{split}
<C_{{\bar a}}, M> &= -32 \Delta^3 \mathbb{D} \delta\\
<C_{{\bar a}}, M'> & = 16 \Delta^2(1-\Delta)\mathbb{D} \delta \\
\end{split}$$
Clearly the theorem is proved.
The above result, though suggestive, does not take us far. This is because the above “Cayley cycle" is not effective, but in fact a linear combination of SL subvarieties with both positive and negative coefficients. (D. Joyce has pointed out that this must be the case given that it represents a $(2,2)$ class.) To make matters worse, a theorem of G. Tian (Theorem 4.3.3 of [@T]) states that blow-up loci of Hermite-Yang-Mills connections are effective *holomorphic* integral cycles consisting of complex subvarieties of codimension two. So any glueing will call for very new techniques.
Adapted complex structures
==========================
In this section we seek translation-invariant complex structures on the eight-torus $V/\Lambda$ such that the classes $A_i$ are of type $(1,1)$ w.r.to these complex structures, and therefore define holomorphic structures on the line bundles ${{\mathcal{L}}}_i$. The original motivation was to exploit twistor techniques for the construction of instantons, but we postpone discussion of possible uses of this investigation to the last section.
Consider a linear change of coordinates of the form $$\begin{split}
z_1&=w_1 + \overline{\alpha_{\bar{1}2}}\bar{w}_2+\overline{\alpha_{\bar{1}4}}\bar{w}_4\\
z_3&=w_3 + \overline{\alpha_{\bar{3}2}}\bar{w}_2+\overline{\alpha_{\bar{3}4}}\bar{w}_4\\
z_2&=w_2 + \overline{\tilde{\alpha}_{\bar{2}1}}\bar{w}_1+\overline{\tilde{\alpha}_{\bar{2}3}}\bar{w}_3\\
z_4&=w_4 + \overline{\tilde{\alpha}_{\bar{4}1}}\bar{w}_1+\overline{\tilde{\alpha}_{\bar{4}3}}\bar{w}_3\\
\end{split}$$ We collect the coefficients into $2 \times 2$ matrices $\alpha$ and $\tilde{\alpha}$ as follows: $$\alpha= \left( \begin{matrix}
\alpha_{\bar{1}2} & \alpha_{\bar{1}4}\\
\alpha_{\bar{3}2} & \alpha_{\bar{3}4} \\
\end{matrix} \right)$$ and $$\tilde\alpha= \left( \begin{matrix}
\tilde\alpha_{\bar{2}1} & \tilde\alpha_{\bar{2}3}\\
\tilde\alpha_{\bar{4}1} & \tilde\alpha_{\bar{4}3} \\
\end{matrix} \right)$$ and rewrite the above change of coordinates as follows: $$\begin{split}
\left ( \begin{matrix}
z_1 \\
z_3 \\
\end{matrix} \right)
&=
\left( \begin{matrix}
w_1 \\
w_3 \\
\end{matrix} \right)
+
\bar\alpha
\left( \begin{matrix}
\bar{w}_2 \\
\bar{w}_4 \\
\end{matrix} \right)\\
\left( \begin{matrix}
z_2 \\
z_4 \\
\end{matrix} \right)
&=
\left( \begin{matrix}
w_2 \\
w_4 \\
\end{matrix} \right)
+
\bar{\tilde\alpha}
\left( \begin{matrix}
\bar{w}_1 \\
\bar{w}_3 \\
\end{matrix} \right)
\end{split}$$
A long but straightforward computation shows $A_i$ will be of type $(1,1)$ provided: $$\begin{split}
h_3(x_1-x_3)(\alpha_{\bar{1}2}\alpha_{\bar{3}4}-\alpha_{\bar{1}4}\alpha_{\bar{3}2})&\\
+h_4(x_1-x_4)\alpha_{\bar{1}2} - h_2(x_1-x_2)\alpha_{\bar{1}4}&\\
+h_2(x_3-x_4)\alpha_{\bar{3}2} - h_4(x_3-x_2)\alpha_{\bar{3}4}&\\
+h_3(x_2-x_4)&=0
\end{split}$$ and $$\begin{split}
h_3(x_2-x_4)(\tilde\alpha_{\bar{2}1}\tilde\alpha_{\bar{4}3}
-\tilde\alpha_{\bar{2}3}\tilde\alpha_{\bar{4}1})&\\
+h_4(x_1-x_4)\tilde\alpha_{\bar{4}3} - h_2(x_3-x_4)\tilde\alpha_{\bar{4}1}&\\
+h_2(x_1-x_2)\tilde\alpha_{\bar{2}3} - h_4(x_3-x_2)\tilde\alpha_{\bar{2}1}&\\
+h_3(x_1-x_3)&=0
\end{split}$$
To rewrite these conditions in a more compact form, we introduce some notation:
1. Given a $2 \times 2$ matrix $A$: $$A=\left( \begin{matrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\\
\end{matrix} \right)$$ let $$\label{hat}
\hat A=\left( \begin{matrix}
a_{22} & -a_{12}\\
-a_{21} & a_{11}\\
\end{matrix} \right)$$ (If $A$ is nonsingular, $\hat{A}=(det\ A)A^{-1}$.)
2. Define the symmetric bilinear form $<,>$ on the space of $2 \times 2$ matrices $$<A,B>=Tr(A\hat{B})=det\ (A+B)-det\ A -det\ B$$
3. Let $$H=\left( \begin{matrix}
-h_4(x_2-x_3) & h_2(x_3-x_4) \\
- h_2(x_1-x_2) & -h_4(x_1-x_4) \\
\end{matrix} \right)$$ so that $$\hat H=\left( \begin{matrix}
-h_4(x_1-x_4) & -h_2(x_3-x_4) \\
h_2(x_1-x_2) & -h_4(x_2-x_3) \\
\end{matrix} \right)$$
The conditions on $\alpha$ and $\tilde\alpha$ can now be rewritten: $$\label{condition1} <\alpha, H>=h_3(x_2-x_4)+h_3(x_1-x_3)det\ \alpha$$ and $$\label{condition2}
<\tilde\alpha, \hat H>=h_3(x_1-x_3)+h_3(x_2-x_4)det\ \tilde\alpha$$
We assume that the inverse coordinate transformation is of the form $$\begin{split}
\left ( \begin{matrix}
w_1 \\
w_3 \\
\end{matrix} \right)
&=
c\left( \begin{matrix}
z_1 \\
z_3 \\
\end{matrix} \right)
+
\bar\beta
\left( \begin{matrix}
\bar{z}_2 \\
\bar{z}_4 \\
\end{matrix} \right)\\
\left( \begin{matrix}
w_2 \\
w_4 \\
\end{matrix} \right)
&=
\tilde{c} \left( \begin{matrix}
z_2 \\
z_4 \\
\end{matrix} \right)
+
\bar{\tilde\beta}
\left( \begin{matrix}
\bar{z}_1 \\
\bar{z}_3 \\
\end{matrix} \right)
\end{split}$$ where $c,\ \tilde c$ are scalars (this will constrain $\alpha$ and $\tilde \alpha$, see below) and $\beta$ and $\tilde \beta$ $2 \times
2$ matrices. One checks that we then need $$\begin{split}
c(1-\bar\alpha \tilde\alpha)&=1\\
\tilde c (1-\bar{\tilde \alpha}\alpha)&=1
\end{split}$$ so that we are requiring that $\bar\alpha \tilde\alpha$ and $\bar{\tilde \alpha}\alpha$ are scalars. Further, $$\begin{split}
\beta &=-{\tilde c} \alpha\\
\tilde \beta &= -c \tilde \alpha
\end{split}$$ Note that either $\bar\alpha \tilde\alpha=\bar{\tilde
\alpha}\alpha=0$ and $c=\tilde{c}=1$ or $$\begin{split}
\bar\alpha \tilde\alpha&=\frac{c-1}{c}\\
\bar{\tilde \alpha}\alpha&=\frac{\tilde{c}-1}{\tilde c}
\end{split}$$ and $\tilde c= \bar{c}$. Note also that once $\alpha$ is chosen to satisfy the equation (\[condition1\]), then (\[condition2\]) is satisfied if we take $$\tilde{\alpha}=\frac{x_1-x_3}{x_2-x_4}\hat{\bar{\alpha}}$$ From now on we will proceed to define $\tilde{\alpha}$ by the above equation. This forces $c$ to satisfy $$c(1-\frac{x_1-x_3}{x_2-x_4} \overline{det\ \alpha})=1$$ Clearly, a necessary condition is $$\label{detalpha}
det\ \alpha \ne \frac{x_2-x_4}{x_1-x_3}$$
We can write down the corresponding almost complex structure. With an obvious schematic notation, $$\begin{split}
J \left ( \begin{matrix}
dz_1 \\
dz_3 \\
\end{matrix} \right)
&=
i(2c-1)\left( \begin{matrix}
dz_1 \\
dz_3 \\
\end{matrix} \right)
+
2i \bar\beta \left( \begin{matrix}
d\bar{z}_2 \\
d\bar{z}_4 \\
\end{matrix} \right)\\
J \left( \begin{matrix}
dz_2 \\
dz_4 \\
\end{matrix} \right)
&=
i(2\tilde{c}-1) \left( \begin{matrix}
dz_2 \\
dz_4 \\
\end{matrix} \right)
+
2i \bar{\tilde\beta} \left( \begin{matrix}
d\bar{z}_1 \\
d\bar{z}_3 \\
\end{matrix} \right)
\end{split}$$
By further restricting $\alpha$ one can ensure that $\omega$ remains of type $(1,1)$. We summarise our results in
Let the co-ordinates $w$ be defined by $$\begin{split}
\left ( \begin{matrix}
z_1 \\
z_3 \\
\end{matrix} \right)
&=
\left( \begin{matrix}
w_1 \\
w_3 \\
\end{matrix} \right)
+
\bar\alpha
\left( \begin{matrix}
\bar{w}_2 \\
\bar{w}_4 \\
\end{matrix} \right)\\
\left( \begin{matrix}
z_2 \\
z_4 \\
\end{matrix} \right)
&=
\left( \begin{matrix}
w_2 \\
w_4 \\
\end{matrix} \right)
+
\bar{\tilde\alpha}
\left( \begin{matrix}
\bar{w}_1 \\
\bar{w}_3 \\
\end{matrix} \right)
\end{split}$$ where the matrix $\alpha$ satisfies $$\label{aoneone} <\alpha, H>=h_3(x_2-x_4)+h_3(x_1-x_3)det\ \alpha$$ and $$\tilde{\alpha}=\frac{x_1-x_3}{x_2-x_4}\hat{\bar{\alpha}}$$ ($\hat{\alpha}$ is defined as in (\[hat\]).) Then the forms $A_i$ are of type $(1,1)$ w.r.to the $w_i$. Further, if $\alpha$ satisfies $$\label{omegaoneone}
\begin{split}
\alpha_{\bar{3}4}&=+\frac{x_1-x_4}{x_2-x_3}\bar{\alpha}_{\bar{1}2}\\
\alpha_{\bar{1}4}&=-\frac{x_3-x_4}{x_1-x_3}\bar{\alpha}_{\bar{3}2}\\
\end{split}$$ then $\omega$ remains of type $(1,1)$.
If $\alpha$ satisfies (\[omegaoneone\]), the condition (\[aoneone\]) becomes $$\begin{split}
h_3(x_1-x_3))(\frac{x_1-x_4}{x_2-x_3}|\alpha_{\bar{1}2}|^2 +
\frac{x_3-x_4}{x_1-x_2}|\alpha_{\bar{3}2}|^2)+h_3(x_2-x_4)&\\+h_4(x_1-x_4)(\alpha_{\bar{1}2}+\bar{\alpha}_{\bar{1}2})
+h_2(x_3-x_4)(\alpha_{\bar{3}2}+\bar{\alpha}_{\bar{3}2})&=0
\end{split}$$ The space of solutions $\tilde{\mathcal{J}}$ is clearly an 3-dimensional ellipsoid in the two-dimensional complex vector space with co-ordinates $(\alpha_{\bar{1}2},\alpha_{\bar{3}2})$. The condition (\[detalpha\]) becomes: $$\frac{x_1-x_4}{x_2-x_3}|\alpha_{\bar{1}2}|^2 +
\frac{x_3-x_4}{x_1-x_2}|\alpha_{\bar{3}2}|^2 \ne 0$$ which corresponds to removing the affine hyperplane $\mathcal H$ given by $$h_3(x_2-x_4)+h_4(x_1-x_4)(\alpha_{\bar{1}2}+\bar{\alpha}_{\bar{1}2})
+h_2(x_3-x_4)(\alpha_{\bar{3}2}+\bar{\alpha}_{\bar{3}2})=0$$ We have therefore to consider $\mathcal J = \tilde{\mathcal J}
\setminus \mathcal H$, which is the union of two open three-discs.
A particular choice of $\alpha$ has remarkable properties. Let $$\alpha^*=\frac{1}{x_1-x_3}\left(\begin{matrix}
(x_2-x_3)(1-\frac{2h_4}{h_3}) & -(x_3-x_4)(1-\frac{2h_2}{h_3})\\
(x_1-x_2)(1-\frac{2h_2}{h_3}) & (x_1-x_4)(1-\frac{2h_4}{h_3}) \\
\end{matrix} \right)$$
With this choice, we have $$\begin{split}
\frac{(x_1 x_3 + x_2 x_4)^2}{4}A_1=&
{(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)}
dw_1 d{\bar w}_2\\
-& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)(x_3 - x_4)^2}
dw_2 d{\bar w}_3 \\
+& {(x_1 - x_2)(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)^2}
dw_3 d{\bar w}_4\\
+& {(x_1 - x_2)^2(x_2 - x_3)(x_1 - x_4)^2(x_3 - x_4)^2}
dw_4 d{\bar w}_1\\
+& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)}
d{\bar w}_1 dw_2 \\
-& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)(x_3 - x_4)^2}
d{\bar w}_2 dw_3 \\
+& {(x_1 - x_2)(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)^2}
d{\bar w}_3 dw_4 \\
+& {(x_1 - x_2)^2(x_2 - x_3)(x_1 - x_4)^2(x_3 - x_4)^2}
d{\bar w}_4 dw_1 \\
\end{split}$$
$$\begin{split}
\frac{(x_1 x_3 + x_2 x_4)^2}{4i\mathbb{D}}A_2=&
{(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)}
dw_1 d{\bar w}_2\\
+& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)(x_3 - x_4)^2}
dw_2 d{\bar w}_3 \\
+& {(x_1 - x_2)(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)^2}
dw_3 d{\bar w}_4\\
-& {(x_1 - x_2)^2(x_2 - x_3)(x_1 - x_4)^2(x_3 - x_4)^2}
dw_4 d{\bar w}_1\\
-& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)}
d{\bar w}_1 dw_2 \\
-& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)(x_3 - x_4)^2}
d{\bar w}_2 dw_3 \\
-& {(x_1 - x_2)(x_2 - x_3)^2(x_1 - x_4)^2(x_3 - x_4)^2}
d{\bar w}_3 dw_4 \\
+& {(x_1 - x_2)^2(x_2 - x_3)(x_1 - x_4)^2(x_3 - x_4)^2}
d{\bar w}_4 dw_1 \\
\end{split}$$
$$\begin{split}
\frac{\Delta(x_1 x_3 + x_2 x_4)^2}{4i\mathbb{D}}\omega\\=-\{ &
{(x_1 - x_2)^2(x_1 - x_3)(x_2 - x_3)(x_1 - x_4)^2(x_3 - x_4)}
dw_1 d{\bar w}_1\\
+
& {(x_1 - x_2)^2(x_2 - x_3)^2(x_1 - x_4)(x_2 - x_4)(x_3 - x_4)}
dw_2 d{\bar w}_2 \\
+
& {(x_1 - x_2)(x_1 - x_3)(x_2 - x_3)^2(x_1 - x_4)(x_3 - x_4)^2}
dw_3 d{\bar w}_3 \\
+
& {(x_1 - x_2)(x_2 - x_3)(x_1 - x_4)^2(x_2 - x_4)(x_3-x_4)^2}
dw_4 d{\bar w}_4\} \\
\end{split}$$
In particular, $-\omega$ is a Kähler form and the corresponding complex structure makes $X_r$ an abelian variety.
\[flip\] It is convenient to consider the consider the *conjugate* complex structure (w.r.to which holomorphic co-ordinates are the ${\bar w}_i$. This has the property that the forms $A_i$ and $\omega$ are of type (1,1), and *in addition, $\omega$ is Kähler.* We let $X'$ denote the corresponding abelian variety.
A strategy
==========
Attempts to invoke twistor methods have not been successful so far. For example, N. Hitchin pointed out that results of M. Verbitsky make hyperkähler twistor spaces quite unsuitable. G. Tian made the following suggestion: construct instantons by deformation (using, say, the continuity method) from a situation when they are known to exist. In fact, the complex structure described in the Remark \[flip\] provides such a starting point. I close with a brief justification for this claim.
With respect to the above complex structure, the bundles ${\hat{\mathcal{V}}}_i$ defined in the statement of Theorem \[bundles\] are holomorphic, and furthermore (using the ampleness of $\omega$), the constant $k$ can be chosen large enough that ${\hat{\mathcal{V}}}_2$ can be embedded as a sub-bundle of ${{\hat{\mathcal{V}}}_1}^{c^2\Delta}$. The quotient bundle can be identified with the difference bundle ${{\mathcal{E}}}$, which therefore has a holomorphic structure depending on the above embedding; *we now show that it is possible to arrange that ${{\mathcal{E}}}$, endowed with this structure, is polystable.* (By stability we shall mean $\mu$-stability w.r.to the polarisation $\omega$.
Let us start by recalling that ${\hat{\mathcal{V}}}_1=L_1 \otimes L_{k\omega}
\oplus L_1^{-1} \otimes L_{k\omega} $, and ${\hat{\mathcal{V}}}_2= L_2 \oplus
L_2^{-1}$. Choose a large enough integer $k_1$ such that $L_{k_1\omega}$ that is very ample, and let $C$ be a general curve cut out by three sections of this line bundle. It follows from Proposition \[miracle2\] that $d \equiv degree\ L_2^{-1}\otimes
L_1 \otimes L_{k\omega}|_C=degree\ L_2\otimes L_1^{-1} \otimes
L_{k\omega}|_C=degree\ L_{k\omega}|_C=kk_1^3<\omega^4>$, and will assume that $k$ is chosen such that $d >
2genus(C)=3k_1^4<\omega^4>+2$. We next make the following assumption: $$\label{linsys}
dim\ H^0(C,L_{k\omega}|_C)=c^2\Delta$$ which we will return to below. Let $W$ denote a subspace of $H^0(X',L_2^{-1}\otimes L_1 \otimes L_{k\omega})$, chosen such that
- the restriction map $W \to H^0(C,L_2^{-1}\otimes L_1 \otimes
L_{k\omega}|_C)$ is an isomorphism, and
- $W$ is base-point free.
Consider now the evaluation map $E:W \otimes {{\mathcal{O}}}_{X'} \to
L_2^{-1}\otimes L_1 \otimes L_{k\omega}$, and let ${{\mathcal{F}}}$ be the kernel; by construction ${{\mathcal{F}}}$ fits in the exact sequence $$0 \to {{\mathcal{F}}}\to W \otimes {{\mathcal{O}}}_{X'} \to L_2^{-1}\otimes L_1 \otimes
L_{k\omega} \to 0\ .$$ By Butler’s Theorem ([@Bu]), the restriction of ${{\mathcal{F}}}$ to $C$ is stable, and this proves that ${{\mathcal{F}}}$ itself is stable. We next choose a subspace $U$ of $H^0(X',L_2\otimes L_1^{-1} \otimes L_{k\omega})$ with similar properties and obtain a second stable bundle ${{\mathcal{G}}}$ that fits in the sequence $$0 \to {{\mathcal{G}}}\to U \otimes {{\mathcal{O}}}_{X'} \to L_2\otimes L_1^{-1} \otimes
L_{k\omega} \to 0$$ Dualising, tensoring by suitable line bundles and adding the two sequences, we get $$0 \to {\hat{\mathcal{V}}}_2 \to {\hat{\mathcal{V}}}_1^{c^2\Delta} \to {\hat{\mathcal{F}}}\otimes L_1 \otimes
L_{k\omega} \oplus {\hat{\mathcal{G}}}\otimes L_1^{-1} \otimes L_{k\omega} \to 0$$ where ${\hat{\mathcal{F}}}$ denotes the dual of ${{\mathcal{F}}}$ and ${\hat{\mathcal{G}}}$ denotes the dual of ${{\mathcal{G}}}$, and we have used the assumption (\[linsys\]), namely, $dim\ W = dim\ U =c^2 \Delta$. Repeatedly using Proposition \[miracle2\] we see that the two summands in the last sum have the same slope. Consider now the assumption (\[linsys\]). By Riemann-Roch, this is equivalent to: $$(kk_1^3-(3/2)k_1^4)<\omega^4>=c^2\Delta$$ This is solved by taking $$k=(\frac{c^2\Delta}{k_1^3}+\frac{3k_1}{2})/<\omega^4>$$ This is where the choice of $c$ comes in - we choose $c$ and $k_1$ such that $k$ is an integer (and large enough). Once this is done
The bundle ${{\mathcal{E}}}$ (on $X'$) can be given a holomorphic structure such that it is polystable.
The above application of Butler’s theorem is inspired by its use in [@M].
By Donaldson-Uhlenbeck-Yau, such a bundle would admit a Hermite-Einstein metric and therefore a Spin(7) instanton.
[EMG]{}
Vector bundles and homogeneous spaces. Prec. Sympos. Pure Math. **3** (1961) 7–38
Selfduality in d=8 dimensional euclidean gravity. Phys. Rev. D **55** (1997) 4521-4524
Special quantum field theories in eight and other dimensions. Commun. Math. Phys. **194** (1998) 149
Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau four-folds. Nuclear Physics B, **480** (1996) 225-238
Complex anti-self-dual instantons and Cayley submanifolds. math.DG/0302094, 2003
Normal generation of vector bundles over a curve. J. Differential Geom. **39** (1994) 1-34.
Gauge theory in higher dimensions, [*in*]{} “The Geometric Universe; Science, Geometry, And The Work Of Roger Penrose”, Oxford University Press, (1998)
Calibrated geometries. Acta Mathematica, **148** (1982) 47-157
Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics (2007)
Spin(7) Instantons. Oxford University D.Phil. thesis (1998) Algebraic cycles on abelian varieties of complex multiplication type. Ann. of Math. **88** (1968) 161-180
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[^1]: It is a pleasure to thank Bobby Acharya, who told me of Spin(7) instantons and gave me a copy of Christopher Lewis’ thesis. I am also grateful to Dominic Joyce, Nigel Hitchin, M.S. Narasimhan, and Gang Tian for very helpful comments.
|
---
author:
- 'Rong-Qiang He'
- 'Zhong-Yi Lu'
title: Quantum Renormalization Groups Based on Natural Orbitals
---
[**Supplementary Material for**]{}
In this Supplementary Material, the results from the shortcut NORG calculations for a cluster-impurity model with $2\times 2$ impurity sites and 60 bath sites are presented to show the practical performance of the shortcut NORG. Then a comparison between NORG and DMRG is made to help to further understand NORG. The notations and conventions follow what the main text has made unless otherwise stated.
Another application example of the shortcut NORG
================================================
The model is schematically shown in Figure \[model2d\], $2\times 2$ impurity sites with 60 bath sites. $t = 1$ and $U/t = 4$. The other parameters of the model are listed in Table \[para2d\]. The results are summarized in Figure \[NO2D4\] and Table \[gseg112d\]. The calculation procedure is slightly different from that in the main text, which is manifested in Table \[gseg112d\]. In Stage II, 32 orbitals with the smallest $w_g$ determined in Stage I are frozen, while the remaining orbitals are subject to a looser restriction than that in Stage I and are further optimized. After Stage II, another stage follows, i.e., Stage III, in which 12 more orbitals with the smallest $w_g$ are frozen, while the remaining orbitals are subject to a looser restriction than that in Stage II and we get more accurate physical quantities.
![(Color online) Visualization of the cluster-impurity model.[]{data-label="model2d"}](figS1){width="0.86\columnwidth"}
![(Color online) Occupancies of the natural orbitals obtained in Stage I of the shortcut NORG. \#1-3 correspond to those in Table \[gseg112d\], respectively. Here $w_g$ with values less than the machine precision don’t show up in the lower panel of the figure. From \#1, \#2, to \#3, the effective Hilbert space is getting systematically larger and larger, and $w_g$ converges fast.[]{data-label="NO2D4"}](figS2){width="\columnwidth"}
$k$ $\epsilon_{1k}$ $V_{1k}$
----- ----------------------- -----------------------
1 1.09432050353993E+00 -3.25861700524933E-01
2 -8.12129452137460E-01 -2.53572949534344E-01
3 2.69973133535642E+00 -2.98856870563701E-01
4 -6.27254332635770E+00 8.40922667355125E-02
5 -6.45481830221366E-01 1.34235217456270E-01
6 4.02420994365745E+00 2.45919037006646E-01
7 -1.63387758294565E+00 3.93233653967598E-01
8 0.00000000000000E+00 -3.76505701495740E-06
: Some of the parameters of the model. $\epsilon_{1k} = \epsilon_{2k} = \epsilon_{3k} = \epsilon_{4k}$ and $V_{1k} = V_{2k} = V_{3k} = V_{4k}$ ($k = 1, 2, ..., 15$) due to the four-fold rotational symmetry. $\epsilon_{1k} = -\epsilon_{1,16-k}$ and $V_{1k} = V_{1,16-k}$ ($k = 1, 2, ..., 8$) due to the particle-hole symmetry imposed.[]{data-label="para2d"}
\# Restriction Pattern IOs Space Size $E_0$ $\epsilon_f$ GF Err. $\epsilon_f$
---- ------------------------------------------- ------ ------------ ----------------- --------- -------------- --------- --------------
1 $ 30^{1+}2^{2+}0^{*}2^{2-}30^{1-} $ Site 71667 -148.5323438591 1.11E-3 0 – 0
2 $ 29^{1+}2^{2+}2^{*}2^{2-}29^{1-} $ Site 641368 -148.6909791787 4.60E-5 0 – 0
3 $ 28^{1+}2^{2+}4^{*}2^{2-}28^{1-} $ Site 7186598 -148.6975369976 1.94E-6 0 – 0
4 $ 16^{0+}15^{2+}2^{*}15^{2-}16^{0-} $ 1 427624 -148.6860293001 7.93E-5 3.71E-11 – 3.75E-8
5 $ 16^{0+}15^{2+}2^{*}15^{2-}16^{0-} $ 2 427624 -148.6860292991 7.93E-5 1.25E-10 – 1.06E-7
6 $ 16^{0+}15^{2+}2^{*}15^{2-}16^{0-} $ 3 427624 -148.6860292987 7.93E-5 1.57E-10 – 7.24E-8
7 $ 16^{0+}14^{2+}4^{*}14^{2-}16^{0-} $ 3 3573606 -148.6977692706 3.83E-7 1.57E-10 – 7.24E-8
8 $ 22^{0+}8^{2+}4^{*}8^{2-}22^{0-} $ 4 404580 -148.6977680385 3.91E-7 4.79E-8 6.52E-4 8.77E-6
9 $ 22^{0+}8^{2+}4^{*}8^{2-}22^{0-} $ 7 404580 -148.6977680646 3.91E-7 9.32E-8 4.79E-4 2.21E-5
10 $ 22^{0+}8^{3+}4^{*}8^{3-}22^{0-} $ 4 9160292 -148.6978260933 – 4.79E-8 – 8.77E-6
11 $ 22^{0+}8^{3+}4^{*}8^{3-}22^{0-} $ 7 9160292 -148.6978261476 – 9.32E-8 – 2.21E-5
\# Restriction Pattern Space Size $E_0$ GF Err.
---- --------------------------------------- ------------ ------------------ --------- ---------
1 $ 25^{0+}5^{2+}4^{*}5^{2-}25^{0-} $ 67656 -148.69749932305 3.90E-7 4.69E-4
2 $ 25^{0+}5^{3+}4^{*}5^{3-}25^{0-} $ 582456 -148.69755653670 4.83E-9 6.03E-6
3 $ 25^{0+}14^{*}25^{0-} $ 11778624 -148.69755725526 0 0
Method NORG DMRG
---------------------------- -------------------------------- --------------------------
Central concept Single-particle density matrix Reduced density matrix
Characteristic quantity Natural-orbital occupancy Entanglement spectrum
Variational state Orbital-frozen state Matrix product state
Favorite interaction form Sparse interactions Short-range interactions
Favorite spatial structure Unrestricted One dimensional chain
Many-body feature Sparse correlation Low entanglement
A comparison between NORG and DMRG
==================================
It is well-known that DMRG works best on 1D systems with short-range interactions. The underlying reason is that such systems often possess low entanglement. Then the entanglement spectrum, namely the eigenvalues of the reduced density matrix, shows a rapid-decay (often exponential-decay) feature, which allows an efficient truncation of many contribution-negligible states and a good description of the ground state by a matrix product state. In comparison, NORG works best on fermionic systems with sparse interactions, namely a small number of interacting sites and a large number of non-interacting sites (quantum multisite and/or multiorbital impurity models are excellent examples). The underlying reason is that such systems often possess ‘sparse correlation’, i.e., the ground state wave function can be expanded by a small number of Slater determinants compared to the dimension of the complete Hilbert space of the systems. (See below for the detailed explanation of the concept of ‘sparse correlation’.) Then the natural-orbital occupancy, namely the eigenvalues of the single-particle density matrix, shows a rapid-decay (often exponential-decay) feature, i.e., the occupancy quickly approaches full occupancy or empty, which allows an efficient truncation of many contribution-negligible orbitals and a good description of the ground state by an orbital-frozen state. These discussions are summarized in Table \[cmpdmrg\].
The NORG described in the left column in the second page of the main text, as a straightforward realization of the spirit above, is an iterative RG. In each iteration some orbitals are added in. After a diagonalization, some orbitals with small contribution are frozen out. In some sense, this process looks like the finite-system DMRG, in which in each iteration the full degrees of freedom of few sites are added in and after a diagonalization some degrees of freedom (states) with small contribution are frozen out (truncated out). This straightforwardly realized NORG works indeed, but not so efficiently as expected. Since we have been aware of the above spirit and reality allowing the existence of an efficient RG approach on fermionic systems with sparse interactions, resembling the case that DMRG works on 1D quantum systems with short-range interactions, and been aware of that the RG procedure of DMRG is merely to find a small structured subspace to accommodate a specific variational wave function (i.e. the matrix product state) with the ability describing accurately the ground state, we don’t have to confine our mind to strict iterative RG. Then the shortcut NORG comes out, in which more sufficient preparatory work (finding a set of relatively good natural orbitals) is done before a tremendous renormalization (freezing many orbitals) takes place. Such a shortcut NORG is so efficient that sometimes only one truncation is needed in solving a system (like the one in the main text), and sometimes more than one (like the one in the Supplementary Material).
Here we explain the concept of ‘sparse correlation’ for fermionic systems qualitatively in terms of Slater determinants. Consider a many-body wave function. If it can be expressed exactly by only one Slater determinant, it possesses no correlation, which means that the Hartree-Fock theory provides an exact description to it. If it can be well approximated by a single Slater determinant, it possesses weak correlation, which means that the Hartree-Fock theory provides a good but approximate description to it. Otherwise, it possesses strong correlation. Then we ask the question how many Slater determinants are needed to (approximately) well describe a wave function possessing strong correlation. A problem that one encounters is that the number of the Slater determinants may depend on the orbital set that one chooses. Therefore, we only consider the minimum possible number of the Slater determinants needed to (approximately) well describe the wave function. If the logarithm of the number is substantially smaller than the number of the single-particle degrees of freedom (orbitals or sites) of the wave function, the wave function possesses sparse correlation. Otherwise the wave function possesses dense correlation. Among all the orbital representations, the minimum possible number appears when the wave function is expanded in the natural-orbital representation, as shown in our manuscript. Through practice we now know that the logarithm of the minimum possible number of the Slater determinants needed to (approximately) well describe the ground state of a general fermionic system is roughly proportional to the number of its interacting sites. If all the sites in a system are interacting, a Hubbard model for example, then the logarithm of the number is roughly proportional to the size of the system. This means that quantum impurity models possess sparse correlation while Hubbard-like models possess dense correlation.
|
---
abstract: |
If you recommend a product to me and I buy it, how much should you be paid by the seller? And if your sole interest is to maximize the amount paid to you by the seller for a sequence of recommendations, how should you recommend optimally if I become more inclined to ignore you with each irrelevant recommendation you make? Finding an answer to these questions is a key challenge in all forms of marketing that rely on and explore social ties; ranging from personal recommendations to viral marketing.
In the first part of this paper, we show that there can be no pricing mechanism that is “truthful” with respect to the seller, and we use solution concepts from coalitional game theory, namely the Core, the Shapley Value, and the Nash Bargaining Solution, to derive provably “fair” prices for settings with one or multiple recommenders. We then investigate pricing mechanisms for the setting where recommenders have different “purchase arguments”. Here we show that it might be beneficial for the recommenders to withhold some of their arguments, unless anonymity-proof solution concepts, such as the anonymity-proof Shapley value, are used.
In the second part of this paper, we analyze the setting where the recommendee loses trust in the recommender for each irrelevant recommendation. Here we prove that even if the recommendee regains her initial trust on each successful recommendation, the expected total profit the recommender can make over an infinite period is bounded. This can only be overcome when the recommendee also incrementally regains trust during periods without any recommendation. Here, we see an interesting connection to “banner blindness”, suggesting that showing fewer ads can lead to a higher long-term profit.
[**Keywords:**]{} recommendations, pricing mechanisms, trust loss in advertising, banner blindness
author:
- 'Paul Dütting[^1]'
- 'Monika Henzinger[^2]'
- 'Ingmar Weber [^3]'
bibliography:
- 'pricing\_recommendations.bib'
title: 'On the Pricing of Recommendations and Recommending Strategically[^4]'
---
Introduction
============
Suppose you buy a new mobile phone and, simply because you like it so much, you recommend it to friends, encouraging them to purchase it as well. Even if you do not recommend it out of monetary reasons, what would be an adequate and *fair* price for the phone manufacturer to pay for your recommendation? If, on the other hand, you recommend a book at Amazon solely due to the monetary incentive given by Amazon’s referral scheme[^5] and your friends realize this, then they are likely to lose trust in your recommendations. Assuming your friends regain trust whenever you make a relevant recommendation, how can you maximize your long-term profit, and is this profit bounded or not? These are the two main research questions we address in this work.
The importance of “word-of-mouth” (WOM) as a marketing channel has long been known [@BR87; @HKK91; @GCD03]. According to [@BR87], “WOM was seven times as effective as newspapers and magazines, four times as effective as personal selling, and twice as effective as radio advertising in influencing consumers to switch brands”. WOM is the causal effect behind “brand congruence” where friends both in offline [@RFBS84] and online [@SW09] social networks tend to use the same products. Recently, a platform called [*Friend Vouch*]{} [@FV08] was founded around the idea of personal recommendations. Users of the service can become “brand ambassadors” who get paid for putting companies in touch with friends. Whether any personal touch is retained in such a system or whether the person in the middle is not simply another marketing channel is debatable and in Section \[sec:classification\] we propose a classification schema to shed light on the differences.
As far as the pricing of recommendations is concerned, one could argue that honest recommendations should always be given without any monetary recompensation and that creating financial incentives could lead to a sell-out of friends. Although this is a valid concern, we argue that it might still be worth paying recommenders, even if these are not asking to be paid. First, even though you might not be profit-maximizing in a strict sense you are probably more inclined to mention a certain product if there is some kind of recompensation: you might be honest enough not to recommend a bad product over a good one for financial reasons, but you are still more likely to recommend a good product if you get reimbursed. Second, a *fair* compensation can lead to increased brand loyalty. If you are already satisfied with the product then the feeling that the company recompensates you in a fair and adequate manner is likely to increase your positive attitude towards the company. On the other hand, if you are only offered \$1 for recommending a particular type of sports car then this might be viewed as “offensive” and is arguably worse than not being offered any recompensation.
Especially, as the issue of trust is of utmost relevance in the realm of personal recommendations, we believe that “fair pricing” is a cornerstone [@R04; @M08a].[^6] Google e.g. advertises its Adsense program by claiming to use a Second-Price Auction to eliminate “that feeling that you’ve paid too much”[^7]. In a similar spirit, our work on pricing recommendations can be viewed as trying to eliminate “the feeling that you’ve been paid too little” for your recommendation.
The second problem we study, relates to scenarios where the recommender is selfish and only makes paid recommendations to maximize her own profit. Here, in a sense, the friend making the recommendation is no more trustworthy or altruistic than a web search engine showing sponsored search results. In these settings we believe the trust between the recommender and the recommendee to be dissipating. More concretely we assume that with every unsuccessful recommendation the recommendee becomes more and more likely to ignore any “advice” given by her friend. We see this as closely related to “banner blindness” [@BL98; @CHN00; @BHNG05], where people have become so overloaded and fed up with advertisement that they stop to notice it completely. Seen from this angle, our findings indicate that advertisers might have to *stop* showing advertisements on a regular basis if they want to retain customers’ trust without seeing click-through-rates converge to zero.
Related Work
------------
Even though recommendations can be seen as just another form of advertising, classical methods for the pricing of advertising, such as sponsored search auctions [@LPSV07], are not directly applicable. This is mainly due to the fact that a true recommendation should be altruistic and so (i) the recommender is not profit maximizing and (ii) there is only a *single* seller, as an altruistic recommender will not accept “bids” from multiple sellers. The differences between various kinds of advertising are described in Section \[sec:classification\].
The work that is most closely related to our paper is [@AMSX09]. There the authors study the sales price of an object as part of a viral marketing campaign. They assume that all “converted” nodes will try to convert all of their neighbors and that the conversion probability depends both on the number of neighbors converted and on the sales price. They do *not* consider the problem of how the recommendation itself should be rewarded. In fact, they mention the problem of finding optimal “cashbacks” in settings where the nodes behave strategically as an open problem.
The problem of optimal pricing with non-social recommender systems, where the recommendations directly come from the potential seller, was studied in [@BO06]. Here by “non-social” we mean “computer-generated” and a typical example would be Amazon’s “Customers who bought X also bought Y”[^8]. The somewhat surprising argument is that customers are willing to *pay* for relevant recommendations as they create “value by reducing product uncertainty for the customers”. In this paper, we consider the case where the recommendations are social and do not come from the seller directly. Though it is imaginable that the recommendee pays the recommender for a good recommendation, we do not investigate the pricing of this possible payment.
It should be clear that we are *not* addressing the problem of *what* to recommend, a problem typically encountered by stores such as Amazon and usually solved using “collaborative filtering” techniques [@SM95; @HKTR04]. In the first part of this paper (Section \[sec:pricing\]), we assume that the recommender recommends an item because she believes this item to be of interest to the recommendee, and the algorithm used by her to determine potential interest is irrelevant. In the second part (Section \[sec:recommending-strategically\]), the recommender is profit maximizing and now only cares about the reward offered to her by the seller and the probability $p$ that the recommendee will buy the item. In this model the “what” is absorbed into $p$ and the recommender simply decides on *when* to recommend.
We are also *not* addressing the topic of how rumors spread through social networks, or how to identify the best nodes to target for a viral marketing campaign [@DR01; @KKT03]. Our work focuses on a single atomic link in the corresponding cascades of conversions and, in the first part, we ask what a fair price should be to pay a node for activating one of her neighbors. In answering this question we limit our attention to the immediate profit of the seller due to the individual sale, and we do *not* consider the additional value due to recommendation cascades caused by the newly activated node. However, given any algorithm to compute this “higher order” profit, it can trivially be incorporated into our results. The question whether a selfish node should actually try to activate her neighbors at all is addressed in Section \[sec:recommending-strategically\].
More generally, in the second part we look at a model where the recommendee loses trust in the recommender, i.e. for each unsuccessful recommendation she becomes less and less inclined to listen to any further suggestion. This is most likely to appear when the recommendee has the feeling that the recommendations are “dishonest”. How honest recommendations can be ensured when there are several recommenders is studied in [@GEB04]. The approach suggested by the authors involves evaluating/ranking recommenders based on the rating given to their recommended items by other people. This motivates recommenders to give good recommendations in a similar way that Ebay’s rating system gives incentives for both buyers and sellers “to behave”. This approach, however, requires a public market where potential buyers can look for recommendations. This is not the setting of personal recommendations considered here.
The problem of trust decay is related to “banner blindness” [@BL98; @CHN00; @BHNG05], where web users become “blind” to banner ads due to overexposure. Cast to this setting our mathematical model suggests that, even if web users’ interest is “refreshed” by a single relevant advertisement that is clicked, the long term profit of advertisers will stagnate as click-through-rates fall to zero. The only possible way out of this dilemma is to *stop* showing banner ads for a while so that users can “unlearn” to ignore all advertising. This approach is also suggested in a recent patent [@PK08].
In typical literature on sponsored search auctions [@LPSV07; @L06] it is assumed that the web search engine is optimizing its expected profit and that its expected profit for showing a particular ad is the ad’s click-through-rate (CTR) multiplied by the price the advertiser will be charged when her ad gets clicked. Usually, only a single round is considered or, when there are budget constraints [@AMT07; @FMPS07], the CTRs are assumed to be *constant* during the duration of the game. If, however, it is assumed that CTRs drop for *all* ads for each unsuccessful advertisement shown then, in the long run, this puts more emphasis on showing ads with high CTR, regardless of how much their advertisers can be charged for a single click. Although different objective functions for the search engine have been considered [@AMT07], the setting of profit maximization with trust decay has not been studied and we deem this an interesting area for future work.
Finally, there is previous work that is relevant on a more technical level. In particular solution concepts such as the Core [@G59; @M97] or the Shapley-Value [@S53; @M97] have been studied extensively before. The exact connection to this group of work will be made clear in the sections with our technical contributions.
Classification of Advertising Schemes {#sec:classification}
-------------------------------------
One could argue that a recommendation is, ultimately, just an advertisement and that an advertisement is just a recommendation. To highlight the differences between different kinds of advertisement in general, we present a simple classification scheme.
1. Addressing: Personal vs. general. A recommendation is per se more personal than an advertisement and should be adapted to reflect the individual needs and interests of the potential buyer. Classic advertisement is not personalized and uses the same “message” for everyone.
2. Trust: High vs. low. A recommendation should come from someone the potential buyer trusts and feels loyal or close to. This can be a personal friend or maybe a well-respected blogger. In classic advertisement the information source is viewed as less reputable, though advertisers try to use trusted icons for their purposes.
3. Intention: Altruistic vs. commercial. The intention of a recommendation by a friend is generally not commercial. She might not get reimbursed at all but she still recommends something as she believes you would profit from it. In ordinary advertising the reason for the act of advertising itself is a commercial one.
The first part of this paper (Section \[sec:pricing\]) considers the setting of personal, highly trusted and altruistic recommendations. The second part (Section \[sec:recommending-strategically\]) then investigates the case of still personal, but commercial recommendations with a decaying amount of trust involved. To demonstrate the general applicability of this schema, we use it to classify a number of different advertising scenarios.
1. Billboards. A chain of pizza restaurants puts up billboards all over the country, without targeting any specific group. Addressing: general, trust: low, intention: commercial.
2. Sponsored search. A web search engine shows targeted sponsored results in addition to “organic” web search results, trying to match the searcher’s intent. Addressing: personal, trust: low, intention: commercial.
3. Testimonial. You liked a book and you write a testimonial on Amazon to convince other unknown readers to read it, too. Addressing: general, trust: high, intention: altruistic.
4. Direct recommendation. A friend asks you for advice on which laptop to buy and you recommend the model which you believe is best for her. Addressing: personal, trust: high, intention: altruistic.
Of course, there are lots of other important differences, e.g. concerning the conversion rates, but we view these differences as consequences of the “axiomatic” differences above and we assume that a personalized, altruistic “advertisement” from a highly trusted source will always have a higher conversion rate than a general, commercial “recommendation” from a disreputable source.
Our Contributions and Outline
-----------------------------
To the best of our knowledge there has been no work focusing on either (i) the pricing of recommendations (our Section \[sec:pricing\]) or (ii) the strategic behavior of recommenders in a setting with decaying trust (our Section \[sec:recommending-strategically\]). We view the introduction of these problems as one of our contributions.
As far as the pricing of recommendations is concerned we prove that there can be no pricing mechanism that is “truthful” with respect to the seller (Section \[sec:truthful\]). This shows that the seller can always pretend to profit less from the recommendations than she actually does to get a larger piece of the pie. We then apply solution concepts from coalitional game theory, namely the Core, the Shapley value, and the Nash Bargaining Solution, to determine provably “fair” prices. For the Core we find that it typically contains all “individual rational” payoff vectors, including the payoff vector where the seller gets everything and the recommenders get nothing. On the one hand, this demonstrates the weakness of the recommenders: They cannot form a coalition with non-zero value without the seller. On the other hand, it shows that the Core is essentially useless for deciding how to distribute the “extra profit” the seller can expect from being recommended among the recommenders (Sections \[sec:one-core\] and \[sec:many-core\]). For the Shapley value we find that it not only defines unique prices, but that these prices are also “fair” in a very intuitive way: The price of a recommendation should be proportional to the “extra profit” the seller can expect from it (Section \[sec:one-shapley\] and \[sec:many-shapley\]). For the Nash Bargaining Solution we find that it yields “fair” prices, namely those obtained by the Shapley value, only if there is a single recommender. Otherwise, especially in situations where the recommenders do [*not*]{} contribute equally to the “extra profit” of the seller, it may lead to “unfair” prices (Section \[sec:one-nash\] and \[sec:many-nash\]). Finally, we also consider the case where each recommendation consists of one or more “purchase arguments”. Here the ordinary Shapley value is no longer the method of choice, as withholding arguments might be beneficial for the recommenders. We show how the anonymity-proof Shapley value from [@OCS08] can be applied to overcome this problem (Section \[sec:many-proofshapley\]).
In the second part on the strategic behavior of profit maximizing recommenders we first show that, not surprisingly, the total expected profit of the recommender is bounded when the recommendee can only lose and does not regain trust (Section \[sec:wo\_reset\_wo\_recovery\]). Then we prove that the total expected profit is still bounded over an infinite (!) sequence of recommendations, even when trust is reset to an initial level on each successful recommendation (Section: \[sec:with\_reset\_wo\_recovery\]). Finally, we show that when trust is regained incrementally when no recommendations are made, the recommender’s optimal total expected profit is unbounded in the long run and that she can recommend both too aggressively and too passively (Section \[sec:with\_reset\_with\_recovery\]). These results are also applicable to the phenomenon of “banner blindness”.
The Pricing of Recommendations {#sec:pricing}
==============================
We model the pricing of recommendations problem as a [*coalitional game with transferable payoff*]{} $\langle N,v \rangle$, where $N$ is a finite set (the set of [*players*]{}) and $v$ is a function that associates with every non-empty subset $S$ of $N$ (a [*coalition*]{}) a real number $v(S)$ (the [*worth*]{} of $S$). We use $s$ to denote the [*seller*]{}, who is paying for recommendations, and $r_i$ to denote the $i$[-th]{} recommender. There is exactly one [*product*]{} for sale.[^9] For each coalition $S$ the number $v(S)$ is the total payoff that is available for division among the members of $S$. We use $\delta \ge 0$ to denote the seller’s [*margin*]{} or [*gain*]{} from selling the product, i.e. the sales price minus the production cost, and distinguish three scenarios for $v$:
1. [*General.*]{} Without any recommendation the product is sold with probability $p \in [0,1].$ If the recommenders $R \subseteq \{r_1, .., r_n\}$ recommend the product, then the probability that the product is sold is $p+f(\{s\} \cup R)$, where $f: 2^N \rightarrow [0,1-p]$ is an arbitrary function with $f(\{s\}) = 0$.
The following two scenarios are special cases of [*General*]{}.
1. [*Linear.*]{} Without any recommendation the product is sold with probability $p \in [0,1].$ The recommendation of the $i$[-th]{} recommender increases this probability by $q_i \in [0, 1 - \sum_{j \neq i} q_j].$ The joint effect of more than one recommendation is the sum of the effect of the individual recommendations. Formally, if the recommenders $R \subseteq \{r_1, .., r_n\}$ recommend the product, then the probability is $p+\sum_{i: r_i \in R } q_i.$
2. [*Threshold.*]{} If less than $k$ recommenders recommend the product, then the product is sold with probability $p \in [0,1].$ If at least $k$ recommenders recommend the product, then it is sold with probability $p+q$ where $q \in [0,1-p].$
We refer to these scenarios as $\langle N, v \rangle$ (General), $\langle N, v \rangle$ (Linear), and $\langle N, v \rangle$ (Threshold). The following table gives the worth $v(S)$ of all $S \subseteq N = \{s, r_1, r_2\}$ for all three scenarios.
$S$ Linear Threshold General
-------------------- -------------------------- ------------------ --------------------------
$\emptyset$ 0 0 0
$\{s\}$ $p \delta$ $p \delta$ $p \delta$
$\{r_1\}$ 0 0 0
$\{r_2\}$ 0 0 0
$\{s, r_1\}$ $(p + q_1) \delta$ $p \delta$ $(p+f(s,r_1))\delta$
$\{s, r_2\}$ $(p + q_2) \delta$ $p \delta$ $(p+f(s,r_2))\delta$
$\{r_1, r_2\}$ 0 0 0
$\{s, r_1, r_2 \}$ $(p + q_1 + q_2) \delta$ $(p + q) \delta$ $(p+f(s,r_1,r_2))\delta$
: Worths $v(S)$ of all possible coalitions $S$ for one seller $s$ and two recommenders $r_1$ and $r_2$ for our three different models. []{data-label="tab:worths"}
Our goal is to find a payoff vector $(x_s,x_{r_1}, \dots x_{r_n})$, where $x_s$ denotes the expected payoff to the seller and $x_{r_i}$ denotes the expected payoff to the $i$[-th]{} recommender. Suppose that the seller $s$ is recommended by all recommenders $r_i \in N \setminus \{s\}$, then the worth of this coalition is $v(N) = (p + f(N)) \cdot \delta.$ We say that the payoff vector $(x_s,x_{r_1}, \dots x_{r_n})$ is [*feasible*]{} if $x_s + \sum_i x_{r_i} = v(N)$. A feasible payoff vector, which prescribes the [*expected*]{} payoff to each player, can be translated into [*prices*]{}, i.e. payments from the seller to the recommenders, as follows:
1. [*Pay-per-Recommendation*]{}: The recommender gets paid by the seller for every recommendation; successful or not. That is, on every recommendation the seller $s$ pays the $i$[-th]{} recommender $r_i$ the money equivalent of $x_{r_i}.$
2. [*Pay-per-Sale*]{}: The recommender gets paid by the seller for successful recommendations only. That is, on every successful recommendation the seller $s$ pays the $i$[-th]{} recommender $r_i$ the money equivalent of $1/(p+f(N)) \cdot x_{r_i}$.
In practice, the Pay-per-Sale approach might be preferable as, on a successful recommendation, one could reasonably assume $p+f(N)=1$, sidestepping the problem of estimating $f(N)$ with very little or no data. Note that the prior probability $p$ is easier to estimate using the seller’s sales record and click-through or conversion-rates.
Impossibility Result {#sec:truthful}
--------------------
Ideally, the payoff vector $(x_s, x_{r_1}, \dots, x_{r_n})$ computed by whatever mechanism should give the seller $s$, who holds the private information on $p$, $f$, and $\delta$, the incentive to reveal her information [*truthfully*]{}. Formally, we want that for all $p'$, $f'$, and $\delta':$ $x_s(p,f,\delta) \ge x_{s}(p',f',\delta')$, where $x_s(p,f,\delta) = (p+f(N)) \cdot \delta - \sum_{i} x_{r_i}(p,f,\delta)$ and $x_{s}(p',f',\delta') = (p+f(N)) \cdot \delta - \sum_{i} x_{r_i}(p',f',\delta').$ Unfortunately, as the following theorem shows, the only truthful payoff vector has $\sum_{i} x_{r_i} = 0$. That is, the seller gets everything and the recommenders get nothing.
\[the:truthful\] There can be no truthful payoff vector $(x_s,$ $x_{r_1}$, $\dots, x_{r_n})$ that has $\sum_{i} x_{r_i} \neq 0$ and ensures participation of the seller $s$ and the recommenders $r_1$ to $r_n.$
To ensure participation for the seller, we must have $x_s(p,f,\delta) = (p+f(N)) \cdot \delta
- \sum_i x_{r_i}(p,f,\delta) \ge 0$ for all $p$, $f$, and $\delta.$ To ensure participation for the recommenders $r_1$ to $r_n$ we must have $\sum_i x_{r_i}(p,f,\delta) \ge 0$ for all $p$, $f$, and $\delta$. Now suppose $(x_s,$ $x_{r_1}$, $\dots, x_{r_n})$ with $\sum_{i} x_{r_i} \neq 0$ was truthful. It follows that $x_s(p,f,\delta) \ge x_s(p',f',\delta')$ for all $p'$, $f'$, and $\delta'$, i.e. $(p+f(N))\cdot \delta - \sum_i x_{r_i}(p,f,\delta) \ge (p+f(N)) \cdot \delta -
\sum_i x_{r_i} (p',f',\delta').$ And hence, $\sum_i x_{r_i} (p',f',\delta') \ge \sum_i x_{r_i}
(p,f,\delta).$ But since $\sum_{i} x_{r_i}(p,f,\delta) > 0$ there must be $p'$, $f'$, $\delta'$ such that$(p'+f'(N)) \cdot \delta' < \sum_i x_{r_i}(p,f,\delta)$ with $\sum_i x_{r_i}(p',f',\delta')
\le (p'+f'(N)) \cdot \delta' < \sum_i x_{r_i}(p,f,\delta).$ Contradiction!
One Recommender {#sec:one}
---------------
We begin by studying the problem of finding “fair” prices in the setting $N = \{s,r\}$, i.e., there is only one seller and one recommender. In this setting the games $\langle N, v \rangle$ (Linear) and $\langle N, v \rangle$ (Threshold) are equivalent. We discuss the solution concepts [*Core*]{}, [*Shapley value*]{}, and [*Nash Bargaining Solution*]{}. For a more detailed discussion of these solution concepts see [@M97; @OR94].
### The Core {#sec:one-core}
The [*Core*]{} [@G59] of a coalitional game is an outcome of cooperation among all players where no coalition of players can obtain higher payoffs for all of its members. A payoff vector in the Core is “fair” in the sense that no subset of players can justifiably argue that they are paid to little, as they are unable to achieve higher payoffs on their own.
More formally, the Core of the game $\langle N,v \rangle$ is the set of [*feasible*]{} payoff vectors $(x_i)_{i \in N}$ for which there is no coalition $S \subseteq N$ and $S$-[*feasible*]{} payoff vector $(y_i)_{i \in N}$ such that $y_i > x_i$ for all $i \in S.$ Recall that a payoff vector $(x_i)_{i \in N}$ is [*feasible*]{} if $\sum_{i \in N} x_i = v(N)$. It is $S$-[*feasible*]{} if $\sum_{i \in S} x_i = V(S)$.
The Core can be shown to be non-empty by means of the Bondareva-Shapley Theorem [@B63; @S67], which states that a game has a non-empty core if and only if it is balanced. A game $\langle N, v \rangle$ is [*balanced*]{} if for every balanced collections of weights $(\lambda_S)_{S \subseteq N}$: $\sum_{S} \lambda_S \cdot v(S) \leq v(N).$ A [*balanced collection of weights*]{} $(\lambda_S)_{S \subseteq N}$ is a collection of numbers $\lambda_S \in [0,1]$ (one for each coalition $S \subseteq N$) such that for all $i$: $\sum_{S \subseteq N: i \in S} \lambda_S = 1.$
\[the:one-core\] The game $\langle \{s,r\},v \rangle$ (General) has a non-empty core.
Let $x \in [0,1].$ All balanced collections of weights $(\lambda_S)_{S \subseteq N}$ are of the form $\lambda_{S} = x$ for $S = \{s\}, \{r\}$ and $\lambda_{S} = 1-x$ for $S = \{s,r\}.$ By the Bondareva-Shapley Theorem, the Core is non-empty if and only if for all values $x \in [0,1]:$ $$\begin{aligned}
x \cdot (v(\{s\}) + v(\{r\})) + (1-x) v(\{s,r\}) & \leq v(\{s,r\}).\end{aligned}$$ For $x = 0$ this is trivially true. Next we we analyze the case $x > 0.$ Since $v(\{r\}) = 0$ and $v(\{s,r\}) - v(\{s\}) = f(s,r) \cdot \delta$, $$\begin{aligned}
& \ x \cdot v(\{s\}) - x \cdot v(\{s,r\}) &&\leq 0\\
\Leftrightarrow & \ v(\{s,r\}) - v(\{s\}) &&\geq 0\\
\Leftrightarrow & \ f(\{s,r\}) \cdot \delta &&\geq 0.\end{aligned}$$ Since $f(\{s,r\}) \ge 0$ and $\delta \ge 0$ this is always true.
Recall that the games $\langle N, v \rangle$ (Linear) and $\langle N, v \rangle$ (Threshold) are special cases of the game $\langle N, v \rangle$ (General) and so Theorem \[the:one-core\] also shows non-emptiness of the Core for these games. Next we give necessary and sufficient conditions for a payoff vector $(x_s, x_{r_1}, \dots, x_{r_n})$ to be in the Core.
The payoff vector $((p+f(\{s,r\}))\delta - x, x)$ is in the Core of the game $\langle \{s,r\}, v\rangle$ (General) if and only if: $$\begin{aligned}
&0 \le x \leq f(\{s,r\}) \cdot \delta.\end{aligned}$$
Let $x_s = (p+f(\{s,r\}))\delta - x$ and let $x_r = x.$ If the vector $(x_s,x_r)$ is in the Core of the game $\langle \{s,r\}, v\rangle$ (General), then there exists no coalition $S \subseteq N = \{s,r\}$ and an $S$[-feasible]{} payoff vector $y=(y_s,y_r)$ such that $y_i > x_i$ for all $i \in S.$ That is, for all $S \subseteq N$ and $S$[-feasible]{} payoff vectors $y=(y_s,y_r)$ we have that $y_i \le x_i$ for all $i \in S.$ For $S = \{s,r\}$ this means that $y_s + y_r = v(\{s,r\}) \le x_s + x_r = v(\{s,r\})$ (which is trivially true). For $S = \{s\}$ this means that $y_s = v(\{s\}) = p \cdot \delta \le x_s = (p + f(\{s,r\})) \cdot \delta - x$, i.e. $x \le f(\{s,r\}) \cdot \delta.$ For $S = \{r\}$ this means that $y_r = v(\{r\}) = 0 \le x_r = x$, i.e. $x \ge 0$. That is, $0 \le x \le f(\{s,r\}) \cdot \delta.$
For the reverse direction assume by contradiction that $0 \le x \le f(\{s,r\}) \cdot \delta$ but that $(x_s,r_r)$ is [*not*]{} in the Core, i.e. there exists a coalition $S \subseteq N = \{s,r\}$ and a $S$[-feasible]{} payoff vector $y=(y_s,y_r)$ such that $y_i > x_i$ for all $i \in S.$ We cannot have $S = \{s,r\}$ as then $y_s+y_r = v(\{s,r\}) > x_s + x_r = v(\{s,r\})$, which gives a contradiction. But if $S = \{s\}$, then $y_s = v(\{s\}) = p \cdot \delta > x_s = (p + f(\{s,r\}))\cdot \delta - x$, i.e. $x > f(\{s,r\}) \cdot \delta$, which gives a contradiction. Finally, if $S = \{r\}$, then $y_r = v(\{r\}) = 0 > x_r = x$, i.e. $x < 0$, which also gives a contradiction.
For the games $\langle \{s,r\}, v\rangle$ (Linear) and $\langle \{s,r\}, v \rangle$ (Threshold) this means that $((p+q)\delta - x, x)$ is in the core if and only if: $$\begin{aligned}
&0 \le x \leq q \cdot \delta.\end{aligned}$$ This implies that any “feasible” payoff vector is in the Core. The only restriction on the payoff vector is that the payoff to the recommender be non-negative and in expectation no higher than the “extra profit” the seller can expect from the recommendation. In particular, a payoff vector that gives everything to the seller and nothing to the recommender would be in the Core. This demonstrates the weakness of the recommenders: They cannot form a coalition with non-zero value without the seller.
### Shapley Value {#sec:one-shapley}
One problem with the Core is that it does [*not*]{} assign a unique payoff vector to a game. This makes it necessary to have another criterion for choosing a payoff vector. The [*Shapley value*]{} [@S53] is a solution concept that assigns to each game a unique, provably fair payoff vector. In general, a [*value*]{} $\phi: v \rightarrow \mathbb{R}^{n+1}$ maps each game $\langle N,v \rangle$ to a unique vector $\phi(v)$; the $i$[-th]{} entry $\phi_i(v)$ of this vector being the [*expected payoff*]{} to player $i.$ The Shapley value is the unique value satisfying the following axioms:
1. [*Symmetry:*]{} If player $i$ and $j$ are interchangeable, then $\phi_i(v) = \phi_j(v).$ Formally, if for every $S \subseteq N$ s.t. $i \in S$, $j \not\in S$: $v( (S \setminus \{i\}) \cup \{j\}) = v(S)$, then $\phi_i(v) = \phi_j(v).$
2. [*Dummy:*]{} If player $i$’s contribution to any coalition $S$ is zero, then $\phi_i(v) = 0.$ Formally, if for every $S \subseteq N \setminus \{i\}$: $v(S \cup \{i\}) = v(S)$, then $\phi_i(v) = 0.$
3. [*Additivity:*]{} Player $i$’s entry $\phi_i(v)$ should be additive in $v.$ Formally, if $\langle N, v + w\rangle$ is derived from $\langle N,v \rangle$ and $\langle N,w \rangle$ by defining $(v+w)(S) = v(S) + w(S)$ for all $S \subseteq N$, then $\phi_i(v+w) = \phi_i(v) + \phi_i(w)$ for all $i \in N$.
These axioms can be interpreted as formalizing a notion of “fairness”, that postulates that the expected payoff to player $i$ be proportional to player $i$’s contribution to the outcome of the game. For an analysis along these lines see [@M91].
\[def:shapley\] The Shapley value $\phi(v) = (\phi_1(v), \dots, \phi_N(v))$ of the game $\langle N, v\rangle$ is defined as follows: $$\begin{aligned}
\phi_i(v) &= \sum_{S \subseteq N \setminus \{ i \}} \frac{|S|!(|N|-1-|S|)!}{|N|!} \cdot (v(S \cup \{ i\}) - v(S)). \end{aligned}$$
One interpretation of this is: Suppose that all the players are arranged in some order, all orders being equally likely, then $\phi_i(v)$ is the [*expected marginal contribution*]{} of player $i$ to the set of players who precede her.
\[the:shapley-one\] Consider the game $\langle \{s,r\}, v \rangle$ (General). The Shapley value $\phi(v) = (\phi_s(v), \phi_{r}(v))$ is given by $$\begin{aligned}
\phi_s(v) = p \delta + \frac{1}{2} f(\{s,r\}) \delta \mbox{\hspace{2mm} and \hspace{2mm}} \phi_{r}(v) = \frac{1}{2} f(\{s,r\}) \delta.\end{aligned}$$
The claim follows from the definition of the game $\langle \{s,r\}, v \rangle$ (General) and Definition \[def:shapley\]. Note that the worths $v(S)$ of all $S \subseteq N = \{s,r\}$ can be read from Table \[tab:worths\] by treating $r_1$ as $r$ and ignoring rows containing $r_2.$
For the game $\langle \{s,r\}, v \rangle$ (Linear, Threshold) this means that $$\begin{aligned}
\phi_s(v) = p \delta + \frac{1}{2} q \delta \mbox{\hspace{2mm} and \hspace{2mm}}\phi_{r}(v) = \frac{1}{2} q \delta.\end{aligned}$$ This shows that the payoff to the recommender should be proportional to her contribution to the seller’s expected “extra profit”. In particular, it shows that the recommender’s payoff should be [*linear*]{} in her contribution to the purchase probability, i.e. $f(\{s,r\})$, and also in the seller’s margin or gain $\delta.$ This is consistent with “real life” pricing schemes that redeem the recommender with a certain percentage of the sales price [@LAH06], assuming that for a given product family the margin is proportional to the sales price.
### The Nash Bargaining Solution {#sec:one-nash}
The last solution concept that we discuss in this section is the [*Nash Bargaining Solution*]{} [@N50].[^10] The basic idea here is to view the game $\langle N,v \rangle$ as a [*bargaining problem*]{} over a set $F$ of feasible payoff vectors $f = (f_0,\dots,f_n)$ and a dedicated payoff vector $d=(d_0,\dots,d_n)$; the payoff vector in case of a [*disagreement*]{}. A [*solution*]{} is a function $\phi: (F,d) \rightarrow F.$ The Nash Bargaining Solution is the unique solution satisfying the following axioms:
1. [*Pareto Efficiency.*]{} There is no $f \in F$ such that $f_i \ge \phi_i(F,d)$ for all $i \in N$ and $f_j > \phi_j(F,d)$ for at least one $j \in N.$
2. [*Individual Rationality.*]{} For all $i \in N:$ $\phi_i(F,d) \ge d_i$.
3. [*Scale Covariance.*]{} If $F' = \{(\lambda_0 \cdot f_0 + \gamma_0, \dots, \lambda_n \cdot f_n + \gamma_n) \ | \ (f_0, \dots, f_n) \in F\}$ and $d' = (\lambda_0 \cdot d_0 + \gamma_0, \dots, \lambda_n \cdot d_n + \gamma_n)$, then $\phi(F',d') = (\lambda_0 \cdot \phi_0(F,d) + \gamma_0, \dots, \lambda_n \cdot \phi_n(F,d) + \gamma_n).$
4. [*Independence of Irrelevant Alternatives.*]{} If $F' \subseteq F$ and $\phi(F,d) \in F'$, then $\phi(F',d) = \phi(F,d)$.
5. [*Symmetry.*]{} If $ (f_0, \dots, f_i, \dots, f_j, \dots, f_n) \in F$ implies $(f_0, \dots, f_j, \dots, f_i, \dots, f_n) \in F$ and $d_i = d_j$, then $\phi_i(F,d) = \phi_j(F,d).$
The Nash Bargaining solution is “fair” in the sense that is [*Pareto effcient*]{}, i.e. it is impossible to improve the payoff of one or more players without hurting that of others. One can show that it satisfies $\phi(F,d) \in \mbox{argmax}_{f \in F} \prod_i (f_i - d_i)$ [@M97]. We use this to prove:
For $\langle \{s,r\}, v \rangle$ (General) let $F = \{ (f_s,f_r) \ |$ $ \ f_s \ge 0, f_r \ge 0 \ \mbox{and} \ f_s + f_r = (p + f(\{s,r\})) \cdot \delta \}$ and $d = (d_s, d_r) = (p \cdot \delta,0)$. Then, $$\begin{aligned}
\phi(F,d) = ((p+\frac{1}{2}f(\{s,r\}) \delta , \frac{1}{2}f(\{s,r\}) \delta ).\end{aligned}$$
For the game $\langle \{s,r\}, v \rangle$ (Linear, Threshold) this means that: $$\begin{aligned}
\phi(F,d) &= \left( \left(p+\frac{1}{2}q\right) \delta , \frac{1}{2}q\delta \right).\end{aligned}$$ This demonstrates that the Nash Bargaining Solution and the Shapley value coincide in our model. On the one hand, this is surprising as the axioms used to define the Shapley value and the Nash Bargaining Solution are quite different. On the other hand, this is intuitive as for two players there is only one non-trivial coalition to be considered for the Shapley value. So if this two-player coalition leads to a bigger payoff $P$ than the sum of its two non-cooperative “atoms”, then the different symmetry axioms present in both solution concepts imply that this surplus should be divided 50-50. This holds as long as the feasible payoff vectors are $F=\{(f_0,f_1) | f_0+f_1 \le P \}$, and so $(f_0,f_1) \in F \Leftrightarrow (f_1,f_0) \in F$, but would stop to hold if, e.g. there were additional constraints on $F$ such as $f_0 \ge 2 \cdot f_1$. In such cases, the symmetry axiom of the Nash Bargaining Solution no longer applies.
Many Recommenders {#sec:many}
-----------------
Next we study the problem of finding “fair” prices in the more general setting $N = \{s, r_1, \dots, r_n\}$, i.e. there is one seller and $n \ge 1$ recommenders. Note that in this setting the games $\langle N,v \rangle$ (Linear) and $\langle N,v \rangle$ (Threshold) are no longer equivalent. As in the setting where $N = \{s,r\}$ we study the solution concepts Core, Shapley value, and Nash Bargaining Solution.
### The Core {#sec:many-core}
Recall that the Core of the game $\langle N, v \rangle$ comprises all feasible payoff vectors with which no coalition $S \subseteq N$ is “unhappy” meaning that the players in $S$ cannot break away to obtain a higher payoff on their own. For a formal definition of the Core (and related definitions) see Section \[sec:one-core\].
\[the:many-core-1\] The game $\langle N,v \rangle$ (General) has a non-empty core iff for every balanced collections of weights $(\lambda_S)_{S \subseteq N}$: $$\begin{aligned}
&\sum_{S \subset N: s \in S} \left[ \lambda_{S} (f(N) - f(S)) \right] \geq 0.\end{aligned}$$
Let $(\lambda_S)_{S \subseteq N}$ be a balanced collection of weights. Since $v(S) = 0$ whenever $s \not\in S$, applying the Bondareva-Shapley Theorem [@B63; @S67] to the game $\langle N,v \rangle$ (General) gives: $$\begin{aligned}
&\sum_{S \subseteq N: s \in S} \left[ \lambda_S (p + f(S)) \right] \leq p + f(N).\end{aligned}$$ Since $(\lambda_S)_{S \subseteq N}$ is a balanced collections of weights, we have $\sum_{S \subseteq N: s \in S} \lambda_S = 1$ and $\lambda_N = 1 - \sum_{S \subset N: s \in S} \lambda_S.$ It follows that: $$\begin{aligned}
& \sum_{S \subseteq N: s \in S} \lambda_S \cdot f(S) & \leq f(N).\\
\Leftrightarrow & \sum_{S \subset N: s \in S} [\lambda_S (f(S) - f(N))] + f(N) & \leq f(N)\\
\Leftrightarrow &\sum_{S \subset N: s \in S} \left[ \lambda_S (f(N) - f(S)) \right] &\geq 0. &&\qedhere\end{aligned}$$
The condition given by Theorem \[the:many-core-1\] holds trivially for $\langle N,v \rangle$ (General) if $f(N) \ge f(S)$ for all $S \subseteq N$ since all the $\lambda_S$ values are non-negative. For the game $\langle N,v \rangle$ (Linear) and $\langle N,v \rangle$ (Threshold) this means that the core is [*always*]{} non-empty since $q_i \ge 0$ for all $i$ respectively $q > 0$.
Consider the game $\langle N, v\rangle$ (General). The payoff vector $(x_s = (p+f(N)) \cdot \delta - \sum_{i} x_{r_i}, x_{r_1}, \dots, x_{r_n})$ is in the Core if and only if for all $T \subseteq N$ s.t. $s \not\in T$: $$\begin{aligned}
&0 \le \sum_{r_i \in T} x_{r_i} \leq (f(N) - f(N \setminus T)) \cdot \delta.\end{aligned}$$
Assume that the payoff vector $(x_s = (p+f(N)) \cdot \delta - \sum_{i} x_{r_i}, x_{r_1}, \dots, x_{r_n})$ is in the Core. Since $v(S) = 0$ for all coalitions $S \subseteq N$ such that $s \not\in S$, it follows that:
1\. For all $S \subseteq N$ such that $s \in S:$ $$\begin{aligned}
&\ (p + f(N)) \delta - \sum_{r_i \in N} x_i + \sum_{r_i \in S} x_i &&\geq v(S)\\
\Leftrightarrow &\ (p + f(N)) \delta - \sum_{r_i \in N \setminus S} x_i &&\geq (p+f(S))\delta\\
\Leftrightarrow &\ (f(N) - f(S)) \delta &&\geq \sum_{r_i \in N \setminus S} x_i \end{aligned}$$ 2. For all $S \subseteq N$ such that $s \not\in S$: $$\begin{aligned}
\sum_{r_i \in S} x_i \geq v(S) \ \Leftrightarrow \ \sum_{r_i \in S} x_i \geq 0.\end{aligned}$$ With $T = N \setminus S$ in 1. and $T = S$ in 2. it follows that $0 \le \sum_{r_i \in T} x_{r_i} \leq (f(N) - f(N \setminus T)) \cdot \delta$ for all $T \subseteq N$ s.t. $s \not\in T$.
For the reverse direction assume by contradiction that $0 \le \sum_{r_i \in T} x_{r_i} \leq (f(N) - f(N \setminus T)) \cdot \delta$ for all $T \subseteq N$ s.t. $s \not\in T$ but that $(x_s = (p+f(N)) \cdot \delta - \sum_i x_{r_i}, x_{r_1}, \dots, x_{r_n})$ is [*not*]{} in the Core, i.e. there exists a coalition $S \subseteq N$ and an $S$[-feasible]{} payoff vector $y=(y_s,y_{r_1}, \dots, y_{r_n})$ in which $y_k > x_k$ for all players $k$ in $S.$ Since $y$ is $S$[-feasible]{}, the total payoff $\sum_{k \in S} y_k$ to the players in $S$ must equal $v(S).$ Since $y_p > x_p$ for all players $p$ in $S$, we must have $\sum_{k \in S} y_k > \sum_{k \in S} x_k.$ Thus, $v(S) > \sum_{k \in S} x_k.$ [*Case 1:*]{} If $s \in S$, since $v(S) = (p + f(S)) \cdot \delta$ and $\sum_{k \in S} x_k = (p+f(N))\delta - \sum_{r_i \in N \setminus S} x_{r_i}$, this means that: $$\begin{aligned}
& (p + f(S)) \cdot \delta &&> (p+f(N))\delta - \sum_{r_i \in N \setminus S} x_{r_i}\\
\Leftrightarrow \ & \sum_{r_i \in N \setminus S} x_{r_i} &&> (f(N) - f(S)) \cdot \delta.\end{aligned}$$ With $T = N \setminus S$ this gives a contradiction to the fact that for all $T \subseteq N$ s.t. $s \not\in T$: $\sum_{r_i \in T} x_{r_i} \leq (f(N) - f(N \setminus T)) \cdot \delta.$
[*Case 2:*]{} If $s \not\in S$, since $v(S) = 0$ and $\sum_{k \in S} x_k = \sum_{r_i \in S} x_{r_i}$, this means that: $$\begin{aligned}
v(S) = 0 &> \sum_{r_i \in S} x_{r_i}.\end{aligned}$$ With $T = S$ this gives a contradiction to the fact that for all $T \subseteq N$ s.t. $s \not\in T$: $\sum_{r_i \in T} x_{r_i} \ge 0.$
For the game $\langle N, v\rangle$ (Linear) this means that the payoff vector $(x_s = (p+\sum_i q_i) \cdot \delta - \sum_i x_{r_i}, x_{r_1}, \dots, x_{r_n})$ is in the Core if and only if for all $T \subseteq N$ s.t. $s \not\in T$: $$\begin{aligned}
&0 \le \sum_{r_i \in T} x_{r_i} \leq \left( \sum_{r_i \in T} q_i \right) \cdot \delta.\end{aligned}$$
For the game $\langle N, v\rangle$ (Threshold) this means that the payoff vector $(x_s = (p+q)\delta - \sum_i x_{r_i}, x_{r_1}, \dots, x_{r_n})$ is in the Core if and only if for all $T \subseteq N$ s.t. $s \not\in T$: $$\begin{aligned}
&0 \le \sum_{r_i \in T} x_i \leq \begin{cases} 0 & \text{, if} \ |T| \le n-k\\ q \cdot \delta& \text{, if} \ |T| > n-k \end{cases}.\end{aligned}$$ This means that for $\langle N, v\rangle$ (Linear) and $\langle N, v\rangle$ (Threshold) with $k = n$ a certain payoff vector is in the core precisely if no coalition of recommenders receives more than their joint contribution to the seller’s expected “extra profit”. For the game $\langle N, v \rangle$ (Threshold) with $k < n$ this means that $x_{r_i} = 0$ for all $i$ (with $T = \{ r_i\}$) and, thus, the only payoff vector in the Core is $((p+q) \delta, 0,
\dots, 0)$, i.e. the seller gets everything and the recommenders get nothing.
### Shapley Value {#sec:many-shapley}
Recall that the Shapley value assigns to each game a unique payoff vector that is “fair” as it satisifies the [*Symmetry*]{}, [*Dummy*]{}, and [*Additivity*]{} axioms. For a formal definition of the Shapley value (and related definitions) see Section \[sec:one-shapley\].
\[the:shapley-many\] Consider the game $\langle N, v \rangle$ (General). The Shapley value $\phi(v) = (\phi_s(v), \phi_{r_1}(v), \dots, \phi_{r_n}(v))$ is given by $$\begin{aligned}
&\phi_s(v) = p \delta + \sum_{S \subseteq N \setminus \{s\}} \frac{|S|!(|N|-1-|S|)!}{|N|!} f(S \cup \{s\}) \delta\\
&\phi_{r_i}(v) = \sum_{\substack{S \subseteq N \setminus \{r_i\}:\\s \in S}} \frac{|S|!(|N|-1-|S|)!}{|N|!} (f(S \cup \{r_i\}) - f(S))\delta.\end{aligned}$$
The claim follows from the definition of the game $\langle N, v \rangle$ (General) and Definition \[def:shapley\]. The worths $v(S)$ of all $S \subseteq N = \{s,r_1,r_2\}$ are given explicitly in Table \[tab:worths\]. For $|N| > 3$ the worths $v(S)$ are obtained similarly.
For the game $\langle N, v \rangle$ (Linear) this means that $$\begin{aligned}
&\phi_s(v) = p \delta + \frac{1}{2}\sum_{i} q_i \delta && \mbox{and}\\
&\phi_{r_i}(v) = \frac{1}{2} q_i \delta && \mbox{for all} \ i.\end{aligned}$$ For the game $\langle N, v \rangle$ (Threshold) this means that $$\begin{aligned}
&\phi_s (v) = p \delta + \left(1-n\frac{k!(n-k)!}{(n+1)!}\right) q \delta && \mbox{and}\\
&\phi_{r_i} (v) = \frac{k!(n-k)!}{(n+1)!} q \delta && \mbox{for all} \ i.\end{aligned}$$ This suggests that in the game $\langle N, v \rangle$ (Linear) each individual recommender should receive a share of exactly one half of her contribution to the expected “extra profit” of the recommender. For the game $\langle N, v \rangle$ (Threshold) the fraction $k!(n-k)!/(n+1)!$ is exactly the fraction of times where this recommender’s recommendation “makes a difference”. So all in all the Shapley value does not only give a unique payoff vector, but it also yields “fair” payoffs in the sense that the payoff to each recommender is proportional to the recommender’s contribution to the “extra profit” the seller can expect.
### Anonymity-Proof Shapley Value {#sec:many-proofshapley}
What would be a “fair” payoff vector if each recommendation was a [*collection of arguments*]{}? A straightforward approach would be to compute the Shapley value on the basis of arguments and to redeem recommender $r_i$ with $\sum_{a \in S_i} \phi_a(v)$, where $a$ is an argument from the set of arguments $A$ and $S_i$ is the set of arguments that recommender $r_i$ possesses; the sets $S_i$ being disjoint. The problem with this approach, however, is that it might be beneficial for a recommender to withhold some of her arguments: Let $A = \{a,b,c\}$, $v(\{a,b\}) = v(\{a,c\}) = v(\{a,b,c\}) = 1$, and $v(\{a\}) = v(\{b\}) = v(\{c\}) = v(\{b,c\}) = 0.$ Let $S_1 = \{a\}$ and $S_2 = \{b,c\}.$ Then $r_1$ gets $\phi_a(v) = \frac{1}{2}$ and $r_2$ gets $\phi_b(v) + \phi_c(v) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}.$
Let $A' = \{a,b\}$, $v(\{a,b\}) = 1$, and $v(\{a\}) = v(\{b\}) = 0.$ Let $S_1 = \{a\}$ and $S_2 = \{b\}.$ Then $r_1$ gets $\phi_a(v) = \frac{1}{2}$ and $r_2$ gets $\phi_b(v) = \frac{1}{2}.$ I.e. $r_2$ would be better off.
The [*anonymity-proof Shapley value*]{} [@OCS08] cannot be “tricked” in this way. It is defined as follows:
For any set $A' \subseteq A$ of declared arguments the [*anonymity-proof Shapley value*]{} $\psi_a(v)$ for $a \in A'$ is: $$\begin{aligned}
&\psi_a(v) = \frac{\phi_a(v)}{\sum_{a' \in A'} \phi_{a'}(v)} v(A').\end{aligned}$$
So a better way to redeem the recommenders would be to compute the anonymity-proof Shapley value $\psi_a(v)$ for each argument $a \in A'$ and to give each recommender $\sum_{a \in S_i} \psi_a(v).$ With this approach $r_1$ and $r_2$ would get $\psi_a = 3/5$ and $\psi_b(v) + \psi_c(v) = 2/5$ in Example 1 and $\psi_a = 3/4$ and $\psi_b(v) = 1/4 < 2/5$ in Example 2.
### The Nash Bargaining Solution {#sec:many-nash}
Recall that the Nash Bargaining Solution is defined as the unique bargaining solution that satisfies the axioms listed in Section \[sec:one-nash\].
For $\langle N, v \rangle$ (General) let $F = \{(f_s,f_{r_1},$ $\dots$, $f_{r_n})$ $| f_s \ge 0$, $f_{r_i} \ge 0 \ (\forall \ i) \ \text{and} \ f_s + \sum_{i} f_{r_i} = (p + f(N)) \cdot \delta\}$ and $d = (d_s, d_{r_1}, \dots, d_{r_n}) = (p \cdot \delta, 0, \dots, 0).$ Then, $$\begin{aligned}
\phi_s(F,d) & = ( p + \frac{1}{n+1} f(N) ) \cdot \delta &&\text{and}\\
\phi_{r_i}(F,d) &= \frac{1}{n+1} f(N) \cdot \delta. &&\text{for all} \ i.\end{aligned}$$
The claim follows from the fact that $\phi(F,d) \in \mbox{argmax}_{f \in F} \prod_i (f_i - d_i)$ [@M97].
One problem with the Nash Bargaining Solution is that it completely ignores the possibility of cooperation among subsets of players. To see that this may lead to “unfair” prices, consider the game $\langle N, v \rangle$ (Linear) with one seller $s$ and two recommenders $r_1$ and $r_2.$ Suppose that $q_1 = 1 - p - \epsilon$ and that $q_2 = \epsilon$ for some some small $\epsilon > 0$. It follows that $\phi_{r_1}(F,d) = \phi_{r_2}(F,d) = \frac{1-p}{2}$, i.e. the expected payoff to both recommenders is the same. But since $r_1$’s contribution to the expected worth of the grand coalition is significantly higher than that of $r_2$ (especially as $\epsilon \to 0$), this cannot be regarded as “fair”. We conclude that for $|N| > 2$ it is not advisable to use the Nash Bargaining Solution to guide the pricing of recommendations.
Recommending Strategically {#sec:recommending-strategically}
==========================
We study the following problem: There are $n$ products. For each product the recommender has two options: “recommend it” or “*not* recommend it”. A recommendation is [*successful*]{} if the buyer buys the product. For a successful recommendation the recommender gets a constant reward of $r$ and this reward is the same for all products. Initially, the probability $p$ of success is $p_0 < 1$. With each unsuccessful recommendation this probability drops from its current value to $p=l\cdot p$, where $l<1$ is the [*loss rate*]{}. The probability $p$ can be seen as an estimate of the recommendee’s *trust* in the recommender and a high value of $l$ corresponds to a slow loss in trust. This basic model is analyzed in Section \[sec:wo\_reset\_wo\_recovery\]. We also consider extensions of this model where trust (= $p$) can increase again in two ways. First, we assume that $p$ is reset to $p=p_0$ on each successful recommendation. This setting we refer to as “with reset” and it is analyzed in Section \[sec:with\_reset\_wo\_recovery\]. Second, we introduce a factor $g\ge 1$ and each time the recommender does not recommend anything trust is regained and $p$ is updated to $p=\min(g\cdot p,p_0)$. This setting we refer to as “with recovery” when $g>1$ and it is analyzed in Section \[sec:with\_reset\_with\_recovery\].
In all settings the recommender’s sole goal is to maximize the overall expected reward $M_n(p_0,l,g)$ for the given parameters $p_0$, $l$ and $g.$ We are interested in the asymptotic behavior of $M_n(p_0,l,g)$, i.e. in $R(p_0,l,g) = \lim_{n \to \infty} M_n(p_0,l,g).$ Before looking at the theoretical analysis, the following section experimentally demonstrates the different behavior of the optimal total expected reward in these settings.
Experimental Results {#sec:experiments}
--------------------
Figure \[fig:experiments\] gives experimental results for $n = 200$, $r = 1$, $p_0 = 0.5$, $l = 0.66$, $g = 1$ (in the setting “without recovery”) and $g=1.33$ (in the setting “with recovery”). It shows that the expected reward of the optimal strategy converges in the setting “without recovery” and diverges in the setting “with recovery”. In the setting “without recovery” the expected reward converges to $2.25$ if the probability of success is not reset and to $5$ if it is reset to $p_0$ on a single successful recommendation. The figure also shows that the expected reward of the heuristic “recommend product $1$, $k+1$, $2k+1$, etc.” converges for $k = 2$ where $l \cdot g^k < 1$ and diverges for $k = 3$ and $4$ where $l \cdot g^k > 1$. Finally, it shows that the expected reward grows faster for $k = 3$ than for $k = 4.$
Without Reset, without Recovery {#sec:wo_reset_wo_recovery}
-------------------------------
Here we consider the case where $g=1$ (= no recovery) and assume that the probability of success is [*not*]{} reset to $p_0$ on a successful recommendation. As the probability of success remains unchanged if no recommendation is given, the optimal strategy is to recommend [*all*]{} products. Therefore we can rewrite $R(p_0,l):=R(p_0,l,1)$ as follows: $R(p_0,l) = p_0 (r + R(p_0,l)) + (1-p_0) R(lp_0,l)$, which we can solve analytically. $$\begin{aligned}
R(p_0,l) &= \frac{p_0}{1-p_0} r + R(p_0 \cdot l,l)\\
&= \frac{p_0}{1-p_0} r + \frac{p_0 \cdot l}{1-p_0} r + R(p_0 \cdot l^2,l)\\
&= \frac{p_0}{1-p_0} \cdot \sum_{i=0}^{\infty} l^i \cdot r = \frac{p_0}{1-p_0} \cdot \frac{1}{1 - l} \cdot r < \infty.
%R(p_0,l) &= \frac{p_0}{1-p_0} r + R(p_0 \cdot l,l) \\
% &= \frac{p_0}{1-p_0} r + \frac{p_0 \cdot l}{1-p_0} r + R(p_0 \cdot l^2,l)\\
% &= \frac{p_0}{1-p_0} \cdot \sum_{i=0}^{\infty} l^i \cdot r \\
% &= \frac{p_0}{1-p_0} \cdot \frac{1}{1 - l} \cdot r < \infty.\end{aligned}$$
So, not surprisingly, if trust can only be lost and if both the initial trust $p_0$ and the loss rate $l$ are strictly smaller than $1$, then the total expected reward the recommender can achieve is finite, even when there is an infinite sequence of items to recommend.
With Reset, without Recovery {#sec:with_reset_wo_recovery}
----------------------------
Now let us analyze the case where still $g = 1$ (= no recovery) but each successful recommendation leads to reset of $p$ to $p_0$. Again, the optimal strategy is to recommend *all* products as there is no gain from not recommending. In this setting, we can rewrite $R(p_0,l)$ as follows: $R(p_0,l) = (1-q) \cdot (r + R(p_0,l))$, where $q$ denotes the probability that there will be not a single successful recommendation over the infinite sequence. This recurrence can be solved (i.e. $\lim_{n\rightarrow \infty} M_n(p_0,l)$ is finite) if and only if $q > 0.$
\[lem:worecov\] Let $\mbox{dilog(x)} = \int_{x}^{1} \frac{\ln(t)}{1-t} \, dt$ and $c = \max(p_0,l)$. Then, for all $1 > p_0 \ge 0$, $q \geq (1-c)\mbox{exp}\left(\frac{\mbox{dilog}(1-c)}{\ln(c)}\right) > 0.$
The probability that there will be not a single successful recommendation is: $$\begin{aligned}
q &= \prod_{k=0}^{\infty} (1-l^k \cdot p_0)\ge \prod_{k=0}^{\infty} (1-c^{k+1}).\end{aligned}$$ Hence it suffices to show that $\prod_{k=0}^{\infty} (1-c^{k+1}) > 0$. Taking the $\ln(\ )$ of both sides we get $$\begin{aligned}
\ln\left( \prod_{k=0}^{\infty} (1-c^{k+1}) \right) &= \sum_{k=0}^{\infty} \ln (1-c^{k+1}) > - \infty,\end{aligned}$$ where we need to prove this inequality. Note that the expression $\ln (1-c^{k+1})$ is strictly increasing in $k$ and hence $\ln (1-c^{k+1}) \ge \int_{k-1}^k \! \ln (1-c^{x+1}) \, dx$. This gives the bound $$\begin{aligned}
\sum_{k=0}^{\infty} \ln (1-c^{k+1})&= \ln (1-c) + \sum_{k=1}^{\infty} \ln (1-c^{k+1}) \\
&\ge \ln(1-c) + \sum_{k=1}^{\infty} \int_{x=k-1}^{k} \! \ln (1-c^{x+1}) \, dx \\
&= \ln(1-c) + \sum_{k=0}^{\infty} \int_{x=k}^{k+1} \! \ln (1-c^{x+1}) \, dx \\
&= \ln(1-c) + \int_{x=0}^{\infty} \! \ln (1-c^{x+1}) \, dx.\end{aligned}$$ Recall that $\mbox{dilog}(x) = \int_{x}^{1} \frac{\ln(t)}{1-t} \, dt.$ The indefinite integral of $\ln(1-x)$ is $-\mbox{dilog}(1-x)/\ln(x).$ We get $$\begin{aligned}
\int_{x=0}^{\infty} \! \ln (1-c^{k+1}) \, dx &= \frac{\mbox{dilog}(1 - c)}{\ln(c)} - \displaystyle{\lim_{x\to\infty}} \frac{\mbox{dilog}(1 - c^{x+1})}{\ln(c)}.\end{aligned}$$ Since $\mbox{dilog}(x)$ is continuous[^11] and $\mbox{dilog}(1) = 0$ (see Lemma \[lem:dilog\_bounds\]), we get $$\begin{aligned}
\int_{x=0}^{\infty} \! \ln (1-c^{k+1}) \, dx &= \frac{\mbox{dilog}(1 - c)}{\ln(c)} - \frac{\mbox{dilog}(1)}{\ln(c)}\\
&= \frac{\mbox{dilog}(1 - c)}{\ln(c)}.\end{aligned}$$ For $0 < x < 1$ we have $0 \le \mbox{dilog}(1-x) < 2e^{-1} + 1$ (see Lemma \[lem:dilog\_bounds\]). For $0 < x < 1$ we have $\ln(x) < 0.$ It follows that $\int_{x=0}^{\infty} \! \ln (1-c^{k+1}) \, dx > - \infty$.
\[lem:dilog\_bounds\] Let $\mbox{dilog}(x) = \int_{1}^{x} \frac{\ln(t)}{1-t} \, dt.$ Then $\mbox{dilog}(x)$ is monotonously decreasing and $0 \le \mbox{dilog}(x) < 2e^{-1} + 1.$
In fact, the tight upper bound of $\mbox{dilog}(x)\le \pi^2/6 < 2e^{-1} + 1$ is known [@AAR01], but we choose to give the following elementary proof of Lemma \[lem:dilog\_bounds\] to have a self-contained argument.
Let $f(t) = -\ln(t)/(1-t)$. Then $f'(t) = -(\frac{1}{t}(1-t)+\ln(t))/(1-t)^2 < 0$ for $0<t<1$. So $\int_{t=x}^1 f(t) dt < \int_{t=x}^{e^{-1}} f(t) dt + (1-e^{-1}) f(e^{-1})$. For $0<t\le e^{-1}$ we also have $f(t) \le -\ln(t)/(1-e^{-1})$. So, $\int_{t=x}^{e^{-1}} f(t) dt \le [t - t\cdot \ln(t)]^{e^{-1}}_{x}$. This is largest when $x\rightarrow 0$ where the whole expression becomes $2e^{-1}$ and so $\int_{t=x}^1 f(t) dt < 2e^{-1} + 1$ for $0\le x < 1$. Note that $f(t)$ is continuous at $t=1$ with $\lim_{x\rightarrow 1} f(t) = 1$ (using e.g. the l’Hopital Rule). So trivially $\mbox{dilog}(1) = 0$. As $f(t) > 0$ this gives the desired lower bound.
Using Lemma \[lem:worecov\] we can prove the following theorem.
\[the:wo-recov\] Let $\mbox{dilog(x)} = \int_{x}^{1} \frac{\ln(t)}{1-t} \, dt$, $c = \max(p_0,l)$, and $\delta(c) = (1-c) \mbox{exp}(\mbox{dilog}(1-c)/\ln(c))$. Then, for all $1 > p_0 \ge 0$, $$\begin{aligned}
R(p_0,l) & \le \frac{1-\delta(c)}{\delta(c)} \cdot r < \infty.\end{aligned}$$
This proves that even if the probability of success is reset to $p_0$ on a single successful recommendation, the total expected reward over an infinite period is bounded.
With Reset, with Recovery {#sec:with_reset_with_recovery}
-------------------------
Finally, we consider the setting where $g > 1$ (= with recovery). Here the probability of success is set to $\min(p_0,g \cdot p)$ if no recommendation was given. Hence it might be better not to recommend all products to avoid that that probability $p$ converges to zero. Let $M_n(p_0,l,g)$ denote the expected reward of the optimal strategy. To obtain bounds for $M_n(p_0,l,g)$, let us consider, as a heuristic, the algorithm $A^{(k)}$ that recommends product $1$, $k+1$, $2k+1$, etc. We write $A^{(k)}_n(p_0,l,g)$ to denote this algorithm’s expected profit.
\[the:w-recov-1\] Let $\psi$ be the smallest integer such that $l \cdot g^{\psi} \ge 1.$ If $k > \psi$, then, for all $1 > p_0, l > 0$ and $\infty > g \ge 1$, $$\begin{aligned}
A^{(k)}_n(p_0,l,g) &= \lfloor \frac{n}{k} \rfloor \cdot p_0 \cdot r.\end{aligned}$$
The expected reward for the first recommendation is $p_0 \cdot r.$ Since $k > \psi$, the expected reward for every other recommendation is also $\min(p_0, p_0 \cdot l \cdot g^{k-1}) = p_0 \cdot r.$ Since there are exactly $\lfloor n/k \rfloor$ recommendations, this shows that $A^{(k)}_n(p_0,l,g) = \lfloor \frac{n}{k} \rfloor \cdot p_0 \cdot r.$
This is instructive as it shows that (a) for $k > \psi$ the expected reward $A^{(k)}_n(p_0,l,g)$ of $A^{(k)}$ does [*not*]{} converge as $n$ tends to infinity and (b) for $k' > k > \psi$ the expected reward $A^{(k')}_n(p_0,l,g)$ of $A^{(k')}$ grows slower (and is ultimately lower) than the expected reward $A^{(k)}_n(p_0,l,g)$ of $A^{(k)}.$ Since the reward $M_n(p_0,l,g)$ of the optimal strategy is at least as high, this also shows non-convergence of $R(p_0,l,g) = \lim_{n \to \infty} M_n(p_0,l,g) = \infty.$
\[the:w-recov-2\] Let $\psi$ be the smallest integer such that $l \cdot g^\psi \ge 1.$ If $k \le \psi$, then, for all $1 > p_0, l > 0$ and $\infty > g \ge 1$, there exist $p'_0$ and $l'$ such that $$\begin{aligned}
\lim_{n \rightarrow \infty} A^{(k)}_n(p_0,l,g) &\le \lim_{n \rightarrow \infty} M_n(p'_0,l') < \infty.\end{aligned}$$
If $k < \psi$, then the profit maximization problem with parameters $p_0$, $l$, and $g$ on the products $1$, $2$, $3$, etc. is equivalent to the profit maximization problem with parameters $p'_0 = p_0$, $l' = l \cdot g^{k-1} < 1$, and $g' = 1$ on the products $1$, $k+1$, $2k+1$, etc. The claim follows from Theorem \[the:wo-recov\].
Whereas Theorem \[the:w-recov-1\] shows that recommending too seldomly is sub-optimal, Theorem \[the:w-recov-2\] shows that recommending too often is even worse.
Discussion and Future Work
==========================
Suppose you recommend a product to a friend and the seller of the product offers to pay you for your recommendation. What would be a “good” price? Our first finding was that the only “truthful” price would be zero. The problem with this, however, is that if you do not get paid, then you might as well decide [*not*]{} to recommend the product. And so the seller might be willing to pay you a “fair” price. We approached the problem of finding “fair” prices by studying solution concepts from coalitional game theory such as the Core, the Shapley value, and the Nash Bargaining Solution. Since each of these solution concepts formalizes some notion of “fairness”, these prices can be regarded as [*provably*]{} “fair”. We view such an “axiomatic” foundation of “fairness” to be the only viable basis for truely “fair” prices in practice.
Now suppose that you get paid for each succesful recommendation you make, and that you want to maximize the amount of money paid to you. At first sight, it might appear that the best strategy for you is to send out as many recommendations to as many friends as possible. But, then, just as you get “blind” when being shown too many ads, your friends will probably start to ignore your “recommendations”. We adressed this problem by modeling the loss in “trust” by a drop in “purchase probability” on each unsuccesful recommendation. Our main finding here was that, even if the “trust” in you is reset to the initial level on a single successful recommendation, the total expected profit you can make over an infinite period of time is bounded. This can only be overcome if the recomendee also incrementally regains “trust” over periods without any recommendation.
We believe that our work motivates a number of interesting research questions. E.g.: What are “good” pricing mechanisms in settings where the seller has objectives such as maintaining the buyer’s “trust”? How exactly do web users respond to being shown irrelevant advertisements? Is it possible to revive their interest in banner ads? What are “optimal” auction mechanisms for sponsored search when the click-through-rates are non-constant and decay with each irrelevant advertisement being shown?
[^1]: Email: [[[email protected]]([email protected])]{}
[^2]: Email: [[[email protected]]([email protected])]{}
[^3]: Email: [[[email protected]]([email protected])]{}
[^4]: This work was conducted as part of a EURYI scheme award. See [<http://www.esf.org/euryi/>]{} for details.
[^5]: <https://affiliate-program.amazon.com/>
[^6]: Somewhat related is the phenomenon of pay-what-you-like pricing where people act “irrationally” and choose to pay an adequate amount [@M08b; @FN09].
[^7]: <http://www.google.com/adsense/afs.pdf>
[^8]: <http://www.amazon.com>
[^9]: Note that this does [*not*]{} restrict the generality of our model. It rather says that each recommendation is for a distinct entity that we refer to as a product.
[^10]: The only connection between this solution concept and the concept of a [*Nash equilibrium*]{} [@M97; @OR94] is John F. Nash.
[^11]: It is even differentiable as it is defined as an indefinite integral.
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---
abstract: 'We describe early success in the evolution of binary black hole spacetimes with a numerical code based on a generalization of harmonic coordinates. Indications are that with sufficient resolution this scheme is capable of evolving binary systems for enough time to extract information about the orbit, merger and gravitational waves emitted during the event. As an example we show results from the evolution of a binary composed of two equal mass, non-spinning black holes, through a single plunge-orbit, merger and ring down. The resultant black hole is estimated to be a Kerr black hole with angular momentum parameter $a\approx 0.70$. At present, lack of resolution far from the binary prevents an accurate estimate of the energy emitted, though a rough calculation suggests on the order of $5\%$ of the initial rest mass of the system is radiated as gravitational waves during the final orbit and ringdown.'
author:
- Frans Pretorius
title: Evolution of Binary Black Hole Spacetimes
---
[**[*[I]{}. Introduction:*]{}**]{} One of the more pressing, unsolved problems in general relativity today is to understand the structure of spacetime describing the evolution and merger of binary black hole systems. Binary black holes are thought to exist in the universe, and the gravitational waves emitted during a merger event are expected to be one of the most promising sources for detection by gravitational wave observatories (LIGO, VIRGO, TAMA, GEO 600, etc.). Detection of such an event would be an unprecedented test of general relativity in the strong-field regime, and could shed light on many issues related to the formation and evolution of black holes and their environments within the universe. Given the design-goal sensitivities of current gravitational wave detectors, matched filtering may be essential to detect the waves from a merger, and extract information about the astrophysical source. During the early stages of a merger, and the later stages of the ringdown, perturbative analytic methods should give a good approximation to the waveform[@blanchet; @price_pullin]; however, during the last several orbits, plunge, and early stages of the ring down, it is thought a numerical solution of the full problem will be needed to provide an accurate waveform.
Smarr[@smarr] pioneered the numerical study of binary black hole spacetimes in the mid-70’s, where he considered the head-on collision process in axisymmetry. The full 3D problem has, for many reasons, proven to be a more challenging undertaking, and only recently has progress been made in the ability of numerical codes to evolve binary systems[@bruegmann; @brandt_et_al; @baker_et_al; @bruegmann_et_al; @alcubierre_et_al]. However, until now no code has been able to simulate a non-axisymmetric collision through coalescence and ringdown. The purpose of this letter is to report on a recently introduce numerical method based on generalized harmonic coordinates[@paper1] that [*can*]{} evolve a binary black hole during these crucial stages of a merger. At a given resolution the code will not run “forever”, though convergence tests suggest that with sufficient resolution the code can evolve the system for as long as needed to extract the desired physics from the problem. As an example we describe an evolution that completes approximately one orbit before coalescence, and runs for long enough afterwards to extract a waveform at large distances from the black hole.
The code has several features of note, some or all of which may be responsible for its stability properties: (1) a formulation of the field equations based on harmonic coordinates as first suggested in [@garfinkle], (2) a discretization scheme where the only evolved quantities are the covariant metric elements, harmonic source and matter functions, thus minimizing the number of constraint equations that need to be solved (which is similar to the discretization scheme used in [@szilagyi_winicour]), (3) the use of a compactified coordinate system where the outer boundaries of the grid are at spatial infinity, hence the physically correct boundary conditions can be placed there, (4) the use of adaptive mesh refinement to adequately resolve the relevant length scales in the problem, (5) dynamical excision that tracks the motion of the black holes through the grid, (6) addition of numerical dissipation to control high-frequency instabilities, (7) a time slicing that slows down the “collapse” of the lapse that would otherwise occur in pure harmonic time slicing, and (8) the addition of “constraint-damping” terms to the field equations[@brodbeck_et_al; @gundlach_et_al]. This final element was not present in the version of the code discussed in [@paper1], and though these terms seem to have little effect when black holes are not present in the numerical domain, they have a significant effect on how long a simulation with black holes can run with reasonable accuracy, at a given resolution.
An outline of the rest of the paper is as follows. In Sec. [II]{} we give a brief overview of the numerical method, focusing on details not present in [@paper1]. Sec. [III]{} gives results from the simulation of one such orbital configuration. We conclude in Sec. [V]{} with a summary of future work. More details, including convergence tests, the effect of constraint damping and a thorough description of the initial data calculation, will be presented elsewhere.
[**[*[II]{}. Overview of the numerical method:*]{}**]{} We briefly summarize the formulation of the field equations, gauge conditions and initial data used here, emphasizing details that are not contained in [@paper1]. We discretize the Einstein field equations expressed in the following form (using units where Newton’s constant and the speed of light are equal to 1): g\^g\_[,]{} + g\^\_[,]{} g\_[,]{} + g\^\_[,]{} g\_[,]{} + 2 H\_[(,)]{}\
- 2 H\_\^\_ +2 \^\_\^\_ = - 8(2 T\_-g\_ T)\
- (n\_C\_+ n\_C\_- g\_ n\^C\_).\[efe\_h\] $H_\mu$ are (arbitrary) source functions encoding the gauge-freedom of the solution, $\Gamma^\delta_{\alpha\beta}$ are the Christoffel symbols, $T_{\alpha\beta}$ is the matter stress tensor with trace $T$, $\kappa$ is a positive constant multiplying the new constraint damping terms following[@gundlach_et_al], $n^\mu = 1/\alpha (\partial/\partial t)^\mu - \beta^i/\alpha (\partial/\partial x^i)^\mu$ is the unit hypersurface normal vector with lapse function $\alpha$ and shift vector $\beta^i$ ($x^0\equiv t$, $x^i \equiv [x^1,x^2,x^3] \equiv [x,y,z]$), and $C_\mu$ are the constraints: C\_H\_- g\_ x\^.\[c\_def\] We use the following to evolve the source functions: H\_t = - \_1 + \_2 H\_[t,]{} n\^\[t\_gauge\], H\_i=0 where $\xi_1$ and $\eta$ are positive constants. Note that (\[t\_gauge\]) is [*not*]{} the usual definition of spatial harmonic gauge, which is defined in terms of contravarient components $H^\mu$.
We use scalar field gravitational collapse to prepare initial data that will evolve towards a binary black hole system. Specifically, at $t=0$ we have two Lorentz boosted scalar field profiles, and choose initial amplitude, separation and boost parameters to approximate the kind of orbit that the black holes (which form as the scalar field collapses) will have. The procedure used to calculate the initial geometry is based on standard techniques[@cook_review], and is a straight forward extension of the method described in[@paper1] to include non-time-symmetric initial data. The initial spatial metric and its first time derivative is conformally flat, and we specify a slice that is maximal and harmonic. The Hamiltonian constraint is used to solve for the conformal factor. The maximal conditions $K=0$ and $\partial_t K=0$ ($K$ is the trace of the extrinsic curvature) give the initial time derivative of the conformal factor and an elliptic equation for the lapse respectively. The momentum constraints are used to solve for the initial values of the shift vectors, and the harmonic conditions $H_\mu=0$ are used to specify the initial first time derivatives of the lapse and shift.
[**[*[III]{}. Results:*]{}**]{} In this section we describe results from the evolution of one example of a scalar field constructed binary system. The present code requires significant computational resources to evolve binary spacetimes[^1], and thus to study the orbital, plunge, and ringdown phases of a binary system in a reasonable amount of simulation time we chose initial data parameters such that the black holes would merge within roughly one orbit—see Fig. \[fig\_orbit\] and Table \[tab\_id\]. The following evolution parameters in (\[efe\_h\]) and (\[t\_gauge\]) were chosen: $\kappa\approx 1.25/M_0,\ \xi_1\approx 19/M_0,\ \xi_2\approx 2.5/M_0,\ \eta=5$ (these parameters did not need to be fine tuned), where $M_0$ is the mass of one black in the binary. This system was evolved using three different grid hierarchies, which we label as “low”, “medium” and “high” resolution. The low resolution simulation has a base grid of $32^3$, with up to 7 additional levels of $2:1$ refinement (giving a resolution in the vicinity of the black holes of $\approx M_0/10$). For computational efficiency we only allowed regridding of level $6$ and higher (at the expense of not being able to accurately track out-going waves). For the medium resolution simulation, we have one additional level of refinement during the inspiral and early phases of the merger, though have the same resolution over the coarser levels and at late times; thus we are able to resolve the initial orbital dynamics more accurately with the medium compared to low resolution run, though both have roughly the same accuracy in the wave zone. The high resolution simulation has up to 10 levels of refinement during the inspiral and early ringdown phase, 9 levels subsequently, and the grid structure of the lower levels is altered so that there is effectively twice the resolution in the wave zone. The reason for this set of hierarchies is again for computational efficiency: doubling (quadrupling) the resolution throughout the low resolution hierarchy would have required 16 (256) times the computer time, which in particular for the higher resolution simulation is impractical to do at this stage.
![A depiction of the orbit for the simulation described in the text (see also Table \[tab\_id\]). The figure shows the coordinate position of the center of one apparent horizon relative to the other, in the orbital plane $z=0$. The units have been scaled to the mass $M_0$ of a single black hole, and curves are shown from simulations with three different resolutions. Overlaid on this figure are reference ellipses of eccentricity $0$, $0.1$ and $0.2$, suggesting that if one were to attribute an initial eccentricity to the orbit it could be in the range $0-0.2$. []{data-label="fig_orbit"}](d2.eps){width="6.5cm"}
low res. med. res. high res.
----------------- ------------------------------- ------------ ------------ ------------
ADM Mass $2.36 M_0$ $2.39 M_0$ $2.39 M_0$
[**initial**]{} BH masses $0.97 M_0$ $0.99 M_0$ $M_0$
orbital eccentricity $0 - 0.2$ $0-0.2$ $0-0.2$
proper separation $16.5 M_0$ $16.6 M_0$ $16.6M_0$
angular velocity $\times M_0$ $0.023$ $0.023$ $0.023$
[**final**]{} BH mass $1.77 M_0$ $1.85 M_0$ $1.90M_0$
BH spin parameter $0.74$ $0.73$ $0.70$
: Some properties of the simulated equal mass binary system described in the text. Where relevant the units have been scaled to the mass $M_0$ of one of the initial black holes, measured from the higher resolution simulation at a time after the majority of scalar field accretion has occurred. The final black hole mass and spin where estimated from data as shown in Fig. \[fig\_am\], a little while after the black hole formed, though not so long after as to be affected by the “drift” from numerical error. The initial proper separation was measured at $t=10 M_0$, and is the proper length of the piece of a coordinate line outside the apparent horizons that connects their coordinate centers. The black holes initially have zero spin. []{data-label="tab_id"}
Fig. \[fig\_am\] shows the horizon masses and final horizon angular momentum as a function of time. The ADM mass of the space time suggests that approximately $15\%$ of the total scalar field energy does not collapse into black holes. The remnant scalar field leaves the vicinity of the orbit quite rapidly (in $t\approx 30 M_0$, which is on the order of the light crossing time of the orbit). Black hole masses are estimated from the horizon area $A$ and angular momentum $J$, and applying the Smarr formula: $$\label{smarr}
M=\sqrt{M_{ir}^2 + J^2/(4 M_{ir}^2)}, \ \ \ M_{ir}\equiv \sqrt{A/16\pi}.$$
![The plot to the left shows the net black hole mass of the spacetime in units of the mass $M_0$ of a single initial black hole, calculated from apparent horizon (AH) properties (using (\[smarr\]) with the dynamical horizon estimate for $J$), and from simulations with three different resolutions. The initial sharp increase in mass is due to scalar field accretion, the small “wiggle” at around $20 M_0$ appears to be a gauge effect, and the “jaggedness” around the time of the merger is due to robustness problems in the AH finder that manifest when the AH shapes are highly distorted. To the right the Kerr parameter $a$ of the final black hole is shown (for clarity we only plot the results from a single simulation), calculated using the ratio $C_r$ of polar to equatorial proper circumference of the AH and applying (\[a\_cr\]), and using the dynamical horizon framework (curve labeled DH). The loss of mass (and similarly increase in $a$) with time after the merger is due to accumulating numerical error. []{data-label="fig_am"}](M.eps "fig:"){width="4.1cm"} ![The plot to the left shows the net black hole mass of the spacetime in units of the mass $M_0$ of a single initial black hole, calculated from apparent horizon (AH) properties (using (\[smarr\]) with the dynamical horizon estimate for $J$), and from simulations with three different resolutions. The initial sharp increase in mass is due to scalar field accretion, the small “wiggle” at around $20 M_0$ appears to be a gauge effect, and the “jaggedness” around the time of the merger is due to robustness problems in the AH finder that manifest when the AH shapes are highly distorted. To the right the Kerr parameter $a$ of the final black hole is shown (for clarity we only plot the results from a single simulation), calculated using the ratio $C_r$ of polar to equatorial proper circumference of the AH and applying (\[a\_cr\]), and using the dynamical horizon framework (curve labeled DH). The loss of mass (and similarly increase in $a$) with time after the merger is due to accumulating numerical error. []{data-label="fig_am"}](a.eps "fig:"){width="4.4cm"}
The horizon angular momentum of the final black hole is calculated using two methods (which [*do*]{} give zero angular momentum when applied to the initial black holes, as expected). First, by using the dynamical horizon framework[@ashtekar_krishnan], though [*assuming*]{} that the rotation axis of the black hole is orthogonal to the $z=0$ orbital plane, and that each closed orbit of the azimuthal vector field (which at late times should become a Killing vector) lies in a $z={\rm constant}$ surface of the simulation. Due to the symmetry of the initial data, these assumptions are probably valid, though this will eventually need to be confirmed. The second method, following [@brandt_seidel], is to measure the ratio $C_r$ of the polar to equatorial proper radius of the horizon, and use the formula that closely approximates the function that is valid for Kerr black holes: $$\label{a_cr}
a \approx \sqrt{1-\left(2.55 C_r -1.55\right)^2}$$ As seen in Fig. \[fig\_am\], the initial ringing of the black hole is quite apparent in the estimate using $C_r$. Remarkably, the dynamical horizon estimate for $a$ and average value obtained using $C_r$ agree quite closely, even shortly after the merger when one might have expected the black hole to still be too far from its stationary state to have either method be applicable.
To estimate the gravitational waves emitted by the binary we use the Newman-Penrose scalar $\Psi_4$, with the null tetrad constructed from the unit timelike normal $n^\mu$, a radial unit spacelike vector normal to $r={\rm constant}$ coordinate spheres, and two additional unit spacelike vectors orthogonal to the radial vector[^2]. Far from the source, the real and imaginary components of $\Psi_4$ are proportional to the second time derivatives of the two polarizations of the emitted gravitational wave. Fig. \[fig\_psi4\] shows an example of the real part of $\Psi_4$. Most of the early, short wavelength burst of waves can be correlated with the passage of the remnant scalar field that did not fall into the black holes (the “noisy” nature of this piece of the waveform is in part due to numerical error). This unwanted radiations leaves the domain quite early on, and so does not significantly affect the subsequent merger waves. Roughly the first long wavelength oscillation in the plot can be associated with orbital motion, and subsequent waves with the ringdown of the final black hole.
![ A sample of the gravitational waves emitted during the merger, as estimated by the Newman-Penrose scalar $\Psi_4$ (from the medium resolution simulation). Here, the real component of $\Psi_4$ multiplied by the coordinate distance $r$ from the center of the grid is shown at a fixed angular location, though several distances $r$. The waveform has also been shifted in time by amounts shown in the plot, so that the oscillations overlap. If the waves are measured far enough from the central black hole then the amplitudes should match, and they should be shifted by the light travel time between the locations (i.e. by $25 M_0$ in this example). That we need to shift the waveforms by more than this suggests the extraction points are still too close to the black hole; the decrease in amplitude is primarily due to numerical error as the wave moves into regions of the grid with relatively low resolution.[]{data-label="fig_psi4"}](psi4.eps){width="8.2cm"}
To estimate the total energy $E$ emitted in gravitational waves, we use the following formula [@smarr_rad] $$\frac{dE}{dt}=\frac{R^2}{4\pi} \int p d\Omega, \ \ \
p=\int_0^t \Psi_4 dt \cdot \int_0^t \bar{\Psi}_4 dt \label{Ep},$$ where $\bar{\Psi}_4$ is the complex conjugate of $\Psi_4$, and the surface integrated over in (\[Ep\]) is a sphere of constant coordinate radius $R$ (in uncompactified coordinates). This method of calculating the energy is quite susceptible to numerical error, as we are summing a positive definite quantity over all time to give a [*change*]{} of energy with respect to time; thus numerical error in $\Psi_4$ will tend to inflate the answer. To reduce some of this error, we filter out the high spherical harmonic components ($\ge\ell=|m|=6$) of $\Psi_4$ before applying (\[Ep\]). Note that the smaller integration radii (as shown in Fig. \[fig\_psi4\]) are not very far from the binary system, and so possibly in a region where (\[Ep\]) is not strictly valid. However, the larger integration radii are in regions of the grid that do not have very good resolution (due both to the mesh refinement structure and the spatially compactified coordinate domain), and so numerical error (mostly dissipation) tends to reduce the amplitude of the waves with distance from the source. With all these caveats in mind, the numbers we obtain from (\[Ep\]) are $4.7\%, 3.2\%, 2.7\%, 2.3\%$ at integration radii of $25M_0, 50M_0, 75M_0$ and $100M_0$ respectively (from the high resolution simulation[^3]), and where the percentage is relative to $2 M_0$. Another estimate of the radiated energy can be obtained by taking the difference between the final and initial horizon masses (Table \[tab\_id\])—this suggests around $5\%$ (high resolution case).
[**[*[V]{}. Conclusion:*]{}**]{} In this letter we have described a numerical method based on generalized harmonic coordinates that can stably evolve (at least a class of) binary black hole spacetimes. As an example, we presented an evolution of a binary system composed of non-spinning black holes of equal mass $M_0$, with an initial proper separation and orbital angular velocity of approximately $16.6M_0$ and $0.023/M_0$ respectively. The binary merged within approximately 1 orbit, leaving behind a blackhole of mass $M_f \approx 1.9 M_0$ and angular momentum $J \approx 0.70 M_f^2$. A calculation of the energy emitted in gravitational waves indicates that roughly $5\%$ of the initial mass (defined as $2 M_0$) is radiated . Future work includes improving the accuracy of simulation (in particular the gravitational waves), exploring a larger class of initial conditions (binaries that are further separated, have different initial masses, non-zero spins, etc.), and attempting to extract more geometric information about the nature of the merger event from the simulations.
[**[*Acknowledgments:*]{}**]{} I would like to thank Carsten Gundlach for describing their constraint damping method for the Z4 system[@gundlach_et_al], and suggesting that it can be applied in a similar fashion with the harmonic scheme. I would also like to thank Matthew Choptuik, Luis Lehner and Lee Lindblom for stimulating discussions related to this work. I gratefully acknowledge research support from NSF PHY-0099568, NSF PHY-0244906 and Caltech’s Richard Chase Tolman Fund. Simulations were performed on UBC’s [**vn**]{} cluster, (supported by CFI and BCKDF), and the [**Westgrid**]{} cluster (supported by CFI, ASRI and BCKDF).
[1]{} L. Blanchet, [*Living Rev.Rel.*]{} [**5**]{}, 3 (2002) R. Price and J. Pullin, [*Phys. Rev. Lett.*]{} [**72**]{}, 3297 (1994) L. Smarr, Ph.D. thesis, University of Texas, Austin, 1975. B. Bruegmann, [*Int.J.Mod.Phys.*]{} [**D8**]{} 85, (1999) S. Brandt et al., [*Phys.Rev.Lett.*]{} [**85**]{} 5496, (2000) J. Baker, B. Bruegmann, M. Campanelli, C. O. Lousto and R. Takahashi [*Phys.Rev.Lett.*]{} [**87**]{} 121103, (2001) B. Bruegmann, W. Tichy and N. Jansen [*Phys.Rev.Lett.*]{} [**92**]{} 211101, (2004) M. Alcubierre et al., gr-qc/0411149 (2004) F. Pretorius, [*Class. Quant. Grav.*]{} [**22**]{} 425, (2005) D. Garfinkle, [*Phys.Rev.*]{} [**D65**]{}, 044029 (2002) B. Szilagyi and J. Winicour, [*Phys.Rev.*]{} [**D68**]{}, 041501 (2003) O. Brodbeck, S. Frittelli, P. Huebner and O. A. Reula, [*J.Math.Phys.*]{} [**40**]{}, 909 (1999) C. Gundlach, J.M. Martin-Garcia, G. Calabrese and I. Hinder, gr-qc/0504114 (2005) G. B. Cook, [*Living Rev.Rel.*]{} [**3**]{}, 5 (2000) A. Ashtekar and B. Krishnan [*Living Rev.Rel.*]{} [**7**]{}, 10 (2004) S.R. Brandt and E. Seidel, [*Phys.Rev.*]{} [**D52**]{}, 870 (1995) L. Smarr, in [*Sources of Gravitational Radiation*]{}, ed. L. Smarr, Seattle, Cambridge University Press (1978)
[^1]: Typical runtimes on 128 nodes of a Xeon Linux cluster are on the order of a few days for the lowest resolutions attempted, to several months at the higher resolutions
[^2]: At this stage we are ignoring all the subtleties in choosing an “appropriate” tetrad
[^3]: the corresponding numbers from the medium(low) res. runs are $5.1(7.1)\%$, $3.5(4.6)\%$, $2.5(3.2)\%$, $1.7(2.1)\%$
|
---
abstract: 'The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from (finite-dimensional) linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems. This approach allows a user to work with functions instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods.'
author:
- 'A. Alqahtani[^1]'
- 'T. Mach[^2]'
- 'L. Reichel[^3]'
bibliography:
- 'longstrings.bib'
- 'bib.bib'
title: 'Continuous Adaptive Cross Approximation for Ill-posed Problems with Chebfun '
---
[ill-posed problem, inverse problem, Chebfun, truncated SVE, Tikhonov regularization]{}
Introduction {#sec:intro}
============
We are interested in the solution of Fredholm integral equations of the first kind, $$\label{inteq} \int_{\Omega_{1}} \kappa(s,t) x(t)\,dt = g(s),\qquad s\in\Omega_{2},$$ with a square integrable kernel $\kappa$. The $\Omega_{i}$ are subsets of ${{\mathbb R}}^{d_{i}}$ for $i=1,2$. Such integral equations are common in numerous applications including remote sensing, computerized tomography, and image restoration. Two major problems arise when solving . The first problem is that the space of functions is of infinite dimensionality. The second problem is that small changes in $g$ may correspond to large changes in $x$ as exemplified by $$\begin{split}
\max_{s\in \Omega_2} {\mbox{$\vert\int_{\Omega_1} \kappa(s,t)\cos(2\pi m t)\,dt\vert$}}, \qquad \Omega_1=\Omega_2=[0,1],
\end{split}$$ where the maximum can be made tiny by choosing $m$ large, despite the maximum of ${\mbox{$\vert\cos(2\pi m t)\vert$}}$ being $1$. This is a consequence of the Riemann–Lebesgue theorem; see, e.g., [@Eng2000; @Han1998] or below for discussions of this result. The second problem is particularly relevant when the right-hand side $g$ is a measured quantity subject to observational errors, as is the case in many applications.
Usually one deals with the first problem by first discretizing the functions $x(t)$ and $g(s)$ in using $n$ piecewise constant, linear, or polynomial basis functions; see e.g., [@Hac95] or [@PCH4]. The kernel $\kappa(s,t)$ is discretized analogously. This transforms the problem into a system of linear equations. The second problem causes the coefficient matrix of said system to be ill-conditioned for sufficiently large $n$. Straightforward solution of these linear systems of equations generally is not meaningful because of severe error propagation. Therefore, this linear system has to be regularized. This can, for instance, be achieved by Tikhonov regularization or truncated singular value decomposition (TSVD). While the first dampens the influence of the small singular values, the latter outright ignores them. One is then often faced with a trade-off between a small discretization error and a small error caused by the regularization; see, e.g., Natterer [@Na77]. In fact, often the more basis functions are used for the discretization, the more ill-conditioned the resulting coefficient matrix becomes, and the larger the need of regularization.
In this paper we will first regularize the problem and then discretize the problem. Regularization will be achieved through a singular value expansion of the kernel. At the same time the singular value expansion provides us with an excellent basis for discretizing the problem. The discretized problem is then a diagonal linear system of equations, which can be solved trivially. Thus, dealing with the second problem first simplifies the other problem.
We will compute the singular value expansion of the kernel using Chebfun [@chebfun]. Hence, our discretization basis will consist of piecewise Chebyshev polynomials. The computed solution is a Chebfun approximation to the function $x(t)$. The advantage of Chebfun is that the solution will feel and behave like a function. Therefore, our approach is arguably closer to directly solving instead of a discretized version.
This paper is organized as follows. In the second section, we will provide basic definitions, introduce our notation, and briefly discuss Chebfun and singular value expansions. Section \[sec:tsve\] discusses the truncated singular value expansion method (TSVE). The Tikhonov regularization method is described in Section \[sec:tik\]. Numerical results that illustrates the performances of the methods of Sections \[sec:tsve\] and \[sec:tik\] are reported in Section \[sec:numerical\_experiments\]. Concluding remarks can be found in Section \[sec:concl\].
Basics {#sec:basics}
======
Let $L^{2}(\Omega_{i})$ for $i=1,2$ be spaces of Lebesgue measurable square integrable functions with inner products $$\label{inner} {\mbox{$\left\langle a(t),b(t) \right\rangle$}}_{\Omega_{i}} = \int_{\Omega_{i}} {{\mbox{$\overline{a(t)}$}}}\, b(t)\,dt, \quad \text{
for } i=1,2,$$ where ${\mbox{$\overline{a}$}}$ represents the complex conjugate of $a\in{{\mathbb C}}$. Based on these inner products, we can define $L^{2}$-norms by $$ {\mbox{$\left\Vertf(t)\right\Vert$}}_{\Omega_{i}}^{2}=\int_{\Omega_{i}} {\mbox{$\vertf(t)\vert$}}^{2}\,dt, \quad \text{
for } i=1,2.$$ Throughout this paper ${\mbox{$\left\Vert\cdot\right\Vert$}}$ stands for an $L^{2}$-norm. We will omit the subscript if the domain is clear from the context. Since the spaces $H_{i}:=L^{2}(\Omega_{i})$ for $i\in\{1,2\}$, with the inner products and norms defined above, are complete vector spaces, they are *Hilbert space*; see, e.g., [@Hac95].
A given kernel $\kappa(\cdot,\cdot) \in L^{2}(\Omega_{1} \times \Omega_{2} )$ induces the bounded linear operator [@Hac95 Thm. 3.2.7] $A : L^{2}(\Omega_{1})\rightarrow L^{2}(\Omega_{2})$ or $H_{1} \rightarrow H_{2}$ defined by $$\label{definition_A} (Ax)(s)=\int_{\Omega_{1}} \kappa(s,t)x(t)\,dt.$$ The operator is sometimes called a *Hilbert-Schmidt integral operator* and the kernel $\kappa$ a *Hilbert-Schmidt kernel*. This allows us to write as $$\label{Axg} Ax = g.$$ In particular, we assume that $g$ is in the range of $A$. Generally, we are interested in the solution of of minimal norm. We refer to this solution as $x_{\text{exact}}$.
In practice, the right-hand side $g$ of is often a measured quantity and therefore is subject to observational errors. Thus, we assume that the error-free function $g$ is not available–only an error contaminated approximation $g^{\delta}\in H_{2}$ of $g$ is known. We assume that $g^{\delta}$ satisfies $$ {\mbox{$\left\Vertg-g^{\delta}\right\Vert$}} \leq \delta,$$ with a *known* bound $\delta>0$. The solution of the equation $$\label{Axgd} Ax=g^{\delta}, \quad \text{ with } x \in H_{1} \text{ and } g^{\delta} \in
H_{2},$$ is generally not a meaningful approximation of the desired solution $x_{\text{exact}}$ of , since $A$ is not continuously invertible. In fact, the equation might not have a solution. The operator $A$ depends on the kernel $\kappa$. We will now have a closer look at known theory about the kernel function $\kappa$. For any square integrable kernel $\kappa$, we define the singular value expansion (SVE) [@Schmidt1989 §4] as $$\label{sve} \kappa(s,t)=\sum_{i} \sigma_{i} \phi_{i}(s)\psi_{i}(t).$$ The functions $\phi_{i}(s)$ and $\psi_{i}(t)$ are referred to as the singular functions. These functions are orthonormal with respect to the usual inner product [@Schmidt1989 §5], i.e., $$\begin{aligned}
{\mbox{$\left\langle \psi_{i},\psi_{j} \right\rangle$}}_{\Omega_{1}} =
{\mbox{$\left\langle \phi_{i},\phi_{j} \right\rangle$}}_{\Omega_{2}}
= \delta_{ij}, \quad \text{ with }i,j=1,2,\dotsc~.\end{aligned}$$ The quantities $\sigma_{i}$ are known as singular values. It can be shown that the only limit point of the singular values for square integrable kernels is zero [@Schmidt1989 §5].[^4] The singular values form a non-increasing sequence: $$\begin{aligned}
\sigma_{1} \geq \sigma_{2} \geq \sigma_{3} \geq \dotsb \geq 0. $$ Let $\sum_{i=1}^{\infty}\sigma_{i}\phi_{i}(s)\psi(t)$ be a uniformly convergent series. Then $$\label{eq:SVE}
\kappa(s,t)=\sum_{i=1}^{\infty}\sigma_{i}\phi_{i}(s)\psi(t),$$ as shown in [@Schmidt1989 §8]. When the summation is finite, then the kernel $\kappa(s,t)$ is said to be *separable* (or *degenerate*). Most applications do not have a separable kernel. However, if the kernel is square integrable, then it can be approximated well by a separable kernel with a suitable number of terms, $\ell$, in . Let $$\label{Aell} \kappa_{\ell}=\sum_{i=1}^{\ell}\sigma_{i} \phi_{i}(s)\psi_{i}(t),$$ with the same ordering of the singular values. Then this is the closest kernel of rank at most $\ell$ to $\kappa$ in the $L^{2}$-norm [@Schmidt1989 §18 Approximation Theorem]. We will use this result to justify the application of the truncated singular value expansion method (TSVE), which will be discussed in Section \[sec:tsve\]. We will also be using this Approximation Theorem to generally restrict our expansion to singular values greater than $\varepsilon$, where $\varepsilon$ is a small enough cut-off–say $10^{-8}$ or $10^{-16}$. Here, there is a trade-off between computing time and approximation accuracy. We try to choose $\varepsilon$ far below the regularization error so that it does not have a significant effect on the accuracy. At the same time, a small $\varepsilon$ means higher cost for computing the singular value expansion and forming the computed approximate solution.
In this paper we will use two regularization methods, TSVE and Tikhonov regularization. The TSVE method is based on the Approximation Theorem mentioned above. We approximate the kernel $\kappa$ by $\kappa_{\ell}$ for some suitable $\ell\geq 0$. This results in an approximation $A_{\ell}$ to $A$ and a solution, denoted by $x_{\ell}$, of the problem $$\label{tsvd} (A_{\ell}x)(s)=\int_{\Omega_{1}} \kappa_{\ell}(s,t) x(t)\,dt = g^{\delta}(s),
\quad s\in\Omega_2.$$ The parameter $\ell$ is a regularization parameter that determines how many singular values and basis functions of $\kappa$ are used to compute the approximate solution $x_{\ell}$ of . The remaining singular values, which are smaller than or equal to $\sigma_{\ell}$, are ignored. The singular value $\sigma_{\ell}$ provides information on the approximation error.
Tikhonov regularization replaces the system by the penalized least-squares problem $$\label{tikhonov}
\min_{x\in H_{1}}\{{\mbox{$\left\VertAx-g^{\delta}\right\Vert$}}^{2}+\lambda^{2}{\mbox{$\left\Vertx\right\Vert$}}^{2}\},$$ which has a unique solution $x_{\lambda}$ for any positive value of the regularization parameter $\lambda$. Substituting the SVE into shows that Tikhonov regularization dampens the contributions to $x_{\lambda}$ of singular values and functions with large index $k$ the most; increasing $\lambda>0$ results in more damping. Since we cannot deal with an infinite series expansion, we will, in practice, first cut-off all singular values that are less than $\varepsilon$ as explained above, and then apply Tikhonov regularization.
The determination of suitable values of the regularization parameters, $\ell$ in and $\lambda$ in , is important for the quality of the computed approximate solution. Several methods have been described in the literature including the discrepancy principle, the L-curve criterion, and generalized cross validation; see [@BRS; @Ki; @KR; @RR] for recent discussions of their properties and illustrations of their performance. Regularization methods typically require that regularized solutions for several parameter values be computed and compared in order to determine a suitable value.
Chebfun {#subsec:chebfun}
-------
We solve by first regularizing followed by discretization. However, we still want to compute the solution numerically. Thus, we need a numerical library that can handle functions in an efficient way. Since a function is representing uncountable many pairs of $x$ and $f(x)$ with $x\mapsto f(x)$, a computer can only handle approximations to functions numerically.[^5]
We chose the Matlab package Chebfun [@chebfun] for this purpose. Chebfun uses piecewise Chebyshev polynomials, so called chebfuns, to approximate functions. All computations within Chebfun’s framework are done with these approximations to the actual function. This in turn means that we project the functions $g\in L^{2}(\Omega_{2})$ onto a space of piecewise Chebyshev polynomials over $\Omega_{2}$. One can argue that this is a discretization. However, Chebfun’s framework is significantly different from other discretizations in the sense that it gives the user the feeling of computing with functions.
Chebfun’s functionality includes the computation of sums and products of functions and derivatives, inner products, norms, and integrals. Chebfun2/3, Chebfun’s extension to functions of two and three variables, can also compute outer products and, most importantly for us here, the singular value expansion [@TowTre13]. The algorithm behind the singular value expansion uses a continuous analogue of adaptive cross approximation. This is where some of the motivation for this work originates, since we recently analyzed the application of adaptive cross approximation to the solution of ill-posed problems [@MaRevBVa15].
The approximation of $\kappa(s,t)$ is computed by an iterative process. First, an approximation of the maximum point $(x,y)$ of $\kappa(s,t)$ is determined. The computation of the exact maximum point is not important. The function is then approximated by $$\begin{aligned}
\kappa_{1}(s,t) = \frac{\kappa(s,y) \kappa(x,t)}{\kappa(s,t)},\end{aligned}$$ where $\kappa(s,y)$ and $\kappa(x,t)$ are one-dimensional chebfuns in $s$ and $t$, respectively.
This process is then repeated for $\kappa(s,t) - \kappa_{1}(s,t)$ to find a rank-1 approximation of the remainder. By recursion one obtains after $k$ steps a rank-$k$ approximation to the original kernel. As soon as the remainder is sufficiently small, the computed rank-$k$ approximation is the sought approximation to $\kappa(s,t)$. At the end we have $\kappa(s,t)\approx C(s)D R(t)^{T}$, with $C(s)$ and $R(t)$ vectors of functions, and $D$ a dense matrix of size $k\times k$.
Based on this approximation it is easy to compute the singular value expansion. Chebfuns continous analogue of the QR factorization can be used to find orthogonal bases for $C(s)$ and $R(t)$. The upper triangular matrices are multiplied by $D$ to form a new matrix $\tilde{D}$. Then a singular value decomposition of $D=U\Sigma V^{T}$ is computed. Finally, the small orthogonal matrices $U$ and $V^{T}$ are combined with $C(s)$ and $R(t)$, respectively; see [@TowTre13]. A very similar process, called adaptive cross approximation [@q467; @q699], was used in [@MaRevBVa15] for the discrete case of matrices and vectors.
Chebfun has some limitations. Currently only functions of at most three variables can be approximated by Chebfun. Hence, we are limited to ill-posed problems in one space-dimension, and to problems in two space-dimensions for which the kernel is separable and also given in a separable representation. This is the case for the kernel that models Gaussian blur in two space-dimensions, making Gaussian blur our only example in two space-dimensions in this paper.
Chebfun2 and Chebfun3 are further limited to domains that are tensor products of intervals. Thus, in this paper all domains are rectangles or rectangular boxes. Chebfun also needs multivariate functions to be of low rank for an efficient approximation, that is there has to exist a sufficiently accurate separable approximation. This is for instance not the case for the kernel $\kappa(s,t) = st - \min(s,t)$ from the deriv2 example of the Regularization Tools package [@PCH4]. This limits the application of the methods described in this paper. However, the Chebfun package is still under development and some of the limitations mentioned might not apply to future releases.
The TSVE method {#sec:tsve}
===============
Assume that the kernel is non-separable and can be expressed as $$\label{kernel}
\kappa(s,t)=\sum_{i=1}^{\infty} \sigma_{i} \phi_{i}(s)\psi_{i}(t),$$ and that the solution can be written as $$\label{solexpress}
x(t)=\sum_{j=1}^{\infty} \beta_{j} \psi_{j}(t).$$ The fact that $\kappa$ is non-separable implies that all $\sigma_i$ are positive, and the assumption that the solution is of the form essentially states that the solution has no component in the null space of $A$. This assumption is justified since the null space of $A$ is orthogonal to all the $\psi_{j}$ and, thus, a component in the direction of the null space would increase the norm of the solution, but not help with the approximation of . Substituting and into , and using the orthonormality of the basis functions yields $$\begin{aligned}
\sum_{i=1}^{\infty} \sigma_{i}\beta_{i}\phi_{i}(s)=g(s).\end{aligned}$$ We further probe the equation with $\phi_{k}(s)$ for all $k$ and use the orthonormality of the basis functions to obtain $$\begin{aligned}
\sigma_{k}\beta_{k}=\int_{\Omega_{2}} \phi_{k}(s) g(s)\,ds,\qquad\forall\, k. \end{aligned}$$ Thus, the exact solution to is given by $$\label{truesol1D}
x(t)=\sum_{j=1}^{\infty} \beta_{j} \psi_{j}(t),\quad \text{ with } \beta_{j}
= \frac{\int_{\Omega_{2}} \phi_{j}(s) g(s)\,ds}{\sigma_{j}}.$$ If we truncate this series after $\ell$ terms and use the noisy right hand side $g^{\delta}$ instead of $g$, then we obtain the TSVE solution to defined by $$\label{TSVDsol1D} x_{\ell}(t)=\sum_{j=1}^{\ell}
\beta^{\delta}_{j} \psi_{j}(t),\quad \text{ with } \beta^{\delta}_{j}=
\frac{\int_{\Omega_{2}} \phi_{j}(s) g^{\delta}(s)\,ds}{\sigma_{j}}.$$ The truncation parameter $\ell$ can be chosen as needed.
In the following lemma, we link the projection of the error onto the space spanned by the $\phi_{i}(s)$ to the norm of the error.
Let $n(s)=g(s)-g^{\delta}(s)$ with $ {\mbox{$\left\Vertn(s)\right\Vert$}} \leq \delta$. Then, $$\label{lemma3.1}
\sum_{i=1}^{\infty}\left(\int_{\Omega_{2}}
\phi_{i}(s) n(s)\,ds\right)^{2}\leq \delta^{2},$$ where $\phi_i(s)$ are orthonormal basis functions.
Using the basis functions $\phi_i(s)$, $n(s)$ can be represented as $$\begin{aligned}
n(s)=\sum\limits_{j=1}^{\infty} \gamma_{j}\phi_{j}(s)+\phi^{\perp}(s),
\end{aligned}$$ for certain coefficients $\gamma_j$, and where $\phi^{\perp}(s)$ is orthogonal to all functions $\phi_j(s)$. Then, $$\begin{aligned}
\int_{\Omega_{2}} \phi_{i}(s) n(s)\,ds&=\int_{\Omega_{2}}
\phi_{i}(s) \left(\sum\limits_{j=1}^{\infty} \gamma_{j}\phi_{j}(s)+\phi^{\perp}(s) \right)\,ds.\\
\intertext{The orthogonality of the basis functions $\phi_{j}$ allows us to
simplify the above expression to}
\int_{\Omega_{2}}
\phi_{i}(s) n(s)\,ds&=\gamma_{i}.
\end{aligned}$$ The same argument can be used to show that $$\sum\limits_{j=1}^{\infty} \gamma_{j}^{2} \leq {\mbox{$\left\Vertn(s)\right\Vert$}}^{2} \leq
\delta^{2}.$$ Combining these results shows . $\square$
We will now use the previous lemma to given an upper bound for the error of the solution obtained with the TSVE regularization method.
\[lemma:tsvd\_error\] Let $x(t)$ and $x_{\ell}(t)$ be the exact solution and the TSVE regularized solutions given by and , respectively. Assume the kernel $\kappa(s,t)$ has finite rank $r$. Then, $$\label{lemmatsve}
{\mbox{$\left\Vertx(t)-x_{\ell}(t)\right\Vert$}} \leq \left(\frac{\delta^{2}}{\sigma_{\ell}^{2}} +\sum\limits_{i=\ell+1}^{r} \beta_{i}^{2}\right)^{1/2}.$$
We will rely on the expansion of the solution in the space spanned by the functions $\psi_{i}(t)$. We have $$\begin{aligned}
{\mbox{$\left\Vertx(t)-x_{\ell}(t)\right\Vert$}}^{2}&={\mbox{$\left\Vert\sum\limits_{i=1}^{r} \beta_{i}\psi_{i}(t)
-\sum\limits_{i=1}^{\ell} \beta^{\delta}_{i} \psi_{i}(t)\right\Vert$}}^{2}.\\
\intertext{Using \eqref{truesol1D} and \eqref{TSVDsol1D} this simplifies to}
{\mbox{$\left\Vertx(t)-x_{\ell}(t)\right\Vert$}}^{2}&={\mbox{$\left\Vert\sum\limits_{i=1}^{\ell} \frac{\int_{\Omega_{2}}
\phi_{i}(s) \left(g(s)-g^{\delta}(s)\right)\,ds}{\sigma_{i}}
\psi_{i}(t)+\sum\limits_{i=\ell+1}^{r} \beta_{i}
\psi_{i}(t)\right\Vert$}}^{2}\\
&\leq {\mbox{$\left\Vert\sum\limits_{i=1}^{\ell} \frac{\int_{\Omega_{2}} \phi_{i}(s)
\left(g(s)-g^{\delta}(s)\right)\,ds}{\sigma_{i}} \psi_{i}(t)\right\Vert$}}^{2}
+ {\mbox{$\left\Vert\sum\limits_{i=\ell+1}^{r} \beta_{i} \psi_{i}(t)\right\Vert$}}^{2}.\\
\intertext{The orthonormality of the basis functions $\psi_{i}$ allows us to simplify the above inequality to}
{\mbox{$\left\Vertx(t)-x_{\ell}(t)\right\Vert$}}^{2}&\leq \sum\limits_{i=1}^{\ell} \left(\frac{\int_{\Omega_{2}} \phi_{i}(s)
\left(g(s)-g^{\delta}(s)\right)\,ds}{\sigma_{i}}\right)^{2}+\sum\limits_{i=\ell+1}^{r} \beta_{i}^{2}.\\
\intertext{Using Lemma \ref{lemma3.1} and the fact the singular values are in non-increasing order gives }
{\mbox{$\left\Vertx(t)-x_{\ell}(t)\right\Vert$}}^{2} &\leq \frac{\delta^{2}}{\sigma_{\ell}^{2}} +\sum\limits_{i=\ell+1}^{r} \beta_{i}^{2}.
\end{aligned}$$ $\square$
coordinates [ (Baart,0.209830752166255) (Foxgood,0.017964227827709) (Gravity,0.036927582968480) (Shaw, 0.086674859374278) (Wing, 0.348049438529407) ]{}; coordinates [ (Baart,0.250507045532956) (Foxgood,1.901536166479170) (Gravity,1.874650936624351) (Shaw, 1.518125686327245) (Wing, 0.423628546907915) ]{};
Lemma \[lemma:tsvd\_error\] provides a justification for chosing $\ell$ such that $$\begin{aligned}
\sigma_{\ell}\leq \eta \delta, \end{aligned}$$ with $\eta$ being a small constant greater than $1$. If $\sigma_{\ell}=\delta$, then the bound from Lemma \[lemma:tsvd\_error\] is at least $1$. Choosing $\eta$ larger means that $\frac{\delta^{2}}{\sigma_{\ell}}$ will be smaller. However, there is a trade-off, since additional $\beta_{i}$ have to be included in the bound. Generally, choosing $\eta$ between to $2$ and $5$ is reasonable. Figure \[fig:1\] illustrates the behavior of the bound for some numerical examples.
Ill-posed problems based on one-dimensional integral equations are arguably less challenging than 2D-problems. Thus, consider the two-dimensional Fredholm integral equations of the first kind, $$\label{integral2d}
\int_{\Omega_{1}} \kappa(s_{1},s_{2},t_{1},t_{2}) x(t_{1},t_{2})\,dt_{1}dt_{2}
=
g^{\delta}(s_{1},s_{2}),\qquad (s_{1},s_{2})\in\Omega_{2}.$$ We employ Chebfun for the numerical solution of the ill-posed problems. Hence, we are limited by Chebfun’s capabilities to deal with higher-dimensional functions. A kernel that can be separated into a product of two functions, i.e., $\kappa(s_{1},s_{2},t_{1},t_{2})=\kappa_{1}(s_{1},t_{1})\times\kappa_{2}(s_{2},t_{2}),$ can be handled by Chebfun. The kernel that models Gaussian blur provides an example and will be used in a numerical illustration. Let the kernel be given by $$\label{kernel2d}
\kappa(s_{1},s_{2},t_{1},t_{2})=\sum\limits_{i=1}^{r_{1}} \sigma_{i}
\phi_{i}^{(1)}(s_{1})\psi_{i}^{(1)}(t_{1})\
\sum\limits_{j=1}^{r_{2}} \mu_{j} \phi_{j}^{(2)}(s_{2})\psi_{j}^{(2)}(t_{2}),$$ where both the $\sigma_i$ and $\mu_j$ denote singular values, and let the solution be of the form $$\label{solexp2d}
x(t_{1},t_{2})= \sum\limits_{k=1}^{r_{1}} \sum\limits_{\ell=1}^{r_{2}} \beta_{k \ell} \psi_{k}^{(1)}(t_{1})\psi_{\ell}^{(2)}(t_{2}).$$ By substituting and into , and using the orthonormality of the basis functions, we get $$\begin{aligned}
\sigma_{i} \mu_{j} \beta_{ij} \phi_{i}^{(1)}(s_{1}) \phi_{j}^{(2)}(s_{2}) =g^{\delta}(s_{1},s_{2}). \end{aligned}$$ We further probe the equation with $\phi_{k}^{(1)}(s_{1}) \phi_{\ell}^{(2)}(s_{2})$ for all $k$ and $\ell$ and use the orthonormality of the basis functions to obtain $$\label{coeff2d}
\beta_{ij} =\frac{\int_{\Omega_{2}} g^{\delta}(s_{1},s_{2})
\phi_{i}^{(1)}(s_{1})\phi_{j}^{(2)}(s_{2})\,ds_{1} ds_{2}}{\sigma_{i} \mu_{j}}.$$ This allows us to implement the solution algorithm using at most functions of three variables and, thus, not exceeding Chebfun3’s capabilities [@HasTre17].
In order to solve problems in two space-dimensions with a non-separable kernel, we would need Chebfun4, which currently is not available.
Tikhonov regularization {#sec:tik}
=======================
For Tikhonov regularization, instead of solving exactly, we solve the functional minimization problem $$\label{tikmineq}
\min_{x \in {H_{1}}}\left\{ {\mbox{$\left\VertAx-g^{\delta}\right\Vert$}}^{2}+\lambda^{2}{\mbox{$\left\Vertx\right\Vert$}}^{2}\right\},$$ where $\lambda$ is a fixed positive number. Using the definition of $L^{2}$-norm, equation can be written as $$\label{3.2}
\underset{x\in {H_{1}}}{\text{min}}\left\{\int_{\Omega_{2}}
{\mbox{$\left\vertAx-g^{\delta}\right\vert$}}^{2}\,ds
+ \lambda^{2} \int_{\Omega_{1}} {\mbox{$\vertx\vert$}}^{2}\,dt\right\}.$$ By substituting and into , and by using the orthonormality of the basis functions, we obtain $$\begin{aligned}
\min_{x(t)\in {H_{1}}} \sum\limits_{i=1}^{\infty} \left(\sigma^{2}_{i}
\beta^{2}_{i} -2\sigma_{i} \beta_{i} \int_{\Omega_{2}} {\phi_{i}(s)}
g^{\delta}(s)\,ds
+ \lambda^{2}\beta^{2}_{i}\right) +\int_{\Omega_{2}}{\mbox{$\vertg^{\delta}(s)\vert$}}^{2}\,ds. \end{aligned}$$ Thus, we can compute the solution as $$\label{solTik1D}
x_{\lambda}(t)=\sum\limits_{j=1}^{\infty} \beta^{(\lambda)}_{j} \psi_{j}(t),\quad
\text{ with } \beta^{(\lambda)}_{j}= \frac{\sigma_{i}
\int_{\Omega_{2}} {\phi_{i}(s)} g^{\delta}(s)\,ds}{(\sigma^{2}_{i} +\lambda^{2})}.$$
For the two-dimensional case, instead of solving exactly, we solve $$\label{Tikeq2d}
\begin{split}
\min_{x(t_1,t_2)\in {H_1}}\bigg\{{\mbox{$\left\Vert\int_{\Omega_1}
\kappa(s_1,s_2,t_1,t_2)x(t_1,t_2) \,dt_1\,dt_2 - g^{\delta}(s_1,s_2) \right\Vert$}}^{2} +\\
\lambda^{2} {\mbox{$\left\Vert x(t_1,t_2)\right\Vert$}}^{2}\bigg\}.
\end{split}$$ By substituting and into , and using the orthonormality of the basis functions, we get $$\begin{split}
\min_{x(t_{1},t_{2})\in {H_{1}}}\sum_{i=1}^{r_{1}} \sum_{j=1}^{r_{2}} \bigg(
\beta^{2}_{ij} \sigma^{2}_{i} \mu^{2}_{j} -2 \beta_{ij}\sigma_{i} \mu_{j}
\int_{\Omega_{2}}\phi_{i}^{(1)}(s_{1})\phi_{j}^{(2)}(s_{2})\
g^{\delta}(s_{1},s_{2})\,ds_{1}\,ds_{2} +\\ \lambda^{2} \beta^{2}_{ij} \bigg)
+\int_{\Omega_{2}} {\mbox{$\vert g^{\delta}(s_{1},s_{2})\vert$}}^{2} \,ds_{1}\,ds_{2},
\end{split}$$ and we can compute the solution by $$\label{solTik2D}
\begin{split}
x_{\lambda}(t_{1},t_{2})&= \sum_{k=1}^{r_{1}} \sum_{\ell=1}^{r_{2}} \beta_{k
\ell} \psi_{k}^{(1)}(t_{1})\psi_{\ell}^{(2)}(t_{2}),\\ \text{ with }
\beta_{k\ell}&= \frac{\sigma_{k}
\mu_{\ell}\int_{\Omega_{2}}\phi_{k}^{(1)}(s_{1})\phi_{\ell}^{(2)}(s_{2})\
g^{\delta}(s_{1},s_{2})\,ds_{1}\,ds_{2}}{\sigma^{2}_{k} \mu^{2}_{\ell} +
\lambda^{2}} .
\end{split}$$
Numerical experiments {#sec:numerical_experiments}
=====================
Conclusion {#sec:concl}
==========
The computed results illustrate the feasibility of using Chebfun to solve linear discrete ill-posed problems and in this way carry out computations in a fashion that is closer to the spirit of the analysis of ill-posed problems found, e.g., in [@Eng2000]. The accuracy and timings of the implementations with Chebfun are competitive.
In the future further extensions to Chebfun including the treatment of functions of four or six variables will allow the application of the Chebfun-based approach discussed in this paper to the solution of linear ill-posed problems in two and three space-dimensions. It would be interesting to see if the observations made here carry over to these classes of problems.
Acknowledgments {#sec:ack .unnumbered}
===============
The authors are grateful for enlightening discussions with Behnam Hashemi (Shiraz University of Technology) about Chebfun and Chebfun3 in particular. We hope that this paper can serve as a motivation for the extension of Chebfun to four and higher dimensional functions.
We also would like to thank Richard Mikaël Slevinsky (University of Manitoba) for first pointing out to us the link between adaptive cross approximation and singular value expansions used in Chebfun2/3.
[^1]: Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia and Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. E-mail: `[email protected]`.
[^2]: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. E-mail: `[email protected]`.
[^3]: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. E-mail: `[email protected]`.
[^4]: Schmidt calls the singular values eigenvalues, since he is mainly concerned with symmetric kernels and the concept of singular values was not developed when he published his paper. We follow modern notation here.
[^5]: There are some notable exceptions like $\sin(x)$ or $x^{2}$. However, we cannot assume that the solution of will fall into this very small set of functions.
|
---
abstract: 'A Szegö-type theorem for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. We give a local version of this result in the setting of positive line bundles on compact symplectic manifolds.'
author:
- 'Roberto Paoletti[^1]'
title: |
A local Szegö-type theorem\
in Toeplitz quantization
---
Introduction
============
The goal of this paper is to establish a local version of a Szegö-type theorem for Toeplitz operators proved by Boutet de Monvel and Guillemin in [@bg] (Theorem 6 in §1, see also §13), which is the analogue of results of Weinstein [@w] and Widom [@wi] for pseudodifferential operators. While [@bg] deals with general Toeplitz structures, here we shall focus on the special case of positive line bundles, which is natural in algebraic geometry and geometric quantization.
We shall adopt the point of view of [@p-weyl], [@p-equiv-asymp], [@p-local-trace], where certain spectral asymptotics for Toeplitz operators are given a local interpretation building on the approach to generalized Szegö kernels in [@z], [@bsz] and [@sz], based on the microlocal theory of [@bs]. Thus the natural context for the present discussion is the category of quantized compact symplectic manifolds and the spaces of almost holomorphic sections described in [@bg] and [@sz]; for ease of exposition, we shall confine ourselves to the complex projective case.
Our setting and normalization conventions are as follows. Let $M$ be connected complex d-dimensional projective manifold, and $A$ an ample line bundle on it. Let $h$ be an Hermitian metric on $A$ such that the unique compatible connection has curvature $\Theta=-2i\omega$, where $\omega$ is a Kähler form on $M$. Let $A^*$ be the dual line bundle and $X\subseteq A^*$ the unit circle bundle, with projection $\pi:X\rightarrow M$. Then the connection 1-form $\alpha$ on $X$ is a contact form, and we shall adopt $d\mu_M=:(1/\mathrm{d}!)\,\omega^{\wedge \mathrm{d}}$ and $d\mu_X=:(1/2\pi)\alpha\wedge \pi^*(d\mu_M)$ as volume forms on $M$ and $X$, respectively. Given these choices, we shall identify (generalized) functions, densities and half-densities on $X$.
Let $H(X)\subseteq L^2(X)$ be the Hardy space, and let $H(X)=\bigoplus _kH_k(X)$ be the decomposition of $H(X)$ into $S^1$-equivariant summands; thus $H_k(X)$ is naturally unitarily isomorphic to the space $H^0\left(M,A^{\otimes k}\right)$ of global holomorphic sections of $A^{\otimes k}$. The orthogonal projector $\Pi:L^2(X)\rightarrow H(X)$ extends to a continuous operator $\Pi:\mathcal{D}'(X)\rightarrow \mathcal{D}'(X)$, and the distributional kernel of the latter is the Szegö kernel studied in [@bs]. Thus, $\Pi=\bigoplus _k\Pi_k$, where $\Pi_k:L^2(X)\rightarrow H_k(X)$ is the level-$k$ Szegö kernel.
A Toeplitz operator on $X$ of order $\nu\in \mathbb{Z}$ is a composition $T=:\Pi\circ Q\circ \Pi$, where $Q$ is a pseudodifferential operator of classical type and order $\nu$ on $X$ [@bg]; we can naturally view $T$ as linear operator on $H(X)$. If in particular $T$ is $S^1$-invariant, then it leaves every summand $H_k(X)$ invariant, inducing equivariantendomorphisms $T_k:H_k(X)\rightarrow H_k(X)$. Our focus will be on $S^1$-invariant zeroth order self-adjoint Toeplitz operators and the (local) spectral asymptotics as $k\rightarrow +\infty$ of their equivariant pieces. A notable example is $T_f=\Pi\circ M_f\circ \Pi$, where $M_f$ is multiplication by a real valued $f\in \mathcal{C}^\infty(M)$; under the unitary isomorphism $H_k(X)\cong H^0\left(M,A^{\otimes k}\right)$, $T_k$ corresponds to the Berezin-Toeplitz quantization of the classical observable $f$.
Thus, let $T$ be an $S^1$-invariant zeroth order self-adjoint Toeplitz operator on $X$, and for $k=1,2,\ldots$ let $T_k:H_k(X)\rightarrow H_k(X)$ be the self-adjoint endomorphism induced by restriction. Let $\lambda_{kj}$, $j=1,\ldots,N_k=\dim\big(H_k(X)\big)$ be the eigenvalues of $T_k$ repeated according to multiplicity, and suppose that $e_{kj}$ is an eigenfunction of $T_k$ for the eigenvalue $\lambda_{kj}$, so chosen that $(e_{kj})_j$ is an orthonormal basis of $H_k(X)$.
We shall study the asymptotics of the following measures on $\mathbb{R}$: $$\mathcal{T}_{x,k}=:\sum_{j=1}^{N_k}\delta_{\lambda_{kj}}\,\big|e_{kj}(x)\big|^2\,\,\,\,\,\,\,\,(x\in X,\,\,k=1,2,\ldots);$$ $\mathcal{T}_{x,k}$ does not depend on the choice of the $e_{kj}$’s. Furthermore, $\big|e_{kj}(x)\big|^2$ only depends on $m=\pi(x)$, hence we may write $\mathcal{T}_{m,k}$ for $\mathcal{T}_{x,k}$.
Before stating our main result, let us recall (after [@bg]) that the connection 1-form generates a closed symplectic cone $$\Sigma=:\big\{(x,r\alpha_x):\,x\in X,r>0\big\}\subseteq T^*X\setminus\{0\},$$ and that the (principal) *symbol* $\sigma_T:\Sigma\rightarrow \mathbb{C}$ of the Toeplitz operator $T=\Pi\circ Q\circ \Pi$ is simply the restriction to $\Sigma$ of the principal symbol $\sigma_Q$ of the pseudodifferential operator $Q$; if $T$ is self-adjoint, then $\sigma_T$ is real valued.
The *reduced symbol* of $T$ is then the $\mathcal{C}^\infty$ function $\varsigma_T(x)=:\sigma_T(x,\alpha_x)$ on $X$; if $T$ is $S^1$-invariant then so is $\varsigma_T$, which may thus be viewed as a $\mathcal{C}^\infty$ function on $M$.
\[thm:maim\] Let $T$ be an $S^1$-invariant zeroth order self-adjoint Toeplitz operator on $X$ and let $\chi\in \mathcal{S}(\mathbb{R})$ be function of rapid decrease. Then uniformly in $m\in M$ the following asymptotic expansion holds as $k\rightarrow +\infty$: $$\begin{aligned}
\langle\mathcal{T}_{m,k},\chi\rangle&\sim& \left(\frac k\pi\right)^\mathrm{d}\,
\left(\chi\big(\varsigma_T(m)\big)\,+\sum_{j\ge 1}k^{-j}P_j(\chi)\big(\varsigma_T(m)\big)\right),\end{aligned}$$ where $P_j$ is a differential operator of degree $2j$ depending on $m$.
We obtain a global asymptotic expansion by integrating over $M$. More precisely, for $k=1,2,\ldots$, define measures on $\mathbb{R}$ given by $\mathcal{T}_{k}=:\sum_{j=1}^{N_k}\delta_{\lambda_{kj}}$.
For any $\chi\in \mathcal{S}(\mathbb{R})$, the following asymptotic expansion holds for $k\rightarrow +\infty$: $$\begin{aligned}
\langle\mathcal{T}_{k},\chi\rangle&\sim& \left(\frac k\pi\right)^\mathrm{d}\,\
\left(\int_M \chi\big(\varsigma_T(m)\big)d\mu_M(m)+\sum_{j\ge 1}k^{-j}C_j(\chi)\right).\end{aligned}$$
In the present setting of positive line bundles, the Szegö-type theorem of [@bg] is the following:
For any $\chi\in \mathcal{S}(\mathbb{R})$, $$\begin{aligned}
\lim _{k\rightarrow +\infty}\left(\frac \pi k\right)^{\mathrm{d}}
\langle\mathcal{T}_{k},\chi\rangle=
\int_M \chi\big(\varsigma_T(m)\big)d\mu_M(m).\end{aligned}$$
Proof of Theorem \[thm:maim\]
=============================
To begin with, we may reduce the proof to the case where $T$ is elliptic with everywhere positive reduced symbol; recall that a Toeplitz operator is called *elliptic* if its symbol is nowhere vanishing, hence if $T$ is self-adjoint then it is elliptic if and only if $\varsigma_T$ has constant sign.
In fact, choose $C>0$ such that $\varsigma_T+C>0$, and define $\widetilde{T}=:T+c\Pi$; then $\widetilde{T}$ is an elliptic zeroth order $S^1$-invariant self-adjoint Toeplitz operator, with reduced symbol $\varsigma_{\widetilde{T}}=\varsigma_T+c>0$ and eigenvalues $\widetilde{\lambda}_{kj}=\lambda_{kj}+c$ with the same eigenfunctions $e_{kj}$. Now suppose that the statement has been proved for $\widetilde{T}$, and define $\chi_c(\tau)=:\chi(\tau-c)$. Then if $\widetilde{\mathcal{T}}_{m,k}$ is the analogue of $\mathcal{T}_{m,k}$ for $\widetilde{T}$ we have $$\begin{aligned}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle =\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2\langle\delta_{\lambda_{kj}},\chi\rangle
=\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2\langle\delta_{\lambda_{kj}+c},\chi_c\rangle }\\
&=&
\left<\widetilde{\mathcal{T}}_{m,k},\chi_c\right>\sim
\left(\frac k\pi\right)^\mathrm{d}\,\left(\chi_c\big(\widetilde{\varsigma}_T(m)\big)+\sum_{j\ge 1}k^{-j}C_j\right)\\
&=&\left(\frac k\pi\right)^\mathrm{d}\,\left(\chi\big(\varsigma_T(m)\big)+\sum_{j\ge 1}k^{-j}C_j\right).\end{aligned}$$
Let us then assume in the following that $\varsigma_T>0$.
We shall denote by either $\mathcal{F}(\gamma)$ or $\widehat{\gamma}$ the Fourier transform of $\gamma\in \mathcal{S}'(\mathbb{R})$ (thus $\widehat{f}(\xi)=:(2\pi)^{-1/2}\int_{-\infty}^{+\infty} f(x)\,e^{-i\xi x}dx$ if $f\in L^1$); for any $\lambda\in \mathbb{R}$, we have $\delta_\lambda=\left(1/\sqrt{2\pi}\right)\,\mathcal{F}\left(e^{i\lambda (\cdot)}\right)$. Define $\chi_k(\tau)=:\chi(\tau/k)$, so that $\langle\delta_\lambda,\chi\rangle=:\langle \delta_{k\lambda},\chi_k\rangle$, $\forall\,\lambda\in \mathbb{R}$. We have if $m=\pi(x)$: $$\begin{aligned}
\label{eqn:fourier-transform}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle =\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2\langle\delta_{\lambda_{kj}},\chi\rangle
=\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2\langle\delta_{k\lambda_{kj}},\chi_k\rangle}\nonumber\\
&=&\frac{1}{\sqrt{2\pi}}\,\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2
\left<\mathcal{F}\left(e^{ik\lambda_{kj} (\cdot)}\right),\chi_k\right>=\frac{1}{\sqrt{2\pi}}\cdot \sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2\left<e^{ik\lambda_{kj} (\cdot)},\widehat{\chi}_k\right>\nonumber\\
&=&\frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^\infty\left(\sum_{j=1}^{N_k}\big|e_{kj}(x)\big|^2
e^{ik\lambda_{kj} \tau}\right)\,\widehat{\chi}_k(\tau)\,d\tau.\end{aligned}$$
To interpret the expression within brackets, let us introduce the first order Toeplitz operator $T'=:D\circ T$, where $D=:-i\,\partial/\partial \theta$ and $\partial/\partial \theta$ is the generator of the $S^1$-action. Then $T'$ is $S^1$-invariant and self-adjoint, and has eigenvalues $\lambda'_{kj}=:k\,\lambda_{kj}$, with eigensections $e_{kj}$. It is furthermore elliptic, with everywhere positive symbol $\sigma_{T'}(x,r\alpha_x)=r\,\varsigma_T(x)$.
By the theory in §2 and §12 of [@bg], there exists an elliptic, self-adjoint $S^1$-invariant operator $Q$ on $X$ satisfying the following conditions: $[\Pi,Q]=0$; $T'=\Pi\circ Q\circ \Pi$; the principal symbol $\sigma_Q$ of $Q$ is everywhere positive on $T^*X\setminus \{0\}$. In particular, $Q$ leaves $H(X)$ invariant, and $T'$ is the restriction of $Q$ to $H(X)$. If $U(\tau)=:e^{i\tau Q}$ ($\tau\in \mathbb{R}$), then $U(\tau)\big(e_{kj}\big)=e^{ik\lambda_{kj}\tau}\cdot e_{kj}$. The distributional kernel of $U_k(\tau)=:U(\tau)\circ \Pi_k$ is then $$U_k(\tau)(x,y)=:\sum_{j=1}^{N_k}e^{ik\lambda_{kj}\tau}\,
e_{kj}(x)\cdot \overline{e_{kj}(y)}\,\,\,\,\,\,\,(x,y\in X);$$ therefore, (\[eqn:fourier-transform\]) may be rewritten $$\begin{aligned}
\label{eqn:fourier-transform-1}
\langle \mathcal{T}_{m,k},\chi\rangle& =&
\frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^\infty\widehat{\chi}_k(\tau)\,U_k(\tau)(x,x)\,d\tau\nonumber\\
&=&\frac{1}{2\pi}\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\chi(\lambda/ k)e^{-i\tau\lambda}
\,U_k(\tau)(x,x)\,d\lambda\,d\tau\nonumber\\
&=&\frac{k}{2\pi}\cdot \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\chi(\lambda)e^{-ik\tau\lambda}
\,U_k(\tau)(x,x)\,d\lambda\,d\tau,\end{aligned}$$ where in the last equality the change of variable $\lambda\rightarrow k\lambda$ has been performed.
Next we choose a sufficiently small $\epsilon>0$, and $\beta\in \mathcal{C}^\infty\big((-\epsilon,\epsilon)\big)$ such that $\beta\ge 0$ and $\beta(\tau)=1$ for $-1/2<\tau<1/2$. Inserting the identity $1=\beta(\tau)+\big(1-\beta(\tau)\big)$ in (\[eqn:fourier-transform-1\]), we obtain $$\langle \mathcal{T}_{m,k},\chi\rangle=\langle \mathcal{T}_{m,k},\chi\rangle '+\langle \mathcal{T}_{m,k},\chi\rangle'',$$ where in the former summand the integrand has been multiplied by $\beta$, and in the latter by $1-\beta$.
As $k\rightarrow +\infty$, we have $\langle \mathcal{T}_{m,k},\chi\rangle''=O\left(k^{-\infty}\right)$.
*Proof.* In view of (\[eqn:fourier-transform-1\]), we have $$\begin{aligned}
\label{eqn:fourier-transform-2}
\langle \mathcal{T}_{m,k},\chi\rangle''=\frac{k}{\sqrt{2\pi}}\,
\int_{-\infty}^{+\infty}\widehat{\chi}(k\tau)\,\big(1-\beta(\tau)\big)\,
\,U_k(\tau)(x,x)\,d\tau.
\end{aligned}$$ Now since $\widehat{\chi}$ is of rapid decrease and $1-\beta$ is bounded and supported where $|\tau|\ge \epsilon/2$, for every $N>0$ we have $$\left|\widehat{\chi}(k\tau)\,\big(1-\beta(\tau)\big)\right|\le \frac{C_N}{k^{2N}\left(\tau^2+\epsilon^2\right)^N}.$$ On the other hand, by the Tian-Zelditch asymptotic expansion one has $\big|U_k(\tau)(x,x)\big|\le C\,k^\mathrm{d}$ uniformly in $\tau\in \mathbb{R}$ and $x\in X$.
Q.E.D.
We thus need only consider the asymptotics of $\langle \mathcal{T}_{m,k},\chi\rangle'$. To proceed further, let us rewrite (\[eqn:fourier-transform-1\]) as $$\begin{aligned}
\label{eqn:fourier-transform-comp}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle}\\
& \sim&
\frac{k}{2\pi}\cdot \int_X\int_{-\infty}^{+\infty}\int_{-\epsilon}^{+\epsilon}\chi(\lambda)\,\beta(\tau)\,e^{-ik\tau\lambda}
\,U(\tau)(x,y)\Pi_k(y,x)\,d\mu_X(y)\,d\lambda\,d\tau,\nonumber\end{aligned}$$ where $\sim$ stands for equal asymptotics for $k\rightarrow +\infty$. Remark that $$\begin{aligned}
\label{eqn:fourier-of-k}
\Pi_k(y,x)&=&\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ik\vartheta}\,\Pi\big(r_\vartheta(y),x\big)\,d\vartheta\nonumber\\
&=&\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ik\vartheta}\,\Pi\big(y,r_{-\vartheta}(x)\big)\,d\vartheta,\end{aligned}$$ where $r_\vartheta:X\rightarrow X$ is the action of $e^{i\vartheta}\in S^1$.
For any $s>0$, let $M_s(m)=:\left\{m'\in M:\mathrm{dist}_M\left(m',m\right)<s\right\}$, where $\mathrm{dist}_M$ is the Riemannian distance of $M$. Choose a sufficiently small $\delta>0$, and let $M_1=:M_{2\delta}(m)$, $M_2=:\overline{M_{\delta}(m)}^c$. Let furthermore $\{\varrho_1,\varrho_2\}$ be a partition of unity on $M$, subordinate to the open cover $\{M_1,M_2\}$, and write $\varrho_j$ for $\varrho_j\circ \pi$. Inserting the identity $\varrho_1(y)+\varrho_2(y)=1$ in (\[eqn:fourier-transform-comp\]), we get $\langle \mathcal{T}_{m,k},\chi\rangle \sim \langle \mathcal{T}_{m,k},\chi\rangle_1 +\langle \mathcal{T}_{m,k},\chi\rangle_2$, where in $\langle\mathcal{T}_{m,k},\chi\rangle_j$ the integrand has been multiplied by $\varrho_j(y)$.
As $k\rightarrow +\infty$, we have $\langle\mathcal{T}_{m,k},\chi\rangle_2=O\left(k^{-\infty}\right)$.
*Proof.* Since the singular support of $\Pi$ is the diagonal in $X\times X$, on the support of $ h\left(y,x'\right)=:\varrho_2(y)\,\Pi\left(y,x'\right)$ we have $\mathrm{dist}_M\big(\pi(y),\pi(x)\big)\ge \delta$; therefore $h$ is $\mathcal{C}^\infty$ on $X_2\times X'$, where $X_2=\pi^{-1}(M_2)$ and $X'=\pi^{-1}\left(M'\right)$, with $M'$ a suitably small open neighborhood of $m$ (say, $M'=M_{\delta/2}(m)$). We may thus regard $h\left(\cdot,x'\right)$ as a $\mathcal{C}^\infty$ family of compactly supported $\mathcal{C}^\infty$-functions on $X_2$, parameterized by $x'\in X'$, where both $X_2$ and $X'$ are $S^1$-invariant.
For each $\tau\in \mathbb{R}$, on the other hand, $U(\tau)$ is a FIO associated to a canonical graph. Therefore, we may also regard $\kappa\left(\tau,x''\right)=:U(\tau)\left(x'',\cdot\right)$ as a $\mathcal{C}^\infty$ family of distributions on $X$, parametrized by $\left(\tau,x''\right)$. Explicitly, $\left<\kappa\left(\tau,x''\right),f\right>=\big(U(\tau)(f)\big)\left(x''\right)$ for $f\in \mathcal{C}^\infty(X)$.
Hence $B\left(\tau,x',x''\right)=:\langle \kappa\left(\tau,x''\right),h\left(\cdot,x'\right)\rangle$ is a $\mathcal{C}^\infty$-function of $\left(\tau,x',x''\right)\in \mathbb{R}\times X'\times X$, and therefore its $l$-th Fourier component $B_l\left(\tau,x',x''\right)$ with respect to $x'$ is $O\left(l^{-\infty}\right)$ as $l\rightarrow \infty$, uniformly on compact subsets of $\mathbb{R}\times X'\times X$.
Now in view of (\[eqn:fourier-transform-comp\]), (\[eqn:fourier-of-k\]) and the above, we have $$\begin{aligned}
%\label{eqn:fourier-transform-comp-2}
\langle \mathcal{T}_{m,k},\chi\rangle_2&=&
\frac{k}{2\pi}\cdot \int_{-\infty}^{+\infty}\int_{-\epsilon}^{+\epsilon}\chi(\lambda)e^{-ik\tau\lambda}
\,\beta(\tau)\,B_{-k}(\tau,x,x)\,d\lambda\,d\tau\nonumber\\
&=&\frac{k}{\sqrt{2\pi}}\cdot \int_{-\epsilon}^{+\epsilon}\widehat{\chi}(k\tau)\,\beta(\tau)
\,B_{-k}(\tau,x,x)\,d\tau=O\left(k^{-\infty}\right).\end{aligned}$$
Q.E.D.
As in [@p-weyl], we shall make use of the microlocal structure of $U(\tau)$ and $\Pi$: up to smoothing terms that contribute negligibly to the asymptotics, these may be represented as Fourier integral operators.
Namely, working in local coordinates near $x$ on the one hand we may write for $\tau\sim 0$: $$\begin{aligned}
\label{eqn:fourier-for-U}
U(\tau)\left(x',x''\right)=\frac{1}{(2\pi)^{2\mathrm{d}+1}}\,\int_{\mathbb{R}^{2\mathrm{d}+1}}
e^{i[\varphi(\tau,x',\eta)-x''\cdot \eta]}a\left(\tau,x',x'',\eta\right)\,d\eta,\end{aligned}$$ where the generating function $\varphi$ and the amplitude $a$ are as follows [@gs]. First, $$\label{eqn:fase-generatrice}
\varphi(\tau,x',\eta)=x'\cdot \eta+\tau\,q\left(x',\eta\right)+O\left(\tau^2\right)\,\|\eta\|;$$ furthermore, $a(\tau,\cdot,\cdot,\cdot)\in S^0_{\mathrm{cl}}$ for every $\tau$, with $a\left(0,x',x'',\eta\right)=1/\mathcal{V}\left(x''\right)$, where $\mathcal{V}$ is the local coordinate expression of the volume form on $X$.
On the other hand, after [@bs], $\Pi$ is a Fourier integral operator with complex phase: $$\label{eqn:fourier-for-pi}
\Pi\left(x',x''\right)=\int_0^{+\infty}e^{i t\psi\left(x',x''\right)}s\left(t,x',x''\right)\,dt,$$ where the Taylor expansion of the phase $\psi$ along the diagonal is determined by the Kähler metric, and $s\left(t,x',x''\right)\sim \sum_{j\ge 0}t^{\mathrm{d}-j}s_j\left(x',x''\right)$.
Inserting (\[eqn:fourier-for-U\]) and (\[eqn:fourier-for-pi\]) in (\[eqn:fourier-transform-1\]), we get $$\begin{aligned}
\label{eqn:insert-foi}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle\sim
\frac{k}{(2\pi)^{2\mathrm{d}+3}}
}\\
&&\cdot \int_{-\infty}^{+\infty}\int_{-\epsilon}^{+\epsilon}\int_{\mathbb{R}^{2\mathrm{d}+1}}\int_{-\pi}^\pi\int_X
e^{i\Phi_k}\,H_k\big(\lambda,\tau,x,y,\eta,t,\vartheta\big)\,d\lambda\,d\tau\,d\eta\,d\vartheta\,d\mu_X(y), \nonumber\end{aligned}$$ where $$\begin{aligned}
\lefteqn{\Phi_k=\Phi_k(t,\vartheta,\lambda,\tau,x,y,\eta)}\\
&=:&\varphi(\tau,x,\eta)-y\cdot \eta-k\lambda \tau+t\,\psi\big(r_\vartheta(y),x\big)-k\vartheta\end{aligned}$$ $$H_k\big(t,\vartheta,\lambda,\tau,x,y,\eta\big)=:\chi(\lambda)\,\beta(\tau)\,\varrho_1(y)\,
\,a\left(\tau,x,y,\eta\right)\,s\big(t,r_\vartheta(y),x\big).$$
With the change of variables $\eta\rightarrow k\eta$ and $t\rightarrow kt$, (\[eqn:insert-foi\]) may be rewritten $$\begin{aligned}
\label{eqn:insert-foi-rescaled}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle\sim
\left(\frac{k}{2\pi}\right)^{2\mathrm{d}+3}}\\
&&\cdot \int_{-\infty}^{+\infty}\int_{-\epsilon}^{+\epsilon}\int_{\mathbb{R}^{2\mathrm{d}+1}}\int_{0}^{+\infty}\int_{-\pi}^\pi\int_X
e^{ik\Phi}\,H_k\big(\lambda,\tau,x,y,k\eta,k t,\vartheta\big)\,d\lambda\,d\tau\,d\eta\,dt\,d\vartheta\,d\mu_X(y), \nonumber\end{aligned}$$ where $$\Phi(t,\vartheta,\lambda,\tau,x,y,\eta)=\varphi(\tau,x,\eta)-y\cdot \eta-\lambda \tau+t\,\psi\big(r_\vartheta(y),x\big)-\vartheta.$$
Implicitly introducing a partition of unity in $\vartheta$, we may assume that the integrand is compactly supported in $\vartheta$. We now make the following remarks, that may proved by an adaptation of the arguments in Lemmata 2.3 - 2.5 of [@p-weyl] (the focus in [@p-weyl] is on the asymptotics with respect to a continuous parameter denoted $\lambda$, while here the asymptotics are with respect to the discrete parameter $k$).
- Integration by parts in $d\vartheta$ shows that only a negligible contribution to the asymptotics is lost if we multiply the integrand by a compactly supported function $\zeta=\zeta (t)$ with $\zeta\in \mathcal{C}^\infty _0\big((1/C,C)\big)$ and $\zeta=1$ on $(2/C,C/2)$.
- We may thus replace $\int_{0}^{+\infty}dt$ by $\int_{0}^{+\infty}\zeta(t)\,dt$ in (\[eqn:insert-foi-rescaled\]).
- Integration by parts in $dt$ then shows that for any $c>0$ and $\xi<1/2$ the contribution to the asymptotics coming from the loci $W_k\subseteq X$ where $\mathrm{dist}_M(y,x)\ge c\,k^{-\xi}$ is negligible.
Let us fix $\xi\in [1/3,1/2)$.
Next let us choose a system of Heisenberg local coordinates for $X$ centered at $x$ (we refer to [@sz] for a definition and discussion of Heisenberg local coordinates); let $x+(\theta,\mathbf{v})$ denote the point with Heisenberg local coordinates $(\theta,\mathbf{v})$. Thus we write $y=x+(\theta,\mathbf{v})$ and $d\mu_X(y)=\mathcal{V}(\theta,\mathbf{v})\,d\theta\,d\mathbf{v}$; in particular, $\mathcal{V}(\theta,\mathbf{0})=1/(2\pi)$ for every $\theta$. By the previous discussion, only a negligible contribution to the asymptotics is lost if we multiply the integrand by $\gamma\left(k^{\xi}\|\mathbf{v}\|\right)$, where $\gamma\in \mathcal{C}^\infty_0(\mathbb{R})$ satisfies $\gamma(b)=1$ if $|b|\le 1$, $\gamma(b)=0$ if $|b|\ge 2$.
Since this amounts to removing a smoothing term from $U(\tau)$, we may multiply the amplitude $a$ by a cut-off function in $\eta$ which vanishes for $\|\eta\|<\delta$ for some small $\delta>0$, and equals $1$ for large $\|\eta\|$. We may then also write $\eta=r\omega$, with $r>0$ and $\omega\in S^{2\mathrm{d}}$, so that $d\eta=r^{2\mathrm{d}}\,dr\,d\omega$, and replace $\int_{\mathbb{R}^{2\mathrm{d}+1}}d\eta$ by $\int_0^{+\infty}\int_{S^{2\mathrm{d}}}r^{2\mathrm{d}}\,dr\,d\omega$. We shall write $\omega=(\omega_0,\omega_1)\in S^{2\mathrm{d}}\subseteq\mathbb{R}\times \mathbb{R}^{2\mathrm{d}}$ with $\omega_0^2+\|\omega_1\|^2=1$.
Now $\big((0,\mathbf{0}),(1,\mathbf{0})\big)$ corresponds to $(x,\alpha_x)$ in Heisenberg local coordinates. An argument similar to the proof of Lemma 2.2 of [@p-weyl] shows that as $k\rightarrow +\infty$ only a rapidly decreasing contribution is lost if integration in $d\omega$ is restricted to a small open neighborhood $S_+\subseteq S^{2\mathrm{d}}$ of $(1,\mathbf{0})$; we may thus multiply the integrand in (\[eqn:insert-foi-rescaled\]) by an appropriate bump function $g\in \mathcal{C}^\infty_0(S_+)$ which is identically $1$ near $(1,\mathbf{0})$ without affecting the asymptotics. We may assume $\omega_0\ge a>0$ for some fixed $a>0$ on $S_+$.
We shall furthermore adopt the rescaling $\mathbf{v}\rightarrow \mathbf{v}/(r\sqrt{k})$, and write $y=x+\big(\theta,\mathbf{v}/(r\sqrt{k})\big)$, so that $d\mu_X(y)=r^{-2\mathrm{d}}k^{-\mathrm{d}}\mathcal{V}(\theta,\mathbf{v})\,d\theta\,d\mathbf{v}$ in the new coordinates. In rescaled coordinates, integration in $d\mathbf{v}$ is over a ball in $\mathbb{C}^\mathrm{d}$ centered at the origin of radius $2\,r\,k^{1/2-\xi}$.
Given this, (\[eqn:insert-foi-rescaled\]) may be rewritten
$$\begin{aligned}
\label{eqn:insert-foi-rescaled-in-v}
\lefteqn{\langle \mathcal{T}_{m,k},\chi\rangle\sim
\frac{k^{\mathrm{d}+3}}{(2\pi)^{2\mathrm{d}+3}}}\\
&&\cdot \int_{-\infty}^{+\infty}\int_{-\epsilon}^{+\epsilon}\int_0^{+\infty}\int_{S^{2\mathrm{d}}}
\int_{1/C}^{C}\int_{-\pi}^\pi\int_{-\pi}^\pi\int_{\mathbb{C}^\mathrm{d}}
e^{ik\Psi_k}\,S_k\,d\lambda\,d\tau\,dr\,d\omega\,dt\,d\vartheta\,d\theta\,d\mathbf{v}, \nonumber\end{aligned}$$
where $$\begin{aligned}
\lefteqn{\Psi_k\big(t,\vartheta,\lambda,\tau,x,\theta,\mathbf{v},r,\omega\big)=:
\Phi\left(\lambda,\tau,x,x+\left(\theta,\frac{\mathbf{v}}{r\sqrt{k}}\right),r\omega,\vartheta\right)}\\
&=&-\frac{1}{\sqrt{k}}\,\mathbf{v}\cdot \omega_1+\Big[-r\omega_0\theta+\tau r\,q(x,\omega)
+O\left(\tau^2\right)\,r\\
&&\left.-\lambda\,\tau+t\,\psi\left(x+\left(\theta+\vartheta,\frac{\mathbf{v}}{r\sqrt{k}}\right),x\right)-\vartheta\right],\end{aligned}$$ and $$\begin{aligned}
S_k\big(\lambda,\tau,x,\theta,\mathbf{v},r,\omega,\vartheta\big)=:H_k\big(\lambda,\tau,x,y,k\eta,k t,\vartheta\big)\,
\zeta(t)\,\gamma\left(k^{\xi-1/2}\|\mathbf{v}\|/r\right)\,g(\omega).\end{aligned}$$
By (65) of [@sz], we have $$\begin{aligned}
\label{eqn:exp-for-psi}
\lefteqn{t\,\psi\left(x+\left(\theta+\vartheta,\frac{\mathbf{v}}{r\sqrt{k}}\right),x\right)}\\
&=&i t\,\left[1-e^{i(\theta+\vartheta)}\right]+\left[i\,\frac{\|\mathbf{v}\|^2 }{2 r^2 k}
+R^\psi_3\left(\frac{\mathbf{v}}{r\sqrt{k}}\right)\right]\,t\,e^{i(\theta+\vartheta)},\nonumber\end{aligned}$$ where $R^\psi_3$ vanishes to third order at the origin.
We can then write $$\begin{aligned}
ik\Psi_k&=&-i\,\sqrt{k}\,\mathbf{v}\cdot \omega_1+i\,k\,\Psi-\frac{\|\mathbf{v}\|^2 }{2 r^2 }\,t\,e^{i(\theta+\vartheta)}
+i\,k\,R^\psi_3\left(\frac{\mathbf{v}}{r\sqrt{k}}\right)\,t\,e^{i(\theta+\vartheta)},\nonumber\end{aligned}$$ where $$\begin{aligned}
\lefteqn{\Psi\big(t,\theta,\vartheta,\lambda,\tau,x,\mathbf{v},r,\omega\big)}\\
&=:&-r\omega_0\theta+\tau r\,q(x,\omega)
+O\left(\tau^2\right)\,r-\lambda\,\tau+i t\,\left[1-e^{i(\theta+\vartheta)}\right]-\vartheta.\end{aligned}$$
Since the exponential $D_k(\mathbf{v},\theta,\vartheta,t)=:\exp\left(i\,k\,R^\psi_3\big(\mathbf{v}/(r\sqrt{k})\big)\,t\,e^{i(\theta+\vartheta)}\right)$ is bounded in the range $\|\mathbf{v}\|<2\,k^{1/2-\xi}$, it may be incorporated into the amplitude (as in §3 and §5 of [@sz]). Furthermore, since $\partial_\theta\Psi=-r\omega_0+t\,e^{i(\theta+\vartheta)}$ and $0<a\le\omega_0<1$, $1/C\le t\le C$, integration by parts in $\theta$ shows that, for some fixed $D\gg 0$, the regions where $r<1/D$ or $r>D$ only give a negligible contribution to the asymptotics for $k\rightarrow +\infty$.
In the same manner, integration by parts in $d\tau$ shows that only a negligible contribution is lost if the integrand is multiplied by $\sigma(\lambda)$, where $\sigma\in \mathcal{C}^{\infty}_{0}\big((-C,C)\big)$ and $\sigma=1$ on $(-C/2,C/2)$.
Replacing $D$ and $C$ with $\max\{D,C\}$, and multiplying the integrand by a suitable bump function $\kappa (r)$, we obtain $$\begin{aligned}
\label{eqn:insert-foi-inner-outer}
\langle \mathcal{T}_{m,k},\chi\rangle\sim
\frac{k^{\mathrm{d}+3}}{(2\pi)^{2\mathrm{d}+3}}\int_{\mathbb{C}^\mathrm{d}}\int_{S^{2\mathrm{d}}}
e^{-i\sqrt{k}\,\mathbf{v}\cdot\omega_1}\,I_k(\mathbf{v},\omega)\,d\mathbf{v}d\omega,\end{aligned}$$ where $$\label{eqn:defn-di-I-k}
I_k(\mathbf{v},\omega)=:\int_{-C}^{C}\int_{-\epsilon}^{+\epsilon}\int_{1/C}^{C}
\int_{1/C}^{C}\int_{-\pi}^\pi\int_{-\pi}^\pi
e^{ik\Psi_{\mathbf{v},\omega}}\,\sigma(\lambda)\cdot T_k\,d\lambda\,d\tau\,dr\,dt\,d\vartheta\,d\theta\, \nonumber$$ with $T_k=: S_k\cdot D_k$ and $$\Psi_{\mathbf{v},\omega}\big(t,\theta,\vartheta,\lambda,\tau,r\big)=:
\Psi\big(t,\theta,\vartheta,\lambda,\tau,x,\mathbf{v},r,\omega\big).$$ Thus we first view $\mathbf{v}$ and $\omega$ as parameters in the inner integral, and evaluate it asymptotically using the stationary phase Lemma. A straightforward computation leads to the following:
\[lem:stationary-point-psi\] $\Psi_{\mathbf{v},\omega}$ has a unique stationary point $$\big(t_0,\theta_0,\vartheta_0,\lambda_0,\tau_0,r_0\big)=
\Big(1,0,0,q(x,\omega)/\omega_0,0,1/\omega_0\Big).$$ The Hessian of $\Psi_{\mathbf{v},\omega}$ at the critical point has determinant $$\det\Big(H(\Psi_{\mathbf{v},\omega})\Big)=-\omega_0^2.$$
In particular, $\Psi_{\mathbf{v},\omega}$ vanishes at the critical point, and $$\frac{1}{\sqrt{\det\big(kH(\phi)/2\pi i\big)}}=\frac{1}{\omega_0}\,\left(\frac{k}{2\pi}\right)^{-3}.$$ Applying the stationary phase Lemma, the asymptotic expansion of the amplitudes of $\Pi$ and $U(\tau)$ and the Taylor expansion of $s_j$, $a_j$, $D_k$ and $\mathcal{V}$, we conclude that for every integer $N\gg 0$ $$\begin{aligned}
\label{eqn:stima-di-I-k}
\lefteqn{I_k(\mathbf{v},\omega)\sim \frac{1}{\omega_0}\,g(\omega)\,\gamma\left(k^{\xi-1/2}\omega_0\,\mathbf{v}\right)
e^{-\omega_0^2\,\|\mathbf{v}\|^2/2}}\\
&&\cdot \left(\frac{k}{2\pi}\right)^{-3}\,\left(\frac k\pi\right)^{\mathrm{d}}\,
\left[\chi\left(\frac{q(x,\omega)}{\omega_0}\right)+\sum_{j= 1}^N k^{-j/2} P_j(\mathbf{v},\omega,\chi)\left(\frac{q(x,\omega)}{\omega_0}\right)\right]
+R_N,\nonumber\end{aligned}$$ where $\big|R_N\big|<C'_N\,k^{-aN}e^{-a\|\mathbf{v}\|^2}Q_N(\mathbf{v})$, for some $C'_N,a>0$ and some polynomial $Q_N$, while each $P_j$ is a differential operator of degree $2j$ in $\chi$, as prescribed by the stationary point Lemma, and is a polynomial in $\mathbf{v}$; we have made made use of the equalities $s_0(x,x)=\pi^{-\mathrm{d}}$, $a(0,x,y,\eta)=1/\mathcal{V}(y)$, $\mathcal{V}(\theta,\mathbf{0})=1/2\pi$.
