The Directed Landscape is a Black Noise
Zoe Himwich Shalin Parekh



Himwich]Zoe Himwich
Parekh]Shalin Parekh

Zoe Himwich: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
himwich@math.columbia.com
Shalin Parekh: Department of Mathematics, University of Maryland, College Park, 4176 Campus Drive, College Park, MD 20742 USA
parekh@umd.edu




    We show that the directed landscape is a black noise in the sense of Tsirelson and Vershik. As a corollary, we show that for any microscopic system in which the height profile converges in law to the directed landscape, the driving noise is asymptotically independent of the height profile. This decoupling result provides one answer to the question of what happens to the driving noise in the limit under the KPZ scaling, and illustrates a type of noise sensitivity for systems in the KPZ universality class. Such decoupling and sensitivity phenomena are not present in the intermediate-disorder or weak-asymmetry regime, thus illustrating a contrast from the weak KPZ scaling regime. Along the way, we prove a strong mixing property for the directed landscape on a bounded time interval under spatial shifts, with a mixing rate α(N)≤ Ce^-dN^3 for some C,d>0. 

The Directed Landscape is a Black Noise
    [
    April 29, 2024
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§ INTRODUCTION



 §.§ Preface

Over the past several decades, mathematicians and physicists who study interacting particle systems, integrable systems, and the Kardar-Parisi-Zhang (KPZ) equation have discovered connections between the large-scale behavior of a wide-ranging class of models. It is known that the height functions of asymmetric exclusion processes and stochastic vertex models exhibit universal scaling behavior in 1+1 dimensions. A common feature in these models is the appearance of 1:2:3 scaling exponents and Tracy-Widom limit laws. Similar results are expected for the Eden model and last passage percolation models with arbitrary weight distributions, as well as Burgers-type stochastic PDEs and ballistic deposition models (e.g. Tetris). This phenomenon is called KPZ universality <cit.>. 

In recent years, there has been a great deal of work involved in constructing and understanding the universal limit objects in the KPZ universality class. Matetski, Quastel, and Remenik <cit.> constructed the KPZ fixed point, the invariant limiting Markov process (conjecturally) of all models in the KPZ universality class. Dauvergne, Ortmann and Virág <cit.> then constructed the directed landscape, a generalization which encodes the natural joint couplings of all KPZ fixed points started from all possible initial data simultaneously.
The directed landscape is a four-parameter field, denoted in this paper as ℒ_s,t(x,y), see <ref>.

It is a longstanding project of mathematical physicists to have a better understanding of this phenomenon of universality. It is conjectured that any model for which it is possible to associate a height function h(t,x) with the basic properties of local dynamics, a smoothing mechanism, lateral growth, and space-time forcing with rapid correlation decay, will converge to the KPZ fixed point under 1:2:3 scaling, ϵ h(ϵ^-3t,ϵ^-2x)-C_ϵt. More generally, the height mechanisms started simultaneously from several distinct initial data should converge to KPZ fixed points coupled through the directed landscape. See the strong and weak KPZ universality conjectures as discussed in the introduction of <cit.>. 
Models which satisfy these conditions, and which have been proved or conjectured to have the KPZ fixed point and directed landscape as a scaling limit, are said to belong to the KPZ universality class.  

The directed landscape (<ref>), which is the main object in this paper, was constructed by Dauvergne, Ortmann, and Virág <cit.> as the scaling limit of the last passage metric between points on a Brownian line ensemble (Brownian LPP). It is a random four-parameter continuous function  ℒ:ℝ^4_↑→ℝ, with ℝ^4_↑:={(x,s;y,t)∈ℝ^4 | s<t}, which may be viewed as a metric between space-time points (s,x) and (t,y) that satisfies a reverse triangle inequality, ℒ_s,r(x,z)≥ℒ_s,t(x,y) + ℒ_t,r(y,z). More precisely the following “metric composition" law holds:

    ℒ_s,r(x,z)     = max_y∈ℝ(ℒ_s,t(x,y) + ℒ_t,r(y,z)),       for all  (x,z)∈ℝ^2,   s<t<r.

