The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better convergence rate for reaching the stationary (uniform) distribution in certain cases. Some experimental evidence for this has been given in the literature. In this note we give a spectral gap comparison between two switch-based chains and the Curveball chain. In particular, this comparison allows us to conclude that the Curveball Markov chain is rapidly mixing whenever one of the two switch chains is rapidly mixing. Our analysis directly extends to the case of sampling binary matrices with forbidden entries (under the assumption of irreducibility). This in particular captures the case of sampling simple directed graphs with given degrees. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tet ali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest. We also obtain an improved bound on the smallest eigenvalue for the switch Markov chain studied by Greenhill for uniformly sampling simple directed regular graphs.
@cite_1: We prove a diffusion scaling limit for the macroscopic densities of colored particles performing the simply excluded random walk, and relate this to the limiting behavior of a test particle in equilibrium.
@cite_0: The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e. , the time taken for it to reach its stationary or equilibrium distribution. The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.
@cite_16: This is an expository paper, focussing on the following scenario. We have two Markov chains, M and M'. By some means, we have obtained a bound on the mixing time of M'. We wish to compare M with M' in order to derive a corresponding bound on the mixing time of M. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.
@cite_20: 
@cite_11: By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of 1, 2, . . ., n with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion
All the results regarding rapid mixing mentioned above rely on the multi-commodity flow method developed by Sinclair @cite_0 . In this work we omit multi-commodity flow techniques in order to compare the switch and Curveball Markov chains, but rather take a more elementary approach based on comparing eigenvalues of transition matrices. One seeming advantage of the eigenvalue comparison is that it allows us to compare the switch and Curveball chains for arbitrary fixed row and column sums. Our spectral gap comparisons are special cases of the classical comparison framework developed largely by Diaconis and Saloff-Coste and is based on so-called Dirichlet form comparisons of Markov chains, see, e.g., @cite_20 @cite_11 , and also Quastel @cite_1 . See also the expository paper by Dyer, Goldberg, Jerrum and Martin @cite_16 . As the stationary distributions are the same for all our Markov chains, we use a more direct, but equivalent, framework based on positive semidefiniteness. We briefly elaborate on this in Appendix for the interested reader.