In the way to estimate the outer integral in (\[eqn:insert-foi-inner-outer\]), let us first remark that, for $N\gg 0$, integration of the remainder $R_N$ over a ball of radius $O\left(k^{1/2-\xi}\right)$ is $O\left(k^{-a'N}\right)$ for some $a'>0$. To estimate the integral of the former summand, we may apply the stationary phase Lemma in $\mu=\sqrt{k}$. The phase $\Upsilon=-\mathbf{v}\cdot \omega_1$ has a unique stationary point for $\mathbf{v}=\omega_1=\mathbf{0}$, and partial integration in $d\omega$ shows that only a negligible contribution to the asymptotics is lost if integration in $d\mathbf{v}$ is restricted to an arbitrary open neighborhood of the origin; we may thus replace $\gamma\left(k^{\xi-1/2}\omega_0\,\mathbf{v}\right)$ with some fixed cut-off $\rho=\rho(\mathbf{v})$ identically one near the origin. At the critical point $\Upsilon$ clearly vanishes, and its Hessian $\Upsilon''$ satisfies $$\det\left(\frac{\mu \Upsilon''}{2\pi i}\right)=\left(\frac{\mu}{2\pi}\right)^{4\mathrm{d}}
=\frac{k^{2\mathrm{d}}}{(2\pi)^{4\mathrm{d}}}.$$ Since $q\big(x,(1,\mathbf{0})\big)=q(x,\alpha_x)=\varsigma_T(m)$, we get $$\begin{aligned}
\label{eqn:final-expansion}
\langle \mathcal{T}_{m,k},\chi\rangle\sim
\left(\frac k\pi\right)^{\mathrm{d}}\left[\chi\big(\varsigma_T(m)\big)+
\sum_{j\ge 1}k^{-j/2}Q_j(\chi)\big(\varsigma_T(m)\big)\right],\end{aligned}$$ where the $Q_j$’s are differential operators acting on $\chi$.
Now we remark that the asymptotic expansions for the amplitudes $s$ and $a$ of $\Pi$ and $U$, respectively, go down by integer steps; therefore, the appearance of fractional powers of $k$ in (\[eqn:stima-di-I-k\]) is due to the asymptotic expansion of the amplitude in $\mathbf{v}/\sqrt{k}$. Hence the general summand in (\[eqn:stima-di-I-k\]) splits as a sum of multiples of $k^{d-3-s/2-l}Q_{s,l}(\mathbf{v},\omega,\chi)$, where $s$ and $l$ are non-negative integers, and $Q_{s,l}(\mathbf{v},\omega,\chi)$ is a homogeneous polynomial of degree $s$ in $\mathbf{v}$, and a differential operator of degree (at most) $2l$ in $\chi$. In turn, when we apply the stationary phase Lemma in $\mu$, and keep track of powers of $k$ involved, this summand gives rise to a linear combination of terms of the form $$k^{\mathrm{d}+3}\, k^{-\mathrm{d}}\, k^{d-3-s/2-l}\,k^{-t/2}\Big(\frac{\partial}{\partial v_{i_1}}\circ
\frac{\partial}{\partial \omega_{1i_1}}\circ \cdots \circ \frac{\partial}{\partial v_{i_t}}\circ
\frac{\partial}{\partial \omega_{1i_t}}\Big)Q_{s,l}(\mathbf{0},(1,\mathbf{0}),\chi),$$ with $t$ a non-negative integer. Since $Q_{s,l}$ is homogeneous of degree $s$ in $\mathbf{v}$ and we are evaluating at the origin, we only get a non-zero contribution for $t=s$. Hence, we get a linear combination of summands of the form $$k^{d-s-l}P_{s,l}(\chi)\big(\varsigma_T(m)\big),$$ with $P_{s,l}$ a differential operator in $\chi$, of degree $2l\le 2(s+l)$.
Q.E.D.
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[^1]: [**Address:**]{} Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; [**e-mail**]{}: [email protected]
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abstract: 'Phase locking dynamics of dipolarly coupled vortices excited by spin-polarized current in two identical nanopillars is studied as a function of the interpillar distance $L$. Numerical study and analytical model have proved the remarkable efficiency of magneto-static interaction to achieve phase locking. Investigating the dynamics in the transient regime towards phase locking, we extract the evolution of the locking time $\tau$, the coupling strength $\mu$ and the interaction energy $W$. Finally, we compare this coupling energy with the one obtained by simple model.'
author:
- 'A.D. Belanovsky'
- 'N. Locatelli'
- 'P.N. Skirdkov'
- 'F. Abreu Araujo'
- 'J. Grollier'
- 'K.A. Zvezdin'
- 'V. Cros'
- 'A.K. Zvezdin'
title: 'Phase locking dynamics of dipolarly coupled vortex-based spin transfer oscillators'
---
Injecting a spin-polarized current through magnetic multilayers leads to new interesting physical phenomena named spin-transfer effect. These interactions between the spins of charge carriers and local magnetic moments create an additional torque exerted on the magnetization [@Stiles2006]. As a result, a complex spin-transfer-driven magnetic dynamics comes out with characteristic bifurcations of the Poincaré-Andronov-Hopf type and limit cycles arises in this highly non-equilibrium medium. The diversity of these new effects is especially true for systems of interacting nanomagnets, penetrated by spin-polarized current. One of the novel effect is the current-driven magnetization oscillations [@Kiselev2003], that might lead to tantalizing possibilities for new nanoscale microwave devices with the frequencies tunable over a wide range using applied currents and fields. While many crucial advances have been made in the fabrication and understanding of such Spin Transfer Nano-Oscillators (STNO), there remain several critical problems yet to be resolved, in particular, the low microwave power and quality factor of a single STNO.
To tackle these issues, particular attention has been recently focused on vortex STNOs that could present a significant output power [@Dussaux2010], a very small spectral linewidth [@Pribiag2007] and/or large frequency agilities at zero field [@Manfrini2009]. Moreover, several encouraging experiments have been reported on the vortices phase-locking through exchange interaction [@Ruotolo2009] and synchronization to external microwave current [@Dussaux2011]. Beyond these practical interests, a magnetic vortex and its dynamical modes [@Guslienko2008], notably the gyrotropic motion of the vortex core, is a model system to investigate deeply the physics of the spin transfer torque acting on a highly non uniform magnetic configuration [@Khvalkovskiy2009; @Khvalkovskiy2010]. Collective gyrotropic modes is a mean to improve drastically the spectral coherence of any oscillator system [@Locatelli2011]. Similarly, vortex based systems can be chosen as a new playground to investigate the influence of the magnetostatic interactions on vortices collective behaviour.
Magnetostatically coupled vortices collective dynamics have been studied both experimentally and theoretically for the case of low amplitude oscillations excited by means of external RF magnetic field [@Otani2003; @Vogel2010; @Awad2010; @Barman2010; @Jung2010] and spin-polarized current [@Sugimoto2011; @Sukhostavets2011]. However all these models are not applicable to the case of interest, i.e. for the large amplitude steady oscillations. The fundamental reason for that is the hypothesis of low amplitude oscillations near the centers of nanodots used by these models. A strong consequence of this approximation is that mathematically it allows to use of the ratio of the vortex orbit to the disk radii as a small parameter. However in the case of the large-amplitute oscillations phase-locking, such linearization is due neither for the of vortex STNOs, nor for the uniform ones [@Slavin2010]. In the letter we propose an original model for the coupled vortices dynamics without using this assumption. This model provides an expression for the coupling energy with the parameters of the transient process, which can be directly determined either through micromagnetic simulations or by experiment.
![(Color online) Schematic representation of two interacting spin-transfer oscillators. Each pillar is composed by a free magnetic layers with vortex, a non magnetic spacer, and a SAF polarizer. Red arrows indicates the direction of spin polarization created by the polarizer. The nanopillars have a diameter $D=200$ nm and are separated by a distance $L$. The parameters $\mathbf{X}_1$ and $\mathbf{X}_2$ define the cores positions.[]{data-label="fig:STNOs"}](fig1){width="2.4"}
The studied system is made of two identical nanopillars with diameters $200~\text{nm}$, each of them being composed by a free magnetic layer, a non magnetic spacer, and a since synthetic antiferromagnet (SAF) polarizer which generates a perpendicular spin polarization $p_{z}$. In our simulations we consider these layers only by the value of spin-polarization like in [@Sugimoto2011], since SAF polarizers that are widely used in vortex STNO experiments, have a negligible magnetostatic field, and thus practically do not influence on the vortices dynamics.
A free layer is $h=10~\text{nm}$ thick Ni$_{81}$Fe$_{19}$ and has a magnetic vortex as a ground state. The magnetic parameters of the free layer are: the magnetization $M_s=800~\text{emu/cm}^3$, the exchange energy $~A=1.3\times 10^{-6}~\text{erg/cm}$ and the damping parameter $~\alpha=0.01$. In order to be above the critical current, a spin polarization $P$ of $0.2$ and a current density $J$ of $7 \times 10^6~\text{A}/\text{cm}^2$ have been chosen. The initial magnetic configuration is two centred vortices with same core polarities and same chiralities. The micromagnetic simulations are performed by numerical integration of the LLG equation using our micromagnetic code SpinPM based on the forth order Runge-Kutta method with an adaptive time-step control for the time integration and a mesh size $2.5 \times 2.5~\text{nm}^2$.
In this work the evolution of the phase-locking dynamics as a function of the interpillar distance has been studied. Therefore a series of micromagnetic simulations with different distances $L$ ($50$, $100$, $200$ and $500$ $~\text{nm}$) has been performed. The results of the simulations are then analyzed to extract the radius of the vortex core trajectory in each free layer as well as the phase difference $\psi$ between core radius-vectors as a function of time.
In Fig. \[fig:50nm\], the simulations results for $L=50$ $~\text{nm}$ are presented. The vortices transient dynamics can be divided in two regimes. At $t=0$ the spin torque is switched on and thus the radii of both cores trajectories increase towards their equilibrium orbits for about $300~\text{ns}$ (see Fig. \[fig:50nm\]a). The phase difference between the two radius vectors shown in Fig. \[fig:50nm\]b remains constant and equal to $-\pi$ because of repulsive core-core interaction. The second regime begins when the two cores have reached orbits close to their steady ones. From this stage, both the intercore distance and the phase difference (see Fig. \[fig:50nm\]a and b) show large oscillations indicating the beginning of the phase locking.
![(Color online) Micromagnetic simulations for $L=50$ nm of the phase locking dynamics. Evolution as a function of time $t$ of the vortex core orbital positions $X_{1}$ and $X_{2}$ (a) and the phase difference $\psi$ (b). In (c), a zoom of the phase difference $\psi$ is presented for the time window in which the fitting with Eq. (\[eq\_transcient\]) has been done.[]{data-label="fig:50nm"}](fig2){width="\linewidth"}
The second regime is one of main interest for this work since the coupling energy can be extracted using the analysis of the cores motion in this transient regime (indicated by the square in Fig. \[fig:50nm\]b). During this time range, the phase difference $\psi$ can be identified as being a low frequency damped oscillation described by the following expression: $$\psi=\mathrm{e}^{-\frac{t}{\tau} + C_1} \cdot \sin(\Omega t + C_2).
\label{eq_transcient}$$ As shown in Fig. \[fig:50nm\]c, the fitting is done for the time window between $500$ and $800~\text{ns}$ in which the mean orbit radii have reached the common equilibrium value $X_{0}$. From the fitting, one can extract for $L=50~\text{nm}$ a frequency $\Omega$ equals to $40.134~\text{MHz}$ and a phase locking time of $82.59~\text{ns}$.
The parameters extracted from the fitting procedure for all the interdot distances are summarized in table \[tab:fit\]. One needs to note that the phase locked equilibrium orbit radius $X_{0}$ does not vary much with $L$.
![(Color online) Phase difference $\psi$ as a function of time $t$ for different interpillar distances $L=100$ (a), $200$ (b) and $500~\text{nm}$ (c).[]{data-label="fig:Psi_vsL"}](fig3){width="\linewidth"}
$L$ (nm) $\Omega$ (MHz) $\tau$ (ns) $X_0$ (nm)
---------- ---------------- ------------- ------------
50 40.134 82.59 63.59
100 28.305 85.28 62.46
200 17.183 89.57 61.82
500 7.018 90.13 61.53
: Values of the phase difference frequency $\Omega$, the phase locking time $\tau$, and the steady-state radius of the core motion $X_0$ at different interpillar distances $L$ using the expression of Eq. (\[eq\_transcient\])[]{data-label="tab:fit"}
To derive the coupling energy between the oscillators from the simulations, we have developed an analytical model based on Thiele equations [@Thiele1973] coupled through the dipolar interaction energy $W_{int}$ [@Metlov2002; @Guslienko2004; @Scholz2003]. Due to the system symmetry, the interaction energy can be expressed as $W_{int}=a_1 x_1 x_2+b_1 y_1 y_2$ that can be reformulated using the cores positions $\mathbf{X_1}$ and $\mathbf{X_2}$ as [@Guslienko2005]: $$W_{int}(\mathbf{X}_1,\mathbf{X}_2)=\mu_1 \mathbf{X}_1\cdot \mathbf{X}_2 + \mu_2(x_1 x_2 - y_1 y_2).
\label{eq_Wint}$$ where $\mu_{1,2}$ are the interaction parameters and $x_{1,2},~y_{1,2}$ are core coordinates. The second term of Eq.(\[eq\_Wint\]) is neglected in our study, since it corresponds only to fast oscillations at double frequency of the gyrotropic modes and thus is averaged over the low frequency dynamics which is responsible for the phase locking. As a consequence, the expression for the interaction energy can be written as $W_{int}(\mathbf{X}_1,\mathbf{X}_2)=\mu \mathbf{X}_1 \cdot \mathbf{X}_2$.
The two Thiele equations of the core dynamics considering both the spin transfer torque and the interaction between the two oscillators are: $$G \left( \mathbf{e}_z \times \dot{\mathbf{X}}_{1,2} \right) - k_{1,2}(\mathbf{X}) \mathbf{X} - \hat{D} \dot{\mathbf{X}}_{1,2} - \mathbf{F}_{STT} - \mathbf{F}_{int} = \mathbf{0}$$ The first three terms are the conventional forces: the gyrotropic force with $G \mathbf{e}_z=-2\pi p \frac{M_s h}{\gamma}\mathbf{e}_z$, the confining force with $k(\mathbf{X})=\omega_0 G \left(1+a\frac{\mathbf{X}^2}{R^2}\right)$ [@Guslienko2006; @Ivanov2007] where the gyrotropic frequency is $\omega_0=\frac{20}{9}\gamma M_s h/R$ and the dyadic damping $\hat{D} = \alpha \eta G,~\eta = \frac{1}{2}\ln \left( \frac{R_D}{2l_0} \right)+\frac{3}{8},~\text{where}~l_0=\sqrt{\frac{A}{2\pi M_s^2}}$. The fourth term $\mathbf{F}_{STT}$ is the spin transfer force. For the case of uniform perpendicularly magnetized polarizer $\mathbf{F}_{STT}= \pi \gamma a_J M_s h \left( \mathbf{e}_z \times \mathbf{X} \right) = \varkappa \left( \mathbf{e}_z \times \mathbf{X} \right)$ [@Khvalkovskiy2009] where the spin torque amplitude is $~a_J=PJ/M_sh$ with $P$ the spin polarization and $J$ the current density. The last term describes the interaction forces and is expressed by $\mathbf{F}_{int}(\mathbf{X}_{1,2})=
- \partial W_{int}(\mathbf{X}_1,\mathbf{X}_2)/\partial \mathbf{X}_{1,2}=-\mu \mathbf{X}_{2,1}.
$ These Thiele equations can be reformulated in polar coordinates as (using $\alpha \eta \ll 1$):
$$\begin{aligned}
\frac{\dot{X}_{1}}{X_{1}} = -\left( \frac{\alpha \eta k_{1}(X_{1}) + \varkappa}{G} \right) + \frac{\mu}{G} \frac{X_{2}}{X_{1}}(\sin \psi-\alpha \eta \cos \psi) \label{eq:X1} \\
\frac{\dot{X}_{2}}{X_{2}} = -\left( \frac{\alpha \eta k_{2}(X_{2}) + \varkappa}{G} \right) - \frac{\mu}{G} \frac{X_{1}}{X_{2}}(\sin \psi+\alpha \eta \cos \psi) \label{eq:X2}\end{aligned}$$
$$\dot{\psi} = a \omega_{0} \left(\frac{X_{1}^2-X_{2}^2}{R^2} \right) - \alpha \eta \left(\frac{\dot{X}_1}{X_1} - \frac{\dot{X}_2}{X_2} \right) - \frac{\mu}{G} \cos \psi \left( \frac{X_2}{X_1} - \frac{X_1}{X_2} \right) \label{eq:psi}$$
These two equations of core motion and the equation of phase difference provide a complete dynamical description of the phase locking. By linearizing Eq.(\[eq:X1\],\[eq:X2\],\[eq:psi\]) with the assumptions that $\psi \ll 1$ and $\varepsilon=\frac{X_1-X_2}{X_1+X_2} \ll 1$, we obtain: $$\begin{aligned}
\dot{\varepsilon} & = & 2\alpha \eta (\tilde{\mu}-\omega_0 a r_{0}^2)\varepsilon +\tilde{\mu}\psi \label{eq:eps_dot} \\
\dot{\psi} & = & 4 (\tilde{\mu} - \omega_0 a r_{0}^2)\varepsilon - 2\alpha \eta \tilde{\mu} \psi \label{eq:psi_dot}\end{aligned}$$ where we used $\tilde{\mu}=\mu /G$ and $r_0 = X_0/R$. The two equations (\[eq:eps\_dot\]) and (\[eq:psi\_dot\]) are linear and their eigenvalues are $$\lambda_{1,2}=\alpha \eta \omega_0 a r_{0}^2 \pm \sqrt{\alpha^2 \eta^2 \omega_0^2 (a r_{0}^2)^2 + 4\tilde{\mu}^2-4\tilde{\mu} \omega_0 a r_{0}^2}.$$
First we consider the case of periodic solutions when $\frac{1}{2}\left(\omega_0 a r_{0}^2-\omega_0 a r_{0}^2 \sqrt{1-\alpha^2 \eta^2}\right)<\tilde{\mu}<\frac{1}{2}\left(\omega_0 a r_{0}^2+\omega_0 a r_{0}^2 \sqrt{1-\alpha^2 \eta^2}\right)$. The eigenvalues can be thus written as: $$\begin{aligned}
1/\tau & = & \alpha \eta \omega_0 a r_{0}^2 \label{eq:tau1}\\
\Omega^2 & = & -\alpha^2 \eta^2 \omega_0^2 (a r_{0}^2)^2 - 4\tilde{\mu}^2 + 4\tilde{\mu}\omega_0 a r_{0}^2 \label{eq:Omega}\end{aligned}$$ The important result of this study is that Eq. (\[eq:tau1\]) and (\[eq:Omega\]) allows to connect the coupling parameter $\mu$ with the phase locking parameters, i.e. $\Omega$ and $\tau$, obtained through micromagnetic simulations. Consequently, an expression of the time-averaged interaction energy $W_{int}$ takes the form: $W_{int}(L) = \mu X_0^2 = \frac{G}{2}\left(1/(\tau \alpha \eta) - \sqrt{1/(\tau \alpha \eta)^2-\Omega(L)^2}\right)X_0^2
\label{eq:Wint_L}.
$ In Fig. \[fig:fig4\], the evolution of this interaction energy $W_{int}$ with the interpillar separation distance $D_{12}$ derived from micromagnetic modelling is displayed by blue square dots. The best fit we obtain if for an energy decay law to be $D^{-3.6} _{12}$. In comparison, in the case of small core amplitudes [@Otani2003; @Vogel2010] this decay law has been found as $D^{-6} _{12}$, however without being confirmed experimentally [@Sugimoto2011].
![(Color online) Absolute values of interaction energy $W_{int}$ as a function of interpillar separation distance $D_{12}=2R+L$ obtained from the micromagnetic simulations (blue square dots) and from the macro-dipole model (red line). Inset: Evolution of the phase locking rate $1/\tau$ versus $D_{12}$ (purple line).[]{data-label="fig:fig4"}](fig4){width="8"}
To get more insights about the origin of this large coupling interaction, the values of the interaction energy $W_{int}$ obtained by simulations are compared with the ones derived from a simple model of two interacting macro dipoles, concentrated at the dot centres and rotating at a frequency $\omega_0$. In such a case, the magnetic dipole interaction energy $ W_{int}^{m-d}$ is defined as $$\displaystyle W_{int}^{m-d}=\frac{(\mathbf{M}_1\cdot\mathbf{M}_2)}{D_{12}^3}V_D^2 - \frac{3(\mathbf{M}_1\cdot\mathbf{D}_{12})(\mathbf{M}_2\cdot\mathbf{D}_{12})}{D_{12}^5}V_D^2,$$ where $D_{12}=2R+D$. The in-plane magnetization $\mathbf{M}_{1,2}$ is perpendicular to the radius-vector of core position, thus one can write $\mathbf{M}_{1,2}=\zeta (\mathbf{X}_{1,2}\times\mathbf{e}_z)$, where $\zeta$ is a constant, that has been numerically calculated: $\zeta \approx 5.6$ G/nm. The interaction energy can be rewritten in the following form: $$W_{int}^{m-d} = \underbrace{A\mathbf{X}_1\mathbf{X}_2}_{\text{low frequency oscillations}}+\underbrace{B X_{1}X_{2}\cos(\varphi_1+\varphi_2)}_{\text{high frequency dynamics}} \label{eq:W_m-d}$$ with $A=-\frac{\zeta^2 V_D^2}{2 D_{12}^3},~B=\frac{3}{2}\frac{\zeta^2 V_D^2}{D_{12}^3}$. As far as the phase locking dynamics is concerned, the second term in (\[eq:W\_m-d\]) corresponding to high frequency oscillations is averaged to zero and thus one can express the mean interaction energy $W_{int}^{m-d}$ in the macro-dipole approximation: $$W_{int}^{m-d}=-\frac{\zeta^2 V_D^2}{2D_{12}^3}\mathbf{X}_1\mathbf{X}_2 = \mu^{m-d} \mathbf{X}_1\mathbf{X}_2$$ In Fig. \[fig:fig4\] we observe that for small interpillar distances $W_{int}$ differs significantly from the macro dipole energy $W_{int}^{m-d}$, this difference demonstrates the importance of the magnetic quadrupole and higher multipoles for the phase-locking dynamics.
Coming back on the Eq. (\[eq:eps\_dot\]) and (\[eq:psi\_dot\]), a second regime has to be considered when $\tilde{\mu} < \frac{1}{2}\left(\omega_0 a r_{0}^2-\omega_0 a r_{0}^2 \sqrt{1-\alpha^2 \eta^2}\right)$ or $\tilde{\mu}>\frac{1}{2}\left(\omega_0 a r_{0}^2+\omega_0 a r_{0}^2 \sqrt{1-\alpha^2 \eta^2}\right)$. In this case, the solutions are aperiodic oscillations and it strongly impact the main features of the phase locking, notably the phase locking rate $1/\tau$. Indeed, in the regime of periodic oscillations, this phase locking parameter is almost independent on the coupling strength $\mu$ for interpillar separation distance $D_{12}$ values as large as $1600~\text{nm}$ (see inset of Fig. \[fig:fig4\]). We emphasize that the weak variation of the phase locking rate obtained in the micromagnetic simulations (see values in table \[tab:fit\]) is in fact solely due to the small variations of the steady orbit radii $X_{0}$ with the interpillar distance $L$ as expected from (\[eq:tau1\]). On the contrary, in the aperiodic regime, the phase locking rate $1/\tau$ depends strongly on the coupling strength $\mu$ with the following expression: $$1/\tau = \eta \omega_0 a r_{0}^2 - \sqrt{\eta^2 \omega_0^2 (a r_{0}^2)^2 + 4\tilde{\mu}^2-4\tilde{\mu}\omega_0 a r_{0}^2}$$
Using the value of the coupling parameter $\mu$ that can be extracted for very large interpillar distance $L$ through the macro-dipole approximation, we obtain a very rapid decrease of $1/\tau$ with interpillar distance $L$ and eventually a phase locking time that tends to $\tau \longrightarrow \infty$ for large distances. It is important to note that interaction energy $W_{int}$ becomes of the same order of magnitude as the room temperature thermal energy $kT$ at the interpillar distances $L$ of about single STNO diameter, thus the role of thermal effects in the phase-locking of vortex STNOs has to be properly investigated.
In conclusion, we have demonstraded an efficient phase-locking between two STNOs through dipolar mechanism. We have succeeded to provide an accurate expression of the interaction energy between two vortices based STNOs by comparing micromagnetic simulations to predictions of an analytical model based on coupled Thiele equations with dipole-dipole interacting forces. A major result is that the phase locking time $\tau$ is almost independent on the separation distances for values up to $1.6~\mu$m before it increases very rapidly at larger distances. We emphasize also the critical importance of higher order multipole terms for a correct description of the interaction energy, especially at shorter separation distances. Finally, our investigation opens ways to design some optimized STNO ensembles for synchronization which is a crucial step toward the development of a new generation of RF devices for telecommunication applications.
The work is supported by the EU Grant MASTER No.NMP-FP7 212257, RFBR Grants No.10-02-01162 and No.11-02-91067, CNRS PICS Russie No. 5743 2011, and the ANR agency (VOICE PNANO-09-P231-36) is also acknowledged. F. Abreu Araujo acknowledges the Research Science Foundation of Belgium (FRS-FNRS) for financial support (FRIA grant).
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abstract: |
We study the formation of magnetic clusters in frustrated magnets in their cooperative paramagnetic regime. For this purpose, we consider the $J_1$-$J_2$-$J_3$ classical Heisenberg model on kagome and pyrochlore lattices with $J_2 = J_3=J$. In the absence of farther-neighbor couplings, $J=0$, the system is in the Coulomb phase with magnetic correlations well characterized by pinch-point singularities. Farther-neighbor couplings lead to the formation of magnetic clusters, which can be interpreted as a counterpart of topological-charge clusters in Ising frustrated magnets \[T. Mizoguchi, L. D. C. Jaubert and M. Udagawa, Phys. Rev. Lett. [**119**]{}, 077207 (2017)\]. The concomitant static and dynamical magnetic structure factors, respectively $\mathcal{S}({\bm{q}})$ and $\mathcal{S}({\bm{q}},\omega)$, develop half-moon patterns. As $J$ increases, the continuous nature of the Heisenberg spins enables the half-moons to coalesce into connected “star” structures spreading across multiple Brillouin zones. These characteristic patterns are a dispersive complement of the pinch point singularities, and signal the proximity to a Coulomb phase. Shadows of the pinch points remain visible at finite energy, $\omega$. This opens the way to observe these clusters through (in)elastic neutron scattering experiments. The origin of these features are clarified by complementary methods: large-$N$ calculations, semi-classical dynamics of the Landau-Lifshitz equation, and Monte Carlo simulations. As promising candidates to observe the clustering states, we revisit the origin of “spin molecules" observed in a family of spinel oxides $AB_2$O$_4$ ($A=$ Zn, Hg, Mg, $B=$ Cr, Fe).
PACS numbers
: 75.10.Kt
author:
- Tomonari Mizoguchi
- 'Ludovic D. C. Jaubert'
- Roderich Moessner
- Masafumi Udagawa
title: |
Magnetic clustering, half-moons, and shadow pinch points as signals of a\
proximate Coulomb phase in frustrated Heisenberg magnets
---
Introduction
============
Geometrically frustrated magnets provide a stage to realize exotic states of matter, ranging from quantum and classical spin liquids [@balents2010; @savary2016; @zhou2017], unconventional magnetic ordering with topological response [@taguchi2001; @onose2010; @nagaosa2013], and states accompanied by exotic phase transitions . Among them, the disordered Coulomb phase is a canonical example for discrete and continuous spins on the three-dimensional pyrochlore lattice when frustration imposes a local divergence-free constraint [@henley2010], whose exotic character has been drawing considerable interest.
The Coulomb phase is based on degenerate classical spin configurations in absence of any spontaneous symmetry breaking. The magnetic correlations due to the local divergence-free constraint are characterized by non-analyticities in the static magnetic structure factor, $\mathcal{S}({\bm{q}})$ [@isakov2004]. These are called pinch points, and have been observed in the canonical spin-ice materials, Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$ [@bramwell2001; @fennell2009].
On top of exotic correlations, the Coulomb phase supports fractional excitations. These excitations are easy to visualize in spin ice, where Ising spins satisfy the so-called ice rules with two spins pointing inwards and two spins pointing outwards on every tetrahedron in the ground state. A tetrahedron in a “three-in-one-out" or “one-in-three-out" configuration, breaking this ice rule, carries a gauge charge and serves as an elementary fractional excitation. In spin ice, these gauge charges are actually effective magnetic charges [@castelnovo2008]. By identifying the spins with their inherent “magnetic field," one can regard the gauge charge as a source or sink of the field, and assign a magnetic charge $-2 (+2)$ for “one-in-three-out" (“three-in-one-out") tetrahedron states. Magnetic charges are defined from the discrete divergence of the magnetic field, i.e., the number of inward spins minus that of outward spins.
The introduction of the concept of magnetic charges turned out to be quite illuminating, carrying over two properties from conventional electromagnetism. Firstly, charge should be conserved. Indeed, the above-mentioned magnetic charge in spin ice satisfies a local conservation in the sense that they are always created/annihilated in pairs of positive and negative charges. And, secondly, opposite charges are expected to attract each other. However, this second property is non-universal. In the canonical spin-ice systems, Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$, opposite charges indeed interact with attractive force, attributed to the long-range dipolar interaction. However, the sign of the force actually depends on the microscopic details of the system.
Indeed, recently, the role of charge interactions is drawing interest in spin ice [@udagawa2016; @rau2016] and its two-dimensional analog [@moller2009; @chern2011; @chern2012; @mizoguchi2017]. If the interaction is chosen “unnaturally," i.e., attractive between same-sign charges, the Coulomb phase is destabilized towards the formation of same-sign-charge hexamer clustering [@udagawa2016; @rau2016; @mizoguchi2017]. The generic tendency to clustering can be naturally understood from the competition of the two-fold properties of charges. Same-sign charges attract each other, but they cannot pair-annihilate due to charge conservation. As a result, they form stable clusters. Their proliferation leads to unconventional classical spin liquids. Accompanying the clustering, the magnetic correlations display a noticeable evolution, characterized by half-moon patterns in $\mathcal{S}({\bm{q}})$, which replace the pinch point singularities [@robert2008; @udagawa2016; @rau2016; @mizoguchi2017].
The rich physics brought by the attraction of charges of the same sign naturally motivates us to generalize its analysis to the system with continuous spins. Indeed, the magnetic charge in the Ising system can be generalized to a conserved magnetic vector in continuous spin systems. With the continuous nature of magnetic vector, one can expect a variety of stable textures beyond hexamer clustering. From this viewpoint, it is interesting to look at a class of spinel oxides with 3$d$ magnetic ions, $AB_2$O$_4$ ($A=$ Zn, Hg, Mg, $B=$ Cr, Fe) [@lee2002; @tomiyasu2011; @tomiyasu2008; @tomiyasu2013; @gao2018; @yamada2002; @kamazawa2003; @conlon2010; @tomiyasu2011_2]. These compounds have weak magnetic anisotropy with small spin-orbit interaction of $3d$ ions, and the classical Heisenberg model with farther-neighbor interactions is expected to give a good starting point of analysis [@conlon2010]. Indeed, according to inelastic neutron scattering experiments, this family lacks “pinch points" in the dynamical structure factor, which are characteristic of the Coulomb phase. Instead, diffuse scattering patterns appear at the corners of the Brillouin zone. The diffuse scatterings are attributed to the clustering of small number of spins, coined as “spin molecules." Depending on materials, molecules take the form of hexamers [@lee2002; @tomiyasu2011; @tomiyasu2008; @tomiyasu2013; @gao2018] and dodecamers [@yamada2002; @kamazawa2003; @tomiyasu2011_2]. In the work by Conlon and Chalker [@conlon2010], the lack of pinch points has been attributed to weak, generic, farther-neighbor exchange, inducing hexagonal cluster scattering as observed in experiments.
To address these issues, in this paper, we consider the classical Heisenberg models on kagome and pyrochlore lattices with farther-neighbor interactions, on the high-symmetry line $J_2=J_3=J$, for arbitrary values of $J>0$. We focus on cooperative paramagnetic region above magnetic ordering temperature, where the magnetic fluctuations reflect the intrinsic nature of the system, in contrast to the ordering pattern itself, which is susceptible to structural changes or other extrinsic effects.
Our main results are summarized as follows: (i) We found three distinct patterns in $\mathcal{S}(\bm{q})$: pinch points, half-moons, and stars. These patterns are counterparts of the topological-charge clusters obtained in the corresponding Ising models. (ii) The three patterns reflect the structure of softest magnetic modes. (iii) The half-moon and star patterns can be interpreted as shadows of pinch points, and serve as a signal of proximity to a Coulomb phase. (iv) These characteristic patterns also appear in the low-energy region of dynamical structure factors, implying the possibility of experimental detection through inelastic neutron scattering.
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The rest of this paper is organized as follows. In Sec. \[sec:model\_and\_formalism\], we first describe the model, namely, the $J_1$-$J_2$-$J_3$ classical Heisenberg Hamiltonian on the kagome and pyrochlore lattices. Next, we introduce the theoretical methods; large-$N$ calculations, Monte Carlo simulations, and semiclassical Landau-Lifshitz (LL) equation. In Sec. \[sec:ssf\], the three distinct patterns of the static structure factor and their origins are discussed. Here, the main arguments are based on band-structure analyses of the large-$N$ approximation, supported by Monte Carlo simulations on the O(3) Heisenberg model. Section \[sec:semiclassical\_dynamics\] is devoted to the spin dynamics, analyzed by the LL equation. In Section \[sec:ising\], we discuss the real-space structure of the clusters, and show that they can be understood as a continuation from the topological charge cluster obtained in the Ising counterpart. Finally, we present discussions and summary in Sec. \[sec:summary\]. Details of the large-$N$ approximation, Monte Carlo simulations, and the quadrupolar order parameter are described in the Appendices.
Model and formalism {#sec:model_and_formalism}
===================
Model
-----
We consider a Heisenberg model on kagome and pyrochlore lattices with up to third-neighbor interactions: $$\begin{aligned}
H = & J_1 \sum_{\langle i,j\rangle_{\mathrm{NN}} } \bm{S}_i \cdot \bm{S}_j + J_2 \sum_{\langle i,j\rangle_{\mathrm{2nd}}} \bm{S}_i \cdot \bm{S}_j \notag \\
+ & J_{3\mathrm{a}} \sum_{\langle i,j\rangle_{\mathrm{3rd,a}}}
\bm{S}_i \cdot \bm{S}_j +J_{3\mathrm{b}}
\sum_{\langle i,j\rangle_{\mathrm{3rd,b}}} \bm{S}_i \cdot \bm{S}_j \notag \\
= & \frac{1}{2}\sum_{n,m}\sum_{\mu,\nu} \sum_{\alpha = x,y,z}
S_n^{\mu,\alpha} H_{n,m}^{\mu,\nu} S_m^{\nu,\alpha}.
\label{eq:hami}\end{aligned}$$
$J_1$, $J_2$, and $J_{3\mathrm{a}, 3\mathrm{b}}$ are the exchange coupling constants connecting nearest-, second-nearest-, and third-nearest neighbors, as shown in Fig. \[fig:bandkagome\]. Notice that two different types of the third-neighbor term ($J_{3\mathrm{a}}$ and $J_{3\mathrm{b}}$) are distinguished in that $J_{3\mathrm{a}}$ connects the sites along edges, while $J_{3\mathrm{b}}$ connects those across a hexagon. Here, we have introduced unit cell indices, $n$, $m$, and sublattice indices, $\mu$, $\nu$, and expressed each site as their combinations: $i=(n,\mu)$ and $j=(m,\nu)$. The unit cells contain respectively triangles (tetrahedra) of one orientation for kagome (pyrochlore) lattice, and the sublattice indices take $\mu= 1, \cdots N_{\rm sub}$, with $N_{\rm sub}=3$ (4). $\bm{S}_n^{\mu} = (S_n^{\mu,x}, S_n^{\mu,y} ,S_n^{\mu,z})$ are (classical) three-component vectors with unit length $|\bm{S}_n^\mu| =1$.
The model with a general parameter set has been intensively studied on both kagome [@chalker1992; @huse1992; @reimers1993; @zhitomirsky2008; @chern2013; @albaracin2018; @li2018] and pyrochlore [@reimers1991; @moessner1998; @canals2001; @chern2008; @conlon2010; @okubo2011; @iqbal2018] lattices, putting an emphasis on the low-temperature ordered states. In the present work, we focus on the parameters $(J_1, J_2, J_{3\mathrm{a}}, J_{3\mathrm{b}}) = (1,J,J,0)$ with $J>0$. The value of $J_{1}=1$ sets the energy and temperature scales of our problem. This parameter set has recently been shown to lead to a clustering of topological charges in the corresponding Ising models for $J>0$ (see Sec. \[sec:ising\] for details [@udagawa2016; @rau2016; @mizoguchi2017]).
Although $J_2=J_3(=J)$ is unlikely to be perfectly satisfied in real materials, the analysis of this isotropic point gives a great insight into the nearby systems. As we will show in the next section, this point allows a rewriting of the Hamiltonian with conserved spins, and simplifies the formulation of the large-$N$ analysis (see Sec. \[sec:largeNformalism\] and Appendix \[sec:largeNanalytics\]). These properties make it easier to grasp the physics of half-moons and magnetic clustering, which are useful to understand the nature of realistic systems around this point.
Conserved spins
---------------
It is instructive to introduce a local magnetic moment for each triangular and tetrahedral unit, $n$, $$\begin{aligned}
{\mathbf M}_n\equiv\zeta_n\sum_{j \in n } {\bm S}_{j},
\label{eq:Mp}\end{aligned}$$ where $\zeta_n=\pm 1$ is a sign factor distinguishing between upward ($+1$) and downward ($-1$) triangles/tetrahedra [@cap1]. Under the condition: $J_2=J_{3a}(=J)$ and $J_{3b}=0$, the Hamiltonian (\[eq:hami\]) can be rewritten as a function of ${\mathbf M}_n$: $$\begin{aligned}
H = \left(\frac{1}{2} - J \right) \sum_n|{\mathbf M}_n|^2 - J\sum_{\langle n,m\rangle}{\mathbf M}_n\cdot {\mathbf M}_m,
\label{Ham_conservedspin}\end{aligned}$$ where the summation over $n$ is taken over both upward and downward triangles/tetrahedra, and the summation over $\langle n,m\rangle$ is over neighboring pairs of triangles/tetrahedra. This expression naturally accounts for the Coulomb phase at $J=0$, with ${\mathbf M}_n=0$ for all $n$, and for its stability for small $J$ as will be discussed in detail in the next sections. Equation (\[Ham\_conservedspin\]) is a generalization of the spin to charge mapping of the corresponding Ising systems [@ishizuka2013; @udagawa2016; @rau2016; @mizoguchi2017] and satisfies a conservation law: $$\begin{aligned}
\sum_{n\in D}{\mathbf M}_n = \sum_{j\in\partial D}{\bm S}_{j}\, ,\end{aligned}$$ where $D$ is a connected ensemble of triangles/tetrahedra, and $\partial D$ is its contour. The contour $\partial D$ is made of all spins shared between two triangles/tetrahedra, $n\in D$ and $m\notin D$. This “Gauss’ law" means that the internal structure of a magnetic cluster is constrained by its boundary spins. Indeed, in the Ising case, the discrete variant of this Gauss’ constraint strictly determines the structure of clusters, and leads to hexamer spin liquids [@mizoguchi2017].
Formalism
---------
We study the static properties of model (\[eq:hami\]) with $(J_1, J_2, J_{3\mathrm{a}}, J_{3\mathrm{b}}) = (1,J,J,0)$ and $J>0$, by combining classical Monte Carlo simulations and analytical large-$N$ method. We also address the dynamics by simulating the semi-classical LL equation. In this section, we introduce the latter two methods, and all details for the classical Monte Carlo simulations will be given in Appendix \[app:MC\].
### Large-$N$ approximation \[sec:largeNformalism\]
To investigate static structure factors, we employ a large-$N$ approximation [@conlon2010; @garanin1999; @isakov2004; @sen2013]. The length of classical Heisenberg spins satisfies a hard constraint $|\bm{S}_n^\mu| = 1$. In the large-$N$ method, Heisenberg spins $\bm{S}_n^\mu $ are replaced by soft-spin variables $s_n^\mu$ whose length is constrained on average: $$\begin{aligned}
\langle (s_n^{\mu})^2 \rangle = \frac{1}{3}.
\label{eq:largeNnorm}\end{aligned}$$ Here disordered phases are assumed with $\langle s_n^{\mu} \rangle =0$. The above constraint (\[eq:largeNnorm\]) is enforced by introducing a Lagrange multiplier $\lambda$ which satisfies $$\begin{aligned}
\frac{1}{N_{\mathrm{site}}}\sum_{\bm{q}} \mathrm{Tr} [\lambda \hat{I} + \beta \hat{H}(\bm{q}) ]^{-1} = \frac{1}{3}, \label{eq:constraint}\end{aligned}$$ where the sum runs over all wavevectors $\bm q$ in the Brillouin zone and $N _{\mathrm{site}}$ is the total number of sites. $\hat{H}(\bm{q})$ represents the Fourier transformation of the exchange matrix: $$\begin{aligned}
[\hat{H}(\bm{q})]_{\mu\nu} = \sum_{m} H_{0,m}^{\mu\nu} e^{i \bm{q} \cdot (\bm{R}_m + \bm{r}_\nu -\bm{r}_\mu)}\, , \label{eq:hami_matrix}\end{aligned}$$ where ${\bm R}_{m}$ is the position of unit cell $m$ with respect to the reference $0$, and $\bm{r}_{\mu}$ is the position of the sublattice $\mu$ within a unit cell. The static structure factor $\mathcal{S}({\bm{q}})$ in this formalism is given as $$\begin{aligned}
\mathcal{S}({\bm{q}}) = & \sum_{\mu, \nu} \langle s^\mu(-\bm{q}) s^\nu(\bm{q}) \rangle \notag = \sum_{\mu, \nu} \left[ \lambda \hat{I} + \beta \hat{H}(\bm{q}) \right]^{-1}_{\mu\nu} \notag \\
= & \sum_{\eta = 1}^{N_{\mathrm{sub}} } \sum_{\mu, \nu} \frac{ [\bm{\psi}^\ast_\eta (\bm{q})]_\mu [\bm{\psi}_\eta (\bm{q})]_\nu }{ \lambda + \beta \varepsilon_\eta(\bm{q}) }, \label{eq:ssf_diag}\end{aligned}$$ where $\varepsilon_\eta(\bm{q})$ and $\bm{\psi}_\eta (\bm{q})$ are, respectively, eigenvalues and eigenvectors of $\hat{H}(\bm{q})$ with a band index $\eta$. Their calculation can be carried out by using the premedial lattices of kagome and pyrochlore, which are respectively the honeycomb and diamond lattices. The main idea of the analytic calculation is to regard the nearest-neighbor (NN) exchange interaction of the original lattice as being mediated by the sites of the premedial lattice located in-between. We describe this method in Appendix A.
### Landau-Lifshitz equation
To investigate the dynamical properties, we numerically solve the following LL equation [@robert2008; @conlon2009; @taillefumier2014]: $$\frac{\partial \bm{S}_i} {\partial t} = - \bm{S}_i \times \bm{H}_{\mathrm{eff},i} , \label{eq:LL}$$ where $\bm{H}_{\mathrm{eff},i}$ is an effective magnetic field at site $i$ given as $$\begin{aligned}
\bm{H}_{\mathrm{eff},i} = & \frac{\partial H}{\partial \bm{S}_i } \notag \\
= & J_1 \left(\sum_{j: \langle i,j \rangle \in {\mathrm{NN} }} \bm{S}_j
\right) + J_2 \left(\sum_{j: \langle i,j \rangle \in {\mathrm{2nd} } } \bm{S}_j \right) \notag \\
+ &J_{3\mathrm{a}} \left(\sum_{j: \langle i,j \rangle \in {\mathrm{3rd,a } }} \bm{S}_j
\right)
+J_{3\mathrm{b}} \left(\sum_{j: \langle i,j \rangle\in {\mathrm{3rd,b} }} \bm{S}_j
\right). \label{eq:eff_field}\end{aligned}$$ In our simulation, we first prepare the initial states, which are well thermalized with temperature $T$, by using single-spin Metropolis updates. Then, we solve Eq. (\[eq:LL\]) by using fourth order Runge-Kutta method. We confirmed the accuracy of this method by checking that conserved quantities, such as the total energy, stay invariant during the simulation time. With this method, we compute $\mathcal{S}({\bm{q}},\omega)$ as $$\begin{aligned}
\mathcal{S}({\bm{q}},\omega) = & \frac{1}{N_t}\sum_{l=0}^{N_{t} } \sum_{n} \sum_{\mu,\nu}
\langle \bm{S}_0^\mu (0) \bm{S}_{n}^\nu (l\delta t) \rangle_{\rm init} \notag \\
\times & e^{i [\omega l \delta t -\bm{q}\cdot(\bm{R}_n+\bm{r}_\nu -\bm{r}_\mu)] },\end{aligned}$$ where $\langle \cdots \rangle_{\rm init}$ represents the sample average of independently-prepared initial states. Numerical details are given in Appendix \[app:MC\].
Fourier-space analysis:\
pinch points, half-moons, and “stars” {#sec:ssf}
=====================================
We focus our attention on the magnetic correlations in the disordered cooperative paramagnetic regime. This is why we shall not go into the details of the low-temperature ordered phases, with the exception of the high-symmetry point $J=1/2$ on pyrochlore in Section \[sec:highsym\], whose nature is particularly enlightening. This approach presents the advantage that, in the cooperative paramagnets, the properties of the kagome and pyrochlore lattices are qualitatively very similar, allowing for a parallel analysis of the two lattices.
The evolution of the correlations in the cooperative paramagnetic regimes are closely linked to the qualitative changes in the band structure obtained by large-$N$ analysis. The discussion in this section relies heavily on the analysis of the low-energy band structure, supported by Monte Carlo simulations at finite temperatures. The outline of the ground-state phase diagram is given in Fig. \[fig:PD\] while the excellent agreement between analytics and numerics is illustrated in Fig. \[fig:SSF\].
To briefly introduce the overall structure of phase diagram (Fig. \[fig:PD\]), the small-$J$ region, which we call region (I), is characterized by the pinch points in the structure factor \[Figs. \[fig:SSF\](a) and \[fig:SSF\](d)\]. Upon increasing $J$, the structure factor shows qualitative changes twice. At intermediate values of $J$, at the beginning of region (II), the structure factor develops a characteristic pattern, which we call “half-moon" after its shape \[Figs. \[fig:SSF\](b) and \[fig:SSF\](e)\]. Further increasing $J$, moving continuously from region (II) to (III), $\mathcal{S}({\bm{q}})$ shows further change into the “star" pattern. Below, we will introduce the nature of each region, separately.
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Flat bands of the Coulomb phase
-------------------------------
The first noticeable outcome of the large-$N$ theory is the persistence of the flat band(s) for all values of $J$ (Figs. \[fig:band\_kagome\] and \[fig:band\_py\]). The flat band persists, as long as $J_2=J_3(=J)$ is satisfied. These flat bands, one for kagome and two for pyrochlore, are well known from the NN model [@garanin1999; @canals2001; @isakov2004]. They represent the Coulomb spin liquid where every unit cell (triangle and tetrahedron) bears zero magnetization, $\{\mathbf{M}_n=0 \;|\;\forall n\}$, and appear in the static structure factor as pinch points. The persistence of the flat bands, and their double degeneracy for pyrochlore, are readily understandable from Eq. (\[Ham\_conservedspin\]) since all configurations of the Coulomb phase with $\mathbf{M}_n=0$ remain degenerate in presence of the farther-neighbor coupling $J$. Mathematically, this persistence takes the form of the exchange matrix $\hat{H}(\bm{q})$ being a polynomial of the NN exchange matrix [@bergman2007] \[see Appendix \[sec:largeNanalytics\], in particular Eqs. (\[eq:2ndhopping\]) and (\[eq:poly\_3\])\]. As such, the two exchange matrices share the same basis of eigenvectors and the flatness of eigenvalues is transmitted from the latter to the former.
Hence, one needs to consider the evolution of the dispersive bands to understand the qualitative changes in the magnetic correlations as a function of $J$.
![Kagome: Top: Heat capacity $C_{h}$ showing the low-temperature bump/kink into the coplanar regime at $T\approx 0.005$ for $J=0$ (black dots) and at $T\approx 0.0004$ for $J_{1c}=1/5$ (colored triangles). Bottom: Temperature evolution of the static structure factor $\mathcal{S}(\mathbf{q})$ for $J=0$ (left) and $J_{1c}=1/5$ (right), obtained by the classical Monte Carlo simulation. []{data-label="fig:ChK"}](kaperChLXJn01Xshort.jpg "fig:"){width="\columnwidth"}\
![Kagome: Top: Heat capacity $C_{h}$ showing the low-temperature bump/kink into the coplanar regime at $T\approx 0.005$ for $J=0$ (black dots) and at $T\approx 0.0004$ for $J_{1c}=1/5$ (colored triangles). Bottom: Temperature evolution of the static structure factor $\mathcal{S}(\mathbf{q})$ for $J=0$ (left) and $J_{1c}=1/5$ (right), obtained by the classical Monte Carlo simulation. []{data-label="fig:ChK"}](kaperSQkL20Jn0Jn02TX.jpg "fig:"){width="\columnwidth"}
![Pyrochlore: (a) Heat capacity $C_{h}$ showing the low-temperature ordering when $J$ is just above $J_{1c}$, but not at $J_{1c}=1/6$, down to $T\sim 10^{-4}$. (b) Static structure factor $\mathcal{S}(\mathbf{q})$ in the \[hhl\] plane at $T=0.005$ showing the pinch points at $J_{1c}$ replaced by very small half moons above $J_{1c}$. []{data-label="fig:ChSqP"}](paperChL8Jn01X.jpg "fig:"){width="0.8\columnwidth"}\
![Pyrochlore: (a) Heat capacity $C_{h}$ showing the low-temperature ordering when $J$ is just above $J_{1c}$, but not at $J_{1c}=1/6$, down to $T\sim 10^{-4}$. (b) Static structure factor $\mathcal{S}(\mathbf{q})$ in the \[hhl\] plane at $T=0.005$ showing the pinch points at $J_{1c}$ replaced by very small half moons above $J_{1c}$. []{data-label="fig:ChSqP"}](paperSQL16Jn01X.jpg "fig:"){width="\columnwidth"}
From pinch points to half-moons, near $J_{1c}$
----------------------------------------------
The flat bands form the ground-state manifold up to $J = J_{1c}=1/5$ for kagome [@li2018] and $1/6$ for pyrochlore. This delimits the region (I) of the phase diagram of Fig. \[fig:PD\]. For $J>J_{1c}$, one of the dispersive bands has a lower energy than the flat bands in parts of the Brillouin zone \[Figs. \[fig:band\_kagome\] and \[fig:band\_py\]\], and the energy minima form a closed line (surface) in a Brillouin zone for a kagome (pyrochlore) lattice \[Fig. \[fig:minima\_kagome\], \[fig:minima\_pyrochlore\]\]. It means that the static structure factor is now dominated by a dispersive band rather than the flat band. As a consequence, the pinch points are smoothed out and their non-analyticality disappears, leaving behind half-moon patterns at the center of the Brillouin zone. The half-moons, and later “star” patterns, are characteristic of the region $J>J_{1c}$, and can be regarded as complementary to the pinch points (see discussion in Sec. \[sec:origin\]).
The kagome NN Heisenberg antiferromagnet ($J=0$) is well known for its Coulomb phase at intermediate temperatures, followed by a coplanar regime at lower temperatures selected by thermal order by disorder [@chalker1992; @huse1992; @reimers1993; @zhitomirsky2008]. The Coulomb phase is marked by a plateau in the heat capacity and pinch points in the structure factor. When the system enters the coplanar regime, the heat capacity exhibits a kink, and sharp peaks of scattering at $\mathbf{q}_{\sqrt{3}}=(4\pi/3,0)$ appear in the structure factor (Fig. \[fig:ChK\]). These peaks represent the onset of the $\sqrt{3}\times\sqrt{3}$ long-range order as $T\rightarrow 0^{+}$ \[\], but are not Bragg peaks since there is no dipolar long-range order at finite temperature.
At $J=J_{1c}$, the softening of the band touching between the lowest dispersive and the flat bands enhances thermal fluctuations compared to $J=0$. As a consequence, the coplanar regime is pushed down to lower temperatures by an order of magnitude (Fig. \[fig:ChK\]). Noticeably, at intermediate temperatures ($T=0.022$), the pinch points visible at $J=0$ have disappeared in favor of the onset of the characteristic half-moons for $J=J_{1c}$. This is why the heat capacity does not show the characteristic plateau of the Coulomb phase at $J_{1c}$.\
As for the pyrochlore lattice, the value of $J_{1c}=1/6$ obtained from large-$N$ is confirmed by simulations down to $T \sim 10^{-4}$ (Fig. \[fig:ChSqP\]). For $J=J_{1c}^{+}$, just above the boundary inside region (II), the system orders, but it remains disordered at the boundary $J_{1c}$. As a consequence, the pinch points of the Coulomb phase are visible up to $J=J_{1c}$, replaced by half-moons as soon as the system enters region (II) \[Fig. \[fig:ChSqP\].(b)\]. Please note that the small thickness of the pinch points for $J=J_{1c}$ is due to the proximity of the half-moon regime at finite temperature.
Origin of the half-moons \[sec:origin\]
---------------------------------------
In region (II), the structure factor develops half-moon patterns \[Fig. \[fig:SSF\](b) and \[fig:SSF\](e)\]. What kind of information can be read from this characteristic magnetic scattering? The intensity of scattering at particular wave vectors $\bm{q}$ is determined by the energy of magnetic modes and the corresponding real-space structure of dominant modes. In our large-$N$ language, they are described by the shape of the energy-minima manifold, $\varepsilon_{\rm min}(\bm{q})$, and the weight of the corresponding eigenfunctions, $\Xi_\eta(\bm{q}) \equiv \sum_{\mu, \nu} [\bm{\psi}^\ast_\eta (\bm{q})]_\mu [\bm{\psi}_\eta (\bm{q})]_\nu $. The weight, $\Xi_\eta(\bm{q})$, satisfies the sum rule: $$\begin{aligned}
\sum_{\eta} \Xi_\eta(\bm{q})= N_{\rm sub},
\label{eq:sum_rule}\end{aligned}$$ due to the normalization of spin length.
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Basically, the high-intensity points of the half-moons follow the position of the energy minima. However, the energy minima do not account for everything. On one hand, the energy-minima manifold, $\varepsilon_{\rm min}(\bm{q})$, is extended in Fourier space. More precisely, it can be defined locally as a hypersurface; a closed line for the two-dimensional kagome and a closed surface for the three-dimensional pyrochlore (see Figs. \[fig:minima\_kagome\] and \[fig:minima\_pyrochlore\], and [Appendices \[sec:appendix\_kagome\] and \[sec:appendix\_pyrochlore\]]{} for more details). On the other hand, it is clear from Figs. \[fig:SSF\](b) and \[fig:SSF\](e) that the extension of the half-moons is finite. They terminate at some point and do not form closed curves as expected from the energy-minima manifold. This vanishing intensity was coined as “ghost” excitations for the kagome Heisenberg antiferromagnet [@robert2008].
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This discrepancy, symbolized by a missing arc that should connect the half-moon pair, can be attributed to the spatial character of the magnetic mode. In Fig. \[fig:ev\_kagome\], we show the intensity maps of $\Xi_\eta (\bm{q})$ for a kagome lattice in the second Brillouin zone, surrounding the wave vector $\left(0, \frac{4\pi}{\sqrt{3}} \right)$. The traditional pinch point resides in the flat band \[Fig. \[fig:ev\_kagome\](c)\]. Half-moons are from the lower-dispersive band \[Fig. \[fig:ev\_kagome\](a)\]. From $\Xi_\eta(\bm{q})$, one finds that the high-intensity regions of flat mode and lower-dispersive mode complement each other. This complementarity is originated in the sum rule, Eq. (\[eq:sum\_rule\]). Around the pinch point, the contribution from the highest-energy band is small, and the sum rule is satisfied only between the flat mode and the lower dispersive mode. The missing arc is attributed to the vanishing weight of lower dispersive band in the bow-tie region, where the flat band contribution is dominant. In this sense, the missing arc of the half-moon can be considered as a shadow of pinch point, thus answering the 10-year old open question about the nature of the “ghost” excitations in the kagome Heisenberg antiferromagnet [@robert2008]. This missing arc signals the proximate presence of a pinch point, and serves as evidence that the system is in the vicinity of a Coulomb phase.
The half-moon formation can be discussed in a more general context not specific to the kagome system. Given the pinch point reflects a singularity of the flat band eigenvectors as a function of momentum, and the completeness of the eigenvector basis, there must be a complementary non-analyticity in (at least) one of the other bands \[Figs. \[fig:illus\] (a) and \[fig:illus\] (b)\]. As these are in general not flat, their constant energy cuts at small radius \[Ref. \] will typically yield the shape of an annulus radially, with an angular modulation characteristic of the longitudinal pinch-point projector \[Figs. \[fig:illus\](c) and \[fig:illus\](d)\]. These combine to yield (a pair of) half moons. Depending on the relative ordering of the bands in energy, these half moon pairs may either appear in the ground state correlations (Fig. \[fig:SSF\]) or in the excitation spectrum (see Sec. \[sec:semiclassical\_dynamics\]).
The same scenario holds for the pyrochlore lattice: the maps are shown in Figs. \[fig:ev\_pyrochlore\](a)-\[fig:ev\_pyrochlore\](c) (centered at $\left[002\right]$) and \[fig:ev\_pyrochlore\](d)-\[fig:ev\_pyrochlore\](f) (centered at $\left[111\right]$). Again, the combination of the energy-minima surface and the intensity map $\Xi_\eta(\bm{q})$ gives rise to the half-moon patterns.
This explanation remains valid throughout the phase diagram for $J>J_{1c}$, and in particular as the half-moons continuously deform into star patterns.
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From half-moons to star patterns for $J > J_{1c}$ {#sec:HMS}
-------------------------------------------------
In the previous subsection, we have seen how the shape of the half-moon is linked to the position of the energy minima in Fourier space. These energy minima continuously move as a function of $J$, and the shape of the half-moon evolves with them, as illustrated in Figs. \[fig:HMK\] and \[fig:HMP\].
![Kagome: evolution from half-moon to star patterns in the static structure factor $\mathcal{S}(\mathbf{q})$ for $J_{1c}<J<J_{2c}$, obtained by Monte Carlo simulation. The simulation temperature is $T=0.32$. []{data-label="fig:HMK"}](kaperSQkL30JnXT032.jpg){width="0.97\columnwidth"}
![Pyrochlore: evolution from half-moon to star patterns in the static structure factor $\mathcal{S}(\mathbf{q})$ for $J_{1c}<J<J_{2c}$, obtained by Monte Carlo simulation. The bottom right panel is below the transition temperature at the high-symmetry point $J=J_{2c}$ with Bragg peaks at $\mathbf{q}_{L}=\left[\frac{1}{2}\frac{1}{2}\frac{1}{2}\right]$. []{data-label="fig:HMP"}](paperSQL20JnXhalfmoon.jpg){width="0.97\columnwidth"}
Upon increasing $J$, the radius of the half-moons increases. Since the radius is limited by the size of the Brillouin zone, neighboring half-moons eventually connect to each other by their extremities, forming star shapes. Please note that while the star shapes are rather obvious for kagome \[see the top panels of Figs. \[fig:SSF\].(c) and \[fig:SSF\](f) and Fig. \[fig:HMK\]\], they are somewhat more figurative for pyrochlore in a \[hhl\] plane \[see the bottom panels of Figs. \[fig:SSF\].(c) amd \[fig:SSF\](f) and Fig. \[fig:HMP\]\]. For convenience, we shall use the name of “star” for both lattices, which shall be understood as the patterns formed by connected half-moons.
The increase of the half-moon radius in Fourier space, $R^{K,P}$, can be calculated analytically thanks to the large-$N$ method $$\begin{aligned}
R^{\mathrm{K}} = \frac{4}{\sqrt{3}} \arccos \sqrt{\frac{1}{8} \left[ \left(\frac{1+J}{2J} \right)^2 -1 \right] },
\label{eq:radiusK}\end{aligned}$$ along the $\Gamma$M direction for kagome, and $$\begin{aligned}
R^{\mathrm{P}} = 2 \arccos \left[ \frac{4J + 1 - 28J^2 }{32J^2} \right],
\label{eq:radiusP}\end{aligned}$$ along the $\Gamma$X direction for pyrochlore. These formulas are in excellent agreement with results obtained from Monte Carlo simulations in the regime of collective paramagnetism, and above any potential transition temperature (Fig. \[fig:radius\]).
![ The evolution of the radius of the half moons in Fourier space (normalized by $2\pi$) for pyrochlore (a) and kagome (b) agrees quantitatively between numerics (red circles) from classical Monte Carlo simulations and analytics (black line) derived from large-$N$ calculations \[Eqs. (\[eq:radiusK\]) and (\[eq:radiusP\])\]. The error bars come from the discretization of the Fourier space in a system of finite size ($L=30$). For kagome, data were taken at $T=0.32$. For pyrochlore, data were taken for a range of temperatures above the transition temperature, $T\in[0.08;0.24]$. []{data-label="fig:radius"}](paperRadiusL30JnX.jpg "fig:"){width="0.8\columnwidth"} ![ The evolution of the radius of the half moons in Fourier space (normalized by $2\pi$) for pyrochlore (a) and kagome (b) agrees quantitatively between numerics (red circles) from classical Monte Carlo simulations and analytics (black line) derived from large-$N$ calculations \[Eqs. (\[eq:radiusK\]) and (\[eq:radiusP\])\]. The error bars come from the discretization of the Fourier space in a system of finite size ($L=30$). For kagome, data were taken at $T=0.32$. For pyrochlore, data were taken for a range of temperatures above the transition temperature, $T\in[0.08;0.24]$. []{data-label="fig:radius"}](kaperRadiusL30JnX.jpg "fig:"){width="0.8\columnwidth"}
Semi-classical dynamics {#sec:semiclassical_dynamics}
=======================
In experiments, signals of anomalous magnetic correlation are sometimes observed in finite-frequency regions, through, e.g., inelastic neutron scattering. For example, it is at finite energy of the kagome Heisenberg antiferromagnet that half-moons were first observed [@robert2008], before being stabilized as signature of the ground state, at low energy, via farther-neighbor interactions [@udagawa2016; @rau2016; @mizoguchi2017]. Accordingly, in order to find the half-moons and stars in a realistic experimental setting, it is desirable to estimate the energy scale of the corresponding magnetic excitations. In the context of the large-$N$ analysis, these magnetic patterns are associated with energy bands (Fig.s \[fig:band\_kagome\] and \[fig:band\_py\]), but this band energy cannot be interpreted as the frequency of experimental probes in itself. A relation connecting them was proposed under the assumption of relaxational dynamics [@conlon2009; @conlon2010]. However, it is not [*a priori*]{} obvious if this assumption holds in our system. To this end, in this section, we address the dynamics of the system by solving the semiclassical LL equation in Eq. (\[eq:LL\]) and calculating the dynamical structure factor $\mathcal{S}(\bm{q},\omega)$. Here, we focus on regions (II) and (III) for both kagome and pyrochlore lattices.
\
Let us first see the results for a kagome lattice. Cuts for several frequencies are shown in Figs. \[fig:llg\_kagome\]. As expected, the characteristic patterns observed for $\mathcal{S}({\bm{q}})$ are obtained in the low-energy sectors in both regions. In region (II), as clearly seen in Fig. \[fig:llg\_kagome\](a), the half-moon signal appears in the quasi-elastic regime: $\omega=0$, showing that this pattern dominates the long-time behavior of magnetic correlation in this region. The pattern sustains with small $\omega$ dependence in the low-frequency region. Upon going to intermediate energy scale comparable to NN coupling, $\omega\sim 1$, the signal smears out. Similarly, in region (III), the star shape appears in the low-frequency part of the dynamical structure factor \[Fig. \[fig:llg\_kagome\](c)\], and it gradually blurs towards higher energy. The remnant of the star pattern remains observable up to higher energy, compared with the vanishing of half-moons in region (II), probably attributed to the growing energy scale of magnetic modes as $J$, as implied by the larger bandwidth obtained in the large-$N$ analysis (Fig.s \[fig:band\_kagome\] and \[fig:band\_py\]).
\
In the Heisenberg antiferromagnet with $J=0$, half-moons have been observed at very low temperature and finite frequency [@robert2008], while pinch points dominate the low-energy physics and are visible in the structure factor (Fig. \[fig:ChK\] and Ref. \[\]). Away from $J=0$, one could have expected the reversed picture: half-moons at $\omega=0$ and pinch points at finite frequency. However, LL dynamics do not show pinch points at any frequency. This is because the LL dynamics is simulated at relatively higher temperatures, where the flat band couples with the dispersive ones and the pinch points are washed out. A clear separation of energies might require a much lower temperature, which is accessible for the Heisenberg antiferromagnet, but not at intermediate and large values of $J$ where simulations, and thus LL dynamics, either order, or are very hard to thermalize.
The same trend is also seen for a pyrochlore lattice, as shown in Fig. \[fig:llg\_pyrochlore\]. The half-moon and the star patterns are clearly visible in each region. The results of both lattices show that the shadow of pinch points can be observed through the excitations in finite-frequency range, i.e., the proximity to Coulomb phase can be captured through the inelastic neutron scattering experiment.
Real-space picture:\
magnetic clustering {#sec:ising}
====================
In this section, we will address the real-space picture, accompanying the characteristic patterns in the structure factors. We will show that half-moons and stars reflect the formation of magnetic clusters. These magnetic clusters can be associated with the cluster of topological charges obtained in the Ising systems, through the analogy of conserved spin introduced in Eq. (\[eq:Mp\]), with the topological charge defined in the Ising system [@udagawa2016; @rau2016; @mizoguchi2017].
Comparison with Ising systems {#sec:Ising}
-----------------------------
The half-moons and stars in $\mathcal{S}({\bm{q}})$ are also seen in the corresponding Ising model [@udagawa2016; @rau2016; @mizoguchi2017]: $$\begin{aligned}
\mathcal{H} & = \sum_{\langle i,j\rangle_{\mathrm{NN} }} \sigma_i^z \sigma_j^z
+J \sum_{\langle i,j\rangle_{\mathrm{2nd} }} \sigma_i^z \sigma_j^z
+J \sum_{\langle i,j\rangle_{\mathrm{3rd,a} }} \sigma_i^z \sigma_j^z. \label{eq:hamiltonain}\end{aligned}$$ For Ising degrees of freedom $\sigma_{i}^{z}$, the Hamiltonian can be re-written in terms of local topological charges on each triangle (for kagome) or tetrahedron (for pyrochlore), $$\begin{aligned}
Q_n = \zeta_n \sum_{i \in n} \sigma_i^z,
\label{eq:Qp}\end{aligned}$$ with $\zeta_n = +(-)1$ for an upward (downward) triangle/tetrahedron. The possible values of charges are $Q_n = \{\pm 3 , \pm 1\}$ for a kagome lattice and $Q_n = \{\pm 4 , \pm 2, 0\}$ for a pyrochlore lattice. The Hamiltonian then becomes [@ishizuka2013; @udagawa2016; @rau2016; @mizoguchi2017] $$\mathcal{H} = \left( \frac{1}{2} - J \right) \sum_n Q_n^2 - J \sum_{\langle n,m \rangle} Q_n Q_m + \mathrm{(const.)}.
\label{eq:hami_charge}$$ The vector field defined in Eq. (\[eq:Mp\]) and the Hamiltonian form of Eq. (\[Ham\_conservedspin\]) were natural extensions of these discrete topological charges to continuous degrees of freedom. We see in Eq. (\[eq:hami\_charge\]) that $J$ couples the NN charges. $J >0 $ means that same-sign charges attract each other.
\
The static structure factors of this Ising model show similar features to those of the Heisenberg model. The half-moons appear for $0 < J < \frac{1}{3}$ on kagome [@mizoguchi2017], and for $J \sim \frac{1}{4}$ on pyrochlore [@udagawa2016; @rau2016]. In both lattices, the origin of the half-moons is to a large extent due to the formation of “hexamers" \[Figs. \[fig:ising\_star\](e) and \[fig:ising\_star\](f)\], which correspond to closed loops made of at least six charges of the same sign. Branches of same-sign charges are then attached to these central hexamers, forming disordered spin-liquid phases made of large clusters of topological charges.
Here, we show that the star patterns discussed in this paper also appear in the large-$J$ region of the Ising models: $J > \frac{1}{3}$ for a kagome lattice \[Fig. \[fig:ising\_star\](a)\] and $J > \frac{1}{4}$ for a pyrochlore lattice \[Fig. \[fig:ising\_star\](b)\]. At lower temperatures, the system orders into phases tiled by small clusters of charges, with a maximal charge at the center, surrounded by smaller charges of the same sign \[Figs. \[fig:ising\_star\](c) and \[fig:ising\_star\](d)\].
From this point of view, the passage from the half-moons to the stars in the static structure factor corresponds to the evolution from a disordered phase made of hexamers to [the ordered phase of]{} smaller clusters centered around a maximal charge. The similarity of half-moons/stars between Ising and Heisenberg models suggests that short-range correlations similar to topological charge clusters also develop in the Heisenberg models, even though topological stability, with a discretized value of the topological charge, cannot be expected for the continuous spin systems. The motivation of the next sections will be to make this idea more quantitative.
Conserved-spin correlator
-------------------------
To characterize the real-space structure in the Heisenberg models, we focus on the conserved spin $\bm{M}_n$ as a vector-field analog of the topological charge $Q_n$ in the Ising models. The momentum-space correlator of $\bm{M}_n$ is defined as $$\begin{aligned}
S_{\bm{M}}(\bm{q}) \equiv \frac{N_{\mathrm{sub}}}{N_{\mathrm{site}}} \sum_{n,m} \langle \bm{M}_n \cdot \bm{M}_m \rangle e^{-i \bm{q} \cdot (\bm{R}_m -\bm{R}_n)},\end{aligned}$$ where $\bm{R}_{n,m}$ is the coordinate at the center of the triangle/tetrahedron where $\bm{M}_{n,m}$ is defined. Within the large-N approximation, $S_{\bm{M}}(\bm{q})$ is represented as $$\begin{aligned}
S_{\bm{M}}(\bm{q}) = &
\frac{N_{\mathrm{sub}}}{N_{\mathrm{site}}}
\sum_{k, k^\prime = \bigtriangleup, \bigtriangledown} \zeta_{k} \zeta_{k^\prime} \sum_{\eta = 1}^{N_{\mathrm{sub}}}
\langle M^{\eta}_{k} (-\bm{q}) M^{\eta}_{k^\prime} (\bm{q}) \rangle \notag \\
= & \sum_{\eta = 1}^{N_{\mathrm{sub}} } \sum_{\mu, \nu} \frac{ [\bm{\psi}^\ast_\eta (\bm{q})]_\mu [\bm{\psi}_\eta (\bm{q})]_\nu }{ [\lambda + \beta \varepsilon_\eta(\bm{q}) ] } \cdot F_{\mu ,\nu}(\bm{q}).\end{aligned}$$ Here, $k= \bigtriangleup, \bigtriangledown$ denotes the direction of triangle/tetrahedron, $\bm{r}_{c} \equiv \frac{1}{N_{\rm sub}} \sum_{\mu} \bm{r}_\mu $ is the coordinate at the center of the upper triangle/tetrahedron, $$M^{\eta}_{k} (\bm{q}) = \sum_{\mu} e^{i \bm{q} \cdot [ \zeta_{k} (\bm{r}_\mu- \bm{r}_c)]} [\psi_\eta(\bm{q})]_{\mu},$$ is the conserved spin of $\eta$-band, and $$F_{\mu ,\nu}(\bm{q}) = \sum_{k, k^\prime = \bigtriangleup, \bigtriangledown } \zeta_{k} \zeta_{k^\prime} e^{i \bm{q} \cdot [ \zeta_{k^\prime} (\bm{r}_\nu- \bm{r}_c) - \zeta_k (\bm{r}_\mu- \bm{r}_c)]},$$ is the additional form factor. Figure \[fig:ccq\_pyrochlore\] shows $S_{\bm{M}}(\bm{q})$ obtained by the large-$N$ approximation. In the region (I), $S_{\bm{M}}(\bm{q})$ becomes very small with decreasing temperature, due to the divergence-free nature of the Coulomb phase. In the large-$N$ sense, the flat band does not contribute to the conserved spin correlator since $M_{k}^{\mu} (\bm{q}) = 0$ for the flat bands. For regions (II) and (III), since $S_{\bm{M}}(\bm{q})$ is written by a linear combination of sublattice-resolved structure factors, it shows the characteristic patterns reminiscent of the static structure factor $\mathcal{S}({\bm{q}})$. While the shadow pinch points are absent due to the additional form factor $F_{\mu ,\nu}^{k,k^\prime}(\bm{q})$, the high-intensity points of $S_{\bm{M}}(\bm{q})$ trace the trajectory of $\varepsilon_{\rm min}(\bm{q})$ in Fourier space.
\
\
\
The real-space correlator, $$S_{\bm{M}}(\bm{R}) \equiv \langle \bm{M}_0 \cdot \bm{M}_{\bm{R}} \rangle,
\label{eq:rsc}$$ is defined on dual honeycomb (diamond) lattice for the kagome (pyrochlore) case, and given by the inverse Fourier transformation of $S_{\bm{M}}(\bm{q})$. We show $S_{\bm{M}}(\bm{R})$ for both cases, in Figs. \[fig:ccr\_kagome\] (kagome) and \[fig:ccr\_pyrochlore\] (pyrochlore). The site $0$ is shown with a white circle. The red (blue) dot means that the correlation $\langle \bm{M}_0 \cdot \bm{M}_{\bm{R}} \rangle$ takes a positive (negative) value, and its radius denotes the rescaled absolute value (see the captions of the figures). Since we are interested in the cooperative, but nonetheless disordered, paramagnetic phase where $\langle \mathbf{M}\rangle=0$, the averaged real-space correlator of Eq. (\[eq:rsc\]) offers an alternative to the spin-configuration snapshot available in the ordered phase of the Ising model. With continuous spins, the real-space representation of this clustering is further complicated by the continuous evolution of the half-moon radius in the phase diagram, which implies an incommensurate wave-vector ordering for generic values of $J$. This issue can, however, be resolved at high-symmetry points of the Hamiltonian, such as exemplified in Sec. \[sec:highsym\].
As shown in Fig. \[fig:ccr\_kagome\] (a), in the region (II), positive correlations develop in NN sites, as can be expected from the $\bm{M}_n$-representation of Hamiltonian \[Eq. (\[Ham\_conservedspin\])\]. Moreover, a noticeable correlation develops beyond n.n sites, especially in surrounding hexagons, implying the clustering of spins reminiscent of the hexamer cluster in the Ising case, made of same-sign charges surrounding a hexagon. In the Heisenberg case, instead of the charge, the conserved spin shows substantial positive correlation around a hexagon.
The qualitative difference of patterns between region (II) with half-moons and region (III) with stars indicates that different types of clusters evolve in these two regions. For region (III) the positive NN correlation is surrounded by the negative correlations, which is reminiscent of the crystalization of double- and triple-charge clusters in the Ising models [@udagawa2016; @mizoguchi2017].
Despite the qualitative similarity of cluster structures between Heisenberg and Ising cases, there is one significant difference. In the Ising case, the cluster shapes are rigidly fixed in regions (II) and (III), respectively, due to the discrete spin nature of Ising spins, and do not change with $J$. Meanwhile in the Heisenberg case, the continuous spins allow continuous modification of clusters, and their amplitudes and cluster sizes also change continuously with $J$. For example, in region (II), the cluster can be considerably long-ranged near the boundary with region (I), as implied by the small half-moon radius in Fourier space \[Fig. \[fig:radius\]\].
Gauge-charge ordering at the high-symmetry point $J=J_{2c}$ on pyrochlore \[sec:highsym\]
-----------------------------------------------------------------------------------------
We next confirm the change of clustering patterns between regions (II) and (III) via Monte Carlo simulations, using the pyrochlore model at the boundary $J=J_{2c}$ as a working example.
What happens at $J_{2c}$ ? Within the large-$N$ approximation, this is where the energy-minima manifold changes its topology, as illustrated in Figs. \[fig:minima\_kagome\] and \[fig:minima\_pyrochlore\]. The manifold moves from enclosing the $\Gamma$ point in region (II) ($J_{1c}<J<J_{2c}$) to the zone corners in region (III) ($J_{2c}<J$). The model at $J_{2c}$ is thus a high-symmetry point of our Hamiltonian. As we will see in this section, it confers to the $J_{2c}$ boundary an advantage of simplicity particularly useful to characterize the low-temperature ordered state.
Since the energy minima cover an extended region in Fourier space, the ordering mechanism is necessarily via thermal order by disorder. For most of the phase diagram when $J>J_{1c}$, the continuous evolution of the energy-minima manifold implies incommensurate order at low temperatures. But, at the high-symmetry model $J_{2c}$, order by disorder selects the $L$ point on the boundary of the Brillouin zone, (h,k,l)$=(1/2,1/2,1/2)$, as suggested by the large-$N$ analysis of Fig. \[fig:band\_py\] and confirmed by Bragg peaks in the Monte Carlo of Fig. \[fig:HMP\] (bottom right panel). The corresponding order parameter $$\begin{aligned}
M_{L}=\left| \dfrac{1}{N}\sum_{i=1}^{N}\mathbf{S}_{i} \textrm{e}^{\imath \mathbf{r}_{i}\cdot \mathbf{q}_{L}}\right|
\label{eq:ML}\end{aligned}$$ displays a clear first-order jump at the transition (Fig. \[fig:J05obs\]). Since there are eight $L$ points for each Brillouin zone with each $L$ point shared between two adjacent Brillouin zones, the saturated value of $M_{L}$ is $2/8=1/4$. Measurements of $M_{L}$ are especially difficult to thermalize below the transition. $M_{L}$ seems to converge towards its saturated value of $1/4$, possibly via a second transition at very low temperature. However, further work is necessary to confirm this point. It is also possible that the order parameter $M_{L}$ does not saturate. Measurements of the quadrupolar order parameter $M_{Q}$, on the other hand, thermalize quite easily to its saturated value of $2/\sqrt{3}$ at zero temperature (see Appendix \[app:Q\] for the definition).
![ First order phase transition at the high-symmetry point $J=J_{2c}$ on pyrochlore, as demonstrated by the discontinuity of (a) the dipolar order parameters $M_{L}$ at wave vector $\mathbf{q}_{L}$ and (b) the quadrupolar order parameter $M_Q$. The black dashed lines indicate the value of saturation for each order parameter. []{data-label="fig:J05obs"}](paperobsLXJn05.jpg){width="\columnwidth"}
{width="18cm"}
In the rest of this section, we will focus on the nature of the ground state. The saturation of the quadrupolar order parameter $M_{Q}$ \[Fig. \[fig:J05obs\].(b)\] indicates that all spins are collinear in the ground state. This collinearity allows to temporarily forget the continuous nature of the classical Heisenberg spins and to consider them as Ising degrees of freedom, pointing either up or down. In analogy with Sec. \[sec:Ising\], each tetrahedron bears an effective charge $Q_{n}=\{0,\pm 2,\pm 4\}$. At $J_{2c}=1/2$, the “chemical potential" (i.e. the first term) of Eq. (\[eq:hami\_charge\]) disappears and the ground-state energy, $E_{GS}$, only comes from the NN coupling between charges $$\begin{aligned}
E_{GS} = - \dfrac{1}{2} \sum_{\langle n,m \rangle} Q_n Q_m .
\label{eq:EGSJ05}\end{aligned}$$ This coupling is attractive between same-sign charges. Because of the staggering of $\zeta_{n}$ between up and down tetrahedra in Eq. (\[eq:Qp\]), this attraction does not give rise to ferromagnetism. The minimization of the energy is done by tiling the lattice with the small clusters of Fig. \[fig:ising\_star\](d): a central charge $+ 4$ or $-4$ surrounded by four charges $+2$ or $-2$, respectively. This state is illustrated in Fig. \[fig:meth\] over four cubic unit cells (32 tetrahedra). In order to avoid contact between charges of opposite sign, each cluster is separated from the other ones by a vacuum (zero-charge tetrahedra). Furthermore, each cluster fits within the eight tetrahedra of a cubic unit cell: one maximal charge $Q_{n}=\pm 4$, four charges $Q_{n}=\pm 2$ and three vacuum tetrahedra $Q_{n}=0$. In order to respect the global neutrality of the system, there must be as many positive as negative clusters. Since the centers of the cubic unit cells form, by definition, a bipartite cubic lattice, the global neutrality is enforced by a staggering arrangement of the clusters, alternatingly positive and negative. The magnetic unit cell of the ground state is made of 32 spins (16 tetrahedra).
This structure is responsible for the peaks at (h,k,l)$=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ in the structure factor of Fig. \[fig:HMP\] (bottom right panel). In the actual simulations, however, the difficulty of thermalization remains at low temperatures, because the order parameter $M_{L}$ does not saturate completely. We believe this is due to the sub-extensive entropy of the ground state, as illustrated in Fig. \[fig:meth\]. We explain the origin of the degeneracy of the $(frac{1}{2},\frac{1}{2},\frac{1}{2})$-state in Appendix \[app:order\].
Summary and discussion {#sec:summary}
======================
We have investigated the magnetic correlations of Heisenberg models with antiferromagnetic farther-neighbor interactions on kagome and pyrochlore lattices in their cooperative paramagnetic regions. For both lattices, we found three distinct patterns of the static structure factor, $\mathcal{S}(\bm{q})$, namely pinch points, half-moons, and stars. We clarified the origin of these patterns by combining the band structure analysis based on a large-$N$ approximation, and Monte Carlo simulations.
Among the above characteristic patterns of the structure factor, the pinch point serves as a direct evidence of a Coulomb phase. Vicinity to a Coulomb phase is signaled by the presence of half-moon patterns. This can be interpreted as complementary to pinch points: they live in the dark regions of the pinch points, and unlike those, generally incorporate a dispersion, so that their radius (distance from the pinch point) changes with energy. As their radius increases, they eventually connect with half moons from neighboring Brillouin zones to generate the star patterns (Figs. \[fig:minima\_kagome\] and \[fig:minima\_pyrochlore\]). Depending on the relative ordering of the bands in energy, these features may even appear in the ground-state correlations.
From a real-space perspective, half-moon and star patterns reflect the formation of magnetic clusters. These clusters involve short-range correlation of the conserved spin, which is analogous to the topological charge defined for the corresponding Ising system. Through this analogy, the half-moon and star cluster can be associated with hexamers and triple-charge clusters obtained in the Ising system, respectively.
The analogy to Ising systems can be extended to the ordering at the high-symmetry point, $J = J_{2c}$ for the pyrochlore system, where the structure of the low-temperature ordered phase can be well understood through the concept of topological charge. In contrast, the difference from the Ising system is found in the rigidity of the cluster structure: While the clusters are rigidly fixed due to the discreteness of the spins in the Ising system, in the Heisenberg system, the cluster shape is flexibly changed upon varying $J$, due to the continuous nature of spin degrees of freedom.
We further addressed the dynamical properties of the model by solving the semiclassical LL equations. As a result, we found that the characteristic half-moon and star patterns appear in the frequency-resolved structure factors, in particular in the low-frequency regime, which means the magnetic clusters dominate the long-time behavior of the dynamics. The patterns in $\mathcal{S}(\bm{q},\omega)$ presented here can be directly accessed through experimental probes, such as inelastic neutron scattering.
In fact, in pyrochlore compounds, several types of magnetic clusterings have been reported. In ZnCr$_2$O$_4$ [@lee2002] and MgCr$_2$O$_4$ [@tomiyasu2008; @tomiyasu2013; @gao2018], six-spin composites dominate the low-energy excitations. While the proposed spatial structure is different from the hexamers obtained in our analysis, our model clearly gives a route to similar clustering around hexagons. It is tempting to point out the possibility that our hexamers may be continuously connected to the low-energy excitations observed for these materials. In this respect, a molybdate pyrochlore material Lu$_2$Mo$_2$O$_5$N$_2$ will also provide an interesting perspective [@clark2014; @iqbal2017]. As for the shadow of pinch points in dispersive bands, it has also been observed in a kagome model for Fe jarosites with Dzyaloshinskii-Moriya and second-neighbor interactions [@chernyshev2015]. Potential connections to half-moon and star patterns in this material have not been investigated yet, and would be an interesting direction to follow. On the other hand, the half-moon signal corresponding to the hexamer-type clusters was recently theoretically proposed for a double-layered kagome material [@pohle2017].
It is also worthwhile to look at another pyrochlore compound, ZnFe$_2$O$_4$. The cluster excitation observed for this compound takes a “dodecamer” form, consisting of 12 spins [@tomiyasu2011_2]. The spatial structure of this excitation is the same as the triple-charge cluster obtained in the region (III) in our model. Interestingly, for this compound, a large farther-neighbor coupling, $J_{3a} > J_1$, is expected [@yamada2002; @kamazawa2003; @tomiyasu2011_2]. A different dodecamer structure reminiscent of the kagome hexamer was also proposed for HgCr$_2$O$_4$ [@tomiyasu2011].
In conclusion, we found new characteristic patterns in magnetic structure factors, complementary to pinch points, which signal the proximity to a Coulomb phase. These patterns signal the formation of magnetic clusters, analogous to the low-energy excitations observed for pyrochlore compounds.
We thank Keisuke Tomiyasu and Jeffrey G. Rau for interesting discussions. This work was supported by the JSPS KAKENHI (Grants No. JP15H05852, No. JP16H04026 and No. JP26400339), MEXT, Japan, and by the Deutsche Forschungsgemeinschaft under Grant No. SFB 1143. TM wishes to thank Max-Planck-Institut fur Physik komplexer Systeme, where part of the present work was done. L.D.C.J. acknowledges support from the University of Bordeaux (IdEx BIS) and from the TQM Unit of the Okinawa Institute of Science and Technology Graduate University, as well as hospitality from Gakushuin University in Tokyo. Part of numerical calculations were carried out on the Supercomputer Center at Institute for Solid State Physics, University of Tokyo.
*Note added:* Recently, we learned of a parallel study by Yan [*et al*]{}., which reports complementary results for a different model [@Yan2018].
Analytical formula for $\mathcal{S}(\bm{q})$ in large-$N$ analysis {#sec:largeNanalytics}
==================================================================
In this appendix, we describe the derivation of $\mathcal{S}(q)$ in large-$N$ approximation, in detail. The $J_1$-$J_2$-$J_3$ model has two special properties at $J_2=J_3\equiv J$, which enables us simple analytical approach. One is the polynomial expression of Hamiltonian in terms of incident matrices, and the other is the line graph correspondence. With the help of graph-theoretical argument, we can construct a simple and systematic way to obtain the analytical expression of $\mathcal{S}(\bm{q})$ in Eq. (\[eq:ssf\_diag\]).
Polynomial expression {#sec:polynomial}
---------------------
We consider kagome and pyrochlore lattices on the same footing, and start with introducing an $N\times N$ incident matrix, $\hat{\delta}^{(1)}\equiv\hat{h}$, where we write $N = N_{\rm site}$ for brevity. Each row $j$, and column indices $j'$ correspond to the sites of the lattice, and the matrix element takes $$\begin{aligned}
[\hat{\delta}^{(1)}]_{jj'} = [\hat{h}]_{jj'} = \left\{\begin{array}{ll}
1 & {\rm if}\ j\ \&\ j' {\rm are\ connected}\\
%
0 & {\rm otherwise}
\end{array}\right.\end{aligned}$$ The Hamiltonian matrix can be expressed as $\hat{H}=\hat{h}$ at $J=0$, supposing $J_1=1$ and $J_2=J_3=J$.
Generalizing $\hat{\delta}^{(1)}$, we introduce a matrix, $\hat{\delta}^{(n)}$, so that the element $[\hat{\delta}^{(n)}]_{jj'}$ takes $1$, if and only if the two sites, $j$ and $j'$ are $n$ Manhattan distance away, and otherwise, $0$. Note that for the kagome and pyrochlore lattices, the Manhattan distance is the minimal number of NN bonds necessary to connect two sites.
Since the squared incident matrix $(\hat{\delta}^{(1)})^2$ connects any two sites, where one can be reached from the other in two hoppings, one can obtain $$\begin{aligned}
(\hat{\delta}^{(1)})^2 = z\hat{I}_{N \times N} + x \hat{\delta}^{(1)} + \hat{\delta}^{(2)}, \label{eq:2ndhopping}\end{aligned}$$ with $z=2(N_{\rm sub} - 1)$ is a number of coordination, and $x = N_{\rm sub} - 2$ is a number of paths through which, starting from a site, one reaches a NN site of that site by two other NN moves. Hereafter, $\hat{I}_{\ell \times \ell }$ represents the $\ell \times \ell$ identity matrix.
Obviously, $\hat{\delta}^{(2)}$ corresponds to the part of Hamiltonian matrix describing the second- and third-neighbor interactions, so we can express $$\begin{aligned}
\hat{H} &= \hat{\delta}^{(1)} + J\hat{\delta}^{(2)} = \hat{\delta}^{(1)} + J(\hat{\delta}^{(1)})^2
- xJ \hat{\delta}^{(1)}
- zJ\hat{I}_{N \times N}
\nonumber\\
%
&= (1-xJ) \hat{h} + J\hat{h}^2 - zJ \hat{I}_{N \times N}. \label{eq:poly_1}\end{aligned}$$ Now the Hamiltonian matrix is expressed as a polynomial of incident matrix, $\hat{h}$, the eigenvalue problem of $\hat{H}$ is reduced to that of $\hat{h}$.
Dual lattice {#sec:duallattice}
------------
To solve the eigenvalue problem of $\hat{h}$, it is convenient to introduce dual lattice. For clarity, we focus on a kagome lattice, first. We start with constructing an intermediate lattice, by placing new sites on the centers of triangles, and connecting the new sites and neighboring old sites, while erasing the original bonds of kagome lattice. Secondly, from this intermediate lattice, we erase the original sites of the kagome lattice, and obtain a honeycomb lattice as a dual lattice \[Fig. \[fig:duallattice\](a)\]. As a dual lattice of pyrochlore lattice, we obtain a diamond lattice in a similar way \[Fig. \[fig:duallattice\](b)\].
The dual lattice shares the same unit cell as the original lattice. Below, we adopt the following lattice conventions. For a kagome lattice, as lattice vectors, we choose $\bm{a}^{(\mathrm{K})}_1 = (1,0)$ and $\bm{a}^{(\mathrm{K})}_2 = \left( \frac{1}{2},\frac{\sqrt{3}}{2} \right)$, and as the coordinates of three sublattices, 1,2, and 3, $\bm{r}^{(\mathrm{K})}_1 = \left( 0,0 \right)$, $\bm{r}^{(\mathrm{K})}_2 = \left( \frac{1}{4} , \frac{\sqrt{3}}{4} \right)$, $\bm{r}^{(\mathrm{K})}_3 = \left( \frac{1}{2} , 0 \right)$. Accordingly, the coordinates of two sublattices, A and B of the dual honeycomb lattice are $\bm{r}^{(\mathrm{H})}_{\mathrm{A}} = \left( \frac{5}{4}, \frac{5\sqrt{3}}{12}\right)$ and $\bm{r}^{(\mathrm{H})}_{\mathrm{B}} = \left( \frac{1}{4} , \frac{\sqrt{3}}{12} \right)$. For a pyrochlore lattice, the lattice vectors are $\bm{a}^{\mathrm{P}}_1 = \left(0, \frac{1}{2} , \frac{1}{2} \right)$, $\bm{a}^{\mathrm{P}}_2 = \left( \frac{1}{2} , 0, \frac{1}{2} \right)$, $\bm{a}^{\mathrm{P}}_3 = \left( \frac{1}{2} , \frac{1}{2}, 0 \right)$, and the positions of four sublattices are $\bm{r}^{\mathrm{P}}_1 = \left(0,0,0 \right)$, $\bm{r}^{\mathrm{P}}_2 = \left(0,\frac{1}{4},\frac{1}{4} \right)$, $\bm{r}^{\mathrm{P}}_3 = \left(\frac{1}{4},0,\frac{1}{4}\right)$, $\bm{r}^{\mathrm{P}}_4 = \left(\frac{1}{4},\frac{1}{4},0 \right)$. For the dual diamond lattice, the coordinates of two sublattices (A and B) are $\bm{r}^{\mathrm{D}}_A = \left( \frac{1}{8} , \frac{1}{8} , \frac{1}{8} \right)$, and $\bm{r}^{\mathrm{D}}_B = \left( \frac{7}{8} , \frac{7}{8} , \frac{7}{8} \right)$.
\
Line-graph correspondence \[sec:linegraph\] {#sec:linegraph}
-------------------------------------------
Here, let us apply the idea of dual lattice to solve the eigenvalue problem of $\hat{h}$. Here, we focus on a kagome lattice, again. First, we look at the intermediate lattice we have introduced in the previous subsection. On this graph, we introduce $N\times N_{\rm H}$ rectangular matrix, $\hat{h}^{\rm K\leftarrow H}$, whose $N$ rows correspond to sites on a kagome lattice, and $N_{\rm H}$ columns correspond to the sites of a honeycomb lattice. We define $\hat{h}^{\rm K\leftarrow H}$ as an incident matrix for the intermediate lattice, i.e., we set $$\begin{aligned}
[\hat{h}^{\rm K\leftarrow H}]_{jl} = \left\{\begin{array}{ll}
1 & {\rm if}\ j\ \&\ l\ {\rm are \ connected}\\
0 & {\rm otherwise}
\end{array}\right.\end{aligned}$$ And we define $N_{\rm H}\times N$ rectangular matrix, $\hat{h}^{\rm H\leftarrow K}$ as $\hat{h}^{\rm H\leftarrow K} = (\hat{h}^{\rm K\leftarrow H})^t$.
The key step to solve the eigenvalue problem is the observation that the matrix $\hat{h}$ can be written as a product of $\hat{h}^{\rm H\leftarrow K}$ and $\hat{h}^{\rm K\leftarrow H}$: $$\begin{aligned}
\hat{h} = \hat{h}^{\rm K\leftarrow H}\hat{h}^{\rm H\leftarrow K} - 2\hat{I}_{N \times N}.
\label{eq:Hamiltonian_decomposition}\end{aligned}$$ The form (\[eq:Hamiltonian\_decomposition\]) immediately tells us significant information on the energy spectrum of $\hat{h}$. For $N\times M$ matrix $\hat{A}$ and $M\times L$ matrix $\hat{B}$, it is known that
- rank$\hat{A}$ $\leq$ min{$N$, $M$}
- rank$\hat{A}\hat{B}$ $\leq$ min{rank$\hat{A}$, rank$\hat{B}$}
Applying these properties to $\hat{h}^{\rm K\leftarrow H}$ and $\hat{h}^{\rm H\leftarrow K}$, we obtain $$\begin{aligned}
{\rm rank}(\hat{h}^{\rm K\leftarrow H}\hat{h}^{\rm H\leftarrow K})\leq N_{\rm H} = (2/3)N.\end{aligned}$$ This inequality results in the existence of at least $N-N_{\rm H}=\frac{1}{3}N$ zero modes, i.e. $\hat{h}$ has $\frac{1}{3}N$ eigenstates with degenerate eigenenergy, $-2$. Moreover, the matrix $\hat{h}^{\rm K\leftarrow H}\hat{h}^{\rm H\leftarrow K}$ and the inverse product, $\hat{h}^{\rm H\leftarrow K}\hat{h}^{\rm K\leftarrow H}$ share the common non-zero eigenvalues. Accordingly, given that the incident matrix of honeycomb lattice is given by $$\begin{aligned}
\hat{h}^{\rm H} = \hat{h}^{\rm H\leftarrow K}\hat{h}^{\rm K\leftarrow H} - N_{\rm sub} \hat{I}_{N^{\mathrm{H}} \times N^{\mathrm{H}}}, \label{eq:poly4}\end{aligned}$$ the eigenspectrum of $\hat{h}$ consists of $N_{\rm H}$ eigenvalues of $\hat{h}^{\rm H} + (N_{\rm sub} - 2)\hat{1}$, and $N-N_{\rm H}$-fold degenerate modes with eigenvalue, $-2$.
Momentum-space expression \[sec:mom\]
-------------------------------------
The translational invariance of the Hamiltonian matrix allows us to block-diagonalize it with respect to the momentum $\bm{q}$. For each $\bm{q}$, we obtain $N_{\rm sub}\times N_{\rm sub}$ Hamiltonian matrix $\hat{H}(\bm{q})$ defined in Eq. (\[eq:hami\_matrix\]). Due to the polynomial expression in Eq. (\[eq:2ndhopping\]), we obtain $$\hat{h}_2 (\bm{q}) = [\hat{h}_1(\bm{q})]^2 - x \hat{h}_1(\bm{q}) - z \hat{I}_{N_{\rm sub}\times N_{\rm sub}}, \label{eq:poly_3}$$ with $$\begin{aligned}
[\hat{h}_i (\bm{q})]_{\mu \nu} = \sum_{m} [\delta^{(i)}]_{(0,\mu),(m,\nu)} e^{-i \bm{q} \cdot (\bm{R}_m + \bm{r}_\nu -\bm{r}_{\mu})}. \end{aligned}$$ Therefore, $\hat{H}(\bm{q})$ is also expressed as a polynomial of the Fourier transformation of $\hat{h}_1$: $$\begin{aligned}
\hat{H}(\bm{q}) = (1-xJ) \hat{h}_1(\bm{q}) + J [\hat{h}_1(\bm{q})]^2 - z J \hat{I}_{N_{\rm sub}\times N_{\rm sub}}, \label{eq:poly_2}\end{aligned}$$ From Eq. (\[eq:poly\_2\]), we see the eigenvalue problem for $\hat{H}(\bm{q})$ is reduced to that for $\hat{h}_1(\bm{q})$. To solve it, one can utilized the momentum space version of Eq. (\[eq:Hamiltonian\_decomposition\]), namely $$\begin{aligned}
\hat{h}_1 (\bm{q}) = \hat{h}^{\rm K\leftarrow H} (\bm{q}) \hat{h}^{\rm H\leftarrow K} (\bm{q}) - 2\hat{I}_{N_{\rm sub}\times N_{\rm sub}}. \label{eq:hami_decomp_2}\end{aligned}$$ Here $\hat{h}^{\rm K\leftarrow H} (\bm{q}) $ is $ N^{\rm H}_{\rm sub} \times N_{\rm sub}$ matrix: $$\begin{aligned}
[\hat{h}^{\rm K\leftarrow H} (\bm{q})]_{\mu,\nu} = \sum_{m}^{N^{\mathrm{H} }_{\rm u.c.}}
[\hat{h}^{\rm K\leftarrow H} ]_{(0,\mu) (m,\nu)} e^{-i \bm{q} \cdot (\bm{R}_{m} + \bm{r}_\nu^{\mathrm{H} } - \bm{r}_{\mu})}, \end{aligned}$$ and $$\begin{aligned}
\hat{h}^{\rm H \leftarrow K} (\bm{q}) = [\hat{h}^{\rm K \leftarrow H}(\bm{q}) ] ^{\dagger}. \end{aligned}$$ Using Eq. (\[eq:hami\_decomp\_2\]) as well as the momentum-space analog of Eq. (\[eq:poly4\]), one can show that the eigenvalues of $\hat{h}_1 (\bm{q})$ consists of eigenvalues of $\hat{h}^{\rm H} (\bm{q})$ (up to a constant) and $(N_{\rm sub} - N_{\rm sub}^{\mathrm{H}})$ flat mode with the eigenvalue $-2$; $\hat{h}^{\rm H} (\bm{q})$ is given by $$\begin{aligned}
[\hat{h}^{\rm H} (\bm{q})]_{\mu,\nu} = \sum_{m}^{N_{\rm u.c.}^{\rm H}} [\hat{h}^{\rm H}]_{(0,\mu),(m,\nu)} e^{-i \bm{q} \cdot (\bm{R}_m + \bm{r}^{\rm H}_\nu -\bm{r}^{\rm H}_{\mu})}. \end{aligned}$$ Note that the similar relations hold in the case of a pyrochlore lattice.
Kagome lattice \[sec:appendix\_kagome\]
---------------------------------------
Utilizing the above idea, we now show the explicit forms of the eigenvalues and eigenvectors of our model. Let us first consider a kagome lattice. As discussed in the previous section, the exchange matrix on a kagome lattice is expressed as $$\hat{H}^{\mathrm{K}}(\bm{q}) = \hat{h}^{\mathrm{K}}_1 (\bm{q}) + J \hat{h}^{\mathrm{K}}_2 (\bm{q}), \label{eq:exK}$$ with
$$\begin{aligned}
[\hat{h}^{\mathrm{K}}_1 (\bm{q})]_{\mu \nu} =\left\{ \begin{array}{ll}
2 \cos\bm{q}\cdot (\bm{r}^{\mathrm{K}}_\mu-\bm{r}^{\mathrm{K}}_\nu)&(\mu \neq \nu)\\
0&(\mu= \nu)\\
\end{array} \right.\end{aligned}$$
$$\begin{aligned}
[\hat{h}^{\mathrm{K}}_2 (\bm{q})]_{\mu \nu} = \left\{ \begin{array}{ll}
2 \cos \bm{q} \cdot \left(\sum_{\rho \neq \mu, \nu} \bm{r}^{\mathrm{K}}_\mu + \bm{r}^{\mathrm{K}}_\nu -2\bm{r}_\rho^{\mathrm{K} } \right) & (\mu \neq \nu) \\
2 \cos \bm{q} \cdot \left[ 2 \sum_{\rho \neq \mu } (\bm{r}^{\mathrm{K}}_\rho - \bm{r}^{\mathrm{K}}_\mu) \right] &(\mu = \nu) \\
\end{array} \right.
%$}
\nonumber \\\end{aligned}$$
To obtain the eigenvalues of $\hat{h}^{\mathrm{H}}_{\mathrm{1} }(\bm{q} )$, we first write down the exchange matrix on a dual honeycomb lattice: $$\hat{h}^{\mathrm{H}}_{\mathrm{1} }(\bm{q} )
= \left(
\begin{array}{cc}
0& G(\bm{q})\\
G^{\ast}(\bm{q}) &0\\
\end{array}
\right),$$ with $$G(\bm{q}) = e^{i\frac{q_y}{\sqrt{3}}} + 2 e^{-i\frac{q_y}{2 \sqrt{3}}} \cos \frac{q_x}{2},$$ The eigenvalues of $\hat{h}^{\mathrm{H}}_{\mathrm{1} }(\bm{q} )$ are given by $\varepsilon_{\pm}^{\rm(H)} (\bm{q}) = \pm |G(\bm{q})|$ and the corresponding eigenvectors are $$\bm{\psi}^{\mathrm{H} (\pm)} (\bm{q} ) = \frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\pm e^{i\theta_G(\bm{q})/2} \\
e^{-i\theta_G(\bm{q})/2} \\
\end{array}
\right),$$ with $\theta_G(\bm{q}) = \mathrm{arg} G(\bm{q})$.
Then, we immediately obtain the eigenvalues and eigenvectors of (\[eq:exK\]) in the following manner. First, the eigenvalues of $h^{\mathrm{K}}_{\mathrm{1} }(\bm{q} )$ are identical with those of $h^{\mathrm{H}}_{\mathrm{1} }(\bm{q})$ up to a constant, and thus, using Eq. (\[eq:poly\_3\]) and the fact that $z=4, x=1$ for a kagome lattice, the two eigenvalues of (\[eq:exK\]) are obtained as $$\varepsilon^{\mathrm{K}}_\pm(\bm{q}) = J |G(\bm{q})|^2 \pm (1+J) |G(\bm{q})| + 1-4J. \label{eq:kagome_eigenvalue}$$ Next, to obtain the corresponding eigenvectors, we consider a rectangular matrix $\hat{h}^{\mathrm{K} \rightarrow \mathrm{H} } (\bm{q})$ as we have discussed int Sec. \[sec:linegraph\]. Its explicit form is given as $$\hat{h}^{\mathrm{K} \rightarrow \mathrm{H} } (\bm{q}) =
\left(
\begin{array}{cc}
e^{i \varphi_1} & e^{-i \varphi_1}\\
e^{i \varphi_2} & e^{-i \varphi_2}\\
e^{i \varphi_3} & e^{-i \varphi_3}\\
\end{array}
\right),$$ with $\varphi_1 = \frac{q_x}{4} + \frac{q_y}{4\sqrt{3}}$, $\varphi_2 =- \frac{q_y}{2\sqrt{3}}$,and $\varphi_3 = - \frac{q_x}{4} + \frac{q_y}{4\sqrt{3}}$. Then, the eigenvectors are obtained as $$\bm{\psi}^{\mathrm{K}}_\pm (\bm{q}) = \frac{ h^{\mathrm{K} \rightarrow \mathrm{H}}(\bm{q} ) \bm{\psi}^{\mathrm{H}}_\pm (\bm{q}) }{|| h^{\mathrm{K} \rightarrow \mathrm{H}}(\bm{q} ) \bm{\psi}^{\mathrm{H} }_\pm(\bm{q}) || }.$$ Note that the remaining flat mode is orthogonal to two dispersive modes, and its eigenvalue is $-2(1-J)$.
[*Energy minima.-*]{} We now obtain the eigenvalues of (\[eq:exK\]), so let us discuss the properties of obtained band structure. The energy minima of the lower dispersive band can be found by solving $$\frac{\partial \varepsilon^{\mathrm{K}}_-(\bm{q})}{\partial q_x} = \frac{\partial \varepsilon^{\mathrm{K}}_-(\bm{q})}{\partial q_y} = 0, \label{eq:minimacond}$$ By using the expression (\[eq:kagome\_eigenvalue\]), one obtains $$\frac{\partial \varepsilon^{\mathrm{K}}_-(\bm{q})}{\partial q_i} = \left[ 2J |G(\bm{q})| -(1+J) \right] \frac{\partial |G(\bm{q})| }{\partial q_i}.$$ Therefore, (\[eq:minimacond\]) is satisfied when
1. $|G(\bm{q})| = \frac{1+J}{2J} $,
2. $\frac{\partial |G(\bm{q})|}{\partial q_x} = \frac{\partial |G(\bm{q})|}{\partial q_y} =0 $.
Condition (ii) is satisfied at $\Gamma$, $K$, and $M$ points, but they do not become energy minima. So let us examine (i). Notice that the solution of (i) in $\bm{q}$ space forms lines, rather than a set of discrete points. The solution evolves as follows. First, when $0\leq J\leq \frac{1}{5}$, (i) does not have solutions: in this region the flat band has the lowest energy and the static structure factors is determined by a flat band, which gives rise to pinch points in $\mathcal{S}(\bm{q})$. Second, when $\frac{1}{5} \leq J \leq 1$, the solution is given by a closed path enclosing $\Gamma$ point \[see Fig. \[fig:minima\_kagome\](a)\]. Finally, when $J \geq 1$, the solution is given by a closed path enclosing $K$ points \[see Fig. \[fig:minima\_kagome\](b)\]. As we have seen in the main text, the shape of energy minima is reflected to the characteristic shape of the static structure factor, namely, half-moons and stars.
[*[Derivation of phase boundaries.- ]{}*]{}
\
The change of topology of energy minima surface can be detected by looking at $\Gamma$M line. On $\Gamma$M line, we can parametrize the momentum as $\bm{q} = (Q, \frac{Q}{\sqrt{3}}) $ with $0\leq Q \leq \pi$. Then, $|G(\bm{q})|$ is given by $$|G(\bm{q})| = \sqrt{8\cos^2 \frac{Q}{2} +1}, \label{eq:min_sol_K}$$ and the solution of (i) is then given by $$Q = 2 \cos^{-1} \sqrt{\frac{1}{8} \left[ \left( \frac{1+J}{2J} \right)^2-1 \right]}. \label{eq:sol_K}$$ Figure \[fig:boundary\_K\] shows $u^{\mathrm{K}} \equiv \sqrt{\frac{1}{8} \left[ \left( \frac{1+J}{2J} \right)^2-1 \right]}$ as a function of $J$. In order that $Q$ is between 0 and $\pi$, $u^{\mathrm{K}}$ has to be between $0$ and 1, which is represented by the shade. One can see that the lower bound is $J=\frac{1}{5}$, and the upper bound is $J=1$, which correspond to $J_{1c}$ and $J_{2c}$, respectively.
[*Nearly isotropic nature of half-moon.-*]{} The energy dispersion of the lower-dispersive band around $\Gamma$ point is isotropic. Indeed, $\varepsilon^{\mathrm{K}}_-(\bm{q})$ can be expanded around $\Gamma$ point as $$\begin{aligned}
\varepsilon^{\mathrm{K}}_-(\bm{q}) \sim & -2(1-J) + \left( \frac{1}{4} - \frac{5J}{4}\right) q^2 +
\left(- \frac{1}{192} +\frac{17J}{192} \right) q^4 \notag \\
+ & \mathcal{O}(q_i^5), \label{eq:en_expand}\end{aligned}$$ with $q =\sqrt{q_x^2 +q_y^2}$. Equation (\[eq:en\_expand\]) shows that the energy dispersion is isotropic up to the order of $q^4$, and this leads to nearly circular shape of the energy-minima surface in region (II) \[see Fig. \[fig:minima\_kagome\](a)\].
Pyrochlore lattice \[sec:appendix\_pyrochlore\]
-----------------------------------------------
The same method can be applied to a pyrochlore lattice, so here we outline the calculations. We consider the exchange matrix on a original pyrochlore lattice: $$\hat{H}^{\mathrm{P}}(\bm{q}) = \hat{h}^{\mathrm{P}}_1 (\bm{q}) + J \hat{h}^{\mathrm{P}}_2 (\bm{q}), \label{eq:exP}$$ where $$\begin{aligned}
[\hat{h}^{\mathrm{P}}_1 (\bm{q})]_{\mu \nu} = \left\{
\begin{array}{ll}
2 \cos \bm{q}\cdot (\bm{r}^{\mathrm{P} }_\mu -\bm{r}^{\mathrm{P} }_\nu) & (\mu \neq \nu) \\
0& (\mu = \nu) \\
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
[\hat{h}^{\mathrm{P}}_2 (\bm{q})]_{\mu \nu} = \left\{
\begin{array}{ll}
2 \cos \bm{q}\cdot \left( \sum_{\rho \neq \mu, \nu}
\bm{r}^{\mathrm{P} }_\mu +\bm{r}^{\mathrm{P} }_\nu - 2 \bm{r}^{\mathrm{P} }_\rho \right) & (\mu \neq \nu) \\
2 \cos \bm{q} \cdot \left[2\sum_{\rho \neq \mu}
(\bm{r}^{\mathrm{P} }_\mu-\bm{r}^{\mathrm{P} }_\rho) \right]& (\mu = \nu) \\
\end{array}
\right. \nonumber \\\end{aligned}$$ The polynomial form of the Hamiltonian (\[eq:exP\]) with respect to $\hat{h}^{\mathrm{P}}_{\mathrm{1} } (\bm{q})$ is obtained as $$\hat{h}^{\mathrm{P}}_{\mathrm{2} } (\bm{q} ) = [\hat{h}^{\mathrm{P}}_{\mathrm{1} } (\bm{q} )]^2-2\hat{h}^{\mathrm{P}}_{\mathrm{1} } (\bm{q} ) -6\hat{I}_{4\times 4}, \label{eq:pnnn}$$ since $z=6$ and $x=2$ for a pyrochlore lattice.
Next, the exchange matrix for the dual diamond lattice is given as $$\hat{h}^{\mathrm{D}}_{\mathrm{1} } (\bm{q} )
= \left(
\begin{array}{cc}
0& g(\bm{q})\\
g^{\ast}(\bm{q}) &0\\
\end{array}
\right),$$ with $$g(\bm{q}) = e^{-i\frac{q_x+q_y+q_z}{4}} + e^{i\frac{q_x+q_y-q_z}{4}} + e^{i\frac{q_x-q_y+q_z}{4}} + e^{i\frac{-q_x+q_y+q_z}{4}}.$$ Its eigenvalues are $\varepsilon_{\pm}^{\rm (P)} (\bm{q}) = \pm |g(\bm{q})|$, and the corresponding eigenvectors are $$\bm{\psi}^{\mathrm{D}}_\pm(\bm{q}) =
\frac{1}{\sqrt{2}}
\left(
\begin{array}{c}
\pm e^{i\theta_g (\bm{q})/2} \\
e^{-i\theta_g (\bm{q})/2}
\end{array}
\right),$$ with $\theta_g(\bm{q}) = \mathrm{arg}\ g(\bm{q})$.
Then, using the argument in Sections \[sec:linegraph\] and \[sec:mom\], we obtain the eigenvalues of the Hamiltonian (\[eq:exP\]) as $$\varepsilon^{\mathrm{P}}_\pm (\bm{q}) = J |g(\bm{q})|^2 \pm (1+2J) |g(\bm{q})| + 2 -6J.$$ The corresponding eigenvectors are given as $$\bm{\psi}^{\mathrm{P}}_{\pm}(\bm{q}) = \frac{ h^{\mathrm{P} \leftarrow \mathrm{D} }(\bm{q}) \bm{\psi}^{\mathrm{D}}_\pm(\bm{q})}{|| h^{\mathrm{P} \leftarrow \mathrm{D} }(\bm{q}) \bm{\psi}^{\mathrm{D}}_\pm(\bm{q}) ||},$$ where $\hat{h}^{\mathrm{P} \leftarrow \mathrm{D}} (\bm{q}) $ is a rectangular matrix $$\hat{h}^{\mathrm{P} \leftarrow \mathrm{D}}(\bm{q})
=
\left(
\begin{array}{cc}
e^{i\phi_1} & e^{-i\phi_1} \\
e^{i\phi_2} & e^{-i\phi_2} \\
e^{i\phi_3} & e^{-i\phi_3} \\
e^{i\phi_4} & e^{-i\phi_4} \\
\end{array}
\right),$$ and $\phi_1 = \frac{q_x + q_y + q_z}{8}$, $\phi_2=\frac{q_x - q_y - q_z}{8} $, $\phi_3= \frac{-q_x + q_y - q_z}{8}$, and $\phi_4= \frac{-q_x - q_y + q_z}{8}$. The rest of eigenvectors, i.e. two flat modes, are orthogonal to $\bm{\psi}^{\mathrm{P}, \pm} (\bm{q})$ and their eigenenergy is $-2(1-J)$.
[*Energy minima.-*]{} The minima of $\varepsilon^{\mathrm{P}}_- (\bm{q})$ is obtained by solving $$\frac{\partial \varepsilon^{\mathrm{P}}_- (\bm{q}) }{\partial q_i }
= [2J |g(\bm{q})| -(1+2J)] \frac{\partial |g( \bm{q})|}{\partial q_i}
= 0. \label{eq:min_P}$$ (\[eq:min\_P\]) is satisfied when
1. $|g(\bm{q})| = \frac{1+2J}{2J}$,
2. $\frac{\partial |g(\bm{q})|}{\partial q_x} = \frac{\partial |g(\bm{q})|}{\partial q_y} =\frac{\partial |g(\bm{q})|}{\partial q_z} =0 $.
Again (ii) is satisfied at several high-symmetry points, which turn out not to be energy minima, so let us focus on (i). When $0\leq J \leq \frac{1}{6}$, (i) does not have solutions, the lowest-energy band in this region is the flat band. Then, for $ \frac{1}{6} \leq J \leq \frac{1}{2} $, the solution is a surface enclosing $\Gamma$ point \[Fig. \[fig:minima\_pyrochlore\](a)\]. Finally, for $J \geq \frac{1}{2}$, a solution is a surface enclosing the zone corners \[Fig. \[fig:minima\_pyrochlore\](b)\].
[*Derivation of the phase boundaries.-*]{}
Similar to the case of a kagome lattice, the phase boundaries for a pyrochlore lattice are determined by the presence/absence of the energy minima on $\Gamma$L line. On $\Gamma$L line, the momentum is parametrized as $\bm{q} = (Q,Q,Q) $ with $0\leq Q \leq \pi$. Then the condition for the energy minima is given by $$|g(\bm{q})| = \sqrt{6\cos Q +10}, \label{eq:sol_min_P}$$ and its solution of (i) is $$Q = 2 \cos^{-1} \left( \frac{-12J^2 +4J-1}{48 J^2} \right).$$ We plot $u^{\mathrm{P}} \equiv \frac{-12J^2 +4J-1}{48 J^2 }$ in Fig. \[fig:boundary\_P\]. We again examine the condition that $Q$ is between 0 and $\pi$ (a shaded area of Fig. \[fig:boundary\_P\]), and find that the lower (upper) bound is $J=\frac{1}{6} \left(\frac{1}{2}\right)$.
Monte Carlo simulations {#app:MC}
=======================
Monte Carlo simulations are performed on systems of classical O(3) spins on the kagome and pyrochlore lattices, whose system sizes are respectively $12L^{2}$ and $16L^{3}$ sites. To decorrelate the system, we use jointly the heatbath method, over-relaxation and parallel tempering. Thermalization is made in two steps: first a slow annealing from high temperature to the temperature of measurement $T$ during $t_{e}$ Monte Carlo steps (MCs) followed by $t_{e}$ MCS at temperature $T$. After thermalization, measurements are done every 10 MCs during $t_{m}=10 t_{e}$ MCs. All temperatures are given in units of $J_{1}=1$. The details of each simulation are as follows:
- Fig. \[fig:SSF\]: $L=30$ for both lattices and $t_{m}=10^{5},10^{6}$ MCs for the pyrochlore and kagome lattice respectively.
- Fig. \[fig:ChK\], top: $t_{m}=10^{6}$ MCs, and the error bars are coming from an average over $n$ runs with different initial configurations, where $n=50$ for $L<15$ and $n=20$ for $L>15$.
- Fig. \[fig:ChK\], bottom: $L=20$ and $t_{m}=10^{6}$ MCs.
- Fig. \[fig:ChSqP\]: $L=8$ (a) and $L=16$ (b) and $t_{m}=10^{6}$ MCs. The error bars in (a) are coming from an average over 6 runs with different initial configurations; when not visible, they are smaller than the dots.
- Fig. \[fig:HMK\]: $L=30$ and $t_{m}=10^{5}$ MCs.
- Fig. \[fig:HMP\]: $L=20$ and $t_{m}=10^{5}$ MCs.
- Fig. \[fig:radius\]: $L=30$ and $t_{m}=10^{5}$ MCs.
- Fig. \[fig:J05obs\]: $L\in\{6,8,10\}$ and $t_{m}\in\{10^{7}, 2.10^{7},10^{7}\}$ MCs respectively.
For LL dynamics on the kagome \[Fig. \[fig:llg\_kagome\]\] and pyrochlore \[Fig. \[fig:llg\_pyrochlore\]\] lattices, we prepared respectively 864 and 432 spin configurations, for system sizes of $3\times 30^2 $ and $4 \times 12^3$ spins. In the parameter region of interest, single-spin-flip Monte Carlo updates were adequate. These spin configurations were then used as seeds for the fourth-order Runge-Kutta method, using $\delta t=0.01$ as the time interval, and $N_t=100000$ as the number of steps of the time evolution. The accuracy of the numerical simulation was confirmed by calculating the energy; indeed, the energy conservation is satisfied during the Landau-Lifshitz dynamics within the accuracy of $10^{-5}$ (Fig. \[figr1\]) .
![ Time evolutions of the energy (per spin) for the fourth-order Runge-Kutta method in the Landau-Lifshitz dynamics for kagome (a), (b) and pyrochlore (c), (d) systems. The conservation of energy is satisfied, within the accuracy of $10^{-5}$. []{data-label="figr1"}](fig_energy.png){width="\columnwidth"}
Rank-two tensor order parameter {#app:Q}
===============================
The rank-two tensor order parameter is time-reversal invariant and measures the on-site quadrupolar order. For a pyrochlore lattice of $N$ sites, the rank-two tensor is defined following Ref. : $$\begin{aligned}
\mathcal{Q}^{\alpha}=\dfrac{1}{N}\sum_{i=1}^{N}\mathcal{Q}_{i}^{\alpha},\end{aligned}$$ where $$\begin{aligned}
\mathcal{Q}^{3z^{2}-r^{2}}_{i}&=&\dfrac{1}{\sqrt{3}}\left[2 (S_{i}^{z})^{2} - (S_{i}^{x})^{2} - (S_{i}^{y})^{2}\right],\\
\mathcal{Q}^{x^{2}-y^{2}}_{i}&=&(S_{i}^{x})^{2} - (S_{i}^{y})^{2},\\
\mathcal{Q}^{xy}_{i}&=&2 S_{i}^{x}\;S_{i}^{y},\\
\mathcal{Q}^{yz}_{i}&=&2 S_{i}^{y}\;S_{i}^{z},\\
\mathcal{Q}^{zx}_{i}&=&2 S_{i}^{z}\;S_{i}^{x}.\end{aligned}$$ The order parameter used in Fig. \[fig:J05obs\].(b) comes from the norm of all quadrupole moments $$\begin{aligned}
M_Q=\sqrt{\sum_{\alpha}\left(\mathcal{Q}^{\alpha}\right)^{2}}\;,
\label{eq:Q}\end{aligned}$$ and is saturated when all spins are collinear, taking the value $2/\sqrt{3}$.
Origin of subextensive entropy in the ordered state at $J=J_{2c}$ for pyrochlore \[app:order\]
===============================================================================================
In this appendix, we explain the origin of the sub-extensive entropy of (1/2,1/2,1/2)-state at $J=J_{2c}$, illustrated in Fig. \[fig:meth\]. For this state, which consists of the double-charge clusters, the energy of the ground state comes from the contact between charges \[Eq. (\[eq:EGSJ05\])\]. These contacts only take place within each cluster. Let us recall there is one double-charge cluster per cubic unit cell, with one double charge, four single charges, and three vacuum tetrahedra. It means that any change which does not affect the integrity and connectivity of the clusters is iso-energetic. Thanks to the vacuum surrounding every cluster, such changes are possible by shifting an entire plane of clusters. An example is given in Fig. \[fig:meth\]. The plane of clusters in the upper part of the figure is shifted in the \[110\] direction between the left and right panels, while the bottom part remains fixed – the thickness of a plane is exactly one cubic unit cell. More precisely all spins along the \[110\] lines are shifted by a distance of two nearest neighbors; the spins along the \[$1\bar{1}0$\] lines are left unchanged. Thanks to the intervening layers of vacuum tetrahedra (the one just below the transparent plane, and the one at the top of the figure), this shift does not affect any NN pair of charges. The resulting state is thus also a ground state. A second shift in the \[110\] direction gives back the initial state. The same is also true if one shifts the spins in the upper plane along the \[$1\bar{1}0$\] lines, leaving the \[110\] lines unchanged. The addition of both shifts makes a fourth possibility. This gives $4^{L}$ ground states for a system of $L$ planes in the \[001\] direction.
The same reasoning applies for planes orthogonal to the \[010\] and \[100\] directions. However, it is not possible to do successively a shift in a (100) plane followed by a shift in a (001) plane. This is because the conservation of the energy depends on the intervening vacuum layers. Visually, a plane of clusters can glide at no energy cost as long as there is a layer of vacuum to isolate it from the two planes above and below. But, the shift of a (100) plane intersects the layers of vacuum orthogonal to the \[001\] direction; the shift of a (001) plane is now forbidden in the ground state. As a result, the entropy of the ground state is sub-extensive, of the order of $L\ln 4$.
The configurations of Fig. \[fig:meth\], as well as the shift of entire planes, have been observed in snapshots of Monte Carlo simulations at low temperatures, up to fluctuations away from collinearity. The presence of the shifting planes favors a given cubic axis but not a given direction. This spontaneously breaks rotational symmetry as measured by the quadrupolar order parameter $M_{Q}$. But, in this picture, the correlations along the preferred cubic axis should be “paramagnetic” and the order parameter $M_{L}$ should vanish like $1/L$ in the thermodynamic limit. This is not what is observed in simulations. The reason is because the system is not made of Ising spins, but of continuous Heisenberg spins. The ground state is selected via order by disorder because of soft modes around the $L$ points of the Brillouin zone, which favor the long-range ordered states where none of the planes are shifted. However, the sub-extensive entropy is probably responsible for the difficulty of the simulations to thermalize at very low temperatures.
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|
---
author:
- Longjie ZHANG
date:
-
-
- |
December, 2015\
Corresponding author University:Graduate School of Mathematical Sciences, The University of Tokyo. Address:3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan. Email:[email protected], [email protected]
title: Mean curvature flow with driving force on fixed extreme points
---
[**Abstract:**]{} In this paper, we consider the mean curvature flow with driving force on fixed extreme points in the plane. We give a general local existence and uniqueness result of this problem with $C^2$ initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.
[**Keywords and phrases:**]{} mean curvature flow, driving force, fixed extreme points
[**2010MSC:**]{} 35A01, 35A02, 35K55, 53C44.
$$$$
Introduction
============
In this paper, we consider the mean curvature with driving force on fixed extreme points given by $$\label{eq:meancur}
V=-\kappa+A,\ \text{on}\ \Gamma(t),$$ $$\label{eq:initial}
\Gamma(0)=\Gamma_0.$$ Here $V$ denotes the upward normal velocity(the definition of “upward” is given by Remark \[rem:direction\]). The sign $\kappa$ is chosen such that the problem is parabolic. $A$ is a positive constant.
If we use the arc length parameter $s$ to represent $\Gamma(t)$ by $\Gamma(t)=\{F(s,t)\in\mathbb{R}^2\mid 0\leq s\leq L(t)\}$, the equation (\[eq:meancur\]), (\[eq:initial\]) can be written as $$\label{eq:meancurpara}
\frac{d}{d t}F(s,t)=\kappa N-AN,\ 0<s<L(t),$$ $$\label{eq:initialpara}
F(s,0)=F_0(s),\ 0\leq s\leq L_0.$$ Here $\Gamma_0=\{F_0(s)\in\mathbb{R}^2\mid 0\leq s\leq L_0\}$; $N$ denotes the unit downward normal vector(the definition of “downward” is given by Remark \[rem:direction\]) and $L(t)$ denotes the length of $\Gamma(t)$. And the notation $\frac{d}{d t}F(s,t)$ denotes the derivative of $t$ by fixing $s$. Noting the assumption that the sign of $\kappa$ is chosen such that the problem (\[eq:meancur\]) is parabolic, combining Frenet formulas, there holds $$\kappa N=\frac{\partial^2}{\partial s^2}F(s,t).$$ Here we give the fixed extreme point boundary condition. $$\label{eq:fixbound}
F(0,t)=P,\ F(L(t),t)=Q,$$ where $P$, $Q$ are two different fixed points in $\mathbb{R}^2$.
[**Main results**]{} Here we give our main theorems.
\[thm:exist\] Assume that $F_0(s)\in C^2([0,L_0]\rightarrow\mathbb{R}^2)$. Then there exist $T>0$ and unique $F(s,t)$ such that $F(s,t)$ satisfies (\[eq:meancurpara\]), (\[eq:fixbound\]), for $0<t<T$ and initial condition (\[eq:initialpara\]).
Assume that $P=(-a,0)$, $Q=(a,0)$, where $0<a\leq 1/A$. Before giving the three categories result, we introduce two equilibrium solutions of (\[eq:meancur\]) with boundary condition (\[eq:fixbound\]). Denote $$\Gamma_*=\{(x,y)\in\mathbb{R}^2\mid y=\sqrt{1/A^2-x^2}-\sqrt{1/A^2-a^2}, -a\leq x\leq a\}$$ and $$\Gamma^*=\partial B_{\frac{1}{A}}\big((0,\sqrt{1/A^2-a^2})\big)\setminus\{(x,-y)\in\mathbb{R}^2\mid (x,y)\in\Gamma_*\}.$$
![Equilibrium solutions of (\[eq:meancur\])[]{data-label="fig:stationaryso"}](stationaryso.pdf){height="6.0cm"}
Obviously, on $\Gamma_*$ and $\Gamma^*$, there holds $\kappa=A$ and the fixed boundary condition. Here we give the three categories theorem. In the following theorem we consider a family of initial curves given by $$\Gamma_{\sigma}=\{(x,y)\in\mathbb{R}^2\mid y=\sigma\varphi(x), -a\leq x\leq a\}.$$ Here $\varphi$ is even, $\varphi\in C^2\big([-a,a]\big)$ and $\varphi^{\prime\prime}(x)\leq0$, $-a<x<a$. And assume that for all $\sigma\in\mathbb{R}$, $\Gamma_{\sigma}$ intersects $\Gamma^*$ at most fourth(including the extreme points). Denote $\Gamma_{\sigma}(t)$ being the solution with $\Gamma_{\sigma}(0)=\Gamma_{\sigma}$.
\[thm:category\] There exists $\sigma^*>0$ such that
(1). for $\sigma>\sigma^*$, there exists $T_{\sigma}^*<T_{\sigma}$ such that $\Gamma_{\sigma}(t)\succ \Gamma^*$, $T_{\sigma}^*<t<T_{\sigma}$;
(2). for $\sigma=\sigma^*$, $T_{\sigma}=\infty$ and $\Gamma_{\sigma}(t)\rightarrow \Gamma^*$ in $C^1$, as $t\rightarrow \infty$;
(3). for $\sigma<\sigma^*$, $T_{\sigma}=\infty$ and $\Gamma_{\sigma}(t)\rightarrow \Gamma_*$ in $C^1$, as $t\rightarrow \infty$.
Where $T_{\sigma}$ denotes the maximal existence time of $\Gamma(t)$.
The notation “$\succ$” can be seen as an order. The precise definition is given in Section 2. We will interpret the sense of $C^1$ convergence in Definition \[def:C1close\].
[**Main method.**]{} Theorem \[thm:exist\] can be easily proved by transport map. The transport map is first used by [@A] to consider the mean curvature flow under the non-graph condition. For the three categories result, we use the intersection number principle to classify the type of the solutions in Lemma \[lem:sigulartime\]. Since $\Gamma_{\sigma}$ intersects $\Gamma^*$ at most fourth, the intersection number between $\Gamma_{\sigma}(t)$ and $\Gamma^*$ can only be two or four. In Lemma \[lem:sigulartime\], one of the following three conditions can hold:
(1). The curve $\Gamma_{\sigma}(t)$ intersects $\Gamma^*$ twice and $\Gamma_{\sigma}(t)\succ \Gamma^*$ eventually;
(2). The curve $\Gamma_{\sigma}(t)$ intersects $\Gamma^*$ fourth for every $t>0$.
(3). The curve $\Gamma_{\sigma}(t)$ intersects $\Gamma^*$ twice and $\Gamma^*\succ\Gamma_{\sigma}(t)$ eventually.
Seeing future, under the condition (2) above, $\Gamma_{\sigma}(t)\rightarrow \Gamma^*$ in $C^1$, as $t\rightarrow \infty$; under the condition (3) above, $\Gamma_{\sigma}(t)\rightarrow \Gamma_*$ in $C^1$, as $t\rightarrow \infty$. In this paper, we prove the asymptotic behavior by using Lyapunov function introduced in Section 5.
[**A short review for mean curvature flow.**]{} For the classical mean curvature flow: $A=0$ in (\[eq:meancur\]), there are many results. Concerning this problem, Huisken [@H] shows that any solution that starts out as a convex, smooth, compact surface remains so until it shrinks to a “round point” and its asymptotic shape is a sphere just before it disappears. He proves this result for hypersurfaces of $\mathbb{R}^{n+1}$ with $n\geq2$, but Gage and Hamilton [@GH] show that it still holds when $n=1$, the curves in the plane. Gage and Hamilton also show that embedded curve remains embedded, i.e. the curve will not intersect itself. Grayson [@Gr] proves the remarkable fact that such family must become convex eventually. Thus, any embedded curve in the plane will shrink to “round point” under curve shortening flow.
For fixed extreme point problem, Forcadel, Imbert and Monneau [@FIM] consider a family of half lines evolves by (\[eq:meancur\]) and one extreme point is fixed at the origin. Precisely, the family of curves given by polar coordinate, $$\left\{
\begin{array}{lcl}
x=\rho\cos\theta(\rho,t),\\
y=\rho\sin\theta(\rho,t),
\end{array}
\right.$$ for $0\leq \rho<\infty$. Therefore, $\theta(\rho,t)$ satisfies $$\label{eq:vistheta}
\rho\theta_t=A\sqrt{1+\rho^2\theta_{\rho}^2}+\theta_{\rho}\big(\frac{2+\rho^2\theta_{\rho}^2}{1+\rho^2\theta_{\rho}^2}\big)+\frac{\rho\theta_{\rho\rho}}{1+\rho^2\theta_{\rho}^2},\ t>0,\rho>0.$$ Obviously, this problem is singular near $\rho=0$. They consider the solution of (\[eq:vistheta\]) in viscosity sense. Since near the fixed extreme point mean curvature flow has singularity by using polar coordinate, there are some papers considering this problem by digging a hole. For example, Giga, Ishimura and Kohsaka [@GIK] consider anisotropic curvature flow equation with driving force in the ring domain $r<\rho<R$. At the boundary, the family of the curves is imposed being perpendicular to the boundary, seeing Figure \[fig:dighole\].
![Research in [@GIK][]{data-label="fig:dighole"}](dighole.pdf){height="6.0cm"}
[**Motivation of this research.**]{} Ohtsuka, Goto and Nakagawa first prove the existence and uniqueness of spiral crystal growth for (\[eq:meancur\]) by level set method in [@GNO] and [@O]. But they also consider this problem by digging a hole near the fixed points. Recently, [@OTG] simulates the level set of the solution given in [@O] by numerical method. In their paper, for $a>1/A$, the level set evolves as in Figure \[fig:introduction1\], \[fig:introduction2\] and \[fig:introduction3\].
![Evolution of level set $\#$1[]{data-label="fig:introduction1"}](introduction1.pdf){height="5.0cm"}
![Evolution of level set $\#$2[]{data-label="fig:introduction2"}](introduction2.pdf){height="5.0cm"}
![Evolution of level set $\#$3[]{data-label="fig:introduction3"}](introduction3.pdf){height="5.0cm"}
Although, in our paper, we only consider the problem under the condition $a\leq 1/A$, the simulated results in [@OTG] give the hit about this research. We are devoted to considering them in analytic way.
The rest of this paper is organized as follows. In Section 2, we give some preliminary knowledge including the definition of semi-order, comparison principle and intersection number principle. In Section 3, we give the existence and uniqueness result for the fixed extreme points problem. Moreover, in Lemma \[lem:regular\], we give a sufficient condition for the solution $\Gamma_{\sigma}(t)$ remaining regular. In Section 4, we give the asymptotic behavior of the solution $\Gamma_{\sigma}(t)$ when $\sigma$ is large or small. Lemma \[lem:sigulartime\] gives an important result for classifying $\Gamma_{\sigma}(t)$ by intersection number. In Section 5, we prove the asymptotic behavior for the condition (3) in Lemma \[lem:sigulartime\] by Lyapunov function. In Section 6, we give the proof of Theorem \[thm:category\].
Preliminary
===========
[**Semi-order**]{} We want to define a semi-order for curves with the same fixed extreme points.
\[def:semiorder\] For any points $P$, $Q\in \mathbb{R}^2$ and $P\neq Q$, assume that maps $F_i(s)\in C([0,l_i]\rightarrow\mathbb{R}^2)$ and $F_i$ are differentiable at $0$ and $l_i$.The curves $\gamma_i$ are given by $\gamma_i=\{F_i(s)\mid 0\leq s\leq l_i, F_i(0)=P,F_i(l_i)=Q\}$, where $l_i$ is the length of $\gamma_i$, $i=1,2$. It is easy to see that $\gamma_i$ have the same extreme points $P$, $Q$, $i=1,2$. We say $\gamma_1\succ\gamma_2$, if
(1). There exists connect, bounded and open domain $\Omega$ such that $\partial \Omega=\gamma_1\cup\gamma_2$;
(2). $\frac{d}{ds}F_1(0)\cdot\frac{d}{ds}F_2(0)\neq 1$ and $\frac{d}{ds}F_1(l_1)\cdot\frac{d}{ds}F_2(l_2)\neq1$;
(3). The domain $\Omega$ is located in the right hand side of $\gamma_1$, when someone walks along $\gamma_1$ from $P$ to $Q$.
Where “$\cdot$” denotes the inner product in $\mathbb{R}^2$. We say $\gamma_1\succeq\gamma_2$, if there exist two sequences of curves $\{\gamma_{in}\}_{n\geq1}$, $i=1,2$ such that
(1). $\lim\limits_{n\rightarrow\infty}d_H(\gamma_{in},\gamma_i)\rightarrow0$, $i=1,2$;
(2). $\gamma_{1n}\succ\gamma_{2n}$, $n\geq 1$.
Where $d_H(A,B)$ denotes the Hausdorff distance for set $A,B\subset \mathbb{R}^2$.
![Definition \[def:semiorder\][]{data-label="fig:semiorder"}](semiorder.pdf){height="6.0cm"}
Let $F(s)\in C^2([0,l]\rightarrow\mathbb{R}^2)$ and $\gamma=\{F(s)\mid s\in[0,l], F(0)=P, F(l)=Q\}$. Using the definition of semi-order, we can define a shuttle neighbourhood of $\gamma$. Seeing the assumption of $\gamma$, we can extend $\gamma$ by $\gamma^*$ such that $\gamma^*$ is $C^1$ curve and divides $\mathbb{R}^2$ into two connect parts denoted by $\Omega_1$ and $\Omega_2$. Moreover, $\Omega_1$ is located in the left hand side when someone walks along $\gamma^*$ from $P$ to $Q$.
\[rem:direction\] We say the normal vector of $\gamma$ is *upward*(*downward*), if the normal vector points to the domain $\Omega_1$($\Omega_2$).
\[def:shuttlenei\] We say $V$ is a *shuttle neighbourhood* of $\gamma$, if there exist $\gamma_1$ and $\gamma_2$ such that
(1). $\gamma_i\subset \Omega_i$, $i=1,2$;
(2). $\gamma_1\succ\gamma\succ\gamma_2$;
(3). $\partial V=\gamma_1\cup \gamma_2$.
[**Comparison principle and intersection number principle**]{} Here we introduce the comparison principle and intersection number principle. The intersection number principle can help us classify the solutions.
For giving comparison principle, we must define sub,super-solution of (\[eq:meancurpara\]).
\[def:subsup\] We say a continuous family of continuous curves $\{\gamma(t)\}$ is a sub(super)-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]), if
(1). $\gamma(t)$ are continuous curves and have the same extreme points $P$, $Q$;
(2). Let $\{S(t)\}$ be a smooth flow with extreme points $P$, $Q$. For some point $P^*$ and some time $t_0>0$ satisfying $P^*\in \gamma(t_0)$ but $P^*\neq P,\ Q$, if near the point $P^*$ and time $t_0$, $\{S(t)\}$ only intersects $\{\gamma(t)\}$ at $P^*$ and time $t_0$ from above(below). Let $V_{S(t)}$ denote the upward normal velocity of $S(t)$ and $\kappa_{S(t)}(P)$ denote the curvature at $P\in S(t)$. Then $$V_{S(t_0)}(P^*)\leq(\geq) -\kappa_{S(t_0)}(P^*)+A.$$
\[thm:comparison\] For two families of curves $\{\gamma_1(t)\}_{0\leq t\leq T}$ and $\{\gamma_2(t)\}_{0\leq t\leq T}$, assume $\{\gamma_1(t)\}_{0\leq t\leq T}$ is a super-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]), $\{\gamma_2(t)\}_{0\leq t\leq T}$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]). If $\gamma_1(0)\succeq\gamma_2(0)$, then $\gamma_1(t)\succ\gamma_2(t)$, $0\leq t\leq T$.
We can prove this theorem by contradiction. Using local coordinate representation, by maximum principle and Hopf lemma, the conclusion can be got easily. Here we omit the detail.
In this paper, besides intersection number $Z[\cdot,\cdot]$, we introduce a related notion $SGN[\cdot,\cdot]$(first used by [@DGM]), which turns out to be exceedingly useful in classifying the types of the solutions.
\[def:signal\] For two curves $\gamma_1$ and $\gamma_2$ satisfying the same conditions in Definition \[def:semiorder\], we define:
(1). $Z[\gamma_1,\gamma_2]$ is the number of the intersections between curves $\gamma_1$ and $\gamma_2$. Noting that $\gamma_1$ and $\gamma_2$ have the same extreme points, then $Z[\gamma_1,\gamma_2]\geq2$;
(2). $SGN[\gamma_1,\gamma_2]$ is defined when $Z[\gamma_1,\gamma_2]<\infty$. Denoting $n+1:=Z[\gamma_1,\gamma_2]<\infty$, let $P=P_0$, $P_1$, $\cdots$, $P_{n-1}$, $P_n=Q$ be the intersections. Here we assume $\wideparen{P_{i+1}P_0}>\wideparen{P_iP_0}$ and $\widetilde{P_{i+1}P_0}>\widetilde{P_iP_0}$, $i=1,\cdots,n$, where $\wideparen{P_iP_j}$ denotes the arc length of $\gamma_1$ between $P_i$ and $P_j$; $\widetilde{P_iP_j}$ denotes the arc length of $\gamma_2$ between $P_i$ and $P_j$. If $\gamma_1\mid_{\wideparen{P_iP_{i-1}}}\succ\gamma_2\mid_{\widetilde{P_iP_{i-1}}}$, we say the sign between $P_i$ and $P_{i-1}$ is “$+$”; Respectively, $\gamma_2\mid_{\widetilde{P_iP_{i-1}}}\succ\gamma_1\mid_{\wideparen{P_iP_{i-1}}}$, we say the sign between $P_i$ and $P_{i-1}$ is “$-$”, $i=1,\cdots,n$. Where $\gamma_1\mid_{\wideparen{P_iP_{i-1}}}$ and $\gamma_2\mid_{\widetilde{P_iP_{i-1}}}$ denote the restriction between $P_{i-1}$ and $P_{i}$.
$SGN[\gamma_1,\gamma_2]$ called *ordered word set* consists the sign between $P_i$ and $P_{i-1}$, $i=1,\cdots,n$.
For explaining Definition \[def:signal\], we give an example. Seeing Figure \[fig:sign\], $Z[\gamma_1,\gamma_2]=6$ and $$SGN[\gamma_1,\gamma_2]=[-\ +\ -\ +\ -].$$
![Example for $SNG[\cdot,\cdot]$[]{data-label="fig:sign"}](sign.pdf){height="6.0cm"}
Let $A$ and $B$ be two ordered word sets, we write $A\triangleright B$, if $B$ is a sub ordered word set of $A$. For example, $$[+\ -]\triangleright B\ \text{for}\ B=[+\ -],\ [+],\ [-],\ \text{but\ not}\ [+\ -]\triangleright[-\ +].$$
\[rem:nothold\] For the curve shortening flow with driving force, even if $\gamma_1(t)$ and $\gamma_2(t)$ satisfy (\[eq:meancurpara\]) and (\[eq:fixbound\]), we can not guarantee that for all $t_1<t_2$, $$Z[\gamma_1(t_2),\gamma_2(t_2)]\leq Z[\gamma_1(t_1),\gamma_2(t_1)],\ SGN[\gamma(t_2),\gamma(t_2)]\triangleleft SGN[\gamma(t_1),\gamma(t_1)].$$
For giving the intersection number principle, we need assume $\gamma_1(t)$ and $\gamma_2(t)$ are homeomorphism to a curve.
\[thm:intersection\] For two families of curves $\{\gamma_1(t)\}_{0\leq t\leq T}$ and $\{\gamma_2(t)\}_{0\leq t\leq T}$ satisfying (\[eq:meancurpara\]) and (\[eq:fixbound\]), assume there exist a $C^1$ curve $M$ with extreme points $P$, $Q$ and two maps $$\varphi_1,\ \varphi_2:M\times[0,T]\rightarrow\mathbb{R}^2$$ such that $$\gamma_i(t)=\{\varphi_i(P,t)\mid P\in M \},\ i=1,2.$$ Then there hold $$Z[\gamma_1(t_2),\gamma_2(t_2)]\leq Z[\gamma_1(t_1),\gamma_2(t_1)],\ SGN[\gamma(t_2),\gamma(t_2)]\triangleleft SGN[\gamma(t_1),\gamma(t_1)],$$ for all $0\leq t_1<t_2\leq T$.
Using the arc length parameter $s$ of $M$, we can express $\gamma_i(t)$ by $$\gamma_i(t)=\{\varphi_i(s,t)\mid 0\leq s\leq L \},\ 0\leq t\leq T,\ i=1,2,$$ where $L$ denotes the length of $M$. Using the local representation and classical intersection number principle, we can prove this results easily. We omit the detail.
\[def:C1close\] For a $C^1$ curve $\gamma$ and a sequence of $C^1$ curves $\gamma_n$ with extreme points $P$, $Q$, we say $\gamma_n\rightarrow\gamma$ in $C^1$, if
\(1) There exist a $C^1$ curve $M$ with extreme points $P$, $Q$ and maps $$\varphi,\ \varphi_n:M\rightarrow\mathbb{R}^2$$ such that $$\gamma=\{\varphi(P)\mid P\in M \},\ \gamma_n=\{\varphi_n(P)\mid P\in M \}.$$
\(2) $$\left\Vert \varphi_n-\varphi \right\Vert_{C^1(M\rightarrow \mathbb{R}^2)}\rightarrow 0,$$ as $n\rightarrow\infty$.
Time local existence and uniqueness of solution
===============================================
In this section, we introduce the the transport map first used by [@A] and prove Theorem \[thm:exist\].
\[lem:vectorfield\] For $\Gamma_0$ satisfying the assumption in Theorem \[thm:exist\], there exist a shuttle neighbourhood $V$ of $\Gamma_0$ and a vector field $X\in C^1(\overline{V}\rightarrow\mathbb{R}^2)$ such that $$X(z)\cdot n(z) >0,\ z\in \Gamma_0$$ and in $V$, there holds $$|X|\geq\delta>0, \text{for\ some}\ \delta>0,$$ where $n$ denotes the unit upward normal vector of $\Gamma_0$.
We extend $\Gamma_0$ by $\Gamma_0^*$ such that $\Gamma_0^*$ is a $C^2$ curve and divide $\mathbb{R}^2$ into two connect parts $\Omega_1$ and $\Omega_2$. Assume $\Omega_1$($\Omega_2$) locates in the left(right) side of $\Gamma_0^*$(“left side” and “right side” are defined as in Section 2).
Let $d(x)$ be the signed distance function defined as following: $$d(x)=d(x,\Omega_2)-d(x,\Omega_1),\ x\in\mathbb{R}^2.$$ Since $\Gamma_0^*$ is $C^2$, as we know, there exists a tubular neighbourhood $U$ of $\Gamma_0^*$ such that $d$ is $C^2$ in $U$. Moreover, there exists a projection map $P$ such that for all $z\in U$ there exists a unique point $z^*\in \Gamma_0^*$ such that $$P z=z^*$$ and $\nabla d(z)=\nabla d(z^*)=n(z^*)$. We choose two curves $\Gamma_1,\Gamma_2\subset U$ and $\Gamma_i\subset \Omega_i$, $i=1,2$, such that $\Gamma_1\succ\Gamma_0\succ\Gamma_2$. Let $V$ be the domain satisfying $\partial V=\Gamma_1\cup\Gamma_2$ and $X(z)=\nabla d(z)$. Obviously $$|X|(z)=1,\ z\in V$$ and $$X(z)\cdot n(z)=1,\ z\in \Gamma_0.$$
[**Transport map**]{} Let $\phi:\Gamma_0\times(-\delta,\delta)\rightarrow V$ be the map generated by vector field $X$, precisely, $$\left\{\begin{array}{lcl}
\frac{d}{d\alpha}\phi(P,\alpha)=X(\phi),\ P\in \Gamma_0, \\
\sigma(P,0)=P,\ P\in\Gamma_0.
\end{array}
\right.$$ Recalling $\Gamma_0=\{F_0(s)\mid 0\leq s\leq L_0\}$ and $F_0(s)\in C^2([0,L_0]\rightarrow\mathbb{R}^2)$, let $$\psi(s,\alpha)=\phi(F_0(s),\alpha).$$ Seeing the assumption of $F_0$ and $X$, $\psi_s$, $\psi_{\alpha}$, $\psi_{ss}$, $\psi_{s\alpha}$, $\psi_{\alpha\alpha}$ are all continuous vectors for $0\leq s\leq L_0$, $-\delta<\alpha<\delta$.
If $\Gamma(t)\subset V$ is $C^1$ close to $\Gamma_0$ and satisfies (\[eq:meancurpara\]), (\[eq:fixbound\]), $0<t<T$ with initial data $\Gamma(0)=\Gamma_0$, then there exists a function $u(\cdot,t):[0,L_0]\rightarrow\mathbb{R}$ such that $$\Gamma(t)=\{\psi(s,u(s,t))\mid 0\leq s\leq L_0\}.$$ Moreover, $u$ satisfies $$\label{eq:localpara}
\left\{
\begin{array}{lcl}
\dis{u_t=\frac{1}{|\psi_s+\psi_{\alpha}u_s|^2}u_{ss}+\frac{\det(\psi_s+\psi_{\alpha}u_s,\psi_{ss}+2u_s\psi_{s\alpha}+\psi_{\alpha\alpha}u_s^2)}{\det(\psi_s,\psi_{\alpha})|\psi_s+\psi_{\alpha}u_s|^2}+A\frac{|\psi_{s\alpha+\psi_{\alpha}u_s}|}{\det(\psi_s,\psi_{\alpha})}},\\
0<s<L_0,\ 0<t<T,\\
u(0,t)=u(L_0,t)=0,\ 0\leq t<T\\
u(s,0)=0,\ 0\leq s\leq L_0
\end{array}
\right.$$ where $\det(\cdot,\cdot)$ denotes the determinant. Indeed, the upward normal velocity $$V=\frac{\det(\psi_s,\psi_{\alpha})u_t}{|\psi_s+\psi_{\alpha}u_s|}$$ and the curvature $$\kappa=\frac{\det(\psi_s,\psi_{\alpha})}{|\psi_s+\psi_{\alpha}u_s|^3}u_{ss}+\frac{\det(\psi_s+\psi_{\alpha}u_s,\psi_{ss}+2u_s\psi_{s\alpha}+\psi_{\alpha\alpha}u_s^2)}{|\psi_s+\psi_{\alpha}u_s|^3}.$$
Following Proposition \[pro:localexist\] implies Theorem \[thm:exist\].
\[pro:localexist\] There exist $T_0>0$ and a unique $u\in C([0,L_0]\times[0,T_0))\cap C^{2+\alpha,1+\alpha/2}([0,L_0]\times(0,T_0))$ such that $u$ satisfies (\[eq:localpara\]) for $T=T_0$.
Since $\psi_{s}(s,0)\cdot\psi_{\alpha}(s,0)=0$, $0\leq s\leq L_0$, then $$|\det(\psi_s,\psi_{\alpha})|(s,0)=1,\ 0\leq s\leq L_0.$$ There exist $\delta_1>0$ and $\alpha_0$ such that for all $-\alpha_0<\alpha<\alpha_0$ and $0\leq s\leq L_0$, $$|\det(\psi_s,\psi_{\alpha})|(s,\alpha)>\delta_1.$$ By the quasi-linear parabolic theory in [@LSU], we can deduce there exist $T_0$ and $u\in C([0,L_0]\times[0,T_0))\cap C^{2+\alpha,1+\alpha/2}([0,L_0]\times(0,T_0))$ such that $u$ satisfies (\[eq:localpara\]) and $|u|\leq \alpha_0$, $0\leq t<T_0$. For the uniqueness, since $\psi_s$, $\psi_{\alpha}$, $\psi_{ss}$, $\psi_{s\alpha}$, $\psi_{\alpha\alpha}$ are all continuous vectors for $0\leq s\leq L_0$, $-\alpha_0<\alpha<\alpha_0$, the unique result can be got easily.
\[rem:weak\] The assumption for initial curve can be weakened. In this paper, we assume $F_0(s)\in C^2([0,L_0]\rightarrow\mathbb{R}^2)$. Indeed, the initial curve can be assumed to be Lipschitz continuous. Recently, [@M] has considered the curve-shortening flow with Lipschitz initial curve, under the Neumann boundary condition. Since the purpose of this paper is to get the three categories of solutions, we do not introduce this part in detail.
\[lem:curvaturee\] For $\Gamma(t)$ satisfying (\[eq:meancurpara\]), (\[eq:fixbound\]), for $0<t<T$, then the curvature $\kappa(s,t)$ satisfies $$\label{eq:curvaturee}
\left\{
\begin{array}{lcl}
\kappa_t=\kappa_{ss}-\kappa\kappa_s^2+\kappa^2(\kappa-A),\ 0<s<L(t),\ 0<t<T\\
\kappa(0,t)=A,\ \kappa(L(t),t)=A, 0<t<T,
\end{array}
\right.$$ where $\kappa_t$ denotes the derivative of $t$ by fixing $s$.
For the proof of the first equation, the calculation can be seen in [@GMSW]. Since at the extreme points, $\Gamma(t)$ does not move, the boundary condition is obvious.
\[lem:regular\] For $\sigma>0$, $\Gamma_{\sigma}(t)$ given in Theorem \[thm:category\], let $F_{\sigma}(s,t)$ satisfy $$\Gamma_{\sigma}(t)=\{F_{\sigma}(s,t)\mid0\leq s\leq L_{\sigma}(t)\},$$ where $L_{\sigma}(t)$ is the length of $\Gamma_{\sigma}(t)$. If $\frac{d}{ds} F_{\sigma}(0,t)\cdot(0,1)>0$, for all $0\leq t\leq t_0$, then $t_0<T_{\sigma}$.
This lemma gives a sufficient condition under which $\Gamma_{\sigma}(t)$ does not become singular. The assumption $\frac{d}{ds} F_{\sigma}(0,t)\cdot(0,1)>0$ means that the $y$-component of the tangential vector $\frac{d}{ds} F_{\sigma}(0,t)$ is positive.
Seeing the choice of $\Gamma_{\sigma}(0)$, then $\kappa_{\sigma}(s,0)\geq0$, $0\leq s\leq L_{\sigma}(0)$, for $\sigma>0$. Combining Lemma \[lem:curvaturee\] and maximum principle, $\kappa_{\sigma}(s,t)>0$, $0<s<L(t)$, $0<t<T_{\sigma}$.
If $T_{\sigma}=\infty$, the result is trivial. We assume $T_{\sigma}<\infty$.
We prove the result by contradiction, assuming $t_0\geq T_{\sigma}$. We claim that every half-line given by $$y=kx,\ y\geq0,\ \text{or}\ x=0,\ y\geq 0$$ intersects $\Gamma_{\sigma}(t)$ only once, $0<t<T_{\sigma}$.
First, for all $0<t<T_{\sigma}$, we prove $x=0$, $y\geq0$ intersects $\Gamma_{\sigma}(t)$ only once. If not, suppose that there exists $t_1<T_{\sigma}$ such that $x=0$, $y\geq0$ intersects $\Gamma_{\sigma}(t_1)$ more than once. Since $\Gamma_{\sigma}(t)$ is symmetric about $y$-axis, it is easy to see that $\Gamma_{\sigma}(t)$ becomes singular at $t_1$. This contradicts to $t_1<T_{\sigma}$.
Next, by contradiction, assume that there exist $t_2<T_{\sigma}$ and $k<0$ such that $y=kx,\ y\geq0$ intersects $\Gamma_{\sigma}(t_2)$ more than once. Combining our assumption $\frac{d}{ds} F_{\sigma}(0,t)\cdot(0,1)>0$, we can choose $k_0$ satisfying $k_0<k<0$ such that half-line $y=k_0x,\ y\geq0$ intersects $\Gamma_{\sigma}(t_2)$ tangentially at some point $P^*$ and near $P^*$, $\Gamma_{\sigma}(t_2)$ is located under the half-line. It is easy to deduce that the curvature at $P^*$, $\kappa_{\sigma}(P^*,t_2)\leq0$. This contradicts to that the curvatures on $\Gamma_{\sigma}(t)$ are all positive, $0<t<T_{\sigma}$. Here we complete the proof of claim.
![Proof of claim[]{data-label="fig:polarcoordinate"}](polarcoordinate.pdf){height="6.0cm"}
Seeing $\frac{d}{ds} F_{\sigma}(0,t)\cdot(0,1)>0$ and the claim above, $\Gamma_{\sigma}(t)\subset \{(x,y)\mid y\geq 0\}$, $t<T_{\sigma}$. The claim implies that we can express $\Gamma_{\sigma}(t)$ by polar coordinate. For $(x,y)\in \Gamma_{\sigma}(t)$, let $$\left\{
\begin{array}{lcl}
x=\rho_{\sigma}(\theta,t)\cos\theta,\\
y=\rho_{\sigma}(\theta,t)\sin\theta,
\end{array}
\right.$$ for $0\leq\theta\leq \pi$, $0\leq t<T_{\sigma}$. Consequently, $\rho_{\sigma}$ satisfies $$\label{eq:polarco}
\left\{
\begin{array}{lcl}
\dis{\rho_{t}=\frac{\rho_{\theta\theta}}{\rho^2+\rho^2_{\theta}}-\frac{2\rho^2_{\theta}+\rho^2}{\rho(\rho^2_{\theta}+\rho^2)}+\frac{1}{\rho}A\sqrt{\rho^2_{\theta}+\rho^2}},\ 0<\theta<\pi,\ 0<t<T_{\sigma},\\
\rho(0,t)=a,\ \rho(\pi,t)=a,\ 0\leq t<T_{\sigma},
\end{array}
\right.$$ recalling $P=(-a,0)$, $Q=(a,0)$.
Since for $\sigma>0$, $\Gamma_{\sigma}(0)\succ \Lambda_0=\{(x,y)\mid y=0,\ -a\leq x\leq a\}$. It is easy to see that $\Lambda_0$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]). By comparison principle, $\Gamma_{\sigma}(t)\succ\Lambda_0$ for $t<T_{\sigma}$. This implies that there exist $t_3>0$ and $\rho_1>0$ such that $\rho_{\sigma}(\theta,t)\geq \rho_1$ for $t_3<t<T_{\sigma}$, $0\leq \theta\leq \pi$. On the other hand, since $T_{\sigma}<\infty$, there exists $\rho_2>0$ such that $\rho_{\sigma}(\theta,t)\leq \rho_2$ for $0<t<T_{\sigma}$, $0\leq \theta\leq \pi$. Therefore, the quasilinear theory in [@LSU] shows that for $\epsilon>0$, there exists $C_{\epsilon}$ such that $$\left\Vert \rho_{\sigma}(\cdot,t)\right\Vert_{C^2[0,\pi]}\leq C_{\epsilon},\ t_3+\epsilon<t<T_{\sigma}.$$ Therefore the curvature of $\Gamma_{\sigma}(t)$ is uniform bounded for $t$ close to $T_{\sigma}$. This implies that the solution $\Gamma_{\sigma}(t)$ can be extended over time $T_{\sigma}$. This contradicts to that $T_{\sigma}$ is the maximal existence time.
\[lem:localconti\] Assume $\rho$ and $\rho_n$ are the solution of (\[eq:polarco\]) for $0\leq \theta\leq \pi$, $0<t<T$. If $\rho$ is bounded from below for some positive constant and $$\lim\limits_{n\rightarrow\infty}\left\Vert\rho_{n}(\cdot,0)-\rho(\cdot,0) \right\Vert_{C^1[0,\pi]}=0,$$ then for all $0<t<T$, $$\lim\limits_{n\rightarrow\infty}\left\Vert\rho_{n}(\cdot,t)-\rho(\cdot,t) \right\Vert_{C^2[0,\pi]}=0.$$
Behavior for $\sigma$ sufficient small or large
===============================================
\[prop:sigmalarge\] There exists $\sigma_1>0$ such that for all $\sigma>\sigma_1$, there exists some time $T_{\sigma}^*<T_{\sigma}$ such that $\Gamma_{\sigma}(t)\succ \Gamma^*$, for $T_{\sigma}^*<t<T_{\sigma}$
For proving this proposition, we introduce the Grim reaper for the curve shortening flow. Grim reaper is given by $$G(x,t)=C-\frac{t}{b}+b\ln \cos\frac{x}{b},\ -\frac{b\pi}{2}<x<\frac{b\pi}{2},\ t>0,$$ where $b>0$ and $C\in\mathbb{R}$. It is easy to see $G(x,t)$ satisfies $$G_t=\frac{G_{xx}}{1+G_x^2}.$$ The Grim reaper $G(x,t)$ is a traveling wave moving downward with speed $1/b$.
\[lem:grimreaper\] If $b<2a/\pi$, the curve $$\gamma_G(t)=\{(x,y)\mid y=\max\{G(x,t),0\},\ |x|<\frac{b\pi}{2}\} \cup \{(x,y)\mid y=0,\ \frac{b\pi}{2}\leq|x|\leq a\}$$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]) in the sense of Definition \[def:subsup\].
When $0<t<bC$, let $x(t)>0$ such that $G(x(t),t)=0$.
For $|x|<x(t)$, $\gamma_G=\{(x,y)\mid y=G(x,t)\}$. Therefore, $$G_{t}\leq \frac{G_{xx}}{1+G_x^2}+A\sqrt{1+G_x^2},\ |x|<x(t).$$ For $x(t)<|x|<a$, $\gamma_G=\{(x,y)\mid y=0\}$. Obviously, $y=0$ is a sub-solution of $$u_{t}\leq \frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2},\ x(t)<|x|<a.$$ At the point $x=x(t)$($x=-x(t)$), it is impossible that for smooth flow $S(t)$, near $x=x(t)$($x=-x(t)$), $S(t)$ touches $\gamma_{G}(t)$ at $x=x(t)$($x=-x(t)$) only once from above.
Therefore, $\gamma_G(t)$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]), for $0<t<bC$.
When $t\geq bC$, $\gamma_G=\Lambda_0=\{(x,y)\mid y=0,\ |x|\leq a\}$(given in the proof of Lemma \[lem:regular\]). Obviously, $\gamma_G$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]), for $t\geq bC$.
Following lemma gives the result for the classification of the solution $\Gamma_{\sigma}(t)$.
\[lem:sigulartime\] For $\Gamma_{\sigma}(t)$ given by Theorem \[thm:category\], for $\sigma>0$, $\Gamma_{\sigma}(t)$ satisfies one of the following four conditions:
(1). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-]$ for $t<T_{\sigma}$. Moreover, $T_{\sigma}=\infty$;
(2). there exists $t^*_{\sigma}$ such that $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<t^*_{\sigma}$ and $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-]$ for $t^*_{\sigma}<t<T_{\sigma}$. Moreover, $T_{\sigma}=\infty$;
(3). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<T_{\sigma}$. Moreover, $T_{\sigma}=\infty$;
(4). there exists $T_{\sigma}^*<T_{\sigma}$ such that $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<T_{\sigma}^*$ and $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[+]$, $T_{\sigma}^*<t<T_{\sigma}$.
Seeing the assumption in Theorem \[thm:category\], there exists $\sigma_0>0$ such that
(a). $0<\sigma\leq\sigma_0$, $\Gamma^*\succeq\Gamma_{\sigma}$;
(b). $\sigma>\sigma_0$, $\Gamma_{\sigma}$ intersects $\Gamma^*$ fourth.
[**Step 1.**]{} For $0<\sigma\leq\sigma_0$, by comparison principle, $\Gamma^*\succ\Gamma_{\sigma}(t)$, for $0<t<T_{\sigma}$. Noting $\sigma>0$, $\Gamma_{\sigma}(t)\succ \Lambda_0$, for $0<t<T_{\sigma}$. Therefore, $\frac{d}{ds}F_{\sigma}(0,t)\cdot(0,1)>0$, $0<t<T_{\sigma}$. By the same method in the proof of Lemma \[lem:regular\], we can prove $T_{\sigma}=\infty$. Therefore, for $0<\sigma\leq\sigma_0$, condition (1) holds.
[**Step 2.**]{} For $\sigma>\sigma_0$, seeing the choice of $\Gamma_{\sigma}$, $SGN(\Gamma_{\sigma},\Gamma^*)=[-\ +\ -]$.
Let $\tau_0$ depending on $\sigma$ satisfy $$\tau_0=\sup\{\tau \mid\frac{d}{ds}F_{\sigma}(0,t)\cdot(0,1)>0, 0<t<\tau\}.$$ Since $\frac{d}{ds}F_{\sigma}(0,0)\cdot(0,1)>0$, we can deduce $\tau_0>0$. Therefore, Lemma \[lem:regular\] implies $T_{\sigma}>\tau_0$. Moreover, $\Gamma_{\sigma}(t)$ can be represented by polar coordinate, $0<t<\tau_0$. This means that $\Gamma_{\sigma}(t)$ satisfies the assumption of Theorem \[thm:intersection\] for $0<t<\tau_0$. Then $$SGN(\Gamma_{\sigma}(t),\Gamma^*)\triangleleft[-\ +\ -],\ 0<t<\tau_0.$$ Seeing the symmetry of $\Gamma_{\sigma}(t)$, then for $t<\tau_0$, one of the following three conditions holds
(i). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[+]$;
(ii). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-]$;
(iii). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ -]$.
(iv). $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$.
[**Step 3.**]{} If for some $t^*_{\sigma}<\tau_0$ (ii) or (iii) holds, then $\Gamma^*\succeq\Gamma_{\sigma}(t^*_{\sigma})$. Then by comparison principle, $\Gamma^*\succ\Gamma(t)\succ\Lambda_0$, $t^*_{\sigma}<t<T_{\sigma}$. Therefore, by the same argument in Step 1, condition (2) holds.
If for some $T_{\sigma}^*<\tau_0$ (i) holds, this means that $\Gamma_{\sigma}(T_{\sigma}^*)\succ\Gamma^*$. By comparison principle, $\Gamma_{\sigma}(t)\succ\Gamma^*$, $T_{\sigma}^*<t<T_{\sigma}$. Therefore, condition (4) holds.
If for every $t<\tau_0$, there holds $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$. Combining $\Gamma_{\sigma}(t)\succ\Lambda_0$, $t<\tau_0$, there exists $\delta>0$ such that $$\frac{d}{ds}F_{\sigma}(0,t)\cdot(0,1)>\delta,\ t<\tau_0.$$ If $\tau_0<\infty$, by the definition of $\tau_0$, $\frac{d}{ds}F_{\sigma}(0,\tau_0)\cdot(0,1)=0$. This yields a contradiction. Therefore, $\tau_0=\infty$. Consequently, $T_{\sigma}=\infty$. Condition (3) holds.
We complete the proof.
In the following, we consider two circles. $$\partial B_1=\big\{(x,y)\mid (x-R)^2+\big(y-(1+R/a)\sqrt{1/A^2-a^2}\big)^2=R^2\big\}$$ and $$\partial B_2=\big\{(x,y)\mid \big(x-R)^2+(y-(1+R/a)\sqrt{1/A^2-a^2}\big)^2=\frac{1}{A^2}(1+R/a)^2\big\},$$ where $R>1/A$. And we let $$(0,K)=\partial B_2\cap\{x=0\mid y>0\}.$$ It is easy to check that $\partial B_2$ intersects $\Gamma^*$ tangentially at $(-a,0)$. Let $R(t)$ be the solution of $$\label{eq:circle}
R^{\prime}(t)=A-1/R(t)$$ with $R(0)=R$. Since $R(0)>1/A$, $R(t)$ is increasing and $\lim\limits_{t\rightarrow\infty}R(t)=\infty$. Noting $a\leq 1/A$ and $(1+R/a)/A>R$, there exists $t^*$ such that $R(t^*)=(1+R/a)/A$.
\[lem:speed\] Let point $(0,y_{\sigma}(t))=\Gamma_{\sigma}(t)\cap\{(x,y)\mid x=0\}$, $t<T_{\sigma}$. There exists $\sigma_1$(indeed $\sigma_1$ in this lemma is the one we want to choose in Proposition \[prop:sigmalarge\]) such that for all $\sigma>\sigma_1$ there holds that if $t^*<T_{\sigma}$, $$y_{\sigma}(t)>K,\ t<t^*.$$
We use the Grim reaper to prove this lemma.
Seeing that Grim reaper given by Lemma \[lem:grimreaper\] $$G(x,t)=C-\frac{t}{b}+b\ln \cos\frac{x}{b}$$ is a traveling wave with uniform speed $1/b$, then choose $C$ large enough such that $G(0,t)=C-t/b>C-t^*/b>K$, $t<t^*$.
We can choose $\sigma_1$ such that for all $\sigma>\sigma_1$, $\Gamma_{\sigma}\succ \gamma_{G}(0)$.
If $t^*<T_{\sigma}$, Lemma \[lem:grimreaper\] implies that $\Gamma_{\sigma}(t)\succ \gamma_{G}(t)$, for $t<t^*$. This means that $y_{\sigma}(t)>K$, $t<t^*$.
Choose $\sigma_1$ as in Lemma \[lem:speed\].
[**Step 1.**]{} For $\sigma>\sigma_1$, if $T_{\sigma}\leq t^*$($t^*$ is given in Lemma \[lem:speed\]), this means that $T_{\sigma}<\infty$. By Lemma \[lem:sigulartime\], only the condition (4) in Lemma \[lem:sigulartime\] can hold. Consequently, the result is true.
[**Step 2.**]{} For $\sigma>\sigma_1$, if $t^*<T_{\sigma}$, by Lemma \[lem:speed\], $y_{\sigma}(t)>K$, $t<t^*$. Here we prove that there holds $$\Gamma_{\sigma}(t)\succ\Gamma^*,\ t^*<t<T_{\sigma}.$$
Let $$\partial B(t)=\big\{(x,y)\mid (x-R)^2+\big(y-(1+R/a)\sqrt{1/A^2-a^2}\big)^2=R(t)^2\big\},$$ where $R(t)$ is given by (\[eq:circle\]). It is easy to see that $\partial B(t)$ evolves by $V=-\kappa+A$.
Let $\Sigma(t)=\partial B(t)\cap\{(x,y)\mid x\leq0,y\geq0\}$. There exists $\delta$ satisfying $0<\delta<t^*$ such that $$R(\delta)=\sqrt{R^2+(1+R/a)^2(1/A^2-a^2)}.$$ Obviously, $\partial B(\delta)$ passes through the origin $(0,0)$. Seeing the Figure \[fig:circlesub1\] and \[fig:circlesub2\] and noting the choice of $t^*$, the boundary of $\Sigma(t)$ does not intersect $\Gamma_{\sigma}(t)$, $t<t^*$. By maximum principle, $\Sigma(t)$ can not intersect $\Gamma_{\sigma}(t)$ interior. Therefore, $\Gamma_{\sigma}(t)$ does not intersect $\Sigma(t)$, $t<t^*$. Here we omit the detail.
![Proof of Proposition \[prop:sigmalarge\][]{data-label="fig:circlesub1"}](circlesub1.pdf){height="6.0cm"}
![Proof of Proposition \[prop:sigmalarge\][]{data-label="fig:circlesub2"}](circlesub2.pdf){height="6.0cm"}
Seeing the Figure \[fig:circlesub3\], $\Sigma(t^*)$ intersects $\Gamma^*$ tangentially at $(-a,0)$. Since $\Gamma_{\sigma}(t)$ does not intersect $\Sigma(t)$ for $t<t^*$, then $\Gamma(t^*)\succ\Gamma^*$. Therefore, $\Gamma(t)\succ\Gamma^*$ for $t^*<t<T_{\sigma}$. Let $T_{\sigma}^*=t^*$, we complete the proof.
![Proof of Proposition \[prop:sigmalarge\][]{data-label="fig:circlesub3"}](circlesub3.pdf){height="6.0cm"}
\[prop:sigmasmall\] There exists $\sigma_2>0$ such that for all $\sigma<\sigma_2$, $T_{\sigma}=\infty$ and $\Gamma_{\sigma}(t)\rightarrow \Gamma_*$ in $C^1$, as $t\rightarrow\infty$.
[**Step 1.**]{} Upper bound.
There exists $\sigma_2$ such that for all $\sigma<\sigma_2$, $\Gamma_*\succ\Gamma_{\sigma}$. Since $\Gamma_{\sigma}$ is represented by the graph of $\sigma\varphi$, then $\Gamma_{\sigma}(t)$ can locally be represented by the graph of some function $u_{\sigma}(x,t)$. Let $T_{\sigma}^g$ be the maximal time such that $$\Gamma_{\sigma}(t)=\{(x,y)\mid y=u_{\sigma}(x,t)\},\ 0\leq t<T_{\sigma}^g.$$ Therefore, $u_{\sigma}(x,t)$ satisfies $$\label{eq:drivinggraph}
\left\{
\begin{array}{lcl}
\dis{u_t=\frac{u_{xx}}{1+u_x^2}+A\sqrt{1+u_x^2}},\ -a<x<a,\ 0<t<T_{\sigma}^g,\\
u(-a,t)=u(a,t)=0,\ 0<t<T_{\sigma}^g,\\
u(x,0)=\sigma\varphi(x),\ -a\leq x\leq a.
\end{array}
\right.$$
Since for all $\sigma<\sigma_2$ there holds $\sigma\varphi(x)\leq \sqrt{1/A^2-x^2}-\sqrt{1/A^2-a^2}$, $-a\leq x\leq a$, by comparison principle, $$\label{eq:upb}
u_{\sigma}(x,t)< \sqrt{1/A^2-x^2}-\sqrt{1/A^2-a^2},\ -a< x< a, \ 0<t<T_{\sigma}^g.$$
[**Step 2.**]{} Lower bound and derivative estimate.
If $0\leq \sigma<\sigma_2$, by comparison principle, $$\label{eq:downb1}
u_{\sigma}(x,t)>0,\ -a< x< a, \ 0<t<T_{\sigma}^g.$$
Combining (\[eq:upb\]) and (\[eq:downb1\]), $$\label{eq:db1}
-\frac{a}{\sqrt{1/A^2-a^2}}< u_{\sigma x}(a,t)<0\ \text{and}\ 0< u_{\sigma x}(-a,t)<\frac{a}{\sqrt{1/A^2-a^2}},\ 0<t<T_{\sigma}^g.$$
Differentiating the first equation in (\[eq:drivinggraph\]) by $x$ and combining boundary condition (\[eq:db1\]), by maximum principle, $$\label{eq:db2}
|u_{\sigma x}(x,t)|<\frac{a}{\sqrt{1/A^2-a^2}},\ -a\leq x\leq a, \ 0<t<T_{\sigma}^g.$$
If $\sigma<0$, let $k>0$ satisfy $k:=\sigma\varphi^{\prime}(a)$. We denote function $$\underline{u}(x)=\max\{-k(x+a),k(x-a)\},\ -a\leq x\leq a.$$ Obviously, $\underline{u}(x)\leq\sigma\varphi$, $-a\leq x\leq a$ and $\underline{u}$ is a sub-solution of (\[eq:drivinggraph\]) in viscosity sense.
Therefore, by maximum principle, $u_{\sigma}(x,t)>\underline{u}(x)$, $-a<x<a$, $0<t<T_{\sigma}^g$. Combining (\[eq:upb\]), we have $$\label{eq:db3}
|u_{\sigma x}(x,t)|<\max\{k,\frac{a}{\sqrt{1/A^2-a^2}}\},\ -a\leq x\leq a, \ 0<t<T_{\sigma}^g.$$
Consequently, (\[eq:db2\]) and (\[eq:db3\]) imply that there exists $C_{\sigma}$ such that $$\label{eq:dbc}
|u_{\sigma x}(x,t)|\leq C_{\sigma},\ -a\leq x\leq a, \ 0<t<T_{\sigma}^g.$$
[**Step 3.**]{} We prove the convergence in this step.
By [@LSU], for $\epsilon>0$, $u_{\sigma xx}(x,t)$ is bounded for all $-a\leq x\leq a,\ \epsilon\leq t<T_{\sigma}^g$. This means that $T_{\sigma}^g=\infty$. Therefore, by [@LSU] again, $u_{\sigma}(x,t)$, $u_{\sigma t}(x,t)$, $u_{\sigma tt}(x,t)$, $u_{\sigma x}(x,t)$, $u_{\sigma xx}(x,t)$ and $u_{\sigma xxx}(x,t)$ are all bounded for some constant $D_{\sigma}>0$, $-a\leq x\leq a,\ \epsilon\leq t<\infty$. For any sequence $t_n\rightarrow\infty$, there exist a subsequence $t_{n_j}$ and function $v(x,t)$ such that $$u_{\sigma}(\cdot,\cdot+t_{n_j})\rightarrow v,\ \text{in}\ C^{2,1}([-a,a]\times[\epsilon,\infty)),$$ as $j\rightarrow\infty$.
[**Step 4.**]{} In this step, we introduce a Lyapunov function.
Let $$J[u]=\int_{-a}^a\sqrt{1+u_x^2}dx.$$ If $u$ is a solution of (\[eq:drivinggraph\]), we calculate $$\begin{aligned}
\frac{d}{dt}J[u]&=&\int_{-a}^a\frac{u_xu_{xt}}{\sqrt{1+u_x^2}}dx=-\int_{-a}^a \frac{u_tu_{xx}}{(1+u_x^2)^{3/2}}dx=-\int_{-a}^a\frac{(u_t)^2}{\sqrt{1+u_x^2}}dx +A\int_{-a}^au_tdx\\
&=&-\int_{-a}^a\frac{(u_t)^2}{\sqrt{1+u_x^2}}dx+A\frac{d}{dt}\int_{-a}^audx.\end{aligned}$$ Therefore, there hold $$J[u(\cdot,t)]\leq J[u(\cdot,\epsilon)]+A\int_{-a}^au(x,t)dx-A\int_{-a}^au(x,\epsilon)dx$$ and $$\int_{\epsilon}^{\infty}\int_{-a}^a\frac{(u_t)^2}{\sqrt{1+u_x^2}}dxdt=A\lim\limits_{t\rightarrow\infty}\int_{-a}^au(x,t)dx-A\int_{-a}^au(x,\epsilon)dx+J[u(\cdot,\epsilon)]-\lim\limits_{t\rightarrow \infty}J[u(\cdot,t)].$$
[**Step 5.**]{} Using the Lyapunov function we complete the proof.
For $u_{\sigma}$ given by above, $u_{\sigma}(x,t)$ is uniformly bounded for $-a\leq x\leq a$, $0<t<\infty$. Then the integral $$|\int_{-a}^au_{\sigma}(x,t)dx|$$ is bounded for $0<t<\infty$. Consequently, $J[u_{\sigma}(\cdot,t)]$ is bounded for $0<t<\infty$. Therefore, the integral $$\int_{\epsilon}^{\infty}\int_{-a}^a\frac{(u_{\sigma t})^2}{\sqrt{1+u_{\sigma x}^2}}dxdt$$ is integrable. Then for all $s_0>0$, $$\int_{s_0}^{s_0+1}\int_{-a}^a\frac{(u_{\sigma t})^2}{\sqrt{1+u_{\sigma x}^2}}(x,t+t_{n_j})dxdt=\int_{s_0+t_{n_j}}^{s_0+1+t_{n_j}}\int_{-a}^a\frac{(u_{\sigma t})^2}{\sqrt{1+u_{\sigma x}^2}}(x,t)dxdt\rightarrow0$$ as $j\rightarrow\infty$. Seeing $u_{\sigma}(\cdot,\cdot+t_{n_j})\rightarrow v$, in $\ C^{2,1}([-a,a]\times[\epsilon,\infty))$, as $j\rightarrow\infty$, we have $$\int_{s_0}^{s_0+1}\int_{-a}^a\frac{(v_{ t})^2}{\sqrt{1+v_x^2}}(x,t)dxdt=0.$$ Then $v_t(x,t)=0$, for all $-a\leq x\leq a$, $s_0\leq t\leq s_0+1$. Seeing that the choice of $s_0$ is arbitrary, $$v_t(x,t)=0,$$ for all $-a\leq x\leq a$, $0<t<\infty$. Therefore, $v$ is independent on $t$ and is a stationary solution of (\[eq:drivinggraph\]). Then $$v=\sqrt{1/A^2-x^2}-\sqrt{1/A^2-a^2},\ -a\leq x\leq a.$$
Here we get that for any sequence $t_n\rightarrow\infty$, there exists a subsequence $t_{n_j}$ such that $$u_{\sigma}(\cdot,\cdot+t_{n_j})\rightarrow v,\ \text{in}\ C^{2,1}([-a,a]\times[\epsilon,\infty)),$$ as $j\rightarrow\infty$. Consequently, $$u_{\sigma}(\cdot,t)\rightarrow v,\ \text{in}\ C^{2}([-a,a]),$$ as $t\rightarrow\infty$.
The proof of this proposition is completed.
Asymptotic behavior for the condition (3) in Lemma \[lem:sigulartime\]
======================================================================
Seeing Lemma \[lem:regular\] and the proof of Lemma \[lem:sigulartime\], under the condition (3) in Lemma \[lem:sigulartime\] we can assume there exists $\rho_{\sigma}$ such that $$\Gamma_{\sigma}(t)=\{(\rho_{\sigma}(\theta,t)\cos\theta,\rho_{\sigma}(\theta,t)\sin\theta)\mid0\leq \theta\leq \pi\},\ 0\leq t<\infty.$$ Moreover $\rho_{\sigma}$ satisfies (\[eq:polarco\]) for $T_{\sigma}=\infty$.
\[lem:ly\] Let $L_{\sigma}(t)$ be the length of $\Gamma_{\sigma}(t)$ and $S_{\sigma}(t)$ be the area of the domain surrounded by $\Gamma_{\sigma}(t)$ and $y=0$. Then, $$\label{eq:ly2}
\frac{d}{dt}L_{\sigma}(t)=-\int_{0}^{L_{\sigma}(t)}(\kappa-A)^2ds+A\frac{d}{dt}S_{\sigma}(t).$$
(1). Noting that under the condition (3) in Lemma \[lem:sigulartime\], $\Gamma_{\sigma}(t)$ located in $\{y\geq0\}$, the definition of $S_{\sigma}(t)$ is well-defined.
(2). The result of this lemma is a general condition for the Lyapunov function in the proof of Proposition \[prop:sigmasmall\].
Seeing the calculation in [@Z2], $$\frac{d}{dt}L_{\sigma}(t)=\int_0^{L_{\sigma}(t)}(A\kappa-\kappa^2)ds.$$ Recall $N$ being the unit downward normal vector. Therefore, $$\begin{aligned}
\frac{d}{dt}L_{\sigma}(t)&=&-\int_0^{L_{\sigma}(t)}(\kappa-A)^2ds+\int_0^{L_{\sigma}(t)}(-A\kappa+A^2)ds=-\int_0^{L_{\sigma}(t)}(\kappa-A)^2ds\\
&+&A\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds,\end{aligned}$$ where $F$ is the point on the curve $\Gamma_{\sigma}(t)$ and for convenience, we omit the subscript of $F_{\sigma}(s,t)$. Let $$\gamma_{\sigma}(t)=\Gamma_{\sigma}(t)\cup\{(x,y)\mid y=0,-a\leq x\leq a\}.$$ By Green’s formula, $$\begin{aligned}
\frac{d}{dt}S_{\sigma}(t)=\frac{1}{2}\frac{d}{dt}\int_{\gamma_{\sigma}(t)}F(s,t)\cdot(-N)ds,\end{aligned}$$ where $F$ is the point on the curve $\gamma_{\sigma}(t)$ and $N$ is the unit inner normal vector. Since the curve $$\{(x,y)\mid y=0,-a\leq x\leq a\}$$ is independent on $t$, $$\frac{1}{2}\frac{d}{dt}\int_{\gamma_{\sigma}(t)}F(s,t)\cdot(-N)ds=\frac{1}{2}\frac{d}{dt}\int_{\Gamma_{\sigma}(t)}F(s,t)\cdot(-N)ds=\frac{1}{2}\frac{d}{dt}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-N)ds.$$ By calculation, $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-N)ds&=&\frac{1}{2}\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds+\frac{1}{2}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-\frac{d}{dt}N)ds\\
&+&\frac{1}{2}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-N)(A\kappa-\kappa^2)ds.\end{aligned}$$ Seeing the calculation in [@Z2], $$\frac{d}{dt}N=-\frac{\partial\kappa}{\partial s}T,$$ where $T=F_s$–the unit tangential vector, then $$\begin{aligned}
\frac{1}{2}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-\frac{d}{dt}N)ds=\frac{1}{2}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(\frac{\partial\kappa}{\partial s}T)ds:=I.\end{aligned}$$ Seeing the symmetry of $\Gamma_{\sigma}(t)$ and $\kappa(0,t)=\kappa(L_{\sigma}(t),t)=A$, at the boundary, $$F(0,t)\cdot T(0,t)\kappa(0,t)=F(L_{\sigma}(t),t)\cdot T(L_{\sigma}(t),t)\kappa(L_{\sigma}(t),t)=AF(0,t)\cdot T(0,t).$$ Integrating $I$ by parts, there holds $$\begin{aligned}
I&=&-\frac{1}{2}\int_0^{L_{\sigma}(t)}F_s\cdot T\kappa ds-\frac{1}{2}\int_0^{L_{\sigma}(t)}F\cdot T_{s}\kappa ds=-\frac{1}{2}\int_0^{L_{\sigma}(t)}\kappa ds\\
&-&\frac{1}{2}\int_0^{L_{\sigma}(t)}F\cdot F_{ss}\kappa ds=-\frac{1}{2}\int_0^{L_{\sigma}(t)}\kappa ds-\frac{1}{2}\int_0^{L_{\sigma}(t)}F\cdot N \kappa^2 ds.\end{aligned}$$ Therefore, $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\int_0^{L_{\sigma}(t)}F(s,t)\cdot(-N)ds&=&\frac{1}{2}\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds-\frac{1}{2}\int_0^{L_{\sigma}(t)}\kappa ds\\
&-&\frac{1}{2}\int_0^{L_{\sigma}(t)}AF(s,t)\cdot(\kappa N)ds=\frac{1}{2}\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds\\
&-&\frac{1}{2}\int_0^{L_{\sigma}(t)}\kappa ds-\frac{1}{2}\int_0^{L_{\sigma}(t)}AF(s,t)\cdot(T_{s})ds\\
&=&\frac{1}{2}\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds+\frac{1}{2}\int_0^{L_{\sigma}(t)}(A-\kappa)ds\\
&=&\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds.\end{aligned}$$ In the last second equality, we use integral by parts. Therefore, $$\int_0^{L_{\sigma}(t)}\frac{d}{dt}F(s,t)\cdot(-N)ds=\frac{d}{dt}S_{\sigma}(t).$$ Consequently, (\[eq:ly2\]) holds.
\[lem:rhob\] Under the condition (3) in Lemma \[lem:sigulartime\], $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<\infty$, there exist $\rho_2>\rho_1>0$ such that $$\rho_1<\rho_{\sigma}(\theta,t)<\rho_2,$$ for $0\leq \theta\leq\pi$, $0<t<\infty$.
First, we prove $\rho_{\sigma}<\rho_2$. We prove this by contradiction, assuming $\rho_{\sigma}$ is not bounded from above.
If $\rho_{\sigma}(\pi/2,t)$ is bounded for all $t$, we can easily prove that there exist some $0<\theta_0<\pi/2$ and $t_0$ such that $\kappa_{\sigma}(\theta_0,t_0)\leq 0$. This contradicts to that $\kappa_{\sigma}(\theta,t)>0$, for all $0<\theta<\pi$, $t<\infty$.