A series of recent works have demonstrated that the directed landscape is the scaling limit of many well-known models with simple microscopic dynamics. 
Works such as <cit.> demonstrated that it is the limit of the Brownian, Poissonian, exponential, and geometric last passage percolation, and Poisson and Seppäläinen-Johansson line ensembles. A work by Wu <cit.> combined with a result of Quastel and Sarkar <cit.> demonstrates that the directed landscape is the scaling limit of the KPZ equation started from all possible initial data but coupled through the same realization of the driving noise. Most recently, a work by Aggarwal, Corwin, and Hegde <cit.> demonstrated a similar convergence for ASEP and stochastic six vertex models. The KPZ fixed point of Matetski, Quastel, and Remenik <cit.> is a marginal of the directed landscape by the variational formula
h_t(y)= sup_x∈ℝ(h_0(x)+ℒ_0,t(x,y)), see <cit.>. 

There are still many basic properties of the directed landscape that have yet to be explored. A natural question to ask is whether it has some simpler Gaussian structure.

 Is the directed landscape the adapted strong solution of a white-noise driven SPDE?

This article was motivated by an interest in better understanding what happens to the driving noise of the KPZ equation under the directed landscape limit. Another natural question in this vein is the following. 

    What happens to the microscopic driving noise of systems like last passage percolation models, asymmetric exclusion processes, and the KPZ equation as we take the limit to the directed landscape? In particular, is any memory of this microscopic noise retained by the limiting directed landscape?

This paper aims to address both of the preceding questions. We study the directed landscape as an abstract noise in the sense of Tsirelson and Vershik <cit.>, demonstrating that it is a black noise (<ref>). Using this property, we show that the directed landscape is independent of any white noise field defined on the same filtered probability space (<ref>). These results have a few corollaries. We conclude that the height function associated to any model which converges to the directed landscape will asymptotically decouple from the random environment (<ref>), which answers <ref>. We can also answer <ref> in the negative (<ref>).





 §.§ Background and Results




  §.§.§ Black Noise
 Black noise was first introduced by Tsirelson and Vershik <cit.>, and subsequently explored at length in a survey by Tsirelson <cit.>. Building towards the definition of black noise, we begin by introducing the concept of a noise.


    A noise, denoted in this paper as a 4-tuple (Ω,(ℱ_s,t)_s<t,ℙ,(θ_h)_h), is a probability space (Ω,ℱ,ℙ) with sub-σ-fields ℱ_s,t⊂ℱ (-∞<s<t<∞) and measurable maps θ_h:Ω→Ω (h∈ℝ) such that the following properties are satisfied.
    
        
  * ℱ_s,t and ℱ_t,u are independent under the probability measure P if s<t<u.
        
  * For all r<s<t, ℱ_s,t and ℱ_r,s together generate ℱ_r,t, i.e., ℱ_r,s∨ℱ_s,t=ℱ_r,t.
        
  * ℱ is generated by the union of all ℱ_s,t.
        
  * If A∈ℱ_s,t then θ_h(A)∈ℱ_s+h,t+h, and P(θ_h(A)) = P(A).
        
  * θ_h θ_k = θ_h+k for all h,k∈ℝ, and θ_0=Id.
    
 This definition of noise uses the language of stochastic flows, in the sense of Tsirelson <cit.>. We can relate this to other, possibly more familiar, definitions of noise. For example, if B is a one-dimensional two-sided Brownian motion, the σ-algebra ℱ_s,t would be generated by random variables of the form B(u)-B(s) for u∈ [s,t], and θ_h B(u) := B(u+h)-B(h). More generally a Gaussian space-time white noise on ℝ×ℝ^d, as discussed in Walsh's book on SPDEs <cit.>, associates a Gaussian random variable to every Borel set, and defines a structure of a noise in a similar and straightforward fashion. These are both examples of white noise.