Therefore, $\rho_{\sigma}(\pi/2,t)$ is not bounded. Assume for some $t_0$, $\rho_{\sigma}(\pi/2,t_0)$ is large enough. We can use the Grim reaper argument as in Proposition \[prop:sigmalarge\] to prove that $\Gamma_{\sigma}(t)\succ\Gamma^*$ in finite time. This contradicts to that $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<\infty$. Therefore, there exists $\rho_2>0$ such that $\rho_{\sigma}(\theta,t)<\rho_2$, for all $0<\theta<\pi$, $t<\infty$.
On the other hand, we note that $\Gamma_{\sigma}\succ \Lambda_0=\{(x,y)\mid y=0,\ -a\leq x\leq a\}$, $0<t<\infty$. Then there exists $\rho_1>0$ such that $\rho_{\sigma}(\theta,t)>\rho_1$, $0<\theta<\pi$, $t<\infty$.
We complete the proof.
Here we give the asymptotic behavior under the condition (3) in Lemma \[lem:sigulartime\].
\[pro:asymup\] Under the condition (3) in Lemma \[lem:sigulartime\], $SGN(\Gamma_{\sigma}(t),\Gamma^*)=[-\ +\ -]$ for $t<\infty$, $$\Gamma_{\sigma}(t)\rightarrow \Gamma^*\ \text{in}\ C^1$$ as $t\rightarrow\infty$.
Since $\rho_1<\rho_{\sigma}<\rho_2$ and $\rho_{\sigma}$ satisfies (\[eq:polarco\]), then there exists $\epsilon>0$ such that $\rho_{\sigma t}$, $\rho_{\sigma tt}$, $\rho_{\sigma\theta}$, $\rho_{\sigma\theta\theta}$ and $\rho_{\sigma\theta\theta\theta}$ are bounded for $0\leq \theta\leq \pi$, $\epsilon \leq t<\infty$. Therefore for any $t_n\rightarrow\infty$, there exist a subsequence $t_{n_j}$ and a function $r(\theta,t)$ such that $$\rho_{\sigma}(\cdot,\cdot+t_{n_j})\rightarrow r\ \text{in}\ C^{2,1}([0,\pi]\times[\epsilon,\infty)),$$ as $j\rightarrow\infty$. Let $$\gamma_{r}(t)=\{(x,y)\mid x=r(\theta,t)\cos\theta,\ y=r(\theta,t)\sin\theta,\ 0\leq\theta\leq \pi\},$$ and the curvature $\kappa_{\sigma}(\cdot,t)$ be the curvature on $\Gamma_{\sigma}(t)$, $\kappa_{r}(\cdot,t)$ be the curvature on $\gamma_{r}(t)$. Therefore, $\kappa_{\sigma}(\cdot,\cdot+t_{n_j})\rightarrow \kappa_{r}$ in $C([0,\pi]\times[\epsilon,\infty))$. Obviously, the length $L_{\sigma}(t)$ and the area $S_{\sigma}(t)$ are bounded. Using the same argument in Proposition \[prop:sigmasmall\] and the Lyapunov function in Lemma \[lem:ly\], we can deduce that $\kappa_{r}\equiv A$. Consequently, $r_t(\theta,t)=0$ for all $0\leq\theta\leq\pi$ and $t>0$. Seeing the curvature of $\gamma_{r}$ is a positive constant, $\gamma_{r}$ can only be a part of circle with radius $1/A$. Seeing $r(0,t)=-a$ and $r(\pi,t)=a$, then $\gamma_{r}=\Gamma^*$ or $\gamma_{r}=\Gamma_*$. But for all $t>0$, $$\Gamma_{\sigma}(t+t_{n_j})\rightarrow \gamma_{r}\ \text{in}\ C^1(\text{indeed,\ the\ convergence\ can\ be\ proved\ in\ $C^2$}),$$ as $j\rightarrow\infty$. If $\gamma_{r}=\Gamma_*$, then for $t$ large enough, $\Gamma^*\succ\Gamma_{\sigma}(t)$. This yields a contradiction. Therefore, $\gamma_{r}=\Gamma^*$. Consequently, $$\Gamma_{\sigma}(t)\rightarrow\Gamma^*\ \text{in}\ C^1(\text{indeed,\ the\ convergence\ can\ be\ proved\ in\ $C^2$}),$$ as $t\rightarrow\infty$.
Here we complete the proof.
Proof of Theorem \[thm:category\]
=================================
\[lem:sigmasmallc\] The set $$B_*=\{\sigma\in \mathbb{R}\mid\Gamma_{\sigma}(t)\rightarrow\Gamma_*\ \text{in}\ C^1,\ \text{as}\ t\rightarrow\infty\}$$ is open and connect.
Proposition \[prop:sigmasmall\] implies that $(-\infty,0]\subset B_*\neq \emptyset$. Therefore we only consider $\sigma_1>0$ and $\sigma_1\in B_*$.
(1). We prove that, for all $\sigma<\sigma_1$, $\Gamma_{\sigma}(t)\rightarrow\Gamma_*\ \text{in}\ C^1$, as $t\rightarrow\infty$.
Since $\Gamma_{\sigma_1}(t)\rightarrow\Gamma_*\ \text{in}\ C^1$, as $t\rightarrow\infty$, then there holds $\Gamma^*\succ\Gamma_{\sigma_1}(t)$ for $t$ large enough. By comparison principle, $\Gamma_{\sigma_1}(t)\succ\Gamma_{\sigma}(t)$, $t<T_{\sigma}$. These imply that the condition (4) in Lemma \[lem:sigulartime\] can not hold. Therefore, $T_{\sigma}=\infty$. By the same argument in Proposition \[prop:sigmasmall\], we can prove $$\Gamma_{\sigma}(t)\rightarrow\Gamma_*\ \text{in}\ C^1,$$ as $t\rightarrow\infty$. Here we prove that $B_*$ is connect.
(2). We are going to prove $B_*$ is open. We only need prove that there exists $\epsilon_0>0$, $(\sigma_1,\sigma_1+\epsilon_0)\subset B_*$. We divide this proof into two steps.
[**Step 1.**]{} Let $$\tau_0(\sigma)=\max\{\tau \mid\frac{d}{ds}F_{\sigma}(0,t)\cdot(0,1)>0, 0<t<\tau\},\ \text{for}\ \sigma>0.$$ By comparison principle, we can prove that $\tau_0(\sigma)$ is non-increasing with respect to $\sigma$. let $\tau^*=\sup\{\tau_0(\sigma)\mid\sigma>\sigma_1\}=\lim\limits_{\sigma\downarrow\sigma^1}\tau_0(\sigma)$. We claim that $\tau^*=\infty$.
For all $t<\tau^*$, there exists $\delta_0$, for $\sigma\in(\sigma_1,\sigma_1+\delta_0)$, $\tau_0(\sigma)>t$. Therefore, for $\sigma\in(\sigma_1,\sigma_1+\delta_0)$, $\Gamma_{\sigma}(t)$ can be represented by polar coordinate and not become singular. By Lemma \[lem:localconti\], $$\Gamma_{\sigma}(t)\rightarrow\Gamma_{\sigma_1}(t),$$ as $\sigma\rightarrow\sigma_1$. Seeing that the condition (1) or (2) in Lemma \[lem:sigulartime\] hold for $\Gamma_{\sigma_1}(t)$, we can prove that there exists $\delta>0$ such that $$\frac{d}{ds}F_{\sigma_1}(0,t)\cdot(0,1)>\delta,\ t<\infty.$$ Consequently, $$\lim\limits_{\sigma\downarrow\sigma^1}\frac{d}{ds}F_{\sigma}(0,t)\cdot(0,1)\geq\delta,\ t<\tau^*.$$ Therefore, $\tau^*=\infty$.
[**Step 2.**]{} We complete the proof.
We choose two curves $\gamma_1$ and $\gamma_2$ such that $\Gamma^*\succ\gamma_1\succ\Gamma_*\succ\gamma_2$ and the domain $V$ be the shuttle neighbourhood of $\Gamma_*$ satisfying $\partial V=\gamma_1\cap\gamma_2$.
Since $$\Gamma_{\sigma_1}(t)\rightarrow\Gamma_*\ \text{in}\ C^1,$$ as $t\rightarrow\infty$, for $t_0$ large enough $\Gamma_{\sigma_1}(t_0)\subset V$. Seeing the result in Step 1, for $\sigma$ close to $\sigma_1$, $\Gamma_{\sigma}(t_0)$ can be represented by polar coordinate and not become singular. By Lemma \[lem:localconti\], $$\Gamma_{\sigma}(t_0)\rightarrow\Gamma_{\sigma_1}(t_0),\ \text{in}\ C^1,$$ as $\sigma\rightarrow\sigma_1$. Then there exists $\epsilon_0$ for $\sigma\in(\sigma_1,\sigma_1+\epsilon_0)$, $\Gamma_{\sigma}(t_0)\subset V$. Using the Lyapunov function given by Lemma \[lem:ly\] and the same argument in Proposition \[pro:asymup\], for all $\sigma\in(\sigma_1,\sigma_1+\epsilon_0)$, $$\Gamma_{\sigma}(t)\rightarrow\Gamma_*\ \text{in}\ C^1,$$ as $t\rightarrow\infty$.
We complete the proof.
\[lem:sigmalargec\] The set $$B^*=\{\sigma\in \mathbb{R}\mid \text{there\ exists}\ T_{\sigma}^*>0\ \text{such\ that}\ \Gamma_{\sigma}(t)\succ\Gamma^*,\ T_{\sigma}^*<t<T_{\sigma}\}$$ is open and connect.
Proposition \[prop:sigmalarge\] and \[prop:sigmasmall\] show that $B^*\subset (0,\infty)$ is not empty. We consider $\sigma_2>0$ and $\sigma_2\in B^*$. Then there exists $T_{\sigma_2}^*>0$ such that $$\Gamma_{\sigma_2}(t)\succ\Gamma^*,\ T_{\sigma_2}^*<t<T_{\sigma_2}.$$
(1). We prove $(\sigma_2,\infty)\subset B^*$.
For $\sigma>\sigma_2$, if $T_{\sigma}<\infty$, then only the condition (4) in Lemma \[lem:sigulartime\] can hold. The result is true. If $T_{\sigma}=\infty$, then by comparison principle, $$\Gamma_{\sigma}(t)\succ\Gamma_{\sigma_2}(t)\succ\Gamma^*,\ T_{\sigma_2}^*<t<T_{\sigma_2}.$$ Here we complete the proof that $B^*$ is connect.
(2). We prove $B^*$ is open. We only need prove that there exists $\epsilon_0>0$ such that $(\sigma_2-\epsilon_0,\sigma_2)\subset B^*$.
We can choose $t_0$ such that $\Gamma_{\sigma_2}(t_0)\succ \Gamma^*$ and $$\frac{d}{ds}F_{\sigma_2}(0,t)\cdot(0,1)>0,\ 0<t\leq t_0.$$ By Lemma \[lem:regular\] and comparison principle, it is easy to see that for all $0<\sigma<\sigma_2$, $T_{\sigma}>t_0$. For $\sigma$ close to $\sigma_2$, $\Gamma_{\sigma}(t)$ can be represented by polar coordinate for $0<t\leq t_0$. By Lemma \[lem:localconti\], $$\Gamma_{\sigma}(t_0)\rightarrow\Gamma_{\sigma_2}(t_0),\ \text{in}\ C^1$$ as $\sigma\rightarrow\sigma_2$. Therefore, there exists $\epsilon_0>0$ such that for all $\sigma\in(\sigma_2-\epsilon_0,\sigma_2)$, $\Gamma_{\sigma}(t_0)\succ\Gamma^*$. By comparison principle, we can get the result easily.
We complete the proof.
\[cor:connect\] There exist $0<\sigma_*\leq \sigma^*$ such that $$B^*=(\sigma^*,\infty)$$ and $$B_*=(-\infty,\sigma_*).$$
Let $\sigma^*=\inf B^*$ and $\sigma_*=\inf B_*$. Obviously, $\sigma_*\leq\sigma^*$.
Lemma \[lem:sigmasmallc\] and \[lem:sigmalargec\] imply that $$B^*=(\sigma^*,\infty)$$ and $$B_*=(-\infty,\sigma_*).$$ Proposition \[prop:sigmasmall\] shows that $\sigma_*>0$. The proof is completed.
\[pro:barrierfunction\] If $\Gamma_{\sigma_0}(t)\rightarrow\Gamma^*$ in $C^1$, for some $\sigma_0$, as $t\rightarrow\infty$, then $(\sigma_0,\infty)\subset B^*$.
For $\sigma>\sigma_0$, if $T_{\sigma}<\infty$, then the result is true. In the following proof, we only consider $T_{\sigma}=\infty$. By comparison principle, $\Gamma_{\sigma}(t)\succ\Gamma_{\sigma_0}(t)$, $t>0$.
Since $$\Gamma_{\sigma_0}(t)\rightarrow\Gamma^*\ \text{in}\ C^1,$$ there exist $t_0$ and $\delta>0$ such that for all $t\geq t_0$, there hold $$\label{eq:ylarz}
\frac{d}{ds}F_{\sigma_0}(0,t)\cdot(0,1)\geq \delta$$ and $$\label{eq:xlesz}
\frac{d}{ds}F_{\sigma_0}(0,t)\cdot(1,0)\leq -\delta.$$ Since $\Gamma_{\sigma}(t_0)\succ\Gamma_{\sigma_0}(t_0)$, then we can choose a small positive constant $c$ such that $$\Gamma_{\sigma}(t_0)\succ\Gamma^{c}(t_0),$$ where $$\Gamma^{c}(t)=\{(x,y+c)\mid (x,y)\in\Gamma_{\sigma_0}(t)\}\cup \{(-a,y)\mid 0\leq y\leq c\}\cup \{(a,y)\mid 0\leq y\leq c\},\ t>0.$$
![Construction of $\Gamma_c(t)$[]{data-label="fig:barrier"}](barrier.pdf){height="6.0cm"}
We claim that $\Gamma^c(t)$ is a sub-solution for $t\geq t_0$. Indeed, the part $\Gamma^c(t)\cap\{y> c\}$ is a translation of $\Gamma_{\sigma_0}(t)$. $\Gamma^c(t)\cap\{y> c\}$ satisfies (\[eq:meancurpara\]). Since the part $\Gamma^c(t)\cap\{y< c\}$ consists of two straight lines, then the part is a sub-solution of (\[eq:meancurpara\]). Next at the points $\{(-a,c),(a,c)\}=\Gamma^c(t)\cap\{y=c\}$, seeing (\[eq:ylarz\]), (\[eq:xlesz\]) for any smooth flow $S(t)$ can not touch $(-a,c)$ or $(a,c)$ above only once. Therefore $\Gamma^c(t)$ is a sub-solution of (\[eq:meancurpara\]) and (\[eq:fixbound\]) in the sense of Definition \[def:subsup\].
By $\Gamma_{\sigma}(t_0)\succ\Gamma^c(t_0)$ and $\Gamma^c(t)$ being sub-solution for $t>t_0$, $$\label{eq:largesub}
\Gamma_{\sigma}(t)\succ\Gamma^c(t),\ t>t_0.$$ Noting that $\Gamma^c(t)\rightarrow \Gamma^{*c}$ in $C$, as $t\rightarrow\infty$, where $$\Gamma^{*c}=\{(x,y+c)\mid (x,y)\in\Gamma^*\}\cup \{(-a,y)\mid 0\leq y\leq c\}\cup \{(a,y)\mid 0\leq y\leq c\}.$$ If $\Gamma_{\sigma}(t)$ satisfies the condition (1) or (2) or (3) in Lemma \[lem:sigulartime\], seeing the proof of Proposition \[pro:asymup\], $\Gamma_{\sigma}(t)\rightarrow \Gamma^*$ or $\Gamma_{\sigma}(t)\rightarrow \Gamma_*$, as $t\rightarrow\infty$.
Combining (\[eq:largesub\]), $\Gamma^*\succeq \Gamma^{*c}$ or $\Gamma_*\succeq \Gamma^{*c}$. But all of these conditions are impossible. Therefore, only the condition (4) in Lemma \[lem:sigulartime\] holds.
The proof is completed.
Let $\sigma_*$ and $\sigma^*$ be given by Corollary \[cor:connect\].
Seeing the definition of $\sigma_*$, $\sigma_*\notin B^*$ and $\sigma_*\notin B_*$. Therefore, $\Gamma_{\sigma_*}(t)$ only satisfies the condition (3) in Lemma \[lem:sigulartime\]. The result in Section 5 shows that $\Gamma_{\sigma_*}(t)\rightarrow \Gamma^*$, as $t\rightarrow\infty$.
By Proposition \[pro:barrierfunction\], $(\sigma_*,\infty)=B^*$. Consequently, $\sigma_*=\sigma^*$.
The proof of Theorem \[thm:category\] is completed.
[**Acknowledge.**]{} The author is grateful to Professor Matano Hiroshi for his inspiring suggestion about Grim reaper argument. He is also grateful to Professor Giga Yoshikazu for letting me know several related useful papers.
[**Conflict of interest.**]{} We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.
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abstract: 'We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we discus some Riesz type results for holomorphic functions.'
address:
- 'Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica, Montenegro'
- 'Department of Mathematics, University of Prishtina, Mother Teresa, No. 5, 10000, Prishtina, Kosovo'
author:
- David Kalaj
- Elver Bajrami
title: On some Riesz and Carleman type inequalities for harmonic functions on the unit disk
---
[^1]
Introduction and statement of main results
==========================================
Throughout the paper we let $\mathbf{U}=\{z\in\Bbb C: |z|<1\}$ be the open unit disk in the complex plane $\Bbb C$, and let $\mathbf{T}=\{z\in\Bbb C: |z|=1\}$ be the unit circle in $\Bbb C$. The normalized area measure on $\mathbf{U}$ will be denoted by $d\sigma$. In terms of real (rectangular and polar) coordinates, we have $$d\sigma=\frac{1}{\pi}dxdy=\frac{1}{\pi}rdrd\theta,\quad
z=x+iy=re^{i\theta}.$$ For $0<p<+\infty$ let $L^p(\mathbf{U},\sigma)=L^p$ denote the familiar Lebesgue space on $\mathbf{U}$ with respect to the measure $\sigma$. The [*Bergman space*]{} $A^p(\mathbf{U})=A^p$ is the space of all holomorphic functions in $L^p(\mathbf{U},\sigma)$. For a fixed $1\le
p<+\infty$, denote by $A^p_0$ the set of all functions $f\in A^p$ for which $f(0)=0$. For a function $f$ in $L^p(\mathbf{U},\sigma)$, we write $$\Vert f\Vert_p=\left(\int_{\mathbf{U}}|f(z)|^p d\sigma\right)^{1/p}.$$
Bergman space $b^p$ of harmonic functions is defined similarly.
The [*Hardy space*]{} $h^p$ is defined as the space of (complex) harmonic functions $f$ such that $$\|f\|_{h^p}:=\sup_{0\le r < 1}(\int_{0}^{2\pi}|f(re^{it})|^p
\frac{dt}{2\pi})^{1/p}<\infty.$$
If $f\in h^p$, then by [@axl Theorem 6.13], there exists $$f(e^{it})=\lim_{r\to 1} f(re^{it}), a.e.$$ and $f\in L^p(\mathbf{T}).$ It can be shown that there hold $$\|f\|^p_{h^p}=\lim_{r\to 1}\int_{0}^{2\pi}|f(re^{it})|^p \frac{dt}{2\pi}
= \int_{0}^{2\pi}|f(e^{it})|^p \frac{dt}{2\pi}.$$
Similarly we define the [*Hardy space*]{} $H^p$ of holomorphic functions.
The starting point of this paper is the well known isoperimetric inequality for Jordan domains and isoperimetric inequality for minimal surfaces due to Carleman [@tc]. In that paper Carleman, among the other results proved that if $u$ is harmonic and smooth in $\overline {\mathbf{U}}$ then $$\int_{\mathbf{U}}e^{2u}dx dy\le
\frac{1}{4\pi}(\int_0^{2\pi}e^u dt)^2.$$ By using a similar approach as Carleman, Strebel ([@ks]) proved the isoperimetric inequality for holomorphic functions; that is if $f\in H^1(\mathbf{U})$ then $$\label{isoper}\int_{\mathbf{U}}|f(z)|^2 dxdy \le \frac{1}{4\pi} (\int_{\mathbf{T}}|f(e^{it})|dt)^2.$$ This inequality has been proved independently by Mateljević and Pavlović ([@mp]). In [@hw] have been done some generalizations for the space.
In this paper we prove the following results.
\[cp\] Let $f$ be a real harmonic function. For $p>1$ we have that $$\label{cep}\| f\|_{b^{2p}}\le C_p\| f\|_{h^{p}}$$ where $$C_p = M_{2p} =\left\{
\begin{array}{ll}
\frac{\cos{\frac{\pi}{4p}}}{\cos\frac{\pi}{2p}}, & \hbox{if $1<p\le 2$;} \\
\frac{\cos{\frac{\pi}{4p}}}{\sin\frac{\pi}{2p}}, & \hbox{if $p\ge 2$.} \\
\end{array}
\right.$$
The inequality extends the main result in [@eb], where Bajrami proved the same result but only for $p=4$. But the wrong constant appear in [@eb], due to a wrong citation to the Duren approach [@duren p. 67-68]. However the same method produces the same constant namely $C_4= \frac{1}{2\sin\frac{\pi}{16}}\approx 2.56292$. We were not able to check if is sharp. We are thankful to professor Hedenmalm who drawn attention to his paper [@hh], where is treated a problem which suggests that the inequalities from Theorem \[hed\], which we use in order to prove , are maybe not sharp. The inequality improves similar results for real harmonic functions proved by Kalaj and Meštrovi' c in [@ka1] and by Chen and Ponnusamy and Wang in [@spw]. We expect that the inequality is true for complex harmonic mappings for every $p>1$. Kalaj and Meštrović in [@ka1] proved it for $p=2$, namely they obtained that $C_2=\frac{1}{2\sin\frac{\pi}{8}}\approx 1.3$. On the same paper the example $f_a(z)=\Re \frac{z}{1-az}$, when $a\uparrow 1$ produces the constant $C_0=(5/2)^{1/4}\approx 1.257$, so the constant $C_2$ is not far from the sharp constant. In this paper we extend it for $p=4$.
Namely we have
\[c4\] If $f\in h^4(\mathbf{U})$ is a nonzero complex harmonic function then $f\in b^8$ and there hold the inequality
$$\|f\|_{b^8}\le \frac{1}{2\sin\frac{\pi}{16}}\|f\|_{h^4}.$$
The main ingredient of the proofs are some sharp M. Riesz type inequalities proved for Hardy space by Verbitsky ([@ver]):
\[verbi\] Let $p> 1$. For every holomorphic mapping there hold the sharp inequalities $${L_p}\|\Re f\|_{h^p}\le \|f\|_{h^p}\le R_p\|\Re f\|_{h^p},$$ provided ${\lvert\arg{f(0)}-\frac{\pi}{2}\rvert}\ge \frac{\pi }{2\bar p}$, or $f(0)=0$, where $\bar{p}=\max\{p,\frac{p}{p-1}\}$ and
$$R_p = \left\{
\begin{array}{ll}
\frac{1}{\cos\frac{\pi}{2p}}, & \hbox{if $1<p\le 2$;} \\
\frac{1}{\sin\frac{\pi}{2p}}, & \hbox{if $p\ge 2$.}
\end{array}
\right.
\text{ and } L_p = \left\{
\begin{array}{ll}
\frac{1}{\sin\frac{\pi}{2p}}, & \hbox{if $1<p\le 2$;} \\
\frac{1}{\cos\frac{\pi}{2p}}, & \hbox{if $p\ge 2$.}
\end{array}
\right.$$
In addition we refer to the references [@verb1] and [@pik] for related results.
By results of Verbitsky we prove some similar inequalities for Bergman space (Theorem \[hed\]).
From Proposition \[verbi\], we obtain that, if $f=u+iv$ is holomorphic with $f(0)=0$, then $$\label{motiv} \|f\|^p_{H^p}\le \frac{R^p_p}{2} (\|u\|^p_{h^p}+\|v\|^p_{h^p})$$ and $$\label{motiv1}\frac{L^p_p}{2} (\|u\|^p_{h^p}+\|v\|^p_{h^p})\le \|f\|^p_{H^p}.$$
Now we formulate a similar result, where $\frac{R^p_p}{2} $ and $\frac{L^p_p}{2} $ are replaced by smaller, respectively bigger constants for $p$ close to $4$.
\[newt\] Let $p\ge 2$ and let $f=u+i v$ be a holomorphic function on the unit disk so that $f(0)=0$ and let $f\in H^p$. Then for $p\in[2,4]$ we have $$\label{fini}\|f\|_{H^p}\le \left(\frac{p}{p-1}\right)^{1/p} \left(\|u\|_{h^p}^p+\|v\|_{h^p}^p\right)^{1/p}$$ and for $p\ge 4$ we have
$$\label{fini1}\|f\|_{H^p}\ge \left(\frac{p}{p-1}\right)^{1/p} \left(\|u\|_{h^p}^p+\|v\|_{h^p}^p\right)^{1/p}.$$
If $C_p= \left(\frac{p}{p-1}\right)^{1/p}$, then $2^{-1/p}L_p\le C_p$ if $ p\ge 4$ and $C_p\le 2^{-1/p}R_p$, if $p_1\le p\le 4$, so inequalities and improve the inequalities and , if $p_1\le p\le 4$, and if $p\ge 4$, respectively. Here $p_1\approx 2.42484$ is the only solution of the equation $$\left(\frac{p}{p-1}\right)^{1/p}= 2^{-1/p}\frac{1}{\sin\frac{\pi}{2p}},\ \ \ p\ge 2.$$
The proofs are presented in sections 2 and 3 and 4.
Proof of Theorem \[cp\]
=======================
From Proposition \[verbi\] we obtain $$L^p_p\int_{\mathbf{T}} |\Re f(re^{it})|^{p}dt\le \int_{\mathbf{T}} |f(re^{it})|^{p}dt \le R^p_p\int_{\mathbf{T}} |\Re f(re^{it})|^{p}dt.$$ So $$\begin{split}L^{p}_p\int_0^1r dr\int_{\mathbf{T}} |\Re f(re^{it})|^{p}dt&\le \int_0^1\int_{\mathbf{T}} r|f(re^{it})|^{p}dt \\&\le R^p_p\int_0^1r dr\int_{\mathbf{T}} |\Re f(re^{it})|^{p}dt.\end{split}$$ Thus we have the following theorem
\[hed\] Let $p> 2$ and $f=\Re f +i \Im f\in b^p(\mathbf{U})$. If ${\lvert\arg{f(0)}-\frac{\pi}{2}\rvert}\ge \frac{\pi }{2p}$, or $f(0)=0$, then we have $$L_p\|\Re f\|_{b^p}\le \|f\|_{b^p}\le R_p\|\Re f\|_{b^p}$$ and $$L_p\|\Im f\|_{b^p}\le \|f\|_{b^p}\le R_p\|\Im f\|_{b^p}.$$
From now on we will use the shorthand notation $$\int_{\mathbf{T}} f:=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{it}) dt$$ and $$\int_{\mathbf{U}}f:=\frac{1}{\pi}\int_{\mathbf{U}}f(z) dxdy, \ \ \ z=x+iy.$$ Thus for $p>2$ we have $$\begin{split}(\int_{\mathbf{U}} |\Re f|^{p})^{1/p}&\le \frac{1}{L_p}(\int_{\mathbf{U}}|f|^p)^{1/p}\\&\le \frac{1}{L_p}(\int_{\mathbf{T}}|f|^{p/2})^{2/p}\\&\le\frac{R_{p/2}}{L_p}(\int_{\mathbf{T}}|\Re f|^{p/2})^{2/p}\\&= M_p(\int_{\mathbf{T}}|\Re f|^{p/2})^{2/p},
\end{split}$$ where $$M_p=\left\{
\begin{array}{ll}
\frac{\cos{\frac{\pi}{2p}}}{\cos\frac{\pi}{p}}, & \hbox{if $2<p\le 4$;} \\
\frac{\cos{\frac{\pi}{2p}}}{\sin\frac{\pi}{p}}, & \hbox{if $p\ge 4$.} \\
\end{array}
\right.$$ This finishes the proof of Theorem \[cp\].
Proof of Theorem \[c4\]
=======================
We assume that $f(z) = g(z)+\overline{h(z)}$, where $h(0)=0$, and $g$ and $h$ are holomorphic on the unit disk.
The function $|a|^2+|b|^2$ is log-subharmonic, provided that $a$ and $b$ are analytic.
We need to show that $f(z)=\log(|a|^2+|b|^2)$ is subharmonic. By calculation we find $$f_z= \frac{a'\bar a+b'\bar b}{|a|^2+|b|^2}$$ and so $$f_{z\bar z}= \frac{a'\bar a'+b'\bar b'}{|a|^2+|b|^2}-\frac{a'\bar a+b'\bar b}{|a|^2+|b|^2}
\frac{a\bar a'+b\bar b'}{|a|^2+|b|^2}$$ $$=\frac{(|a'|^2+|b'|^2)(|a|^2+|b|^2)-|\bar aa'+\bar bb'|^2}{(|a|^2+|b|^2)^2}$$ which is clearly positive.
Now from the isoperimetric inequality for log-subharmonic functions (e.g. [@ka Lemma 2.2]), we have
\[ipl\] For every positive number $p$ and analytic functions $a$ and $b$ defined on the unit disk $U$ we have that $$\int_{\mathbf{U}}(|a|^2+|b|^2)^{2p}\le \left(\int_{\mathbf{T}}(|a|^2+|b|^2)^{p}\right)^2.$$
Further let $$L=\int_{\mathbf{U}} (|g + \bar h|^2)^4=\int_{\mathbf{U}} (|g|^2 + |h|^2+2\Re (gh))^4.$$ Then
$$\begin{split}
L&
= \sum_{k=0}^4 \binom{4}{k}\int_{\mathbf{U}} (|g|^2+|h|^2)^k (2\Re(gh))^{4-k}\\&\le \sum_{k=0}^4\binom{4}{k}\int_{\mathbf{U}} ((|g|^2+|h|^2)^4)^{k/4} (\int_{\mathbf{U}} |2\Re(gh)|^4)^{(4-k)/4}.
\end{split}$$ Let $E_4 =
{\cos\frac{\pi}{8}}
$. From Lemma \[ipl\] and Theorem \[hed\] we have $$\begin{split}L & \le \sum_{k=0}^4 \binom{4}{k} E_4^{4-k} (\int_{\mathbf{U}} (|g|^2+|h|^2)^4)^{k/4} (\int_{\mathbf{U}} (2|gh|)^4)^{(4-k)/4}
\\&\le \sum_{k=0}^4 \binom{4}{k} E_4^{4-k} (\int_{\mathbf{T}} (|g|^2+|h|^2)^{4/2})^{2k/4} (\int_{\mathbf{T}} (2|gh|)^{4/2})^{2(4-k)/4}.\end{split}$$ Further we have $$\begin{split}
X:&=\int_{\mathbf{T}} |g + \bar h|^4\\&=\int_{\mathbf{T}} (|g|^2 + |h|^2+2\Re (gh))^2\\&=\int_{\mathbf{T}} (|g|^2 + |h|^2)^2+4(|g|^2 + |h|^2)\Re (gh)^2+4(\Re (gh))^2
\\&=\int_{\mathbf{T}} (|g|^2 + |h|^2)^2+4(|g|^2 + |h|^2)\Re (gh)+2|gh|^2.\end{split}$$ Let $$A= \left(\int_{\mathbf{T}} (|g|^2 + |h|^2)^2\right)^{1/2}$$ and $$B = \left(\int_{\mathbf{T}} 4 |g|^2|h|^2\right)^{1/2}.$$ Then $$\begin{split}\left|\int_{\mathbf{T}} (|g|^2 + |h|^2)\Re (gh)\right|&\le |(\int_{\mathbf{T}} ((|g|^2 + |h|^2))^2)^{1/2} (\int_{\mathbf{T}} |gh|^2/2)^{1/2}
\\&=A B\cdot \frac{\sqrt{2}}{4}.\end{split}$$ Further we have $$X\ge A^2+\frac{B^2}{2}-\sqrt{2}AB=(A-\frac{\sqrt{2}}{2}B)^2.$$ Furthermore $$A^2- B^2=\int_{\mathbf{T}} (|g|^2-|h|^2)^2\ge 0$$ and thus $$\label{x1}X\ge (\frac{2-\sqrt{2}}{2})^2 B^2$$ and similarly $$\label{x2}
X\ge \left(\frac{2-\sqrt{2}}{2}\right)^2 A^2.$$ Hence $$\begin{split}\int_{\mathbf{U}} (|g + \bar h|)^8&\le \sum_{k=0}^4 \binom{4}{k}E_4^{4-k}\left(\frac{2}{2-\sqrt{2}}\right)^4 \left(\int_{\mathbf{T}} |g + \bar h|^4\right)^2
\\& = \left(\frac{2(1+E_4)}{2-\sqrt{2}}\right)^4 \left(\int_{\mathbf{T}} |g + \bar h|^4\right)^2.\end{split}$$ So $$\|g+\bar h\|_{b^8}\le \sqrt{\frac{2(1+E_4)}{2-\sqrt{2}}}\|g+\bar h\|_{h^4},$$ where $E_4=\cos\left[\frac{\pi }{8}\right]=\frac{\sqrt{2+\sqrt{2}}}{2}$. Finally we have that $$\|g+\bar h\|_{b^8}\le \frac{1}{2\sin\frac{\pi}{16}}\|g+\bar h\|_{h^4}.$$ Here $\frac{1}{2\sin\frac{\pi}{16}}\approx 2.56292$. This finishes the proof of Theorem \[c4\].
Proof of Theorem \[newt\]
=========================
We use the following form of Green formula
$$\label{green}r \int_0^{2\pi} \frac{\partial G(re^{it})}{\partial r} dt = \int_{|z|\le r} \Delta G(z) dx dy.$$
Let $p>2$ and let $ f = u + i v$ be an analytic function. Let $q=\frac{p}{p-1}$ and $\epsilon>0$ and define $$F_\epsilon(z)=|q\epsilon+|f(z)|^2|^{p/2}$$
$$U_\epsilon(z)=|\epsilon+u(z)^2|^{p/2}$$
$$V_\epsilon(z)=|\epsilon+v(z)^2|^{p/2}$$
Then by direct calculation we obtain $$\label{d1}\Delta F_\epsilon = p(q\epsilon+|f|^2)^{p/2-2} (2q\epsilon + p |f|^2)|f'|^2,$$
$$\label{d2}\Delta U_\epsilon=p(u^2+\epsilon)^{p/2-2} |f'|^2(\epsilon+(p-1) u^2),$$
and
$$\label{d3}\Delta V_\epsilon=p(v^2+\epsilon)^{p/2-2} |f'|^2(\epsilon+(p-1) v^2).$$
If $p=4$, then $$\Delta U_\epsilon+\Delta V_\epsilon=\frac{3}{4}\Delta F_\epsilon .$$
\[new\] Let $f$ be a holomorphic function defined on the unit disk $\mathbf{U}$. For $p>4$ and $z\in\mathbf{U}$ and $\epsilon>0$ we have $$\Delta (U_\epsilon + V_\epsilon)\le \left(\frac{p-1}{p}\right)\Delta F_\epsilon .$$ For $1\le p\le 4$ and $z\in\mathbf{U}$ and $\epsilon>0$ we have $$\Delta (U_\epsilon + V_\epsilon)\ge \left(\frac{p-1}{p}\right)\Delta F_\epsilon .$$
Let $f=u+iv=r e^{is}$ and define $$Q(s)=\frac{\Delta (U_\epsilon + V_\epsilon)}{\Delta F_\epsilon }.$$ Then from , and we have $$Q(s)=\frac{ \left(\epsilon+r^2 \cos^2 s\right)^{-2+\frac{p}{2}} \left(\epsilon+(p-1) r^2 \cos^2 s\right)}{ \left(2 \epsilon+(p-1) r^2\right)\left(\frac{\epsilon p}{-1+p}+r^2\right)^{-2+\frac{p}{2}}}$$ $$+\frac{ \left(\epsilon+r^2 \sin^2 s\right)^{-2+\frac{p}{2}} \left(\epsilon+(p-1) r^2 \sin^2 s\right)}{ \left(2 \epsilon+(p-1) r^2\right)\left(\frac{\epsilon p}{-1+p}+r^2\right)^{-2+\frac{p}{2}}}.$$ We should prove that
$$Q(s)\left\{
\begin{array}{ll}
\ge & \hbox{if $p<4$;} \\
\le & \hbox{if $p\ge 4$.}
\end{array}
\right.$$ First of all $$Q'(s)=\frac{(-2+p) r^2 \left(\frac{\epsilon p}{p-1}+r^2\right)^{2-\frac{p}{2}} \cos s \sin s }{2 \epsilon+(p-1) r^2}T$$ where
$$\begin{split}T&=\left(\epsilon+r^2 \cos^2 s\right)^{-3+\frac{p}{2}} \left(-3 \epsilon+(1-p) r^2 \cos^2 s\right)\\&+\left(\epsilon+r^2 \sin^2 s\right)^{-3+\frac{p}{2}} \left(3 \epsilon+(p-1) r^2 \sin^2 s\right).\end{split}$$ Then $T=0$, if and only if $$L=S^{\frac{1}{2} (6-p)}=R:=\frac{3 \epsilon+(p-1) r^2 \sin^2 s}{3 \epsilon+(p-1) r^2 \cos^2 s},$$ where $$S=\frac{\epsilon+r^2 \sin^2 s}{\epsilon+r^2 \cos^2 s}.$$ If $p=6$, then $\cos^2 s=\sin^2 s$. If $p>6$ then we also have $\cos^2 s=\sin^2 s$, because if for example $\cos^2 s>\sin^2 s $, then $L>1>R$. Similarly $\cos^2 s<\sin^2 s $ implies $L<1<R$. If $4<p<6$ and $\cos^2 s>\sin^2 s $ , then $$R-S=\frac{\epsilon (4-p) r^2 \cos(2 s)}{\left(\epsilon+r^2 \cos^2 s\right) \left(3 \epsilon+(p-1) r^2 \cos^2 s\right)}<0.$$ Thus
$$S<R<1.$$ Since $0<\frac{6-p}{2}<1$, it follows that
$$S<R^{\frac{6-p}{2}}<1.$$ So $L\neq R$. If $4<p<6$ and $\cos^2 s<\sin^2 s $, then
$$S>R>1.$$ So $$S>R^{\frac{6-p}{2}}.$$ Thus $L\neq R$.
If $p<4$, then $\cos^2 s>\sin^2 s $ implies that $$1>S>R>R^{\frac{6-p}{2}}.$$ Finally if $p<4$ and $\cos^2 s>\sin^2 s $. Then $$1<S<R<R^{\frac{6-p}{2}}.$$ We proved that the only stationary points of $Q$ are $$s_j = \frac{\pi j }{4},\ \ \ j=0,\dots, 7.$$ So wee need to show that $Q(s_j)\le 1$ for $p\le 4$ and $Q(s_j)\ge 1$ for $p\ge 4$.
Let $s=0$. Show that $$Q(0)=\frac{ \left(\epsilon^{-1+\frac{p}{2}} p+p \left(\epsilon+r^2\right)^{-2+\frac{p}{2}} \left(\epsilon+(p-1) r^2\right)\right)}{p \left(2 \epsilon+(p-1) r^2\right)\left(\frac{\epsilon p}{p-1}+r^2\right)^{\frac{p}{2}-2}}\left\{
\begin{array}{ll}
\ge 1, & \hbox{if $p\le 4$;} \\
\le 1, & \hbox{if $p\ge 4$.}
\end{array}
\right.$$
Let $\epsilon= t r^2$, $t>0$. Then $$Q(0)= \frac{ \left( \left(1+\frac{p t}{{p-1}}\right)\right)^{2-\frac{p}{2}} \left(t^{-1+\frac{p}{2}}+(1+t)^{-2+\frac{p}{2}} ({p-1}+t)\right)}{{p-1}+2 t}.$$ By using convexity of the function $k(x)=x^{p/2-2}$, for $p<4$ we have $$\frac{t^{-1+\frac{p}{2}}}{{p-1}+2 t}+\frac{(1+t)^{-2+\frac{p}{2}} ({p-1}+t)}{{p-1}+2 t}>\left(\frac{{p-1}+p t+2 t^2}{{p-1}+2 t}\right)^{-2+\frac{p}{2}}.$$ Further $$\left(\frac{{p-1}+p t+2 t^2}{{p-1}+2 t}\right)^{-2+\frac{p}{2}}>\left(\frac{{p-1}+p t}{{p-1}}\right)^{-2+\frac{p}{2}}.$$ So $Q(0)\ge 1$.
For $4\le p$ we need to show that
$$\frac{t^{-1+\frac{p}{2}}}{{p-1}+2 t}+\frac{(1+t)^{-2+\frac{p}{2}} ({p-1}+t)}{{p-1}+2 t}<\left(\frac{{p-1}+p t}{{p-1}}\right)^{-2+\frac{p}{2}},$$ i.e. $$\frac{t^{-1+\frac{p}{2}}}{{p-1}+2 t}+\frac{(1+t)^{-2+\frac{p}{2}} ({p-1}+t)}{{p-1}+2 t}<\left(1+t+\frac{t}{{p-1}}\right)^{-2+\frac{p}{2}}.$$ The last inequality is equivalent with the trivial inequalities $$\frac{t(t^{-2+\frac{p}{2}} -(t+1)^{-2+\frac{p}{2}})}{{p-1}+2 t}<0<\left(1+t+\frac{t}{{p-1}}\right)^{-2+\frac{p}{2}}-(1+t)^{-2+\frac{p}{2}}.$$ For $s=\pi/4$, and $\epsilon=r^2 t$, $$Q(s)=2^{2-\frac{p}{2}} \left(\frac{{p-1}+p t}{(p-1) (1+2 t)}\right)^{2-\frac{p}{2}}.$$ So $$Q(s)\left\{
\begin{array}{ll}
\ge 1, & \hbox{if $p\le 4$;} \\
\le 1, & \hbox{if $p\ge 4$.}
\end{array}
\right.$$
The other cases can be treated in a similar way.
Assume that $p\le 4$. By using the Green formula , and Lemma \[new\] we obtain $$\begin{split}r\int_0^{2\pi}\frac{\partial}{\partial r} F_\epsilon dt&=\int_0^r\int_0^{2\pi}\rho\Delta F_\epsilon d\rho dt.
\\&
\le \frac{p}{p-1}\int_0^r\int_0^{2\pi}\rho (\Delta U_\epsilon +\Delta V_\epsilon)d\rho dt\\&= r \frac{p}{p-1}\int_0^{2\pi}\frac{\partial}{\partial r} (U_\epsilon+V_\epsilon) dt.
\end{split}$$ Dividing by $r$ and integrating in $[0,1]$ with respect to $r$ we obtain $$\begin{split}\int_0^{2\pi} [F_\epsilon(e^{it})&-F_\epsilon(0)] dt \\&\le \frac{p}{p-1}\bigg(\int_0^{2\pi} [U_\epsilon(e^{it})-U_\epsilon(0)] dt+\int_0^{2\pi} [V_\epsilon(e^{it})-V_\epsilon(0)] \bigg)dt.\end{split}$$ Letting $\epsilon\to 0$ we obtain $$\int_0^{2\pi} |f(e^{it})|^p dt \le \frac{p}{p-1}\left(\int_0^{2\pi} |u(e^{it})|^p dt+\int_0^{2\pi} |v(e^{it})|^p \right)dt.$$ Similarly we prove the related inequality for $p\ge 4$.
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<span style="font-variant:small-caps;">D. Kalaj,</span> *Isoperimetric inequality for the polydisk.* Ann. Mat. Pura Appl. (4) 190 (2011), no. 2, 355–369 <span style="font-variant:small-caps;">D. Kalaj, R. Meštrovi' c,</span> *Isoperimetric type inequalities for harmonic functions.* J. Math. Anal. Appl. 373 (2011), no. 2, 439–448.
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[^1]: 2010 *Mathematics Subject Classification*: Primary 47B35
|
---
abstract: |
Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category $X$ are considered. In particular, we illustrate the formulas $$(\down P)x = \ten(x/X,P) \qv (P\rdown)x = \hom(X/x,P)$$ which give the reflection $\down P$ and the coreflection $P\rdown$ of a category $P$ over $X$ in discrete fibrations. The explicit use of the ťensor functor" $$\ten := \comp(-\times -):\CatX\times\CatX \to \Set$$ given by the components of products, allows a vast generalization of the corresponding analysis in the two-valued context. For any df $A$, the functor $\ten(A,-):\CatX \to \Set$ has a right adjoint $\neg A$ valued in dof’s (and vice versa); such a complement operator, which in the two-valued case reduces to the classical complementation between lower and upper parts of a poset, turns out to be an effective tool in the set-valued context as well.
Various applications of the formulas and of the accompanying conceptual frame are presented.
address: |
via Gioberti 86,\
10128 Torino, Italy.
author:
- Claudio Pisani
title: |
Components, complements\
and reflection formulas
---
Introduction
============
We introduce the subject by a simple example. Consider an equivalence relation $\sim$ on a set $X$. (To fix ideas, let $X$ be the plane and let the equivalence classes of $\sim$ be given by a tiling of $X$.) Then, among the parts of $X$ we have the closed parts, that is those subsets $P\sub X$ such that $$x\in P \andq x\sim y \iimp y\in P$$ so that $P$ is an union of equivalence classes. The inclusion $i$ of closed parts in all the parts of $X$, has both left and right adjoints $$\dbf{(-)} \adj i \adj \ul{(-)}$$ giving the best“ outer and inner approximations of a part $P$ by closed parts: $$\ul P \sub P \sub \dbf P$$ Note that for any singleton part $x\in X$, $\dbf x$ is the equivalence class of $x$. It is intuitively clear that the coreflection and the reflection are given explicitly by \[eq1\] x P x P \[eq2\] x P x P where $\meets$ is the meets” predicate: $P$ meets $Q$ if they have a non-void intersection: $$P \meets Q \iff P \cap Q \not\sub \emptyset$$ (warning: the similarity of this symbol with the cotensor operator $\,\cap \:\!\!\!\! |\,$ is fortuitous).
Dropping the symmetry condition, let $X$ be a poset. Now, a part $P$ of $X$ may be donward or upward closed. So we have the inclusions $$i : \df X \inc \P X \qv j : \dof X \inc \P X$$ of lower and upper parts in all the parts of $X$, which have left and right adjoints $$\down(-) \adj i \adj (-)\rdown \qv \up(-) \adj j \adj (-)\rup$$ explicitly given by \[eq3\] x Px P x Px P\
\[eq4\] x P x P x P x P which are the non-symmetrical version of the (\[eq1\]) and (\[eq2\]) above.
In Section \[posets\] we prove the formulas (\[eq3\]) and (\[eq4\]) in such a way that they can be generalized in a natural way to the set-valued context. In fact, after preparing the setting in Section \[cats\] and Section \[complements\], we show in Section \[proof\] that the coreflection and the reflection of a category $P$ over $X$ in df’s and dof’s over $X$ have the following form \[eq5\] (P)x = (x,P) (P)x = (x,P)\
\[eq6\] (P)x = (x,P) (P)x = (x,P) which are exactly the set-valued versions of (\[eq3\]) and (\[eq4\]). (Note that $\down x \cong X/x$ and $\up x \cong x/X$ correspond to the representable presheaves.) While the former are immediate, the latter rest on the (partially defined) čomplement operator" given by exponentials on discrete basis: $$\neg:=(\disc -)^- : (\CatX)\op\times\Set \to \CatX$$ which takes a df $A$ in the functor $\neg A : \Set \to \CatX$ valued in dof (and vice versa) and allows to prove the almost obvious two-valued formula P Q P Q in the set-valued context too: (P,Q) (P,Q) That the formulas (\[eq5\]) and (\[eq6\]) actually give the desired coreflection and reflection is also checked in Section \[proof\].
Among the various applications of the theory discussed in Section \[applications\] there are:
- A treatment of limits and colimits which makes clear the relevance of the reflection in df’s.
- An analysis of (weak) atoms in $\CatX$, that is those categories $x$ over $X$ such that (x,A) (x,A) for any df or dof $A$. Among them we find not only the objects of $X$, but also the idempotent arrows in $X$. The reflections of atoms are, as presheaves on $X$, the retracts of representable functors; so they generate the Cauchy completion of $X$, displaying it as the Karoubi envelope of $X$.
- A proof of the Kan extensions formulas for set functors.
- Applications of the formulas in ǒther" contexts; e.g. we show how they give in a very pictorial way the reflection of graphs in evolutive sets. In fact, one of the themes of the present work is that of treating on an equal footing the coreflections and the reflections: while the former are easily analyzed by figures via the $\hom$ functor, a similar direct analysis is now possible for the latter using the tensor functor instead. The symmetry or duality of the two approaches is due to the fact that we use atomic shapes.
The present paper has evolved from the preprint [@bip] and the presentation of a part of it at the International Conference on Category Theory (CT06) held at White Point in June 2006. While [@bip] aimed toward a more general setting, we here develop several new items. Among them there is a natural presentation of strong dinaturality and a corresponding analysis of ends and coends in Section \[cats\]; a more careful study of the ďuality" in $\CatX$ in Section \[complements\], which is obtained by isolating some key features of the classical duality in $\P X$ and between sieves and cosieves of a poset in Section \[posets\]; and the treatment of limits in Section \[applications\], which gives a clearer perspective on some points studied in [@par73].
It is the author’s opinion that basic category theory still needs clarification, both for divulgation purposes and as a basis for further research. We hope to show that a careful analysis of the dialectical relationships between categories over a base and discrete fibrations, along with an explicit consideration of the tensor functor, is a step in this direction.
[*Acknoweledgements.*]{} Useful discussions with R.J. Wood and Vincent Schmitt are gratefully acknoweledged. I also acknoweledge the deep influence of the work of Lawvere and of Paré (in particular [@par73]). My general attitude in preparing the present work was that of ňoticing, cultivating, and teaching helpful examples of an elementary nature" in category theory, as advocated in [@law86].
The two-valued context {#posets}
======================
In this section we derive in several ways the formulas for the reflection and coreflection of the parts of a poset in lower and upper parts. In so doing, we introduce the conceptual frame that will serve for the corresponding analysis in the set-valued context as well: the reflection and coreflection of categories over a base in discrete fibrations.
Note though that we are not claiming that there is a general enriched theory subsuming the two-valued and set-valued cases as particular instances. On the contrary, the question of the existence of such a theory is one of the points that may deserve further research.
Truth values, posets, lower and upper parts {#2sub1}
-------------------------------------------
In this section we consider $\2$ as the truth values category, and so denote its arrows by $$\false \vdash \false \qv \false \vdash \true \qv \true \vdash \true$$ The category $\2$ is cartesian closed, the product being ǎnd", and exponentials being all equal to $\true$ except $$\true \imp \false\q = \q\false$$ Following [@law73], a poset $X$ is a $\2$-category, that is $$x\leq y \iff X(x,y) \in \2$$ is the truth value of the dominance of $x$ by $y$. (Note that $\iff$ is a way to express equality in $\2$.)
A functor $A:X\op\to\2$ corresponds to a subset $A\sub |X|$ of the underlying set of $X$ by $$x\in A \iff Ax$$ Such a subet is a ľower set“ or šieve” or ǒrder ideal“: $$x\in A \andq y\leq x \q\vdash\q y\in A$$ Dually, a functor $A:X\to\2$ corresponds to an ǔpper set” or čosieve": $$x\in A \andq x\leq y \q\vdash\q y\in A$$ If we denote by $$\df X = \2^{X\op} \qv \dof X = \2^X$$ the categories of lower and upper sets, we have the inclusions $$i : \df X \inc \P X \qv j : \dof X \inc \P X$$ that have left and right adjoints: $$\down(-) \adj i \adj (-)\rdown \qv \up(-) \adj j \adj (-)\rup$$ Observe that each singleton part $x\in X$ has reflections given by the representable (or principal) sieve $\down x = X(-,x)$ $$y\in\,\down x\iff y\leq x$$ and the representable cosieve $\up x = X(x,-)$ $$y\in\,\up x\iff x\leq y$$ that is (Yoneda lemma) $$x\in A \iff x\sub A \iff \down x\sub A \qv x\in D \iff x\sub D \iff \up x\sub D$$ for any lower part $A$ and any upper part $D$.
The formulas via Kan extensions {#2sub2}
-------------------------------
If we denote by $$\disc \adj \pt : \Cat\to\Set$$ the ďiscrete“ and ǒbjects” functors (see section \[3sub3\]), the counit of the adjunction $$|X| = \disc\pt X \to X$$ is the functor that includes the discrete category associated to $X$ in $X$ itself. Note that $|X\op| = |X|$. In our case, we can pose $$\P X = \2^{|X|} = \2^{|X\op|} = \P X\op$$ and consider the above inclusions as induced by $|X|\to X$ and $|X|\to X\op$, so that the left and right Kan extensions along these functors give their left and right adjoints.
Thus, the reflection $\down(-)$ can be obtained by the coend formula for left Kan extensions: $$(\down P)x = \Coend_{|X|}(X(x,-)\,\,\and\,\, P) \qv (\up P)x = \Coend_{|X|}(X(-,x)\,\,\and\,\, P)$$ that is $$x\in\,\down P \iff \exists y\, (x\leq y \,\,\and\,\, y\in P) \qv x\in\,\up P \iff \exists y\, (y\leq x \,\,\and\,\, y\in P)$$ So the coend in question (the tensor product of functors on $|X|$) reduces to the meets" predicate $\varmeets$ on $\P X$ ($P$ meets $Q$ iff they have a non-void intersection) giving the reflection formula (\[eq4\]): $$x \in\,\down P \iff \up x \meets P \qv x \in\,\up P \iff \down x \meets P$$ Similarly the coreflection is given by the end formula: $$(P\rdown)x = \End_{|X|}(X(-,x)\imp P) \qv (P\rup)x = \End_{|X|}(X(x,-)\imp P)$$ that is $$x\in P\rdown \iff \forall y\, (y\leq x \,\imp\, y\in P) \qv x\in P\rup \iff \forall y\, (x\leq y \,\imp\ y\in P)$$ so that the end in question reduces to the inclusion predicate on $\P X$, giving the coreflection formula (\[eq3\]): $$x \in P\rdown \iff \down x \sub P \qv x \in P\rup \iff \up x \sub P$$ This method would fail in the set-valued context, since our main concern, the inclusion of df’s in categories over a base is not a Kan-like functor (note e.g. that $\CatX$ is not a presheaf category).
The coreflection formula via figures {#2sub3}
------------------------------------
One can obtain the coreflection formula in a more straightforward manner by using the singleton parts as figures: $$\begin{array}{c}
x\in P\rdown \\ \hline
x\sub P\rdown \\ \hline
\down x\sub P\rdown \\ \hline
\down x\sub P
\end{array}
\qq\qq
\begin{array}{c}
x\in P\rup \\ \hline
x\sub P\rup \\ \hline
\down x\sub P\rup \\ \hline
\down x\sub P
\end{array}$$ where only the adjunction laws have been used.
The meets" predicate and complements {#2sub4}
------------------------------------
To derive the reflection formula it seems natural to use duality. This can be done in a simple-minded way (see section \[2sub7\]) or in a more wise" manner that allows a similar derivation in the set-valued context (see section \[2sub9\]). Duality has two aspects: complementation, that is related to exponentiation, and negation, that is complementation in the truth-values category which lifts to functors valued therein. Of course the two aspects are strictly related.
Let $\A$ be any bounded poset, that is a poset with a maximum (a terminal object) $\top$ and a minimum (an initial object) $\bot$. There are functors $$\comp \adj \disc \adj \pt :\A\to\2$$ given by
1. $\disc \true = \top \qv \disc \false = \bot$
2. $\pt x = \true {\q\rm iff\q} \top\leq x$
3. $\comp x = \false {\q\rm iff\q} x\leq\bot$
So we may interpret $\pt$ as giving the truth value of the proposition “$x$ is a maximum of $\A$", and $\comp$ as giving the truth value of the proposition “$x$ is not a minimum of $\A$" or “$x$ is not empty".
If $\A$ is also a meet semilattice (that is, it has products) then it is natural to define the meets“ predicate as follows: $$x\meets y := \comp (x\wedge y)$$ that is v$x$ meets $y$ iff their meet is not empty”.
If the functor $x\varmeets - :\A\to\2$ has a right adjoint, we call it the čomplement" of $x$ and denote it by $\neg x$: $$x\meets -\adj\neg x :\2\to\A$$
To motivate the above definition, recall that if the exponential $x\imp y$, characterized by $ \q z\leq x\imp y \iff z\wedge x \leq y \,$, exists then it is called the řelative pseudocomplement“ of $x$ with respect to $y$. If $\,\neg x := x\imp\bot$ exists in the bounded semilattice $\A$ then it is called the pseudocomplement” of $x$.
\[24prop1\] An element $x$ of a bounded semilattice $\A$ has complement $\neg x:\2\to\A$ iff the pseudocomplement $x\imp\bot$ exists. If this is the case, $(\neg x)\false$ is the pseudocompement of $x$ itself, while $(\neg x)\true$ is a obviously a maximum of $\A$.
In one direction, if $x\imp\bot$ exists (and since $x\imp\top = \top$ always exists), the desired right adjoint can be obtained by composition of adjoints: $\neg x = x\imp\disc -$. In the other direction $$\begin{array}{c}
y\leq (\neg x)\false \\ \hline
x\meets y \vdash \false \\ \hline
x\wedge y \leq \bot
\end{array}$$ which gives the required adjunction law. So the pseudocomplement $\neg x$ is only a trace of a functor $\neg x : \2\to\A$, namely its value at $\false\in\2$. The fact that the other value is $(\neg x)\true = \top$ conceals its significance, making it superfluous in the two-valued context. On the contrary, it becomes essential in the set-valued context.
Atoms in a poset {#2sub5}
----------------
Let $\A$ be a bounded semilattice. The meets“ predicate therein allows a very natural definition of atom: an element $x\in\A$ is an ǎtom” of $\A$ when for any $y\in\A$ $$x\meets y \iff x\leq y$$ That is, if we see the elements of $\A$ as parts“, an atom is šo small” to be included in any part that it meets, but also big enough" to meet any part in which it is included (in particular it meets itself). The second condition excludes the bottom $\bot$.
Let $\A$ be the poset of positive natural numbers ordered by divisibility. Since $\bot = 1$ and $n\wedge m =\gcd(n,m)$, two numbers meet each other in $\A$ iff they have a common divisor greater than $1$. Thus the atoms of $\A$ are the prime numbers. So in this case the atoms coincide with the prime elements, that is those elemets such that the representable $\q x\leq -\q$ preserves the finite sups (i.e. the finite ’s).
\[25prop1\] If an atom of $\A$ has complement then it is a completely prime element. In particular, the atoms of a Heyting algebra are completely prime.
Indeed, the complement $\neg x:\2\to\A$ of the atom $x\in\A$ is the right adjoint of $\q x\meets -\q =\q x\leq -\q$ so that $\q x\leq -\q$ preserves all sups (colimits).
\[25prop2\] The singleton parts $x\in X$ are atoms in $\P X$: $$x\meets P \iff x\sub P \iff x\in P$$ for any part $P\in\P X$.
Negation and complements in $\P X$ {#2sub6}
----------------------------------
If $\A$ is a bounded meet semilattice, pseudocomplementation $$-\imp\bot:\A\to\A\op$$ is in general only a partially defined functor. If $\A=\2$, pseudocomplementation is usually called ňegation“ $$\no := -\imp\false:\2\to\2\op$$ and is an isomorphism which induces $$\no^{X\op} : \2^{X\op}\to(\2\op)^{X\op} \cong (\2^X)\op \qv \no^X : \2^X\to(\2\op)^X \cong {(\2^{X\op})}\op$$ giving isomorphisms $$\neg : \df X{\mathop{\longrightarrow}\limits}(\dof X)\op \qv \neg : \dof X{\mathop{\longrightarrow}\limits}(\df X)\op$$ Explicitly, for any $A:X\op\to\2$, $\neg A$ is (the opposite of) $\no\circ A$, and similarly for any $D:X\to\2$: \[26eq1\] A : x Ax D : x Dx In $\P X$, relative pseudocomplemets (exponentials) are computed pointwise”: $$(P\imp Q)x \iff Px\imp Qx$$ In particular, pseudocomplementation in $\P X$ \[26eq2\] (-)’ := -: ¶X (¶X) is the isomorphism given by \[26eq3\] P’ : x Px= Px By comparing (\[26eq1\]) and (\[26eq3\]) we have the following commutative diagrams, whose rows are isomorphisms: \[26eq4\] Each one of the diagrams (\[26eq4\]) can be obtained from the other by substituting $X$ with $X\op$ (the lower row remains unchanged, since $\P X = \P(X\op)$); furthermore each one can be merged with the opposite of the other giving \[26eq5\] whose horizontal compositions are identities. For future reference, we summarize the above considerations in the following proposition
\[26prop1\] The pseudocomplement of a lower part is an upper part, and conversely. Equivalently, the complement of a lower part is valued in upper parts, and conversely.
More generally, note that if $A$ and $D$ are a lower and an upper part, then their exponential $A\imp D$ in $\P X$ is an upper part (and dually $D\imp A$ is a lower part). Indeed, since $\P X$ is in fact a boolean algebra, $A\imp D$ is the union of two upper parts $$A\imp D = A'\cup D$$ Alternatively, observe that, since exponentials are computed pointwise", if $x\leq y$ $$(A\imp D)x \iff Ax\imp Dx \iimp Ay\imp Dy \iff (A\imp D)y$$ Remarkably, this holds also for discrete fibrations and discrete opfibrations, when the exponential is taken in $\CatX$ (see Proposition \[31prop1\]).
The reflection formula via duality {#2sub7}
----------------------------------
The duality summarized by diagrams (\[26eq4\]) and (\[26eq5\]) can be used to reduce the reflection to the known coreflection of section \[2sub3\]. Indeed, in general a left adjoint of a functor $F:\A\to\B$ is a right adjoint of $F\op:\A\op\to\B\op$. So to find the left adjoint to $j : \dof X \inc\P X$ we look for the right adjoint to $j\op : (\dof X)\op \inc(\P X)\op$. But by diagram (\[26eq4\]) the latter is the šame" as $i : \df X \inc \P X$ via the isomorphisms $\neg$ and $(-)'$. Explicitly, we have $\up P = \neg(P'\rdown)$, that is $$\begin{array}{c}
x\in\,\up P \\ \hline
x\in \neg(P'\rdown) \\ \hline
\no\,\,\, x\in P'\rdown \\ \hline
\no\, \down x\sub P' \\ \hline
\no\,\,\, \down x \cap P \sub \emptyset \\ \hline
\down x\meets P
\end{array}$$ This method too would fail in the set-valued context, since we have no such duality between df’s and dof’s.
Properties of meets" and complements in $\P X$ {#2sub8}
----------------------------------------------
We now prove an ǎdjunction-like" property of the meets predicate on $\P X$ which is based on the complement functor and allows us to derive directly the reflection formula.
\[28prop1\] Let $X$ be a poset, $A$ a lower part and $D$ an upper part of $X$. Then for any part of $P$ of $X$ $$P\meets A \iff \up P\meets A \qv P\meets D \iff \down P\meets D$$
We prove the first one, the other being symmetrical: $$\begin{array}{c}
P\meets A \vdash \false \\ \hline
P \sub (\neg A)\false \\ \hline
\up P \sub (\neg A)\false \\ \hline
\up P\meets A \vdash \false
\end{array}$$ where we have used Proposition \[24prop1\] and Proposition \[26prop1\].
The following proposition is almost obvious, but we record it since it will be generalized in the set-valued context:
\[28prop2\] If $A$ and $B$ are both lower sets (or both upper sets), then $$A\leq B \iff \neg B \leq \neg A$$ where the right-hand condition may be seen in $\P X$ or equivalently in $(\P X)^\2$, as a natural transformation between the complements $\neg B, \neg A:\2\to\P X$. Briefly, the štrong contraposition law“ holds. Furthermore, lower parts are properly analyzed by upper parts (and conversely) via the meets” predicate: $$A\leq B \iff A\meets - \,\leq\, B\meets-$$ where the right-hand side refers to functors $\dof X\to\2$: $$\forall D\in\dof X \quad (A\meets D\q\imp\q B\meets D)$$
The reflection formula again {#2sub9}
----------------------------
We are now in a position to derive the reflection formula in a way straightforwardly generalizable to the set-valued context. Since singleton parts are atoms in $\P X$ (see Proposition \[25prop2\]), we can analyze parts $P$ not only with figures like in section \[2sub3\] but also with the meets" predicate: $$\begin{array}{c}
x\in\,\,\down P \\ \hline
x\meets\!\down P \\ \hline
\up x\meets\!\down P \\ \hline
\up x\meets P
\end{array}
\qq\qq
\begin{array}{c}
x\in\,\,\up P \\ \hline
x\meets\!\up P \\ \hline
\down x\meets\!\up P \\ \hline
\down x\meets P
\end{array}$$ where we used twice the adjunction-like properties of Proposition (\[28prop1\]). We stress the similarity between the derivations of the reflection and coreflection formulas by posing them side by side: $$\begin{array}{c}
x\in\,\,\down P \\ \hline
x\meets\!\down P \\ \hline
\up x\meets\!\down P \\ \hline
\up x\meets P
\end{array}
\qq\qq
\begin{array}{c}
x\in P\rdown \\ \hline
x\sub P\rdown \\ \hline
\down x\sub P\rdown \\ \hline
\down x\sub P
\end{array}$$
Categories over a base and components {#cats}
=====================================
We collect some facts that will be needed in the sequel. But in so doing we hopefully also cast some new light on categorical concepts such as strong dinaturality, components, coends and exponentials in $\CatX$.
Categories over a base and discrete fibrations {#3sub1}
----------------------------------------------
We denote by $\df X$ and $\dof X$ the full subcategories of $\CatX$ of discrete fibrations and discrete opfibrations: @1@C=3pc[[X]{} @[\^[(]{}->]{}\[r\]\^-i & & [X]{} @[\_[(]{}->]{}\[l\]\_-j ]{} By a well-known argument, we have equivalences \[31eq1\] \^[X]{} X \^X X and so also full and faithful functors \[31eq2\] @1@C=3pc[ \^[X]{} @[\^[(]{}->]{}\[r\] & & \^X @[\_[(]{}->]{}\[l\] ]{} When opportune, we do not distinguish notationally a df and the corresponding presheaf; e.g., if $A$ is a presheaf we may say v$A$ over $X$“ to emphasize that we are thinking of it as a category over $X$ via the above equivalence (the čategory of elements”).
Any object $x:\1\to X$ of the base category $X$ is a category over $X$. If $\pi:P\to X$ is a category over $X$, the product $P\times x$ in $\CatX$ has as total category the fiber $Px$ over $x$. Similarly, for any arrow $f:\2\to X$ in $X$, we have a fiber" $Pf$ which is in fact a category over $\2$, as displayed by the right-hand pullback: @R=3pc@C=3pc[ Px & P \^\
\^x & X ]{} @R=3pc@C=3pc[ Pf & P \^\
\^f & X ]{} Exponentials in $\CatX$ may not exist (for example, $x:\1\to X$ is not exponentiable if there is another object $y\cong x$ in $X$; see e.g. [@bun00] for an account on exponentiability in $\CatX$). When they do, the objects of $P^Q$ over $x$ are the functors $Px\to Qx$, while the arrows of $P^Q$ over $f$ are the functors in $Pf\to Qf$ over $\2$.
Discrete fibrations and opfibrations are exponentiable. The following particular case (which corresponds to Proposition \[26prop1\]) is worth noticing:
\[31prop1\] If $A$ is a df and $D$ a dof over $X$ then the exponential $D^A$ in $\CatX$ is a dof (and, by duality, $A^D$ is a df).
By the above description of exponentials, the objects over $x$ of $D^A$ are the [*mappings*]{} $Ax\to Dx$. Furthermore, arrows over $f$ in $D^A$ are morphisms $Af\to Df$ in $\Cat/\2$. But $Af$ and $Df$ have discrete fibers and no non-trivial compositions; they look like these:
$$Af = \qq
\xymatrix@R=0.5pc@C=3pc{
{a_1} &&&& \ar[lllld]{b_1} \\
{a_2} &&&& \ar[lllld]{b_2} \\
{a_3} &&&& \ar[lllluu] {b_3} \\
{a_4} &&&& \ar[lllluuu]{b_4} } \\$$ $$Df = \qq
\xymatrix@R=0.5pc@C=3pc{
{c_1}\ar[rrrrdd] &&&& {d_1} \\
{c_2}\ar[rrrru] &&&& {d_2} \\
{c_3}\ar[rrrr] &&&& {d_3} \\
{c_4}\ar[rrrruuu] &&&& }$$ and the morphisms $Af\to Df$ over $\2$ are uniquely determined by the mappings $Ax\to Dx$; indeed, any such a mapping forces an arrow mapping which, on its own turn, forces $Ay\to Dy$. For example, if $a_2\mapsto c_3$ then the arrow in $Af$ starting from $b_1$ is forced to be mapped to the arrow in $Df$ starting from $c_3$, so that in particular $b_1\mapsto d_3$, and so on. Then $D^A$ is indeed a dof. Explicitly, as a covariant presheaf $D^A$ is the composite $$\xymatrix@1@C=4pc{ X \ar[r]^-\Delta & X\times X \ar[r]^-{A\times D} & \Set\op\times\Set \ar[r]^-\hom & \Set }$$ that is, $(D^A)x = Dx^{Ax}$ and (D\^A)f = Df\^[Af]{}&:&Dx\^[Ax]{}Dy\^[Ay]{}\
& & hDfhAf The categories over $X$ that are both df’s and dof’s are called discrete bifibrations (dbf’s) and correspond to presheaves or to covariant presheaves that act by bijections; the direct link between these associates to $A:X\op\to\Set$ the $A':X\to\Set$ defined by $A'x = Ax$ and $A'f = (Af)\inv$. In particular, if $X$ is a grupoid any df or dof is a bifibration, and there is an isomorphism $\Set^{X\op} \cong \Set^X$ defined by $A'f = A(f\inv)$.
\[31prop2\] If $B$ and $C$ are discrete bifibrations over $X$ then also the exponential $B^C$ in $\CatX$ is a discrete bifibration. In particular, if $X$ is a grupoid, exponentials in $$\Set^{X\op}\cong\Set^X\simeq\df X = \dof X$$ are computed as in $\CatX$. For example, if $B,C : X\to\Set$, their exponential in $\Set^X$ is given by $(B^C)x = Bx^{Cx}$ and (B\^C)f = Bf\^[Cf]{}&:&Bx\^[Cx]{}By\^[Cy]{}\
& & hBfhCf
Among the dbf’s there are the constant ones: $\disc S$ with $S\in\Set$.
\[31prop3\] If $A$ is a df and $D$ a dof over $X$ then the exponential $(\disc S)^A$ in $\CatX$ is a dof and $(\disc S)^D$ is a df. Explicitly, as a covariant presheaf $(\disc S)^A = \Set(A-,S)\,$, that is: $(\disc S)^A x = S^{Ax}$ and (S)\^A f = S\^[Af]{}&:&S\^[Ax]{}S\^[Ay]{}\
& & hhAf
Strong dinaturality {#3sub2}
-------------------
So as the concept of natural transformation between presheaves $A,B:X\op\to\Set$ is subsumed by that of morphism between the corresponding df’s over $X$: $$\Nat(A,B) \cong \CatX(A,B)$$ we now show how also the strong dinatural transformations (also known as Barr dinatural", see [@par98] and [@din]) between set-functors $H,K:X\op\times X\to\Set$ can be seen as morphisms between suitably constructed categories over $X$: $$\Dinat^*(H,K) \cong \CatX(H,K)$$ giving them a very natural status.
Given a functor $H:X\op\times X \to \Set$, we define a category over $X$ as follows:
- the objects over $x\in X$ are the elements of $H(x,x)$;
- given $f:x\to y$ in $X$, there is at most one arrow from $a\in H(x,x)$ to $b\in H(y,y)$ over $f$, and this is the case iff $H(x,f)a = H(f,y)b \in H(x,y)$. So we can name this arrow also by $f$.
We only have to check composition: if $f:a\to b$ and $g:b\to c$ then $H(x,f)a = H(f,y)b$ and $H(y,g)b = H(g,z)c$. So H(x,gf)a = H(x,g)(H(x,f)a) = H(x,g)(H(f,y)b) = H(f,g)b\
= H(f,z)(H(y,g)b) = H(f,z)(H(g,z)c) = H(gf,z)c As in the case of presheaves and df’s, we often refer to this category as v$H$ over $X$" (or simply as $H$).
\[32ex1\] The following particular cases are worth noting:
1. If $H$ is constant, then $H$ over $X$ is the corresponding constant dbf.
2. If $H$ is dummy in one variable, that is if it factors through a projection, then we find again the df or dof of section \[3sub1\].
3. If $X$ is a grupoid, $H$ over $X$ is the bifibration corresponding to $$\xymatrix@1{X\ar[rr]^-{\Delta_X} && X\times X \ar[rrr]^-{(-)\inv\times X} &&& X\op\times X \ar[rr]^-H && \Set}$$
4. Given $A:X\op\to\Set$ and $D:X\to\Set$, let $H(x,y)=Ax\times Dy$, that is $H$ is the composite $$\xymatrix@1{X\op\times X \ar[rr]^-{A\times D} && \Set\times\Set \ar[rr]^-\times && \Set}$$ Then $H$ over $X$ is the product of $A$ and $D$ over $X$ (that is, of the df $A$ and the dof $D$ in $\CatX$).
5. Given $A,B:X\to\Set$, let $H(x,y) = By^{Ax} = \Set(Ax,By):X\op\times X\to\Set$, that is the composite $$\xymatrix@1{X\op\times X \ar[rr]^-{A\times B} && \Set\op\times\Set \ar[rr]^-\hom && \Set}$$ Then it is easy to see that $H$ over $X$ is the exponential $B^A$ in $\CatX$.
6. If $H$ is $\hom_X:X\op\times X\to\Set$ then $H$ over $X$ has the endomorphisms in $X$ as objects, and given $h:x\to x$ and $k:y\to y$, $f:x\to x'$ is an arrow $h\to k$ in $H$ iff $f\circ h = k\circ f$. So $H$ over $X$ is the endomorphism category $X^\N$, where $\N$ is the monoid of natural numbers.
7. If $F:X\to X$ is an endofunctor on $X$, and $H$ is $\hom_X(F-,-):X\op\times X\to\Set$ then $H$ over $X$ is the category of v$F$-algebras" with the forgetful functor to $X$. (The commutative squares below correspond to the arrows in $H$ in the last three examples.) @R=3.5pc@C=3.5pc[ Ax \^[Af]{}\_h & Ay \^k\
Bx \^[Bf]{} & By ]{} @R=3.5pc@C=3.5pc[ x \^f\_h & y \^k\
x \^f & y ]{} @R=3.5pc@C=3.5pc[ Fx \^[Ff]{}\_h & Fy \^k\
x \^f & y ]{}
While in the first four examples above, $H$ over $X$ is exponentiable in $\CatX$, this is not true in general. E.g., let $X$ be the category with three objects and the following non-identity arrows $$\xymatrix@1{x\ar@(ul,dl)[]_l\ar@<0.5ex>[rr]^{f_1}\ar@<-0.5ex>[rr]_{f_2}\ar@/^1.5pc/@<0.5ex>[rrrr]^g && y \ar[rr]^h && z }$$ with $l^2=x$, $f_1 l = f_2$, $f_2 l = f_1$. If $H$ is the endomorphism category of $X$ as in example 6 above, then $g:l\to z$ in $H$ and $g=h f_1$ in $X$, but this factorization cannot be lifted to $H$, since $f_1$ is not an arrow $l\to y$.
Now, given $H,K:X\op\times X \to \Set$, what is a morphism in $\CatX$ among the corresponding categories over $X$? First of all, we need to give a family of mappings $\alpha_x : H(x,x) \to K(x,x)$. If there is a functor over $X$ with such a family as object mapping, it is uniquely determined. And this is the case if and only if for any $f:x\to y$ in $X$, whenever $f:a\to b$ is an arrow in $H$, $f:\alpha_x a \to \alpha_x b$ is an arrow in $K$ too. Explicitly, the $\alpha_x$ define a morphism $H\to K$ in $\CatX$ if and only if the following diagram commutes for any $f:x\to y$ in $X$: $$\xymatrix@R=2.5pc{
& H(x,x)\ar[rr]^{\alpha_x} && K(x,x)\ar[dr]^{K(x,f)} & \\
\pb_f\ar[ur]\ar[dr] & && & K(x,y) \\
& H(y,y)\ar[rr]^{\alpha_y} && K(y,y)\ar[ur]_{K(f,y)} & \\ }$$ where the pullback $\pb_f$ \[32eq1\] @R=3.5pc@C=3.5pc[ \_f & H(x,x) \^[H(x,f)]{}\
H(y,y) \^[H(f,y)]{} & H(x,y) ]{} expresses the pairs of objects $\langle a,b \rangle$ of $H$ such that $f:a\to b$ is an arrow in $H$. So morphisms $H\to K$ over $X$ correspond to the strong dinatural transformations $H\to K$. Of course, if \[32eq2\] @R=3.5pc@C=3.5pc[ H(y,x) \^[H(f,x)]{}\_[H(y,f)]{} & H(x,x) \^[H(x,f)]{}\
H(y,y) \^[H(f,y)]{} & H(x,y) ]{} is a pullback for any $f:x\to y$, then dinatural transformations with domain $H$ are already strongly dinatural. Here are some typical cases when this is the case for any codomain category:
\[32prop1\] Let $H:X\op\times X \to \Set$. Each of the following conditions implies that [(\[32eq2\])]{} is a pullback for any $f:x\to y$, and so also that $$\Dinat(H,K) \cong \Dinat^*(H,K) \cong \CatX(H,K)$$ for any $K:X\op\times X \to \Set\,$:
1. $H$ is constant.
2. $H$ is dummy in one variable.
3. $X$ is a grupoid.
4. $H(x,y)=Ax\times Dy\q$ [(see Examples \[32ex1\])]{}.
For example, let’s check the last case (the other ones are simpler): \[32eq3\] Let’s denote by ${\mathop{\pi}\limits}^{x,x}{}\!\!_i$ the projections of $Ax\times Dx$, and similarly the other projections. Given $\tau_x$ and $\tau_y$ which make the diagram (\[32eq3\]) commute, if $\tau$ exists we have $${\mathop{\pi}\limits}^{y,x}\!\!_2\circ\tau = Dx\circ{\mathop{\pi}\limits}^{y,x}\!\!_2\circ\tau = {\mathop{\pi}\limits}^{x,x}\!\!_2\circ (Af\times Dx)\circ\tau = {\mathop{\pi}\limits}^{x,x}\!\!_2\circ\tau_x$$ and similarly, ${\mathop{\pi}\limits}^{y,x}\!\!_1\circ\tau = {\mathop{\pi}\limits}^{y,y}\!\!_1\circ\tau_y$ so that $\tau$ must have the form $\tau = \langle {\mathop{\pi}\limits}^{y,y}\!\!_1\circ\tau_y , {\mathop{\pi}\limits}^{x,x}\!\!_2\circ\tau_x \rangle$. Now to check e.g. that $(Af\times Dx)\circ\tau = \tau_x$ we compose with projections: $${\mathop{\pi}\limits}^{x,x}\!\!_2\circ (Af\times Dx)\circ\langle {\mathop{\pi}\limits}^{y,y}\!\!_1\circ\tau_y , {\mathop{\pi}\limits}^{x,x}\!\!_2\circ\tau_x \rangle
= Dx\circ{\mathop{\pi}\limits}^{y,x}\!\!_2\circ\langle {\mathop{\pi}\limits}^{y,y}\!\!_1\circ\tau_y , {\mathop{\pi}\limits}^{x,x}\!\!_2\circ\tau_x \rangle = {\mathop{\pi}\limits}^{x,x}\!\!_2\circ\tau_x$$ \^[x,x]{}\_1(AfDx)\^[y,y]{}\_1\_y , \^[x,x]{}\_2\_x = Af\^[y,x]{}\_1\^[y,y]{}\_1\_y , \^[x,x]{}\_2\_x = Af\^[y,y]{}\_1\_y\
= \^[x,y]{}\_1(AfDy)\_y = \^[x,y]{}\_1(AxDf)\_x = \^[x,x]{}\_1\_x The third case of the above proposition, that is when $X$ is a grupoid, can be reversed in the following sense:
If the [(\[32eq2\])]{} are pullbacks for any $H$ and $f$ then $X$ is a grupoid.
Let $H = \hom_X:X\op\times X\to\Set$; the pair $\langle \id_x , \id_y \rangle$ is in the pullback (\[32eq1\]) for any $f:x\to y$. Then, if (\[32eq2\]) is such a pullback, there is an arrow $f':y\to x$ in $\hom_X(y,x)$ such that $$f'\circ f = \hom_X(f,x)f' = \id_x \qv f\circ f' = \hom_X(y,f)f' = \id_y$$ In section \[3sub5\] we shall apply some of the results illustrated to the calculus of ends and coends.
Components of graphs and categories {#3sub3}
-----------------------------------
In order to properly generalize the meets“ operator of section \[2sub4\] to the set-valued context, we need to consider categories $\A$ that have components, that is there exist functors $$\comp \adj \disc \adj \pt : \A\to\Set$$ called the čomponents”, the ďiscrete“ and the points” functors. The objects of $\A$ isomorphic to those of form $\disc S$ (for a set $S\in\Set$) are said to be discrete or constant, while elements of $\pt A$ are called points and those of $\comp A$ components of the object $A\in \A$. If $\A$ is a category with components, then it is such in an essentially unique way, since in this case $\disc 1$ is terminal in $\A$, $\disc S$ is the copower $S\cdot 1$ (that is the sum of $S$ copies of $1\in \A$) and the points functor is represented by $1\in \A$.
Among categories with components, of pivotal importance is the category of (irreflexive) graphs: $$\comp \adj \disc \adj \pt : \Grph\to\Set$$ Since the terminal graph is the loop, $\pt X$ is the set of the loops of the graph $X$, while the discrete graphs are sums of loops.
Our attitude in the present paper is of considering the category $\Grph$ as primitive, rather than as a presheaf category: consider a graph as made up of nodes and arrows between them; arrows are immaterial, so that they can cross each other, and yet each one bounds his domain and codomain nodes. The set $\comp X$ of the components of a graph $X$ is what you get by shrinking its arrows to a zero length. The adjunction $\comp\adj\disc$ expresses the fact that in order to map a graph to a discrete graph, one has to perform a similar process, before eventually doing further identifications (now the arrows are shrinked to loops, but the effect on the nodes is the same). Usually one says: ťhe components of $X$ are the equivalence classes in $X/\!\!\sim$ of the least equivalence relation $\sim$ on the nodes of $X$ which contains the relation $R$ defined by $x R y$ iff there exists an arrow $f:x\to y$ in $X$“. This follows from the fact that, since a discrete graph is in fact an equivalence relation, $\disc:\Set\to\Grph$ factors through $\Set\to\Eq\to\Rel\to\Grph$ and its left adjoint $\comp:\Grph\to\Set$ also factors through $\Grph\to\Rel$ (ťhere exists an arrow”), $\Rel\to\Eq$ (ťhe least equivalence relation“, which in turn could be factorized explicitly though symmetrical relations: v$x\sim y$ iff there is a sequence $x_0=x,x_1,\dots,x_n=y$ such that $x_i R x_{i+1}$ or $x_{i+1} R x_i$”) and $\Eq\to\Set$ (ťhe equivalence classes“). Of course, other factorizations are possible, such as that through categories with involution (reversible categories):$\q\comp:\Grph\to\Cat^{\bf rev}\to\Eq\to\Set$ (see [@hig71]); then one would say ťhe components of $X$ are the equivalence classes of the equivalence relation on the nodes of $X$ given by $x\sim y$ iff there exists an undirected path $x\to y$ in $X$”.
While these factorizations of the left adjoint may be useful to compute it (e.g. to prove that two nodes ǎre in the same component", that is are mapped to the same element by the unit $\eta:X\to\disc\comp X$ of the adjunction), from a conceptual point of view they conceal, rather than explain, the simplicity of the above sketched idea of the direct adjunction. (On the contrary, one can see the above categories as categories of special graphs, or graphs with structure, so that their own components functors are subsumed by that on graphs.)
Of course, one can say that graphs are presheaves on the category $\D = \xymatrix@1{\bullet \ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \bullet}$, so that components are given by the colimit (coequalizer) functor $\Set^{\D\op}\to\Set$. But concretely colimits in $\Set$ are given by the components of a graph (of the corresponding discrete fibration, see section \[3sub4\]). So we would reduce the components of a graph to those of a more involved graph (the šubdivision graph" of Kan).
If $X$ is a graph, $\Grph/X$ also has components: if $\pi:P\to X$ is a graph over $X$, its points are the sections of $\pi$, while its components are those of the total graph $P$.
The category $\Cat$ of categories also has components: $$\comp \adj \disc \adj \pt : \Cat\to\Set$$ the points being the objects of the category, while the components are those of the underlying graph. Similarly we have $$\comp \adj \disc \adj \pt : \CatX\to\Set$$ components being given by those of the total category. The discrete objects $\disc S$ of $\CatX$ are those discrete bifibrations that correspond to the constant presheaves with value $S\in\Set$.
Limits and colimits of set functors {#3sub4}
-----------------------------------
Of course, any presheaf category $\Set^{X\op}$ has components $$\comp \adj \disc \adj \pt : \Set^{X\op}\to\Set$$ Usually $\pt$ and $\comp$ are known as the limit and the colimit functors, while as $\disc$ one can use the diagonal functor: $$\Colim \adj \Delta \adj \Lim : \Set^{X\op}\to\Set$$ The important fact is that, since $\disc_{\CatX}$ factors (up to an isomorphism) through $\disc_{\Set^{X\op}}$, \[34eq0\] @R=4pc@C=4pc[ [\^[X]{}]{} @[\_[(]{}->]{}\[d\] & \^-@2[-]{}\[d\]\
& \_-]{} these functors can be obtained by restricting the corresponding ones of $\CatX$: \[34eq1\] so that the limit of $A:X\to\Set$ is the set of sections of $A$ over $X$, while its colimit is the set components of $A$ over $X$ (that is, the components of the graph of the category of elements of $A$). More explicitly, we have the following chain of natural isomorphisms: $$\begin{array}{c}
\Cone(S,A) \\ \hline
\Nat(\Delta S,A) \\ \hline
\CatX(\disc S,A) \\ \hline
\Set(S,\pt A)
\end{array}
\qq\qq
\begin{array}{c}
\Cone(A,S) \\ \hline
\Nat(A,\Delta S) \\ \hline
\CatX(A,\disc S) \\ \hline
\Set(\comp A,S)
\end{array}$$ thus $\pt A = \Lim A$ and $\comp A = \Colim A$.
\[34rmk1\] We have just seen that colimits of set functors $A:X\op\to\Set$ are given by the components of the df $A$ over $X$. Conversely, the components of any category $P\in\Cat$ can be expressed as the colimit of a functor $X\op\to\Set$, for [*any*]{} df $\, P\to X$ with total category $P$. In particular $P$ is a df $\, P{\mathop{\to}\limits}^\id P$ on itself, corresponding to the functor $\Delta 1 : P\op\to\Set$, the terminal object in $\Set^{P\op}$, so that $$\comp P \,\cong\, \Colim \Delta 1$$
Ends and coends {#3sub5}
---------------
As remarked in Proposition \[32prop1\], if the domain is constant strong dinaturality reduces to standard dinaturality: $$\Dinat(\Delta S,H) \cong \Dinat^*(\Delta S,H)$$ So we have $$\begin{array}{c}
\Wedge(S,H) \\ \hline
\Dinat(\Delta S,H) \\ \hline
\Dinat^*(\Delta S,H) \\ \hline
\CatX(\disc S,H) \\ \hline
\Set(S,\pt H)
\end{array}$$ and we conclude $$\pt H = \End H$$ For coends in general the situation is slightly different: $$\begin{array}{c}
\Wedge^*(H,S) \\ \hline
\Dinat^*(H,\Delta S) \\ \hline
\CatX(H,\disc S) \\ \hline
\Set(\comp H,S)
\end{array}$$ so that we can only conclude that $$\comp H = \Coend^* H$$ where $\Coend^* H$ is a representing object for $\Wedge^*(H,S)$. But, as shown in the example below, in general $\Coend^* H \ne \Coend H$.
Let us test the above concepts in the case $H = \hom_X$. Recall that $\hom_X$ over $X$ is the endomorphism category of $X$ (see the sixth of Examples \[32ex1\]).
- $\End\,\hom_X = \End^*\hom_X = \pt\hom_X$ is the set of its sections: families of endomorphisms $\, e_x:x\to x,\, x\in X$, such that $f\circ e_x = e_y\circ f$ for any $f:x\to y$ in $X$. This is the well-known čenter" of $X$.
- $\Coend^*\hom_X = \comp\hom_X$, wherein two endomorphisms $e_x$ and $e_y$ are identified whenever $f\circ e_x = e_y\circ f$, with $f:x\to y$ in $X$.
- $\Coend\,\hom_X$ is obtained by identifying the two compositions $f\circ g$ and $g\circ f$ of two cycles of arrows in $X$: $$\xymatrix{ x\ar@/^/[rr]^f && y\ar@/^/[ll]^g }$$
That $\Coend\,\hom_X \ne \Coend^*\hom_X$ is then seen already in the simple case $X=\2$ (denote by $f:x\to y$ its non-identity arrow): $\id_x$ and $\id_y$ are identified in the strong coend by $f$, while they are not identified in the standard coend since there are no non-trivial two-cycles. So $\Coend^*\hom_X$ is a terminal set, while $\Coend\,\hom_X$ has two elements. A similar conclusion holds for the commutative monoid $\{ 1,e \}$ with $e$ idempotent, while in general if $X$ is a commutative monoid $\Coend\,\hom_X \cong \End\,\hom_X$ is the set of arrows of $X$. If $X$ is a grupoid, we know from Proposition \[32prop1\] that $\Coend\,\hom_X = \Coend^*\hom_X$; in fact they give the conjugacy classes of $X$.
\[35rmk1\] Following the fifth of Examples \[32ex1\], we have $$\Nat(A,B) = \CatX(A,B) = \pt(B^A) = \End\,\Set(A-,B-)$$ that is we find again the well-known expression of the set of natural transformations as an end.
The above considerations are summarized by the following commutative diagrams (which subsume diagrams (\[34eq1\]) about limits and colimits), where $\Set^{X\op\times X}_*$ is the category whose arrows are strong dinatural transformations:
The tensor functor and complements of categories over a base {#complements}
============================================================
Following Section \[posets\], we now generalize some aspects of the two-valued duality to the set-valued context: the meets“ predicate becomes a ťensor” functor, $\ten:\A\times\A\to\Set$, which allows to define the čomplement“ of $A\in\A$ as the right adjoint to $\ten(A,-)$. Once more, it appears that $\CatX$ encloses and clarifies well-known concepts: in this case, the tensor functor on $\CatX$ extends the classical tensor product $\otimes:\Set^{X\op}\times\Set^X\to\Set$, while each complement functor $\neg A:\Set\to\CatX$ ěxtends” the right adjoints of $A\otimes -:\Set^X\to\Set$.
The tensor functor and complements {#4sub1}
----------------------------------
If $\A$ has components and products we can naturally generalize the meets“ predicate of Section \[posets\] obtaining the functor: $$\ten_\A := \comp (-\times -) : \A\times \A\to\Set$$ We may call functors isomorphic to some $\ten(A,-):\A\to\Set$ ť-representable” functors. It is worth stressing that non isomorphic objects may t-represent isomorphic functors; an example is given by two graphs in $\Grph$ with the same thin reflection or by any two [*reflexive*]{} graphs with the same number of components. On the other hand, in the category of presheaves on a grupoid it is true that $$\ten(A,-) \,\cong\, \ten(B,-) \iimp A\cong B$$ If $\A$ is also (cartesian) closed, then the dual or complementary roles played by the functors $\ten_\A$ and $\hom_\A$ are more evident, as summarized in the following proposition:
\[41prop1\] Let $\A$ be a cartesian closed category with components, then $$\begin{aligned}
\ten(A,B) := \comp (A\times B) \nn \\
\hom(A,B) \cong \pt (B^A) \nn \\
\comp \cong \ten(1,-) \nn \\
\pt \cong \hom(1,-) \nn \\
\label{41eq1}
\ten(A,-) \adj (\disc -)^A \\
\label{41eq2}
(\disc -)\times A \adj \hom(A,-) \end{aligned}$$ the last two being adjunctions with parameter $A\in\A$.
In general, the copower functor $-\cdot A:\Set\to \A$, left adjoint to the representable $\hom(A,-)$, may exist without being given by the formula displayed in equation (\[41eq2\]) (as in the case of categories with a zero object, on which the functors $\comp$, $\disc$ and $\pt$ become constant). On the other hand, we will shortly show that if $\ten(A,-)$ has a right adjoint then it has the form displayed in (\[41eq1\]).
\[41def1\] If there is a universal arrow $\ten(A,B)\to S$ from $\ten(A,-)$ to the set $S$, we denote $B$ by $(\neg A)S$ and call it ťhe $S$-complement of $A$“. In particular, if $(\neg A)S$ exists for any set $S$, we say that $A$ ȟas complement” and call functor $\neg A:\Set\to\A$, right adjoint to $\ten(A,-)$, ťhe complement of $A$".
\[41prop2\] If $\A$ has products and components, an object $A\in\A$ has an $S$-complement iff the exponential $(\disc S)^A$ exists (that is, iff there is a universal arrow from $A\times -$ to the set $\disc S$). If this is the case, $$(\neg A)S \cong (\disc S)^A$$ In particular, $A$ has complement iff the exponential $(\disc S)^A$ exists for any set $S$. If this is the case (that is, if $A$ is ěxponentiable for discrete basis") then $$\neg A \cong (\disc -)^A$$
If $(\disc S)^A$ exists, its universality with respect to $\ten(A,-)$ and $S$ is given by the following chain of bijections, natural in $B\in\A$: $$\begin{array}{c}
\ten(A,B) \to S \\ \hline
\comp(A\times B) \to S \\ \hline
A\times B \to \disc S \\ \hline
B\to (\disc S)^A
\end{array}$$ In the other direction, if $(\neg A)S$ exists, we have $$\begin{array}{c}
B\to (\neg A)S \\ \hline
\ten(A,B) \to S \\ \hline
\comp(A\times B) \to S \\ \hline
A\times B \to \disc S
\end{array}$$ which gives the universality of it with respect to $A\times -$ and $\disc S$. In general, the adjunction with parameter $A\in\A$: $$\ten(A,-) \adj \neg A = (\disc -)^A$$ is only partially defined.
Atoms {#4sub2}
-----
Let $\A$ be a category with products and components. Following section \[2sub5\], we say that an object $A\in\A$ is a štrong atom" if there are bijections $$\ten(A,B) \,\cong\, \hom(A,B)$$ natural in $B\in\A$. For example, if $\A=\Grph$ is the category of graphs, the dot graph (one object and no arrows) is a strong atom. The set-valued correspective of Proposition \[25prop1\] (to which we refer for the proof) is the following:
\[42prop1\] If a strong atom $A$ of $\A$ has complement then it is a absolutely presentable object (that is, $\hom(A,-)$ preserves all colimits). In particular, if $\A$ is cartesian closed then its strong atoms are absolutely presentable objects.
Often the above condition is too strong to be interesting. But it can be relativized by restricting the class of objects against which to test atomicity.
In our main concern, $\A=\CatX$, we say that a category $P$ over $X$ is a ľeft atom“ if there is a natural bijection as above $$\ten(P,A) \,\cong\, \hom(P,A)$$ but restricted to the $A\in\df X$. Dually, $P$ is a řight atom” if $$\ten(P,D) \,\cong\, \hom(P,D)$$ natural in $D\in\dof X$. We say that $P\in\CatX$ is an atom if it is both a left and a right atom. We shall study thoroughly atoms in section \[6sub2\], where we shall also see that the distinction between left and right atoms is illusory. For now we just record
\[42prop2\] Any object $x$ of the base category $X$ is an atom: for any df or dof $A$ $$\ten(x,A) \cong \hom(x,A) \cong Ax$$
In general $\ten(x,P)=\comp Px$ and $\hom(x,P)=\pt Px$ are the components and the objects of the fiber category over $x$. But a category $\A$ is discrete iff $\comp\A \cong \pt\A$, and so if $P$ has discrete fibers then $\comp Px \cong \pt Px$.
The tensor functor on $\CatX$ and tensor products {#4sub3}
-------------------------------------------------
We now show that $\ten_{\CatX}:\CatX\times\CatX\to\Set$ extends the usual tensor product of set functors. Recall that given $A:X\op\to\Set$ and $D:X\to\Set$, if $H(x,y)=Ax\times Dy$ like in Examples \[32ex1\], then \[43eq1\] AD := H is called the tensor product of $A$ and $D$. We have seen that $H$ over $X$ is the product of $A$ and $D$ over $X$, and that in this case $$\Wedge(H,S) = \Wedge^*(H,S)$$ (see Proposition \[32prop1\]). So we have $$\Coend H = \Coend^* H = \comp H = \comp(A\times B) = \ten_{\CatX}(A,B)$$ that is, the following diagram commutes: \[43eq2\] So the tensor product is reduced directly to the components of a category over $X$. Usually it is reduced to the components of the same (total) category but seen as a dof over a df over $X$: a colimit of the set functor $$A {\mathop{\longrightarrow}\limits}^{\pi_A} X {\mathop{\longrightarrow}\limits}^D \Set$$ on the category of elements of $A$. The two constructions are clearly equivalent, but the former is more direct and symmetrical. The other possibility would be the reduction of the coend (\[43eq1\]) to a colimit via the šubdivision category“ or the ťwisted arrow category” (see [@mac71]). But so doing would mean to consider the components of a more involved total category (whose objects are given by all the $Ax\times Dy$, not only those with $x = y$).
Observe that ďually" the diagrams $$\xymatrix{
{\Set^{X\op}\!\!\times\Set^{X\op}} \ar[rr]^-\Nat\ar@{_{(}->}[dd] && {\Set}\ar@2{-}[dd] \\ \\
{\CatX\times\CatX} \ar[rr]^-\hom && {\Set} }
\qq\qq
\xymatrix{
{\Set^X\!\!\times\Set^X} \ar[rr]^-\Nat\ar@{_{(}->}[dd] && {\Set}\ar@2{-}[dd] \\ \\
{\CatX\times\CatX} \ar[rr]^-\hom && {\Set} }$$ express the fullness of the inclusions (\[31eq2\]).
Negation and complements in $\CatX$ {#4sub4}
-----------------------------------
Now we turn to complements in $\CatX$ and will see in what sense the dualities of Section \[posets\] can be generalized to the set-valued context. We have seen in section \[2sub6\] that the truth value category $\2$ is very peculiar in that the functor $\q\no := -\imp\false:\2\to\2\op\q$ is an isomorphism, which we usually call ňegation“ and which in turn induces a duality between lower and upper parts of a poset $\neg:\df X\to(\dof X)\op$. The point to be made presently is that for the category of sets there is not a single dualizing object playing the role of false”; rather, we must consider the whole bunch of functors $$\no_S := S^- : \Set\to\Set\op$$ as the ňegation“ in $\Set$, which in turn induce a bunch of negation operators $$\neg_S:\df X\to(\dof X)\op$$ These are not isomorphisms, but negation is classical in that a presheaf $A$ is recoverable from the bunch (in fact a functor on $\Set$) of covariant presheaves $\neg_S A$. Furthermore, once again, the negation operator extends, at least partially, to the environment $\CatX$ making it possible to derive the formulas for the reflection in discrete fibrations: the $S$-negation of a presheaf is the šame” as the $S$-complement in $\CatX$ of the corresponding df, as in the two-valued case. (In [@law96], it is proposed a definition of negation in a locally cartesian closed category or in any hyperdoctrine, that is similar to our definition of complement but is parametrized by objects over 1, rather than by sets).
Let us begin with complements in the category of sets. Since $\Set$ is cartesian closed and $$\comp \adj \disc \adj \pt : \Set\to\Set$$ are the identity functors, any set $S$ has the complement functor $$(-)^S : \Set\to\Set$$ which we call also the ňegation“ of $S$, since we are thinking to $\Set$ as a ťruth-values” category. So the $T$-negation of $S$ is simply $\no_T S = T^S$, where $\no_T$ is the functor $$\no_T = T^- : \Set\to\Set\op$$ As for the complement operator on $\CatX$, $$\neg P = (\disc -)^P$$ in general it is only partially defined. Anyway Corollary \[31prop2\] can now be rephrased as follows:
\[44prop1\] Any df $A$ on $X$ has a complement $$\neg A = (\disc-)^A:\Set\to\CatX$$ which is valued in dof’s (and conversely). So the $S$-complement of a df on $X$ is a dof (and conversely).
On the other hand, as sketched before, the dof $(\neg A)S$ can also be seen as induced directly by negation in $\Set$: $$(\no_S)^{X\op} : \Set^{X\op}\to(\Set\op)^{X\op} \cong (\Set^X)\op$$ gives a functor $$\neg_S : \Set^{X\op}\to(\Set^X)\op$$ Expicitly, for any $A:X\op\to\Set$, $\neg_S A$ is the opposite of $(S^-)\circ A$, that is $$\neg_S A = \Set(A-,S)$$ So $\neg_S A$ corresponds to the exponential $(\disc S)^A$ in $\CatX$ for the df $A$, as described in Corollary \[31prop2\], and we can construct exponentials in $\CatX$ in such a way that the following diagram commutes for all $S\in\Set$: \[44eq1\] where the lower arrow is the partially defined functor which takes a category over $X$ in its $S$-complement. Incorporating the parameter $S$ explicitly in the diagram, we get so, recalling diagram (\[43eq2\]), we conclude that $$A\otimes - \adj \neg_- A : \Set\to\Set^X$$ is an adjunction with parameter $A\in\Set^{X\op}$. Indeed, this is the usual closed structure on a fragment of the closed bicategory of distributors. So, the point to be stressed is that the above člassical" adjunction extends to the adjunction $$\ten(A,-) \adj \neg A : \Set\to\CatX$$ of section \[4sub1\], with parameter $A$ in $\df X$.
Properties of the tensor functor and of complements in $\CatX$ {#4sub5}
--------------------------------------------------------------
Like in section \[2sub8\] we now prove an ǎdjunction-like" property of the tensor functor on $\CatX$ which allows to derive directly the reflection formula.
\[45prop1\] Let $X$ be a category, $A$ a df and $D$ a dof on $X$. Then for any $P\in\CatX$ with reflections $\down P$ in df’s and $\up P$ in dof’s, there are natural bijections $$\ten(P,A) \cong \ten(\up P,A) \qv \ten(P,D) \cong \ten(\down P,D)$$
We prove the first one, the other being symmetrical. For any set $S$ $$\begin{array}{c}
\ten(P,A) \to S \\ \hline
P \to (\neg A)S \\ \hline
\up P \to (\neg A)S \\ \hline
\ten(\up P,A) \to S
\end{array}$$ where we have used Proposition \[44prop1\]. The result follows by Yoneda.
The following proposition can be interpreted as the fact that complementation in $\CatX$ is classical, when restricted to df’s and dof’s (compare it with Proposition \[28prop2\]); in particular, it is possible to řecover" a df (or presheaf) from its negation:
\[45prop2\] If $A$ and $B$ are both df’s (or both dof’s), then there is a natural bijection \[45eq1\] (A,B) (B,A) Briefly, the štrong contraposition law" holds. Furthermore, df’s are properly analyzed by dof’s (and conversely) via the tensor functor: there is a natural bijection $$\hom(A,B) \cong \Nat(\ten(A,-),\ten(B,-))$$
By section \[4sub4\], the isomorphism (\[45eq1\]) may be interpreted equivalently using preshaves instead of df’s: $$\Nat(A,B) \cong \Nat(\neg B,\neg A)$$ where $\neg B,\neg A:\Set\to\Set^X$ are given by $(\neg A)S = \neg_S A = \Set(A-,S)$. By the isomorphism $(\Set^X)^\Set \cong \Set^{X\times\Set}$, $\neg A$ corresponds to the functor $\neg A : X\times\Set \to \Set$ given by $$\neg A (x,S) = (\neg A S)x = \Set(Ax,S)$$ and natural transformations $\alpha:\neg B\to\neg A:\Set\to\Set^X$ correspond to natural transformations $\alpha:\neg B\to\neg A:X\times\Set \to \Set$, that is families of mappings $$\alpha_S^x:\Set(Bx,S)\to\Set(Ax,S) \q x\in X\, ,\, S\in\Set$$ such that both the diagrams \[45eq2\] commute for any $u:x\to y$ in $X$ and any $f:S\to T$ in $\Set$. By Yoneda, the former of (\[45eq2\]) implies that $\alpha_S^x = \Set(h_x,S)$ for a unique map $h_x:Ax\to Bx$, so that the latter becomes \[45eq3\] But if $S$ has more than one element, then for any $h,k:U\to V$ in $\Set$, $$\Set(h,S) = \Set(k,S) : \Set(V,S) \to \Set(U,S) \iimp h = k$$ and so the commutativity of (\[45eq3\]) for any $S$ and $u:x\to y$ implies that also $$\xymatrix{
Ay \ar[rr]^-{h_y}\ar[dd]_{Au} && By\ar[dd]^{Bu} \\ \\
Ax \ar[rr]^-{h_x} && Bx }$$ commutes for any $u:x\to y$ in $X$, giving a natural transformation $a:A\to B$. In the other direction, any such $a$ induces $\alpha:\neg B \to \neg A$, and the two processes are clearly each other inverses. The last part of the proposition follows from the fact that $$\Nat(\ten(A,-),\ten(B,-)) \cong \Nat(\neg B,\neg A)$$ by a general property of adjunctions with parameters (see e.g. [@bip]).
The reflection and the coreflection of categories over a base in discrete fibrations {#proof}
====================================================================================
We begin by showing that the reflections of the objects of the base category in df’s have a very familiar form. Then we derive the reflection and coreflection formulas for general categories over $X$. Finally, we verify that the formulas give indeed the desired reflection and coreflection.
The reflections of objects and the Yoneda Lemma {#5sub1}
-----------------------------------------------
First note that the representable functors on $X$ correspond to slices of $X$: $X(-,x)$ over $X$ is $X/x$, and dually $X(x,-)$ over $X$ is $x/X$. Thus for any df $A$, $\,\Nat(X(-,x),A) \cong \CatX(X/x,A)$, and the Yoneda Lemma may then be rephrased as $$\hom(X/x,A) \,\cong\, Ax$$ On the other hand, since also (see section \[4sub2\]) $\, \hom(x,A) \cong Ax\, $, we have $$\hom(X/x,A) \,\cong\, \hom(x,A)$$ that is
\[51prop1\] The reflections of an object-atom $x$ of the base category $X$ in df’s and dof’s are given by: $$\down x = X/x \qv \up x = x/X$$
Since also $ \, \ten(x,A) \cong Ax \,$ (see Proposition \[42prop2\]), by Proposition \[45prop1\] we get $$\ten(\up x,A) \,\cong\, Ax$$ that is, recalling diagram (\[43eq2\]), we have proved the čo-Yoneda Lemma":
\[51prop2\] For any presheaf $A:X\op\to\Set$ and any $x\in X$ $$X(x,-) \otimes A \,\cong\, Ax$$
Moreover, if $\Y$ is the functor $$\begin{aligned}
\label{51eq1}
\Y:X\to\CatX \\
x\mapsto X/x \nn\end{aligned}$$ then the ǎddendum“ to the Yoneda lemma (see [@mac71]) may be rephrased by saying that the equivalence (\[31eq1\]) can be obtained as $$\begin{aligned}
\label{51eq2}
\hom(\Y-,-):\df X \to \Set^{X\op} \\
A\mapsto\hom(\Y-,A):X\op\to\Set \nn\end{aligned}$$ Ďually” if we define $\,\Y':X\op\to\CatX\,$ by $\, x\mapsto x/X\,$, the co-Yoneda lemma may be rephrased by saying that $$\begin{aligned}
\label{51eq4}
\ten(\Y'-,-):\df X \to \Set^{X\op} \\
A\mapsto\ten(\Y'-,A):X\op\to\Set \nn \end{aligned}$$ is an equivalence. In fact, the functors (\[51eq2\]) and (\[51eq4\]) are isomorphic (see [@bip]).
The coreflection and reflection formulas {#5sub2}
----------------------------------------
We are now in a position to derive the (coreflection and the) reflection formula in a straightforward manner, exactly like in section \[2sub9\], using the object-atoms $x\in X$ (see section \[4sub2\]) as test shapes both for figures (the elements of $\hom(x,P)$) and for ť-figures" (the elements of $\ten(x,P)$).
Though in the following propositions we should use the hypothetical form ǐf the reflection (coreflection) exists then it has the form …", we do not since it will be proved in the next section that these formulas actually give the desired coreflection and reflection for any category over $X$.
\[52prop1\] The fibers of the coreflections $P\rdown$ and $P\rup$ of a category $P$ over $X$ in df’s and dof’s have the form $$(P\rdown)x = \hom(\down x,P) \qv (P\rup)x = \hom(\up x,P)$$
$$\begin{array}{c}
(P\rdown)x \\ \hline
\hom(x,P\rdown) \\ \hline
\hom(\down x,P\rdown) \\ \hline
\hom(\down x,P)
\end{array}
\qq\qq
\begin{array}{c}
(P\rup)x \\ \hline
\hom(x,P\rup) \\ \hline
\hom(\up x,P\rup) \\ \hline
\hom(\up x,P)
\end{array}$$ where only the adjunction laws have been used.
\[52prop2\] The fibers of the reflections $\down P$ and $\up P$ of a category $P$ over $X$ in df’s and dof’s have the form $$(\down P)x = \ten(\up x,P) \qv (\up P)x = \ten(\down x,P)$$
$$\begin{array}{c}
(\down P)x \\ \hline
\ten(x,\down P) \\ \hline
\ten(\up x,\down P) \\ \hline
\ten(\up x,P)
\end{array}
\qq\qq
\begin{array}{c}
(\up P)x \\ \hline
\ten(x,\up P) \\ \hline
\ten(\down x,\up P) \\ \hline
\ten(\down x,P)
\end{array}$$ where we used twice the adjunction-like properties of Proposition \[45prop1\]. It is worth stressing the strong similarity between the derivations of the reflection and coreflection formulas.
To obtain the arrows of, say, $\up P$ (that is, the action of $X$ on its fibers) we should look at the reflections in $\df X$ of the arrows in $X$. If $f:x\to y$ is an arrow in $X$, its reflection $\down f$ as a category $f:\2\to X$ over $X$ is $X/x$ again; and the domain and codomain inclusions over $X$, $x\to f$ and $y\to f$ become $$\id : X/x \to X/x \qv -\circ f : X/x \to X/y$$ So we get the following proposition, of which the Yoneda and co-Yoneda Lemmas are particular cases (see section \[5sub1\]):
\[52prop3\] The coreflection $P\rdown$ and the reflection $\down P$ are given as presheaves by $$P\rdown\, = \hom(\Y-,P) \qv \down P = \ten(\Y'-,P)$$
The formulas work {#5sub3}
-----------------
As it often happens, after having derived the form of a supposed adjoint functor, one has to check that what he got really does the job. We concentrate on dof’s, though by duality one clearly has specular results for df’s.
We begin by analyzing morphisms in $\CatX$ to and from dof’s. If the codomain is a dof, such a morphism is easily seen to be determined by its object mapping:
\[53prop1\] Given a category $P$ over $X$ and a dof $D$ on $X$, a mapping $\phi$ over $X$ between the objects of the total categories of $P$ and $D$ extends (uniquely) to a functor over $X$ iff $$f(\phi a) = \phi b$$ for any $u:a\to b$ over $f:x\to y$.
On the other hand, a morphism in $\CatX$ with a dof as domain has the following form, which is simply the general form adapted to the arrows of a dof:
\[53prop2\] A functor $\phi$ over $X$ from the dof $D$ to $P$ is given by its object mapping over $X$: $$a \,\mapsto\, \phi a$$ and by its arrow mapping over $X$: $$\la a,x{\mathop{\to}\limits}^f y \ra\, \mapsto\, \phi(a,x{\mathop{\to}\limits}^f y)$$ (with $a\in Dx$) which is a graph morphism and respects identities and compositions:
1. $\phi(a,x{\mathop{\to}\limits}^f y):\phi a\to \phi(fa)$
2. $\phi(a,x{\mathop{\to}\limits}^\id x) = \id_{\phi a}\,\,\,$ and $\,\,\,\,\phi(a,x{\mathop{\to}\limits}^{gf}z) = \phi(fa,y{\mathop{\to}\limits}^g z)\circ \phi(a,x{\mathop{\to}\limits}^f y)$
Now we describe more concretely $\up P$ and $P\rup$. Observing that $P/x = X/x\times P$ and so $\ten(\down x,P) = \comp(P/x)$, we get the following more familiar version of Proposition \[52prop2\] and Corollary \[52prop3\] (see [@law73] and [@par73]):
\[53prop3\] The reflection of a category $P$ over $X$ in discrete opfibrations is given by $$(\up P)x = \comp(P/x)$$ for any $x\in X$, and $$f[\la a , \pi a{\mathop{\to}\limits}^h x\ra] = [\la a , \pi a{\mathop{\to}\limits}^{fh}y\ra] \in \comp(P/y)$$ for any $f:x\to y$ in $X$ and $[\la a , \pi a{\mathop{\to}\limits}^h x\ra] \in \comp(P/x)$.
As for the coreflection, from Proposition \[52prop1\], Corollary \[52prop3\] and Proposition \[53prop2\] we straightforwardly get:
\[53prop4\] The coreflection of a category $P$ over $X$ in discrete opfibrations is given by $(P\rup)x = \CatX(\up x,P)$ for any $x\in X$; hence each $\xi\in(P\rup)x$ is a mapping $$x{\mathop{\to}\limits}^h y \,\,\mapsto\,\xi(x{\mathop{\to}\limits}^h y)\, \in Py$$ for any arrow in $X$ with domain $x$, and a mapping $$\la x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z\ra\, \mapsto\, \xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)$$ for any two consecutive arrows in $X$ with domain $x$, such that $\xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)$ is an arrow in $P$ over $k$ and
1. $\xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z):\xi(x{\mathop{\to}\limits}^h y)\to\xi(x{\mathop{\to}\limits}^{kh} z)$
2. $
\xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^\id y) = \id_{\xi h} \\
\xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^{lk}w) = \xi(x{\mathop{\to}\limits}^{kh}z,z{\mathop{\to}\limits}^l w)\circ\xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)
$
Furthermore, for any $f:x\to y$ in $X$, $f\xi :\,\up y \to P$ is given by $$(f\xi)(y{\mathop{\to}\limits}^h z) = \xi(x{\mathop{\to}\limits}^{hf}z) \qv (f\xi)(y{\mathop{\to}\limits}^h z,z{\mathop{\to}\limits}^k w) = \xi(x{\mathop{\to}\limits}^{hf}z,z{\mathop{\to}\limits}^k w)$$
\[53prop5\] The morphism $\phi_P:P\to\,\up P$ which takes an object $a$ over $x$ to the component $[\la a , x {\mathop{\to}\limits}^\id x\ra]$ in $\comp(P/x)$ is a universal morphism to a dof. So the formula of Proposition \[52prop2\] gives indeed the reflection of $P$.
First, we have to check that $\phi_P$ is indeed a morphism. By Proposition \[53prop3\], for any $u:a\to b$ in $P$ over $f:x\to y$, $$f[\la a , x {\mathop{\to}\limits}^\id x\ra] = [\la a , x {\mathop{\to}\limits}^f y\ra] = [\la b , y {\mathop{\to}\limits}^\id y\ra]$$ where the last equality follows from the arrow $u : \la b , y {\mathop{\to}\limits}^\id y\ra \to \la a , x {\mathop{\to}\limits}^f y\ra$ in $P/y$. So the condition of Proposition \[53prop1\] is fulfilled.
As for universality, given any morphism $\phi : P\to D$ to a dof, suppose that $\dbf\phi\circ\phi_p = \phi$. Then $\phi a = \dbf\phi(\phi_p a) = \dbf\phi[\la a , x {\mathop{\to}\limits}^\id x\ra]$, so that $$\dbf\phi[\la a , x{\mathop{\to}\limits}^h y\ra] = \dbf\phi(h[\la a , x {\mathop{\to}\limits}^\id x\ra]) = h(\dbf\phi[\la a , x {\mathop{\to}\limits}^\id x\ra]) = h(\phi a)$$ which gives the unicity of $\dbf\phi$. $$\xymatrix@R=3.5pc@C=3.5pc{
P \ar[r]^{\phi_P}\ar[dr]_\phi & \up P \ar@{..>}[d]^{\dbf\phi} \\
& D }$$ So we have to check that $\dbf\phi$ defined by $$\dbf\phi[\la a , x{\mathop{\to}\limits}^h y\ra] = h(\phi a)$$ is indeed a morphism $D\to P\rup$ in $\CatX$:
1. $\dbf\phi$ is well-defined: if $u:a\to b$ in $P$ over $g:x\to z$ is an arrow $\la a , x{\mathop{\to}\limits}^h y\ra$ to $\la a , z{\mathop{\to}\limits}^k y\ra$ in $P/y$ (that is $kg = h$) then by Proposition \[53prop1\] $$h(\phi a) = kg(\phi a) = k(\phi b)$$
2. $\dbf\phi$ is a morphism in $\CatX$ (or in $\dof X\simeq \Set^X$), since for any $f:y\to x$ in $X$ $$\dbf\phi f[\la a , x{\mathop{\to}\limits}^h y\ra] = \dbf\phi [\la a , x{\mathop{\to}\limits}^{fh} z\ra] = fh(\phi a) = f\dbf\phi[\la a , x{\mathop{\to}\limits}^h y\ra]$$
3. $\dbf\phi\circ\phi_p = \phi$, since for any $x\in Px$, $\, \dbf\phi(\phi_p a) = \dbf\phi([\la a , x {\mathop{\to}\limits}^\id x\ra]) = \id_x(\phi a) = \phi a $.
\[53prop6\] The morphism $\phi_P:P\rup\,\to P$ which takes any $\xi\in(P\rup)x$ to the object $\xi(x{\mathop{\to}\limits}^\id x)$ in $P$ over $x$, and any arrow $\la \xi , x{\mathop{\to}\limits}^f y \ra$ in $P\rup$ over $f$ to the arrow $\xi(x{\mathop{\to}\limits}^\id x,x{\mathop{\to}\limits}^f y)$ in $P$ over $f$ is a universal morphism from a dof. So the formula of Proposition \[52prop1\] gives indeed the coreflection of $P$.
First, we have to check that $\phi_P$ is indeed a morphism in $\CatX$, that is that the conditions of Proposition \[53prop2\] are fulfilled. But these follow directly from the corresponding ones about the $\xi$ in $(P\rup)x$ (see Proposition \[53prop4\]):
1. $ \phi_P(\xi,x{\mathop{\to}\limits}^f y) = \xi(x{\mathop{\to}\limits}^\id x,x{\mathop{\to}\limits}^f y):\xi(x{\mathop{\to}\limits}^\id x)\to \xi(x{\mathop{\to}\limits}^f y)\,$; but\
$ \xi(x{\mathop{\to}\limits}^\id x) = \phi_P\xi \,\,$ and $ \,\xi(x{\mathop{\to}\limits}^f y) = (f\xi)(y{\mathop{\to}\limits}^\id y) = \phi_P(f\xi) \,$ as required.
2. $ \phi_P(\xi,x{\mathop{\to}\limits}^\id x) = \xi(x{\mathop{\to}\limits}^\id x,x{\mathop{\to}\limits}^\id x) = \id_{\xi(\id_x)} = \id_{\phi_P\xi} \,\, $ and\
$
\phi_P(\xi,x{\mathop{\to}\limits}^{gf}z) = \xi(x{\mathop{\to}\limits}^\id x,x{\mathop{\to}\limits}^{gf}z) = \xi(x{\mathop{\to}\limits}^f y,y{\mathop{\to}\limits}^g z)\circ\xi(x{\mathop{\to}\limits}^\id x,f) \\
= (f\xi)(y{\mathop{\to}\limits}^\id y,y{\mathop{\to}\limits}^g z)\circ\xi(x{\mathop{\to}\limits}^\id x,f) = \phi_P(f\xi,y{\mathop{\to}\limits}^g z)\circ\phi_P(\xi,x{\mathop{\to}\limits}^f y)
$
As for universality, given any morphism $\phi : D\to P$ from a dof, suppose that $\phi_P\circ\dbf\phi = \phi$. Then $ \phi a = \phi_P(\dbf\phi a) = (\dbf\phi a)(x{\mathop{\to}\limits}^\id x) $, so that $$(\dbf\phi a)(x{\mathop{\to}\limits}^h y) = (h(\dbf\phi a))(y{\mathop{\to}\limits}^\id y) = (\dbf\phi(ha))(y{\mathop{\to}\limits}^\id y) = \phi(ha)$$ and similarly $(\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z) = \phi(ha,y{\mathop{\to}\limits}^k z)$, giving the unicity of $\dbf\phi$. $$\xymatrix@R=3.5pc@C=3.5pc{
D \ar@{..>}[d]_{\dbf\phi}\ar[dr]^\phi & \\
P\rup \ar[r]^{\phi_P} & P }$$ So we have to check that $\dbf\phi$ defined by $$(\dbf\phi a)(x{\mathop{\to}\limits}^h y) = \phi(ha) \qv (\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z) = \phi(ha,y{\mathop{\to}\limits}^k z)$$ is indeed a morphism $D\to P\rup$ in $\CatX$:
1. $\dbf\phi$ is well-defined, that is $\dbf\phi a$ is indeed a morphism $\up x \to P$, that is the conditions of Proposition \[53prop4\]: are fulfilled. But these follow directly from the corresponding ones about $\phi$ (see Proposition \[53prop2\]):
1. $ (\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z) = \phi(ha,y{\mathop{\to}\limits}^k z):\phi(ha)\to\phi(kha) \,$; but\
$ \phi(ha) = (\dbf\phi a)(x{\mathop{\to}\limits}^h y) \,\,$ and $ \,\phi(kha) = (\dbf\phi a)(x{\mathop{\to}\limits}^{kh}z) $, as required.
2. $ (\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^\id y) = \phi(ha,y{\mathop{\to}\limits}^\id y) = \id_{\phi(ha)} = \id_{(\dbf\phi a)h}\,\, $ and\
$
(\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^{lk}w) = \phi(ha,y{\mathop{\to}\limits}^{lk}w) = \phi(kha,z{\mathop{\to}\limits}^l w)\circ\phi(ha,y{\mathop{\to}\limits}^k z) \\
= (\dbf\phi a)(x{\mathop{\to}\limits}^{kh}z,z{\mathop{\to}\limits}^l w)\circ(\dbf\phi a)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)
$
2. $\dbf\phi$ is a morphism in $\CatX$ (or in $\dof X\simeq \Set^X$), since for any $f:x\to y$ in $X$\
$ \dbf\phi(fa)(x{\mathop{\to}\limits}^h y) = \phi(hfa) = (\dbf\phi a)(x{\mathop{\to}\limits}^{hf}y) = (f(\dbf\phi a))(x{\mathop{\to}\limits}^h y) \,\, $ and similarly\
$
\dbf\phi(fa)(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z) = (f(\dbf\phi a))(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)
$
3. $\phi_P\circ\dbf\phi = \phi$, since for any $a\in Dx$, $ \,\phi_p(\dbf\phi a) = (\dbf\phi a)\id_x = \phi(\id_x a) = \phi a \,$, and similarly for any arrow $\la a , x{\mathop{\to}\limits}^ f y\ra$ in $D$.
\[53rmk1\] As suggested by Richard Wood, the reflection and coreflection of a category over $X$ in discrete fibrations can be factorized through fibrations over $X$. Indeed, the inclusion $i:\dof X \inc \CatX$ factors (up to equivalence) through fibrations as follows $$\dof X \simeq \Set^X \inc \Cat^X \inc \CatX$$ where the last inclusion is given by the Grothendieck construction. While the functor $(\disc)^X : \Set^X \inc \Cat^X$ clearly has both left and right adjoints $$\comp^X\adj(\disc)^X\adj~\pt^X$$ given pointwise" by components and points on $\Cat$, the same is true also for the Grothendieck functor. Its left adjoint $\CatX\to\Cat^X$ is simply given by slicing: $$P\mapsto P/-:X\to\Cat$$ (see also [@her98]). Its right adjoint takes $P$ to the functor $\dbf P:X\to\Cat$ such that $\dbf P x$ is the category whose objects are the morphisms $\xi : \,\up x \to P$ like in Proposition \[53prop4\], while an arrow $\varphi : \xi \to \xi'$ is given by mappings $x{\mathop{\to}\limits}^h y\,\, \mapsto\, \varphi(x{\mathop{\to}\limits}^h y)$ such that
1. $\varphi(x{\mathop{\to}\limits}^h y)\,$ is an arrow in $P$ over $\id_y$
2. $\varphi(x{\mathop{\to}\limits}^h y):\xi(x{\mathop{\to}\limits}^h y)\to \xi'(x{\mathop{\to}\limits}^h y)$
3. $\xi'(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)\circ\varphi(x{\mathop{\to}\limits}^h y) = \varphi(x{\mathop{\to}\limits}^{kh}z)\circ \xi(x{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z)$.
while for any $f:x\to x'$, the functor $\dbf P f:\dbf P x \to \dbf P x'$ is given by\
$ (f\xi)(x'{\mathop{\to}\limits}^h y) = \xi(x{\mathop{\to}\limits}^{hf}y) $, $ (f\xi)(x'{\mathop{\to}\limits}^h y,y{\mathop{\to}\limits}^k z) = \xi(x{\mathop{\to}\limits}^{hf}y,y{\mathop{\to}\limits}^k z) $ on objects, and\
$\, (f\varphi)(x'{\mathop{\to}\limits}^h y) = \varphi(x{\mathop{\to}\limits}^{hf}y) $ on arrows.
Applications
============
In section \[6sub1\] we analyze reflections and coreflections in df’s in relation with a čhange of base" given by a functor $X\to Y$, showing also their interdependence with Kan extensions.
The relevance of the reflection $\down(-)$ in df’s over $X$ for the study of colimits of functors $\Phi:P\to X$ was shown thoroughly in [@par73]. In section \[6sub2\], we present those facts in a more conceptual way, resting on the universal property of $\down(-)$. We begin with an unusual operative definition of (co)limits: the colimit of $\Phi\in\CatX$ is its reflection in the full subcategory of principal" (or representable) df’s: those with a terminal object. Thus $\down(-):\CatX\to\df X$ can be seen as an intermediate step of this reflection.
In section \[4sub2\] we have seen that any object $x\in X$ is an atom in $\CatX$. In section \[6sub3\] we show that the same is true for any idempotent $e$ in $X$, and that the reflections of atoms are the retracts of representable functors.
In section \[6sub4\] we present ǎnother“ context in which one can apply the reflection formula, which allows in particular a direct and intuitive calculation of the best left approximation” of a graph with an (idempotent, bijective or $n$-periodic) evolutive set.
Whenever opportune, we will be more careful about notations regarding categories over a base: referring to a category over $X$ we shall write e.g. $P{\mathop{\to}\limits}^p X$, rather than only $P$ as before. Accordingly, its reflection will be denoted by $\down p$ rather than by $\down P$.
Dual statements, obtained by dualizing the base category, are often left to the reader. Note that $\,\Phi\,\mapsto\,\Phi\op\,$ gives an isomorphism $\CatX\to\CatX\op$, such that the following diagrams commute: $$\xymatrix@R=1.5pc@C=3pc{
{\df X} \ar[r]^-i \ar@2{-}[d] & \CatX \ar@2{-}[d] & {\dof X} \ar[l]_-j \ar@2{-}[d] \\
{\dof{X\op}} \ar[r]^-j & \CatX\op & {\df{X\op}} \ar[l]_-i }
\qq\qq
\xymatrix@R=1.5pc@C=3pc{
{\df X} \ar@2{-}[d] & \CatX \ar@2{-}[d]\ar[l]_-{\down\,(-)}\ar[r]^-{\up\,(-)} & {\dof X} \ar@2{-}[d] \\
{\dof{X\op}} & \CatX\op \ar[l]_-{\up\,(-)} \ar[r]^-{\down\,(-)} & {\df{X\op}} }$$ Briefly, $\up\Phi\op = \,\down\Phi$ and $\down\Phi\op = \,\up\Phi$. Furthermore, $\,\Lim\Phi\op = \Colim\Phi\,$, $\,\Colim\Phi\op = \Lim\Phi\,$, $\,x/X\op = X/x\,$ and $\,X\op/x = x/X\,$.
Kan extensions via reflection {#6sub1}
-----------------------------
Recall that any functor $X\to Y$ gives rise to a pair of adjoint functors $$f_!\adj f^*:\CatY\to\CatX$$ where $f_!$ is given by composition with $f$, while $f^*$ is obtained by the following pullback in $\Cat$: $$\xymatrix@R=3.5pc@C=3.5pc{
f^*P \ar[r] \ar[d]_{f^* p} & P \ar[d]^ p \\
X \ar[r]^f & Y }$$ Note that $\comp^X:\CatX\to\Set$ factors through $f_!:\CatX\to\CatY$ and $\comp^Y$ up to isomorphisms; indeed, the components of a category over a base are given by those of the total category (see section \[3sub3\]) and $P$ and $f_!P$ have the same total category. As a consequence also $\disc_X$ factors through $f^*$ and $\disc_Y$ (up to isomorphisms): \[61eq2\] @R=3.5pc@C=3.5pc[ \^[f\_!]{} \^& \^\
@2[-]{}\[r\] & ]{} @R=3.5pc@C=3.5pc[ & \_-[f\^\*]{}\
\_@2[-]{}\[r\] & \_]{} Furthermore, the pair $f_!\adj f^*$ satisfies the Frobenius law, that is the morphism \[61eq1\] f\_!\_1(\_Qf\_!\_2) : f\_!(Pf\^\*Q) f\_! PQ is an isomorphism for any $P\in\CatX$ and $Q\in\CatY$, $\eps$ being the counit of the adjunction. This follows essentially from the fact that the composition of the two pullback squares below, expressed by $f_!(P\times f^*Q)$, gives another pullback, expressed by $f_! P\times Q$: $$\xymatrix@R=3.5pc@C=3.5pc{
P\times f^*Q\ar[r]\ar[d] & f^*Q \ar[r] \ar[d]_{f^*q} & Q\ar[d]^q \\
P\ar[r]^p & X \ar[r]^f & Y }$$ As it is easily checked (see [@law70]), this implies that $f^*$ preserves existing exponentials. In particular, by (\[61eq2\]), $f^*$ preserves complements: \[61eq0\] @R=3.5pc@C=3.5pc[ \^[(S)\^-]{} & \_[f\^\*]{}\^[(S)\^-]{}\
()& ()\_[f\^\*]{} ]{} The following adjunction-like property holds:
\[61prop1\] For any functor $f:X\to Y$ there is a bijection $$\ten_X(P,f^*Q) \cong \ten_Y(f_!P,Q)$$ natural in $P\in\CatX$ and $Q\in\CatY$.
We have \_X(P,f\^\*Q) = \^X(Pf\^\*Q) \^Y(f\_!(Pf\^\*Q))\
\^Y(f\_!PQ) = \_Y(f\_!P,Q) where we have used the first of (\[61eq2\]) and the Frobenius law.
\[61prop12\] Let $f:X\to Y$ any functor. Then for any atom $T{\mathop{\to}\limits}^t X$ in $\CatX$, $f_! t:T\to Y$ is an atom in $\CatY$.
We have the following chain of natural bijections: $$\ten_{\CatY}(f_!x,D) \cong \ten_{\CatX}(x,f^*D) \cong \hom_{\CatX}(x,f^*D) \cong \hom_{\CatY}(f_!x,D)$$ Now we turn to Kan extensions of set functors. First recall that pullbacks preserve df’s and dof’s: if $D$ over $Y$ is in $\dof Y$ then $f^*D$ is in $\dof X$. So $f^*:\CatY\to\CatX$ restricts to $f^*:\df Y\to\df X$ and $f^*:\dof Y\to\dof X$: $$\xymatrix@R=3.5pc@C=3.5pc{
{\df X} \ar[d]^i & {\df Y}\ar[l]_{f^*}\ar[d]^i \\
\CatX & \CatY\ar[l]_{f^*} }
\qq\qq
\xymatrix@R=3.5pc@C=3.5pc{
{\dof X} \ar[d]^j & {\dof Y}\ar[l]_{f^*}\ar[d]^j \\
\CatX & \CatY\ar[l]_{f^*} }$$ In terms of presheaves $f^*:\Set^Y\to\Set^X$ can be defined by composition: $$f^*D = D\circ f : X\to\Set$$
\[61rmk1\] Thus also diagram (\[61eq0\]) restricts to \[61eq01\] @R=3.5pc@C=3.5pc[ \^[X]{} \^[\_S]{} & \^[Y]{} \_[f\^\*]{}\^[\_S]{}\
(\^X)& (\^Y)\_[f\^\*]{} ]{} E.g., if $f:X\to Y$ is a quotient obtained by adding ľaws“ (commutative diagrams) to $X$, then $f^*$ is full and faithful and we can say that complementation in $\CatX$ preserves (or better, reverses) algebraic laws”. For example, let $X$ be the free category on $\xymatrix@1{x \ar@<-1ex>[r]\ar@<1ex>[r] & y \ar[l]}$, and let $D\in\dof X$ be a dof which happens to be a reflexive graph. Then its complements (in particular $(\neg D)(1+1)$ given by $\xymatrix@1{\P(Dy) \ar@<-1ex>[r]\ar@<1ex>[r] & \P(Dx) \ar[l]}$) are cylinders (mappings with two sections); conversely, the complements of cylinders are reflexive graphs (see [@law89]).
We are interested in the left and right adjoints of $f^*$: $$\exists_f \adj f^* \adj \forall_f :\dof X \to \dof Y$$ also known as Kan extensions. We begin with the following
\[61prop2\] For any functor $f:X\to Y$ and any $P$ over $X$, $$\exists_f(\up P) \cong \,\up(f_!P)$$
Since in the diagram below the square of right adjoint arrows commutes up to isomorphisms, so does also the square of left adjoint arrows: @R=4pc@C=4pc[ @<1ex>\[r\]\^[f\_!]{}\_@<-1ex>\[d\]\_[(-)]{}\^ & @<1ex>\[l\]\^[f\^\*]{}@<-1ex>\[d\]\_[(-)]{}\^\
[X]{} @<1ex>\[r\]\^[\_f]{}\_@<-1ex>\[u\]\_[j]{} & [Y]{}@<1ex>\[l\]\^[f\^\*]{}@<-1ex>\[u\]\_[j]{} ]{}
\[61prop3\] The following formulas hold: $$\exists_f D = \,\up(f_!D) \qv \up f = \exists_f \id_X$$
For any $D\in\dof X$ and $f:X\to Y$, $ \,\exists_f D = \exists_f \up D = \,\up(f_!D) $\
while for any category $X{\mathop{\to}\limits}^f Y$ over $Y$, $$\up f = \,\up(f_!\id_X) = \exists_f(\up\id_X) = \exists_f \id_X$$ since $\id_X:X\to X$ is the terminal df $\,\disc 1$.
\[61prop4\] The left Kan extension of a set functor $D:X\to\Set$ along $f:X\to Y$ is given by the čoend formula" $$(\exists_f D)y = Y(f-,y)\otimes D = \Coend\,(Y(f-,y)\times D-)$$
By the above corollary, $\exists_f D = \,\up(f_!D)$ so that $$(\exists_f D)y = \,\up(f_!D)y = \ten_{\CatY}(\down y,f_! D) = \ten_{\CatX}(f^*\down y,D) = \Coend\,(Y(f-,y)\times D-)$$ where Proposition \[61prop1\] and property (\[43eq2\]) have been used.
Supposing on the contrary that the coend formula of Corollary \[61prop4\] is known, one gets again the reflection formula using Corollary \[61prop3\] as follows: $$\begin{array}{c}
(\up f)y \\ \hline
(\exists_f \id_X)y \\ \hline
\ten_X(f^*\down y,\id_X) \\ \hline
\comp((f^*\down y)\times\id_X) \\ \hline
\comp(f^*\down y) \\ \hline
\comp(f\times\down y) \\ \hline
\comp(f/y)
\end{array}$$
As it was the case for the [*coreflection*]{} formula of Proposition \[52prop1\], the ěnd formula" for [*right*]{} Kan extensions can be obtained directly as follows (we use Remark \[35rmk1\] in the last step): $$\begin{array}{c}
(\forall_f D)y \\ \hline
\Nat(\up y,\forall_f D) \\ \hline
\Nat(f^*\up y,D) \\ \hline
\End\,\Set(Y(y,f-),D- )
\end{array}$$
Limits and colimits {#6sub2}
-------------------
Given a functor $\Phi:P\to X$ and an object $x\in X$, a cone from $\Phi$ to $x$ is a family of arrows $\, f_i:\Phi i \to x\, , i\in P\,$, such that $\Phi u\circ f_i = f_j$ for any $u:i\to j$ in $P$. If $f:x\to y$ is an arrow in $X$, the family $\, f\circ f_i:\Phi i \to y\,$ a cone with vertex $y$. So we have a functor $$\Cone(\Phi,-) : X\to\Set$$ This functor can be substantiated in two ways, that lead to two different points of view on limits and colimits. The usual way is to see a cone as a natural transformation from $\Phi$ to $\Delta x$, so that one defines $$\Cone(\Phi,x) := X^P(\Phi,\Delta x)$$ But a cone from $\Phi$ to $x$ is also the object mapping $\, i\,\mapsto\, f_i \,$ of a functor $\Phi\to X/x$ over $X$ (see Proposition \[52prop1\]), and we can define the above functor equivalently as $$\Cone(\Phi,x) := \CatX(\Phi,X/x)$$ Thus, while for any fixed $\Phi$ we get isomorphic functors \[62eq0\] (,X/-) (,-) we in fact have two different modules čone" $$\Cone:X^P\to X \qv \Cone:\CatX\to X$$ both representable on the right.
Since a colimit of $\Phi$ is a representing object for $\Cone(\Phi,-)$, we have also two different operative definitions of the colimit [*functor*]{}: either as the partially defined left adjoint $$\Colim : X^P\to X$$ to the functor $\,\Delta:X\to X^P\,$, or as the partially defined left adjoint $$\Colim : \CatX\to X$$ to the functor $\, X/- : X \to \CatX\,$. Dually, we have two limit functors $$\Lim : X^P\to X \qv \Lim : \CatX\to X\op$$ respectively right adjoint to $\,\Delta:X\to X^P\,$ and left adjoint to $\, -/X : X\op\to \CatX\,$.
- Since $X/-$, unlike $\Delta$, is always full and faithful, the counit $\eps_x:\Colim X/x\to x$ of the adjunction $\Colim\adj X/-$ is an isomorphism. (Indeed, this follows also from the fact that $X/x$ has a terminal object over $x$, that is $x{\mathop{\to}\limits}^\id x$.) The unit $\eta_\Phi:\Phi\to X/\Colim\Phi$ is again the universal cone.
- The two colimit functors (which agree on each single $\Phi:P\to X$ as an object of two different categories) can be merged in a single $$\Colim : \CatX^*\to X$$ (see [@mac71], [@par73] and [@kan58]) where the šupercomma category“ $\CatX^*$ is the Grothendieck category associated to the 2-representable functor $\Cat(-,X):\Cat\to\Cat$ (which includes both $\CatX$ and all the $X^P$ as subcategories). Indeed, we have the further module čone” (which includes the other ones) $$\Cone(\Phi,x) := \CatX^*(\Phi,\delta x)$$ where $\delta:X\to\CatX^*$ is the full and faithful functor which sends an object $x\in X$ to the corresponding category over $X$.
- If $H=\hom_X(F-,-)$, then $H$ over $X$ is the forgetful functor $U$ from $F$-algebras to $X$ (see the last of Examples \[32ex1\]); then the strong dinatural transformations $H\to X/x$ are the cones $U\to x$, as in [@din].
Since $X/x$ is a df (associated to the representable $X(-,x)$), the functor $X/-$ factors through $i:\df X\inc\CatX$. Thus its left adjoint $\Colim$ also factors through $\down(-):\CatX\to\df X$. More explicitly, we have the following chain of isomorphisms: $$\Cone(\Phi,-) \cong \CatX(\Phi,X/-) \cong \CatX(\down\Phi,X/-) \cong \Cone(\down\Phi,-)$$ so that $\Colim\,\Phi$ exists iff $\Colim\down\Phi$ exists, and if this is the case $$\Colim\,\Phi \,\cong\, \Colim\down\Phi$$ where $\down\Phi:P'\to X$ is the reflection of $\Phi:P\to X$ in discrete fibrations; dually $\, \Lim\,\Phi \,\cong\, \Lim\up\Phi \,$, either one existing if the other one does.
\[62prop1\] The colimit of a functor depends only on its reflection in df’s. Dually, the limit of a functor depends only on its reflection in dof’s.
The above considerations can be parametrized: given a functor $f:X\to Y$, one can compose the adjunctions $f_!\adj f^*$ and $\Colim\adj Y\-$ as follows: $$\xymatrix@C=3.5pc{ \CatX \ar@<1ex>[r]^{f_!}_\bot\ar@/^2pc/@<1ex>[rr]_{\Colim(f\circ-)}
& \CatY \ar@<1ex>[l]^{f^*} \ar@<1ex>[r]^\Colim_\bot & Y \ar@<1ex>[l]^{Y/-}\ar@/^2pc/@<1ex>[ll]_{f/-} }$$ So we have the partially defined adjunction with parameter $f:X\to Y$: \[62eq1\] (f-) f/-: Y (Alternatively, one easily checks directly that $\,\Cone(f\circ\Phi,y) \cong \CatX(\Phi,f/y)\,$.) Since the category $\, f/y = f^*(Y/y)\,$ over $X$ is the df associated to $Y(f-,y):X\op\to\Set$, the functor $f/-$ factors through $i:\df X\inc\CatX$. Thus its left adjoint $\Colim(f\circ-)$ also factors through $\down(-):\CatX\to\df X$. More explicitly, we have (f,-) (f,Y/-) (,f/-) (,f/-)\
(f,Y/-) (f,-) Thus $\Colim\,f\circ\Phi$ exists iff $\Colim\,f\circ\down\Phi$ exists, and if this is the case $$\Colim(f\circ\Phi) \cong \Colim(f\circ\down\Phi)$$ Furthermore, the weak colimits of $f\circ\Phi$ coincide with those of $f\circ\down\Phi$. In particular we have proved in a very direct manner the following proposition:
\[62prop2\] If two functors $\Phi:P\to X$ and $\Psi:Q\to X$ with the same codomain have isomorphic reflections $\,\down\Phi\,\, \cong\,\, \down\Psi\,$, then for any $f:X\to Y$ $$\Colim(f\circ\Phi) \,\cong\, \Colim(f\circ\Psi)$$ either side existing if the other one does. Furthermore, $f\circ\Phi:P\to Y$ and $f\circ\Psi:Q\to Y$ have the same weak colimits.
To prove the converse of Proposition \[62prop2\], we begin with the following
\[62prop21\] Let $P{\mathop{\to}\limits}^\Phi X$ be a category over $X$, $D$ a dof over $X$, and $\y$ the Yoneda embedding" $\,\, -/X:X\op\to\dof X\,$. Then there is a bijection $$\CatX(\Phi,D) \,\cong\, \Cone(\y\circ\Phi\op,D)$$ natural in $D\in\dof X$.
Rephrasing Proposition \[53prop1\], a morphism $\phi:\Phi\to D$ in $\CatX$ is given by a family of morphisms $\phi_i:\Phi i\to D,\, i\in P$, in $\CatX$, such that for any $u:i\to j$ in $P$, $\Phi u:\phi_i(\Phi i)\,\mapsto\,\phi_j(\Phi j)$ in $D$. By the universality of $\up(\Phi i) = \Phi i/X$, the latter corresponds to a family of morphisms $\phi_i:\Phi i/X\to D,\, i\in P$, in $\CatX$, such that for any $u:i\to j$ in $P$, $\Phi u:\phi_i(\Phi i{\mathop{\to}\limits}^\id\Phi i)\,\mapsto\,\phi_j(\Phi j{\mathop{\to}\limits}^\id\Phi j)$ in $D$; that is $\phi_i(\Phi i{\mathop{\to}\limits}^{\Phi u}\Phi j) = \phi_j(\Phi j{\mathop{\to}\limits}^\id\Phi j)$, as required by the cone condition, since $(\y\circ\Phi\op)u:\Phi j/X\to\Phi i/X:\Phi j{\mathop{\to}\limits}^\id\Phi j\mapsto\Phi i{\mathop{\to}\limits}^{\Phi u}\Phi j$.
\[62prop3\] The cone $\,\y\circ\Phi\op\to\,\up\Phi\,$ in $\dof X$, corresponding to the unit $\eta:\Phi\to\,\up\Phi$ of the reflection $\up(-):\CatX\to\dof X$ in dof’s, is a universal cone. Briefly, $\up\Phi$ is the colimit of $\,\y\circ\Phi\op:P\op\to\dof X$ (or the limit of $\,\y\op\circ\Phi:P\to(\dof X)\op$).
Putting together Proposition \[62prop2\] and Corollary \[62prop3\], we get (see [@par73]):
\[62prop4\] Let $\Phi:P\to X$ and $\Psi:Q\to X$ be two functors with the same codomain. The following are equivalent:
- $ \up\Phi\, \cong\,\, \up\Psi \,$;
- $\, \Lim(f\circ\Phi) \cong \Lim(f\circ\Psi) \,$, for any $f:X\to Y$, either side existing if the other one does.
\[62rmk1\]
- If in Lemma \[62prop21\] we take $\,D = x/X = \y x$, then we have a bijection $$\Cone(\Phi\op,x) \,\cong\, \Cone(x,\Phi) \,\cong\, \CatX(\Phi,x/X) \,\cong\, \Cone(\y\circ\Phi\op,\y x)$$ which is the same that we would obtain by applying directly the Yoneda embedding $\,\y : X\op\to\dof X\,$ to the cones $\,\Phi\op \to x$.
- Alternatively, following [@par73], one can prove Corollary \[62prop3\] by noting that since colimits in $\Set^{X\op}$ are computed pointwise, \[62eq3\] (())x (()x) X(-,x) (/x) ()x Conversely, by Corollary \[62prop3\], the (\[62eq3\]) above gives a new proof of the reflection formula.
- A further proof of Corollary \[62prop3\] can be obtained noticing that morphisms $\Phi\to D$ in $\CatX$ correspond also to cones $1\to D\circ\Phi$ in $\Set$, considering $D$ as a presheaf. (This fact also allows another proof of the second assertion in Proposition \[61prop3\], as explained in [@law70].) Then we have $$\begin{array}{c}
\dof X(\Colim(\y\circ\Phi\op),D) \\ \hline
\Lim\,\dof X(\y\circ\Phi\op -,D) \\ \hline
\Lim(D\circ\Phi) \\ \hline
\Cone(1,D\circ\Phi) \\ \hline
\CatX(\Phi,D)
\end{array}$$
Among the consequences of Corollary \[62prop4\], the following are particularly significant (see [@par73]):
\[62prop5\]
1. A category over a base and its reflections $\,\up\Phi$ and $\,\down\Phi$ have the same components.
2. The functor $\Phi$ is initial iff $\,\,\up\Phi \,\cong\, \id_X\,$.
3. The unit $\, \eta:\Phi\to\up\Phi \,$ is an initial functor between the respective total categories.
4. The object $x\in X$ is the absolute limit of $\Phi$ iff $\,\,\up\Phi\,\cong\, x/X\,$.
5. The object $x\in X$ is an absolute weak limit of $\Phi$ iff $\,\up\Phi$ is a retract of $\,x/X\,$ in $\dof X$.
<!-- -->
1. By Remark \[34rmk1\], for any $P{\mathop{\to}\limits}^p X$ in $\CatX$ we have $$\comp P \cong \Colim \Delta 1 \cong \Colim (\Delta 1\circ p) \cong \Colim (\Delta 1\circ\down p) \cong \comp(\down P)$$ The same result follows from the fact that since in the diagram below the square of right adjoint arrows commutes up to isomorphisms, so does also the square of left adjoint arrows: $$\xymatrix@R=4pc@C=4pc{
{\CatX} \ar@<1ex>[r]^{\comp}_{\bot}\ar@<-1ex>[d]_{\down\,(-)}^{\adj} & {\Set}\ar@<1ex>[l]^{\disc}\ar@2{-}[d] \\
{\df X} \ar@<1ex>[r]^{\comp}_{\bot}\ar@<-1ex>[u]_{i} & {\Set}\ar@<1ex>[l]^{\disc} }$$
2. Apply Corollary \[62prop4\] with $\Psi = X{\mathop{\to}\limits}^\id X$, the terminal df.
3. By point 2 just proved, we must show that for any dof $D$ over $\up P$ and any $\phi:\eta\to d$ over $\up P$, $\phi$ factors uniquely over $\up P$ as in the diagram below: $$\xymatrix@R=3pc@C=3pc{
P \ar[r]^\eta \ar@/^1.5pc/[rr]^\phi \ar[dr]_\eta \ar@/_1.5pc/[ddr]^p & \up P \ar[d]^\id \ar@{..>}[r]^s & D \ar[dl]^d \ar@/^1.5pc/[ddl] \\
& \up P \ar[d]^{\up\,\,p} & \\
& X & }$$ Since $\,\up p\circ d\,$ is a dof over $X$, and by the universality of $\eta$ over $X$, $\,\phi:p\to\,\,\up p\circ d\,$ factors over $X$ as $\,\phi = s\circ\eta$. Then $d\circ s\circ\eta \cong d\circ\phi \cong \eta$, so that $d\circ s \cong \id_{\up\,P}$, again by the universality of $\eta$ over $X$. Thus $s$ is a morphism over $\up\Phi$, giving the existence of the desired factorization. As for unicity, another such $s$ would give another factorization through $\eta$ of $\phi$ over $X$. (Note that this proof is not specifically linked to the situation now considered, unlike those given in [@par73], [@her98], and [@kel82].)
4. Apply Corollary \[62prop4\] with $\Psi = 1{\mathop{\to}\limits}^x X$. Note that each cone $f_i:x\to\Phi i$ in $X$, corresponding to $\Phi\to x/X$ in $\CatX$, induces by universality \[62eq4\] @R=3pc@C=3pc[ & @[..>]{}\[d\]\^\
\^& x/X ]{} Recalling Corollary \[62prop3\] and the first of Remarks \[62rmk1\], we may see $\alpha$ as the universally induced morphism $$\Colim(\y\circ\Phi\op) \to x/X$$ in $\CatX$, through which the cone $\y f_i = f_i/X : \Phi i/X \to x/X$ factors. Then the cone $f_i:x\to\Phi i$ in $X$ is an absolute limit iff the cone $f_i/X :\Phi i/X \to x/X$ is a limit in $\dof X$ (or in $\CatX$) iff $\alpha$ is an isomorphism.
5. If the cone $f_i:x\to\Phi i$ in $X$ is an absolute weak limit in $X$, then the cone $f_i/X:\Phi i/X \to x/X$ is a weak limit. Thus the universally induced $\alpha$, as in (\[62eq4\]) above, has a retraction. The reverse implication will be proved in \[63rmk3\].
- If $Y=\Set$, the adjunction (\[62eq1\]) becomes the adjunction $$\ten(A,-) \adj \neg A : \Set\to\CatX$$ of section \[4sub4\], with parameter $A$ in $\df X$.
- Since $f/y$ is a df over $X$, the adjunction (\[62eq1\]) restricts to \[62eq2\] (f-) f/-: YX\^[X]{} which displays the weighted colimit construction (as a functor of the weight $A\in\Set^{X\op}$) $$A \,\mapsto\, A*f \in Y$$ as the left adjoint to the functor $$y \,\mapsto\, Y(f-,y) \in \Set^{X\op}$$ (of course the $A$ in $\Colim(f\circ A)$ is intended over $X$ so that the colimit is taken on the čategory of elements“ of the corresponding presheaf). Thus, at least in the non-enriched context, the paradigmatical adjunction (\[62eq2\]) (as illustrated e.g. in [@mam91]) is nothing but a particular case of the more general (\[62eq1\]). (See also [@mac71], where the restriction of (\[62eq0\]) to df’s is referred to as the čoyoneda lemma” on page 62.)
Note that from Proposition \[62prop2\] we have $$\Colim(f\circ\Phi) \cong \Colim(f\circ\down\Phi) = \,\down\Phi*f$$
Atoms in $\CatX$ and their reflections {#6sub3}
--------------------------------------
We begin by recalling some well-known facts about idempotents and their splitting in $\Set$ and $\Set^X$. Let $\e$ be the monoid which represents the idempotent arrows of categories, that is the one whose unique non-identity arrow is idempotent.
\[63prop0\] For any category $X$ and any functor $\,\e{\mathop{\to}\limits}^e X$, the following are equivalent:
1. The idempotent $e:x\to x$ in $X$ splits.
2. The functor $e$ has a limit.
3. The functor $e$ has a colimit.
4. The pair $e,\id:x\to x$ has an equalizer.
5. The pair $e,\id:x\to x$ has a coequalizer.
If this is the case, all the above limits and colimits are absolute and canonically isomorphic.
Now let $e:D\to D$ an idempotent mapping, and $\,\e{\mathop{\to}\limits}^e \Set$ the corresponding functor. As a dof over $\e$, the category $e$ is (apart from the identities) exactly what we would call the ǧraph of the endomapping $e$".
\[63prop01\] For any functor $\,\e{\mathop{\to}\limits}^e \Set$, its limit, given by the $\fix\, e$ (the set of fixed points of the endomapping $e:D\to D$), and its colimit, given by the components of $e$ over $\e$, are canonically isomorphic.
Explicitly, the bijection $\beta :\Lim\,e \to \Colim\,e$ is given by $\beta x = [x]$ and $\beta\inv [x] = ex$.
\[63prop1\] Any idempotent $\,\e{\mathop{\to}\limits}^e X$ in $X$ is an atom in $\CatX$.
For any dof $D$, $\hom(\e,D)$ gives $\fix De$, while $\ten(\e,D)$ gives the components $\e\times D$ which, over $\e$, is the graph of $De:Dx\to Dx$ as in Corollary \[63prop01\]. Dually, $e$ is also a left atom.
\[63prop11\] For any atom $\, T{\mathop{\to}\limits}^t X$ of $\CatX$, $$\down T \otimes - \,\cong\, \Nat(\up T,-):\Set^X\to\Set \qv -\,\otimes\up T \,\cong\, \Nat(\down T,-):\Set^{X\op}\to\Set$$ For any idempotent atom $\,\e{\mathop{\to}\limits}^e X$, $$\down e \otimes D \cong \Nat(\up e, D) \cong \fix De \qv A\,\otimes\up e \cong \Nat(\down e, A) \cong \fix Ae$$
Since $T$ is a right atom, for any dof $D$ one has the following bijections, natural in $D\in\dof X$: $$\q \Nat(\up T,D) \cong \hom(\up T, D) \cong \hom(T, D) \cong \ten(T, D) \cong \ten(\down T, D) \cong\, \down T \otimes D$$ and similarly for any df $A$. The second part follows from Corollary \[63prop1\].
\[63rmk1\]
1. In particular, for any idempotent $\, e:x_0\to x_0$ in $X$, $$\hom(\up e, \up x) \cong \ten(\down e, \up x) \cong \fix X(x,e)$$ where $X(x,e) = e\circ- : X(x,x_0)\to X(x,x_0)$. But the reflection formula (or co-Yoneda) gives $\down e \cong \ten(\down e, \up -)$, so that $$\down e \cong \hom(\up e, \up -) \cong \fix X(-,e)$$ that is, $\down e$ is the subfunctor of $\,\down x_0 = X(-,x_0)\,$ given by the arrows fixed by composition with $e$: $$x{\mathop{\to}\limits}^f x_0 \in \fix X(x,e) \iff e\circ f = f$$ Dually, $$\up e \cong \hom(\down e, \down -) \cong \fix X(e,-)$$
2. In particular, for any idempotents $\, e:x\to x$ in $X$ and $\, e':y\to y$, $$\hom(\up e, \up e') \cong \ten(\down e, \up e') \cong \fix (\up e')e$$ is the subset of $(\up e')x$ fixed by $(\up e')e$. By the above remark, this is the set of arrows in $\fix X(e',x)$ fixed also by $e$, that is the arrows $f:y\to x$ such that $e\circ f = f = f\circ e'$. Dually $\, \hom(\up e, \up e')\, $ is the set of arrows $f:x\to y$ such that $f\circ e = f = e'\circ f$.
By Yoneda, any idempotent in $\Set^X$ on a representable $X(x_0,-)$ has the form $$X(e,-):X(x_0,-)\to X(x_0,-)$$ for a unique idempotent $\, e:x_0\to x_0$ in $X$.
\[63prop02\] Let $D:X\to\Set$. The following are equivalent:
1. $D$ is a retract in $\Set^X$ of the representable functor $X(x_0,-)$, associated to the idempotent $X(e,-):X(x_0,-)\to X(x_0,-)$.
2. $D$ is isomorphic to the subfunctor $\fix X(e,-)$ of $X(x_0,-)$.
3. $D$ is isomorphic to the quotient of $X(x_0,-)$ given by $\, f\circ e \sim f $.
4. $D$ is the reflection of the idempotent atom $e$: $\, D\,\cong\,\,\up e$.
By Proposition \[63prop0\], the retract $D$ is the limit of $\e{\mathop{\longrightarrow}\limits}^{X(e,-)} \Set^X$; but the latter is computed pointwise, so that 1 is equivalent to 2 and 3 by Corollary \[63prop01\]. That 4 is equivalent to 2 follows from the first of Remarks \[63rmk1\]; alternatively note that the functor in 3 is $\comp (\e/-)$, and the latter is $\up e$ by the reflection formula.
\[63prop12\] Let $A$ be a df and $D$ a dof on $X$ such that $$A\otimes - \cong \Nat(D,-):\Set^X\to\Set$$ Then $D$ is a retract in $\Set^X$ of a representable functor.
Let $u\in A\otimes D = \comp(A\times D)$ be an universal element of $A\otimes -$. Then $u$ is the component of a pair $\la a,d \ra$ over, say, $x_0\in X$: $$u = [\la a,d \ra],\q a\in Ax_0,\, d\in Dx_0$$ Let $\iota:D\to X(x_0,-)$ be the unique morphism in $\Set^X$ such that $(A\otimes\iota)u =[\la a,\id_{x_0} \ra]$: $$[\la a,\id_{x_0} \ra] = (A\otimes\iota)u = \comp(A\times\iota)[\la a,d \ra] = [\la a,\iota d \ra]$$ and let $\rho:X(x_0,-)\to D$ be the unique morphism in $\Set^X$ such that $\rho\,\id_{x_0} = d$. Then $$A\otimes\rho\circ\iota = \comp(A\times\rho\circ\iota): [\la a,d \ra] \,\mapsto\, [\la a,d \ra]$$ so that $\rho\circ\iota = \id_D$, as required. Note that $D$ splits the idempotent $\iota\circ\rho:X(x_0,-)\to X(x_0,-)$ which corresponds to the idempotent $e = \iota d:x_0\to x_0$ in $X$.
\[63prop2\] Let $A$ be a df and $D$ a dof on $X$. The following are equivalent:
1. $A\otimes - \cong \Nat(D,-):\Set^X\to\Set$.
2. $-\otimes D \cong \Nat(A,-):\Set^{X\op}\to\Set$.
3. $A\,\cong\,\,\down e$ and $D\,\cong\,\,\up e$, for an idempotent atom $e$ in $\CatX$.
4. $A\,\cong\,\,\down T$ and $D\,\cong\,\,\up T$, for an atom $T$ in $\CatX$.
Trivially 3 implies 4, and 4 implies 1 and 2 by Proposition \[63prop11\]. From Proposition \[63prop12\] and Corollary \[63prop02\] it follows that 1 implies that $D\,\cong\,\up e$. Furthermore, $$A\,\cong\, A\,\otimes \up - \,\cong\,\Nat(D,\up -)\,\cong\,\Nat(\up e,\up -)\,\cong\,\,\down e\,\otimes\up - \,\cong\,\,\down e$$ where Proposition \[63prop11\] have been used again. Thus 1 implies 3. By duality we also have the equivalence between 2 and 3 (note that 3 is autodual).
In [@kel05] it is shown, in an enriched context, the equivalence between 1 and 2 and the fact that $A$ and $D$ are adjoint modules.
Any right atom is also a left atom, and vice versa.
Suppose that $T$ is a right atom. As in Proposition \[63prop11\], $\, \down T \otimes - \,\cong\, \Nat(\up T,-) \,$, and so by Proposition \[63prop2\] $\, -\,\otimes\up T \,\cong\, \Nat(\down T,-) \,$. Then, for any df $A$, $$\ten(A,T) \,\cong\, A\,\otimes\up T \,\cong\, \Nat(\down T,A) \,\cong\, \hom(T,A)$$ that is, $T$ is also a left atom.
\[63prop3\] Let $A$ be a df on $X$. The following are equivalent:
1. $A$ is a retract of a representable functor.
2. $A \,\cong\,\,\down e$ for an idempotent atom $e$ in $\CatX$.
3. $A \,\cong\,\,\down T$ for an atom $T$ in $\CatX$.
4. $A\otimes -:\Set^X\to\Set$ is representable.
5. $\Nat(A,-)\cong -\otimes D:\Set^{X\op}\to\Set$ for a $D\in\Set^X$.
6. $A\otimes -:\Set^X\to\Set$ has a left adjoint.
7. $\Nat(A,-):\Set^{X\op}\to\Set$ has a right adjoint.
8. $A\otimes - :\Set^X\to\Set$ preserves all limits.
9. $\Nat(A,-):\Set^{X\op}\to\Set$ preserves all colimits.
10. $A$ is a left adjoint module.
We have already seen the equivalence between the first five properties. If $A\otimes - \cong \Nat(D,-)$, then $(-)\cdot D\adj A\otimes -$. If $\Nat(A,-)\cong -\otimes D$, then $\Nat(A,-)\adj\neg D$. For the equivalence between the last three properties and 4 and 5, we refer again to [@kel05], where they are proved in an enriched context.
\[63rmk3\] We can now complete the proof of Proposition \[62prop5\], showing that if $\up\Phi$ is a retract of a representable functor $X(x,-)$ then $x$ is an absolute weak limit of $\Phi$. By Corollary \[63prop02\], $\up\Phi \cong\, \up e$, for an idempotent $e:x\to x$. But $e:x\to x$ itself is a cone from $x$ to $e:\e\to X$, and it is weakly universal: $$\xymatrix@R=2pc@C=2pc{ y \ar@{..>}[d]_f \ar[dr]^f & \\
x \ar[r]^e & x \ar@(ur,dr)^e }$$ Furthermore, this weak limit is clearly absolute. Then, by Proposition \[62prop2\], $x$ is also an absolute weak limit of $\Phi$.
By Proposition \[63prop2\], for any atom $T{\mathop{\to}\limits}^t X$ there exists an idempotent $e:x\to x$ in $X$ such that $\up T = \,\up e$ and $\down T = \,\down e$. This idempotent is not unique, since e.g. if $e$ splits as $x{\mathop{\to}\limits}^r y{\mathop{\to}\limits}^i x$, then $\up e = \,\up y$. Anyway, by the above remark, any atom has an absolute weak (co)limit.
For an atom $T{\mathop{\to}\limits}^t X$ the following are equivalent:
- $T$ has a limit.
- $T$ has a colimit.
- $T$ has an absolute limit.
- $T$ has an absolute colimit.
Indeed, by Proposition \[63prop0\], all the above are equivalent for any $e:\e\to X$, and so by Corollary \[62prop4\] they are equivalent also for $T$.
If the conditions of the above proposition are satisfied, we could say that $t$ čonverges“ to its (co)limit. In particular, any split idempotent in $X$ converges to its retracts, and a category is Cauchy complete iff any atom converges. Furthermore, any functor $f:X\to Y$ is čontinuous”: if $t$ converges to $x$, then $f_! t$ converges to $fx$ (see Corollary \[61prop12\]).
Recall that the Cauchy completion of a category can be obtained as the full subcategory of $\Set^{X\op}$ generated by the retracts of representable functors (see e.g. [@bor94]). But we have seen that the latter have the form $\down T$ or $\down e$, and so we get the first part of the following proposition. The second part follows from the second of Remarks \[63rmk1\].
The reflections of atoms (or of idempotent atoms) in df’s generate the Cauchy completion of $X$. Furthermore, given two idempotent arrows $e:x\to x$ and $e':y\to y$ in $X$, $\hom(\down e,\down e')$ is the set of arrows $f:x\to y$ such that $f\circ e = f = e'\circ f$. So the Cauchy completion is the same thing as the Ǩaroubi envelope" of $X$ [(see [@rey04], [@lam86] and [@law89]).]{}
Graphs and evolutive sets {#6sub4}
-------------------------
Let $X$ be a graph in $\Grph$. A graph $P$ over $X$ is a (graph) opfibration if for any $f$ in $X$, the fiber $Pf$ (defined like in section \[3sub1\]) ǐs" a mapping. Fibrations are defined dually. So we have the inclusions $$i : \df X \inc \GrphX \qv j : \dof X \inc \GrphX$$ of fibrations and opfibrations in graphs over $X$. These have left and right adjoints $$\down(-) \adj i \adj (-)\rdown \qv \up(-) \adj j \adj (-)\rup$$ as can be shown in two ways:
- Since $\Grph$ is a presheaf category, $\GrphX$ is also a presheaf category, namely on the total category of the df $X$; the above inclusions may be seen as induced by functors from this category, so that the reflection and coreflection are given by the left and the right Kan extensions along these functors.
- The functor $\,\GrphX \to \CatFX\,$ given by $$P{\mathop{\longrightarrow}\limits}^p X \q\mapsto\q {\cal F}\!P{\mathop{\longrightarrow}\limits}^{{\cal F}\! p}{\cal F}\!X$$ where $\,{\cal F}:\Grph\to\Cat\,$ is the free category“ functor, is a full and faithful functor whose values are (up to isomorphisms) the ǓFL functors” to ${\cal F}\!X$ (see e.g. [@bun00]); and $p$ is a graph fibration iff ${\cal F}\! p$ is a df. So the reflection of the category ${\cal F}\!P{\mathop{\longrightarrow}\limits}^{{\cal F}\! p}{\cal F}\!X$ over ${\cal F}\!X$ in df’s (being itself a UFL functor) gives also the reflection of $P{\mathop{\longrightarrow}\limits}^p X$ in graphs fibrations: @R=4pc@C=4pc[ \_[(-)]{} & \^[(-)]{}\
[X]{} \^\~& ]{}
But perhaps it is more interesting to see why and how the reflection formulas can be applied directly to this context:
- Every node $x\in X$, as a graph over $X$, is a strong atom in $\GrphX$ since $$\hom(x,P) \cong \ten(x,P) \cong Px$$ (note that the fiber $Px$ is [*always*]{} discrete).
- The reflections $\up x$ and $\down x$ are given by $(\up x)y = {\cal F}\!X(x,y)$ and $(\down x)y = {\cal F}\!X(y,x)$.
- Any graph over $X$ has a complement (since $\GrphX$ is cartesian closed); like in section \[3sub1\], one sees that the complement of a fibration is valued in opfibrations, and vice versa.
- The above facts are all what is needed to prove the reflection formulas $$(\down P)x = \ten(\up x,P) \qv (\up P)x = \ten(\down x,P)$$ as was done for categories over a base. Of course, one has also the coreflection formulas $$(P\rdown)x = \hom(\down x,P) \qv (P\rup)x = \hom(\up x,P)$$
In particular, if $X$ is the terminal graph (the ľoop"), we have the inclusions $$i : \Lendo \inc \Grph \qv j : \Rendo \inc \Grph$$ of anti-evolutive sets and evolutive sets (or left and right endomappings) in graphs. The reflections of the dot atom $\d$ (see section \[4sub2\]) are the infinite chain and anti-chain: $$\up \d = \q \xymatrix@1@C=2pc{
{\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar@{..>}[r] & }$$ $$\down \d = \q \xymatrix@1@C=2pc{
{\bullet} & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & \ar@{..>}[l] }$$ So the reflection $\up P$ of a graph $P$ in evolutive sets is given by the set $\comp(\down \d \times P)$, with the action given by the right shift of the anti-chain $\down\d$: $$[n,x]\,\mapsto\, [n+1,x]$$
\[64ex1\]
1. Let $P$ be the graph $ \q \xymatrix@1@C=2pc{{\bullet}\ar[r] & {\bullet} }\q $ which is not a (right) endomapping because there are no arrows out of one node. Applying the reflection formula, we multiply $P$ with the anti-chain $\down \d$, getting the graph $$\xymatrix@R=2pc@C=2pc{
{\bullet} & {\bullet} & {\bullet} & {\bullet} & {\bullet} & \\
{\bullet} & {\bullet}\ar[ul] & {\bullet}\ar[ul] & {\bullet}\ar[ul] & {\bullet}\ar[ul] & \ar@{..>}[ul] }$$ with an infinite number of components, which are the nodes of the reflection of $P$. Furthermore, the action on it is given by translation, which sends any component in the one on its right. So $\up P$ is the chain $ \, \xymatrix@1@C=2pc{
{\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar[r] & {\bullet}\ar@{..>}[r] & } \, $, wherein the missing codomains in $P$ have been added. On the other hand, there are no chains in $P$ (that is, $\hom(\up\d,P) = \emptyset$), so that the coreflection $P\rup$ is the void endomapping: the nodes with no codomains have been deleted.
2. Let $P$ be the graph $ \qq \xymatrix@1@C=2pc{{\bullet}\ar@(ul,dl) & {\bullet}\ar[l]\ar[r] & {\bullet}\ar@(ur,dr) }\qq $ which is not a (right) endomapping because there are two arrows out of one node. Applying the reflection formula, we multiply $P$ with the antichain $\down \d$, getting the two-components graph $$\xymatrix@R=2pc@C=2pc{
{\bullet} & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & \ar@{..>}[l] \\
{\bullet} & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & {\bullet}\ar[l] & \ar@{..>}[l] \\
{\bullet} & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & \ar@{..>}[ul]\ar@{..>}[uul] }$$ So the reflection of $P$ has two nodes. Furthermore, the action on it is given again by right translation, so that the second component is a fixed point and we get $$\up P = \xymatrix@1@C=2pc{{\bullet}\ar[r] & {\bullet}\ar@(ur,dr) }$$ wherein the multiple codomains in $P$ have been identified. On the other hand, there are four chains in $P$ so that $P\rup$ has four nodes: $$P\rup\q = \qq\xymatrix@1@C=2pc{{\bullet}\ar@(ul,dl) & {\bullet}\ar[l] & {\bullet}\ar[r] & {\bullet}\ar@(ur,dr) }$$ that is, the nodes with multiple codomains have been splitted.
3. If $ P = \, \xymatrix@1@C=2pc{{\bullet}\ar@/^1pc/[rr] & {\bullet}\ar[l]\ar[r] & {\bullet} }\, $, $\down\d\times P$ is the graph $$\xymatrix@R=2pc@C=2pc{
{\bullet} & {\bullet} & {\bullet} & {\bullet} & {\bullet} & \\
{\bullet} & {\bullet}\ar[ul] & {\bullet}\ar[ul] & {\bullet}\ar[ul] & {\bullet}\ar[ul] & \ar@{..>}[ul] \\
{\bullet} & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & {\bullet}\ar[ul]\ar[uul] & \ar@{..>}[ul]\ar@{..>}[uul] }$$ so that $\, \up P = \,\xymatrix@1@C=2pc{{\bullet}\ar[r] & {\bullet}\ar@(ur,dr) }\q\,\, $ again. One should compare this example with the technique used in [@rey04] to compute the same reflection.
4. An example where both the phenomena of adding and identifying codomains, are present is given by the graph $ \q \xymatrix@1@C=2pc{{\bullet} & {\bullet}\ar[l]\ar[r] & {\bullet} }\q $, whose reflection is again the chain.
5. If $ \, P_1 = \q\,\, \xymatrix@1@C=2pc{{\bullet}\ar@(ul,dl)\ar[r] & {\bullet} } \,$, $ \, P_2 = \q\,\, \xymatrix@1@C=2pc{{\bullet}\ar@(ul,dl)\ar[r] & {\bullet}\ar@(ur,dr) }\q\,\, $ and $ \, P_3 = \q\,\, \xymatrix@1@C=2pc{{\bullet}\ar@(ul,dl)\ar@(ur,dr) }\q\,\, $, then $\up P_1$, $\up P_2$ and $\up P_3$ are all the loop (identification prevails in $\up P_1$). As for coreflections, $P_1\rup$ is the loop, $P_2\rup$ is the sum of a loop and a left infinite chain ending with a loop on the right, and $P_3\rup$ is the set $2^\N$ of sequences in a two-element set, under the action $f\,\mapsto\, f(1+ -)$.
$\Rendo$ and $\Lendo$ can be defined as the full subcategories of $\Grph$ generated by the objects orthogonal respectively to \[64eq1\] @1@C=2pc[ = & & & @[..>]{}\[r\] & ]{} @1@C=2pc[ = & & & & @[..>]{}\[l\] ]{} meaning the morphism of graphs from the dot $\d$ to the chain (respectively, the anti-chain) which sends the unique node of $\d$ to the highlighted node. Similarly, orthogonality with respect to \[64eq3\] = @1@C=2pc[& @(ur,dr) ]{} = @1@C=2pc[ & @(ur,dr) ]{} \[64eq4\] @1@C=2pc[ =@[..>]{}\[r\] & & & & & @[..>]{}\[r\] & ]{} \[64eq5\] \_n =@R=1.5pc@C=1.5pc[ & @/\^/@[..>]{}\[dl\]\
& ]{} (n [arrows]{}) defines respectively the full subcategories $\Lidem$ and $\Ridem$ of left and right idempotent endomappings, and the full subcategories $\Bij$ of bijective endomappings and $\Bij_n$ of n-periodic bijective endomappings. So we have the following inclusions: \[64eq6\] @R=2pc@C=2pc[ & &\
@<.5ex>@[..>]{}\[rr\] & & @<.5ex>@[..>]{}\[ll\]\
@<.5ex>@/\_1pc/@[..>]{}\[rr\] & @[..>]{}@(ul,ur) & @<.5ex>@/\^1pc/@[..>]{}\[ll\]\
& \_n@[..>]{}@(dl,dr) & ]{}
\[64rmk1\] All the above categories are equivalent to presheaf categories, and all the above inclusions correspond to functors induced by epimorphism between the corresponding categories of shapes. So they have left and right adjoints given by Kan extensions. But also in this case, a direct application of the formulas gives easier computations of these adjoints.
- The effect of complementation in $\Grph$ is displayed by the dotted arrows in diagram (\[64eq6\]): the complement of a left (right) idempotent endomapping is valued in right (left) idempotent endomappings; the complement of a bijective endomapping is valued in bijective endomappings; the complement of a $n$-periodic endomapping is valued in $n$-periodic endomappings. These facts may be seen as a consequence of Remark \[61rmk1\] and Remark \[64rmk1\].
- Since the (\[64eq1\]), (\[64eq3\]), (\[64eq4\]) and (\[64eq5\]) are all orthogonal to themselves, they serve also as the reflections of the dot graph $\d$ in the corresponding subcategories (where the highlighted morphism from $\d$ now gives universal arrow).
The above facts are all what is needed to prove the reflection formulas of graphs in each of the above subcategories; e.g., $\, \ten(\ee\op,P) = \comp(\ee\op \times P) \,$ gives the nodes of the reflection of the graph $P$ in (right) idempotent mappings. Of course, one has also the corresponding coreflection formulas; e.g., $\, \Grph(\ee,P) \,$ gives the nodes of the coreflection of the graph $P$ in (right) idempotent mappings.
- It is easy to see that if $P$ is connected (that is, $\comp P = 1$) then $\ee\op\times P$ has $n+1$ components, where $n$ is the number of nodes that are not codomains of any arrow. So the reflection in (right) idempotent mappings acts on each component mantaining such nodes, and collapsing the rest of it to the fixed point.
- The reflection $\up P = \,\down P$ of a graph $P$ in bijective endomappings is obtained by taking the components of $P\times\z$. For the graphs of Examples \[64ex1\] we obtain either $\z$ itself or the fixed point $\z_1$. If $\, P\, =\, \xymatrix@R=2pc@C=2pc{ \bullet \ar[r] & \bullet \ar@/^/[r] & \bullet \ar@/^/[l] } \,$, then $\up P$ and $P\rup$ are both $\z_2$.
- The reflection $\up P = \,\down P$ of a graph $P$ in $n$-periodic endomappings is obtained by taking the components of $P\times\z_n$. Since, as is easily checked, $$\z_k\times \z_n = \gcd(k,n)\cdot \z_{\lcm(k,n)}$$ we deduce that $\up \z_k$ has $\gcd(k,n)$ nodes and so, being connected by Corollary \[62prop5\], $$\up \z_k = \z_{\gcd(k,n)}$$ Any bijective endomapping $P$ is a sum of cycles: $P = \sum_{k=1}^\infty S_k\cdot \z_k$, so that $$\up P = \up\Big(\sum_{k=1}^\infty S_k\cdot \z_k \Big) = \sum_{k=1}^\infty S_k\cdot \up \z_k = \sum_{k=1}^\infty S_k\cdot \z_{\gcd(k,n)}$$
<!-- -->
- The graphs (\[64eq3\]) and (\[64eq5\]) are instances of the following $$\ee(n,m) \, =\q \xymatrix@R=1.5pc@C=1.5pc{ \circ\ar[r] & \bullet\ar@{..>}[r] & \bullet\ar[r] & \bullet\ar@{..>}@/^/[dl] \\
& & \bullet\ar[u] & }
\qv
\ee(n,m)\op \, =\q \xymatrix@R=1.5pc@C=1.5pc{ \circ & \bullet\ar[l] & \bullet\ar@{..>}[l]\ar[d] & \bullet\ar[l] \\
& & \bullet\ar@{..>}@/_/[ur] & }$$ where $n\geq 1$ is the length of the cycle, while $m\geq 0$ is the length of the chain. As before, these graphs allow to compute the reflection (and coreflection) in the corresponding full subcategories of $\Grph$, with the usual formulas.
- Obviously, when the graph $P$ grows bigger, it becomes more and more difficult to visualize the components of $\down\d\times P$. So an appropriate software would be helpful. Note that if $P$ is finite, then also when $\down\d$ is infinite, only a finite portion of the product is needed to capture the reflection. The size of this portion depends on that of $P$ itself.
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---
abstract: 'We investigate the class of the edge-shelling convex geometries of trees. The edge-shelling convex geometry of a tree is the convex geometry consisting of the sets of edges of the subtrees. For the edge-shelling convex geometry of a tree, the size of the stem of any rooted circuit is two. The class of the edge-shelling convex geometry of a tree is closed under trace operation. We characterize the class of the edge-shelling convex geometry of a tree in terms of trace-minimal forbidden minors. Moreover, the trace-minimal forbidden minors are specified for the class of convex geometries such that the size of any stem is two.'
author:
- |
Kenji Kashiwabara and Masataka Nakamura\
Department of Systems Science, The University of Tokyo
title: 'A characterization of the edge-shelling convex geometries of trees'
---
Introduction
============
A convex geometry is a combinatorial abstraction of convexity whereas a matroid is an abstraction of linear dependency. Convex geometries are derived from a variety of combinatorial objects, such as posets, affine point configurations, chordal graphs, semi-lattices, and so on [@EJ]. The complement of a convex geometry is known as an antimatroid. We investigate convex geometries whose rooted circuits have stems of size 2.
A tree is a connected graph containing no cycle. A tree provides two types of convex geometries, that is, the edge-shelling convex geometry of the tree and the vertex-shelling convex geometry of the tree. The edge-shelling convex geometry of a tree is the convex geometry consisting of the sets of edges of the subtrees. A subtree of a tree is a connected subgraph. The vertex-shelling convex geometry of a tree is the convex geometry consisting of the sets of vertices of the subtrees. The size of any stem of the edge-shelling convex geometry of a tree is two. Every trace of the edge-shelling convex geometry of a tree is also the edge-shelling convex geometry of some tree. On the other hand, the size of some stem of some vertex-shelling convex geometry is not two. Some trace of some vertex-shelling convex geometry is not a vertex-shelling convex geometry. We compare an edge-shelling convex geometry and a vertex-shelling convex geometry in Subsection \[subsec:edgevertex\].
In this paper, we characterize the class of the edge-shelling convex geometries of trees. We specify all the trace-minimal forbidden minors. This is our main result, presented in Section \[sec:main\].
A simplicial-shelling convex geometry is the convex geometry arising from a chordal graph. Any rooted circuit of a simplicial-shelling convex geometry has stems of size 2. We show an example of a simplicial-shelling convex geometry whose trace does not belong to the class of a simplicial-shelling convex geometry. This will be presented in Section \[sec:simplicial\].
Moreover, the trace-minimal forbidden minors are determined for the class of convex geometries such that the size of any stem is two. This class is characterized in the families of rooted sets whereas the class of an edge-shelling convex geometry is characterized in the convex geometries. This will be presented in Section \[sec:eminor\].
Although there are a rich number of forbidden-minor type characterization theorems in matroid theory, there are a few characterization theorems in the theory of convex geometries. Nakamura [@N] gives a forbidden minor characterization for the class of convex geometries of the graph search with respect to the operations of deletion and contraction. Okamoto and Nakamura [@ON] gives that of the convex geometries of the graph line search.
Preliminaries
=============
Closure systems
---------------
Let $E$ be a finite ground set throughout this papar.
Since a convex geometry is a special case of closure systems, we begin with defining the closure system. The closure operator and the extreme operator are defined for a closure system. These operators play an important role for convex geometries.
A family $\cal K$ of sets on $E$ is a closure system if the following conditions hold.
1. $E\in {\cal K}$.
2. $A,B\in {\cal K}$ implies $A\cap B \in {\cal K}$.
An element of a closure system is called a [*closed set*]{}.
For a closure system, define $\tau(X)=\bigcap\{A \in {\cal K}|X\subseteq A\}$ for $X\subseteq E$. Since $\tau(X)$ is a closed set, it is the smallest set in all the closed sets including $X$. $\tau:2^E\to 2^E$ is called the [*closure operator*]{} of the closure system.
Define $ex(X) = \{x\in X|x \notin \tau(X-x)\}$ for $X\subseteq E$. ${\mbox{ex}}:2^E\to 2^E$ is called the [*extreme operator*]{} of the closure system.
For $e\in V$ and $X\subseteq E$, $(X,e)$ with $e\notin X$ is called a [*rooted set*]{}. $e$ is called the [*root*]{} of the rooted set, and $X$ is called the [*stem*]{} of the rooted set. In the literature, $(X\cup\{e\},e)$, other than $(X,e)$, is often called a rooted set. But when we write a rooted set $(X,e)$, we assume that $X$ does not include $e$.
For a closure system ${\cal K}$ on $E$, a rooted set $(X,e)$ is called a [*rooted circuit*]{} if $X$ is a minimal set satisfying $e\in \tau(X)$.
The next lemma gives the relation between the extreme operator and the rooted circuits.
\[lem:exstem\] For a closure system, $e\in ex(X)$ if and only if there exists no rooted circuit $(A,e)$ with $A\subseteq X$.
For a family $\cal C$ of rooted circuits and $T\subseteq E$, the [*trace*]{} ${\cal C}:T$ from $\cal C$ on $T$ is defined as $\{(X,e)\in {\cal C}|X \subseteq T, e\in T\}$.
While trace operation is usually defined for a family of sets, we define trace operation for a family of rooted circuits. We will see in Lemma \[lem:tracerooted\] that both definitions coincide with each other for convex geometries. This definition plays an important role in Section \[sec:eminor\].
Convex geometries
-----------------
A convex geometry is a special case of a closure system. We can define that a closure system is a convex geometry in various ways.
A closure system ${\cal K}$ is a [*convex geometry*]{} if, for any non-empty closed set $X\in {\cal K}$, ${\mbox{ex}}(X)$ is non-empty.
An element of a convex geometry is called a [*convex set*]{}.
It is known that a convex geometry is also defined in terms of the axiom of rooted circuits.
[@D]\[lem:rootedaxiom\] A family $\cal C$ of rooted sets becomes the set of rooted circuits of a convex geometry if and only if the following two conditions hold.
\(1) If $(X,e)$ and $(Y,e)$ belong to $\cal C$ with $X\subseteq Y$, then $X=Y$.
\(2) If $(X,e)$ and $(Y,f)$ belong to $\cal C$ with $e\in Y$, then there exists $Z\subseteq X\cup Y-\{e\}$ with $(Z,f) \in \cal C$.
The next lemma follows from the definitions of trace and rooted circuits.
\[lem:tracerooted\] The trace ${\cal C}:T$ from the rooted circuits $\cal C$ of a convex geometry $\cal K$ on $T\subseteq E$ is the rooted circuits of a convex geometry $\{T \cap X|X \in {\cal K}\}$. Note that $T$ may not be a convex set.
So the class of a convex geometry is closed under trace operation.
Convex geometries of stem size 2
--------------------------------
The class of edge-shelling convex geometries of trees is a subclass of convex geometries with stems of size 2. In this subsection, we consider convex geometries with stems of size 2. We call such a convex geometry a [*convex geometry of stem size 2*]{}. There are many important classes of convex geometries which belong to this class, e.g., the double-shelling convex geometry of a poset, the simplicial-shelling convex geometry of a chordal graph, the edge-shelling convex geometry of a tree, the vertex-shelling convex geometry of a tree, and so on.
The existence of a rooted circuit $(\{x,y\},z)$ may be interpreted as that $z$ is between $x$ and $y$ generally.
We consider a family of rooted circuits $(\{x,y\},z)$ and we discuss whether it satisfies the axiom of the convex geometry or not. At that time, the trace of such a family on $X\subseteq E$ is defined as the collection of $(\{x,y\},z)$ with $x,y,z\in X$.
The next lemma follows from the case where the size of stem is 2 in Lemma \[lem:rootedaxiom\].
\[lem:t\] For any convex geometry with the stem size 2, when $(\{x,y\},z)\in {\cal C}$ and $(\{v,z\},u)\in {\cal C}$, then $(\{x,v\},u)\in {\cal C}$, $(\{y,v\},u)\in {\cal C}$, or $(\{x,y\},u)\in {\cal C}$ holds.
By Lemma \[lem:rootedaxiom\], there exists a rooted circuit $(X,u) \in {\cal C}$ such that $X\subseteq \{x,y,v\}$. Since the size of the stem is 2, $X=\{x,v\}, \{y,v\},$ or $\{x,y\}$.
The following lemmas are special cases of Lemma \[lem:t\].
\[lem:p\] For any convex geometry with the stem size 2, when $(\{x,y\},z)\in {\cal C}$ and $(\{x,z\},u)\in {\cal C}$, then $(\{x,y\},u)\in {\cal C}$ holds.
\[lem:i4\] For any convex geometry with the stem size 2, when $(\{x,y\},z)\in {\cal C}$ and $(\{u,z\},y)\in {\cal C}$, $(\{x,u\},y)\in {\cal C}$ and $(\{x,u\},z)\in {\cal C}$ hold.
![Lemma \[lem:i4\][]{data-label="fig:rule-i"}](rule-t.pdf)
![Lemma \[lem:i4\][]{data-label="fig:rule-i"}](rule-p.pdf)
![Lemma \[lem:i4\][]{data-label="fig:rule-i"}](rule-i.pdf)
In this paper, we illustrate a rooted circuit as a line connecting three points as in Figures \[fig:rule-t\], \[fig:rule-p\], and \[fig:rule-i\]. Since these figures do not depict Hasse diagrams, the vertical position of points in these figures has no meaning. When a figure depicts the Hasse diagram of a poset, we write explicitly ‘the Hasse diagram’ in its caption to avoid confusion.
We consider the ordering arising from a convex geometry of stem size 2 and an extreme element as follows.
Fix an element $x\in {\mbox{ex}}(E)$ since ${\mbox{ex}}(E)$ is not empty.
When $x$ is contained in some rooted circuit, $x$ is contained in the stem of the rooted circuit since $x\in {\mbox{ex}}(E)$ by Lemma \[lem:exstem\].
For $y,z \in E-\{x\}$, define a binary relation $y>_{x} z$ when $(\{x,z\},y)$ is a rooted circuit. Define $x >_{x} y$ for any $y\in E-\{x\}$.
\[lem:strict\] For a convex geometry of stem size 2 and $x \in {\mbox{ex}}(E)$, binary relation $>_x$ is a strict partial order.
It suffices to show asymmetry and transitivity.
First we show asymmetry. Since the case where $x$ is involved in is easy, we prove the asymmetry which does not involve $x$. Suppose that $y>z$ and $z>y$. Then we have $(\{x,z\},y)\in {\cal C}$ and $(\{x,y\},z)\in {\cal C}$. This contradicts that the trace on $\{x,y,z\}$ is a convex geometry.
Next, we show transitivity. Since the case where $x$ is involved in is easy, we prove the transitivity which does not involve $x$. Suppose that $y>z$ and $z>u$. Then we have $(\{x,z\},y)\in {\cal C}$ and $(\{x,u\},z)\in {\cal C}$. By Lemma \[lem:p\], we have $(\{x,u\},y)\in {\cal C}$ and $y>u$.
Edge-shelling convex geometry of a tree {#subsec:edgevertex}
---------------------------------------
The convex geometry arising from the edge shelling of a tree is a convex geometry defined on the edges of the tree. Its convex sets consist of a subset of the edges of the given tree that forms a tree. The class of convex geometries arising from the edge shelling of a tree is closed under trace operation. The size of any stem of the edge-shelling convex geometry of a tree is two.
For the edge-shelling convex geometry of a tree with rooted circuits ${\cal C}$, $(\{x,y\},z)\in {\cal C}$ if and only if $z$ is on the path between $x$ and $y$ on the tree.
This lemma follows from that $\tau(\{x,y\})$ consists of the edges on the path between $x$ and $y$.
The vertex-shelling convex geometry of a tree is the convex geometry consisting of the sets of vertices of the subtrees. The size of any stem of the edge-shelling convex geometry of a tree is two. Every trace of the edge-shelling convex geometry of a tree is also the edge-shelling convex geometry of some tree. On the other hand, the size of some stem of some vertex-shelling convex geometry is not two. Some trace of some vertex-shelling convex geometry is not a vertex-shelling convex geometry.
We give an example of the edge-shelling convex geometry and the vertex-shelling convex geometry of a tree. The tree shown in Figure \[fig:treeshelling\](left) gives arise to the edge-shelling convex geometry shown in Figure \[fig:treeshelling\](middle) and the vertex-shelling convex geometry shown in Figure \[fig:treeshelling\](right). A curve connecting three points in these figures depicts a rooted circuit of stem size 2. For example, edge b is on the path between edge a and edge c in the tree. So the edge-shelling convex geometry has rooted circuits $(\{a,c\},b)$ and $(\{a,d\},b)$. The vertex-shelling convex geometry has rooted circuits $$(\{A,C\},B), (\{A,D\},C), (\{A,D\},B), (\{B,D\},C), (\{A,E\},C), (\{A,E\},B), (\{B,E\},C).$$
![Tree, edge-shelling convex geometry, and vertex-shelling convex geometry[]{data-label="fig:treeshelling"}](treeshelling.pdf)
The next lemma means that the class of edge-shelling convex geometries is closed under trace operation.
The trace of the edge-shelling convex geometry of a tree is also the edge-shelling convex geometry of some tree. The trace on $A\subseteq E$ corresponds to the contraction of the edges in $E-A$ from the graph.
It is because the contracting the graph does not affect whether $z$ is on the path between $x$ and $y$.
Characterization of edge-shelling convex geometries {#sec:main}
===================================================
Main theorem
------------
In this section, we present trace-minimal forbidden minors for the class of the edge-shelling convex geometry of a tree. The next theorem is the main theorem of this paper, which gives a forbidden-minor-type characterization for edge-shelling convex geometries.
\[thm:main\] For a convex geometry with the rooted circuits ${\cal C}$, it is the edge-shelling convex geometry of a tree if and only if any trace of it coincides with none of following convex geometries.
- Type i:$(\{y,z\},u)$ on $\{x,y,z,u\}$.
- Type Y:$(\{x,y\},u),(\{y,z\},u),(\{x,z\},u)$.
- Type O:$(\{x,z\},y),(\{x,z\},u)$.
- Type OC:$(\{x,z\},y),(\{x,z\},u),(\{u,z\},y)$.
- Type B:$(\{x,z\},y),(\{x,y\},u),(\{y,z\},u),(\{x,z\},u)$.
![Trace-minimal non-edge-shelling convex geometries[]{data-label="fig:forbiddenedge"}](forbidden4.pdf)
We call each of convex geometries Type i, Type Y, Type O, Type OC, and Type B a [*trace-minimal non-edge-shelling convex geometry*]{}. In fact, each of them is not an edge-shelling convex geometry but any proper minor is an edge-shelling convex geometry.
For a convex geometry of stem size 2, it does not have any trace-minimal non-edge-shelling convex geometry if and only if the following rule is satisfied.
Rule Y: For distinct four elements $x,y,z,u$, $(\{y,z\},u)\in {\cal C}$ implies that exactly one of $(\{x,y\},u)\in {\cal C}$ and $(\{x,z\},u)\in {\cal C}$ holds.
Since no trace-minimal non-edge-shelling convex geometry in Figure ¥ref[fig:forbiddenedge]{} satisfies Rule Y, Rule Y is not satisfied when the convex geometry has a trace-minimal non-edge-shelling convex geometry.
Conversely, suppose that the convex geometry does not satisfy Rule Y. Then there exist some four elements which satisfy the assumption of Rule Y and does not satisfy the conclusion of Rule Y. There exist two cases, that is, both satisfy the conclusion and both do not satisfy the conclusion.
We first consider the case where both conditions satisfy the conclusion. In this case, we may assume that the convex geometry has at least $(\{x,y\},u), (\{y,z\},u), $ or $(\{x,z\},u)$. It may have other rooted circuits in $\{x,y,z,u\}$. For a convex geometry each three set included in $\{x,y,z,u\}$ cannot allow more than one rooted circuit. So if it has an additional rooted circuit on $\{x,y,z,u\}$, the ground set of the rooted circuit must be $\{x,y,z\}$. In this case, the convex geometry obtained by adding each of $(\{x,y\},u),(\{y,u\},x)$ and $(\{u,x\},y)$ is isomorphic to Type B. So the convex geometry must have Type Y or Type B as a trace minor.
Next we consider the case where neither of the two conditions satisfies the conclusion. The convex geometry has at least $(\{y,z\},u)$. It may have other rooted circuits in $\{x,y,z,u\}$. When $(\{x,u\},y)$ is a rooted circuit, $(\{x,z\},u)$ is a rooted circuit by Lemma \[lem:i4\], which contradicts the assumption of this case. When $(\{x,z\},u)$ is a rooted circuit, $(\{x,z\},u)$ is a rooted circuit by Lemma \[lem:p\], which contradicts the assumption of this case. When $(\{y,u\},x)$ is a rooted circuit, $(\{y,z\},x)$ should be a rooted circuit by Lemma \[lem:p\]. So the convex geometry has Type i, Type O, Type OC, or Type B.
For a convex geometry of stem size 2, Rule Y implies Rule O.
Rule O: When $(\{x,z\},y)\in {\cal C}$ and $(\{x,z\},u)\in {\cal C}$ hold, $(\{x,y\},u)\in {\cal C}$ or $(\{x,u\},y)\in {\cal C}$ holds.
Suppose that $(\{x,y\},u)\notin {\cal C}$ and $(\{x,u\},y)\notin {\cal C}$. By applying Rule Y to $y$ and $(\{x,z\},u)$, we have $(\{x,z\},u)\in {\cal C}$. By applying Rule Y to $u$ and $(\{x,z\},y)$, we have $(\{x,y\},u)\in {\cal C}$. By $(\{x,z\},u)\in {\cal C}$ and $(\{x,y\},u)\in {\cal C}$, the trace on $\{x,z,u\}$ is not a convex geometry, a contradiction.
Proof of the main theorem
-------------------------
Each trace-minimal non-edge-shelling convex geometry does not satisfy Rule Y. The necessity of Theorem \[thm:main\] follows from the fact that any trace-minimal non-edge-shelling convex geometry is not an edge-shelling convex geometry.
We prove the sufficiency of Theorem \[thm:main\]. So assume that a convex geometry of stem size 2 satisfying Rule Y is given. Fix an element $x\in {\mbox{ex}}(E)$.
\[lem:tree\] For a convex geometry of stem size 2 satisfying Rule O and $x\in {\mbox{ex}}(E)$, the Hasse diagram of the ordering $>_x$ appeared in Lemma \[lem:strict\] is a tree except isolated vertices.
Suppose that the Hasse diagram has a cycle. Take a cycle of minimum length. Assign the direction to each edge on the cycle between covering pair of vertices from the smaller vertex to the larger vertex with respect to $>$. Since $>$ is a strict partial order, there exists a source. Note that $u$ is not covered by $x$. Let $y$ and $z$ be vertices covering $u$. By the assumption of the minimality of the cycle, neither $y>_x z$ nor $z>_x y$ holds. Hence we have $(\{x,z\},y)\notin {\cal C}$ and $(\{x,y\},z)\notin {\cal C}$. By applying Rule O to $x,y,z,u$, one of $(\{x,z\},y)\in {\cal C}$ and $(\{x,y\},z)\in {\cal C}$ holds, a contradiction.
Some element in $E$ may be contained in no rooted circuits containing $x$.
\[lem:parallel\] Consider a convex geometry satisfying Rule Y. Let $y\in E-\{x\}$ be an element such that no rooted circuit contains both $x$ and $y$. Then $(\{x,z\},u)\in {\cal C}$ if and only if $(\{y,z\},u)\in {\cal C}$.
By applying Rule Y to $y$ and $(\{x,z\},u)\in {\cal C}$, either $(\{y,z\},u)\in {\cal C}$ or $(\{y,x\},u)\in {\cal C}$ holds. Since $y$ is not contained in any rooted circuit containing $x$, we have $(\{y,z\},u)\in {\cal C}$.
The converse direction follows similarly.
We call such an element $y$ an [*element parallel to*]{} $x$. Denote the set of elements in $E$ except $x$ and the elements parallel to $x$ by $E_x$ .
Although the vertices in the Hasse diagram correspond to $E_x \cup \{x\}$, we want to make a new tree whose edges correspond to $E$ by using the Hasse diagram.
The Hasse diagram has the maximum element $x$. Every element $y \in E_x$ is covered by a unique element $y'$ by Lemma \[lem:tree\]. So we can associate every element $y \in E_x$ to the edge $\{y,y'\}$. Add an edge corresponding to $x$ at the maximum element of the graph. Moreover add edges corresponding to the parallel elements to $x$ at vertex $x$. Then any element in $E$ corresponds to some edge in the graph.
For example, we consider a convex geometry satisfying Rule Y with $(\{x,b\},a), (\{x,c\},a)\in {\cal C}$ as shown in Figure \[fig:edgeshelling\](left). The ordering $>_x$, appeared in Lemma \[lem:strict\], satisfies $x>_x a>_x b$ and $a>_x c$. Moreover assume that it has $y \in E$ parallel to $x$. By Lemma \[lem:parallel\], it must have $(\{y,b\},a), (\{y,c\},a)\in {\cal C}$. Then we have the corresponding tree as shown in Figure \[fig:edgeshelling\](right).
![Hasse diagram and the corresponding tree[]{data-label="fig:edgeshelling"}](edgeshelling.pdf)
We show that this tree gives rise to the given convex geometry. For that purpose, consider whether $(\{y,z\},u)$ is a rooted circuit or not for three edges $y,z,u \in E_x$.
Let $u\in E_x$ satisfy that the deletion of $u$ from the tree has two connected components. Then $y$ does not belong to the component containing $x$ if and only if $u>y$. So by definition, it is equivalent to $(\{x,y\},u)\in {\cal C}$. So $y\in E-\{x,u\}$ belongs to the component containing $x$ if and only if $(\{x,y\},u)\notin {\cal C}$.
We say that $y$ belongs to the opposite component to $x$ in this case.
\[lem:oo\] When both $y$ and $z$ belong to the opposite component to $x$, $(\{y,z\},u)$ is not a rooted circuit.
In this case, $(\{x,z\},u)\in {\cal C}$, $(\{x,y\},u)\in {\cal C}$ hold. If $(\{y,z\},u)\in {\cal C}$, it contradicts Rule Y to $x$ and $(\{y,z\},u)\in {\cal C}$.
\[lem:ss\] When both $y$ and $z$ belong to the same component to $x$, $(\{y,z\},u)$ is not a rooted circuit.
In this case, $(\{x,y\},u)\notin {\cal C}$ and $(\{x,z\},u)\notin {\cal C}$ hold. Suppose that $(\{y,z\},u)\in {\cal C}$. By applying Rule Y to $x$ and $(\{y,z\},u)\in {\cal C}$, either $(\{z,x\},u)\in {\cal C}$ or $(\{y,x\},u)\in {\cal C}$ holds, a contradiction.
\[lem:os\] When $y$ belongs the same component to $x$, and $z$ belongs to the opposite component, $(\{y,z\},u)\in {\cal C}$ holds.
Assume that $(\{x,z\},u)\in {\cal C}$ and $(\{x,y\},u)\notin {\cal C}$. By applying Rule Y to $y$ and $(\{x,z\},u)\in {\cal C}$, we have $(\{y,z\},u)\in {\cal C}$.
By Lemmas \[lem:oo\], \[lem:ss\], \[lem:os\], and \[lem:parallel\], the tree gives rise to the given convex geometry. We have completed the proof of the sufficiency.
Other classes of convex geometries of stem size 2 {#sec:simplicial}
=================================================
There are many classes of the convex geometries of stem size 2 besides the edge-shelling convex geometry of a tree, for example, a double-shelling convex geometry, the vertex-shelling convex geometry of a tree, the simplicial-shelling convex geometry of a chordal graph and so on.
A simplicial-shelling convex geometry is the convex geometry arising from a chordal graph. A graph is [*chordal*]{} if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle. A vertex is called [*simplicial*]{} if its neighbors are pairwise adjacent. The extreme set ${\mbox{ex}}(E)$ of the simplicial-shelling convex geometry of a chordal graph consists of all the simplicial vertices of the graph. It is known that the simplicial-shelling convex geometry of a chordal graph is a convex geometry.
The class of simplicial-shelling convex geometries is not closed under trace operation.
We show an example of a simplicial-shelling convex geometry whose trace is not a simplicial-shelling convex geometry. Consider the simplicial-shelling convex geometry induced from the chordal graph shown in Figure \[fig:chordal\](left).
![A simplicial shelling whose trace is not a simplicial shelling[]{data-label="fig:chordal"}](chordal.pdf)
Assume that the chordal graph gives rise to a simplicial-shelling convex geometry with a family $\cal C$ of rooted circuits. Consider the trace ${\cal C}:T$ of $\cal C$ on $T=E-\{e\}$. Suppose that ${\cal C}:T$ is the rooted circuits of the simplicial-shelling convex geometry of some graph $G'$. Note that ${\cal C}:T$ is $\{(X,e)\in {\cal C}:X \subseteq T, e\in T\}$. Therefore, we can obtain the edges of $G'$ from ${\cal C}:T$. For example, since $(\{a,b\},c) \in {\cal C}$, $(\{a,b\},c) \in {\cal C}:T$. This means that $G'$ has edges $\{a,c\}$ and $\{b,c\}$. By easy calculation, the graph $G'$ must be the graph shown in Figure \[fig:chordal\](right). Because $G'$ has a chordless cycle $b \to f \to d \to h \to b$, $G'$ is not a chordal graph. So ${\cal C}:T$ is not a simplicial-shelling convex geometry.
So a simplicial-shelling convex geometry cannot be characterized by trace-minimal forbidden minors.
The class of convex geometries arising from the vertex shelling of a tree is not closed under trace operation. This class is a subclass of simplicial-shelling convex geometries.
The class of double-shelling convex geometries is another important class of convex geometries of stem size 2. A double-shelling convex geometry is induced from a poset. This class is closed under trace operation. Kashiwabara and Nakamura [@KN] succeeded in characterizing double-shelling convex geometries in terms of trace-minimal forbidden minors.
Characterization of the convex geometries of stem size 2 {#sec:eminor}
========================================================
In this section, we characterize the class of convex geometries of stem size 2 in terms of the trace-minimal forbidden minors.
A family of rooted circuits of stem size 2 is called a [*trace-minimal non-convex-geometry*]{} of stem size 2 if it satisfies the following two conditions: (1) It is the rooted circuits of no convex geometry, (2) Its trace on any proper subset is the rooted circuits of a convex geometry.
\[lem:stemclose\] The class of convex geometries of stem size 2 is closed under trace operation. As for the class of rooted circuits of stem size 2, any trace-minimal non-convex-geometry has a ground set of at most size 5.
It follows from Lemma \[lem:tracerooted\] that the class is closed under trace operation.
Let $\cal C$ be a set of rooted circuits of stem size 2. Assume that $\cal C$ is the rooted circuits of no convex geometry. Moreover, suppose that there exists a trace-minimal non-convex-geometry of size more than 5. Then, by the axiom of convex geometries in terms of rooted circuits, there exist 5 points $\{x,y,z,u,v\}$ in the trace-minimal non-convex-geometry such that $(\{x,y\},z)$ and $(\{z,v\},u)$ belong to $\cal C$ by Lemma \[lem:rootedaxiom\], and neither $(\{x,v\},u)$ nor $(\{y,v\},u)$ belongs to $\cal C$. The trace of $\cal C$ on $\{x,y,z,u,v\}$ is not a convex geometry, a contradiction.
By specifying all trace-minimal non-convex geometries of stem size 2, we characterize the class of convex geometries of stem size 2. After that, we may consider the non-convex-geometries of stem size 2 as the trace-minimal forbidden minors.
The trace-minimal non-convex-geometries for the class of convex geometries of stem size 2 are all identified as listed in Figures \[fig:e3\], \[fig:e4-2\], \[fig:e4-3\], \[fig:e4-4\], and \[fig:e5\].
Any trace-minimal non-convex-geometry of stem size 2 has size at most 5 by Lemma \[lem:stemclose\].
We make the list up to permutation of the ground set.
First, we make families of rooted circuits of small size. We obtain a new family by adding a rooted circuit to them iteratively, identify isomorphic families, and pick up trace-minimal non-convex-geometries.
There exist two of size 3 shown in Figure \[fig:e3\]. Note that when there exist more than one rooted circuit in three points, the family contains a trace-minimal non-convex-geometry of size 3. So since any trace-minimal non-convex-geometry of size at least 4 does not contain these minors, the ground set $\{x,y,z\}$ of each rooted circuit $(\{x,y\},z)$ is different from one another. This fact leads us to identify trace-minimal non-convex-geometries easier.
There are many trace-minimal non-convex-geometries of size 4. So we classify them according to the number of their rooted circuits.
There are two families which have two rooted circuits, shown in Figure \[fig:e4-2\].
![Trace-minimal non-convex-geometries of size 3[]{data-label="fig:e3"}](e3.pdf)
![Trace-minimal non-convex-geometries of size 4 with two rooted circuits[]{data-label="fig:e4-2"}](e4-2.pdf)
![Trace-minimal non-convex-geometries of size 4 with three rooted circuits[]{data-label="fig:e4-3"}](e4-3.pdf)
There are five families which have three rooted circuits, shown in Figure \[fig:e4-3\].
![Trace-minimal non-convex-geometries of size 4 with four rooted circuits[]{data-label="fig:e4-4"}](e4-4.pdf)
There are four families which have four rooted circuits, shown in Figure \[fig:e4-4\].
![Trace-minimal non-convex-geometries of size 5[]{data-label="fig:e5"}](e5.pdf)
There are two families of size 5, shown in Figure \[fig:e5\].
[99]{} G. Birkhoff and M. K. Bennett, The convexity lattice of a poset, Order 2 (1985), 223 – 242. P.H. Edelman and R. Jamison, The theory of convex geometries, Geom. Dedicata 19(1985), 247–274. B. L. Dietrich, A circuit set characterization of antimatroids, J. Combin. Theory Ser. B 43 (1987), 314–321. B. Korte, L. Lovász, and R. Schrader, Greedoids, Springer, Berlin, 1991. K. Kashiwabara and M. Nakamura, Characterizations of the convex geometries arising from the double shellings of posets(2009). M. Nakamura, Excluded-minor characterizations of antimatroids arisen from posets and graph searches, Discrete Applied Mathematics 129, Issues 2-3 (2003) 487–498. Y. Okamoto and M. Nakamura, The forbidden minor characterization of line-search antimatroids of rooted digraphs, Discrete Applied Mathematics 131 (2003) 523–533.
|
---
author:
- 'M. Billò,'
- 'F. Fucito,'
- 'G.P. Korchemsky,'
- 'A. Lerda'
- 'and J. F. Morales'
title: 'Two-point correlators in non-conformal $\mathcal{N}=2$ gauge theories'
---
Introduction {#sec:intro}
============
Supersymmetric Yang-Mills theories (SYM) have long been considered as an ideal play-ground to get exact results in Quantum Field Theory. In recent times, exact formulae for special observables in theories with extended supersymmetries have been found. In four-dimensional theories with maximal ${{{\mathcal{N}}}}=4$ supersymmetry, the exact resummation of the infinite series of perturbative corrections to the expectation value of circular Wilson loops, also in presence of chiral operators, has been performed [@Erickson:2000af; @Berenstein:1998ij; @Drukker:2000rr; @Semenoff:2001xp; @Fucito:2015ofa]. These results are based on the counting of the relevant rainbow-like Feynman diagrams by means of a matrix model. The introduction of this matrix model has been considered *ad hoc* until it was shown [@Pestun:2007rz] that localization for the ${{{\mathcal{N}}}}=2^*$ theory on the four-sphere $S^4$, after having performed the ${{{\mathcal{N}}}}=4$ limit, predicts its existence. Moreover, localization provides a non-perturbative formula for the circular Wilson loop in a general ${{{\mathcal{N}}}}=2$ theory which takes into account both perturbative and non-perturbative, instanton and anti-instanton, corrections in an interacting matrix model. A two-loop test of this formula against perturbation theory was presented in [@Andree:2010na] in the case of superconformal QCD.
It is natural to ask whether localization on $S^4$ can be used to compute non-trivial quantities other than the Wilson loop expectation value. In [@Baggio:2014sna; @Baggio:2015vxa; @Gerchkovitz:2016gxx; @Baggio:2016skg; @Rodriguez-Gomez:2016ijh; @Rodriguez-Gomez:2016cem; @Baggio:2014ioa] it has been proposed that two-point correlators between chiral and anti-chiral operators in a superconformal ${{{\mathcal{N}}}}=2$ theory on $\mathbb R^4$ can be computed from the partition function of the theory on the four-sphere with chiral and anti-chiral insertions at the north and south pole respectively; localization expresses this partition function as a matrix model. In a conformal ${{\mathcal{N}}}=2$ theory, two-point correlators between a chiral operator $O_{\vec n}={\rm tr} (\varphi^{n_1}) {\rm tr} (\varphi^{n_2}) \cdots\,$, where $\vec n= (n_1,n_2,\dots)$ and $\varphi$ is the complex scalar of the gauge vector multiplet, and an anti-chiral operator ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}$ made out of the complex conjugate field ${\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}$, take the form $$\label{2pt-CFT}
\big\langle O_{\vec n}(x) \,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(0)\big\rangle
= \frac{G_{\vec{n}, \vec{m}} (g_0) }{(4 \pi^2 x^2)^ n} \,\delta_{n,m}\,,$$ where $n=\sum_i n_i$ and $m=\sum_j m_j$ are the [scaling]{} dimensions of the two operators. $G_{\vec{n}, \vec{m}} (g_0)$ is a non-trivial function of the coupling constant $g_0$, but bears no dependence on the distance $x$ since chiral and anti-chiral operators are protected in conformal ${{{\mathcal{N}}}}=2$ theory. The two-point functions on a four-sphere also take the form (\[2pt-CFT\]) but with $x^2$ being the chordal distance on $S^4$. The function $G_{\vec{n}, \vec{m}} (g_0)$ is the same on the sphere and in flat space, and it is given by a two-point function in a matrix model obtained from localization.
Explicit tests of the match between the field theory and the matrix model descriptions of the correlator (\[2pt-CFT\]) have been performed up to two loops for low-dimensional operators in SU(2) and SU(3) gauge theories with $N_f=4$ and $N_f=6$ matter hypermultiplets. The results where extended in [@Billo:2017glv] to generic chiral operators in a superconformal SU($N$) theory with $N_f=2N$. Also the one-point functions of chiral operators in presence of a circular Wilson loop can be expressed in terms of the matrix model obtained via localization of the Wilson loop on $S^4$, as checked up to two loops in [@Billo:2018oog].
It is of obvious importance to investigate to what extent the matrix model description of the correlation function persists [in ${{\mathcal{N}}}=2$ theory]{} in non-conformal set-ups. A first step in this direction was carried out in [@Billo:2017glv] for a SU$(N)$ theory with $N_f$ flavors, where suitable operators were chosen in such a way that their two-point correlators vanish in perturbation theory up to a given loop order, leaving a finite contribution at the next leading loop order. In such a situation, a perfect match between the perturbation theory and localization was shown for an arbitrary rank and any number of flavors at two and three loops. This strongly hints that chiral/anti-chiral correlators are related to the $S^4$ matrix model also beyond the conformal case.
Considering generic chiral/anti-chiral correlation functions away from the conformal point $N_f= 2N$, we encounter important differences. For $N_f\neq 2N$, the gauge coupling and the operators are not anymore protected from quantum corrections and have to be renormalized to account for the ultraviolet (UV) divergences. As a consequence, the two-point correlation functions of renormalized operators $O^R_{\vec n}$ and ${{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}}^R_{\vec m}$ depend on the renormalization scale $\mu$ and are no longer forced to have just a power-like dependence on the distance $x$ as in (\[2pt-CFT\]). Instead, they take the general form $$\label{2pt-nonCFT}
\big\langle O^R_{\vec n}(x) \,{{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}}^R_{\vec m}(0)\big\rangle
= \frac{G_{\vec n, \vec m}^R (g,\nu)}{(4\pi^2 x^2)^n}\,\delta_{n,m}~,$$ where $g= g(\mu)$ is the renormalized coupling and the dimensionless quantity $$\label{defnu}
\nu= 2 + \gamma_{\text{E}} + \ln \pi \mu^2 x^2$$ parametrizes the distance separation. Here $\gamma_{\rm E}$ denotes the Euler-Mascheroni constant. In contrast to the conformal case, the function $G_{\vec n, \vec m}^R(g,\nu)$ depends non-trivially on the distance $x^2$ through the quantity $\nu$. Such a dependence cannot be obtained from localization on $S^4$, because in this case the operators are inserted at the opposite poles of the four-sphere and the distance between them is fixed in terms of the sphere radius. Still our results show that, up to two loops, the $\nu$-dependence is very simple and can be put in the factorized form $$G_{\vec n, \vec m}^R (g,\nu) = \frac{G_{\vec n, \vec m}^R (g, 0)}{\big(1+\frac{1}{2}\beta_0\,g^2\nu\big)^n} \delta_{nm}
+ O(g^6) \,,
\label{gfactorn}$$ where $\beta_0$ is the expansion coefficient of the exact one-loop $\beta$-function of the theory, namely $$\label{defbetaintro}
\beta_0 = \frac{N_f-2N}{8\pi^2}~.$$ Remarkably, the $\nu$-dependent prefactor in (\[gfactorn\]) depends only on the scaling dimension $n$ and not on the details of the operators. As a consequence, up to two loops at least, ratios of correlators of the same scaling dimension are actually $\nu$-independent and can be compared directly against the matrix model results. We show that the field theory results for such observables are indeed in perfect agreement with the predictions from the localization matrix model. This is consistent with the fact that all Feynman diagrams contributing to the ratios are finite in four dimensions, and finite loop integrals on $S^4$ and $\mathbb{R}^4$ yield the same result after replacing propagators in flat space by those on the four-sphere, as we will show in Section \[secn:sphere\].
Furthermore, one can ask whether the results for the renormalized correlators at a given renormalization scale can be directly matched against those coming from localization. At one loop, we show that this is indeed the case if we choose $\mu^2 x^2={\mathrm{e}}^{\gamma_{\text{E}}}/\pi$ in the minimal subtraction scheme. Moreover, by considering the field theory on $S^D$ and evaluating the relevant one-loop integrals in dimensional regularization, we find that, apart from the obvious replacement of propagators, not only the divergent parts but also the finite parts agree with the results on $\mathbb{R}^D$ for $D\to 4$. While the agreement of the divergent part is expected, since the divergences are sensible to short distances and do not distinguish between the sphere and flat space, the agreement of the finite part is neither expected nor guaranteed [*[a priori]{}*]{}, but nevertheless it holds. At two loops, we find that the matrix model results reproduce the field theory ones, up to a term proportional to $(2N-N_f)$ and to the dimension $n$ of the operators. This suggests that the difference could be interpreted as a conformal anomaly which, in non-conformal theories, affects the correlation functions in going from the four-sphere to the flat space.
In this paper we keep the numbers of colors $N$ and flavors $N_f$ arbitrary and compute the two-point correlators for a general choice of chiral/anti-chiral operators. On the field theory side we do this at the two-loop level. We summarize our findings in Section \[secn:pert\] and give a detailed account of the Feynman diagram computations in Appendices \[app:loop\_integrals\] and \[app:Feynman diagrams\]. To keep track of the various particles exchanged in the loops we use a superfield formalism. The loop integrals are evaluated using the integration methods pioneered in [@Chetyrkin:1981qh] (see also [@Grozin:2005yg] for a review). In Section \[secn:rencorr\] we compute the renormalized correlators and their anomalous dimensions. Our results suggest that the anomalous dimensions $\gamma_{\vec n,0}$ of the chiral operators are one-loop exact and are given by the simple formula $\gamma_{\vec n,0}= \frac{n}{2}\,\beta_0$. In Section \[secn:loc\] we discuss the computation of the correlators on the matrix model side, building on the techniques described in [@Billo:2017glv], and compare the results with those previously obtained from the field theory side. To facilitate the comparison, we show that, up to two loops, the localization matrix model can be re-expressed as a complex matrix model encoding the color factors and the combinatorics of the Feynman diagrams that contribute to the chiral/anti-chiral correlators. Finally, in Section \[secn:sphere\] we discuss the field theory calculation of the two-point correlators on the four-sphere $S^4$, and in Section \[secn:concl\] we present our conclusions. Several technical details for such calculations are provided in Appendix \[app:feynsphere\].
Two-point correlators from perturbation theory {#secn:pert}
==============================================
We consider a $\mathcal{N}=2$ SYM theory with gauge group SU($N$) and $N_f$ hypermultiplets in the fundamental representation. For $N_f=2N$ the theory is conformally invariant also at the quantum level. We denote by $\varphi(x)$ the complex scalar field of the ${{\mathcal{N}}}=2$ vector multiplet which, in ${{\mathcal{N}}}=1$ notation, is the lowest component of a chiral superfield $\Phi$. In this theory a basis of chiral operators can be given in terms of the multi-trace operators $$\label{On}
O_{\vec n}(x) = {\mathrm{tr}\,}\left(\varphi^{n_1}(x) \right)\, {\mathrm{tr}\,}\left(\varphi^{n_2}(x)\right) \,
\ldots \,{\mathrm{tr}\,}\left(\varphi^{n_\ell}(x)\right)\,,$$ where $\vec n = (n_1,n_2,\ldots,n_\ell)$. The scaling (bare) dimension of $O_{\vec n}(x)$ is $$n = \sum_{k=1}^\ell n_k ~.$$ We expand the scalar field $\varphi(x) = \varphi^a(x)\, T^a$ over the SU$(N)$ generators $T^a$ ($a = 1,\ldots, N ^2-1)$ in the fundamental representation, normalized as $$\label{tadef}
{\mathrm{tr}\,}(T^a T^b) = \frac 12 \,\delta_{ab}\,.$$ In terms of the components $\varphi^a(x)$, the operators (\[On\]) become $$O_{\vec n}(x) = R_{\vec n}^{\,a_1\ldots a_n}\, \varphi^{a_1}(x)\ldots \varphi^{a_n}(x)\,,
\label{Ona}$$ where $R_{\vec n}^{a_1\ldots a_n}$ is a completely symmetric tensor [^1]. In an analogous way, we define the anti-chiral operators ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec n}(x)$ using the complex conjugate field ${\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}(x)$ instead of $\varphi(x)$.
We are interested in computing the two-point correlation functions $$\big\langle\,O_{\vec n}(x) \,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(0)\,\big\rangle
\label{OnOm}$$ in non-conformal $\mathcal{N}=2$ theories using standard perturbative techniques. We perform our calculations at the origin of moduli space, where the scalar fields have vanishing vacuum expectation values. This is a preferred point in the sense that here the breaking of conformal invariance occurs only at the quantum level, as a consequence of the dimensional transmutation phenomenon. Therefore, this is the natural place in which to test whether the matrix model approach based on localization agrees with the standard perturbative field-theory calculations also in the non-conformal case. Unlike the $\mathcal{N}=4$ theory where the correlators (\[OnOm\]) are exact at tree-level, in $\mathcal{N}=2$ theories they receive quantum corrections, starting from one loop for $N_f\not=2N$ and from two loops in the conformal case $N_f=2N$. Moreover, in the non-conformal theories, the loop integrals are UV divergent, in general, and need to be regularized. Here, we use the dimensional regularization taking the space-time dimension to be $D=4-2\epsilon$. As a consequence, the bare gauge coupling constant, $g_0$, becomes dimensionful.
In general, the bare two-point functions (\[OnOm\]) take the form $$\label{formOO}
\big \langle \,O_{\vec n}(x) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(0)\,\big\rangle
= \Delta^n(x) \, G_{\vec n,\vec m}(g_0,\epsilon,x) \, \delta_{nm}$$ where $n= m$ is the common scaling dimension of the two operators, and $$\Delta(x) = \int \frac{d^D k}{(2\pi)^D} \frac{{\mathrm{e}}^{{\mathrm{i}}k\cdot x}}{k^2} =
\frac{\Gamma(1-\epsilon)}{4\pi\,(\pi x^2)^{1-\epsilon}}~.
\label{Delta}$$ is the massless scalar propagator in $D$-dimensions. The correlator $G_{\vec n,\vec m}(g_0,\epsilon,x)$ can be computed at weak coupling as an expansion in powers of $g_0^2$. We refer to [@Billo:2017glv] for details on the Feynman rules that are needed to perform this calculation; they are summarized for convenience in Appendix \[app:Feynman diagrams\].
The diagrams which contribute to the two-point functions (\[OnOm\]) up to order $g_0^4$ are schematically represented in Fig. \[fig:diagrams\].
The diagram $(a)$ is the tree-level contribution, the diagram $(b)$ is the one-loop correction, while the other four diagrams represent the two-loop part. The blobs labeled by $v_{k,\ell}$ stand for the sum of all irreducible diagrams of order $g_0^{2\ell}$ with $k$ external lines - half of them connected to the chiral fields $\varphi$ of $O_{\vec n}$, half to the anti-chiral ones of ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}$.
A convenient way to organize the computation of these diagrams is to consider the $\mathcal{N}=4$ theory, remove all contributions from Feynman diagrams involving loops of the adjoint hypermultiplet (which we call $H$) and add those with loops of the fundamental matter multiplets (which we call $Q$ and $\widetilde{Q}$) [@Andree:2010na]. Since the two-point correlators in $\mathcal{N}=4$ theory are exact at tree-level, we can write $$\begin{aligned}
G_{\vec n,\vec m}\,&=\,G_{\vec n,\vec m}\big|_{\mathcal{N}=4}
-G_{\vec n,\vec m}\big|_{H}+G_{\vec n,\vec m}\big|_{Q,\widetilde{Q}} \phantom{\Big|}\\
\,&=\, G_{\vec n,\vec m}\big|_{\mathrm{tree}}
-G_{\vec n,\vec m}\big|_{H}+G_{\vec n,\vec m}\big|_{Q,\widetilde{Q}} \phantom{\Big|}\,,
\end{aligned}
\label{ampldec}$$ where, in an obvious notation, $G_{\vec n,\vec m}\big|_{H}$ stands for all diagrams in the $\mathcal{N}=4$ theory with the adjoint hypermultiplet $H$ circulating in the loops, and $G_{\vec n,\vec m}\big|_{Q,\widetilde{Q}}$ stands for the same diagrams in the $\mathcal{N}=2$ theory with loops of fundamental matter multiplets $Q$ and $\widetilde{Q}$. In the following, we will sometimes refer to this method as “performing the computation in the difference theory”. We stress that in the difference theory one should take into account only diagrams involving loops of the adjoint hypermultiplet or loops of the fundamental ones, but not both.
Tree-level
----------
The tree-level contribution to the correlator (\[formOO\]) comes from the diagram in Fig. \[fig:diagrams\]$(a)$. To obtain its explicit expression, one contracts the fields $\varphi$ in $O_{\vec n}$ with the fields ${\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}$ in ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}$ by means of a free scalar propagator $$\big\langle\,\varphi^a(x)\,{\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}^{\,b}(0)\,\big\rangle
= \parbox[c]{60mm}{\hspace*{-20mm}\PandocStartInclude{propphi.pdf_tex}\PandocEndInclude{input}{539}{58}}~.
\label{propagator-tree}$$ In this way one finds that the correlator $\big \langle O_{\vec n}(x) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(0)\big\rangle$ at tree-level takes the form [(\[formOO\])]{} with $$\label{Gtree}
G_{\vec n,\vec m}\,\big|_{\text{tree}}
= n!\, R_{\vec n}^{\,a_1\ldots a_n}\, R_{\vec m}^{\,a_1 \ldots a_n}$$ being a constant that is determined by the color structure of the two operators. For example, for the first operators of even dimension, one finds [@Billo:2017glv] $$\label{Trees}
\begin{aligned}
G_{(2),(2)}\big|_{\text{tree}}\,&=\frac{N^2-1}{2}~,\\
G_{(2,2),(2,2)}\big|_{\text{tree}}\,&=\frac{N^4-1}{2}~,\\
G_{(4),(2,2)}\big|_{\text{tree}}\,&=\frac{(N^2-1)(2N^2-3)}{2N}~,\\
G_{(4),(4)}\big|_{\text{tree}}\,&=\frac{(N^2-1)(N^4-6N^2+18)}{4N^2}~.
\end{aligned}$$ Explicit expressions can be easily found also for operators with higher dimension.
One-loop diagrams
-----------------
We first observe that in the ${{\mathcal{N}}}=4$ theory there are no one-loop corrections to the propagators of the adjoint scalars $\Phi_I$ ($I=1,2,3$). At one loop, this is schematically represented in Fig. \[fig:pphi23-1l\].
This implies that there is no one-loop correction to the propagator of the adjoint hypermultiplet $H$ in the difference theory.
The one-loop correction to the propagator of the fundamental matter superfields $Q$ and $\widetilde{Q}$ vanishes as well. Indeed, as shown in Fig. \[fig:pQ1l\], the contribution of this diagram is similar to the previous one upon replacing the generators in the adjoint with those in the fundamental representation, so that the same cancellation mechanism at work for the adjoint scalars applies here as well.
To find the contribution of the diagram shown in Fig. \[fig:diagrams\]$(b)$, we need to compute the one-loop correction to the propagator of the scalar field in the $\mathcal{N}=2$ theory. This is represented in Fig. \[fig:onelooppphi-iniz\], where in the second line we have used the $\mathcal{N}=4$ result of Fig. \[fig:pphi23-1l\] for $I=1$, to replace the diagram with the vector propagator by the diagram with a scalar loop.
Explicitly computing these superdiagrams (see Appendix \[app:Feynman diagrams\] for details), we find $${\parbox[c]{.25\textwidth}{\includegraphics[width = .25\textwidth]{v21.png}}}
\equiv ~v_{2,1}\, \Delta(x)\, \delta^{ab}\,,
\label{oneloopa2}$$ where $$v_{2,1}
= \frac{g_0^2}{8 \pi^2}\, (2N - N_f)\, \frac{(\pi x^2)^\epsilon\,\Gamma(1-\epsilon)}{2\epsilon(1-2\epsilon)}
~.
\label{v1loop}$$ The correction (\[oneloopa2\]) can be of course inserted in any of the $n$ propagators connecting $O_{\vec{n}}$ and ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}$, so that the one-loop contribution to the two-point correlator [(\[formOO\])]{} corresponding to Fig. \[fig:diagrams\]$(b)$ is $$\label{onelres}
G_{\vec n,\vec m}\,\big|_{\text{1-loop}} \,=\, n\,v_{2,1}
\, G_{\vec n,\vec m}\,\big|_{\text{tree}}~.$$
Two-loop diagrams
-----------------
At order $g_0^4$ there are several diagrams that contribute to the correlator (\[formOO\]). They are schematically represented by the last four diagrams, from $(c)$ to $(f)$, of Fig. \[fig:diagrams\]. A detailed derivation of the various contributions and the evaluation of the corresponding loop integrals can be found in Appendix \[app:Feynman diagrams\]. Here we simply summarize the results for the building blocks of each of these diagrams.
### $v_{2,1}^2\,$- contributions
The two-loop reducible contributions proportional to $v_{2,1}^2$ arise from two insertions of the one-loop effective interaction vertex (\[oneloopa2\]). These can occur either on two different scalar propagators connecting the operators $O_{\vec{n}}$ and ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}$, or on a single propagator. These two possibilities correspond, respectively, to the diagrams $(c)$ and $(d)$ of Fig. \[fig:diagrams\].
In the difference theory, the diagram $(c)$ contains as building blocks the diagrams represented in the left-hand side of Fig. \[fig:2p1l\].
If we exploit the identity of Fig. \[fig:pphi23-1l\] to replace the diagrams with the vector field propagator in favor of the ones with a scalar loop, we can easily realize that the diagrams in the left-hand side Fig. \[fig:2p1l\] precisely reconstruct the square represented in the right-hand side. Using (\[oneloopa2\]), and taking into account the appropriate multiplicity factor of the diagrams, this gives $$\label{2p1lres}
{\parbox[c]{.25\textwidth}{ \includegraphics[width = .25\textwidth]{v21square1}}}
~= ~\frac 12\, v_{2,1}^2\, \Delta^2(x)\,\delta^{a_1 b_1}\delta^{a_2 b_2}~.$$ Let us now consider the contribution corresponding to the two-loop diagram in Fig. \[fig:diagrams\]$(d)$. In this case, the insertion of two one-loop corrections on the same scalar propagator leads to the diagrams displayed in the first two lines of Fig. \[fig:pphi2l\].
If we use again the identity of Fig. \[fig:pphi23-1l\] to replace the diagrams containing the vector field propagator with those with a scalar loop, we reconstruct the square of the one-loop correction, as shown in the last line of Fig. \[fig:pphi2l\]. Evaluating explicitly the loop integrals in this case, we obtain that the result can be written as the square of the one-loop up to terms of order $\epsilon$ (see Appendix \[app:Feynman diagrams\] for details), namely $${\parbox[c]{.35\textwidth}{ \includegraphics[width = .35\textwidth]{v21square2.png}}}
\approx ~v_{2,1}^2\, \Delta(x)\, \delta^{ab} ~.
\label{2loopv212}$$ Here and in the following, we use the approximate symbol $\,\approx\,$ for equations that hold up to terms vanishing in the limit $\epsilon \to 0$.
### $v_{2,2}\,$- and $v_{4,2}\,$- contributions
Let us now consider the two-loop irreducible corrections to the scalar propagator which appear in Fig. \[fig:diagrams\]$(e)$. A first class of contributions arises from correcting one of the internal lines in the one-loop diagrams of Fig. \[fig:onelooppphi-iniz\] by using the one-loop propagator of the matter superfields $Q$ and $\widetilde{Q}$, or of the adjoint hypermultiplet $H$. However, as we have seen before, these contributions vanish.
Other terms that correct the scalar propagator at two loops in the difference theory are those represented in Fig. \[newfig\].
Here we have introduced the dashed double-line notation as a convenient way to represent the difference between the loop with fundamental flavors and the loop with the adjoint hypermultiplet. Actually there are other three classes of irreducible diagrams that correct the scalar propagator at two loops. In Fig. \[fig:pphi2l-irred\] we have drawn all such diagrams, whose evaluation is presented in Appendix \[app:Feynman diagrams\] to which we refer for details.
Summing all contributions, we find that the irreducible two-loop correction to the scalar propagator is $$\label{2loop2}
{\parbox[c]{.25\textwidth}{ \includegraphics[width = .25\textwidth]{v22}}}
\equiv ~ v_{2,2}\, \Delta(x)\, \delta^{ab}\,,$$ where $$v_{2,2}\approx -\Big(\frac{g_0^2}{8\pi^2}\Big)^2\left[3\,\zeta(3)
\Big(\frac{N_f}{2N}+N^2\Big)-N(2N-N_f)\frac{\Gamma^2(1-\epsilon)}{4\epsilon^2(1-2\epsilon)
(1+\epsilon)}\right](\pi x^2)^{2\epsilon}~.
\label{v2}$$
In the difference theory, there are irreducible two-loop contributions that involve two chiral and two anti-chiral fields and give rise to the diagram of Fig. \[fig:diagrams\]$(f)$ . These contributions can be further distinguished according to their overall color structure and can be split into two terms which we denote as $v_{4,2}^{(A)}$ and $v_{4,2}^{(B)}$.
The diagrams yielding $v_{4,2}^{(A)}$ are drawn in Fig. \[fig:vh42diag\], where again we have used the dashed double-line notation to represent the difference between $Q$ and $H$ loops.
They are explicitly evaluated in Appendix \[app:Feynman diagrams\], and the final result is $$\label{v42Aform}
{\parbox[c]{.25\textwidth}{ \includegraphics[width = .25\textwidth]{v4a}}} \equiv ~ v_{4,2}^{(A)} \,\Delta^2(x) \, C_{4}^{(A)\,a_1 a_2 b_1 b_2}+ \cdots~,$$ where $$\begin{aligned}
\label{v42Ares}
{}& v_{4,2}^{(A)} \approx \left(\frac{g_0^2}{8\pi^2}\right)^2 \,N\,(2N-N_f)
\left[\frac{21}{2} \zeta(3) + \frac{\Gamma^2(1-\epsilon)}{4\epsilon^2(1-2\epsilon)(1 + \epsilon)}\right]
(\pi x^2)^{2\epsilon}~,
\\[3mm]
\label{C4Ais}
{}& C_{4}^{(A)\,a_1 a_2 b_1 b_2}
= -\frac{1}{N} \,f^{c\,a_1 b_1}\, f^{c\,a_2 b_2} ~,\end{aligned}$$ with $f^{abc}$ being the SU$(N)$ structure constants. In (\[v42Aform\]) the ellipses stand for terms with color tensors that are anti-symmetric in $(a_1,a_2)$ and $(b_1,b_2)$. Such terms do not contribute to the two-point correlation functions (\[OnOm\]) because they are contracted with the symmetric tensors $R_{\vec{n}}$ and ${\@ifnextchar^{{\wide@bar{R}{0}}}{\wide@bar{R}{1}}}_{\vec{m}}$ defined in (\[Ona\]).
The last two-loop diagram we have to consider is the one represented in Fig. \[fig:v42diag\].
This diagram was already computed in [@Billo:2017glv] and the result, which is also reviewed in Appendix \[app:Feynman diagrams\], is $$\label{2loop4}
{\parbox[c]{.22\textwidth}{ \includegraphics[width
= .22\textwidth]{v4b.png}}} \equiv ~ {v}_{4,2}^{(B)}\, \Delta^2(x) \,{C}_{4}^{(B)\,a_1a_2b_1b_2}
+\cdots~,$$ where $$\begin{aligned}
\label{v42is}
{}& v_{4,2}^{(B)}\approx \Big(\frac{g_0^2}{8\pi^2}\Big)^2 \,3\,\zeta(3) (\pi x^2)^{2\epsilon}~,
\\[3mm]\notag
{}& {C}_{4}^{(B)\,a_1a_2b_1b_2}= -(2N-N_f)\, {\mathrm{tr}\,}T^{a_1}T^{b_1}T^{a_2}T^{b_2}\,
\\
{}& ~~\qquad\qquad\qquad
-\,\frac{1}{2}\big(
\delta^{a_1b_1}\delta^{a_2 b_2}+\delta^{a_1a_2}\delta^{b_1 b_2}
+\delta^{a_1b_2}\delta^{a_2 b_1}\big)~.
\label{C4}\end{aligned}$$ Again, the ellipses in (\[2loop4\]) stand for terms with color tensors which are anti-symmetric in $(a_1,a_2)$ and $(b_1,b_2)$ and, therefore, vanish when inserted in the two-point correlation function.
Effective vertices
------------------
In order to later compare the results of perturbation theory to those of the matrix model, we find it convenient to introduce effective vertices following the ideas of [@Billo:2017glv]. In particular, to obtain the color dependence of the two-point function $G_{\vec{n},\vec{m}}$ we strip the $x$-dependence of the scalar fields and introduce the adjoint matrix ${\bm{\varphi}}= {\bm{\varphi}}^a T^a$ and its conjugate ${\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}={\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^aT^a$, such that $$\label{treemc}
\big\langle{\bm{\varphi}}^a\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^b\big\rangle=\delta^{ab}~,\qquad\qquad
\big\langle{\bm{\varphi}}^a\,{\bm{\varphi}}^b\big\rangle=\big\langle{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^a\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^b\big\rangle=0~.$$ We denote by $O_{\vec n}({\bm{\varphi}})$ the operator obtained by replacing in (\[On\]) the field $\varphi(x)$ with the constant matrix ${\bm{\varphi}}$, and by ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})$ the same with ${\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}(x)$ replaced with ${\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}$.
With these definitions, it is straightforward to see that the tree-level correlator (\[Gtree\]) can be written as $$\label{Gtreemm}
G_{\vec n,\vec m}\big|_{\text{tree}} = \big\langle O_{\vec n}({\bm{\varphi}})
\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle~.$$ Also the one-loop correlator (\[onelres\]) can be written in a simple way using this formalism. Indeed, we have $$\label{onelresmm}
G_{\vec n,\vec m}\big|_{\text{1-loop}}
= \,v_{2,1}\,\big\langle V_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,
O_{\vec n}({\bm{\varphi}})\, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle~,$$ where $$\label{defV21vphi}
V_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) = \,\,\delta^{ab}{:\! {\bm{\varphi}}^a\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^b \!:}
\,\,=2{:\! {\mathrm{tr}\,}{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}} \!:} ~.$$ As usual, the notation $:\,\,:$ stands for normal ordering. Using (\[treemc\]), it is easy to check that inside a vacuum expectation value (\[onelresmm\]) we can use the relation $$V_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,O_{\vec n}({\bm{\varphi}}) = n\,O_{\vec n}({\bm{\varphi}})
\label{V2On}$$ that follows from the SU$(N)$ fusion/fission rules $$\begin{aligned}
{\mathrm{tr}\,}\, T^a B_1 T^a B_2 &=\frac{1}{2}\, {\mathrm{tr}\,}\, B_1\,{\mathrm{tr}\,}\, B_2 -\frac{1}{2N}\,{\mathrm{tr}\,}\,B_1 B_2~,\\
{\mathrm{tr}\,}\,T^a B_1~{\mathrm{tr}\,}\, T^a B_2&=\frac{1}{2}\,{\mathrm{tr}\,}\,B_1 B_2 -\frac{1}{2N}\,{\mathrm{tr}\,}\,B_1~{\mathrm{tr}\,}\,B_2~,
\end{aligned}
\label{identities}$$ valid for two arbitrary $N\times N$ matrices $B_1$ and $B_2$.
Let us now compute the two-loop contribution to the two-point correlation function from diagrams shown in Fig. \[fig:diagrams\]. Using the reducible term (\[2p1lres\]) we find that the contribution of the diagram in Fig. \[fig:diagrams\]$(c)$ is $$G_{\vec n,\vec m}^{(c)}=\frac 12 n(n-1)\, v_{2,1}^2\, G_{\vec n,\vec m}\big|_{\text{tree}}~.
\label{G2lc}$$ In terms of the effective vertex (\[defV21vphi\]), this result can be rewritten as follows: $$\begin{aligned}
G_{\vec n,\vec m}^{(c)}&=\frac 12\, v_{2,1}^2\, \big\langle\! :\!
\left[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\right]^2 \!:
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle\\
&=\frac{1}{2}\, v_{2,1}^2\Big[\big\langle
\left[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\right]^2
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle
-2\,\big\langle {V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\\
&\qquad\qquad-\big\langle
\left[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\right]^2\big\rangle\,\big\langle
O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle
\Big]~,
\end{aligned}
\label{G2lc1}$$ where in the second line we have used the identity $$:\! {V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) {V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \!: \,\,=\left[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\right]^2-
2\,{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})-\big\langle
\left[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\right]^2\big\rangle$$ that follows from Wick’s theorem.
The two-loop reducible correction (\[2loopv212\]) to the scalar propagator can be inserted in any of the $n$ internal lines. Thus, the contribution of the diagram in Fig. \[fig:diagrams\]$(d)$ is $$G_{\vec n,\vec m}^{(d)} \approx n\, v_{2,1}^2\, G_{\vec n,\vec m}\big|_{\text{tree}} = v_{2,1}^2 \, \big\langle {V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle ~,
\label{G2ld1}$$ where we used (\[V2On\]). In a similar way, the irreducible two-loop correction (\[2loop2\]) produces the following contribution to the diagram in Fig. \[fig:diagrams\]$(e)$: $$G_{\vec n,\vec m}^{(e)}=n \, v_{2,2} \,G_{\vec n,\vec m}\big|_{\text{tree}} =v_{2,2}\,\big\langle V_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle~.
\label{V22Onb1}$$
Let us now consider the two-loop contributions proportional to $v_{4,2}^{(A)}$ and $v_{4,2}^{(B)}$ given, respectively, in (\[v42Aform\]) and (\[2loop4\]). To write the results in a compact form, it is convenient to introduce the quartic vertices $$\begin{aligned}
V_{4}^{(A)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) & = C_{4}^{(A)\,a_1 a_2 b_1 b_2} \,:{\bm{\varphi}}^{a_1}\,{\bm{\varphi}}^{a_2}\, {\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^{b_1}\, {\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^{b_2}: \phantom{\Big|}\notag\\
& = \frac{2}{N} \,:{\mathrm{tr}\,}{\left[\,{\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}\,\right]}^2\!: \,\,
= \frac{4}{N}\,\big(\!:{\mathrm{tr}\,}{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}\,{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}} : - :{\mathrm{tr}\,}{\bm{\varphi}}^2\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^2:\big)~,\phantom{\Big|}\ \label{V4Ais}\\[3mm]
V_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
&= {C}_{4}^{(B)\,a_1a_2b_1b_2}\,:{\bm{\varphi}}^{a_1}\,{\bm{\varphi}}^{a_2}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^{b_1}\,
{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}^{b_2}:\phantom{\Big|}\notag\\
&=-(2N-N_f):{\mathrm{tr}\,}{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}\,{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}:
-\,4:\big({\mathrm{tr}\,}{\bm{\varphi}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}\big)^2\!:
-\,2:{\mathrm{tr}\,}{\bm{\varphi}}\,{\bm{\varphi}}\,\,{\mathrm{tr}\,}{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}\,{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}:~,\phantom{\Big|}\label{V4}\end{aligned}$$ Then, the contribution of the effective vertex (\[v42Aform\]) to the correlator can be written as $$\label{v42V4A}
G^{(A)}_{\vec n,\vec m} =
v_{4,2}^{(A)} \,\big\langle V_{4}^{(A)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}}) \,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle~.$$ Repeatedly using the fission/fusion identities (\[identities\]), it is possible to show that, inside the vacuum expectation value [(\[v42V4A\])]{}, the following relation holds: $$V_{4}^{(A)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,\,{\mathrm{tr}\,}{\bm{\varphi}}^n
=\frac{n}{N}\,\sum_{\ell=0}^{n-2}\big({\mathrm{tr}\,}{\bm{\varphi}}^{\ell+1}~{\mathrm{tr}\,}{\bm{\varphi}}^{n-\ell-1}-
{\mathrm{tr}\,}{\bm{\varphi}}^\ell~{\mathrm{tr}\,}{\bm{\varphi}}^{n-\ell}\,\big) =-n\,{\mathrm{tr}\,}{\bm{\varphi}}^n~.
\label{identity0V4hat}$$ More generally, one can prove that $$V_{4}^{(A)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,\,O_{\vec n}({\bm{\varphi}})=-n\,O_{\vec n}({\bm{\varphi}})~.
\label{identityV4hat}$$ By comparing with (\[V2On\]) we conclude that the quartic vertex $V_{4}^{(A)}$ can be effectively replaced by $(-V_2)$ inside a vacuum expectation value, so that (\[v42V4A\]) becomes $$\label{G42la}
G^{(A)}_{\vec n,\vec m}= -v_{4,2}^{(A)}\,\big\langle V_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle=
-n \, v_{4,2}^{(A)}\, G_{\vec n,\vec m}\big|_{\text{tree}}~.$$
The contribution of the effective vertex (\[2loop4\]) to the correlator can be treated in an analogous way, and it reads $$G^{(B)}_{\vec n,\vec m}={v}_{4,2}^{(B)}\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle~.
\label{V4On}$$ Notice that, in distinction to the other two-loop contributions, the expectation value (\[V4On\]) is not proportional, in general, to the tree-level correlator $G_{\vec n,\vec m}\big|_{\text{tree}}$ due to the structure of the vertex $V_4^{(B)}$, and it has to be computed case by case. A few explicit examples with operators of even dimensions are: $$\begin{aligned}
\!\!&\big\langle V_4^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \,
O_{(2)}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{(2)}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \big\rangle=
-\frac{N^2-1}{2}\,\Big(N^2+\frac{N_f}{2N}\Big)~,\\[2mm]
\!\!\!\!&\big\langle V_4^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \,
O_{(2,2)}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{(2,2)}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \big\rangle=-
\frac{N^2-1}{2}\,\Big(2 N^4+22 N^2-3 N N_f+\frac{7N_f}{N}\Big)
~,\\[2mm]
\!\!&\big\langle V_4^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \,
O_{(4)}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{(2,2)}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \big\rangle=-\frac{N^2-1}{2}\,\Big(6 N^3- N^2 N_f + 6 N + 8 N_f - \frac{21 N_f}{N^2} \Big)~,\\[2mm]
\!\!&\big\langle V_4^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \,
O_{(4)}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{(4)}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}) \big\rangle=
-\frac{N^2-1}{2}\,\Big(12 N^2+ N N_f-18-\frac{21N_f}{N}+\frac{63N_f}{N^3}\Big)~.
\end{aligned}
\label{V4onO}$$
The contribution to the two-point correlator from the two-loop diagram of Fig.
\[fig:diagrams\]$(f)$ is given by the sum of (\[G42la\]) and (\[V4On\]), namely $$G^{(f)}_{\vec n,\vec m}=-n \, v_{4,2}^{(A)}\, G_{\vec n,\vec m}\big|_{\text{tree}}+
{v}_{4,2}^{(B)}\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\,,
\label{Gfmn}$$ where $ v_{4,2}^{(A)}$ and $ v_{4,2}^{(B)}$ are defined in [(\[v42Ares\])]{} and [(\[v42is\])]{}, respectively,
Summary of results
------------------
Collecting our findings, up to two loops the bare correlator is given by $$\begin{aligned}
G_{\vec n,\vec m}
& \approx \,\Big[1+n\,v_{2,1}+
\frac{n(n+1)}{2} \, v_{2,1}^2\Big]\,G_{\vec n,\vec m}\big|_{\text{tree}}\phantom{\Big|}\\
&\quad+ n\,\big(v_{2,2}-{v}_{4,2}^{(A)} \big)
\,G_{\vec n,\vec m}\big|_{\text{tree}}
+{v}_{4,2}^{(B)}\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle \phantom{\Big|}~.
\end{aligned}
\label{Gnmtot}$$ The first line contains the tree-level term, the one-loop correction and the reducible two-loop part, while the irreducible two-loop terms are written in the second line.
Eq. (\[Gnmtot\]) is the main result of this section. It expresses the bare two-point correlator between chiral and anti-chiral operators up to order $g_0^4$, in terms of the tree-level correlator and the [matrix model]{} correlator with the insertion of the quartic effective vertex $V_4^{(B)}$. Moreover, it [exhibits]{} a particularly simple structure of the UV divergences of $G_{\vec n,\vec m}$. First, we notice that all divergent terms in the difference $\big(v_{2,2}-{v}_{4,2}^{(A)}\big)$ exactly cancel. Indeed, using (\[v2\]) and (\[v42Ares\]), we have $$v_{2,2}-{v}_{4,2}^{(A)} \approx -3\zeta(3) \Big(\frac{g_0^2}{8\pi^2}\Big)^2\,
\Big(8N^2-\frac{7N N_f}{2}+\frac{N_f}{2N}\Big)~.
\label{v42etc}$$ This, together with the fact that $$v_{4,2}^{(B)} \approx 3\,\zeta(3) \,\Big(\frac{g_0^2}{8\pi^2}\Big)^2 ~,
\label{v42Bexp}$$ implies that the total two-loop irreducible contribution in the second line of (\[Gnmtot\]) is finite for $\epsilon\to 0$. The only divergences remaining at two loops come from the square of those present at one-loop. At the given loop order, they can be nicely combined into an overall factor (see the first line of (\[Gnmtot\])) $$1+n\,v_{2,1}+\frac{n(n+1)}{2} \, v_{2,1}^2+ \cdots =\frac{1}{(1-v_{2,1})^n}~,$$ which depends only on the [bare]{} dimension $n$ of the operators but not on their detailed structure.
We conclude this section by showing that the two-loop result (\[Gnmtot\]) can be rewritten in an alternative and elegant form as a correlator in the matrix model. [Combining]{} the tree-level, one-loop and two-loop contributions given in (\[Gtreemm\]), (\[onelresmm\]), (\[G2lc1\]), (\[G2ld1\]), (\[V22Onb1\]) and (\[Gfmn\]), we obtain $$\begin{aligned}
G_{\vec n,\vec m} &\approx \big\langle O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\big\rangle+\big(v_{2,1}+v_{2,2}-{v}_{4,2}^{(A)}\big)\,\big\langle
V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle
\phantom{\Big|}\\
&~~~+\frac{1}{2}\, v_{2,1}^2\Big[\big\langle
[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})]^2
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle
-\big\langle
[{V}_{2}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})]^2\big\rangle\,\big\langle
O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\Big]\\
&~~~+{v}_{4,2}^{(B)}\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle~.\phantom{\Big|}
\end{aligned}
\label{Gnmfin1}$$ Then, defining the effective interaction vertex $$V_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})=-\big(v_{2,1}+v_{2,2}-{v}_{4,2}^{(A)}\big)\,
V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})-v_{4,2}^{(B)}\,V_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})~,
\label{Veff}$$ we can recast (\[Gnmfin1\]) in a very compact way as follows: $$G_{\vec n,\vec m} \approx \frac{\displaystyle{\big\langle {\mathrm{e}}^{-V_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}~O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\phantom{\Big|}}}
{\displaystyle{\big\langle
{\mathrm{e}}^{-V_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}\big\rangle}\phantom{\Big|}}~.
\label{Gnmfin}$$ Indeed, expanding the exponentials up to order $g_0^4$, we precisely recover all terms of (\[Gnmfin1\]). In particular, from the insertion of a single $V_{\rm eff}$ in the correlator we obtain the linear terms in the $v_{k\,\ell}$’s appearing in the first and third line of (\[Gnmfin1\]), while the quadratic terms in the second line arise from two insertions of $V_{\rm eff}$. Notice that since this effective vertex is normal-ordered, the denominator in (\[Gnmfin\]) contributes up to order $g_0^4$ only with the term proportional to $\big\langle [V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})]^2\big\rangle$ appearing in the second line of (\[Gnmfin1\]).
In Section \[secn:loc\] we show that localization on a four sphere produces an expression similar to (\[Gnmfin\]). However, in order to compare the two expressions, we should first get rid of the UV divergences and scheme ambiguities that are present in the bare correlator $G_{\vec n,\vec m} $. This is the content of the next section.
Renormalization {#secn:rencorr}
===============
The dimensionally regularized bare correlators $G_{\vec n, \vec m}$ given in (\[Gnmtot\]) are divergent for $\epsilon \to 0$, since the one-loop coefficient $v_{2,1}$ defined in (\[v1loop\]) behaves for small $\epsilon$ as $$v_{2,1}\approx\frac{g_0^2}{16 \pi^2}\,(\pi x^2)^\epsilon\, (2N - N_f)\,\Big(\frac{1}{\epsilon}+2+\gamma_{\text{E}}\Big)\,.
\label{v21exp}$$ As we have remarked before, the UV divergence due to $v_{2,1}$ is the only one that plagues the correlators, since all other terms in $G_{\vec n, \vec m}$ are finite for $\epsilon\to 0$. To get rid of this divergence, we have to apply the standard renormalization procedure. First, we introduce the dimensionless renormalized gauge coupling constant $g$ through the relation $$\label{gtog0}
g_0^2 = \mu^{2\epsilon}g^2 \, Z( g^2, \epsilon) \,,$$ where $\mu$ is an arbitrary scale, and $Z$ is a suitable function to be determined. Then, we define the renormalized operators $O^R_{\vec n}(x)$ according to $$O^R_{\vec n}(x) = \sum_{\vec{m}} Z_{\vec n}^{~\vec m}(g^2,\epsilon)\,O_{\vec m}(x) \,,
\label{Onren0}$$ where $Z_{\vec n}^{~\vec m}$ is a matrix-valued function. However, in the previous section we have shown that the divergences of the two-point functions depend only on the scaling dimensions of the operators and not on the operator details; therefore a block-diagonal matrix can do the job, and we can simplify (\[Onren0\]) by setting $$\label{Onren}
O^R_{\vec{n}}(x)
= Z_n(g^2,\epsilon) \, O_{\vec{n}}(x)~.$$ A similar formula holds for the anti-chiral renormalized operators ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}^R_{\vec{n}}(x)$.
The singular terms of the functions $Z(g^2, \epsilon)$ and $Z_n(g^2,\epsilon)$ are determined by requiring that the correlator $\big\langle O^R_{\vec{n}}(x)\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}^R_{\vec{m}}(0)\big\rangle$ should be finite when expressed in terms of the renormalized coupling $g$. This means that the renormalized correlator $$\label{renG}
G^R_{\vec{n},\vec{m}} = Z_n^2( g^2, \epsilon)\, G_{\vec n,\vec m}
\Big|_{g_0^2=\mu^{2\epsilon}g^2\,Z(g^2, \epsilon)}$$ is well-defined and free of divergences in the limit $\epsilon\to 0$.
The $\beta$-function and anomalous dimensions
---------------------------------------------
The dependence of the renormalized coupling $g^2$ and of the renormalization constant $Z_n(g^2,\epsilon)$ on the energy scale $\mu$ is described, respectively, by the $\beta$-function and by the anomalous dimensions $\gamma_n(g^2)$ of the operators $O_{\vec n}$. They are defined as follows: $$\label{defbeta}
\beta(g^2) \equiv \mu\,\frac{d g^2}{d\mu}=-2\epsilon g^2
- g^2 \,\mu\,\frac{d \ln Z(g^2,\epsilon)}{d\mu}\,,$$ where the last equality stems from the $\mu$-independence of $g_0$, and $$\label{defgamma}
\gamma_n(g^2) \equiv - \mu\,\frac{d \ln Z_n (g^2,\epsilon)}{d\mu}
= - \beta(g^2)\,\frac{d \ln Z_n(g^2,\epsilon)}{d g^2}\,,$$ where in the second step we used (\[defbeta\]). Using the perturbative expansions $$\begin{aligned}
\beta(g^2) &= -2\epsilon g^2+\beta_0\, g^4 + \beta_1\, g^6 + \ldots~,\\
\gamma_n(g^2)&=\gamma_{n,0}\, g^2 + \gamma_{n,1}\, g^4+\ldots~,
\end{aligned}
\label{betagammaexp}$$ we can explicitly integrate (\[defbeta\]) and (\[defgamma\]) and get in the minimal subtraction (MS) scheme $$\begin{aligned}
Z(g^2,\epsilon)&= \exp\left[-\int_0^{g^2 } \!\frac{d t}{t}
\, \frac{\beta(t)+2\epsilon t}{\beta(t)}\right] =
1+ g^2 \,\frac{\beta_0}{2\epsilon} + g^4 \,\Big( \frac{\beta_0^2}{4\epsilon^2}
+ \frac{\beta_1}{4\epsilon}\Big)+
\ldots~,\\[3mm]
Z_n(g^2,\epsilon)
& =\exp\left[-\int_0^{g^2} \!d t \,\frac{\gamma_n(t)}{\beta(t)}\right]
= 1+ g^2 \,\frac{\gamma_{n,0}}{2\epsilon} + g^4 \,\Big(\frac{\beta_0 \,\gamma_{n,0}+\gamma_{n,0}^2}{8\epsilon^2}+\frac{\gamma_{n,1}}{4\epsilon}\Big) +\ldots~.
\end{aligned}
\label{ZZnexp}$$ This shows that the expansion coefficients of the $\beta$-function and of the anomalous dimensions $\gamma_n$ are directly related to the coefficients of the $1/\epsilon\,$-terms in $Z$ and $Z_n$ respectively.
Differentiating (\[renG\]) and using $d G_{\vec n, \vec m}/d\mu=0$, one obtains the Callan-Symanzik equation $$\label{CS}
\left( \mu\,\frac{\partial}{\partial \mu} + \beta(g^2)\,\frac{\partial}{\partial g^2}
+2 \gamma_n(g^2) \right)\,G^R_{\vec n,\vec m}=0$$ on which we can safely take the limit $\epsilon\to 0$.
We now determine the coefficients of the $\beta$-function and the anomalous dimensions, using the explicit results of the previous section. To do so, we expand the bare correlator (\[Gnmtot\]) for small $\epsilon$ using (\[v21exp\]), and write $$G_{\vec n,\vec m}=\alpha_0+g_0^2\Big(\frac{\alpha_{1,1}}{\epsilon}+\alpha_{1,0}+\ldots\Big)+
g_0^4\Big(\frac{\alpha_{2,2}}{\epsilon^2}+\frac{\alpha_{2,1}}{\epsilon}+\ldots\Big)+O(g_0^6)\,,
\label{Gbareexp}$$ where $\alpha_0= G_{\vec n,\vec m}\big|_{\mathrm{tree}}$ and $$\begin{aligned}
\alpha_{1,1}&=n\,\frac{2N-N_f}{16\pi^2}\,\alpha_0
~,
&& \alpha_{1,0}=n\,\frac{(2N-N_f)(2+\gamma_{\text{E}}+\ln \pi x^2)}{16\pi^2}\,\alpha_0
~,\label{alphas}\\[2mm]
\alpha_{2,2}&=n(n+1)\,\frac{(2N-N_f)^2}{512\pi^4}\,\alpha_0~,
&&\alpha_{2,1}=n(n+1)\,\frac{(2N-N_f)^2(2+\gamma_{\text{E}}+\ln \pi x^2)}{256\pi^4}\,\alpha_0~.
\notag\end{aligned}$$ Plugging the expansion (\[Gbareexp\]) into the renormalized correlator (\[renG\]), using (\[ZZnexp\]) and requiring that all divergent terms cancel, one finds $$\begin{aligned}
\label{betagammares0}
\beta_0= \frac{2\,\big(\alpha_{1,1}^2-2\,\alpha_0 \,\alpha_{2,2}\big)}{\alpha_0\, \alpha_{1,1}}
,\qquad
\gamma_{n,0} = -\frac{\alpha_{1,1}}{\alpha_0}
,\qquad
\gamma_{n,1} =\frac{2\,\big(2\,\alpha_{1,0} \,\alpha_{2,2}-\alpha_{1,1} \,\alpha_{2,1} \big)}
{\alpha_0\,\alpha_{1,1}},\end{aligned}$$ leading to $$\begin{aligned}
\beta_0 =-\frac{2N-N_f}{8\pi^2}~,\qquad
\gamma_{n,0} & = -n \frac{2N-N_f}{16\pi^2}~,\qquad
\gamma_{n,1} = 0~.
\end{aligned}
\label{betagammares}$$ This value of $\beta_0$ is in agreement with the well-known result for the one-loop coefficient of the $\beta$-function in ${{\mathcal{N}}}=2$ SQCD. We also notice that $\gamma_{2,0}=\beta_0$. This is consistent with $\mathcal{N}=2$ supersymmetry, since the chiral operator $O_{(2)} = {\mathrm{tr}\,}\varphi^2$ and the Yang-Mills Lagrangian $-{\mathrm{tr}\,}(F^2/4)+\dots $ belong to the same supermultiplet, and thus should renormalize in the same way, that is $$\label{Z2isZ}
Z_2(g^2,\epsilon) = Z(g^2,\epsilon)~.$$ Moreover, using the fact that in $\mathcal{N}=2$ SYM theories the $\beta$-function receives only one-loop correction[^2], [*[i.e.]{}*]{} $\beta_\ell=0$ for all $\ell\geq 1$, we conclude that also the anomalous dimensions of ${\mathrm{tr}\,}\varphi^2$ are corrected only at one loop, [*[i.e.]{}*]{} $\gamma_{2,\ell}=0$ for all $\ell\geq 1$. This implies that $$\label{wavefunctions}
Z_2(g^2,\epsilon) = Z(g^2,\epsilon) =1-\frac{g^2(2N-N_f) } {16 \pi^2\, \epsilon}+
\frac{g^4(2N-N_f)^2} {256 \pi^4\, \epsilon^2}+\ldots
=\frac{1}{1+\frac{g^2(2N-N_f)} {16 \pi^2\, \epsilon}}~.$$ Furthermore, we deduce from (\[betagammares\]) that the following relation $$\label{ZnisZ2n2}
Z_n(g^2,\epsilon) = \left[Z(g^2,\epsilon)\right] ^{\frac n2}$$ holds up to two loops. It would be very interesting to investigate whether this relation holds also at higher loops. While this issue is not relevant for the two-loop analysis of the present paper, it is tempting to speculate that (\[ZnisZ2n2\]) might actually be true in general. Indeed, in our set-up the anomalous dimensions of the chiral and anti-chiral operators arise because of the breaking of conformal invariance at the quantum level due to dimensional transmutation. The fact that the coefficients $\gamma_{n,0}$ and $\beta_0$ are proportional to each other and that the proportionality factor is $n/2$ (see (\[betagammares\])), together with $\mathcal{N}=2$ supersymmetry, naturally leads one to propose the relation (\[ZnisZ2n2\]). Notice that in the conformal case $N_f=2N$, the renormalization functions simply reduce to 1, due to the absence of divergences, so that (\[ZnisZ2n2\]) is trivially satisfied in this case.
Renormalized correlators
------------------------
Using the previous results, it is easy to see that up to two loops and in the limit $\epsilon\to 0$, the renormalized correlators (\[renG\]) take a form completely analogous to the bare correlators (\[Gnmtot\]), namely $$\begin{aligned}
G^R_{\vec n,\vec m}& =\,\Big[1+n\,c_1+
\frac{n(n+1)}{2} \, c_1^2+ n\,c_2\,\Big]\,G_{\vec n,\vec m}\big|_{\text{tree}}\phantom{\Big|}
+c_3\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec n}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\phantom{\Big|} ~,
\end{aligned}
\label{GnmRtot}$$ where $$\begin{aligned}
c_1&=\frac{g^2}{16\pi^2}\,\nu\,(2N-N_f)~,\\
c_2&= -3\,\zeta(3) \Big(\frac{g^2}{8\pi^2}\Big)^2\,
\Big(8N^2-\frac{7N N_f}{2}+\frac{N_f}{2N}\Big)~,\\
c_3&=3\,\zeta(3) \,\Big(\frac{g^2}{8\pi^2}\Big)^2~,
\end{aligned}
\label{c123}$$ and $$\label{nudef}
\nu= 2 + \gamma_{\text{E}} + \ln \pi \mu^2 x^2~.$$ The coefficients $c_i$ are obtained from (\[v21exp\]), (\[v42etc\]) and (\[v42Bexp\]). In the conformal case $c_1$ vanishes and the first perturbative correction to the correlator appears at order $g^4$.
Performing the same manipulations as described in Section \[secn:pert\] for the bare correlators, we can rewrite (\[GnmRtot\]) in the following form: $$\begin{aligned}
{}& G^R_{\vec n,\vec m} = \frac{\displaystyle{\big\langle {\mathrm{e}}^{-V^R_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}
~O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\phantom{\Big|}}}
{\displaystyle{\big\langle {\mathrm{e}}^{-V^R_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}\big\rangle}\phantom{\bigg|}}
+O(g^6)\,,
\label{GRnmfin}
\\[2mm]
{}& V^R_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})=-\big(c_1+c_2\big)\,V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
-c_3\,V_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\,,
\label{VReff}\end{aligned}$$ where the two- and four-point vertices $V_2$ and $V_4^{(B)}$ are defined, respectively, in (\[defV21vphi\]) and (\[V4\]).
We remark that for $N_f\neq 2N$, the renormalized correlator $G^R_{\vec{n},\vec{m}}$ is *not* a constant, but it depends on $x$ through the $\ln \pi \mu^2 x^2$ term contained in $\nu$. At first sight, this fact makes it unlikely that the correlator can be encoded in a matrix model. However, the dependence of $G^R_{\vec{n},\vec{m}}$ on $\nu$ is determined by the Callan-Symanzik equation (\[CS\]) [^3] and by its value at a reference point $\widehat\nu$. For instance, setting $$\label{msbar}
\mu^2 x^2= \frac{{\mathrm{e}}^{\gamma_{\text{E}}}}{\pi}~,$$ we get [from (\[nudef\])]{} $$\label{Lmsbar}
\widehat{\nu} = 2(1+\gamma_{\text{E}})~,$$ which, as we will see in Section \[secn:loc\], is the combination that matches the one-loop matrix model results from localization. In particular, with the choice [(\[msbar\])]{} the coefficient $c_1$ becomes $$c_1=\frac{g^2}{8\pi^2}\,(1+\gamma_{\text{E}})\,(2N-N_f) ~.
\label{c1msbar}$$ Using [(\[Trees\])]{} and [(\[V4onO\])]{} we find from (\[GnmRtot\]) $$\begin{aligned}
G^R_{(2), (2)} &=
\frac{N^2-1}{2}
\bigg[1+ 2\, c_1+3\, c_1^2+2 \,c_2-c_3\Big(N^2+\frac{N_f}{2N}\Big) \bigg]+
O(g^6)~.
\end{aligned}
\label{GR22}$$ Similar explicit formulae can be worked out for correlators involving higher dimensional operators. For example, at dimension 4 we find
$$\begin{aligned}
G^R_{(2,2),(2,2)}&=\frac{N^4-1}{2}
\Big(1+4\, c_1+10\, c_1^2 +4 \,c_2 \Big)\notag\\
&\qquad
-c_3\,\frac{(N^2-1)}{2}\,\Big(2 N^4+22 N^2-3 N N_f+\frac{7N_f}{N}\Big)+O(g^6)~,\\[3mm]
G^R_{(4),(2,2)}&=\frac{2N^4-5N^2+3}{2N}
\Big(1+4\, c_1+10\, c_1^2 +4 \,c_2 \Big)\notag\\
&\qquad
-c_3\,\frac{(N^2-1)}{2}\,\Big(6 N^3- N^2 N_f + 6 N + 8 N_f - \frac{21 N_f}{N^2} \Big)+O(g^6)
~,\\[3mm]
G^R_{(4),(4)}&=\frac{N^6- 7 N^4+ 24 N^2-18}{4 N^2}
\Big(1+4\, c_1+10\, c_1^2 +4 \,c_2 \Big)\notag\\
&\qquad
-c_3\,\frac{(N^2-1)}{2}\,\Big(12 N^2+ N N_f-18-\frac{21N_f}{N}+\frac{63N_f}{N^3}\Big)+O(g^6) ~,\end{aligned}$$
\[GR44\]
where the coefficients $c_1$, $c_2$ and $c_3$ are defined in [(\[c123\])]{}. Notice that $c_1$ and $c_2$ enter into these expressions in the same combination.
Normalized correlators {#subsecn:normcorr}
----------------------
The renormalized correlators $G^R_{\vec{n},\vec{m}}$ are finite but they cannot be considered as physical observables since they depend on the choice of the renormalization scheme. In particular, given a renormalized correlator at a certain normalization scale $\mu$, one can always perform a finite renormalization of the operators by multiplying the renormalization factors $Z$ and $Z_n$ defined in (\[gtog0\]) and (\[Onren\]), respectively, by an arbitrary finite function of the coupling. This transformation preserves the UV finiteness of the correlator and corresponds to a change of the renormalization scheme (see, for example, [@Sint:1998iq] for a discussion of this point in a different context).
Since, up to two loops we have $Z_n=(Z_2)^{\frac{n}{2}}$, we can eliminate the scheme dependence by considering dimensionless ratios of correlators. In fact, the renormalized correlators can be written as $$\label{GRexp3}
G^R_{\vec{n},\vec{m}} = \frac{1}{(1-c_1-c_2)^n}
\bigg[G_{\vec{n},\vec{m}}\big|_{\text{tree}}+
c_3\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle \bigg]
+ O(g^6)~.$$ This relation shows that the dependence on $x^2$ and on the renormalization scale $\mu$ coming from the coefficient $c_1$ is entirely encoded in a prefactor which only depends on the bare scaling dimension $n$ of the operators but not on their specific form. Therefore, this prefactor cancels in the ratio of correlators of operators of the same dimension. Choosing, for example, as a reference the correlator between two operators $O_{(2)}$, which are the only ones with dimension 2, we are led to introduce the normalized correlators $${A}^R_{\vec{n},\vec{m}} =
\frac{G^R_{\vec{n},\vec{m}}}{ \big[ G^R_{(2),(2)} \big]^{\frac{n}{2}}\phantom{\Big|}}~.
\label{defARratio}$$ These ratios are independent of the choice of the renormalization scale $\mu$ and scheme, and as such they represent physical quantities.
It is interesting to observe that the two-loop contribution to the ratio ${A}^R_{\vec{n},\vec{m}}$ only comes from the irreducible diagram represented in Fig. \[fig:v42diag\], which is finite in the limit $\epsilon\to 0$ (see (\[2loop4\]) and (\[v42is\])). This shows that also the bare ratios $${A}_{\vec{n},\vec{m}} =
\frac{G^{\phantom{R}}_{\vec{n},\vec{m}}}{ \big[ G_{(2),(2)} \big]^{\frac{n}{2}}
\phantom{\Big|}}={A}^R_{\vec{n},\vec{m}}
\label{defAratio}$$ are finite. The equality between ${A}_{\vec{n},\vec{m}}$ and ${A}^R_{\vec{n},\vec{m}} $ comes from the fact that the $Z_n$-factors cancel between the numerator and denominators, and the $Z$-renormalization of the gauge coupling starts to contribute at the next order. One can check this explicitly, by writing the bare correlators (\[Gnmtot\]) as $$\label{GRexp2}
G_{\vec{n},\vec{m}} = \frac{1}{\big(1-v_{2,1}-v_{2,2}+v_{4,2}^{(A)}\big)^n}
\bigg[G_{\vec{n},\vec{m}}\big|_{\text{tree}}
+{v}_{4,2}^{(B)}\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle \bigg]
+ O(g_0^6)~,$$ which shows that the divergence encoded in the one-loop coefficient $v_{2,1}$ cancels in the ratios $A_{\vec{n},\vec{m}}$. Furthermore, by comparing (\[GRexp2\]) and (\[GRexp3\]), we can realize that the bare and the renormalized ratios match at two loops, apart from the obvious replacement of $g_0$ with $g$.
The explicit expressions of the normalized correlators for operators of dimension 4 are:
$$\begin{aligned}
{A}^R_{(22),(22)} &= \frac{2 \left(N^2+1\right)}{N^2-1}-
\frac{3g^4\zeta(3)}{16 \pi^4} \frac{ \left(10 N^3 -2\, N^2 N_f +3 N_f\right)}
{N \left(N^2-1\right)}~,\\[3mm]
{A}^R_{(4),(22)} &=\frac{ 2(2 N^2-3) }{N(N^2-1) }- \frac{3g^4\zeta(3)}{32 \pi^4 }
\frac{ \left(2 N^5 +12 N^3-N^4 N_f+
6 N^2 N_f -18 N_f \right)}{N^2 \left(N^2-1\right)} ~,\\[3mm]
{A}^R_{(4),(4)} &= \frac{N^4-6 N^2+18}{N^2 \left(N^2-1\right)}\notag\\
&\qquad+ \frac{3 g^4 \zeta (3)}{64 \pi^4 }
\frac{\left(2 N^7-36 N^5 +72 N^3-N^4N_f +
36 N^2N_f -108N_f \right)}{N^3\left(N^2-1\right)}~.
\label{aqft}\end{aligned}$$
Similar formulae can be easily found also for operators of other dimensions.
In the next section we will recover this same result from the matrix model obtained by applying localization on the sphere $S^4$.
Matrix model approach {#secn:loc}
=====================
In [@Pestun:2007rz] it was shown, using localization techniques, that the partition function of a $\mathcal{N}=2$ SYM theory with gauge group SU$(N)$ defined on a four-sphere $S^4$ can be written in terms of a traceless Hermitian $N\times N$ matrix $a$ in the following way: $${{\mathcal{Z}}}_{S^4}=\int \prod_{u=1}^{N}\! da_u~\Delta(a)\, \big| Z({\mathrm{i}}a,\tau)\big|^2
\,\delta\Big(\sum_{v=1}^Na_v\Big)~.
\label{ws444}$$ Here we have denoted by $a_u$ the (real) eigenvalues of $a$, by $\Delta(a)$ the Vandermonde determinant $$\Delta(a) = \prod _{u<v=1}^{N}a_{uv}^2~,
\label{vandermon}$$ where $a_{uv}=a_u-a_v$, and for simplicity have set to 1 the radius $R$ of the four-sphere[^4]. Furthermore, $Z({\mathrm{i}}a,\tau)$ is the gauge theory partition function with $\tau$ being the complexified gauge coupling: $$\tau=\frac{\theta}{2\pi }+{\mathrm{i}}\frac{4\pi}{g^2}\,.
\label{tau}$$ In this paper we actually set the $\theta$-angle to zero. We remark that in the non-conformal theories the coupling $g$ appearing in the matrix model has to be interpreted as the renormalized gauge coupling at a scale proportional to the inverse radius of the four-sphere [@Pestun:2007rz].
The gauge theory partition function $Z({\mathrm{i}}a,\tau)$ is computed using the localization techniques of [@Nekrasov:2002qd; @Nekrasov:2003rj] with a purely imaginary vacuum expectation value ${\langle \varphi \rangle}={\mathrm{i}}\,a$ for the adjoint scalar, and an $\Omega$-background with parameters $\epsilon_1=\epsilon_2=1/R$. This partition function can be written as a product of the classical, one-loop and instanton contributions, namely $$Z({\mathrm{i}}a,\tau) = Z_{\mathrm{class}}({\mathrm{i}}a,\tau)\, Z_{\mathrm{one-loop}} ({\mathrm{i}}a)\,
Z_{\mathrm{inst}} ({\mathrm{i}}a,\tau) \,.$$ Since we work at weak coupling $g^2 \ll 1$, where instantons are exponentially suppressed, we can set $Z_{\mathrm{inst}}({\mathrm{i}}a)=1$. The classical part produces a simple Gaussian term in the matrix model: $$\left|Z_{\mathrm{class}}({\mathrm{i}}a,\tau)\right|^2
= {\mathrm{e}}^{-\frac{8\pi^2}{g^2} \sum_u a_u^2 }
\, = \, {\mathrm{e}}^{-\frac{8\pi^2}{g^2} \,{\mathrm{tr}\,}a^2 }~.$$ The one-loop contribution arising from the gauge multiplet and $N_f$ matter multiplets can be written as [@Pestun:2007rz] (see also [@Billo:2017glv] for details) $$\left|Z_{\mathrm{1-loop}}({\mathrm{i}}a)\right|^2 \,=\,
{\mathrm{e}}^{-S_2(a) -S_4(a) +\,\cdots}
\label{zloops4}$$ where $S_n(a)$ are homogeneous polynomials in $a$ of order $n$. The first few of them are:
$$\begin{aligned}
S_2(a) &= -(1+\gamma_{\text{E}})
\bigg(\sum_{u,v=1}^N a_{uv}^2-N_f \sum_{u=1}^N a_{u}^2 \bigg)
=\, -(1+\gamma_{\text{E}}) \,(2N-N_f)\, \mathrm{tr}\,a^2 ~, \label{S2intexp}\\[1mm]
S_4(a) &= \frac{\zeta(3)}{2} \, \bigg(\sum_{u,v=1}^N
a_{uv}^4-N_f \sum_{u=1}^N a_{u}^4 \bigg) =\,\frac{\zeta(3)}{2} \, \Big[ (2 N-N_f) \,
{\mathrm{tr}}\,a^4 + 6 \left(\mathrm{tr}\,a^2\right)^2 \Big] ~, \label{S4intexp}\\[1mm]
S_6(a) &= -\frac{\zeta(5)}{3}\, \bigg(\sum_{u,v=1}^N a_{uv}^6-N_f \sum_{u=1}^N a_{u}^6 \bigg)
=\,- \frac{\zeta(5)}{3} \, \Big[ (2 N-N_f) \, \mathrm{tr} \,a^6\label{S6intexp}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad~
+30\, \mathrm{tr} \,a^4 \, \mathrm{tr} \,a^2 -20 \,\left(\mathrm{tr} \,a^3\right)^2
\Big]~. \notag
\end{aligned}$$
Performing the rescaling $$\label{resca}
a \to \Big(\frac{g^2}{8\pi^2}\Big)^{\frac 12}\, a~,$$ the matrix model gets a canonically normalized Gaussian factor and the sphere partition function (\[ws444\]) becomes $$\label{rescaledmm}
{{\mathcal{Z}}}_{S^4} = \Big(\frac{g^2}{8\pi^2}\Big)^{\frac{N^2-1}{2}} \, \int \prod_{u=1}^{N}
da_u~\Delta(a)~{\mathrm{e}}^{-{\mathrm{tr}\,}a^2 - S_{\mathrm{int}}(a)}\,\delta\Big(\sum_{v=1}^Na_v\Big)$$ with $$\label{Sintresc}
S_{\mathrm{int}}(a) = \frac{g^2}{8\pi^2}~S_2(a) + \Big(\frac{g^2}{8\pi^2}\Big)^2 \, S_4(a) +
\Big(\frac{g^2}{8\pi^2}\Big)^3\, S_6(a) + \cdots ~.$$ The term of order $g^{2\ell}$ in $S_{\mathrm{int}}(a)$ accounts for effects that take place at $\ell$ loops in the corresponding field theory computation. Therefore we will refer to the $g^2$-expansion of $S_{\mathrm{int}}$ as a loop expansion.
Exploiting the Vandermonde determinant $\Delta(a)$ and writing $a=a^b\,T^b$, we can alternatively express the integral (\[rescaledmm\]) using a flat integration measure $da$ over all matrix components $a^b$ as follows $${{\mathcal{Z}}}_{S^4}= c_N\,\Big(\frac{g^2}{8\pi^2}\Big)^{\frac{N^2-1}{2}} \,
\int da~{\mathrm{e}}^{-{\mathrm{tr}\,}a^2 - S_{\mathrm{int}}(a)}
\label{ZS41}$$ where $c_N$ is a $g$-independent constant and $da\propto \prod_b da^b$. The overall prefactors in (\[ZS41\]) are irrelevant when computing correlators and thus can be neglected.
Given any function $f(a)$, its vacuum expectation value in the matrix model described above is defined as follows $$\label{vevmat}
\begin{aligned}
\big\langle\,f(a)\,\big\rangle &=
\frac{1}{{{\mathcal{Z}}}_{S^4}}
\int \prod_{u=1}^{N}da_u~\Delta(a)\, \big| Z({\mathrm{i}}a,\tau)\big|^2
\,\delta\Big(\sum_{v=1}^N a_v\Big) \, f(a) \\[3mm]
&=\,\frac{\displaystyle{ \int da ~{\mathrm{e}}^{-{\mathrm{tr}\,}a^2-S_{\mathrm{int}}(a)}\,
f(a)}}{ \displaystyle{\int da~{\mathrm{e}}^{-{\mathrm{tr}\,}a^2-S_{\mathrm{int}}(a)}} }
\,=\,
\frac{\displaystyle{\big\langle\,{\mathrm{e}}^{-S_{\mathrm{int}}(a)}\,f(a)\,\big\rangle_0}}
{\displaystyle{\big\langle\,{\mathrm{e}}^{-S_{\mathrm{int}}(a)}\,\big\rangle_0}}~,
\end{aligned}$$ where in the second line we have used (\[ZS41\]). The subscript “0” denotes the vacuum expectation values taken with respect to the Gaussian measure, which can be computed by repeatedly using Wick’s theorem to reduce them to the basic contraction $$\label{2pfa}
\big\langle\,a^b\, a^c\,\big\rangle_{0} = \delta^{bc}~.$$
Chiral and anti-chiral operators in the matrix model
----------------------------------------------------
We are interested in extracting from the matrix model (\[vevmat\]) the two-point functions (\[OnOm\]). To this aim we have first to find counter-partners of the chiral and anti-chiral operators in the matrix model. It would seem natural to associate to the multi-trace operator $O_{\vec{n}}(x)$ defined in (\[On\]), an analogous function $O_{\vec{n}}(a)$ in the matrix model, given by the same expression (\[On\]) but with $\varphi(x)$ replaced by $a$, namely $$O_{\vec{n}}(a) = {\mathrm{tr}\,}a^{n_1}\, {\mathrm{tr}\,}a^{n_2} \, \ldots \,{\mathrm{tr}\,}a^{n_\ell}~.$$ However, the operator $O_{\vec{n}}(x)$ has vanishing vacuum expectation value in the field theory, while in the matrix model ${\langle O_{\vec{n}}(a) \rangle}\neq 0$ due to the self-contractions of $a$. This means that we have to refine the dictionary and make $O_{\vec{n}}(a)$ normal-ordered. This can be done by subtracting from $O_{\vec{n}}(a)$ all possible self-contractions and making it orthogonal to all operators with lower dimensions.
As discussed in [@Billo:2017glv; @Billo:2018oog; @Sysoeva:2017fhr], the prescription to define the normal ordering of any operator $O(a)$ in the matrix model is the following. Let be $\Delta$ the dimension of $O(a)$ and $\big\{O_p(a)\big\}$ a basis in the finite-dimensional space of matrix operators with dimension smaller than $\Delta$. Denoting by $C_\Delta$ the (finite-dimensional) matrix of correlators $$\big(C_\Delta\big)_{pq} = \big\langle\,O_p(a)\,O_q(a)\,\big\rangle\,,
\label{Cpq}$$ which are computed according to (\[vevmat\]), we define the normal-ordered operator as $${:\! O(a) \!:_g}\,\,
=\, O(a) - \sum_{p,q} \big\langle\,O(a)\, O_{p}(a)\,\big\rangle\, (C_\Delta^{-1})^{pq}\, O_q(a)~.
\label{normalo}$$ Our notation stresses the fact that this normal-ordering is $g$-dependent, since the correlators on the right hand side of (\[normalo\]) are computed in the interacting matrix model. The proposal is then to associate to the field theory operators the corresponding normal-ordered matrix operators, namely $$\label{corrOcO}
O_{\vec{n}}(x) \,\quad \to\,\quad {{\mathcal{O}}}_{\vec{n}}(a) =\,\, {:\! O_{\vec{n}}(a) \!:_g}~.$$ A similar replacement holds for the anti-chiral operators.
For example, using the definition [(\[normalo\])]{} we find $$\begin{aligned}
{{\mathcal{O}}}_{(2)}(a)&=& {\mathrm{tr}\,}a^2 -\frac{N^2-1}{2}\bigg[1+\frac{g^2}{8\pi^2}\,
(1+\gamma_{\text{E}})(2N-N_f)+
\Big(\frac{g^2}{8\pi^2}\Big)^2
(1+\gamma_{\text{E}})^2(2N-N_f)^2\nonumber \\
&&\qquad\qquad\qquad
-\Big(\frac{g^2}{8\pi^2}\Big)^2\zeta(3)\Big(5N^2-N\,N_f
+\frac{3N_f}{2N}\Big)\bigg]+O(g^6)~.
\label{O2g}\end{aligned}$$ The term of order $g^0$ inside the square brackets represents the self-contraction of ${\mathrm{tr}\,}a^2$, while the terms of higher order in $g^2$ represent the self-contractions of the operator through the interaction vertices coming from the matrix model action. Analogous expressions can be worked out for operators of higher dimension.
Correlators in the matrix model
-------------------------------
Once the operators have been identified, their correlators can be computed in a straightforward way using the definition (\[vevmat\]). In particular the two-point correlators are defined as $$\label{Gmmis}
{{\mathcal{G}}}_{\vec{n}, \vec{m}} = \big\langle\,{{\mathcal{O}}}_{\vec{n}}(a)\,{{\mathcal{O}}}_{\vec{m}}(a)\,\big\rangle
= \frac{\displaystyle{\big\langle\,{\mathrm{e}}^{-S_{\mathrm{int}}}
\,{{\mathcal{O}}}_{\vec{n}}(a)\,{{\mathcal{O}}}_{\vec{m}}(a)\,\big\rangle_0}}
{\displaystyle{\big\langle\,{\mathrm{e}}^{-S_{\mathrm{int}}}\,\big\rangle_0}}~.$$ Since normal-ordered operators [with different dimensions]{} are orthogonal to each other, ${{\mathcal{G}}}_{\vec{n}, \vec{m}}$ vanishes for $n\neq m$.
For instance, for the simplest operator ${{\mathcal{O}}}_{(2)}(a)$ defined in (\[O2g\]) we find $$\begin{aligned}
{{\mathcal{G}}}_{(2), (2)} &=
\frac{N^2-1}{2}
\bigg[1+\frac{g^2}{8\pi^2}\,2\,(1+\gamma_{\text{E}})(2N-N_f)+
\Big(\frac{g^2}{8\pi^2}\Big)^2 3\,(1+\gamma_{\text{E}})^2(2N-N_f)^2\\
&\quad-\Big(\frac{g^2}{8\pi^2}\Big)^2\zeta(3)\Big(15N^2-3N N_f+\frac{9N_f}{2N}\Big)\bigg]+
O(g^6)~.
\end{aligned}
\label{Gmm22}$$ The explicit expressions of correlators with higher dimensional operators can be computed in a similar way. At dimension 4 we find
$$\begin{aligned}
{{\mathcal{G}}}_{(2,2),(2,2)}&=\frac{N^4-1}{2}
\bigg[1+\Big(\frac{g^2}{8\pi^2}\Big)4\,(1+\gamma_{\text{E}})(2N-N_f)+
\Big(\frac{g^2}{8\pi^2}\Big)^2 10\,(1+\gamma_{\text{E}})^2(2N-N_f)^2\notag\\
&\quad-\Big(\frac{g^2}{8\pi^2}\Big)^2\,12\,
\zeta(3)\Big(2N^2-\frac{N N_f}{2}+\frac{N_f}{2N}\Big)\bigg]\notag\\
&\quad
-\Big(\frac{g^2}{8\pi^2}\Big)^2\,\frac{3}{2}\,\zeta(3)\,(N^2-1)\,\Big(2 N^4+22 N^2-3 N N_f+\frac{7N_f}{N}\Big)+O(g^6)~,\\[4mm]
{{\mathcal{G}}}_{(4),(2,2)}&=\frac{2N^4-5N^2+3}{2N}
\bigg[1+\Big(\frac{g^2}{8\pi^2}\Big)4\,(1+\gamma_{\text{E}})(2N-N_f)\notag\\
&\quad+
\Big(\frac{g^2}{8\pi^2}\Big)^2 10\,(1+\gamma_{\text{E}})^2(2N-N_f)^2
-\Big(\frac{g^2}{8\pi^2}\Big)^2\,12\,
\zeta(3)\Big(2N^2-\frac{N N_f}{2}+\frac{N_f}{2N}\Big)\bigg]\notag\\
&\quad
-\Big(\frac{g^2}{8\pi^2}\Big)^2\,\frac{3}{2}\,\zeta(3)(N^2-1)\,\Big(6 N^3- N^2 N_f + 6 N + 8 N_f - \frac{21 N_f}{N^2} \Big)+O(g^6)~,\\
{{\mathcal{G}}}_{(4),(4)}&=\frac{N^6- 7 N^4+ 24 N^2-18}{4 N^2}
\bigg[1+\Big(\frac{g^2}{8\pi^2}\Big)4\,(1+\gamma_{\text{E}})(2N-N_f)\notag\\
&\quad+
\Big(\frac{g^2}{8\pi^2}\Big)^2 10\,(1+\gamma_{\text{E}})^2(2N-N_f)^2
-\Big(\frac{g^2}{8\pi^2}\Big)^2\,12\,
\zeta(3)\Big(2N^2-\frac{N N_f}{2}+\frac{N_f}{2N}\Big)\bigg]\notag\\
&\quad
-\Big(\frac{g^2}{8\pi^2}\Big)^2\,\frac{3}{2}\,\zeta(3)(N^2-1)\,\Big(12 N^2+ N N_f-18-\frac{21N_f}{N}+\frac{63N_f}{N^3}\Big)+O(g^6) ~.
\end{aligned}$$
\[Gmm44\]
Here we have split the $O(g^4)$ contribution into the sum of a few terms in order to facilitate a comparison with the field theory calculation.
It is easy to check that for $N_f = 2N$ the matrix model correlators ${{\mathcal{G}}}_{\vec{n}, \vec{m}}$ exactly match, up to two loops, the correlators $G^R_{\vec{n}, \vec{m}}$ computed in perturbation theory (see (\[GR22\]) and (\[GR44\])), thus confirming the general results obtained in [@Baggio:2014sna; @Baggio:2015vxa; @Gerchkovitz:2016gxx; @Baggio:2016skg; @Rodriguez-Gomez:2016ijh; @Rodriguez-Gomez:2016cem; @Baggio:2014ioa; @Billo:2017glv] [^5]. The fact that the partition function on the sphere $S^4$ and its associated matrix model contain information on the correlators in the flat space $\mathbb{R}^4$ is not too surprising in the conformal case. We now want to investigate to what extent this relation holds in the non-conformal theories with $N_f\neq 2N$.
Comparison between matrix model and field theory correlators
------------------------------------------------------------
Comparing (\[GR22\]) with (\[Gmm22\]), and (\[GR44\]) with (\[Gmm44\]), we see that they have the same structure and that many terms exactly match. However, for $N_f\neq 2N$ there are some differences in the terms proportional to $\zeta(3)$. To make the comparison simpler, it is convenient to write ${{\mathcal{G}}}_{\vec{n},\vec{m}}$ in terms of the complex matrices ${\bm{\varphi}}$ and ${\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}}$ using the formalism introduced in Section \[secn:pert\]. Indeed, it is possible to explicitly check that, up to two loops, the matrix model correlators (\[Gmmis\]) can be expressed as follows $${{\mathcal{G}}}_{\vec n,\vec m} = \frac{\displaystyle{\big\langle {\mathrm{e}}^{-\widehat{V}_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}
~O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle\phantom{\Big|}}}
{\displaystyle{\big\langle {\mathrm{e}}^{-\widehat{V}_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})}\big\rangle}\phantom{\bigg|}}
+O(g^6)\,,
\label{Gmmfin}$$ where $$\widehat{V}_{\text{eff}}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})=-\big(\widehat{c}_1+
\widehat{c}_2\big)\,V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
-\widehat{c}_3\,V_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\label{Vmmeff}$$ with $V_2$ and $V_4^{(B)}$ defined, respectively, in (\[defV21vphi\]) and (\[V4\]), and $$\begin{aligned}
\widehat{c}_1& =\frac{g^2}{8\pi^2}\,(1+\gamma_{\text{E}})(2N-N_f)~,\\
\widehat{c}_2&= -3\,\zeta(3) \Big(\frac{g^2}{8\pi^2}\Big)^2\,
\Big(2N^2-\frac{N N_f}{2}+\frac{N_f}{2N}\Big)~,\\
\widehat{c}_3&= 3\,\zeta(3) \,\Big(\frac{g^2}{8\pi^2}\Big)^2~.
\end{aligned}
\label{chat123}$$ Notice that the effective vertex (\[Vmmeff\]) has the same form as the renormalized vertex (\[VReff\]) obtained from perturbation theory. Comparing (\[chat123\]) with (\[c123\]) and (\[c1msbar\]), we find $$\widehat{c}_1=c_1~,~~\widehat{c}_2=c_2+9\,\zeta(3)
\Big(\frac{g^2}{8\pi^2}\Big)^2 \,N(2N-N_f)~,~~\widehat{c}_3
=c_3~.$$ Therefore, the difference between the effective vertex $\widehat{V}_{\text{eff}}$ of the matrix model and the renormalized effective vertex ${V}^R_{\text{eff}}$ is $$\delta=\widehat{V}_{\text{eff}}-{V}^R_{\text{eff}}=9\,\zeta(3)
\Big(\frac{g^2}{8\pi^2}\Big)^2 \,N(2N-N_f)\,V_2({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})~.
\label{delta}$$ It is interesting to observe that $\delta$ vanishes in the conformal case. Moreover, it is proportional to $V_2$ which, as follows from (\[V2On\]), computes the scaling dimension of the operators [^6]. This fact suggests that it might be interpreted as due to a conformal anomaly which, in non-conformal theories, affects the correlation functions in going from the four-sphere $S^4$ to the flat space $\mathbb{R}^4$, or vice versa.
In the two-loop approximation, we can rewrite (\[Gmmfin\]) as follows: $$\label{Gmmexp}
\begin{aligned}
{{\mathcal{G}}}_{\vec{n},\vec{m}} &= \frac{1}{(1-\widehat{c}_1-\widehat{c}_2)^n}
\bigg[G_{\vec{n},\vec{m}}\big|_{\text{tree}}+
\widehat{c}_3\,\big\langle {V}_{4}^{(B)}({\bm{\varphi}},{\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})
\,O_{\vec{n}}({\bm{\varphi}})\,{\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec{m}}({\@ifnextchar^{{\wide@bar{{\bm{\varphi}}}{0}}}{\wide@bar{{\bm{\varphi}}}{1}}})\big\rangle \bigg] + O(g^6)~.
\end{aligned}$$ This formula clearly shows that the dependence on $\widehat{c}_1$ and $\widehat{c}_2$ drops out in the ratio between correlators of operators with the same scaling dimensions. Thus, in analogy with (\[defARratio\]), we are led to define the ratio of correlators in the matrix model $${{\mathcal{A}}}_{\vec{n},\vec{m}} =
\frac{{{\mathcal{G}}}_{\vec{n},\vec{m}}}{ \big[ {{\mathcal{G}}}_{(2),(2)} \big]^{\frac{n}{2}}\phantom{\Big|}}~.
\label{defAmmratio}$$ Since $\widehat{c}_3=c_3$, it exactly matches the normalized correlator $A^R_{\vec{n},\vec{m}}$, namely $${{\mathcal{A}}}_{\vec{n},\vec{m}} = A^R_{\vec{n},\vec{m}} ~.
\label{Amatch}$$ We have checked this relation in many explicit examples, with operators of dimensions up to 6.
Two-point correlators on the four-sphere {#secn:sphere}
========================================
In this section we study in more detail the relation between the correlators in flat space, discussed in Sections \[secn:pert\] and \[secn:rencorr\], and those on the four-sphere $S^4$. The latter are closely related to the correlators derived from matrix model presented in Section \[secn:loc\]. In particular we consider the one-loop correction to the scalar propagator on $S^4$ and compare it with the one-loop propagator in flat space defined in (\[oneloopa2\]).
To this aim, it is convenient to describe a sphere in $D-$dimensions by using flat embedding coordinates $\{\eta_0,\eta_\mu\}$ satisfying the quadratic constraint $$\begin{aligned}
\label{sp}
\eta_0^2 + \sum_{\mu=1}^D \eta_\mu^2 = R^2~,\end{aligned}$$ where $R$ is the radius of the $D$-sphere. Following [@Adler:1972qq; @Adler:1973ty; @Drummond:1975yc; @Drummond:1977dn; @Drummond:1977uy], we use the stereographic projection $$\label{map}
\eta_0= R\, \frac{x^2-R^2}{x^2+R^2}~,\qquad \eta_\mu = R^2\, \frac{2x_\mu}{x^2+R^2}
\qquad\mbox{with}~~x^2=\sum_{\mu=1}^Dx_\mu^2~,$$ to relate a theory defined on a $D$-sphere to a theory in $\mathbb{R}^D$, parametrized by the flat coordinates $x_\mu$. One of the advantages of this formalism is that the scalar propagator on the sphere, denoted by a subscript ${{{S}}}$, takes a very simple form given by $$\label{propsphere}
\big\langle\,\varphi^a(\eta_1)\,{\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}^{\,b}(\eta_2)\,\big\rangle_{S}=
\Delta_{{{{S}}}}(\eta_{12})\,\delta^{ab}~,$$ where $\eta_{12}=\eta_1-\eta_2$ and $$\label{DeltaS}
\Delta_{{{{S}}}}(\eta_{12}) = \frac{\Gamma(1-\epsilon)}{
4\pi\,(\pi \eta_{12}^2)^{1-\epsilon}}~.$$ Here we have used $D=4-2\epsilon$ and defined $$\eta_{12}^2= \frac{(x_1-x_2)^2}{\kappa(x_1)\kappa(x_2)}~,\qquad
\kappa(x)= \frac{x^2+R^2}{2R^2}~.
\label{kappa}$$ Inserting this into (\[DeltaS\]) and comparing with (\[Delta\]), we get $$\Delta_{{{{S}}}}(\eta_{12}) = \big[\kappa(x_1)\,\kappa(x_2)\big]^{1-\epsilon}\,\Delta(x_{12})~.
\label{DeltaS1}$$ The scalar propagator on the sphere is thus proportional to the one in flat space, with a scaling factor raised to the engineering dimensions of the scalar fields. Notice that this is the same scaling factor that defines the induced metric on the sphere through the conformal map (\[map\]); indeed $$ds^2 = d\eta_0^2 +\sum_{\mu=1}^D d\eta_\mu^2 = \frac{1}{\kappa^2(x)}
\sum_{\mu=1}^D dx_\mu^2~.$$
Let us now consider the correlators between two operators on the sphere. They have a structure similar to the ones in flat space given in (\[formOO\]), namely $$\label{formOOsphere}
\big \langle \,O_{\vec n}(\eta_1) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(\eta_2)\,\big\rangle_{{{{S}}}}
= \Delta^n_{{{{S}}}}(\eta_{12}) \, G^{({{{S}}})}_{\vec n,\vec m}(g_0,\epsilon,\eta_{12})
\, \delta_{nm} ~.$$ The correlators $G^{({{{S}}})}_{\vec n,\vec m}(g_0,\epsilon,\eta_{12})$, which we will simply denote as $G^{({{{S}}})}_{\vec n,\vec m}$, can be computed order by order in perturbation theory. At tree level, we have just to contract the color indices of the constituent fields, so that $$G^{({{{S}}})}_{\vec n,\vec m}\Big|_{\text{tree}} =G_{\vec n,\vec m}\Big|_{\text{tree}}~.
\label{GStree}$$ Inserting this into (\[formOOsphere\]), using the propagator (\[DeltaS\]) and taking the limit $\epsilon\to 0$, we can easily obtain $$\label{OOspheretree}
\big \langle \,O_{\vec n}(\eta_1) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(\eta_2)\,\big\rangle_{{{{S}}}}
\,\Big|_{\text{tree}}
= \kappa^n(x_1)\,\kappa^m(x_2)\,\big \langle \,O_{\vec n}(x_1) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(x_2)\,\big\rangle\,\Big|_{\text{tree}}~.$$ This is the expected relation between correlators on the sphere and correlators in flat space that follows from the conformal map (\[map\]).
Let us now consider the one-loop correction. Before analyzing the correlators on the sphere, it is convenient to revisit the calculation of one-loop correction to the scalar propagator in flat space, given in (\[oneloopa2\]) in coordinate space. The one-loop correction to $\big\langle \varphi^a(x_1)\,{\@ifnextchar^{{\wide@bar{\varphi}{0}}}{\wide@bar{\varphi}{1}}}^{\,b}(x_2) \big\rangle$ can be written as $$\label{WR1}
W_1^{ab}(x_{12}) = -g_0^2\, (2N-N_f)\,W_1(x_{12})\,\delta^{ab} ~,$$ where $$W_1(x_{12})=\!\int \!
d^D x_3\, d^2\bar\theta_3 \,d^D x_4\, d^2\theta_4 \,\Delta(x_{13})
\big( {\mathrm{e}}^{-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3}\Delta(x_{43})\big)^2
\Delta(x_{42})~.
\label{Wintis}$$ Its Fourier transform is the function ${{\mathcal{W}}}_1(p)$ discussed in Appendix \[app:Feynman diagrams\] (see in particular (\[cW1is\]) and (\[cW1nc\])). Computing the integrals, we find $$W_1(x_{12})=-\frac{(\pi x_{12}^2)^\epsilon\,\Gamma(1-\epsilon)}{(4\pi)^2\,\epsilon(1-2\epsilon)}
\,\Delta(x_{12})~.
\label{Wintexpl}$$ Using this in (\[WR1\]), one recovers the result presented in (\[oneloopa2\]) and (\[v1loop\]).
Going to the sphere, we find that the one-loop correction to the scalar propagator has a form similar to (\[WR1\]), that is $$\label{WS1}
W_{1\,{{{S}}}}^{ab}(\eta_{12}) = -g_0^2\, (2N-N_f)\,W_{1\,{{{S}}}}(\eta_{12})\,\delta^{ab} ~,$$ where the function $W_{1\,{{{S}}}}$ is the sphere generalization of $W_1$. Applying the embedding formalism [@Adler:1972qq; @Adler:1973ty; @Drummond:1975yc; @Drummond:1977dn; @Drummond:1977uy], the expression of $W_{1\,{{{S}}}}$ can be obtained by performing the conformal transformation (\[map\]) to (\[Wintis\]). Under this map, both the integration measure and the scalar propagators acquire scale factors according to $$\label{scaling}
\begin{aligned}
\int \! d^D x_i \,d^2\theta_i {}& \to \int \! d^D x_i \,d^2\theta_i \, \kappa^{-D+1}(x_i)~,\\
\Delta(x_{ij}) {}& \to \Delta(x_{ij})\, \big[\kappa(x_i)\,\kappa(x_j)\big]^{\frac{D-2}{2}}~,
\end{aligned}$$ so that $W_1(x_{12})$ becomes $$\label{WS2}
W_{1\,{{{S}}}}(\eta_{12}) = \big[\kappa(x_1)\,\kappa(x_2)\big]^{1-\epsilon}\,I(x_1,x_2)~,$$ where $$I(x_1,x_2)=\!\int \!
d^D x_3\, d^2\bar\theta_3 \,d^D x_4\, d^2\theta_4 \,\Delta(x_{13})
\big({\mathrm{e}}^{-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3}\Delta(x_{43})\big)^2
\Delta(x_{42})\,\big[\kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}~.
\label{Iis}$$ Comparing this integral with (\[Wintis\]), we notice the presence of the additional scaling factor $\big[ \kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}$, which clearly becomes 1 in four dimensions.
Therefore, if the integrals in (\[Wintis\]) and (\[WS2\]) were finite, $W_{1\,{{{S}}}}$ and $W_1$ would only differ by the overall scaling factor $ \kappa(x_1)\,\kappa(x_2)$. In other words, if no UV divergences are present, one can safely perform the limit $\epsilon\to 0$ inside the integrals. However, the integral in (\[Iis\]) is divergent, and thus the scaling factor $\big[\kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}$ in the integrand cannot be neglected. The evaluation of this integral is presented in Appendix \[app:feynsphere\], and the result is $$\label{WS3}
W_{1\,{{{S}}}}(\eta_{12}) \approx -\frac{(\pi \eta_{12}^2)^\epsilon\,\Gamma(1-\epsilon)}{(4\pi)^2\,\epsilon(1-2\epsilon)}
\,\Delta_{{{{S}}}}(\eta_{12})~.$$ Comparing with (\[Wintexpl\]), we see that, up to terms $O(\epsilon)$, the two expressions coincide upon replacing $\Delta_{{{{S}}}}$ with $\Delta$, and $\eta_{12}^2$ with $x_{12}^2$. The fact that the divergent parts of $W_{1\,{{{S}}}}$ and $W_1$ coincide, is not surprising since the UV divergences come from integration at short distances where there is no distinction between the sphere and flat space. What is non trivial, however, is that the finite parts coincide, modulo the obvious replacement of $x_{12}$ with $\eta_{12}$.
Putting everything together, we see that the one-loop correction to the scalar propagator on the sphere is $$W^{ab}_{1\,{{{S}}}}(\eta_{12})=v^{({{{S}}})}_{2,1}\, \Delta_{{{{S}}}}(\eta_{12})\, \delta^{ab}~,
\label{WSfin}$$ with $$v^{({{{S}}})}_{2,1}
\approx \frac{g_0^2}{8 \pi^2}\, (2N - N_f)\, \frac{(\pi \eta_{12}^2)^\epsilon\,\Gamma(1-\epsilon)}{2\epsilon(1-2\epsilon)}~,
\label{v1loopS}$$ in full analogy with (\[oneloopa2\]) and (\[v1loop\]). This implies that $$\label{onelresS}
G^{({{{S}}})}_{\vec n,\vec m}\,\big|_{\text{1-loop}} \,=\, n\,v^{({{{S}}})}_{2,1}
\, G_{\vec n,\vec m}\,\big|_{\text{tree}}~.$$ Thus, the renormalization procedure can be done following the same steps we described in Section \[secn:rencorr\]. Choosing the renormalization scale $\mu^2$ as in (\[msbar\]) with $x^2$ replaced by $\eta_{12}^2$ on the sphere and by $x_{12}^2$ in flat space, then $$\label{OOsphere1loop}
\big \langle \,O^R_{\vec n}(\eta_1) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}^R_{\vec m}(\eta_2)\,\big\rangle_{{{{S}}}}
\,\Big|_{\text{1-loop}}
= \kappa^n(x_1) \,\kappa^m(x_2) \,\big \langle \,O^R_{\vec n}(x_1) \, {\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}^R_{\vec m}(x_2)\,\big\rangle\,\Big|_{\text{1-loop}}~.$$ The relation (\[onelresS\]) and the explicit expression of $v^{({{{S}}})}_{2,1}$ explain why the correlators ${{\mathcal{G}}}_{\vec{n},\vec{m}}$ obtained from the matrix model perfectly agree with those computed in field theory at one loop.
The same analysis can be carried out at two loops, even though the resulting integrals on the sphere become way more complicated. Most of the two-loop diagrams develop UV divergences and need to be regularized. As in the one-loop case, the integrals on the sphere differ from those in flat space because of scaling factors $[\kappa(x_i)]^{-\epsilon}$ appearing in the integrands. Such factors do not modify the leading UV divergent contribution but they do affect the finite part. As a result, there is no reason [*a priori*]{} to expect that the finite part of the correlation functions on the sphere and in flat space should coincide. However, we stress the fact that the finite part of the two-point correlator is not a physical observable since it depends on the regularization scheme. The explicit one-loop calculation shows a perfect agreement between the matrix model and the field theory results for the two-point correlators for the special choice of the renormalization scale. It is natural to ask whether such identification holds also at higher loops. At two loops, the results of Sections \[secn:rencorr\] and \[secn:loc\] reveal that the finite part of the correlation functions are different in flat space and in the matrix model. Still, a perfect match is found for physical observables that are independent of the renormalization scheme, such as the ratios of correlators with operators of the same dimension. In such ratios, all divergent two-loop diagrams cancel and the whole contribution at order $g^4$ is due to a single and finite Feynman diagram, namely the irreducible diagram represented in Fig. \[fig:v42diag\]. The corresponding Feynman integral is finite in $\mathbb R^4$ and does not require a regularization. As a consequence, it possesses the four-dimensional conformal symmetry and takes the same form in $\mathbb R^4$ and $S^4$. This explains why the ratios of the correlation functions match the prediction from localization at two loops, as shown in (\[Amatch\]).
Summary and conclusions {#secn:concl}
=======================
We have explicitly computed the two-point correlation functions between chiral and anti-chiral operators in the ${{\mathcal{N}}}=2$ SYM theory with gauge group SU($N$) and $N_f$ fundamental flavors up to two loops, using standard (super) Feynman diagrams in dimensional regularization. Our results show that these correlators have a remarkably simple structure of UV divergences stemming from the fact that the anomalous dimensions of the operators are proportional to the $\beta$-function. We demonstrated that when the renormalization scale $\mu$ and the separation $x$ between the operators are inversely proportional to each other, these correlators can be obtained via a matrix model which is strikingly similar to the matrix model that computes the partition function and the chiral/anti-chiral correlators on the four-sphere using localization. Up to two loops, the difference between the two matrix models is just a term of order $g^4$ proportional to $(2N-n_f)\,V_2$, which acting on the operators gives their anomalous dimensions. This suggests that this difference that vanishes in the conformal theories, might be interpreted as a conformal anomaly. In the non-conformal cases this could explain the difference between the correlators on the four-sphere and those in flat space.
We have also constructed normalized correlators, which are scheme independent and, as such, represent physical quantities. Up to two-loops, these normalized correlators are the same on the four-sphere and in flat space, and can be computed either using the field theory approach with Feynman diagrams, or using localization methods via a simple matrix model.
Our analysis clarifies the relation between the perturbative field calculations and the localization results in ${{\mathcal{N}}}=2$ SYM theories. It would be interesting to generalize it in various directions, for example to compute the correlators at three or more loops, or to compute the one-point or higher-point correlation functions in presence of Wilson loops. In particular it would be interesting to explore in detail the two-loop calculations of the correlators using Feynman diagrams on the sphere, and/or obtain a “first-principle" derivation of the difference between the two matrix models that yield the correlators on the four-sphere and in flat space. We hope to be able to return to some of these points in future works.
0.2cm We would like to thank A. Belitsky, L. Bianchi, M. Frau, R. Frezzotti, F. Galvagno, P. Gregori, K. Papadodimas, N. Tantalo and T. Vladikas for many useful discussions.
The work of M.B., A.L., F.F., J.F.M. is partially supported by the MIUR PRIN Contract 2015MP2CX4 “Non-perturbative Aspects Of Gauge Theories And Strings”. The work of G.P.K. is supported by the French National Agency for Research grant ANR-17-CE31-0001-01. The work of A.L. is partially supported by the “Fondi Ricerca Locale dell’Università del Piemonte Orientale”. All authors would like thank the Galileo Galilei Institute for Theoretical Physics for hospitality during the course of this work. G.P.K. is grateful to INFN and the Simons Foundation for partial support.
Loop integrals {#app:loop_integrals}
==============
In this appendix we follow closely [@Chetyrkin:1981qh] (see also [@Grozin:2005yg] for a review) and collect some useful formulae necessary to evaluate the Feynman integrals. We work in $D=4-2\epsilon$ dimensions and use the propagator of a massless scalar field given in (\[Delta\]), namely $$\Delta(x) = \int \frac{d^D k}{(2\pi)^D} \frac{{\mathrm{e}}^{{\mathrm{i}}k \cdot x}}{k^2} =
\frac{\Gamma(1-\epsilon)}{(4\pi)\,(\pi x^2)^{1-\epsilon}}~.
\label{DeltaApp}$$ For later convenience, we introduce the graphical notation for Feynman integrals in the momentum representation $$\parbox[c]{.08\textwidth}{\PandocStartInclude{Ialfabeta.pdf_tex}\PandocEndInclude{input}{2137}{52}} ~\equiv~
\int \frac{d^D k}{(2\pi)^D}\frac{1}{(k^2)^{\alpha}\,((p-k)^2)^{\beta}}
\,=\, \frac{I_{\alpha,\beta}}{(p^2)^{\alpha+\beta-2+\epsilon}}~,
\label{int1}$$ where $$I_{\alpha,\beta}=
\frac{\Gamma(2-\epsilon-\alpha)\,\Gamma(2-\epsilon-\beta)\,\Gamma(\alpha
+\beta-2+\epsilon)}{(4\pi)^{2-\epsilon}\,\Gamma(\alpha)\,\Gamma(\beta)\,\Gamma(4-2\epsilon-\alpha
-\beta)}~.
\label{Ialphabeta}$$ The black dots on the left and the right of the diagram in (\[int1\]) denote, respectively, the incoming and outgoing momentum $p$. Furthermore, in each interaction vertex the momentum conservation is enforced. When $\alpha$ or $\beta$ is 1, for simplicity we do not write the labels. With these notations, we then have $${\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Iab.jpg}}}~=~
\frac{\Gamma^2(1-\epsilon) \,\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}\,\Gamma(2-2\epsilon)}\,\frac{1}{(p^2)^{\epsilon}}~.
\label{int11}$$ We will also make use of the Fourier transform integral $$\begin{aligned}
\Pi_{\alpha}(x)=\int \frac{d^D k}{(2\pi)^D}\frac{{\mathrm{e}}^{{\mathrm{i}}k\cdot x}}{(k^2)^{\alpha}}&=
\frac{(x^2)^{\alpha+\epsilon-2}\,\Gamma(2-\epsilon-\alpha)}
{4^\alpha\pi^{2-\epsilon}\Gamma(\alpha)}
=\frac{(x^2)^{\alpha-1}\,\Gamma(2-\epsilon-\alpha)}{4^{\alpha-1}\Gamma(\alpha)\,
\Gamma(1-\epsilon)}\,\Delta(x)~,
\end{aligned}
\label{Pialpha}$$ which for $\alpha=1$ reduces to (\[DeltaApp\]). In particular we will need the following explicit formulae
$$\begin{aligned}
\Pi_{1+\epsilon}(x)&=\frac{(x^2)^{\epsilon}\,\Gamma(1-2\epsilon)}{4^\epsilon\,\Gamma(1+\epsilon)
\,\Gamma(1-\epsilon)}\,\Delta(x)~,\label{Pialphaa}\\[2mm]
\Pi_{1+2\epsilon}(x)&=\frac{(x^2)^{2\epsilon}\,\Gamma(1-3\epsilon)}{4^{2\epsilon}\,\Gamma(1+2\epsilon)
\,\Gamma(1-\epsilon)}\,\Delta(x)~,\label{Pialphab}\\[2mm]
\Pi_{3\epsilon}(x)&=\frac{(x^2)^{2\epsilon}\,\Gamma(2-4\epsilon)}{4^{3\epsilon-2}\,
\pi^{\epsilon-2}\,\Gamma(3\epsilon)
\,\Gamma(1-\epsilon)^2}\,\Delta^2(x) ~.\label{Pialpha1c}
\end{aligned}$$
\[Pialpha1\]
Triangle identity
-----------------
Let us consider the integral $$J\big(\{\alpha_i\}\big)=\int\! \frac{d^D k}{(2\pi)^D}\,
\frac{1}{(k^2)^{\alpha_1}\,((k-q)^2)^{\alpha_2}\,((k-p)^2)^{\alpha_3}\,(q^2)^{\alpha_4}
\,((q-p)^2)^{\alpha_5}\,(p^2)^{\alpha_6}}~,
\label{Jalpha1}$$ which corresponds to the triangle diagram of Fig. \[fig:triangle\].
Following [@Chetyrkin:1981qh; @Grozin:2005yg], we have $$\begin{aligned}
0&=\int\! \frac{d^D k}{(2\pi)^D}\,
\frac{\partial}{\partial k^\mu}
\bigg[\frac{(k-q)^\mu}{(k^2)^{\alpha_1}\,((k-q)^2)^{\alpha_2}\,((k-p)^2)^{\alpha_3}\,(q^2)^{\alpha_4}\,((q-p)^2)^{\alpha_5}\,(p^2)^{\alpha_6}}\bigg]
\phantom{\Bigg|}\\[2mm]
&=\int\! \frac{d^D k}{(2\pi)^D}\,
\frac{D-\alpha_1\,\frac{2k\cdot(k-q)}{k^2}-2\alpha_2
-\alpha_3\,\frac{2(k-p)\cdot(k-q)}{(k-p)^2}}{(k^2)^{\alpha_1}\,((k-q)^2)^{\alpha_2}\,((k-p)^2)^{\alpha_3}\,(q^2)^{\alpha_4}\,((q-p)^2)^{\alpha_5}\,
(p^2)^{\alpha_6}}\phantom{\Bigg|}\\[2mm]
&=\int\! \frac{d^D k}{(2\pi)^D}\,
\frac{D-\alpha_1-\alpha_1\,\frac{(k-q)^2-q^2}{k^2}-2\alpha_2-\alpha_3
-\alpha_3\,\frac{(k-q)^2-(q-p)^2}{(k-p)^2}}{(k^2)^{\alpha_1}\,((k-q)^2)^{\alpha_2}
\,((k-p)^2)^{\alpha_3}\,(q^2)^{\alpha_4}\,((q-p)^2)^{\alpha_5}\,(p^2)^{\alpha_6}}~.\phantom{\Bigg|}
\end{aligned}$$ [From]{} this, it is easy to obtain the so-called triangle identity: $$(D-\alpha_1-2\alpha_2-\alpha_3)\,J\big(\{\alpha_i\}\big)
=\Big[\alpha_1 \, \mathbf{1}^+ \big(\mathbf{2}^- \!-\mathbf{4}^-\big)
+\alpha_3 \, \mathbf{3}^+ \big(\mathbf{2}^- \!- \mathbf{5}^-\big)\Big] J\big(\{\alpha_i\}\big)\,,
\label{triangle}$$ where the notation $\mathbf{n}^\pm J\big(\{\alpha_i\}\big)$ means the integral (\[Jalpha1\]) with $\alpha_n$ replaced by $\alpha_n\pm 1$. For example, we have $$\mathbf{1}^+\mathbf{2}^- J\big(\{\alpha_i\}\big)=\!\int\! \frac{d^D k}{(2\pi)^D}\,
\frac{1}{(k^2)^{\alpha_1+1}\,((k-q)^2)^{\alpha_2-1}
\,((k-p)^2)^{\alpha_3}\,(q^2)^{\alpha_4}\,((q-p)^2)^{\alpha_5}\,(p^2)^{\alpha_6}}~.$$ Repeated applications of the triangle identity allow us to reduce the power of one of the propagators to zero and to express in the end the result in terms of the basic integrals (\[int1\]). A few examples are described in the next subsection.
Scalar integrals
----------------
Let us consider the two-loop integral $$\int\! \frac{d^D k\,d^D q}{(2\pi)^{2D}}\,\frac{1}{k^2\,(k-q)^2\,(k-p)^2\,q^2\,(q-p)^2}
~\equiv~{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}}$$ Here we have adopted the same graphical conventions as in (\[int11\]).
Applying the triangle identity (\[triangle\]), we obtain $${\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}}
~=~\frac{1}{\epsilon}~\Bigg[{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{D1.pdf}}} -{\parbox[c]{.17\textwidth}{ \includegraphics[width = .17\textwidth]{D2.pdf}}}\Bigg]~,
\label{int1app}$$ where
$$\begin{aligned}
& {\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{D1.pdf}}} \equiv\,
\int\! \frac{d^D k\,d^D q}{(2\pi)^{2D}}\,\frac{1}{(k^2)^2
\,(k-q)^2\,(k-p)^2\,(q-p)^2}=\frac{I_{1,1}\, I_{2,1+\epsilon}}{(p^2)^{1+2\epsilon}}
~,\\
& {\parbox[c]{.17\textwidth}{ \includegraphics[width = .17\textwidth]{D2.pdf}}}
\equiv\,\int\! \frac{d^D k\,d^D q}{(2\pi)^{2D}}\,\frac{1}{(k^2)^2\,(k-p)^2\,q^2\,(q-p)^2}
=\frac{I_{2,1}\, I_{1,1}}{(p^2)^{1+2\epsilon}}~.
\end{aligned}$$
The last steps in these equations follow from (\[int1\]). Inserting these expressions into (\[int1app\]) and expanding for $\epsilon\to 0$, we obtain $${\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}}
=\,\frac{I_{1,1}\,\big[I_{2,1+\epsilon}-I_{2,1}\big]}{\epsilon}\,\frac{1}{(p^2)^{1+2\epsilon}}=
\frac{6\zeta(3)}{(4\pi)^4}\,\frac{1}{(p^2)^{1+2\epsilon}}+\cdots~.$$ After Fourier transforming and using (\[Pialphab\]), we get $${\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}} \longrightarrow ~~ \frac{6\zeta(3)}{(4\pi)^4}\, \Pi_{1+2\epsilon} (x) +\cdots
~=~ \frac{6\zeta(3)}{(4\pi)^4}\,(\pi x^2)^{2\epsilon}\,\Delta(x)+\cdots~.
\label{int2x}$$
The same procedure can be applied to express other two-loop integrals in terms of $I_{\alpha,\beta}$ defined in (\[Ialphabeta\]). For example, we have $${\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}} = ~
\frac{I_{1,1} I_{1,2}\,\big[ I_{1+\epsilon,1+\epsilon}-I_{1,1+2\epsilon}
\big]}{\epsilon}\,\frac{1}{(p^2)^{3\epsilon}}
=\frac{2\zeta(3)}{\epsilon\,(4\pi)^6}\,\frac{1}{(p^2)^{3\epsilon}}+\cdots~.
\label{int3}$$ Computing the Fourier transform and using (\[Pialpha1c\]), we find $${\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
\longrightarrow ~~
\frac{2\zeta(3)}{\epsilon\,(4\pi)^6}\,\Pi_{3\epsilon}(x)+\cdots=
\frac{6\zeta(3)}{(4\pi)^4}\,(\pi x^2)^{2\epsilon}\,\Delta(x)^2 +\cdots
~.
\label{int3x}$$
Another scalar integral that will be needed is the one represented by the diagram $$p^2\,{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}}\,~.$$ Using (\[int11\]) and expanding for small $\epsilon$, one can prove that $${\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}} \,
=\,\frac{I_{1,1}}{(p^2)^{\epsilon}}{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}} +\cdots\, =
\frac{6\zeta(3)}{\epsilon\,(4\pi)^6}\,\frac{1}{(p^2)^{1+3\epsilon}}+\cdots
\label{int4}$$ where the ellipses stand for terms that vanish for $\epsilon\to 0$. Comparing with (\[int3\]), we easily conclude that $$p^2\,{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}} \,=
3\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}+\cdots
\label{int4bis}$$ so that, after Fourier transform, we have $$p^2\,{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}} ~
\longrightarrow ~~
\frac{18\zeta(3)}{(4\pi)^4}\,(\pi x^2)^{2\epsilon}\,\Delta(x)^2 +\cdots ~.
\label{int4x}$$
In a similar way one can derive the following relation $${\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot8}}} \,
=\,\frac{I_{1,1}}{(p^2)^{\epsilon}}{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}} +\cdots\, =
\frac{6\zeta(3)}{\epsilon\,(4\pi)^6}\,\frac{1}{(p^2)^{1+3\epsilon}}+\cdots
\label{int50}$$ from which we get $$p^2 {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot8}}} \,=
3\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}+\cdots~.
\label{int51app}$$ Performing the Fourier transform we obtain $$p^2 {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot8}}} ~
\longrightarrow ~~
\frac{18\zeta(3)}{(4\pi)^4}\,(\pi x^2)^{2\epsilon}\,\Delta(x)^2 +\cdots ~.
\label{int51x}$$
The following divergent integrals also appear in the two-loop calculation
$$\begin{aligned}
\frac{1}{p^2}{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Iab.jpg}}}
~~&~~=~ \frac{ I_{1,1}}{(p^2)^{1+\epsilon} } =\frac{\Gamma^2(1-\epsilon)\,
\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}\,\Gamma(2-2\epsilon)}\,\frac{1}{(p^2)^{1+\epsilon} }~,\label{int11app}\\[3mm]
{\parbox[c]{.18\textwidth}{ \includegraphics[width = .18\textwidth]{Ydot4.jpg}}}
&~~=~ \frac{ I_{1,1}^2}{(p^2)^{1+2\epsilon} }=\bigg[\frac{\Gamma^2(1-\epsilon) \,
\Gamma(\epsilon)}{(4\pi)^{2-\epsilon}\,\Gamma(2-2\epsilon)}\bigg]^2
\frac{1}{(p^2)^{1+2\epsilon}}~,\label{int22app}\\[3mm]
\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}
\!\!\!\!\!&~~=~\frac{ I_{1,1}\, I_{1,1+\epsilon}}{(p^2)^{1+2\epsilon}}=
\frac{\Gamma^3(1-\epsilon)\,\Gamma(2\epsilon)}
{(4\pi)^{4-2\epsilon}\,\epsilon (1-2\epsilon)\,\Gamma(2-3\epsilon)}
\,\frac{1}{(p^2)^{1+2\epsilon}}~,
\label{int33app}\\[3mm]
{\parbox[c]{.085\textwidth}{ \includegraphics[width = .085\textwidth]{Ydot3.jpg}}}
\,\,&~~=~ \frac{ I_{1,1}\, I_{1,2+\epsilon}}{(p^2)^{1+2\epsilon}}=-
\frac{\Gamma^3(1-\epsilon)\,\Gamma(2\epsilon)}
{(4\pi)^{4-2\epsilon}\epsilon(1-2\epsilon)(1+\epsilon)\Gamma(1-3\epsilon)}
\,\frac{1}{(p^2)^{1+2\epsilon}}~,
\label{int44app}\\[3mm]
{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}\!&~~=~
\frac{I_{1,1}^2 I_{\epsilon,2+\epsilon}}{(p^2)^{3\epsilon}}=
-\frac{\Gamma^4(1-\epsilon)\,\Gamma(3\epsilon)}
{2(4\pi)^{6-3\epsilon}\,\epsilon^2(1-2\epsilon)\,(1+\epsilon)\,\Gamma(2-4\epsilon)}
\,\frac{1}{(p^2)^{3\epsilon}}~.
\label{int55app}
\end{aligned}$$
After Fourier transforming, we get using (\[Pialpha1\])
$$\begin{aligned}
\frac{1}{p^2}{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Iab.jpg}}}\quad
& ~~\longrightarrow ~~ I_{1,1}\, \Pi_{1+\epsilon} (x) =
\frac{(\pi x^2)^\epsilon\,\Gamma(1-\epsilon)}{(4\pi)^2\,(1-2\epsilon)}\,\Delta(x)~,
\label{int11x}\\[3mm]
{\parbox[c]{.18\textwidth}{ \includegraphics[width = .18\textwidth]{Ydot4.jpg}}}
& ~~\longrightarrow ~~
I_{1,1}^2 \Pi_{1+2\epsilon}(x)=\bigg[\frac{
(\pi x^2)^\epsilon\,\Gamma(1-\epsilon)}{(4\pi)^2\,\epsilon(1-2\epsilon)}\bigg]^2\,\Delta(x)+O(\epsilon)~,\label{int22x}\\[3mm]
\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}
\!\!\!&~~ \longrightarrow ~~
I_{1,1}\, I_{1,1+\epsilon} \, \Pi_{1+2\epsilon}(x) =
\frac{(\pi x^2)^{2\epsilon}\,\Gamma^2(1-\epsilon)}
{2(4\pi)^4\,\epsilon^2(1-2\epsilon)(1-3\epsilon)}\,\Delta(x)~,\label{int33x}\\[3mm]
{\parbox[c]{.085\textwidth}{ \includegraphics[width = .085\textwidth]{Ydot3.jpg}}}
~~&~~ \longrightarrow ~~
I_{1,1}\, I_{1,2+\epsilon} \, \Pi_{1+2\epsilon}(x)=-
\frac{(\pi x^2)^{2\epsilon}\,\Gamma^2(1-\epsilon)}
{2(4\pi)^4\,\epsilon^2(1-2\epsilon)(1+\epsilon)}\,\Delta(x)~,\label{int44x}\\[3mm]
{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}
& ~~\longrightarrow ~~
I_{1,1}^2 I_{\epsilon,2+\epsilon} \, \Pi_{3\epsilon}(x)=-
\frac{(\pi x^2)^{2\epsilon}\,\Gamma^2(1-\epsilon)}
{2(4\pi)^4\,\epsilon^2(1-2\epsilon)(1+\epsilon)}\,\Delta(x)^2~.
\label{int55x}
\end{aligned}$$
Evaluation of the relevant (super)diagrams {#app:Feynman diagrams}
==========================================
In this appendix we explicitly compute the diagrams discussed in Section \[secn:pert\]. We use dimensional regularization and the ${{\mathcal{N}}}=1$ superspace formalism in the Feynman gauge (we refer to [@Billo:2017glv] for more details).
Feynman rules
-------------
We first summarize the momentum-space Feynman rules in the chosen formalism. Let us start from the propagators for the chiral multiplets. We use a continuous line for the superfields $\Phi_I$ ($I=1,2,3$) of the ${{\mathcal{N}}}=4$ gauge multiplet, which carry SU($N$) adjoint indices $a,b,\ldots$, a dashed line for the superfields $Q_A$ ($A=1,\ldots N_f$), which carry SU($N$) fundamental indices $u,v,\ldots$, and a dotted line for the superfields $\widetilde Q_A$, also carrying fundamental indices, which form a ${{\mathcal{N}}}=2$ hypermultiplet together with $Q_A$. We have\
$$\begin{aligned}
\label{chiralprops}
\parbox[c]{.35\textwidth}{\PandocStartInclude{chirprop.pdf_tex}\PandocEndInclude{input}{2423}{55} } & \!\!\!\!=~~
\delta^{ab}\,\delta^{IJ}\, {\mathrm{e}}^{-\theta_1 k \bar \theta_1 -\theta_2 k \bar \theta_2
+2\theta_1 k \bar \theta_2 } \, \frac{1}{k^2}~,\\ \nonumber \\
\parbox[c]{.35\textwidth}{\PandocStartInclude{chirpropQ.pdf_tex}\PandocEndInclude{input}{2426}{56} } & \!\!\!\!=~~
\delta^{uv}\,\delta^{AB}\, {\mathrm{e}}^{-\theta_1 k \bar \theta_1 -\theta_2 k \bar \theta_2
+2\theta_1 k \bar \theta_2 } \, \frac{1}{k^2}~,\\ \nonumber\\
\parbox[c]{.35\textwidth}{\PandocStartInclude{chirpropQt.pdf_tex}\PandocEndInclude{input}{2429}{57} } & \!\!\!\!=~~
\delta^{uv}\,\delta^{AB}\, {\mathrm{e}}^{-\theta_1 k \bar \theta_1 -\theta_2 k \bar \theta_2
+2\theta_1 k \bar \theta_2 } \, \frac{1}{k^2}~. \\ \nonumber
\end{aligned}$$
Note that the arrow indicates both the orientation of the chiral propagator and the flow of the momentum. In (\[chiralprops\]) we have used the notation $$\label{convtheta}
\theta k \bar\theta = \theta^T\sigma^\mu \,\bar{\theta}\, k_\mu = \theta^{\alpha}\,
(\sigma^\mu)_{\alpha\dot{\beta}}\,\bar{\theta}^{\dot{\beta}}\,k_\mu~.$$ Our conventions on spinor indices and Pauli matrices are the same as those explained in Appendix A of [@Billo:2017glv].
The propagator for the ${{\mathcal{N}}}=1$ vector superfield is given by $$\label{vectorprop}
\parbox[c]{.3\textwidth}{\PandocStartInclude{vectorprop.pdf_tex}\PandocEndInclude{input}{2446}{52} }~~~
=~\, - \frac{\delta^{ab}}{2} \, \theta_{12}^2 \, \bar\theta_{12}^2 \, \frac{1}{k^2}\,,$$ where $\theta_{12} \equiv \theta_1 - \theta_2$.
The diagrams we have to compute only contain three-point vertices. These are given by the following rules:
$$\begin{aligned}
\label{vertex1}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex1.pdf_tex}\PandocEndInclude{input}{2456}{55} }
= ~\frac{1}{3!} \,\epsilon_{IJK} \, \sqrt{2} g_0 \, \theta^2\, (T^a)^{bc}~,\\
\notag\\
\label{vertex2}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex2.pdf_tex}\PandocEndInclude{input}{2460}{55} }
= ~- \frac{1}{3!}\, \epsilon_{IJK} \, \sqrt{2}g_0 \, \bar\theta^2\, (T^a)^{bc}~, \\
\notag\\
\label{vertex3}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex3.pdf_tex}\PandocEndInclude{input}{2464}{55} }
= ~- {\mathrm{i}}\,\delta_{AB}\, \delta_{I1}\, \sqrt{2} g_0\, \theta^2\, (T^a)_{uv}~, \\
\notag\\
\label{vertex4}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex4.pdf_tex}\PandocEndInclude{input}{2468}{55} }
=~ {\mathrm{i}}\,\delta_{AB}\, \delta_{I1}\, \sqrt{2} g_0\, \bar\theta^2\, (T^a)_{uv}~,\\
\notag\\
\label{vertex5}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex5.pdf_tex}\PandocEndInclude{input}{2472}{55} }
~= ~ \delta_{IJ}\, 2g_0\, (T^a)^{bc}~,\\
\notag\\
\label{vertex6}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex6.pdf_tex}\PandocEndInclude{input}{2476}{55} }
~= ~\delta_{AB}\, 2g_0\, (T^a)_{uv}~,\\
\notag\\
\label{vertex7}
&\parbox[c]{.24\textwidth}{\PandocStartInclude{vertex7.pdf_tex}\PandocEndInclude{input}{2480}{55} }
~= ~-\delta_{AB}\, 2g_0\, (T^a)_{uv}~.
\end{aligned}$$
Here $(T^a)^{bc}=-{\mathrm{i}}\,f^{abc}$ are the generators in the adjoint representation, and $(T^a)_{uv}$ those in the fundamental representation. The $\theta$ variables appearing in the vertices are those associated to the vertex point.
One-loop diagrams
-----------------
At one loop, we have to compute the diagrams in Fig. \[fig:onelooppphi-iniz\]. Denoting $$\label{v21is}
{\parbox[c]{.25\textwidth}{\includegraphics[width = .25\textwidth]{v21}}} {} = ~
W_1^{ab}(x)$$ and isolating a prefactor containing the combinatorial and color factors, we have $$\label{W1is}
\begin{aligned}
W_1^{ab}(x) & = (\sqrt{2} g_0)^2\,
\big(N_f\,{\mathrm{tr}\,}T^a T^b - {\mathrm{tr}\,}_{\text{adj}} T^a T^b\big)\, W_1(x)\\[2mm]
& = g_0^2\, (N_f - 2N)\, W_1(x)\,\delta^{ab}~,
\end{aligned}$$ where $W_1(x)$ is given by the one-loop Feynman diagram shown in (\[cW1is\]) below. The first term in the color factor arises from the loop diagram of $N_f$ fundamental $Q,\tilde Q$ superfields, while the second term originates from the loop of the adjoint hypermultiplet $H$.
It is convenient to Fourier transform $W_1(x)$ and write $$\label{W1ft}
W_1(x) = \int \!\frac{d^Dp}{(2\pi)^D} \, {{\mathcal{W}}}_1(p)\,{\mathrm{e}}^{{\mathrm{i}}p\cdot x}~,$$ where ${{\mathcal{W}}}_1(p)$ is described by the following diagram in the momentum space: $$\label{cW1is}
{{\mathcal{W}}}_1(p) \,=~ \parbox[c]{.40\textwidth}{\PandocStartInclude{w1.pdf_tex}\PandocEndInclude{input}{2515}{59} }~.$$ Here the numbers label the external points and the interaction vertices. Note that the two external points $1$ and $2$ are connected to a bosonic scalar field, so that the propagators from $1$ to $3$ and from $4$ to $2$ are in fact free scalar propagators with no $\theta$ dependence. Taking into account the Feynman rules given above, we get $$\begin{aligned}
\label{cW1nc}
{{\mathcal{W}}}_1(p) & = \int \!\frac{d^Dk}{(2\pi)^D}\, \frac{1}{p^4\, k^2\,(k-p)^2} \int d^2\bar\theta_3\, d^2\theta_4\, {\mathrm{e}}^{-2 \theta_4 p\, \bar{\theta_3} }\nonumber\\
& = - \frac{1}{p^2} \int \frac{d^Dk}{(2\pi)^D} \frac{1}{k^2\,(k-p)^2} =
-\frac{1}{p^2}{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Iab.jpg}}}
~.\end{aligned}$$ In the second step we used the Grassmann integral identity $$\int d^2\theta_1 \,d^2\bar \theta_2 \,{\mathrm{e}}^{\,\alpha\,\theta_1 k\, \bar\theta_2} = -
\frac{\alpha^2}{4}\, k^2~,$$ and then we exploited the graphical representation introduced in Appendix \[app:loop\_integrals\]. Using (\[int11app\]) and the Fourier transform (\[int11x\]), we finally obtain $$\label{W1fin}
W_1(x) =- \frac{(\pi x^2)^\epsilon \, \Gamma(1-\epsilon)}{(4\pi)^2\,\epsilon(1 - 2\epsilon)}\,\Delta(x) ~,$$ leading to the relation (\[v1loop\]) in the main text: $$\label{W1abfin}
W_1^{ab}(x) = \frac{g_0^2}{8\pi^2} (2N-N_f) \,\frac{(\pi x^2)^\epsilon\,\Gamma(1-\epsilon)}{2\epsilon(1 - 2\epsilon)}\,
\Delta(x)\,\delta^{ab} \equiv v_{2,1} \, \Delta(x)\,\delta^{ab}~.$$
Two-loop diagrams
-----------------
At two loops we have to compute diagrams that correct either a two-point or a four-point vertex. Such diagrams have been displayed in Section \[secn:pert\]. All of them have two external points, corresponding to the positions of the two operators $O_{\vec n}(x)$ and ${\@ifnextchar^{{\wide@bar{O}{0}}}{\wide@bar{O}{1}}}_{\vec m}(0)$, four internal points, corresponding to the interaction vertices, and either seven or eight propagators, for the corrections to the two-point or the four-point vertex respectively.
Some notations {#some-notations .unnumbered}
--------------
It is useful to introduce some notation that allows us to write the various diagrams in a uniform way. In each two-loop diagram labeled by an index $I$, we label by $i=1,2$ the external points and by $i=3,\ldots, 6$ the internal ones. We denote by ${{\mathcal{E}}}_I$ the set of propagators of the diagram, and label each propagator in ${{\mathcal{E}}}_I$ by $s$. Any propagator connects a point $i$ to a point $j$, and in general there can be a number $r(i,j)$ of propagators connecting the same two points. A possible way of expressing the label $s$ of the propagators is thus $$\label{slabelis}
s \to (i,j;r)$$ where $r=1,\ldots, r(i,j)$; in the following we will omit the index $r$ if $r(i,j)=1$. The momentum $k_s$ associated to the propagator will then be denoted as $k_{ij;r}$, with the convention that we take it to flow from $i$ to $j$. This is useful to write the delta-functions of momentum conservation at internal vertices, which take the form $$\label{consk}
\delta_{\text{int}}(k) \equiv \prod_{i=3}^6\delta^D\Big(\sum_{j}\sum_{r=1}^{r(i,j)} k_{ij;r}\Big)~.$$ Similarly, the relation between the internal momenta and the external momentum $p$ is enforced by $$\label{pis}
\delta_{\text{ext}}(p,k) \equiv \delta^D\Big(p -\sum_{j}\sum_{r=1}^{r(1,j)} k_{1j}\Big)~.$$
Just as in the one-loop case, any two-loop diagram will be written as the product of a factor containing the weights in the vertices, the combinatorial and color factors, and of a colorless diagram $W_I(x)$, which we will obtain from its Fourier transform ${{\mathcal{W}}}_I(p)$. The latter has the following structure $$\label{WYD}
{{\mathcal{W}}}_I(p) = \int \!\prod_{s\in{{\mathcal{E}}}_I} \frac{dk_{s}}{(2\pi)^D}~{{\mathcal{Y}}}_I(p,k)~{{\mathcal{Z}}}_I(p,k)~,$$ where $$\begin{aligned}
{{\mathcal{Y}}}_I(p,k) &= \prod_{s\in{{\mathcal{E}}}_I}\frac{1}{k_s^2}\, \delta_{\rm int}(k)\,\delta_{\rm ext}(p,k) ~,\\
{{\mathcal{Z}}}_I(p,k) &=\int \!\prod_{i=3}^6 d^2\theta_i\,d^2\bar\theta_i\,\, \mathfrak D_I (p,k,\theta,\bar\theta)~.
\end{aligned}
\label{Y}$$ The factor ${{\mathcal{Y}}}_I$ contains the contribution of the propagators and the conditions for the momentum conservation at each vertex of the $I$-th diagram, while the factor $\mathfrak D_I (p,k,\theta,\bar\theta)$ contains all $\theta$ and $\bar{\theta}$ terms coming from the vertices and from the superfield propagators. Some of the Grassmann integrations yielding might be obvious, in which case we will indicate only the non-trivial integrals and denote the integrand of ${{\mathcal{Z}}}_I$ as $\widetilde{\mathfrak D}_I$.
Reducible diagrams {#reducible-diagrams .unnumbered}
------------------
The reducible two-loop diagrams are represented in Fig. \[fig:2p1l\] and Fig. \[fig:pphi2l\]. In the diagrams of Fig. \[fig:2p1l\] there are two independent one-loop corrections to a propagator line. Hence the result follows simply from the one-loop computation presented in the previous subsection, and has been given in (\[2p1lres\]) of the main text.
Let us consider then the diagrams of Fig. \[fig:pphi2l\]. The overall factors are simply the square of those of (\[W1is\]), and thus we get $$\label{W2def}
\parbox[c]{.35\textwidth}{ \includegraphics[width = .35\textwidth]{v21square2.png}} =
g_0^4 (N_f - 2N)^2 \, W_2(x) \, \delta^{ab}~.$$ The Fourier transform of $W_2(x)$ is given by the diagram $$\label{W2pis}
{{\mathcal{W}}}_2(p) =~~ \parbox[c]{.50\textwidth}{\PandocStartInclude{w2.pdf_tex}\PandocEndInclude{input}{2613}{58} }~.
\vspace{0.2cm}$$ According to the conventions described earlier, the labeling of the momenta is determined by that of the vertices. In the following, therefore, in drawing momentum space diagrams we will only exhibit the labeling of the vertices. This diagram can be expressed in the form (\[WYD\]), with $$\label{w2D}
{{\mathcal{Z}}}_2(p,k)=
\int \! d^2\bar\theta_3\, d^2\theta_4\, d^2\bar\theta_5\, d^2\theta_6\, \widetilde{\mathfrak D }_2~,$$ where $$\label{w2D1}
\begin{aligned}
\widetilde{\mathfrak D }_2
& = \exp \left( 2 \theta_4 (k_{43;1} +k_{43;2}) \bar\theta_3
+ 2 \theta_6 (k_{65;1} +k_{65;2}) \bar\theta_5
+ 2 \theta_4 k_{45} \bar\theta_5 \right) \\
& = \exp \left(\!-2 \theta_4 p\, \bar\theta_3 - 2 \theta_6 p \,\bar\theta_5
+ 2 \theta_4 p\, \bar\theta_{5} \right)~.
\end{aligned}$$ In the second step we used the momentum conservation $\delta$-functions that are present in the factor ${{\mathcal{Y}}}$ defined in (\[WYD\]). Performing the Grassmann integrals, one finds ${{\mathcal{Z}}}_2(p,k)= p^4$. This factor cancels the two “external” propagators in ${{\mathcal{Y}}}_2$ and the integral over internal momenta can be represented in the graphical notation of Appendix \[app:loop\_integrals\] as follows: $$\label{w2D3}
{{\mathcal{W}}}_2(p) = \parbox[c]{.18\textwidth}{ \includegraphics[width = .18\textwidth]{Ydot4.jpg}}~.$$ Taking the Fourier transform of this expression via (\[int22x\]) and inserting it into (\[W2def\]), we finally find $$\begin{aligned}
\parbox[c]{.35\textwidth}{ \includegraphics[width = .35\textwidth]{v21square2.png}} &=&
g_0^4 (2N-N_f )^2\, \bigg[\frac{(\pi x^2)^\epsilon\,\Gamma(1-\epsilon)}{(4\pi)^2\,\epsilon(1 - 2\epsilon)}
\bigg]^2 \Delta(x) \, \delta^{ab}+ O(\epsilon)\nonumber\\
&=& v_{2,1}^2\, \Delta(x)\, \delta^{ab} + O(\epsilon)~, \label{w2final}\end{aligned}$$ in agreement with the formula (\[2loopv212\]) in the main text.
Irreducible diagrams: the $v_{2,2}$ part {#irreducible-diagrams-the-v_22-part .unnumbered}
----------------------------------------
Let us now consider the irreducible two-loop corrections to the scalar propagator, namely the diagrams represented in Fig. \[fig:pphi2l-irred\]. We start from $$\label{w3x}
\begin{aligned}
W_3^{ab}(x) & \equiv ~\parbox[c]{.35\textwidth}{\PandocStartInclude{w3x.pdf_tex}\PandocEndInclude{input}{2661}{68}} \\
& = \frac{1}{2} (\sqrt{2} g_0)^2 (2 g_0)^2
\big(N_f\,{\mathrm{tr}\,}T^a T^c T^b T^c - {\mathrm{tr}\,}_{\text{adj}}T^a T^c T^b T^c\big)\, W_3(x)\\[2mm]
& = - 2 g_0^4\,\Big(\frac{N_f}{2N}+ N^2\Big)\,W_3(x)\,\delta^{ab}~.
\end{aligned}$$ In momentum space, we have to compute $$\label{w3}
{{\mathcal{W}}}_3(p) ~=~ \parbox[c]{.45\textwidth}{\PandocStartInclude{w3.pdf_tex}\PandocEndInclude{input}{2670}{59}}$$ which has the general form (\[WYD\]). The $\theta$-factors present in the two chiral vertices saturate the integrations over $\theta_3$ and $\bar{\theta}_4$, while those in the gluon propagator set $\theta_{6}$ and $\bar\theta_{6}$ equal to $\theta_{5}$ and $\bar\theta_{5}$ respectively. The remaining Grassmann integrations are $$\label{w3-1}
{{\mathcal{Z}}}_3(p,k)=\int d^2\bar\theta_3\, d^2\theta_4\, d^2\theta_5\, d^2\bar\theta_5\, \,
\widetilde{\mathfrak D}_3$$ with $$\begin{aligned}
\label{w3-2}
\widetilde{\mathfrak D}_3
& = \exp\left(\!-\theta_5 (k_{53} + k_{63} + k_{45} + k_{46})\,\bar{\theta}_5
+ 2 \theta_5 (k_{53} + k_{63}) \bar{\theta_3} + 2 \theta_4 (k_{45} + k_{46})
\,\bar{\theta_5}\right)
\nonumber\\
& = \exp\left(2 \theta_5 p\, \bar{\theta}_5 - 2 \theta_5 p\, \bar\theta_3 - 2 \theta_4 p \,\bar{\theta}_5\right)~, \end{aligned}$$ where in the second step we used momentum conservation. Performing the $\theta$-integrals, we get ${{\mathcal{Z}}}_3(p,k)=p^4$, which cancels the two “external” propagators in ${{\mathcal{Y}}}_3$; the remaining integral over internal momenta can be represented in the graphical notation of Appendix \[app:loop\_integrals\] as follows: $$\label{w3-4}
{{\mathcal{W}}}_3(p) = {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot1}}}
~.$$ Taking the Fourier transform via (\[int2x\]), and inserting the result into (\[w3x\]), we obtain $$\label{w3-5}
W_3^{ab}(x) =
-\Big(\frac{g_0^2}{8\pi^2}\Big)^2\,3\,\zeta(3)\,\Big(\frac{N_f}{2N}+N^2\Big)\,(\pi x^2)^{2\epsilon}\,\Delta(x)
\, \delta^{ab} + \cdots$$
Let us now consider the diagram $$\begin{aligned}
\label{w4x}
W_4^{ab}(x) & ~\equiv~~ \parbox[c]{.29\textwidth}{\PandocStartInclude{w4x.pdf_tex}\PandocEndInclude{input}{2714}{70}}\\
& = 4\times \frac{1}{2} (\sqrt{2} g_0)^2 (2 g_0)^2
\big(N_f \,{\mathrm{tr}\,}T^a T^d T^c - {\mathrm{tr}\,}_{\text{adj}} T^a T^d T^c\big)\,
(T^c)^{db}\, W_4(x)~.
\end{aligned}$$ Using the relations $$\label{3traces}
{\mathrm{tr}\,}T^{a} T^{d} T^c = \frac{1}{4} \,\big(d^{a d c} + {\mathrm{i}}\, f^{a d c}\big)~,~~~
{\mathrm{tr}\,}_{\text{adj}} T^{a} T^{d} T^c= {\mathrm{i}}\,\frac{N}{2}\, f^{a d c}$$ we get $$\begin{aligned}
\label{w4xbis}
W_4^{ab}(x) & = 4 g_0^4 \,\big(N_f \big(d^{adc}+ {\mathrm{i}}\,f^{adc}\big) - 2 N {\mathrm{i}}f^{adc}\big)
\,{\mathrm{i}}\,f^{cdb}\, W_4(x) \\[2mm]
& = - 4 g_0^4\, N(2N-N_f)\, W_4(x)\,\delta^{ab}~,
\end{aligned}$$ where in the second step we took advantage of the identities $$d^{adc}\, f^{cdb} = 0~, ~~~ f^{adc}\, f^{cdb} = - {\mathrm{tr}\,}_{\text{adj}}T^a T^b = - N \,\delta^{ab}~.$$ In momentum space, we have to compute $$\label{w4}
{{\mathcal{W}}}_4(p) \,= \!\!\!\!\!\!\!\!\!\!\!\parbox[c]{.45\textwidth}{\PandocStartInclude{w4.pdf_tex}\PandocEndInclude{input}{2742}{81}}
\vspace{-0.2cm}$$ which has the general form (\[WYD\]). Again, the $\theta$-factors of the vertices saturate the integrations over $\theta_3$ and $\bar{\theta}_4$, while the gluon propagator sets $\theta_{6}$ and $\bar\theta_{6}$ equal to $\theta_{5}$ and $\bar\theta_{5}$. The remaining Grassmann integrations are as in (\[w3-1\]), but now with $$\begin{aligned}
\label{w4-2}
\widetilde{\mathfrak D}_4 & = \exp\left(-\theta_5 (k_{53} + k_{45} + k_{46} + k_{62})\,\bar{\theta}_5
+ 2 \theta_5 k_{53}\, \bar{\theta_3} + 2 \theta_4 (k_{45} + k_{46})\, \bar{\theta_5}
+ 2 \theta_4 k_{43}\,\bar{\theta}_3\right)
\nonumber\\[2mm]
& = \exp\left(- 2 \theta_5 p\, \bar{\theta}_5 - 2 \theta_5 k_{53}\, \bar\theta_{53} - 2 \theta_4 k_{43}\, \bar{\theta}_{53}\right)\end{aligned}$$ where in the second step we used momentum conservation. The Grassmann integrations yield ${{\mathcal{Z}}}_4(p,k)= p^2\, k_{43}^2$. This factor cancels one external and one internal propagator in ${{\mathcal{Y}}}_4$ and we remain with $$\label{w4-4}
{{\mathcal{W}}}_4(p) =
\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}
\!\!\!\!\!\!.$$ Thus, the momentum space contribution corresponding to (\[w4xbis\]) is $$\label{w4-5}
{{\mathcal{W}}}_4^{ab}(p) = - 4 g_0^4\, N(2N-N_f)\,\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}} \!\!\!\!\!\!\delta^{ab} ~.$$
We now consider the diagram $$\begin{aligned}
\label{w5x}
W_5^{ab}(x) & \equiv ~~\parbox[c]{.25\textwidth}{\PandocStartInclude{w5x.pdf_tex}\PandocEndInclude{input}{2777}{69}}\\
& = \, - \frac{1}{2} (\sqrt{2} g_0)^2 (2 g_0)^2
\big(T^a T^b\big)^{cd} \,\big(N_f \,{\mathrm{tr}\,}T^c T^d - {\mathrm{tr}\,}_{\text{adj}} T^c T^d\big) \,
W_5(x)\\[2mm]
& =\, 2 g_0^4 \,N(2N-N_f)\,W_5(x)\,\delta^{ab} ~.
\end{aligned}$$ In momentum space, we have to compute $$\label{w5}
\vspace{0.1cm}
{{\mathcal{W}}}_5(p) = \parbox[c]{.45\textwidth}{\PandocStartInclude{w5.pdf_tex}\PandocEndInclude{input}{2788}{57}}
\vspace{-0.4cm}$$ which again has the form (\[WYD\]) with $$\begin{aligned}
\label{w5-2}
\widetilde{\mathfrak D}_5 & = \exp\left(
-2 \theta_5 (p + k_{53})\,\bar{\theta}_5 + 2 \theta_5 k_{53} \,\bar{\theta}_3
- 2 \theta_4 k_{53}\, \bar{\theta}_3+ 2 \theta_4 k_{53}\, \bar{\theta}_5 \right)\\[2mm]
& = \exp\left(- 2 \theta_5 p\, \bar{\theta}_5 + 2 \theta_5 k_{53} \,\bar{\theta}_{35} - 2 \theta_4 k_{53} \,\bar{\theta}_{35}\right)~.
\end{aligned}$$ The Grassmann integration leads to ${{\mathcal{Z}}}_5(p,k)=p^2\, k_{53}^2$, which cancels one external and one internal propagator in ${{\mathcal{Y}}}_5$. We then remain with $$\label{w5-4}
{{\mathcal{W}}}_5(p) =
\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}\!\!
\!\!\!\!.$$ Using this in (\[w5x\]), we find that the total diagram in momentum space is given by $$\label{w5-5}
{{\mathcal{W}}}_5^{ab}(p) = 2 g_0^4\, N(2N-N_f)\,\frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}} \!\!\!\!\!\!\delta^{ab} ~.$$
Finally, we consider the diagram $$\begin{aligned}
\label{w6x}
W_6^{ab}(x) & \equiv ~~\parbox[c]{.29\textwidth}{\PandocStartInclude{w6x.pdf_tex}\PandocEndInclude{input}{2820}{69}}\\
& = 2\times \Big(\!-\frac{1}{2}\Big)^2 (2 g_0)^4
\big(T^a T^b\big)^{cd}\,\big(N_f \,{\mathrm{tr}\,}T^c T^d - {\mathrm{tr}\,}_{\text{adj}}T^c T^d\big) \,W_6(x)
\\[2mm]
& = -4 g_0^2\, N(2N-N_f)\,W_6(x)\,\delta^{ab}~.
\end{aligned}$$ In momentum space, we have to compute $$\label{w6}
{{\mathcal{W}}}_6(p) = \parbox[c]{.35\textwidth}{\PandocStartInclude{w6.pdf_tex}\PandocEndInclude{input}{2830}{57}}$$ which, once again, is of the form (\[WYD\]) with $$\label{w6-2}
\begin{aligned}
\widetilde{\mathfrak D}_6 & = \exp\left(
-\theta_3(p + k_{56} + k_{65}+ k_{34})\,\bar{\theta}_3
-\theta_4(p + k_{56} + k_{65}+ k_{34})\,\bar{\theta}_4 \right.\\[2mm]
& \phantom{+ \exp (}~ \left. + 2 \theta_3(k_{34} + k_{56})\,\bar{\theta}_4 + 2 \theta_4 k_{65}
\,\bar{\theta}_3\right)\\[2mm]
& = \exp\left(- 2 \theta_3(k_{34}+ k_{56})\,\bar{\theta}_3 + 2 \theta_{34} (k_{34}+ k_{56})
\,\bar{\theta}_4 + 2 \theta_4 k_{65}\, \bar{\theta}_3
\right) ~,
\end{aligned}$$ having used momentum conservation in the second step. The Grassmann integral we have to compute in this case is $$\label{w6-2bis}
{{\mathcal{Z}}}_6(p,k)=\int d^2\theta_3 \,d^2\bar{\theta}_2\, d^2\theta_4\, d^2\bar{\theta}_4\,\,
\widetilde{\mathfrak D}_6~.$$ Integrating over $\bar{\theta}_4$ produces a factor of $ (k_{34}+ k_{56})^2 \, \theta_{34}^2$ which sets $\theta_4 = \theta_3$, so that in the end we remain with $$\label{w6-2quater}
{{\mathcal{Z}}}_6(p,k)= (k_{34}+ k_{56})^2 \int d^2\theta_3\, d^2\bar{\theta}_3
\,\exp\left(-2 \theta_3 p \,\bar{\theta}_3\right) = (k_{34}+ k_{56})^2\, p^2
\sim 2 \left(k_{34}\cdot k_{56}\right)\, p^2~.$$ The last step follows from the fact that when we integrate this expression over momenta, both the $k_{34}^2$ and the $k_{56}^2$ terms, canceling the corresponding propagator, give rise to tadpole-like integrals, which vanish in dimensional regularization. The symmetry of the diagram under $k_{56} \leftrightarrow - k_{65}$ allows us to rewrite the above result as $$\label{w6-3}
{{\mathcal{Z}}}_6(p,k)=\big(k_{34}\cdot(k_{56}- k_{65})\big) \,p^2 = (k_{34}\cdot k_{35})\, p^2
= \frac 12 \,(p^2 - k_{34}^2 - k_{35}^2)\, p^2
\sim \frac 12 \,(p^2 - k_{35}^2) \, p^2~,$$ where again we discarded the tadpole originating from $k_{34}^2$. The first term cancels the two external propagators of ${{\mathcal{Y}}}_6$, while the second term cancels one external and one internal propagator. Using the graphical notation of Appendix \[app:loop\_integrals\], we can write $${{\mathcal{W}}}_6(p) = \frac{1}{2} \,\bigg[ \parbox[c]{.085\textwidth}{ \includegraphics[width = .085\textwidth]{Ydot3.jpg}}
- \frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}
\!\!\!\!\!\!\!\bigg]~,
\label{w6-4}$$ so that, from (\[w6x\]) we see that the total diagram in momentum space is $$\label{w6-5}
{{\mathcal{W}}}_6^{ab}(p) = -2 g_0^4 \,N(2N-N_f) \,
\bigg[ \parbox[c]{.085\textwidth}{ \includegraphics[width = .085\textwidth]{Ydot3.jpg}}
- \frac{1}{p^2}{\parbox[c]{.12\textwidth}{ \includegraphics[width = .12\textwidth]{Ydot2a.png}}}
\!\!\!\!\!\!\!\bigg]\,\delta^{ab}~.$$
Summing the three diagrams (\[w4-5\]), (\[w5-5\]) and (\[w6-5\]), a simplification takes place and we are left with $$\label{w456}
\sum_{I=4}^6 {{\mathcal{W}}}_I^{ab}(p) = - 2 g_0^4 \,N(2N-N_f) \,
\parbox[c]{.085\textwidth}{ \includegraphics[width = .085\textwidth]{Ydot3.jpg}}\,\delta^{ab} ~.$$ Taking the Fourier transform via (\[int44app\]), we then have $$\label{w456x}
\sum_{I=4}^6 W_I^{ab}(x) =\Big(\frac{g_0^2}{8\pi^2}\Big)^2\,N(2N-N_f)
\frac{\Gamma^2(1-\epsilon)}{4\epsilon^2(1-2\epsilon)(1+\epsilon)}\,(\pi x^2)^{2\epsilon}
\,\Delta(x)\, \delta^{ab}~.$$ If we include also the $W_3^{ab}(x)$ diagram given in (\[w3-5\]), we obtain the two-loop irreducible corrections to the propagator: $$\label{w3456}
\sum_{I=3}^6 W_I^{ab}(x)\equiv \,v_{2,2}\, \Delta(x)\, \delta^{ab}$$ with $$\label{v22isapp}
v_{2,2}=-\Big(\frac{g_0^2}{8\pi^2}\Big)^2\left[3\,\zeta(3)
\Big(\frac{N_f}{2N}+N^2\Big)-N(2N-N_f)\frac{\Gamma^2(1-\epsilon)}{4\epsilon^2(1-2\epsilon)
(1+\epsilon)}\right](\pi x^2)^{2\epsilon}~,$$ as reported in the formula (\[v2\]) of the main text.
Irreducible diagrams: the $v_{4,2}$ part {#irreducible-diagrams-the-v_42-part .unnumbered}
----------------------------------------
We now evaluate the irreducible two-loop diagrams that give rise to the contribution $(c)$ in Fig. \[fig:diagrams\]. We start from the diagrams represented in Fig. \[fig:vh42diag\]. The first of these is $$\begin{aligned}
W_7^{a_1a_2b_1b_2}(x) & \equiv ~~\parbox[c]{.25\textwidth}{\PandocStartInclude{w7x.pdf_tex}\PandocEndInclude{input}{2926}{80}}\label{w7x}\\
& = 2\!\times\! \Big(\!-\frac{1}{2}\Big) (\sqrt{2} g_0)^2 (2 g_0)^2
\big(N_f \, {\mathrm{tr}\,}T^c T^{a_2} T^{b_2} -
{\mathrm{tr}\,}_{\text{adj}}T^c T^{a_2} T^{b_2} \big)\,(T^c)^{a_1 b_1}\, W_7(x)~.\nonumber\end{aligned}$$ Using the relations (\[3traces\]), we find $$\label{w7xbis}
W_7^{a_1a_2b_1b_2}(x) = -2 g_0^4\,
\big({\mathrm{i}}\,f^{c a_2 b_2}
(N_f - 2 N)+ N_f \,d^{c a_2 b_2} \big)\big(-{\mathrm{i}}\, f^{c a_1 b_1}\big)\, W_7(x)~.$$ Defining the tensor (see (\[C4Ais\])) $$\label{defC4A}
C_{4}^{(A)\,a_1 a_2 b_1 b_2}
= -\frac{1}{N}\,f^{c\,a_1 b_1}\, f^{c\,a_2 b_2} ~,$$ we can write $$\label{w7xtris}
W_7^{a_1a_2b_1b_2}(x) =- 2 g_0^4 \big( N(2N-N_f)\,C_4^{(A)a_1 a_2 b_1 b_2} +
{\mathrm{i}}\, N_f\, d^{a_2 b_2 c}\, f^{a_1 b_1 c} \big)\, W_7(x)~.$$ Note that the term proportional to $d^{a_2 b_2 c} f^{a_1 b_1 c}$ is actually anti-symmetric in $(a_1,a_2)$ and in $(b_1, b_2)$, and thus it vanishes when we insert this sub-diagram in a chiral/anti-chiral correlator. Therefore in the following we discard this term.
In momentum space, we have to compute $$\label{w7}
{{\mathcal{W}}}_7(p) = ~~\parbox[c]{.40\textwidth}{\PandocStartInclude{w7.pdf_tex}\PandocEndInclude{input}{2958}{59}}$$ which has the canonical form (\[WYD\]). In this case we have $$\label{w7-2}
\begin{aligned}
\widetilde{\mathfrak D}_7 & = \exp\left(
-\theta_6(k_{63} + k_{46} + k_{15}+ k_{52})\,\bar{\theta}_6
+ 2 \theta_6 k_{63}\, \bar{\theta}_3 + 2 \theta_4 k_{46}\, \bar{\theta}_6
+ 2 \theta_4 k_{43}\,\bar{\theta}_3 \right)\\[2mm]
& = \exp\left(- 2 \theta_6(k_{46}+ k_{15})\,\bar{\theta}_6
+ 2 \theta_6 k_{63}\, \bar{\theta}_3 + 2 \theta_4 k_{46} \,
\bar{\theta}_6 + 2 \theta_4 k_{43}\,\bar{\theta}_3 \right) ~,
\end{aligned}$$ where in the second step we used momentum conservation. To perform the Grassmann integral $$\label{w7-3}
{{\mathcal{Z}}}_7(p,k)=
\int d^2\bar{\theta}_3\, d^2\theta_4 \,d^2\theta_6\, d^2\bar{\theta}_6 \,\,
\widetilde{\mathfrak D}_7 ~,$$ we use the formula $$\label{4trace}
\int d^2\theta_i \,d^2\bar{\theta}_j \,d^2\theta_k \,d^2\bar{\theta}_l \,
\,{\mathrm{e}}^{2\theta_i A \bar{\theta}_j + 2 \theta_i B \bar{\theta}_l + 2 \theta_k C \bar{\theta}_l
+ 2 \theta_k D \bar{\theta}_j}
= A^2 C^2 + B^2 D^2 - {\mathrm{tr}\,}\big(A D C B\big)$$ where $$\begin{aligned}
{\mathrm{tr}\,}\big(A D C B\big) &= {\mathrm{tr}\,}\big(\sigma^\mu\,\bar{\sigma}^\nu\,\sigma^\lambda\,\bar{\sigma}^\rho\big)\,A_\mu\,D_\nu\,C_\lambda\,B_\rho\\
&=2\,A\!\cdot\! D\,\,C\!\cdot\! B-2A\!\cdot\! C\,\,D\!\cdot\! B+2\,A\!\cdot\! B\,\,D\!\cdot\! C-\varepsilon^{\mu\nu\lambda\rho}\,
A_\mu\,D_\nu\,C_\lambda\,B_\rho~.
\end{aligned}
\label{trace4sigma}$$ In this way we obtain $$\begin{aligned}
{{\mathcal{Z}}}_7(p,k)
& =
(k_{46} + k_{15})^2\, k_{43}^2 + k_{63}^2\, k_{46}^2 +
{\mathrm{tr}\,}\big((k_{46} + k_{15})\, k_{46}\, k_{43}\, k_{63}\big)\nonumber\\[2mm]
& = (k_{46} + k_{15})^2 \,k_{43}^2 + 2 \big((k_{46} + k_{15})\cdot k_{46}\big)
\,(k_{43}\cdot k_{63})\label{w7-3bis}
\\[2mm]
& ~~~- 2 \big((k_{46} + k_{15})\cdot k_{43}\big)\,(k_{46}\cdot k_{63})
+ 2 \big((k_{46} + k_{15})\cdot k_{63}\big)\,(k_{43}\cdot k_{46})~,
\nonumber\end{aligned}$$ where we have discarded a term proportional to the anti-symmetric $\varepsilon$-tensor coming from the trace of four Pauli matrices given in (\[trace4sigma\]), that will not contribute to the correlator for symmetry reasons. Using momentum conservation, after some algebra the polynomial ${{\mathcal{Z}}}_7(p,k)$ can be rewritten as $$\begin{aligned}
\label{w7-4}
&~(p\cdot k_{43})\,(k_{56}^2 - k_{13}^2 - k_{42}^2)
-\frac{p^2}{2} \,(k_{13}^2 + k_{42}^2)
\nonumber\\
&+ \frac 12 \,(k_{15}^2 k_{43}^2 - k_{13}^2 k_{43}^2)
+ \frac 12 \,(k_{52}^2 k_{43}^2 - k_{42}^2 k_{43}^2)
+ \frac 12 \,(k_{13}^2 k_{46}^2 - k_{15}^2 k_{46}^2)+
\frac 12 \,(k_{42}^2 k_{63}^2 - k_{52}^2 k_{63}^2)\nonumber\\
& +
\frac{p^2}{2}\,(k_{63}^2 + k_{46}^2)+ \frac 12 k_{52}^2 k_{13}^2 + \frac 12 k_{15}^2 k_{42}^2~.\end{aligned}$$ This polynomial has to be multiplied by the factor ${{\mathcal{Y}}}_7$ containing all propagators and then integrated over the momenta. It is not difficult to show that the terms in the first line, proportional to $p\cdot k_{43}$ and to $p^2 k_{13}^2$ or $p^2 k_{42}^2$, yield contributions that vanish for $\epsilon \to 0$ after Fourier transform. The terms in the second line in each brackets cancel each other owing to the symmetries of the diagram. The remaining terms in the third line of (\[w7-4\]) give a non-vanishing contributions. Thus, we can effectively use $${{\mathcal{Z}}}_7(p,k)=\frac{p^2}{2}\,(k_{63}^2 + k_{46}^2)+ \frac 12 k_{52}^2 k_{13}^2 + \frac 12 k_{15}^2 k_{42}^2~.$$ All these terms lead to cancellations of some of the propagators of ${{\mathcal{Y}}}_7$ and the result can be written in the graphical notation of Appendix \[app:loop\_integrals\]. Altogether we find, $$\label{w7-5}
{{\mathcal{W}}}_7(p) =\,
p^2\,{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}} \,+
\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
=\,4\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}+\cdots$$ where the last step follows from (\[int4bis\]). Using this result, we find that the momentum space expression corresponding to $W_7^{a_1a_2b_1b_2}(x)$ given in (\[w7xtris\]) is $${{\mathcal{W}}}_7^{a_1a_2b_1b_2}(p) =
-8 g_0^4 \, N(2N-N_f) \,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
\,C_4^{(A)a_1 a_2 b_1 b_2}+\cdots
\label{w7-6}$$ where the dots stand for terms that do not contribute in the correlators due to their colour factors or that vanish for $\epsilon\to 0$.
The second diagram we have to consider is $$\label{w8x}
\begin{aligned}
W_8^{a_1a_2b_1b_2}(x) & \equiv ~~\parbox[c]{.25\textwidth}{\PandocStartInclude{w8x.pdf_tex}\PandocEndInclude{input}{3060}{80}}\\
& = 2\times \Big(\!-\frac{1}{2}\Big) (2 g_0)^2\, g_0^2\, (N_f - 2 N) \,(T^c)^{a_1 b_1}
(T^c)^{a_2 b_2}\, W_8(x)\\[2mm]
& = 4 g_0^2\, N(2N-N_f)\, C_4^{(A)a_1 a_2 b_1 b_2}\, W_8(x)~.
\end{aligned}$$ In momentum space, we must compute $$\label{w8}
{{\mathcal{W}}}_8(p) =~~ \parbox[c]{.40\textwidth}{\PandocStartInclude{w8.pdf_tex}\PandocEndInclude{input}{3069}{59}}$$ which again is of the form (\[WYD\]). In this case we have $$\begin{aligned}
\label{w8-2}
\widetilde{\mathfrak D} & = \exp\left(
-\theta_6\left(k_{15} + k_{16} + k_{52} + k_{63}\right)\bar{\theta}_6
+ 2 \theta_6\, k_{63}\, \bar{\theta}_3 + 2 \theta_4\, k_{63}\, \bar{\theta}_3\right)\\[2mm]
& = \exp\left(- 2 \theta_6 \,p\, \bar{\theta}_6 + 2 \theta_6\, k_{63}\, \bar{\theta}_3 + 2 \theta_4 \,k_{63}\, \bar{\theta}_3\right)~,
\end{aligned}$$ while the Grassmann integration yields ${{\mathcal{Z}}}_8(p,k)= p^2 \,k_{63}^2$. Inserting this into the momentum integrals, we remain with $$\label{w8-5}
{{\mathcal{W}}}_8(p) = p^2\,{\parbox[c]{.08\textwidth}{ \includegraphics[width = .08\textwidth]{Ydot9}}} =
\,3\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}+\cdots$$ where the last step follows from (\[int4bis\]). Using this result, we find that the momentum space expression corresponding to $W_8^{a_1a_2b_1b_2}(x)$ given in (\[w8x\]) is $${{\mathcal{W}}}_8^{a_1a_2b_1b_2}(p) =
12 g_0^4 \, N(2N-N_f) \,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
\,C_4^{(A)a_1 a_2 b_1 b_2}+\cdots
\label{w8-6}$$ where the dots stand for terms that vanish for $\epsilon\to 0$.
The third diagram we need to consider is $$\begin{aligned}
\label{w9x}
W_9^{a_1a_2b_1b_2}(x) & \equiv ~~\parbox[c]{.29\textwidth}{\PandocStartInclude{w9x.pdf_tex}\PandocEndInclude{input}{3103}{80}}\nonumber\\
& = 2\!\times\!\Big(\!-\frac{1}{2}\Big)^2 (2 g_0)^2 \,g_0^2\,(N_f - 2 N) (T^c)_{a_1 b_1} (T^c)_{a_2 b_2}\, W_9(x)\nonumber\\[2mm]
& = -2 g_0^4\, N(2N-N_f)\, C_4^{(A)a_1 a_2 b_1 b_2} \, W_9(x)~.
\end{aligned}$$ In momentum space, we have to compute $$\label{w9}
{{\mathcal{W}}}_9(p) = ~~\parbox[c]{.40\textwidth}{\PandocStartInclude{w9.pdf_tex}\PandocEndInclude{input}{3111}{59}}$$ which has again the form (\[WYD\]) with $$\label{w9-2}
\begin{aligned}
\widetilde{\mathfrak D}_9 & = \exp \bigl(
-\theta_3\left(k_{13} + k_{32} + k_{56} + k_{65}\right)\bar{\theta}_3
-\theta_4\left(k_{14} + k_{42} + k_{56} + k_{65}\right)\bar{\theta}_4
\\[1mm]
& \phantom{= \exp( }\,
+ 2 \theta_3\, k_{56} \,\bar{\theta}_4 + 2 \theta_4 \,k_{65}\, \bar{\theta_3}
\bigr)
\\[1mm]
& = \exp\left(- 2 \theta_3\left(k_{13} + k_{65}\right)\bar{\theta}_3
- 2 \theta_4\left(k_{14} + k_{56}\right)\bar{\theta}_4
+ 2 \theta_3\, k_{56}\, \bar{\theta}_4 + 2 \theta_4\, k_{65}\, \bar{\theta_3}
\right)~.
\end{aligned}$$ The Grassmann integration is carried out using (\[4trace\]) and gives $$\label{w9-3}
{{\mathcal{Z}}}_9(p,k) = (k_{13} + k_{65})^2 (k_{14} + k_{56})^2 + k_{56}^2 k_{65}^2
- {\mathrm{tr}\,}\left((k_{13} + k_{65}) k_{65} (k_{14} + k_{56}) k_{56}\right)~.$$ We expand the trace according to (\[trace4sigma\]), and take into account the part proportional to the $\varepsilon$-tensor does not contribute. The terms proportional to $k_{56}^2$ and/or to $k_{65}^2$, as well as the term $k_{13}^2 k_{14}^2$, are tadpole-like and vanish in dimensional regularization, and thus we remain with $$\begin{aligned}
\label{w9-4}
{{\mathcal{Z}}}_9(p,k)=&\,
2 k_{14}^2 (k_{65}\cdot k_{13}) + 2 k_{13}^2 (k_{56}\cdot k_{14})
- 2 (k_{13}\cdot k_{65})(k_{14}\cdot k_{56})
\\[1mm]
&+ 2 (k_{13}\cdot k_{56})(k_{14}\cdot k_{65})+ 2 (k_{13}\cdot k_{14})(k_{56}\cdot k_{65})~.
\end{aligned}$$ The diagram is symmetric under the exchange $k_{56} \leftrightarrow k_{65}$. Symmetrizing ${{\mathcal{Z}}}_9(p,k)$ with respect to this exchange, exploiting momentum conservation and discarding tadpole-like terms proportional to $k_{56}^2$, $k_{65}^2$ or $k_{13}^2 k_{14}^2$ we can rewrite (\[w9-4\]) as $$\begin{aligned}
\label{w9-5}
{{\mathcal{Z}}}_9(p,k)&=
- k_{14}^2 (k_{35}\cdot k_{13}) + k_{13}^2 (k_{64}\cdot k_{14}) - k_{64}^2 (k_{13}\cdot k_{14})
\\
&= \frac 12 k_{14}^2\, k_{32}^2 + \frac 12 k_{13}^2 \,k_{42}^2 - \frac 12 p^2\, k_{64}^2~.
\end{aligned}$$ The first two terms in the last expression give the same result and cancel two internal propagators, while the last term cancels one external and one internal propagator of ${{\mathcal{Y}}}_9$. In the end, adopting the graphical notation of Appendix \[app:loop\_integrals\], we have $$\begin{aligned}
\label{w9-6}
{{\mathcal{W}}}_9(p) = {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}
- \,\frac 12\, p^2 {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot8}}} =
{\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}
- \,\frac 32\,{\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
+\cdots\end{aligned}$$ where the second step follows from (\[int51app\]). Inserting this result in (\[w9x\]), we see that the momentum space expression corresponding to $W_9^{a_1a_2b_1b_2}(x)$ is $$\label{w9-9}
{{\mathcal{W}}}_9^{a_1a_2b_1b_2}(p) = g_0^4\, N(2N-N_f)
\bigg[-2 {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}
+3\, {\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
\bigg] C_4^{(A)a_1 a_2 b_1 b_2}+\cdots~.$$ Summing the three diagrams (\[w7-6\]), (\[w8-6\]) and (\[w9-9\]), we find $$\label{w789p}
\sum_{I=7}^9 {{\mathcal{W}}}_I^{a_1a_2b_1b_2}(p) = g_0^4\, N(2N-N_f)
\bigg[7\, {\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}
-2 {\parbox[c]{.10\textwidth}{ \includegraphics[width = .10\textwidth]{Ydot7.jpg}}}
\bigg] C_4^{(A)a_1 a_2 b_1 b_2}+\cdots~.$$ Performing the Fourier transform using (\[int2x\]) and (\[int51x\]), we finally obtain $$\sum_{I=7}^9 W_I^{a_1a_2b_1b_2}(x) =
v_{4,2}^{(A)} \,\Delta(x)^2 \, C_{4}^{(A)\,a_1 a_2 b_1 b_2}+ \cdots
\label{w789x}$$ with $$v_{4,2}^{(A)} =\left(\frac{g_0^2}{8\pi^2}\right)^2 N(2N-N_f)
\left[\frac{21}{2} \zeta(3) + \frac{\Gamma^2(1-\epsilon)}{4\epsilon^2(1-2\epsilon)(1 + \epsilon)}\right]
(\pi x^2)^{2\epsilon}+\ldots
\label{v42Aapp}$$ in agreement with the formula (\[v42Ares\]) of the main text.