We can now give a definition of black noise. There are several equivalent definitions; Tsirelson <cit.> uses the notion of a stable σ-algebra to define white noise and black noise. The definition we use in this paper is more direct. 


Let (Ω, (ℱ_s,t)_s<t, ℙ, (θ_h)_h∈ℝ) be a noise. A random variable F:Ω→ℝ is called linear if 

    𝔼[F|ℱ_s,t] + 𝔼[F|ℱ_t,u] = 𝔼[F|ℱ_s,u]           for   all     s<t<u.

We say that (Ω, (ℱ_s,t)_s<t, ℙ, (θ_h)_h∈ℝ) is a white noise if ℱ is generated by linear random variables. Likewise, we say that (Ω, (ℱ_s,t)_s<t, ℙ, (θ_h)_h∈ℝ) is a black noise 
if 0 is the only linear random variable.


In general, white noise consists of Gaussian and Poissonian components, whereas black noise can be much more complex and admits no simple description. White noise and black noise do not form a dichotomy: there are noises that satisfy <ref> which have both a “white noise part” and a “black noise part," see <cit.>. 



  §.§.§ The KPZ Universality Class and the Directed Landscape
 In this section, we define the directed landscape and several other important objects in the KPZ universality class. We begin by discussing the Airy process and Airy line ensemble.



    The Airy Line Ensemble is a collection of random functions {𝒜_i}_i∈ℕ with law determined by the Airy kernel (<ref>). For any finite set I⊂ℝ, we define the point process {(x,𝒜_i(x))|x∈ I} on I×ℝ as the determinantal point process <cit.> 
    with kernel K(x_1,t_1;x_2,t_2) given by the extended Airy kernel
    
    K(x_1,t_1;x_2,t_2)     := ∫_0^∞e^-r(t_1-t_2)Ai(x_1+r)Ai(x_2+r)dr     t_1≥ t_2
    
                - ∫_-∞^0e^-r(t_1-t_2)Ai(x_1+r)Ai(x_2+r)dr     t_1< t_2

    In this formula, Ai(·) is the Airy function,
    
    Ai(x)     = 1/πlim_b→∞∫_0^bcos(t^3/3 + xt)dt

    The parabolic Airy line ensemble is defined by 𝒫_i(x)=𝒜_i(x)-x^2. We commonly refer to the top curve 𝒫_1(·) as the parabolic Airy_2 process. 
 
The parabolic Airy line ensemble can also be defined as the unique line ensemble with a Brownian Gibbs resampling property and with the parabolic Airy_2 process as its top line. Existence of such a line ensemble was shown by Corwin and Hammond <cit.> and uniqueness was shown by Dimitrov and Matetski <cit.>. 

For a metric space X, we let C(X) denote the space of continuous functions X→ℝ, equipped with the topology of uniform convergence on compacts. For the choices of X used in this paper, the space C(X) is always separable and metrizable, hence Polish. We can now introduce the Airy sheet, a two-variable random function that can be viewed as the universal limit of the last passage times for pairs of points of models in the KPZ class. 


 The Airy sheet is a random continuous function 𝒮:ℝ^2→ℝ, i.e., a C(ℝ^2)-valued random variable, such that the following properties hold: 

    
  * 𝒮(·,·) has the same law as 𝒮(· +t,· +t) for all t∈ℝ, as C(ℝ^2)-valued random variables. 
    
    
  * 𝒮 can be coupled with a parabolic Airy line ensemble so that 𝒮(0,·)=𝒫_1(·) and for all (x,y,z)∈ℝ^+×ℝ^2, 
    
    lim_k→∞𝒫[(-√(k/2x),k)→(z,1)]-𝒫[(-√(k/2x),k)→(y,1)] = 𝒮(x,z)-𝒮(x,y),
 
    where 𝒫[(a,k)→(b,1)] is the last passage time between points (a,k) and (b,1) on the Airy line ensemble,
    
    𝒫[(a,k)→(b,1)] :=sup_a=t_k<...<t_0=b∑_i=1^k(𝒫_i(t_i-1)-𝒫_i(t_i)).
 


Dauvergne, Ortmann, and Virág <cit.> proved that the Airy sheet exists and that it is unique in law, in fact, it was later shown to be a deterministic functional of the Airy line ensemble <cit.>. With these definitions in hand, we can discuss the directed landscape. As explained in <cit.>, the directed landscape can be interpreted as a random directed spacetime metric of negative sign. We can think of the directed landscape as a universal limit of the metric defined by last passage times on a last passage model. We begin by defining the canonical probability space for the directed landscape. 


We define the canonical probability space of the directed landscape (Ω,ℱ^ℒ). Let ℝ^4_↑:={(x,s;y,t)∈ℝ^4 | s<t}. Define the underlying probability space Ω=C(ℝ^4_↑), equipped with its Borel sets from the topology of uniform convergence on compact sets. The canonical process on this space is the identity map ℒ:C(ℝ^4_↑)→ C(ℝ^4_↑). The evaluation map at a point (x,s,y,t) applied to ℒ will be denoted in this paper as ℒ_s,t(x,y). In this notation, we can define ℱ^ℒ as the σ-algebra generated by 
    ℱ^ℒ_s,t:=σ({ℒ_a,b(x,y):  s≤ a<b≤ t, (x,y)∈ℝ^2}),        -∞<s<t<∞.
 

 
    The directed landscape is a four-parameter field of random variables ℒ_s,t(x,y), indexed by -∞<s<t<∞ and x,y∈ℝ, such that

    
  * ℒ_s,t and ℒ_t,u are independent for s<t<u. 

    
  * ℒ_s,t∘ℒ_t,u = ℒ_s,u for all s<t<u where the “metric composition" is defined for two functions f,g:ℝ^2→ℝ as 
    (f∘ g)(x,z) = sup_y∈ℝ f(x,y)+g(y,z).


    
  * For all s<t, ℒ_s,t has the same law as R_t-sℒ_0,1 where R_ϵ f(x,y) := ϵ^1/3 f(ϵ^-2/3x,ϵ^-2/3 y).

    
  * ℒ_0,1 is distributed as the Airy Sheet.


As an immediate consequence of this definition, we conclude that the directed landscape defines a noise. 
 The directed landscape defines a noise (Ω,(ℱ^ℒ_s,t)_s<t,ℙ,(θ_h)_h) in the sense of <ref>.

 Let Ω and ℱ^ℒ_s,t be as in <ref>. Define the measurable maps θ_h:Ω→Ω by (θ_h ℒ)_s,t(x,y):= ℒ_s+h,t+h(x,y). 
Properties (1) and (2) of <ref> verify properties (1) and (2) of <ref>, respectively. Properties (4) and (5) of <ref> are satisfied directly by the definition of the map θ_h. Finally, we note that ℱ^ℒ=⋁_-∞<s<t<∞ℱ^ℒ_s,t by definition.


We can now state our main theorem.


 The noise associated to the directed landscape is a black noise in the sense of <ref>. 


There are a few equivalent conditions for demonstrating that a noise is a black noise (we use the one stated in <ref>). These conditions usually involve a variance bound which resembles a noise sensitivity condition with respect to all functions in L^2(Ω). Noise sensitivity, which was studied in the context of Boolean functions in foundational works such as <cit.>, and then for its applications to physics and computer science in works such as <cit.>, is closely related to black noise. In fact, another way to understand the concept of black noise is to consider it as a system with respect to which all observables (functionals) are noise sensitive. 
It would be interesting to explore these noise sensitivity properties further and in a more precise fashion. Ganguly and Hammond <cit.>, for instance, study the asymptotic noise sensitivity of Brownian last passage, using the notion of the overlap of perturbed geodesics to measure noise sensitivity. 