The last two-loop diagram we have to compute is $$\begin{aligned}
\label{w10x}
W_{10}^{a_1a_2b_1b_2}(x) & \equiv~~ \parbox[c]{.29\textwidth}{\PandocStartInclude{w10x.pdf_tex}\PandocEndInclude{input}{3208}{84}}\\[1mm]
& = \frac{2}{2!^2} (\sqrt{2}g_0)^4\,
\big(N_f\,{\mathrm{tr}\,}T^{a_1} T^{b_1} T^{a_2} T^{b_2}
- {\mathrm{tr}\,}_{\text{adj}}T^{a_1} T^{b_1} T^{a_2} T^{b_2}\big)\, W_{10}(x)~.
\nonumber\end{aligned}$$ This diagram was already computed in [@Billo:2017glv] in configuration space. For completeness we report here its evaluation in momentum space. Using the relation $$\begin{aligned}
\label{tr4adj}
{\mathrm{tr}\,}_{\text{adj}}T^{a_1} T^{b_1} T^{a_2} T^{b_2} & =
2 N \,{\mathrm{tr}\,}T^{a_1} T^{b_1} T^{a_2} T^{b_2} + \frac 12
\big(\delta^{a_1b_1}\delta^{a_2 b_2}+\delta^{a_1a_2}\delta^{b_1 b_2}
+\delta^{a_1b_2}\delta^{a_2 b_1}\big)\nonumber\\
& + \frac{{\mathrm{i}}\,N}{4}\big(f^{a_1b_1c}\,d^{a_2b_2c}+f^{a_2b_2c}\,d^{a_1b_1c}\big)~,\end{aligned}$$ and introducing the tensor (see (\[C4\])) $${C}_{4}^{(B)\,a_1a_2b_1b_2}= -(2N-N_f)\, {\mathrm{tr}\,}T^{a_1}T^{b_1}T^{a_2}T^{b_2}\,-\,\frac{1}{2}\big(
\delta^{a_1b_1}\delta^{a_2 b_2}+\delta^{a_1a_2}\delta^{b_1 b_2}
+\delta^{a_1b_2}\delta^{a_2 b_1}\big)~,
\label{C4B}$$ we can rewrite (\[w10x\]) as $$\label{w10xbis}
W_{10}^{a_1a_2b_1b_2}(x) = 2 g_0^4 \,\Big[
{C}_{4}^{(B)\,a_1a_2b_1b_2} - \frac{{\mathrm{i}}\,N}{4}\,\big(f^{a_1b_1c}\,d^{a_2b_2c}+f^{a_2b_2c}\,d^{a_1b_1c}\big)\Big]\, W_{10}(x)~.$$ As noted after (\[w7xtris\]), the last two terms in the square brackets are anti-symmetric in $(a_1,a_2)$ and $(b_1,b_2)$. Therefore they vanish when inserted in a chiral/anti-chiral two-point function and can be discarded. The momentum space diagram corresponding to $W_{10}(x)$ is $$\label{w10}
{{\mathcal{W}}}_{10}(p) =~~\parbox[c]{.40\textwidth}{\PandocStartInclude{w10.pdf_tex}\PandocEndInclude{input}{3243}{62}}$$ which has the form (\[WYD\]) with $$\label{w10-2}
\widetilde{\mathfrak D}_{10} =
\exp \bigl(
2 \theta_4\, k_{43}\, \bar{\theta}_3 + 2 \theta_4 \,k_{45}\, \bar{\theta}_5 + 2 \theta_6\, k_{65}
\,\bar{\theta}_5 + 2 \theta_6 \,k_{63}\, \bar{\theta}_3
\bigr)~.$$ The corresponding Grassmann integration is easily carried out using (\[4trace\]): $$\begin{aligned}
\label{w10-3}
\mathcal{Z}_{10}(p,k) & = \int d^2\bar{\theta}_3\,d^2\theta_4\, d^2\bar{\theta}_5 \,
d^2\theta_6\, \,\widetilde{\mathfrak D}_{10}
= k_{43}^2\, k_{65}^2 + k_{45}^2\, k_{63}^2
\\
\notag
&- 2 \left(k_{43}\cdot k_{63}\right) \left(k_{45}\cdot k_{65}\right)
+ 2 \left(k_{43}\cdot k_{65}\right) \left(k_{45}\cdot k_{63}\right)
- 2 \left(k_{43}\cdot k_{45}\right) \left(k_{63}\cdot k_{65}\right)~.\end{aligned}$$ Here we have neglected a term proportional to the to the anti-symmetric $\varepsilon$-tensor coming from the trace of four Pauli matrices given in (\[trace4sigma\]) which does not contribute for symmetry reasons. Recalling that $p = k_{13}+ k_{15}= k_{42}+ k_{62}$, we can exploit momentum conservation and discard terms involving $p^2$ and $p\cdot k_{43}$, which give contributions that vanish for $\epsilon\to 0$ after Fourier transform. After some algebra we are left with $$\label{w10-4}
{{\mathcal{Z}}}_{10}(p,k) = k_{13}^2\, k_{62}^2~.$$ When inserted in the momentum integral ${{\mathcal{Y}}}_{10}$, this cancels two propagators so that $$\label{w10-5}
{{\mathcal{W}}}_{10}(p) = {\parbox[c]{.09\textwidth}{ \includegraphics[width = .09\textwidth]{Ydot5}}}~.$$ Going back to configuration space using (\[int3x\]) and inserting the result in (\[w10xbis\]), up to terms that do not contribute for their color structure or that vanish in the limit $\epsilon\to 0$, we get $$W_{10}^{a_1a_2b_1b_2}(x) ={v}_{4,2}^{(B)}\, \Delta(x)^2\,{C}_{4}^{(B)\,a_1a_2b_1b_2}
+\cdots
\label{w10-6}$$ with $$\label{v42Bapp}
v_{4,2}^{(B)}=\Big(\frac{g_0^2}{8\pi^2}\Big)^2 \,3\,\zeta(3) (\pi x^2)^{2\epsilon}+\ldots$$ in agreement with formula (\[v42is\]) of the main text and with the findings of [@Billo:2017glv].
This completes the calculation of the two-loop diagrams contributing to the chiral/anti-chiral correlators.
Feynman integral on the sphere {#app:feynsphere}
==============================
In this appendix, we evaluate the integral $I(x_1,x_2)$ that appears in the expression for the one-loop correlation function on the sphere, given in (\[WS1\]) and (\[WS2\]).
For convenience we first rewrite here the definition (\[Iis\]) of the integral $I(x_1,x_2)$, namely $$I(x_1,x_2)=\!\int \!
d^D x_3\, d^2\bar\theta_3 \,d^D x_4\, d^2\theta_4 \,\Delta(x_{13})
\big({\mathrm{e}}^{-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3}\Delta(x_{43})\big)^2
\Delta(x_{42})\,\big[\kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}~,
\label{Iisapp}$$ where $\Delta(x)$ is the scalar propagator in $D-$dimension and $\kappa(x)$ is the scaling factor (\[kappa\]). We then observe that $$\begin{aligned}
{\mathrm{e}}^{-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3}\Delta(x_{43})=
\Delta(x_{43})-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3\,\Delta(x_{43})
-\theta_4^2\,\bar\theta_3^2\,\delta^{(D)}(x_{43})~,
\end{aligned}$$ where the last term follows from $\square_x\,\Delta(x)=-\delta^{(D)}(x)$. Using this relation and performing the Grassmann integrations over $\bar{\theta}_3$ and $\theta_4$, we get $$\int\! d^2\bar\theta_3\, d^2\theta_4 \,
\big({\mathrm{e}}^{-2{\mathrm{i}}\theta_4 \partial_{x_{43}}\bar\theta_3}\Delta(x_{43})\big)^2
= 2 \,\partial_{x_{43}}\Delta(x_{43})\cdot \partial_{x_{43}}\Delta(x_{43})
-2\Delta(x_{43})\,\delta^{(D)}(x_{43})~.
\label{int43}$$ Inserting this expression in the integral (\[Iisapp\]), we see that the term proportional to the $\delta$-function yields a tadpole-like contribution, which vanishes in dimensional regularization and thus can be discarded. We then remain with $$\begin{aligned}
I(x_1,x_2)&=2\int \!
d^D x_3 \,d^D x_4 \,\Delta(x_{13})\,
\partial_{x_{43}}\Delta(x_{43})\cdot \partial_{x_{43}}\Delta(x_{43})\,
\Delta(x_{42})\,\big[\kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}\\
&=8\,\bigg(\frac{\Gamma(1-\epsilon)}{4\pi^{2-\epsilon}}\bigg)^4(1-\epsilon)^2
\int \! d^D x_3 \,d^D x_4 \,\frac{\big[\kappa(x_3)\,\kappa(x_4)\big]^{-\epsilon}}{
(x_{13}^2)^{1-\epsilon}\,(x_{43}^2)^{3-2\epsilon}\,(x_{42}^2)^{1-\epsilon}}~,
\end{aligned}$$ where in the second step we used the explicit expression (\[Delta\]) of the scalar propagator.
To simplify the calculation, without any loss of generality, we set $R=1$ and choose the point $\eta_2$ to be at the north pole on the sphere, namely $\eta_2=(1,0,\ldots,0)$. According to the stereographic projection (\[map\]), this corresponds to sending $x_2\to \infty$. We therefore find $$\big[\kappa(x_1)\,\kappa(x_2)\big]^{1-\epsilon}\,I(x_1,x_2)
~\stackrel{x_2\to \infty}{\approx } ~ 2^{2+3\epsilon}
\,\bigg(\frac{\Gamma(1-\epsilon)}{4\pi^{2-\epsilon}}\bigg)^2
\,\bigg(\frac{x_1^2+1}{2}\bigg)^{1-\epsilon}\,Y(x_1^2)~,
\label{W-int2}$$ where $$Y(x_1^2)= \bigg(\frac{\Gamma(2-\epsilon)}{4\pi^{2-\epsilon}}\bigg)^2
\int \! d^D x_3 \,d^D x_4 \,
\frac{1}{(x_{13}^2)^{1-\epsilon}\,(x_{43}^2)^{3-2\epsilon}\,(x_3^2+1)^{\epsilon}\,(x_4^2+1)^{\epsilon}}~.
\label{Yis}$$ It is not difficult to realize that this function is regular for $x_1^2\to 0$ and satisfies the following differential equation $$\square_{x_1} Y(x_1^2)=-\frac{\Gamma(2-\epsilon)}{4\pi^{2-\epsilon}}\,(1-\epsilon)
\,(x_1^2+1)^{-\epsilon}\int\!d^D x_4 \,\frac{1}{
(x_{41}^2)^{3-2\epsilon}\,(x_4^2+1)^{\epsilon}}~.
\label{boxI}$$ We rewrite the right hand side of (\[boxI\]) using the Schwinger parametrization $$\frac{1}{(x^2+a^2)^\alpha}=\frac{1}{\Gamma(\alpha)}\int_0^\infty\!ds\,s^{\alpha-1}
\,{\mathrm{e}}^{-s(x^2+a^2)}~,
\label{Schwinger}$$ and, after computing the resulting Gaussian integral over $x_4$, we obtain $$\begin{aligned}
\square_{x_1} Y(x_1^2)\,&\,=-\frac{\Gamma(2-\epsilon)}{8 \Gamma(2-2\epsilon)\,\Gamma(\epsilon)}
\,(x_1^2+1)^{-\epsilon}
\int_0^\infty\!ds_1\int_0^\infty\!ds_2\,\,\frac{s_1^{-1+\epsilon}\,s_2^{2-2\epsilon}}{(s_1+s_2)^{2-\epsilon}}\,\,{\mathrm{e}}^{-s_1\frac{s_1+s_2(x_1^2+1)}{s_1+s_2}}\\[2mm]
&\,=-\frac{\Gamma(2-\epsilon)}{8\,\Gamma(2-2\epsilon)\,\Gamma(\epsilon)}\,(x_1^2+1)^{-\epsilon}
\int_0^\infty\!dt\,t^{-2+\epsilon}\,(1+t)^{-1+\epsilon}\,\frac{1}{t+x_1^2+1}~,
\end{aligned}
\label{boxI1}$$ where the last step follows from changing the integration variable according to $s_1\to t \,s_2$ and performing the resulting integral over $s_2$. With the further change of integration variable $t\to\frac{1-y}{y}$, we can rewrite the $t$-integral as $$\int_0^1\!dy\,y^{2-2\epsilon}\,(1-y)^{-2+\epsilon}\,\frac{1}{1+x_1^2\,y}=\frac{\Gamma(3-2\epsilon)
\,\Gamma(\epsilon-1)}{\Gamma(2-\epsilon)}\,{}_2F_1(1,3-2\epsilon,2-\epsilon;-x_1^2)~.
\label{hyper}$$ Substituting this into (\[boxI1\]), in the end we find $$\begin{aligned}
\square_{x_1} Y(x_1^2) &= \frac{1}{4}\,(x_1^2+1)^{-\epsilon}\,{}_2F_1(1,3-2\epsilon,2-\epsilon;-x_1^2)
=\frac{x_1^2+2}{8(x_1^2+1)^2}+O(\epsilon)~.
\end{aligned}
\label{boxI2}$$ The general solution to this differential equation which is regular for $x_1^2\to 0$ is $$Y(x_1^2) = \frac{1}{32}\,\big(c_0+\ln(x_1^2+1)+O(\epsilon)\big)
\label{Yfin}$$ with $c_0$ an arbitrary constant. To fix it, we examine $Y(x_1^2)$ for $x_1\to \infty$, corresponding to the short-distance limit on the sphere in which also $\eta_1$ is sent to the north pole. In this limit the leading contribution to (\[Yis\]) comes from large $x_3^2$ and $x_4^2$, allowing us to replace the scaling factors $(1+x_i^2)^\epsilon$ with $(x_i^2)^\epsilon$. This leads to $$\begin{aligned}
Y(x_1^2)
{}&
\stackrel{x_1^2\to \infty}{\simeq }
\bigg(\frac{\Gamma(1-\epsilon)}{4\pi^{2-\epsilon}}\bigg)^2\,(1-\epsilon)^2
\int\! d^{4-2\epsilon} x_3\, \frac{1}{(x_{13}^2)^{1-\epsilon} (x_3^2 )^{\epsilon}}
\,\int\! d^{4-2\epsilon} x_4 \, \frac{1}{(x_{34}^2)^{3-2\epsilon} (x_4^2)^{\epsilon}}
\notag\\[2mm]
&~~\,\simeq - \frac{(x_1^2)^{-\epsilon}}{32\epsilon\,(1-2 \epsilon )}
\simeq
\frac{1}{32}\,\Big(\!-\frac{1}{\epsilon\,(1-2 \epsilon )}+\ln x_1^2+O(\epsilon)\Big)~.\end{aligned}$$ Comparing with (\[Yfin\]) in the limit $x_1^2\to\infty$, we deduce that $$c_0=-\frac{1}{\epsilon\,(1-2 \epsilon )}~.$$ Therefore, we can write $$Y(x_1^2)=-\frac{(x_1^2+1)^{-\epsilon}}{32\epsilon\,(1-2\epsilon)}+O(\epsilon)~.
\label{Yfinal}$$
The $x_2$-dependence can be easily restored by noticing that $\eta_{12}^2 \simeq 4/(x_1^2+1)$ at large $x_2$; this means that at finite $x_2$, the variable $x_1^2$ must be replaced by $$r_{12}^2 = \frac{4}{\eta_{12}^2} -1$$ and the function $Y(x_1^2)$ by $$Y(r_{12}^2)= -\frac{2^{-2\epsilon}\,(\eta_{12}^2)^\epsilon}{
32\epsilon\,(1-2\epsilon)}+O(\epsilon)~.
\label{Yfinal1}$$ We now use this information in (\[W-int2\]) and find $$\begin{aligned}
W_{1\,{{{S}}}}(\eta_{12})&\equiv
\big[\kappa(x_1)\,\kappa(x_2)\big]^{1-\epsilon}\,I(x_1,x_2) \\[1mm]
&= 2^{3+2\epsilon}\,
\frac{\Gamma(1-\epsilon)}{4\pi^{2-\epsilon}}\,\Delta_{{{{S}}}}(\eta_{12})\,Y(r_{12}^2)\\[1mm]
&= \frac{(\pi \eta_{12}^2)^\epsilon\,\Gamma(-\epsilon)}{(4\pi)^2\,(1-2\epsilon)}
\,\Delta_{{{{S}}}}(\eta_{12}) + O(\epsilon)~,
\end{aligned}
\label{finalresult}$$ where we used (\[DeltaS\]) in the second line, and (\[Yfinal1\]) in the final step. This is the formula (\[WS3\]) of the main text. We have also computed the $O(\epsilon)$ terms, finding $$W_{1\,{{{S}}}}(\eta_{12})
= \frac{(\pi \eta_{12}^2)^\epsilon\,\Gamma(-\epsilon)}{(4\pi)^2\,(1-2\epsilon)}
\,\Delta_{{{{S}}}}(\eta_{12}) \,\Big(1-\epsilon^2\,\phi(r_{12}^2)+ O(\epsilon^3)\Big)
\label{finalresult1}$$ with $$\phi(x^2)=\mathrm{Li}_2(-x^2)+\frac{1}{2}\,\ln^2(x^2+1)+\frac{\ln(x^2+1)}{x^2}+\frac{\pi^2}{6}~.$$ It is straightforward to verify that $\phi(x^2)$ vanishes at large $x^2$ and approaches a finite value for $x^2\to 0$.
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[^1]: Explicitly, $$R_{\vec{n}}^{\,a_1\dots a_n}
= {\mathrm{tr}\,}\big(T^{(a_1}\cdots T^{a_{n_1}}\big)~
{\mathrm{tr}\,}\big(T^{a_{n_1+1}}\cdots T^{a_{n_1+n_2}}\big)\ldots
{\mathrm{tr}\,}\big(T^{a_{n_1 + \ldots + n_{\ell-1}+1}}\cdots T^{a_n)}\big)$$ where the indices are symmetrized with strength 1. \[footnote:R\]
[^2]: This fact has been tested by explicit computations at two loops, see for example [@Jones:1974mm; @Capper:1979ns; @Howe:1984xq], and then extended to all loops using non-renormalization and anomaly arguments, and further strengthened at the non-perturbative level [@Novikov:1983uc; @Novikov:1985rd; @Seiberg:1988ur].
[^3]: This equation requires that the correlator must have the form $$\label{GRfinite}
G^R_{\vec{n},\vec{m}} = d_0 + g^2 (d_1-d_0 \gamma_{n,0}\,\nu) + g^4
\left[d_{2}-\nu\,d_0 \gamma_{n,1}+\frac{\nu}{4}(\nu\,d_0 \gamma_{n,0} -2 d_1) (\beta_0+2\gamma_{n,0}) \right] + O(g^6)~.$$ It is easy to check that (\[GnmRtot\]) satisfies this requirement. Notice that the whole $\nu$-dependence of $G^R_{\vec{n},\vec{m}}$ can be reconstructed order by order in $g^2$ from the correlator at a given value $\widehat{\nu}$, for instance $\widehat\nu=0$, and the coefficients $\beta_0$, $\gamma_{n,0}$ and $\gamma_{n,1}$.
[^4]: The dependence on $R$ can be easily restored by replacing $a_u$ with $a_u\,R$.
[^5]: In the recent paper [@Bourget:2018fhe] a discrepancy at six loops, proportional to $\zeta^2(5) g^{12}$, has been pointed out in the comparison between the matrix model results and the correlators obtained by solving Toda equations.
[^6]: Notice that a non-zero value of $\delta$ can be compensated by performing a finite renormalization of the scalar operators.
|
---
abstract: 'In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, Bregman proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal conditional gradient method and universal method for variational inequalities with composite structure. These method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. As a particular case of our general framework, we introduce relative smoothness for operators and propose an algorithm for VIs with such operator. We also generalize our framework for relatively strongly convex objectives and strongly monotone variational inequalities.'
author:
-
bibliography:
- 'main.bib'
title: 'Inexact Relative Smoothness and Strong Convexity for Optimization and Variational Inequalities by Inexact Model[^1]. '
---
Convex optimization; composite optimization; proximal method; level-set method; variational inequality; universal method; mirror prox; acceleration; relative smoothness; saddle-point problem.
Introduction
============
In this paper we consider the following convex optimization problem $$\label{eq:Problem}
\min_{x\in Q} f(x),$$ where $Q$ is a convex subset of finite-dimensional vector space $E$, $f$ is generally a non-convex function.
Most of minimization methods for such problems are constructed using some model of the objective $f$ at the current iterate $x_k$. This can be a quadratic model based on the $L$-smoothness of the objective $$\label{eq:quadr_model}
f(x_k) + {\langle}\nabla f(x_k), x-x_k {\rangle}+ \frac{L}{2}\|x-x_k\|_2^2.$$ The step of gradient method is obtained by the minimization of this model [@nesterov2018lectures]. More general models are constructed based on regularized second-order Taylor expansion [@nesterov2006cubic] or other Taylor-like models [@drusvyatskiy2019nonsmooth] as well as other objective surrogates [@mairal2013optimization]. Another example is the conditional gradient method [@frank1956algorithm], where a linear model of the objective is minimized on every iteration. Adaptive choice of the parameter of the model with provably small computational overhead was proposed in [@nesterov2006cubic] and applied to first-order methods in [@nesterov2013gradient; @nesterov2015universal; @dvurechensky2018computational]. Recently, first-order optimization methods were generalized to the so-called relative smoothness framework [@bauschke2016descent; @lu2018relatively; @ochs2017non], where $\frac{1}{2} \|x-x_k\|_2^2$ in the quadratic model for the objective is replaced with general Bregman divergence.
The literature on first-order methods [@devolder2014first; @bogolubsky2016learning; @dvurechensky2017universal] considers also gradient methods with inexact information, relaxing the model to $$\label{eq:inex_quadr_model}
f_{\delta}(x_k) + {\langle}\nabla f_\delta(x_k), x-x_k {\rangle}+ \frac{L}{2}\|x-x_k\|^2_2 + \delta,$$ with $(f_\delta,\nabla f_\delta)$ called inexact oracle and this model being an upper bound for the objective. In particular, this relaxation allows to obtain universal gradient methods [@nesterov2015universal].
One of the goals of this paper is to describe and analyze first-order optimization methods which use a very general *inexact model* of the objective function, the idea being to replace the linear part in by a general function $\psi_{\delta}(x,x_k)$ and the squared norm by general Bregman divergence. The resulting model includes as a particular case inexact oracle model and relative smoothness framework, and allows to obtain many optimization methods as a particular case, including conditional gradient method [@frank1956algorithm], Bregman proximal gradient method [@chen1993convergence] and its application to optimal transport [@xie2018fast] and Wasserstein barycenter [@stonyakin2019gradient] problems, general Catalyst acceleration technique [@lin2015universal], (accelerated) composite gradient methods [@nesterov2013gradient; @beck2009fast], (accelerated) level methods [@nemirovskii1985optimal; @lan2015bundle]. First attempts to propose this generalization were made in [@stonyakin2019gradient; @gasnikov2017universal] for non-accelerated methods and in [@tyurin2017fast] for accelerated methods, yet without relative smoothness paradigm. In this paper we propose the inexact model in a very general setting including adaptivity of the algorithms to the parameter $L$, possible relative strong convexity and relative smoothness. We also provide convergence rates for the gradient method and accelerated gradient method using inexact model of the objective. As an application of our general framework, we develop a universal conditional gradient method, providing a parameter-free generalization of the results in [@nesterov2018complexity].
We believe that our model is flexible enough to be extended for problems with primal-dual structure[^2] [@nesterov2009primal-dual; @nemirovski2010accuracy; @nesterov2018complexity], e.g. for problems with linear constraints [@anikin2017dual; @chernov2016fast; @guminov2019accelerated; @nesterov2018primal-dual]; for random block-coordinate descent [@dvurechensky2017randomized]; for tensor methods [@nesterov2018implementable; @gasnikov2019near]; for distributed optimization setting [@scaman2017optimal; @uribe2018distributed; @dvurechensky2018decentralize; @dvinskikh2019primal]; and adaptive stochastic optimization [@iusem2019variance; @ogaltsov2019adaptive].
Optimization problem is tightly connected with variational inequality (VI) $$\label{eq:VI}
\text{Find} \; x_* \in Q \; \text{s.t.} \; {\langle}g(x_*),x_*-x{\rangle}\leq 0, \; \forall x \in Q,$$ where $g(x) = \nabla f(x)$. A special VI is also equivalent to finding a saddle-point of a convex-concave function $$\label{eq:SP}
\min_{u\in Q_1}\max_{v\in Q_2} f(u,v)$$ for $x=(u,v)$ and $g(x) = (\nabla_u f(u,v),-\nabla_v f(u,v))$. This motivates the second part of this paper, which consists in generalization of the inexact model of the objective function to an inexact model for an operator in variational inequality. In particular, we extend the relative smoothness paradigm to variational inequalities with monotone and strongly monotone operators and provide a generalization of Mirror-Prox method [@nemirovski2004prox], its adaptive version [@gasnikov2019adaptive] and universal version [@dvurechensky2018generalized] to variational inequalities with such general inexact model of the operator. As a partucular case, our approach allows to partially the results of [@chambolle2011first-order]. We also apply the general framework for variational inequalities to saddle-point problems.
To sum up, we present a unified view on inexact models for convex optimization problems, variational inequalities, and saddle-point problems.
The structure of the paper is the following. In Section \[S:Inexact\_min\] we introduce inexact model of the objective in optimization and provide several examples to illustrate the flexibility and generality of the proposed model. In particular, we demonstrate that relative smoothness and strong convexity are particular cases of our general framework.
In Section \[GM\] we consider adaptive gradient method (GM) and adaptive fast gradient method (FGM). FGM has better convergence rate, yet it is not adapted to the relative smoothness paradigm. In section \[univ\_frank\_wolfe\], we construct universal conditional gradient (Frank–Wolfe) method using FGM with inexact projection. To the best of our knowledge, this is the first attempt to combine Frank–Wolfe method [@har2015cond; @jaggi2013revisiting] and universal method [@nesterov2015universal]. In Section \[VI\] we generalize inexact model to variational inequalities and saddle-point problems for the case of monotone and strongly monotode operators. In the former case, we construct an adaptive generalization of the Mirror-Prox algorithm for variational inequalities and saddle-point problems with such inexact model. In the latter case the proposed algorithm is accelerated by the restart technique to have linear rate of convergence. We especially consider the case of $m$-strong convexity of the model. The natural motivation for such a formulation are composite saddle problems, and mixed variational inequalities with a $m$-strongly convex composite.
The contribution of this paper is follows:
1. 2. 3.
Inexact Model in Minimization Problems. Definition and Examples {#S:Inexact_min}
===============================================================
We start with the general notation. Let $E$ be an $n$-dimensional real vector space and $E^*$ be its dual. We denote the value of a linear function $g \in E^*$ at $x\in E$ by ${\langle}g, x {\rangle}$. Let $\|\cdot\|$ be some norm on $E$, $\|\cdot\|_{*}$ be its dual, defined by $\|g\|_{*} = \max\limits_{x} \big\{ {\langle}g, x {\rangle}, \| x \| \leq 1 \big\}$. We use $\nabla f(x)$ to denote any (sub)gradient of a function $f$ at a point $x \in {\rm dom} f$. We define a continuous convex on $Q$ function $d(x)$ to be distance generating function and $V[y](x) = d(x) - d(y) - {\langle}\nabla d(y), x - y {\rangle}$ to be the corresponding Bregman divergence. Most typically it is assumed that $d$ is $1$-strongly convex on $Q$ w.r.t. $\|\cdot\|$-norm, which we refer to as (1-SC) assumption w.r.t. $\|\cdot\|$-norm. Namely, for all $x,y \in Q$, $d(x) - d(y) - {\langle}\nabla d(y), x - y {\rangle}\geqslant \frac{1}{2}\|x-y\|^2$. We underline that, in general, we do not make this assumption, and, in what follows, we explicitly write if this assumption is made.
Note that in Definition \[defRelStronglyConvex\] we allow $L$ to depend on $\delta$. Definition \[defRelStronglyConvex\] is a generalization of $(\delta, L)$-model from [@gasnikov2017universal; @tyurin2017fast; @stonyakin2019gradient], where $\mu=0$ and $m = 0$. Further, we denote $(\delta, L,0,0,V)$-model as $(\delta, L)$-model.
Let us illustrate the above definitions by several examples.
[**Composite optimization, [@beck2009fast; @nesterov2013gradient].**]{} Assume that in , $f(x) = g(x) + h(x)$ with $L$-smooth w.r.t. norm $\|\cdot\|$ part $g$ and simple convex part $h$. In this case we assume that $V[y](x)$ satisfies (1-SC) condition w.r.t $\|\cdot\|$, and define $f_{\delta}(x) = f(x) + h(x)$ and $\psi_\delta(x,y) = \langle \nabla g(y), x - y \rangle + h(x) - h(y)$. It is clear that holds with $\delta=0$ and $\mu=0$ and we are in the situation of Definition \[defRelStronglyConvex\] with $m=0$. If $h$ turns out to be relatively $m$-strongly convex [@lu2018relatively] relatively to $d$, i.e. $h(x)-h(y)-{\langle}\nabla h(y),x-y{\rangle}\geq mV[y](x)$, then holds, but in $\mu=0$. On the other hand, if $g$ turns out to be relatively $\mu$-strongly convex [@lu2018relatively] relatively to $d$, i.e. $g(x)-g(y)-{\langle}\nabla g(y),x-y{\rangle}\geq \mu V[y](x)$, then holds with $\delta=0$, but in $m=0$.
A particular example is the following minimization problem [@anikin2015modern] motivated by traffic demands matrix estimation from link loads $$f(x) = \frac{1}{2}\|Ax - b\|^2_2 + m \sum \limits_{k=1}^n x_k\ln{x_k} \rightarrow \min \limits_{x\in S_n(1)}.$$ In this case $g(x) = \frac{1}{2}\|Ax - b\|^2_2$ and $h(x) = m \sum \limits_{k=1}^n x_k\ln{x_k}$. Choosing $\|\cdot\| = \|\cdot\|_1$ and $d(x) = \sum \limits_{k=1}^n x_k\ln{x_k}$, $V[y](x) = \sum \limits_{k=1}^n x_k\ln{(x_k/y_k)}$, we obtain that $g$ is has Lipschitz gradient w.r.t. $\|\cdot\|_1$ with the constant $L=\max_{\|h\|_1 \leq 1} {\langle}h,A^TA h{\rangle}=\max_{k=1,...,n}\|A_k\|_2^2$, where $A_k$ is the $k$-th column of $A$. Finally, $\psi_\delta(x,y) = \langle \nabla g(y), x - y \rangle + h(x) - h(y)$ is a $(0,L,0,m,V)$-model. At the same time, the part $g$ is not necessarily strongly convex. Thus, our framework allows to obtain (accelerated) gradient method for composite optimization and their counterparts for inexact oracle models.
[**Relative smoothness and relative strong convexity, [@bauschke2016descent; @lu2018relatively].**]{} Assume that in , the objective $f$ is relatively smooth [@bauschke2016descent; @lu2018relatively] relative to $d$, i.e. $$f(x)-f(y) -{\langle}\nabla f(y), x-y {\rangle}\leq \left(d(x)-d(y) -{\langle}\nabla d(y), x-y {\rangle}\right) = LV[y](x), \; \forall x,y \in Q$$ and relatively strongly convex [@lu2018relatively] relative to $d$, i.e. $$\mu V[y](x) = \mu\left(d(x)-d(y) -{\langle}\nabla d(y), x-y {\rangle}\right) \leq f(x)-f(y) -{\langle}\nabla f(y), x-y {\rangle}, \; \forall x,y \in Q.$$ Then, clearly, Definition \[defRelStronglyConvex\] holds with $m = 0$, $\delta = 0$, $\psi_\delta(x,y) = \langle \nabla f(y), x - y \rangle$. Importantly, the function $d$ is not necessarily strongly convex.
Note that if $V[y](x) \leq C_n{\left\lVertx-y\right\rVert}^2$ for some constant $C_n = O(\log n)$, the condition of $(\mu C_n)$-strong convexity w.r.t. norm ${\left\lVert\cdot\right\rVert}$, namely $\mu C_n {\left\lVertx-y\right\rVert}^2 + f_\delta(y) + \psi_\delta(x,y) \leqslant f(x)$ implies the left inequality in .
One of the main applications of general relative smoothness and strong convexity is the step of tensor methods which use the derivatives of the objective of the order higher than 2 [@nesterov2018implementable; @gasnikov2019near]. Thus, our framework allows to obtain gradient method for optimization with relative smoothness and strong convexity and extend them to the case of inexact oracle setting.
[**Superposition of functions, [@nemirovskii1985optimal]**]{}. Assume that in [@nemirovskii1985optimal; @lan2015bundle] $f(x) := g(g_1(x), \dots, g_m(x)) \rightarrow \min_{x \in Q}$, where each function $g_k(x)$ is a smooth convex function with $L_k$-Lipschitz gradient w.r.t. $\|\cdot\|$-norm for all $k$. Function $g(x)$ is a $M$-Lipschitz convex function w.r.t 1-norm, non-decreasing each of its arguments. The chosen Bregman divergence $V[y](x)$ is assumed to satisfy (1-SC). From these assumptions we have [@boyd2004convex; @lan2015bundle] that function $f(x)$ is also convex and $$\begin{gathered}
0 \leq f(x) - f(y) - g(g_1(y) + \langle\nabla g_1(y), x - y \rangle, \dots, g_m(y)+\langle\nabla g_m(y), x - y \rangle) + f(y) \leq\\\leq M\frac{\sum_{i=1}^{m}L_i}{2}{\left\lVertx - y\right\rVert}^2 \leq MV[y](x)\sum_{i=1}^{m}L_i, \,\,\,\, \forall x,y \in Q.\end{gathered}$$ Therefore, $$\psi_{\delta}(x,y) = g(g_1(y) + \langle\nabla g_1(y), x - y \rangle, \dots, g_m(y)+\langle\nabla g_m(y), x - y \rangle) - f(y),$$ is $\left(0,M\cdot\left(\sum_{i=1}^{m}L_i\right) \right)$-model of $f$ with $f_{\delta}(y) = f(y)$ at a given point $y$. Thus, our framework allows to obtain (accelerated) level gradient methods considered in [@nemirovskii1985optimal; @lan2015bundle] as a special case. Moreover, we generalize these methods for the case of inexact oracle information.
\[E:Proximal\][**Proximal method, [@chen1993convergence].**]{} \[prox\_ex\] Let us consider optimization problem , where $f$ is an arbitrary convex function (not necessarily smooth). Then, for arbitrary $L \ge 0$, $\psi_{\delta}(x,y) = f(x) - f(y)$ is $(0, L)$-model of $f$ with $f_{\delta}(y) = f(y)$ at a given point $y$. Thus, our framework allows to obtain (Bregman) proximal gradient methods [@chen1993convergence; @parikh2014prox] as a special case and extend them to the case of inexact oracle setting. In particular, based on this model (with Bregman divergence to be Kullback–Leibler divergence) we propose in [@stonyakin2019gradient] proximal Sinkhorn’s algorithm for Wasserstein distance calculation problem and in [@kroshnin2019complexity] proximal IBP for Wasserstein barycenter problem.
[**Min-min problem.**]{} Assume that in $f(x) := \min_{z \in Q}F(z,x)$, the set $Q$ is convex and bounded, function $F$ is smooth and convex w.r.t. all variables. Moreover, assume that $${\left\lVert\nabla F(z',x') -\nabla F(z,x)\right\rVert}_2 \leq L {\left\lVert(z',x') -(z,x)\right\rVert}_2,\,\,\forall z,z'\in Q,\,x,x'\in {{\mathbb R}}^n.$$ Let $V[y](x)=\frac{1}{2}\|x-y\|_2^2$. If we can find a point $\widetilde{z}_\delta(y) \in Q$ such that $$\begin{gathered}
\langle\nabla_z F(\widetilde{z}_\delta(y), y), z - \widetilde{z}_\delta(y)\rangle \geq -\delta, \,\,\,\forall z \in Q,\end{gathered}$$ then $F(\widetilde{z}_\delta(y), y) - f(y) \leq \delta$ and $\psi_{\delta}(x,y) = \langle\nabla_z F(\widetilde{z}_\delta(y), y), x - y\rangle$ is $(6\delta, 2L,0,0,V)$-model of $f$ with $f_\delta(y) = F(\widetilde{z}_\delta(y), y) - 2\delta$ at a given point $y$.
[**Saddle point problem, [@devolder2014first]**]{}
Assume that in $
f(x) = \max_{z \in Q}\left[\langle x, b - Az\rangle - \phi(z)\right] \rightarrow \min_{x \in {{\mathbb R}}^n}$, where $\phi(z)$ is a $\mu$-strongly convex function w.r.t. $p$-norm ($1\leq p\leq2$). Then $f$ is smooth and convex and its gradient is Lipschitz continuous with constant $L = \frac{1}{\mu}\max_{{\left\lVertz\right\rVert}_p\leq1}{\left\lVertAz\right\rVert}_2^2$. If $z_\delta(y)\in Q$ is an approximate solution to auxiliary max-problem, i.e. $$\max_{z \in Q}\left[\langle y, b - Az\rangle - \phi(z)\right] - \left[\langle y, b - Az_{\delta}(y)\rangle - \phi(z_{\delta}(y))\right]\le \delta,$$ then $
\psi_{\delta}(x,y) =\langle b - Az_\delta(y), x - y\rangle
$ is $(\delta, 2L,0,0,V)$-model of $f$ with $f_\delta(y) = \langle y, b - Az_\delta(y)\rangle - \phi(z_\delta(y))$ at the point $y$ if we define $V[y](x)=\frac{1}{2}\|x-y\|_2^2$.
[**Augmented Lagrangians, [@devolder2014first].**]{} Let us consider $$\begin{gathered}
\min_{Az=b,\,z\in Q} \phi(z) + \frac{\mu}{2}{\left\lVertAz -b\right\rVert}_2^2
\end{gathered}$$ and the corresponding dual problem $$\begin{aligned}
\min_{x \in {{\mathbb R}}^n} \left\{f(x) = \max_{z \in Q}\underbrace{\left(\langle x, b - Az\rangle - \phi(z) - \frac{\mu}{2}{\left\lVertAz -b\right\rVert}_2^2\right)}_{\Lambda(x,z)}\right\} .
\end{aligned}$$ If $z_\delta(y)$ is an approximate solution of auxiliary max-problem, i.e. then $ \psi_{\delta}(x,y) =\langle b - Az_\delta(y), x - y\rangle$ is $(\delta, \mu^{-1},0,0,V)$-model of $f$ with $$f_\delta(y) = \langle y, b - Az_\delta(y)\rangle - \phi(z_\delta(y))- \frac{\mu}{2}{\left\lVertAz_\delta(y) -b\right\rVert}_2^2$$ at the point $y$ if we take $V[y](x)=\frac{1}{2}\|x-y\|_2^2$.
[**Moreau envelope of the objective function, [@devolder2014first].**]{} \[Moreau\_ex\] Let us consider optimization problem: $$\begin{aligned}
\min_{x \in {{\mathbb R}}^n} \left\{f_L(x) := \min_{z \in Q}\underbrace{\left\{f(z) + \frac{L}{2}{\left\lVertz - x\right\rVert}^2_2\right\}}_{\Lambda(x,z)}\right\}.
\label{prox}
\end{aligned}$$ Assume that $f$ is convex and, for some $z_L(y)$, $$\begin{gathered}
\max_{z \in Q}\left\{\Lambda(y,z_L(y)) - \Lambda(y,z) + \frac{L}{2}{\left\lVerty - z_L(y)\right\rVert}^2_2\right\} \leq \delta.
\end{gathered}$$ Then $ \psi_{\delta}(x,y) =\langle L(y - z_L(y)), x - y\rangle$ is $(\delta, L,0,0,V)$-model of $f$ with $$f_\delta(y) = f(z_L(y)) + \frac{L}{2}{\left\lVertz_L(y) - y\right\rVert}^2_2 - \delta$$ at the point $y$ if we take $V[y](x)=\frac{1}{2}\|x-y\|_2^2$.
[**Clustering by Electorial Model, [@stonyakin2019gradient].**]{} \[Clustering\_electorial\]
Another example of an optimization problem that allows for $(\delta, L, 0, m, V)$-model with strong convexity of the function $\psi_\delta(x, y)$ is proposed in [@stonyakin2019gradient] to address a *non-convex* optimization problem which arises in an electoral model for clustering introduced in [@RePEc:cor:louvco:2018001]. In this model, voters (data points) select a party (cluster) iteratively by minimizing the following function $$\label{Nesterov_Electoral_Model}
\min_{z \in S_n(1), p \in \mathbb{R}^m_{+}} \left\{f_{\mu_1, \mu_2}(x = (z, p)) = g(x) + \mu_1 \sum\limits_{k=1}^{n}z_k \ln z_k + \frac{\mu_2}{2} \|p\|^2_2 \right\}.$$ Let us choose ${\left\lVertx\right\rVert}^2 = {\left\lVertz\right\rVert}^2_1 + {\left\lVertp\right\rVert}^2_2$ and assume that, in general non-convex, $g(x)$ has $L_g$–Lipschitz continuous gradient $${\left\lVert\nabla g(x) - \nabla g(y)\right\rVert}_* \leq L_g {\left\lVertx - y\right\rVert} \quad \forall x, y \in S_n(1) \times \mathbb{R}^m_{+}$$ and $L_g \leq \mu_1$ and $L_g \leq \mu_2$. It can be shown (see [@stonyakin2019gradient]) that $$\begin{gathered}
\psi_{\delta}(x, y) = \langle \nabla g(y), x-y \rangle - L_g \cdot {\rm KL}(z_x|z_y) - \frac{L_g }{2}\|p_x-p_y\|^2_2 \\ + \mu_1 ({\rm KL}(z_x | \textbf{1}) - {\rm KL}(z_y | \textbf{1})) + \frac{\mu_2 }{2} \left(\|p_x\|^2_2-\|p_y\|^2_2\right)
\end{gathered}$$ is a $(0, 2L_g,0,\min\{\mu_1, \mu_2\} - L_g,V)$-model of $f_{\mu_1, \mu_2}(x)$. Here ${\rm KL}(z_x | z_y) = \sum_{i=1}^m [z_x]_i \ln ([z_x]_i/[z_y]_i)$ and $$\begin{aligned}
V[y](x) = {\rm KL}(z_x | z_y) + \frac{1}{2}\|p_x - p_y\|^2_2.\end{aligned}$$
We finish this section by defining an approximate solution to an optimization problem. This definition will be used to allow inexact solutions of auxiliary minimization problems on each iteration of our algorithms.
\[solNemirovskiy\] For a convex optimization problem $\min_{x \in Q} \Psi(x)$, we denote by $\text{Arg}\min_{x \in Q}^{\widetilde{\delta}}\Psi(x)$ a set of such $\widetilde{x}$ that $$\begin{gathered}
\label{eqv_inex_sol}
\gav{\exists} h \in \partial\Psi(\widetilde{x})\gav{:\forall x \in Q} \,\, \gav{\to}\, \langle h, x - \widetilde{x} \rangle \geq -\widetilde{\delta}.
$$ We denote by $\operatorname*{argmin}_{x \in Q}^{\widetilde{\delta}}\Psi(x)$ some element of $\text{Arg}\min_{x \in Q}^{\widetilde{\delta}}\Psi(x)$.
Gradient Method with Inexact Model. {#GM}
===================================
We note that non-accelerated Algorithm \[Alg2\] is suitable for the problems with relative smoothness and relative strong convexity, also there is no accumulation of errors. Accelerated Algorithm \[FastAlg2\_strong\] gives a better estimate with errors close to zero, however, accumulation of errors is possible. We consider Algorithm \[FastAlg2\_strong\] for the narrower class of problems with $(\delta, L, \mu, m, V,\|\cdot \|)$-models (see Definition \[defRelStronglyConvexFast\]) w.r.t norm $\|\cdot\|$. It means, that non-accelerated method (Algorithm \[Alg2\]) is suitable for a wider class of problems.
Adaptive Gradient Method with $(\delta, L, \mu, m, V)$-Model {#SC}
------------------------------------------------------------
In this section we consider adaptive gradient method for problem , which uses a $(\delta, L, \mu, m, V)$-model of the objective. For the case when $\mu + m > 0$ our method has linear convergence .
We assume that in each iteration $k$, the method has access to $(\delta, \bar{L}_{k+1}, \mu, m, V)$-model of $f$ .
**Input:** $x_0$ is the starting point, $\mu \geq 0$ and $\delta$. Set $S_0 := 0 $ Find the smallest $i_k\geq 0$ such that $$\label{exitLDL_G_S}
f_{\delta}(x_{k+1}) \leq f_{\delta}(x_{k}) + \psi_{\delta}(x_{k+1}, x_{k}) +L_{k+1}V[x_{k}](x_{k+1}) + \delta,$$ where $L_{k+1} = 2^{i_k-1}L_k$ for $L_k > 2 \mu$ and $L_{k+1} = 2^{i_k}L_k$ for $L_k \leq 2 \mu$,\
$\alpha_{k+1}:= \frac{1}{L_{k+1}}$, $S_{k+1} := S_k + \alpha_{k+1}$. $$\label{equmir2DL_G_S}
\phi_{k+1}(x) := \psi_{\delta}(x, x_k)+L_{k+1}V[x_k](x), \quad
x_{k+1} := {\arg\min_{x \in Q}}^{\widetilde{\delta}} \phi_{k+1}(x).$$
We consider the case of $m$-strong convexity of the function $\psi_\delta(x, y)$ Algorithm \[Alg2\], a linear convergence for $\mu > 0$ or $m>0$.
For and all $k \geq 0$, we denote $$q_k {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{L_{k}-\mu}{L_{k} + m}, \quad Q_j^k {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\prod_{i=j}^{k}q_{i}.$$ We assume that $Q_j^k = 1$ for $j > k$.
\[Th:str\_conv\_adap\] Assume that $\psi_{\delta}(x, y)$ is a $(\delta, L, \mu, m, V)$-model according to Definition \[defRelStronglyConvex\]. Denote by $y_{N} = \operatorname*{argmin}_{k = 1,\dots, N}f(x_k)$. Then, after $N$ iterations of Algorithm \[Alg2\] we have $$\begin{aligned}
f(y_{N}) - f(x_*)
&\leq \min\left\{(L_{N} + m) Q_1^{N}, \frac{1}{\sum_{i=1}^{N}\frac{1}{L_{i} + m}}\right\}V[x_{0}](x_*) + \widetilde{\delta} + 2\delta, \label{analysis_algorithm_grad:ineq_2}\\
V[x_{N}](x_*) &\leq Q_1^{N} V[x_0](x_*) + (\widetilde{\delta} + 2\delta)\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}. \label{analysis_algorithm_grad:ineq_1}
$$
To prove Theorem \[Th:str\_conv\_adap\] we need the following lemma.
\[lemma:str\_conv\] Let $\psi(x)$ be a $m$-strongly convex function, $m \geq 0$, and $$\begin{gathered}
y = {\operatorname*{argmin}_{x \in Q}}^{\widetilde{\delta}} \{\psi(x) + \beta V[z](x)\},
\end{gathered}$$ where $\beta \geq 0$. Then $$\psi(x) + \beta V[z](x) \geq \psi(y) + \beta V[z](y) + (\beta + m)V[y](x) - \widetilde{\delta} ,\,\,\, \forall x \in Q.$$
By Definition \[solNemirovskiy\]: $$\begin{gathered}
\exists g \in \partial\psi(y), \,\,\, \langle g + \att{\beta}\nabla_y V[z](y), x - y \rangle \geq -\widetilde{\delta} ,\,\,\, \forall x \in Q.
\end{gathered}$$ Then inequality $$\begin{gathered}
\psi(x) - \psi(y) \geq \langle g, x - y \rangle + m V[y](x) \geq \langle \att{\beta}\nabla_y V[z](y), y - x \rangle - \widetilde{\delta} + m V[y](x)
\end{gathered}$$ and equality $$\begin{gathered}
\langle \nabla_y V[z](y), y - x \rangle=\langle \nabla d(y) - \nabla d(z), y - x \rangle=d(y) - d(z) - \langle \nabla d(z), y - z \rangle +\\ + d(x) - d(y) - \langle \nabla d(y), x - y \rangle - d(x) + d(z) + \langle \nabla d(z), x - z \rangle=\\=
V[z](y) + V[y](x) - V[z](x)
\end{gathered}$$ complete the proof.
Since by Definition \[defRelStronglyConvex\] with $x = y$, $f(x) - \delta \leqslant f_{\delta}(x) \leqslant f(x)$, and , we have the following series of inequalities $$\begin{aligned}
f(x_{N})
&\leq f_{\delta}(x_{N}) + \delta \leq f_{\delta}(x_{N-1}) + \psi_{\delta}(x_{N}, x_{N-1}) +L_{N}V[x_{N-1}](x_{N}) + 2\delta.\end{aligned}$$ Using Lemma \[lemma:str\_conv\] for we have $$\begin{aligned}
f(x_{N})
&\leq f_{\delta}(x_{N-1}) + \psi_{\delta}(x, x_{N-1}) +L_{N}V[x_{N-1}](x) - \left(L_{N} + m\right)V[x_{N}](x) + \widetilde{\delta} + 2\delta.\end{aligned}$$ In view of the left inequality , we have $$\begin{aligned}
\label{analysis_algorithm_grad:prove_1}
f(x_{N}) \leq f(x) + (L_{N} - \mu)V[x_{N-1}](x) - \left(L_{N} + m\right)V[x_{N}](x) + \widetilde{\delta} + 2\delta.\end{aligned}$$ Taking $x = x_*$ and using inequality $f(x_*) \leq f(x_{N})$, we obtain $$\begin{aligned}
\left(L_{N} + m\right)V[x_{N}](x_*) &\leq (L_{N} - \mu)V[x_{N-1}](x_*) + \widetilde{\delta} + 2\delta.\end{aligned}$$ Thus, we have that $$\begin{aligned}
V[x_{N}](x_*) &\leq q_{N} V[x_{N-1}](x_*) + \frac{\widetilde{\delta} + 2\delta}{L_{N} + m}\leq Q_1^{N} V[x_0](x_*) + (\widetilde{\delta} + 2\delta)\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}.\end{aligned}$$ The last inequality proves . Now we rewrite for $x = x_*$ as $$\begin{aligned}
V[x_{N}](x_*) \leq \frac{1}{L_{N} + m}(f(x_*) - f(x_{N}) + \widetilde{\delta} + 2\delta) + q_{N}V[x_{N-1}](x_*).\end{aligned}$$ Recursively, we have $$\begin{aligned}
V[x_{N}](x_*) \leq \sum_{i=1}^{N}\left(\frac{Q_{i+1}^{N}}{L_{i} + m}(f(x_*) - f(x_{i})+ \widetilde{\delta} + 2\delta)\right) + Q_{1}^{N} V[x_{0}](x_*).\end{aligned}$$ Using that $V[x_{N}](x_*) \geq 0$ and the definition of $y_{N}$, we get $$\begin{aligned}
Q_{1}^{N} V[x_{0}](x_*)
&\geq \sum_{i=1}^{N}\left(\frac{Q_{i+1}^{N}}{L_{i} + m}(f(x_i) - f(x_*) - \widetilde{\delta} - 2\delta)\right)\\
&\geq (f(y_{N}) - f(x_*))\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m} - (\widetilde{\delta} + 2\delta)\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}.\end{aligned}$$ by $\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}$, $$\begin{aligned}
f(y_{N}) - f(x_*)
&\leq \frac{Q_{1}^{N}}{\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}} V[x_{0}](x_*) + \widetilde{\delta} + 2\delta.\end{aligned}$$ $\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m} \geq \frac{1}{L_{N} + m}$ and $Q_{i}^{N} \geq Q_{1}^{N}$ for all $i \geq 1$, we get $$\begin{aligned}
f(y_{N}) - f(x_*) \leq \min\left\{(L_{N} + m) Q_1^{N}, \frac{1}{\sum_{i=1}^{N}\frac{1}{L_{i} + m}}\right\}V[x_{0}](x_*) + \widetilde{\delta} + 2\delta.\end{aligned}$$
This proves .
\[remark:grad\_method\] Let us assume that $L_0 \leq L$, and we know that $\bar{L}_{k+1} \leq L$ for all $k \geq 0$ (or in other words, $(\delta, L, \mu, m, V)$-model exists for all $k \geq 0$). This means that $L_k \leq 2L$ for all $k \geq 0$ due to $(\delta, L, \mu, m, V)$–model definition and $L_k$ selection rule. From this fact we can obtain that $\sum_{i=1}^{N}\frac{1}{L_{i} + m} \geq \frac{N}{2L + m}$ and $q_k \leq q {\mathrel{\stackrel{\makebox[0pt]{\mbox{\normalfont\tiny def}}}{=}}}\frac{2L - \mu}{2L + m}$. In view of the last two inequalities, we have $$\begin{aligned}
f(y_N) - f(x_*) &\leq \min\left\{\frac{2L + m}{N}, (2L + m) q^{N}\right\}V[x_0](x_*) + \widetilde{\delta} + 2\delta,\end{aligned}$$ $$\begin{aligned}
V[x_{N}](x_*) &\leq q^{N} V[x_0](x_*) + (\widetilde{\delta} + 2\delta)\sum_{i=1}^{N}\frac{Q_{i+1}^{N}}{L_{i} + m}.\end{aligned}$$
We can see that convergence rates for the objective and for the argument are obtained for different sequences in Theorem \[Th:str\_conv\_adap\]. With an additional assumption on the Bregman divergence $V$ and the objective $f$ we can prove a result which guarantees convergence for the objective and for the argument with the same sequence. Namely, in the next theorem we assume that $V[y](x)$ satisfies (1-SC) condition w.r.t. $\|\cdot\|$-norm (see Remark \[NormModel\]) and function $f$ is a convex function.
\[Th:str\_conv\_adap\_pos\_m\] . Denote by $\bar{x}_N= \frac{1}{S_N}\sum_{k=0}^{N-1}\frac{x_{k+1}}{L_{k+1}}$. Then, after $N$ iterations of Algorithm \[Alg2\] we have $$\begin{aligned}
\label{Th:str_conv_adap_pos_m:ineq_1}
f(\bar{x}_N) - f(x_*) + \frac{m}{2}{\left\lVert\bar{x}_N - x_*\right\rVert}^2 \leq \frac{V[x_0](x_*)}{S_N} + \widetilde{\delta} + 2\delta.\end{aligned}$$
$V[x_k](x) \geq 0$ for all $k \geq 0$, we get $$\begin{aligned}
f(x_{k+1}) \leq f(x_*) + L_{k+1}V[x_k](x_*) - \left(L_{k+1} + m\right)V[x_{k+1}](x_*) + \widetilde{\delta} + 2\delta.\end{aligned}$$ Dividing by $L_{k+1}$, we obtain $$\begin{aligned}
&\frac{f(x_{k+1}) - f(x_*)}{L_{k+1}} + V[x_{k+1}](x_*) + \frac{m}{L_{k+1}}V[x_{k+1}](x_*) \leq V[x_k](x_*) + \frac{\widetilde{\delta}+2\delta}{L_{k+1}}.\end{aligned}$$ Next, we telescope this inequality for $k$ from $0$ to $N-1$ $$\begin{aligned}
&\sum_{k=0}^{N-1}\frac{f(x_{k+1}) - f(x_*)}{L_{k+1}} + V[x_{N}](x_*) + \sum_{k=0}^{N-1}\frac{mV[x_{k+1}](x_*)}{L_{k+1}}\leq V[x_0](x_*) + (\widetilde{\delta} + 2\delta) S_N.\end{aligned}$$ In view of $V[x_{N}](x_*) \geq 0$ and (1-SC) condition, we have $$\begin{aligned}
&\sum_{k=0}^{N-1}\frac{f(x_{k+1}) - f(x_*)}{L_{k+1}} + \sum_{k=0}^{N-1}\frac{m{\left\lVertx_{k+1} - x_*\right\rVert}^2}{2L_{k+1}}\leq V[x_0](x_*) + (\widetilde{\delta} + 2\delta S_N).\end{aligned}$$ Dividing inequality by $S_N$ and using the convexity of square norm and $f$, we obtain $$\begin{aligned}
f(\bar{x}_N) - f(x_*) + \frac{m}{2}{\left\lVert\bar{x}_N - x_*\right\rVert}^2 \leq \frac{V[x_0](x_*)}{S_N} + \widetilde{\delta} + 2\delta.\end{aligned}$$
Let us assume that $\bar{L}_{k+1} \leq L$ for all $k \geq 0$ which means thah $L_{k+1} \leq 2L$ (see Remark \[remark:grad\_method\]) and $S_N \geq \frac{N}{2L}$. From Theorem \[Th:str\_conv\_adap\_pos\_m\] we get $$\begin{aligned}
f(\bar{x}_N) - f(x_*) + \frac{m}{2}{\left\lVert\bar{x}_N - x_*\right\rVert}^2 \leq \frac{2LV[x_0](x_*)}{N} + \widetilde{\delta} + 2\delta.\end{aligned}$$
Adaptive Fast Gradient Method with $(\delta, L, \mu, m, V,\|\cdot \|)$-model
----------------------------------------------------------------------------
In this section we consider accelerated method for problems with $(\delta, L, \mu, m, V,\|\cdot \|)$-model of the objective (see Definition \[defRelStronglyConvexFast\]). The method is close to accelerated mirror-descent type of methods (see [@tseng2008accelerated; @lan2012optimal; @dvurechensky2018computational]). As in the previous subsection we assume that some constant $\bar{L}_{k+1}$ such that $(\delta_k, \bar{L}_{k+1}, \mu, m, V,\|\cdot \|)$-model of $f$ exists at $k$-th step ($k=0,..,N-1$) of Algorithm \[FastAlg2\_strong\]. . For instance, this allows to obtain Universal Fast Gradient Method in which different values of $\{\delta_k\}_{k\geq 0}$ are required (see [@nesterov2015universal; @baimurzina2017universal]).
\[Th:fast\_str\_conv\_adap\] $$\begin{aligned}
&f(x_N) - f(x_*) \leq \frac{V[u_0](x_*)}{A_N} + \frac{2\sum_{k=0}^{N-1}A_{k+1}\delta_k}{A_N} + \frac{\sum_{k=0}^{N-1}\widetilde{\delta}_k}{A_N}, \label{Th:fast_str_conv_adap:result_1} \\
&V[u_N](x_*)\leq \frac{V[u_0](x_*)}{(1 + A_N\mu + \att{A_Nm})} + \frac{2\sum_{k=0}^{N-1}A_{k+1}\delta_k}{(1 + A_N\mu + \att{A_Nm})} + \frac{\sum_{k=0}^{N-1}\widetilde{\delta}_k}{(1 + A_N\mu + \att{A_Nm})}. \label{Th:fast_str_conv_adap:result_2}
\end{aligned}$$
Despite the adaptive structure of Algorithm \[FastAlg2\_strong\] as in [@nesterov2015universal] it can be shown that in average the algorithm up to logarithmic terms requires four computations of function and two computations of $(\delta, L, \mu, m, V,\|\cdot \|)$-model per iteration.
**Input:** $x_0$ is the starting point, $\mu \geq 0$, $m \geq 0$, $\{\widetilde{\delta}_k\}_{k\geq 0}$, $\{\delta_k\}_{k\geq 0}$ and $ L_{0} > 0$. Set $y_0 := x_0$, $u_0 := x_0$, $\alpha_0 := 0$, $A_0 := \alpha_0$ Find the smallest $i_k \geq 0$ such that $$\begin{gathered}
f_{\delta_k}(x_{k+1}) \leq f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(x_{k+1}, y_{k+1}) +\frac{L_{k+1}}{2}{\left\lVertx_{k+1} - y_{k+1}\right\rVert}^2 + \delta_k,
\label{exitLDL_strong}
\end{gathered}$$ where $L_{k+1} = 2^{i_k-1}L_k$, $\alpha_{k+1}$ is the largest root of $$\begin{gathered}
\label{alpha_def_strong}
A_{k+1}{(1 + A_k \mu + \att{A_k m})}=L_{k+1}\alpha^2_{k+1},\quad A_{k+1} := A_k + \alpha_{k+1}.\end{gathered}$$ $$\begin{gathered}
y_{k+1} := \frac{\alpha_{k+1}u_k + A_k x_k}{A_{k+1}}. \label{eqymir2DL_strong}\end{gathered}$$ $$\phi_{k+1}(x)=\alpha_{k+1}\psi_{\delta_k}(x, y_{k+1}) + {(1 + A_k\mu + \att{A_k m})} V[u_k](x) + {\alpha_{k+1} \mu V[y_{k+1}](x)}.$$ $$\label{equmir2DL_strong}
u_{k+1} := {\operatorname*{argmin}_{x \in Q}}^{\widetilde{\delta}_k}\phi_{k+1}(x).$$ $$\begin{gathered}
x_{k+1} := \frac{\alpha_{k+1}u_{k+1} + A_k x_k}{A_{k+1}}. \label{eqxmir2DL_strong}\end{gathered}$$
In order to prove Theorem \[Th:fast\_str\_conv\_adap\] we need the following lemma.
\[lemma:fast\_str\_conv\] Let $\psi(x)$ be a $m$-strongly convex function, $m \geq 0$, and $$\begin{gathered}
y = {\operatorname*{argmin}_{x \in Q}}^{\widetilde{\delta}} \{\psi(x) + \beta V[z](x) + \gamma V[u](x)\},
\end{gathered}$$ where $\beta \geq 0$ and $\gamma \geq 0$. Then $$\psi(x) + \beta V[z](x) + \gamma V[u](x) \geq \psi(y) + \beta V[z](y) + \gamma V[u](y) + (\beta + \gamma + m)V[y](x) - \widetilde{\delta} ,\,\,\, \forall x \in Q.$$
the proof of Lemma \[lemma:str\_conv\].
\[lemma:fast\_strong\_main\] For all $x \in Q$, we have $$\begin{aligned}
&A_{k+1} f(x_{k+1}) - A_{k} f(x_{k}) + ({1 + A_{k+1} \mu + \att{A_{k+1} m}}) V[u_{k+1}](x) - ({1 + A_k \mu + \att{A_k m}})V[u_{k}](x)\\
&\leq \alpha_{k+1}f(x) + 2\delta_k A_{k+1} + \widetilde{\delta}_k.
\end{aligned}$$
Since by Definition \[defRelStronglyConvexFast\] with $x = y$, $f(x) - \delta \leqslant f_{\delta}(x) \leqslant f(x)$, and , we have $$\begin{aligned}
f(x_{k+1}) \leq f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(x_{k+1},y_{k+1}) + \frac{L_{k+1}}{2}{\left\lVertx_{k+1} - y_{k+1}\right\rVert}^2 + 2\delta_k.
\end{aligned}$$ Using definitions and of sequences $x_{k+1}$ and $y_{k+1}$ we can show that $$\begin{aligned}
f(x_{k+1})
&\leq f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}\left(\frac{\alpha_{k+1}u_{k+1} + A_k x_k}{A_{k+1}},y_{k+1}\right)\\
&\hspace{2em}+ \frac{L_{k+1}}{2}{\left\lVert\frac{\alpha_{k+1}u_{k+1} + A_k x_k}{A_{k+1}} - y_{k+1}\right\rVert}^2 + 2\delta_k\\
&= f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}\left(\frac{\alpha_{k+1}u_{k+1} + A_k x_k}{A_{k+1}},y_{k+1}\right)+\frac{L_{k+1} \alpha^2_{k+1}}{2 A^2_{k+1}}{\left\lVertu_{k+1} - u_k\right\rVert}^2 + 2\delta_k.
\end{aligned}$$ Since $\psi_{\delta_k}(\cdot,y)$ is convex, we have $$\begin{aligned}
f(x_{k+1})
&\leq\frac{A_k}{A_{k+1}}\left(f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(x_k, y_{k+1})\right)+\frac{\alpha_{k+1}}{A_{k+1}}\left(f_{\delta_k}(y_{k+1}) +
\psi_{\delta_k}(u_{k+1}, y_{k+1})\right)\\
&\hspace{2em}+ \frac{L_{k+1} \alpha^2_{k+1}}{2 A^2_{k+1}}{\left\lVertu_{k+1} - u_k\right\rVert}^2 + 2\delta_k.
\end{aligned}$$ In view of definition for the sequence $\alpha_{k+1}$, we obtain $$\begin{aligned}
f(x_{k+1})&\leq\frac{A_k}{A_{k+1}}\left(f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(x_k,y_{k+1})\right) +\frac{\alpha_{k+1}}{A_{k+1}}\Big(f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(u_{k+1},y_{k+1})\\
&\hspace{2em}+ \frac{{1 + A_k \mu + \att{A_k m}}}{2 \alpha_{k+1}}{\left\lVertu_{k+1} - u_k\right\rVert}^2\Big) + 2\delta_k.
\end{aligned}$$ Using (1-SC) condition w.r.t. norm for $V$ and the left inequality in , we get $$\begin{aligned}
\begin{split}
\label{lemma:fast_str_conv:ineq_1}
f(x_{k+1}) &\leq \frac{A_k}{A_{k+1}}f_{\delta_k}(x_{k})+\frac{\alpha_{k+1}}{A_{k+1}}\Big(f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(u_{k+1},y_{k+1})\\
&\hspace{2em}+ \frac{{1 + A_k \mu + \att{A_k m}}}{\alpha_{k+1}}V[u_k](u_{k+1})\Big) + 2\delta_k.
\end{split}
\end{aligned}$$ By Lemma \[lemma:fast\_str\_conv\] for the optimization problem in , it holds that $$\begin{aligned}
&\alpha_{k+1}\psi_{\delta_k}(u_{k+1}, y_{k+1}) + (1 + A_k \mu + \att{A_k m})V[u_{k}](u_{k+1}) + \alpha_{k+1} \mu V[y_{k+1}](u_{k+1}) \\
&\hspace{2em}+ (1 + A_{k+1}\mu + \att{A_{k+1} m})V[u_{k+1}](x) - \widetilde{\delta}_k \\
&\leq \alpha_{k+1}\psi_{\delta_k}(x, y_{k+1}) + (1 + A_k \mu + \att{A_k m})V[u_{k}](x) + \alpha_{k+1} \mu V[y_{k+1}](x).\end{aligned}$$ From the fact that $V[y_{k+1}](u_{k+1}) \geq 0$, we have $$\begin{aligned}
\begin{split}
\label{lemma:fast_str_conv:ineq_2}
&\alpha_{k+1}\psi_{\delta_k}(u_{k+1}, y_{k+1}) + (1 + A_k\mu + \att{A_k m})V[u_{k}](u_{k+1})\\
&\leq \alpha_{k+1}\psi_{\delta_k}(x, y_{k+1}) + (1 + A_k\mu + \att{A_k m})V[u_{k}](x)\\
&\hspace{2em}- (1 + A_{k+1}\mu + \att{A_{k+1} m})V[u_{k+1}](x) + \alpha_{k+1} \mu V[y_{k+1}](x) + \widetilde{\delta}_k.
\end{split}\end{aligned}$$ Combining and , we obtain $$\begin{aligned}
f(x_{k+1})
&\leq \frac{A_k}{A_{k+1}} f(x_k)+\frac{\alpha_{k+1}}{A_{k+1}}\Big(f_{\delta_k}(y_{k+1}) + \psi_{\delta_k}(x,y_{k+1}) + {\mu V[y_{k+1}](x)}\\
&\hspace{2em}+ \frac{{1 + A_k \mu + \att{A_k m}}}{\alpha_{k+1}}V[u_k](x) - \frac{{1 + A_{k+1} \mu + \att{A_{k+1} m}}}{\alpha_{k+1}}V[u_{k+1}](x) + \frac{\widetilde{\delta}_k}{\alpha_{k+1}}\Big) + 2\delta_k.\end{aligned}$$ We finish the proof of Lemma \[lemma:fast\_strong\_main\] applying left inequality in $$\begin{aligned}
f(x_{k+1})
&\leq \frac{A_k}{A_{k+1}} f(x_k) +
\frac{\alpha_{k+1}}{A_{k+1}} f(x) \\
&\hspace{2em}+ \frac{{1 + A_k \mu + \att{A_k m}}}{A_{k+1}}V[u_k](x) - \frac{{1 + A_{k+1} \mu + \att{A_{k+1} m}}}{A_{k+1}}V[u_{k+1}](x) + 2\delta_k + \frac{\widetilde{\delta}_k}{A_{k+1}}.
\end{aligned}$$
We telescope the inequality in Lemma \[lemma:fast\_strong\_main\] for $k$ from $0$ to $N-1$ and take $x = x_*$: $$\begin{aligned}
\begin{split}
\label{Th:str_conv_adap:proof:ineq_1}
A_{N} f(x_{N})\leq A_{N}f(x_*) + V[u_0](x_*) - (1 + A_N(\mu + m)) V[u_N](x_*) + 2\sum_{k=0}^{N-1}A_{k+1}\delta_k + \sum_{k=0}^{N-1}\widetilde{\delta}_k.
\end{split}\end{aligned}$$ Since $V[u_{k+1}](x_*) \geq 0$ for all $k \geq 0$, we have $$\begin{aligned}
&A_{N} f(x_{N}) - A_{N}f(x) \leq V[u_0](x_*) + 2\sum_{k=0}^{N-1}A_{k+1}\delta_k + \sum_{k=0}^{N-1}\widetilde{\delta}_k.\end{aligned}$$ The last inequality proves .
\[lemma:a\_n\_sequence\] For all $N\geq 0$, $$\begin{aligned}
A_N \geq \max\left\{\frac{1}{2}\left(\sum_{k=0}^{N-1}\frac{1}{\sqrt{L_{k+1}}}\right)^2, \frac{1}{L_1}\prod_{k=1}^{N-1}\left(1 + \sqrt{\frac{\mu + m}{2L_{k+1}}}\right)^2\right\}.\end{aligned}$$
\[remark:fast\_strong\_lip\] Let us assume that function $f$ has $L$-Lipschitz continuous gradient. This means that for $L_k \geq L$ inequality always holds, whence, $L_k \leq 2L$, assuming that $L_0 \leq L$. From Lemma \[lemma:a\_n\_sequence\] we have $A_N \geq \frac{N^2}{4L}$ and $$\begin{aligned}
A_N \geq \frac{1}{2L} \left(1 + \frac{1}{2}\sqrt{\frac{\mu + m}{L}}\right)^{2(N-1)} \geq \frac{1}{2L}\exp\left(\frac{N-1}{2}\sqrt{\frac{\mu + m}{L}}\right).\end{aligned}$$ In the last inequality we used inequality $\log(1+2x) \geq x$ for all $x \in [0,\frac{1}{4}]$. Combining Theorem \[Th:fast\_str\_conv\_adap\] and Lemma \[lemma:a\_n\_sequence\] we have $$\begin{aligned}
\begin{split}
\label{remark:optimal_conv_rate}
f(x_N) - f(x_*) &\leq \min\left\{\frac{4L}{N^2}, 2L\exp\left(-\frac{N-1}{2}\sqrt{\frac{\mu + m}{L}}\right)\right\}V[u_0](x_*)\\
&\hspace{2em}+ \frac{2\sum_{k=0}^{N-1}A_{k+1}\delta_k}{A_N} + \frac{\sum_{k=0}^{N-1}\widetilde{\delta}_k}{A_N}.
\end{split}\end{aligned}$$ The first term in the right hand side of ineqaulity up to a constant factor is optimal for $\mu$–strongly convex functions with $L$-Lipschitz continuous gradient.
Note that in [@devolder2013firstCORE] for non-adaptive fast gradient method with $(\delta, L, \mu)$–oracle for the case when $\delta_k$ is a constant it is shown that $$\frac{\sum_{k=0}^{N-1}A_{k+1}\delta}{A_N} \leq \min\left\{\left(\frac{1}{3}k + 2.4\right), \left(1 + \sqrt{\frac{L}{\mu}}\right)\right\}\delta.$$ This means that for $\mu > 0$ error $\delta$ does not accumulate.
\[remark:FastGradConvRate\] In view of assumptions from Remark \[remark:fast\_strong\_lip\]. For the case when $\mu = m = 0$ Algorithm \[FastAlg2\_strong\] can guarantee the following convergence rate $$\begin{aligned}
\label{eq:FastGradConvRate}
f(x_N) - f_* &\leq \frac{4LV[x_0](x_*)}{N^2} + 2N\delta + \frac{4L\widetilde{\delta}}{N}.\end{aligned}$$ A similar result was shown in [@tyurin2017fast].
Let us analyze the convergence rate of the argument from Theorem \[Th:fast\_str\_conv\_adap\]. There are two different scenarios:
1. $\mu = m = 0$. In this case we have: $$\begin{aligned}
V[u_N](x_*)\leq V[u_0](x_*) + 2\sum_{k=0}^{N-1}A_{k+1}\delta_k +\sum_{k=0}^{N-1}\widetilde{\delta}_k.
\end{aligned}$$ For non-strongly convex case we can only bound $V[u_N](x_*)$ by $V[u_0](x_*)$ up to additive noise.
2. $\mu + m > 0$. Using Lemma \[lemma:a\_n\_sequence\] we can see that Theorem \[Th:fast\_str\_conv\_adap\] guarantees linear convergence in argument up to additive noise.
Note that convergence rates for the objective and for the argument are obtained for different sequences $x_N$ and $u_N$, respectively.
Universal conditional gradient (Frank–Wolfe) method {#univ_frank_wolfe}
---------------------------------------------------
Let us show an example of $(\delta, L, \mu, m, V,\|\cdot \|)$–model application. We use Algorithm \[FastAlg2\_strong\] as a proxy method for universal Frank–Wolfe method with $\mu = 0$ and $m = 0$. In order to construct universal Frank–Wolfe method let us introduce the following constraints to the optimization problem :
1. The set $Q$ is bounded w.r.t $V[y](x)$: $\exists R_Q \in \mathbb{R}:~ V[y](x) \leq R_Q^2\quad \forall x, y \in Q.$
2. The function $f(x)$ has Holder-continuous subgradients: $${\left\lVert\nabla f(x) - \nabla f(y)\right\rVert}_* \leq L_\nu{\left\lVertx - y\right\rVert}^\nu\,\,\,\,\forall x,y \in Q.$$ From this we can get an inequality (see [@nesterov2015universal]): $$\begin{gathered}
0 \leq f(x) - f(y) - \langle\nabla f(y), x - y \rangle \leq \frac{L(\delta)}{2}{\left\lVertx - y\right\rVert}^2 + \delta \,\,\,\, \forall x,y \in Q,
\end{gathered}$$ where $$\begin{gathered}
L(\delta)=L_\nu\left[\frac{L_\nu}{2\delta}\frac{1-\nu}{1+\nu}\right]^\frac{1-\nu}{1+\nu}\end{gathered}$$ and $\delta > 0$ is a free parameter.
First, let us take $\delta_k = \epsilon \frac{\alpha_{k+1}}{4A_{k+1}}$. With this choice of $\delta_k$ and the fact that the objective function has Holder continues subgradient as in Theorem 3 from [@nesterov2015universal] we can get the following inequality for $A_N$: $$\label{AN_universal2} A_N \geq \frac{N^\frac{1+3\nu}{1+\nu}\epsilon^\frac{1-\nu}{1+\nu}}{2^\frac{3+5\nu}{1+\nu}L_\nu^\frac{2}{1+\nu}}.$$
It it shown in [@tyurin2017fast] that in order to construct the classical Frank–Wolfe method instead of an auxiliary problem $\phi_{k+1}(x)=\alpha_{k+1}\psi_{\delta_k}(x, y_{k+1}) + V[u_k](x)$ in Algorithm \[FastAlg2\_strong\] for $m = 0$ and $\mu = 0$ (see also section 3, [@tyurin2017fast]) we can take an auxiliary problem $\widetilde{\phi}_{k+1}(x) = \alpha_{k+1}\psi_{\delta_k}(x, y_{k+1})$. Let us look at this substitution from the view of $\widetilde{\delta}_k$–precision from Definition \[solNemirovskiy\]. As in [@tyurin2017fast] we can show that an error in sense of Definition \[solNemirovskiy\] would not be greater than $2R^2_Q$. Therefore, we can take $\widetilde{\delta}_k = 2R^2_Q$. From Theorem \[Th:fast\_str\_conv\_adap\] we can get the following inequality: $$f(x_N) - f(x_*) \leq \frac{R_Q^2}{A_N} + \frac{{\varepsilon}}{2} + \frac{2R^2_QN}{A_N} \leq \frac{3R^2_QN}{A_N} + \frac{{\varepsilon}}{2}.$$
Using inequality , we can finally get the following upper bound for the number of steps in order to get ${\varepsilon}$-solution:
$$\begin{gathered}
N \leq \inf_{\nu\in(0,1]}\left[2^\frac{3+4\nu}{\nu}\left(\frac{L_\nu R_Q^{1+\nu}}{{\varepsilon}}\right)^\frac{1}{\nu}\right].\end{gathered}$$
This inequality for $\nu = 1$ has the same convergence rate as in the classical Frank–Wolfe method, however, universal Frank–Wolfe method can work with any function that has Holder continuous subgradients with constant $\nu > 0$. Note that in the classical Frank–Wolfe method $\psi_{\delta_k}(x, y_{k+1}) = \langle \nabla f(y_{k+1}), x - y_{k+1} \rangle$. However, here we assume that $\psi_{\delta_k}(x, y_{k+1})$ can have a more general representation (see Definition \[defRelStronglyConvexFast\]).
Inexact Model for Variational Inequalities {#VI}
==========================================
In this section, we go beyond minimization problems and propose an abstract inexact model counterpart for variational inequalities. we propose a universal method for variational inequalities with complexity $
O\left(\inf_{\nu \in [0,1]}\left(\frac{1}{\varepsilon}\right)^{\frac{2}{1+\nu}}\right)$, where $\varepsilon$ is the desired accuracy of the solution and $\nu$ is the Hölder exponent of the operator. According to the lower bounds in [@Optimal], this algorithm is optimal for $\nu = 0$ (bounded variation of the operator) and $\nu = 1$ (Lipschitz continuity of the operator). Based on the model for VI and functions, we introduce inexact model for saddle-point problems (see Definition \[DefSaddleModel\]). We are also motivated by mixed variational inequalities [@Konnov_2017; @Bao_Khanh] and composite saddle-point problems [@chambolle2011first-order].
Formally speaking, we consider the problem of finding the solution $x_*\in Q$ for VI in the following abstract form $$\label{eq13}
\psi(x,x_*)\geqslant 0 \quad \forall x \in Q$$ for some convex compact set $Q\subset\mathbb{R}^n$ and some function $\psi:Q\times Q\rightarrow\mathbb{R}$. Assuming the abstract monotonicity of the function $\psi$ $$\label{eq:abstr_monot}
\psi(x,y)+\psi(y,x)\leqslant0\;\;\;\forall x,y\in Q,$$ any solution to is a solution of the following inequality $$\label{eq115}
\max_{x\in Q}\psi(x_*,x)\leqslant 0 $$
In the general case, we make an assumption about the existence of a solution $x_*$ of the problem . As a particular case, if for some operator $g: Q \rightarrow\mathbb{R}^n$ we set $\psi(x,y)=\langle g(y),x-y\rangle\;\;\forall x,y\in Q$, then and are equivalent, respectively, to a standard strong and weak variational inequality with the operator $g$.
We propose an adaptive proximal method for the problems and . We start with a concept of $(\delta, L, V)$-model for such problems.
\[Def\_Model\_VI\] We say that function $\psi$ has $(\delta, L\pd{,V})$-model $\psi_{\delta} (x, y)$ for some fixed values if the following properties hold for each $x, y, z \in Q$:
1. $\psi(x, y) \leq \psi_{\delta}(x, y) + \delta$;
2. $\psi_{\delta} (x, y)$ convex in the first variable;
3. $\psi_{\delta}(x,x)=0$;
4. ([*abstract $\delta$-monotonicity*]{}) $$\label{eq:abstr_monot}
\psi_{\delta}(x,y)+\psi_{\delta}(y,x)\leq \delta;$$
5. ([*generalized relative smoothness*]{}) $$\label{VIeq20}
\psi_{\delta}(x,y)\leqslant\psi_{\delta}(x,z)+\psi_{\delta}(z,y)+ LV[z](x)+ LV[y](z)+\delta.$$
For some operator $g:Q\rightarrow\mathbb{R}^n$ and a convex function $h:Q\rightarrow\mathbb{R}^n$ choice $$\label{eq17}
\psi(x,y)=\langle g(y),x-y\rangle+h(x)-h(y)$$ leads to a [*mixed variational inequality from*]{} [@Konnov_2017; @Bao_Khanh] $$\label{eq18}
\langle g(y),y-x\rangle+h(y)-h(x)\leqslant 0,$$ which in the case of the monotonicity of the operator $g$ implies $$\label{eq19}
\langle g(x),y-x\rangle+h(y)-h(x)\leqslant 0.$$
\[BregmanVI\] Similarly to Definition \[defRelStronglyConvex\] above, in general case, we do not need the (1-SC) assumption for $V[y](x)$ in Definition \[Def\_Model\_VI\]. In some situations we make (1-SC) assumption for $V[y](x)$ (see Example \[UMP\_Example\] and Section \[UMPStrongApp\]).
Note that for $\delta=0$ the following analogue of for some fixed $a, b > 0$ $$\label{eq200}
\psi(x,y)\leqslant\psi(x,z)+\psi(z,y)+ a\|z - y\|^2 + b\|x - z\|^2 \quad \forall x, y, z \in Q$$ was introduced in [@Mastroeni]. Condition is used in many works on equilibrium programming. Our approach allows us to work with non-Euclidean set-up without (1-SC) assumption and inexactness $\delta$, that is important for the ideology of universal methods [@nesterov2015universal] (see Example \[UMP\_Example\] below).
One can directly verify that if $\psi_{\delta}(x,y)$ is $(\delta/3, L, 0, 0, V)$-model of the function $f$ at a given point $y$ then $\psi_{\delta}(x,y)$ is $(\delta, L, V)$-model in the sense of Definition \[Def\_Model\_VI\].
Let us consider some examples.
\[Examp\_Rel\_Smooth\_VI\]
\[UMP\_Example\] [**Variational Inequalities with monotone Hölder continuous operator.**]{} \[example\_universal\_g\] and for a monotone operator $g$ there exists $\nu\in[0,1]$ such that $$\label{Hold_cont_g}
{\left\lVertg(x) - g(y)\right\rVert}_* \leq L_{\nu}{\left\lVertx - y\right\rVert}^\nu,\,\,\,\forall x,y \in Q.$$
Then we have: $
\langle g(z)-g(y), z-x\rangle\leq
\|g(z) - g(y) \|_* \|z-x\| \leq L_{\nu}\|z-y\|^{\nu} \|z-x\|
$ $$\label{Hold_interpol}
\leq \frac{L(\delta)}{2}||z-x||^2+\frac{L(\delta)}{2}||z-y||^2+\delta \leq LV[z](x)+ LV[y](z) +\delta$$ for $$\label{UMP_constant} L(\delta) = \left(\frac{1}{2\delta}\right)^\frac{1-\nu}{1+\nu} L_{\nu}^{\frac{2}{1+\nu}}$$ with arbitrary $\delta > 0$. In this case $\psi_{\delta}(x, y):= \langle g(y), x - y \rangle$ is $(\delta, L,V)$-model.
Note that for the previous two examples in Algorithm \[Alg:UMPModel\] and Theorem \[thmm1inexact\] we need $V[z](x)$ to satisfy (1-SC) condition.
Next, we introduce our novel adaptive method (Algorithm \[Alg:UMPModel\]) for abstract variational inequalities with inexact $(\delta, L, V)$-model. This method adapts to the local values of $L$ and allows us to construct universal method for variational inequalities by applying it to VI with Hölder interpolation for $\delta = \frac{\varepsilon}{2}$ and $L = L\left(\frac{\varepsilon}{2}\right)$.
accuracy ${\varepsilon}> 0$, oracle error $\delta >0$, initial guess $L_{0} >0$, prox set-up: $d(x)$, $V[z] (x)$. Set $k=0$, $z_0 = \arg \min_{u \in Q} d(u)$. Find the smallest integer $i_k \geq 0$ such that $$\label{eqUMP23}
\begin{split}
\hspace{-3em}\psi_{\delta}(z_{k+1}, z_{k})\leq \psi_{\delta}(z_{k+1}, w_{k})+\psi_{\delta}(w_k,z_k)+ L_{k+1}(V[z_k](w_k) + V[w_k](z_{k+1})) + \delta,
\end{split}$$ where $L_{k+1}=2^{i_k-1}L_{k}$ and $$\begin{aligned}
w_k&={\mathop {\operatorname*{argmin}}\limits_{x\in Q}} \left\{\psi_{\delta}(x, z_k)+ L_{k+1}V[z_k](x) \right\}.
\label{eq:UMPwStepMod} \\
z_{k+1}&={\mathop {\operatorname*{argmin}}\limits_{x\in Q}} \left\{\psi_{\delta}(x, w_k) + L_{k+1}V[z_k](x) \right\}. \label{eq:UMPzStepMod}
\end{aligned}$$ $S_N:= \sum_{k=0}^{N-1}\frac{1}{L_{k+1}}\geqslant \frac{\max\limits_{x \in Q}V[x^0](x)}{\varepsilon}.$ $\widehat{w}_N = \frac{1}{\sum_{k=0}^{N-1}\frac{1}{L_{k+1}}}\sum_{k=0}^{N-1}\frac{1}{L_{k+1}}w_k$.
.
\[thmm1inexact\] For Algorithm \[Alg:UMPModel\] the following inequality holds $$- \frac{1}{S_N}\sum_{k=0}^{N-1}\frac{\psi_{\delta}(x,w_{k})}{L_{k+1}} \leq \frac{V[z_0](x)}{S_N} + \delta + 2\widetilde{\delta} \quad \forall x \in Q.$$ It means that: $$\max\limits_{u \in Q}\psi(\widehat{w}_N,u)
\leq \frac{2L \max_{u\in Q}V[z_0](u)}{N} + 3\delta + 2\widetilde{\delta}$$
Note that the Algorithm \[Alg:UMPModel\] works no more than $$\label{est_th4.5}
\left\lceil\frac{2L \max_{u\in Q}V[z_0](u)}{\varepsilon}\right\rceil$$ iterations.
After $(k+1)$-th iteration ($k=0,1,2\ldots$) from and we have for each $u \in Q$: $$\psi_{\delta}(w_k, z_k)\leqslant\psi_{\delta}(u, z_k) +L_{k+1}V[z_k](u)-L_{k+1}V[w_k](u)- L_{k+1}V[z_k](w_k) + \widetilde{\delta}$$ and $$\psi_{\delta}(z_{k+1}, w_k) \leq \psi_{\delta}(u, w_k)+L_{k+1}V[z_k](u)-L_{k+1}V[z_{k+1}](u)-L_{k+1}V[z_k](z_{k+1}) + \widetilde{\delta}.$$ The first inequality means that $$\psi_{\delta}(w_k, z_k)\leqslant\psi_{\delta}(z_{k+1}, z_k) +L_{k+1}V[z_k](z_{k+1})-L_{k+1}V[w_k](z_{k+1})- L_{k+1}V[z_k](w_k) + \widetilde{\delta}.$$ Taking into account , we obtain for all $u \in Q$ $$-\psi_{\delta}(u, w_k) \leq L_{k+1}V[z_k](u)-L_{k+1}V[z_{k+1}](u) + \delta + 2\widetilde{\delta}.$$ So, the following inequality holds: $$-\sum_{k=0}^{N-1} \frac{\psi_{\delta}(u,w_{k})}{L_{k+1}} \leq V[z_0](u) - V[z_N](u) + S_N (\delta + 2\widetilde{\delta}).$$ By virtue of and the choice of $L_{0}\leqslant 2L$, it is guaranteed that $L_{k+1}\leqslant 2L\;\;\forall k=\overline{0,N-1}$ and we have from Definition \[Def\_Model\_VI\] $$\begin{aligned}
\label{eq30}
& \max\limits_{u \in Q} \psi(\widehat{w}_N, u) \leqslant \max\limits_{u \in Q} \psi_{\delta}(\widehat{w}_N, u)+\delta \\
&\leqslant
-\frac{1}{S_N}\sum_{k=0}^{N-1}\frac{\psi_{\delta}(u,w_{k})}{L_{k+1}} + 2\delta \leqslant \frac{2L \max_{u\in Q}V[z_0](u)}{N} + 3\delta + 2\widetilde{\delta}. \notag \end{aligned}$$
For universal method to obtain precision $\varepsilon$ we can choose $\delta = \frac{\varepsilon}{2}$ and $L = L\left(\frac{\varepsilon}{2}\right)$ according to and and the estimate reduces to $$\label{est_rem8}
\left\lceil 2 \inf_{\nu\in[0,1]}\left(\frac{2L_{\nu}}{{\varepsilon}} \right)^{\frac{2}{1+\nu}} \cdot \max_{u\in Q}V[z_0](u)\right\rceil.$$
Note that estimate is optimal for variational inequalities and saddle-point problems in the cases $\nu = 0$ and $\nu = 1$.
Thus, the introduced concept of the $(\delta, L, V)$-model for variational inequalities allows us to extend the previously proposed universal method for VI a wider class of problems, including [*mixed variational inequalities*]{} [@Konnov_2017; @Bao_Khanh] and [*composite saddle-point problems*]{} [@chambolle2011first-order].
Now we introduce inexact model for saddle-point problems. The solution of variational inequalities reduces the so-called saddle points problems, in which for a convex in $u$ and concave in $v$ functional $f(u,v):\mathbb{R}^{n_1+n_2}\rightarrow\mathbb{R}$ ($u\in Q_1\subset\mathbb{R}^{n_1}$ and $v\in Q_2\subset\mathbb{R}^{n_2}$) needs to be found the point $(u_*, v_*)$ such that: $$\label{eq31}
f(u_*,v)\leqslant f(u_*,v_*)\leqslant f(u,v_*)$$ for arbitrary $u\in Q_1$ and $v\in Q_2$. Let $Q=Q_1\times Q_2\subset\mathbb{R}^{n_1+n_2}$. For $x=(u,v)\in Q$, we assume that $||x||=\sqrt{||u||_1^2+||v||_2^2}$ ($||\cdot||_1$ and $||\cdot||_2$ are the norms in the spaces $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$). We agree to denote $x=(u_x,v_x),\;y=(u_y,v_y)\in Q$.
It is well known that for a sufficiently smooth function $f$ with respect to $u$ and $v$ the problem reduces to VI with an operator $
g(x)= (f_u'(u_x,v_x), \; - f_v'(u_x,v_x))$.
For saddle-point problems we propose some adaptation of the concept of the $(\delta, L, V)$-model for abstract variational inequality.
\[DefSaddleModel\] We say that the function $\psi_{\delta}(x,y)$ $(\psi_{\delta}:\mathbb{R}^{n_1+n_2}\times\mathbb{R}^{n_1\times n_2}\rightarrow\mathbb{R})$ is a $(\delta,L,V)$-model for the saddle-point problem if the conditions (ii) – (v) of Definition \[Def\_Model\_VI\] hold and in addition $$\label{eq33}
f(u_y,v_x)-f(u_x,v_y)\leqslant-\psi_{\delta}(x,y) + \delta \quad \forall x, y \in Q.$$
The proposed concept of the $(\delta,L,V)$-model for saddle-point problems is quite applicable, for example, for composite saddle point problems of the form considered in the popular article [@chambolle2011first-order]: $$\label{eq34}
f(u,v)=\tilde{f}(u,v)+h(u)-\varphi(v)$$ for some convex in $u$ and concave in $v$ subdifferentiable functions $\tilde{f}$, as well as convex functions $h$ and $\varphi$. In this case, we can put $$\label{eq35}
\psi_{\delta}(x,y)=\langle\tilde{g}(y),x-y\rangle+h(u_x)+\varphi(v_x)-h(u_y)-\varphi(v_y),$$ where $$\tilde{g}(y)=
\begin{pmatrix}
\tilde{f}_u'(u_y,v_y)\\
-\tilde{f}_v'(u_y,v_y)
\end{pmatrix}.$$
Theorem \[thmm1inexact\] implies
If for the saddle problem there is a $(\delta,L,V)$-model $\psi_{\delta}(x,y)$, then after stopping the algorithm we get a point $$\label{eq36}
\widehat{y}_N=(u_{\widehat{y}_N},v_{\widehat{y}_N}):=(\widehat{u}_N, \widehat{v}_N):=\frac{1}{S_N}\sum_{k=0}^{N-1}\frac{y_{k}}{L_{k+1}},$$ for which the following inequality is true: $$\label{eq37}
\max_{v\in Q_2}f(\widehat{u}_N, v)-\min_{u\in Q_1}f(u, \widehat{v}_N)\leqslant \frac{2L \max_{(u, v) \in Q} V[u_0, v_0](u, v)}{N} +2\tilde{\delta}+2\delta.$$
Inexact Model for Strongly Monotone VI {#UMPStrongApp}
======================================
In this section similarly with the concept of ($\delta, L, \mu, m, V$)-model in optimization we consider inexact model for VI with a stronger version of monotonicity condition .
\[Def\_Model\_VI\_2\] We say that functional $\psi$ has $(\delta, L, \mu, V)$-model $\psi_{\delta} (x, y)$ at a given point $y$ if the following properties hold for each $x, y, z \in Q$:
1. $\psi(x, y) \leq \psi_{\delta}(x, y) + \delta$;
2. $\psi_{\delta} (x, y)$ convex in the first variable;
3. $\psi_{\delta} (x, y)$ continuous in $x$ and $y$;
4. $\psi_{\delta}(x,x)=0$;
5. ([*$\mu$-strong $\delta$-monotonicity*]{}) $$\label{eq:abstr_trong_monot}
\psi_{\delta}(x,y)+\psi_{\delta}(y,x)+\mu \|x-y\|^2\leq \delta;$$
6. ([*generalized relative smoothness*]{}) $$\label{Strongeq20}
\psi_{\delta}(x,y)\leqslant\psi_{\delta}(x,z)+\psi_{\delta}(z,y)+ LV[z](x)+ LV[y](z)+\delta$$ for some fixed values $L>0$, $\delta>0$.
We note that we cannot replace $\|x-y\|^2$ on $V[y](x)$ in since it is essentially used in the proof of Theorem \[Th:RUMPCompl\].
Now we propose method with linear rate of convergence for VI with $(\delta, L, \mu, V)$-model. We slightly modify the assumptions on prox-function $d(x)$. Namely, we assume that $\operatorname*{argmin}_{x \in Q} d(x) = 0$ and that $d$ is bounded on the unit ball in the chosen norm $\|\cdot\|$, that is $$d(x) \leq \frac{\Omega}{2}, \quad \forall x\in Q : \|x \| \leq 1,
\label{eq:dUpBound}$$ where $\Omega$ is some known constant. Note that for standard proximal setups $\Omega = O(\ln \text{dim}E)$. Finally, we assume that we are given a starting point $x_0 \in Q$ and a number $R_0 >0$ such that $\| x_0 - x_* \|^2 \leq R_0^2$, where $x^*$ is the solution to abstract VI. The procedure of restating of Algorithm \[Alg:UMPModel\] is applicable for abstract strongly monotone variational inequalities.
accuracy ${\varepsilon}> 0$, $\mu >0$, $\Omega$ s.t. $d(x) \leq \frac{\Omega}{2} \ \forall x\in Q: \|x\| \leq 1$; $x_0, R_0 \ s.t. \|x_0-x_*\|^2 \leq R_0^2.$ Set $p=0,d_0(x)=d\left(\frac{x-x_0}{R_0}\right)$. Set $x_{p+1}$ as the output of Algorithm \[Alg:UMPModel\] after $N_p$ iterations of Algorithm \[Alg:UMPModel\] with prox-function $d_{p}(\cdot)$ and stopping criterion $\sum_{k=0}^{N_p-1}\frac{1}{L_{k+1}} \geq \frac{\Omega}{\mu}$. Set $R_{p+1}^2 = R_0^2 \cdot 2^{-(p+1)}$. Set $d_{p+1}(x) \leftarrow d\left(\frac{x-x_{p+1}}{R_{p+1}}\right)$. Set $p=p+1$. $p > \log_2\frac{R_0^2}{{\varepsilon}}$ $x_{p+1}$.
\[Th:RUMPCompl\] Assume that $\psi_{\delta}$ is a $(\delta, L, \mu, V)$-model for $\psi$. Also assume that the prox function $d(x)$ satisfies and the starting point $x_0 \in Q$ and a number $R_0 >0$ are such that $\| x_0 - x_* \|^2 \leq R_0^2$, where $x_*$ is the solution to . Then, for each $p\geq 0$ $$\|x_p - x_*\|^2 \leq R_0^2\cdot 2^{-p} +\frac{\delta}{\mu}+\frac{2\widetilde{\delta}}{\mu} \leq \varepsilon + \frac{\delta}{\mu}+\frac{2\widetilde{\delta}}{\mu}.$$ The total number of iterations of the inner Algorithm \[Alg:UMPModel\] does not exceed $$\label{eq_abst_strong_monot}
\left\lceil \frac{2L\Omega}{\mu}\cdot \log_2 \frac{R_0^2}{{\varepsilon}}\right\rceil,$$ where $\Omega$ satisfies .
We show by induction that for $p \geq 0$ $$\|x_p - x_*\|^2 \leq R_0^2\cdot 2^{-p} + \frac{\delta}{\mu}+\frac{2\widetilde{\delta}}{\mu},$$ which leads to the statement of the Theorem. For $p=0$ this inequality holds by the Theorem assumption. Assuming that it holds for some $p\geq 0$, our goal is to prove it for $p+1$ considering the outer iteration $p+1$. Observe that the function $d_{p}(x)$ defined in Algorithm \[Alg:RUMP\] is 1-strongly convex w.r.t. the norm $\|\cdot\| / R_{p}$.
This means that, at each step $k$ of inner Algorithm \[Alg:UMPModel\], $L_{N_p}$ changes to $L_{N_p} \cdot R_{p}^2$. Using the definition of $d_{p}(\cdot)$ and , we have, since $x_p = \operatorname*{argmin}_{x \in Q} d_p(x)$ $$V_{p}[x_{p}](x_*) = d_{p}(x_{*}) - d_{p}(x_{p}) - {\langle}\nabla d_{p}(x_{p}), x_{*} - x_{p} {\rangle}\leq d_{p}(x_{*}) \leq \frac{\Omega}{2}.$$ Denote by $$S_{N_p}:= \sum_{k=0}^{N_p-1}\frac{1}{L_{k+1}}.$$
Thus, by Theorem \[thmm1inexact\], taking $u = x_*$, we obtain $$- \frac{1}{S_{N_p}} \sum_{k=0}^{N_p-1} \frac{\psi_{\delta}(x_*, w_k)}{L_{k+1}} \leq \frac{R_{p}^2V_{p}[x_{p}](x_{*})}{S_{N_p}} + \delta + 2\widetilde{\delta} \leq \frac{\Omega R_p^2}{2S_{N_{p}}} + \delta + 2\widetilde{\delta}.$$ Since the operator $\psi$ is continuous and abstract monotone, we can assume that the solution to weak VI is also a strong solution and $- \psi(w_k, x_*) \leq 0$, $k=0,...,N_p-1$ and, by Definition \[Def\_Model\_VI\_2\] (i), $-\psi_{\delta}(\omega_k,x_*)\leq\delta$ ($k=0,\ldots,N_p-1$). This and gives, that for each $k=0,...,N_p-1$, $$\begin{aligned}
- \psi_{\delta} (x_*, w_k) &\geq - \delta - \psi_{\delta}(x_*, w_k) - \psi_{\delta}(w_k, x_*) \geq - \delta + \mu\|w_k-x_*\|^2, \notag \\
-\psi_{\delta}(x_*,\omega_k)&\geq-\delta -\psi_{\delta}(x_*,\omega_k)-\psi_{\delta}(\omega_k,x_*)\geq - \delta +\mu\|\omega_k-x_*\|^2. \notag\end{aligned}$$
Thus, by convexity of the squared norm, we obtain $$\begin{aligned}
- 2\delta + \mu \|x_{p+1}-x_*\|^2 & = - 2 \delta + \mu \left\|\frac{1}{S_{N_p}} \sum_{k=0}^{N_p-1} \frac{w_k}{L_{k+1}}-x_*\right\|^2 \leq - 2\delta + \frac{\mu}{S_{N_p}} \sum_{k=0}^{N_p-1} \frac{\|w_k-x_*\|^2}{L_{k+1}} \notag \\
& \leq - 2 \delta - \frac{1}{S_{N_p}} \sum_{k=0}^{N_p-1} \frac{\psi_{\delta}(x_*, w_k)}{L_{k+1}} \leq \frac{\Omega R_p^2}{2 S_{N_p}} - \delta + 2\widetilde{\delta}. \notag\end{aligned}$$ Using the stopping criterion $S_{N_p} \geq \frac{\Omega}{\mu}$ we have $$\begin{aligned}
\|x_{p+1}-x_*\|^2 &\leq \frac{R_p^2}{2} +\frac{\delta+2\widetilde{\delta}}{\mu}= \frac{1}{2} R_0^2 \cdot 2^{-p} +\frac{\delta+2\widetilde{\delta}}{\mu} \notag \\
& = R_0^2 \cdot 2^{-(p+1)} + \frac{\delta+2\widetilde{\delta}}{\mu}, \notag\end{aligned}$$ which finishes proof by induction.
If for some $m>0$ $\psi_{\delta}(x,y)$ is a $m$-strongly convex function in $x$ then for Algorithm \[Alg:RUMP\] we can prove estimate $$\|x_p - x_*\|^2 \leq R_0^2\cdot 2^{-p} +\frac{\delta}{m+\mu}+\frac{2\widetilde{\delta}}{m+\mu} \leq \varepsilon + \frac{\delta}{m+\mu}+\frac{2\widetilde{\delta}}{m+\mu}$$ for each $p\geq 0$ and instead of we obtain $$\label{eq_abst_strong_monot_1}
\left\lceil \frac{2L\Omega}{m+\mu}\cdot \log_2 \frac{R_0^2}{{\varepsilon}}\right\rceil.$$
Conclusion {#S:Conclusion}
==========
we consider convex optimization problem . It is well known (see [@devolder2014first; @dvurechensky2017universal; @gorbunov2019optimal]) that if , for all $x,y \in Q$, $$\label{eq:inexact_oracle_DGN}
f(y) + {\langle}\nabla_{\delta} f(y), x-y {\rangle}- \delta_1 \le f(x) \le f(y) + {\langle}\nabla_{\delta} f(y), x-y {\rangle}+ \frac{L}{2}\|x-y\|^2_2 + \delta_2,$$ then Gradient Method (GM) and Fast Gradient Method (FGM) $$\label{estimate}
f(x_N) - f(x_*) = O\left(\frac{LR^2}{N^p} + \delta_1 + N^{p-1}\delta_2\right),$$ where $p=1$ [corresponds to]{} GM and $p=2$ [corresponds to]{} FGM, $x_*$ – is a solution of , in the case of $\mu$-strongly convex objective (model) the estimate can be improved to $$f(x_N) - f(x_*) = O\left(\Delta f \exp\left(-O(1)\left(\frac{\mu}{L}\right)^{\frac{1}{p}}N\right) + \delta_1 + \left(\frac{L}{\mu}\right)^{\frac{p-1}{2}}\delta_2 \right),$$ where $\Delta f = f(x^0) - f(x_*)$, $p=1$ for GM and $p=2$ for restarted FGM.
In this paper saddle-point problems and variational inequalities. We consider universal (adaptive) generalizations [@nesterov2015universal] and relative smoothness generalizations, generalizing [@bauschke2016descent; @lu2018relatively] . We also investigate the sensitivity of the convergence results to the accuracy
For the we indicate that if we additionally assume that $\delta_1, \delta_2$ are independently chosen at each iteration random variables such that $\mathds{E} \delta_1 = 0$ and $\delta_1,\sqrt{\delta_2}$ have correspondingly $\left(\delta_1'\right)^2$-subgaussian variance and $\delta_2'$-subgaussian second moment [@gorbunov2019optimal] then with high probability changes to $$f(x_N) - f(x_*) = \tilde{O}\left(\frac{LR^2}{N^p} + \frac{\delta_1'}{\sqrt{N}} + N^{p-1}\delta_2'\right).$$ From this result and mini-batch trick [@gasnikov2017universal] one can obtain the main estimates for convex and strongly convex stochastic optimization problems [@dvurechensky2016stochastic; @gasnikov2016stochasticInter; @gorbunov2019optimal; @kulunchakov2019estimate].
Auxiliary facts
===============
Let us commenton the inexact solution of the auxiliary problem.
\[RemarkInexact\] We can show that if $\widetilde{x} \in \text{Arg}\min_{x \in Q}^{\widetilde{\delta}}\Psi(x)$, then $\Psi(\widetilde{x}) - \Psi(x_*) \leq \delta$. Indeed, we have $\Psi(x_*) \geq \Psi(\widetilde{x}) + \langle h, x_* - \widetilde{x} \rangle \geq \Psi(\widetilde{x}) - \widetilde{\delta}$. The converse statement is not always true. However, for some general cases we can resolve the problem (see [@tyurin2017fast] and Example \[appendix\_inexact\]).
\[appendix\_inexact\] Let us show an example, how we can resolve the problem in Remark \[RemarkInexact\].
Proof for Lemma \[lemma:a\_n\_sequence\] {#proof:a_n_sequence}
========================================
In view of definition of sequence $\alpha_{k+1}$, we have: $$\begin{aligned}
A_N &\leq A_N(1 + \mu A_{N-1} + m A_{N-1}) = L_{N}(A_N - A_{N-1})^2\\
&\leq L_{N}(A_N^{1/2} - A_{N-1}^{1/2})^2(A_N^{1/2} + A_{N-1}^{1/2})^2 \leq 2L_{N} A_N (A_N^{1/2} - A_{N-1}^{1/2})^2.\end{aligned}$$ We can see that $$\begin{aligned}
A_N^{1/2} \geq A_{N-1}^{1/2} + \frac{1}{2L_N}\end{aligned}$$ and $$\begin{aligned}
A_N \geq \frac{1}{2}\left(\sum_{k=0}^{N-1}\frac{1}{\sqrt{L_{k+1}}}\right)^2.\end{aligned}$$ For the case when $\mu + m> 0$ we obtain: $$\begin{aligned}
(\mu + m) A_{N-1} A_{N} \leq A_N(1 + \mu A_{N-1} + m A_{N-1}) \leq 2L_{N} A_N (A_N^{1/2} - A_{N-1}^{1/2})^2.\end{aligned}$$ From the fact that $A_1 = 1 / L_1$ and the last inequality we can show that $$\begin{aligned}
A_N^{1/2} \geq \left(1 + \sqrt{\frac{\mu + m}{2L_N}}\right)A_{N-1}^{1/2} \geq \frac{1}{\sqrt{L_1}}\prod_{k=1}^{N-1}\left(1 + \sqrt{\frac{\mu + m}{2L_{k+1}}}\right).\end{aligned}$$
Fast gradient method with $(\delta, L, \mu, m, V, \|\cdot\|)$-model. Restart technique.
=======================================================================================
Let us consider the case of a strongly convex functional $f$ and show how to accelerate the work of the Algorithm \[Alg2\] using the restart technique. Let us assume that $$\psi_{\delta}(x,x_*) \ge 0\,\,\, \forall x \in Q.$$ Note that this assumption is natural, e.g. $\psi_{\delta}(x, y):= \langle\nabla f(y), x - y \rangle \,\,\, \forall x,y \in Q$. We also modify the concept of $\mu$-strong convexity in the following way
\[defRelStronglyConvexRest\] Say that the function $f$ is a left relative $\mu$-strongly convex if the following inequality $$\begin{gathered}
\mu V[x](y) \leq f(x) - f(y)- \psi_{\delta}(x, y) \quad \forall x, y \in Q
\end{gathered}$$ holds.
\[RemEuclidBreg\] Let us remind that if $d(x-y) \leq C_n{\left\lVertx-y\right\rVert}^2$ for $C_n = O(\log n)$, (where $n$ is dimension of vectors from $Q$) then $V[y](x) \leq C_n{\left\lVertx-y\right\rVert}^2$. This assumption is true for many standard proximal setups. In this case the condition of $(\mu C_n)$-strong convexity $$\mu C_n {\left\lVertx-y\right\rVert}^2 + f_\delta(y) + \psi_\delta(x,y) \leqslant f(x)$$ entails right relative strong convexity: $$\mu V[y](x) + f_\delta(y) + \psi_\delta(x,y) \leqslant f(x).$$
Note that concepts of right and left relative strongly convexity from Definitions \[defRelStronglyConvex\] and \[defRelStronglyConvexRest\] are equivalent in the case of an assumption from Remark \[RemEuclidBreg\] ($V[x](y) \leq C_n \|x - y\|^2$ for each $x, y \in Q$).
We show that using the restart technique can also accelerate the work of non-adaptive version of Algorithm \[Alg2\] ($L_{k+1} = L$) for $(\delta, L, 0, 0, V, \|\cdot\|)$-model and relative $\mu$-strogly convex function $f$ in sense Definition \[defRelStronglyConvexRest\]: $$\begin{gathered}
\mu V[x](y) + f(y)+ \psi_{\delta}(x, y) - \delta \leq f(x) \leq f(y) + \psi_{\delta}(x, y) + \frac{L}{2}\|x - y\|^2 + \delta.\end{gathered}$$ for each $x, y \in Q$. By Theorem \[Th:fast\_str\_conv\_adap\] and Remark \[remark:FastGradConvRate\]:
$$\label{FGeq1}
f(x_N) - f(x_*) \leqslant \dfrac{4LV[x_0](x_*)}{N^2}+\dfrac{4L\widetilde{\delta}}{N} + 2N\delta.$$
Consider the case of relatively $\mu$-strongly convex function $f$. We will use the restart technique to obtain the method for strongly convex functions.
Let $f$ be a left relative $\mu$-strongly convex function and $ \psi_{\delta}(x,y)$ is a\
$(\delta, L, 0, 0, V, \|\cdot\|)$-model. Let $\delta$ and $\widetilde{\delta}$ satisfy $\frac{4\mu\sqrt{10}}{L}\left(5\delta{\left \lceil{\sqrt{\frac{L}{\mu}}}\right \rceil}^3+{\widetilde{\delta}}L{\left \lceil{\sqrt{\frac{L}{\mu}}}\right \rceil} \right) \leqslant \varepsilon.$ Then, using the restarts of Algorithm \[Alg2\], we need $$N = \left \lceil \log_2 \dfrac{\mu R^2}{\varepsilon} \right\rceil \cdot \left \lceil \sqrt{\dfrac{10L}{\mu}}\right \rceil.$$ iterations to achieve $\varepsilon$ accuracy by function: $f(x_N) - f(x_*) \leqslant \varepsilon$.
By $\eqref{FGeq1}$ and Definition \[defRelStronglyConvexRest\]: $$\label{FGeq2}
\mu V[x_{N_1}](x_*) \le f(x_{N_1}) - f(x_*) \le \dfrac{4LV[x_0](x_*)}{N^2} + \dfrac{4L\widetilde{\delta}}{N} +2N\delta.$$ Let’s choose $N_1$ so that the following inequality holds: $$\label{FGeq4}
\dfrac{4L\widetilde{\delta}}{N_1}+2N_1\delta \leq \dfrac{LV[x_0](x_*)}{N_1^2}.$$ We restart method as $V[x_{N_1}](x_*)\leq\frac{V[x_0](x_*)}{2}.$ Using , we obtain an estimation for the number of iterations on the first restart: $\dfrac{5L}{\mu N_1^2} \le \dfrac{1}{2}.$ Therefore, let’s choose $$\label{FGeq3}
N_1 = \left \lceil \sqrt{\dfrac{10L}{\mu}}\right \rceil.$$ Then after $N_1$ iterations we restart method. Similarly, we restart after $N_2$ iterations, such that $V[x_{N_2}](x_*)\leq \frac{V[x_{N_1}](x_*)}{2}$. We obtain $N_2 = \left \lceil \sqrt{\frac{10L}{\mu}}\right \rceil.$ So, after $p$-th restart the total number of iterations is $M = p \cdot \left \lceil \sqrt{\frac{10L}{\mu}}\right \rceil.$
Now let’s consider how many iterations is needed to achieve accuracy $\varepsilon = f(x_{N_p}) - f(x_*)$. From and we take $p = \left \lceil \log_2 \dfrac{\mu R^2}{\varepsilon} \right\rceil$ and the total number of iterations is $M = \left \lceil \log_2 \dfrac{\mu R^2}{\varepsilon} \right\rceil \cdot \left \lceil \sqrt{\dfrac{10L}{\mu}}\right \rceil.$
We have chosen our errors as $\delta$ and $\widetilde{\delta}$ to satisfy . Indeed, from using $N_k = \left \lceil \sqrt{\dfrac{10L}{\mu}}\right \rceil$ we can deduce the following inequality: $$\varepsilon \ge \dfrac{4\sqrt{10}\mu}{L}\left(5\delta\left \lceil \sqrt{\dfrac{L}{\mu}}\right \rceil^3 + \widetilde{\delta} L \left \lceil \sqrt{\dfrac{L}{\mu}}\right \rceil \right).$$ One can see that such a choice of $\delta$ and $\widetilde{\delta}$ as above satisfies that inequality.
Partially, we can choose $\delta = O\left(\frac{\varepsilon L}{\mu \left \lceil \sqrt{\frac{10L}{\mu}}\right \rceil^3} \right)$ and $\widetilde{\delta} = O\left(\frac{\varepsilon}{\mu \left \lceil \sqrt{\frac{10L}{\mu}}\right \rceil}\right)$.
[^1]: The work in Sections 2, 3.1, Definitions 4.1 and 4.6, Algorithm 3, Theorem 4.5, Example 4.7 and Appendix C was supported by Russian Foundation for Basic Research, project number 18-31-20005 mol-a-ved. The work in Sections 3.2 and 3.3 was funded by Russian Foundation for Basic Research grant, project number 19-31-90062. The work of F. Stonyakin in Example 4.4, Remark 8 and Theorem 4.8 was supported by Russian Science Foundation, project number 18-71-00048. The work of F. Stonyakin in Section 5 was supported by Russian Science Foundation, project number 18-71-10044. The work in Section 6 was supported by Russian Foundation for Basic Research, project number 18-31-03071 mk
[^2]: see recent results on this generalization in [@tyurin2019primal].
|
---
abstract: 'Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a PDE model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modelling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing is discussed.'
author:
- Sean Kramer and Erik Bollt
bibliography:
- 'AutoSync.bib'
title: Spatially Dependent Parameter Estimation and Nonlinear Data Assimilation by Autosynchronization of a System of Partial Differential Equations
---
Introduction {#sec:intro}
============
Parameter estimation in ODEs and PDEs has developed into a vast field in applied mathematics and control engineering. For models representing important physical processes, accurate estimates of appropriate model parameters may help inform short-term management decisions by model forecasting. However, to forecast a system one requires not only accurate parameter estimates, but also full knowledge of the initial state of the system. There are widely varying and powerful methods for parameter estimation of spatio-temporal systems including, but certainly not limited to Kalman filter methods [@schiff08; @annan05; @wan00], multiple shooting methods [@Muller04; @muller02], and adjoint methods [@navon98]. A method of parameter estimation based on synchronization has drawn substantial interest [@PC90; @P96; @PJK96; @SKP96; @YCCLP07; @YP08; @SO09; @QBCKA09; @SSLP10; @BLP11; @SP11]. Applications include communications and cryptography [@SKP96], electronics and circuit dynamics [@PJK96; @KTP97], and cardiac cell dynamics [@SSLP10] to name just a few. There are currently several methods to estimate parameters based on synchronization. One approach is to optimize a time-averaged synchronization error on which synchronization acts as a regularizing force; the optimization problem of finding the minimum synchronization error in parameter space is well-posed [@PJK96; @QBCKA09; @SSLP10; @BLP11; @SP11]. Our interest here will be based on an approach to force a response model to adapt to observed data by developing additional equations for the parameters that depend on the synchronization error [@P96; @SO09].
To estimate model parameters by synchronization, we exploit a special variation of synchronization called “autosynchronization”. For systems of ODEs, an observed scalar time series is coupled to a response system during model simulation. The goal of this feedback is to cause the response system to synchronize to the drive system. Ideally a proof of convergence follows by demonstration of an appropriate Lyapunov function [@P96; @YCCLP07]. In [@YP08; @SO09] we see some generalizations of how to derive synchronization schemes for many systems including the case where, apriori, we do not know the model form of the drive system [@SO09]. By autosynchronization, we can recover the model parameters, the current model state, and in some cases, a model form for an observed system.
Stating an autosynchronization problem in the ODE setting, we require a drive system $$\label{eq:DriveModel}
\mathbf{u_t} = \mathbf{f(u,p)},$$ from which we are able to sample data with (unknown to us) parameters $\mathbf{p} \in \mathbf{R^m}$. Then we must state a response system $$\label{eq:ResponseModel}
\mathbf{v_t} = \mathbf{g(u,v,q)}$$ with the same model form as the drive system if $\mathbf{q = p}$. By “same” we mean in as far as possible by our understanding of the underlying physics. Then the goal is that when $\mathbf{u}$ is coupled forward into Eq , then Eq will synchronize with Eq and $\mathbf{u \rightarrow v}$. Furthermore, parameter ODEs are given by $$\label{eq:ParameterODE}
\mathbf{q_t} = \mathbf{h(u,v,q)}$$ so that $\mathbf{(v,q)} \rightarrow \mathbf{(u,p)}$ as $t \rightarrow \infty$.
The idea of synchronization was extended to one-dimensional systems of PDEs in [@KTP97] and two-dimensional systems in [@BLP11], where the authors considered the Grey-Scott and Barkely reaction-diffusion systems respectively. In these works, the authors observed synchronization of an infinite-dimensional system by coupling the drive and response systems at only a finite number of grid points. Further work examines parameter estimation for given PDE systems using optimization over the synchronization error surface as discussed above. The authors observe single-species assimilation as they drive the PDE system to synchronization while coupling with only one species [@BLP11]. However, none of these utilize autosynchronization.
In many systems, it is very reasonable to expect that model parameters need not be spatially homogeneous. For example, taking our problem of interest, spatial inhomogeneity in parameter values may be central when constructing models for coastal algal blooms, since plankton growthrate is affected by near-shore nutrient runoff and upwelling [@M02]. More to that point, ocean fronts and eddies cause flow-induced long-term inhomogeneities in the ocean which results in a formidable spatial structure for density profiles in the ocean [@M02]. Whether inhomogeneities be the result of the flow dynamics or of “boundary conditions" from nutrient runoff, they are an important consideration for modelling ecology over large coastal domains. Thus it is reasonable to argue that a biophysics-based model should accept spatially dependent parameters.
A drawback to the aforementioned methods, including Kalman filter and multiple shooting methods, is that they do not consider spatially dependent parameter values, a priority noted in [@schiff08]. Parameter estimation by filtering methods adapted for PDEs can be computationally expensive [@muller02]. Furthermore, we have found that some filtering methods have trouble during periods of exponential growth, such as might be expected during plankton blooms. Optimizing the time-averaged synchronization error in some function space is far more complicated than the finite-dimensional alternative with scalar parameters as in [@BLP11]; optimization methods may not be practical.
Our work aims to extend the method of parameter estimation for PDE systems by synchronization to autosynchronization, especially including autosynchronization with spatially dependent parameters. Thus, we investigate observed data from the PDE drive system $$\label{eq:DriveModelPDE}
\mathbf{u_t}(x,y,t) = \mathbf{f(u}(x,y),\mathbf{p}(x,y))$$ with parameters $\mathbf{p}(x,y) \in C^0(\Omega)$ and a response system $$\label{eq:ResponseModelPDE}
\mathbf{v_t}(x,y,t) = \mathbf{g(u}(x,y),\mathbf{v}(x,y),\mathbf{q}(x,y)).$$ We formulate an associated system of PDEs for the parameters of Eq $$\label{eq:ParameterEqn}
\mathbf{q_t}(x,y,t) = \mathbf{h(u}(x,y),\mathbf{v}(x,y))$$ with the goal that $\mathbf{(v,q)} \rightarrow \mathbf{(u,p)}$ as $t \rightarrow \infty$. We design our methods considering a benchmark system of reaction-diffusion PDEs. Since we know the model form of the drive system and the parameters used to build the observed data, we can compare our estimated parameters with the exact parameters.
In preview of the paper layout, we begin by introducing the reaction-diffusion equations that we will use as the drive system. We discuss how this system is solved numerically and the parameters used to simulate complex spatiotemporal dynamics. Next, we implement the response system and show the parameter PDEs used to find autosynchronization. We demonstrate the power of the systems of PDEs to autosynchronize by employing three different spatial functions for the parameters. Next, we show the estimated parameters and the convergence plots for both state variables and parameters to the correct values. Finally, we give an improvement on the response system that admits autosynchronization wherein only one species is sampled, which is an important breakthrough for applications since generally only the phytoplankton is easily observable.
The Parameter Estimation Method {#sec:RD}
===============================
Consider the system of two PDEs as given in [@M02], $$\begin{aligned}
\label{eq:Fish}
\frac{\partial P}{\partial t} &=& \triangle P + P(1-P) - \frac{P Z}{P + h},\\
\nonumber \frac{\partial Z}{\partial t} &=& \triangle Z + k\frac{P Z}{P+h} - mZ,\end{aligned}$$ on a compact connected two-dimensional domain, $\Omega$, with zero-flux boundary conditions.
In terms of the biology of the model, the system represents a dimensionless reaction-diffusion model for phytoplankton-zooplankton predator prey dynamics in a horizontal layer where vertical distributions of plankton are considered uniform. For simulations shown here, we choose $\Omega$ to be a rectangle of size $864 \times 288$. Although shown in dimensionless form, the model is derived from principles in which phytoplankton concentrations obey a logistic growth and are grazed upon by zooplankton, following a Holling-type [slowromancap2@]{} functional response. First classified by Holling, [@Holling59], the Holling-type [slowromancap2@]{} functional response assumes a decelerating growth rate such that the predator, or consumer, is limited by its ability to efficiently process food. Zooplankton grow at a rate, k, proportional to phytoplankton mortality and are subject to a natural mortality rate m. For certain parameters, this system gives rise to transient spiral pattern behaviour on its way to spatially irregular patchy patterns [@M02]. In [@M02], parameters are set to $k = 2$, $h = 0.4$, and $m = 0.6$. For homogeneous initial plankton distributions, the system remains in a homogeneous state for all time so we use the perturbed initial conditions found in [@M02].
We solve this system using a finite difference method, with a nine-point center difference stencil for spatial derivatives and forward Euler time stepping. Our simulations use spatial discretization with $dx = 2$ and and Euler time step of $dt = .2$. The model output is treated as an image sequence given by a particular (known) model form but with *unknown* parameters $k$ and $m$, to be determined. Thus we will mimic our target application of remote sensing oceanographic images of hyperspectral images filtered to reveal plankton blooms. Further, we will be interested to allow $k$ and $m$ to vary spatially as functions, $k(x,y)$ and $m(x,y)$.
Our interest in this PDE model stems from our work in remote sensing, to build a better understanding of our ocean’s ecology. Particularly, we aim to predict short term behavior of coastal algal blooms. Such a system may in principle be modelled by estimating parameters directly from observed data in the field. However, hyperspectral satellite imagery provides the observed data to which we would synchronize a response model in hopes of autosynchronization providing good parameter estimates for forecasting. Since phytoplankton are largely affected by spatial inhomogeneities in the ocean such as nitrogen runoff, regions of hypoxia, or upwelling, to name a few parameter inhomogeneity-inducing effects, we wish to allow model parameters to vary spatially. These considerations are especially important since our models will be built over coastal domains where large changes in ocean biology occur spatially, leading naturally to spatially dynamic parameters.
We are only able to observe time series data Eq (\[eq:DriveModelPDE\]) as a movie and we know the model form of Eq (\[eq:Fish\]), but want to estimate the parameters used to create the observed data. The system of Eq (\[eq:Fish\]) will be taken as the drive system and we form a response system to be synchronized to the observations as,
$$\begin{aligned}
\label{eq:Fishr}
\frac{\partial \hat{P}}{\partial t} &=& \triangle \hat{P} + \hat{P}(1-\hat{P}) - \frac{\hat{P} \hat{Z}}{\hat{P} + h} + \kappa(P-\hat{P}),\\
\nonumber \frac{\partial\hat{Z}}{\partial t} &=& \triangle \hat{Z} + \hat{k}\frac{\hat{P} \hat{Z}}{\hat{P}+h} - \hat{m}\hat{Z} + \kappa(Z-\hat{Z}),\end{aligned}$$
where we assume $\hat{P}(x,y,0) \neq P(x,y,0)$, $\hat{Z}(x,y,0) \neq Z(x,y,0)$, $\hat{k}(x,y,0) \neq k(x,y)$, and $\hat{m}(x,y,0) \neq m(x,y)$. Thus, we do not know the initial model states, and wish to recover the spatially varying parameters $m(x,y)$ and $k(x,y)$. To derive Eq (\[eq:Fishr\]), a diffusive coupling term is added to each equation in Eq (\[eq:Fish\]) accounting for the error between the drive and response values with a coupling strength, $\kappa$. These additional terms drive $\hat{P} \rightarrow P$ and $\hat{Z} \rightarrow Z$, so that the PDEs will synchronize after a short time. The synchronization is of identical type and dependent upon the choice of $\kappa$, as is the synchronization speed.
Results and Simulations of Autosynchronization Parameter Estimation
===================================================================
We modify the system Eq (\[eq:Fish\]) as found in [@M02] by allowing the parameters to be nonnegative $C^0(\Omega)$ functions. Here $\Omega$ is the domain, which in the case of our simulations, $\Omega \subset \mathbf{R}^2$ is a compact domain such as a rectangle or even a domain shaped as the Gulf of Mexico. Parameters are updated as diffusively coupled PDEs during the synchronization process as $$\begin{aligned}
\label{eq:ParUpdate}
\frac{\partial \hat{k}}{\partial t} &=& -s(P - \hat{P}) \ \ \ \ s > 0, \\
\nonumber \frac{\partial\hat{m}}{\partial t} &=& -s(Z - \hat{Z}),\end{aligned}$$ where we choose $s = 30$ for specificity and for which we observe good convergence results. The parameter equations are evolved simultaneously with Eq with a forward Euler discretization and the same time step. The model form of Eq was chosen after testing several forms and there may exist other forms for which synchronization is possible. Once the model form was chosen, a parameter search was performed to find $s=30$. As we vary $s$ and $\kappa$, autosynchronization can fail, a common situation with diffusively coupled systems. Parameters may be updated as reaction-diffusion PDEs, by adding a diffusion term, however we need to restrict parameters to be nonnegative $C^2(\Omega)$ functions and stability may be affected. To begin the simulation, parameters are initialized as the constant function $$\begin{aligned}
\label{eq:ParInit}
\hat{k}(x,y,0) &=& 10,\\
\hat{m}(x,y,0) &=& 10. \end{aligned}$$ We evolve Eq forward and count the model output as observed data. Initial conditions for the response system are $\hat{P}(x,y,0) = 0$ and $\hat{Z}(x,y,0) = 0$. Furthermore, to avoid values outside the normal range of Eq, we enforce that $$\begin{aligned}
\nonumber \hat{P} = \Bigg\{ \begin{matrix} \hat{P}&:& 0 < \hat{P} < 2 \\0&:& \hat{P} \leq 0\\2&:& \hat{P} \geq 2 \end{matrix},\end{aligned}$$ during the simulation.
First, we develop synthetic datasets with spatially varying parameters to challenge our methods. Spatially dependent parameters are chosen to be in the range given in [@M02] for spatially irregular behavior. Three different functional forms for the parameters are tested. First, we use a linearly varying parameter set $$\begin{aligned}
\label{eq:Linear}
k_1(x,y) &=& a \left(\frac{x}{n} + \frac{y}{m} + l\right) + b, \\
\nonumber m_1(x,y) &=& c \left(\frac{x}{n} + \frac{y}{m} + l\right) + d,\end{aligned}$$ where $m$ and $n$ represent the size of the domain shown in Figure \[fig: Domain\] and $a = 2, b = 0.14, c = 0.6, d = 0.1$, and $l = 5$. Thus, appropriate parameters are chosen to maintain $m(x,y)$ and $k(x,y)$ in the target range. Next, we define a Gaussian parameter function as
$$\begin{aligned}
\label{eq:Gauss}
k_2(x,y) &=& a e^{-\left(\frac{(x-n/2)^2}{2\sigma^2} + \frac{(y-m/2)^2}{2\sigma^2}\right)}, \\
\nonumber m_2(x,y) &=& c e^{-\left(\frac{(x-n/2)^2}{2\sigma^2} + \frac{(y-m/2)^2}{2\sigma^2}\right)},\end{aligned}$$
where $a,b,m,n$ have the same values. For example, $k_2(x,y)$ is shown on the top half of (a) in Figure \[fig:figP2\]. Finally, we define
$$\begin{aligned}
\label{eq:Sineplot}
k_3(x,y) &=& a \ \cos(b x + d)\sin(by) + s, \\
\nonumber m_3(x,y) &=& c \ \cos(b x + d)\sin(by) + t,\end{aligned}$$
where $a = 0.2, b = \pi/(m/2), c = 0.6, d = \pi/2, s = 0.5$, and $t = 1.5$, to test the quality of the autosynchronization scheme to resolve fine structures in model parameters. The surface produced by $k_3(x,y)$ is displayed on the top half of (b) in Figure \[fig:figP2\]. We observe solution data at every time step relative to the response system, Eq , and the parameter system, Eq , to drive $(\hat{P},\hat{Z}) \rightarrow (P,Z)$ and $(\hat{m}(x,y),\hat{k}(x,y)) \rightarrow (m(x,y),k(x,y))$ as $t \rightarrow \infty$. For brevity, only the parameters defined by Eq and Eq are shown and compared with their estimated counterparts.
We observe autosynchronization for each test set of parameters and the spatial inhomogeneities in each case are effectively resolved.
In Figure (\[fig:figP1\]), the globally averaged error between the drive and response PDEs (Phytoplankton density) has been driven to less than $1.0 \times 10^{-10}$ and the globally averaged error between true and estimated parameters has been driven to below $1.0 \times 10^{-6}$. Interestingly, the results show a change of convergence rate midway through the simulation. For these results, we choose $\kappa = .3625$.
![Autosynchronization of spatially dependent parameters at $t=2000$. Drive (top) vs response (bottom). Left: Model parameters given by $k_2(x,y)$ (shown) and $m_2(x,y)$. Right: Model parameters given by $k_3(x,y)$ (shown) and $m_3(x,y)$.[]{data-label="fig:figP2"}](AutoSync_k_Surfaces_Sample_Both_Gaussian.png "fig:"){width="3.0in"}(a)
![Autosynchronization of spatially dependent parameters at $t=2000$. Drive (top) vs response (bottom). Left: Model parameters given by $k_2(x,y)$ (shown) and $m_2(x,y)$. Right: Model parameters given by $k_3(x,y)$ (shown) and $m_3(x,y)$.[]{data-label="fig:figP2"}](AutoSync_k_Surfaces_Sample_Both_Sine.png "fig:"){width="3.0in"}(b)
In, Figure (\[fig:figP2\]), the results of autosynchronization after 2000 seconds of simulation are shown where reconstructed parameters are compared with their true counterparts. Both plots demonstrate how effectively the parameters have been reconstructed. Similar results were found by testing parameters that vary spatially according to Eq .
Syncronization by Sampling Only One Species
===========================================
To this point, an important criticism of our work is that we need to sample both species to drive the response model and parameters. As mentioned above, our interest in autosynchronization for parameter estimation stems from work with ocean models for phytoplankton-zooplankton ecology. In fact, hyperspectral satellite imagery provides phytoplankton density inferences but provides no data for zooplankton. Certainly, parameter estimation using the response model above will fail. Even given correct model parameters, it is impossible to forecast the model since zooplankton initial conditions are not supplied. Our problem of interest requires that we somehow estimate zooplankton initial conditions based on phytoplankton observations.
We find that, by a modification of Eq , it is possible to drive zooplankton density to its true state by sampling phytoplankton alone. This is a first demonstration of the possibility of simulating this system with only partial knowledge. As an added bonus we observe autosynchronization. Thus this technique gives us a tool to estimate parameters and to initialize a model for short term forecasts. The response model that drives these results is
$$\begin{aligned}
\label{eq:FishPhyto}
\frac{\partial \hat{P}}{\partial t} &=& \triangle \hat{P} + \hat{P}(1-\hat{P}) - \frac{\hat{P} \hat{Z}}{\hat{P} + h} + \kappa(P-\hat{P}),\\
\nonumber \frac{\partial\hat{Z}}{\partial t} &=& \triangle \hat{Z} + \hat{k}\frac{\hat{P} \hat{Z}}{P+h} - \hat{m}\hat{Z},\end{aligned}$$
where the absence of hats denotes where observation data is coupled directly into the PDE. We use a combination of diffusive coupling and complete replacement coupling in the response PDE to observe autosynchronization. Note that zooplankton density is no longer observed in Eq . The parameter update equations are $$\begin{aligned}
\label{eq:ParUpdatePhytoOnly}
\frac{\partial \hat{k}}{\partial t} &=& s_1(P - \hat{P}) \\
\nonumber \frac{\partial\hat{m}}{\partial t} &=& s_2(P - \hat{P})\hat{P},\end{aligned}$$ with $s_1 = 0.2$, $s_2 = 0.6$, and $\kappa = 0.6$. Here, as above, the parameter equations are evolved simultaneously with the using a forward Euler discretization. Figure (\[figParkmPhytoOnly\]) shows results obtained when autosynchronizing with Eq at $t = 4434$, a substantially longer time epoch. Results obtained for the Gaussian form, Eq , for parameters show that both $\hat{k}(x,y,t)$ and $\hat{m}(x,y,t)$ converge, as before, to their true values $k_2(x,y)$ and $m_2(x,y)$. Here, all initial conditions are set to $\hat{P}(x,y,0) = \hat{Z}(x,y,0) = \hat{k}(x,y,0) = \hat{m}(x,y,0) = 1$.
In Figure (\[figErrorsPhytoOnly\]), globally averaged errors are shown to diminish over time as the synchronization scheme evolves. Both parameters converge to within about $1.0 \times 10^{-6}$ after 4434 seconds. Importantly, we note zooplankton convergence to nearly $1.0 \times 10^{-7}$ of ground truth. Therefore, we need not sample zooplankton to observe autosynchronization and we find the true zooplankton density profile such that model simulations may be initialized.
Conclusion
==========
In this paper, we have shown that it is possible to derive an autosynchronization scheme for a system of PDEs. We emphasize here the improvements we have made upon past synchronization methods in that we use autosynchronization as a means of parameter estimation of parameters that exist in a function space. We assume that we know the model form of the true observed system, but have no prior knowledge of the parameters. By sampling at every time step, we observed identical synchronization between the response and drive systems as described in [@BLP11]. As a first attempt, we have given a model form for adaptive parameters in the response system such that we observe identical synchronization between response model parameters and true parameters, or autosynchronization. Our techniques were implemented on a benchmark model and results converge to ground truth. Thus, autosynchronization is observed for PDEs with spatially homogeneous parameters.
Next, we considered the same system of PDEs wherein the parameters were spatially dependent. We provided a scheme with which we observe autosychronization of spatially dependent parameters. We tested our results against several different functional forms for parameters and found the method to be robust.
We markedly improved upon these results once more with an autosynchronization scheme that requires sampling of only one species (phytoplankton). We noted that in order to evolve a system of PDEs for forecasting, we need initial conditions for both species; this is a serious problem when dealing with remote sensing data with which we can only observe one of the species. This concern was addressed by providing a response system that autosynchronizes parameters and synchronizes zooplankton using only phytoplankton data. These methods are therefore plausible for use in remote sensing problems.
As discussed above, synchronization schemes can be proven to work for a given range of coupling parameters using, for example, Lyapunov functions. It remains to be shown why this scheme works on this system, and to perhaps derive autosynchronization model forms for a wider class of reaction-diffusion PDEs.
A drawback of this technique with application to hyperspectral satellite data is that data may be noisy; this is where filtering techniques have a built-in advantage. Data may also be occluded because of cloud cover. Since there is no hope to synchronize PDEs without data, we require techniques to fill in that which is missing, for example, inpainting.
Another problem with applying these techniques to satellite imagery is temporal data resolution. There may be several days between images and autosynchronization requires ample data observations. The need for frequent observables is perhaps the main drawback to this method. However, autosynchronization may be advantageous for parameter estimation or model building in situations where spatiotemporal data are abundant and especially where parameters are expected to vary spatially.
Acknowledgements
================
This work was supported by the Office of Naval Research under grant \#N00014-09-1-0647. The authors would also like to the thank the anonymous referee for helpful comments and suggestions on the manuscript.
|
---
abstract: 'We will argument how infalling information can be chaotized inside realistic quantum black holes.'
author:
- Andrea Addazi$^1$
title: Aspects of quantum chaos inside black holes
---
\#1[| \#1|]{}
Introduction and Conclusions
============================
Theoretical physicists are all agreed that Semiclassical Black holes are paradoxical objects (as nicely reconfirmed by several discussions during the Karl Schwarzschild meeting 2015). However, a clear strategy in order to solve this problem is still unknown.
In this paper, we would like to suggest that infalling information could be chaotized inside a black hole . Our claim is related to a different picture about quantum black holes’ nature: we retained unmotivated to think seriously about a quantum black hole as a conformal Penrose’s diagram, [*i.e*]{} as a smoothed semiclassical geometry with a singularity in its center (eventually cutoff at the planck scale). In particular, one could expect that, in a “window” of lenght scales among the Schwarzschild radius and the Planck scale, there is a non-topologically trivial region of space-time rather than a smoothed one. A realistic black hole could be a superposition of different horizonless solutions, perhaps associated GR gravitational instantons or “exotic” gravitational instantons [^1]. In this picture, a black hole’ horizon is an approximated Chauchy null-like surface (for energy scales closed to an inverse Schwarzschild radius). However, for lenght scales $L$ in the range $l_{Pl}<<L<<R$, geometrical deviations and asperities with respect to semiclassical smoothed geometries are reasonable expected. In this regime, gravitational interactions among horizonless geometries can be neglected as well as microscopical exchanges of matter and gauge fields among their surfaces. In this sense, a black hole cannot be described by a single Penrose’s diagram at all the lenght scales. In particular, in the “middle region” a black hole would be described by a superposition of a large number of Penrose’s diagrams.
Such a black hole can be rigorously defined in an euclidean path integral formulation. It emits a thermal radiation like a semiclassical BH, with small corrections on Bekenstein-Hawking entropy (see section II) [@Addazi:2015gna].
At this point, a further question is the following: what happen to infalling informations in such a “scale variant” system? Let us consider the usual thought experiment of a infalling radiation in a quantum pure state, with a very small initial frequency $\omega \simeq R^{-1}$. Such a radiation will start to probe a smoothed semiclassical geometry of a black hole, near the horizon. However, radiation will be inevitably blueshifted inside the gravitational potential of a black hole, [*i.e*]{} its De Broglie wave length starts to be smaller than the Schwarzschild’s radius. So that, infalling radiation will start to probe the middle region before than the full quantum quantum regime. In the middle region, radiation is scattered back and forth among asperities that usually are not present at all in semiclassical BH solutions. As a consequence, radiation will be chaotically diffracted inside this system. At that middle scales, a black hole is a sort of [*space-temporal chaotic Sinai billiard*]{} rather than a smoothed manifold. Usually, in simpler classical chaotic billiards than our one, chaotic zones of unstable orbits trapped forever in the system are formed. Simple examples of such a trapped paths: i) an orbit trapped in back and forth scatterings among the asperity A and the asperity B (AB segments); ii) one trapped among $A,B,C$ asperities (triangular orbits); and so on. Considering quantum fields rather than classical trajectories, one has also to consider quantum transitions induced by inelastic scatterings on gravitational backgrounds $<g,...,g>$ (thought as a vacuum expectation value of gravitons). $\phi +<g,...,g>\rightarrow X+<g,...,g>$ where $\phi$ is a generic gauge/matter field, and $X$ is a collection of $N$ fields. For example a process like a photon-background scattering $$\gamma +<g,...,g> \rightarrow q\bar{q}+<g,...,g>\rightarrow hadronization +<g,...,g>$$ will lead to a complicated hadronic cascade of entangled fields. As a consequence, such a system is even more chaotic than classical one. So that, a part of the initial infalling information is effectively fractioned in a “forever” (black hole lifetime or so) trapped part and another one, so that $$|IN\rangle =a|OUT\rangle +b|TRAPPED\rangle$$ where $a,b$ parametrize our ignorance about the space-time billiard, $|OUT \rangle$ is emitted as Bekestein-Hawking radiation. As a consequence, the in-going information is a linear combination of outgoing informations and trapped informations during $0<<t<<t_{Evaporation}$. In this picture, information paradoxes are understood as an [*apparent*]{} losing of unitarity. In fact, $| IN \rangle \rightarrow |OUT \rangle$ is not allowed by quantum mechanics: $|IN\rangle$ is a pure state, while $|OUT \rangle$ is a mixed one. However, also $|TRAPPED \rangle$ is a mixed state, and a linear combination of two mixed states can be a pure one. In this approach, a $|IN\rangle \rightarrow |OUT \rangle$ transition can be effectively described in a density matrix approach, with an effective non-unitary evolution. However, unitarity is not lost at fundamental level because of the real transition $|IN\rangle \rightarrow a|OUT \rangle+b|TRAPPED\rangle$ is not contradicting unitarity. Let us consider, for example, a (famous) Bekenstein-Hawking particle-antiparticle pair created nearby the black hole horizon. As usual, one the two is captured inside the black hole space-like interior, while the second one can tunnel outside the horizon. As well known, the two particles are entangled, and this will lead to the undesired firewall paradox. However, in a frizzy black hole, the infalling pair will start to be blueshifted so that it will start to scatter back and forth inside the system, giving rise to an exponentially growing cascade of N particles continuing to scatter and to scatter in the billiard. The process will be even more chaotic in a realistic case in which a large number of infalling partners from a large number of Bekenstein-Hawking pairs have to be considered. As a consequence, P outgoing pairs will be entangled with a total number $N>>P$ of particles inside the system. This practically disentangles the $P$ outgoing pairs from the original ones, as a quantum decoherence effect induced by the non-trivial space-time topology. In other words, the space-time topology is collapsing the entangled wave function as a quantum decoherence phenomena, as well as two entangled pairs are disentangled by a an experimental apparatus. The entanglement entropy is linearly growing with the number of back and forth scatterings $n$ of a particle, because of the density matrix of the internal black states are exponentially growing with $n$: $$S_{interior}=-Tr \rho_{interior}log S_{interior}\sim n$$ so that is growing with time. On the other hand, for $P$ Bekenstein-Hawking particles $S_{int}\sim n \,log\,P$. Our model predicts $S_{B.H.}\sim P$ from entanglement entropy definition.
However, if a frizzy black hole emits a Bekenstein-Hawking radiation with small deviations from thermality, it cannot have an infinite life-time. On the other hand, the non-trivial topological space-time configuration of a frizzy black hole is sourced by the black hole mass. The final configuration after the complete black hole evaporation is a Minkowski space-time with a dilute residual radiation. As a consequence, a space-time phase transition from the “frizzy” topology to the Minkowski space-time is expected at the Page time or so. As a consequence, chaotic saddles of trapped information will be emitted in the environment as a [*final information burst*]{}. For this motivation, the S-matrix describing BH evolution from the initial collapse/formation to its complete evaporation is unitary: $$\langle COLLAPSE|S|EVAPORATION\rangle$$ $$=\langle TOTAL INFALLING| S| (a|TRAPPED\rangle +b |OUT\rangle)$$ The trapped probability density $\rho(T)$ is approximately described by $$\frac{d\rho(T)}{dT}\sim -\frac{1}{T^{2}}e^{-\Gamma T }$$ In fact, $\rho(T)$ is dependent by the number of asperities $N_{s}$ as $\rho \sim N_{s}e^{-\Gamma T}$, where $\Gamma$ is proportional to [*effective average deepness*]{} of asperities (trapping $\rho$). But the number of asperities is depending by the Black hole mass. In turn, the black hole mass decreases with the temperature as $dM/dT=-1/8\pi T^{2}$.
To conclude, chaotic aspects of quantum black holes could be relevantly connected to the information paradoxes. In particular, a semiclassical black hole could be reinterpreted as a superposition of horizonless geometries, chaotizing infalling informations. Such an approach could have surprising connections with recent results in contest of AdS/CFT correspondence [@C0].
Euclidean path integral of a frizzy black hole
==============================================
[**Definition**]{}: let us consider a generic system of N horizonless background metrics (suppose to be eliminated at the Planck scale) inside a box with a surface $\partial \mathcal{M}$. This system is defined [*frizzy black hole*]{} if it satisfies the following hypothesis:
i\) A formal definition of partition functions $Z_{I}$ for each metric tensor $g^{I=1,...,N}$ cab be defined. The N metrics are in thermal equilibrium with the box.
ii\) In semiclassical regime, the leading order of the total partition function associated to this system is the product of the single partition function: $$Z_{TOT}= \prod_{I=1}^{N} Z_{I}$$ [^2]
iii\) The total average partition function has a form $$\langle Z_{TOT}\rangle=e^{-\frac{\beta^{2}}{16\pi}-\frac{\sigma_{\beta}^{2}}{16\pi}}=Z_{E}e^{\frac{-\sigma_{\beta}^{2}}{16\pi}}$$ where $Z_{E}$ is the usual semiclassical euclidean partition function, $\sigma_{\beta}$ the variance of $\beta$-variable in the system. This corresponds to an entropy $$\langle S \rangle=\frac{\beta^{2}}{16\pi}+\frac{\sigma_{\beta}^{2}}{16\pi}$$
Semiclassical chaotic scattering on a Space-time Sinai Biliard
==============================================================
An effective non-relativistic quantum mechanical approach is not fully valid and motivated. However, we will show some general proprieties suggested by this approach: these can be extended to the more realistic problem.
Let us remind, that $\psi_{t}({\mbox{\boldmath$r$}})$ is obtained by an initial $\psi_{0}({\mbox{\boldmath$r$}}_{0})$ by the unitary evolution $$\psi_{t}({\mbox{\boldmath$r$}})=\int d{\mbox{\boldmath$r$}}_{0}K({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t)\psi_{0}({\mbox{\boldmath$r$}}_{0})$$ where $K$ is $$K({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t)=\int \mathcal{D}{\mbox{\boldmath$r$}} e^{\frac{i}{\hbar}I}$$ and $$I=\int_{0}^{t} dt L({\mbox{\boldmath$r$}},\dot{{\mbox{\boldmath$r$}}})$$ and $L$ is the lagrangian of a particle. The semiclassical limit corresponds to $$I=\int_{0}^{t}[{\mbox{\boldmath$p$}}\cdot d{\mbox{\boldmath$r$}}-Hd\tau]>>\hbar$$ In this regime, the leading contribution to the path integral is just given by classical chaotic trapped orbits. The semiclassical propagator can be written as $$K_{WKB}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t)\simeq \sum_{n}\mathcal{A}_{n}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t)e^{\frac{i}{\hbar}I_{n}}$$ where we are summing on all over the classical orbits of the system, while amplitudes $\mathcal{A}_{n}$ are $$\mathcal{A}_{n}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t)=\frac{1}{(2\pi i \hbar)^{\nu/2}}\sqrt{|det[\partial_{{\mbox{\boldmath$r$}}_{0}}\partial_{{\mbox{\boldmath$r$}}_{0}}I_{n}[{\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}_{0},t]]|}e^{-\frac{i\pi h_{n}}{2}}$$ with $h_{n}$ counting the number of conjugate points along the n-th orbit. The probability amplitude is related to Lyapunov exponents: $$|\mathcal{A}_{n}|\sim exp\left(-\frac{1}{2}\sum_{\lambda_{k}>0}\lambda_{k}t \right)$$ along unstable orbits; while $$|\mathcal{A}_{n}|\sim |t|^{-\nu/2}$$ along stable ones.
The level density of bounded quantum states is described by the trace of the propagator. In semiclassical limit, the trace over the propagator is peaked on around the periodic orbits and stationary saddle points. This allows to semiclassically quantize semiclassical unstable periodic orbits that are densely sited in the invariant set.
In our chaotic system, we expect many resonances. So that, transitions’ probabilities can be averaged over the large number resonances’ peaks. So that, a wavepacket $\psi_{t}({\mbox{\boldmath$r$}})$ in a region $R$ ($\nu$-dimensional space) has a quantum survival probability $$P(t)=\int_{R}|\psi_{t}({\bf r})|^{2}d{\mbox{\boldmath$r$}}$$ This can be rexpressed in terms of the initial density matrix $\rho_{0}=|\psi_{0}\rangle \langle \psi_{0}|$ as $$P(t)=tr \mathcal{I}_{D}({\bf r})e^{-\frac{iHt}{\hbar}}\rho_{0}e^{+\frac{iHt}{\hbar}}$$ where $\mathcal{I}_{D}$ is zero for resonances ${\bf r}$ out of the region $D$ and$1$ into $D$. We can express the survival probability as $$P(t)\simeq \int \frac{d\Gamma_{ph}}{(2\pi \hbar)^{f}}\mathcal{I}_{D}e^{{\bf L}_{cl}t}\tilde{\rho}_{0}+O(\hbar^{-\nu+1})$$ $$+\frac{1}{\pi \hbar}\int dE\sum_{p}\sum_{a}\frac{\cos\left(a\frac{S_{p}}{\hbar}-a\frac{\pi}{2}{\bf m}_{p} \right)}{\sqrt{|det({\bf m}_{p}^{a}-{\bf 1})|}}\int_{p}\mathcal{I}_{D}e^{{\bf L}_{cl}t}\tilde{\rho}_{0}dt$$ $$+O(\hbar^{0})$$ where $d\Gamma_{ph}=d{\mbox{\boldmath$p$}}d{\mbox{\boldmath$r$}}$, the sum is on all the periodic orbits (primary periodic orbits $p$ and the number of their repetitions $a$); $S_{p}(E)=\int {\mbox{\boldmath$p$}}\cdot d{\mbox{\boldmath$r$}}$, $\tau_{p}=\int_{E}S_{p}(E)$, ${\bf m}_{p}$ is the Maslov index, and $\mathcal{M}$ is a $(2\nu-2)\times (2\nu-2)$ matrix associated to the Poincaré map in the neighborhood of the a-orbit; ${\bf L}_{cl}$ is the classical Liouvillian operator, defined in terms of classical Poisson brackets as ${\bf L}_{cl}=\{H_{cl},...\}_{Poisson}$; $\tilde{\rho}_{0}$ is the Wigner transform of the initial density state. The operator ${\bf L}_{cl}$ have Pollicott-Ruelle resonances as eigenvalues $${\bf L}_{cl}\phi_{n}=\{H_{cl},\phi_{n} \}_{Poisson}=\lambda_{n}\phi_{n}$$ where eigenstates $\phi_{n}$ are Gelfald-Schwartz distributions. On the other hand, the adjoint problem $${\bf L}_{cl}^{\dagger}\tilde{\phi}_{n}=\tilde{\lambda}_{n}\tilde{\phi}_{n}$$ The eigenvalues $\lambda_{n}$ are in general complex: their real part $Re(\lambda_{n})\leq 0$ because of they correspond to an ensamble bounded periodic orbits; their $Im(\lambda_{n})$ correspond to decays in the ensambles. Expanding the survival probability over resonances as $$P(t)\simeq \int \sum_{n}\langle \mathcal{I}_{D}|\phi_{n}(E)\rangle \langle \tilde{\phi}_{n}(E)|e^{\lambda_{n}(E)t}| \phi_{n}(E)\rangle \langle \tilde{\phi}_{n}(E)|\tilde{\rho}_{0}\rangle$$ we can get the 0-th leading order $\sim e^{\lambda_{0}(E)t}$. So that, the survival probability is behaving like $P(t)\sim e^{-\gamma(E)t}$, [*i.e*]{} $s_{0}=-\gamma(E)$: the decay of the system is related to the classical escape rate $\gamma(E)$.
[**Acknowledgments**]{}
A. A would like to thank Gia Dvali, Gerard t’Hooft and Carlo Rovelli for useful discussions on these aspects. A. A work was supported in part by the MIUR research grant “Theoretical Astroparticle Physics” PRIN 2012CPPYP7.
[99]{}
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[^1]: In string theory, the class of instantons is much larger than in field theories. Applications of a particular class of these solutions in particle physics were recently studied in [@Addazi:2015gna; @Addazi:2014ila; @Addazi:2015ata; @Addazi:2015rwa; @Addazi:2015hka; @Addazi:2015fua; @Addazi:2015oba; @Addazi:2015goa; @Addazi:2015yna; @Addazi:2015ewa] .
[^2]: This approximation can be trusted if and only of intergeometries’ interactions are small with respect to the temperature inside the box.
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abstract: 'The rising complexity of our terrestrial surrounding is an empirical fact. Details of this process evaded description in terms of physics for long time attracting attention and creating myriad of ideas including non-scientific ones. In this essay we explain the phenomenon of the growth of complexity by combining our up to date understanding of cosmology, non-equilibrium physics and thermodynamics. We argue that the observed increase of complexity is causal in nature, stands in agreement with the second law of thermodynamics and has it’s origin in the cosmological expansion. Moreover, we highlight the connection between the leader of complexity growth in localized areas of space with free energy rate density, starting from the largest scales towards the smaller ones. Finally, in the light of recent advances in non-equilibrium statistical mechanics, our belief in the causal structure of modern scientific theories is transferred to biological systems. On relevant scales, adaptation and complexity growth follows a similar pattern, in which free energy rate density is provided by an external electromagnetic radiation. The presented, holistic approach, arms us with predicting power about variety of attributes of complex systems and leads to a chain of successful explanations on all scales of the Universe.'
author:
- |
Micha[ł]{} Mandrysz and Jakub Mielczarek\
Institute of Physics, Jagiellonian University, [Ł]{}ojasiewicza 11, 30-348 Cracow, Poland
bibliography:
- 'bibliography.bib'
title: 'The Top-Down Complexity'
---
The complexity of life on Earth can appear perplexing to our scientific apparatus. Simple microscopic laws that performed amazingly well in cases of small inanimate systems were in many cases less successful in the description of biological forms, let alone the ones with information processing capabilities (agency). This, however, does not mean that we need to discard this approach. Rather, we realize that emergence involves *history* and that we cannot understand microcosmos without comprehending the cosmos. Therefore, let us begin our discussion towards the origin of complexity rise on Earth from the largest scales perceived by the human mind.
The Universe (See Fig. \[Fig1\]), which we understand as basically everything that exist; is forming the largest, perfectly isolated system with no environment.
![Top-Down trend of the complexity growth.[]{data-label="Fig1"}](triangle){width="17cm"}
Therefore, if quantum mechanics applies, the Universe as a whole is described by a pure state $|\Psi\rangle$ satisfying the timeless Wheeler-DeWitt equation $\hat{H}|\Psi\rangle=0$, such that there is no evolution of the state with respect to some external time parameter[^1]. However, *internal times* are possible to introduce, which encode relative evolution between subsystems of the Universe.
Since the state $|\Psi\rangle$ is pure it contains all information available in the universe. Consequently, for such a state with density matrix is $\rho=|\Psi\rangle\langle\Psi|$ the von Neumann entropy $S=-\text{tr}\left(\rho\ln\rho\right)$ remains constant. In the information theoretical picture, the von Neumann entropy can be considered as a measure of our lack of information about the system. Therefore, for a fictitious observer, with an access to the whole state $|\Psi\rangle$ the total amount of information in the Universe might be considered a conserved quantity.
However, in general we deal with situations in which the observer can collect information only about a limited number of degrees of freedom. Such subset of the Universe we call the *system*. The rest we call the *environment*.
In the cosmological context, the Universe can be decomposed into background (the system) and inhomogeneities (the environment). As the studies for perturbative inhomogeneities suggest, in such a case the value of entanglement entropy and the value of the scale factor $a$ are positively correlated [@Zeh:2007uw; @Kiefer:2017iy]. Consequently, in the present phase of the Universe expansion the entanglement entropy $S$ is increasing, satisfying the second law of thermodynamics. This, however, does not mean that the Universe will become homogeneous in the whole space, on the contrary islands of complexity are indeed possible as will shortly become clear.
At the end of the radiation epoch the inhomogeneity of the Universe was at the level $\Delta T/T \sim 10^{-5}$ and baryonic matter and radiation remained in the thermal equilibrium forming a primordial plasma (see Fig. \[Fig2\]a). Then, as a consequence of the temperature drop, the recombination process (formation of neutral matter) occurred leading to departure of the system from the thermal equilibrium. This process can be perceived as a decay of the system into two equilibrium states each characterized by a different temperature (thermal decoupling). For radiation, temperature gradually decreased following the $T_r\sim1/a$ trend, which was valid also for the relativistic plasma before the recombination.
The cosmic expansion enhanced inhomogeneities of the matter density, which ultimately became unstable initiating the process of gravitational collapse. This process lead to formation of dense inhomogeneous structures (stars) in which gravitational potential energy was transformed into the kinetic energy, increasing their temperatures. Quite often the details of this process (concerning both the formation of stars and planets) are neglected and lead to confusion whether matter localized as stars or planets represents a state of higher entropy. Therefore, let us articulate this clearly - in case of non-interacting particles the state of maximum entropy is a homogenous distribution. However, in case of gravitationally interacting particles the entropic “price" for localizing particles in one place is “paid" by the heat emitted to the environment, such that the total entropy in this process also rises. For stars the accumulated energy due to this process leads to ignition of the nuclear reactions increasing further the temperature of stars.
![a) Thermal decoupling of radiation and matter. b) Schema of a heat engine consisting of the system (S) and its environment in thermal equilibrium (Heater $H$, Cooler $C$).[]{data-label="Fig2"}](rec-eng){width="18cm"}
The maximal temperature of localized matter systems (stars) $T_{m,max}$ eventually became much higher[^2] than the temperature of the background radiation $T_r$ filling the Universe (See Fig. \[Fig2\]a).
This difference of temperatures between the hot spots (stars) and cold surrounding allows for work or the free energy $\Delta F$ (depicted as cogs in the illustrations) to be extracted as it happens in case of a heat engine (See Fig. \[Fig2\]b). The stars are the heat reservoirs here while the cold space plays the role of the cooler. Objects existing close to the stars can obviously benefit by absorbing the incoming radiation and transforming it into usable work. These objects are called planets.
The planets or rather planetary atmospheres are open systems resembling heat engines powered by solar radiation. Simple calculation yields that incoming solar photon may give rise to about 20 outgoing terrestrial infrared photons. Each photon has approximately the same entropy of order - one bit, so the entropy carried by the photons increases 20-fold which is more than enough to allow some decrease of the entropy of the life forms. The total entropy will go up even if the organisms manage to reduce their disorder. This of course only settles the consistence of the second law of thermodynamics, but does little to answer the question of why we observe the emergence of complex structures. This is why we will now perform an analysis of a simple non-equilibrium model that resembles the systems described in previous paragraphs and demonstrate that a system within that model lowers it’s entropy.
Consider a system $S$, stacked in between a heater $H$ and a cooler $C$, much bigger than the system and with constant temperatures $T_H$ and $T_C$ (See Fig. \[Fig2\]b) such that a steady amount of heat flows from the heater through the system to the cooler.
Following Prigogine’s approach [@Prigogine:1978kz] we will analyze it from the perspective of internal entropy produced $(i)$ and external entropy transferred $(e)$ *to* the system of interest $S$. Of course the entropy change of the system $dS_S$ would be the sum of those two contributions:
$\frac{dS_S}{dt}=\frac{dS_i}{dt}+\frac{dS_e}{dt}$.
As we have mentioned, in our analysis we limit ourselves to the scenario of a *stable state* in which the same amount of heat that goes in also goes out. In other words $dQ_C=-dQ_H$, using which we get the following equation
$dS_e=\frac{dQ_H}{T_H}+\frac{dQ_C}{T_C}=dQ_H\left(\frac{1}{T_H}-\frac{1}{T_C}\right)
=dQ_H\left(\frac{T_C-T_H}{T_HT_C}\right)<0$.
Therefore, the heat flux transfers some of the entropy outside the system (or negative entropy in). Now, since we require the state to be stable and the system’s entropy can’t grow ad infinitum we have $dS_S=0$ and thus the external negative entropy flow balances the internal entropy production $dS_i=-dS_e>0$ (In some sense we could say that the rate of internal entropy production $dS_i$ is a function of the entropy inflow $dS_e$).
Our system is peculiar in one simple manner, namely, thanks to the constant heat flow, the system entropy decreases[^3]! Non-equilibrium conditions indeed let us escape the tyranny of the entropy.
When we turn on the driving force (here temperature difference) the entropy smoothly lowers to minimum entropy $S_{min}$. Moreover, this model does not store energy, hence internal energy stays constant, which means that the free energy, defined by $F~=~U~-~T~S$ gets maximized. Free energy allows us to perform work which can be realized in different ways depending on the nature of the system. It can mean the formation of stars, emergence of complex behaviour in the atmosphere or circular convective motion in the Benard cell.
In fact, Chaisson hypothesized [@Chaisson:2001] that for an object of mass $m$ the *free energy rate density* $\frac{1}{m} \frac{\Delta F}{\Delta t}$ is a universal indicator of complexity of the system and that cosmic evolution is correlated with the growth of localized free energy rate density. Indeed, this trend is visible from our top-bottom approach in which cosmic expansion leads to thermal decoupling (formation of a heat engine). Then, going to smaller and smaller scales this free energy is lent and localized in the subsystems resembling a bifurcation phenomena (See Fig. \[Fig3\]a). First the free energy is passed from the solar systems to planetary atmospheres and then from planetary atmospheres to the biospheres on human scales. This trend does not seem to stop here and the free energy continues to get localized in human societies.
![a) Emergence of heat engines powering complexity growth as a result of departure from equilibrium. b) Timeline of free energy rate density growth adopter from Ref. [@Chaisson:2001].[]{data-label="Fig3"}](biff-exp){width="18cm"}
At first it may appear counterintuitive, but the free energy rate density is indeed smaller for stars than it is for planetary atmospheres, and it is smaller for planetary atmospheres than it is for animals. The XXI century inventions, like engines and computers also have their place in it and might be considered a part of our modernised society, amounting to our complexity. Human brains take the top places in this ladder of complexity and continuously climb it; through augmentation with technology.
We did not prove the association made by Chaisson by any rigorous way. At the end some further improvements might be possible, but the trend is clearly visible, almost as in Darwinian evolution. The inflows of free energy localize on smaller and smaller scales in tandem with growth of complexity.
On the other hand, one shall not be tempted to try to find the explanation in terms of teleology, a way which has a certain appeal for human beings. A great example can be given by trying to explain the convection currents in Benard cell. The sentence “Benard cell activates another mean of transport to efficiently transport heat from the heater to the cooler" implies an idea proposed by the Maximum Entropy Production (MEP) principle which validity is under recent theoretical investigations [@Ozawa:2003jt]. Precisely because the MEP principle is currently teleological in nature it does not get a very wide approval among scientific community. Laymen might think that it’s some close-minded thinking of the scientific community, but this insistence on causal explanations is one of main reasons of triumph and an enormous success of science in explaining natural phenomena. Advancement in our understanding of say, gravity, was made by the retreat from teleological explanations. For example the ideas of ancient greeks who believed that the cause of the downward motion of heavy bodies on Earth was related to their nature, which caused them to move downward, toward the center of the Universe, which was their natural place. Conversely, light bodies such as the element fire, moved by their nature upward toward the inner surface of the sphere of the Moon [@Pedersen:99571; @Grant:2009ue].
Nowadays we recognize the naivety of those explanations and the superiority of Newtonian gravity. Even though the benefits of adopting Newtonian ideas were enormous as they allowed us to predict various events, they still left some room for improvements (force had instantaneous effect on distant bodies) as Newton himself noted [@Janiak:2006uw]. This and more weaknesses were brilliantly noticed by Einstein, whose theory not only opened new areas of inquiry (black holes and studies of the beginning of the Universe) and made gravitational predictions more precise (allowing us to develop GPS systems etc.), but also strengthened the *causal* nature of gravity.
It’s not as to say that teleology can not be a vestibule to physics. If the idea is further developed, a teleological explanation may be untangled to “theory of principle" or a “constructive theory", classification proposed by Einstein in 1919 [@Einstein:104771]. With enough work, MEP principle (in one form or another) might become a part of a valid scientific framework leading us to a correct and complete description of non-equilibrium phenomena.
So far we have learned that teleological explanations have little place in physics, but what might be less obvious is that it is also the case for evolutionary biology. To see this, let’s try to discriminate the content of various statements. The everyday statement, we usually consider true “birds have wings *to* fly” has little scientific relevance. A slightly modified version of this statement “birds evolved wings to fly” is much better, because (even though it contains the grammatical particle “to") it underlines it’s causal ingredient, namely evolution. Natural selection is all about discovering the evolutionary pressures that led to formation of complex features; there is no goal in this process. Of course we talk about “survival of the fittest", but this is just a poetic way of the description and the ultimate end state (goal) is not known. This feature of biology convinces us that it can, and will be ultimately described by physics and chemistry.
To make the case stronger, just recently J. England [@England:2015hl] proposed a physical process by which evolution can take place at the microscopic scale. In fact his approach is another facade of the non-equilibrium theory, taste of which we experienced in the previous paragraphs. In his paper, J. England employs a peculiar family of laws known as Fluctuation Theorems. In contrast to known laws of Statistical Mechanics those laws take into account the history of the process and discriminate between the more likely histories according to the entropy transferred to the environment. The more entropy (or in other words heat) is produced the more likely a microscopic path ($\Rightarrow$) is relative to it’s reverse ($\Leftarrow$):
$\frac{\Rightarrow}{\Leftarrow}=\exp\Delta S_{env}$.
Once again, the end states are determined by microscopic evolution, the end goal is not set. Moreover, this last equation can be easily expanded to macrostates which give us a generalized form of second law of thermodynamics
$\Delta S_{tot}+\ln(\frac{\Leftarrow}{\Rightarrow})>0$,
(note the change of direction of the arrows) where $\Delta S_{tot}$ is the sum of entropy change of the environment and the system under consideration.
What this tells us is that the more irreversible a transition is, the less entropy has to be produced *inside* the system (in a similar way to our previous considerations). However, the form of this equation makes the link between irreversibility and entropy much more evident and fruitful, thus we will mention here a few results to which it leads.
If we apply it to a system under periodic driving force, then it was shown [@Perunov:2016hl] how the system can transform or remodel to a form in which it dissipates energy more efficiently. This result gave a pioneer example of evolution taking place at the microscale. On the other hand applying this equation to self-replicators [@doi:10.1063/1.4818538] an inverse relation between the maximal growth rate of a self-replicator and it’s durability was found. This implies that there must be a biological tradeoff between the pace of reproduction and durability of self-replicators. Those advancements push us towards understanding the origin of biological complexity and it is likely that the next step, with the help of information theory, will unravel the emergence of agency. In fact, purpose directed behaviour could be seen as a result of information processing and storage which in itself relies on availability of free energy as shown by Landauer [@Landauer:2002wc]. This accumulation of knowledge, being synonymous to the *free energy rate density* growth is also expected to follow the double exponent rule [@Kurzweil:2007vy]. Those and other insights guide as in understanding of complex phenomena and allow us to extrapolate the trends and prognose the future.
To sum up, we have presented a raw mechanism, by which localized complexity (stars, atmosphere, life, technology) emerges from physical laws of Nature. The crucial role was played by the free energy rate density, given by the chain of spontaneously forming heat engines initiated by cosmic expansion. We then argued that teleological reasoning can also be thought as an imperative result of the growing complexity which at about XVI century has been superseded by modern, causal, scientific reasoning in a similar way in which walking on land superseded swimming in sea for our predecessors. Obviously we still use it (as we still enjoy swimming), but it’s purpose is different. Finally, thanks to recent advancement in non-equilibrium statistical mechanics, our belief in the causal structure and the illuminating power of modern scientific theories was transferred to the smaller scales governed by biology. There, adaptation took place under the influence of an external periodic force, in case of Earth - solar radiation.
The upcoming years will certainly bring us closer towards understanding the emergence of life and agency, in the context of causal frameworks of physics. Therefore, the insistence on finding purpose in the laws of Nature is illusionary and futile. This, however, should not be perceived negatively or discourage us, as we believe that through ingenuity and the pursuit of truth we will find ourselves a purpose.
*“The significance of our lives and our fragile planet is then determined by our own wisdom and courage. We are the custodians of life’s meaning. ...If we crave some cosmic purpose, then let us find ourselves a worthy goal." — Carl Sagan, 1997 [@Sagan:1997up]*
[^1]: This is just a reflection of the fact that there is no environment with respect to which the evolution of the system could be measured.
[^2]: Nevertheless, the averaged matter temperature remained below the temperature of radiation.
[^3]: To see this, Taylor expand $\frac{dS_i}{dt}=
j_i(S_S)=j_i\left(S_0\right)+\left(S_S-S_0\right)C_1+\mathcal{O}\left(S_S^2\right)$ and solve differential equation $\frac{dS_S}{dt}=j_e + j_i\left(S_S\right)=j_e +\frac{S_0-S_S}{\tau}$, getting $S_S(t)=S_0+j_e\tau \left(1-e^{-t/\tau }\right)$, where $j_e$ is a negative constant. In the $t\rightarrow \infty$ limit the system’s entropy falls from the initial value $S(t=0)=S_0$ to the minimal value $S_{min}=S(t\rightarrow \infty) =S_0+j_e \tau < S_0$.
|
---
abstract: 'Mass measurements of gravitational microlenses require one to determine the microlens parallax ${\pi_{\rm E}}$, but precise ${\pi_{\rm E}}$ measurement, in many cases, is hampered due to the subtlety of the microlens-parallax signal combined with the difficulty of distinguishing the signal from those induced by other higher-order effects. In this work, we present the analysis of the binary-lens event OGLE-2017-BLG-0329, for which ${\pi_{\rm E}}$ is measured with a dramatically improved precision using additional data from space-based $Spitzer$ observations. We find that while the parallax model based on the ground-based data cannot be distinguished from a zero-${\pi_{\rm E}}$ model at 2$\sigma$ level, the addition of the $Spitzer$ data enables us to identify 2 classes of solutions, each composed of a pair of solutions according to the well-known ecliptic degeneracy. It is found that the space-based data reduce the measurement uncertainties of the north and east components of the microlens-parallax vector ${\mbox{\boldmath $\pi$}}_{\rm E}$ by factors $\sim 18$ and $\sim 4$, respectively. With the measured microlens parallax combined with the angular Einstein radius measured from the resolved caustic crossings, we find that the lens is composed of a binary with components masses of either $(M_1,M_2)\sim (1.1,0.8)\ M_\odot$ or $\sim (0.4,0.3)\ M_\odot$ according to the two solution classes. The first solution is significantly favored but the second cannot be securely ruled out based on the microlensing data alone. However, the degeneracy can be resolved from adaptive optics observations taken $\sim 10$ years after the event.'
author:
- |
C. Han, S. Calchi Novati, A. Udalski, C.-U. Lee, A. Gould, V. Bozza,\
and\
P. Mróz, P. Pietrukowicz, J. Skowron, M. K. Szymański, R. Poleski, I. Soszyński, S. Koz[ł]{}owski, K. Ulaczyk, M. Pawlak, K. Rybicki, P. Iwanek,\
(The OGLE Collaboration)\
M. D. Albrow, S.-J. Chung, K.-H. Hwang, Y. K. Jung, Y.-H. Ryu, I.-G. Shin, Y. Shvartzvald, J. C. Yee, W. Zang, W. Zhu, S.-M. Cha, D.-J. Kim, H.-W. Kim, S.-L. Kim, D.-J. Lee, Y. Lee, B.-G. Park, R. W. Pogge, W.-T. Kim\
(The KMTNet Collaboration),\
C. Beichman, G. Bryden, S. Carey, B. S. Gaudi, C. B. Henderson,\
(The $Spitzer$ Team)\
M. Dominik, C. Helling, M. Hundertmark, U. G. J[ø]{}rgensen, P. Longa-Pe[ñ]{}a, S. Lowry, S. Sajadian, M. J. Burgdorf, J. Campbell-White, S. Ciceri, D. F. Evans, L. K. Haikala, T. C. Hinse, S. Rahvar, M. Rabus, C. Snodgrass\
(The MiNDSTEp Collaboration)
title: 'OGLE-2017-BLG-0329L: A Microlensing Binary Characterized with Dramatically Enhanced Precision Using Data from Space-based Observations'
---
Introduction {#sec:one}
============
Microlensing phenomena occur by the gravitational field of lensing objects regardless of their luminosity. Due to this nature, microlensing can, in principle, provide an important tool to determine the mass spectrum of Galactic objects based on samples that are unbiased by luminosity [@Han1995].
Construction of the mass spectrum requires one to determine the masses of individual lenses. For most microlensing events, the only observable related to the physical parameters of the lens is the Einstein timescale. However, the Einstein timescale is related to not only the lens mass but also the relative lens-source parallax, $\pi_{\rm rel}$, and the proper motion, $\mu_{\rm rel}$, by $$t_{\rm E}= {{\theta_{\rm E}}\over \mu_{\rm rel}},\qquad {\theta_{\rm E}}=(\kappa M \pi_{\rm rel})^{1/2},$$ where ${\theta_{\rm E}}$ is the angular Einstein radius, $\kappa=4G/(c^2 {\rm au})\sim 8.14\ {\rm mas}/M_\odot$, $\pi_{\rm rel}={\rm au}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$, and $D_{\rm L}$ and $D_{\rm S}$ denote the distances to the lens and source, respectively. As a result, the lens mass cannot be uniquely determined from the event timescale alone. For the unique determination of the lens mass, one needs to measure two additional quantities: the angular Einstein radius ${\theta_{\rm E}}$ and the microlens-parallax ${\pi_{\rm E}}$ with which the mass and distance to the lens are determined by [@Gould2000] $$M={{\theta_{\rm E}}\over \kappa {\pi_{\rm E}}};\qquad
D_{\rm L}={{\rm au}\over {\pi_{\rm E}}{\theta_{\rm E}}+\pi_{\rm S}}$$ where $\pi_{\rm S}={\rm au}/D_{\rm S}$.
The angular Einstein radius can be measured from deviations in lensing lightcurves affected by finite-source effects. Finite-source effects occur when a source star is located in the region within which the gradient of lensing magnifications is significant and thus different parts of the source are differentially magnified. For a lensing event produced by a single mass, this corresponds to the very tiny region around the lens, and thus finite-source effects can be effectively detected only for a very small fraction of events for which the lens transits the surface of the source [@Witt1994; @Nemiroff1994; @Gould1994]. For events produced by binary lenses, on the other hand, the chance to detect finite-source effects is relatively high because the lens systems form extended caustics around which the magnification gradient is high. Analysis of deviations affected by finite-source effects yields the normalized source radius $\rho$, which is defined as the ratio of the angular source radius $\theta_*$ to the angular Einstein radius. By estimating $\theta_*$ from external information of the source color, the angular Einstein radius is determined by ${\theta_{\rm E}}=\theta_*/\rho$.
The microlens-parallax can be measured from deviations in lensing lightcurves caused by the positional change of an observer. In the single frame of Earth, such deviations occur due to the acceleration of Earth induced by the orbital motion: “annual microlens parallax” [@Gould1992]. However, precise ${\pi_{\rm E}}$ measurement from the deviations induced by the annual microlens-parallax effect is difficult because the positional change of an observer during $\sim (O)10$-day durations of typical lensing events is, in most cases, very minor. As a result, ${\pi_{\rm E}}$ measurements have been confined to a small fraction of all events, preferentially events with long timescales and/or events caused by relatively nearby lenses. For binary-lens events, ${\pi_{\rm E}}$ measurement becomes further complicated because the orbital motion of the binary lens produces deviations in lensing light curves similar to those induced by microlens-parallax effects [@Batista2011; @Skowron2011; @Han2016].
Microlens parallaxes of lensing events can also be measured if events are simultaneously observed using ground-based telescopes and space-based satellite in a heliocentric orbit: “space-based microlens parallax” [@Refsdal1966; @Gould1994]. For typical lensing events with physical Einstein radii of order au, the separation between Earth and a satellite can comprise a significant fraction of the Einstein radius. Then, the lensing lightcurves observed from the ground and from the satellite appear to be different due to the difference in the relative lens-source positions, and the comparison of the two lightcurves leads to the determination of ${\pi_{\rm E}}$.
In this work, we present the analysis of the binary microlensing event OGLE-2017-BLG-0329 that was observed both from the ground and in space using the $Spitzer$ telescope. We show that while the parallax model based on the ground-based data cannot be distinguished from a zero-${\pi_{\rm E}}$ model, the addition of the $Spitzer$ data leads to the firm identification of two classes of microlens-parallax solutions
Observations and Data {#sec:two}
=====================
The microlensing event OGLE-2017-BLG-0329 occurred on a star located toward the Galactic bulge. The equatorial coordinates of the event are $({\rm RA},{\rm DEC})_{\rm J2000}=$(17:53:43.20, -32:55:27.4), which correspond to the Galactic coordinates $(l,b)=(-2.53^\circ,-3.54^\circ)$. The baseline magnitude of the event before lensing magnification was $I_{\rm base}\sim 15.84$.
Figure \[fig:one\] shows the light curve of the event. The light curve is characterized by 3 peaks. The first smooth peak occurred at ${\rm HJD}'={\rm HJD}-2450000 \sim 7882$ and the other two sharp peaks occurred at ${\rm HJD}' \sim 7900$ and 7927. The smooth and sharp peaks are typical features that occur when a source approaches the cusp and passes over the fold of a binary-lens caustic, respectively. The event was already in progress before the 2017 microlensing season and lasted for more than 100 days.
The lensing event was observed from the ground by two microlensing surveys of the Optical Gravitational Lensing Experiment [OGLE: @Udalski2015a] and the Korea Microlensing Telescope Network [KMTNet: @Kim2016]. OGLE observations of the event were conducted using the 1.3m telescope located at the Las Campanas Observatory in Chile. The OGLE survey first identified the event from its Early Warning System on 2017 March 14 (${\rm HJD}'=7828.4$). KMTNet observations were carried out using 3 globally distributed 1.6m telescopes located at the Cerro Tololo Inter-American Observatory in Chile (KMTC), the South African Astronomical Observatory in South Africa (KMTS), and the Siding Spring Observatory in Australia (KMTA). The event was identified by KMTNet as BLG22K0103.001613. Observations by both surveys were conducted mainly in $I$ band and some $V$-band images were obtained for the color measurement of the source star. The event was located in the OGLE BLG502 and KMTNet BLG22 fields, which were observed with cadences of 0.17/hr and 1/hr by the OGLE and KMTNet surveys, respectively. With the high cadence of the surveys, both the caustic entrance and exit were resolved. See the upper panels of Figure \[fig:one\]. Besides the survey experiments, the event was additionally observed from follow-up experiment conducted by the MiNDSTEp Collaboration during the period $7887.9 < {\rm HJD}' < 7954.7$ using the 1.5m Danish Telescope at La Silla Observatory in Chile. Photometry of the data were conducted using the pipelines developed by the individual groups based on the difference imaging analysis method [@Alard1998]. Since the data sets were taken using different instruments and reduced based on different softwares, we normalize the error bars of the individual data sets using the method described in @Yee2012.
The event was also observed in space. At the time that it was originally evaluated for [*Spitzer*]{} observations (2017 May 1; ${\rm HJD}' = 7874$), it was believed to be a point-lens event, and hence the decision was made in accordance with the protocols of Yee et al. (2015), which are designed to obtain an unbiased sample of events to probe the Galactic distribution of planets. The [*Spitzer*]{} team specified that the event should be observed provided that it reached $I<15.65$ at ${\rm HJD}'=7924$, i.e., the time of the first upload. Since this requirement was met, these observations were initiated, and were ultimately conducted during the period 7930.5 – 7966.9 ($\sim 36.4$ days), with both dates set essentially by the spacecraft’s Sun-angle restrictions. $Spitzer$ images were taken in the 3.6 $\mu$m channel of the IRAC camera, and the data were reduced with a specially developed version of point response function photometry [@Calchi2015b].
Analysis {#label:three}
========
OGLE-2017-BLG-0329 is of scientific importance because it may be possible to measure the microlens parallax not only from the ground-based data (annual microlens parallax) but also from the combined ground+space data (space-based microlens parallax). For this event, the chance to measure the annual microlens parallax is high due to its long timescale. Since the event was additionally observed by the $Spitzer$ telescope, the microlens parallax can also be measured from the combined ground+space data. Therefore, the event provides a test bed in which one can check the consistency of the ${\pi_{\rm E}}$ values and compare the precision of ${\pi_{\rm E}}$ measurements by the individual methods. We note that there exist four cases for which ground-based ${\pi_{\rm E}}$ measurements have been confirmed by space-based data: OGLE-2014-BLG-0124 [@Udalski2015b], OGLE-2015-BLG-0196 [@Han2017], OGLE-2016-BLG-0168 [@Shin2017], and MOA-2015-BLG-020 [@Wang2017].
Ground-based Data {#sec:three-one}
-----------------
We first conduct analysis of the event based on the data obtained from ground-based observations. We start modeling the light curve under the approximation that the relative lens-source motion is rectilinear (“standard model”). For this modeling, one needs 7 principal lensing parameters. These parameters include the time of the closest source approach to a reference position of the binary lens, $t_0$, the lens-source separation at that time, $u_0$ (impact parameter), the event timescale, $t_{\rm E}$, the projected separation $s$ (normalized to ${\theta_{\rm E}}$), and the mass ratio $q$ between the binary-lens components, the angle between the source trajectory and the binary-lens axis, $\alpha$ (source trajectory angle), and the normalized source radius $\rho$. We choose the barycenter of the binary lens as the reference position of the lens.
Since both the caustic crossings of the light curve were resolved, we consider finite-source effects. Finite-source magnifications are computed using the ray-shooting method [@Schneider1986; @Kayser1986; @Wambsganss1997] In computing lensing magnifications, we consider the surface-brightness variation of the source star caused by limb darkening. For this, we model the surface brightness profile of the source star as $S \propto 1-\Gamma(1-3\cos \phi/2)$, where $\Gamma$ is the linear limb-darkening coefficient and $\phi$ is the angle between the line of sight toward the center of the source star and the normal to the surface of the source star. Based on the spectral type of the source star (see Section \[sec:four-one\]), we adopt $\Gamma_I=0.53$.
To find the solution of the lensing parameters, we first conduct a grid search for the parameters $\log s$ and $\log q$, while the other parameters $(t_0, u_0, t_{\rm E}, \rho, \alpha)$ at each point on the ($\log s,\log q$) plane are searched for by minimizing $\chi^2$ using the Markov Chain Monte Carlo (MCMC) method. This first-round search yields local minima in the ($\log s,\log q$) plane. For each local minimum, we then refine the solution by allowing all parameters to vary. We identify a global minimum by comparing $\chi^2$ values of the individual local solutions. From this modeling, we find a unique solution of the event. According to this solution, the event was produced by a binary with a mass ratio between the components of $q\sim 0.7$ and a projected separation of $s\sim 1.4$. Due to the similar masses of the binary components and the proximity of the separation to unity, the caustic forms a single big closed curve (resonant caustic).
[lcc]{} Standard & 2392.1 & -\
Orbit & 2338.2 & -\
Parallax & 2356.8 & 2363.1\
Orbit+Parallax & 2336.1 & 2330.8
Since the event can be subject to higher-order effects due to its long timescale, we conduct additional modeling considering two such effects. In the “parallax model” and “lens-orbital model”, we separately consider the microlens-parallax and lens-orbital effects, respectively. We also conduct modeling by simultaneously considering both higher-order effects (“orbit+parallax model”). Consideration of the microlens-parallax effects requires to include 2 additional parameters of $\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$ , which represent the north and east components of the microlens parallax vector ${\mbox{\boldmath $\pi$}}_{\rm E}$, projected on the sky in the north and east equatorial coordinates, respectively. Under the approximation that the positional change of the lens is small, the lens-orbital effects are described by two parameters $ds/dt$ and $d\alpha/dt$, which represent the change rates of the binary separation and the source trajectory angle, respectively. For parallax solutions, it is known that there can exist a pair of degenerate solutions with $u_0>0$ and $u_0<0$ due to the mirror symmetry of the lens system geometry [@Smith2003; @Skowron2011]. We check this so-called “ecliptic degeneracy” whenever we consider microlens-parallax effects in modeling. The lensing parameters of the two solutions resulting from the ecliptic degeneracy are approximately in the relations of $(u_0, \alpha, \pi_{{\rm E},N}, d\alpha/dt) \leftrightarrow -(u_0, \alpha, \pi_{{\rm E},N}, d\alpha/dt)$.
In Table \[table:one\], we present the goodness of the fits expressed in terms of $\chi^2$ values for the individual tested models. From the comparison of $\chi^2$ values, it is found that the model fit improves with the consideration of the higher-order effects. The improvement by the microlens-parallax and lens-orbital effects are $\chi^2=35.3$ and 53.9, respectively. When both higher-order effects are simultaneously considered, on the other hand, it is found that the fit improvement is merely $\chi^2=7.4$ with respect to the orbital model. From the fact that (1) the fit improvement by the lens-orbital effect is bigger than the improvement by the microlens-parallax effect and (2) the further improvement from the orbital model by additionally considering the microlens-parallax effect is small, we judge that the dominant higher-order effect is the lens-orbital effect and the microlens-parallax effect is relatively small.
The weakness of the microlens-parallax effect can also be seen in Figure \[fig:two\], where we present the $\Delta\chi^2$ distribution in the ($\pi_{{\rm E},N},\pi_{{\rm E},E}$) plane obtained from the modeling considering both microlens-lens and lens-orbital effects. It shows that the model is consistent with the zero-${\pi_{\rm E}}$ model by $\Delta\chi^2\lesssim 4$. For the validation of the weak microlens-parallax interpretation, the lens parameters resulting from the orbit+parallax model should be physically permitted. For this, we estimate the ranges of the lens mass ($M=M_1+M_2$) and the projected kinetic-to-potential energy ratio, which is computed by $$\left({{\rm KE} \over {\rm PE}} \right)_\perp =
{(a_\perp/{\rm au})^3 \over 8\pi (M/M_\odot)}
\left[ \left( {1\over s} {ds\over dt}{\rm yr} \right)^2 + \left( { d\alpha\over dt} {\rm yr}\right)^2 \right] .$$ We describe the procedure to measure the angular Einstein radius ${\theta_{\rm E}}$, which is needed to determine $M$ and $({\rm KE/PE})_\perp$, in Section \[sec:four-one\]. We find that the ranges of the lens mass and the energy ratio are $0.9 \leq M/M_\odot \leq 4.6 $ and $0.04 \leq ({\rm KE/PE})_\perp \leq 0.1$, respectively. The estimated lens mass roughly corresponds to those of binaries composed of stars. The kinetic-to-potential energy ratio also meets the condition $({\rm KE}/{\rm PE})_\perp < {\rm KE}/{\rm PE}< 1$, that is required for the binary lens to be a gravitationally bound system. Therefore, the solution based on the ground-based data is physically permitted, although the range of the estimated lens mass is very wide due to the large uncertainty of the measured ${\pi_{\rm E}}$.