While we do not study more specific noise sensitivity phenomena in this paper, we do obtain a few corollaries of <ref>. The next result, <ref>, demonstrates the independence of the directed landscape from any white noise defined on the same probability space. For the purpose of this text, a Gaussian white noise on an open subset U⊂ℝ^d is a family of jointly Gaussian random variables {ξ(f)}_f∈ L^2(U) all defined on the same probability space, satisfying 𝔼[ξ(f)ξ(g)] = (f,g)_L^2(U). We often write ξ(f):= ∫_U f(x)ξ(dx) or similar. A Gaussian space-time white noise is just a special case U=ℝ^2, where the first coordinate is interpreted as “time" and the second as “space."


    Consider a noise (Ω, (ℱ_s,t)_s<t,ℙ, (θ_h)_h). Let ℒ and ξ respectively be a copy of a directed landscape and a Gaussian space-time white noise, defined on this same probability space, which together generate the entire σ-algebra ℱ:=⋁_s<tℱ_s,t, and which are both adapted to the filtration (ℱ_s,t)_s<t, in the sense that
    
        
  * ∫_ℝ^2ϕ(s,y) ξ(ds,dy) is ℱ_u,t-measurable if ϕ is supported on [u,t]×ℝ.
    
        
  * ℒ_a,b(x,y) is ℱ_s,t measurable for all s≤ a<b≤ t and x,y∈ℝ.
     Then ℒ and ξ are independent under ℙ, and all linear random variables (<ref>) are measurable with respect to ξ.


The directed landscape is not unique in this respect: the proof will show that one could replace ℒ by any space-time field that is known to be a black noise. This theorem may be surprising, since it illustrates a contrast with many prelimiting models in the KPZ class, in which case one often has ℒ_N=F_N(ξ_N) for some adapted path functionals F_N of the discrete driving noise ξ_N. As we take the limit N→∞, <ref> shows that this dependence no longer holds. In fact, the two fields ℒ_N and ξ_N become independent. The following corollary illustrates this phenomenon in the context of the KPZ equation.

  Let ξ be a Gaussian space-time white noise on ℝ_+×ℝ, and let H^ε(t,x) denote the Hopf-Cole solution of the KPZ equation ∂_t H^ε = 1/4(∂_xH^ε)^2 + ε/4∂_x^2 H^ε +ε^1/2ξ + 1/12, started from any deterministic sequence of upper semicontinuous initial data H^ε(0,x) satisfying H^ε(0,x) ≤ C(1+|x|) for some C>0, and converging as ε→ 0 to some upper semicontinuous profile H_0(x) in the sense of local hypograph convergence <cit.>. Consider any joint limit point (𝔥,ξ) as ε→ 0 of the pair (H^ε,ξ) in the topology of C((0,∞)×ℝ)×𝒮'(ℝ^2). Then 𝔥 is distributed as the KPZ fixed point started from H_0(x), independent of the space-time white noise ξ.


Before proving the <ref>, we note that joint limit points do exist because each coordinate is tight, which implies that the pair is tight. This theorem characterizes any joint limit point uniquely, and therefore it is a limit theorem for the pair (H^ε,ξ). The local hypograph convergence for the sequence of initial data is a weaker assumption than uniform convergence on compacts if all profiles are continuous. 


    We define a four-parameter field ℋ^ε_s,t(x,y) for each ε>0 such that (t,y)↦ℋ^ε_s,t(x,y) is the Hopf-Cole solution of the same equation solved by H^ε, with the same realization of the noise ξ, started at time s from initial condition given by a narrow wedge at position x (meaning that the exponential is started from a Dirac mass centered at location x). The main results of <cit.> imply that this four-parameter field ℋ^ε converges in law (in the topology of C(ℝ^4_↑)) to the directed landscape. 
    
    <ref> implies that any joint limit point of (ℋ^ε,ξ) must be of the form (ℒ,ξ) where ℒ is a directed landscape independent of ξ. Indeed, one can define the σ-algebras ℱ_s,t to be precisely the ones generated by random variables of the two forms given by the two bullet points in <ref>. The noise property needed to apply <ref> holds in that limit simply because it holds trivially in the prelimit and any limit point will still respect this property.
    