[lcccc]{} $\chi^2$ & 2373.1 (3.1) & 2398.4 (12.7) & 2395.1 (5.2) & 2384.4 (1.7)\
$t_0$ (HJD’) & 7904.908 $\pm$ 0.098 & 7904.885 $\pm$ 0.106 & 7904.873 $\pm$ 0.057 & 7905.071 $\pm$ 0.057\
$u_0$ & -0.151 $\pm$ 0.002 & 0.152 $\pm$ 0.002 & -0.151 $\pm$ 0.001 & 0.149 $\pm$ 0.001\
$t_{\rm E}$ (days) & 41.73 $\pm$ 0.06 & 41.73 $\pm$ 0.05 & 41.71 $\pm$ 0.04 & 41.64 $\pm$ 0.04\
$s$ & 1.438 $\pm$ 0.002 & 1.438 $\pm$ 0.002 & 1.440 $\pm$ 0.001 & 1.438 $\pm$ 0.001\
$q$ & 0.704 $\pm$ 0.005 & 0.702 $\pm$ 0.006 & 0.701 $\pm$ 0.002 & 0.712 $\pm$ 0.003\
$\alpha$ (rad) & -0.642 $\pm$ 0.001 & 0.642 $\pm$ 0.001 & -0.648 $\pm$ 0.001 & 0.647 $\pm$ 0.001\
$\rho$ (10$^{-3}$) & 8.76 $\pm$ 0.07 & 8.69 $\pm$ 0.07 & 8.68 $\pm$ 0.05 & 8.78 $\pm$ 0.06\
$\pi_{{\rm E},N}$ & 0.034 $\pm$ 0.003 & -0.030 $\pm$ 0.004 & -0.100 $\pm$ 0.006 & 0.121 $\pm$ 0.007\
$\pi_{{\rm E},E}$ & 0.040 $\pm$ 0.009 & 0.031 $\pm$ 0.007 & 0.070 $\pm$ 0.007 & 0.065 $\pm$ 0.009\
$ds/dt$ (yr$^{-1}$) & 0.215 $\pm$ 0.058 & 0.197 $\pm$ 0.059 & 0.122 $\pm$ 0.013 & 0.175 $\pm$ 0.021\
$d\alpha/dt$ (yr$^{-1}$) & -0.165 $\pm$ 0.031 & 0.158 $\pm$ 0.030 & -0.168 $\pm$ 0.020 & 0.009 $\pm$ 0.019\
$f_{{\rm S},I}$ & 7.35 & 7.36 & 7.37 & 7.35\
$f_{{\rm b},I}$ & -0.06 & -0.07 & -0.08 & -0.06
Additional Space-based Data {#sec:three-two}
---------------------------
Knowing the difficulty of secure ${\pi_{\rm E}}$ measurement based on only the ground-based data, we test the possibility of ${\pi_{\rm E}}$ measurement with the additional data obtained from $Spitzer$ observations. To compute lensing magnifications seen from the $Spitzer$ telescope, one needs the position and the distance to the satellite. The position of the $Spitzer$ telescope was in the ranges of $110^\circ \lesssim {\rm RA}\lesssim 194^\circ$ and $-7^\circ\lesssim {\rm DEC}\lesssim 21^\circ$ and the distance was in the range of $1.566\lesssim d_{\rm sat}/{\rm au}\lesssim 1.584$ during the 2017 bulge season.
The $Spitzer$ data partially covered the event light curve. Furthermore, they do not cover major features such as those produced by caustic crossings. See the $Spitzer$ data presented in Figure \[fig:one\]. In such a case, it is known that external information of the color between the passbands used for observations from Earth and from the $Spitzer$ telescope can be useful in finding a correct model [@Yee2015; @Shin2017]. We, therefore, apply a color constraint with the measured instrumental color of $I-L=2.33\pm 0.012$ The color constraint is imposed by giving penalty $\chi^2$ defined in Eq. (2) of @Shin2017.
For single-lensing events observed both from Earth and from a satellite, it is known that there exist four sets of degenerate solutions [@Refsdal1966; @Gould1994]. This degeneracy among these solutions, referred to as $(+,+)$, $(-,-)$, $(+,-)$, and $(-,+)$ solutions, arises due to the ambiguity in the signs of the lens-source impact parameters as seen from Earth (the former sign in the parenthesis) and from the satellite (the latter sign in the parenthesis). In many case of binary-lens events, this four-fold degeneracy reduces into a two-fold degeneracy [@Han2017] due to the lack of lensing magnification symmetry. The remaining two degenerate solutions, $(+,+)$ and $(-,-)$ solutions, are caused by the mirror symmetry of the source trajectory with respect to the binary axis, and thus the degeneracy corresponds to the ‘ecliptic degeneracy’. Besides the known types, binary events can be subject to various other types of degeneracy.
In order to check the existence of degenerate solutions, we explore the space of the lensing parameters using two methods. First, we conduct a grid search over the ($\pi_{{\rm E},N},\pi_{{\rm E},E}$) plane. Second, we search for local solutions using a downhill approach from various starting points that are obtained from the analysis based on the ground-based data. From these searches, we identify 2 classes of solutions, in which each class is composed of 2 solutions arising from the ecliptic degeneracy.
![ The lens system geometry and the portion of the light curve in the vicinity of the $Spitzer$ data for the 4 degenerate solutions. For each lens system geometry, the source trajectories seen from Earth and the $Spitzer$ telescope are marked by blue and red curves (with arrows), respectively. The cuspy closed curve represents the caustic. The coordinates are centered at the barycenter of the binary lens. The blue and red curves superposed on the data points represent the model light curves for the ground and $Spitzer$ data, respectively. []{data-label="fig:four"}](f4.eps){width="\columnwidth"}
In Figure \[fig:three\], we present the locations of the local solutions in the ($\pi_{{\rm E},N},\pi_{{\rm E},E}$) plane. It is found that one pair of solutions have $\pi_{\rm E}=(\pi_{{\rm E},N}^2 + \pi_{{\rm E},E})^{1/2}\gtrsim 0.1$ (‘big-${\pi_{\rm E}}$’ solutions) and the other pair have ${\pi_{\rm E}}\lesssim 0.1$ (‘small-${\pi_{\rm E}}$’ solutions). For each pair, the lensing parameters of the two degenerate solutions are approximately in the relation of $(u_0, \alpha, \pi_{{\rm E},N}, d\alpha/dt) \leftrightarrow -(u_0, \alpha, \pi_{{\rm E},N}, d\alpha/dt)$, and thus we refer to the solutions as $u_0<0$ and $u_0>0$ solutions. We note that although the higher-order parameters $(\pi_{{\rm E},N}, \pi_{{\rm E},E}, ds/dt, d\alpha/dt)$ of these degenerate solutions are different from one another, the other lensing parameters are similar because they are mostly determined from the ground-based data. By comparing the ranges of the $\Delta\chi^2$ distributions with and without the [*Spitzer*]{} data, one finds that the uncertainties of the determined microlens-parallax parameters are greatly reduced with the use of the $Spitzer$ data.
In Table \[table:two\], we list the lensing parameters of the 4 degenerate solutions along with their $\chi^2$ values. We find that the (small-${\pi_{\rm E}}$)/($u_0<0$) solution is preferred over the other solutions for two major reasons. First, the (small-${\pi_{\rm E}}$)/($u_0<0$) solution provides a better fit to the observed data than the other solutions by $11.3 < \Delta\chi^2 <25.3$. Second, the small-${\pi_{\rm E}}$ solutions are preferred over the big-${\pi_{\rm E}}$ solution according to the “Rich argument” [@Calchi2015a], which states that, other factors being equal, small parallax solutions are preferred over large ones by a probability factor $(\pi_{\rm E, big}/\pi_{\rm E,small})^2\gtrsim 6$. Although the (small-${\pi_{\rm E}}$)/($u_0<0$) solution is favored, one cannot completely rule out the other degenerate solutions. We, therefore, discuss the methods that can firmly resolve the degeneracy in Section \[sec:five\]. Also presented in Table \[table:two\] are the fluxes of the source, $f_{{\rm S},I}$, and the blend, $f_{{\rm b},I}$, estimated based on the OGLE data. The small $f_{{\rm b},I}$ indicates that the blend flux is small. We note that the small negative blending is quite common for point-spread-function photometry in crowded fields [@Park2004].
![ Contour map of lensing magnification in the outer region of the caustic. Contours are drawn at every $\Delta A=0.05$ step from $A=1.1$ to $A=2.0$. The lines with arrows represent the source trajectories of the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions seen from the $Spitzer$ telescope. The crosses on each trajectory represent the expected positions of the source when $Spitzer$ data were taken. []{data-label="fig:five"}](f5.eps){width="\columnwidth"}
In Figure \[fig:four\], we present the lens-system geometry of the 4 degenerate solutions. For each geometry, we present the source trajectories with respect to the lens components (small empty dots marked by $M_1$ and $M_2$) and the caustic (cuspy closed curve). For each geometry, the source trajectories seen from Earth and the $Spitzer$ telescope are marked by blue and red curves (with arrows), respectively. We also present the portion of the light curve in the vicinity of the $Spitzer$ data and the model light curve.
As mentioned, the degeneracy between the $u_0<0$ and $u_0>0$ solutions is caused by the mirror symmetry of the lens system geometry. On the other hand, the degeneracy between the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions is caused by the difference in the source trajectory angles seen from the $Spitzer$ telescope. For the small-${\pi_{\rm E}}$ solution, the source trajectory angle as seen from the $Spitzer$ telescope is bigger than the angle of the source trajectory seen from the ground. In contrast, the $Spitzer$ trajectory angle of the big-${\pi_{\rm E}}$ solution is smaller than the angle of the ground trajectory. We note that the latter degeneracy is different from the degeneracy between $(+,+)$ and $(+,-)$ solutions because both ground and satellite trajectories pass on the same side with respect to the barycenter of the binary lens. Such a degeneracy is not previously known.
In order to further investigate the cause of the degeneracy between the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions, in Figure \[fig:five\], we present the magnification contours in the outer region of the caustic. On the contour map, we plot the $Spitzer$ source trajectories of the two degenerate solutions. From the map, it is found that the magnification patterns along the source trajectories of the two degenerate solutions are similar to each other, suggesting that the degeneracy is caused by the symmetry of magnification pattern in the outer region of the caustic. We note that the degeneracy could have been resolved if the caustic exit part of the light curve had been covered by $Spitzer$ data because the times of the caustic exit (seen from the $Spitzer$ telescope) expected from the two degenerate solutions are different from each other. We find that the caustic-exit times for the small-${\pi_{\rm E}}$ solutions are in the range of $7926~({\rm for}~u_0<0) \lesssim {\rm HJD}' \lesssim 7928~({\rm for}~u_0>0)$. On the other hand, the range for the big-${\pi_{\rm E}}$ solutions is $7922~({\rm for}~u_0<0) \lesssim {\rm HJD}' \lesssim 7924~({\rm for}~u_0>0)$. With the $\sim 4$ day time gap between the caustic-crossing times of the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions, the degeneracy could have been easily lifted. In conclusion, we find that the new type of degeneracy is caused by the partial symmetry of the magnification pattern outside the caustic combined with the fragmentary coverage of the $Spitzer$ data.
From the comparison of the analyses conducted with and without the space-based data, we find two important results.
1. First, while the microlens parallax cannot be securely determined based on only the ground-based data, the addition of the $Spitzer$ data enables us to clearly identify two classes of microlens-parallax solutions. The degeneracy is either intrinsic to lensing systems ($u_0<0$ versus $u_0>0$ degeneracy) or due to the combination of the partial symmetry of magnification pattern combined with the fragmentary $Spitzer$ coverage of the event (small-${\pi_{\rm E}}$ versus big-${\pi_{\rm E}}$ degeneracy).
2. Second, the space-based data greatly improve the precision of the ${\pi_{\rm E}}$ measurement. We find that the measurement uncertainties of the north and east components of ${\mbox{\boldmath $\pi$}}_{\rm E}$ are reduced by factors $\sim 18$ and $\sim 4$, respectively, with the use of the $Spitzer$ data. Since the lens mass is directly proportional to $1/{\pi_{\rm E}}$, the uncertainty of the mass measurement reduces by the same factors.
[lcccc]{} $M_1$ ($M_\odot$) & 1.09 $\pm$ 0.15 & 1.33 $\pm$ 0.21 & 0.47 $\pm$ 0.04 & 0.41 $\pm$ 0.04\
$M_2$ ($M_\odot$) & 0.77 $\pm$ 0.11 & 0.93 $\pm$ 0.15 & 0.33 $\pm$ 0.03 & 0.29 $\pm$ 0.03\
$D_{\rm L}$ (kpc) & 6.38 $\pm$ 0.79 & 6.66 $\pm$ 0.86 & 4.69 $\pm$ 0.45 & 4.48 $\pm$ 0.42\
$a_\perp$ (au) & 7.20 $\pm$ 0.89 & 7.59 $\pm$ 0.98 & 5.35 $\pm$ 0.52 & 5.04 $\pm$ 0.47\
$({\rm KE}/{\rm PE})_\perp$ & 0.12 $\pm$ 0.02 & 0.12 $\pm$ 0.02 & 0.09 $\pm$ 0.01 & 0.04 $\pm$ 0.01\
$\psi$ (deg) & $51$ & $132$ & $141$ & $32$
Physical Lens Parameters {#sec:four}
========================
Source Star and Angular Einstein Radius {#sec:four-one}
---------------------------------------
For the unique determination of the lens mass and distance, one needs to estimate the angular Einstein radius in addition to the microlens parallax. Since the angular Einstein radius is determined by ${\theta_{\rm E}}=\theta_*/\rho$, it is required to estimate the angular radius of the source star.
We estimate $\theta_*$ from the dereddened color $(V-I)_0$ and brightness $I_0$ of the source. In order to derive $(V-I)_0$ from the instrumental color-magnitude diagram, we use the method of @Yoo2004, which uses the centroid of red giant clump (RGC) as a reference. In Figure \[fig:six\], we present the location of the source and the RGC centroid in the instrumental color-magnitude diagram constructed from the $I$- and $V$-band DoPHOT photometry of the KMTC data set. It is found that the offsets in color and brightness of the source with respect to the RGC centroid are $\Delta (V-I,I)=(0.16,-0.07)$. With the known dereddened color and magnitude of RGC centroid, $(V-I,I)_{\rm RGC}=(1.06,14.5)$ [@Bensby2011; @Nataf2013], we find that the dereddened color and brightness of the source star are $(V-I,I)_0= (V-I,I)_{\rm RGC}+\Delta(V-I,I)=(1.22\pm 0.07,14.48\pm 0.09)$. This indicates that the source is a K-type giant star. Using the color-color relation provided by @Bessell1988, we convert $V-I$ into $V-K$. Using the relation between $V-K$ and the surface brightness [@Kervella2004], we estimate the angular source radius. The estimated angular source radius is $\theta_* = 6.9 \pm 0.6\ \mu{\rm as}$. Combined with value of $\rho$, we estimate that the angular Einstein radius of the lens system is $${\theta_{\rm E}}= 0.79 \pm 0.06\ {\rm mas}.$$ With the measured angular Einstein radius, the relative lens-source proper motion is estimated by $$\mu = 6.89 \pm 0.56\ {\rm mas}\ {\rm yr}^{-1}.$$
![ Location of the source with respect to the centroid of red giant clump (RGC) in the instrumental color-magnitude diagram. []{data-label="fig:six"}](f6.eps){width="\columnwidth"}
Lens Parameters {#sec:four-two}
---------------
With the measured microlens parallax and the angular Einstein radius, we estimate the mass and distance to the lens using the relations in Equation (2). In Table \[table:three\], we list the determined physical parameters, including masses of the primary, $M_1$, and the companion, $M_2$, the distance to the lens, $D_{\rm L}$, and the projected separation between the lens components, $a_\perp=sD_{\rm L}{\theta_{\rm E}}$, for the individual degenerate lensing solutions. To check the physical validity of the parameters, we also present the ratio between the projected potential energy to the kinetic energy, i.e., $({\rm KE}/{\rm PE})_\perp$.
Due to the difference in the microlens-parallax values between the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solution classes, the estimated lens masses and distances for the two classes of solutions are substantially different from each other. On the other hand, the physical parameters for the pair of the $u_0>0$ and $u_0<0$ solutions are similar to each other. We find that the masses of the primary and companion are $1.1\lesssim M_1/M_\odot \lesssim 1.3$ and $0.8\lesssim M_2/M_\odot \lesssim 0.9$ for the small-${\pi_{\rm E}}$ solutions. For the big-${\pi_{\rm E}}$ solutions, the masses of the lens components are $0.4\lesssim M_1/M_\odot \lesssim 0.5$ and $M_2\sim 0.3\ M_\odot$. The estimated distances to the lens are $7.2\lesssim D_{\rm L}/{\rm kpc}\lesssim 7.6$ and $5.0\lesssim D_{\rm L}/{\rm kpc}\lesssim 5.4$ according to the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions, respectively.
[lccccc]{} Small ${\pi_{\rm E}}$ & $u_0<0$ & 18.8 & 17.0 & 15.8 & 13.5\
& $u_0>0$ & 17.7 & 16.5 & - & -\
Big ${\pi_{\rm E}}$ & $u_0<0$ & 21.5 & 18.7 & - & -\
& $u_0>0$ & 22.1 & 19.1 & - & -
Resolving Degeneracy {#sec:five}
====================
Lens Brightness
---------------
The estimated masses of the lens for the small-${\pi_{\rm E}}$ and big-${\pi_{\rm E}}$ solutions are considerably different due to the difference in the measured microlens-parallax values. Then, if the lens-source can be resolved from future high-resolution imaging observations, the degeneracy can be resolved from the lens brightness.
If the proposed follow-up high-resolution observations are conducted, the observations will likely be conducted in the near-IR band. We, therefore, estimate the $H$-band magnitudes of the source and lens. From the dereddened $I$-band magnitude $I_0\sim 14.5$, the dereddened $H$-band source magnitude of the source is $H_0\sim 13.1$ [@Bessell1988]. The $V$-band extinction of $A_V\sim 2.6$ in combination with the extinction ratio $(A_H/A_V)\sim 0.108$ [@Nishiyama2008] toward the bulge field yields the $H$-band extinction of $A_H\sim 0.28$. Then, the apparent $H$-band magnitude of the source is $H_{\rm S}=H_0 + A_H\sim 13.4$. We compute the lens brightness based on the mass and distance under the assumption that the lens and source experience the same amount of extinction. In Table \[table:four\], we present the expected combined (primary plus companion) $I$- and $H$-band magnitudes of the lens and source. The brightness of the lens varies depending on the solution. For the small-${\pi_{\rm E}}$ solutions, the apparent $H$-band magnitude of the lens is $H_{\rm L}\sim 16.5$ – 17.0. For the big-${\pi_{\rm E}}$ solutions, on the other hand, the expected $H$-band lens brightness is $H_{\rm L}\sim 18.7$ –19.1.
According to the estimated $I$-band lens brightness, the lens-to-source flux ratio for the (small-${\pi_{\rm E}}$)/($u_0>0$) solution is $f_{{\rm L},I}/f_{{\rm S},I}\sim 17\%$. Since the light from the lens contributes to blended light, then, this ratio is too big to be consistent with the small amount of the measured blended flux, even considering the uncertainties of the lens mass and distance. Therefore, the solution is unlikely to be the correct solution not only because of its worst $\chi^2$ value among the degenerate solutions but also because of the limits on blended light. The lens-to-source flux ratio for the (small-${\pi_{\rm E}}$)/($u_0<0$) is about 6%, but with the lens mass and distance at the $1\sigma$ ($2\sigma$) level, the ratio is $\sim 2\%$ ($1\%$), which is consistent with the blending.
Relative Lens-source Proper Motion
----------------------------------
The degeneracy between $u_0<0$ and $u_0>0$ solutions can also be lifted once the lens and source are resolved. The relative lens-source proper motion vector is related to ${t_{\rm E}}$, ${\theta_{\rm E}}$, and $(\pi_{{\rm E},N}, \pi_{{\rm E},E})$ by $${\mbox{\boldmath $\mu$}}= (\mu_N,\mu_E)=
\left(
{{\theta_{\rm E}}\over t_{\rm E}} {\pi_{{\rm E},N} \over {\pi_{\rm E}}},
{{\theta_{\rm E}}\over t_{\rm E}} {\pi_{{\rm E},E} \over {\pi_{\rm E}}}
\right).$$ For the pair of the degenerate solutions with $u_0<0$ and $u_0>0$, the north components of ${\mbox{\boldmath $\pi$}}_{\rm E}$ have opposite signs. This implies that the relative motion vectors of the two degenerate solutions are directed in substantially different directions and thus the degeneracy can be resolved from the lens motion with respect to the source.
The heliocentric lens-source proper motion is $\mu_{\rm helio}\sim 7~{\rm mas}~{\rm yr}^{-1}$, which is about what is expected for a disk lens. In this case, the expected direction of ${\mbox{\boldmath $\mu$}}_{\rm helio}$ (i.e., the direction of Galactic rotation $\psi\sim 30^\circ$) is roughly $30^\circ$ East of North. In Table \[table:three\], we list the orientation angles $\psi$ of ${\mbox{\boldmath $\mu$}}_{\rm helio}$, as measured from North to East, corresponding to the individual solutions. The heliocentric proper motion is related to the geocentric proper motion ${\mbox{\boldmath $\mu$}}_{\rm geo}$ by $${\mbox{\boldmath $\mu$}}_{\rm helio}={\mbox{\boldmath $\mu$}}_{\rm geo}+{\bf v}_{\oplus,\perp}{ \pi_{\rm rel}\over {\rm au}},$$ where ${\bf v}_{\oplus,\perp}$ represents the projected Earth motion at $t_0$. One finds that the expected direction ${\mbox{\boldmath $\mu$}}_{\rm helio}$ is consistent with the (small-${\pi_{\rm E}}$)/($u_0<0$) and the (big-${\pi_{\rm E}}$)/($u_0>0$) solutions but inconsistent with the others.
From Keck adaptive optics observations, @Batista2015 resolved the lens from the source $\sim 8.2$ years after the event OGLE-2005-BLG-169, for which the relative lens-source proper motion is $\mu \sim 7.4\ {\rm mas}\ {\rm yr}^{-1}$. The estimated proper motion of OGLE-2017-BLG-0329 ($\mu \sim 6.9\ {\rm mas}\ {\rm yr}^{-1}$) is similar to that of OGLE-2005-BLG-169. Considering the large lens/source flux ratio, the lens-source resolution by Keck observations will take $\sim 10$ years, which is somewhat longer than the time for OGLE-2005-BLG-169. We note that GMT/TMT/ELT, which will have better resolution than Keck, may be available before Keck can resolve the event and thus the time for follow-up observations can be shortened.
Conclusion {#sec:six}
==========
We presented the analysis of the binary microlensing event OGLE-2017-BLG-0329, which was observed both from the ground and in space using the $Spitzer$ telescope. We found that the parallax model based on the ground-based data could not be distinguished from a zero-${\pi_{\rm E}}$ model at 2$\sigma$ level. However, with the use of the additional $Spitzer$ data, we could identify 2 classes of microlens-parallax solutions, each composed of a pair of solutions according to the well-known ecliptic degeneracy. We also found that the space-based data helped to greatly reduce the measurement uncertainties of the microlens-parallax vector ${\mbox{\boldmath $\pi$}}_{\rm E}$. With the measured microlens parallax combined with the angular Einstein radius measured from the resolved caustics, we found that the lens was composed of a binary with components masses of either $(M_1,M_2)\sim (1.1,0.8)\ M_\odot$ or $\sim (0.4,0.3)\ M_\odot$ according to the two solution classes. The degeneracy among the solution would be resolved from adaptive optics observations taken $\sim 10$ years after the event.
Work by C. Han was supported by the grant (2017R1A4A1015178) of National Research Foundation of Korea. Work by WZ, YKJ, and AG were supported by AST-1516842 from the US NSF. WZ, IGS, and AG were supported by JPL grant 1500811. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to A. Udalski. Work by YS was supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Universities Space Research Association through a contract with NASA. This work was (partially) supported by NASA contract NNG16PJ32C. Work by S. Rahvar and S. Sajadian is supported by INSF-95843339. This research has made use of the KMTNet system operated by the Korea Astronomy and Space Science Institute (KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO in South Africa, and SSO in Australia. We acknowledge the high-speed internet service (KREONET) provided by Korea Institute of Science and Technology Information (KISTI).
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---
abstract: 'We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelisation and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretised diffusions. Our experimental results are promising and show that the IS type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelisation.'
address:
- 'University of Jyväskylä, Department of Mathematics and Statistics, P.O.Box 35, FI-40014 University of Jyväskylä, Finland'
- 'Linköping University, Department of Science and Technology, Media and Information Technology, Campus Norrköping SE-601 74 Norrköping, Sweden'
- 'University of Jyväskylä, Department of Mathematics and Statistics, P.O.Box 35, FI-40014 University of Jyväskylä, Finland'
author:
- Matti Vihola
- Jouni Helske
- Jordan Franks
title: Importance sampling type estimators based on approximate marginal MCMC
---
Introduction
============
Markov chain Monte Carlo (MCMC) has become a standard tool in Bayesian analysis. The greatest benefit of MCMC is its general applicability — it is guaranteed to be consistent with virtually no assumptions on the underlying model. However, the practical applicability of MCMC generally depends on the dimension of the unknown variables, the number of data, and the computational resources available. Because MCMC is only asymptotically unbiased, and sequential in nature, it can be difficult to implement efficiently with modern parallel and distributed computing facilities [@lee-yau-giles-doucet-holmes; @green-latuszynski-pereyra-robert; @wilkinson].
We promote a simple two-phase inference approach, based on importance sampling (IS), which is well-suited for parallel implementation. It combines a typically low-dimensional MCMC targeting an approximate marginal distribution with independently calculated estimators, which yield exact inference over the full posterior. The estimator is similar to self-normalised importance sampling, but is more general, allowing for sequential Monte Carlo and multilevel type corrections. The method is naturally applicable in a latent variable models context, where the MCMC operates on the hyperparameter distribution using an approximate marginal likelihood, and re-weighting is based on a sampling scheme on the latent variables. We detail the application of the method with Bayesian state space models, where we use importance sampling and particle filters for correction.
Related work
------------
We consider a framework which combines and generalises upon various previously suggested methods, which, to our knowledge, has not been systematically explored before. Importance sampling correction of MCMC has been suggested early in the MCMC literature [e.g. @hastings; @glynn-iglehart; @doss], and used, for instance, to estimate Bayes factors using a single MCMC output [@doss-bayes-factors]. Related confidence intervals have been suggested based on regeneration [@bhattacharya] and in case of multiple Markov chains [@tan-doss-hobert]. Using unbiased estimators of importance weights in this context has been suggested at least in [@lin-liu-sloan; @lyne-girolami-atchade-strathmann-simpson], who consider marginal inference with a generalisation of the pseudo-marginal method, allowing for likelihood estimators that may take negative values, and in [@quiroz-villani-kohn] with data sub-sampling.
Nested or compound sampling has also appeared in many forms in the Monte Carlo literature. The SMC^2^ algorithm [@chopin-jacob-papaspiliopoulos] is based on an application of nested sequential Monte Carlo steps, which has similarities with our framework, and the IS^2^ method [@tran-scharth-pitt-kohn] focuses on the case where the preliminary inference is based on independent sampling. We focus on the MCMC approximation of the marginal distribution, which we believe often to be easily implementable in practice, also when the marginal distribution has a non-standard form. The Markov dependence in the marginal Monte Carlo approximation comes with some extra theoretical issues, which we address in detail.
Our setting highlights explicitly the connection of IS type correction and delayed acceptance (DA) [@fox-nicholls; @liu-mc; @christen-fox], and recently developed pseudo-marginal type MCMC [@andrieu-roberts; @lin-liu-sloan] such as particle MCMC [@andrieu-doucet-holenstein], grouped independence Metropolis-Hastings [@beaumont], approximate Bayesian computation (ABC) MCMC [@marjoram-molitor-plagnol-tavare], the algorithm for estimation of discretely observed diffusions suggested in [@beskos-papaspiliopoulos-roberts-fearnhead], and annealed IS [@karagiannis-andrieu; @neal-annealed]. Theoretical advances of pseudo-marginal methods [@andrieu-vihola-pseudo; @andrieu-vihola-order; @doucet-pitt-deligiannidis-kohn; @sherlock-thiery-roberts-rosenthal; @andrieu-lee-vihola; @chopin-singh; @lindsten-douc-moulines] have already led to more efficient implementation of such methods, but have also revealed fundamental limitations. For instance, the methods may suffer from slow (non-geometric) convergence in practically interesting scenarios [@andrieu-roberts; @lee-latuszynski]. Adding dependence to the estimators [cf. @andrieu-vihola-order], such as using the recently proposed correlated version of the pseudo-marginal MCMC [@deligiannidis-doucet-pitt-kohn], may help in more efficient implementation in certain scenarios, but a successful implementation of such a method may not always be possible, and the question of efficient parallelisability remains a challenge. The blocked parallelisable particle Gibbs [@singh-lindsten-moulines] has appealing limiting properties, but its implementation still requires synchronisation between every update cycle, which may be costly in some computing environments.
The IS approach which we propose may assuage some of the aforementioned challenges of the pseudo-marginal framework; see Section \[sec:algorithmic-da\].
Outline
-------
We introduce a generic Bayesian latent variable model in Section \[sec:latent\], detail our approach algorithmically, and compare it with DA. We also discuss practical implications, modifications and possible extensions. After introducing notation in Section \[sec:notation\], we formulate general IS type correction of MCMC and related consistency results in Section \[sec:simple\]. We detail the general case (Theorem \[thm:proper-consistency\]), based on a concept (Definition \[def:proper\]), which we call a ‘proper weighting’ scheme (following the terminology of Liu [@liu-mc]), which is natural and convenient in many contexts. In Section \[sec:asvar-clt\], we state central limit theorems and expressions for asymptotic variances. Section \[sec:block\] focuses on estimators which calculate IS correction once for each accepted state, stemming from a so-called ‘jump chain’ representation. Section \[sec:pseudo\] details consistency of our estimators in case the approximate chain is pseudo-marginal.
We detail proper weighting schemes in the state space models (SSMs) using sequential Monte Carlo (SMC) in Section \[sec:ssm\]. We then focus on SSMs with linear-Gaussian state dynamics in Section \[sec:lin-gauss-ssms\], and show how a Laplace approximation can be used both for approximate inference, and for construction of efficient proper weighting schemes. Section \[sec:diffusion-ssms\] describes an instance of our approach in the context of discretely observed diffusions, with an approximate pseudo-marginal chain. We compare empirically several algorithmic variations in Section \[sec:exp\] with Poisson observations, with a stochastic volatility model and with a discretely observed geometric Brownian motion. Section \[sec:discussion\] concludes, with discussion.
The proposed latent variable model inference methodology {#sec:latent}
========================================================
A generic Bayesian latent variable model is defined in terms of three random vector, and corresponding conditional densities:
- ${\Theta}\sim {\mathrm{pr}}({\,\cdot\,})$ — prior density of (hyper)parameters,
- ${X}\mid {\Theta}={\theta}\sim
\mu^{({\theta})}({\,\cdot\,})$ — prior of latent variables given parameters, and
- ${Y}\mid ({\Theta}={\theta},{X}={x})
\sim g^{({\theta})}({\,\cdot\,}\mid
{x})$ — the observation model.
The aim is inference over the posterior of $({\Theta},{X})$ given observations ${Y}={y}$, with density $\pi({\theta},{x}) \propto {\mathrm{pr}}({\theta})
\mu^{({\theta})}({x}) g^{({\theta})}({y}\mid{x})$. Standard MCMC algorithms may, in principle, be applied directly for inference, but the typical high dimension of the latent variable ${x}$ and the common strong dependency structures often lead to poor performance of generic algorithms.
Our inference approach focuses on the specific structure of the model, based on the factorisation ${\pi}({\theta},{x})
= {\pi_m}({\theta}) {r}({x}\mid{\theta})$, where the marginal posterior density ${\pi_m}$ and the corresponding conditional ${r}$ are: $$\begin{aligned}
{\pi_m}({\theta}) {\mathrel{\mathop:}=}\int {\pi}({\theta},{x}){\mathrm{d}}{x}\propto {\mathrm{pr}}({\theta}) L({\theta}) \qquad\text{and}\qquad
{r}({x}\mid {\theta}) {\mathrel{\mathop:}=}\frac{p^{({\theta})}({x},{y})}{L({\theta})},\end{aligned}$$ with the joint density of the latent and the observed $p^{({\theta})}({x}, {y})$, and the marginal likelihood $L({\theta})$ given as follows: $$\begin{aligned}
p^{({\theta})}({x}, {y}) &{\mathrel{\mathop:}=}\mu^{({\theta})}({x}) g^{({\theta})}({y}\mid
{x}) &\qquad&\text{and}\qquad&
L({\theta}) &{\mathrel{\mathop:}=}\int
p^{({\theta})}({x},{y}) {\mathrm{d}}{x}.\end{aligned}$$ Two particularly successful latent variable model inference methods, the integrated nested Laplace approximation (INLA) [@rue-martino-chopin] and the particle MCMC methods (PMCMC) [@andrieu-doucet-holenstein], rely on this structure. In essence, the INLA is based on an efficient Laplace approximation $p_a^{({\theta})}({x},{y})$ of $p^{({\theta})}({x},{y})$, determining an approximate marginal likelihood $L_a({\theta})$ and approximate conditional distribution ${r}_a({x}\mid{y})$. Particle MCMC uses a specialised SMC algorithm, which provides an unbiased approximation of expectations with respect to $p^{({\theta})}({x},{y})$ allowing for exact inference, and which is particularly efficient in the state space models context.
An algorithmic description {#sec:algorithmic-method}
--------------------------
The primary aim of this paper is the efficient use of an approximate marginal likelihood $L_a({\theta})$ within a Monte Carlo framework that leads to efficient, parallelisable and exact inference. For instance, Laplace approximations often lead to a natural choice for $L_a({\theta})$. The inference method which we propose comprises two algorithmic phases, which are summarised below:
1. \[item:approx-phase\] Simulate a Markov chain $({\Theta}_k)_{k=1,\ldots,n}$ targeting an approximate hyperparameter posterior $
{\pi_a}({\theta}) \propto {\mathrm{pr}}({\theta}) L_a({\theta}).
$
2. \[item:is-phase\] For each ${\Theta}_k$, sample $({V}_k^{(i)},{X}_k^{(i)})_{i=1,\ldots,m}$ where $V_k^{(i)}\in{\mathbb{R}}$ and ${X}_k^{(i)}$ are in the latent variable space, and calculate ${W}_k^{(i)} {\mathrel{\mathop:}=}{V}_k^{(i)}/L_a({\Theta}_k)$, which determine a weighted estimator $$E_n(f) {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n \sum_{i=1}^m {W}_k^{(i)}
f({\Theta}_k, {X}_k^{(i)})}{\sum_{j=1}^n \sum_{\ell=1}^m
{W}_j^{(\ell)}}
\label{eq:corrected-estimator}$$ of the full posterior expectation $ {\mathbb{E}}_{\pi}[f({\Theta},{X})]
= \int
f({\theta},{x}){\pi}({\theta},{x}){\mathrm{d}}{\theta}{\mathrm{d}}{x}$.
The essential conditions required for the validity of the estimator are:
1. \[cond:consistency\] The approximation is consistent, in the sense that $L_a({\theta})>0$ whenever $L({\theta})>0$, and $\int {\mathrm{pr}}({\theta})L_a({\theta}){\mathrm{d}}{\theta}<\infty$.
2. \[cond:harris\] The Markov chain $({\Theta}_k)_{k\ge n}$ is Harris ergodic (Definition \[def:harris\]) with respect to ${\pi_a}$.
3. \[cond:unbiasedness\] Denoting $f^*({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}_{\pi}[f({\Theta},{X})\mid{\Theta}={\theta}]
= \int r({x}\mid{\theta}) f({\theta},{x}){\mathrm{d}}{x}$, there exists a constant $c_w>0$ such that $$\begin{aligned}
{\mathbb{E}}\bigg[ \sum_{i=1}^m {V}_k^{(i)}
f({\Theta}_k,{X}_k^{(i)}){\mathrel{\bigg|}}{\Theta}_k={\theta}\bigg]
&= c_w L({\theta}) f^*({\theta}),
\label{eq:simple-unbiasedness}\end{aligned}$$ for all ${\theta}\in{{\mathsf{T}}}$, all functions $f$ of interest, and for $f\equiv 1$ (i.e. holds with $f({\,\cdot\,})$ and $f^*({\,\cdot\,})$ omitted). The value of $c_w$ need not be known.
Both C\[cond:consistency\] and C\[cond:harris\] are easily satisfied by construction of the approximation, and C\[cond:unbiasedness\] is satisfied by many schemes. Section \[sec:ssm\] reviews how (unnormalised) importance sampling and particle filter lead to such schemes. There is also a (mild) integrability condition, which $({W}_k^{(i)},{X}_k^{(i)})$ must satisfy in order to guarantee a strong convergence $E_n(f) \to
{\mathbb{E}}_{\pi}[f({\Theta},{X})]$. When ${V}_k^{(i)}\ge 0$ almost surely, it suffices that $|f|$ satisfies ; see Section \[sec:simple\] for details. Further conditions ensure a central limit theorem $\sqrt{n}\{E_n(f) - {\mathbb{E}}_{\pi}[f({\Theta},{X})]\}\to
N(0,\sigma^2)$, as detailed in Section \[sec:asvar-clt\].
When Phase \[item:approx-phase\] is a Metropolis-Hastings algorithm, it is possible to generate only one batch of $(\tilde{{V}}_k^{(i)},\tilde{{X}}_k^{(i)})_{i=1,\ldots,m}$ for each *accepted* state $(\tilde{{\Theta}}_k)$. If $N_k$ stands for the time spent at $\tilde{{\Theta}}_k$, then the corresponding weights are determined as $\tilde{W}_k {\mathrel{\mathop:}=}N_k
V_k^{(i)}/L_a(\tilde{{\Theta}}_k)$; see Section \[sec:block\] for details about such ‘jump chain’ estimators.
Use with approximate pseudo-marginal MCMC {#sec:algorithmic-pseudo}
-----------------------------------------
In many scenarios, such as with time-discretised diffusions, the latent variable prior density $\mu^{({\theta})}$ cannot be evaluated, and exact simulation is impossible or very expensive. Simulation is also expensive with a fine enough time-discretisation.
A coarsely discretised model leads to a natural cheap approximation $\hat{\mu}^{({\theta})}$, but in Phase \[item:approx-phase\], the Markov chain will often be a pseudo-marginal MCMC [cf. @andrieu-roberts], in which case our scheme would have the following form:
1. \[item:approx-phase2\] Simulate a pseudo-marginal Metropolis-Hastings chain $({\Theta}_k,U_k)$ for $k=1,\ldots,n$, following
(i) Draw a proposal $\tilde{{\Theta}}_k$ from $q({\Theta}_{k-1},{\,\cdot\,})$ and given $\tilde{{\Theta}}_k$, construct an estimator $\tilde{U}_k\ge 0$ such that ${\mathbb{E}}[\tilde{U}_k\mid
\tilde{{\Theta}}_k={\theta}] = L_a({\theta})$.
(ii) With probability $
\min\Big\{1,\frac{{\mathrm{pr}}(\tilde{{\Theta}}_k)
\tilde{U}_k q(\tilde{{\Theta}}_k,{\Theta}_{k-1})
}{
{\mathrm{pr}}({\Theta}_{k-1})U_{k-1}
q({\Theta}_{k-1},\tilde{{\Theta}}_k)}
\Big\}$, accept and set $({\Theta}_k,U_k) =
(\tilde{{\Theta}}_k,\tilde{U}_k)$; otherwise reject the move.
2. \[item:pseudo-correct-phase2\] For each $({\Theta}_k,U_k)$, sample $({V}_k^{(i)},{X}_k^{(i)})_{i=1,\ldots,m}$ and set ${W}_k^{(i)} {\mathrel{\mathop:}=}{V}_k^{(i)}/U_k$, which determine the estimator as in .
Algorithmically, the pseudo-marginal version above is similar to the method in Section \[sec:algorithmic-method\], with the likelihood $L_a({\Theta}_k)$ replaced with its estimator $U_k$. The requirements for the approximate likelihood C\[cond:consistency\] and its estimator C\[cond:unbiasedness\] remain identical, and C\[cond:harris\] must hold for the pseudo-marginal chain $({\Theta}_k,U_k)$, together with the following condition:
1. \[cond:positivity\] The estimators $\tilde{U}_k$ are strictly positive, almost surely, for all $\tilde{{\Theta}}_k\in{{\mathsf{T}}}$.
These are enough to guarantee consistency; see Section \[sec:pseudo\], and in particular Proposition \[prop:pseudo-approx-is\] for details, which also justifies why C\[cond:positivity\] is needed for consistency. In practice it may be easily satisfied, because the likelihood estimators $\tilde{U}_k$ may be inflated, if necessary (see Section \[sec:discussion\]).
Note that the variables $({V}_k^{(i)},{X}_k^{(i)})$ may depend on both ${\Theta}_k$ and the related likelihood estimate $U_k$. The dependency may be useful, if positively correlated ${V}_k^{(i)}$ and $U_k$ are available, leading to lower variance weights ${W}_k^{(i)}={V}_k^{(i)}/U_k$. This is similar to the correlated pseudo-marginal algorithm [@deligiannidis-doucet-pitt-kohn], which relies on a particular form of ${V}_k^{(i)}$ and $U_k$. If positively correlated structure is unavailable, $({V}_k^{(i)},{X}_k^{(i)})$ may be constructed independent of $U_k$.
Comparison with delayed acceptance {#sec:algorithmic-da}
----------------------------------
The key condition, under which we believe our method to be useful, is that the Phase \[item:approx-phase\] Markov chain is computationally relatively cheap compared to construction of the random variables $({W}_k^{(i)},{X}_k^{(i)})$ computed in Phase \[item:da-step\]. Similar rationale, and similar building blocks — a ${\pi_a}$-reversible Markov chain and random variables analogous to $({W}_k^{(i)},{X}_k^{(i)})$ — have been suggested earlier for construction of a delayed acceptance (DA) pseudo-marginal MCMC scheme [cf. @golightly-henderson-sherlock]. Such an algorithm defines a Markov chain $({\Theta}_k,{W}_k^{(i)},{X}_k^{(i)})_{k\ge 1}$, with one iteration consisting of the following steps:
1. \[item:da-proposal\] Draw $\tilde{{\Theta}}_k\sim P({\Theta}_{k-1},{\,\cdot\,})$. If $\tilde{{\Theta}}_k = {\Theta}_{k-1}$ reject, otherwise go to (DA \[item:da-step\]).
2. \[item:da-step\] Conditional on $\tilde{{\Theta}}_k$, draw $(\tilde{{V}}_k^{(i)},\tilde{{X}}_k^{(i)})$ which satisfies with $\tilde{{\Theta}}_k$ in place of ${\Theta}_k$, and set $\tilde{{W}}_k^{(i)} {\mathrel{\mathop:}=}\tilde{{V}}_k^{(i)}/L_a(\tilde{{\Theta}}_k)$. With probability $ \min\Big\{1,\frac{\sum_{i=1}^m
\tilde{{W}}_k^{(i)}}{\sum_{\ell=1}^m
{W}_{k-1}^{(\ell)}}\Big\}$, accept $(\tilde{{\Theta}}_k,\tilde{{W}}_k^{(i)},\tilde{{X}}_k^{(i)})$, otherwise reject.
If the pseudo-marginal method is used in DA \[item:da-proposal\], the value $L_a({\Theta}_k)$ is replaced with the related likelihood estimator. Under essentially the same assumptions as required by our scheme, and additionally requiring that $\tilde{{W}}_k^{(i)}\ge 0$, the DA scheme described above leads to a consistent estimator: $$\frac{1}{n}\sum_{k=1}^n \sum_{i=1}^m
\bigg(\frac{{W}_k^{(i)}}{\sum_{\ell=1}^m
{W}_k^{(\ell)}}\bigg)
f({\Theta}_k,{X}_k^{(i)})
\xrightarrow{n\to\infty} {\mathbb{E}}_\pi[f({\Theta},{X})]
$$
Our IS scheme is a natural alternative to such a DA scheme, replacing the independent Metropolis-Hastings type accept-reject step DA \[item:da-step\] with analogous weighting. This relatively small algorithmic change brings many, potentially substantial, benefits over DA, which we note next.
(i) Phase \[item:is-phase\] corrections are entirely independent ‘post-processing’ of Phase \[item:approx-phase\] MCMC output $({\Theta}_k)_{k=1,\ldots,n}$, which is easy to implement efficiently using parallel or distributed computing. This is unlike DA \[item:da-proposal\] and DA \[item:da-step\], which must be iterated sequentially.
(ii) \[item:is-da-comparison\] If Phase \[item:is-phase\] correction variables are calculated only once for each accepted ${\Theta}_k$ (so-called ‘jump chain’ representation, see Section \[sec:block\]), the IS method will typically be computationally less expensive than DA with the same number of iterations, even without parallelisation.
(iii) \[item:is-da-comparison-thinning\] The Phase \[item:approx-phase\] MCMC chain $({\Theta}_k)$ may be (further) thinned before applying (much more computationally demanding) Phase \[item:is-phase\]. Thinning of the DA chain is less likely beneficial [cf. @owen-thinning].
(iv) In case the approximate marginal MCMC $({\Theta}_k)$ is based on a deterministic likelihood approximation, it is generally ‘safer’ than (pseudo-marginal) DA using likelihood estimators, because pseudo-marginal MCMC may have issues with mixing [cf. @andrieu-vihola-pseudo]. It is also easier to implement efficiently. For instance, popular adaptive MCMC methods which rely on acceptance rate optimisation [@andrieu-thoms and references therein] are directly applicable.
(v) Reversibility of the MCMC kernel $P$ in DA \[item:da-proposal\] is necessary, but not required for the Phase \[item:approx-phase\] MCMC.
(vi) Non-negativity of ${W}_k^{(i)}$ is required in DA \[item:da-step\], but not in Phase \[item:is-phase\]. This may be useful in certain contexts, where multilevel [@heinrich; @giles-or] or debiasing [@mcleish; @rhee-glynn; @vihola-unbiased] are applicable. (See also the discussion in [@jacob-thiery] why pseudo-marginal method may not be applicable at all in such a context.)
(vii) The separation of ‘approximate’ Phase \[item:approx-phase\] and ‘exact’ Phase \[item:is-phase\] allows for two-level inference. In statistical practice, preliminary analysis could be based on (fast) purely approximate inference, and the (computationally demanding) exact method could be applied only as a final verification to ensure that the approximation did not affect the findings.
To elaborate the last point, the approximate likelihood $L_a({\theta})$ is usually based on an approximation $p_a^{({\theta})}({x},{y})$ of the latent model $p^{({\theta})}({x},{y})$. If the approximate model admits tractable expectations of functions $f$ of interest or exact simulation, direct approximate inference is possible, because $$\frac{1}{n} \sum_{k=1}^n
f_a^* ({\Theta}_k) \to
{\mathbb{E}}_{\tilde{\pi}}[f({\Theta},{X})],
\qquad \text{where}\qquad
f_a^*({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}_{\tilde{\pi}}[f({\Theta},{X})\mid{\Theta}={\theta}],$$ with approximate joint posterior $\tilde{\pi}({\theta},{x})\propto
{\mathrm{pr}}({\theta})p_a^{({\theta})}({x},{y})$. Then, Phase \[item:is-phase\] allows for quantification of the bias ${\mathbb{E}}_{\tilde{\pi}}[f({\Theta},{X})]-{\mathbb{E}}_{\pi}[f({\Theta},{X})]$, and confirmation that both inferences lead to the same conclusions.
The further work [@franks-vihola] considers the relationship between IS and DA in terms of the asymptotic variance.
Notation and preliminaries {#sec:notation}
==========================
Throughout the paper, we consider general state spaces while using standard integral notation. If the model at hand is given in terms of standard probability densities, the rest of this paragraph can be skipped. Each space ${\mathsf{X}}$ is assumed to be equipped with a $\sigma$-finite dominating measure ‘${\mathrm{d}}x$’ on a $\sigma$-algebra denoted with a corresponding calligraphic letter, such as ${\mathcal{X}}$. Product spaces are equipped with the related product $\sigma$-algebras and product dominating measures. If ${\mathsf{X}}$ is a subset of an Euclidean space ${\mathbb{R}}^d$, ${\mathrm{d}}x$ is taken by default as the Lebesgue measure and ${\mathcal{X}}$ as the Borel subsets of ${\mathsf{X}}$. ${\mathbb{R}}_+$ stands for the non-negative real numbers, and constant unit function is denoted by ${\mathbf{1}}$.
If $\nu$ is a probability density on ${\mathsf{X}}$, we define the support of $\nu$ as ${\mathop{\mathrm{supp}}}(\nu) {\mathrel{\mathop:}=}\{x\in{\mathsf{X}}{\,:\,}\nu(x)>0\}$, and the probability measure corresponding to $\nu$ with the same symbol $\nu({\mathrm{d}}x) {\mathrel{\mathop:}=}\nu(x){\mathrm{d}}x$.[^1] If $g:{\mathsf{X}}\to{\mathbb{R}}$, we denote $\nu(g){\mathrel{\mathop:}=}\int
g(x)\nu({\mathrm{d}}x)$, whenever well-defined. For a probability density or measure $\nu$ on ${\mathsf{X}}$ and $p\in[1,\infty)$, we denote by $L^p(\nu)$ the set of measurable $g:{\mathsf{X}}\to{\mathbb{R}}$ with $\nu(|g|^p)<\infty$, and by $L_0^p(\nu){\mathrel{\mathop:}=}\{g\in
L^p(\nu){\,:\,}\nu(g)=0\}$ the corresponding set of zero-mean functions. If $P$ is a Markov transition probability, we denote the probability measure $(\nu P)(A)
{\mathrel{\mathop:}=}\int \nu({\mathrm{d}}x) P(x,A)$, and the function $(Pg)(x) {\mathrel{\mathop:}=}\int
P(x,{\mathrm{d}}y) g(y)$. Iterates of transition probabilities are defined recursively through $P^n(x,A) {\mathrel{\mathop:}=}\int P(x,{\mathrm{d}}y) P^{n-1}(y,A)$ for $n\ge 1$, where $P^0(y,A) {\mathrel{\mathop:}=}{\mathbb{I}\left(y\in A\right)}$.
We follow the conventions $0/0{\mathrel{\mathop:}=}0$ and ${\mathbb{N}}{\mathrel{\mathop:}=}\{1,2,\ldots\}$. For integers $a\le b$, we denote by $a{:}b$ the integers within the interval $[a,b]$. We use this notation in indexing, so that $x_{a:b} = (x_a,\ldots,x_b)$, $x^{(a:b)}
= (x^{(a)},\ldots,x^{(b)})$. If $a>b$, then $x_{a:b}$ or $x^{(a:b)}$ is void, so that for example $g(x,y_{1:0})$ is interpreted as $g(x)$. Similarly, if $i_{1:T}$ is a vector, then $x^{(i_{1:T})} =
(x^{(i_1)},\ldots,x^{(i_T)})$ and $x_{1:T}^{(i_{1:T})}=(x_1^{(i_1)},\ldots,x_T^{(i_T)})$. We also use double-indexing, such as $x_k^{(1:m,1:n)}=(x_k^{(1,1)}$, $\!\ldots\,$, $x_k^{(1,m)}$, $x_k^{(2,1)}$, $\!\ldots\,$, $x_k^{(m,n)})$.
Throughout the paper, we assume the underlying MCMC scheme to satisfy the following standard condition.
\[def:harris\] A Markov chain is called *Harris ergodic* with respect to $\nu$, if it is $\psi$-irreducible, Harris recurrent and with invariant probability $\nu$.
Virtually all MCMC schemes are Harris ergodic [cf. @tierney; @nummelin-mcmcs], although in some cases careless implementation could lead to a non-Harris chain [cf. @roberts-rosenthal-harris]. Thanks to the Harris assumption, all the limit theorems which we give hold for any initial distribution of the related Markov chain.
General importance sampling type correction of MCMC {#sec:simple}
===================================================
Hereafter, ${\pi_a}$ is a probability density on ${{\mathsf{T}}}$ and represents an approximation of a probability density ${\pi_m}$ of interest. The consistency of IS type correction relies on the following mild assumption.
\[a:mcmc-is\] The Markov chain $({\Theta}_k)_{k\ge
1}$ and the density ${\pi_a}$ satisfy:
(i) $({\Theta}_k)_{k\ge 1}$ is Harris ergodic with respect to ${\pi_a}$.
(ii) \[item:support\] ${\mathop{\mathrm{supp}}}({\pi_m}) \subset {\mathop{\mathrm{supp}}}({\pi_a})$.
(iii) $w_u({\theta}) {\mathrel{\mathop:}=}c_w {\pi_m}({\theta})/{\pi_a}({\theta})$, where $c_w>0$ is a constant.
If Assumption \[a:mcmc-is\] holds and it is possible to calculate the unnormalised importance weight $w_u({\theta})$ pointwise, the chain $({\Theta}_k)_{k\ge
1}$ can be weighted in order to approximate ${\pi_m}(g)$ for every $g\in L^1({\pi_m})$, using (self-normalised) importance sampling [e.g. @glynn-iglehart; @doss] $$\frac{\sum_{k=1}^n w_u({\Theta}_k) g({\Theta}_k)}{\sum_{j=1}^n
w_u({\Theta}_j)}
= \frac{n^{-1}\sum_{k=1}^n w_u({\Theta}_k)
g({\Theta}_k)}{n^{-1}\sum_{j=1}^n w_u({\Theta}_j)}
\xrightarrow[\text{a.s.}]{n\to\infty} \frac{{\pi_a}(w_u g)}{{\pi_a}(w_u)}
= {\pi_m}(g),$$ as Harris ergodicity guarantees the almost sure convergence of both the numerator and the denominator.
In case ${\pi_m}$ is a marginal density, which we will focus on, both the ratio $w_u({\theta})$ and the function $g$ (which will be a conditional expectation) are typically intractable. Instead, it is often possible to construct unbiased estimators, which may be used in order to estimate the numerator and the denominator, in place of $w_u({\Theta}_k)$ and $g({\Theta}_k)$, under mild conditions. In order to formalise such a setting, we give the following generic condition for ratio estimators, which resemble the IS correction above.
\[a:super-general\] Suppose Assumption \[a:mcmc-is\] holds, and let $(S_k)_{k\ge 1}$, where $S_k = \big(A_k,B_k\big)\in{\mathbb{R}}^2$, be conditionally independent given $({\Theta}_k)_{k\ge 1}$, such that the distribution of $S_k$ depends only on the value of ${\Theta}_k$, and
(i) $f_A({\theta}){\mathrel{\mathop:}=}{\mathbb{E}}[A_k\mid {\Theta}_k={\theta}]$ satisfies ${\pi_a}(f_A) = c_w {\pi_m}(g)$,
(ii) $f_B({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}[B_k\mid {\Theta}_k={\theta}]$ satisfies ${\pi_a}(f_B) = c_w$, and
(iii) ${\pi_a}(m^{(1)})<\infty$ where $m^{(1)}({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}\big[ |A_k| + |B_k| \;\big|\;
{\Theta}_k = {\theta}\big]$.
We record the following simple statement which guarantees consistency under Assumption \[a:super-general\].
\[lem:super-general\] If Assumption \[a:super-general\] holds for some $g\in L^1({\pi_m})$, then $$E_n(g) {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n A_k}{\sum_{j=1}^n B_j}
\xrightarrow[\mathrm{a.s.}]{n\to\infty} {\pi_m}(g).$$
The proof of Lemma \[lem:super-general\] follows by observing that $({\Theta}_k,S_k)_{k\ge 1}$ is Harris ergodic, where $S_k=(A_k,B_k)$, and the functions $h_1({\theta},a,b) = a$ and $h_2({\theta},a,b)=b$ are integrable with respect to its invariant distribution $\check{\pi}({\mathrm{d}}{\theta}\times{\mathrm{d}}s) {\mathrel{\mathop:}=}{\pi_a}({\mathrm{d}}{\theta})
Q({\theta},{\mathrm{d}}s)$, where $Q({\theta},A) {\mathrel{\mathop:}=}{\mathbb{P}}(S_k\in A\mid {\Theta}_k={\theta})$; see Lemma \[lem:aug-prop\] in Appendix \[app:aug\].
In the latent variable model discussed in Section \[sec:latent\], the aim is inference over a joint target density ${\pi}({\theta},{x})
{\mathrel{\mathop:}=}{\pi_m}({\theta}) {r}({x}\mid {\theta})$ on an extended state space ${{\mathsf{T}}}\times{{\mathsf{X}}}$. For every function $f\in L^1({\pi})$, we denote by $f^*({\theta}){\mathrel{\mathop:}=}\int
{r}({x}\mid {\theta}) f({\theta},{x}){\mathrm{d}}{x}$ the conditional expectation of $f$ given ${\theta}$, so ${\pi}(f) = {\pi_m}(f^*)$. The following formalises a scheme which satisfies Assumption \[a:super-general\] with $g = f^*$ and therefore guarantees consistency for a class of functions $f\in {\mathcal{L}}\subset L^1({\pi})$.
\[def:proper\] Suppose Assumption \[a:mcmc-is\] holds, and let $(P_k)_{k\ge 1}$ be conditionally independent given $({\Theta}_k)_{k\ge 1}$, such that the distribution of each $P_k= (M_k,{W}_k^{(1:M_k)},{X}_k^{(1:M_k)})$ depends only on the value of ${\Theta}_k$, where $M_k\in{\mathbb{N}}$, ${W}_k^{(i)}\in{\mathbb{R}}$ and ${X}_k^{(i)}\in{{\mathsf{X}}}$. Define for any $f\in L^1({\pi})$, $${\xi}_k(f) {\mathrel{\mathop:}=}\sum_{i=1}^{M_k} {W}_k^{(i)}
f({\Theta}_k,{X}_k^{(i)}).$$ Let ${\mathcal{L}}\subset L^1({\pi})$ be all the functions for which
(i) ${\mu}_f({\theta}){\mathrel{\mathop:}=}{\mathbb{E}}[{\xi}_k(f)\mid {\Theta}_k={\theta}]$ satisfies $\pi_a({\mu}_f) = c_w
\pi(f)$, and \[item:proper-unbiased\]
(ii) ${\pi_a}(m_f^{(1)})<\infty$ where $m_f^{(1)}({\theta})
{\mathrel{\mathop:}=}{\mathbb{E}}\big[ |{\xi}_k(f)| {\mathrel{\big|}}{\Theta}_k={\theta}\big]$. \[item:proper-general\]
If ${\mathbf{1}}\in {\mathcal{L}}$, then $({W}_k^{(1:M_k)},{X}_k^{(1:M_k)})_{k\ge 1}$ or equivalently $({\xi}_k)_{k\ge 1}$, form a ${\mathcal{L}}$-*proper weighting scheme*.
Regarding Definition \[def:proper\]:
(i) In case of non-negative weights, that is, ${W}_k^{(i)}\ge 0$ almost surely, we have $|{\xi}_k({\mathbf{1}})| =
{\xi}_k({\mathbf{1}})$, so $f\equiv {\mathbf{1}}\in{\mathcal{L}}$ if and only if . Further, if holds for both $f$ and $|f|$, then holds, because $|{\xi}_k(f)| \le {\xi}_k(|f|)$.
(ii) When certain multilevel [@heinrich; @giles-or] or debiasing methods [cf. @mcleish; @rhee-glynn; @glynn-rhee] are applied, ${W}_k^{(i)}$ generally take also negative values. In such a case, an extra integrability condition is necessary, and we believe is required for consistency in general.
(iii) Note that ${\mathcal{L}}$ is closed under linear operations, that is, if $a,b\in{\mathbb{R}}$ and $f,g\in{\mathcal{L}}$, then $af+bg\in{\mathcal{L}}$. This, together with ${\mathcal{L}}$ containing constant functions, implies that if $f\in{\mathcal{L}}$, then $\bar{f} {\mathrel{\mathop:}=}f - \pi(f)\in{\mathcal{L}}$.
(iv) In fact, ${\xi}_k$ may be interpreted as a *random (signed) measure*. Our results extend also to such generalisation, which may be a useful interpretation for instance in the context of Rao-Blackwellisation, where ${\xi}_k$ could be mixtures of Gaussians.
The following consistency result is a direct consequence of Lemma \[lem:super-general\].
\[thm:proper-consistency\] If $({\xi}_k)_{k\ge 1}$ form a ${\mathcal{L}}$-proper weighting scheme, then the IS type estimator is consistent, that is, $$E_n(f) {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n
{\xi}_k(f) }{\sum_{j=1}^n
{\xi}_j({\mathbf{1}}) }
\xrightarrow{n\to\infty} {\pi}(f),
\qquad \text{almost surely}.
\label{eq:proper-estimator}$$
Let us next exemplify a ‘canonical’ setting of a proper weighting scheme, stemming from standard unnormalised importance sampling.
\[prop:augmented-is\] Suppose Assumption \[a:mcmc-is\] holds and $q^{({\theta})}({\,\cdot\,})$ defines a probability density on ${{\mathsf{X}}}$ for each ${\theta}\in{{\mathsf{T}}}$ and ${\mathop{\mathrm{supp}}}({\pi}) \subset \{({\theta},{x}){\,:\,}{\pi_a}({\theta})q^{({\theta})}({x})>0\}$. Let $${X}_k^{(1:m)}{\overset{\mathrm{i.i.d.}}{\sim}}q^{({\Theta}_k)},
\quad
{V}_k^{(i)} {\mathrel{\mathop:}=}\frac{1}{m}\cdot
\frac{c_w{\pi}({\Theta}_k,
{X}_k^{(i)})}{q^{({\Theta}_k)}({X}_k^{(i)})}
\quad\text{and}\quad
{W}_k^{(i)} {\mathrel{\mathop:}=}\frac{{V}_k^{(i)}}{{\pi_a}({\Theta}_k)},$$ where $c_w>0$ a constant. Then, $({W}_k^{(1:m)},{X}_k^{(1:m)})_{k\ge 1}$ form a $L^1({\pi})$-proper weighting scheme.
When the weights are all positive, we record the following simple observations how a proper weighting property is inherited in sub-sampling, which may be useful for instance due to memory constraints.
\[prop:randomise-proper\] Suppose that $({W}_k^{(1:M_k)}, {X}_k^{(1:M_k)})_{k\ge 1}$ forms a ${\mathcal{L}}$-proper weighting scheme with non-negative ${W}_k^{(1:M_k)}\ge 0$ (a.s.). Let ${W}_k {\mathrel{\mathop:}=}\sum_{i=1}^{M_k} {W}_k^{(i)}$ and let $(I_k)$ be random variables conditionally independent of $({\Theta}_k,{X}_k^{(i)})$ such that ${\mathbb{P}}(I_k = i) = {W}_k^{(i)}/{W}_k$ (and let $I_k=1$ if $W_k=0$). Then, $(W_k, {X}_k^{(I_k)})_{k\ge 1}$ forms a ${\mathcal{L}}$-proper weighting scheme.
The sub-sampling estimator simplifies to $$E_n(f) =
\frac{\sum_{k=1}^n W_k f({\Theta}_k,{X}_k^{(I_k)})
}{
\sum_{k=1}^n W_k }.$$ We conclude by recording a complementary statement about convex combinations, allowing to merge multiple proper sampling schemes.
\[prop:convex-proper\] Suppose $({\xi}_{k,j})_{k\ge 1}$ forms a ${\mathcal{L}}$-proper weighting scheme for each $j\in\{1{:}N\}$, then, for any constants $\beta_{1},\ldots,\beta_N\ge 0$ with $\sum_{j=1}^N
\beta_j = 1$, the convex combinations ${\xi}_k(f) {\mathrel{\mathop:}=}\sum_{j=1}^N \beta_j {\xi}_{k,j}(f)$ form a ${\mathcal{L}}$-proper sampling scheme.
Asymptotic variance and a central limit theorem {#sec:asvar-clt}
===============================================
The asymptotic variance is a common efficiency measure for Markov chains, which coincides with the limiting variance of related estimators in case a central limit theorem (CLT) holds.
\[def:asvar\] Suppose the Markov chain $({\Theta}_k)_{k\ge 1}$ on ${{\mathsf{T}}}$ has transition probability $P$ which is Harris ergodic with respect to invariant probability ${\pi_a}$. For $f\in L^2({\pi_a})$, the asymptotic variance of $f$ with respect to $P$ is $${\mathrm{Var}}(f,P) {\mathrel{\mathop:}=}\lim_{n\to\infty} {\mathbb{E}}\bigg(
\frac{1}{\sqrt{n}}\sum_{k=1}^n \big[f({\Theta}_k^{(s)}) - {\pi_a}(f)\big]\bigg)^2,$$ whenever the limit exists in $[0,\infty]$, where $({\Theta}_k^{(s)})_{k\ge 1}$ stands for the *stationary Markov chain* with transition probability $P$, that is, with ${\Theta}_1^{(s)}\sim {\pi_a}$.
In what follows, we denote by $\bar{f}({\theta},{x}) =
f({\theta},{x}) - {\pi}(f)$ the centred version of any $f\in L^1({\pi})$, and recall that if $f\in{\mathcal{L}}$, then $\bar{f}\in{\mathcal{L}}$. We also denote $m_f^{(2)}({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}[|{\xi}_k(f)|^2\mid{\Theta}_k={\theta}]$ for any $f\in{\mathcal{L}}$. The proof of the following CLT is given in Appendix \[app:clt\].
\[thm:proper-clt-rev\] Suppose that the conditions of Theorem \[thm:proper-consistency\] are satisfied, and $({\Theta}_k)_{k\ge 1}$ is aperiodic. Let $f\in
{\mathcal{L}}\cap L^2({\pi})$ and denote $\bar{f}({\theta},{x}) {\mathrel{\mathop:}=}f({\theta},{x}) - {\pi}(f)$. If ${\pi_a}(m_{\bar{f}}^{(2)})<\infty$ and either of the following hold:
(i) \[item:proper-kv\] $({\Theta}_k)_{k\ge 1}$ is reversible and ${\mathrm{Var}}({\mu}_{\bar{f}}, P)<\infty$, or
(ii) \[item:proper-mw\] $\sum_{n=1}^\infty n^{-3/2}
\big\{{\pi_m}\big(
\big[\sum_{k=0}^{n-1} P^{k} ({\mu}_{\bar{f}}
)\big]^2\big)\big\}^{1/2}<\infty$,
then, the estimator $E_n(f)$ defined in satisfies a CLT $$\sqrt{n}[E_n(f)-{\pi}(f)] \xrightarrow{n\to\infty}
N\big(0,\sigma_f^2\big),\quad\text{where}\quad
\sigma_f^2{\mathrel{\mathop:}=}\frac{{\mathrm{Var}}({\mu}_{\bar{f}}, P) + {\pi_a}(v)}{c_w^2}$$ in distribution, where $v({\theta}) {\mathrel{\mathop:}=}{\mathrm{Var}}\big({\xi}_k(\bar{f}){\mathrel{\big|}}{\Theta}_k={\theta}\big)$.
In case of reversible chains, the condition in Theorem \[thm:proper-clt-rev\] is essentially optimal, and the CLT relies on a result due to Kipnis and Varadhan [@kipnis-varadhan]. The condition always holds when $({\Theta}_k)_{k\ge
1}$ is geometrically ergodic, for instance $({\Theta}_k)_{k\ge 1}$ is a random-walk Metropolis algorithm and ${\pi_a}$ is light-tailed [@jarner-hansen; @roberts-tweedie]. In case $({\Theta}_k)_{k\ge 1}$ is sub-geometric, such as polynomial, extra conditions are required; see for instance [@jarner-roberts-heavytailed]. The condition which applies for non-reversible chains is also nearly optimal, and relies on a result due to Maxwell and Woodroofe [@maxwell-woodroofe]. See also the review on Markov chain CLTs by Jones [@jones].
Note that the latter term ${\pi_a}(v)$ in the asymptotic variance expression contains the contribution of the ‘noise’ in the IS estimates. If the estimators ${\xi}_k(f)$ are made increasingly accurate, in the sense that ${\pi_a}(v)$ becomes negligible, the limiting case corresponds to an IS corrected approximate MCMC and calculating averages over conditional expectations ${\mu}_{\bar{f}}({\theta})$. We conclude by relating the asymptotic variance with a straightforward estimator.
\[thm:importance-var\] Suppose $f\in{\mathcal{L}}\cap L^2({\pi})$ and ${\pi_a}(v)<\infty$ where $v$ is defined in Theorem \[thm:proper-clt-rev\], and also ${\pi_a}(m_{{\mathbf{1}}}^{(2)})<\infty$. Then, the estimator $$v_n {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n \big({\xi}_k(f) -
{\xi}_k({\mathbf{1}})E_n(f)\big)^2 }{
\big(\sum_{j=1}^n {\xi}_j({\mathbf{1}}) \big)^2}$$ satisfies $n v_n \to {\pi_a}(v+{\mu}_{\bar{f}}^2)/c_w^2$ almost surely as $n\to\infty$.
Proof of Theorem \[thm:importance-var\] is given in Appendix \[app:clt\].
The estimator $nv_n$ in Theorem \[thm:importance-var\] provides a consistent estimate for the CLT variance $\sigma_f^2/n$ when $P$ corresponds to i.i.d. sampling, in which case ${\mathrm{Var}}({\mu}_{\bar{f}}, P)= \pi_a({\mu}_{\bar{f}}^2)$. Typically, ${\mathrm{Var}}({\mu}_{\bar{f}}, P)\ge
\pi_a({\mu}_{\bar{f}}^2)$ (which is always true when $P$ is positive), and then $n v_n$ provides a lower bound of the variance. It can provide useful information about the importance sampling noise contribution, and may be used as an optimisation criteria when adjusting the accuracy of the related estimators. Generic Markov chain asymptotic variance estimators (see, e.g., the review [@flegal-jones] and references therein) may also be used with IS correction, by estimating the asymptotic variance of $n^{-1} \sum_{k=1}^n {\xi}_k(f)$ and dividing it by $[n^{-1}
\sum_{k=1}^n {\xi}_k({\mathbf{1}})]^2$.
Jump chain estimators {#sec:block}
=====================
Many MCMC algorithms such as the Metropolis-Hastings include an accept-reject mechanism, which results in blocks of repeated values ${\Theta}_k = \ldots = {\Theta}_{k+b}$. In the context of IS type correction, and when the computational cost of each estimate ${\xi}_k$ is high, it may be desirable to construct only one estimator per each *accepted* state. To formalise such an algorithm we consider the ‘jump chain’ representation of the approximate marginal chain [cf. @douc-robert; @doucet-pitt-deligiannidis-kohn; @deligiannidis-lee].
\[def:jump\] Suppose that $({\Theta}_k)_{k\ge 1}$ is Harris ergodic with respect to $\pi_a$. The corresponding jump chain $(\tilde{{\Theta}}_k)_{k\ge 1}$ with holding times $(N_k)_{k\ge 1}$ is defined as follows: $$\tilde{{\Theta}}_k{\mathrel{\mathop:}=}{\Theta}_{\bar{N}_{k-1}+1}
\qquad\text{and}\qquad
N_k {\mathrel{\mathop:}=}\inf\big\{j\ge 1{\,:\,}{\Theta}_{\bar{N}_{k-1}+j+1}\neq
\tilde{{\Theta}}_{k}\big\},$$ where $\bar{N}_k {\mathrel{\mathop:}=}\sum_{j=1}^k N_j$, and with $\bar{N}_0\equiv 0$.
If $({\Theta}_k)_{k\ge 1}$ corresponds to a Metropolis-Hastings chain, with non-diagonal proposal distributions $q$ (that is, $q({\theta},\{{\theta}\})=0$ for every ${\theta}\in{{\mathsf{T}}}$), then the jump chain $(\tilde{{\Theta}}_k)$ consists of the accepted states, and $N_k-1$ is the number of rejections occurred at state $(\tilde{{\Theta}}_k)$.
Hereafter, we denote by $\alpha({\theta}) {\mathrel{\mathop:}=}{\mathbb{P}}({\Theta}_{k+1}\neq
{\Theta}_{k}\mid
{\Theta}_{k}={\theta})$ the overall acceptance probability at ${\theta}$. We consider next the practically important ‘jump IS’ estimator, involving a proper weighting for each accepted state.
\[a:block-natural-general\] Suppose that Assumption \[a:mcmc-is\] holds, and let $(\tilde{{\Theta}}_k,N_k)_{k\ge 1}$ denote the corresponding jump chain (Definition \[def:jump\]). Let $({\xi}_k)_{k\ge 1}$ be a ${\mathcal{L}}$-proper weighting scheme, where the variables $(M_k,{W}_k^{(1:M_k)},{X}_k^{(1:M_k)})$ in the scheme are now allowed to depend on both $\tilde{{\Theta}}_k$ and $N_k$, and the conditions and in Definition \[def:proper\] are replaced with the following:
(i) ${\mathbb{E}}[{\xi}_k(f)\mid {\Theta}_k={\theta},N_k=n] = {\mu}_f({\theta})$ for all $n\in {\mathbb{N}}$ and ${\pi_a}({\mu}_f) = c_w \pi(f)$, \[item:proper-unbiased-block\]
(ii) ${\pi_a}(\bar{m}^{(1)})<\infty$ where $\bar{m}^{(1)}({\theta})
{\mathrel{\mathop:}=}\sup_{n\in{\mathbb{N}}} {\mathbb{E}}\big[ |{\xi}_k(f)| {\mathrel{\big|}}{\Theta}_k={\theta}, N_k=n\big]$. \[item:proper-block\]
\[thm:block-natural-general\] Suppose Assumption \[a:block-natural-general\] holds, then, $$E_n(f) {\mathrel{\mathop:}=}\frac{
\sum_{k=1}^n N_k {\xi}_k(f)
}{
\sum_{j=1}^n
N_j {\xi}_j({\mathbf{1}})}
\xrightarrow[\mathrm{a.s.}]{n\to\infty} {\pi}(f).
\label{eq:block-natural}$$
The proof follows from Lemma \[lem:super-general\] because $(\tilde{{\Theta}}_k)$ is Harris ergodic with invariant probability $\tilde\pi_a({\theta}) \propto {\pi_a}({\theta})\alpha({\theta})$; see Proposition \[prop:jump-properties\] in Appendix \[app:jump\]. Furthermore, the holding times $N_k\ge 0$ are, conditional on $(\tilde{{\Theta}}_k)$, independent geometric random variables with parameter $\alpha(\tilde{{\Theta}}_k)$ (Proposition \[prop:jump-properties\]), and therefore ${\mathbb{E}}[N_k\mid \tilde{{\Theta}}_k={\theta}] =
1/\alpha({\theta})$.
\[rem:jump\] Regarding Assumption \[a:block-natural-general\]:
(i) Condition in Assumption \[a:block-natural-general\] is practically convenient, because $\xi_k$ are usually chosen either as independent of $N_k$, or increasingly accurate in $N_k$ (often taking $M_k$ proportional to $N_k$); see the discussion below. However, is not optimal: it is not hard to find examples where the estimator is strongly consistent, even though $\bar{m}^{(1)}({\theta})=\infty$ for some ${\theta}\in{{\mathsf{T}}}$.
(ii) \[item:jump-batch\] In case each ${\xi}_k$ is constructed as a mean of independent $({\xi}_{k,1},\ldots,{\xi}_{k,N_k})$ (cf. Proposition \[prop:convex-proper\]), the jump chain estimator coincides with the simple estimator discussed in Section \[sec:asvar-clt\] (at jump times). However, the jump chain estimator offers more flexibility, which may allow for variance reduction, for instance by using a single $m N_k$ particle filter (cf. Section \[sec:ssm\]) instead of an average of $N_k$ independent $m$-particle filters, or by stratification or control variates.
(iii) Even though we believe that the estimators of the form are often appropriate, we note that in some cases Rao-Blackwellised lower-variance estimators of $1/\alpha(\tilde{{\Theta}}_j)$ may be used instead of $N_k$, as suggested in [@douc-robert].
Let us finally consider a central limit theorem corresponding to the estimator in Theorem \[thm:block-natural-general\], whose proof is given in Appendix \[app:jump\].
\[thm:block-clt\] Suppose Assumption \[a:block-natural-general\] holds, $(\tilde{{\Theta}}_k)_{k\ge 1}$ is aperiodic, $f\in{\mathcal{L}}\cap
L^2({\pi})$, $${\pi_a}\big(\alpha \tilde{m}^{(2)}\big)<\infty,
\qquad\text{where}\qquad
\tilde{m}^{(2)}({\theta}){\mathrel{\mathop:}=}{\mathbb{E}}\big[N_k^2 |{\xi}_k(\bar{f})|^2{\mathrel{\big|}}\tilde{{\Theta}}_k={\theta}\big],
\label{eq:clt-tight}$$ and one of the following holds:
(i) \[item:block-clt-rev\] $({\Theta}_k)_{k\ge 1}$ is reversible and ${\mathrm{Var}}({\mu}_{\bar{f}} ,
P)<\infty$.
(ii) \[item:block-clt-poisson\] There exists $g\in L^2({\pi_a})$ satisfying the Poisson equation $g- Pg =
{\mu}_{\bar{f}}$.
Then, the estimator $E_n(f)$ in satisfies $$\sqrt{n}\big[E_n(f)-{\pi}(f)\big] \xrightarrow{n\to\infty} N(0,\sigma^2)
\qquad\text{in distribution},$$ where the limiting variance can be given as: $$\begin{aligned}
\sigma^2 &=
\frac{{\pi_a}(\alpha)}{c_w^2}
\Big[ {\mathrm{Var}}\big({\mu}_{\bar{f}}, P\big) +
{\pi_a}(\alpha \tilde{v})
\Big],
\label{eq:block-clt-var}
\end{aligned}$$ where $\tilde{v}({\theta}) {\mathrel{\mathop:}=}{\mathbb{E}}\big[ N_k^2
{\mathrm{Var}}\big({\xi}_k(\bar{f}) {\mathrel{\big|}}\tilde{{\Theta}}_k={\theta}, N_k\big){\mathrel{\big|}}\tilde{{\Theta}}_k={\theta}\big]$.
Let us briefly discuss the conditions and implications of Theorem \[thm:block-clt\] under certain specific cases. When the acceptance probability is bounded from below, $\inf_{{\theta}} \alpha({\theta})>0$, using a proper weighting ${\xi}_k$ independent of $N_k$ is ‘safe’, because $$\tilde{v}({\theta})\le \tilde{m}^{(2)}({\theta}) \le
\frac{2-\alpha({\theta})}{\alpha^2({\theta})} b({\theta});\quad
b({\theta}){\mathrel{\mathop:}=}\sup_{n\ge 1} {\mathbb{E}}\big[|{\xi}_k (\bar{f})|^2{\mathrel{\big|}}\tilde{{\Theta}}_k={\theta},N_k=n\big],$$ and so ${\pi_a}(b)<\infty$ guarantees . For example, if $({\Theta}_k)_{k\ge 1}$ is $L^2$-geometrically ergodic, then the acceptance probability is (essentially) bounded away from zero [@roberts-tweedie], and $g{\mathrel{\mathop:}=}\sum_{k\ge 0} P^k {\mu}_{\bar{f}} \in L^2({\pi_a})$ satisfies $g - Pg = {\mu}_{\bar{f}}$, so that is satisfied.
When ${\xi}_k$ corresponds to an average of i.i.d. ${\xi}_{k,1},\ldots$, ${\xi}_{k,N_k}$ (cf. Proposition \[prop:convex-proper\]) which do not depend on $N_k$, $${\mathrm{Var}}\big({\xi}_k(\bar{f}) {\mathrel{\big|}}\tilde{{\Theta}}_k={\theta},N_k\big)
=\hat{v}({\theta})/N_k;\quad
\hat{v}({\theta}) {\mathrel{\mathop:}=}{\mathrm{Var}}\big({\xi}_{k,1}(\bar{f}){\mathrel{\big|}}\tilde{{\Theta}}_k={\theta}).$$ Then, ${\pi_a}(\alpha \tilde{v}) = {\pi_a}(\hat{v})$, which leads to an asymptotic variance that coincides with simple IS correction (cf. Theorem \[thm:proper-clt-rev\]).
In the non-reversible case, our CLT only applies when a solution $g\in
L^2({\pi_a})$ to the Poisson equation $g - Pg = {\mu}_{\bar{f}}$ exists. We believe that the result holds more generally, but this requires showing that the jump chain $(\tilde{{\Theta}}_k)_{k\ge 1}$ inherits a central limit theorem from the base chain $({\Theta}_k)_{k\ge
1}$ under more general conditions.
Pseudo-marginal approximate chain {#sec:pseudo}
=================================
We next discuss how our limiting results still apply, in case the approximate chain is a pseudo-marginal MCMC, as discussed in Section \[sec:algorithmic-pseudo\]. Let us formalise next a pseudo-marginal Markov chain $({\Theta}_k,\Phi_k)_{k\ge 1}$ on ${{\mathsf{T}}}\times{\mathsf{S}}_\Phi$. Let ${\Theta}_0\in{{\mathsf{T}}}$ and $\Phi_0\in{\mathsf{S}}_\Phi$ such that $U(\Phi_0)>0$, and for $k\ge 1$, iterate
(i) Generate $\tilde{{\Theta}}_k \sim q({\Theta}_{k-1}, {\,\cdot\,})$ and $\tilde{\Phi}_k \sim Q_a(\tilde{{\Theta}}_k, {\,\cdot\,})$.
(ii) With probability $
\min\big\{1,\frac{U(\tilde{\Phi}_k)
q(\tilde{{\Theta}}_k,{\Theta}_{k-1})}{U(\Phi_{k-1})
q({\Theta}_{k-1},\tilde{{\Theta}}_k)}
\big\},
$ accept and set $({\Theta}_k,\Phi_k)=(\tilde{{\Theta}}_k,\tilde{\Phi}_k)$; otherwise reject and set $({\Theta}_k,\Phi_k) =
({\Theta}_{k-1},\Phi_{k-1})$.
Above, $Q_a({\theta},{\,\cdot\,})$ defines a (regular conditional) distribution on (a measurable space) ${\mathsf{S}}_\Phi$, and $U:{\mathsf{S}}_\Phi\to{\mathbb{R}}_+$ is a (measurable) function. Under the following condition, the Markov chain $({\Theta}_k,\Phi_k)_{k\ge 1}$ is reversible with respect to the probability measure $
{\pi_a}^{\circ}({\mathrm{d}}{\theta}, {\mathrm{d}}\phi)
{\mathrel{\mathop:}=}{\mathrm{d}}{\theta}Q_a({\theta}, {\mathrm{d}}\phi)U(\phi)/c_a,
$ which admits the marginal ${\pi_a}({\theta})$ [e.g. @andrieu-vihola-pseudo]:
\[a:approx-pseudo\] There exists a constant $c_a>0$ such that for each ${\theta}$, the random variable $\Phi_{\theta}\sim Q_a({\theta}, {\,\cdot\,})$ satisfies ${\mathbb{E}}[U(\Phi_{\theta})] = c_a {\pi_a}({\theta})$.
In addition, $({\Theta}_k,\Phi_k)_{k\ge 1}$ is easily shown to be Harris ergodic under minimal conditions.
Let us consider next an abstract minimal condition which ensures consistency of an IS type estimator. We discuss practically relevant sufficient conditions later in Proposition \[prop:sufficient-pseudo-is\].
\[a:pseudo-proper\] Suppose Assumption \[a:mcmc-is\] holds, $({\Theta}_k,\Phi_k)_{k\ge 1}$ is Harris ergodic, $c_m>0$ is a constant, and let $(P_k)_{k\ge 1}$ be conditionally independent given $({\Theta}_k,\Phi_k)_{k\ge 1}$, such that the distribution of each $P_k= (M_k,{V}_k^{(1:M_k)},{X}_k^{(1:M_k)})$ depends only on $({\Theta}_k,\Phi_k)$, where $M_k\in{\mathbb{N}}$, ${V}_k^{(i)}\in{\mathbb{R}}$ and ${X}_k^{(i)}\in{{\mathsf{X}}}$. Define for any $f\in L^1({\pi})$, ${\zeta}_k(f) {\mathrel{\mathop:}=}\sum_{i=1}^{M_k} {V}_k^{(i)}
f({\Theta}_k,{X}_k^{(i)})$, and let ${\mathcal{L}}\subset L^1({\pi})$ stand for all the functions for which
(i) $
\textstyle\iint Q_a(\theta, {\mathrm{d}}\phi) {\mathbb{I}\left(U(\phi)>0\right)}
{\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}, \Phi_k=\phi]
{\mathrm{d}}\theta = c_m {\pi}(f),
$ and \[item:proper-pseudo-unbiased\]
(ii) $
\textstyle\iint Q_a(\theta, {\mathrm{d}}\phi) {\mathbb{I}\left(U(\phi)>0\right)}
{\mathbb{E}}\big[ |{\zeta}_k(f)| {\mathrel{\big|}}{\Theta}_k={\theta},\Phi_k=\phi\big]
{\mathrm{d}}\theta <\infty.
$ \[item:proper-pseudo-general\]
\[prop:pseudo-approx-is\] Suppose Assumption \[a:approx-pseudo\] and \[a:pseudo-proper\] hold, and ${\mathbf{1}}\in {\mathcal{L}}$. Then, Theorem \[thm:proper-consistency\] holds with $${\xi}_k(f) {\mathrel{\mathop:}=}\sum_{i=1}^{M_k} {W}_k^{(i)}
f({\Theta}_k,{X}_k^{(i)}) \qquad\text{where}\qquad
{W}_k^{(i)} = \frac{{V}_k^{(i)}}{U(\Phi_k)}.$$
The proof of Proposition \[prop:pseudo-approx-is\] follows by noting a proper weighting scheme involving the augmented approximate marginal distribution ${\pi_a}^\circ$ and target distribution ${\pi}^\circ$ (Lemma \[lem:pseudo-approx-is\]), and Theorem \[thm:proper-consistency\].
\[lem:pseudo-approx-is\] Suppose the conditions of Proposition \[prop:pseudo-approx-is\] hold. Then, ${\xi}_k$ form a ${\mathcal{L}}^\circ$-proper weighting scheme, with ${\mathcal{L}}^\circ {\mathrel{\mathop:}=}\{f^\circ({\theta},\phi,{x}) =
f({\theta},{x}){\,:\,}f\in {\mathcal{L}}\}$, in the sense of Proposition \[def:proper\], corresponding to
(i) approximate marginal ${\pi_a}^\circ({\mathrm{d}}{\theta}, {\mathrm{d}}\phi)
= {\mathrm{d}}{\theta}Q_a({\theta}, {\mathrm{d}}\phi)U(\phi)/c_a$,
(ii) target ${\pi}^{\circ}\big(({\mathrm{d}}{\theta}, {\mathrm{d}}\phi), {\mathrm{d}}{x}\big)$ which admits the marginal ${\pi}({\theta}, {x}){\mathrm{d}}{\theta}{\mathrm{d}}{x}$.
For any $f^\circ \in L^\circ$ and $\phi\in{\mathsf{S}}_\Phi$, let $\nu_f({\theta},\phi) {\mathrel{\mathop:}=}{\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}, \Phi_k=\phi]$. Whenever $U(\phi)>0$, define $${\mu}_{f^\circ}^\circ({\theta},\phi) {\mathrel{\mathop:}=}{\mathbb{E}}[{\xi}_k(f^\circ)\mid {\Theta}_k={\theta},\Phi_k=\phi]
= \nu_f({\theta},\phi)/U(\phi),$$ and ${\mu}_{f^\circ}^\circ({\theta},\phi) {\mathrel{\mathop:}=}0$ otherwise. We have $${\pi_a}^\circ({\mu}_{f^\circ}^\circ) = c_a^{-1}
\textstyle \iint Q_a({\theta},{\mathrm{d}}\phi) {\mathbb{I}\left(U(\phi)>0\right)}
\nu_f({\theta},\phi) {\mathrm{d}}{\theta}= c_w {\pi}(f),$$ by Assumption \[a:pseudo-proper\] , where $c_w = c_m/c_a$. We also have $$m_{f^\circ}^{\circ(1)}({\theta},\phi) {\mathrel{\mathop:}=}{\mathbb{E}}[{\xi}_k(f^\circ)\mid {\Theta}_k={\theta},\phi_k=\phi]
= |\nu_f({\theta},\phi)|/U(\phi),$$ so ${\pi_a}^\circ(m_{f^\circ}^{\circ(1)}) <\infty$ by Assumption \[a:pseudo-proper\] .
Let us finally consider different conditions, which guarantee Assumption \[a:pseudo-proper\] ; the integrability Assumption \[a:pseudo-proper\] may be shown similarly.
\[prop:sufficient-pseudo-is\] Assumption \[a:pseudo-proper\] holds if one of the following hold:
(i) \[item:pseudo-is-positive\] For ${\pi_a}$-a.e. ${\theta}\in{{\mathsf{T}}}$, $U(\Phi_{\theta})>0$ a.s. and $${\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}]
= c_m {\pi_m}({\theta}) f^*({\theta}),
\label{eq:unbiasedness-general-pseudo}$$ where ${\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}]
= \textstyle \int Q_a({\theta},{\mathrm{d}}\phi)
{\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}, \Phi_k=\phi]$.
(ii) \[item:pseudo-is-independent\] ${\zeta}_k$ only depend on ${\Theta}_k$, and for ${\pi_a}$-a.e. ${\theta}\in{{\mathsf{T}}}$, $${\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}]
= c_m {\pi_m}({\theta}) f^*({\theta})/p({\theta}),$$ where $p({\theta}) {\mathrel{\mathop:}=}{\mathbb{P}}(U(\Phi_{\theta})>0)$ with $\Phi_{\theta}\sim
Q_a({\theta},{\,\cdot\,})$.
(iii) \[item:pseudo-is-support\] For ${\pi_a}$-a.e. ${\theta}\in{{\mathsf{T}}}$ holds, and $U(\phi)=0$ implies ${\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta}, \Phi_k=\phi]=0$.
Note that implies , under which $$\textstyle
\iint Q_a({\theta},{\mathrm{d}}\phi) {\mathbb{I}\left(U(\phi)>0\right)}
\nu_f({\theta},\phi) {\mathrm{d}}{\theta}= c_m \int
{\pi_m}({\theta}) f^*({\theta}) {\mathrm{d}}{\theta}= c_m \pi(f),$$ where $\nu_f({\theta},\phi) = {\mathbb{E}}[{\zeta}_k(f)\mid
{\Theta}_k={\theta},\Phi_k=\phi]$.
In case of , we have $\nu_f({\theta},\phi) = {\mathbb{E}}[{\zeta}_k(f)\mid {\Theta}_k={\theta}]$ and so $$\textstyle \int Q_a({\theta},{\mathrm{d}}\phi) {\mathbb{I}\left(U(\phi)>0\right)}
\nu_f({\theta},\phi)
= c_m {\pi_m}({\theta}) f^*({\theta}).\qedhere$$
\[rem:positivity-essential\] Proposition \[prop:sufficient-pseudo-is\] is the most straightforward in the latent variable context, and often sufficient, since we may choose a positive $U(\phi)$ (e.g. by considering inflated $\tilde{U}((\phi) = U(\phi)+\epsilon$ instead). Proposition \[prop:sufficient-pseudo-is\] may be used directly to verify the validity of an MCMC version of the lazy ABC algorithm [@prangle]. It also demonstrates why positivity plays a key role: if only is assumed and $p({\theta})$ is non-constant, then $p({\theta})$ must be accounted for, or else we end up with biased estimators targeting a marginal proportional to ${\pi_m}({\theta})p({\theta})$. Proposition \[prop:sufficient-pseudo-is\] demonstrates that strict positivity is not necessary, but in this case a delicate dependency structure is required.
General state space models and sequential Monte Carlo {#sec:ssm}
=====================================================
State space models (SSM) are latent variable models which are commonly used in time series analysis [cf. @cappe-moulines-ryden]. In the setting of Section \[sec:latent\], SSMs are parametrised by ${\theta}\in{{\mathsf{T}}}$, and ${x}= {z}_{1:T} \in
{{\mathsf{X}}}= {\mathsf{S}}_{z}^T$ and ${y}= {y}_{1:T}\in {{\mathsf{Y}}}= {\mathsf{S}}_{y}^T$, and $$\mu^{({\theta})}({x})
= \prod_{t=1}^T
\mu_t^{({\theta})}({z}_t\mid {z}_{t-1})
\qquad\text{and}\qquad
g^{({\theta})}({y}\mid{x}) =
\prod_{t=1}^T g_t^{({\theta})}({y}_t\mid{z}_t),$$ where, by convention, $\mu_1^{({\theta})}({z}_1\mid {z}_0) {\mathrel{\mathop:}=}\mu_1^{({\theta})}({z}_1)$. That is, the latent states ${Z}_{1:T}$ form a Markov chain with initial density $\mu_1^{({\theta})}$ and state transition densities $\mu_t^{({\theta})}$, and the observations ${Y}_{1:T}$ are conditionally independent with $Y_i \sim g_t^{({\theta})}({\,\cdot\,}\mid {Z}_t)$.
This section reviews general techniques to generate random variables ${V}^{(1:m)}_{\theta}$ and ${X}^{(1:m)}_{\theta}$ for which ${\zeta}_{\theta}(h) {\mathrel{\mathop:}=}\sum_{i=1}^m
{V}^{(i)}_{\theta}h({X}^{(i)})$ satisfy: $$\begin{aligned}
{\mathbb{E}}[{\zeta}_{\theta}(h)]
&= \int p^{({\theta})}({z}_{1:T}, {y}_{1:T})
h({z}_{1:T}) {\mathrm{d}}{z}_{1:T}.
\label{eq:ssm-proper}\end{aligned}$$ for any ${\theta}$ and for some class of functions $h:{\mathsf{S}}_{z}^T\to{\mathbb{R}}$. These random variables may be used in order to construct a proper weighting; see Corollary \[cor:proper-ssm\] below.
Simple IS correction may be applied directly (see Proposition \[prop:augmented-is\]). Note that is satisfied for all integrable $h$, so ${\mathcal{L}}= L^1({\pi})$. It is often useful to combine such schemes as in Proposition \[prop:convex-proper\], allowing for instance variance reduction by using pairs of antithetic variables [@durbin-koopman2000].
For the rest of the section, we focus on the particle filter (PF) algorithm [@gordon-salmond-smith]; see also the monographs [@doucet-freitas-gordon; @del-moral; @cappe-moulines-ryden]. We consider a generic version of the algorithm, with the following components [cf. @del-moral]:
(i) Proposal distributions: $M_1$ is a probability density on ${\mathsf{S}}_{z}$ and $M_t({\,\cdot\,}\mid {z}_{1:t-1})$ defines conditional densities on ${\mathsf{S}}_{z}$ given ${z}_{1:t-1}\in{\mathsf{S}}_{z}^{t-1}$.
(ii) Potential functions: $G_t:{\mathsf{S}}_{z}^t\to{\mathbb{R}}_+$.
(iii) \[item:resampling\] Resampling laws: $\mathrm{Res}({\,\cdot\,}\mid \bar{\omega}^{(1:m)})$ defines a probability distribution on $\{1{:}m\}^m$ for every discrete probability mass $\bar{\omega}^{(1:m)}$.
The following two conditions are minimal for consistency:
\[a:smc-consistency\] Suppose that the following hold:
(i) \[a:fk-proper\] $\prod_{t=1}^T
M_t({z}_t\mid {z}_{1:t-1})G_t({z}_{1:t}) \
= p^{({\theta})}({z}_{1:T},y_{1:T})$ for all ${z}_{1:T}\in{\mathsf{S}}_{z}^T$.
(ii) \[a:resampling\] ${\mathbb{E}}\big[\sum_{i=1}^m
{\mathbb{I}\left(\vphantom{(}\smash{A^{(i)}=j}\right)}\big]=m\bar{\omega}^{(j)}$, where $A^{(1:m)} \sim \mathrm{Res}({\,\cdot\,}\mid \bar{\omega}^{(1:m)})$, for any $j\in\{1{:}m\}$ and any probability mass vector $\bar{\omega}^{(1:m)}$.
Assumption \[a:smc-consistency\] holds with traditionally used ‘filtering’ potentials $G_t({z}_{1:t})
{\mathrel{\mathop:}=}g_t^{({\theta})}({y}_t\mid {z}_t)
\mu_t^{({\theta})}({z}_t\mid
{z}_{t-1})/
M_t({z}_t\mid
{z}_{1:t-1})$, assuming a suitable support condition. We discuss another choice of $M_t$ and $G_t$ in Section \[sec:lin-gauss-ssms\], inspired by the ‘twisted SSM’ approach of [@guarniero-johansen-lee]. It allows a ‘look-ahead’ strategy based on approximations of the full smoothing distributions $q^{({\theta})}({z}_{1:T}\mid {y}_{1:T})$. Assumption \[a:smc-consistency\] allows for multinomial resampling, where $A_t^{(i)}$ are independent draws from $\bar{\omega}_t^{(1:m)}$, but also for lower variance schemes, including stratified, residual and systematic resampling methods [cf. @douc-cappe-moulines].
Below, whenever the index ‘$i$’ appears, it takes values $i=1,\ldots,m$.
\[alg:pf\] Initial state:
(i) Sample ${Z}_1^{(i)}\sim M_1$ and set $\bar{{Z}}_1^{(i)} = {Z}_1^{(i)}$.
(ii) Calculate $\omega_1^{(i)} {\mathrel{\mathop:}=}G_1({Z}_1^{(i)})$ and set $\bar{\omega}_1^{(i)} {\mathrel{\mathop:}=}\omega_1^{(i)}/\omega_1^*$ where $\omega_1^* = \sum_{j=1}^m
\omega_{1}^{(j)}$.
For $t=2,\ldots,T$, do:
(i) Sample $A_{t-1}^{(1:m)} \sim
\mathrm{Res}({\,\cdot\,}\mid \bar{\omega}_{t-1}^{(1:m)})$.
(ii) Sample ${Z}_t^{(i)}\sim
M_t({\,\cdot\,}\mid \bar{{Z}}_{t-1}^{(A_{t-1}^{(i)})})$ and set $\bar{{Z}}_{t}^{(i)} = (
\bar{{Z}}_{t-1}^{(A_{t-1}^{(i)})}, {Z}_t^{(i)})$.
(iii) Calculate $\omega_t^{(i)} {\mathrel{\mathop:}=}G_t(\bar{{Z}}_{t-1}^{(A_{t-1}^{(i)})}, {Z}_t)$ and set $\bar{\omega}_t^{(i)} {\mathrel{\mathop:}=}\omega_t^{(i)}/\omega_t^*$ where $\omega_t^* = \sum_{j=1}^m
\omega_{t}^{(j)}$.
If $\omega_{t}^*=0$, then Algorithm \[alg:pf\] is terminated immediately, and all the estimators considered (cf. Proposition \[prop:particle-proper\]) equal zero.
The following result summarises alternative ways how the random variables $({V}^{(1:m)}_{\theta},{X}^{(1:m)}_{\theta})$ may be constructed from the PF output, in order to satisfy . The results stated below are scattered in the literature [e.g. @del-moral; @pitt-dossantossilva-giordani], and some may be stated under slightly more stringent conditions, but a self-contained and concise proof of Proposition \[prop:particle-proper\] may be found in [@vihola-helske-franks-preprint].
\[prop:particle-proper\] Let ${\theta}\in{{\mathsf{T}}}$ be fixed, assume $\mathrm{Res}$, $M_t$ and $G_t$ satisfy Assumption \[a:smc-consistency\], and let $h:{\mathsf{S}}_{z}^T\to{\mathbb{R}}$ be such that the integral in is well-defined and finite. Consider the random variables generated by Algorithm \[alg:pf\], and let $U{\mathrel{\mathop:}=}\prod_{t=1}^T
\big(\frac{1}{m}\omega_t^*\big)$. Then,
(i) \[item:filter-smoother\] the random variables $({V}^{(1:m)}_{\theta},{X}^{(1:m)}_{\theta})$ where $
{V}^{(i)}_{\theta}=
U \bar{\omega}_T^{(i)}$ and ${X}^{(i)}_{\theta}= \bar{{Z}}_T^{(i)}$ satisfy .
Suppose in addition that $M_t({z}_t\mid {z}_{1:t-1})
G_t({z}_{1:t}) = C_t({z}_{t-1:t})$ for all $t\in\{1{:}T\}$ and all ${z}_{1:T}\in{\mathsf{S}}_{z}^T$. Define for $t\in\{2{:}T\}$, and any $i_t,i_{t-1}\in\{1{:}m\}$, the backwards sampling probabilities $$b_{t-1}(i_{t-1}\mid i_t)
{\mathrel{\mathop:}=}\frac{
\bar{\omega}_{t-1}^{(i_{t-1})}
C_t({Z}_{t-1}^{(i_{t-1})}, {Z}_t^{(i_t)})
}
{
\sum_{\ell=1}^m
\bar{\omega}_{t-1}^{(\ell )}
C_t({Z}_{t-1}^{(\ell)}, {Z}_t^{(i_t)})
},
\quad\text{and}\quad
b_T(i_T\mid i_{T+1})
=\bar{\omega}_T^{(i_T)}.$$
(i) \[item:backwards-sampling\] Let $I_{1:T}$ be random indices generated recursively backwards by $I_T\sim
b_T$ and $I_{t}\sim b_t({\,\cdot\,}\mid I_{t+1})$. The random variables $({V}^{(1)}_{\theta}, {X}^{(1)}_{\theta})$ satisfy , where ${V}^{(1)}_{\theta}=U$ and ${X}^{(1)}_{\theta}=
{Z}_{1:T}^{(I_{1:T})}$.
(ii) \[item:fwd-bwd-smoothing\] If $h({z}_{1:T}) =
\hat{h}({z}_{t-1},{z}_{t})$ for some $t\in \{2{:}T\}$, that is, $h$ is constant in all coordinates except $t-1$ and $t$, then, the random variables $({V}^{(1:m,1:m)}_{\theta}$, ${X}^{(1:m,1:m)}_{\theta})$ satisfy (with $\hat{h}$ on the left), where
(a) ${X}^{(i,j)}_{\theta}{\mathrel{\mathop:}=}({Z}_{t-1}^{(i)},{Z}_{t}^{(j)})$,
(b) ${V}^{(i,j)}_{\theta}{\mathrel{\mathop:}=}U b_{t-1}(i\mid j)\hat{\omega}_{t}^{(j)}$, and where
(c) \[item:smoothing-weights\] $\hat{\omega}_T^{(i)} {\mathrel{\mathop:}=}\bar{\omega}_T^{(i)}$ and $\hat{\omega}_{t}^{(i)} {\mathrel{\mathop:}=}\sum_{k=1}^m
\hat{\omega}_{t+1}^{(k)} b_{t}(i\mid k)$ for $t=T-1,\ldots,t$.
(iii) \[item:fwd-bwd-smoothing2\] If $h({z}_{1:T}) = \hat{h}({z}_t)$ for some $t\in
\{1{:}T\}$, then the random variables $({V}^{(1:m)}_{\theta},{X}^{(1:m)}_{\theta})$ satisfy (with $\hat{h}$ on the left), where ${X}^{(i)}_{\theta}= {Z}_t^{(i)}$ and ${V}^{(i)}_{\theta}=
U \hat{\omega}_t^{(i)}$ are defined in .
The estimator in Proposition \[prop:particle-proper\] was called the filter-smoother in [@kitagawa]. This property was shown in [@del-moral Theorem 7.4.2] in case of multinomial resampling, and extended later [cf. @andrieu-doucet-holenstein]. The statement holds also when the PF is applied with a general sequence of distributions rather than the SSM [@del-moral]. Proposition \[prop:particle-proper\] corresponds to backwards simulation smoothing [@godsill-doucet-west]. Drawing a single backward trajectory is, perhaps surprisingly, probabilistically equivalent to subsampling one trajectory from the filter-smoother estimate in Proposition \[prop:particle-proper\] [@doucet-lee-smc-gm]. However, drawing several trajectories independently as in Proposition \[prop:particle-proper\] may lead to lower variance estimators. Proposition \[prop:particle-proper\] and its special case correspond to the forward-backward smoother [@doucet-godsill-andrieu]; see also [@cappe-moulines-ryden]. It is a Rao-Blackwellised version of , but applicable only when considering estimates of a single marginal (pair). This scheme can lead to lower variance, but its square complexity in $m$ makes it inefficient with large $m$.
We next formally state how Proposition \[prop:particle-proper\] allows to use Algorithm \[alg:pf\] to derive a proper weighting scheme.
\[cor:proper-ssm\] Let $({\Theta}_k)_{k\ge 1}$ be a Markov chain which is Harris ergodic with respect to ${\pi_a}$. Suppose each $({V}^{(1:m)}_k,{X}^{(1:m)}_k)$ corresponds to an independent run of Algorithm \[alg:pf\] with ${\theta}={\Theta}_k$, as defined in Proposition \[prop:particle-proper\] , , or . Then, $({W}_k^{(1:m)}, {X}^{(1:m)}_k)_{k\ge 1}$ with ${W}_k^{(i)} {\mathrel{\mathop:}=}{\mathrm{pr}}({\theta}_k) {V}_k^{(i)}/{\pi_a}({\theta}_k)$ provide a proper weighting scheme for target distribution ${\pi}({\theta},{x}_{1:T}) =
p({\theta}, {x}_{1:T}\mid {y}_{1:T})$ (Definition \[def:proper\]), for the following classes of functions, respectively: $$\begin{aligned}
\eqref{item:filter-smoother}\;
{\mathcal{L}}& = L^1({\pi}), &
\eqref{item:fwd-bwd-smoothing}\;
{\mathcal{L}}&= \{f\in L^1({\pi}){\,:\,}f({\theta},{x}_{1:T}) =
\hat{f}({\theta},{x}_{t-1:t})\}, \\
\eqref{item:backwards-sampling}\;
{\mathcal{L}}& = L^1({\pi}), &
\eqref{item:fwd-bwd-smoothing2}\;
{\mathcal{L}}&= \{f\in L^1({\pi}){\,:\,}f({\theta},{x}_{1:T}) =
\hat{f}({\theta},{x}_{t})\}.\end{aligned}$$ In case $({\Theta}_k,U_k)_{k\ge 1}$ is a pseudo-marginal algorithm, ${W}_k {\mathrel{\mathop:}=}{\mathrm{pr}}({\theta}_k) {V}_k^{(i)}/U_k$.
The latter two cases in Corollary \[cor:proper-ssm\] are stated for a single marginal (pair), but it is clear that we may calculate estimates simultaneously for several marginal (pairs), so that Proposition \[prop:particle-proper\] is applicable for every function which is of the form $\sum_{t=1}^{T-1} f_t({\theta},{x}_{t:t+1})$ and Proposition \[prop:particle-proper\] for a function of the form $\sum_{t=1}^T f_t({\theta},{x}_t)$. See also the general discussion of smoothing functionals in [@cappe-moulines-ryden §4.1.2].
We state finally an implication of Proposition \[prop:particle-proper\] outside the main focus of this paper, in general SSM smoothing context (with fixed $\theta$). This result is widely known among particle filtering experts, but appears not to be widely adopted.
\[prop:particle-smooth-confidence\] Suppose ${\theta}\in{{\mathsf{T}}}$ is fixed, and let $({V}_k^{(1:m)}, {X}^{(1:m)}_k)_{k\ge 1}$ correspond to independent realisations of random variables defined in Proposition \[prop:particle-proper\].
(i) If the conditions of Proposition \[prop:particle-proper\] are satisfied, then the estimator $$E_n(h) {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n {\zeta}_k(h)}{
\sum_{j=1}^n {\zeta}_j({\mathbf{1}})}
\xrightarrow[\mathrm{a.s.}]{n\to\infty} \mu_h {\mathrel{\mathop:}=}\int
p^{({\theta})}({x}_{1:T}\mid {y}_{1:T}) h({x}_{1:T}){\mathrm{d}}{x}_{1:T}.$$
(ii) If also $\sigma_*^2{\mathrel{\mathop:}=}{\mathbb{E}}\big[|{\zeta}_1(\bar{h})|^2\big]<\infty$, where $\bar{h} = h -
\mu_h$, then $$\sqrt{n} \big[ E_n(h) - \mu_h\big] \xrightarrow[\mathrm{d}]{n\to\infty}
N(0,\sigma^2),
\qquad\text{where}\qquad
\sigma^2 {\mathrel{\mathop:}=}\frac{\sigma_*^2}{p^{({\theta})}({y}_{1:T})^2}.$$
(iii) If in addition ${\mathbb{E}}\big[|{\zeta}_1({\mathbf{1}})|^2]<\infty$, then $n v_n \to \sigma^2$, almost surely, where $$v_n {\mathrel{\mathop:}=}\frac{\sum_{k=1}^n \big({\zeta}_k(h) -
{\zeta}_k({\mathbf{1}}) E_n(h)\big)^2}{
\big(\sum_{j=1}^n {\zeta}_k({\mathbf{1}})\big)^2}.$$
Proof is similar to Theorem \[thm:importance-var\] in Appendix \[app:clt\].
The estimator $E_n(h)$ in Proposition \[prop:particle-smooth-confidence\] is an importance sampling analogue of the particle independent Metropolis-Hastings (PIMH) algorithm suggested in [@andrieu-doucet-holenstein]. Unlike the PIMH, calculation of $E_n(h)$ is parallelisable, and allows for straightforward consistent confidence intervals $\big[E_n(f) \pm \beta
\sqrt{v_n}\big]$, where $\beta$ corresponds to the desired standard Gaussian quantile. Calculation of consistent confidence intervals for a single realisation of a particle smoothing algorithm requires sophisticated techniques [@lee-whiteley-variance]. Another promising method recently suggested in [@jacob-lindsten-schon] relies on unbiased estimators obtained by coupling of conditional sequential Monte Carlo and debiasing tricks as in [@glynn-rhee; @mcleish; @rhee-glynn].
State space models with linear-Gaussian state dynamics {#sec:lin-gauss-ssms}
======================================================
We consider a special case of the general SSM in Section \[sec:ssm\], where both ${\mathsf{S}}_{z}$ and ${\mathsf{S}}_{y}$ are Euclidean and $\mu_t^{({\theta})}$ are linear-Gaussian, but the observation models $g_t^{({\theta})}$ may be non-linear and/or non-Gaussian, taking the form $$g_t^{({\theta})}({y}_t \mid {z}_t) =
\eta_t^{({\theta})}({y}_t\mid H_t^{({\theta})} {z}_t).$$ Our setting covers exponential family observation models with Gaussian, Poisson, binomial, negative binomial, and Gamma distributions, and a stochastic volatility model. This class contains a large number of commonly used models, such as structural time series models, cubic splines, generalised linear mixed models, and classical autoregressive integrated moving average models.
Marginal approximation {#sec:lin-gauss-marginal}
----------------------
The scheme we consider here is based on [@shephard-pitt; @durbin-koopman1997], and relies on a Laplace approximation $p^{({\theta})}_a({z}_{1:T}, \tilde{{y}}_{1:T}^{({\theta})})
= \mu^{({\theta})}({z}_{1:T})
\tilde{g}^{({\theta})}(\tilde{{y}}_{1:T}^{({\theta})}\mid
{z}_{1:T})$, where $\tilde{g}^{({\theta})}(\tilde{{y}}_{1:T}^{({\theta})}\mid {z}_{1:T})
{\mathrel{\mathop:}=}\prod_{t=1}^T
\tilde{g}_t^{({\theta})}(\tilde{{y}}_t^{({\theta})}\mid {z}_t)$. The linear-Gaussian terms $\tilde{g}_t$ approximate $g_t$ in terms of pseudo-observations $\tilde{{y}}_{t}^{({\theta})}$ and pseudo-covariances $R_t^{({\theta})}$, which are found by an iterative process, which we detail next for a fixed ${\theta}$. Denote $D_t^{(n)}({z}_t) {\mathrel{\mathop:}=}\frac{\partial^n}{\partial^n z_t}
\log \eta_t^{({\theta})}({y}_t \mid
{z}_t)$, and assume that $\tilde {z}_{1:T}$ is an initial estimate for the mode $\hat{{z}}_{1:T}^{({\theta})}$ of $p^{({\theta})}({z}_{1:T} \mid {y}_{1:T})$. following:
(i) $R_t^{({\theta})} = - [D_t^{(2)}(H_t^{({\theta})}\tilde{{z}}_t)]^{-1}$ and $\tilde {y}_t^{({\theta})} = H_t^{({\theta})} \tilde {z}_t + R_t^{({\theta})}
D_t^{(1)}(H_t^{({\theta})} \tilde{{z}}_t)$
(ii) Run the Kalman filter and smoother for the model with $g_t^{({\theta})}({y}_t\mid {z}_t)$ replaced by $ \tilde{g}_t^{({\theta})}(\tilde{{y}}_t^{({\theta})}\mid {z}_t)
= N(\tilde{{y}}_t^{({\theta})}; H_t^{({\theta})}{z}_t, R_t^{({\theta})})$ and set $\tilde{{z}}_{1:T}$ to the smoothed mean.
These steps are then repeated until convergence, which typically take less than 10 iterations [@durbin-koopman2000].
Consider the following decomposition of the marginal likelihood: $$\label{logp}
L({\theta})
= \tilde{L}_a({\theta})
\frac{g^{({\theta})}({y}_{1:T} \mid \hat
{z}_{1:T}^{({\theta})})}{
\tilde g^{({\theta})}(\tilde{{y}}_{1:T}^{({\theta})}\mid
\hat {z}_{1:T}^{({\theta})})}
{\mathbb{E}}\left[\frac{g^{({\theta})}({y}_{1:T} \mid
{Z}_{1:T})
/ g^{({\theta})}({y}_{1:T} \mid \hat
{z}_{1:T}^{({\theta})})
}{\tilde g^{({\theta})}(\tilde{{y}}_{1:T}^{({\theta})}\mid {Z}_{1:T})
/ \tilde g^{({\theta})}(\tilde{{y}}_{1:T}^{({\theta})} \mid \hat
{z}_{1:T}^{({\theta})})
}\right],$$ where $\tilde{L}_a({\theta}) {\mathrel{\mathop:}=}\int p^{({\theta})}_a({z}_{1:T}, \tilde
{y}_{1:T}^{({\theta})})
{\mathrm{d}}{z}_{1:T}$ is the marginal likelihood (from the Kalman filter), and the expectation is taken with respect to the approximate smoothing distribution $p_a^{({\theta})}({z}_{1:T}\mid \tilde{{y}}_{1:T}^{({\theta})}) =
p^{({\theta})}_a({z}_{1:T}, \tilde{y}_{1:T}^{({\theta})})/\tilde{L}_a({\theta})$. If the pseudo-likelihoods $\tilde{g}_t^{({\theta})}$ are nearly proportional to the true likelihoods $g_t^{({\theta})}$ around the mode of $p_a^{({\theta})}({z}_{1:T}\mid {y}_{1:T})$, the expectation in is close to one. Our approximation is based on dropping the expectation in : $ L_a({\theta}) {\mathrel{\mathop:}=}\tilde{L}_a({\theta})
g^{({\theta})}({y}_{1:T} \mid \hat {z}_{1:T})/\tilde
g^{({\theta})}(\tilde {y}_{1:T}^{({\theta})}\mid \hat
{z}_{1:T}).$ The same approximate likelihood $L_a({\theta})$ was also used in a maximum likelihood setting by [@durbin-koopman2012] as an initial objective function before more expensive importance sampling based maximisation was done. The evaluation of the approximation $L_a(\theta)$ above requires a reconstruction of the Laplace approximation for each value of $\theta$. We call this *local approximation*, and consider also a faster *global approximation* variant, where the pseudo-observations and covariances are constructed only once, at the maximum likelihood estimate of ${\theta}$.
Proper weighting schemes {#sec:lin-gauss-proper}
------------------------
The simplest approach to construct a proper weighting scheme based on the Laplace approximations is to use the approximate smoothing distribution $p_a^{({\theta})}({z}_{1:T}\mid {y}_{1:T})$ as IS proposal. Such a scheme using the simulation smoother [@durbin-koopman2002] antithetic variables, we call **SPDK**, following [@shephard-pitt].
We consider also several variants of $M_t$ and $G_t$ in the particle filter discussed in Section \[sec:ssm\]. The bootstrap filter [@gordon-salmond-smith], abbreviated as **BSF**, uses $M_t=\mu_t^{({\theta})}$ and $G_t=g_t^{({\theta})}({y}_t\mid {\,\cdot\,})$, and hence does not rely on an approximation. Inspired by the developments in [@whiteley-lee; @guarniero-johansen-lee], we consider also the choice $$M_t({z}_t\mid {z}_{1:t-1}) = p_a^{({\theta})}({z}_t\mid
{z}_{t-1},{y}_{1:T}) ,\;\text{and}\; G_t({z}_{1:t})
= g_t^{({\theta})}({y}_t\mid {z}_t) /
\tilde{g}_t^{({\theta})}(\tilde{{y}}_t\mid {z}_t),$$ where $p_a^{({\theta})}({z}_t\mid
{z}_{t-1},{y}_{1:T}) = p_a^{({\theta})}({z}_t\mid
{z}_{1:t-1},{y}_{1:T})$ are conditionals of $p_a^{({\theta})}({z}_{1:T}\mid {y}_{1:T})$. This would be optimal in our setting if the $G_t$ were constants [@guarniero-johansen-lee]. As they are often approximately so, we believe that this choice, which we call **$\psi$-APF** following [@guarniero-johansen-lee], can provide substantial benefits over BSF.
Discretely observed diffusions {#sec:diffusion-ssms}
==============================
In many applications, for instance in finance or physical systems modelling, the SSM state transitions arise naturally from a continuous time diffusion model, such as $${\mathrm{d}}\tilde{{Z}}_t =
m^{(\theta)}(t,\tilde{{Z}}_t) {\mathrm{d}}t +
\sigma^{(\theta)}(t,\tilde{{Z}}_t) {\mathrm{d}}B_t,$$ where $B_t$ is a (vector valued) Brownian motion and where $m^{(\theta)}$ and $\sigma^{(\theta)}$ are functions (vector and matrix valued, respectively). The latent variables ${X}=
({Z}_1,\ldots,{Z}_T)$ are assumed to follow the law of $(\tilde{{Z}}_{t_1},\ldots,\tilde{{Z}}_{t_T})$, so $\mu_k^{({\theta})}$ would ideally be the transition densities of $\tilde{{Z}}_{t_k}$ given $\tilde{{Z}}_{t_{k-1}}$. These transition densities are generally unavailable (for non-linear diffusions), but standard time-discretisation schemes allow for straightforward approximate simulation [cf. @kloeden-platen]. The denser the time-discretisation mesh used, the less bias introduced. However, the computational complexity of the simulation is higher — generally proportional to the size of the mesh.
The MCMC-IS may be applied to speed up the inference of discretely observed diffusions by the following simple two-level approach. The ‘true’ state transition $\mu_t^{({\theta})}$ are based on ‘fine enough’ discretisations, which are assumed to ensure a negligible bias, but which are expensive to simulate. Cheaper ‘coarse’ discretisation corresponds to transitions $\hat{\mu}_t^{({\theta})}$.
Because neither of the models admit exact calculations, we may only use a pseudo-marginal approximate chain as discussed in Sections \[sec:algorithmic-pseudo\] and \[sec:pseudo\]). More specifically, we may use the bootstrap filter (Section \[sec:ssm\]) with SSM $(\hat{\mu}_t^{(\tilde{{\Theta}}_k)}, g_t^{(\tilde{{\Theta}}_k)})$ to generate the likelihood estimators $\tilde{U}_k$ in Phase \[item:approx-phase2\]’, and in Phase \[item:pseudo-correct-phase2\]’, we may use bootstrap filters for SSM $(\mu_t^{({\Theta}_k)},g_t^{({\Theta}_k)})$ to generate $({V}_k^{(i)},{X}_k^{(i)})$.
Assuming that the observation model satisfies $g_t^{({\theta})}>0$ guarantees the validity of this scheme, because then $\tilde{U}_k>0$ (see Proposition \[prop:sufficient-pseudo-is\] ). It is most straightforward to simulate the bootstrap filters in Phases \[item:approx-phase2\]’ and \[item:pseudo-correct-phase2\]’ independent of each other, but they may be made dependent as well, by using a coupling strategy [cf. @sen-thiery-jasra]. The correction phase could be also based on exact sampling for diffusions [@beskos-papaspiliopoulos-roberts-fearnhead], which allow for elimination of the discretisation bias entirely.
The recent work [@franks-jasra-law-vihola] details how unbiased inference is also possible with IS type correction, using randomised multilevel Monte Carlo.
Experiments {#sec:exp}
===========
We did experiments for our generic framework with SSMs, using Laplace approximations (Section \[sec:lin-gauss-ssms\]) and an approximation based on coarsely discretised diffusions (Section \[sec:diffusion-ssms\]). We compared several approaches in our experiments:
AI
: Approximate inference with MCMC targeting ${\pi_a}({\theta})$, and for each accepted $\tilde{{\Theta}}_k$, sampling one realisation from $\tilde p^{({\Theta}_k)}({z}_{1:T} \mid {y}_{1:T})$.
PM
: Pseudo-marginal MCMC with $m$ samples targeting directly ${\pi}({\theta},{x})$.
DA
: Two-level delayed acceptance pseudo-marginal MCMC with first stage acceptance based on ${\pi_a}({\theta})$ and with target ${\pi}({\theta},{x})$.
IS1
: Jump chain IS correction with $m N_k$ samples for each accepted $\tilde{{\Theta}}_k$.
IS2
: Jump chain IS correction with $m$ samples for each accepted $\tilde{{\Theta}}_k$.
The IS1 algorithm is similar to simple IS estimator , but is expected to be generally safer; see Remark \[rem:jump\] . Except for AI, all the algorithms are asymptotically exact. Ignoring the effects of parallel implementation, the average computational complexity, or cost, of DA and IS2 are roughly comparable, and we have similar pairing between PM and IS1. However, as the weighting in IS methods is based only on the post-burn-in chain, the IS methods are generally somewhat faster.
We used a random walk Metropolis algorithm for ${\pi_a}$ with a Gaussian proposal distribution, whose covariance was adapted during burn-in following [@vihola-ram], targeting the acceptance rate 0.234. In DA, the adaptation was based on the first stage acceptance probability.
All the experiments were conducted in `R` [@r-core] using the `bssm` package which is available online [@helske-vihola]. The experiments were run on a Linux server with eight octa-core Intel Xeon E7-8837 2.67GHz processors with total 1TB of RAM.
In each experiment, we calculated the Monte Carlo estimates several times independently, and the inverse relative efficiency (IRE) was reported. The IRE, defined as the mean square error (MSE) of the estimate multiplied by the average computation time, provides a justified way to compare Monte Carlo algorithms with different costs [@glynn-whitt].
Further details and results of the experiments may be found in the preprint version of our article [@vihola-helske-franks-preprint].
Laplace approximations
----------------------
In case of Laplace approximations, the maximum likelihood estimate of ${\theta}$ was always used as the starting value of MCMC. We used sub-sampling as in Proposition \[prop:randomise-proper\], and sampled one trajectory ${Z}_{1:T}$ per each accepted state. We tested the exact methods with three different IS correction schemes, SPDK, BSF and $\psi$-APF, described in Section \[sec:lin-gauss-proper\]. For BSF and $\psi$-APF, the filter-smoother estimates as in Proposition \[prop:particle-proper\] were used. When calculating the MSE, we used the average over all estimates from all unbiased algorithms as the ground truth.
For all the exact methods, we chose the IS accuracy parameter $m$ based on a pilot experiment, following the guidelines for optimal tuning of pseudo-marginal MCMC in [@doucet-pitt-deligiannidis-kohn]. More specifically, $m$ was set so that the standard deviation of the logarithm of the likelihood estimate, denoted with $\delta$, was around 1.2 in the neighbourhood of the posterior mean of ${\theta}$. We kept the same $m$ for all methods, for comparability, even though in some cases optimal choice might differ [@sherlock-thiery-lee].
### Poisson observations {#poisson}
Our first model is of the following form: $$g_t^{({\theta})}({y}_t \mid {z}_t) =
\mathrm{Poisson}({y}_t; e^{u_t}), \quad\text{and}\quad
\begin{pmatrix}
u_{t+1} \\
v_{t+1}\end{pmatrix}
= \begin{pmatrix}
u_t + v_t + \sigma_{\eta}\eta_t \\
v_t + \sigma_{\xi}\xi_t
\end{pmatrix},
$$ with ${Z}_1=(U_1,V_1)\sim N(0, 0.1 I)$, where $\xi_t,\eta_t\sim N(0,1)$. For testing our algorithms, we simulated a single set of observations ${y}_{1:100}$ from this model with ${Z}_1=(0, 0)$ and ${\theta}= (\sigma_{\eta},
\sigma_{\xi})= (0.1, 0.01)$. We used a uniform prior $U(0, 2s)$ for the parameters, where the cut-off parameter $s$ was set to $1.6$ based on the sample standard deviation of $\log({y}_{1:T})$, where zeros were replaced with 0.1. Results were not sensitive to this upper bound.
Based on a pilot optimisation, we set $m=10$ for SPDK and $\psi$-APF, leading to $\delta\approx 0.1$, and $m=200$ for BSF with $\delta\approx 1.2$. For all algorithms, we used 100,000 MCMC iterations with the first half discarded as burn-in. We ran all the algorithms independently 1000 times.
Table \[table:poisson\] shows the IREs, which are re-scaled such that all IREs of PM-BSF equal one. The overall acceptance rate of DA-BSF was around 0.104, and 0.234 for all others. All exact methods led to essentially the same overall mean estimate $(0.093, 0.016, -0.075, 2.618$–$2.619)$ for ($\sigma_{\eta}$, $\sigma_{\xi}$, $u_1$, $u_{100}$), in contrast with AI showing some bias on $(u_1, u_{100})$, with overall mean estimates $(-0.064, 2.629)$ and $(-0.065, 2.631)$ with local and global approximation, respectively. IS2-BSF outperformed DA-BSF by about a factor of two in terms of IRE, because of the burn-in benefit. Similarly, IS1-BSF outperformed PM-BSF by a clear margin. With SPDK and $\psi$-APF, the IS1 and IS2 outperformed the PM and DA alternatives, but with a smaller margin because of smaller overall execution times. There were no significant differences between the SEs of local and global variants, but the global one was faster leading to smaller IREs.
--------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --
(lr)[3-5]{} (lr)[6-9]{} (lr)[10-13]{} AI DA IS1 IS2 PM DA IS1 IS2 PM DA IS1 IS2
Time 54 281 600 166 78 61 71 53 115 78 83 62
$\sigma_{\eta}$ 0.039 0.721 0.535 0.336 0.060 0.047 0.056 0.042 0.082 0.068 0.065 0.049
$\sigma_{\xi}$ 0.042 0.676 0.537 0.278 0.064 0.052 0.059 0.044 0.091 0.068 0.069 0.051
$u_1$ 0.561 0.911 0.609 0.406 0.063 0.055 0.057 0.042 0.097 0.071 0.076 0.053
$u_{100}$ 1.211 1.049 0.623 0.441 0.072 0.059 0.067 0.052 0.106 0.075 0.074 0.060
Time 11 235 549 120 35 17 28 10 72 34 38 19
$\sigma_{\eta}$ 0.012 0.596 0.476 0.218 0.025 0.013 0.022 0.008 0.052 0.028 0.030 0.015
$\sigma_{\xi}$ 0.052 0.564 0.530 0.197 0.029 0.015 0.025 0.009 0.061 0.031 0.034 0.017
$u_1$ 0.085 0.779 0.527 0.273 0.027 0.016 0.023 0.009 0.056 0.030 0.033 0.015
$u_{100}$ 0.333 0.804 0.563 0.305 0.034 0.016 0.027 0.010 0.068 0.034 0.036 0.019
--------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --
: IREs for the Poisson model, with local (top) and global (bottom) approximations. Times are in seconds. For PM-BSF, IREs are one and time 676s.[]{data-label="table:poisson"}
### Stochastic volatility model {#sv-model}
Our second illustration is more challenging, involving analysis of real time series: the daily log-returns for the S&P index from 4/1/1995 to 28/9/2016, with total number of observations $T =
5473$. The data was analysed using the following stochastic volatility (SV) model: $${Y}_t\mid{Z}_t \sim N(0,e^{{Z}_t}), \qquad
{Z}_{t+1}\mid {Z}_t \sim N(\nu + \phi ({Z}_t - \nu),
\sigma_{\eta}^2),$$ with ${Z}_1 \sim N(\nu, \sigma^2_{\eta} / (1-\phi^2))$. We used a uniform prior on $[-0.9999, 0.9999]$ for $\phi$, a half-Gaussian prior with standard deviation 5 for $\sigma_{\eta}$, and a zero-mean Gaussian prior with standard deviation 5 for $\nu$. SPDK was expected to be problematic, due to its well-known exponential deterioration in $T$, unlike the particle filter which often scales much better in $T$ [@whiteley-stability]. In addition, it is known that for this particular model, the importance weights may have large variability [@pitt-tran-scharth-kohn2013; @koopman-shephard-creal2009]. While in principle $\psi$-APF may also be affected by such fluctuations, we did not observe any problems with it in our experiments.
Based on our pilot experiment, we chose $m=10$ for $\psi$-APF, $m=70$ for SPDK and $m=300$ for BSF, which all led to $\delta\approx 1.1$. We used 100,000 MCMC iterations with the first half discarded as burn-in, and 100 independent replications. the IREs re-scaled here with respect to DA-BSF are shown in Table \[table:sv\]. The PM and IS1 were not tested because of their high costs. The results with global approximation are shown only for AI, and indicate significant computational savings. The parallelisation with 8 cores dropped the execution time nearly ideally. The total acceptance rates were 0.1 for DA-BSF, PM-SPDK and DA-$\psi$-APF, 0.06 for DA-SPDK, and 0.15 for PM-$\psi$-APF.
--------------------------------------- ------- ------- ------- -------- ------- ------- ------- ------- ------- ------- ------- -------
(lr)[4-5]{} (lr)[6-9]{} (lr)[10-13]{} AI AI^G^ IS2 IS2^8^ PM DA IS1 IS2 PM DA IS1 IS2
Time 1.3 0.2 25.2 4.6 4.4 1.9 2.8 1.5 2.4 1.4 1.5 1.3
$\phi$ 0.083 0.062 0.304 0.050 1.015 0.696 0.684 0.483 0.021 0.024 0.009 0.017
$\sigma_{\eta}$ 0.726 0.298 0.483 0.096 3.090 3.307 0.603 0.710 0.044 0.055 0.016 0.028
$\nu$ 0.008 0.747 0.287 0.042 1.208 2.544 0.228 0.404 0.026 0.027 0.010 0.020
${Z}_{1}$ 0.133 0.035 0.321 0.071 3.054 1.883 0.346 0.373 0.029 0.026 0.007 0.018
${Z}_{5473}$ 1.887 0.417 0.540 0.112 6.574 1.871 0.444 0.810 0.057 0.064 0.012 0.039
--------------------------------------- ------- ------- ------- -------- ------- ------- ------- ------- ------- ------- ------- -------
: IREs for SV model. Times are in hours. AI^G^ is with global approximation and IS2^8^ is with 8 parallel cores. For DA-BSF, IREs are one and time 46.3h. []{data-label="table:sv"}
Like in the Poisson experiment, the overall means of the exact methods were close to each other, but AI had some bias, this time also with some of the hyperparameters ($\sigma_\eta$ and $\nu$). The IS1 and IS2 methods outperformed the PM and DA methods similarly as in the Poisson experiment. Due to a much smaller $m$, the DA-SPDK and DA-$\psi$-APF were an order of magnitude faster than DA-BSF. Diagnostics from the individual runs of PM-SPDK and DA-SPDK sometimes showed poor mixing, and despite the large reductions in execution time, the IREs were worse than PM-BSF. We observed also cases with a few very large correction weights in IS1-SPDK and IS2-SPDK, which had some impact also on their efficiencies. The SEs of DA-$\psi$-APF were comparable with the DA-BSF. We did not experience problems with mixing or overly large weights with $\psi$-APF, which suggests $\psi$-APF being more robust than SPDK. There were no significant differences in the SEs between the exact methods when using the local and global approximation schemes.
Discretely observed Geometric Brownian motion
---------------------------------------------
Our last experiment was about a discretely observed diffusion as discussed in Section \[sec:diffusion-ssms\]. The model was a geometric Brownian motion, with noisy log-observations: $${\mathrm{d}}\tilde{Z}_t = \nu \tilde{Z}_t {\mathrm{d}}t + \sigma_{z}\tilde{Z}_t
{\mathrm{d}}B_t, \qquad {Y}_k \mid ({Z}_k={z}) \sim N(\log({z}),
\sigma^2_{y}),$$ with $\tilde{Z}_0\equiv 1$, where $(B_t)_{t\ge 1}$ stands for the standard Brownian motion, and where ${Z}_k = \tilde{Z}_k$. The discretisations $\mu_t^{({\theta})}$ and $\hat{\mu}_t^{({\theta})}$ were based on a Milstein discretisation with uniform meshes of sizes $2^{L_F}$ and $2^{L_C}$, respectively, with $L_C=4$ and $L_F=16$, reflected to positive values. We did not consider optimising $L_C$ and $L_F$, but rather aimed for illustrating the potential gains for the IS2 algorithm from parallelisation. The data was a single simulated realisation of length $50$ from the exact model, with $\nu=0.05$, $\sigma_x = 0.3$, and $\sigma_y=1$. We used a half-Gaussian prior with s.d. $0.1$ for $\nu$, a half-Gaussian prior with s.d. $0.5$ for $\sigma_x$, and $N(1.5,0.5^2)$ prior truncated to $>0.5$ for $\sigma_y$. For both IS2 and DA, and both levels, we used $m=50$ which led to $\delta \approx 0.6$.
Assuming a unit cost for each step in the BSF, the total average cost of a parallel IS2 run is $n 2^{L_C} + \alpha (n - n_b) 2^{L_F}/M$, where $\alpha$ is the mean acceptance rate of the approximate MCMC, $n_b$ is the length of burn-in and $M$ is the number of parallel cores used for the weighting. We chose $n =
5000$, $n_b = 2500$, $M=48$, and the target acceptance rate $\alpha=0.234$, leading to an expected 43-fold speed-up due to the parallelisation of IS2.
Single run of DA cannot be easily parallelised, but we ran instead multiple independent DA chains in parallel, and averaged their outputs for inference. While such parallelisation may not be optimal, it allows for utilisation of all of the available computational resources. The running time of each DA chain was constrained to be similar to the time required by IS2, leading to $n = 100$ with $n_b = 50$. Because of the short runs, we suspected that initial bias could play a significant role, which was explored by running two experiments, with MCMC initialised either to the prior mean $\theta_0 = (0.08, 0.4, 1.5)$, or to an independent sample from the prior. We experimented also with further thinning, by forming the IS2 correction based on every other accepted state.
Table \[table:gbm-results\] summarises the results from 100 replications. The run time of the parallel DA algorithms was defined as the maximum run time of all parallel chains. The parallelisation speedup of IS2 was nearly optimal, as well as the further speedup from thinning. The SEs with prior mean initialisation were similar between DA and IS2, but DA produced slightly biased results, leading to 9.5 to 13.0 times higher IREs. The efficiency gains of thinning were inconclusive, indicating some gains for the hyperparameters $\theta$, but not for the state variables. The smaller memory requirements and smaller absolute time requirements for the thinning make it still appealing. With prior sample initialisation, DA behaved sometimes poorly, in contrast with IS2 which behaved similarly with both initialisation strategies.
---------------------------------------------------- ------- ------- ------- -------- -------- ------- -------- ------- -------- ---------------- -------
(lr)[2-7]{} (lr)[8-12]{} [Init.]{}
(lr)[3-5]{} (lr)[6-7]{} (lr)[8-10]{} (lr)[11-12]{} GT DA IS2 IS2^t^ DA IS2 DA IS2 IS2^t^ DA IS2
[Time]{} — 12.3 3.4 1.9 14.0 3.3 12.3 3.4 1.9 14.0 3.3
$\nu$ 0.053 0.061 0.053 0.053 0.064 0.053 0.069 0.004 0.002 0.135 0.004
$\sigma_x$ 0.253 0.278 0.253 0.253 0.251 0.252 0.576 0.029 0.019 0.336 0.022
$\sigma_y$ 1.058 1.054 1.058 1.058 1.083 1.058 0.088 0.020 0.014 1.010 0.022
${Z}_{1}$ 1.254 1.273 1.254 1.246 1.243 1.252 0.670 0.109 0.119 0.805 0.103
${Z}_{50}$ 2.960 2.953 2.966 2.935 20.773 2.971 12.605 1.880 2.074 4$\times 10^6$ 2.308
---------------------------------------------------- ------- ------- ------- -------- -------- ------- -------- ------- -------- ---------------- -------
: Results for the geometric Brownian motion experiment using 48 cores. IS2^t^ is with thinning, and time is in minutes. Ground truth (GT) was calculated with MCMC using exact latent inference.[]{data-label="table:gbm-results"}
Summary of results
------------------
In our experiments with Laplace approximations, IS1 and IS2 were competitive alternatives to PM and DA, respectively, even without parallelisation. The differences were more emphasised when the cost of correction (number of samples $m$) was higher. The $\psi$-APF was generally preferable over SPDK, and BSF was the least efficient. The global approximation gave additional performance boost in our experiments, without compromising the accuracy of the estimates, but we stress that it may not be stable in all scenarios.
As noted earlier, the use of the guidelines by [@doucet-pitt-deligiannidis-kohn] were not necessarily optimal in our setting. We did an additional experiment to inspect how the choice of $m$ affects the IRE with BSF in the Poisson model, and with $\psi$-APF in the SV model. Figure \[fig:poisson-ire\] shows the average IREs as a function of $m$. Both IS2 and DA behaved similarly, and IS2 was less than DA uniformly in terms of IRE. In the Poisson-BSF case, the choice $m=200$ based on [@doucet-pitt-deligiannidis-kohn] appears nearly optimal. In case of the SV-$\psi$-APF, the optimal $m$ for DA and IS2 was around 50, which was higher than $m=10$ based on [@doucet-pitt-deligiannidis-kohn]. This is likely because of the initial overhead cost of the approximation.
![Average IRE of $(\sigma_\eta,\sigma_\xi,{Z}_1,{Z}_{100})$ in the Poisson model with BSF (left) and of $(\phi,\sigma_\eta, \nu, {Z}_1,{Z}_{5473})$ in the SV model with $\psi$-APF (right). DA is shown in black and IS2 in red.[]{data-label="fig:poisson-ire"}](poisson_ire.pdf "fig:"){width="45.00000%"} ![Average IRE of $(\sigma_\eta,\sigma_\xi,{Z}_1,{Z}_{100})$ in the Poisson model with BSF (left) and of $(\phi,\sigma_\eta, \nu, {Z}_1,{Z}_{5473})$ in the SV model with $\psi$-APF (right). DA is shown in black and IS2 in red.[]{data-label="fig:poisson-ire"}](sv_ire.pdf "fig:"){width="45.00000%"}
The discretely observed geometric Brownian motion example illustrated the potential gains which may be achieved by using the IS2 method in a parallel environment. While we admit that our experiment is academic, we believe that it is indicative, and shows that IS2 can provide substantial gains, and makes reliable inference possible in a much shorter time than DA. The IS framework is less prone to issues with burn-in bias, which can be problematic with naive MCMC parallelisation based on independent chains.
Discussion {#sec:discussion}
==========
Our framework of IS type estimators based on approximate marginal MCMC provides a general way to construct consistent estimators. Our experiments demonstrate that the IS estimator can provide substantial speedup relative to a delayed acceptance (DA) analogue with parallel computing, and appears to be competitive to DA even without parallelisation. We believe that IS is often better than DA in practice, but it is not hard to find simple examples where DA can be arbitrarily better than IS (and vice versa) [@franks-vihola]. Our followup work [@franks-vihola] complements our findings by theoretical considerations, with guaranteed asymptotic variance bounds between IS and DA.
IS is known to be difficult to implement efficiently in high dimensions, but this is not a major concern in most latent variable models, where the hyperparameters are low-dimensional. The IS weight may also be regularised easily by inflating the (estimated) approximate likelihood, for instance with $L_a({\theta})+\epsilon$, with some $\epsilon>0$. If the likelihood $L$ is bounded, then $w_u({\theta}) \propto
L({\theta})/(L_a({\theta})+\epsilon)$ is bounded as well. The latter approach can be seen as an instance of defensive importance sampling [@hesterberg]. Other generic safe IS schemes may also be useful [cf. @owen-zhou], and tempering may be applied for the likelihood as well.
We used adaptive MCMC in order to construct the marginal chain $({\Theta}_k)_{k\ge 1}$ in our experiments, and believe that it is often useful [cf. @andrieu-thoms]. Note, however, that our theoretical results do not apply directly with adaptive MCMC, unless the adaptation is stopped after suitable burn-in. Our results could be extended to hold with continuous adaptation, under certain technical conditions. We detailed proper weighting schemes based on standard IS and particle filters. We note that various PF variations, such as Rao-Blackwellisation, alternative resampling strategies [@cappe-moulines-ryden], or quasi-Monte Carlo updates [@gerber-chopin], apply directly. PFs can also be useful beyond the state space models context [@delmoral-doucet-jasra]. Twisted particle filters [@whiteley-lee; @alaluhtala-whiteley-heine-piche] could also be applied, instead of the $\psi$-APF.
In a diffusion context, a proper weighting can be constructed based on randomised multilevel Monte Carlo, as recently described in [@franks-jasra-law-vihola]. We are currently investigating various other instances of our framework. Laplace approximations are available for a wider class of Gaussian latent variable models beyond SSMs [cf. @rue-martino-chopin]. Variational approximations [@beal; @jordan-gm] and expectation propagation [@minka] have been found useful in a wide variety of models. In the SSM context, various non-linear filters could also be applied [cf. @sarkka-filtering-smoothing]. Our framework provides a generic validation mechanism for approximate inference, where assessment of bias is difficult in general [cf. @ogden]. Contrary to purely approximate inference, our approach only requires moderately accurate approximations, as demonstrated by our experiment with global Laplace approximations. Debiased MCMC, as suggested in [@glynn-rhee] and further explored in [@jacob-lindsten-schon; @jacob-oleary-atchade], may also lead to useful proper weighting schemes.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors have been supported by an Academy of Finland research fellowship (grants 274740, 284513 and 312605). We thank Christophe Andrieu, Arnaud Doucet, Anthony Lee and Chris Sherlock for many insightful remarks.
Properties of augmented Markov chains {#app:aug}
=====================================
Throughout this section, suppose that $K$ is a Markov kernel on ${\mathsf{X}}$ and $Q(x,B)$ is a kernel from ${\mathsf{X}}$ to a space ${\mathsf{S}}$. We consider here properties of an augmented Markov kernel $\check{K}$ defined on ${\mathsf{X}}\times{\mathsf{S}}$ as follows: $$\begin{aligned}
\check{K}\big((x,s),{\mathrm{d}}x'\times {\mathrm{d}}s'\big)
{\mathrel{\mathop:}=}K(x,{\mathrm{d}}x') Q(x',{\mathrm{d}}s').\end{aligned}$$ We first state the following basic result.
\[lem:aug-prop\] The properties of $K$ and the augmented chain $\check{K}$ are related as follows:
(i) \[lem:aug-mim-irr\] Let $\mathrm{irr}(K)$ denote the set of $\phi$-irreducibility measures of a Markov kernel $K$, then
- ${{\varphi_{P}}}\in \mathrm{irr}(K) \implies
{{\varphi_{\check{P}}}}({\mathrm{d}}x\times {\mathrm{d}}s)
{\mathrel{\mathop:}=}{{\varphi_{P}}}({\mathrm{d}}x)Q(x,{\mathrm{d}}s)
\in \mathrm{irr}(\check{K})$,
- ${{\varphi_{\check{P}}}}\in \mathrm{irr}(\check{K}) \implies
{{\varphi_{P}}}({\mathrm{d}}x){\mathrel{\mathop:}=}{{\varphi_{\check{P}}}}({\mathrm{d}}x \times {\mathsf{S}}) \in \mathrm{irr}(K)$.
(ii) \[lem:aug-mim-mim\] The implications in hold when $\mathrm{irr}(K)$ and $\mathrm{irr}(\check{K})$ are replaced with sets of maximal irreducibility measures of $K$ and $\check{K}$, respectively.
(iii) \[item:aug-invar\] The invariant probabilities of $K$ and $\check{K}$ satisfy:
- $\nu K = \nu
\implies \check{\nu} \check{K} =
\check{\nu}\quad\text{where}\quad
\check{\nu}({\mathrm{d}}x\times{\mathrm{d}}y) {\mathrel{\mathop:}=}\nu({\mathrm{d}}x)
Q(x,{\mathrm{d}}y)$,
- $\check{\nu} \check{K} = \check{\nu}
\implies \nu K =\nu\quad\text{where}\quad
\nu({\mathrm{d}}x){\mathrel{\mathop:}=}\check{\nu}({\mathrm{d}}x \times {\mathsf{S}})$.
These implications hold also with invariance replaced by reversibility.
(iv) \[item:aug-harris\] $K$ is Harris recurrent if and only if $\check{K}$ is Harris recurrent.
(v) \[item:aug-iterates\] Suppose $h:{\mathsf{X}}\times{\mathsf{S}}\to{\mathbb{R}}$ is measurable and such that $m_h(x) {\mathrel{\mathop:}=}\int Q(x,{\mathrm{d}}s) h(x,s)$ and $(K^n m_h)(x)$ are well-defined. Then, for any $n\ge 1$, $$(\check{K}^n h)(x,s) = (K^n m_h)(x).$$
The inheritance of irreducibility measures (\[lem:aug-mim-irr\]), maximal irreducibility measures (\[lem:aug-mim-mim\]), invariant measures , and reversibility is straightforward.
For Harris recurrence , let the probability $\phi_K$ be a maximal irreducibility measure for $K$, then $\phi_{\check{K}}({\mathrm{d}}x\times
{\mathrm{d}}s) {\mathrel{\mathop:}=}\phi_K({\mathrm{d}}x) Q(x,{\mathrm{d}}s)$ is the maximal irreducibility measure for $\check{K}$. Let $C\in {\mathcal{X}}\otimes{\mathcal{S}}$ with $\phi_{\check{K}}(C)>0$, and choose $\epsilon>0$ such that $\phi_K(C(\epsilon))>0$, where $C(\epsilon) {\mathrel{\mathop:}=}\{x\in{\mathsf{X}}{\,:\,}Q(x,C_x)>\epsilon\}$ with $C_x {\mathrel{\mathop:}=}\{s\in {\mathsf{S}}{\,:\,}(x,s)\in C\}$. Notice that $$\begin{aligned}
{\mathbb{P}}\bigg(\sum_{k=1}^\infty {\mathbb{I}\left((X_k,S_k)\in C\right)} =\infty \bigg)
&\ge {\mathbb{P}}\bigg(\sum_{k=1}^\infty {\mathbb{I}\left(S_{\tau_k}\in
C_{X_{\tau_k}}\right)}=\infty \bigg),
\end{aligned}$$ where $\tau_k$ are the hitting times of $(X_k)$ to $C(\epsilon)$. This concludes the proof because ${\mathbb{I}\left(S_{\tau_k} \in C_{X_{\tau_k}}\right)}$ are independent Bernoulli random variables with success probability at least $\epsilon$. The converse statement is similar.
For , it is enough to notice that for any $(x,s)\in{\mathsf{X}}\times{\mathsf{S}}$ and $n\ge 1$, it holds that $\check{K}^n\big((x,s),{\mathrm{d}}x'\times{\mathrm{d}}s'\big) = K^n(x,{\mathrm{d}}x')Q(x',{\mathrm{d}}s)$.
We next state the following generic results about the asymptotic variance and the central limit theorem of an augmented Markov chain. For $h\in L_0^2(\check{\nu})$, we denote as above the conditional mean $m_h(x) {\mathrel{\mathop:}=}\int Q(x,{\mathrm{d}}s)h(x,s)$ and the conditional variance $v_h(x) {\mathrel{\mathop:}=}\int Q(x,{\mathrm{d}}s)h^2(x,s) - m_h^2(x)$.
\[lem:aug-asvar\] Let $h\in L_0^2(\check{\nu})$. The asymptotic variance of an augmented Markov chain satisfies $${\mathrm{Var}}(h,\check{K}) = {\mathrm{Var}}(m_h, K) + \nu(v_h),$$ whenever ${\mathrm{Var}}(m_h, K)$ is well-defined.
Let $(X_k,S_k)$ be a stationary Markov chain with transition probability $\check{K}$. $$\begin{aligned}
{\mathrm{Var}}\bigg(\frac{1}{\sqrt{n}} \sum_{k=1}^n h(X_k,S_k)\bigg)
&= \check{\nu}(h^2) + \frac{2}{n} \sum_{i=1}^{n-1}
\sum_{\ell=1}^{n-i} {\mathbb{E}}[h(X_0,S_0)h(X_\ell,S_\ell)],\end{aligned}$$ by stationarity. For $\ell\ge 1$, Lemma \[lem:aug-prop\] implies $$\begin{aligned}
{\mathbb{E}}[h(X_0,S_0)h(X_\ell,S_\ell)]
&= {\mathbb{E}}[m_h(X_0) m_h(X_\ell)].\end{aligned}$$ We deduce for any $n\ge 1$ $${\mathrm{Var}}\bigg(\frac{1}{\sqrt{n}} \sum_{k=1}^n h(X_k,S_k)\bigg)
= {\mathrm{Var}}\bigg(\frac{1}{\sqrt{n}} \sum_{k=1}^n m_h(X_k) \bigg)
+ \nu(v_h),$$ because $\check{\nu}(h^2) - \nu(m_h^2) = \nu(v_h)$. The claim follows by taking limit $n\to\infty$.
\[lem:aug-clt\] Suppose $K$ is Harris ergodic and aperiodic, and $h\in L_0^2(\check{\nu})$. The CLT $$\frac{1}{\sqrt{n}} \sum_{k=1}^n h(X_k,S_k)
\xrightarrow{n\to\infty} N\big(0,{\mathrm{Var}}(m_h, K) + \nu(v_h)\big)
\label{eq:aug-clt}$$ holds for every initial distribution, if one of the following holds:
(i) \[item:kipnis-varadhan\] $K$ is reversible and ${\mathrm{Var}}(m_h, K)<\infty$.
(ii) \[item:maxwell-woodroofe\] $\sum_{n=1}^\infty n^{-3/2} \big\{\nu\big( \big[
\sum_{i=0}^{n-1}
K^i m_h\big]^2\big)\big\}^{1/2}<\infty$.
(iii) \[item:poisson-clt\] There exists $g\in L^2(\nu)$ which solves the Poisson equation $g - Kg = m_h$.\
In this case, ${\mathrm{Var}}(m_h,K) = \nu(g^2 - (Kg)^2)$.
The reversible case follows from Lemma \[lem:aug-asvar\] and the Kipnis and Varadhan CLT [@kipnis-varadhan], which implies for the initial distribution $\check{\nu}$. The jump chain is Harris by Lemma \[lem:aug-prop\] , so [@meyn-tweedie Proposition 17.1.6] guarantees for every initial distribution.
The case follows similarly, but relies on a result due to Maxwell and Woodroofe [@maxwell-woodroofe], which guarantees from $\check{\nu}$-almost every starting point, if $
\sum_{n=1}^\infty n^{-3/2}
\big\{\check{\nu}\big( \big[
\sum_{i=0}^{n-1}
\check{K}^i h\big]^2\big)\big\}^{1/2}
<\infty.
$ Notice that for $n\ge 2$ by Lemma \[lem:aug-prop\] , $$\begin{aligned}
\check{\nu}\bigg( \bigg[
\sum_{i=0}^{n-1}
\check{K}^i h\bigg]^2\bigg)
&=
\check{\nu}\bigg( \bigg[ (h - m_h) +
\sum_{i=0}^{n-1}
K^i m_h\bigg]^2\bigg)
= \nu(v_h) +
\nu\bigg( \bigg[
\sum_{i=0}^{n-1}
K^i m_h\bigg]^2\bigg).\end{aligned}$$ Because $(a+b)^{1/2} \le a^{1/2}+b^{1/2}$ for $a,b\ge 0$ and $\nu(v_h)<\infty$, the claim follows.
For , we first observe that $$\check{g} - \check{K}\check{g} = h
\qquad\text{where}\qquad
\check{g}(x,s) {\mathrel{\mathop:}=}g(x) + h(x,s)-m_h(x)\in L^2(\check{\nu}).$$ Indeed, it is clear that $\check{g}\in L^2(\check{\nu})$ and because $(\check{K} \check{g})(x,s) = (Kg)(x)$, $$\check{g}(x,s) - (\check{K} \check{g})(x,s)
= g(x) - (Kg)(x) + h(x,s) - m_h(x) = h(x,s).$$ The CLT and asymptotic variance follow from [@meyn-tweedie Theorem 17.4.4].
Proofs about CLT and asymptotic variance {#app:clt}
========================================
Whenever $\sum_{i=1}^n {\xi}_i({\mathbf{1}}) >0$, we may write $$\sqrt{n}\big[E_n(f)-{\pi}(f)\big]
= \frac{n^{-1/2} \sum_{k=1}^n {\xi}_k(\bar{f}) }{
n^{-1} \sum_{j=1}^n {\xi}_k({\mathbf{1}})}.$$ The denominator converges to $c_w>0$ almost surely, som by Slutsky’s lemma, it is enough to show that the numerator converges in distribution to $N\big(0,{\mathrm{Var}}(\nu_{\bar{f}}, P) + {\pi_a}(v)\big)$. This follows from Lemma \[lem:aug-clt\] and , under conditions and , respectively.
For $n$ large enough such that $\sum_{j=1}^n {\xi}_j({\mathbf{1}})>0$, we may write $$n v_n
= \frac{\frac{1}{n} \sum_{k=1}^n \big({\xi}_k(f)
- {\xi}_k({\mathbf{1}}) E_n(f) \big)^2}{\big(\frac{1}{n}\sum_{j=1}^n
{\xi}_j({\mathbf{1}})\big)^2}.$$ The denominator converges to $c_w^2$, and the numerator can be written as $$\frac{1}{n} \sum_{k=1}^n \big[{\xi}_k^2(\bar{f})
+ {\xi}_k^2({\mathbf{1}})D_n^2 + 2
{\xi}_k({\mathbf{1}}){\xi}_k(\bar{f})D_n \big]
\qquad\text{with}\qquad
D_n {\mathrel{\mathop:}=}{\pi}(f)-E_n(f).$$ The term $n^{-1} \sum_{k=1}^n {\xi}_k^2(\bar{f})\to
{\pi_a}(v+{\mu}_{\bar{f}}^2)$, and because $D_n\to 0$, the remainder terms $D_n^2 \big(n^{-1}\sum_{k=1}^n {\xi}_k^2({\mathbf{1}})\big)\to 0$ and $2D_n \big(n^{-1} \sum_{k=1}^n {\xi}_k({\mathbf{1}}){\xi}_k(\bar{f})\big)
\to 0$.
Proofs about jump chain estimators {#app:jump}
==================================
In this section, $K$ is assumed to be a Markov kernel on ${\mathsf{X}}$ which is non-degenerate, that is, $a(x) {\mathrel{\mathop:}=}K(x,{\mathsf{X}}\setminus\{x\}) >0$ for all $x\in{\mathsf{X}}$. The following proposition complements [@douc-robert Lemma 1] and [@deligiannidis-lee], which are stated for more specific cases.
\[prop:jump-properties\] Suppose $(X_k)$ is a Markov chain with kernel $K$ and $(\tilde{X}_k)$ the corresponding jump chain with holding times $(N_k)$ (Definition \[def:jump\]). Then, the following hold:
(i) \[item:jump-trans\] $(\tilde{X}_k)$ is Markov with transition kernel $\tilde{K}(x,A) = K(x,A\setminus\{x\})/a(x)$.
(ii) \[item:jump-joint-trans\] The holding times $(N_k)$ are conditionally independent given $(\tilde{X}_k)$, and each $N_k$ has geometric distribution with parameter $a(\tilde{X}_k)$.
(iii) \[item:jump-invar\] If $K$ admits invariant probability $\nu({\mathrm{d}}x)$, then $\tilde{K}$ admits invariant probability $
\tilde{\nu}({\mathrm{d}}x) {\mathrel{\mathop:}=}\nu({\mathrm{d}}x) a(x)/\nu(a).
$ In addition, if $K$ is reversible with respect to $\nu$, then $\tilde{K}$ is reversible with respect to $\tilde{\nu}$.
(iv) \[item:psi-irreducibility\] $(X_k)$ is $\psi$-irreducible if and only if $(\tilde{X}_k)$ is $\psi$-irreducible, with the same maximal irreducibility measure.
(v) \[item:harris-recurrence\] $(X_k)$ is Harris recurrent if and only if $(\tilde{X}_k)$ is Harris recurrent.
The expression of the kernel is due to straightforward conditioning, and was observed in [@douc-robert]. The invariance follows from $$\begin{aligned}
\int \tilde{\nu}({\mathrm{d}}x) \tilde{K}(x,A)
&= \frac{1}{\nu(a)} \int \nu({\mathrm{d}}x) \big[K(x,A) -
{\mathbb{I}\left(x\in A\right)} K(x,\{x\})\big] \\
&= \frac{1}{\nu(a)}\bigg[ \nu(A) - \int_A \nu({\mathrm{d}}x)\big(1-a(x)\big)\bigg]
= \tilde{\nu}(A),
\end{aligned}$$ and the reversibility is shown in [@douc-robert]. For it is sufficient to observe that $$\forall x\in {\mathsf{X}}: \sum_{n\ge 1} {\mathbb{P}}_x(X_n\in A)>0
\iff
\forall x\in {\mathsf{X}}: \sum_{n\ge 1} {\mathbb{P}}_x(\tilde{X}_n\in A)>0,$$ where ${\mathbb{P}}_x({\,\cdot\,}) = {\mathbb{P}}({\,\cdot\,}\mid X_0=\tilde{X}_0=x)$, which holds because the sets $\{X_k\}_{k\ge 0}$ and $\{\tilde{X}_k\}_{k\ge 0}$ coincide. Similarly, holds because $$\forall x\in {\mathsf{X}}:
{\mathbb{P}}_x(\eta_A =\infty) = 1 \iff
\forall x\in{\mathsf{X}}:
{\mathbb{P}}_x(\tilde{\eta}_A =\infty) = 1,$$ where $\eta_A {\mathrel{\mathop:}=}\sum_{k=1}^\infty {\mathbb{I}\left(X_k\in A\right)}$ and $\tilde{\eta}_A {\mathrel{\mathop:}=}\sum_{k=1}^\infty
{\mathbb{I}\left(\smash{\tilde{X}_k\in A}\right)}$.
We now state results about the asymptotic variance of the jump chain, complementing the reversible case characterisation of [@deligiannidis-lee; @doucet-pitt-deligiannidis-kohn].
\[prop:jump-variance\] Let $f\in L^2_0(\tilde{\nu})$. With the notation of Proposition \[prop:jump-properties\],
(i) \[item:deligiannidis-lee\] If $K$ is reversible, then ${\mathrm{Var}}(f,\tilde{K})<\infty$ iff $af\in L^2(\nu)$ and ${\mathrm{Var}}(af,K)<\infty$, and $${\mathrm{Var}}(f,\tilde{K}) = \nu(a)^{-1}
\big[ {\mathrm{Var}}(a f, K) - \nu\big(a(1-a) f^2\big)\big].
\label{eq:jump-variance}$$
(ii) \[item:jump-poisson\] If there exists a function $g\in L^2(\nu)$ which satisfies $g - Kg = af$, then ${\mathrm{Var}}(f,\tilde{K})<\infty$, ${\mathrm{Var}}(af,K)<\infty$, holds, $g - \tilde{K} g = f$ and $g\in L^2(\tilde{\nu})$.
The reversible case is a restatement of [@deligiannidis-lee Theorem 1].
Consider then . By Proposition \[prop:jump-properties\] , we obtain for any $h:{\mathsf{X}}\to{\mathbb{R}}$ with $Kh$ well-defined, $$(\tilde{K} h)(x) = \frac{(Kh)(x) -
\big(1-a(x)\big)h(x)}{a(x)}
= \frac{(Kh)(x) -h(x)}{a(x)} + h(x).$$ Consequently, we observe that $g - \tilde{K} g = a^{-1} \big( g - Kg\big) =
f$ implying . Because $g\in L^2(\tilde{\nu})$, Lemma \[lem:aug-clt\] and a straightforward calculation yield $$\begin{aligned}
{\mathrm{Var}}(f, \tilde{K}) &= \tilde{\nu}\big(g^2 - (\tilde{K}g)^2\big) \\
&= 2 \langle g, g - \tilde{K} g\rangle_{\tilde{\nu}}
- \langle g-\tilde{K}g, g - \tilde{K}g\rangle_{\tilde{\nu}} \\
&= \nu(a)^{-1} \big[
2 \langle g, g - Kg \rangle_{\nu} - \nu( a f^2)
\big],
\end{aligned}$$ where $\langle f,g\rangle_{\nu} {\mathrel{\mathop:}=}\int f(x) g(x)
\nu({\mathrm{d}}x)$. Similarly, by Lemma \[lem:aug-clt\] $${\mathrm{Var}}(a f, K) = \nu\big(g^2 -
(Kg)^2\big) = 2 \langle g, g - Kg \rangle_\nu - \nu( a^2 f^2),$$ which allows us to conclude.
Whenever $\sum_{j=1}^n {\xi}_j({\mathbf{1}})>0$, we may write $$\sqrt{n}\big[E_n(f)-{\pi}(f)\big] = \frac{n^{-1/2}
\sum_{k=1}^n N_k {\xi}_k(\bar{f}) }{n^{-1}
\sum_{j=1}^n N_j {\xi}_j({\mathbf{1}})}.$$ We shall show below that the CLT holds for the numerator, with asymptotic variance $\sigma^2 {\mathrel{\mathop:}=}\big[{\mathrm{Var}}({\mu}_{\bar{f}}, P)+{\pi_a}(\alpha
\tilde{v})\big]/ {\pi_a}(\alpha)$. This implies the claim by Slutsky’s lemma, as the denominator converges to $c_w/{\pi_a}(\alpha)$. For the rest of the proof, let $\tilde{P}$ and $\check{P}$ be the Markov kernels of $(\tilde{{\Theta}}_k)_{k\ge 1}$ and $(\tilde{{\Theta}}_k$,$N_k$,${\xi}_k(\bar{f}))_{k\ge 1}$, respectively, and let $\tilde{\pi}$ and $\check{\pi}$ be the corresponding invariant probabilities. Note that the function $h({\theta},n,{\xi}) {\mathrel{\mathop:}=}n {\xi}$ is in $L^2(\check{\pi})$ by assumption .
In case holds, also $\tilde{P}$ and $\check{P}$ are reversible by Proposition \[prop:jump-properties\] and Lemma \[lem:aug-prop\] . Lemma \[lem:aug-clt\] with $K = \tilde{P}$, $\check{K}=\check{P}$, $\nu =
\tilde{\pi}$ and $\check{\nu}=\check{\pi}$ implies that a CLT holds for $h$ whenever the asymptotic variance is finite: $${\mathrm{Var}}(h,\check{P})
= {\mathrm{Var}}\big({\mu}_{\bar{f}}/\alpha, \tilde{P}\big)
+ {\pi_a}(\alpha \tilde{v}_{N{\xi}})/{\pi_a}(\alpha),$$ where, by the variance decomposition formula, $$\begin{aligned}
\tilde{v}_{N{\xi}}({\theta}) &{\mathrel{\mathop:}=}{\mathrm{Var}}(N_k {\xi}_k(\bar{f}) \mid
\tilde{{\Theta}}_k={\theta}) \\
&= \tilde{v}({\theta}) + {\mathrm{Var}}\big(N_k {\mathbb{E}}[{\xi}_k(\bar{f}) \mid
\tilde{{\Theta}}_k={\theta}, N_k]{\mathrel{\big|}}\tilde{{\Theta}}_k={\theta}\big) \\
&= \tilde{v}({\theta}) + {\mu}_{\bar{f}}^2({\theta})
\big(1-\alpha({\theta})\big)/\alpha^2({\theta}).\end{aligned}$$ Proposition \[prop:jump-variance\] implies that $${\mathrm{Var}}\big({\mu}_{\bar{f}}/\alpha, \tilde{P}\big)
= {\pi_a}(\alpha)^{-1}
\big[ {\mathrm{Var}}({\mu}_{\bar{f}}, P)
- {\pi_a}\big((1-\alpha)
{\mu}_{\bar{f}}^2/\alpha\big)\big],$$ which implies ${\mathrm{Var}}(h,\check{P}) = \sigma^2$.
Consider then . Proposition \[prop:jump-variance\] implies that $g - \tilde{P} g = {\mu}_{\bar{f}}/\alpha$, and $g\in L^2(\tilde{\pi})$. Lemma \[lem:aug-clt\] implies the CLT, and together with Proposition \[prop:jump-variance\] leads to ${\mathrm{Var}}(h,\check{P})=\sigma^2$.
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[^1]: Note that our definition is set-theoretic support of the density, and differs in general from the support of the measure $\nu$ (on a topological space ${\mathsf{X}}$).
|
---
abstract: 'In the standard $q$-voter model, a given agent can change its opinion only if there is a full consensus of the opposite opinion within a group of influence of size $q$. A more realistic extension is the threshold $q$-voter, where a minimal agreement (at least $0<q_0\le q$ opposite opinions) is sufficient to flip the central agent’s opinion, including also the possibility of independent (non conformist) choices. Variants of this model including non-conformist behavior have been previously studied in fully connected networks (mean-field limit). Here we investigate its dynamics in random networks. Particularly, while in the mean-field case it is irrelevant whether repetitions in the influence group are allowed, we show that this is not the case in networks, and we study the impact of both cases, with or without repetition. Furthermore, the results of computer simulations are compared with the predictions of the pair approximation derived for uncorrelated networks of arbitrary degree distributions.'
address: |
$^1$Department of Physics, PUC-Rio, Rua Marquês de São Vicente, 225, 22451-900, Rio de Janeiro, Brazil\
$^2$IFISC, Instituto de F[í]{}sica Interdisciplinar y Sistemas Complejos (CSIC-UIB), Campus Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain\
$^3$National Institute of Science and Technology for Complex Systems (INCT-SC), Rio de Janeiro, Brazil
author:
- 'A. R. Vieira$^1$, Antonio F. Peralta$^2$, Raul Toral$^2$, Maxi San Miguel$^2$, C. Anteneodo$^{3}$'
title: 'Pair approximation for the noisy threshold $q$-voter model'
---
Introduction
============
The voter model is a paradigmatic exponent of opinion dynamics in a binary scenario, where individuals have to choose between two alternatives, e.g., for or against. According to this model, a randomly chosen agent can flip its opinion by imitation, copying the choice of one of its contacts [@review; @voter1; @voter2].
An enriched variant of the voter model is the $q$-voter proposed by Castellano et al. [@q-voter], where an agent is persuaded not by one but by $q$ neighbors in the network of contacts. However, the consensus among $q$ contacts required for changing opinion seems too restrictive. In a real scenario, a simple majority or even a minimal number may be enough. This can be expressed by the introduction of a threshold [@threshold-sznajd; @threshold-celia-allan; @sznajd-plos], which allows a flip of opinion when there is a minimal number of $q_0$ contacts sharing the opposite opinion within the influence group of size $q$.
It is also realistic to include the possibility of stochastic behavior given by anti-conformist or nonconformist trends [@q-voter-noises; @q-voter-zealots]. In the original version of the $q$-voter model [@q-voter], a parameter $\varepsilon$ is introduced which has a hybrid interpretation in terms of nonconformism. But alternative rules have been proposed [@sznajd-plos]. For instance, the chosen agent can flip its opinion state ignoring the neighbors, with a given probability $p$, while the conformist response following the threshold dynamics occurs with probability $1-p$. This is the variant of the threshold $q$-voter that we consider in this paper.
Another relevant ingredient in opinion dynamics is the structure of the network of contacts. However, few works on $q$-voter models go beyond the mean-field dynamics. Among them, we find the standard $q$-voter in complex networks [@qvoterRRN; @qvoterPA] and in multiplex networks [@qvoter_multiplex; @qvoter_multiplexPA], as well as the non-linear noisy voter [@antonioChaos; @antonioNJP], related to the $q$-voter, for which analytical results were obtained through the pair approximation (PA). In order to investigate the threshold $q$-voter on random networks we use the PA approximation complemented by Monte Carlo simulations.
Furthermore, a feature that is not significant in the mean-field (or globally coupled) case but relevant when we introduce network structure is the possibility of repetition or not when choosing $q$ agents amongst $k$ neighbors. Therefore, we also investigate the effects of repetitions on the critical portrait of this voter model.
In Sec. \[sec:model\] we describe the opinion dynamics, and revisit the mean-field limit in Sec. \[sec:MF\]. We present analytical results using the PA in Sec. \[sec:PA\], and comparison with simulations in Sec. \[sec:results\]. Finally, concluding remarks are addressed in Sec. \[sec:remarks\]. Details of the calculations can be found in Appendix A and Appendix B.
Model {#sec:model}
=====
Let us consider that individuals in a network of size $N$ can hold any of two opinions ($\oplus$ and $\ominus$), $n$ of them with opinion $\oplus$ and $N-n$ the opposite one $\ominus$. The state of each individual can change according to the following algorithm: We randomly select an individual (node) $i$ in the network. With probability $1-p$ this opinion is subject to change in conformity with an influence group, while with probability $p$ it acts independently of its neighbors, flipping its state with chance 1/2. In the conformity rule, we select $q$ neighbors of the central node $i$. If in this set there are at least $q_0$ elements that share the opposite opinion of node $i$, then its opinion state is flipped. The particular case $q_0=q$ represents the $q$-voter model, where unanimity is required in order to influence opinions [@q-voter]. The still more particular case $q_0=q=1$ corresponds to the standard well-known “noisy-voter model” (Kirman model) [@noisy-voter1; @noisy-voter2]. The noise-parameter $p$ is a measure of the degree of independence of a node with respect to its neighbors, and we name this particular set of rules as the “noisy threshold $q$-voter model”.
Let us remark that in the selection of the influence set, the $q$ nodes can be either distinct (i.e. chosen without repetitions) or one can allow for repetitions, which reflects interactions that are more frequent or intense. As we will see, the existence or not of repetitions in the selection of the influence set is a relevant ingredient of the model.
We note also that, at variance with the version discussed in [@threshold-celia-allan], where a different type of noise was considered [@q-voter], the systems discussed here do not show the existence of absorbing states ($n=0$ and $n=N$) where the systems get trapped.
In the following sections, analytical results are obtained for networks of arbitrary degree distribution. Simulations are carried out mainly in random-regular networks [@survey-rrn; @algorithm-rrn] but, Erdős-Rényi graphs [@Erdos1959a; @Barabasi2002a] and networks with a power-law degree distribution [@config1; @config2] are also considered.
Mean-field solution {#sec:MF}
===================
In this Section we present the mean-field (MF) scenario, with the idea of defining the quantities needed to develop the more sophisticated pair approximation scheme in Sec. \[sec:PA\] and to provide a framework of comparison for our analytical and numerical results in structured networks.
Let us then consider a fully connected network of size $N$. The stochastic evolution of the collective (or average) opinion can be described by knowing the transition rates $w(n\to n^\prime)$ between collective states characterized, respectively, by $n$ and $n^\prime$ agents with opinion $\oplus$. The (non-null) transition rates are $$\begin{aligned}
\label{eq: nonnull tax}
w(n\to n-1) & = & n \, G(1-x) , \\ \nonumber
w(n\to n+1) & = & (N-n) \, G(x) , \end{aligned}$$ where $x \equiv n/N$ and $ G(x)$ \[resp. $G(1-x)$\] is the conditional probability that a sorted agent with opinion $\ominus$ \[resp. $\oplus$\] flips its opinion. In the extension of the $q$-voter dynamics with threshold $q_0$ and independence level $p$, this flipping probability is given by the governing rules specified in Section \[sec:model\], see for instance, reference [@sznajd-plos], $$\label{eq:G}
G(x;q,q_0,p)=(1 - p)g_1(x;q,q_0) + p/2 \, ,$$ where $$\begin{aligned}
\label{eq:gx}
&&g_1(x;q,q_0) = \sum_{j = q_0}^{q} {{q}\choose{j}} \, x^j (1-x)^{q-j} .\end{aligned}$$ is the cumulative probability of sorting at least $q_0$ agents with opposite opinion amongst $q$ neighbors. As explained in Appendix A, it turns out that $g_1(x;q,q_0)$ can be written in terms of hypergeometric functions. Eq. (\[eq:gx\]) refers to the case where one allows repetitions in the selection of the $q$ neighbors. However, in the fully-connected network with $N\gg q$, the probability of repetition is very small and hence in the thermodynamic limit $N\to\infty$ the results coincide for both selection options. Function $g_1(x;q,q_0)$ is depicted in Fig. \[fig:gx\] for several values of $q$ and $q_0$.
\
The time evolution of the average value $\langle x \rangle$ can be obtained from the rate equation $$\label{eq:drift}
\frac{dx}{dt} = (1-x)G(x) - x G(1-x) \equiv \upsilon(x;p,q,q_0) \, ,$$ where we have used the deterministic approximation $\langle x^{m} \rangle = \langle x \rangle^{m}$ (which is of general validity in the thermodynamic limit $N \rightarrow \infty$ [@peralta-moments]) and short notation $\langle x \rangle \equiv x$. Note that, due to the symmetries of the model, the function $\upsilon(x)$ can be expanded in a Taylor series containing only odd powers of $(x-1/2)$. Furthermore, the steady-state probability of finding $n$ agents in state $\oplus$ can be written, in the limit of large $N$, in a large-deviation form $P_\textrm{st}(n)\propto e^{-NV(n/N)}$, with a potential function [@peralta:JSTAT] $$V(x)=\int^xdx\log\left[\frac{xG(1-x)}{(1-x)G(x)}\right].$$ The fixed points of Eq. (\[eq:drift\]), satisfying $\upsilon(x;p) =0$ coincide with the extrema of the potential $V(x)$, such that stable (resp. unstable) fixed points are minima (resp. maxima) of $V(x)$. In the case of coexistence of multiple stable fixed points, the macroscopically observed configurations (phases) correspond to the absolute minimum of the potential, while relative minima are metastable phases. The fixed points are obtained as the roots $x(p)$ of the equation $$\label{eq:extreme}
p = \frac{x {g_1}(1-x) - (1-x) {g_1}(x)}{x {g_1}(1-x) - (1-x) {g_1}(x) + (1/2-x)} \, .$$ We now discuss the different solutions and their stability.
The trivial solution $x(p)=1/2$, that corresponds to a disordered (D) phase, exists for all parameter values and it is stable for $p>p_c$ and unstable for $p<p_c$. The stability boundary can be obtained by taking the limit $x\to 1/2$ in Eq. (\[eq:extreme\]), yielding $$\label{pc_MF}
p_c^{-1} = 1 + \dfrac{2^{q-1} \Gamma(q_0+1)\Gamma(q - q_0 +1) }
{\Gamma(q+1) [q_0- {_2F_1}(1, q_0-q, q_0+1, -1)]} ,$$ This equation generalizes the mean-field result for the $q$-voter model [@qvoterPA], $p_c^{-1} = 1+ \frac{2^{q-1}}{q-1}$, recovered by setting $q_0=q$ in Eq. (\[pc\_MF\]). We note that $p_c$ becomes equal to zero for $q_0\le q_0^*$ where $q_0^*$ is the value that cancels the denominator of the right-hand-side of Eq. (\[pc\_MF\]), namely $q_0^*={_2F_1}(1, q_0^*-q, q_0^*+1, -1)$. This equation yielding the value $q_0^*(q)$ has to be solved numerically.
Below $p<p_c$, $x=\frac12$ becomes unstable and two new stable solutions $x_\pm$, symmetric around $x=\frac12$, appear. These solutions correspond to ordered (O) phases such that the transition at $p=p_c$ between the order and disordered phases is continuous, see panel A) of Fig. \[fig:xst\]. For some values of $(q,q_0)$, Eq. (\[eq:extreme\]) has, besides $x=\frac12$ and $x_\pm$, two additional real roots $x^*_\pm$ symmetric around $x=\frac12$ such that $x_-<x^*_-<1/2<x^*_+<x_+$. Again, the solutions $x_\pm$ are stable, while the $x^*_\pm$ are unstable. These four additional solutions appear in an interval $p_c<p<p^*$, being $p^*(q,q_0)$ a function of $q$ and $q_0$, while only the $x_\pm$ solutions exist for $p<p_c$. The value $p^*$ is found as the solution of $\upsilon'(x_\pm(p^*);p^* )=0$, an equation that has to be solved numerically. When several fixed points coexist, the potential $V(x)$ displays the absolute minimum at $x=1/2$ for $p\in(p_\mathrm{M},p^*)$ and at $x_\pm$ for $p\in(p_c,p_\mathrm{M})$. The first-order phase transition between the ordered and the disordered phases occurs at the Maxwell point $p_\mathrm{M}$ where the potential takes equal values $V(1/2)=V(x_\pm)$ at the minima, while $p_c$ and $p^*$ set the boundaries of the hysteresis cycle, see panel B) of Fig. \[fig:xst\]. Again, it does not seem to be possible to obtain an analytical expression for $p_\mathrm{M}$, and this value has to be obtained numerically.
The limiting condition $p_c=p^*$ necessary for the appearance of the ordered phases can be shown to be equivalent to $\upsilon'(1/2;q,q_0,p)=\upsilon'''(1/2;q,q_0,p)=0$. Eliminating $p$ from these equations one finds that the first-order phase transition only exists for $q_{0} < q_{0}^{-}$ and $q_{0} > q_{0}^{+}$, where the tricritical points are obtained as $$\label{q0c}
q_{0}^{\pm}(q) = \frac{1}{4} \left(5+2q\pm \sqrt{5+4q}\right).$$
Take as an example, $q=12$ that yields $q_0^*\approx 4.75$, $q_{0}^{-} \approx 5.43$ and $q_{0}^{+} \approx 9.07$ indicating that first-order transitions occur for $q_0<q_0^-$ or $q_0>q_0^+$, namely $q_0=2,3,4,5,10,11,12$, while continuous transitions occur for the other values $q_0=6,7,8,9$. For $q_0<q_0^*$, namely $q_0=2,3,4$, the disordered phase is either stable or metastable, but never unstable. For $q=6$, continuous transitions occur for $q_0=3,4,5$ while first-order transitions occur for $q_0=2,6$. In the case $q_{0}=1$, the only observed fixed point is $x=1/2$ (stable) which presents no transitions as a function of $p$.
An alternative representation of these results is presented in the phase diagrams depicted in panel (a) of Fig. \[fig:diagrams\] for $q=6$ (left) and $q=12$ (right). Gray regions are disordered phases where $x=1/2$ is the only stable solution. The lilac region corresponds to the existence of stable symmetric fixed points $x_\pm$, while in the green region $x=1/2$ and $x_\pm$ coexist as stable fixed points. The upper limit of the lilac region is the value $p_c$ and the upper limit of the green region is the value $p^*$. If the lilac region goes directly into the gray, it indicates a continuous transition between order and disordered phases. When the green region goes into gray, it indicates a first-order phase transition occurring at the Maxwell point (not depicted) somewhere in the green region.
![Schematic representation of the possible bifurcation diagrams of the model at the mean-field level. The black solid lines are the stable solutions, i.e. absolute minima of the potential $V(x)$ (macroscopic phases), dashed lines are metastable solutions (relative minima of the potential) and dotted lines are unstable solutions (maxima of the potential). The critical point (or lower limit of the hysteresis cycle), $p_c$, is given by Eq. (\[pc\_MF\]) while the upper limit $p^*$ and the transition (Maxwell) point $p_\mathrm{M}$ are found numerically. Panel A) corresponds to a continuous phase transition occurring for $q_0\in(q_0^-,q_0^+)$. Panel B) corresponds to a first-order phase transition in the cases $q_0<q_0^-$, and $q_0>q_0^+$, and panel C) is a particular case of first-order transition occurring for $q_0<q_0^*$. The figure only shows the interval $x\in(1/2,1)$ and all lines have a mirror image in the interval $x\in(0,1/2)$, not shown for the sake of clarity. []{data-label="fig:xst"}](q0_7 "fig:")\
![Schematic representation of the possible bifurcation diagrams of the model at the mean-field level. The black solid lines are the stable solutions, i.e. absolute minima of the potential $V(x)$ (macroscopic phases), dashed lines are metastable solutions (relative minima of the potential) and dotted lines are unstable solutions (maxima of the potential). The critical point (or lower limit of the hysteresis cycle), $p_c$, is given by Eq. (\[pc\_MF\]) while the upper limit $p^*$ and the transition (Maxwell) point $p_\mathrm{M}$ are found numerically. Panel A) corresponds to a continuous phase transition occurring for $q_0\in(q_0^-,q_0^+)$. Panel B) corresponds to a first-order phase transition in the cases $q_0<q_0^-$, and $q_0>q_0^+$, and panel C) is a particular case of first-order transition occurring for $q_0<q_0^*$. The figure only shows the interval $x\in(1/2,1)$ and all lines have a mirror image in the interval $x\in(0,1/2)$, not shown for the sake of clarity. []{data-label="fig:xst"}](q0_11 "fig:")\
![Schematic representation of the possible bifurcation diagrams of the model at the mean-field level. The black solid lines are the stable solutions, i.e. absolute minima of the potential $V(x)$ (macroscopic phases), dashed lines are metastable solutions (relative minima of the potential) and dotted lines are unstable solutions (maxima of the potential). The critical point (or lower limit of the hysteresis cycle), $p_c$, is given by Eq. (\[pc\_MF\]) while the upper limit $p^*$ and the transition (Maxwell) point $p_\mathrm{M}$ are found numerically. Panel A) corresponds to a continuous phase transition occurring for $q_0\in(q_0^-,q_0^+)$. Panel B) corresponds to a first-order phase transition in the cases $q_0<q_0^-$, and $q_0>q_0^+$, and panel C) is a particular case of first-order transition occurring for $q_0<q_0^*$. The figure only shows the interval $x\in(1/2,1)$ and all lines have a mirror image in the interval $x\in(0,1/2)$, not shown for the sake of clarity. []{data-label="fig:xst"}](q0_4 "fig:")
\(a) Mean-field (MF)\
\(b) Without repetition, in random regular networks\
\
(c) With repetition, in random regular networks\
\
Pair approximation {#sec:PA}
===================
Now, we wish to extend the results of the previous section to more complex network topologies. To this end, let us consider an arbitrary network with $N$ nodes where each node has a different number of connections, or degree, $k$ and we denote by $P(k)$ the degree distribution. An active link is defined as joining two nodes in different states. While in the all-to-all connected network the fraction of active links $\rho$ is related to the density $x$ of nodes in the $\oplus$ state by $\rho=2x(1-x)$, in a general network $\rho(t)$ and $x(t)$ must be considered as independent variables for which one needs to write down evolution equations. The pair approximation (henceforth, PA) is one of the simplest, yet very successful approaches, to derive such a set of equations taking into consideration the underlying network structure. One of the main differences between the all-to-all connectivity and a non-fully connected network is that in the latter the flipping probability depends on whether in the random selection of the $q$ neighbors one allows or not for repetitions of the chosen nodes. Recall that in the large $N$ limit, this difference was irrelevant in the all-to-all connectivity as the probability of repetition was negligible. If the chosen node has $k$ links amongst which $\ell\in(0,k)$ of them are active, the probability that this node flips its state is: $$\label{eq:F}
F(\ell;k,q,q_0,p)=(1-p)f(\ell;k,q,q_0)+p/2,$$ where $$f(\ell;k,q,q_0) \equiv
\left \{ \begin{array}{lr}
\displaystyle \sum_{j = q_0}^{q} {{q}\choose{j}}{{k-q}\choose{\ell-j}}\left/{{k}\choose{\ell}}\right., &\text{(a)} \\[5mm]
\displaystyle \sum_{j = q_0}^{q} {{q}\choose{j}} \,
\left(\frac{\ell}{k} \right)^j \left(1-\frac{\ell}{k}\right)^{q-j}, &\text{(b)}
\end{array}
\right.
\label{eq:f}$$ depending on whether repetition in the selection is forbidden, case (a), or allowed, case (b). In the former case, it is understood that $f(\ell;k,q,q_0)=0$ if $k<q$. The combinatorial number in (a) is also understood to be equal to zero if $\ell<j$.
We focus in this work in a version of the PA based on the single degree distribution $P(k)$ as developed in Refs. [@vazquezNJP; @vazquezJST] and in more detail in Ref. [@antonioNJP]. This is expected to work well for random networks that are not highly clustered nor correlated, otherwise a heterogeneous version of the PA considering joint degree distributions should be used. In this approach, we consider as description variables the fraction of nodes $x_{k}$ with degree $k$ that hold $\oplus$ opinions, and the fraction $\rho$ of active links. Following Ref. [@antonioNJP], the rate equations for the description variables $\lbrace x_{k}, \rho \rbrace$ can be derived as follows: $$\begin{aligned}
\notag
\frac{d\rho}{dt} &=& \frac{2}{\mu}\sum_k\sum_{i=\oplus,\ominus} P(k) P_{i,k} \langle (k-2\ell)F(\ell;k,q,q_0,p) \rangle_{\rho_i},\\
\label{eq:rho_peralta}\\
\label{eq:x_peralta}
\frac{dx_{k}}{dt} &=& -\sum_{i=\oplus,\ominus} S_i P(k) P_{i,k} \langle F(\ell;k,q,q_0,p) \rangle_{\rho_i},\end{aligned}$$ where $S_\oplus=1$, $S_\ominus=-1$, $\mu \equiv \sum_{k} P(k) k$ is the mean degree and $\langle \cdots \rangle_{\rho_i}$ is the average calculated over the binomial probability ${k\choose \ell}\rho_i^\ell(1-\rho_i)^{k-\ell}$. Moreover, we identify $P_{\oplus,k}=x_{k}$, $P_{\ominus,k}=1-x_{k}$, and $\rho_\oplus=P(\ominus|\oplus)=\rho/(2 x_{L})$, $\rho_\ominus=P(\oplus|\ominus)=\rho/[2(1-x_{L})]$, where $x_{L}=\sum_{k} P(k) k x_{k}/\mu$ is the so-called [*link-magnetization*]{}, and $P(i'|i)$ is the conditional probability of selecting a neighbor with opinion $i'$ if the node chosen for updating holds opinion $i$.
It is straightforward to show that a particular solution of Eqs. (\[eq:x\_peralta\]) is $x_{k}(t) = x_{L}(t)$ for all degrees $k$. As shown in [@antonioNJP] the use of this particular solution is appropriate as far as we restrict the analysis to steady-state deterministic values. Using this simplification it is possible to reduce the problem to two closed rate equations for the density of active links $\rho(t)$ and the density of nodes in the $\oplus$ state, $x(t)$, namely: $$\begin{aligned}
\frac{d\rho}{dt} &=& \frac{2}{\mu}\sum_k\sum_{i=\oplus,\ominus}
P(k)\, P_{i} \langle (k-2\ell)F(\ell;k,q,q_0,p) \rangle_{\rho_i},\notag\\
\label{eq:rho}\\
\label{eq:x}
\frac{dx}{dt} &=& -\sum_k\sum_{i=\oplus,\ominus} \frac{k}{\mu} P(k)\,S_i\,P_{i} \langle F(\ell;k,q,q_0,p) \rangle_{\rho_i},\end{aligned}$$ where $P_{\oplus}=x$, $P_{\ominus}=1-x$, $\rho_\oplus=\rho/(2 x)$, $\rho_\ominus=\rho/[2(1-x)]$. We now determine the fixed points of these equations and their stability for different values of $q$ and $q_0$ in the cases with and without repetition in the selection of the neighbors.
Without repetition
------------------
In this case, we replace Eqs. (\[eq:F\],\[eq:f\]a) into the rate equations (\[eq:rho\],\[eq:x\]). It turns out that, due to some cancellation of the combinatorial numbers, the averages $\langle \dots\rangle_{\rho_i}$ over the binomial distributions lead to relatively simple expressions which are linear in $k$, see Appendix A for details. Therefore, the further average over the degree distribution $\sum_k P(k)$ leads to expressions that depend only on the first moment $\langle k \rangle \equiv \mu$, namely $$\begin{aligned}
&&\frac{d \rho}{dt} = p(1-2\rho) +\label{eq:rho_without}\\
&&+ \frac{2(1-p)}{\mu} \biggl[
(1-x) G_2(\rho_\ominus;q,q_0,\mu)+xG_2(\rho_\oplus;q,q_0,\mu)\biggl], \nonumber \\
&& \frac{dx}{dt} = (1-x)G(\rho_\ominus;q,q_0,p) -
x G(\rho_\oplus;q,q_0,p) \, .
\label{eq:without}\end{aligned}$$ Where the function $G$ has been defined in Eqs. (\[eq:G\])-(\[eq:gx\]) and $G_2$ is defined in Appendix A.
It can be easily checked that $x(p)=1/2$ (the disordered phase) is always a fixed point of Eqs. (\[eq:rho\_without\],\[eq:without\]) for any value of $p,\,q,\,q_0$. This is a straightforward consequence of the symmetry of the rates around $x=1/2$. The corresponding fixed point $\rho(p)$, obtained from $\sum_kP(k)\langle (k-2\ell)F(\ell;k,q,q_0,p)\rangle_{\rho(p)}=0$, can not be found analytically for general $(p,q,q_0)$. However, at the special point $p_c(q,q_0)$ where the fixed point $x=1/2$ loses its stability, a general argument [@gleeson] valid for any model with up-down symmetric rates in random regular networks defined by $P(k)=\delta(k-\mu)$, leads to $\rho_c\equiv\rho(p_c)=\frac{\mu-2}{2(\mu-1)}$. The condition to obtain $p_c$ is thus reduced to the single equation $\langle (\mu-2\ell)F(\ell; \mu, q, q_0, p_c)\rangle_{\rho_c} = 0$. This leads to the analytical result: $$\label{eq:pc}
p_c^{-1} = %\dfrac{1}{
1+\dfrac{2^{q-1} \bigl(\frac{\mu}{\mu-2}\bigr)^{q_0}\bigl(\frac{\mu-1}{\mu}\bigr)^{q}
\Gamma(q_0+1)\Gamma(q - q_0 +1) }
{\Gamma(q+1) [q_0- {_2F_1}(1, q_0-q, q_0+1,2/\mu -1)]},$$ that generalizes the result for the $q$-voter model [@qvoterPA] $p_c^{-1} = 1+ \frac{2^{q-1}}{q-1} (\frac{\mu-1}{\mu-2})^q$ and for the MF Eq. (\[pc\_MF\]), recovered in the limit $\mu\to\infty$. Since for the case without repetition the rate equations (\[eq:without\]) depend only on the average degree, the values of $p_c$ and $\rho_c$ are also valid for networks with an arbitrary degree distribution.
Figure \[fig:keffect\] shows the effect of increasing $\mu$, predicted by Eq. (\[eq:pc\]). The examples correspond to (a) simple majority ($q_0=q/2$), (b) two-thirds majority or [*byzantine agreement*]{} ($q_0=2q/3$), and (c) the $q$-voter ($q_0=q$). In case (a), the quantity of elements in the influence group with opposite opinions is equal or greater than the quantity of elements sharing the central node opinion, in case (b) the opposite opinions are at least twice, while in (c) total consensus of opposite opinions is required.
![Critical point $p_c$ vs. $\mu/q$ predicted by Eq. (\[eq:pc\]), in networks with average connectivity $\mu$, for (a) $q_0=q/2$, (b) $q_0=2q/3$ and (c) $q_0=q$, without repetition. Each symbol/color corresponds to a different value of $q$. The shaded region emphasizes the region where the ordered state (in either O or OD phases) is stable when $q=6$ as a reference. []{data-label="fig:keffect"}](pc-vs-kq3.eps)
We first note that to attain an ordered collective state where one of the opinions dominates is facilitated in cases (a) and (b) with respect to the $q$-voter model (c). Moreover, the two-thirds condition is optimal for small size $q$. For increasing $q$, dominance of one of the opinions is facilitated in case (a) and degraded in case (c) while (b) is more robust against changes in group size. In (b) and (c), increasing the relative connectivity $\mu/q$ favors order, but the opposite effect is observed in (a). For $\mu/q\to \infty$, $p_c$ tends to the MF value in all cases, as expected.
The level of the threshold $q_0$ appears to have a larger impact on the collective state of the system than the structure parameter $\mu$. The effect of $q_0$ predicted by Eq. (\[eq:pc\]) is shown in Fig. \[fig:q0effect\], where the critical curves (below which the system orders) are given for the extreme cases $\mu=q$ (solid lines) and MF limit (dashed lines) given by Eq. (\[pc\_MF\]). For increasing $q$, both critical curves tend to collapse, and the optimal value of $q_0$ to attain ordered configurations is shifted from $q_0\simeq 2q/3$ towards $q_0=q/2$, as observed also in Fig. \[fig:keffect\]. Moreover, order occurs for increasing level of independence $p$, but becomes restricted to the vicinity of $q_0/q=1/2$.
![Critical point $p_c$ vs. $q_0/q$, predicted by Eq. (\[eq:pc\]), in networks with average connectivity $\mu/q=1$ (solid lines) and mean field limit (dashed lines). The system orders below the curves. Values of $q$ are indicated in the legend. []{data-label="fig:q0effect"}](pc-vs-q0.eps)
Additional stable fixed points of the rate equations (\[eq:rho\_without\],\[eq:without\]) where $x\ne 1/2$ (ordered phases) can appear depending on the value of the parameters $(p,q,q_0)$ and lead to first-order phase transitions for $q_0<q_0^-$ and $q_0>q_0^+$, in a similar fashion to the one depicted in Fig. \[fig:xst\]. It is possible to obtain an equivalent expression of $q_{0}^{\pm}$ as Eq. (\[q0c\]) in the case of networks, but an additional approximation is needed in Eq. (\[eq:without\]). Following the reasoning of Refs. [@vazquezNJP; @vazquezJST; @antonioChaos; @antonioNJP], at the critical point $p=p_{c}$ we can eliminate the $\rho$ variable as follows $\rho \approx 4 \rho_{c} x (1-x)$. This leads to a closed equation $\frac{dx}{dt}=\tilde{\upsilon}(x;q,q_0,p)$ for the variable $x$ that can be analyzed in the same way than in the mean-field scenario. The result is that the limits for the existence of first-order transitions are: $$\begin{aligned}
\label{q0c_net}
& q_{0}^{\pm}(q,\mu) \approx \frac{1}{2} \bigg[6+(2q-7)\rho_{c} \bigg.\notag\\
&\bigg. \pm \sqrt{16+\rho_{c} \left(-44+4 q (1-\rho_{c})+29 \rho_{c} \right)} \bigg].\end{aligned}$$
In the case $\mu=q=12$, Eq. (\[q0c\_net\]) yields $q_{0}^{-} \approx 5.00$ and $q_{0}^{+} \approx 8.73$. Therefore a discontinuous transition is predicted for $q_0=9$, contrarily to the mean-field case (equivalent to $\mu\to\infty$) where the transition at $q_0=9$ was continuous. The change of nature of the transition from continuous to discontinuous when decreasing the value of $\mu$ occurring for $q_0=9$ is shown in the upper panel of Fig. \[fig:MFlimit\]. Note that, additionally to the MF-like phenomenology (where one, three or five fixed point solutions for $x$ were obtained), it is found that in the case with repetition up to seven fixed points (out of which four are stable) can be found in a narrow range of parameters, see, e.g. Fig. \[fig:MFlimit\] for $q_0=9$ and $\mu=q$. However, these multistability solutions with four stable points appear to be spurious as they are not found in the numerical simulations as we will show in the next sections. Other features of the phase diagram will be discussed in Section \[sec:results\] in the context of the comparison with numerical simulations.
Finally, let us mention that for $\mu\gg q$, we can use the mean-field relation $\rho=2x(1-x)$ to show that both the rate equation (\[eq:without\]) for $x$ and the expression (\[q0c\_net\]) for the limits for the existence of discontinuous transitions, reduce to the corresponding expressions in the mean-field limit Eqs. (\[eq:drift\]) and (\[q0c\]), respectively.
\
With repetition
---------------
When repetition in the choosing of the $q$ neighbors is allowed, we must use, inside the rate Eqs. (\[eq:x\]), the corresponding definition of $f(\ell;k,q,q_0)$ given in Eq. (\[eq:f\]b). For this modality, we did not manage to find general closed expressions for the averages $\langle \dots\rangle_{\rho_i}$ over the binomial distribution. Moreover, while without repetition the binomial averages depend only on the average degree $\mu$, independently of other features of the degree distribution $P(k)$, in the variant with repetition, also negative moments $\langle k^{-m} \rangle$, with $1 \le m \le q-1$, contribute. It is still true that $x=1/2$ is a fixed point and that, as in the case without repetition, in the limit $\mu \gg q$, it is $\rho=2x(1-x)$ and the rate equation for $x$ becomes Eq. (\[eq:drift\]). However, in general, the whole analysis of the existence and stability of the fixed points must be performed numerically. A difficulty of the numerical calculation is that the binomial coefficients appearing in the rate equations take huge values for large $k$, while $\rho^{k}$ takes very small ones, leading to a very delicate numerical evaluation. To circumvent this problem when solving numerically equations ((\[eq:rho\],\[eq:x\])) in order to find the fixed points [@software], we made expansions using the moments of the binomial distribution and the negative moments of the degree distribution, instead of evaluating the bare expressions.
Before making a detailed comparison of the theoretical predictions of the pair approximation with numerical simulations for different underlying networks, let us stress the main differences that appear when allowing or not repetition in the selection of the $q$ neighbors in the influence group.
Comparison “with vs. without” repetition
----------------------------------------
We have found numerically the fixed points of the rate equations (\[eq:rho\],\[eq:x\]) in order to obtain the stability diagrams for networks of increasing connectivity $\mu=m\,q$ (with $m \in \mathbb{N}$ and $q=12$), predicted by the PA, for the rules (a) without and (b) with repetition. The results are displayed in Fig. \[fig:MFlimit\], together with the mean-field results discussed above. In the case without repetition, PA calculations depend only on the average degree $\mu$. With repetition, the PA depends on other moments, then we choose for this example $P(k)=\delta(k-\mu)$, which is the distribution of random regular networks to be considered in detail in another section. The values of $\mu$ are indicated in the figure as multiples of $q$. Values of $q_0$ were selected to illustrate qualitatively different behaviors. For $q_0\simeq q/2$, the order/disorder transition is typically continuous, while for $q_0\simeq q$, it is typically discontinuous. In all cases, the MF result is analytically recovered in the limit of very large $\mu$ and achieved in the simulations performed (with or without repetition) in fully connected networks (not shown). In most cases, the nature of the transitions is not altered when changing $\mu$, except for some cases, as those displayed in the figure.
\(a) Without repetition, the PA prediction remains close to the MF limit when $q_0\simeq q/2$, even for small $\mu$. Note that the critical point slightly increases when $\mu$ decreases, meaning that, surprisingly, the network favors order with respect to the MF. The deviation from MF is more pronounced when $q_0$ approaches $q$. Even the nature of the transition changes from continuous to discontinuous when departing from the MF (bear in mind that the multistability of the case $q_0=9$ around $p\simeq 0.3$ is spurious, absent in simulations although the discontinuous character is still observed, as we will see in Sec. \[sec:results\].)
\(b) Differently, with repetition, a noticeable deviation from MF occurs for $q_0\simeq q/2$. The critical point increases with $\mu$ but the transition remains always continuous. When $q_0\simeq q$, although the variation of the critical point is smaller, there is a change in the nature of the transition, from discontinuous to continuous, when $\mu$ decreases (at $q<\mu <2q$), contrarily to the case without repetition.
Comparison with numerical simulations {#sec:results}
=====================================
Simulations of the noisy threshold $q$-voter opinion dynamical rules were extensively performed. In the following, we compare the predictions of the pair approximation with the numerical simulations over different topologies of the network of contacts. We focus on the case $\mu=q$, which is the one more distant from the MF limit, when repetitions are forbidden.
Random regular networks
-----------------------
\
\
In a random regular network each node has the same number of neighbors, $P(k)=\delta(k-\mu)$. It is a suitable network to implement the dynamics based on groups of size $q$, specially in the case without repetition, which imposes constraints in other topologies (i.e. $P(k<q)=0$). Moreover, it is a random network characterized by average clustering coefficient that decays with network size as $C\sim (\mu-1)^2/(N\mu)$), for which the pair approximation is expected to apply well, allowing comparison of simulations with a theoretical framework. The outcomes are illustrated in Figs. \[fig:12\] and \[fig:12rep\], setting $\mu=q=12$, for the versions without and with repetition, respectively.
In connection with Fig. \[fig:12\], which shows the results without repetition, the agreement between PA prediction (lilac full line) and numerical simulations (light symbols) is very good in all cases, mainly for small $q_0\simeq q/2$, while disagreements are more noticeable when $q_0$ increases. Note, for instance, for $q_0=10$, that the multistability predicted by PA is not observed in numerical simulations. In fact, there is a narrow interval around $p\simeq 0.11$, with 4 stable steady states in the PA curve (while $x=0.5$ is unstable), however, in simulations we observe only tristability (the two more ordered PA states and also the disordered one are reached). In contrast, for $q_0=9$, the transition is also discontinuous, but the state $x=0.5$ is not reached below the transition, in agreement with the PA prediction. When $q_0=q$ (standard $q$-voter) the system is disordered for any $p$, in accordance with the theoretical PA approach, highlighting the crucial role of $q_0$ to promote order.
In the case without “noise" in the rates $p=0$, we observe a freezing of the dynamics, when starting from initial conditions regularly distributed in the interval \[0,1\]. The freezing (zero active links) occurs at earlier times when $q_0$ approaches $q$, giving rise to a spread of points at $p=0$, which become more concentrated when $q_0$ decreases.
When repetition is allowed (see Fig. \[fig:12rep\]), the agreement of PA with numerical simulations is still better than in the case without repetition. Note that the critical point decreases with respect to the MF result when $q_0$ approaches $q/2$, in contrast with the case without repetitions where the discrepancy with MF is stronger when $q_0$ approaches $q$. The system orders more easily with repetitions, except when $q_0\simeq q/2$. Also note that discontinuous transitions are less common than in the case without repetitions.
A summary of the results in random regular networks is presented in panels B) and C) if Fig. \[fig:diagrams\] that shows the phase diagrams obtained through numerical simulations and PA. The critical points were evaluated from the simulations using finite-size scaling analysis with precision $\delta p < 0.002$, as illustrated in the Appendix B. The MF case is also shown for comparison. It is evident that the presence of a network structure has a stronger influence in the case with repetition, where the discontinuous transitions are less common than in fully connected networks. This influence is observed for $\mu=q=6$ and even for $\mu=q=12$ (closer to the MF). Also note that the PA prediction of critical points is very good, which validates our use of analytical expressions to obtain some of the reported results (Figs. \[fig:q0effect\]-\[fig:MFlimit\]).
Other networks
--------------
We also considered networks such as Erdős-Rényi, where $P(k)$ follows a Poisson distribution $P(k)={\rm e}^{-\mu} \mu^k/k!$, and power-law, where $P(k)\sim k^{-a}$.
Recall that in the case without repetition, the pair approximation predicts dependence only on the first moment $\mu$ of the degree distribution. Indeed, in the numerical simulations, we did not detect any significant discrepancy between random regular, Erdős-Rényi and power-law networks, with the same $\mu$, in agreement with PA predictions. However, we cannot guarantee that it is not due to the large values of $\mu$ used to fulfill the condition $k \ge q$ for all $k$. In order to go far away from the MF limit where the effects of repetitions and other features related to structure vanish, we must decrease $\mu/q$. But, then, the probability of finding nodes with degree $k<q$ increases. A lower bound $k_{min} \ge q$ can be imposed, but this restricts the minimal value of $\mu$ in Erdős-Rényi networks or the shape of $P(k)$, like heavy-tailed ones, whose exponent is approximately given by $a \simeq (2\mu-k_{min})/(\mu - k_{min})$. Therefore, the accessible values of the parameters may remain limited to a region where the effects of the degree distribution beyond the first moment are not significant. This is the case of the results reported by Jȩdrzejewski [@qvoterPA] about the standard $q$-voter model without repetitions, keeping $\mu$ much larger than $q$. Then no significant differences were reported for random regular, Erdős-Rényi or even scale-free networks. Only for Watts-Strogatz [@WS] networks with low rewiring probability, hence higher clustering, differences were observed as expected.
\
\
But such limitations are absent if repetition is allowed. Moreover, in such case, PA predicts a dependency on further details of the network structure beyond $\mu$, which makes more motivating the study of the dynamics in different topologies. Then analytical calculations and simulations for Erdős-Rényi and power-law networks, with repetition, were performed.
As shown in Fig. \[fig:RRER\], small, but visible, discrepancies between Erdős-Rényi and random regular results exist for $q=\mu=6$. These differences become negligible if we increase the average degree, for example, to $\mu=3q=18$.
In Erdős-Rényi networks, whose average clustering behaves as $C\sim \mu/N$, there is still a good agreement between simulations and the corresponding analytical results of the pair approximation. Note that a change of the nature of the transition, from discontinuous (in MF, equivalent to $\mu\to \infty$) to continuous, occurs for $q=q_0=6$, in both networks. The same change is observed in random regular networks for $q=12$ and $q_0=10$ (see Figs. \[fig:MFlimit\] and \[fig:12\]). The critical points depart more from the MF value at $q_0=4$, and are always smaller than in MF, consistent with the intuition that the system has more difficulty in ordering when networks have structure, compared to the MF scenario. Note also that the deviation between the critical points for random regular and Erdős-Rényi is not systematic but changes sign, between $q_0=3$ and $6$.
The results for random regular and Erdős-Rényi are very close, even for $\mu=6$, and become even closer when $\mu$ increases (because the MF is approached). In fact, the influence of the negative moments becomes irrelevant when the network loses structure. We expect that the impact of these negative moments increases for power-law decaying $P(k)$. Then we considered $P(k) ={\cal N}/k^a$, for $k_{min}\le k \le k_{max}$, and null otherwise, where ${\cal N}$ is a normalization factor. For given values of the minimal degree $k_{min}$, we adjusted $a$ to obtain the desired average degree $\mu$. Results are shown in Fig. \[fig:PL\]. For $k_{min}=5$, the power-law exponent is large and the results resemble more those of random regular and Erdős-Rényi networks. As $k_{min}$ decreases (hence exponent $a$ decreases), we observe that in all cases the curves separate from random regular but two distinct behaviors are observed. While for $q_0=3$ and 4, the critical point shifts towards the MF limit, for $q_0=5$ and 6, contrarily the critical values decrease, becoming $p_c=0$ at the $q$-voter limit $q_0=q=6$. This may be due to the increasing presence of nodes with low connectivity concomitantly with the fattest tail. Nodes with low connectivity have influence only on few nodes, hampering order, while for highly connected nodes, the requirement of local consensus for change (when $q_0\simeq q$) is more difficult to be attained than for the simple majority (when $q_0\simeq q/2$).
Final remarks {#sec:remarks}
=============
We investigated the effect of network topology on the dynamics of the threshold $q$-voter model with level of independence $p$. The introduction of structure makes relevant the study of the possibility of repetitions in the selection of $q_0$ amongst $q$ opinions. Then, we considered two implementations, either allowing repetition or not.
Analytical results were derived using the PA, for random networks with arbitrary degree distribution $P(k)$. Simulations of the threshold $q$-voter dynamics were obtained mainly on random regular networks, which is more adequate to implement the group dynamics. But, simulations on Erdős-Rényi and power-law networks were also performed.
The structure of random-regular networks has a stronger influence in the case with repetition, where the discontinuous transitions are less common than in fully connected networks (see Fig. \[fig:diagrams\]). The threshold parameter $q_0$ has a crucial interplay with network structure and repetition effects. Without repetition, deviations from MF behavior are more pronounced when $q_0\simeq q$. Differently, with repetition, the maximal deviation from MF occurs for $q_0\simeq q/2$. (See Figs. \[fig:diagrams\] and \[fig:MFlimit\].) Furthermore, depending on the values of $q,q_0$, while transitions which are continuous in the MF can become discontinuous in networks without repetition, the opposite behavior can be observed with repetition (see Fig. \[fig:MFlimit\]). As can be seen in the comparative phase diagrams of Fig. \[fig:diagrams\], and also in the bifurcation diagrams of Fig. \[fig:12rep\], the critical points estimated through the PA are in very good agreement with those obtained through finite-size scaling analysis of the outcomes of simulations in random regular networks. This is specially true when $\mu$ increases approaching the exact MF result where repetition and other issues related to structure become irrelevant. Forbidding repetition, PA works better when $q_0$ approaches $q/2$, and spurious results, such as multistability beyond 3 states are observed in cases with $q_0 \simeq q$ and $\mu\simeq q$. Larger deviations are expected in networks with higher correlations.
For Erdős-Rényi and power-law networks, there is also a good agreement between the simulations and the predictions of the pair approximation. When repetitions are forbidden, we observed agreement of the results of simulations for (Erdős-Rényi and power-law) networks with same values of $\mu$ as predicted by the PA. However, the structures that can be visited are limited by the constraint $k\ge q$. This is in agreement with results reported for the $q$-voter [@qvoterPA], where discrepancies were observed only for networks with large clustering coefficient. Allowing repetitions, the PA predicts dependency on network structure beyond the average degree $\mu$, in accord with simulations. These effects are weak in Erdős-Rényi networks (see Fig. \[fig:RRER\]) but stronger in networks with power-law degree distribution (see Fig. \[fig:PL\]), where long tails are concomitant with high probability of poorly connected nodes to realize small values of $\mu$, far away from the MF. Also in this case $q_0/q$ plays a crucial role. While for $q_0\simeq q/2$, by decreasing the power-law exponent, the critical point increases towards the MF value, for $q_0 \simeq q$, the opposite effect occurs, and the appearance of highly connected nodes promotes disorder.
Acknowledgements {#acknowledgements .unnumbered}
================
Partial financial support has been received from the Agencia Estatal de Investigación (AEI, Spain) and Fondo Europeo de Desarrollo Regional (FEDER, UE), under Project PACSS (RTI2018-093732-B-C21/C22) and the Maria de Maeztu Program for units of Excellence in R&D (MDM-2017-0711). A.F.P. acknowledges support by the Formación de Profesorado Universitario (FPU14/00554) program of Ministerio de Educación, Cultura y Deportes (MECD) (Spain). A.R.V. and C.A. received partial financial support from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). C.A. also acknowledge partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ).
Appendix A. Some mathematical expressions {#appB .unnumbered}
=========================================
We start from the sum: $$\label{app_suma1}
\sum_{j=q_0}^q {q\choose j}a^jb^{q-j}={q\choose q_0}a^{q_0}b^{q-q_0}{_2F_1}\left(1,q_0-q;q_0+1;\frac{-a}{b}\right).$$ From which it follows $$\begin{aligned}
\sum_{j=q_0}^q {q\choose j}x^j(1-x)^{q-j}=g_1(x;q,q_0),\end{aligned}$$ where we have defined the family of functions $$\begin{aligned}
\label{eq:gs}
g_s(x;q,q_0)={q\choose q_0}x^{q_0}(1-x)^{q-q_0}{_2F_1}\left(s,q_0-q;q_0+1;\frac{-x}{1-x}\right).\end{aligned}$$ Taking the derivative of Eq. (\[app\_suma1\]) with respect to $a$ we obtain: $$\label{app_suma2}
\sum_{j=q_0}^q {q\choose j}ja^jb^{q-j}=a\frac{d}{da}\sum_{j=q_0}^q {q\choose j}a^jb^{q-j}.$$ Using Eq. (\[app\_suma1\]), the known properties of the derivative of the hypergeometric function, and replacing $a\to x$, $b\to1-x$ we arrive after some algebra at: $$\begin{aligned}
\sum_{j=q_0}^q {q\choose j}jx^j(1-x)^{q-j}=q_0g_1(x;q,q_0)+g_2(x;q,1+q_0).\end{aligned}$$
In order to perform the averages $\langle F(\ell;k,q,q_0,p)\rangle_{\rho_i}$ and $\langle \ell F(\ell;k,q,q_0,p)\rangle_{\rho_i}$ appearing in Eqs. (\[eq:rho\],\[eq:x\]) after replacing the probabilities Eq. (\[eq:f\]) using the binomial distribution ${k\choose \ell}\rho_i^l(1-\rho_i)^{k-\ell}$, we use the following results: $$\begin{aligned}
\sum_{\ell=0}^k{k-q\choose \ell-j}\rho^{\ell}(1-\rho)^{k-\ell}=\rho^j(1-\rho)^{q-j},\end{aligned}$$ $$\begin{aligned}
\sum_{\ell=0}^k\ell{k-q\choose \ell-j}\rho^{\ell}(1-\rho)^{k-\ell}=\rho^j(1-\rho)^{q-j}\left[j+(k-q)\rho\right].\end{aligned}$$ Replacement of these results in Eq. (\[eq:x\]) and the definition of the average $\mu=\sum_k kP(k)$ leads straightforwardly to Eq. (\[eq:without\]) where $$\begin{aligned}
G_2(z;q,q_0,\mu)\equiv\left[\mu-2q_0-2(\mu-q)z\right]g_1(z;q,q_0)+g_2(z;q,1+q_0).\end{aligned}$$
Appendix B. Evaluation of critical points from numerical simulations {#app .unnumbered}
====================================================================
For constructing the diagrams in Fig. \[fig:diagrams\] for random regular networks, the critical points were determined with accuracy of order $\pm 0.001$. Finite-size scaling analysis was performed, as illustrated in Figs. \[fig:fss-continuous\] and \[fig:fss-discontinuous\], for continuous and discontinuous transitions, respectively.
\
When the O-D transition is continuous (lilac-orange) the critical values and critical exponents were determined by defining the following quantities. The order parameter $$\label{order3}
O = \left\langle \frac{1}{N}\left|\sum_{i=1}^{N} o_{i}\right|\right\rangle ~,$$ where $o_i=2x_i-1$, $\langle\, ...\, \rangle$ denotes a configurational average. It is sensitive to the unbalance between extreme opinions and plays the role of the “magnetization per spin” in magnetic systems. In addition, we also consider the fluctuations $V$ of the order parameter (or “susceptibility”)
$$\label{chi}
V = N\,(\langle O^{2}\rangle - \langle O \rangle^{2})$$
and the Binder cumulant $U$, defined as [@binder] $$\label{eq5}
U = 1 - \frac{\langle O^{4}\rangle}{3\,\langle O^{2}\rangle^{2}} \,.$$
The critical value of $p$, the exponents $\beta$, $\gamma$ and $\nu$, were obtained by means of the usual scaling equations $$\begin{aligned}
\label{eq6}
O(N) & \sim & N^{-\beta/\nu}, \\ \label{eq7}
V(N) & \sim & N^{\gamma/\nu}, \\ \label{eq8}
U(N) & \sim & {\rm constant}, \\ \label{eq9}
p_{c}(N) - p_{c}& \sim & N^{-1/\nu} ~,\end{aligned}$$ that are valid in the vicinity of the transition.
The procedure is illustrated in Fig. \[fig:fss-continuous\], for the case $k=q=12$, $q_0=7$, in random regular networks. We obtained mean-field values of the exponents $\beta=1/2$, $\gamma=1$, $\nu=2$.
The procedure for discontinuous transitions is illustrated in Fig. \[fig:fss-discontinuous\] [@discontinuous1; @discontinuous2]. We considered the same parameters as before. In particular the critical point was obtained by extrapolating the extreme values of $U$ and $V$ to the limit $N\to \infty$.
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---
abstract: 'In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios there is no longer any guarantee that we can correctly estimate generative low-rank matrices of interest, theoretical special cases notwithstanding. Consequently, this paper proposes an alternative empirical Bayesian procedure build upon a variational approximation that, unlike the nuclear norm, retains the same globally minimizing point estimate as the rank function under many useful constraints. However, locally minimizing solutions are largely smoothed away via marginalization, allowing the algorithm to succeed when standard convex relaxations completely fail. While the proposed methodology is generally applicable to a wide range of low-rank applications, we focus our attention on the robust principal component analysis problem (RPCA), which involves estimating an unknown low-rank matrix with unknown sparse corruptions. Theoretical and empirical evidence are presented to show that our method is potentially superior to related MAP-based approaches, for which the convex principle component pursuit (PCP) algorithm [@Candes11] can be viewed as a special case.'
author:
- |
David Wipf\
Visual Computing Group, Microsoft Research Asia\
`[email protected]`
title: |
Non-Convex Rank Minimization via\
an Empirical Bayesian Approach
---
INTRODUCTION {#sec:intro}
============
Recently there has been a surge of interest in finding low-rank decompositions of matrix-valued data subject to some problem-specific constraints [@Babacan11; @Candes08b; @Candes11; @Chandrasekaran11; @Ding11]. While the methodology proposed herein is applicable to a wide range of low-rank applications, we will focus our attention on the robust principal component analysis problem (RPCA) described in [@Candes11]. We begin with the observation model $$Y = X + S + E,$$ where $Y \in \mathbb{R}^{m \times n}$ is an observed data matrix, $X$ is an unknown low-rank component, $S$ is a sparse corruption matrix, and $E$ is diffuse noise, with iid elements distributed as $N(0,\lambda)$. Without loss of generality, we will assume throughout that $n \geq m$. To estimate $X$ and $S$ given $Y$, one possibility is to solve $$\label{eq:rpca_prob}
\min_{X,S} \frac{1}{\lambda} \|Y- X - S \|_{\mathcal{F}}^2 + n \mbox{Rank}\left[X\right] + \|S\|_0,$$ where $\|S\|_0$ denotes the matrix $\ell_0$ norm of $S$, or a count of the nonzero elements in $S$. The reason for the factor of $n$ is to ensure that the rank and sparsity terms are properly balanced, meaning that both terms range from $0$ to $nm$, which reflects our balanced uncertainty regarding their relative contributions to $Y$. Unfortunately, solving (\[eq:rpca\_prob\]) is problematic because the objective function is both discontinuous and non-convex. In general, the only way to guarantee that the global minimum is found is to conduct an exhaustive combinatorial search, which is intractable in all but the simplest cases.
The most common alternative, sometimes called principle component pursuit (PCP) [@Candes11], is to replace (\[eq:rpca\_prob\]) with a convex surrogate such as $$\label{eq:pcp_prob}
\min_{X,S} \frac{1}{\lambda} \|Y- X - S \|_{\mathcal{F}}^2 + \sqrt{n} \| X\|_* + \|S\|_1,$$ where $\| X\|_*$ denotes the nuclear norm of $X$ (or the sum of its singular values). Note that the scale factor from (\[eq:rpca\_prob\]) has been changed from $n$ to $\sqrt{n}$; this is an artifact of the relaxation mechanism balancing the nuclear and $\ell_1$ norms.[^1] A variety of recent theoretical results stipulate when solutions of (\[eq:pcp\_prob\]), particularly in the limiting case where $\lambda \rightarrow 0$ (reflecting the assumption that $E = 0$), will produce reliable estimates of $X$ and $S$ [@Candes11; @Chandrasekaran11]. However, in practice these results have marginal value since they are based upon strong, typically unverifiable assumptions on the support of $S$ and the structure of $X$. In general, the allowable support of $S$ may be prohibitively small and unstructured (possibly random); related assumptions are required for the rank and structure of $X$. Thus there potentially remains a sizable gap between what can be achieved by minimizing the ‘ideal’ cost function (\[eq:rpca\_prob\]) and the convex relaxation (\[eq:pcp\_prob\]).
In Section \[sec:map\_estimation\], as a motivational tool we discuss a simple non-convex scheme based on a variational majorization-minimization approach for locally minimizing (\[eq:rpca\_prob\]). Then in Section \[sec:empirical\_bayes\] we reinterpret this method as *maximum a posteriori* (MAP) estimation and use this perspective to design an alternative empirical Bayesian algorithm that avoids the major shortcomings of MAP estimation. Section \[sec:analysis\] investigates analytical properties of this empirical Bayesian alternative with respect to globally and locally minimizing solutions. Later in Section \[sec:empirical\] we compare a state-of-the-art PCP algorithm with the proposed approach on simulated data as well as a photometric stereo problem.
NON-CONVEX MAJORIZATION-MINIMIZATION {#sec:map_estimation}
====================================
One possible alternative to (\[eq:pcp\_prob\]) is to replace (\[eq:rpca\_prob\]) with a non-convex yet smooth approximation that can at least be locally minimized. In the sparse estimation literature, one common substitution for the $\ell_0$ norm is the Gaussian entropy measure $\sum_i \log |s_i|$, which may also sometimes include a small regularizer to avoid taking the log of zero.[^2] This can be justified (in part) by the fact that $$\sum_i \log |s_i| \equiv \lim_{p \rightarrow 0} \frac{1}{p} \sum_i \left( |s_i|^p - 1 \right) \propto \|{{ \bm s }}\|_0.$$ An analogous approximation suggests replacing the rank penalty with $\log |X X^T|$ as has been suggested for related rank minimization problems [@Mohan10], since $\log |X X^T| = \sum_i \log \sigma_i$, where $\sigma_i$ are the singular values of $XX^T$. This leads to the alternative cost function $$\label{eq:map_prob}
\min_{X,S} \frac{1}{\lambda} \|Y- X - S \|_{\mathcal{F}}^2 + n \log |X X^T| + 2 \sum_{i,j} \log|s_{ij}|.$$ Optimization of (\[eq:map\_prob\]) can be accomplished by a straightforward majorization-minimization approach based upon variational bounds on the non-convex penalty terms [@Jordan99]. For example, because $\log |s|$ is a concave function of $s^2$, it can be expressed using duality theory [@Boyd04] as the minimum of a particular set of upper-bounding lines: $$\label{eq:bound1}
\log s^2 = \min_{\gamma \geq 0} \frac{s^2}{\gamma} + \log \gamma - 1.$$ Here $\gamma$ is a non-negative variational parameter controlling the slope. Therefore, for any fixed $\gamma$ we have a strict, convex upper-bound on the concave log function. Likewise, for the rank term we can use the analogous representation [@Mohan10] $$\label{eq:bound2}
n \log \left| X X^T \right| = \min_{\Psi \succeq 0} \mbox{Trace}\left[X X^T \Psi^{-1} \right] + n\log| \Psi | + C,$$ where $C$ is an irrelevant constant and $\Psi$ is a positive semi-definite matrix of variational parameters.[^3] Combining these bounds we obtain an equivalent optimization problem $$\min_{X,S,\Gamma \geq 0,\Psi \succeq 0 } \frac{1}{\lambda} \|Y- X - S \|_{\mathcal{F}}^2 + \sum_{ij} \left(\frac{s^2_{ij}}{\gamma_{ij}} + \log \gamma_{ij} \right) \nonumber$$ $$\label{eq:map_prob_expand}
\hspace*{2cm} + \hspace*{0.2cm} \mbox{Trace}\left[X X^T \Psi^{-1} \right] + n\log| \Psi |,$$ where $\Gamma$ is a matrix of non-negative elements composed of the variational parameters corresponding to each $s_{ij}$. With $\Gamma$ and $\Psi$ fixed, (\[eq:map\_prob\_expand\]) is quadratic in $X$ and $S$ and can be minimized in closed form via $$\begin{aligned}
\label{eq:means}
{{ \bm x }}_j & \rightarrow & \Psi \left(\Psi + \bar{\Gamma}_j \right)^{-1} {{ \bm y }}_j, \nonumber \\
{{ \bm s }}_j & \rightarrow & \bar{\Gamma}_j \left(\Psi + \bar{\Gamma}_j \right)^{-1} {{ \bm y }}_j, \hspace*{0.2cm} \forall j\end{aligned}$$ where ${{ \bm y }}_j$, ${{ \bm x }}_j$, and ${{ \bm s }}_j$ represent the $j$-th columns of $Y$, $X$, and $S$ respectively and $\bar{\Gamma}_j$ is a diagonal matrix formed from the $j$-th column of $\Gamma$. Likewise, with $X$ and $S$ fixed, $\Gamma$ and $\Psi$ can also be obtained in closed form using the updates $$\begin{aligned}
\Psi & \rightarrow & \frac{1}{n} X X^T \nonumber \\
\gamma_{ij} & \rightarrow & s^2_{ij}, \hspace*{0.2cm} \forall i,j.\end{aligned}$$ While local minimization of (\[eq:map\_prob\]) is clear cut, finding global solutions can still be highly problematic just as before. Whenever any coefficient of $S$ goes to zero, or whenever the rank of $X$ is reduced, we are necessarily at a local minimum with respect to this quantity such that we can never increase the rank or a zero-valued coefficient magnitude in search of the global optimum. (This point will be examined in further detail in Section \[sec:analysis\].) Thus the algorithm may quickly converge to one of a combinatorial number of local solutions.
VARIATIONAL EMPIRICAL BAYESIAN ALGORITHM {#sec:empirical_bayes}
========================================
From a Bayesian perspective we can formulate (\[eq:map\_prob\]) as a MAP estimation problem based on the distributions $$\begin{aligned}
p(Y|X,S) & \propto & \exp \left[-\frac{1}{2\lambda} \|Y- X -S \|_{\mathcal{F}}^2 \right] \nonumber \\
p(X) & \propto & \frac{1}{\left| X X^T \right|^{n/2}} \nonumber \\
p(S) & \propto & \prod_{i,j} \frac{1}{\left|s_{ij}\right|}.\end{aligned}$$ It is then transparent that solving $$\max_{X,S} p(X,S|Y) \equiv \max_{X,S} p(Y|X,S)p(X)p(S)$$ is equivalent to solving (\[eq:map\_prob\]) after an inconsequential $-2\log(\cdot)$ transformation. But as implied above, this strategy is problematic because the effective posterior is characterized by numerous spurious peaks rendering MAP estimation intractable. A more desirable approach would ignore most of these peaks and focus only on regions with significant posterior mass, regions that hopefully also include the posterior mode. One way to accomplish this involves using the bounds from (\[eq:bound1\]) and (\[eq:bound2\]) to construct a simple approximate posterior that reflects the mass of the original $p(X,S|Y)$ sans spurious peaks. We approach this task as follows.
From (\[eq:bound1\]) and (\[eq:bound2\])) we can infer that $$\begin{aligned}
p(S) & \propto & \max_{\Gamma \geq 0} \hat{p}(S;\Gamma) \\
p(X) & \propto & \max_{\Psi \succeq 0} \hat{p}(X;\Psi)\end{aligned}$$ where $$\begin{aligned}
\hat{p}(S;\Gamma) & \triangleq & \exp\left[ -\frac{1}{2} \sum_{ij} \left(\frac{s^2_{ij}}{\gamma_{ij}} + \log \gamma_{ij} \right) \right] \\
\hat{p}(X;\Psi) & \triangleq & \exp\left[ -\frac{1}{2} \mbox{Trace}\left[X X^T \Psi^{-1} \right] - \frac{n}{2} \log| \Psi | \right], \nonumber\end{aligned}$$ which can be viewed as unnormalized approximate priors offering strict lower bounds on $p(S)$ and $p(X)$. We also then obtain a tractable posterior approximation given by $$\label{eq:approx_posterior}
\hat{p}(X,S|Y;\Gamma,\Psi) \triangleq \frac{p(Y|S,X) \hat{p}(S;\Gamma)\hat{p}(X;\Psi)}{\int p(Y|S,X) \hat{p}(S;\Gamma)\hat{p}(X;\Psi) dS dX}.$$ Here $\hat{p}(X,S|Y;\Gamma,\Psi)$ is a Gaussian distribution with closed-form first and second moments, e.g., the means of $S$ and $X$ are actually given by the righthand sides of (\[eq:means\]). The question remains how to choose $\Gamma$ and $\Psi$. With the goal of reflecting the mass of the true distribution $p(Y,X,S)$, we adopt the approach from [@Wipf11] and attempt to solve $$\label{eq:min_error_mass1}
\hspace*{-0.1cm} \min_{\Psi,\Gamma} \int \left|p(Y,X,S) - p(Y|S,X) \hat{p}(S;\Gamma)\hat{p}(X;\Psi) \right| dX dS$$ $$\label{eq:min_error_mass2}
\hspace*{0.0cm} = \min_{\Psi,\Gamma} \int p(Y|S,X) \left|p(X)p(S) - \hat{p}(S;\Gamma)\hat{p}(X;\Psi) \right| dX dS.$$
The basic idea here is that we only care that the approximate priors match the true ones in regions where the likelihood function $p(Y|X,S)$ is significant; in other regions the mismatch is more or less irrelevant. Moreover, by virtue of the strict lower variational bound, (\[eq:min\_error\_mass2\]) reduces to $$\label{eq:min_error_mass1}
\max_{\Psi,\Gamma} \int p(Y|S,X) \hat{p}(S;\Gamma)\hat{p}(X;\Psi) dX dS \equiv \min_{\Psi,\Gamma} \mathcal{L}(\Psi,\Gamma)$$ where $$\label{eq:emp_bayes_cost}
\mathcal{L}(\Psi,\Gamma) \triangleq \sum_{j=1}^n \left[ {{ \bm y }}_j^T \Sigma_{y_j}^{-1} {{ \bm y }}_j + \log \left| \Sigma_{y_j} \right| \right]$$ with $$\label{eq:covariance}
\Sigma_{y_j} \triangleq \Psi + \bar{\Gamma}_j + \lambda I.$$ This $\Sigma_{y_j}$ can be viewed as the covariance of the $j$-th column of $Y$ given fixed values of $\Psi$ and $\Gamma$. To recap then, we need now minimize $\mathcal{L}(\Psi,\Gamma)$ with respect to $\Psi$ and $\Gamma$, and then plug these estimates into (\[eq:approx\_posterior\]) giving the approximate posterior. The mean of this distribution (see below) can then be used as a point estimate for $X$ and $S$. This process is sometimes referred to as *empirical Bayes* because we are using the data to guide our search for an optimal prior distribution [@Berger85; @Tipping01].
UPDATE RULE DERIVATIONS
-----------------------
It turns out that minimization of (\[eq:emp\_bayes\_cost\]) can be accomplished concurrently with computation of the posterior mean leading to simple, efficient update rules. While (\[eq:emp\_bayes\_cost\]) is non-convex, we can use a majorization-minimization approach analogous to that used for MAP estimation. For this purpose, we utilize simplifying upper bounds on both terms of the cost function as has been done for related sparse estimation problems [@Wipf10].
First, the data-dependent term is concave with respect to $\Psi^{-1}$ and $\Gamma^{-1}$ and hence can be expressed as a minimization over $(\Psi^{-1},\Gamma^{-1})$-dependent hyperplanes. With some linear algebra, it can be shown that $$\label{eq:data_decomp}
{{ \bm y }}_j^T \Sigma_{y_j}^{-1} {{ \bm y }}_j = \min_{{{ \bm x }}_j,{{ \bm s }}_j} \frac{1}{\lambda} \|{{ \bm y }}_j - {{ \bm x }}_j - {{ \bm s }}_j \|_{\mathcal{F}}^2 + {{ \bm x }}_j^T \Psi^{-1} {{ \bm x }}_j + \sum_i \frac{s_{ij}^2}{\gamma_{ij}}$$ for all $j$. With a slight abuse of notation, we adopt $X = [{{ \bm x }}_1,\ldots,{{ \bm x }}_n]$ and $S = [{{ \bm s }}_1,\ldots,{{ \bm s }}_n]$ as the variational parameters in (\[eq:data\_decomp\]) because they end up playing the same role as the unknown low-rank and sparse coefficients and provide a direct link to the MAP estimates. Additionally, the ${{ \bm x }}_j$ and ${{ \bm s }}_j$ which minimize (\[eq:data\_decomp\]) turn out to be equivalent to the posterior means of (\[eq:approx\_posterior\]) given $\Psi$ and $\Gamma$ and will serve as our point estimates.
Secondly, for the log-det term, we first use the determinant identity $$\label{eq:det_ident}
\log\left|\Psi + \bar{\Gamma}_j + \lambda I\right| = \log\left|\Psi \right| + \log\left|\bar{\Gamma}_j \right| + \log |A_j| + C,$$ where $$A_j \triangleq \lambda^{-1}
\left[ \begin{array}{cc}
I & I \\
I & I
\end{array} \right] + \left[ \begin{array}{cc}
\Psi^{-1} & {\bf 0} \\
{\bf 0} & \bar{\Gamma}_j^{-1}
\end{array} \right]$$ and $C$ is an irrelevant constant. The term $\log |A_j|$ is jointly concave in both $\Psi^{-1}$ and $\bar{\Gamma}_j^{-1}$ and thus can be bounded in a similar fashion as (\[eq:data\_decomp\]), although a closed-form solution is no longer available. (Other decompositions lead to different bounds and different candidate update rules.) Here we use $$\hspace*{-6cm} \log\left| A_j \right| =$$ $$\hspace*{1.0cm} \min_{U_j,V_j \succeq 0} \mbox{Trace} \left[U_j^T \Psi^{-1} + V_j^T \bar{\Gamma}_j^{-1}\right] - h^*(U_j,V_j) \nonumber$$ where $h^*(U_j,V_j)$ is the concave conjugate function of $\log |A_j|$ with respect to $\Psi^{-1}$ and $\bar{\Gamma}_i^{-1}$. Note that while $h^*(U_j,V_j)$ has no closed-form solution, the minimizing values of $U_j$ and $V_j$ can be computed in closed-form via $$U_j = \frac{\partial \log\left| A_j \right| }{\partial \Psi^{-1}} , \hspace*{1.0cm} V_j = \frac{\partial \log\left| A_j \right| }{\partial \bar{\Gamma}_j^{-1}}.$$
When we drop the minimizations over the variational parameters ${{ \bm x }}_j$, ${{ \bm s }}_j$, $U_j$, and $V_j$ for all $j$, we arrive at a convenient family of upper bounds on the cost function $\mathcal{L}(\Psi,\Gamma)$. Given some estimate of $\Psi$ and $\Gamma$, we can evaluate all variational parameters in closed form (see below). Likewise, given all of the variational parameters we can solve directly for $\Psi$ and $\Gamma$ because now $\mathcal{L}(\Psi,\Gamma)$ has been conveniently decoupled and we need only compute $$\min_{\Psi \succeq 0} \sum_j \left( {{ \bm x }}_j^T \Psi^{-1} {{ \bm x }}_j + \mbox{Trace} \left[U_j^T \Psi^{-1} \right] \right) + n\log\left|\Psi \right|$$ and $$\min_{\gamma_{ij}\geq 0} \frac{s_{ij}^2}{\gamma_{ij}} + \frac{[V_j]_{ii}}{\gamma_{ij}} + \log \gamma_{ij}, \hspace*{0.3cm} \forall i,j.$$
We summarize the overall procedure next.
ALGORITHM SUMMARY {#eq:algorithm}
-----------------
1. Compute $\kappa \triangleq \frac{1}{nm} \| Y \|^2_{\mathcal{F}}$
2. Initialize $\Psi^{(0)} \rightarrow \kappa I$, and $\bar{\Gamma}_j^{(0)} \rightarrow \kappa I$ for all $j$.
3. For the $(k+1)$-th iteration, compute the optimal ${{ \bm x }}_j$ and ${{ \bm s }}_j$ via $${{ \bm x }}_j^{(k+1)} \rightarrow \Psi^{(k)} \left( \Psi^{(k)} + \bar{\Gamma}^{(k)}_j + \lambda I \right)^{-1} {{ \bm y }}_j \nonumber$$ $${{ \bm s }}_j^{(k+1)} \rightarrow \bar{\Gamma}^{(k)} \left( \Psi^{(k)} + \bar{\Gamma}^{(k)}_j + \lambda I \right)^{-1} {{ \bm y }}_j$$
4. Likewise, compute the optimal $U_j$ and $V_j$ via $$U_j^{(k+1)} \rightarrow \Psi^{(k)} - \Psi^{(k)}\left( \Psi^{(k)} + \bar{\Gamma}^{(k)}_j + \lambda I \right)^{-1}\Psi^{(k)} \nonumber$$ $$V_j^{(k+1)} \rightarrow \bar{\Gamma}^{(k)}_j - \bar{\Gamma}^{(k)}_j\left( \Psi^{(k)} + \bar{\Gamma}^{(k)}_j + \lambda I \right)^{-1}\bar{\Gamma}^{(k)}_j$$
5. Update $\Psi$ and $\Gamma$ using the new variational parameters via
$$\Psi^{(k+1)} \rightarrow \frac{1}{n} \sum_j \left[ {{ \bm x }}_j^{(k+1)} \left( {{ \bm x }}_j^{(k+1)} \right)^T + U_j^{(k+1)} \right] \nonumber$$
$$\gamma_{ij}^{(k+1)} \rightarrow \left( s_{ij}^{(k+1)} \right)^2 + \left[ V_j^{(k+1)} \right]_{ii} , \forall i,j$$
6. Repeat steps 3 through 5 until convergence. (Recall that $\bar{\Gamma}_j$ is a diagonal matrix formed from the $j$-th column of $\Gamma$.) This process is guaranteed to reduce or leave unchanged the cost function at each iteration.
Note that if we set $U_j^{(k+1)}, V_j^{(k+1)} \rightarrow 0$ for all $j$, then the algorithm above is guaranteed to (at least locally) minimize the MAP cost function from (\[eq:map\_prob\]). Additionally, for matrix completion problems [@Candes08b], where the support of the sparse errors is known a priori, we need only set each $\gamma_{ij}$ corresponding to a corrupted entry to $\infty$. This limiting case can easily be handled with efficient reduced rank updates.
One positive aspect of this algorithm is that it is largely parameter free. We must of course choose some stopping criteria, such as a maximum number of iterations or a convergence tolerance. (For all experiments in Section \[sec:empirical\] we simply set the maximum number of iterations at 100.) We must also choose some value for $\lambda$, which balances allowable contributions from a diffuse error matrix $E$, although frequently methods have some version of this parameter, including the PCP algorithm. For all of our experiments we simply choose $\lambda = 10^{-6}$ since we did not include an $E$ component consistent with the original RPCA formulation from [@Candes11].
From a complexity standpoint, each iteration of the above algorithm can be computed in $O(m^3n)$, where $n\geq m$, so it is linear in the larger dimension of $Y$ and cubic in the smaller dimension. For many computer vision applications (see Section \[sec:empirical\] for one example), images are vectorized and then stacked, so $Y$ may be $m = $number-of-images by $n = $ number-of-pixels. This is relatively efficient, since the number of images may be on the order of 100 or fewer (see [@LWu10]). However, when $Y$ is a large square matrix, the updates are more expensive to compute. In the future we plan to investigate various approximation techniques to handle this scenario.
As a final implementation-related point, when given access to a priori knowledge regarding the rank of $X$ and/or sparsity of $S$, it is possible to bias the algorithm’s initialization (from Step 1 above) and improve the estimation accuracy. However, we emphasize that for all of the experiments reported in Section \[sec:empirical\] we assumed no such knowledge.
Alternative Bayesian Methods
----------------------------
Two other Bayesian-inspired methods have recently been proposed for solving the RPCA problem. The first from [@Ding11] is a hierarchical model with conjugate prior densities on model parameters at each level such that inference can be performed using a Gibbs sampler. This method is useful in that the $\lambda$ parameter balancing the contribution from diffuse errors $E$ is estimated directly from the data. Moreover, the authors report significant improvement over PCP on example problems. A potential downside of this model is that theoretical analysis is difficult because of the underlying complexity. Additionally, a large number of MCMC steps are required to obtain good estimates leading to a significant computational cost even when $Y$ is small. It also uses an estimate of $\mbox{Rank}[X]$ which can effect the convergence rate of the Gibbs sampler.
A second method from [@Babacan11] similarly employs a hierarchial Bayesian model but uses a factorized mean-field variational approximation for inference [@Attias00]. Note that this is an entirely different type of variational method than ours, relying on a posterior distribution that factorizes over $X$ and $S$, meaning $p(X,S|Y) \approx q(X|Y)q(S|Y)$, where $q(X|Y)$ and $q(S|Y)$ are approximating distributions learned by minimizing a free energy-based cost function.[^4] Unlike our model, this factorization implicitly decouples $X$ and $S$ in a manner akin to MAP estimation, and may potentially produce more locally minimizing solutions (see analysis below). Moreover, while this approach also has a mechanism for estimating $\lambda$, there is no comprehensive evidence given that it can robustly expand upon the range of corruptions and rank that can already be handled by PCP.
To summarize both of these methods then, we would argue that while they offer a compelling avenue for computing $\lambda$ automatically, the underlying cost functions are substantially more complex than PCP or our method rendering more formal analyses somewhat difficult. As we shall see in Sections \[sec:analysis\] and \[sec:empirical\], the empirical Bayesian cost function we propose is analytically principled and advantageous, and empirically outperforms PCP by a wide margin.
ANALYSIS {#sec:analysis}
========
In this section we will examine global and local minima properties of the proposed method and highlight potential advantages over MAP, of which PCP can also be interpreted as a special case. For analysis purposes and comparisons with MAP estimation, it is helpful to convert the empirical Bayes cost function (\[eq:emp\_bayes\_cost\]) into $(X,S)$-space by first optimizing over $U_j$, $V_j$, $\Psi$ and $\Gamma$, leaving only the unknown coefficient matrices $X$ and $S$. Using this process, it is easily shown that the estimates of $X$ and $S$ obtained by globally (or locally) minimizing (\[eq:emp\_bayes\_cost\]) will also globally (or locally) minimize $$\label{eq:x_space_type_ii}
\min_{X,S} \|Y - X - S \|_{\mathcal{F}}^2 + \lambda g_{EB}(X,S;\lambda),$$ where the penalty function is given by $$\hspace*{-6.0cm} g_{EB}(X,S;\lambda) \triangleq$$ $$\hspace*{0.1cm} \min_{\Gamma \geq 0,\Psi \succeq 0} \sum_{i=1}^n {{ \bm x }}_j \Psi^{-1} {{ \bm x }}_j + {{ \bm s }}_j^T \bar{\Gamma}_j^{-1} {{ \bm s }}_j + \log\left|\Psi + \bar{\Gamma}_j + \lambda I \right|. \nonumber$$ Note that the implicit MAP penalty from (\[eq:map\_prob\]) is nearly identical: $$\hspace*{-6.0cm} g_{map}(X,S) \triangleq$$ $$\hspace*{0.6cm} \min_{\Gamma \geq 0,\Psi \succeq 0} \sum_{i=1}^n {{ \bm x }}_j \Psi^{-1} {{ \bm x }}_j + {{ \bm s }}_j^T \bar{\Gamma}_j^{-1} {{ \bm s }}_j + \log\left|\Psi\right| + \log\left|\bar{\Gamma}_j\right|. \nonumber$$ The primary distinction is that in the MAP case the variational parameters separate whereas in empirical Bayesian case they do not. (Note that, as discussed below, we can apply a small regularizer analogous to $\lambda$ to the log terms in the MAP case as well.) This implies that $g_{map}(X,S)$ can be expressed as some $g_{map}(X) + g_{map}(S)$ whereas $g_{EB}(X,S;\lambda)$ cannot. A related form of non-separability has been shown to be advantageous in the context of sparse estimation from overcomplete dictionaries [@Wipf11].
We now examine how this crucial distinction can be beneficial in producing maximally sparse, low-rank solutions that optimize (\[eq:rpca\_prob\]). We first demonstrate how (\[eq:x\_space\_type\_ii\]) mimics the global minima profile of (\[eq:rpca\_prob\]). Later we show how the smoothing mechanism of the empirical Bayesian marginalization can mitigate spurious locally minimizing solutions.
The original RPCA development from [@Candes11] assumes that $E = 0$, which is somewhat easier to analyze. We consider this scenario first.
\[thm:global\_min\_match\] Assume that there exists at least one solution to $Y = X + S$ such that $\mbox{Rank}[X] + \max_j \|{{ \bm s }}_j\|_0 < m$. Then in the limit as $\lambda \rightarrow 0$, any solution that globally minimizes (\[eq:x\_space\_type\_ii\]) will globally minimize (\[eq:rpca\_prob\]).
Proofs will be deferred to a subsequent journal publication. Note that the requirement $\mbox{Rank}[X] + \max_j \|{{ \bm s }}_j\|_0 < m$ is a relatively benign assumption, because without it the matrices $X$ and $S$ are formally unidentifiable even if we are able to globally solve (\[eq:rpca\_prob\]). For $E > 0$, we may still draw direct comparisons between (\[eq:x\_space\_type\_ii\]) and (\[eq:rpca\_prob\]) when we deviate slightly from the Bayesian development and treat $g_{EB}(X,S;\lambda)$ as an abstract, stand-alone penalty function. In this context we may consider $g_{EB}(X,S;\alpha)$, with $\alpha \neq \lambda$ as a more general candidate for estimating RPCA solutions.
\[cor:global\_min\_match2\] Assume that $X_{(\lambda)}$ and $S_{(\lambda)}$ are a unique, optimal solution to (\[eq:rpca\_prob\]) and that $\mbox{Rank}\left[X_{(\lambda)}\right] + \max_j \|\left[{{ \bm s }}_{(\lambda)}\right]_j\|_0 < m$. Then there will always exist some $\lambda'$ and $\alpha'$ such that the global minimum of $$\min_{X,S} \|Y - X - S \|_{\mathcal{F}}^2 + \lambda' g_{EB}(X,S;\alpha'),$$ denoted $X_{(\lambda',\alpha')}$ and $S_{(\lambda',\alpha')}$, satisfies the conditions $\left\|X_{(\lambda',\alpha')} - X_{(\lambda)} \right\| < \epsilon$ and $\left\|S_{(\lambda',\alpha')} - S_{(\lambda)} \right\| < \epsilon$, where $\epsilon$ can be arbitrarily small.
Of course MAP estimation can satisfy a similar property as Theorem \[thm:global\_min\_match\] and Corollary \[cor:global\_min\_match2\] after a minor modification. Specifically, we may define $$\hspace*{-5.8cm} g_{map}(X,S;\alpha) \triangleq$$ $$\hspace*{0.0cm} \min_{\Gamma \geq 0,\Psi \succeq 0} \sum_{j=1}^n {{ \bm x }}_j \Psi^{-1} {{ \bm x }}_j + {{ \bm s }}_j^T \bar{\Gamma}_j^{-1} {{ \bm s }}_j + \log\left|\Psi + \alpha \right| + \log\left|\bar{\Gamma}_j + \alpha \right| \nonumber$$ and then achieve a comparable result to the above using $g_{map}(X,S;\alpha')$. The advantage of empirical Bayes then is not with respect to global minima, but rather with respect to local minima. The separable, additive low-rank plus sparsity penalties that emerge from MAP estimation will always suffer from the following limitation:
\[thm:global\_min\_match2\] Let $S_{ij}^{(a)}$ denote any matrix $S$ with $s_{ij} = a$. Now consider any optimization problem of the form $$\label{eq:general_rpca_map}
\min_{X,S} g_1(X) + g_2(S), \hspace*{0.3cm} \mbox{s.t. } Y = X + S,$$ where $g_1$ is an arbitrary function of the singular values of $X$ and $g_2$ is an arbitrary function of the magnitudes of the elements in $S$. Then to ensure that a global minimum of (\[eq:general\_rpca\_map\]) is a global minimum of (\[eq:rpca\_prob\]) for all possible $Y$, we require that $$\lim_{\epsilon \rightarrow 0} \frac{g_2\left[S_{ij}^{(\epsilon)}\right] - g_2\left[S_{ij}^{(0)}\right]}{\epsilon} = \infty$$ for all $i$ and $j$ and $S$. An analogous condition holds for the function $g_1$.
This result implies that whenever an element of $S$ approaches zero, it will require increasing the associated penalty $g_2(S)$ against an arbitrarily large gradient to escape in cases where this coefficient was incorrectly pruned. Likewise, if the rank of $X$ is prematurely reduced in the wrong subspace, there may be no chance to ever recover since this could require increasing $g_1(X)$ against an arbitrarily large gradient factor. In general, Theorem \[thm:global\_min\_match2\] stipulates that if we would like to retain the same global minimum as (\[eq:rpca\_prob\]) using a MAP estimation-based cost function, then we will necessarily enter an inescapable basin of attraction whenever *either* $\mbox{Rank}[X] < m$ *or* $\|{{ \bm s }}_j\|_0 < m$ for some $j$. This is indeed a heavy price to pay.
Crucially, because of the coupling of low-rank and sparsity regularizers, the penalty function $g_{EB}(X,S;\lambda)$ does not have this limitation. In fact, we only encounter insurmountable gradient barriers when $\mbox{Rank}[X] + \|{{ \bm s }}_j\|_0 < m$ for some $j$, in which case the covariance $\Sigma_{y_j}$ from (\[eq:covariance\]) becomes degenerate (with $\lambda$ small), a much weaker condition. To summarize (emphasize) this point then, MAP can be viewed as heavily dependent on degeneracy of the matrices $\Psi$ and $\Gamma$ in isolation, whereas empirical Bayes is only sensitive to degeneracy of their summation.
This distinction can also be observed in how the effective penalties on $X$ and $S$, as imbedded in $g_{EB}(X,S;\lambda)$, vary given fixed values of $\Gamma$ or $\Psi$ respectively. For example, when $\Psi$ is close to being full rank and orthogonal (such as when the algorithm is initialized), then the implicit penalty on $S$ is minimally non-convex (only slightly concave). In fact, as $\Psi$ becomes large and orthogonal, the penalty converges to a scaled version of the $\ell_1$ norm. In contrast, as $\Psi$ becomes smaller and low-rank, the penalty approaches a scaled version of the $\ell_0$ norm, implying that maximally sparse corruptions will be favored. Thus, we do not aggressively favor maximally sparse $S$ until the rank has already been reduced and we are in the basin of attraction of a good solution. Of course no heuristic annealing strategy is necessary, the transition is handled automatically by the algorithm.
Additionally, whenever $\Psi$ is fixed, the resulting cost function formally decouples into $n$ separate, canonical sparse estimation problems on each ${{ \bm s }}_j$ in isolation. With $\lambda = 0$, it not difficult to show that each of these subproblems is equivalent to solving $$\min_{{{ \bm s }}_j} \| {{ \bm y }}_j - \Phi {{ \bm s }}_j \|_2^2 + g_{EB}({{ \bm s }}_j)$$ where $$g_{EB}({{ \bm s }}_j) \triangleq \min_{{{ \bm \gamma }}_j \geq 0} \sum_{j=1}^n {{ \bm s }}_j^T \bar{\Gamma}_j^{-1} {{ \bm s }}_j + \log\left|\Phi \bar{\Gamma}_j \Phi^T + I \right|$$ is a concave sparsity penalty on ${{ \bm s }}_j$ and $\Phi$ is any matrix such that $\Phi \Psi \Phi^T = I$.[^5] When $\Phi$ is nearly orthogonal, this problem has no local minima and a global solution that approximates the hard thresholding of the $\ell_0$ norm; however, direct minimization of the $\ell_0$ norm will have $2^n$ local minima [@Wipf11]. In contrast, when $\Phi$ is poorly conditioned (with approximately low-rank structure, it has been argued in [@Wipf11] that penalties such as $g_{EB}({{ \bm s }}_j)$ are particularly appropriate for avoiding local minima.
Something similar occurs when $\Gamma$ is now fixed and we evaluate the penalty on $X$. This penalty approaches something like a scaled version of the nuclear norm (less concave) when elements of $\Gamma$ are set to a large constant and it behaves more like the rank function when $\Gamma$ is small. At initialization, when $\Gamma$ is all ones, we are relatively free to move between solutions of various rank without incurring a heavy penalty. Later as $\Gamma$ becomes sparse, solutions satisfying $\mbox{Rank}[X] + \|{{ \bm s }}_j\|_0 < m$ for some $j$ become heavily favored.
As a final point, the proposed empirical Bayesian approach can be implemented with alternative variational bounds and possibly optimized with something akin to simultaneous reweighted nuclear and $\ell_1$ norm minimization, a perspective that naturally suggests further performance analyses such as those applied to sparse estimation in [@Wipf10].
EMPIRICAL RESULTS {#sec:empirical}
=================
This section provides some empirical evidence for the efficacy of our RPCA method. First, we present comparisons with PCP recovering random subspaces from corrupted measurements. Later we discuss a photometric stereo application. In all cases we used the the augmented lagrangian method (ALM) from [@Lin10] to implement PCP. This algorithm has efficient, guaranteed convergence and in previous empirical tests ALM has outperformed a variety of other methods in computing minimum nuclear norm plus $\ell_1$ norm solutions.
RANDOM SUBSPACE SIMULATIONS
---------------------------
Here we demonstrate that the empirical Bayesian algorithm from Section \[eq:algorithm\], which we will refer to as EB, can recovery unknown subspaces from corrupted measurements in a much broader range of operating conditions compared to the convex PCP. In particular, for a given value of $\mbox{Rank}[X]$, our method can handle a substantially larger fraction of corruptions as measured by $\rho = \|S\|_0/(nm)$. Likewise, for a given value of $\rho$, we can accurately estimate an $X$ with much higher rank. Consistent with [@Candes11], we consider the case where $E = 0$, such that all the error is modeled by $S$. This allows us to use the stable, convergent ALM code available online.[^6]
The first experiment proceeds as follows. We generate a low-rank matrix $X$ with dimensions reflective of many computer vision problems: *number-of-images* $\times$ *number-of-pixels*. Here we choose $m = 20$ and $n = 10^4$, the later dimension equivalent to a $100 \times 100$ pixel image. For each trial, we compute an $m \times n$ matrix with iid $\mathcal{N}(0,1)$ entries. We then compute the SVD of this matrix and set all but the $r$ largest singular values to zero to produce a low-rank $X$. $S$ is generated with nonzero entries selected uniformly with probability $\rho = 0.2$. Nonzero values are sampled from an iid Uniform\[-10,10\] distribution. We then compute $Y = X + S$ and try to estimate $X$ and $S$ using the EB and PCP algorithms. Estimation results averaged over multiple trials as $r$ is varied from $1$ to $10$ are depicted in Figure \[fig:empirical\_results\_1\]. We plot normalized mean-squared error (MSE) as computed via $\left< \|X - \hat{X} \|_{\mathcal{F}}^2/\|X \|^2_{\mathcal{F}} \right>$ as well as the average angular error between the estimated and true subspaces. In both cases the average is across 10 trials.
From Figure \[fig:empirical\_results\_1\] we observe that EB can accurately estimate $X$ for substantially higher values of the rank. Interestingly, we are also still able to estimate the correct subspace spanned by columns of $X$ perfectly even when the MSE of estimating $X$ starts to rise (compare Figure \[fig:empirical\_results\_1\](*Top*) with Figure \[fig:empirical\_results\_1\](*Bottom*)). Basically, this occurs because, even if we have estimated the subspace perfectly, reducing the MSE to zero implicitly requires solving a challenging sparse estimation problem for every observation column ${{ \bm y }}_j$. For each column, this problem requires learning $d_j \triangleq \mbox{Rank}[X] + \|{{ \bm s }}_j\|_0$ nonzero entries given only $m = 20$ observations. For our experiment, we can have $d_j > 14$ with high probability for some columns when the rank is high, and thus we may expect some errors in $\hat{S}$ (not shown). However, the encouraging evidence here is that EB is able to keep these corrupting errors at a minimum and estimate the subspace accurately long after PCP has failed. Moreover, if an accurate estimate of $X$ is needed, as opposed to just the correct spanning subspace, then a postprocessing error correction step can potentially be applied to each column individually to improve performance.
The second experiment is similar to the first only now we hold $\mbox{Rank}[X]$ fixed at 4, meaning $\mbox{Rank}[X]/m = 0.2$, and vary the fraction of corrupted entries in $S$ from $0.1$ to 0.8. Figure \[fig:empirical\_results\_2\] shows that EB is again able to drastically expand the range whereby successful estimates are obtained. Notably it is able to recover the correct subspace even with 70% corrupted entries.
As a final comparison, we tested PCP and EB on a $400 \times 400$ observation matrix $Y$ generated as above with a $\mbox{Rank}[X]/m = 0.1$ and $\rho = 0.5$. The estimation results are reported in Table \[tab:400\_by\_400\_result\]. PCP performs poorly since the normalized MSE is above one, meaning we would have been better off simply choosing $\hat{X} = 0$ in this regard. Additionally, the angular error is very near 90 degrees, consistent with the error from a randomly chosen subspace in high dimensions. In contrast, EB provides a reasonably good estimate considering the difficulty of the problem.
\[sample-table\]
[rrrr]{} & &\
\
MSE (norm.) & 1.235 & 0.066\
Angular Error & 88.50 & 5.01\
PHOTOMETRIC STEREO
------------------
Photometric stereo is a method for estimating surface normals of an object or scene by capturing multiple images from a fixed viewpoint under different lighting conditions [@Woodham80]. At a basic level, this methodology assumes a Lambertian object surface, point light sources at infinity, an orthographic camera view, and a linear sensor response function. Under these conditions, it has been shown that the intensities of a vectorized stack of images $Y$ can be expressed as $$\label{eq:lambertian}
Y = L^T N \Upsilon,$$ where $L$ is a $3 \times m$ matrix of $m$ normalized lighting directions, $N$ is a $3 \times n$ matrix of surface normals at $n$ pixel locations, and $\Upsilon$ is a diagonal matrix of diffuse albedo values [@Woodham80]. Thus, if we were to capture at least 3 images with known, linearly independent lighting directions we can solve for $N$ using least squares. Of course in reality many common non-Lambertian effects can disrupt this process, such as specularities, cast or attached shadows, and image noise, etc. In many cases, these effects can be modeled as an additive sparse error term $S$ applied to (\[eq:lambertian\]).
As proposed in [@LWu10], we can estimate the subspace containing $N$ by solving (\[eq:rpca\_prob\]) assuming $X = L^T N \Upsilon$ and $E=0$. The resulting $\hat{X}$, combined with possibly other *a priori* information regarding the lighting directions $L$ can lead to an estimate of $N$. [@LWu10] propose using a modified version of PCP for this task, where a shadow mask is included to simplify the sparse error correction problem. However, in practical situations it may not always be possible to accurately locate all shadow regions in this manner so it is desirable to treat them as unknown sparse corruptions.
For this experiment we consider the synthetic Caesar image from the INRIA 3D Meshes Research Database[^7] with known surface normals. Multiple 2D images with different known lighting conditions can easily be generated using the Cook-Torrance reflectance model [@Cook81]. These images are then stacked to produce $Y$. Because shadows are extremely difficult to handle in general, as a preprocessing step we remove rows of $Y$ corresponding to pixel locations with more than 10% shadow coverage. Specular corruptions were left unfiltered. We tested our algorithm as the number of images, drawn randomly from a batch of 40 total, was varied from 10 to 40. Results averaged across 5 trials are presented in Figure \[fig:empirical\_results\_photo\]. The error metrics have been redefined to accommodate the photometric stereo problem. We now define the normalized MSE as $\left< \|X - \hat{X} \|_{\mathcal{F}}^2/\| X - Y \|^2_{\mathcal{F}} \right>$, which measures how much improvement we obtain beyond just using the observation matrix $Y$ directly. Similarly we normalized the angular error by dividing by the angle between $Y$ and the true $X$ for each trial.
From Figure \[fig:empirical\_results\_photo\] it is clear that EB outperforms PCP in both MSE and angular error, especially when there are fewer images present. It is not entirely clear however why the MSE and angular error are relatively flat for EB as opposed to dropping lower as $m$ increases. Of course these are errors relative to using $Y$ directly to predict $X$, which could play a role in this counterintuitive effect.
CONCLUSIONS {#sec:conclusions}
===========
In this paper we have analyzed a new empirical Bayesian approach for matrix rank minimization in the context of RPCA, where the goal is to decompose a given data matrix into low-rank and sparse components. Using a variational approximation and subsequent marginalization, we ultimately arrive at a novel regularization term that couples low-rank and sparsity penalties in such a way that locally minimizing solutions are effectively smoothed while the global optimum matches that of the ideal RPCA cost function.
[17]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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[^1]: Actually, different scaling factors can be adopted to reflect different assumptions about the relative contributions of low-rank and sparse terms. But we assume throughout that no such knowledge is available.
[^2]: Algorithms and analysis follow through much the same regardless.
[^3]: If $X$ is full rank, then $\Psi$ must be positive definite.
[^4]: Additional factorizations are also included in the model.
[^5]: We have assumed here that $\Psi$ is full rank.
[^6]: <http://perception.csl.uiuc.edu/matrix-rank/>
[^7]: <http://www-roc.inria.fr/gamma/gamma/download/download.php>
|
---
abstract: 'We study the stability of the equilibrium points of a skew product system. We analyze the possibility to construct a Lyapunov function using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case.'
author:
- |
Dan Comănescu\
[Department of Mathematics, West University of Timişoara]{}\
[Bd. V. P\^ arvan, No 4, 300223 Timişoara, Rom\^ ania]{}\
[E-mail addresses: [email protected]]{}
title: Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods
---
[**MSC 2010**]{}: 34D20, 37B25, 70E50, 70H14.
**Keywords:** rigid body, gyrostat, stability.
Introduction
============
A classical problem in mechanics is the problem of the rotation of a heavy rigid body with a fixed point. If the rigid body is also acted by a gyrostatic torque we obtain the problem of the heavy gyrostat with a fixed point. The rotation of a heavy gyrostat is governed by the differential system, see [@birtea-casu-comanescu] and [@gavrilov], $$\label{sistem general}
\left\{\begin{array}{ll}
\dot{\vec{M}}=(\vec{M}+\vec{\mu})\times \mathbb{I}^{-1}\vec{M} +m\vec{\gamma}\times\vec{r}_G \\
\dot{\vec{\gamma}}=\vec{\gamma}\times\mathbb{I}^{-1}\vec{M},\end{array}\right.$$ where $m$ is the mass of the gyrostat, $\vec{r}_G$ is the vector with the initial point in the fixed point $O$ and the terminal point in the center of gravity $G$, $\mathbb{I}$ is the moment of inertia tensor at the point $O$, $\vec{M}$ is the angular momentum vector, $\vec{\gamma}$ is the direction of the gravitational field and $\vec{\mu}$ is the constant vector of gyrostatic moment. In the case when $G$ coincide with the fixed point $O$ we have $\vec{r}_G=\vec{0}$ and we obtain the well-known Zhukovski case, see [@gavrilov]. The mathematical model of the rotation of a heavy gyrostat in the Zhukovski case is $$\label{M}
\left\{\begin{array}{ll}
\dot{\vec{M}}=(\vec{M}+\vec{\mu})\times \mathbb{I}^{-1}\vec{M}\\
\dot{\vec{\gamma}}=\vec{\gamma}\times\mathbb{I}^{-1}\vec{M}.\end{array}\right.$$ This system admits a Hamilton-Poisson formulation, see [@birtea-casu-comanescu] and [@gavrilov]. The Poisson bracket is a modified “$-$” Kirillov-Kostant-Souriau bracket on the dual Lie algebra $e(3)^*$. In the coordinates $(M_1,M_2,M_3,\gamma_1,\gamma_2,\gamma_3)$ the matrix associated with this Poisson bracket has the expression $$\Pi_{\vec{\mu}}(\vec{M},\vec{\gamma})=\left[\begin{array}{cc}
\widehat{\vec{M}+\vec{\mu}} & \widehat{\vec{\gamma}} \\
\widehat{\vec{\gamma}} & \mathcal{O}_3
\end{array}\right],$$ where the matrix $\widehat{v}=\left(
\begin{array}{ccc}
0 & -r & q \\
r & 0 & -p \\
-q & p & 0 \\
\end{array}
\right)$ is defined by the vector $\vec{v}$ which has the components $(p,q,r)$. If “$\,\cdot\,$” is the scalar product, then we have four conserved quantities: the Hamiltonian function $H$, the Casimir functions $C_1$ and $C_2$ and a fourth conserved quantity $F$. These functions are: $$H=\frac{1}{2}\vec{M}\cdot \mathbb{I}^{-1}\vec{M},\,\,\,C_1=\frac{1}{2}\vec{\gamma}\cdot\vec{\gamma},\,\,\,C_2=(\vec{M}+\vec{\mu})\cdot \vec{\gamma},\,\,\,\text{and}\,\,\,F=\frac{1}{2}(\vec{M}+\vec{\mu})\cdot (\vec{M}+\vec{\mu}).$$
The system of the rotation of a heavy gyrostat in the Zhukovski case is a skew product system of differential equations of the form $$\label{semi-decoupled}
\left\{
\begin{array}{ll}
\dot{y}=g(y) \\
\dot{z}=h(y,z)
\end{array}
\right.$$ where $g:D_1\subset \mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ and $h:D_1\times D_2\subset \mathbb{R}^{m}\times \mathbb{R}^{n-m}\rightarrow \mathbb{R}^{n-m}$ are locally Lipschitz functions and $D_1, D_2$ are open sets. By analogy with the case of the rotations of heavy gyrostat in the Zhukovski case, we suppose that we have complete information on the stability for the reduced differential system $$\label{reduced-system}
\dot{y}=g(y).$$ Assume that we have $q\geq 1$ conserved quantities $F_1=F_1(y),...,F_q=F_q(y)$ for . These functions are conserved quantities for . We suppose that we have also $k-q\geq 1$ conserved quantities $F_{q+1}=F_{q+1}(y,z),..., F_{k}=F_{k}(y,z)$ for the system .
In Section 2 we study the stability of the equilibrium points of a differential system of the form . First, we present a result about the spectral stability of equilibrium points of a differential system of the form . Using the theory described in [@comanescu], we prove a theorem about the stability with respect to the set of conserved quantities $\{F_1,...,F_k\}$ of an equilibrium point of a system of the form . An equilibrium point is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$ if and only if we can construct, in a neighborhood of the equilibrium point, a Lyapunov function using the conserved quantities. At the end of this section we study the partial stability of an equilibrium point. We present a theorem about $y$-instability and $z$-instability of an equilibrium point $(y_e,z_e)$ in the case when the equilibrium point belongs to a certain invariant subset.
In Section 3 we present stability results of the uniform rotations of a torque-free gyrostat. These results have been proved in [@comanescu].
In Section 4 we apply the theoretical results proved in Section 2 for studying the stability of the equilibrium states of a heavy gyrostat. We consider the Zhukovski case and suppose that $\vec{\mu}$, the vector of gyrostatic moment, is situated along a principal axis of inertia of the gyrostat. First, we give a list of equilibrium states and their spectral stability. Second, we analyze the stability of the equilibrium states with respect to the set of conserved quantities $\{H,C_1,C_2,F\}$. The results of spectral stability and stability with respect to the set of conserved quantities are used to decide the Lyapunov stability of the equilibrium states. There exists some equilibrium states which are spectrally stable and they are not stable with respect to the set of conserved quantities. For a part of them we can prove the Lyapunov instability by using the exact expression of some solutions of . For the complementary part of them we can not decide the Lyapunov stability or instability. At the end of the section we study the $\vec{M}$-stability and $\vec{\gamma}$-stability of the equilibrium states which is an important practical problem.
The stability of equilibria of a skew product system
====================================================
In this section we study the stability of the equilibrium points ofa differential system of the form . The spectral stability of an equilibrium point is a necessary condition for Lyapunov stability. According to [@marsden-ratiu], pp. 32, an equilibrium point $(y_e,z_e)$ is spectrally stable if the linearization of $(g,h)$ at $(y_e,z_e)$ has all the eigenvalues with nonpositive real parts. For our differential system the linearization at $(y_e,z_e)$ is $$\frac{\partial (g,h)}{\partial (y,z)}(y_e,z_e)=\left(
\begin{array}{cc}
\frac{\partial g}{\partial y}(y_e) & \mathbf{O}_{m,n-m} \\
\frac{\partial h}{\partial y}(y_e,z_e) & \frac{\partial h}{\partial z}(y_e,z_e) \\
\end{array}
\right).$$ If $A$ is a square matrix we note by $Sp(A)$ the set of eigenvalues of $A$.
Let $(y_e,z_e)$ be an equilibrium point of , then we have $Sp(\frac{\partial (g,h)}{\partial (y,z)}(y_e,z_e))=Sp(\frac{\partial g}{\partial y}(y_e))\cup Sp(\frac{\partial h}{\partial z}(y_e,z_e))$.
We remind that if $A$ is a square matrix with $m$ columns, $C$ is a square matrix with $n-m$ columns and $B$ is a matrix with $m$ columns and $n-m$ rows then $\det \left(
\begin{array}{cc}
A & \mathbf{O}_{m,n-m} \\
B & C \\
\end{array}
\right)=\det A\det C.$
In the following we study the stability of equilibrium points with respect to a set of conserved quantities. The stability of an equilibrium point with respect to a set of conserved quantities is a sufficient condition for Lyapunov stability. If an equilibrium point is not stable with respect to a set of conserved quantities, then we cannot construct a Lyapunov function by using this set of conserved quantities. We remind some theoretical considerations, from the paper [@comanescu]. We consider an open set $D\subset\mathbb{R}^n$ and the locally Lipschitz function $f:D\rightarrow \mathbb{R}^n$ which generates the differential equation $$\label{ecuatie-diferentiala}
\dot{x}=f(x)$$ Let $x_e$ be an equilibrium point. A continuous function $V:D\rightarrow \mathbb{R}$ which satisfies $V(x_e)=0$ and $V(x)>0$ for every $x$ in a neighborhood of $x_e$ and $x\neq x_e$ is called a positive definite function in the equilibrium point $x_e$. [*The equilibrium point $x_e$ of is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$ if there exists a continuous function $\Phi:\mathbb{R}^k\rightarrow \mathbb{R}$ such that $x\rightarrow \Phi(F_1,....,F_k)(x)-\Phi(F_1,....,F_k)(x_e)$ is a positive definite function in $x_e$.*]{} In the conditions of the above definition the function $x\rightarrow \Phi(F_1,....,F_k)(x)-\Phi(F_1,....,F_k)(x_e)$ is a Lyapunov function in the equilibrium point $x_e$ and we have the following results.
\[implication-stability\] If the equilibrium point $x_e$ of is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$ then it is stable in the sense of Lyapunov.
\[stability\] Let $x_e$ be an equilibrium point of and $\{F_1,...,F_k\}$ a set of conserved quantities. The following statements are equivalent:
- $x_e$ is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$;
- $x\rightarrow ||(F_1,...,F_k)(x)-(F_1,...,F_k)(x_e)||$ is a positive definite function in $x_e$;
- the system $F_1(x)=F_1(x_e),...,F_k(x)=F_k(x_e)$ has no root besides $x_e$ in some neighborhood of $x_e$.
The Theorem \[stability\] $(iii)$ offer an algebraic method to prove the Lyapunov stability of an equilibrium point. An interesting proof of this implication is presented in [@aeyels]. This method was used in [@comanescu] in the problem of stability of uniform rotations of a torque-free gyrostat.
In this paper we study a differential system of the form .
\[semi-decoupled-stability-theory\] Let $(y_e,z_e)$ be an equilibrium point of .
- Suppose that the equilibrium point $y_e$ of the system is stable with respect to the set of conserved quantities $\{F_1,...,F_q\}$. The following statements are equivalent:
- The equilibrium point $(y_e,z_e)$ of the system is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$;
- The solution $z_e$ of the algebraic system $F_{q+1}(y_e,z)=F_{q+1}(y_e,z_e),...,F_k(y_e,z)=F_k(y_e,z_e)$, with the unknown $z$, is isolated in the set of all the solutions.
- If the equilibrium point $y_e$ of the system is Lyapunov unstable then the equilibrium point $(y_e,z_e)$ of the system is not stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$.
$a)$ The implication $a.1.)\Rightarrow a.2.)$ is trivial.
Suppose that the proposition $a.2.)$ is true and $(y_e,z_e)$ is not stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$. By using the Theorem \[stability\] there exists a sequence $(y_n,z_n)_{n\in\mathbb{N}}$ of solutions of the algebraic system $F_1(y,z)=F_1(y_e,z_e),...,F_k(y,z)=F_k(y_e,z_e)$ such that $(y_n,z_n)\rightarrow (y_e,z_e)$ and $(y_n,z_n)\neq (y_e,z_e)$. All the terms of the sequence $(y_n)_{n\in\mathbb{N}}$ are solutions of the algebraic system $F_1(y)=F_1(y_e),...,F_q(y)=F_q(y_e)$. Because the equilibrium point $y_e$ of the system is stable with respect to the set of conserved quantities $\{F_1,...,F_q\}$ we deduce the existence of an index $n_1\in \mathbb{N}$ such that $y_n=y_e$ if $n\geq n_1$. We have that $z_n$, with $n>n_1$, are solutions of the algebraic system $F_{q+1}(y_e,z)=F_{q+1}(y_e,z_e),...,F_k(y_e,z)=F_k(y_e,z_e)$ and $z_n\neq z_e$. We have a contradiction and consequently we obtain the enounced result.
b\) By using the definition of the Lyapunov stability it is easy to see the following result: “if the equilibrium point $y_e$ of the system is Lyapunov unstable then, the equilibrium point $(y_e,z_e)$ of the system is Lyapunov unstable”. Suppose that the equilibrium point $(y_e,z_e)$ of the system is stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$ and we deduce, by using the Theorem \[implication-stability\], the Lyapunov stability of $(y_e,z_e)$. We have a contradiction and consequently, the equilibrium point $(y_e,z_e)$ of the system is not stable with respect to the set of conserved quantities $\{F_1,...,F_k\}$.
The above theorem does not offer information about the stability with respect to the set of conserved quantities $F_1,...,F_k$ of an equilibrium point $(y_e,z_e)$ in the case when $y_e$ is a Lyapunov stable equilibrium of but it is not stable with respect to the set of conserved quantities $F_1,...,F_q$.
In practical problems, the study of stability of an equilibrium point of a differential system implies the study of partial stability with respect to $y$ and $z$. According to [@rouche], pp. 15, the equilibrium point $(y_e,z_e)$ is $y$-stable (respectively $z$-stable) if for all $\varepsilon>0$ there exists $\delta>0$ such that $||y(t, y_0,z_0)-y_e||<\varepsilon $ (respectively $||z(t, y_0,z_0)-z_e||<\varepsilon$) for all initial conditions $||(y_0,z_0)-(y_e,z_e)||<\delta$ and all $t\geq 0$. About the partial stability of the equilibrium points of a skew product system we have the following properties.
\[partial-semi-decoupled-generality\] Let $(y_e,z_e)$ be an equilibrium point of .
- If $(y_e,z_e)$ is Lyapunov stable, then it is $y$-stable and $z$-stable.
- The equilibrium point $y_e$ of is Lyapunov stable if and only if $(y_e,z_e)$ is $y$-stable.
- If the equilibrium point $y_e$ of is Lyapunov stable, then $(y_e,z_e)$ is Lyapunov stable if and only if it is $z$-stable.
The above theorem does not offer sufficient information about $z$-stability of an equilibrium points. We present a method to prove $z$-instability in a particular case which will appear in our concrete problem. This method assumes the existence of a certain invariant set of the dynamics generated by . The invariant set can by found by the method presented in the paper [@birtea-comanescu]. For the following considerations we suppose that $n=2 m$ and there exists the $C^1$-function $s:\mathbb{R}^{2m}\rightarrow \mathbb{R}^m$ such that the set $$\label{}
\mathcal{M}=\{(y,z)\,|\,s(y,z)=0\}$$ is invariant under the dynamics generated by .
\[instability\] Let $(y_e,z_e)\in \mathcal{M}$ be an equilibrium point of . Suppose that the matrices $\frac{\partial s}{\partial y}(y_e,z_e)$ and $\frac{\partial s}{\partial z}(y_e,z_e)$ are invertible. If $y_e$ is not Lyapunov stable for the dynamics generated by then $(y_e,z_e)$ is not $y$-stable and it is not $z$-stable.
The fact that $(y_e,z_e)$ of is not $y$-stable is a consequence of the Theorem \[partial-semi-decoupled-generality\] $(ii)$.
By using the implicit function theorem we have an open set $U\subset \mathbb{R}^m$ containing $y_e$, an open set $V\subset \mathbb{R}^m$ containing $z_e$ and $C^1$ function $r:U\rightarrow V$ such that $r(y_e)=z_e$ and $\{(y,r(y))\,|\,y\in U\}=\{(y,z)\in U\times V\,|\,s(y,z)=0\}$. If $y\in U$, then $s(y,r(y))=0$ and we deduce that $\frac{\partial r}{\partial y}(y_e)=-[\frac{\partial s}{\partial z}(y_e,z_e)]^{-1}\frac{\partial s}{\partial y}(y_e,z_e)$. By hypotheses we have that $\frac{\partial r}{\partial y}(y_e)$ is an invertible matrix. By local inversion theorem we have that exists an open set $W\subset \mathbb{R}^m$ containing $y_e$ such that the function $(y\in W)\rightarrow ||r(y)-r(y_e)||$ is a positive definite function and we deduce the existence of a constant $k>0$ such that $||r(y)-r(y_e)||\geq k||y-y_e||$ for $y\in W$. By hypotheses $y_e$ is not Lyapunov stable for the dynamics generated by . There exists $\varepsilon>0$, a sequence $y_i\rightarrow y_e$ and a sequence $t_i>0$ such that $B(y_e,\frac{\varepsilon}{k})\subset U$ and $||y(t_i,y_i)-y_e||=\frac{\varepsilon}{k}$. We use the notation $y(\cdot,y_0)$ for the solution of which start from $y_0$ and an analogous notation for the solutions of . We denote by $z_i=r(y_i)$ and by continuity we $z_i\rightarrow z_e$. Because $(y_i,z_i)\in \mathcal{M}$ and $\mathcal{M}$ is invariant under the dynamics generated by we deduce that $(y(t,y_i,z_i),z(t,y_i,z_i))\in \mathcal{M}$ for all $t$. With the above notations we have $$||z(t_i,y_i,z_i)-z_e||=||r(y(t_i,y_i,z_i))-r(y_e)||>k||y(t_i,y_i)-y_e||=\varepsilon,$$ which implies the $z$-instability of the equilibrium point $(y_e,z_e)$.
The stability of the uniform rotations of a torque-free gyrostat {#torque-free gyrostat}
================================================================
The problem of the rotations of a torque-free gyrostat is a classical problem in the theory of the motion of a rigid body (see [@wittenburg]). The study of stability of uniform rotations appears in many papers. In [@puta-comanescu] is studied an equivalent differential system and the results of this paper are used in [@comanescu] to study the stability with respect to a set of conserved quantities and Lyapunov stability of uniform rotations. The torque-free gyrostat equation is $$\label{torque-free}
\dot{\vec{M}}=(\vec{M}+\vec{\mu})\times \mathbb{I}^{-1}\vec{M}$$ This equation has two conserved quantities $H=\frac{1}{2}\vec{M}\cdot \mathbb{I}^{-1}\vec{M}\,\,\text{and}\,\,F=\frac{1}{2}(\vec{M}+\vec{\mu})\cdot (\vec{M}+\vec{\mu}).$ We observe that if we replace the vector of gyrostatic moment $\vec{\mu}$ with $-\vec{\mu}$ then we obtain the equation $$\label{minus-torque-free}
\dot{\vec{M}}=(\vec{M}-\vec{\mu})\times \mathbb{I}^{-1}\vec{M}.$$ A function $\vec{M}:\mathbb{R}\rightarrow \mathbb{R}^3$ is a solution of if and only if the function $-\vec{M}$ is a solution of . We have the same stability properties for an uniform rotations $\vec{M}_e$ of and for an uniform rotations $-\vec{M}_e$ of .
We denote by $I_1,I_2$ and $I_3$ the principal moment of inertia and suppose that $I_1>I_2>I_3$. For the following considerations we use a body frame for which the axes are principal axes of inertia. The matrix of the moment of inertia tensor in this body frame has the form $\mathbb{I}=\hbox{diag}(I_1,I_2,I_3)$. Also, we denote by $\mu_1,\mu_2$ and $\mu_3$ the components of the vector $\vec{\mu}$ with respect to this frame. We remind, see [@comanescu], some results of stability for the uniform rotations in the cases in which the gyrostatic moment is directed in the positive sense along a principal axis of inertia. Using the above observations we have analogous results for the cases in which the gyrostatic moment is directed in the negative sense along a principal axis of inertia.
[**I.**]{} An uniform rotation is Lyapunov stable if and only if it is stable with respect to the set of conserved quantities $\{H,F\}$.
[**II.**]{} For the case $\mu_1>0,\,\mu_2=0,\,\mu_3=0$ we have:
- An uniform rotation of the form $\vec{M}_{1-2}=(q,0,0)$, with $q\in \mathbb{R}$, has the properties:
- It is spectrally stable if and only if $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2}]\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$.
- It is stable with respect to the set of conserved quantities $\{H,F\}$ if and only if $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$.
- An uniform rotation of the form $\vec{M}_{4}=(\frac{I_1}{I_2-I_1}\mu_1, q,0)$, $q\neq 0$, is spectrally unstable.
- An uniform rotation of the form $\vec{M}_{5}=(\frac{I_1}{I_3-I_1}\mu_1, 0, q)$, $q\neq 0$, is stable with respect to the set of conserved quantities $\{H,F\}$.
[**III.**]{} For the case $\mu_1=0,\mu_2>0,\,\mu_3=0$ we have:
- An uniform rotation of the form $\vec{M}_{1-2}=(0,q,0)$ is stable with respect to the set of conserved quantities $\{H,F\}$ if and only if $q\in [-\frac{I_2\mu_2}{I_2-I_3},\frac{I_2\mu_2}{I_1-I_2}]$.
- An uniform rotation of the form $\vec{M}_{3}=(q, \frac{I_2}{I_1-I_2}\mu_2, 0)$ or $\vec{M}_{5}=(0, \frac{I_2}{I_3-I_2}\mu_2, q)$, $q\neq 0$, is stable with respect to the set of conserved quantities $\{H,F\}$.
[**IV.**]{} For the case $\mu_1=0,\,\mu_2=0,\,\mu_3>0$ we have:
- An uniform rotation of the form $\vec{M}_{1-2}=(0,0,q)$, with $q\in \mathbb{R}$, has the properties:
- It is spectrally stable if and only if $q\in (-\infty, \frac{I_3\mu_3}{I_1-I_3}]\cup [\frac{I_3\mu_3}{I_2-I_3},\infty)$.
- It is stable with respect to the set of conserved quantities $\{H,F\}$ if and only if $q\in (-\infty, \frac{I_3\mu_3}{I_1-I_3}]\cup (\frac{I_3\mu_3}{I_2-I_3},\infty)$.
- An uniform rotation of type $\vec{M}_3=(q,0,\frac{I_3}{I_1-I_3}\mu_3)$, $q\neq 0$, is stable with respect to the set of conserved quantities $\{H,F\}$.
- An uniform rotation of type $\vec{M}_4=(0,q,\frac{I_3}{I_2-I_3}\mu_3)$, $q\neq 0$, is spectrally unstable.
The stability of the equilibrium states in the Zhukovski case of integrability of a heavy gyrostat
==================================================================================================
In this section we apply the results from Section 2 for the Zhukovski case of a heavy gyrostat. We consider the cases in which the vector of gyrostatic moment is situated along a principal axis of inertia of the gyrostat. We denote, also, by $I_1,I_2$ and $I_3$ the principal moment of inertia and suppose that $I_1>I_2>I_3$. The matrix of the moment of inertia tensor in our body frame has the form $\mathbb{I}=\hbox{diag}(I_1,I_2,I_3)$ and $\mu_1,\mu_2,\mu_3$ are the components of the vector $\vec{\mu}$.
The equilibrium states
----------------------
The equations of the equilibrium states are $$(\vec{M}+\vec{\mu})\times \mathbb{I}^{-1}\vec{M}=\vec{0},\,\,\,\vec{\gamma}\times \mathbb{I}^{-1}\vec{M}=\vec{0}.$$ It is easy to see that we have the following result.
\[echilibre-M\] The equilibrium states of are characterized by the following properties.
- If $(\vec{M}_e,\vec{\gamma}_e)$ is an equilibrium state for the system then $\vec{M}_e$ is an equilibrium state for .
- For all $\vec{\gamma}_e\in \mathbb{R}^3$, the states $(\vec{0},\vec{\gamma}_e)$ are equilibrium states for .
- If $\vec{M}_e\neq \vec{0}$ is an equilibrium state for then $(\vec{M}_e,\vec{\gamma}_e)$ is an equilibrium state for the system if and only if there exists $\theta \in \mathbb{R}$ such that $\vec{\gamma}_e=\theta\mathbb{I}^{-1}\vec{M}_e$.
According to the results of Section \[torque-free gyrostat\] and using Lemma we obtain:
- If $\mu_2=\mu_3=0$ we have the equilibrium states: $$(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3),\,\,(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0),$$ $$(\vec{M},\vec{\gamma})_4=(\frac{I_1}{I_2-I_1}\mu_1, \beta,0,\frac{\theta\mu_1}{I_2-I_1},\frac{\theta\beta}{I_2},0),\,\,(\vec{M},\vec{\gamma})_5=(\frac{I_1}{I_3-I_1}\mu_1, 0, \beta,\frac{\theta\mu_1}{I_3-I_1},0,
\frac{\theta\beta}{I_3}).$$
- If $\mu_1=\mu_3=0$ we have the equilibrium states: $$(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3),\,\,(\vec{M},\vec{\gamma})_{1-2}=(0,q,0,0,\alpha,0),$$ $$(\vec{M},\vec{\gamma})_3=(\beta, \frac{I_2}{I_1-I_2}\mu_2, 0,\frac{\theta\beta}{I_1},\frac{\theta\mu_2}{I_1-I_2},
0),\,\,\,(\vec{M},\vec{\gamma})_5=(0, \frac{I_2}{I_3-I_2}\mu_2, \beta,0,\frac{\theta\mu_2}{I_3-I_2},
\frac{\theta\beta}{I_3}).$$
- If $\mu_1=\mu_2=0$ we have the equilibrium states: $$(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3),\,\,(\vec{M},\vec{\gamma})_{1-2}=(0,0,q,0,0,\alpha),$$ $$(\vec{M},\vec{\gamma})_3=(\beta, 0, \frac{I_3}{I_1-I_3}\mu_3,\frac{\theta\beta}{I_1},0,
\frac{\theta\mu_3}{I_1-I_3}),\,\,\,(\vec{M},\vec{\gamma})_4=(0, \beta, \frac{I_3}{I_2-I_3}\mu_3,0,\frac{\theta\beta}{I_2},
\frac{\theta\mu_3}{I_2-I_3}).$$
In the above expressions we have $\alpha,\alpha_1,\alpha_2,\alpha_3,\theta\in \mathbb{R}$ and $q,\beta\in\mathbb{R}^*$.
Spectral stability of the equilibrium states
--------------------------------------------
We prove that spectral stability of the equilibrium states is characterized by spectral stability of the uniform rotations of a torque-free gyrostat. Spectral stability of an uniform rotation of the torque-free gyrostat is studied in paper [@comanescu] using some computations which appear in paper [@puta-comanescu]. These results are presented in Section \[torque-free gyrostat\].
\[spectral-stability\] An equilibrium state $(\vec{M}_e,\vec{\gamma}_e)$ of the system is spectrally stable if and only if the uniform rotation $\vec{M}_e$ of the torque-free gyrostat equation is spectrally stable.
The linearization of the right side of the equation is $$\mathcal{L}(\vec{M},\vec{\gamma})=\left(
\begin{array}{cc}
\mathcal{L}_M(\vec{M}) & \mathcal{O}_3 \\
\widehat{\vec{\gamma}}\cdot\mathbb{I}^{-1} & \widehat{\mathbb{I}^{-1}\vec{M}} \\
\end{array}
\right),$$ where $\mathcal{L}_M(\vec{M})$ is the linearization of the right side of . The characteristic polynomial of $\mathcal{L}(\vec{M}_e,\vec{\gamma}_e)$ is $$\mathcal{P}_{(\vec{M}_e,\vec{\gamma}_e)}(t)=-t(t^2+||\mathbb{I}^{-1}\vec{M}_e||^2)\mathcal{P}_{\vec{M}_e}(t),$$ where $\mathcal{P}_{\vec{M}_e}(t)$ is the characteristic polynomial of $\mathcal{L}_M(\vec{M}_e)$.
Stability with respect to the set of conserved quantities
---------------------------------------------------------
In this section we study the stability of an equilibrium state of the system with respect to the set $\mathcal{CQ}=\{H,C_1,C_2,F\}$ of conserved quantities. To apply Theorem \[semi-decoupled-stability-theory\] it is necessary to study the algebraic system $$\label{reduced}
\left\{
\begin{array}{ll}
C_1(\vec{\gamma})= C_1(\vec{\gamma}_0)\\
C_2(\vec{\gamma})= C_2(\vec{\gamma}_0)
\end{array}
\right.
\Leftrightarrow
\left\{
\begin{array}{ll}
\vec{\gamma}\cdot\vec{\gamma}= \vec{\gamma}_0\cdot\vec{\gamma}_0\\
(\vec{M}_0+\vec{\mu})\cdot\vec{\gamma}= (\vec{M}_0+\vec{\mu})\cdot\vec{\gamma}_0
\end{array}
\right.,$$ where $\vec{M}_0,\vec{\gamma}_0\in \mathbb{R}^3$ and $\vec{\gamma}$ is the unknown.
\[solve-system-reduced\] The solutions of the system have the properties:
- if $\vec{\gamma}_0=\vec{0}$, then $\vec{\gamma}_0$ is the unique solution of the system ;
- if $\vec{M}_0=-\vec{\mu}$ and $\vec{\gamma}_0\neq\vec{0}$, then the set of solutions of is $\{\vec{\gamma}\,|\,\vec{\gamma}\cdot\vec{\gamma}= \vec{\gamma}_0\cdot\vec{\gamma}_0\}$;
- if $\vec{M}_0\neq -\vec{\mu}$ and $\vec{\gamma}_0\neq\vec{0}$, then the solution $\vec{\gamma}_0$ is isolated in the set of all the solutions of the system if and only if the vectors $\vec{M}_0+\vec{\mu}$ and $\vec{\gamma}_0$ are linearly dependent.
It is easy to see that the affirmations $(i)$ and $(ii)$ are true.
In the hypotheses of $(iii)$ the solutions are the intersection of a sphere with a plane. The sphere and the plane have a common point $\vec{\gamma}_0$. In the case in which the plane is tangent to the sphere the intersection is formed by the unique point $\vec{\gamma}_0$. In the case in which the plane is not tangent to the sphere, then the intersection is a circle and $\vec{\gamma}_0$ is not an isolated point in the set of all the solutions. We observe that the plane is tangent to the sphere if and only if the vectors $\vec{M}_0+\vec{\mu}$ and $\vec{\gamma}_0$ are linearly dependent and we deduce the enounced result.
\[stability-conserved-quantities-first-axis\] In the case of gyrostatic moment along the first axis of inertia, $\mu_1>0,\,\mu_2=0,\,\mu_3=0$, we have the following results about the equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $\alpha_2=\alpha_3=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$ and $q\neq -\mu_1$ or $q=-\mu_1$ and $\alpha=0$.
- $(\vec{M},\vec{\gamma})_4=(\frac{I_1}{I_2-I_1}\mu_1, \beta,0,\frac{\theta\mu_1}{I_2-I_1},\frac{\theta\beta}{I_2},0)$ is not stable with respect to the set of conserved quantities $\mathcal{CQ}$.
- $(\vec{M},\vec{\gamma})_5=(\frac{I_1}{I_3-I_1}\mu_1, 0, \beta,\frac{\theta\mu_1}{I_3-I_1},0,
\frac{\theta\beta}{I_3})$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
$a)$ Using the results of Section \[torque-free gyrostat\] we have that $(0,0,0)$ is an uniform rotation of the torque-free gyrostat system which is stable with respect to the set of conserved quantities $\{H,F\}$. We can apply Theorem \[semi-decoupled-stability-theory\] (a.) and consequently we have that the equilibrium state $(\vec{M},\vec{\gamma})_0$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if the vector $(\alpha_1,\alpha_2,\alpha_3)$ is an isolated solution of the algebraic system , with $\vec{M}_0=(0,0,0)$ and $\vec{\gamma}_0=(\alpha_1,\alpha_2,\alpha_3)$. By using Lemma \[solve-system-reduced\] $(i)$ and $(iii)$ we have that $(\vec{M},\vec{\gamma})_0$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if the vectors $(\mu_1,0,0)$ and $(\alpha_1,\alpha_2,\alpha_3)$ are linearly dependent and consequently, $\alpha_2=\alpha_3=0$.
b\) If $q\notin (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$, then from Section \[torque-free gyrostat\] we have that the uniform rotation $(q,0,0)$ of the torque-free gyrostat system is Lyapunov unstable. We apply Theorem \[semi-decoupled-stability-theory\] (b.) and consequently, the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}$ is not stable with respect to the set of conserved quantities $\mathcal{CQ}$.
If $q=-\mu_1$, then the uniform rotation $(q,0,0)$ of the torque-free gyrostat system is stable with respect to the set of conserved quantities $\{H,F\}$. If $\alpha=0$, then by Lemma \[solve-system-reduced\] $(i)$ the algebraic system , with $\vec{M}_0=(-\mu_1,0,0)$ and $\vec{\gamma}_0=(0,0,0)$, has a unique solution and we deduce that the equilibrium state $(-\mu_1,0,0,0,0,0)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$. If $\alpha\neq 0$, then by Lemma \[solve-system-reduced\] $(ii)$ the solution $(\alpha,0,0)$ of the algebraic system , with $\vec{M}_0=(-\mu_1,0,0)$ and $\vec{\gamma}_0=(\alpha,0,0)$, is not isolated in the set of all the solutions and we deduce that the equilibrium state $(-\mu_1,0,0,\alpha,0,0)$ is not stable with respect to the set of conserved quantities $\mathcal{CQ}$.
If $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$ and $q\neq -\mu_1$, then from Section \[torque-free gyrostat\] we have that the uniform rotation $(q,0,0)$ of the torque-free gyrostat system is stable with respect to the set of conserved quantities $\{H,F\}$. We observe that the vectors $\vec{M}_{1-2}+\vec{\mu}=(q+\mu_1,0,0)$ and $\vec{\gamma}_{1-2}=(\alpha,0,0)$ are linearly dependent. By using Lemma \[solve-system-reduced\] and Theorem \[semi-decoupled-stability-theory\] (a.) we deduce that the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
c\) From Section \[torque-free gyrostat\] we know that the uniform rotation $(\frac{I_1}{I_2-I_1}\mu_1, \beta,0)$ of the torque-free gyrostat system is Lyapunov unstable. We apply Theorem \[semi-decoupled-stability-theory\] (b.) and consequently, the equilibrium state $(\vec{M},\vec{\gamma})_{4}$ is not stable with respect to the set of conserved quantities $\mathcal{CQ}$.
d\) The uniform rotation $(\frac{I_1}{I_3-I_1}\mu_1, 0, \beta)$ of the torque-free gyrostat system is stable with respect to the set of conserved quantities $\{H,F\}$. We observe that the vectors $\vec{M}_{5}+\vec{\mu}=(\frac{I_1}{I_3-I_1}\mu_1+\mu_1,0,\beta)$ and $\vec{\gamma}_{5}=(\frac{\theta\mu_1}{I_3-I_1},0,
\frac{\theta\beta}{I_3})$ are linearly dependent. By using Lemma \[solve-system-reduced\] and Theorem \[semi-decoupled-stability-theory\] (a.) we deduce that the equilibrium state $(\vec{M},\vec{\gamma})_{5}$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
Using analogous considerations we can prove the following results.
In the case of gyrostatic moment along the second axis of inertia, $\mu_1=0,\,\mu_2>0,\,\mu_3=0$, we have the following results about the equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $\alpha_1=\alpha_3=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,q,0,0,\alpha,0)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $q\in [-\frac{I_2\mu_2}{I_2-I_3},\frac{I_2\mu_2}{I_1-I_2}]$ and $q\neq -\mu_2$ or $q=-\mu_2$ and $\alpha=0$.
- $(\vec{M},\vec{\gamma})_3=(\beta, \frac{I_2}{I_1-I_2}\mu_2, 0,\frac{\theta\beta}{I_1},\frac{\theta\mu_2}{I_1-I_2},
0)$ and $(\vec{M},\vec{\gamma})_5=(0, \frac{I_2}{I_3-I_2}\mu_2, \beta,0,\frac{\theta\mu_2}{I_3-I_2},
\frac{\theta\beta}{I_3})$ are stable with respect to the set of conserved quantities $\mathcal{CQ}$.
In the case of gyrostatic moment along the third axis of inertia, $\mu_1=0,\,\mu_2=0,\,\mu_3>0$, we have the following results about the equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $\alpha_1=\alpha_2=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,0,q,0,0,\alpha)$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$ if and only if $q\in (-\infty, \frac{I_3\mu_3}{I_1-I_3}]\cup (\frac{I_3\mu_3}{I_2-I_3},\infty)$ and $q\neq -\mu_3$ or $q=-\mu_3$ and $\alpha=0$.
- $(\vec{M},\vec{\gamma})_3=(\beta, 0, \frac{I_3}{I_1-I_3}\mu_3,\frac{\theta\beta}{I_1},0,
\frac{\theta\mu_3}{I_1-I_3})$ is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
- $(\vec{M},\vec{\gamma})_4=(0, \beta, \frac{I_3}{I_2-I_3}\mu_3,0,\frac{\theta\beta}{I_2},
\frac{\theta\mu_3}{I_2-I_3})$ is not stable with respect to the set of conserved quantities $\mathcal{CQ}$.
The Lyapunov stability of the equilibrium states
------------------------------------------------
In this section we study the Lyapunov stability of the equilibrium states. To prove the Lyapunov stability of an equilibrium state we use its stability with respect to the set of conserved quantities $\mathcal{CQ}=\{H,C_1,C_2,F\}$. To prove that an equilibrium state is not Lyapunov stable we use its spectrally instability or we apply Theorem \[instability\]. There exists some cases of equilibrium points for which we can not decide the Lyapunov stability.
\[Lyapunov\] Let $(\vec{M}_e,\vec{\gamma}_e)$ be an equilibrium state of .
- In the case of a gyrostatic moment along the first axis of inertia, $\mu_1>0,\,\mu_2=0,\,\mu_3=0$, and if $(\vec{M}_e,\vec{\gamma}_e)\notin \{(-\mu_1,0,0,\alpha,0,0)\,|\, \alpha\neq 0\}$, then the equilibrium state $(\vec{M}_e,\vec{\gamma}_e)$ is Lyapunov stable if and only if it is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
- In the case of a gyrostatic moment along the second axis of inertia, $\mu_1=0,\,\mu_2>0,\,\mu_3=0$, and if $(\vec{M}_e,\vec{\gamma}_e)\notin \{(0,-\mu_2,0,0,\alpha,0)\,|\, \alpha\neq 0\}$, then the equilibrium state $(\vec{M}_e,\vec{\gamma}_e)$ is Lyapunov stable if and only if it is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
- In the case of a gyrostatic moment along the third axis of inertia, $\mu_1=0,\,\mu_2=0,\,\mu_3>0$, and if $(\vec{M}_e,\vec{\gamma}_e)\notin \{(0,0,-\mu_3,0,0,\alpha)\,|\, \alpha\neq 0\}$, then the equilibrium state $(\vec{M}_e,\vec{\gamma}_e)$ is Lyapunov stable if and only if it is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
$(i)$ An equilibrium point of the form $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$, with $\alpha_2=\alpha_3=0$, is Lyapunov stable because it is stable with respect to the set of conserved quantities $\mathcal{CQ}$ (see Theorems \[stability-conserved-quantities-first-axis\] and \[implication-stability\]).
We prove that if $\alpha_2\neq 0$ or $\alpha_3\neq 0$, then the equilibrium state $(\vec{M},\vec{\gamma})_0$ is not Lyapunov stable. We study the solution of the system which verify the initial condition $(q,0,0,\alpha_1,\alpha_2,\alpha_3)$. It is easy to see that $M_1(t)=q,\,M_2(t)=M_3(t)=0$. The components $\gamma_1,\,\gamma_2$ and $\gamma_3$ are the solution of the Cauchy problem $$\dot{\gamma}_1=0,\,\,\dot{\gamma}_2=\frac{q}{I_1}\gamma_3,\,\,\dot{\gamma}_3=-\frac{q}{I_1}\gamma_2,\,\,\gamma_1(0)=\alpha_1,\,\,\gamma_2(0)=\alpha_2,\,\,\gamma_3(0)=\alpha_3.$$ We obtain $$\gamma_1(t)=\alpha_1,\,\,\gamma_2(t)=\alpha_2\cos(\frac{q}{I_1}t)+\alpha_3\sin(\frac{q}{I_1}t),\,\,\gamma_3(t)=\alpha_3\cos(\frac{q}{I_1}t)-\alpha_2\sin(\frac{q}{I_1}t).$$ We observe that for all $q\in \mathbb{R}^*$ we have $[-|\alpha_2|,|\alpha_2|]\subset \gamma_2(\mathbb{R})$ and $[-|\alpha_3|,|\alpha_3|]\subset \gamma_3(\mathbb{R})$ which implies that the equilibrium state $(\vec{M},\vec{\gamma})_0$ is not Lyapunov stable.
If $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$ and $q\neq -\mu_1$ or $q=-\mu_1$ and $\alpha=0$, then by using Theorems \[stability-conserved-quantities-first-axis\] and \[implication-stability\] we deduce that the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0)$ is Lyapunov stable.
If $q\in (-\frac{I_1\mu_1}{I_1-I_2}, -\frac{I_1\mu_1}{I_1-I_3})$ then the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0)$ is not spectrally stable and consequently, it is not Lyapunov stable.
If $q=-\frac{I_1\mu_1}{I_1-I_2}$, then by using the results of Section \[torque-free gyrostat\] we have that the uniform rotation $(q,0,0)$ of the torque-free gyrostat equation is not Lyapunov stable and, consequently the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}=(-\frac{I_1\mu_1}{I_1-I_2},0,0,\alpha,0,0)$ is not Lyapunov stable for the system .
According to Theorem \[spectral-stability\] and the results of Section \[torque-free gyrostat\] the equilibrium state $(\vec{M},\vec{\gamma})_4$ is not spectrally stable and consequently, it is not Lyapunov stable.
An equilibrium state $(\vec{M},\vec{\gamma})_5$ is Lyapunov stable because it is stable with respect to the set of conserved quantities $\mathcal{CQ}$.
Analogously we obtain the results enounced in $(ii)$ and $(iii)$.
In the above theorem and for the case of a gyrostatic moment along the first axis of inertia, $\mu_1>0,\,\mu_2=0,\,\mu_3=0$, the Lyapunov stability of the equilibrium states of the form $(-\mu_1,0,0,\alpha,0,0)$ with $\alpha\neq 0$ cannot be decided. This equilibrium states have the properties:
- They are spectrally stable, see Theorem \[spectral-stability\] and the results of Section \[torque-free gyrostat\].
- They are not stable with respect to the set of conserved quantities $\mathcal{CQ}$ (see Theorem \[stability-conserved-quantities-first-axis\]).
- All of them are Lyapunov stable or all of them are not Lyapunov stable. To prove this affirmation we observe that if $(\vec{M},\vec{\gamma})$ is a solution of and $a$ is a real number, then $(\vec{M},a\cdot\vec{\gamma})$ is also a solution of .
- All o them are Lyapunov stable on the leaf $(H,F,C_1,C_2)(\vec{M},\vec{\gamma})=(H,F,C_1,C_2)(-\mu_1,0,0,\alpha,0,0)$. This leaf is characterized by the relations $\vec{M}=-\vec{\mu}$ and $\vec{\gamma}^2=\alpha^2$. The solution of which start from $\vec{M}_0=-\vec{\mu},\,\,\vec{\gamma}_0=(\gamma_{10},\gamma_{20},\gamma_{30})$ are $\vec{M}(t)=-\vec{\mu}$ and $\gamma_1(t)=\gamma_{10}$, $\gamma_2(t)=\gamma_{20}\cos(\frac{\mu_1}{I_1}t)-\gamma_{30}\sin(\frac{\mu_1}{I_1}t)$, $\gamma_3(t)=\gamma_{30}\cos(\frac{\mu_1}{I_1}t)
+\gamma_{20}\sin(\frac{\mu_1}{I_1}t)$.
For the case of a gyrostatic moment along the second axis of inertia, $\mu_1=0,\,\mu_2>0,\,\mu_3=0$ the Lyapunov stability of the equilibrium states of the form $(0,-\mu_2,0,0,\alpha,0)$ with $\alpha\neq 0$ cannot be decided.
For the case of a gyrostatic moment along the second axis of inertia, $\mu_1=0,\,\mu_2=0,\,\mu_3>0$ the Lyapunov stability of the equilibrium states of the form $(0,0,-\mu_3,0,0,\alpha)$ with $\alpha\neq 0$ cannot be decided.
Partial stability of the equilibrium states
-------------------------------------------
A practical interest is for the following problems: “is an equilibrium state $\vec{M}$-stable?” or “is it $\vec{\gamma}$-stable?”. The problem of $\vec{M}$-stability of an equilibrium state is solved by using Theorem \[partial-semi-decoupled-generality\] and the results of the paper [@comanescu] (see Section \[torque-free gyrostat\]). In this section we study $\vec{\gamma}$-stability of an equilibrium state of the system . We use Theorem 2.3 from [@birtea-comanescu] for the system and the vectorial conserved quantity $(C_1,C_2,F)$. We have that the set $$\mathcal{M}=\{(\vec{M},\vec{\gamma})\,|\,\text{rank} \left(
\begin{array}{c}
\nabla C_1 \\
\nabla C_2 \\
\nabla F \\
\end{array}
\right)
(\vec{M},\vec{\gamma})=2\}$$ is an invariant set for the dynamics generated by . By a direct computation we have that $$\nabla C_1(\vec{M},\vec{\gamma})=(\vec{0},\vec{\gamma}),\,\,\nabla C_2(\vec{M},\vec{\gamma})=(\vec{\gamma},\vec{M}+\vec{\mu}),\,\,\nabla F(\vec{M},\vec{\gamma})=(\vec{M}+\vec{\mu},\vec{0}),$$ and consequently, we have $$\mathcal{M}=\{(\vec{M},\vec{\gamma})\,|\,\vec{M}+\vec{\mu} \,\,\text{and}\,\,\vec{\gamma}\,\,\text{are linear dependent}\}\setminus \{(-\vec{\mu},\vec{0})\}.$$ The dynamics in the invariant set $\mathcal{M}$ has the following property.
\[M-invariant\] If the point $(\vec{M}_0,\vec{\gamma}_0)\in \mathcal{M}$ satisfy $\vec{M}_0+\vec{\mu}=\delta\vec{\gamma}_0$ with $\delta\in \mathbb{R}^*$, then the set $$\mathcal{M}_{(\vec{M}_0,\vec{\gamma}_0)}=\{(\vec{M},\vec{\gamma})\,|\,\vec{M}-\delta\vec{\gamma}+\vec{\mu}=\vec{0}\}$$ is an invariant set of the dynamics generated by .
We observe that $\vec{\gamma}_0\neq \vec{0}$ and $(\vec{M}_0,\vec{\gamma}_0)\in\mathcal{M}_{(\vec{M}_0,\vec{\gamma}_0)}\subset \mathcal{M}$. A solution $(\vec{M}(t),\vec{\gamma}(t))$ of which start from $\mathcal{M}_{(\vec{M}_0,\vec{\gamma}_0)}$ verify $\vec{M}(t)+\vec{\mu}=\delta\vec{\gamma}(t)$. By the construction of $\mathcal{M}$ we have that there exists a function $h$ such that $\vec{M}(t)+\vec{\mu}=h(t)\vec{\gamma}(t)$. We deduce $(\vec{M}(t)+\vec{\mu})\cdot\vec{\gamma}(t)=h(t)\vec{\gamma}^2(t)$ and using the conserved quantities $C_1$ and $C_2$ we obtain $h(t)=\frac{C_2(\vec{M}(0),\vec{\gamma}(0)}{C_1(\vec{M}(0),\vec{\gamma}(0))}=\delta.$
In the case of a gyrostatic moment along the first axis of inertia, $\mu_1>0,\,\mu_2=0,\,\mu_3=0$, we have the following results about the $\vec{\gamma}$-stability of equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is $\vec{\gamma}$-stable if and only if $\alpha_2=\alpha_3=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0)$ is $\vec{\gamma}$-stable if $\alpha=0$ or $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$ and $q\neq -\mu_1$.
- $(\vec{M},\vec{\gamma})_{1-2}=(q,0,0,\alpha,0,0)$ is not $\vec{\gamma}$-stable if $\alpha\neq 0$ and $q\in [-\frac{I_1\mu_1}{I_1-I_2}, -\frac{I_1\mu_1}{I_1-I_3})$.
- $(\vec{M},\vec{\gamma})_4=(\frac{I_1}{I_2-I_1}\mu_1, \beta,0,\frac{\theta\mu_1}{I_2-I_1},\frac{\theta\beta}{I_2},0)$ is $\vec{\gamma}$-stable if and only if $\theta=0$.
- $(\vec{M},\vec{\gamma})_5=(\frac{I_1}{I_3-I_1}\mu_1, 0, \beta,\frac{\theta\mu_1}{I_3-I_1},0,
\frac{\theta\beta}{I_3})$ is $\vec{\gamma}$-stable.
$a)$ The uniform rotation $(0,0,0)$ is Lyapunov stable for the system . Using Theorem \[partial-semi-decoupled-generality\] we deduce that $(\vec{M},\vec{\gamma})_0$ is Lyapunov stable if and only if it is $\vec{\gamma}$-stable. Using Theorem \[Lyapunov\] we obtain the enounced results.
[*b.1)*]{} If $\alpha=0$, then we obtain the $\vec{\gamma}$-stability of $(\vec{M},\vec{\gamma})_{1-2}$ by definition and using the conserved quantity $C_1$. If $q\in (-\infty, -\frac{I_1\mu_1}{I_1-I_2})\cup [-\frac{I_1\mu_1}{I_1-I_3},\infty)$ and $q\neq -\mu_1$, then we obtain the $\vec{\gamma}$-stability of $(\vec{M},\vec{\gamma})_{1-2}$ by using Theorem \[Lyapunov\].
[*b.2)*]{} For the cases in which $\alpha\neq 0$ and $q\in [-\frac{I_1\mu_1}{I_1-I_2}, -\frac{I_1\mu_1}{I_1-I_3})$ the equilibrium state $(\vec{M},\vec{\gamma})_{1-2}$ is not $\vec{M}$-stable. We observe that $(\vec{M},\vec{\gamma})_{1-2}\in \mathcal{M}$ and by using Lemma \[M-invariant\] we deduce that $\mathcal{M}_{(\vec{M}_{1-2},\vec{\gamma}_{1-2})}$ is an invariant set of the dynamics generated by . We can apply Theorem \[instability\] for the function $s(\vec{M},\vec{\gamma})=\vec{M}-\frac{\alpha}{q+\mu_1}\vec{\gamma}+\vec{\mu}$ to deduce the $\vec{\gamma}$-instability of the equilibrium state.
[*c)*]{} The $\vec{\gamma}$-stability of $(\vec{M},\vec{\gamma})_4$ for $\theta=0$ is proved by using the conserved quantity $C_1$. If $\theta\neq 0$ we have that the equilibrium state $(\vec{M},\vec{\gamma})_4$ is not $\vec{M}$-stable. We use the Lemma \[M-invariant\] to decide that the equilibrium state $(\vec{M},\vec{\gamma})_4$ is not $\vec{\gamma}$-stable.
[*d)*]{} The enounced result is the consequence of Theorem \[Lyapunov\].
In the case of a gyrostatic moment along the first axis of inertia, $\mu_1>0,\,\mu_2=0,\,\mu_3=0$, we do not have a result for the $\vec{\gamma}$-stability of the equilibrium states $(-\mu_1,0,0,\alpha,0,0)$ with $\alpha\neq 0$.
In the case of gyrostatic moment along the second axis of inertia, $\mu_1=0,\,\mu_2>0,\,\mu_3=0$, we have the following results about the equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is $\vec{\gamma}$-stable if and only if $\alpha_1=\alpha_3=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,q,0,0,\alpha,0)$ is $\vec{\gamma}$-stable if $\alpha=0$ or $q\in [-\frac{I_2\mu_2}{I_2-I_3},\frac{I_2\mu_2}{I_1-I_2}]$ and $q\neq -\mu_2$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,q,0,0,\alpha,0)$ is not $\vec{\gamma}$-stable if $\alpha\neq 0$ and $q\notin [-\frac{I_2\mu_2}{I_2-I_3},\frac{I_2\mu_2}{I_1-I_2}]$.
- $(\vec{M},\vec{\gamma})_3=(\beta, \frac{I_2}{I_1-I_2}\mu_2, 0,\frac{\theta\beta}{I_1},\frac{\theta\mu_2}{I_1-I_2},
0)$ is $\vec{\gamma}$-stable.
- $(\vec{M},\vec{\gamma})_5=(0, \frac{I_2}{I_3-I_2}\mu_2, \beta,0,\frac{\theta\mu_2}{I_3-I_2},
\frac{\theta\beta}{I_3})$ is $\vec{\gamma}$-stable.
We do not have a result for the $\vec{\gamma}$-stability of the equilibrium states $(0,-\mu_2,0,0,\alpha,0)$ with $\alpha\neq 0$.
In the case of gyrostatic moment along the third axis of inertia, $\mu_1=0,\,\mu_2=0,\,\mu_3>0$, we have the following results about the equilibrium states.
- $(\vec{M},\vec{\gamma})_0=(0,0,0,\alpha_1,\alpha_2,\alpha_3)$ is $\vec{\gamma}$-stable if and only if $\alpha_1=\alpha_2=0$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,0,q,0,0,\alpha)$ is $\vec{\gamma}$-stable if $\alpha=0$ or $q\in (-\infty, \frac{I_3\mu_3}{I_1-I_3}]\cup (\frac{I_3\mu_3}{I_2-I_3},\infty)$ and $q\neq -\mu_3$.
- $(\vec{M},\vec{\gamma})_{1-2}=(0,0,q,0,0,\alpha)$ is not $\vec{\gamma}$-stable if $\alpha\neq 0$ and $q\in (\frac{I_3\mu_3}{I_1-I_3}, \frac{I_3\mu_3}{I_2-I_3}]$.
- $(\vec{M},\vec{\gamma})_3=(\beta, 0, \frac{I_3}{I_1-I_3}\mu_3,\frac{\theta\beta}{I_1},0,
\frac{\theta\mu_3}{I_1-I_3})$ is $\vec{\gamma}$-stable.
- $(\vec{M},\vec{\gamma})_4=(0, \beta, \frac{I_3}{I_2-I_3}\mu_3,0,\frac{\theta\beta}{I_2},
\frac{\theta\mu_3}{I_2-I_3})$ is $\vec{\gamma}$-stable if and only if $\theta=0$.
We do not have a result for the $\vec{\gamma}$-stability of the equilibrium states $(0,0,-\mu_3,0,0,\alpha)$ with $\alpha\neq 0$.
[**Acknowledgments.**]{} This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-RU-TE-2011-3-0006.
[99]{} Birtea P., Caşu I., Comănescu D. Hamilton-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque. [*Physics Letters A.*]{} 2011; [**375**]{}: 3941-3945. DOI: 10.1016/j.physleta.2011.08.075 Gavrilov L. Non-integrability of the equations of heavy gyrostat. [*Compositio Mathematica*]{}. 1992; [**3**]{}: 257-291. Comănescu D. Stability problem for the torque-free gyrostat by using algebraic methods. http://arxiv.org/PS$\_$cache/arxiv/pdf/1108/1108.6148v1.pdf Marsden J.E., Ratiu T.S. [*Introduction to Mechanics and Symmetry, Second Edition*]{}. Springer. 1999 Aeyels D. On stabilization by means of the Energy-Casimir method. [*Systems $\&$ Control Lett.*]{} 1992; [**18**]{}: 325-328. Rouche N., Habets P., Laloy M. [*Stability Theory by Liapunov’s Direct Method.*]{} Springer-Verlag. 1977. Birtea P., Comănescu D. Invariant critical sets of conserved quantities. [*Chaos, Solitons $\&$ Fractals*]{}. 2011; [**44**]{} (9): 693-701. Wittenburg J. [*Dynamics of Multibody Systems, Second Edition.*]{} Springer, 2008. Puta M., Comănescu D. On the Rigid Body with Three Linear Controls. [*Analele Univ. Timişoara, Ser. Matematică-Informatică*]{}. 1997; [**XXXV**]{} (1): 63-74.
|
---
abstract: 'We present the crumpling transition in three-dimensional Euclidian space of dynamically triangulated random surfaces with edge extrinsic curvature and fixed topology of a sphere as well as simulations of a dynamically triangulated torus. We used longer runs than previous simulations and give new and more accurate estimates of critical exponents. Our data indicate a cusp singularity in the specific heat. The transition temperature, as well as the exponents are topology dependent.'
author:
- |
\
\
Christian Münkel and Dieter W. Heermann\
\
date: |
[ *Institut für theoretische Physik\
Universität Heidelberg\
Philosophenweg 19\
6900 Heidelberg\
and\
Interdisziplinäres Zentrum\
für wissenschaftliches Rechnen\
der Universität Heidelberg\
Germany* ]{}
title: The Crumpling Transition of Dynamically Triangulated Random Surfaces
---
=0.75cm =0.5cm
PACS: $05.90$, $64.60A$, $64.60F$, $68.10$, $87.20C$, $12.90$
Introduction
============
In this paper we are concerned with the statistical mechanics of surfaces. A possible approach in space dimensions $D>1$ is to discretize the surface using a triangulation \[1-4\]. For this we need to specify the Hamiltonian ${\cal H}$. The surface $S$ is replaced by a simplical triangulation $T$, specified by the number of nodes $N$, of links and triangles, and the $X$-coordinate field by the coordinates $X$ of the nodes. The metrical fluctuations of the manifold are modeled by summing over triangulations induced by link-flips \[5-7\]. The Hamiltonian is now choosen, such that the partition function is not dominated by configurations with spikes. In order to surpress these spikes one adds a term with extrinsic curvature. As a function of the extrinsic curvature the model shows a transition (crumpling transition) at finite rigidity of the surface \[8-18\]. In a previous paper, we discussed the extrinsic and intrinsic geometrical properties of dynamically triangulated random surfaces (for example the Hausdorff-dimension, spectral and spreading dimension) above, below and at the transition [@muenkel92a].
The partition function of the above model can be written as
$${\cal Z}_N = \int{d^DX_0 {\int {\prod_{i = 1}^{N-1}{d^DX_i e^{-{\cal H}}}}}}$$
where the translational mode is integrated out. The Hamiltonian ${\cal H}$ is defined as
$${\cal H} = \ \beta \cdot \underbrace{\sum_{\langle i,j \rangle}^{N}{(X_i^\mu -
X_j^\mu)^2}}_{{\cal H}_g}
+ \ \lambda \cdot \underbrace{\sum_{\bigtriangleup_i,\bigtriangleup_j}{(1 -
\hat{n}_{\bigtriangleup_i} \cdot \hat{n}_{\bigtriangleup_j})}}_{{\cal
H}_e}
- \ \alpha \cdot \underbrace{\sum_{i = 0}^{N}{\log{\sigma_i}}}_{{\cal H}_m} \
\;.
\label{equ:Hamiltonian_dtrs}$$
The Gaussian part of the Hamiltonian ${\cal H}_g$ is a sum over the positions $X$ in embedding Euclidian space of all nearest neighbours nodes, i.e. all links of the triangulation. We shall use $\beta=1$ because of the rescaling invariance.
${\cal H}_e$ is an edge extrinsic curvature term \[19-26\]. $\sum_{\bigtriangleup_i,\bigtriangleup_j}$ denotes a summation over all adjacent triangles which share an edge and $\hat{n}_{\bigtriangleup_i} \cdot \hat{n}_{\bigtriangleup_j}$ is the scalar product of the vectors normal to a triangle.
The third part of the Hamiltonian ${\cal H}_m$ is the discretization of the square root of the metric. $\sigma_i$ denotes the number of nearest neighbours of node $i$. $\alpha$ depends on the measure. We used $\alpha=D/2$ for the simulations in $D=3$ dimensional embedding space.
Here we investigate such a model numerically (Monte Carlo simulations), applying finite-size scaling ideas. These ideas have proven successful in other contexts for the analysis of phase transitions. We expect that the above model shows a phase transition at some critical value of the coupling $\lambda$. There are several questions which we want to address: What is the order of the transition? If the transition is of second order, what are the exponents? Are the exponents topology dependent? Is the transition temperature topology dependent?
Simulation Method and Autocorrelations
======================================
First let us study the effect of the extrinsic curvature on the auto-correlation time of several observables, i.e., we are interested in the dynamics of a random surface induced by a Monte Carlo process. Such a study is of course a prerequisite for a detailed numerical study of the apparent transition which such surfaces undergo as a function of a bending rigidity. From our results which we present below we must conclude that extensive computer resources must be applied because of the very long relaxation times which even for moderate surface sizes can reach several tenthousand sweeps.
We want to look at the closed dynamically triangulated random surfaces without self-avoidance. As models for such surfaces we take the first two topologically closed surfaces: The sphere and the torus (c.f. Figure \[fig:show\]). The torus is specified by identifying edges of the parameter space ${\cal P}$ whereas the sphere, because of its genus needs a slightly more complicated set-up procedure.
What we are interested in is to calculate the auto-correlation function of an observable ${\cal O}$ as a function of time $$\rho_{\cal O }(t,N) = \frac{<{\cal O}(0){\cal O}(t)>_N - <{\cal O}>_N^2}
{<{\cal O}^2>_N - <{\cal O}>_N^2}$$ and its dependence on the number of nodes $N$ of the surface. From the auto-correlation function we are then able to extract the auto-correlation time $\tau_{\cal O} (N)$. This is done calculating the integrated auto-correlation function, i.e., $$\tau_{int,{\cal O}} = \frac{1}{2} \int_{-\infty}^{+\infty} dt\ \ \rho_{\cal
O}(t,N) \ \ \ .$$ We also compared the integrated auto-correlation time with the exponential correlation time and the statistical inefficiency, but found no disagreement within the errors. If the successively generated configurations can be considered as non-interacting beads as is done in the Rouse theory [@rouse53; @edwards86] we will find for example for the radius of gyration $$\tau_{{\cal R}_{gyr}^2} \propto N$$ and in general we expect $$\tau_{\cal O} \propto N^a \ \ \ .$$
To calculate observables such as the gaussian part ${\cal H}_g$ of the Hamiltonian or the radius of gyration $R_{gyr}$, we use the standard Monte Carlo algorithm \[29-31\]. Such an algorithm induces a stochastic dynamics and all our results apply only to such an induced dynamics. One Monte Carlo sweep is completed when each triangulation point was given the chance for a displacement from its previous position and the edges of the triangulation were given the chance to re-connected or flip to an orthogonal position (c.f. Figure \[fig:mc\]. The flip operation implements the sum over all possible triangulation). This corresponds to one new configuration.
Table \[tab\_sph\] summarizes our results for the exponent $a$ for the sphere. The exponents where extracted from the data of system sizes from $36$ up to $288$. At this point we should mention that the expected transition is at $\lambda_c
\approx
1.5$. Out data was collected above and below the transition.
Above and below the transition the radius of gyration shows the Rouse behaviour. The gaussian part of the Hamiltonian shows clearly a dependence on the bending rigidity. Above the transition point the auto-correlation exponent changes to one and below it is almost zero. The same holds for the edge part of the Hamiltonian.
[|c||c|c|c||c|c|]{} $\lambda$ & &\
& $\tau_{{\cal H}_{gaussian}}$ & $\tau_{{\cal H}_{edge}}$ & $R_{gyr}^2$ & shift & flips\
0.00 & $0.2 \pm 0.3$ & $0.2 \pm 0.2$ & $1.1 \pm 0.2$ & 0.51 & 0.76\
1.00 & $0.1 \pm 0.3$ & $0.4 \pm 0.2$ & $1.1 \pm 0.4$ & 0.51 & 0.36\
1.75 & $1.0 \pm 0.1$ & $0.8 \pm 0.2$ & $1.1 \pm 0.2$ & 0.47 & 0.28–0.34\
3.00 & $1.0 \pm 0.1$ & $1.0 \pm 0.1$ & $1.0 \pm 0.3$ & 0.50 & 0.23–0.33\
Almost all of the observations for the spherical case carry over to the case of the torus.
Results of the Simulations
==========================
Finite size scaling assumes, that there is only one relevant linear length scale, which is compared to the correlation length. To apply finite size scaling to the crumpling transition of dynamically triangulated random surfaces (DTRS) one must therefore assume a single length scale determined by the number of nodes $N$ and the internal dimension $d$ of the surface
$$L \propto N^{1/d} \ \;.$$
This internal dimension $d$ also depends on the external properties of the surface and $\lambda$ [@muenkel92a].
So let us first look at the specific heat. If the transition is of second order we would have
$$C(\lambda ,L) = L^{\alpha / \nu} \hat{C}\left[{ (\lambda -
\lambda_c)L^{-\nu}}\right]$$
where $\hat{C}$ is a scaling function, which depends on how one implements the surface. At the critical $\lambda$ the scaling function is regular and the scaling hypothese leads to $$C_N^{max} \propto A N^{\alpha/\nu{d}} + \ldots$$
for the scaling of the peak in the specific heat. If we assume a first order transition then $C_N^{max}$ diverges as $L^d$ because of the $\delta$-distribution of $C_\infty(T)$ [@binder86; @binder84; @landau88].
An evaluation of the specific heat $C$ of DTRS (neglecting the metric contribution ${\cal H}_m$) gives the following expression
$$C_{all} = \frac{D}{2} + \frac{{\lambda}^2}{N} (<{\cal H}_e^2> - <{\cal
H}_e>^2) \ \ \; .
\label{equ:c_fluc}$$
The first part is related to the Gaussian Hamiltonian ${\cal H}_g$ and the second to the specific heat $C$ of the edge extrinsic curvature ${\cal H}_e$. The specific heat for the edge extrinsic curvature is shown in Figure \[fig:ceall\] for the two topologies considered. The interpolation was done by the method of Ferrenberg and Swendsen [@ferrenberg89; @ferrenberg88] using histograms of ${\cal H}_e$ .
Using the data of the specfic heat obtained by applying the extrapolation method, we can get a very accurate estimate of the positions of the maxima and of the heights. Figure \[fig:cmax\_scale\] shows $C_N^{max}$ of the torus and the sphere. A change to $L^d$ behaviour is very unlikely and for that reason the data strongly suggest a continuous phase transition in agreement with previous work \[8,14-17,38,39\]. From the data shown in Figure \[fig:cmax\_scale\] we can obtain the following upper boundaries of critical exponents
$$\hbox{Sphere:} \frac{\alpha}{\nu{d}} \leq 0.00 \pm 0.04 \ \ \ \
\hbox{Torus:} \frac{\alpha}{\nu{d}} \leq 0.06 \pm 0.02
\label{equ:alphnud_res}$$
Using this simple method, we cannot distinguish a diverging specific heat with very small but positiv $\alpha$, a logarithmic divergence $\alpha_s=0$ and a power law cusp $\alpha_s < 0$, where $\alpha_s$ denotes the exponent of the singular part of the specific heat. We will use the abbreviation $\alpha$ instead of $\alpha_s$.
Following Fisher [@fisher71], these three cases may be distinguished with a fit of the form
$$C(\Delta\lambda) = A \cdot \frac{1}{\alpha} \left({{\Delta\lambda}^{-\alpha} -
1}\right) + B
\ \ \ \; , \ \ \; \Delta\lambda = \left|\lambda_c^N - \lambda\right|
\label{equ:fit_alpha}$$
Figure \[fig:ce\_fit\_alpha\] shows such fits for $\alpha=0.1$ (power law), $\alpha=-0.01$ (near logarithmic) and $\alpha=-1.2,-2.0$ (power law cusp). The fits clearly favour a power law cusp of the specific heat with a large negative value of $\alpha$, although we were not able to estimate the exponent $\alpha$ precisely.
Our results are in contrast to the estimates of Renken and Kogut [@renken91] $\alpha/\nu{d}=0.14(3)$ for the sphere. They assumed scaling above $N=72$, but the more accurate data in Figure \[fig:cmax\_scale\] shows that this assumption is not true. The decreasing slope in Figure \[fig:cmax\_scale\] is also present in their Figure 5 [@renken91] although this is slightly masked by larger error barrs and less data points. Catteral et all. [@catteral91] reported an estimate of $\alpha/\nu{d}~=~0.185(50)$, but they used a different discretization based on $\phi^3$ graphs with a flip acceptance rate of only O(1%) at $\lambda_c\approx1.5$ and larger correlation times of the edge extrinsic curvature than in our discretization.
Another possibility to determine the order of the transition is the cumulant ${\cal V}_N$ of the edge extrinsic curvature
$${\cal V}_N := 1 - \frac{1}{3}
\frac{{\langle{{{\cal H}_e}^4}\rangle}_N}{{\langle{{{\cal
H}_e}^2}\rangle}_N^2} \;,
\label{equ:VL_def}$$
which behaves quite differently at temperature driven first- and second-order transitions [@binder86; @binder84] :
$$\begin{aligned}
\hbox{1. and 2.order:} & {\left.{{\cal V}_N}\right|}_{min}
\stackrel{N\rightarrow\infty}{=} \frac{2}{3} & T{\neq}T_c \hbox{fixed}
\label{equ:VL_TneqTc}\\
\hbox{2.order:} & {\left.{{\cal V}_N}\right|}_{min}
\stackrel{N\rightarrow\infty}{=} \frac{2}{3} & T=T_c(N) \label{equ:VL_TeqTc2}\\
\hbox{1.order:} & {\left.{{\cal V}_N}\right|}_{min}
\stackrel{N\rightarrow\infty}{=}
1-\frac{2\left({E_+^4+E_-^4}\right)}{3{\left({E_+^2+E_-^2}\right)}^2} &
T=T_c(N)\label{equ:VL_TeqTc1}\end{aligned}$$
$E_+$ and $E_-$ are the energies of the system above and below the transition. For a very weak first-order transition ($E_+\approx E_-$) we also have ${\left.{{\cal V}_N}\right|}_{min} \approx 2/3$.
We computed ${\cal V}_N$ defined by equation (\[equ:VL\_def\]) using again the method of Ferrenberg and Swendsen [@ferrenberg89; @ferrenberg88]. The resulting figures show the predicted single peak minima with wings following equation (\[equ:VL\_TneqTc\]). The finite size dependence of the minima of $ {\left.{{\cal V}_N}\right|}_{min} $ shown in Figure \[fig:VL\_min\] distinctly favours the asymptotic behaviour in equation (\[equ:VL\_TeqTc2\]) and therefore a continuous phase transition or a very weak first order transition.
In general, finite size scaling predicts also a shift $\Delta\lambda=\lambda_c^N-\lambda_c^\infty$ of the effective transition ‘temperature’ $\lambda_c^N$ proportional to $N^{-1}(=L^{-d})$ for a first- and proportional to $N^{-1/{\nu}d}(=L^{-1/\nu})$ for a second-order transition. Unfortunatly $\lambda_c^\infty$ of dynamically triangulated random surfaces is not known. For that reason, we have to use a two-parameter fit with unknowns $\lambda_c^\infty$ and ${\nu}d$. But we can improve the reliability of this fit, because we know more. First, the fit has to be a straight line (neglecting corrections to scaling) and we have $\lim_{N\rightarrow\infty} \Delta\lambda(N) = 0$. Second, the shift $\Delta\lambda(N)$ is different for the specific heat and the cumulant in general. Therefore the slope of the fitted lines will be different in general, but we still have $\lim_{N\rightarrow\infty} \Delta\lambda(N) = 0$ for both observables.
Figure \[fig:fit\_dlambda\] shows the best fits for the sphere and the torus. The estimates of the parameters are $$\lambda_c^\infty = 1.51 \pm 0.04 \ \ \ , \ \ \ {{\nu}d} = 3.2 \pm 0.5
\label{equ:s_alphaandnud_res}$$ for the sphere and $$\lambda_c^\infty = 1.47 \pm 0.02 \ \ \ , \ \ \ {{\nu}d} = 2.5 \pm 0.5
\label{equ:t_alphaandnud_res}$$ for the torus.
We turn now to the order parameter itself. Common practise is to take $\zeta = R/L$ ($R$ is the typical radius, $L$ the linear size of the membrane) to be a suitable order parameter \[41-47\]. Initially [@kantor87a] it was also defined as $R_g(L)={\zeta}L$ ($L\rightarrow\infty$), with the linear size $L$ of the hexagon and the radius of gyration $R_g^2{\propto}\sum_{ij}{\left\langle{{|X_i-X_j|}^2}\right\rangle}$. A suitable choice for the order parameter therefore is $$\zeta := R^2 / N$$
with
$$R^2 = \frac{1}{N(N - 1)} \left\langle
\sum_{i,j}^{N}{ \sigma_i \sigma_j {(\vec{X}_i - \vec{X}_j)}^2}
\right\rangle
\label{equ:r_def_dtrs}$$
$\sigma_i$ denotes the number of nearest neighbours, i.e. the number of links connected to a node $i$. The associated susceptibility is
$$\chi_{R^2} := L^d \left( {\langle{\zeta^2}\rangle - {\langle{\zeta}\rangle}^2}
\right)\nonumber\
= \frac{1}{N} \left( {\langle{R^4}\rangle - {\langle{R^2}\rangle}^2}\right)
\label{equ:chi_def}$$
Figure \[fig:s\_zeta\] shows the order parameter $\zeta$ and Figure \[fig:s\_chi\] the susceptibilty of the sphere. The results for the torus are similar.
We also measured another possible order paramter $\zeta'=\langle{{R'}^2}\rangle / N$, $$\langle{{R'}^2}\rangle = \frac{1}{N} \left\langle
\sum_{i}^{N}{{\left({\vec{X}_i - \overline{\vec{X}}}\right)}^2}
\right\rangle$$ which exhibits a small increase near the phase transition and a slower decay of $\zeta'$ for $\lambda{\rightarrow}0$. The difference is caused by a change of the internal geometry near the phase transition [@muenkel92a].
With the data for the sphere in Figure \[fig:s\_zeta\] and the corresponding data of the torus one can estimate the critical exponent $\beta/{\nu}d$ of the order parameter assuming a second order transition. Here we use as the effective critical temperature the position of the peak of the specific heat. $C^{max}$ (c.f. Figure \[fig:rg2\_scale\]).
An estimate of $\beta/{\nu}d$ using
$$\langle{R^2}(N,\lambda_c^{eff})\rangle \propto N^{\beta/({\nu}d)\ +\ 1}$$
results in
$$\hbox{Torus:}\ \beta/{\nu}d = 0.28 \pm 0.02\ \;,\; \ \hbox{Sphere:}\
\beta/{\nu}d = 0.35 \pm 0.04
\label{equ:betanud_res}$$
and with the values of ${\nu}d$ in equation (\[equ:s\_alphaandnud\_res\]) and (\[equ:t\_alphaandnud\_res\])
$$\hbox{Torus:}\ \beta = 0.7 \pm 0.2\ \;,\; \ \hbox{Sphere:}\ \beta = 1.1 \pm
0.2
\label{equ:beta_res}$$
Figure \[fig:chi\_scale\] shows the scaling of the maxima of the susceptiblity $\chi$ (equation (\[equ:chi\_def\]), Figure \[fig:s\_chi\]).
The results are $$\hbox{Torus:}\ \gamma/{\nu}d = 0.62 \pm 0.06\ \;,\; \ \hbox{Sphere:}\
\gamma/{\nu}d = 0.66 \pm 0.06
\label{equ:gammanud_res}$$
and, using the values of ${\nu}d$ in equation (\[equ:s\_alphaandnud\_res\]) and (\[equ:t\_alphaandnud\_res\]),
$$\hbox{Torus:}\ \gamma = 1.6 \pm 0.5\ \;,\; \ \hbox{Sphere:}\ \gamma = 2.1 \pm
0.5\ .
\label{equ:gamma_res}$$
With these large error bars and estimated values $\alpha\approx-1.5$ the results are almost compatible with the scaling relation
$$\alpha + 2 \beta + \gamma = 2
\label{equ:abg_scaling}$$
[**Acknowledgements**]{}
Partial support by the SFB 123 and the BMFT project $0326657\mbox{D}$ and $031240284$ is gratefully acknowledged.
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|
---
abstract: 'Circuit quantization links a physical circuit to its corresponding quantum Hamiltonian. The standard quantization procedure generally assumes any external magnetic flux to be static. Time dependence naturally arises, however, when flux is modulated or when flux noise is considered. In this case, application of the existing quantization procedure can lead to inconsistencies. To resolve these, we generalize circuit quantization to incorporate time-dependent external flux.'
author:
- Xinyuan You
- 'J. A. Sauls'
- Jens Koch
bibliography:
- 'Manuscript.bib'
title: 'Circuit quantization in the presence of time-dependent external flux'
---
Introduction {#sec:introduction}
============
Superconducting circuits are electrical circuits fabricated from superconducting materials. Due to the flexibility in circuit design, such circuits hold substantial promise as qubits [@Devoret2013], exhibiting coherence times now approaching the millisecond scale [@Nguyen2018]. The link between a physical circuit and its quantum mechanical behavior is provided by circuit quantization [@qnetwork; @devoret1995quantum; @Burkard2004; @Nigg2012; @Ulrich2016; @vool; @Parra-Rodriguez2019]. This implementation of canonical quantization has been widely used in the context of superconducting qubits and circuit quantum electrodynamics (cQED), but is restricted to circuits enclosing time-independent external magnetic flux[^1]. Recently, there have been a number of studies utilizing time-varying external flux. Alternating-current flux modulation has been proposed for realizing fast two-qubit gates via first-order sideband transitions [@Beaudoin2012; @Strand2013]. Parametric flux modulation also enables random access multi-qubit control, thus improving qubit connectivity in quantum processors [@Naik2017; @Reagor2018; @Didier2018]. Furthermore, modulating the coupling strength between a qubit and a resonator by means of a time-dependent flux threading a superconducting quantum interference device (SQUID) allows for universal stabilization of arbitrary single-qubit states [@Lu2017a; @huang2018]. In addition to parametric modulation, time-dependence is also crucial for the modeling of flux-induced qubit dissipation and dephasing [@Wellstood1987; @Ithier2005; @Yoshihara2006; @Kumar2016]. Therefore, it is timely to examine the general framework of circuit quantization for time-varying external flux.
In standard circuit quantization there exists a gauge freedom when formulating the Hamiltonian for a circuit threaded by external flux. Depending on the choice of gauge, the external flux may be associated with any of the potential energy terms. For example, in a fluxonium qubit [@Manucharyan2009] the external flux may be associated with the potential energy of the Josephson junction [@Spilla2015; @Sete2017; @Nguyen2018], or that of the superinductor [@Diggins1997; @Koch2009]. However, this freedom leads to inconsistent predictions of the rate of qubit relaxation induced by fluctuations of the external flux, as we show below. Our generalization of circuit quantization to include time-dependent external flux resolves these inconsistencies and provides a formulation capable of handling a wider variety of circuits.
The paper is organized as follows. In Sec. \[sec:time\_independent\], we review the standard procedure of circuit quantization that is valid for time-independent external flux. We show in Sec. \[sec:inconsistent\] that inconsistencies arise if this procedure is applied to circuits coupled to time-varying flux; we specifically analyze qubit relaxation due to flux noise. In Sec. \[sec:time\_dependent\], we generalize the circuit-quantization framework for two concrete examples: the dc SQUID circuit and the fluxonium qubit with time-dependent external flux. The concept of irrotational degrees of freedom is introduced, and the previous inconsistency in the relaxation time is resolved. Canonical quantization for general single-loop and multi-loop circuits with time-dependent flux is presented in Sec. \[sec:general\]. We summarize our work in Sec. \[sec:summary\].
Circuit quantization with time-independent external flux {#sec:time_independent}
========================================================
We begin with a brief review of standard circuit quantization, valid for time-independent flux. The starting point is the construction of the Lagrangian for the circuit. Kinetic energy terms in this Lagrangian correspond to capacitive energies. The potential energy is composed of the energies of all inductive elements, including those associated with Josephson junctions. As a concrete example, we apply circuit quantization to an asymmetric dc SQUID consisting of two Josephson junctions with junction capacitances $C_\text{l}$, $C_\text{r}$, and Josephson energies $E_\text{Jl}$, $E_\text{Jr}$ \[Fig. \[fig:squid\](a)\]. The static external flux, $\Phi_\text{e}$, which threads the circuit loop, is treated as a classical variable.
![\[fig:squid\] (a) Asymmetric SQUID formed by two Josephson junctions. $\Phi_\text{l}$ and $\Phi_\text{r}$ are branch flux variables for the left and right junction. The node flux $\phi$ is associated with the upper node, while the lower node is chosen to be the ground node. An external flux $\Phi_\text{e}$ threads the loop. (b) and (c) depict the two equivalent choices of spanning trees (solid lines) and closure branches (dashed lines).](squid.pdf){width="38.00000%"}
Each branch flux variable is defined as the time integral of the voltage across the corresponding circuit element. The dc SQUID has two branch variables, $\Phi_\text{l}$ and $\Phi_\text{r}$, for the left and right arm of the SQUID, respectively. Following Ref. , we also define flux variables associated with each circuit node. Setting the lower node in Fig. \[fig:squid\](a) to be the ground node, the system can be described by one active node flux $\phi$ corresponding to the upper node. If we choose the left arm to form the spanning tree, then the right arm becomes the closure branch \[see Fig. \[fig:squid\](b)\]. By convention, one allocates the external flux to the closure branch [@devoret1995quantum]. This step guarantees fluxoid quantization for time-independent external flux [@barone1982physics; @orlando1991foundations]. The relation between the node variable $\phi$ and the branch variables is thus $$\label{node_l}
\Phi_\text{l}=\phi\,, \qquad \Phi_\text{r}=-\phi+\Phi_\text{e}\,.$$
In terms of the node flux, the Lagrangian can be written as a function of one independent degree of freedom, $$\mathcal{L}_\text{l} = \dfrac{1}{2} C_\Sigma \dot{\phi}^2
+ E_\text{Jl}\cos{\dfrac{2\pi}{\Phi_0}\phi}
+ E_\text{Jr}\cos{\dfrac{2\pi}{\Phi_0}(-\phi+\Phi_\text{e})}\,,$$ where $C_\Sigma=C_\text{l}+C_\text{r}$ is the total capacitance, and $\Phi_0=h/2e$ the flux quantum. The Hamiltonian is obtained by performing a Legendre transformation with generalized momentum $Q = \partial \mathcal{L}_\text{l}/\partial \dot{\phi} = C_\Sigma \dot{\phi}$. Employing canonical quantization in the coordinate representation, $Q=-i\hbar\,d/d\phi \equiv 2en$, and denoting $E_\text{C} = e^2/2C_\Sigma$, the Hamiltonian takes the form $$\label{inleft}
\mathcal{H}_\text{l} = 4E_\text{C}n^2
- E_\text{Jl}\cos{\dfrac{2\pi}{\Phi_0}\phi}
- E_\text{Jr}\cos{\dfrac{2\pi}{\Phi_0}(\phi-\Phi_\text{e})}
\,.$$
In Eq. we chose the right arm to be the closure branch. Equivalently, we may choose the left arm as the closure branch \[Fig. \[fig:squid\](c)\]. This choice groups the external flux with the left arm. The two branch fluxes then take on the form $$\label{node_r}
\Phi_\text{l}=\phi + \Phi_\text{e}\,, \qquad \Phi_\text{r}=-\phi\,,$$ resulting in the Hamiltonian $$\label{inright}
\mathcal{H}_\text{r} = 4E_\text{C}n^2 - E_\text{Jl} \cos{\dfrac{2\pi}{\Phi_0}(\phi+\Phi_\text{e})} - E_\text{Jr}\cos{\dfrac{2\pi}{\Phi_0}\phi}\,.$$ The two Hamiltonians $\mathcal{H}_\text{l}$ and $\mathcal{H}_\text{r}$ are related by the unitary transformation $$\label{transformation1}
\mathcal{H}_\text{r} = U^\dag \mathcal{H}_\text{l} U\,, \qquad U=\exp(-\dfrac{i}{\hbar}Q\Phi_\text{e})$$ which implies they have the same eigenvalue spectrum, and two sets of eigenfunctions shifted by $\Phi_\text{e}$ in the variable $\phi$.
Inconsistent predictions of qubit relaxation times {#sec:inconsistent}
==================================================
While the two Hamiltonians obtained for different choices of closure branches predict equivalent results for static flux, inconsistencies arise if we directly apply standard quantization to study qubit relaxation caused by a fluctuating external flux. We will trace back this issue to the proper treatment of the time-dependent holonomic constraint imposed by fluxoid quantization in Sec. [\[sec:time\_dependent\]]{}. Suppose the flux is randomly varying around some fixed, but controllable, value $\Phi_\text{e}^0$, $$\label{noise}
\Phi_\text{e}(t) = \Phi_\text{e}^0 + \delta \Phi_\text{e}(t)\,.$$ In general fluctuations induce transitions between the qubit states $|g\rangle$ and $|e\rangle$. According to Fermi’s golden rule, the decay rate of the qubit state $\ket{e}$ is proportional to [@Ithier2005] $$\label{fermi}
T_1^{-1} \propto |\langle g| \partial_{\Phi_\text{e}} \mathcal{H} | e\rangle|^2\,,$$ where the partial derivative is evaluated at $\Phi_\text{e} (t) = \Phi_\text{e}^0$. Equation implies that the qubit relaxation rate would depend on the choice of closure branch because of the association of the external flux with different terms in the Hamiltonian. In particular, the Hamiltonians $\mathcal{H}_\text{l}$ and $\mathcal{H}_\text{r}$ exhibit two different instances of flux grouping: The former allocates flux to the right branch, and the resulting relaxation rate would be proportional to $E_\text{Jr}^2$. In the latter case, flux is grouped with the left branch, and the relaxation rate would be proportional to $E_\text{Jl}^2$. In an asymmetric SQUID the two Josephson energies differ, $E_\text{Jl}\neq E_\text{Jr}$, and thus the qubit relaxation rate seemingly depends on how we allocate the flux in the Hamiltonian[^2]. This inconsistency is not limited to an asymmetric SQUID loop, but generally applies to circuit loops with distinct potential energy coefficients. For example, the external flux in a fluxonium qubit [@Manucharyan2009] can be allocated either to the inductor [@Diggins1997; @Koch2009] or the Josephson junction [@Spilla2015; @Sete2017; @Nguyen2018] which would lead to different predictions for the relaxation time due to flux noise[^3]. We emphasize that the transformation is merely a variable displacement, and hence unrelated to rotating-frame transformations that may affect decoherence rates [[@Brown2007; @Jing2014]]{}. This inconsistency rather hinges upon the correct treatment of time-dependent fluxoid quantization, as we will see next.
Circuit quantization for time-dependent external flux: dc SQUID and Fluxonium {#sec:time_dependent}
=============================================================================
We next formulate circuit quantization in the presence of time-dependent flux for the concrete examples of the dc SQUID circuit and the fluxonium qubit. The general discussion of arbitrary circuits is presented in the subsequent section.
From fluxoid quantization to the time-dependent Hamiltonian: dc SQUID {#sec-DC-SQUID}
---------------------------------------------------------------------
We begin with the constraint imposed by fluxoid quantization. Most naturally, it is written in terms of branch variables, $$\label{fluxoid}
\Phi_\text{l} + \Phi_\text{r} = \Phi_\text{e}(t)
\,,$$ which is a time-dependent holonomic constraint. According to it, the sum of the branch variables is fixed by the external flux, and hence is not a dynamical variable. The single degree of freedom, denoted by the variable $\widetilde{\Phi}$, can be expressed as a linear combination of the two branch variables, $$\label{dofvar}
\widetilde{\Phi} = m_\text{l} \Phi_\text{l} + m_\text{r} \Phi_\text{r}
\,,$$ where $m_\text{l} \neq m_\text{r}$. The node variables defined in Eqs. and are obtained by setting $(m_\text{l},m_\text{r})=(1,0)$ and $(0,-1)$, respectively.
To derive the time-dependent Hamiltonian, we first express the circuit Lagrangian in terms of the two branch variables, $$\label{lagbranch}
\mathcal{L} = \dfrac{1}{2}C_\text{l}\dot{\Phi}_\text{l}^2
+ \dfrac{1}{2}C_\text{r}\dot{\Phi}_\text{r}^2
+ E_\text{Jl}\cos{\varphi_\text{l}}
+ E_\text{Jr}\cos{\varphi_\text{r}}
\,,$$ where all lower-case $\varphi$’s denote reduced flux variables, $\varphi=2\pi \Phi/\Phi_0$. Next, we transform to the variable [$\tilde\Phi$]{} and the external flux [$\Phi_\text{e}$]{} by inverting Eqs. and . This results in $$\label{branch_transformations}
{
\Phi_\text{l} = (\widetilde{\Phi} - m_\text{r}\Phi_\text{e})/m_\Delta\,,\qquad
\Phi_\text{r} = -(\widetilde{\Phi} - m_\text{l}\Phi_\text{e})/m_\Delta\,,
}$$ where $m_\Delta=m_\text{l}-m_\text{r}$. We now see that proper implementation of the time-dependent holonomic constraint yields $$\begin{aligned}
\label{lag}
\mathcal{L}(m_\text{l},m_\text{r})
=
& \dfrac{C_\Sigma}{2m_\Delta^2}\dot{\widetilde{\Phi}}^2 - \dfrac{C_\text{r}m_\text{l}+C_\text{l}m_\text{r}}{m_\Delta^2}\dot{\widetilde{\Phi}}\dot{\Phi}_\text{e}
\\
& + E_\text{Jl}\cos{\dfrac{\widetilde{\varphi}-m_\text{r}\varphi_\text{e}}{m_\Delta}} + E_\text{Jr}\cos{\dfrac{\widetilde{\varphi}-m_\text{l}\varphi_\text{e}}{m_\Delta}}\,,
\nonumber\end{aligned}$$ where we dropped the term $\propto \dot{\Phi}_\text{e}^2$. Performing a Legendre transformation leads to the Hamiltonian $$\begin{aligned}
\label{ham}
\mathcal{H}(m_\text{l},m_\text{r}) = & 4E_\text{C}m_\Delta^2 n^2 + \dfrac{C_\text{r}m_\text{l}+C_\text{l}m_\text{r}}{C_\Sigma}2en\dot{\Phi}_\text{e}
\\
\nonumber
&-E_\text{Jl}\cos{\dfrac{\widetilde{\varphi}-m_\text{r}\varphi_\text{e}}{m_\Delta}} - E_\text{Jr}\cos{\dfrac{\widetilde{\varphi}-m_\text{l}\varphi_\text{e}}{m_\Delta}}\,. \end{aligned}$$ The term $\propto\dot{\Phi}_\text{e}$ is missing in standard circuit quantization for Hamiltonians parametrized only by the time-independent flux, $\Phi_\text{e}$. For special choices of $(m_\text{l},m_\text{r})$ obeying[^4] $$\label{irrcondition}
C_\text{r}m_\text{l}+C_\text{l}m_\text{r} = 0
\,,$$ the linear coupling of the charge operator to the fluctuating EMF generated by $\dot\Phi_\text{e}$ vanishes. We refer to Eq. as the *irrotational* constraint[^5] on the variable $\widetilde{\Phi}$, and to $\widetilde{\Phi}$ with the constraint imposed as the irrotational operator, variable or degree of freedom.
In general, Hamiltonians for different choices $(m_\text{l},m_\text{r})$ and $(m_\text{l}',m_\text{r}')$ are related by a time-dependent unitary transformation $\mathcal{U}(t) = S\,U(t)$, consisting of a gauge transformation and a scale transformation. The gauge transformation is given by $$U(t)=\exp(\dfrac{i}{\hbar}Q
\dfrac{m_\text{l}m_\text{r}'-m_\text{l}'m_\text{r}}{m_\Delta}
\Phi_\text{e}(t))$$ and the scaling transformation by $$S = \exp(\dfrac{i}{2\hbar}\ln(\dfrac{m_\Delta'}{m_\Delta})G)
\,,$$ where the generator is the anti-commutator of the conjugate coordinate and momentum, $G = \widetilde{\Phi}Q + Q\widetilde{\Phi}$. The action of $S$ on the conjugate variables is $$S^\dag Q S = \dfrac{m_\Delta'}{m_\Delta} Q\,,
\qquad
S^\dag \widetilde{\Phi} S = \dfrac{m_\Delta}{m_\Delta'}\widetilde{\Phi}
\,.$$ Since operators and states transform according to $$\mathcal{A}' = \mathcal{U}^\dag(t) \mathcal{A}\,\mathcal{U}(t)\,, \qquad |\psi'\rangle = \mathcal{U}^\dag(t) |\psi\rangle\,,$$ all expectation values, measured at one particular time, are invariant under the unitary transformation $\mathcal{U}(t)$: $\langle \psi'| \mathcal{A}'|\psi'\rangle
%=\langle\psi|\mathcal{U}(t)\mathcal{U}^\dag(t)\mathcal{A}\,\mathcal{U}(t)\mathcal{U}^\dag(t)| \psi \rangle
=\langle \psi| \mathcal{A} | \psi \rangle$. The Schr[ö]{}dinger equation also remains form invariant, with the transformed Hamiltonian given by $$\label{Utransform}
\mathcal{H}(m_\text{l}',m_\text{r}')
=
\mathcal{U}^\dag(t)\mathcal{H}(m_\text{l},m_\text{r})\mathcal{U}(t)
-i\hbar\,\mathcal{U}^\dag(t)\partial_t \mathcal{U}(t)\,.$$ However, as we discuss below, multi-time correlation functions are in general not invariant under the time-dependent unitary transformation.
Multi-time observables\[sec:correlator\]
----------------------------------------
Consider the probability $$C(0,t) = |\langle \psi(0)|\psi(t)\rangle|^2
\,,$$ for transitions induced by flux noise. The decay rate of the transition probability averaged over noise realizations, $\langle C(0,t)\rangle_\text{av}$, is directly related to the spontaneous relaxation rate. For simplicity, we will ignore the usual 1/$f$ character of flux noise. Instead, we consider stationary Gaussian noise with short correlation time $t_\text{c}$, standard deviation $\sigma$, and noise power spectrum $S_{\Phi_\text{e}}(\omega)$, defined as the Fourier transform of the autocorrelation function.
To calculate the noise-averaged transition probability, fluctuations in both $\Phi_\text{e}$ and $\dot{\Phi}_\text{e}$ must be taken into account. We prepare the system in the first excited eigenstate $|\psi(0)\rangle=|e^0(m_\text{l},m_\text{r})\rangle$ of the unperturbed Hamiltonian $$\begin{aligned}
\label{ham0}
\mathcal{H}^0 (m_\text{l},m_\text{r}) = & 4E_\text{C}m_\Delta^2 n^2
\\
\nonumber
& - E_\text{Jl}\cos{\dfrac{\widetilde{\varphi}-m_\text{r}\varphi^0_\text{e}}{m_\Delta}} - E_\text{Jr}\cos{\dfrac{\widetilde{\varphi}-m_\text{l}\varphi^0_\text{e}}{m_\Delta}}\,,\end{aligned}$$ obtained by omitting all fluctuating terms from $\mathcal{H}(m_\text{l},m_\text{r})$ in Eq. . We proceed by evolving the initial state under the full Hamiltonian $\mathcal{H}(m_\text{l},m_\text{r})$ to some later time $t$, then carry out the average over noise realizations. As shown in Appendix \[sec:correlation\], for $t_\text{c}\ll t\ll T_1$ one obtains the perturbative result $$\begin{aligned}
\langle C(0,t)\rangle_\text{av} = & 1 -|\langle g^0_\text{irr} | \partial_{\Phi_\text{e}}
\mathcal{H}_\text{irr} | e^0_\text{irr}\rangle|^2 S_{\Phi_\text{e}}(\omega_{\text{eg}})t/\hbar^2
\nonumber
\\
&-\eta^2(m_\text{l}, m_\text{r})|\langle e^0_\text{irr} | n | g^0_\text{irr} \rangle|^2 8\sigma^2e^2/\hbar^2
\,.
\label{corr}\end{aligned}$$ Here, $\mathcal{H}^0_\text{irr}$ and $\mathcal{H}_\text{irr}$ are obtained from $\mathcal{H}^0$ \[Eq. \] and $\mathcal{H}$ \[Eq. \] by satisfying the irrotational constraint with $$\label{irrot}
m_\text{l} = \dfrac{C_\text{l}}{C_\Sigma}\,,\qquad m_\text{r} = -\dfrac{C_\text{r}}{C_\Sigma}\,,$$ where we take $m_\Delta=1$ for simplicity. $|g_\text{irr}^0\rangle$ and $|e_\text{irr}^0\rangle$ are the lowest two eigenstates of $\mathcal{H}^0_\text{irr}$, with energy difference $\omega_\text{eg}$. While Eq. is expressed in terms of quantities with irrotational constraints, it is valid for general choice of $(m_\text{l},m_\text{r})$.
There are two key features to the result for the transition probability in Eq. . First, the averaged transition rate is $\Gamma_1=|\langle g^0_\text{irr}|\partial_{\Phi_\text{e}}\mathcal{H}_\text{irr}|e^0_\text{irr}\rangle|^2
S_{\Phi_\text{e}}(\omega_{\text{eg}})/\hbar^2$. Hence there is no ambiguity in the relaxation rate with respect to the choice of $(m_\text{l},m_\text{r})$. Secondly, the calculation yields an offset term which depends on the choice of gauge. This unphysical offset is eliminated if we impose the irrotational constraint of Eq. . A similar gauge-dependent term appears in the expression for pure dephasing of the qubit, as discussed in Appendix \[sec:pure\_dephasing\_time\]. While the procedure outlined above provides a satisfactory theory for the transition rate, the gauge-fixing requirement used to eliminate the unphysical offset in the transition probability requires further analysis and discussion.
![(color online) The overlap between two states at different times is not conserved under a time-dependent unitary transformation, since $\langle\psi(0)|U(0)U^\dag(t)|\psi(t)\rangle$$\neq$$\langle\psi(0)|\psi(t)\rangle$ in general.[]{data-label="fig:correlator"}](spheres.pdf){width="45.00000%"}
The fact that time-dependent unitary transformations can affect the values of multi-time correlation functions is not surprising. For states defined as points on the Bloch sphere, Fig. \[fig:correlator\] illustrates how a time-dependent rotating-frame transformation, carried out at two different times, generally leads to different state-vector overlaps. In the following we examine the origin of the unphysical offset term in the transition probability in Eq. .
Consider the limit in which the system parameters $E_\text{C}$, $E_\text{Jl}$, $E_\text{Jr}$ tend to zero. Despite the fact that the system Hamiltonian should vanish in this limit, Eq. leads to $$\label{Hlimit}
\mathcal{H}(m_\text{l},m_\text{r}) \longrightarrow 2e\,\overline{\eta}(m_\text{l},m_\text{r})n\dot{\Phi}_\text{e}\,,$$ where $\overline{\eta}(m_\text{l},m_\text{r})=\eta(m_\text{l},m_\text{r})m_\Delta$ in general remains finite when taking the limit with fixed capacitance ratios. A non-vanishing value of for $\bar\eta(m_{\text{l}},m_{\text{r}})$ not only conflicts with the vanishing of the system Hamiltonian, but implies a gauge-dependent term proportional to $\dot\Phi_\text{e}$ in the Hamiltonian. One easily confirms that this term originates from the derivative term in the time-dependent unitary transformation, $-i\hbar\, \mathcal{U}^\dag \partial_t \mathcal{U}$, and that it vanishes when the irrotational constraint is imposed. Thus, the irrotational constraint eliminates the unphysical term, and renders the Hamiltonian form invariant.
Finally, we show that the spurious offset term in Eq. is generated by the unphysical term in the Hamiltonian, Eq. . Consider the time evolution of the excited state generated by this Hamiltonian: $$|\psi(t)\rangle = \exp[\dfrac{1}{i\hbar}\int_0^t dt' 2e\overline{\eta} n\dot{\Phi}_\text{e}(t')]
| e^0(m_\text{l},m_\text{r}) \rangle\,.$$ Expanding to order $O(\delta \Phi_\text{e})$ in the fluctuations of $\Phi_{\text{e}}(t)$, the overlap with the ground state becomes $$\begin{aligned}
&\langle g^0(m_\text{l},m_\text{r})|\psi(t)\rangle\\
&\qquad = \dfrac{2e}{i\hbar} \overline{\eta}\, \langle g^0(m_\text{l},m_\text{r})
|n|e^0(m_\text{l},m_\text{r})\rangle [\delta\Phi_\text{e}(t) - \delta\Phi_\text{e}(0)]
\nonumber
\\
\nonumber
&\qquad = \dfrac{2e}{i\hbar} \eta\,
\langle g^0_\text{irr} |n|e^0_\text{irr} \rangle[\delta\Phi_\text{e}(t) - \delta\Phi_\text{e}(0)]\,.\end{aligned}$$ The corresponding transition probability, averaged over the noise realizations, yields $$\label{offsetterm}
\left\langle |\langle g^0(m_\text{l},m_\text{r})|\psi(t)\rangle|^2 \right\rangle_\text{av} = \eta^2|\langle e^0_\text{irr} | n | g^0_\text{irr} \rangle|^2 8\sigma^2e^2/\hbar^2\,,$$ where we have approximated $\langle \delta \Phi_\text{e}(0) \delta \Phi_\text{e}(t) \rangle \approx 0$, valid for $t \gg t_\text{c}$ [@Clerk2010]. Equation reproduces the unphysical offset term we obtained in Eq. for $\langle C(0,t)\rangle_\text{av}$.
In summary, we find that fluctuations of the external flux lead to decay of the qubit with a gauge-invariant rate. In addition, the spurious offset term is also eliminated by enforcing the irrotational constraint. The latter removes the term $\propto\dot{\Phi}_\text{e}$ in the Lagrangian and Hamiltonian. A similar step to avoid $\propto\dot{\Phi}_\text{e}$ terms was taken for a four-junction circuit by Qiu *et al.* [@Qiu2016]. We conclude that Hamiltonians defined in terms of irrotational degrees of freedom represent the correct formulation for consistent calculations of multi-time correlation functions.
For the asymmetric dc SQUID, employing the irrotational variable specified by Eq. yields the Hamiltonian $$\begin{aligned}
\label{irrHSQUID}
\mathcal{H}_\text{irr} =& 4E_\text{C}n^2 \\\nonumber
&- E_\text{Jl} \cos{\left(\widetilde{\varphi}-\dfrac{C_\text{r}}{C_\Sigma}\varphi_\text{e}\right)} - E_\text{Jr} \cos{\left(-\widetilde{\varphi}-\dfrac{C_\text{l}}{C_\Sigma}\varphi_\text{e}\right)}\,. \end{aligned}$$ Note that the external flux is grouped with both junctions’ potential energy terms, with weights given by capacitance ratios.
Circuit with inductor: fluxonium qubit
--------------------------------------
![\[fig:fluxonium\] The fluxonium qubit is formed by a Josephson junction and a superinductor with corresponding branch variables $\Phi_\text{J}$ and $\Phi_\text{L}$. An external flux $\Phi_\text{e}$ threads the loop.](fluxonium.pdf){width="30.00000%"}
The SQUID circuit analyzed in Sec. \[sec-DC-SQUID\] is an example of a single-loop circuit composed of Josephson junctions only. To see how flux distributes among Hamiltonian terms if the loop contains an inductor $L$, we consider the fluxonium qubit [@Manucharyan2009] (Fig. \[fig:fluxonium\]). We can obtain the irrotational degree of freedom by associating an auxiliary parallel capacitance $C_\text{L}$ with the inductor, and subsequently taking the limit $C_\text{L}\to0$.
The analysis of the circuit then proceeds in a manner analogous to that of the SQUID circuit. Following the same steps as above, we arrive at the Hamiltonian $$\mathcal{H}_\text{aux} = 4 E_\text{C} n^2 - E_\text{J} \cos{\left(\widetilde{\varphi}-\dfrac{C_\text{L}}{C_\Sigma}\varphi_\text{e}\right)}
+ \dfrac{1}{2}E_\text{L} \left(-\widetilde{\varphi}-\dfrac{C_\text{J}}{C_\Sigma}\varphi_\text{e}\right)^2$$ with $E_\text{L}=(\Phi_0/2\pi)^2/L$. Taking the limit $C_\text{L}\to0$ yields the proper irrotational Hamiltonian, $$\mathcal{H}_\text{irr} = 4 E_{\text{CJ}}n^2
- E_\text{J}\cos{\widetilde{\varphi}}
+ \dfrac{1}{2}E_\text{L}(-\widetilde{\varphi}-\varphi_\text{e})^2\,,$$ where $E_{\text{CJ}}=e^2/2C_\text{J}$. Thus, in a single-loop circuit with an inductor, the flux is entirely associated with the potential energy of the inductor.
Canonical quantization of a general circuit network with external flux $\Phi_\text{e}(t)$ {#sec:general}
=========================================================================================
In this section, we generalize the results obtained in the previous section to cover arbitrary circuits threaded by time-dependent flux. We start with single-loop circuits, and then extend to multi-loop circuits.
Single-loop circuits\[singleloop\]
----------------------------------
For a general single-loop circuit with $N$ elements, we define the vector $\bm{\Phi} = (\Phi_1,\Phi_2,\ldots,\Phi_N)^\intercal$ collecting all branch fluxes in the circuit. Due to fluxoid quantization there are $N-1$ degrees of freedom denoted by $\widetilde{\bm{\Phi}}=(\widetilde{\Phi}_1,\widetilde{\Phi}_2,\ldots,\widetilde{\Phi}_{N-1})^\intercal$. In general, they are linear combinations of the original branch variables, $$\widetilde{\bm{\Phi}} = \mathbf{M} \bm{\Phi}\,,$$ where $\mathbf{M}$ is an $(N{-}1){\times}N$ matrix with elements yet to be determined. It is useful to cast the fluxoid quantization constraint into matrix form as well, $$\label{fluxconstraint}
\Phi_\text{e} = \mathbf{R} \bm{\Phi}\,,$$ where $\mathbf{R}=(1,1,\dots,1)$ is a $1{\times} N$ matrix. We further introduce the augmented vector $\widetilde{\bm{\Phi}}_+$ and the augmented $N{\times} N$ matrix $\mathbf{M}_+$ by defining $$\bm{\widetilde{\Phi}}_+
=
\begin{pmatrix}
\widetilde{\bm{\Phi}} \\
\Phi_\text{e}
\end{pmatrix}\,,
\qquad
\mathbf{M}_+
=
\begin{pmatrix}
\mathbf{M}\\
\mathbf{R}
\end{pmatrix}\,.$$ Both the information on degrees of freedom and on the fluxoid quantization constraint are now compactly written as $$\widetilde{\bm{\Phi}}_+ = \mathbf{M}_+ \bm{\Phi}\,.$$ We note that $\det(\mathbf{M}_+){\neq} 0$ which is seen as follows. All rows of $\mathbf{M}$ generate genuine degrees of freedom which by definition must be linearly independent. Moreover, rows of $\mathbf{M}$ must be linearly independent of the row vector $\mathbf{R}$, since $\mathbf{R}$ does not generate a dynamical variable. Hence all rows of $\mathbf{M}_+$ are linearly independent.
We first consider a circuit loop composed of junctions only, and include capacitors and inductors subsequently. To construct irrotational degrees of freedom and find $\mathcal{H}_\text{irr}$, we inspect the kinetic energy, $$\mathcal{L}_\text{k} = \dfrac{1}{2}\dot{\bm{\Phi}}^\intercal \mathbf{C} \dot{\bm{\Phi}}\,,$$ where $\mathbf{C}=\text{diag}(C_\text{1},\,C_\text{2},\ldots,\,C_\text{N})$ is a diagonal matrix composed of all capacitances in the circuit. Since $\mathbf{M}_+$ is non-singular, we have $$\label{phiphitilde}
\bm{\Phi} = \mathbf{M}_+^{-1} \widetilde{\bm{\Phi}}_+\,,$$ in which case the kinetic energy can be rewritten as $$\mathcal{L}_\text{k} = \dfrac{1}{2}\dot{\bm{\widetilde{\Phi}}}_+^\intercal \mathbf{C}_\text{eff} \dot{\bm{\widetilde{\Phi}}}_+
\,,$$ where the effective capacitance matrix is given by $\mathbf{C}_\text{eff}=(\mathbf{M}_+^{-1})^\intercal \mathbf{C} \mathbf{M}_+^{-1}$. For a loop composed of junctions both $\mathbf{C}$ and $\mathbf{C}_\text{eff}$ are invertible.
Following the discussion in Sec. \[sec:time\_dependent\], we obtain irrotational degrees of freedom by demanding that all Lagrangian terms $\propto\dot{\Phi}_\text{e}$ vanish. This is equivalent to requiring $\mathbf{C}_\text{eff}$ and, hence, its inverse to be block-diagonal. Specifically, by block matrix multiplication one finds $$\mathbf{C}_\text{eff}^{-1} =
\begin{pmatrix}
\mathbf{M}\\
\mathbf{R}
\end{pmatrix}\!
\mathbf{C}^{-1}\!
\begin{pmatrix}
\mathbf{M}^\intercal & \mathbf{R}^\intercal
\end{pmatrix}
=
\begin{pmatrix}
\mathbf{M}\mathbf{C}^{-1}\mathbf{M}^\intercal & \mathbf{M}\mathbf{C}^{-1}\mathbf{R}^\intercal \\
\mathbf{R}\mathbf{C}^{-1}\mathbf{M}^\intercal & \mathbf{R}\mathbf{C}^{-1}\mathbf{R}^\intercal \\
\end{pmatrix},\vspace*{1mm}$$ so that the condition for irrotational degrees of freedom can be written as $$\label{m_eq}
\mathbf{R}\mathbf{C}^{-1}\mathbf{M}^\intercal=(C_1^{-1},C_2^{-1},\ldots,C_N^{-1})\mathbf{M}^\intercal=0\,.$$ This generalizes Eq. . One readily verifies that a particular solution for $\mathbf{M}$ is given by $$(\overline{\mathbf{M}})_{ij}= \delta_{ij} - \dfrac{C_i^{-1}}{\sum_{k=1}^N C_k^{-1}}\,.$$ Generally speaking, solutions $\mathbf{M}^\intercal$ to Eq. are matrices whose columns form a basis of the null space of $\mathbf{R}\mathbf{C}^{-1}$. Since the columns of $\overline{\mathbf{M}}^\intercal$ evidently form one particular basis of the null space, all possible solutions can be expressed as $$\mathbf{M} = \mathbf{A} \overline{\mathbf{M}}\,.$$ Here, $\mathbf{A}$ is an arbitrary non-singular $(N-1){\times}(N-1)$ matrix.
The Hamiltonian of the circuit is now obtained from the branch-flux Lagrangian by calculating the inverse $\mathbf{M}_+^{-1}$ and employing Eq. . For $\mathbf{M}=\overline{\mathbf{M}}$, the inverse $\mathbf{M}_+^{-1}$ can be evaluated analytically, and Legendre transform of the Lagrangian yields
$$\label{bigH}
\mathcal{H}_\text{irr} = \dfrac{1}{2}\sum_{i,j=1}^{N-1}\left[\dfrac{\delta_{ij}}{C_i}-\dfrac{1}{C_iC_j\sum_k C_k^{-1}}\right]Q_iQ_j
-\sum_{i=1}^{N-1} E_{\text{J}i} \cos{\left(\widetilde{\varphi}_i-\dfrac{C_i^{-1}}{\sum_k C_k^{-1}}\varphi_\text{e}\right)}
- E_{\text{J}N} \cos{\left( -\sum_{i=1}^{N-1}\widetilde{\varphi}_i -\dfrac{C_N^{-1}}{\sum_k C_k^{-1}}\varphi_\text{e} \right)}\,.$$
Hence, the weighting coefficient of the external flux for the $i$th element in the circuit loop is $C_i^{-1}/\sum_{k=1}^N C_k^{-1}$. (Note that the matrix $\mathbf{A}$ cannot affect the flux allocation.) It is simple to verify that the above Hamiltonian reduces to the Hamiltonian of the asymmetric SQUID Eq. when choosing $N=2$.
While our previous discussion only covers single-loop circuits consisting of Josephson junctions, it is not difficult to generalize to single-loop circuits including capacitors and inductors. If the loop includes capacitors, then $\mathbf{C}_\text{eff}$ remains invertible and we can proceed as before. As the only change, we must eliminate the corresponding junction potential terms from Eq. . (Details regarding the nature of the constraint replacing fluxoid quantization are provided in Appendix \[sec:Kirchhoff\].)
In the presence of an inductor $L$ in a single-loop circuit, our results can be adapted by taking an appropriate limit as follows. Let the inductor be the $N$th element in the loop[^6]. We temporarily associate an auxiliary parallel capacitor $C_N$ with the inductor, perform time-dependent circuit quantization as described above, and then let the capacitance $C_N$ go to zero. This limit can be directly performed on the Hamiltonian . Since the flux-grouping coefficients satisfy $$\dfrac{C_i^{-1}}{\sum_{k=1}^N C_k^{-1}} \xrightarrow{C_N\to0} \delta_{iN}\,,$$ the external flux is entirely grouped with the inductor in the single-loop circuit, and the resulting $N$th potential energy term reads $$\dfrac{1}{2}E_\text{L}\left(-\sum_{i=1}^{N-1}\widetilde{\varphi}_i-\varphi_\text{e}\right)^2$$ with $E_\text{L} = (\Phi_0/2\pi)^2/L$.
Strictly speaking, every circuit loop will have some finite geometric inductance $L$. Hence, one might wonder whether this inductance has to be included as it leads to different flux grouping. We show in Appendix \[sec:geoinductance\] that inclusion of negligible $L$ is not necessary and that the limit $L\to0$ is well-behaved. Specifically, the limit $L\to0$ leads back to the inductor-less case discussed above, with flux allocated to the various junction terms.
The above results are formulated for single-loop circuits, but actually apply more broadly. Often, only a single loop $\mathsf{L}$ inside a multi-loop circuit is threaded by a time-varying flux, while the flux through the remaining loops is zero. In cases where each element outside $\mathsf{L}$ is in parallel to an element inside $\mathsf{L}$, the multi-loop circuit effectively reduces to a single-loop circuit.
Multi-loop circuits
-------------------
Next, we extend our discussion to the case of multi-loop circuits threaded by time-dependent fluxes through each loop. Some of our development here resembles the discussion by Burkard *et al.* [@Burkard2004]. We begin by labeling all circuit elements (i.e., Josephson junctions, capacitors, inductors) in the network, and construct a vector containing the $N$ branch flux variables associated with the elements, $\bm{\Phi} = (\Phi_1,\Phi_2,\ldots,\Phi_N)^\intercal$. Due to the constraints from fluxoid quantization and Faraday’s law, these variables are in general not independent of each other.
Constraints apply to loops in the circuit. To eliminate ambiguity in the choice of loops, we select all meshes, which are loops containing no other loops. Suppose the circuit encompasses $F$ meshes, then we define $\bm{\Phi}_{\text{e}}=(\Phi_{\text{e}}^1,\Phi_{\text{e}}^2,\ldots,\Phi_{\text{e}}^F)^\intercal$ as the vector containing all time-dependent external fluxes threading each mesh. The positive directions of branch fluxes and external fluxes is defined in the beginning in one consistent way maintained throughout the calculation. The number of constraints applying to the $N$ variables is $F$, leading to $N{-}F$ degrees of freedom in the circuit.
We denote the vector composed of the corresponding dynamical variables by $\bm{\widetilde{\Phi}}=(\widetilde{\Phi}_1,\widetilde{\Phi}_2,\ldots,\widetilde{\Phi}_{N-F})^\intercal$. These $N{-}F$ variables are given by appropriate linear combinations of the $N$ branch variables, $$\bm{\widetilde{\Phi}} = \mathbf{M} \bm{\Phi}\,.$$ Here, $\mathbf{M}$ is an $(N{-}F){\times} N$ matrix with elements $(\mathbf{M})_{ij}$ to be determined. The constraints from fluxoid quantization and Faraday’s law can jointly be expressed as $$\bm{\Phi}_\text{e} = \mathbf{R} \bm{\Phi}\,,$$ where the $F{\times} N$ mesh matrix $\mathbf{R}$ is defined as follows. Let $\mathsf{L}_i$ be the $i$th mesh threaded by flux $\Phi_\text{e}^i$. Then the elements of $\mathbf{R}$ are given by $$(\mathbf{R})_{ij}=
\begin{cases}
\phantom{-}1\,, & \text{$\Phi_j \in \mathsf{L}_i$ with same orientation as $\Phi_\text{e}^i$} \\
-1\,, & \text{$\Phi_j \in \mathsf{L}_i$ with orientation opposite to $\Phi_\text{e}^i$} \\
\phantom{-}0\,, & \text{$\Phi_j \notin \mathsf{L}_i$}
\end{cases}\,,$$ where the orientation of $\Phi_\text{e}^i$ is clockwise (counterclockwise) when the local magnetic field points into (out of) the plane of the loop (see, e.g. the circular arrows in Fig. [\[fig:dsquid\]]{}). Analogous to the single-loop case, we further introduce the augmented vector $\bm{\widetilde{\Phi}}_+$ and the $N{\times} N$ matrix $\mathbf{M}_+$ via $$\bm{\widetilde{\Phi}}_+
=
\begin{pmatrix}
\bm{\widetilde{\Phi}} \\
\bm{\Phi}_\text{e}
\end{pmatrix}\,,
\qquad
\mathbf{M}_+
=
\begin{pmatrix}
\mathbf{M}\\
\mathbf{R}
\end{pmatrix}\,,$$ so that both information on degrees of freedom and constraints is again compactly captured by $$\bm{\widetilde{\Phi}}_+ = \mathbf{M}_+ \bm{\Phi}\,.$$ For the same reasons as in the single-loop case, we have $\det(\mathbf{M}_+){\neq} 0$.