    Now we consider any joint limit point (𝔥,ℒ,ξ) of the triple (H^ε,ℋ^ε,ξ). On one hand, we already know that ℒ and ξ must be independent under this limit point. On the other hand, the results of <cit.> imply that 𝔥(t,y) = sup_x∈ℝ H_0(x) + ℒ_0,t(x,y) . Therefore 𝔥 is ℒ-measurable, which implies that it is independent of ξ. Projecting onto the first and third marginals 𝔥 and ξ gives the result.


While <ref> was formulated purely in terms of the KPZ equation, the result of <ref> implies something much stronger: in any system for which there a height process which converges to the directed landscape, the underlying field of environment variables necessarily decouples from the height process under that limit. In the case of ASEP, the environment consists of the Poisson clocks that result in executed jumps for the system. In the case of last passage percolation or directed polymers, the underlying environment consists of the independent, identically distributed weights through which the last passage paths travel. <ref> implies that for any system that converges to the directed landscape, the environment and the height function asymptotically decouple as they respectively scale to the white noise and to the landscape. 
Such decoupling and sensitivity phenomena are not present in the intermediate-disorder or weak-asymmetry regime. 

This phenomenon illustrates a contrast between the intermediate-disorder regime and the strong-disorder regime for (1+1)-dimensional systems in the KPZ universality class. For example, in scaling limits where particle systems, SPDEs, or directed polymers are known to converge to the KPZ equation <cit.> as opposed to the KPZ fixed point, it is straightforward to show that the field of environment variables and the height process converge jointly in law to the driving noise of the limiting KPZ equation. This is always the case in the presence of intermediate disorder or weak asymmetry scaling of models in the KPZ university class. There are a few intuitive reasons why the same phenomenon does not happen when scaling to the landscape, i.e., in the strong disorder or fixed asymmetry regime. At the level of last passage percolation or directed polymers, the height process only sees the largest variables in the environment field, whereas most of the smaller weights are ignored. The height process thus ignores “most" of the environment under the strong disorder scaling, with only a few special exceptions. In contrast, all of the noise variables contribute to some extent for the intermediate disorder scaling, as individual large contributions have a less dramatic effect on the overall height process.

We discuss another, perhaps more precise, conjecture about why the strong disorder scaling is so different from the intermediate disorder regime with regard to retaining the memory of the microscopic environment. As before, we work through the lens of polymers (for the purpose of this discussion, they may be discrete or continuous as the reader prefers). Polymers admit a chaos expansion in terms of multilinear polynomials of the underlying environment field. Distinct terms in this expansion are always orthogonal to one another in L^2 of the probability space associated to the underlying environment. Under intermediate disorder scaling, it is well understood from the work of Alberts, Khanin, and Quastel <cit.> that every individual term of the prelimiting chaos expansion contributes meaningfully to the scaling limit. In fact, they prove that the KPZ equation limit appears by proving term-by-term convergence of the polynomial chaos expansion of the polymer partition function to that of the exponential of the KPZ equation. In contrast, in the strong disorder regime, we conjecture that most of the L^2-mass of the terms of the chaos expansion of the height process escapes into the tails of the chaos expansion. And in fact, we conjecture that this “mass escape” phenomenon will not just occur for the height process itself, but also for every bounded measurable functional of the height process. The “mass escape” phenomenon is another way to formulate noise sensitivity <cit.>. 

Consider the special case of this mass escape conjecture, where the polymer model is itself given by the KPZ equation, i.e., the setting of <ref>. For a given observable of interest for the height process H^ε (e.g. evaluation at (t,x)=(1,0)), it would be interesting to study how much mass escapes into the k^th chaos at fixed but very small
ϵ>0 under the scaling of <ref>. 