To identify the irrotational degrees of freedom, we turn to the kinetic energy $$\mathcal{L}_\text{k} = \dfrac{1}{2}\dot{\bm{\widetilde{\Phi}}}_+^\intercal \mathbf{C}_\text{eff} \dot{\bm{\widetilde{\Phi}}}_+\,,$$ written in terms of $\mathbf{C}_\text{eff} = (\mathbf{M}_+^{-1})^\intercal \mathbf{C} \mathbf{M}_+^{-1}$. Irrotational degrees of freedom are obtained by making the terms $\propto\dot{\Phi}_\text{e}$ in the Lagrangian vanish. As before, this leads to the condition $$\mathbf{R}\mathbf{C}^{-1}\mathbf{M}^\intercal=0\,.$$
While $\mathbf{R}$ is very simple in the single-loop case, its structure is now more complex as it encodes the geometry of the multi-loop circuit. Different from the single-loop problem, this added complexity makes it impractical to obtain general closed expressions for $\mathbf{M}$. Instead, a particular solution for $\mathbf{M}$ can be obtained on a case-by-case basis as follows. Perform the singular-value decomposition $$\mathbf{R}\mathbf{C}^{-1} = \mathbf{U} \mathbf{S} \mathbf{V}\,,$$ where $\mathbf{V}$ is an $N{\times} N$ matrix. $\overline{\mathbf{M}}$ can be obtained from the transpose of the last $(N{-}F)$ columns of $\mathbf{V}$, and the general solution satisfies $\mathbf{M} = \mathbf{A} \overline{\mathbf{M}}$ with $\mathbf{A}$ an arbitrary non-singular $(N{-}F){\times}(N{-}F)$ matrix.
As before, Lagrangian and Hamiltonian are constructed by calculating $\mathbf{M}_+^{-1}$ and expressing branch variables in terms of the irrotational degrees of freedom. The grouping of flux $\Phi_\text{e}^j$ with the $i$th potential energy term can be obtained from $(\mathbf{M}_+^{-1})_{i,N-F+j}$. For capacitors and inductors in the multi-loop circuit we follow the same procedures discussed in the single-loop analysis. The above formulation likewise applies to multi-loop circuits threaded by both time-dependent and time-independent flux, considering static flux a special case of time-varying flux.
![\[fig:dsquid\] Multi-loop circuit formed by two SQUID loops with one shared junction. $\Phi_\text{1}$, $\Phi_\text{2}$, and $\Phi_\text{3}$ are branch flux variables for the three Josephson junctions. External fluxes $\Phi_\text{e}^1$ and $\Phi_\text{e}^2$ thread the left and right loops.](dsquid.pdf){width="38.00000%"}
For illustration, we apply the above formalism to a concrete multi-loop circuit: two SQUID loops with a shared junction (Fig. \[fig:dsquid\]). The junction capacitances and Josephson energies are denoted $C_k$ and $E_{\text{J}k}$ $(k=1,2,3)$. The left and right loops are threaded by external flux $\Phi_\text{e}^1$ and $\Phi_\text{e}^2$, respectively. The capacitance matrix and the mesh matrix for the given circuit-element orientations are $$\mathbf{C}
= \text{diag}(C_1,C_2,C_3)\,, \qquad
\mathbf{R}
=
\begin{pmatrix}
1 & -1 & 0 \\
0 & 1 & -1 \\
\end{pmatrix}\,.$$ Singular-value decomposition and re-scaling yields a particular solution of Eq. , namely $$\mathbf{M} = C_\Sigma^{-1}
\begin{pmatrix}
C_1 & C_2 & C_3
\end{pmatrix}\,,$$ where $C_\Sigma=\sum_{k=1}^3 C_k$. With this, one obtains $$\mathbf{M}_+^{-1}=
\begin{pmatrix}
1 & (C_2+C_3)C_\Sigma^{-1} & C_3C_\Sigma^{-1} \\
1 & -C_1C_\Sigma^{-1} & C_3C_\Sigma^{-1} \\
1 & -C_1C_\Sigma^{-1} & -(C_1+C_2)C_\Sigma^{-1} \\
\end{pmatrix}\,,$$ and $$\label{multi_cond}
\mathbf{C}_\text{eff} =
\begin{pmatrix}
C_\Sigma & 0 & 0 \\
0 & C_1(C_2+C_3)C_\Sigma^{-1} & C_1C_3C_\Sigma^{-1} \\
0 & C_1C_3C_\Sigma^{-1} & (C_1+C_2)C_3C_\Sigma^{-1} \\
\end{pmatrix}\,.$$ This now allows us to construct the irrotational Hamiltonian $$\begin{aligned}
\mathcal{H}_\text{irr} = & 4E_\text{C}n^2 \nonumber \\
& - E_\text{J1} \cos{\left(\widetilde{\varphi}+\dfrac{C_2+C_3}{C_\Sigma}\varphi_\text{e}^1+\dfrac{C_3}{C_\Sigma}\varphi_\text{e}^2\right)} \nonumber \\
& - E_\text{J2} \cos{\left(\widetilde{\varphi}-\dfrac{C_1}{C_\Sigma}\varphi_\text{e}^1+\dfrac{C_3}{C_\Sigma}\varphi_\text{e}^2\right)} \nonumber \\
& - E_\text{J3} \cos{\left(\widetilde{\varphi}-\dfrac{C_1}{C_\Sigma}\varphi_\text{e}^1-\dfrac{C_1+C_2}{C_\Sigma}\varphi_\text{e}^2\right)}\,,\end{aligned}$$ where $E_\text{C} = e^2/2C_\Sigma$.
Conclusions {#sec:summary}
===========
Standard circuit quantization [@qnetwork; @devoret1995quantum; @Burkard2004; @vool; @Ulrich2016] applies to circuits threaded by static external flux, but leads to inconsistencies when applied to circuits subject to time-dependent flux. Here, we have presented a generalization of circuit quantization for single-loop and multi-loop circuits that incorporates time-dependent flux. Our treatment shows that time-varying flux generally produces Lagrangian and Hamiltonian terms $\propto\dot{\Phi}_\text{e}$ which are crucial for resolving inconsistencies. We find that the freedom in choosing independent circuit variables leaves expectation values of single-time observables invariant, but generates gauge-dependent offsets in multi-time correlation functions. We have identified irrotational degrees of freedom which both eliminate the terms $\propto\dot{\Phi}_\text{e}$ in the Hamiltonian and remove the spurious offset terms in correlation functions associated with relaxation and pure-dephasing dynamics. Finally, we note that the irrotational constraint enforces a unique allocation of external flux to the various potential energy terms in the Hamiltonian.
We are indebted to Peter Groszkowski, Andy C. Y. Li, Yao Lu, and Jay Lawrence for illuminating discussions. This work was supported by the Northwestern–Fermilab Center for Applied Physics and Superconducting Technologies. The work of J. A. Sauls was supported by National Science Foundation Grant DMR-1508730.
Evaluation of $\langle C(0,t)\rangle_\text{av}$ \[sec:correlation\]
===================================================================
This appendix details the evaluation of the noise-averaged transition probability introduced in Sec. \[sec:correlator\], $$\label{cavg1}
\langle C(0,t) \rangle_\text{av} = \Big\langle |\langle \psi(0)|\psi(t)\rangle|^2 \Big\rangle_\text{av}\,.$$ This quantity is defined for a particular choice of the SQUID’s degree of freedom, specified by $(m_\text{l},m_\text{r})$. As in the main text, the initial state is the unperturbed excited state $|e^0(m_\text{l},m_\text{r})\rangle$, an eigenstate of $\mathcal{H}^0(m_l,m_r)$. (In the following, we will simplify notation and frequently suppress the dependence on $m_\text{l}$, $m_\text{r}$ where context allows.)
Truncating the Hilbert space to the subspace spanned by ground and first excited state, we can rewrite $$C(0,t) = |\langle \psi(0) | \psi(t) \rangle|^2 = 1 - |\langle g^0 | \psi(t) \rangle|^2\,,$$ and calculate the overlap with the ground state perturbatively. This yields $$\langle g^0 | \psi(t) \rangle = -\dfrac{i}{\hbar}\int_0^t e^{-i\omega_\text{eg}t'} V(t') dt'\,,$$ where the perturbation term in the integrand is $$\begin{aligned}
V(t) = & \langle g^0 | \partial_{\Phi_\text{e}} \mathcal{H}(m_\text{l},m_\text{r})| e^0\rangle \delta\Phi_\text{e}(t) \nonumber\\
& + \bar{\eta} \langle g^0 | Q |e^0\rangle \delta\dot{\Phi}_\text{e} (t) \nonumber\\
= & \underbrace{\langle g^0_\text{irr} | \partial_{\Phi_\text{e}} \mathcal{H}_\text{irr}| e^0_\text{irr}\rangle \delta\Phi_\text{e}(t)}_{V_\text{irr}(t)} \nonumber\\
& + \underbrace{\eta\langle g^0_\text{irr} | Q | e^0_\text{irr} \rangle [- i\omega_\text{eg}\delta\Phi_\text{e}(t) + \delta\dot{\Phi}_\text{e} (t)]}_{W(t)}\,.\end{aligned}$$ The last expression follows from applying Eq. . We note immediately that $V_\text{irr}(t)$ is precisely the perturbation term obtained within the irrotational frame of reference. Plugging back into Eq. , we obtain
$$\langle C(0,t) \rangle_\text{av} = 1-\dfrac{1}{\hbar^2}\int_0^{t}\!\!\!dt_1\int_0^{t}\!\!\!dt_2\,e^{-i\omega_\text{eg}(t_1-t_2)}\left[
\langle V_\text{irr}(t_1)V^*_\text{irr}(t_2) \rangle_\text{av} + \langle V_\text{irr}(t_1)W^*(t_2) + W(t_1) V_\text{irr}^*(t_2) \rangle_\text{av}
+ \langle W(t_1)W^*(t_2) \rangle_\text{av}
\right]\,.$$
Here, the term proportional to $\delta\Phi_\text{e}(t)$ in $W$ is canceled upon performing integration by parts on the $\delta\dot{\Phi}_\text{e}(t)$ term. For classical noise with zero mean we have $\langle \delta\Phi_\text{e}(t)\rangle_\text{av}=0$, and the noise spectrum is symmetric, $S_{\Phi_\text{e}}(\omega)=S_{\Phi_\text{e}}(-\omega)$. As a consequence, one finds that the terms mixing $V_\text{irr}$ and $W$ cancel out. We evaluate the remaining two terms separately. For the $V_\text{irr}$ correlator we find $$\begin{aligned}
\dfrac{1}{\hbar^2}\int_0^{t}\!\!\!dt_1\int_0^{t}\!\!\!dt_2\,e^{-i\omega_\text{eg}(t_1-t_2)}
\langle V_\text{irr}(t_1)V^*_\text{irr}(t_2) \rangle_\text{av}
&=\dfrac{1}{\hbar^2} |\langle g^0_\text{irr} | \partial_{\Phi_\text{e}} \mathcal{H}_\text{irr} | e^0_\text{irr}\rangle|^2 \int_0^{t}\!\!\!dt_1\int_0^{t}\!\!\!dt_2\, e^{-i\omega_\text{eg}(t_2-t_1)}\langle \delta\Phi_\text{e}(t_2-t_1)\delta\Phi_\text{e}(0)\rangle \nonumber\\
&\approx \dfrac{1}{\hbar^2}|\langle g^0_\text{irr} | \partial_{\Phi_\text{e}} \mathcal{H}_\text{irr} | e^0_\text{irr}\rangle|^2 S_{\Phi_\text{e}}(\omega_{\text{eg}})\,t\,,
\end{aligned}$$ where we have used $t \gg t_\text{c}$ (correlation time of the noise) for the approximation in the last step [@Clerk2010]. The $W$ correlator yields $$\begin{aligned}
\dfrac{1}{\hbar^2}\int_0^{t}\!\!\!dt_1\int_0^{t}\!\!\!dt_2\,e^{-i\omega_\text{eg}(t_1-t_2)}
\langle W_\text{irr}(t_1)W^*_\text{irr}(t_2) \rangle_\text{av} = & \dfrac{1}{\hbar^2}\eta^2|\langle g^0_\text{irr} | Q | e^0_\text{irr} \rangle|^2
\langle [\delta\Phi_\text{e}(t) e^{-i\omega t}-\delta\Phi_\text{e}(0)]\times[\text{c.c.}] \rangle_\text{av}\nonumber\\
\approx & \dfrac{1}{\hbar^2}\eta^2|\langle g^0_\text{irr} | Q | e^0_\text{irr} \rangle|^2 2\sigma^2\,,
\end{aligned}$$
using the same type of approximation as above. Together, these results confirm the validity of Eq. for $t_\text{c}\ll t\ll T_1$.
Constraint in single-loop circuits with capacitors\[sec:Kirchhoff\]
===================================================================
Fluxoid quantization applies to closed loops, consisting of Josephson junctions and inductors. In general, a loop may be interrupted by a capacitor. In this case, fluxoid quantization is not applicable, but Faraday’s law leads to a similar constraint for time-dependent flux [@Burkard2004]. To see this, consider the example of an LC circuit threaded by an external flux $\Phi_\text{e}$. If this flux is time-dependent, Faraday’s law results in the relation $$V_\text{C}+V_\text{L} = -\dot{\Phi}_\text{e}(t)\,,$$ where $V_\text{C}$ and $V_\text{L}$ are the voltages across the capacitor and inductor, respectively. The time integral of this equation leads to the following constraint on the branch flux variables: $$\Phi_\text{C} + \Phi_\text{L} = -\Phi_\text{e}(t)\,.$$ Once we redefine $\Phi_\text{e}$ to absorb the sign from Lenz’s rule, this constraint has the same form as the fluxoid-quantization constraint, see Eq. . It is straightforward to extend this result from the LC circuit to a general mesh in a circuit network. Thus, open loops involving capacitors can be treated on equal footing with the closed loops subject to fluxoid quantization.
Time of pure dephasing due to flux noise {#sec:pure_dephasing_time}
========================================
In the main text, we focused on flux-noise–induced relaxation of the qubit, and illustrated that inconsistencies may arise if flux is not grouped as prescribed by the irrotational frame of reference. We here briefly note that calculations of pure-dephasing times are likewise affected by the choice of the degree of freedom. For simplicity, we again ignore the $1/f$ nature of flux noise. Working in the irrotational frame, we indeed find the standard expression [@Clerk2010; @Groszkowski2017b] for the decay of the off-diagonal density matrix elements, $$\langle \rho_\text{eg} (t) \rangle \approx \exp[-(\partial_{\Phi_\text{e}}\omega_\text{eg})^2 S_{\Phi_\text{e}}(0)\,t/2 ]\,.$$ By contrast, when choosing other degrees of freedom specified by $(m_\text{l},m_\text{r})$, an additional frame-dependent term arises, $$\begin{aligned}
\langle \rho'_\text{eg} (t) \rangle \approx & \exp[-(\partial_{\Phi_\text{e}}\omega_\text{eg})^2 S_{\Phi_\text{e}}(0)\,t/2 ] \\\nonumber
& \times \exp[-(\langle g | Q | g \rangle - \langle e | Q | e \rangle)^2\eta^2(m_\text{l},m_\text{r})\sigma^2/\hbar^2]\,.\end{aligned}$$ As for the relaxation-time results, the spurious offset is eliminated by choosing the irrotational frame where the Lagrangian term $\propto\dot{\Phi}_\text{e}$ vanishes and $\eta=0$.
Geometric inductance and $L\to0$ limit {#sec:geoinductance}
======================================
Section \[sec:general\] showed that flux through loops composed of Josephson junctions is distributed across the various junction terms. By contrast, for loops including an inductor the flux is grouped with the inductor term. This situation raises two questions: Does the finite geometric inductance associated with any loop have to be included in circuit quantization even if it is negligibly small, and is the limit $L\to0$ singular? In this appendix we prove that both questions can be answered in the negative.
Consider a circuit loop composed of geometric inductance $L$, and $N$ Josephson junctions. Denote by $\Phi_\text{L}$ the branch flux across the inductor, and by $\Phi_1,\ldots,\Phi_N$ the junction branch variables. Fluxoid quantization leaves us with $N$ degrees of freedom. We may associate one of them with the inductor variable, $\widetilde{\Phi}_N=-\Phi_\text{L}+\Phi_\text{e}$, grouping the external flux exclusively with the inductor. The remaining $N{-}1$ degrees of freedom are composed of linear combinations of $\Phi_1,\ldots,\Phi_N$.
Generally, the resulting Lagrangian will have kinetic terms $\propto$ $\dot{\widetilde{\Phi}}_i\dot{\widetilde{\Phi}}_N$ ($i{\neq}N$) which couple the inductor to the remaining degrees of freedom. To evaluate the $L{\to}0$ limit, it is crucial to eliminate this coupling [@Qiu2016]. We can do so in a way analogous to Sec. \[singleloop\]. The fluxoid quantization condition is here replaced by $\sum_{i=1}^N \Phi_i=\widetilde{\Phi}_N$, i.e., the role played by $\Phi_\text{e}$ in Eq. is now played by $\widetilde{\Phi}_N$. Defining shorthands $\bm{\widetilde{\Phi}}=(\widetilde{\Phi}_1,\ldots,\widetilde{\Phi}_{N-1})$ and $\mathbf{Q}=(Q_1,\ldots,Q_{N-1})$, we obtain for the Hamiltonian $$\mathcal{H} = \mathcal{H}_\text{irr}(\mathbf{Q},\bm{\widetilde{\Phi}}, \widetilde{\Phi}_N) + \mathcal{H}_\text{osc}\,,$$ where $\mathcal{H}_\text{irr}(\mathbf{Q},\bm{\widetilde{\Phi}}, \widetilde{\Phi}_N)=[\text{Eq.}\ \eqref{bigH}]|_{\Phi_\text{e}=\widetilde{\Phi}_N}$. The oscillator Hamiltonian composed of inductor and series combination of all junction capacitors is $$\mathcal{H}_\text{osc} = 4E_\text{C}n_N^2 + \dfrac{1}{2}E_\text{L}(\widetilde{\varphi}_N-\varphi_\text{e})^2 - E_0$$ with $E_\text{C} = \dfrac{1}{2}e^2\sum_k C_k^{-1}$ and $E_\text{L} = (\Phi_0/2\pi)^2/L$. To prevent divergence of the oscillator ground state energy in the $L\to0$ limit, we explicitly subtract its value $E_0=\dfrac{1}{2}\sqrt{8E_\text{L}E_\text{C}}$.
In the limit $L\to0$ where $E_\text{L}\to\infty$, the oscillator will occupy its ground state. Moreover, position fluctuations around $\widetilde{\varphi}_N=\varphi_\text{e}$ will become negligible. As a result, we find the limiting behavior $$\mathcal{P}_0\mathcal{H}\mathcal{P}_0
\xrightarrow{E_\text{L}\to \infty}
\mathcal{H}_\text{irr}(\mathbf{Q},\bm{\widetilde{\Phi}},\Phi_\text{e})\,,$$ where $\mathcal{P}_0$ projects onto the oscillator ground state. We thus find that the $L\to0$ limit reproduces the irrotational Hamiltonian we obtained in Sec. \[singleloop\] without the inductor. In other words, the $L\to0$ limit is non-singular, and sufficiently small loop inductances may be neglected in the circuit quantization.
As a final check we consider the decay rate $\Gamma_1 \propto |\langle g| \partial_{\Phi_\text{e}} \mathcal{H} | e\rangle|^2$. It is tempting to reason that for a circuit loop with an inductor (where flux is grouped with the inductor term), one simply finds $\Gamma_1\propto E_\text{L}^2$ (wrong). To see that the $E_\text{L}\to\infty$ limit is well-behaved, we consider flux noise as denoted in Eq. . Taking the flux derivative, we see that the decay rate is proportional to $E_\text{L}^2 |\langle g^0 | \widetilde{\varphi}_N | e^0 \rangle|^2$, where $|g^0\rangle$ and $|e^0\rangle$ are eigenstates of $\mathcal{H}$ for $\Phi_\text{e} = \Phi_\text{e}^0$. Despite this form, $\Gamma_1$ does not diverge for $E_\text{L}\to \infty$, since both eigenstates and the operator $\widetilde{\varphi}_N$ depend on $E_\text{L}$ in such a way that $E_\text{L}$ factors cancel. We show in the following that the decay rate reaches a proper limit as the geometric inductance approaches zero.
We saw above that fluctuations of $\widetilde{\varphi}_N$ around $\varphi_\text{e}$ become negligible in the $E_\text{L}\to0$ limit. Hence, $\widetilde{\varphi}_N - \varphi_\text{e}$ is a small parameter. We rewrite $\widetilde{\varphi}_N$ as $\widetilde{\varphi}_N = \varphi_\text{e} + (\widetilde{\varphi}_N - \varphi_\text{e})$ and plug this into the Hamiltonian to obtain $$\mathcal{H} \approx \mathcal{H}_\text{irr}(\mathbf{Q},\bm{\widetilde{\Phi}},\Phi_\text{e}) + \mathcal{H}_\text{osc}
+ V\,,$$ with the perturbation $V=(\partial_{\varphi_\text{e}}\mathcal{H}_\text{irr})(\widetilde{\varphi}_N - \varphi_\text{e})$. Employing perturbation theory, the eigenstates $|g^0\rangle$ and $|e^0\rangle$ can now be expanded to first order in $V$. One finds that the resulting corrections are inversely proportional to $\hbar\omega_\text{osc}=\sqrt{8E_\text{C}E_\text{L}}$ for large $E_\text{L}$. Dependence on $E_\text{L}$ is also hiding in $\widetilde{\varphi}_N$ when expressed in terms of oscillator ladder operators, $\widetilde{\varphi}_N \propto (a+a^\dag)E_\text{L}^{-1/4}$. With these $E_\text{L}$ dependencies, the decay rate is seen to have a finite value in the limit $E_\text{L}\to\infty$. Furthermore, one finds $$\langle g^0|E_\text{L}\widetilde{\varphi}_N|e^0\rangle|^2 \xrightarrow{E_\text{L}\to \infty} |\langle g_\text{irr}^0|\partial_{\varphi_\text{e}}\mathcal{H}_\text{irr}|e_\text{irr}^0\rangle|^2\,,$$ where $|g_\text{irr}^0\rangle$ and $|e_\text{irr}^0\rangle$ are the eigenstates of the irrotational Hamiltonian $\mathcal{H}_\text{irr}(\mathbf{Q},\bm{\widetilde{\Phi}},\Phi_\text{e}^0)$. The RHS of the above equation reproduces the result obtained by neglecting the geometric inductance in the first place.
[^1]: Time-dependent flux is mentioned by Burkard *et al.* [@Burkard2004], but kinetic terms $\propto\dot{\Phi}_\text{e}$ are not discussed in that work.
[^2]: We note that the switch from eigenstates of $\mathcal{H}_\text{r}$ to those of $\mathcal{H}_\text{l}$ does not compensate the observed different scaling with $E_\text{Jl}$ and $E_\text{Jr}$, respectively.
[^3]: Typically, relaxation due to flux noise is subdominant due to the 1/$f$ nature of the noise, so that conclusions in the literature are not changed to the best of our knowledge.
[^4]: We note that there remains a continuous family of choices for $(m_\text{l},m_\text{r})$ satisfying Eq. . This freedom is characterized by the scaling factor $m_\Delta$. In Sec. [\[sec:general\]]{}, the remaining freedom is expressed by the matrix $\mathbf{A}$.\[foot\]
[^5]: As the $\dot\Phi_\text{e}$ term is related to a time-dependent unitary transformation (rotating-frame transformation), we call the constraint eliminating this term “irrotational".
[^6]: Just as in Ref. [@vool], we focus on circuits with a simply-connected capacitive sub-network. This avoids the situation where a generalized velocity is absent from the Lagrangian. With this requirement, there can only be *one* inductor in a single-loop circuit.
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---
abstract: 'K. Nikodem and the present author proved in [@NikWas95] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems connected with separation of $n$-convex function from $n$-concave function by a polynomial of degree at most $n$ and a stability result of Hyers-Ulam type for polynomials.'
author:
-
bibliography:
- 'was\_own.bib'
- 'was\_pub.bib'
date: January 1996
title: Polynomial selections and separation by polynomials
---
Introduction
============
By $\R$, $\N$ we denote the set of all reals and positive integers, respectively. Let $I\subset\R$ be an interval. In this paper we present a necessary and sufficient condition under which two functions $f,g:I\to\R$ can be separated by a polynomial of degree at most $n$, where $n\in\N$ is a fixed number. Our main result is a generalization of the theorem concerning separation by affine functions obtained recently by K. Nikodem and the present author in [@NikWas95]. To get it we use Behrends and Nikodem’s abstract selection theorem (cf. [@BerNik95 Theorem 1]). It is a variation of Helly’s theorem (cf. [@Val64 Theorem 6.1]).
By $\ccR$ we denote the family of all non-empty compact real intervals. Recall that if $F:I\to \ccR$ is a set-valued function then a function $f:I\to\R$ is called a *selection* of $F$ iff $f(x)\in F(x)$ for every $x\in I$.
Behrends and Nikodem’s theorem states that if $\mathscr{W}$ is an $n$-dimensional space of functions mapping $I$ into $\R$ then a set-valued function $F:I\to \ccR$ has a selection belonging to $\mathscr{W}$ if and only if for every $n+1$ points $x_{1},\dots,x_{n+1}\in I$ there exists a function $f\in\mathscr{W}$ such that $f(x_{i})\in F(x_{i})$ for $i=1,\dots,n+1$.
Let us start with the notation used in this paper. Let $n\in\N$. If $x_{1},\dots,x_{n}\in I$ are different points then for $i=1,\dots,n$ we define $$c_{i}(x;x_{1},\dots,x_{n})=
\prod_{\substack{j=1\\ j\ne i}}^{n}
\frac{x-x_{j}}{x_{i}-x_{j}}.$$
Note that $c_{i}(x_{j};x_{1},\dots,x_{n})$ is equal to 0 if $i\ne j$ and to 1 if $i=j$, $i,j=1,\dots,n$. ${\mathscr{P}}_{n}$ stands for the family of all polynomials of degree at most $n$. If $x_{1},\dots,
x_{n+1}\in I$ are different points then the Lagrange interpolating polynomial going through the points $(x_{i},y_{i})$, $i=1,\dots,n+1$, is given by the following formula: $$\label{Lagrange}
w(x)=\sum_{i=1}^{n+1} c_{i}(x;x_{1},\dots,x_{n+1})y_{i}.$$ This uniquely determined polynomial is a member of ${\mathscr{P}}_{n}$. Moreover, if $x<x_{1}<\dots<x_{n+1}$ then $c_{i}(x;x_{1},\dots,x_{n+1})$ is positive if $i$ is odd and negative if $i$ is even.
Polynomial selections of set-valued functions
=============================================
Now we shall prove a selection theorem which will be used to obtain our main result. If $n\in\N$ and $A_{i}\subset\R$, $i=1,\dots,n$, then $$\sum_{i=1}^{n} A_{i}$$ denotes the algebraic sum of the sets $A_{i}$, $i=1,\dots,n$.
\[tw1\] Let $n\in\N$. The set-valued function $F:I\to \ccR$ has a selection belonging to ${\mathscr{P}}_{n}$ if and only if for every $x_{0},x_{1},\dots,
x_{n+1}\in I$ such that $x_{0}<x_{1}<\dots<x_{n+1}$ the following condition holds: $$\label{selekcja}
F(x_{0})\cap \biggl(\sum_{i=1}^{n+1} c_{i}(x_{0};x_{1},\dots,x_{n+1})
F(x_{i})\biggr)\ne\emptyset.$$
If $F$ has a selection belonging to ${\mathscr{W}}_{n}$ then is obvious. We prove that implies an existence of a polynomial selection of $F$. First we note that ${\mathscr{P}}_{n}$ is an $(n+1)$-dimensional space of functions. If we prove that for every $n+2$ points $x_{0},x_{1},\dots,x_{n+1}\in I$ there exists a $w\in {\mathscr{P}}_{n}$ such that $w(x_{i})\in F(x_{i})$, $i=0,1,\dots,n+1$, then by Behrends and Nikodem’s theorem $F$ will have a desired selection. (For another Helly-type theorem which may be used here cf. also [@Val64 Theorem 6.9].)
Fix any different points $x_{0},x_{1},\dots,x_{n+1}\in I$ and assume $x_{0}< x_{1}<\dots<x_{n+1}$. Let $L_{i}=c_{i}(x_{0};x_{1},\dots,x_{n+1})$, $i=1,\dots,n+1$. Thus has the form $$\label{sel1}
F(x_{0})\cap\biggl(\sum_{i=1}^{n+1} L_{i}F(x_{i})\biggr)\ne\emptyset.$$ It is easy to observe that $L_{i}$ is positive if $i$ is odd and negative if $i$ is even.
Put $$y_{0}=\inf F(x_{0}),\;\;\;z_{0}=\sup F(x_{0})$$ and for $i=1,\dots,n+1$, $$y_{i}=\begin{cases}
\inf F(x_{i})&\text{if }L_{i}>0,\\
\sup F(x_{i})&\text{if }L_{i}<0,
\end{cases}\qquad
z_{i}=\begin{cases}
\sup F(x_{i})&\text{if }L_{i}>0,\\
\inf F(x_{i})&\text{if }L_{i}<0.
\end{cases}$$
Therefore $F(x_{0})=\left[ y_{0},z_{0}\right]$ and for $i=1,\dots,n+1$, $$F(x_{i})=\begin{cases}
[y_{i},z_{i}]&\text{if }L_{i}>0,\\
[z_{i},y_{i}]&\text{if }L_{i}<0.
\end{cases}$$ Since $-\left[\alpha,\beta\right] =\left[-\beta,-\alpha\right]$ for all $\alpha,\beta\in\R$, we have $L_{i}F(x_{i})=\left[ L_{i}y_{i}, L_{i}z_{i}
\right]$, $i=1,\dots,n+1$. If $u=L_{1}y_{1}+\dots+L_{n+1}y_{n+1}$ and $v=L_{1}z_{1}+\dots+L_{n+1}z_{n+1}$ then $u\le v$. Furthermore,
$$\begin{gathered}
\sum_{i=1}^{n+1} L_{i}F(x_{i})
=[L_{1}y_{1}, L_{1}z_{1}]+\dots+[L_{n+1}y_{n+1},L_{n+1}z_{n+1}]\\
=[ L_{1}y_{1}+\dots+L_{n+1}y_{n+1},L_{1}z_{1}+\dots+L_{n+1}z_{n+1}]=[u,v]\end{gathered}$$
and by we get $$\label{sel2}
\left[ y_{0},z_{0}\right]\cap\left[ u,v\right]\ne\emptyset.$$
There are three cases of the location of the above intervals:
(a) $u\in[y_{0},z_{0}]$,
(b) $v\in[y_{0},z_{0}]$,
(c) $[y_{0},z_{0}]\subset[u,v]$.
Fix $t\in\left[ 0,1\right]$ and consider the polynomial $\varphi _{t}\in{\mathscr{P}}_{n}$ going through $n+1$ different points: $$\bigl(x_{0},tu+(1-t)v\bigr)\text{\;\;and\;\;}
\bigl(x_{i},ty_{i}+(1-t)z_{i}\bigr)\text{\;\;for\;\;}i=1,\dots,n-1,n+1.$$ We shall show later that $$\label{fi_t}
\varphi _{t}(x_{n})=ty_{n}+(1-t)z_{n}.$$
Hence, in the case (a) for $w=\varphi _{1}$ we have $$\begin{aligned}
w(x_{0})&=u\in\left[ y_{0},z_{0}\right] =F(x_{0}),\\
w(x_{i})&=y_{i}\in F(x_{i}),\;\;i=1,\dots,n-1,n,n+1\end{aligned}$$ and similarly in the case (b) for $w=\varphi _{0}$. In the case (c) $y_{0}=\lambda u+(1-\lambda)v$ for some $\lambda\in\left[
0,1\right]$. For $w=\varphi _{\lambda}$ we obtain $$\begin{aligned}
w(x_{0})&=y_{0}\in F(x_{0}),\\
w(x_{i})&=\lambda y_{i}+(1-\lambda)z_{i}\in F(x_{i}),\;\;i=1,\dots,n-1,n,n+1.\end{aligned}$$ So in all cases there exists a $w\in {\mathscr{P}}_{n}$ such that $w(x_{i})\in
F(x_{i})$, $i=0,\dots,n+1$, which will complete the proof if we show that holds true.
By we get $$\begin{aligned}
\varphi_{t}(x)&=c_{0}(x;x_{0},x_{1},\dots,x_{n-1},x_{n+1})\bigl(tu+(1-t)v\bigr)\\
&+\sum_{i=1}^{n-1}c_{i}(x;x_{0},x_{1},\dots,x_{n-1},x_{n+1})\bigl(ty_{i}+(1-t)z_{i}\bigr)\\
&+c_{n+1}(x;x_{0},x_{1},\dots,x_{n-1},x_{n+1})\bigl(ty_{n+1}+(1-t)z_{n+1}\bigr).\end{aligned}$$ If $M_{i}=c_{i}(x_{n};x_{0},x_{1},\dots,x_{n-1},x_{n+1})$, $i=0,1,\dots,
n-1,n+1$, then after a bit of computation $$\varphi _{t}(x_{n})=\sum_{\stackrel{\scriptstyle i=1}{i\ne n}}^{n+1}
(M_{0}L_{i}+M_{i})\bigl(ty_{i}+(1-t)z_{i}\bigr)+M_{0}L_{n}\bigl(ty_{n}+(1-t)z_{n}\bigr).$$
One can verify (using the product formula given in Introduction) that $M_{0}L_{n}=1$ and $M_{0}L_{i}+M_{i}=0$ for $i=1,\dots,n-1,n+1$. Hence holds and this finishes the proof.
As a consequence of Theorem \[tw1\] we obtain
\[wn1\][[@Was95 Theorem 1]]{} A set-valued function $F:I\to \ccR$ has an affine selection iff for every $x,y\in I$, $t\in \left[ 0,1\right]$ $$F\bigl(tx+(1-t)y\bigr)\cap\bigl(tF(x)+(1-t)F(y)\bigr)\ne\emptyset.$$
The above condition is equivalent to for $n=1$, $x<y$, $x_{0}=x$, $x_{2}=y$, $x_{1}=tx_{0}+(1-t)x_{2}$, where $t=\frac{x_{1}-x_{2}}{x_{0}-x_{2}}$.
Separation by polynomials
=========================
The main result of this paper reads as follows
\[tw2\] Let $n\in\N$, $f,g:I\to\R$. The following conditions are equivalent:
1. \[tw2:i\] there exists $w\in {\mathscr{P}}_{n}$ such that $f(x)\le w(x)\le g(x)$, $x\in I$;
2. \[tw2:ii\] $f(b)\le g(b)$, where $b\in I$ is the right side endpoint of $I$ (if exists) and for every $x_{0},x_{1},\dots,x_{n+1}\in I$ such that $x_{0}\le x_{1}<\dots<x_{n+1}$
$$\label{oddziel}
\begin{aligned}
f(x_{0})&\le \sum_{\substack{i=1\\ i\text{\ \normalfont{odd}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})g(x_{i})+
\sum_{\substack{i=1\\ i\text{\ \normalfont{even}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i}),\\
g(x_{0})&\ge \sum_{\substack{i=1\\ i\text{\ \normalfont{odd}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i})+
\sum_{\substack{i=1\\ i\text{\ \normalfont{even}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})g(x_{i}).
\end{aligned}$$
To prove that implies fix any $x_{0},x_{1},\dots,x_{n+1}\in I$ such that $x_{0}\le x_{1}<\dots<x_{n+1}$. Since the polynomial $w$ goes through the points $(x_{i},w(x_{i}))$, $i=1,\dots,n+1$, we have $$w(x_{0})=\sum_{i=1}^{n+1} c_{i}(x_{0};x_{1},\dots,x_{n+1})w(x_{i}).$$ Then the ineualities are obvious.
To prove the converse implication first note that replacing $x_{0}$ by $x_{1}$ in we have $f(x_{1})\le g(x_{1})$ in both ineualities, i.e. (ii) yields $f\le g$ on $I$. Let $$F(x)=\left[ f(x),g(x)\right],\;\;\;x\in I.$$ We shall show that $F:I\to \ccR$ fulfils . Fix any $x_{0},
x_{1},\dots,x_{n+1}\in I$ such that $x_{0}<x_{1}<\dots<x_{n+1}$. Let $u,v$ be equal to the right hand sides of the upper and lower ineualities , respectively. Therefore $v\le u$ and $$\label{oddz1}
\left[ f(x_{0}),g(x_{0})\right]\cap\left[ v,u\right]\ne\emptyset$$ (otherwise $g(x_{0})<v$ or $u<f(x_{0})$ – contradiction with ). Let $L_{i}=c_{i}(x_{0};x_{1},\dots,x_{n+1})$, $i=1,\dots,n+1$. Then $$L_{i}F(x_{i})=\begin{cases}
[L_{i}f(x_{i}),L_{i}g(x_{i})]&\text{if $i$ is odd},\\
[L_{i}g(x_{i}),L_{i}f(x_{i})]&\text{if $i$ is even}
\end{cases}$$ and $$\left[ v,u\right] =\sum_{i=1}^{n+1} L_{i}F(x_{i}).$$ Thus implies . By Theorem \[tw1\] $F$ has a selection $w\in {\mathscr{P}}_{n}$. This finishes the proof.
Ineualities in Theorem \[tw2\] do not guarantee $f(b)\le g(b)$, where $b\in I$ is the right side endpoint of $I$ (if exists). The following two functions $$f(x)=\begin{cases}
\frac{1}{2}x&\text{for }0\le x<1,\\
1&\text{for }x=1
\end{cases}\qquad\text{and}\qquad
g(x)=\begin{cases}
x&\text{for }0\le x<1,\\
\frac{1}{2}&\text{for }x=1
\end{cases}$$ fulfil for $n=1$ but $f(1)>g(1)$. Of course, $f$ and $g$ can not be separated by a straight line.
As a consequence of Theorem \[tw2\] we obtain
[@NikWas95 Theorem 1] Let $f,g:I\to\R$. The following conditions are equivalent:
1. there exists an affine function $h:I\to\R$ such that $f(x)\le h(x)\le g(x)$, $x\in I$;
2. for every $x,y\in I$, $t\in\left[ 0,1\right]$ $$\begin{aligned}
f\bigl(tx+(1-t)y\bigr)&\le tg(x)+(1-t)g(y)\\
\intertext{and}
g\bigl(tx+(1-t)y\bigr)&\ge tf(x)+(1-t)f(y).
\end{aligned}$$
The above ineualities are equivalent to in Theorem \[tw2\] (cf. the proof of Corollary \[wn1\]).
Applications
============
One can verify that $f:I\to\R$ is convex iff for every $x_{0},x_{1},x_{2}\in I$ such that $x_{0}<x_{1}<x_{2}$ $$f(x_{0})\ge c_{1}(x_{0};x_{1},x_{2})f(x_{1})+
c_{2}(x_{0};x_{1},x_{2})f(x_{2}).$$ We adopt the following definition (cf. [@RobVar73 §83], [@Cie59], [@Pop44], [@Pop34]).
Let $n\in\N$. The function $f:I\to\R$ is $n$-*convex* iff for every $x_{0},x_{1},\dots,x_{n+1}\in I$ such that $x_{0}<x_{1}<\dots<x_{n+1}$ $$(-1)^{n}f(x_{0})\le (-1)^{n}\sum_{i=1}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i}).$$ $f$ is $n$-*concave* iff $(-f)$ is $n$-convex.
If $f$ is both $n$-convex and $n$-concave then $f$ is a polynomial belonging to ${\mathscr{P}}_{n}$ (going through the points $(x_{i},f(x_{i}))$, $i=0,1,\dots,n+1$).
\[wn3\] Let $n\in\N$. If $f:I\to\R$ is $n$-convex, $g:I\to\R$ is $n$-concave and $f(x)\le g(x)$, $x\in I$, then there exists a polynomial $w\in {\mathscr{P}}
_{n}$ such that $f(x)\le w(x)\le g(x)$, $x\in I$.
Fix any $x_{0},x_{1},\dots,x_{n+1}\in I$ such that $x_{0}\le
x_{1}<\dots<x_{n+1}$. If $n$ is even then by $n$-convexity of $f$ $$\begin{aligned}
f(x_{0})&\le\sum_{i=1}^{n+1}c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i})\\
&\le\sum_{\substack{i=1\\ i\text{\ \normalfont{odd}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})g(x_{i})
+\sum_{\substack{i=1\\ i\text{\ \normalfont{even}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i}).\end{aligned}$$ If $n$ is odd then by $n$-concavity of $g$ $$\begin{aligned}
f(x_{0})&\le g(x_{0})\le\sum_{i=1}^{n+1}c_{i}(x_{0};x_{1},\dots,x_{n+1})g(x_{i})\\
&\le\sum_{\substack{i=1\\ i\text{\ \normalfont{odd}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})g(x_{i})
+\sum_{\substack{i=1\\ i\text{\ \normalfont{even}}}}^{n+1}
c_{i}(x_{0};x_{1},\dots,x_{n+1})f(x_{i}).\end{aligned}$$ The proof of the second ineuality in is analogous. Theorem \[tw2\] completes the proof.
In the same way we get
\[wn4\] Let $n\in\N$. If $f:I\to\R$ is $n$-concave, $g:I\to\R$ is $n$-convex and $f(x)\le g(x)$, $x\in I$, then there exists a polynomial $w\in {\mathscr{P}}
_{n}$ such that $f(x)\le w(x)\le g(x)$, $x\in I$.
For $n=1$ the above two results are well known and Corollaries \[wn3\] and \[wn4\] generalize them.
Finally we prove a stability result for polynomials (cf. a Hyers-Ulam stability theorem for affine functions in [@NikWas95]). First observe that if $n\in\N$ and $w(x)=1$, $x\in I$, then $w\in {\mathscr{P}}_{n}$ and for every different points $x_{1},\dots,x_{n+1}\in I$ has the form $$\sum_{i=1}^{n+1} c_{i}(x;x_{1},\dots,x_{n+1})=1,\;\;x\in I.$$
Let $n\in\N$, $\varepsilon >0$ and $f:I\to\R$. If for every $x_{0},x_{1},\dots,$\
$x_{n+1}\in I$ such that $x_{0}\le x_{1}<\dots<x_{n+1}$ $$\label{modul1}
\biggl| f(x_{0})-\sum_{i=1}^{n+1} c_{i}(x_{0};x_{1},\dots,x_{n+1})
f(x_{i})\biggr| \le\varepsilon$$ then there exists a polynomial $w\in {\mathscr{P}}_{n}$ such that $$\label{modul2}
\left| f(x)-w(x)\right| \le\frac{\varepsilon}{2},\;\;x\in I.$$
If $f$ satisfies then $(ii)$ in Theorem \[tw2\] holds for $g(x)=f(x)+\varepsilon$, $x\in I$. So there exists a polynomial $\varphi\in {\mathscr{P}}_{n}$ such that $f(x)\le\varphi (x)\le f(x)+\varepsilon$, $x\in I$. For $$w(x)=\varphi (x)-\frac{\varepsilon}{2},\;\;x\in I$$ we obtain .
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abstract: 'Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models. The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations. In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established. In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain. In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles.'
address:
- |
R. Douc\
Télécom SudParis\
9 rue Charles Fourier\
91011 Evry-Cedex\
France\
- |
A. Garivier\
E. Moulines\
Télécom ParisTech\
46 rue Barrault\
75634 Paris, Cedex 13\
France\
\
- |
J. Olsson\
Center for Mathematical Sciences\
Lund University\
Box 118\
SE-22100 Lund\
Sweden\
author:
-
-
-
-
title: 'Sequential [M]{}onte [C]{}arlo smoothing for general state space hidden Markov models'
---
,
,
and
.
.
Introduction {#sec1}
============
Statistical inference in general state space hidden Markov models (HMM) involves computation of the *posterior distribution* of a set ${\ifthenelse{\equal{}{}}{\ensuremath{{X}_{s:s'}}}{
\ensuremath{X^}_{s:s'}}
} {\stackrel{\mathrm{def}}{=}}[X_s, \dots, X_{s'}]$ of state variables conditional on a record ${\ifthenelse{\equal{}{}}{\ensuremath{{Y}_{0:T}}}{
\ensuremath{Y^}_{0:T}}
} = {\ifthenelse{\equal{}{}}{\ensuremath{{y}_{0:t}}}{
\ensuremath{y^}_{0:t}}
}$ of observations. This distribution will, in the following, be denoted by ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s'|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s'|T}}}}
}
}
}$ where the dependence of this measure on the observed values ${\ifthenelse{\equal{}{}}{\ensuremath{{y}_{0:T}}}{
\ensuremath{y^}_{0:T}}
}$ is implicit. The posterior distribution can be expressed in closed-form only in very specific cases, principally, when the state space model is linear and Gaussian or when the state space of the hidden Markov chain is a finite set. In the vast majority of cases, nonlinearity or non-Gaussianity render analytic solutions intractable [@kailathsayedhassibi2000; @risticarulampalamgordon2004; @cappemoulinesryden2005; @vanhandel2007].
This limitation has led to an increase of interest in alternative computational strategies handling more general state and measurement equations without constraining a priori the behavior of the posterior distributions. Among these, *sequential [M]{}onte [C]{}arlo* (SMC) *methods* play a central role. SMC methods—in which the *sequential importance sampling* and *sampling importance resampling* methods proposed by [@handschinmayne1969] and [@rubin1987], respectively, are combined—refer to a class of algorithms approximating a *sequence of probability distributions*, defined on a *sequence of probability spaces*, by updating recursively a set of random *particles* with associated nonnegative *importance weights*. The SMC methodology has emerged as a key tool for approximating state posterior distribution flows in general state space models; see [@delmoralmiclo2000; @delmoral2004; @delmoraldoucet2009] for general introductions as well as theoretical results for SMC methods and [@liu2001; @doucetdefreitasgordon2001; @risticarulampalamgordon2004] for applications of SMC within a variety of scientific fields.
The recursive formulas generating the *filter distributions* $\phi_{T}$ (short-hand notation for ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{T:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{T:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{T:T|T}}}}
}
}
}$) and the *joint smoothing distributions* ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}$ are closely related; thus, executing the standard SMC scheme in the filtering mode provides, as a by-product, approximations of the joint smoothing distributions. More specifically, the branches of the genealogical tree associated with the historical evolution of the filtering particles up to time step $T$ form, when combined with the corresponding importance weights of these filtering particles, a weighted sample approximating the joint smoothing distribution ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}$; see [@delmoral2004], Section 3.4, for details. From these paths, one may readily obtain a weighted sample targeting the fixed lag or fixed interval smoothing distribution by extracting the required subsequence of states while retaining the weights. This appealingly simple scheme can be used successfully for estimating the joint smoothing distribution for small values of $T$ or any marginal smoothing distribution ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}$, with $s \leq T$, when $s$ and $T$ are close; however, when $T$ is large and $s \ll T$, the associated particle approximations are inaccurate since the genealogical tree degenerates gradually as the interacting particle system evolves [@godsilldoucetwest2004; @fearnheadwyncolltawn2010].
In this article, we thus give attention to more sophisticated approaches and consider instead the *forward filtering backward smoothing* (FFBSm) *algorithm* and the *forward filtering backward simulation* (FFBSi) *sampler*. These algorithms share some similarities with the Baum–Welch algorithm for finite state space models and the Kalman filter-based smoother and simulation smoother for linear Gaussian state space models [@dejongshephard1995]. In the FFBSm algorithm, the particle weights obtained when approximating the filter distributions in a forward filtering pass are modified in a backward pass; see [@kitagawa1996; @huerzelerkuensch1998; @doucetgodsillandrieu2000]. The FFBSi algorithm simulates, conditionally independently given the particles and particle weights produced in a similar forward filtering pass, state trajectories being approximately distributed according to the joint smoothing distribution; see [@godsilldoucetwest2004].
The computational complexity of the FFBSm algorithm when used for estimating marginal fixed interval smoothing distributions or of the original formulation of the FFBSi sampler grows (in most situations) as the square of the number $N$ of particles multiplied by the time horizon $T$. To alleviate this potentially very large computational cost, some methods using intricate data structures for storing the particles have been developed; see, for example, [@klaasbriersdefreitasdoucetmaskelllang2006]. These algorithms have a complexity of order $O(N \log(N))$ and are thus amenable to practical applications; however, this reduction in complexity comes at the cost of introducing some level of approximation.
In this paper, a modification of the original FFBSi algorithm is presented. The proposed scheme has a complexity that grows only *linearly* in $N$ and does not involve any numerical approximation techniques. This algorithm may be seen as an alternative to a recent proposal by [@fearnheadwyncolltawn2010] which is based on the so-called *two-filter algorithm* [@briersdoucetmaskell2010].
The smoothing weights computed in the backward pass of the FFBSm algorithm at a given time instant $s$ (or the law of the FFBSi algorithm) are statistically dependent on all forward filtering pass particles and weights computed before and after this time instant. This intricate dependence structure makes the analysis of the resulting particle approximation challenging; up to our best knowledge, only a single consistency result is available in [@godsilldoucetwest2004], but its proof is plagued by a (subtle) mistake that seems difficult to correct. Therefore, very little is known about the convergence of the schemes under consideration, and the second purpose of this paper is to fill this gap.[^1] In this contribution, we focus first on finite time horizon approximations. Given a finite time horizon $T$, we derive *exponential deviation inequalities* stating that the probability of obtaining, when replacing ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}$ by the corresponding FFBSm or FFBSi estimator, a [M]{}onte [C]{}arlo error exceeding a given $\epsilon
> 0$ is bounded by a quantity of order $O(\exp(- c N \epsilon^2))$ where $c$ is positive constant depending on $T$ as well as the target function under consideration. The obtained inequalities, which are presented in Theorem \[thm:Hoeffding-FFBS\] (FFBSm) and Corollary \[cor:Hoeffding-FFBSi\] (FFBSi), hold for any given number $N$ of particles and are obtained by combining a novel backward error decomposition with an adaptation of the Hoeffding inequality to statistics expressed as ratios of random variables. We then consider the asymptotic (as the number $N$ of particles tends to infinity) regime and establish a central limit theorem (CLT) with rate $\sqrt
{N}$ and with an explicit expression of the asymptotic variance; see Theorem \[thm:FFBS-CLT\]. The proof of our CLT relies on a technique, developed gradually in [@chopin2004; @kuensch2005; @doucmoulines2008], which is based on a CLT for triangular arrays of dependent random variables; however, since we are required to take the complex dependence structure of the smoothing weights into account, our proof is significantly more involved than in the standard filtering framework considered in the mentioned works.
The second part of the paper is devoted to time uniform results, and we here study the behavior of the particle-based marginal smoothing distribution approximations as the time horizon $T$ tends to infinity. In this setting, we first establish, under the assumption that the Markov transition kernel ${M}$ of the latent signal is strongly mixing (Assumption \[assum:strong-mixing-condition\]), time uniform deviation bounds of the type described above which hold for any particle population size $N$ and where the constant $c$ is *independent* of $T$; see Theorem \[theo:Hoeffding-uniform\]. This result may seem surprising, and the nonobvious reason for its validity stems from the fact that the underlying Markov chain forgets, when evolving conditionally on the observations, its initial conditions in the forward *as well as* the backward directions. Finally, we prove (see Theorem \[theo:CLT-uniform\]), under the same uniform mixing assumption, that the asymptotic variance of the CLT for the particle-based marginal smoothing distribution approximations remains bounded as $T$ tends to infinity. The uniform mixing assumption in Assumption \[assum:strong-mixing-condition\] points typically to applications where the state space of the latent signal is compact; nevertheless, in the light of recent results on filtering stability [@kleptsynaveretennikov2008; @doucfortmoulinespriouret2009] one may expect the geometrical contraction of the backward kernel to hold for a significantly larger class of nonuniformly mixing models (see [@doucfortmoulinespriouret2009] for examples from, e.g., financial economics). But even though the geometrical mixing rate is supposed to be constant in this more general case, applying the mentioned results will yield a bound of contraction containing a multiplicative constant depending highly on the initial distributions as well as the observation record under consideration. Since there are currently no available results describing this dependence, applying such bounds to the instrumental decomposition used in the proof of Theorem \[thm:Hoeffding-FFBS\] seems technically involved. Recently, [@vanhandel2009] managed to derive *qualitative* time average convergence results for standard (bootstrap-type) particle filters under a mild tightness assumption being satisfied also in the noncompact case when the hidden chain is geometrically ergodic. Even though this technique does not (on the contrary to our approach) supply a rate of convergence, it could possibly be adopted to our framework in order to establish time average convergence of the particle-based marginal smoothing distribution approximations in a noncompact setting.
The paper is organized as follows. In Section \[sec:FFBS\], the FFBSm algorithm and the FFBSi sampler are introduced. An exponential deviation inequality for the fixed interval joint smoothing distribution is derived in Section \[sec:exponentialFFBS\], and a CLT is established in Section \[sec:CLTFFBS\]. In Section \[sec:TimeUniformExponentialFFBS\], time uniform exponential bounds on the error of the FFBSm marginal smoothing distribution estimator are computed under the mentioned mixing condition on the kernel ${M}$. Finally, under the same mixing condition, an explicit bound on the asymptotic variance of the marginal smoothing distribution estimator is derived in Section \[sec:TimeUniformCLTFFBS\].
Notation and definitions {#sec:Notations and Definitions .unnumbered}
------------------------
For any sequence $\{a_n\}_{n \geq0}$ and any pair of integers $0 \leq
m \leq n$, we denote ${\ifthenelse{\equal{}{}}{\ensuremath{{a}_{m:n}}}{
\ensuremath{a^}_{m:n}}
} {\stackrel{\mathrm{def}}{=}}(a_m, \dots, a_n)$. We assume in the following that all random variables are defined on a common probability space $(\Omega, {\mathcal{F}_{}^{}}, {\mathbb{P}})$. The sets ${\mathbb{X}}$ and ${\mathbb{Y}}$ are supposed to be Polish spaces and we denote by ${\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}$ and ${\mathcal{B}({\mathbb{Y}})}$ the associated Borel $\sigma$-algebras. ${\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$ denotes the set of all bounded ${\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}/ \mathcal{B}({\mathbb{R}})$-measurable functions from ${\mathbb{X}}$ to ${\mathbb{R}}$. For any measure $\zeta$ on $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$ and any $\zeta$-integrable function $f$, we set $\zeta(f) {\stackrel{\mathrm{def}}{=}}\int_{\mathbb{X}}f(x) \zeta({d}x)$. Two measures $\zeta$ and $\zeta'$ are said to be *proportional* (written $\zeta\propto\zeta'$) if they differ only by a normalization constant.
A kernel $V$ from $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$ to $({\mathbb{Y}}, {\mathcal{B}({\mathbb{Y}})})$ is a mapping from ${\mathbb{X}}\times{\mathcal{B}({\mathbb{Y}})}$ into $[0, 1]$ such that, for each $A \in{\mathcal{B}({\mathbb{Y}})}$, $x \mapsto V (x, A)$ is a nonnegative, bounded, and measurable function on ${\mathbb{X}}$, and, for each $x \in{\mathbb{X}}$, $A
\mapsto V (x, A)$ is a measure on ${\mathcal{B}({\mathbb{Y}})}$. For $f \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$ and $x \in{\mathbb{X}}$, denote by $V(x, f) {\stackrel{\mathrm{def}}{=}}\int V(x, {d}x')
f(x')$; we will sometimes also use the abridged notation $Vf(x)$ instead of $V(x, f)$. For a measure $\nu$ on $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$, we denote by $\nu V$ the measure on $({\mathbb{Y}}, {\mathcal{B}({\mathbb{Y}})})$ defined by, for any $A \in{\mathcal{B}({\mathbb{Y}})}$, $\nu V(A) {\stackrel{\mathrm{def}}{=}}\int_{\mathbb{X}}V(x,A) \nu({d}x)$.
Consider now a possibly nonlinear state space model, where the *state process* $\{ X_t \}_{t \geq0}$ is a Markov chain on the state space $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$. Even though $t$ is not necessarily a temporal index, we will often refer to this index as “time.” We denote by ${\chi}$ and ${M}$ the initial distribution and transition kernel, respectively, of this process. The state process is assumed to be hidden but partially observed through the *observations* $\{ Y_t
\}_{t \geq0}$ which are ${\mathbb{Y}}$-valued random variables being conditionally independent given the latent state sequence $\{ X_t \}_{t
\geq0}$; in addition, there exists a $\sigma$-finite measure $\lambda
$ on $({\mathbb{Y}}, {\mathcal{B}({\mathbb{Y}})})$ and a nonnegative transition density function $g$ on ${\mathbb{X}}\times{\mathbb{Y}}$ such that ${\ifthenelse{\equal{}{}}{\mathbb{P}[ Y_t \in A | X_t
]}{\mathbb{P}_{}[ Y_t \in A | X_t ]}} = \int_A
g(X_t, y) \lambda({d}y)$ for all $A \in{\mathcal{B}({\mathbb{Y}})}$. The mapping $x
\mapsto g(x, y)$ is referred to as the *likelihood function* of the state given an observed value $y \in{\mathbb{Y}}$. The kernel ${M}$ as well as the transition density $g$ are supposed to be known. In the setting of this paper, we assume that we have access to a record of arbitrary but fixed observations ${\ifthenelse{\equal{}{}}{\ensuremath{{y}_{0:T}}}{
\ensuremath{y^}_{0:T}}
} {\stackrel{\mathrm{def}}{=}}[y_0, \dots,
y_T]$, and our main task is to estimate the posterior distribution of (different subsets of) the state vector $X_{0:T}$ given these observations. For any $t \geq0$, we denote by $g_t(x) {\stackrel{\mathrm{def}}{=}}g(x,
y_t)$ (where the dependence on $y_t$ is implicit) the likelihood function of the state $X_t$ given the observation $y_t$.
For simplicity, we consider a *fully dominated* state space model for which there exists a $\sigma$-finite measure $\nu$ on $({\mathbb{X}},
{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$ such that, for all $x \in{\mathbb{X}}$, ${M}(x, \cdot)$ has a transition probability density $\ensuremath{m}(x, \cdot)$ [with respect to]{} $\nu
$. For notational simplicity, $\nu({d}x)$ will sometimes be replaced by ${d}x$.
For any initial distribution ${\chi}$ on $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$ and any $0 \leq s \leq s'\leq T$, denote by ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s'|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s'|T}}}}
}
}
}$ the posterior distribution of the state vector ${\ifthenelse{\equal{}{}}{\ensuremath{{X}_{s:s'}}}{
\ensuremath{X^}_{s:s'}}
}$ given the observations ${\ifthenelse{\equal{}{}}{\ensuremath{{y}_{0:T}}}{
\ensuremath{y^}_{0:T}}
}$. For lucidity, the dependence of ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s'|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s'|T}}}}
}
}
}$ on the initial distribution ${\chi}$ is omitted. Assuming that ${\int\cdots\int}{\chi}({d}x_0) \prod_{u = 1}^T g_{u-1}(x_{u-1}) {M}(x_{u-1},\allowbreak {d}x_u) g_T(x_T)\,{>}\,0$, this distribution may be expressed as, for all $h\,{\in}\,{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{s'-s+1})}$, $${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s'|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s'|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s'|T}}}}
}
}
}(h) =
\frac{{\int\cdots\int}{\chi}({d}x_0) \prod_{u = 1}^T g_{u-1}(x_{u-1}) {M}(x_{u-1}, {d}x_u) g_T(x_T) h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s:s'}}}{
\ensuremath{x^}_{s:s'}}
})}
{{\int\cdots\int}{\chi}({d}x_0) \prod_{v = 1}^T g_{v-1}(x_{v-1}) {M}(x_{v-1}, {d}x_v) g_T(x_T)} .$$ In the expression above, the dependence on the observation sequence is implicit. If $s = s'$, we use ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}$ (the marginal smoothing distribution at time $s$) as shorthand for ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s|T}}}}
}
}
}$. If $s = s'
= T$, we denote by ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
} {\stackrel{\mathrm{def}}{=}}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s}}}}
}
}
}$ the filtering distribution at time $s$.
Algorithms {#sec:FFBS}
==========
Conditionally on the observations ${\ifthenelse{\equal{}{}}{\ensuremath{{y}_{0:T}}}{
\ensuremath{y^}_{0:T}}
}$, the state sequence $\{ X_s \}_{s \geq0}$ is a time inhomogeneous Markov chain. This property remains true in the *time-reversed* direction. Denote by ${\mathrm{B}_{\eta}}$ the so-called *backward kernel* given by, for any probability measure $\eta$ on $({\mathbb{X}}, {\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$, $$\label{eq:backward-kernel}
{\mathrm{B}_{\eta}}(x, h) {\stackrel{\mathrm{def}}{=}}\frac{\int\eta({d}x') \ensuremath{m}(x',
x) h(x')}
{\int\eta({d}x') \ensuremath{m}(x', x)} ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})} .$$ The posterior distribution ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}$ may be expressed as, for any integers $T > 0$, $s \in\{0, \dots, T-1\}
$ and any $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T-s+1})}$, $$\label{eq:smoothing:backw_decomposition}
\quad {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h) = {\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}({d}x_T) {\mathrm{B}_{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{T-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{T-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T-1}}}
}
}
}
}
}}}(x_T, {d}x_{T-1}) \cdots{\mathrm{B}_{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}}}(x_{s+1}, {d}x_s)
h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s:T}}}{
\ensuremath{x^}_{s:T}}
}) .\hspace*{-10pt}$$ Therefore, the joint smoothing distribution may be computed recursively, backward in time, according to $$\label{eq:smoothing:backw_decomposition_recursion}
{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h) = {\int\cdots\int}{\mathrm{B}_{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}}}(x_{s+1}, {d}x_s) {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s+1:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s+1:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s+1:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s+1:T|T}}}}
}
}
}({d}{\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s+1:T}}}{
\ensuremath{x^}_{s+1:T}}
}) h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s:T}}}{
\ensuremath{x^}_{s:T}}
}) .$$
The forward filtering backward smoothing algorithm
--------------------------------------------------
As mentioned in the , the method proposed by [@huerzelerkuensch1998; @doucetgodsillandrieu2000] for approximating the smoothing distribution is a two pass procedure. In the forward pass, particle approximations ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ of the filter distributions ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ are computed recursively for all time steps from $s = 0$ up to $s = T$. The filter distribution flow $\{
{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
} \}_{s \geq0}$ satisfies the forward recursion $$\label{eq:forward-filtering-recursion}
\quad {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h) = \frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h)}{{
\ifthenelse{\equal{}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}({\mathbf{1}})} \qquad \mbox{where } {
\ifthenelse{\equal{}{}}{\ensuremath{\gamma_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\gamma^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{0}}}
}
}
}
}
}(h) = {\chi}(g_0 h) , {
\ifthenelse{\equal{}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h) {\stackrel{\mathrm{def}}{=}}{
\ifthenelse{\equal{}{}}{\ensuremath{\gamma_{s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\gamma^{N}_{s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s-1}}}
}
}
}
}
} {M}(g_s h) , s \geq1 ,$$ for $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, with ${\mathbf{1}}$ being the unity function $x
\mapsto1$ on ${\mathbb{X}}$. In terms of SMC, each filter distribution ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ is approximated by means of a set of particles $\{ {\ensuremath{\xi_{s}^{i}}} \}_{i = 1}^N$ and associated importance weights $\{ {\ensuremath{\omega_{s}^{i}}} \}_{i = 1}^N$ according to $$\label{eq:approximation-filtering-distribution}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h) {\stackrel{\mathrm{def}}{=}}\frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h)}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}({\mathbf{1}})} \qquad \mbox{where } {
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h) {\stackrel{\mathrm{def}}{=}}N^{-1} \sum_{i =
1}^N {\ensuremath{\omega_{s}^{i}}} h({\ensuremath{\xi_{s}^{i}}}) .$$ Having produced, using methods described in Section \[section:APF\] below, a sequence of such weighted samples $\{ ({\ensuremath{\xi_{t}^{i}}}, {\ensuremath{\omega_{t}^{i}}}) \}_{i = 1}^N$, $1 \leq t \leq T$, an approximation of the smoothing distribution is constructed in a backward pass by replacing, in [(\[eq:smoothing:backw\_decomposition\])]{}, the filtering distribution by its particle approximation. This yields $$\label{eq:smoothing:backw_decomposition_sample}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h) {\stackrel{\mathrm{def}}{=}}\int\!\!\cdots\!\!\int\!{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}({d}x_T) {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{T-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{T-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T-1}}}
}
}
}
}
}}}(x_T,{d}x_{T-1})\cdots{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}}}(x_{s+1}, {d}x_s) h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s:T}}}{
\ensuremath{x^}_{s:T}}
})\hspace*{-30pt}$$ for any $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T-s+1})}$. The approximation above can be computed recursively in the backward direction according to $$\label{eq:smoothing:backw_decomposition_recursion_sample}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h) = \int\!\!\cdots\!\!\int{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}}}(x_{s+1}, {d}x_s) {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s+1:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s+1:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s+1:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s+1:T|T}}}}
}
}
}({d}{\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s+1:T}}}{
\ensuremath{x^}_{s+1:T}}
}) h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{s:T}}}{
\ensuremath{x^}_{s:T}}
}).$$ Now, by definition, $${\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}}}(x, h) = \sum_{i = 1}^N \frac{{\ensuremath{\omega_{s}^{i}}}
\ensuremath{m}
({\ensuremath{\xi_{s}^{i}}}, x)}{\sum_{\ell= 1}^N {\ensuremath{\omega_{s}^{\ell}}} \ensuremath
{m}({\ensuremath{\xi_{s}^{\ell}}},x)} h ( {\ensuremath{\xi_{s}^{i}}} ) ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})} ,$$ and inserting this expression into [(\[eq:smoothing:backw\_decomposition\_sample\])]{} gives $$\label{eq:forward-filtering-backward-smoothing}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h) \!=\! \sum_{i_s = 1}^N\!\!\cdots\!\!\sum_{i_T = 1}^N
\Biggl(\!\prod_{u=s+1}^T \frac{{\ensuremath{\omega_{u-1}^{i_{u-1}}}} \ensuremath{m}({\ensuremath{\xi_{u-1}^{i_{u-1}}}},{\ensuremath{\xi_{u}^{i_u}}})}{\sum_{\ell=1}^N {\ensuremath{\omega_{u-1}^{\ell}}}
\ensuremath{m}({\ensuremath{\xi_{u-1}^{\ell}}},{\ensuremath{\xi_{u}^{i_u}}})} \!\Biggr) \frac{{\ensuremath{\omega_{T}^{i_T}}}}{{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{T}}}{\ensuremath{
\Omega_{T}^{()}}}}} h ({\ensuremath{\xi_{s}^{i_s}}}, \dots, {\ensuremath{\xi_{T}^{i_T}}} )
,\hspace*{-30pt}$$ of ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h)$, where $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T-s-1})}$ and $$\label{eq:defOmega}
{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t}}}{\ensuremath{
\Omega_{t}^{()}}}} {\stackrel{\mathrm{def}}{=}}\sum_{i=1}^N {\ensuremath{\omega_{t}^{i}}} .$$ The estimator ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}$ is impractical since the cardinality of its support grows exponentially with the number $T-s$ of time steps; nevertheless, it plays a key role in the theoretical developments that follow. A more practical approximation of this quantity will be defined in the next section. When the dimension of the input space is moderate, the computational cost of evaluating the estimator can be reduced to $O(N \log N)$ by using the *fast multipole method* as suggested in [@klaasbriersdefreitasdoucetmaskelllang2006]; note, however, that this method involves approximations that introduce some bias. On the other hand, in certain specific scenarios, such as discrete Markov chains with sparse transition matrices over large state spaces, the complexity can even be reduced to $O(N T)$ without any truncation; see [@barembruchgariviermoulines2009].
The forward filtering backward simulation algorithm {#subsec:FFBSi}
---------------------------------------------------
The estimator [(\[eq:forward-filtering-backward-smoothing\])]{} may be understood alternatively by noting that the normalized smoothing weights define a probability distribution on the set $\{1, \dots, N\}
^{T - s}$ of trajectories associated with an inhomogeneous Markov chain. Indeed, consider, for $t \in\{0, \dots, T-1\}$, the Markov transition matrix $\{ \Lambda_t^N(i,j) \}_{i,j = 1}^N$ given by $$\label{eq:definition-transition-matrix-W}
\Lambda^N_{t}(i,j) = \frac{{\ensuremath{\omega_{t}^{j}}} \ensuremath{m}({\ensuremath{\xi_{t}^{j}}}, {\ensuremath{\xi_{t+1}^{i}}})}{\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{t}^{\ell}}},
{\ensuremath{\xi_{t+1}^{i}}})} ,\qquad (i,j) \in\{1, \dots, N\}^2 .$$ For $1 \leq t \leq T$, denote by $$\label{eq:definition-mcf}
{\mathcal{F}_{t}^{N}} {\stackrel{\mathrm{def}}{=}}\sigma\{{\ifthenelse{\equal{}{}}{\ensuremath{{Y}_{0:T}}}{
\ensuremath{Y^}_{0:T}}
}, ({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\omega_{s}^{i}}}); 0 \leq s \leq t, 1 \leq i \leq N\}$$ the $\sigma$-algebra generated by the observations from time $0$ to time $T$ as well as the particles and importance weights produced in the forward pass up to time $t$. The transition probabilities defined in [(\[eq:definition-transition-matrix-W\])]{} induce an inhomogeneous Markov chain $\{ J_{u} \}_{u = 0}^T$ evolving backward in time as follows. At time $T$, the random index $J_T$ is drawn from the set $\{1, \dots, N\}
$ such that $J_T$ takes the value $i$ with a probability proportional to ${\ensuremath{\omega_{T}^{i}}}$. At time $t \leq T-1$ and given that the index $J_{t + 1}$ was drawn at time step $t + 1$, the index $J_t$ is drawn from the set $\{1,\dots,N\}$ such that $J_t$ takes the value $j$ with probability $\Lambda
^N_{t}(J_t, j)$. The joint distribution of ${\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
}$ is therefore given by, for ${\ifthenelse{\equal{}{}}{\ensuremath{{j}_{0:T}}}{
\ensuremath{j^}_{0:T}}
} \in\{1, \dots, N\}^{T + 1}$, $$\label{eq:distribution-non-homogeneous}
\mathbb{P} [ {\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
} = {\ifthenelse{\equal{}{}}{\ensuremath{{j}_{0:T}}}{
\ensuremath{j^}_{0:T}}
} | {\mathcal{F}_{T}^{N}} ] =
\frac{{\ensuremath{\omega_{T}^{j_T}}}}{{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{T}}}{\ensuremath{
\Omega_{T}^{()}}}}} \Lambda_T^N(J_T, j_{T-1}) \cdots
\Lambda_0^N(j_1, j_0) .$$ Thus, and this is a key observation, the FFBS estimator [(\[eq:forward-filtering-backward-smoothing\])]{} of the joint smoothing distribution may be written as the conditional expectation $$\label{eq:EspCond}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) = {\ifthenelse{\equal{}{}}{\mathbb{E}[ h ({\ensuremath{\xi_{0}^{J_0}}}, \dots, {\ensuremath{\xi_{T}^{J_T}}} ) | {\mathcal{F}_{T}^{N}}
]}{\mathbb{E}_{}[ h ({\ensuremath{\xi_{0}^{J_0}}}, \dots, {\ensuremath{\xi_{T}^{J_T}}} ) | {\mathcal{F}_{T}^{N}} ]}} ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})} .$$ We may therefore construct an unbiased estimator of the FFBS estimator by drawing, conditionally independently given ${\mathcal{F}_{T}^{N}}$, $N$ paths of $\{{\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
}^\ell\}_{\ell=1}^N$ of the inhomogeneous Markov chain introduced above and then forming the (practical) estimator $$\label{eq:FFBSi:estimator}
{
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) = N^{-1} \sum_{\ell= 1}^N h ( {\ensuremath{\xi_{0}^{J_0^\ell}}}, \dots, {\ensuremath{\xi_{T}^{J_T^\ell}}} ) ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})} .$$ This practical estimator was introduced in [@godsilldoucetwest2004] (Algorithm 1, page 158). For ease of notation, we have here simulated $N$ replicates of the backward, index-valued Markov chain, but it would of course also be possible to sample a number of paths that is either larger or smaller than $N$. The estimator ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}$ may be seen as a Rao–Blackwellized version of ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}$. The variance of the latter is increased, but the gain in computational complexity is significant. The associated algorithm is referred in the sequel to as the forward filtering backward simulation (FFBSi) algorithm. In Section \[sec:TimeUniformExponentialFFBS\], forgetting properties of the inhomogeneous backward chain will play a key role when establishing time uniform stability properties of the proposed smoothing algorithm.
The computational complexity for sampling a single path of ${\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
}$ is $O(NT)$; therefore, the overall computational effort spent when estimating ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
} $ using the FFBSi sampler is $O(N^2T)$. Following [@klaasbriersdefreitasdoucetmaskelllang2006], this complexity can be reduced further to $O(N \log(N) T)$ by means of the fast multipole method; however, here again computational work is gained at the cost of introducing additional approximations.
A fast version of the forward filtering backward simulation algorithm
---------------------------------------------------------------------
We are now ready to describe one of the main contributions of this paper, namely a novel version of the FFBSi algorithm that can be proved to reach linear computational complexity under appropriate assumptions. At the end of the filtering phase of the FFBSi algorithm, all weighted particle samples $\{({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\omega_{s}^{i}}})\}_{i=1}^N$, $0 \leq
s \leq T$, are available, and it remains to sample efficiently index paths $\{ {\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
}^\ell\}_{\ell= 1}^N$ under the distribution [(\[eq:distribution-non-homogeneous\])]{}. When the transition kernel $\ensuremath{m}$ is bounded from above in the sense that $\ensuremath{m}(x,x') \leq\sigma_+$ for all $(x,x') \in{\mathbb{X}}\times{\mathbb{X}}$, the paths can be simulated recursively backward in time using the following accept–reject procedure. As in the standard FFBSi algorithm, the recursion is initiated by sampling $J_T^1, \dots, J_T^N$ multinomially with probabilities proportional to $\{ {\ensuremath{\omega_{T}^{i}}} \}
_{i = 1}^N$. For $s \in\{0, \dots, T\}$, let $\mathcal{G}_s^N$ the smallest $\sigma$-field containing ${\mathcal{F}_{T}^{N}}$ and $\sigma(J_t^\ell
\dvtx 1\leq l\leq N, t\geq s)$; then in order to draw $J_s^\ell$ conditionally on $\mathcal{G}_{s+1}^N$, we draw, first, an index proposal $I^{\ell}_s$ taking the value $i \in\{1, \ldots, N\}$ with a probability proportional to ${\ensuremath{\omega_{t}^{i}}}$ and, second, an independent uniform random variable $U_s^{\ell}$ on $[0,1]$. Then we set $J_s^\ell= I_s^\ell$ if $U_s^\ell\leq\ensuremath{m}({\ensuremath{\xi_{s}^{I_s^\ell}}},
{\ensuremath{\xi_{s+1}^{J_{s+1}^\ell}}}) / \sigma_+$; otherwise, we reject the proposed index and make another trial. To create samples of size $n \in\{1, \dots, N\}$ from a multinomial distribution on a set of $N$ elements at lines 1 and 6, Algorithm \[alg:smooth\] relies on an efficient procedure described in Appendix \[subsec:multisampling\] that requires $O (
n(1+\log(1+N/n)) )$ elementary operations; see Proposition \[prop:multisample\]. Using this technique, the computational complexity of Algorithm \[alg:smooth\] can be upper-bounded as follows.
\[alg:samp1\]sample $J_T^1, \dots, J_T^N$ multinomially with probabilities proportional to $\{ {\ensuremath{\omega_{T}^{i}}} \}_{i = 1}^N$ $L \gets(1, \dots, N)$ $n\gets$ size(L) \[alg:updatem\] \[alg:samp2\] sample $I_1, \dots, I_n$ multinomially with probabilities proportional to$\{ {\ensuremath{\omega_{s}^{i}}} \}_{i = 1}^N$ sample $U_1,\dots,U_n$ independently and uniformly over $[0,1]$ $nL \gets\varnothing$ $J_{s}^{L(k)} \gets I_k$ $nL \gets nL \cup\{L(k)\}$ $L \gets nL$
For the bootstrap particle filter as well as the fully adapted auxiliary particle filter (see Section \[section:APF\] for precise descriptions of these SMC filters), it is possible to derive an asymptotic expression for the number of simulations required at line 8 of Algorithm \[alg:smooth\] even if the kernel $m$ is not bounded from below. The following result is obtained using theory derived in the coming section.
\[prop:complexityBoostrap\] Assume that the transition kernel is bounded from above, $m(x, x') \leq\sigma_+$ for all $(x, x') \in{\mathbb{X}}\times{\mathbb{X}}$. At each iteration $s \in\{0, \dots, T-1\}$, let $Z_s^N$ be the number of simulations required in the accept–reject procedure of Algorithm \[alg:smooth\].
- For the bootstrap auxiliary filter, $Z_s^N /N$ converges in probability to $$\alpha(s) {\stackrel{\mathrm{def}}{=}}\sigma_+ {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}(g_s) \frac{{\int\cdots\int}{d}x_{s+1} \prod_{u = s + 2}^T \int m(x_{u-1}, {d}x_u)
g_u(x_u)}{{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}({d}x_s) g_s(x_s) \prod_{u = s +
1}^T m(x_{u-1}, {d}x_u) g_u(x_u)}$$ as $N$ goes to infinity.
- In the fully adapted case, $Z_s^N / N$ converges in probability to $$\beta(s) {\stackrel{\mathrm{def}}{=}}\sigma_+ \frac{{\int\cdots\int}{d}x_{s+1}\prod
_{u=s+2}^T \int m(x_{u-1}, {d}x_u) g_u(x_u)}{{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}({d}x_s) g_s(x_s) \prod_{u = s + 1}^T m(x_{u-1}, {d}x_u) g_u(x_u)}$$ as $N$ goes to infinity.
[=0.2em plus 0.05em minus 0.02em A sufficient condition for ensuring finiteness of $\alpha(s)$ and $\beta(s)$ is that$\int g_u(x_u)\, {d}x_u <$]{} $ \infty$ for all $u \geq0$.
If the transition kernel satisfies stronger mixing conditions, it is possible to derive an upper-bound on the computational complexity of the FFBSi for any auxiliary particle filter, that is, the total number of computations (and not only the total number of simulations). Note that this result is not limited to the bootstrap and the fully adapted cases.
\[prop:complexity\] Assume that the transition kernel is bounded from below and above, that is, $\sigma_- \leq m(x, x') \leq\sigma_+$ for all $(x, x') \in
{\mathbb{X}}\times{\mathbb{X}}$. Let $C(N,T)$ denote the number of elementary operations required in Algorithm \[alg:smooth\]. Then, there exists a constant $K$ such that such that $ {\mathbb{E}}[C(N,T)] \leq K N T \sigma_+ / \sigma_- $.
The proofs of Propositions \[prop:complexityBoostrap\] and \[prop:complexity\] involve theory developed in the coming section and are postponed to Section \[sec:complexity:proofs\].
Before concluding this section on reduced complexity, let us mention that efficient smoothing strategies have been considered by [@fearnhead2005] using quasi-[M]{}onte [C]{}arlo methods. The smoother (restricted to be one-dimensional) presented in this work has a complexity that grows quadraticly in the number of particles $N$; nevertheless, since the variance of the same decays as $O(N^{-2})$ (or faster) thanks to the use of quasi-random numbers, the method is equivalent to methods with complexity growing linearly in $N$ \[since the standard [M]{}onte [C]{}arlo variance is $O(N^{-1})$\]. This solution is of course attractive; we are however not aware of extensions of this approach to multiple dimensions.
Auxiliary particle filters {#section:APF}
--------------------------
It remains to describe in detail how to produce sequentially the weighted samples $\{({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\omega_{s}^{i}}})\}_{i = 1}^N$, $0
\leq s \leq T$, which can be done in several different ways (see [@doucetdefreitasgordon2001; @liu2001; @cappemoulinesryden2005] and the references therein). Still, most algorithms may be formulated within the unifying framework of the [*auxiliary particle filter*]{} described in the following. Let $\{{\ensuremath{\xi_{0}^{i}}}\}_{i = 1}^N$ be [i.i.d.]{} random variables such that ${\ensuremath{\xi_{0}^{i}}} \sim{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}$ and set ${\ensuremath{\omega_{0}^{i}}} {\stackrel{\mathrm{def}}{=}}{d}{\chi}/ {d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}})$. The weighted sample $\{ ({\ensuremath{\xi_{0}^{i}}}, {\ensuremath{\omega_{0}^{i}}})\}_{i =
1}^N$ then targets the initial filter ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}$ in the sense that ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}(h)$ estimates ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}(h)$ for $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$. In order to describe the sequential structure of the auxiliary particle filter, we proceed inductively and assume that we have at hand a weighted sample $\{ ({\ensuremath{\xi_{s-1}^{i}}}, {\ensuremath{\omega_{s-1}^{i}}}) \}_{i = 1}^N$ targeting ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s - 1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s - 1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s - 1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s - 1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s - 1}}}
}
}
}
}
}$ in the same sense. Next, we aim at simulating new particles from the target ${
\ifthenelse{\equal{tar}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{tar}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{tar}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{tar}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{tar}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ defined as $$\label{eq:target-forward-filtering}
{
\ifthenelse{\equal{tar}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{tar}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{tar}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{tar}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{tar}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}(h) = \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{s-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{s-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s-1}}}
}
}
}
}
} {M}(g_s h)}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{s-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{s-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{s-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{s-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{s-1}}}
}
}
}
}
} {M}(g_s)} ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})} ,$$ in order to produce an updated particle sample approximating the subsequent filter ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$. Following [@pittshephard1999], this may be done by considering the *auxiliary* target distribution $$\label{eq:target-auxiliary}
{
\ifthenelse{\equal{aux}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{aux}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{aux}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{aux}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{aux}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}{} (i, h) {\stackrel{\mathrm{def}}{=}}\frac{{\ensuremath{\omega_{s-1}^{i}}} {M}({\ensuremath{\xi_{s-1}^{i}}}, g_s h)}{\sum_{\ell= 1}^N {\ensuremath{\omega_{s-1}^{\ell}}} {M}({\ensuremath{\xi_{s-1}^{\ell}}}, g_s h)} ,\qquad h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})} ,$$ on the product space $\{ 1, \dots, N \} \times{\mathbb{X}}$ equipped with the product $\sigma$-algebra $\mathcal{P}(\{ 1, \dots, N \}) \otimes
{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}$. By construction, ${
\ifthenelse{\equal{tar}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{tar}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{tar}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{tar}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{tar}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ is the marginal distribution of $\phi
_s^{N, \mathrm{a}}$ with respect to the particle index. Therefore, we may approximate the target distribution ${
\ifthenelse{\equal{tar}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{tar}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{tar}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{tar}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{tar}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$ on $({\mathbb{X}},
{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$ by simulating from the auxiliary distribution and then discarding the indices. More specifically, we first simulate pairs $\{
(I_s^i, {\ensuremath{\xi_{s}^{i}}}) \}_{i = 1}^N$ of indices and particles from the instrumental distribution $$\label{eq:instrumental-distribution-filtering}
{\pi_{s|s}}(i, h) \propto{\ensuremath{\omega_{s-1}^{i}}} {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\vartheta_{s}}{
\vartheta_{s}({\ensuremath{\xi_{s-1}^{i}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}({\ensuremath{\xi_{s-1}^{i}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}({\ensuremath{\xi_{s-1}^{i}}})}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{P_{s}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{s}}{\mathrm{erreur}}}}}({\ensuremath{\xi_{s-1}^{i}}},h) ,\qquad h \in
{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})} ,$$ on the product space $\{1, \dots, N\} \times{\mathbb{X}}$, where $\{
{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\vartheta_{s}}{
\vartheta_{s}({\ensuremath{\xi_{s-1}^{i}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}({\ensuremath{\xi_{s-1}^{i}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{i}}}}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}({\ensuremath{\xi_{s-1}^{i}}})}}{\mathrm{erreur}}}}} \}_{i = 1}^N$ are so-called *adjustment multiplier weights* and ${\ifthenelse{\equal{}{}}{P_{s}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{s}}{\mathrm{erreur}}}}}$ is a Markovian *proposal* transition kernel. In the sequel, we assume for simplicity that ${\ifthenelse{\equal{}{}}{P_{s}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{s}}{\mathrm{erreur}}}}}(x, \cdot)$ has, for any $x
\in{\mathbb{X}}$, a density ${\ifthenelse{\equal{}{}}{p_{s}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{s}}{\mathrm{erreur}}}}}(x, \cdot)$ with respect to the reference measure $\nu$. For each draw $(I_s^i,
{\ensuremath{\xi_{s}^{i}}})$, $i = 1, \dots, N$, we compute the importance weight $$\label{eq:weight-update-filtering}
{\ensuremath{\omega_{s}^{i}}} {\stackrel{\mathrm{def}}{=}}\frac{\ensuremath{m}({\ensuremath{\xi_{s-1}^{I_s^i}}},{\ensuremath{\xi_{s}^{i}}})
g_s({\ensuremath{\xi_{s}^{i}}})}{{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{I_s^i}}}}{}}{\vartheta_{s}}{
\vartheta_{s}({\ensuremath{\xi_{s-1}^{I_s^i}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{I_s^i}}}}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}({\ensuremath{\xi_{s-1}^{I_s^i}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{s-1}^{I_s^i}}}}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}({\ensuremath{\xi_{s-1}^{I_s^i}}})}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{p_{s}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{s}}{\mathrm{erreur}}}}}({\ensuremath{\xi_{s-1}^{I_s^i}}},{\ensuremath{\xi_{s}^{i}}})} ,$$ such that ${\ensuremath{\omega_{s}^{i}}} \propto{d}\phi_s^{N, \mathrm{a}} / {d}{\pi_{s|s}}(I_s^i, {\ensuremath{\xi_{s}^{i}}})$, and associate it to the corresponding particle position ${\ensuremath{\xi_{s}^{i}}}$. Finally, the indices $\{I_s^i \}_{i = 1}^N$ are discarded whereupon $ \{ ({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\omega_{s}^{i}}} ) \}_{i=1}^N$ is taken as an approximation of ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}$. The simplest choice, yielding to the so-called *bootstrap particle filter algorithm* proposed by [@gordonsalmondsmith1993], consists of setting, for all $x \in{\mathbb{X}}$, ${\ifthenelse{\equal{}{}}{\ifthenelse{\equal{x}{}}{\vartheta_{s}}{
\vartheta_{s}(x)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{x}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}(x)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{x}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}(x)}}{\mathrm{erreur}}}}} \equiv1$ and ${\ifthenelse{\equal{}{}}{p_{s}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{s}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{s}}{\mathrm{erreur}}}}}(x, \cdot) \equiv
\ensuremath{m}
(x, \cdot)$. A more appealing—but often computationally costly—choice consists of using the adjustment weights ${\ifthenelse{\equal{}{}}{\ifthenelse{\equal{x}{}}{\vartheta_{s}}{
\vartheta_{s}(x)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{x}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}(x)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{x}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}(x)}}{\mathrm{erreur}}}}} \equiv{\ifthenelse{\equal{fully}{}}{\ifthenelse{\equal{x}{}}{\vartheta_{s}}{
\vartheta_{s}(x)}}
{\ifthenelse{\equal{fully}{smooth}}{\ifthenelse{\equal{x}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}(x)}}
{\ifthenelse{\equal{fully}{fully}}{\ifthenelse{\equal{x}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}(x)}}{\mathrm{erreur}}}}} {\stackrel{\mathrm{def}}{=}}\int
\ensuremath{m}(x,
x') g_s(x') \,{d}x'$, $x \in{\mathbb{X}}$, and the proposal transition density $${\ifthenelse{\equal{fully}{}}{p_{s}}
{\ifthenelse{\equal{fully}{fully}}{p^{\star}_{s}}
{\ifthenelse{\equal{fully}{smooth}}{\tilde{r}_{s}}{\mathrm{erreur}}}}}(x, x') {\stackrel{\mathrm{def}}{=}}\frac{\ensuremath{m}(x,x')
g_s(x')}{{\ifthenelse{\equal{fully}{}}{\ifthenelse{\equal{x}{}}{\vartheta_{s}}{
\vartheta_{s}(x)}}
{\ifthenelse{\equal{fully}{smooth}}{\ifthenelse{\equal{x}{}}{\tilde{
\vartheta}_{s}}{\tilde{\vartheta}_{s}(x)}}
{\ifthenelse{\equal{fully}{fully}}{\ifthenelse{\equal{x}{}}{\vartheta^
\star_{s}}{\vartheta^\star_{s}(x)}}{\mathrm{erreur}}}}}} ,\qquad (x, x') \in{\mathbb{X}}\times{\mathbb{X}}.$$ In this case, the auxiliary particle filter is referred to as *fully adapted*. Other choices are discussed in [@doucmoulinesolsson2008] and [@cornebisemoulinesolsson2008].
Convergence of the FFBS and FFBSi algorithms
============================================
In this section, the convergence of the FFBS and FFBSi algorithms are studied. For these two algorithms, nonasymptotic Hoeffding-type deviation inequalities and CLTs are obtained. We also introduce a decomposition, serving as a basis for most results obtained in this paper, of the error ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
} - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}$ and some technical conditions under which the results are derived.
For any function $f\dvtx {\mathbb{X}}^d \to{\mathbb{R}}$, we define by ${\ifthenelse{\equal{}{}}{| f |_\infty}{| f|^2_{\infty}}}
{\stackrel{\mathrm{def}}{=}}\sup_{x\in{\mathbb{X}}^d} |f(x)|$ and ${\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(f
)}}{\ensuremath{\operatorname{osc}^{}(f)}}} {\stackrel{\mathrm{def}}{=}}\sup
_{(x,x') \in{\mathbb{X}}^d \times{\mathbb{X}}^d} |f(x)-f(x')|$ the supremum and oscillator norms, respectively. Denote $\bar{{\mathbb{N}}} {\stackrel{\mathrm{def}}{=}}{\mathbb{N}}\cup
\{\infty\}$ and consider the following assumptions where $T$ is the time horizon which can be either a finite integer or infinity.
\[assum:bound-likelihood\] For all $0 \leq t \leq T$, $g_t(\cdot) > 0$ and $\sup_{0 \leq t
\leq T} {\ifthenelse{\equal{}{}}{| g_t |_\infty}{| g_t|^2_{\infty}}} <\infty$.
Define for $t \geq0$ the importance weight functions $$\label{eq:definition-weightfunction-forward}
\quad {\ensuremath{\omega_{0}}}(x) {\stackrel{\mathrm{def}}{=}}\frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}(x)
g_0(x) \quad \mbox{and}\quad {\ensuremath{\omega_{t}}}(x,x') {\stackrel{\mathrm{def}}{=}}\frac{\ensuremath{m}(x,x')
g_t(x')}{{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{x}{}}{\vartheta_{t}}{
\vartheta_{t}(x)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{x}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(x)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{x}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(x)}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(x,x')} ,\qquad t \geq1 .$$
\[assum:borne-FFBS\] $\sup_{1 \leq t \leq T} {\ifthenelse{\equal{}{}}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}} |_\infty}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|^2_{\infty}}} < \infty$ and $\sup_{0 \leq t \leq T} {\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}} < \infty$.
The latter assumption is rather mild; it holds in particular under Assumption \[assum:bound-likelihood\] for the bootstrap filter (${\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}=\ensuremath{m}$ and ${\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}\equiv1$) and is automatically fulfilled in the fully adapted case (${\ensuremath{\omega_{t}}} \equiv1$).
The coming proofs are based on a decomposition of the joint smoothing distribution that we introduce below. For $0 \leq t < T$ and $h \in
{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, define the kernel $L_{t,T}: {\mathbb{X}}^{t+1}\times
{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}^{\otimes T+1} \to[0,1]$ by $$\label{eq:defLt}
L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h) {\stackrel{\mathrm{def}}{=}}{\int\cdots\int}\Biggl(\prod_{u=t+1}^{T} {M}(x_{u-1},{d}x_{u}) g_{u}(x_{u}) \Biggr) h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:T}}}{
\ensuremath{x^}_{0:T}}
})$$ and set $L_{T,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:T}}}{
\ensuremath{x^}_{0:T}}
},h){\stackrel{\mathrm{def}}{=}}h({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:T}}}{
\ensuremath{x^}_{0:T}}
})$. By construction, for every $t \in\{0,\dots,T\}$, the joint smoothing distribution may be expressed as $$\label{eq:smooth:recursion}
{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)=\frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,h)]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,{\mathbf{1}})]} .$$ This expression extends the classical forward–backward decomposition to the joint smoothing distribution; here $L_{t,T}(\cdot, h)$ plays the role of the so-called backward variable. This suggests to decompose the error ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)$ as the following telescoping sum: $$\begin{aligned}
\label{eq:decomp_Smooth}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) &=& \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,{\mathbf{1}})]} -
\frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,h)]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,{\mathbf{1}})]}
\nonumber\\[-8pt]\\[-8pt]
&&{} + \sum_{t=1}^{T} \biggl\{ \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot
,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,{\mathbf{1}})]} - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,{\mathbf{1}})]} \biggr\} .\hspace*{-16pt}\nonumber\end{aligned}$$ The first term on RHS of the decomposition above can be easily dealt with since ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}$ is a weighted empirical distribution associated to [i.i.d.]{} random variables.
To cope with the terms in the sum of the RHS in [(\[eq:decomp\_Smooth\])]{}, we introduce some kernels (depending on the [*past*]{} particles) that stress the dependence [with respect to]{} the [*current*]{} particules. More precisely, ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,h)]$ is expressed as $$\label{eq:randomDenom}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,h)] = {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t}}}
}
}
}
}
}[{\mathcal L}^N_{t,T}(\cdot,h)] = \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{t}}}
}
}
}
}
}[{\mathcal L}^N_{t,T}(\cdot
,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{t}}}
}
}
}
}
}({\mathbf{1}})} ,$$ where the random kernels ${\mathcal L}^N_{t,T}: {\mathbb{X}}\times{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}^{\otimes
(T+1)}\to[0,1]$ are defined by: for all $0 < t \leq T$, and $x_t \in
{\mathbb{X}}$, $$\label{eq:definition-Ft}
{\mathcal L}^N_{t,T}(x_t,h) {\stackrel{\mathrm{def}}{=}}{\int\cdots\int}{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}}}(x_t, {d}x_{t-1}) \cdots{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}}}(x_{1}, {d}x_{0}) L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}, h) ,\hspace*{-25pt}$$ and $$\label{eq:definition-Fs-As}
{\mathcal L}^N_{0,T}(x,h) {\stackrel{\mathrm{def}}{=}}L_{0,T}(x,h) .$$ We stress that the kernels ${\mathcal L}^N_{t,T}$ depend on the particles and weights $({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\omega_{s}^{i}}})_{i=1}^N$, $0 \leq s \leq t-1$, through the particle approximations ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}, \ldots, {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}$ of the filter distributions. When proving the CLT for the FFBS algorithm, it will be crucial to establish that for any $h \in
{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, ${\mathcal L}^N_{t,T}(\cdot, h)$ converges (see Lemma \[lem:limLG\] below), as the number $N$ of particles tends to infinity, to a deterministic function ${\mathcal L}_{t,T}(\cdot, h)$ given by $$\label{eq:definition-Ft-lim}
{\mathcal L}_{t,T}(x_t, h) {\stackrel{\mathrm{def}}{=}}{\int\cdots\int}{\mathrm{B}_{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}}}(x_t, {d}x_{t-1})
\cdots{\mathrm{B}_{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}}}(x_{1}, {d}x_{0}) L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h).\hspace*{-25pt}$$ In the sequel, the case $h={\mathbf{1}}$ will be of particular importance; in that case, $L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},{\mathbf{1}})$ does not depend on ${\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t-1}}}{
\ensuremath{x^}_{0:t-1}}
}$, yielding $$\label{eq:Lone}
{\mathcal L}^N_{t,T}(x_t, {\mathbf{1}}) = {\mathcal L}_{t,T}(x_t, {\mathbf{1}}) = L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}, {\mathbf{1}})$$ for all $x_{0:t} \in{\mathbb{X}}^{t + 1}$. Using these functions, the difference appearing in the sum in [(\[eq:decomp\_Smooth\])]{} may then be rewritten as $$\begin{aligned}
\label{eq:definition-A}
\nonumber
&&\frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,{\mathbf{1}})]} - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,{\mathbf{1}})]} \hspace*{-25pt}\\
&&\qquad = \frac{1}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{t}}}
}
}
}
}
}[{\mathcal L}^N_{t,T}(\cdot,{\mathbf{1}})]} \biggl({
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{t}}}
}
}
}
}
}[{\mathcal L}^N_{t,T}(\cdot,h)] - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{\mathbf{1}})]}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\gamma_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\gamma^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\gamma}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\gamma^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\gamma^{N,\mathrm{a}}_{t}}}
}
}
}
}
}[{\mathcal L}^N_{t,T}(\cdot,{\mathbf{1}})] \biggr)\hspace*{-25pt} \\
&&\qquad = \frac{N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, h)}{N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, {\mathbf{1}})} ,\nonumber\hspace*{-25pt}\end{aligned}$$ where the kernel $G^N_{t,T}: {\mathbb{X}}\times{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}^{T+1} \to[0,1]$ is defined by, for $x \in{\mathbb{X}}$, $$\label{eq:definition-G}
G^N_{t,T}(x,h) {\stackrel{\mathrm{def}}{=}}{\mathcal L}^N_{t,T}(x,h) - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{\mathbf{1}})]} {\mathcal L}^N_{t,T}(x,{\mathbf{1}}) .$$ Similarly to ${\mathcal L}^N_{t,T}(\cdot, h)$, the functions $G^N_{t,T}(\cdot,
h)$ depend on the past particles; it will however be shown (see Lemma \[lem:limLG\] below) that $G^N_{t,T}(\cdot, h)$ converges to the deterministic function given by, for $x \in{\mathbb{X}}$, $$\label{eq:definition-G-lim}
G_{t,T}(x, h) {\stackrel{\mathrm{def}}{=}}{\mathcal L}_{t, T}\bigl(x, h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)\bigr) .$$ The key property of this decomposition is stated in the following lemma.
\[lem:GstTiszeromean\] Assume that Assumptions \[assum:bound-likelihood\]–\[assum:borne-FFBS\] hold for some $T < \infty$. Then, for any $0 \leq t\leq T$, the variables $\{{\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h)\}_{\ell=
1}^N$ are, conditionally on the $\sigma$-field ${\mathcal{F}_{t-1}^{N}}$, [i.i.d.]{}with zero mean. Moreover, there exists a constant $C$ (that may depend on $t$ and $T$) such that, for all $N \geq1$, $\ell\in\{1, \dots,
N\}$, and $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, $$| {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, h) | \leq{|{\ensuremath{\omega_{t}^{}}}|_{\infty}} |G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, h) | \leq C {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}}
.$$
By construction, all pairs of particles and weights of the weighted sample $\{({\ensuremath{\xi_{t}^{\ell}}}, {\ensuremath{\omega_{t}^{\ell}}})\}_{\ell=1}^N$ are [i.i.d.]{} conditionally on the $\sigma$-field ${\mathcal{F}_{t-1}^{N}}$. This implies immediately that the variables $\{{\ensuremath{\omega_{t}^{\ell}}}
G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h)\}_{\ell= 1}^N$ are also [i.i.d.]{}conditionally on the same $\sigma$-field ${\mathcal{F}_{t-1}^{N}}$. We now show that $\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}}, h) | {\mathcal{F}_{t-1}^{N}}] = 0$. Using the definition of $G^N_{t,T}$ and the fact that ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot, h)]$ and ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{\mathbf{1}})]$ are ${\mathcal{F}_{t-1}^{N}}$-measurable, we have $$\begin{aligned}
&&{\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) | {\mathcal{F}_{t-1}^{N}} ]}} \\
&&\qquad ={\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}(x,h) | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}(x,h) | {\mathcal{F}_{t-1}^{N}} ]}}- \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{\mathbf{1}})]} {\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}(x,{\mathbf{1}}) | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}(x,{\mathbf{1}}) | {\mathcal{F}_{t-1}^{N}} ]}} ,\end{aligned}$$ which is equal to zero provided that the relation $$\label{eq:espCondA}
{\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} {\mathcal L}^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) | {\mathcal{F}_{t-1}^{N}} ]}}=\frac
{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})}$$ holds for any $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$. We now turn to the proof of [(\[eq:espCondA\])]{}. Note that for any $f \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, $$\begin{aligned}
\label{eq:technique}
{\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} f({\ensuremath{\xi_{t}^{1}}}) | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} f({\ensuremath{\xi_{t}^{1}}}) | {\mathcal{F}_{t-1}^{N}} ]}} &=& \frac{\sum
_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} \int{M}({\ensuremath{\xi_{t-1}^{\ell}}}, {d}x)
g_t(x) f(x)}{\sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\vartheta_{t}}{
\vartheta_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}{\mathrm{erreur}}}}}} \nonumber\\[-8pt]\\[-8pt]
&=& \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{M}(\cdot,g_t f)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} .\nonumber\end{aligned}$$ It turns out that [(\[eq:espCondA\])]{} is a consequence of [(\[eq:technique\])]{} with $f(\cdot)={\mathcal L}_{t,T}^N(\cdot,h)$, but since ${\mathcal L}^N_{t-1,T}(\cdot,h)$ is in general different from ${M}(\cdot, g_t {\mathcal L}_{t,T}^N(\cdot,h))$, we have to prove directly that $$\label{eq:cequonveut}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [ {\mathcal L}^N_{t-1,T}(\cdot,h) ] = {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [ {M}(\cdot, g_t {\mathcal L}_{t,T}^N(\cdot,h)) ] .$$ Write $$\begin{aligned}
\label{eq:technique1}
&&{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [{M}(\cdot,g_t {\mathcal L}_{t,T}^N(\cdot,h)) ] \nonumber \\
&&\qquad = {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t}}}{\ensuremath{
\Omega_{t}^{()}}}}^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}\ensuremath{m}
({\ensuremath{\xi_{t-1}^{\ell}}},x_t) g_t(x_t) \Biggl(\prod_{u=1}^{t} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) \Biggr)\\
&&\hphantom{\qquad = {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t}}}{\ensuremath{
\Omega_{t}^{()}}}}^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}}
{}\times L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h) \,{d}x_t.\nonumber\end{aligned}$$ To simplify the expression in the RHS, we will use the two following equalities: $$\begin{aligned}
\label{eq:backRelat}
\Biggl(\!\sum_{\ell=1}^N\! {\ensuremath{\omega_{t-1}^{\ell}}}
\ensuremath{m}({\ensuremath{\xi_{t-1}^{\ell}}},x_t)\!\!\Biggr)\!
{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}}}(x_t,{d}x_{t-1})\! &=&\!\sum_{\ell=1}^N\! {\ensuremath{\omega_{t-1}^{\ell}}} \ensuremath{m}(x_{t-1},x_t) \delta_{{\ensuremath{\xi_{t-1}^{\ell
}}}}({d}x_{t-1}) , \hspace*{-30pt}\\
\label{eq:recurL}
\int{M}(x_{t-1}, {d}x_t) g_t(x_t)L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}, h) &=&
L_{t-1,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t-1}}}{
\ensuremath{x^}_{0:t-1}}
}, h) .\end{aligned}$$ The first relation is derived directly from the definition ${(\ref{eq:backward-kernel})}$ of the backward kernel, the second is a recursive expression of $L_{t,T}$ which is straightforward from the definition [(\[eq:defLt\])]{}. Now, [(\[eq:backRelat\])]{} and [(\[eq:recurL\])]{} allow for writing $$\begin{aligned}
&& \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}m({\ensuremath{\xi_{t-1}^{\ell
}}},x_t) g_t(x_t) \prod_{u=1}^{t} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h) \,{d}x_t \\
&&\qquad = \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}{M}(x_{t-1}, {d}x_t) g_t(x_t) \delta_{{\ensuremath{\xi_{t-1}^{\ell}}}}({d}x_{t-1}) \\
&&\hphantom{\qquad = \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}}
{}\times\prod
_{u=1}^{t-1} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h) \\
&&\qquad = \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\int\cdots\int}\delta_{{\ensuremath{\xi_{t-1}^{\ell}}}}({d}x_{t-1}) \prod_{u=1}^{t-1} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) L_{t-1,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t-1}}}{
\ensuremath{x^}_{0:t-1}}
},h) \\
&&\qquad = \sum_{\ell=1}^N {\ensuremath{\omega_{t-1}^{\ell}}} {\mathcal L}^N_{t-1}({\ensuremath{\xi_{t-1}^{\ell
}}},h).\end{aligned}$$ By plugging this expression into [(\[eq:technique1\])]{}, we obtain [(\[eq:cequonveut\])]{} from which [(\[eq:espCondA\])]{} follows via [(\[eq:technique\])]{}. Finally, $\mathbb{E}[{\ensuremath{\omega_{t}^{1}}}
G^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) | {\mathcal{F}_{t-1}^{N}} ] = 0$. It remains to check that the random variable ${\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}}, h)$ is bounded. But this is immediate since $$\begin{aligned}
\label{eq:borne-ewghtfuncxG}
|{\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}}, h) | &=& {\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}}
\biggl|{\mathcal L}^N_{t,T}(\cdot, h) - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot, h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot, {\mathbf{1}})]} {\mathcal L}_{t,T}(\cdot,{\mathbf{1}})\biggr|_{\infty} \nonumber\hspace*{-30pt}\\[-8pt]\\[-8pt]
&\leq&2 {\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}} {\ifthenelse{\equal{}{}}{| {\mathcal L}^N_{t,T}(\cdot,{\mathbf{1}}) |_\infty}{| {\mathcal L}^N_{t,T}(\cdot,{\mathbf{1}})|^2_{\infty}}}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} \leq2 {\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}} {\ifthenelse{\equal{}{}}{| L_{t,T}(\cdot,
{\mathbf{1}}) |_\infty}{| L_{t,T}(\cdot,
{\mathbf{1}})|^2_{\infty}}} {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} .\hspace*{-30pt}\nonumber\end{aligned}$$
Exponential deviation inequality {#sec:exponentialFFBS}
--------------------------------
We first establish a nonasymptotic deviation inequality. Considering [(\[eq:definition-A\])]{}, we are led to prove a Hoeffding inequality for ratios. For this purpose, we use the following elementary lemma which will play a key role in the sequel. The proof is postponed to Appendix \[sec:proof:lem:inegEssentielle\].
\[lem:inegessentielle\] Assume that $a_N$, $b_N$ and $b$ are random variables defined on the same probability space such that there exist positive constants $\beta
$, $B$, $C$ and $M$ satisfying:
(I) $|a_N/b_N|\leq M$, ${\mathbb{P}}$-[a.s.]{} and $b \geq\beta$, ${\mathbb{P}}$-[a.s.]{},
(II) for all $\epsilon>0$ and all $N\geq1$, ${\mathbb{P}}[|b_N-b|>\epsilon]\leq
B {e}^{-C N \epsilon^2}$,
(III) for all $\epsilon>0$ and all $N\geq1$, ${\mathbb{P}}[ |a_N|>\epsilon]\leq
B {e}^{-C N (\epsilon/M )^2}$.
Then $${\mathbb{P}}\biggl( \biggl| \frac{a_N}{b_N} \biggr| > \epsilon\biggr) \leq B \exp \biggl(-C N \biggl(\frac
{\epsilon\beta}{2M} \biggr)^2 \biggr) .$$
\[thm:Hoeffding-FFBS\] Assume that Assumptions \[assum:bound-likelihood\]–\[assum:borne-FFBS\] hold for some $T < \infty$. Then, there exist constants $0 < B$ and $C
< \infty$ (depending on $T$) such that for all $N$, $\epsilon> 0$, and all measurable functions $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, $$\label{eq:Hoeffding-1}
{\mathbb{P}}[ | {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) | \geq\epsilon]
\leq B {e}^{-C N \epsilon^2 / {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}} .$$ In addition, $$\label{eq:LGN-unnormalised-L}
N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell
}}},{\mathbf{1}}) {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \rightarrow\infty} \frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} .$$
As a by-product, Theorem \[thm:Hoeffding-FFBS\] provides an exponential inequality for the particle approximation of the filter. For any $h\in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, define the function ${\ifthenelse{\equal{}{}}{\ensuremath{{h}_{0:T}}}{
\ensuremath{h^}_{0:T}}
}:
{\mathbb{X}}^{T+1} \to{\mathbb{R}}$ by ${\ifthenelse{\equal{}{}}{\ensuremath{{h}_{0:T}}}{
\ensuremath{h^}_{0:T}}
}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:T}}}{
\ensuremath{x^}_{0:T}}
})=
h(x_T)$. By construction, ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}({\ifthenelse{\equal{}{}}{\ensuremath{{h}_{0:T}}}{
\ensuremath{h^}_{0:T}}
})= {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}(h)$ and ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}({\ifthenelse{\equal{}{}}{\ensuremath{{h}_{0:T}}}{
\ensuremath{h^}_{0:T}}
})= {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}(h)$. With this notation, equation [(\[eq:Hoeffding-1\])]{} may be rewritten as $${\mathbb{P}}[ | {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{T}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{T}}}
}
}
}
}
}(h) | \geq\epsilon] \leq B {e}^{-C N \epsilon^2 / {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}} .$$ An inequality of this form was first obtained by [@delmoralmiclo2000] (see also [@delmoral2004],Chapter 7).
We prove [(\[eq:Hoeffding-1\])]{} by induction on $T$ using the decomposition [(\[eq:decomp\_Smooth\])]{}. Assume that [(\[eq:Hoeffding-1\])]{} holds at time $T - 1$, for ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T
- 1|T - 1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T
- 1|T - 1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T
- 1|T - 1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T
- 1|T - 1}}}}
}
}
}(h)$. Let $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$ and assume without loss of generality that ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) = 0$. Then [(\[eq:smooth:recursion\])]{} implies that ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot, h)] = 0$ and the first term of the decomposition [(\[eq:decomp\_Smooth\])]{} thus becomes $$\label{eq:initialRecurInegExpo}
\frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot
,{\mathbf{1}})]} = \frac{N^{-1}\sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) L_{0,T}({\ensuremath{\xi_{0}^{i}}},h)}{N^{-1}
\sum_{\ell=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{\ell}}}) g_0({\ensuremath{\xi_{0}^{\ell}}})L_{0,T}({\ensuremath{\xi_{0}^{\ell}}},{\mathbf{1}})} ,$$ where $\{{\ensuremath{\xi_{0}^{i}}}\}_{i = 1}^N$ are [i.i.d.]{} random variables with distribution ${\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}$. We obtain an exponential inequality for [(\[eq:initialRecurInegExpo\])]{} by applying Lemma \[lem:inegessentielle\] with $$\cases{
\displaystyle a_N = N^{-1} \sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) L_{0,T}({\ensuremath{\xi_{0}^{i}}},h) ,
\cr
\displaystyle b_N = N^{-1} \sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) L_{0,T}({\ensuremath{\xi_{0}^{i}}},{\mathbf{1}})
, \cr
b = \beta= {\chi}[g_0(\cdot) L_{0,T}(\cdot,{\mathbf{1}})] .
}
$$ Condition is trivially satisfied and conditions and follow from the Hoeffding inequality for [i.i.d.]{} variables.
By [(\[eq:decomp\_Smooth\])]{} and [(\[eq:definition-A\])]{}, it is now enough to establish an exponential inequality for $$\label{eq:relation0}
\frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t|t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t|t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t|t}}}}
}
}
}[L_{t,T}(\cdot,{\mathbf{1}})]} - \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[L_{t-1,T}(\cdot,{\mathbf{1}})]} = \frac{N^{-1} \sum_{\ell
=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h)}{N^{-1} \sum
_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}})} ,\hspace*{-25pt}$$ where $0 < t \leq T$. For that purpose, we use again Lemma \[lem:inegessentielle\] with $$\label{eq:definition-b-1}
\cases{
\displaystyle a_N = N^{-1}\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) ,\cr
\displaystyle b_N = N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}}),\cr
\displaystyle b = \beta= \frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} .
}
$$ By considering the LHS of [(\[eq:relation0\])]{}, $ |{a_N}/{b_N} |\leq
2 {\ifthenelse{\equal{}{}}{| h |_\infty}{| h|^2_{\infty}}}$, verifying condition in Lemma \[lem:inegessentielle\]. By Lemma \[lem:GstTiszeromean\], Hoeffding’s inequality implies that there exist constants $B$ and $C$ such that for all $N$, $\epsilon> 0$, and all measurable function $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, $$\begin{aligned}
&&{\mathbb{P}}\Biggl[ \Biggl|N^{-1}\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) \Biggr| \geq\epsilon\Biggr]\\
&&\qquad = {\mathbb{E}}\Biggl[\mathbb{P}\Biggl[ \Biggl|N^{-1}\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}}
G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) \Biggr| \geq\epsilon\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] \Biggr] \leq
B {e}^{-C N \epsilon^2/{\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}},\end{aligned}$$ verifying condition in Lemma \[lem:inegessentielle\]. It remains to verify condition . Since the pairs of particles and weights of the weighted sample $\{({\ensuremath{\xi_{t}^{\ell}}},{\ensuremath{\omega_{t}^{\ell}}})\}_{\ell
=1}^N$ are [i.i.d.]{} conditionally on ${\mathcal{F}_{t-1}^{N}}$, Hoeffding’s inequality implies that $$\label{eq:relat0}
{\mathbb{P}}\Biggl[ \Biggl| b_N - \mathbb{E}\Biggl[N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}}{\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}})\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] \Biggr| \geq\epsilon\Biggr] \leq B
{e}^{-CN \epsilon^2} .$$ Moreover, by [(\[eq:technique\])]{}, [(\[eq:Lone\])]{}, and the definition [(\[eq:defLt\])]{}, we have $$\begin{aligned}
\label{eq:relat2}
&&\mathbb{E}\Biggl[N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, {\mathbf{1}})\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] - b \nonumber\\[-8pt]\\[-8pt]
&&\qquad = \frac{ {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} - \frac{ {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot
,{\mathbf{1}})]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} = \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(H)}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} ,\nonumber\end{aligned}$$ with $H(\cdot) {\stackrel{\mathrm{def}}{=}}{\mathcal L}_{t-1,T}(\cdot, {\mathbf{1}}) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})] {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}(\cdot) / {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})$. To obtain an exponential deviation inequality for [(\[eq:relat2\])]{}, we apply again Lemma \[lem:inegessentielle\] with $$\cases{
a'_N = {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(H) , \cr
b'_N={
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}) , \cr
b' = \beta'= {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}) .
}
$$ By using the inequality $$\begin{aligned}
&&{\mathcal L}_{t-1,T}(x_{t - 1}, {\mathbf{1}}) \\
&&\qquad = {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}(x_{t - 1}) \int\frac{\ensuremath{m}(x_{t - 1}, x_t)
g_{t}(x_t)}{{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}(x_{t - 1}){\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(x_{t - 1}, x_t) }
{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(x_{t - 1}, x_t) {\mathcal L}_{t,T}(x_t, {\mathbf{1}}) \,{d}x_t \\
&&\qquad \leq{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}(x_{t - 1}) {|{\ensuremath{\omega_{t}}}|_{\infty}} {|{\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|_{\infty}} ,\end{aligned}$$ we obtain the bound $| {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(H)/{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})|\leq2 {|{\ensuremath{\omega_{t}}}|_{\infty}} {|{\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|_{\infty}}$ which verifies condition . Now, since $t-1 \leq T-1$ and ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(H)=0$, the induction assumption implies that conditions and are satisfied for $|b'_N-b'|$ and $|a'_N|$. Hence, Lemma \[lem:inegessentielle\] shows that $$\label{eq:relat1}
{\mathbb{P}}\Biggl[ \Biggl| \mathbb{E}\Biggl[ N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}})\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] - b \Biggr| > \epsilon\Biggr] \leq B
{e}^{-C N \epsilon^2} .$$ Finally, [(\[eq:relat0\])]{} and [(\[eq:relat1\])]{} ensure that condition in Lemma \[lem:inegessentielle\] is satisfied and an exponential deviation inequality for [(\[eq:relation0\])]{} follows. The proof of [(\[eq:Hoeffding-1\])]{} is complete. The last statement [(\[eq:LGN-unnormalised-L\])]{} of the theorem is a consequence of [(\[eq:relat0\])]{} and [(\[eq:relat1\])]{}.
The exponential inequality of Theorem \[thm:Hoeffding-FFBS\] may be more or less immediately extended to the FFBSi estimator.
\[cor:Hoeffding-FFBSi\] Under the assumptions of Theorem \[thm:Hoeffding-FFBS\] there exist constants $0 < B$ and $C < \infty$ (depending on $T$) such that for all $N$, $\epsilon> 0$, and all measurable functions $h$, $$\label{eq:Hoeffding-2}
{\mathbb{P}}[ | {
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) | \geq\epsilon
] \leq B {e}^{-C N \epsilon^2 / {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}},$$ where ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)$ is defined in [(\[eq:FFBSi:estimator\])]{}.
Using [(\[eq:EspCond\])]{} and the definition of ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{s:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{s:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:T|T}}}}
}
}
}(h)$, we may write $$\begin{aligned}
&&{
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) \\
&&\qquad = N^{-1} \sum_{\ell=1}^N \bigl[ h ({\ensuremath{\xi_{0}^{J_0^\ell}}},\dots, {\ensuremath{\xi_{T}^{J_T^\ell}}} ) - {\ifthenelse{\equal{}{}}{\mathbb{E}[ h ({\ensuremath{\xi_{0}^{J_0}}},\dots,{\ensuremath{\xi_{T}^{J_T}}}
) | {\mathcal{F}_{T}^{N}}
]}{\mathbb{E}_{}[ h ({\ensuremath{\xi_{0}^{J_0}}},\dots,{\ensuremath{\xi_{T}^{J_T}}}
) | {\mathcal{F}_{T}^{N}} ]}} \bigr] ,\end{aligned}$$ which implies [(\[eq:Hoeffding-2\])]{} by the Hoeffding inequality and [(\[eq:Hoeffding-1\])]{}.
Asymptotic normality {#sec:CLTFFBS}
--------------------
We now extend the theoretical analysis of the forward-filtering backward-smoothing estimator [(\[eq:smoothing:backw\_decomposition\_sample\])]{} to a CLT. Consider the following mild assumption on the proposal distribution.
\[assum:bound-proposal-kernel\] ${|\ensuremath{m}|_{\infty}} < \infty$ and $\sup_{0 \leq t \leq T}
{|{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}|_{\infty}} < \infty$.
CLTs for interacting particle models have been established in [@delmoralmiclo2000; @delmoral2004; @doucmoulines2008]; the application to these results to auxiliary particle filters is presented in [@johansendoucet2008] and [@doucmoulinesolsson2008], Theorem 3.2. Here, we base our proof on techniques developed in [@doucmoulines2008] (extending [@chopin2004] and [@kuensch2005]). As noted in the previous section, it turns out crucial that $G^N_{t,T}(\cdot, h)$ converges to a deterministic function as $N
\to\infty$. This convergence is stated in the following lemma.
\[lem:limLG\] Assume Assumptions \[assum:bound-likelihood\]–\[assum:bound-proposal-kernel\]. Then, for any $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$ and , $$\begin{aligned}
\lim_{N \to\infty} {\mathcal L}^N_{t,T}(x,h) &=& {\mathcal L}_{t,T}(x,h) ,\qquad {\mathbb{P}}\mbox{-{a.s.}}, \\
\lim_{N \to\infty} G^N_{t,T} (x,h) &=& G_{t,T}(x,h) ,\qquad {\mathbb{P}}\mbox{-{a.s.}},\end{aligned}$$ where ${\mathcal L}^N_{t,T}$, ${\mathcal L}_{t,T}$, $G^N_{t,T}$ and $G_{t,T}$ are defined in [(\[eq:definition-Ft\])]{}, [(\[eq:definition-Ft-lim\])]{}, [(\[eq:definition-G\])]{} and [(\[eq:definition-G-lim\])]{}. Moreover, there exists a constant $C$ (that may depend on $t$ and $T$) such that for all $N \geq1$, $\ell\in\{1, \dots, N\}$, and $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, $$| {\ensuremath{\omega_{t}^{\ell}}} G_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) | \leq{|{\ensuremath{\omega_{t}^{}}}|_{\infty}} |G_{t,T}({\ensuremath{\xi_{t}^{\ell}}}, h) |\leq C {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}}
,\qquad {\mathbb{P}}\mbox{-{a.s.}}$$
[Proof of Lemma \[lem:limLG\]]{} Let $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$ and $x_t \in{\mathbb{X}}$. By plugging [(\[eq:backward-kernel\])]{} with $\eta={
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}$ into the definition [(\[eq:definition-Ft\])]{} of ${\mathcal L}^N_{t,T}(x_t,h)$, we obtain immediately $$\begin{aligned}
&&{\mathcal L}^N_{t,T}(x_t, h) \\
&&\qquad = \frac{{\int\cdots\int}{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({d}x_{t-1}) \prod_{u = 0}^{t-2}
{\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u}}}
}
}
}
}
}}}(x_{u+1},{d}x_{u}) \ensuremath{m}(x_{t-1}, x_t)
L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},h)}{\int{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({d}x_{t-1})
\ensuremath{m}
(x_{t-1}, x_{t})} \\
&&\qquad = \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:t-1|t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:t-1|t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:t-1|t-1}}}}
}
}
}[H([\cdot,x_t])]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[\ensuremath{m}
(\cdot, x_t)]} \qquad \mbox{with } H({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}){\stackrel{\mathrm{def}}{=}}\ensuremath{m}
(x_{t-1}, x_t) L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}, h) .\end{aligned}$$ The convergence of ${\mathcal L}^N_{t,T}(\cdot, h)$ follows from Theorem \[thm:Hoeffding-FFBS\]. The proof of the convergence of $G^N_{t,T}(\cdot
, h)$ follows the same lines. Finally, the final statement of the lemma is derived from Lemma \[lem:GstTiszeromean\] and the almost sure convergence of $G^N_{t,T}(\cdot, h)$ to $G_{t,T}(\cdot, h)$.
Now, we may state the CLT with an asymptotic variance given by a finite sum of terms involving the limiting kernel $G_{t,T}$.
\[thm:FFBS-CLT\] Assume Assumptions \[assum:bound-likelihood\]–\[assum:bound-proposal-kernel\]. Then, for any $h\in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$, $$\label{eq:clt}
\sqrt{N} \bigl({
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) \bigr) {\stackrel{\mathcal{D}}{\longrightarrow}}\mathcal
{N} (0, {
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{h}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[h]}}}
{\ifthenelse{\equal{h}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[h]}}}
} )$$ with $$\begin{aligned}
\label{eq:expression-covariance}
{
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{h}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[h]}}}
{\ifthenelse{\equal{h}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[h]}}}
} &{\stackrel{\mathrm{def}}{=}}&\frac{{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}[{\ensuremath{\omega_{0}}}^2(\cdot
) G_{0,T}^2(\cdot,h)]}{{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}^2 [{\ensuremath{\omega_{0}}}(\cdot) {\mathcal L}_{0,
T}(\cdot, {\mathbf{1}})]} + \sum_{t=1}^T \frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[\upsilon
_{t,T}(\cdot,h)] {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}^2[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})]}, \\
\label{eq:definition-upsilon}
\upsilon_{t,T}(\cdot, h) &{\stackrel{\mathrm{def}}{=}}&{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} \int{\ifthenelse{\equal{}{}}{P_{t}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{t}}{\mathrm{erreur}}}}}(\cdot, {d}x) {\ensuremath{\omega_{t}}}^2(\cdot, x) G^2_{t,T}(x, h).\end{aligned}$$
Without loss of generality, we assume that ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)=0$. We show that $\sqrt{N} {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)$ may be expressed as $$\label{eq:decomp_Smooth_1}
\sqrt{N} {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)= \sum_{t=0}^T \frac
{V^N_{t,T}(h)}{W^N_{t,T}} ,$$ where the sequence of random vectors $[V^N_{0,T}(h), \dots,
V^N_{T,T}(h)]$ is asymptotically normal and $[W^N_{0,T}, \dots,
W^N_{T,T}]$ converge in probability to a deterministic vector. The proof of [(\[eq:clt\])]{} then follows from Slutsky’s lemma. Actually, the decomposition [(\[eq:decomp\_Smooth\_1\])]{} follows immediately from the backward decomposition [(\[eq:decomp\_Smooth\])]{} by setting, for $t
\in\{1, \dots, T\}$, $$\begin{aligned}
V_{0,T}^N(h) &{\stackrel{\mathrm{def}}{=}}& N^{-1/2} \sum_{\ell=1}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{\ell}}})g_0({\ensuremath{\xi_{0}^{\ell}}})
G_{0,T}({\ensuremath{\xi_{0}^{\ell}}},h) , \\
V_{t,T}^N(h) &{\stackrel{\mathrm{def}}{=}}& N^{-1/2}\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}}
G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) , \\
W_{0,T}^N &{\stackrel{\mathrm{def}}{=}}& N^{-1} \sum_{\ell=1}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{\ell}}})g_0({\ensuremath{\xi_{0}^{\ell}}}) {\mathcal L}_{0,T}({\ensuremath{\xi_{0}^{\ell}}},{\mathbf{1}}) , \\
W_{t,T}^N &{\stackrel{\mathrm{def}}{=}}& N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}}) .\end{aligned}$$ The convergence $$\begin{aligned}
W_{0,T}^N &{\stackrel{\mathrm{P}}{\longrightarrow}}_{N \to\infty}& {\chi}[g_0(\cdot) {\mathcal L}_{0,T}(\cdot
,{\mathbf{1}}) ] , \\
W_{t,T}^N &{\stackrel{\mathrm{P}}{\longrightarrow}}_{N \to\infty}& \frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}_{t-1,T}(\cdot
,{\mathbf{1}})]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})}\end{aligned}$$ of $[W^N_{0,T}, \dots, W^N_{T,T}]$ to a deterministic vector is established immediately using [(\[eq:LGN-unnormalised-L\])]{} and noting that the initial particles $({\ensuremath{\xi_{0}^{i}}})_{i=1}^N$ are [i.i.d.]{} We devote the rest of the proof to showing that the sequence of random vectors $[V^N_{0,T}(h), \dots,\break V^N_{T,T}(h)]$ is asymptotically normal. Proceeding recursively in time, we prove by induction over $t \in
\{0, \dots, T\}$ (starting with $t=0$) that $[V^N_{0,T}(h), \dots,
V^N_{t,T}(h)]$ is asymptotically normal. More precisely, using the Cramér–Wold device, it is enough to show that for all scalars $(\alpha_0,\dots,\alpha_t) \in{\mathbb{R}}^{t+1}$, $$\label{eq:multivariate-CLT-FFBS}
\sum_{r=0}^t \alpha_r V_{r,T}^N(h) {\stackrel{\mathcal{D}}{\longrightarrow}}_{N \to\infty} \mathcal
{N} \Biggl(0, \sum_{r=0}^t \alpha_r^2 {
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{r,T}[h
]}}{\ensuremath{\sigma^2_{,r,T}[h]}}
} \Biggr) ,$$ where, for $r \geq1$, $${
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{0,T}[h
]}}{\ensuremath{\sigma^2_{,0,T}[h]}}
}{\stackrel{\mathrm{def}}{=}}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}[{\ensuremath{\omega_{0}}}^2
G_{0,T}^2(\cdot,h)] ,\qquad {
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{t,T}[h
]}}{\ensuremath{\sigma^2_{,t,T}[h]}}
} {\stackrel{\mathrm{def}}{=}}\frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[\upsilon_{t,T}(\cdot,h)]}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} .$$ The case $t = 0$ is elementary since the initial particles $\{ {\ensuremath{\xi_{0}^{i}}} \}_{i=1}^N$ are [i.i.d.]{} Assume now that [(\[eq:multivariate-CLT-FFBS\])]{} holds for some $t-1 \leq T$; for all scalars $(\alpha_1,\dots,\break\alpha_{t-1}) \in{\mathbb{R}}^{t-1}$, $$\label{eq:induction-assumption}
\sum_{r=s}^{t-1} \alpha_r V_{r,T}^N(h) {\stackrel{\mathcal{D}}{\longrightarrow}}_{N \to\infty}
\mathcal{N} \Biggl(0, \sum_{r=s}^{t-1} \alpha_r^2 {
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{r,T}[h
]}}{\ensuremath{\sigma^2_{,r,T}[h]}}
} \Biggr)
.$$ The sequence of random variable $V_{t,T}^N(h)$ may be expressed as an additive function of a triangular array of random variables, $$V_{t,T}^N(h)= \sum_{\ell=1}^N U_{N,\ell} ,\qquad U_{N,\ell}{\stackrel{\mathrm{def}}{=}}{\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},h) / \sqrt{N} ,$$ where $G^N_{t,T}(x,h)$ is defined in [(\[eq:definition-G\])]{}. Lemma \[lem:GstTiszeromean\] implies that $\mathbb{E}[ V_{t,T}^N(h) |
{\mathcal{F}_{t-1}^{N}}] = 0$, yielding $$\mathbb{E}\Biggl[\sum_{r=0}^t \alpha_r V_{r,T}^N(h)\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] = \sum
_{r=0}^{t-1} \alpha_r V_{r,T}^N(h) {\stackrel{\mathcal{D}}{\longrightarrow}}_{N \to\infty}
\mathcal{N} \Biggl(0, \sum_{r=1}^{t-1} \alpha_r^2 {
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{r,T}[h
]}}{\ensuremath{\sigma^2_{,r,T}[h]}}
} \Biggr),$$ where the last limit follows by the induction assumption hypothesis [(\[eq:induction-assumption\])]{}. By [@doucmoulines2008], Theorem A.3, page 2360, as the random variables $\{ U_{N,\ell} \}
_{\ell=1}^N$ are centered and conditionally independent given ${\mathcal{F}_{t-1}^{N}}$, [(\[eq:multivariate-CLT-FFBS\])]{} holds provided that the asymptotic smallness condition $$\label{eq:triangular-3}
\sum_{\ell=1}^N \mathbb{E}\bigl[U_{N,\ell}^2 \mathbh{1}_{\{|U_{N,\ell}|
\geq\epsilon\}}|{\mathcal{F}_{t-1}^{N}}\bigr] {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \to\infty} 0$$ holds for any $\epsilon> 0$ and that the conditional variance converges: $$\label{eq:triangular-2}
\sum_{\ell=1}^N {\ifthenelse{\equal{}{}}{\mathbb{E}[ U_{N,\ell}^2 | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ U_{N,\ell}^2 | {\mathcal{F}_{t-1}^{N}} ]}} {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \to
\infty} {
\ifthenelse{\equal{}{}}{\ensuremath{\sigma^2_{t,T}[h
]}}{\ensuremath{\sigma^2_{,t,T}[h]}}
} .$$ Lemma \[lem:GstTiszeromean\] implies that $|U_{N,\ell}|\leq C
{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} / \sqrt{N}$, verifying immediately the asymptotic smallness condition [(\[eq:triangular-3\])]{}. To conclude the proof, we thus only need to establish the convergence [(\[eq:triangular-2\])]{} of the asymptotic variance. Via Lemma \[lem:GstTiszeromean\] and straightforward computations, we conclude that $$\begin{aligned}
\label{eq:espCondU}
\sum_{\ell=1}^N {\ifthenelse{\equal{}{}}{\mathbb{E}[ U^2_{N,\ell} | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ U^2_{N,\ell} | {\mathcal{F}_{t-1}^{N}} ]}} &=& {\ifthenelse{\equal{}{}}{\mathbb{E}[ ( {\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) )^2 | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ ( {\ensuremath{\omega_{t}^{1}}} G^N_{t,T}({\ensuremath{\xi_{t}^{1}}},h) )^2 | {\mathcal{F}_{t-1}^{N}} ]}}\nonumber
\\
&=& \int\sum_{\ell=1}^N \frac{{\ensuremath{\omega_{t-1}^{\ell}}} {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\vartheta_{t}}{
\vartheta_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{\ell}}}}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}({\ensuremath{\xi_{t-1}^{\ell}}})}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{P_{t}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{t}}{\mathrm{erreur}}}}}({\ensuremath{\xi_{t-1}^{\ell}}}, {d}x)}
{\sum_{j=1}^N {\ensuremath{\omega_{t-1}^{j}}} {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\vartheta_{t}}{
\vartheta_{t}({\ensuremath{\xi_{t-1}^{j}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}({\ensuremath{\xi_{t-1}^{j}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}({\ensuremath{\xi_{t-1}^{j}}})}}{\mathrm{erreur}}}}}} (
{\ensuremath{\omega_{t}}}({\ensuremath{\xi_{t-1}^{\ell}}},x) G^N_{t,T}(x,h) )^2\hspace*{-20pt}
\nonumber\\[-8pt]\\[-8pt]
&=& \biggl(\frac{{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t-1}}}{\ensuremath{
\Omega_{t-1}^{()}}}}}{\sum_{j=1}^N {\ensuremath{\omega_{t-1}^{j}}} {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\vartheta_{t}}{
\vartheta_{t}({\ensuremath{\xi_{t-1}^{j}}})}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}({\ensuremath{\xi_{t-1}^{j}}})}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{{\ensuremath{\xi_{t-1}^{j}}}}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}({\ensuremath{\xi_{t-1}^{j}}})}}{\mathrm{erreur}}}}}} \biggr) \Biggl(\frac{1}{{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t-1}}}{\ensuremath{
\Omega_{t-1}^{()}}}}}\sum_{\ell=1}^N
{\ensuremath{\omega_{t-1}^{\ell}}} \upsilon^N_{t,T}({\ensuremath{\xi_{t-1}^{\ell}}},h) \Biggr) \nonumber\\
&=& \frac
{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[\upsilon^N_{t,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})} ,\nonumber\end{aligned}$$ where ${\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{t}}}{\ensuremath{
\Omega_{t}^{()}}}}$ is defined in [(\[eq:defOmega\])]{} and $$\upsilon^N_{t,T}(\cdot,h) {\stackrel{\mathrm{def}}{=}}{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} \int{\ifthenelse{\equal{}{}}{P_{t}}
{\ifthenelse{\equal{}{fully}}{P^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{R}_{t}}{\mathrm{erreur}}}}}(\cdot, {d}x) {\ensuremath{\omega_{t}}}^2(\cdot,x) [ G^N_{t,T}(x,h) ]^2
.$$ The denominator in on RHS of [(\[eq:espCondU\])]{} converges evidently in probability to ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})$ by Theorem \[thm:Hoeffding-FFBS\]. The numerator is more complex since $\upsilon
^N_{t,T}$ depends on $G^N_{t,T}$ whose definition involves all the approximations ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
},\ldots,{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}$ of the past filters. To obtain its convergence, note that, by Theorem \[thm:Hoeffding-FFBS\], ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(\upsilon_{t,T}(\cdot,h))
{\stackrel{\mathrm{P}}{\longrightarrow}}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(\upsilon_{t,T}(\cdot,h))$ as $N$ tends to infinity; hence, it only remains to prove that $$\label{eq:technicos}
{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [\upsilon^N_{t,T}(\cdot, h) - \upsilon_{t,T}(\cdot
, h) ] {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \to\infty} 0 .$$ For that purpose, introduce the following notation: for all $x \in
{\mathbb{X}}$, $$\begin{aligned}
A_N(x) &{\stackrel{\mathrm{def}}{=}}&{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x) {\ensuremath{\omega_{t}}}^2(\cdot,x)| (G^N_{t,T}(x,h))^2 -
G_{t,T}^2(x,h)| ] , \\
B_N(x) &{\stackrel{\mathrm{def}}{=}}&{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x) ] .\end{aligned}$$ Applying Fubini’s theorem, $$\label{eq:intAn}
\lim_{N \to\infty} {\mathbb{E}}\biggl[ \int A_N(x) \,{d}x \biggr] = \lim_{N \to\infty
} \int{\mathbb{E}}[A_N(x)] \,{d}x = 0 ,$$ where the last equality is due to the generalized Lebesgue convergence theorem [@royden1988], Proposition 18, page 270, with $f_N(x) = {\mathbb{E}}[A_N(x)]$ and $g_N(x) = 2 C {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} {\mathbb{E}}[B_N(x)]$ provided that the following conditions hold:
(i) for any $x\in{\mathbb{X}}$, ${\mathbb{E}}[A_N(x)] \leq2
C^2 {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}} {\mathbb{E}}[B_N(x)]$,
(ii) for any $x \in{\mathbb{X}}$, $\lim_{N \to\infty}
{\mathbb{E}}[A_N(x)] = 0$, ${\mathbb{P}}$-[a.s.]{},
(iii) $\lim_{N \to\infty} \int{\mathbb{E}}[B_N(x)]
\,{d}x=\int\lim_{N \to\infty} {\mathbb{E}}[B_N(x)] \,{d}x$.
*Proof of* . The bound follows directly from Lemmas \[lem:limLG\] and \[lem:GstTiszeromean\].
*Proof of* . Using again Lemmas \[lem:limLG\] and \[lem:GstTiszeromean\], for any $x \in{\mathbb{X}}$, $$\begin{aligned}
A_N(x) &\leq&2C^2 {|{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|_{\infty}} {|{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}|_{\infty}}
{\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}} , \\
\limsup_{N \to\infty} A_N(x) &\leq&{|{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}{\ensuremath{\omega_{t}}}^2|_{\infty}} \limsup_{N \to\infty} | (G^N_{t,T}(x, h))^2
- G_{t,T}^2(x,h) | = 0 ,\qquad {\mathbb{P}}\mbox{-{a.s.}}\end{aligned}$$ These two inequalities combined with $A_N(x)\geq0$ allow for applying the Lebesgue dominated convergence theorem, verifying condition .
*Proof of* . We have $$\begin{aligned}
\lim_{N \to\infty} \int{\mathbb{E}}[B_N(x)] \,{d}x &\stackrel{\mathrm{(a)}}{=} &\lim
_{N \to\infty} {\mathbb{E}}\biggl[ {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} \biggl( {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} \int
{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x) \,{d}x \biggr) \biggr] \\
&\stackrel{\mathrm{(b)}}{=}& {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} ({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}} ) \stackrel{\mathrm{(c)}}{=}
\int{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} ({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}} {\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x) )
\,{d}x \\
&\stackrel{\mathrm{(d)}}{=}& \int\lim_{N \to\infty} {\mathbb{E}}[{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} ({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}}{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x) ) ] \,{d}x\\
& =&
\int\lim_{N \to\infty} {\mathbb{E}}[B_N(x)] \,{d}x ,\end{aligned}$$ where (a) and (c) are consequences of Fubini’s theorem and (b) and (d) follows from the ${\mathsf{L}^1}$-convergence of ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t}}}
}
}
}
}
}(h)$ to ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t}}}
}
}
}
}
}(h)$ (see Theorem \[thm:Hoeffding-FFBS\]) with $h(\cdot
)={\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}}$ and $h(\cdot)={\ifthenelse{\equal{}{}}{\ifthenelse{\equal{\cdot}{}}{\vartheta_{t}}{
\vartheta_{t}(\cdot)}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{\cdot}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}(\cdot)}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{\cdot}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}(\cdot)}}{\mathrm{erreur}}}}}{\ifthenelse{\equal{}{}}{p_{t}}
{\ifthenelse{\equal{}{fully}}{p^{\star}_{t}}
{\ifthenelse{\equal{}{smooth}}{\tilde{r}_{t}}{\mathrm{erreur}}}}}(\cdot, x)$.
Thus, [(\[eq:intAn\])]{} holds, yielding that $\int A_N(x) \,{d}x {\stackrel{\mathrm{P}}{\longrightarrow}}0$ as $N$ tends to infinity. This in turn implies [(\[eq:technicos\])]{} via the inequality $$| {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [\upsilon^N_{t,T}(\cdot,h)-\upsilon_{t,T}(\cdot
,h) ] | \leq\int A_N(x) \,{d}x .$$ This establishes [(\[eq:multivariate-CLT-FFBS\])]{} and therefore completes the proof.
The weak convergence of $\sqrt{N} ( {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h))$ for the FFBS algorithm implies more or less immediately the one of $\sqrt{N} ({
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) )$ for the FFBSi algorithm.
Under the assumptions of Theorem \[thm:FFBS-CLT\], $$\begin{aligned}
\label{eq:cltSi}
&&\sqrt{N} \bigl({
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) \bigr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad {\stackrel{\mathcal{D}}{\longrightarrow}}\mathcal{N} \bigl(0, {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}^2 [h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) ] +
{
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)]}}}
{\ifthenelse{\equal{h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)]}}}
} \bigr) .\nonumber\end{aligned}$$
Using [(\[eq:EspCond\])]{} and the definition of ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)$, we may write $$\begin{aligned}
&&\sqrt{N} \bigl({
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) \bigr) \\
&&\qquad = N^{-1/2} \sum_{\ell=1}^N \bigl[ h ({\ensuremath{\xi_{0}^{J_0^\ell}}}, \dots, {\ensuremath{\xi_{T}^{J_T^\ell}}} ) - {\ifthenelse{\equal{}{}}{\mathbb{E}[ h ({\ensuremath{\xi_{0}^{J_0}}}, \dots, {\ensuremath{\xi_{T}^{J_T}}}
) | {\mathcal{F}_{T}^{N}}
]}{\mathbb{E}_{}[ h ({\ensuremath{\xi_{0}^{J_0}}}, \dots, {\ensuremath{\xi_{T}^{J_T}}}
) | {\mathcal{F}_{T}^{N}} ]}} \bigr] \\
&&\qquad\quad {} + \sqrt{N} \bigl( {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) \bigr) .\end{aligned}$$ Note that since $\{ {\ifthenelse{\equal{}{}}{\ensuremath{{J}_{0:T}}}{
\ensuremath{J^}_{0:T}}
}^\ell\}_{\ell= 1}^N$ are [i.i.d.]{}conditional on ${\mathcal{F}_{T}^{N}}$, [(\[eq:cltSi\])]{} follows from [(\[eq:clt\])]{} and direct application of [@doucmoulines2008], Theorem A.3, page 2360, by noting that $$\begin{aligned}
&&N^{-1} \sum_{\ell=1}^N \mathbb{E}\bigl[ \{h ({\ensuremath{\xi_{0}^{J_0^\ell}}},\dots,
{\ensuremath{\xi_{T}^{J_T^\ell}}} ) - {\ifthenelse{\equal{}{}}{\mathbb{E}[ h ({\ensuremath{\xi_{0}^{J_0}}},\dots,{\ensuremath{\xi_{T}^{J_T}}} ) | {\mathcal{F}_{T}^{N}}
]}{\mathbb{E}_{}[ h ({\ensuremath{\xi_{0}^{J_0}}},\dots,{\ensuremath{\xi_{T}^{J_T}}} ) | {\mathcal{F}_{T}^{N}} ]}} \}^2|{\mathcal{F}_{T}^{N}}\bigr] \\
&&\qquad = \bigl({
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
} [h - {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) ] \bigr)^2
{\stackrel{\mathrm{P}}{\longrightarrow}}\bigl({
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
} [h - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h) ] \bigr)^2 .\end{aligned}$$
Time uniform bounds {#sec:TimeUniformExponentialFFBS}
===================
Most often, it is not required to compute the joint smoothing distribution but rather the marginal smoothing distributions ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}$. Considering [(\[eq:forward-filtering-backward-smoothing\])]{} for a function $h$ that depends on the component $x_s$ only, we obtain particle approximations of the marginal smoothing distributions by associating the set $\{ {\ensuremath{\xi_{s}^{j}}} \}_{j = 1}^N$ of particles with weights obtained by marginalizing the joint smoothing weights according to $${\ensuremath{\omega_{s|T}^{i_s}}} = \sum_{i_{s+1} = 1}^N \cdots\sum_{i_T = 1}^N \prod
_{u=s+1}^t \frac{{\ensuremath{\omega_{u-1}^{i_{u-1}}}} \ensuremath{m}({\ensuremath{\xi_{u-1}^{i_{u-1}}}},
{\ensuremath{\xi_{u}^{i_u}}})}{\sum_{\ell=1}^N {\ensuremath{\omega_{u-1}^{\ell}}} \ensuremath
{m}({\ensuremath{\xi_{u-1}^{\ell}}}, {\ensuremath{\xi_{u}^{i_u}}})} \frac{{\ensuremath{\omega_{T}^{i_T}}}}{{\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{T}}}{\ensuremath{
\Omega_{T}^{()}}}}}.$$ It is easily seen that these marginal weights may be recursively updated backward in time as $$\label{eq:FFBS-marginal-weight-update}
{\ensuremath{\omega_{s|T}^{i}}} = \sum_{j = 1}^N \frac{
{\ensuremath{\omega_{s}^{i}}} \ensuremath{m}({\ensuremath{\xi_{s}^{i}}}, {\ensuremath{\xi_{s+1}^{j}}})}{\sum
_{\ell=1}^N
{\ensuremath{\omega_{s}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{s}^{\ell}}}, {\ensuremath{\xi_{s+1}^{j}}})}
{\ensuremath{\omega_{s+1|T}^{j}}} .$$ In this section, we study the long-term behavior of the marginal fixed-interval smoothing distribution estimator. For that purpose, it is required to impose a type of mixing condition on the Markov transition kernel; see [@chiganskylipstervanhandel2008] and the references therein. For simplicity, we consider elementary but strong conditions which are similar to the ones used in [@delmoral2004], Chapter 7.4, or [@cappemoulinesryden2005], Chapter 4; these conditions, which points to applications where the state space ${\mathbb{X}}$ is compact, can be relaxed, but at the expense of many technical difficulties [@chiganskylipster2004; @vanhandel2008a; @vanhandel2008b; @vanhandel2009a].
\[assum:strong-mixing-condition\] There exist two constants $0 < \sigma_- \leq\sigma_+ < \infty$, such that, for any $(x,x') \in{\mathbb{X}}\times{\mathbb{X}}$, $$\label{eq:minorq}
\sigma_- \leq\ensuremath{m}(x,x') \leq\sigma_+.$$ In addition, there exists a constant $c_- > 0$ such that, $\int{\chi}({d}x_0) g_0(x_0) \geq c_-$ and for all $t \geq1$, $$\label{eq:minorg}
\inf_{x \in{\mathbb{X}}} \int{M}(x, {d}x') g_t(x') \geq c_- > 0 .$$
Assumption \[assum:strong-mixing-condition\] implies that $\nu
({\mathbb{X}})<\infty$; in the sequel, we will consider without loss of generality that $\nu({\mathbb{X}})=1$. Note also that, under Assumption \[assum:strong-mixing-condition\], the average number of simulations required in the accept–reject mechanism per sample of the FFBSi algorithm is bounded by $\sigma_+/\sigma_-$.
The goal of this section consists in establishing, under the assumptions mentioned above, that the FFBS approximation of the *marginal* fixed interval smoothing probability satisfies an exponential deviation inequality with constants that are uniform in time and, under the same assumptions, that the variance of the CLT is uniformly bounded in time.
For obtaining these results, we will need upper-bounds on $G^N_{t,T}$ and $G_{t,T}$ that are more precise than the ones stated in Lemmas \[lem:GstTiszeromean\] and \[lem:limLG\]. For any function $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$ and $s\leq T$, define the extension ${{\Pi_{s,T} h}} \in
{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$ of $h$ to ${\mathbb{X}}^{T + 1}$ by $$\label{eq:defH}
{{\Pi_{s,T} h}}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:T}}}{
\ensuremath{x^}_{0:T}}
}) {\stackrel{\mathrm{def}}{=}}h(x_s) ,\qquad x_{0:T} \in
{\mathbb{X}}^{T + 1} .$$
\[lem:G-uniform\] Assume that Assumptions \[assum:bound-likelihood\]–\[assum:strong-mixing-condition\] hold with $T = \infty$. Let $s \leq
T$. Then, for all $t$,$T$, $N \geq1$, and $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, $$\label{eq:time-unif-G}
{|G^N_{t,T}(\cdot, {{\Pi_{s,T} h}})|_{\infty}} \leq\rho
^{|t-s|}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} {|{\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|_{\infty}},$$ where ${\mathcal L}_{t,T}$ is defined in [(\[eq:definition-Ft-lim\])]{} and $$\label{eq:definition-rho}
\rho= 1 - \frac{\sigma_-}{\sigma_+} .$$ Moreover, for all $t$, $T \geq1$, and $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, $$\label{eq:time-unif-G-lim}
{|G_{t,T}(\cdot, {{\Pi_{s,T} h}})|_{\infty}} \leq\rho
^{|t-s|}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} {|{\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|_{\infty}}.$$
Using [(\[eq:Lone\])]{} and [(\[eq:definition-G\])]{}, $$\label{eq:expressionG}
\frac{G^N_{t,T}(x, {{\Pi_{s,T} h}})}{{\mathcal L}_{t,T}(x,{\mathbf{1}})}=\frac{{\mathcal L}^N_{t,T}(x,{{\Pi_{s,T} h}})}{{\mathcal L}^N_{t,T}(x,{\mathbf{1}})}- \frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{{\Pi_{s,T} h}})]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{\mathbf{1}})]} .$$ To prove [(\[eq:time-unif-G\])]{}, we will rewrite [(\[eq:expressionG\])]{} and obtain an exponential bound by either using ergodicity properties of the “a posteriori” chain (when $t \leq s$), or by using ergodicity properties of the backward kernel (when $t>s)$.
Assume first that $t \leq s$. The quantity $L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},
{{\Pi_{s,T} h}})$ does not depend on ${\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t-1}}}{
\ensuremath{x^}_{0:t-1}}
}$ so that by [(\[eq:definition-Ft\])]{} and definition [(\[eq:defLt\])]{} of $L_{t,T}$, $$\begin{aligned}
\label{eq:one}
{\mathcal L}^N_{t,T}(x_t,{{\Pi_{s,T} h}}) &=& L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
}, {{\Pi_{s,T} h}}) \nonumber\\
&=& {\int\cdots\int}\Biggl( \prod_{u=t+1}^T {M}(x_{u-1},{d}x_u) g_u(x_u) \Biggr)
h(x_s)\\
&=& {\mathcal L}_{t,T}(x_t, {{\Pi_{s,T} h}}) .\nonumber\end{aligned}$$ Now, by construction, for any $t \leq s $, $$\label{eq:two}
{\mathcal L}_{t-1,T}(x_{t-1},{{\Pi_{s,T} h}}) = \int{M}(x_{t-1},{d}x_t)
g_t(x_t) {\mathcal L}_{t,T}(x_t,{{\Pi_{s,T} h}}) .$$ The relations [(\[eq:expressionG\])]{}, [(\[eq:one\])]{} and [(\[eq:two\])]{} imply that $$\frac{G^N_{t,T}(x, {{\Pi_{s,T} h}})}{{\mathcal L}_{t,T}(x,{\mathbf{1}})} = \frac{\mu
[{\mathcal L}_{t,T}(\cdot, {{\Pi_{s,T} h}})]}{\mu[{\mathcal L}_{t,T}(\cdot,{\mathbf{1}})]} -
\frac{\mu'[{\mathcal L}_{t,T}(\cdot,{{\Pi_{s,T} h}})]}{\mu'[ {\mathcal L}_{t,T}(\cdot,{\mathbf{1}})]} , \label{eq:Gn-t-inf-s}$$ where $\mu{\stackrel{\mathrm{def}}{=}}\delta_x$ and $\mu'$ is the nonnegative finite measure defined by $$\mu'(A) {\stackrel{\mathrm{def}}{=}}{\int\!\!\!\int}{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({d}x_{t-1}) {M}(x_{t-1},{d}x_t) g_{t}(x_t){\mathbf{1}}_A(x_t) ,\qquad A \in{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}.$$ Now, for any finite measure $\mu$ on $({\mathbb{X}},{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$, the quantity $$\begin{aligned}
&&\frac{\mu[{\mathcal L}_{t,T}(\cdot,{{\Pi_{s,T} h}})]}{\mu[{\mathcal L}_{t,T}(\cdot
,{\mathbf{1}})]} \\
&&\qquad = \frac{{\int\cdots\int}\mu({d}x_t) \prod_{u=t+1}^T {M}(x_{u-1},{d}x_u) g_u(x_{u}) h(x_s)}{{\int\cdots\int}\mu({d}x_t) \prod
_{u=t+1}^T {M}(x_{u-1}, {d}x_u) g_u(x_u)} \\[2pt]
&&\qquad = \frac{{\int\cdots\int}\mu({d}x_t) \prod_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u(x_{u}) h(x_s) {\mathcal L}_{s,T}(x_s,{\mathbf{1}})}{{\int\cdots\int}\mu({d}x_t)
\prod_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u(x_u) {\mathcal L}_{s,T}(x_s,{\mathbf{1}})}\end{aligned}$$ may be seen as the expectation of $h(X_s)$ conditionally on ${\ifthenelse{\equal{}{}}{\ensuremath{{Y}_{t:T}}}{
\ensuremath{Y^}_{t:T}}
}$, where $X_t$ is distributed according to $A \mapsto\mu(A)/
\mu({\mathbb{X}})$. Under the strong mixing condition (Assumption \[assum:strong-mixing-condition\]), it is shown in [@delmoralmiclo2000] (see also [@delmoral2004]) that, for any $t
\leq s \leq T$, any finite measure $\mu$ and $\mu'$ on $({\mathbb{X}},{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$, any function $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}})}$, that $$\begin{aligned}
&&\biggl| \frac{{\int\cdots\int}\mu({d}x_t) \prod_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u (x_u) h(x_s) {\mathcal L}_{s,T}(x_s,{\mathbf{1}})}{{\int\cdots\int}\mu({d}x_t) \prod
_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u (x_u) {\mathcal L}_{s,T}(x_s,\mathbf{1})}
\\[2pt]
&&\quad {}- \frac{{\int\cdots\int}\mu'({d}x_t) \prod_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u (x_u) h(x_s) {\mathcal L}_{s,T}(x_s,{\mathbf{1}})}{{\int\cdots\int}\mu'({d}x_t)
\prod_{u=t+1}^s {M}(x_{u-1},{d}x_u) g_u (x_u) {\mathcal L}_{s,T}(x_s,{\mathbf{1}})}
\biggr|\\
&&\qquad \leq\rho^{s-t} {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} ,\end{aligned}$$ where $\rho$ is defined in [(\[eq:definition-rho\])]{}. This shows [(\[eq:time-unif-G\])]{} when $t$ is smaller than $s$.
Consider now the case $s < t \leq T$. By definition, $$\begin{aligned}
\label{eq:cas2premiere}
{\mathcal L}^N_{t,T}(x_t,{{\Pi_{s,T} h}})&=&{\int\cdots\int}L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},{{\Pi_{s,T} h}}) \prod_{u=s+1}^{t} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) \nonumber\\[-8pt]\\[-8pt]
&=& {\int\cdots\int}{\mathcal L}_{t,T}(x_t,{\mathbf{1}}) \prod_{u=s+1}^{t} {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) h(x_s) ,\nonumber\end{aligned}$$ where the last expression is obtained from the following equality, valid for $s < t$: $$\begin{aligned}
L_{t,T}({\ifthenelse{\equal{}{}}{\ensuremath{{x}_{0:t}}}{
\ensuremath{x^}_{0:t}}
},{{\Pi_{s,T} h}})&=&h(x_s) {\int\cdots\int}\prod
_{u=t+1}^T {M}(x_{u-1},{d}x_u) g_u(x_u) \\
&=& h(x_s) {\mathcal L}_{t,T}(x_t,{\mathbf{1}}).\end{aligned}$$ Moreover, combining [(\[eq:cequonveut\])]{} and [(\[eq:cas2premiere\])]{}, $$\begin{aligned}
&&{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[{\mathcal L}^N_{t-1,T}(\cdot,{{\Pi_{s,T} h}})]\\
&&\qquad = {\int\cdots\int}{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({d}u_{t-1}) M(u_{t-1},{d}x_t) g_t(x_t) {\mathcal L}^N_{t,T}(x_t,{{\Pi_{s,T} h}}) \\
&&\qquad = {\int\cdots\int}{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({d}u_{t-1}) M(u_{t-1},{d}x_t) g_t(x_t)
{\mathcal L}_{t,T}(x_t,{\mathbf{1}}) \\
&&\hphantom{\qquad = {\int\cdots\int}}
{}\times\prod_{u=s+1}^t {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) h(x_s) .\end{aligned}$$ By plugging this expression and [(\[eq:cas2premiere\])]{} into [(\[eq:expressionG\])]{}, we obtain $$\frac{G^N_{t,T}(x, {{\Pi_{s,T} h}})}{{\mathcal L}_{t,T}(x,{\mathbf{1}})}={\int\cdots\int}\biggl\{
\frac{\mu({d}x_{t})}{\mu({\mathbb{X}})} - \frac{\mu'({d}x_{t})}{\mu
'({\mathbb{X}})} \biggr\} \prod_{u=s+1}^t {\mathrm{B}_{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{u-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{u-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{u-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{u-1}}}
}
}
}
}
}}}(x_u, {d}x_{u-1}) h(x_s),$$ with $\mu({d}x_t) = \delta_x({d}x_t) {\mathcal L}_{t,T}(x_t,{\mathbf{1}})$ and $\mu
'$ being the nonnegative measure defined by $$\mu'(A) = \int{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}[\ensuremath{m}(\cdot,x_t)] g_t(x_t)
{\mathcal L}_{t,T}(x_{t},{\mathbf{1}}) {\mathbf{1}}_A(x_{t}) \,{d}x_t .$$ Under the uniform ergodicity condition (Assumption \[assum:strong-mixing-condition\]) it holds, for any probability measure $\eta$ on $({\mathbb{X}},{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
})$, and any $A \in{\ifthenelse{\equal{}{}}{\ensuremath{\mathcal{B}({\mathbb{X}})}}{\ensuremath{
\mathcal{B}({\mathbb{X}}^{})}}
}$, $${\mathrm{B}_{\eta}}(x,A) = \frac{\int_A \eta( {d}x') \ensuremath{m}(x',x)
}{\int\eta
( {d}x') \ensuremath{m}(x',x) } \geq\frac{\sigma_-}{\sigma_+}
\eta(A) ;$$ thus, the transition kernel ${\mathrm{B}_{\eta}}$ is uniformly Doeblin with minorizing constant $\sigma_-/\sigma_+$ and the proof of [(\[eq:time-unif-G\])]{} for $ s < t \leq T$ follows. The last statement of the Lemma follows from [(\[eq:time-unif-G\])]{} and the almost-sure convergence $$\lim_{N \to\infty}G^N_{t,T}(x,h) = G_{t,T}(x,h) ,\qquad {\mathbb{P}}\mbox{-{a.s.}},$$ for all $x \in{\mathbb{X}}$, which was established in Lemma \[lem:limLG\].
A time uniform exponential deviation inequality
-----------------------------------------------
Under the strong mixing Assumption \[assum:strong-mixing-condition\], a time uniform deviation inequality for the *marginal smoothing* approximation can be derived using the exponentially decreasing bound on the quantity $G^N_{t,T}$ obtained in Lemma \[lem:G-uniform\].
\[theo:Hoeffding-uniform\] Assume Assumptions \[assum:bound-likelihood\]–\[assum:strong-mixing-condition\] hold with $T= \infty$. Then, there exist constants $0\leq B,\ C< \infty$ such that for all integers $N$, $s$, $T$, with $s \leq T$, and for all $\epsilon> 0$, $$\begin{aligned}
\label{eq:TU-Hoeffding-smoothing-normalized}
{\mathbb{P}}[ | {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h) | \geq\epsilon] &\leq&
B{e}^{-C N \epsilon^2 / {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}} , \\
\label{eq:TU-Hoeffding-smoothing-normalized-2}
{\mathbb{P}}[ | {
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h) - {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h) | \geq\epsilon]
&\leq& B{e}^{-C N \epsilon^2 / {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}} ,\end{aligned}$$ where ${
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h)$ and ${
\ifthenelse{\equal{tilde}{}}{\ensuremath{\phi_{s|T}}}{\ifthenelse{\equal{tilde}{hat}}{\ensuremath{\phi^{N}_{s|T}}}
{\ifthenelse{\equal{tilde}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|T}}}
{\ifthenelse{\equal{tilde}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|T}}}}
}
}
}(h)$ are defined in [(\[eq:smoothing:backw\_decomposition\_sample\])]{} and [(\[eq:FFBSi:estimator\])]{}.
Letting $s = T$ in Theorem \[theo:Hoeffding-uniform\] provides, as a special case, the (already known) time uniform deviation inequality for the [*filter*]{} approximation; however, the novelty of the bounds obtained here is that these confirm the stability of the FFBSm and FFBSi marginal smoothing approximations also when $s$ is fixed and $T$ tends to infinity (see [@delmoral2004] for further discussion).
[Proof of Theorem \[theo:Hoeffding-uniform\]]{} Combining [(\[eq:Lone\])]{} with the definition [(\[eq:defLt\])]{} and Assumption \[assum:strong-mixing-condition\] yields, for all $x \in{\mathbb{X}}$, $$\label{eq:majorationL}
\frac{\sigma_-}{\sigma_+} \leq\frac{{\mathcal L}_{t,T}(x,{\mathbf{1}})}{{\ifthenelse{\equal{}{}}{| {\mathcal L}_{t,T}(\cdot, {\mathbf{1}}) |_\infty}{| {\mathcal L}_{t,T}(\cdot, {\mathbf{1}})|^2_{\infty}}}} \leq1 .$$ Let $h \in{\mathcal{F}_{\mathrm{b}}({\mathbb{X}}^{T+1})}$ and assume without loss of generality that ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0:T|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0:T|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{0:T|T}}}}
}
}
}(h)=0$. Then, [(\[eq:smooth:recursion\])]{} implies that ${
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,h)] = 0$ and the first term of the decomposition [(\[eq:decomp\_Smooth\])]{} thus becomes $$\label{eq:initialRecurInegExpo-1}
\frac{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot,h)]}{{
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{0}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{0}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{0}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{0}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{0}}}
}
}
}
}
}[L_{0,T}(\cdot
,{\mathbf{1}})]} = \frac{N^{-1}\sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) L_{0,T}({\ensuremath{\xi_{0}^{i}}},h)}{N^{-1}
\sum_{\ell= 0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{\ell}}}) g_0({\ensuremath{\xi_{0}^{\ell}}}) L_{0,T}({\ensuremath{\xi_{0}^{\ell}}},{\mathbf{1}})} ,$$ where $({\ensuremath{\xi_{0}^{\ell}}})_{\ell= 1}^N$ are [i.i.d.]{} random variables with distribution ${\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}$. Noting that $L_{0,T} = {\mathcal L}_{0,T}$ we obtain an exponential deviation inequality for [(\[eq:initialRecurInegExpo-1\])]{} by applying Lemma \[lem:inegessentielle\] with $$\cases{
\displaystyle a_N = N^{-1}\sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) {\mathcal L}_{0,T}({\ensuremath{\xi_{0}^{i}}},h)/{|{\mathcal L}_{0,T}(\cdot,h)|_{\infty}} , \cr
\displaystyle b_N = N^{-1}\sum_{i=0}^N \frac{{d}{\chi}}{{d}{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}}({\ensuremath{\xi_{0}^{i}}}) g_0({\ensuremath{\xi_{0}^{i}}}) {\mathcal L}_{0,T}({\ensuremath{\xi_{0}^{i}}},{\mathbf{1}})/{|{\mathcal L}_{0,T}(\cdot,h)|_{\infty}} , \cr
b = {\chi}[g_0(\cdot) {\mathcal L}_{0,T}(\cdot,{\mathbf{1}})]/{|{\mathcal L}_{0,T}(\cdot
,h)|_{\infty}} , \cr
\beta= {\chi}(g_0) \sigma_- / \sigma_+ .
}
$$ Here, condition is trivially satisfied and conditions and follow from the Hoeffding inequality for [i.i.d.]{} variables.
According to [(\[eq:decomp\_Smooth\])]{} and [(\[eq:definition-A\])]{}, it is now required, for any $1\leq t\leq T$, to derive an exponential inequality for $$A^N_{t,T} {\stackrel{\mathrm{def}}{=}}\frac{N^{-1}\sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}}
G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{{\Pi_{s,T} h}})}{N^{-1} \sum_{\ell
=1}^N {\ensuremath{\omega_{t}^{\ell}}} {\mathcal L}_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{\mathbf{1}})} .$$ Note first that, using [(\[eq:majorationL\])]{}, we have $$|A^N_{t,T}| \leq\biggl( \frac{\sigma_+}{\sigma_-} \biggr) \frac{N^{-1} \sum
_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{{\Pi_{s,T} h}}) / {\ifthenelse{\equal{}{}}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}}) |_\infty}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|^2_{\infty}}}}{N^{-1} \sum_{\ell=1}^N
{\ensuremath{\omega_{t}^{\ell}}}} .$$ We use again Lemma \[lem:inegessentielle\] with $$\cases{
\displaystyle a_N = N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{{\Pi_{s,T} h}})/{\ifthenelse{\equal{}{}}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}}) |_\infty}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}})|^2_{\infty}}} , \cr
\displaystyle b_N = N^{-1} \sum_{\ell=1}^N {\ensuremath{\omega_{t}^{\ell}}} , \cr
b = \mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} | {\mathcal{F}_{t-1}^{N}}] = {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
} [ {M}(\cdot, g_t) ] / {
\ifthenelse{\equal{hat}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{hat}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{hat}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{hat}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}) , \cr
\beta= c_- / {\ifthenelse{\equal{}{}}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}} |_\infty}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|^2_{\infty}}} .
}
$$ Assumption \[assum:strong-mixing-condition\] shows that $b \geq
\beta$ and Lemma \[lem:G-uniform\] shows that $|a_N/b_N| \leq M
{\stackrel{\mathrm{def}}{=}}\rho^{|t-s|} {\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}}$, where $\rho$ is defined in [(\[eq:definition-rho\])]{}. Therefore, condition of Lemma \[lem:inegessentielle\] is satisfied and the Hoeffding inequality gives $$\begin{aligned}
{\mathbb{P}}[ | b_N - b | \geq\epsilon] &\leq&{\mathbb{E}}\Biggl[ \mathbb{P}\Biggl[ \Biggl|N^{-1} \sum
_{\ell= 1}^N ( {\ensuremath{\omega_{t}^{\ell}}} - {\ifthenelse{\equal{}{}}{\mathbb{E}[ {\ensuremath{\omega_{t}^{1}}} | {\mathcal{F}_{t-1}^{N}}
]}{\mathbb{E}_{}[ {\ensuremath{\omega_{t}^{1}}} | {\mathcal{F}_{t-1}^{N}} ]}}
) \Biggr| \geq\epsilon\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] \Biggr]\\
&\leq&2 \exp( -2 N \epsilon^2 / {\ifthenelse{\equal{2}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}} ),\end{aligned}$$ establishing condition in Lemma \[lem:inegessentielle\]. Finally, Lemma \[lem:G-uniform\] and the Hoeffding inequality imply that $$\begin{aligned}
{\mathbb{P}}[ |a_N | \geq\epsilon] &\leq&{\mathbb{E}}\Biggl[ \mathbb{P}\Biggl[ \Biggl|N^{-1} \sum_{\ell
=1}^N {\ensuremath{\omega_{t}^{\ell}}} G^N_{t,T}({\ensuremath{\xi_{t}^{\ell}}},{{\Pi_{s,T} h}})/{\ifthenelse{\equal{}{}}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}}) |_\infty}{| {\mathcal L}_{t,T}(\cdot,{\mathbf{1}}) |^2_{\infty}}} \Biggr| \geq\epsilon\Big|{\mathcal{F}_{t-1}^{N}}\Biggr] \Biggr]\\
&\leq&2 \exp\biggl( - 2 \frac{N \epsilon^2}{{\ifthenelse{\equal{2}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}}
\rho^{2|t-s|} {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}}} \biggr) = 2 \exp\biggl( - 2 \frac{N \epsilon
^2}{{\ifthenelse{\equal{2}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}} M^2} \biggr) .\end{aligned}$$ Lemma \[lem:inegessentielle\] therefore yields $${\mathbb{P}}\biggl( \biggl| \frac{a_N}{b_N} \biggr| \geq\epsilon\biggr) \leq2 \exp\biggl( - \frac{N
\epsilon^2 c_-^2}{2 {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}} \rho^{2|t-s|}{\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}}^2{\ifthenelse{\equal{}{}}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}} |_\infty}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|^2_{\infty}}}^2} \biggr) ,$$ so that $${\mathbb{P}}( | A^N_{t,T} | \geq\epsilon) \leq2 \exp\biggl( - \frac{N \epsilon
^2 c_-^2 \sigma_-^2}{2 {\ifthenelse{\equal{2}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{2}(h)}}} \rho^{2|t-s|}{\ifthenelse{\equal{}{}}{| {\ensuremath{\omega_{t}}} |_\infty}{| {\ensuremath{\omega_{t}}}|^2_{\infty}}}^2{\ifthenelse{\equal{}{}}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}} |_\infty}{| {\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|^2_{\infty}}}^2 \sigma_+^2} \biggr) .$$ A time uniform exponential deviation inequality for $\sum_{t=1}^T
A_{t,T}$ then follows from Lemma \[lem:pasfor\] and the proof is complete.
A time uniform bound on the variance of the marginal smoothing distribution {#sec:TimeUniformCLTFFBS}
---------------------------------------------------------------------------
Analogous to the result obtained in the previous section, a time uniform bound on the asymptotic variance in the CLT for the [*marginal smoothing*]{} approximations can, again under the strong mixing Assumption \[assum:strong-mixing-condition\], be easily obtained from the exponentially decreasing bound on $G_{t,T}$ stated and proved in Lemma \[lem:G-uniform\] for the quantity.
\[theo:CLT-uniform\] Assume Assumptions \[assum:bound-likelihood\]–\[assum:strong-mixing-condition\] hold with $T = \infty$. Then, for all ${s\leq T}$, $${
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{{\Pi_{s,T} h}}}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[{{\Pi_{s,T} h}}]}}}
{\ifthenelse{\equal{{{\Pi_{s,T} h}}}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[{{\Pi_{s,T} h}}]}}}
} \leq\biggl(\frac{\sigma_+}{\sigma
_-} \Bigl( 1 \vee\sup_{t \geq1} {|{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|_{\infty}} \Bigr) \sup_{t
\geq0} {|{\ensuremath{\omega_{t}}}|_{\infty}}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} \biggr)^2 \frac{1+\rho
^2}{1-\rho^2} ,$$ where ${
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[]}}}
{\ifthenelse{\equal{}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[]}}}
}$ is defined in [(\[eq:expression-covariance\])]{}.
In accordance with the results of the previous section, letting $s = T$ in the previous theorem provides a time uniform bound on the asymptotic variance for the [*filter*]{} approximation; nevertheless, as mentioned previously, the situation of interest for us is when $s$ is fixed and $T$ goes to infinity.
[Proof of Theorem \[theo:CLT-uniform\]]{} Combining [(\[eq:majorationL\])]{} and [(\[eq:time-unif-G-lim\])]{} with ${\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}({\ensuremath{\omega_{0}}})=1$ yields $$\frac{{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}({\ensuremath{\omega_{0}}}^2(\cdot) G_{0,T}^2(\cdot,{{\Pi_{s,T} h}}))}{{\ifthenelse{\equal{}{}}{\ensuremath{\rho_{0}}}{\ensuremath{\check{
\rho}_{0}}}}^2 [{\ensuremath{\omega_{0}}}(\cdot) {\mathcal L}_{0,T}(\cdot,{\mathbf{1}}) ]} \leq\biggl(\frac{\sigma_+}{\sigma_-}{|{\ensuremath{\omega_{0}}}|_{\infty}}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}}\rho^s \biggr)^2 .$$ Moreover, by inserting, for any $0 < t \leq T$, the bound obtained in [(\[eq:time-unif-G-lim\])]{} into the expression [(\[eq:definition-upsilon\])]{} of $\upsilon_{t,T}$ we obtain $$\frac{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}(\upsilon_{t,T}(\cdot,{{\Pi_{s,T} h}}))) {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}({\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}})}{{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{t-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{t-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{t-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{t-1}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{t-1}}}
}
}
}
}
}^2[{\mathcal L}_{t-1,T}(\cdot,{\mathbf{1}})]} \leq
\biggl(\frac{\sigma_+}{\sigma_-} {|{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|_{\infty}}{|{\ensuremath{\omega_{t}}}|_{\infty}}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} \rho^{|t-s|} \biggr)^2 .$$ Finally, plugging the two bounds above into [(\[eq:expression-covariance\])]{} gives $${
\ifthenelse{\equal{}{}}{\ifthenelse{\equal{{{\Pi_{s,T} h}}}{}}{\ensuremath{
\Gamma_{0:T|T}}}{\ensuremath{\Gamma_{0:T|T}[{{\Pi_{s,T} h}}]}}}
{\ifthenelse{\equal{{{\Pi_{s,T} h}}}{}}{\ensuremath{\Gamma_{,0:T|T}}}{\ensuremath{
\Gamma_{,0:T|T}[{{\Pi_{s,T} h}}]}}}
}\leq\biggl(\frac{\sigma_+}{\sigma_-}
\Bigl( 1 \vee\sup_{t \geq1} {|{\ifthenelse{\equal{}{}}{\ifthenelse{\equal{}{}}{\vartheta_{t}}{
\vartheta_{t}()}}
{\ifthenelse{\equal{}{smooth}}{\ifthenelse{\equal{}{}}{\tilde{
\vartheta}_{t}}{\tilde{\vartheta}_{t}()}}
{\ifthenelse{\equal{}{fully}}{\ifthenelse{\equal{}{}}{\vartheta^
\star_{t}}{\vartheta^\star_{t}()}}{\mathrm{erreur}}}}}|_{\infty}} \Bigr) \sup_{t \geq
0} {|{\ensuremath{\omega_{t}}}|_{\infty}}{\ifthenelse{\equal{}{}}{\ensuremath{\operatorname{osc}(h
)}}{\ensuremath{\operatorname{osc}^{}(h)}}} \biggr)^2 \Biggl(\sum_{t=0}^\infty\rho
^{2|t-s|} \Biggr) ,$$ which completes the proof.
Proofs of Propositions 1 and 2 {#sec:complexity:proofs}
==============================
Having at hand the theory established in the previous sections, we are now ready to present the proofs of Propositions \[prop:complexityBoostrap\] and \[prop:complexity\].
[Proof of Proposition \[prop:complexityBoostrap\]]{} The average number of simulations required to sample $J_s^\ell$ conditionally on $\mathcal{G}_{s+1}^N$ is $\sigma_+ {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}} /
\sum_{u = 1}^N {\ensuremath{\omega_{s}^{u}}} \ensuremath{m}({\ensuremath{\xi_{s}^{u}}}, {\ensuremath{\xi_{s+1}^{J_{s+1}^\ell}}})$. Hence, the number of simulations $Z_s^N$ required to sample $\{ J_s^\ell\}_{\ell= 1}^N$ has conditional expectation $$\mathbb{E} [ Z_s^N | \mathcal{G}_{s+1}^N ] = \sum_{\ell= 1}^N \frac
{\sigma_+ {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}}}{\sum_{i = 1}^N {\ensuremath{\omega_{s}^{i}}} \ensuremath
{m}({\ensuremath{\xi_{s}^{i}}},{\ensuremath{\xi_{s+1}^{J_{s+1}^\ell}}})} .$$ We denote ${\ensuremath{\omega_{s|T}^{i}}} {\stackrel{\mathrm{def}}{=}}\mathbb{P}[J^1_s = i|{\mathcal{F}_{T}^{N}}]$ and ${\ensuremath{\omega_{s:s+1|T}^{\ell i}}} {\stackrel{\mathrm{def}}{=}}\mathbb{P}[J^1_s = \ell,
J^1_{s+1} = i|{\mathcal{F}_{T}^{N}}]$ and write $$\begin{aligned}
\mathbb{E} [ Z_s^N | {\mathcal{F}_{T}^{N}} ] &=& \sum_{i=1}^N {\ensuremath{\omega_{s+1|T}^{i}}}\frac{\sigma_+ {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}}}{\sum_{j=1}^N {\ensuremath{\omega_{s}^{j}}}
\ensuremath{m}
({\ensuremath{\xi_{s}^{j}}}, {\ensuremath{\xi_{s+1}^{i}}})} \\
&=& \sigma_+ {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}} \sum_{i=1}^N \sum_{\ell= 1}^N \frac
{{\ensuremath{\omega_{s+1|T}^{i}}} {\ensuremath{\omega_{s}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{s}^{\ell}}},
{\ensuremath{\xi_{s+1}^{i}}})}{\sum_{j=1}^N {\ensuremath{\omega_{s}^{j}}} \ensuremath{m}({\ensuremath{\xi_{s}^{j}}},
{\ensuremath{\xi_{s+1}^{i}}})} \times\frac{1}{{\ensuremath{\omega_{s}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{s}^{\ell
}}},{\ensuremath{\xi_{s+1}^{i}}})} \\
&=& \sigma_+ {\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}} \sum_{i=1}^N \sum_{\ell= 1}^N {\ensuremath{\omega_{s:s+1|T}^{\ell i}}} \frac{1}{{\ensuremath{\omega_{s}^{\ell}}} m({\ensuremath{\xi_{s}^{\ell}}},
{\ensuremath{\xi_{s+1}^{i}}})} .\end{aligned}$$ For the bootstrap particle filter, ${\ensuremath{\omega_{s}^{\ell}}} \equiv g_s({\ensuremath{\xi_{s}^{\ell}}})$; Theorem \[thm:Hoeffding-FFBS\] then implies that ${\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}}/ N {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \rightarrow\infty} {
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}(g_s)$ and $$\begin{aligned}
&&\sum_{i=1}^N \sum_{\ell= 1}^N {\ensuremath{\omega_{s:s+1|T}^{\ell i}}} \frac
{1}{{\ensuremath{\omega_{s}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{s}^{\ell}}}, {\ensuremath{\xi_{s+1}^{i}}})} \\
&&\qquad {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \rightarrow\infty} {\int\!\!\!\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s+1|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s+1|T}}}}
}
}
}({d}x_{s:s + 1}) \frac
{1}{g_s(x_s) m(x_s, x_{s+1})} .\end{aligned}$$ Besides, $$\begin{aligned}
&&{\int\!\!\!\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s+1|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s+1|T}}}}
}
}
}({d}x_{s:s + 1}) \frac{1}{g_s(x_s)
\ensuremath{m}(x_s, x_{s+1})} \\
&&\qquad = {\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}({d}x_s) \frac{g_s(x_s) {M}(x_s,
{d}x_{s+1})}{g_s(x_s) \ensuremath{m}(x_s, x_{s+1})}
g_{s+1}(x_{s+1})\\
&&\qquad \quad \hphantom{{\int\cdots\int}}
{}\times\prod_{u=s+2}^T {M}(x_{u-1}, {d}x_u) g_u(x_u)\\
&&\qquad \quad {}\Big/{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}({d}x_s) g_{s}(x_{s}) \prod_{u=s+1}^T {M}(x_{u-1}, {d}x_u)
g_u(x_u) \\
&&\qquad = \frac{{\int\cdots\int}{d}x_{s+1} \prod_{u=s+2}^T \int{M}(x_{u-1},
{d}x_u) g_u(x_u)}{{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}({d}x_s) g_s(x_s) \prod
_{u=s+1}^T {M}(x_{u-1}, {d}x_u) g_u(x_u)} .\end{aligned}$$ Similarly, in the fully adapted case we have ${\ensuremath{\omega_{s}^{i}}} \equiv1$ for all $i \in\{1, \dots, N\}$; thus, ${\ifthenelse{\equal{}{}}{\ensuremath{\Omega_{s}}}{\ensuremath{
\Omega_{s}^{()}}}} = N$ and $$\begin{aligned}
&&\sum_{i=1}^N \sum_{\ell= 1}^N {\ensuremath{\omega_{s:s+1|T}^{\ell i}}} \frac
{1}{{\ensuremath{\omega_{s}^{\ell}}} \ensuremath{m}({\ensuremath{\xi_{s}^{\ell}}}, {\ensuremath{\xi_{s+1}^{i}}})} \\
&&\qquad {\stackrel{\mathrm{P}}{\longrightarrow}}_{N \rightarrow\infty} {\int\!\!\!\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s:s+1|T}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s:s+1|T}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s:s+1|T}}}}
}
}
}({d}x_{s:s + 1}) \frac
{1}{m(x_s, x_{s+1})}
\\
&&\qquad = \frac{{\int\cdots\int}g_{s+1}(x_{s+1})\, {d}x_{s+1}\prod_{u=s+2}^T
\int{M}(x_{u-1}, {d}x_u) g_u(x_u)}{{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,\mathrm{t}}_{s}}}
{\ifthenelse{\equal{}{aux}}{\ensuremath{\phi^{N,\mathrm{a}}_{s}}}
}
}
}
}
}({d}x_s)
\prod_{u=s+1}^T {M}(x_{u-1}, {d}x_u) g_u(x_u)} .\end{aligned}$$ In both cases, the numerator can be bounded from above by $$\frac{\sigma_+^{T-s-1} \prod_{u=s+1}^T \int g_u(x_u)\, {d}x_u}{{\int\cdots\int}{
\ifthenelse{\equal{}{}}{\ensuremath{\phi_{s|s-1}}}{\ifthenelse{\equal{}{hat}}{\ensuremath{\phi^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tilde}}{\ensuremath{\tilde{\phi}^{N}_{s|s-1}}}
{\ifthenelse{\equal{}{tar}}{\ensuremath{\phi^{N,
\mathrm{t}}_{s|s-1}}}}
}
}
}({d}x_s) g_s(x_s) \prod_{u=s+1}^T {M}(x_{u-1}, {d}x_u) g_u(x_u)}$$ if $\int g_u(x_u) \,{d}x_u < \infty$ for all $u \geq0$.
[Proof of Proposition \[prop:complexity\]]{} Fix a time step $s$ of the algorithm and denote by $C_s$ the number of elementary operations required for this step. For $k \in\{1, \dots,
n\}$, let $T_s^k$ be the number of times that $k$ appears in list $L$ at time $s$ in the ‘while’ loop. Let also $N_s^u {\stackrel{\mathrm{def}}{=}}\sum_{k=1}^N \mathbh{1}_{\{T_s^k \geq u\}}$ be the size of $L$ (i.e., the value of $n$ at line 6) after $u$ iterations of the ‘while’ loop, with $N_s^0 {\stackrel{\mathrm{def}}{=}}N$. Then, using Proposition \[prop:multisample\] there exists a constant C such that $$C_s \leqC \sum_{u=0}^\infty N_s^u \biggl( 1 + \log\biggl(1 + \frac{N}{N_s^u} \biggr) \biggr) .$$ As $n\to n(1+\log(1+N/n))$ is a concave, increasing function, it holds by Jensen’s inequality that $${\mathbb{E}}[C_s] \leq C \sum_{u=0}^\infty{\mathbb{E}}[N_s^u] \biggl( 1+\log\biggl(1 + \frac
{N}{{\mathbb{E}}[N_s^u]} \biggr) \biggr) .$$ Besides, $${\mathbb{E}}[N_s^u] = \sum_{k = 1}^N {\mathbb{P}}(T_s^k \geq u) \leq N \biggl(1 - \frac
{\sigma_-}{\sigma_+} \biggr)^u$$ as $\sigma_-/\sigma_+$ is a lower bound on the acceptation probability. Thus, $${\mathbb{E}}[C_s] \leq C N \sum_{u=0}^\infty\biggl(1-\frac{\sigma_-}{\sigma_+}
\biggr)^{u} \biggl( 1+\log\biggl(1+\frac{1}{ (1-\sigma_-/\sigma_+ )^u} \biggr) \biggr)\leq
\frac{KN\sigma_+}{\sigma_-} .$$
Appendix A: Proof of Lemma 4 {#sec:proof:lem:inegEssentielle}
============================
Write $$\biggl|\frac{a_N}{b_N} \biggr| \leq b^{-1} \biggl| \frac{a_N}{b_N} \biggr| |b-b_N| + b^{-1} |
a_N | \leq\beta^{-1}M |b-b_N|+ \beta^{-1} | a_N | ,\qquad
\mathbb{P}\mbox{-a.s.}$$ Thus, $$\biggl\{ \biggl|\frac{a_N}{b_N} \biggr| \geq\epsilon\biggr\} \subseteq\biggl\{|b-b_N| \geq\frac
{\epsilon\beta}{2M} \biggr\} \cup\biggl\{ |a_N | \geq\frac{\epsilon\beta
}{2} \biggr\} ,$$ from which the proof follows.
Appendix B: Technical results {#sec:TechnicalResults}
=============================
\[lem:pasfor\] Let $\{ Y_{n,i} \}_{i=1}^n$ be a triangular array of random variables such that there exist constants $B > 0$, $C > 0$, and $\rho$ with $0 <
\rho< 1$ satisfying, for all $n$, $i \in\{1, \dots, n \}$, and $\epsilon> 0$, $${\mathbb{P}}( |Y_{n,i}| \geq\epsilon) \leq B \exp( -C \epsilon^2 \rho
^{-2i} ) .$$ Then, there exist constants $\bar{B} > 0$ and $\bar{C} > 0$ such that, for any $n$ and $\epsilon> 0$, $${\mathbb{P}}\Biggl(\Biggl | \sum_{i=1}^n Y_{n,i} \Biggr| \geq\epsilon\Biggr) \leq\bar{B} {e}^{- \bar{C} \epsilon^2} .$$
Set $S {\stackrel{\mathrm{def}}{=}}\sum_{\ell= 1}^\infty\sqrt{\ell} \rho^\ell$; one easily concludes that $${\mathbb{P}}\Biggl( \Biggl| \sum_{i=1}^n Y_{n,i} \Biggr| \geq\epsilon\Biggr) \leq\sum_{i=1}^n {\mathbb{P}}\bigl( |Y_{n,i}| \geq\epsilon S^{-1} \sqrt{i} \rho^i \bigr) \leq B \sum
_{i=1}^n \exp( -C S^{-1} \epsilon^2 i ) .$$ Set $\epsilon_0 > 0$. The proof follows by noting that, for any $\epsilon\geq\epsilon_0$, $$\sum_{i=1}^n \exp( -C S^{-1} \epsilon^2 i ) \leq\exp( C S^{-1}
\epsilon_0^2 ) \exp( -C S^{-1} \epsilon^2 ) / \bigl( 1 - \exp( C S^{-1}
\epsilon_0^2 ) \bigr) .\qquad$$
Description of the sampling procedure {#subsec:multisampling}
-------------------------------------
In this section, we describe and analyze an efficient multinomial sampling procedure, detailed in Algorithm \[alg:multisample\]. Given a probability distribution $(p_1,\dots,p_N)$ on the set $\{1, \dots,
N\}$, it returns a sample of size $n$ of that distribution. Compared to the procedure described in Section 7.4.1 in [@cappemoulinesryden2005], its main virtue is to be efficient for both large and small samples sizes: if $n = 1$, the complexity is $O(\log
(N))$, while if $n = N$, the complexity is $O(N)$.
\[prop:multisample\] The number of elementary operations required by Algorithm \[alg:multisample\] is $O (n + n \log(1 + N/n) )$.
The order statistics at line 5 and the permutation at line 6 can be sampled using $O(n)$ operations; see [@devroye1986], Chapter V and XIII. For each value of $k$ between $1$ and $n$, denote by $G_k$ the number of times lines 11–13 are executed. Observe that line 18 is executed the same number of times, and thus the number of elementary operations required by call to Algorithm \[alg:multisample\] is $O(n + \sum
_{k=1}^nG_k)$. But the value of $l$ is increased during iteration $k$ by at least $2^{G_k}-1$, and as the final value of $l$ is at most equal to $N$, it holds that $$\sum_{k=1}^n 2^{G_k} \leq N + n .$$ By convexity, $$\exp\Biggl(\frac{\log(2)}{n} \sum_{k=1}^n G_k \Biggr) \leq\frac{1}{n} \sum
_{k=1}^n 2^{G_k} \leq1 + \frac{N}{n} ,$$ which implies that $$\sum_{k=1}^nG_k \leqn \log\biggl(1 + \frac{N}{n} \biggr) / \log(2) .$$
$q_1 \gets p_1$ $q_k \gets q_{k-1}+p_k$ sample an order statistics $U_{(1)},\dots,U_{(n)}$ of an i.i.d. uniform distribution uniformly sample a permutation $\sigma$ on $\{1, \dots, n\}$ $l \gets0, r\gets1$ $d \gets1$ $l \gets r$ $r \gets\min(r + 2^d, N)$ $d \gets d+1$ $m \gets\lfloor(l+r)/2 \rfloor$ $l \gets m$ $r \gets m$ $I_{\sigma(k)} \gets r$
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[^1]: Since the first version of this paper has been released, an article [@delmoraldoucetsingh2010] has been published. This work, developed completely independently from ours, complement the results presented in this manuscript. In particular, this paper presents a functional central limit theorems as well as nonasymptotic variance bounds. Additionally, this work shows how the forward filtering backward smoothing estimates of additive functionals can be computed using a forward only recursion.
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