Although this conjecture seems beyond the scope of our methods, we can provide some motivation. The first reason for posing the conjecture is that, at the level of the KPZ equation, <ref> implies the slightly weaker asymptotic claim that lim_ε→ 0Cov(F(H^ε),G(ξ)) = 0 for all bounded continuous functions F of the height process and all bounded measurable functions G of the noise. Our conjecture essentially amounts to showing that for any orthonormal basis {G_r(ξ)}_r≥ 1 of the k^th homogeneous chaos of ξ, we have lim_ε→ 0∑_r=1^∞Cov(F(H^ε),G_r(ξ))^2 = 0. It seems to be a challenging problem to interchange the limit and sum in this setting. The second reason for this conjecture is that <ref> implies, at the level of the continuum limit, that any observable of the landscape ℒ must be orthogonal in L^2 to every homogeneous chaos generated by the noise ξ (this is equivalent to ℒ being independent of ξ).


As another corollary of <ref>, we show that the directed landscape is not an SPDE driven by space-time Gaussian white noise (in fact, any Gaussian noise that is white in time would not be sufficient to generate the landscape). To formalize the notion of being an SPDE, we note that any probabilistic strong solution of an SPDE defines an adapted path functional on the probability space of the white noise. We therefore show that no such adapted functional can exist.

 Let ξ be a Gaussian space-time white noise on ℝ^2, defined on some probability space (Ω, ℱ^ξ,ℙ). For s<t let ℱ_s,t^ξ denote the σ-algebra generated by the random variables ∫_ℝ^2ϕ(s,y)ξ(ds,dy) as ϕ ranges over all smooth functions of support contained in [s,t]×ℝ. Then there does not exist any random variable ℒ taking values in C(ℝ^4_↑), defined on the same probability space Ω, satisfying the following properties:

    
  * ℒ is distributed as the directed landscape.
    
  * ℒ_a,b(x,y) is ℱ^ξ_s,t-measurable whenever s≤ a<b≤ t and x,y∈ℝ.



The proof is immediate from <ref>, as ℒ cannot be simultaneously independent of ξ and adapted to its filtration: this would imply ℒ is independent of itself and thus deterministic. 

Next, we discuss a few open questions which are related to the content of this paper. The two-dimensional critical stochastic heat flow is a continuum model recently constructed by <cit.>. It is a flow of random measures on ℝ^2, written as u_s,t(x⃗,dy⃗) indexed by -∞<s<t<∞ and x⃗,y⃗∈ℝ^2. We can think of it as the random field obtained from the universal scaling limit of partition functions of intermediate-disorder directed polymers in a space-time random environment in spatial dimension d=2. This is another example of a random field which is a noise in the sense of <ref>, and it exhibits certain properties which strongly suggest that it will not be a white noise <cit.>. 

 The two-dimensional critical stochastic heat flow constructed in <cit.> is a black noise.


By the remark after <ref>, if this conjecture is true, it would be possible to prove decoupling theorems analogous to <ref> for two-dimensional polymers. Our next conjecture is about the directed landscape being a noise in its space variables as well as its time variables. To formulate this precisely, one needs to define the notion of being a two-dimensional noise. The following is adapted from the definition of a d-dimensional noise stated in Tsirelson <cit.>, or in Ellis and Feldheim <cit.>. 
 A two-dimensional noise is a probability space (Ω,ℱ,ℙ) with a collection of sub-σ-algebras ℱ_x⃗,y⃗⊂ℱ associated to all open two-dimensional rectangles in ℝ^2, and a collection of measurable maps θ_h⃗:Ω→Ω indexed by h⃗∈ℝ^2, such that the following properties hold.

    
  * ℱ_x⃗_1,y⃗_1 and ℱ_x⃗_2,y⃗_2 are independent under ℙ when R_1:=(x_1^(1),x_1^(2))× (y_1^(1),y_1^(2)) and R_2:=(x_2^(1),x_2^(2))× (y_2^(1),y_2^(2)) satisfy R_1∩ R_2=∅.
    
  * ℱ_x⃗_1,y⃗_1∨ℱ_x⃗_2,y⃗_2=ℱ_x⃗_3,y⃗_3 whenever the rectangles R_1 and R_2 satisfy R_1∩ R_2=∅ and R_1∪ R_2=R_3.
    
  * ℱ is generated by the union of all of the ℱ_x⃗,y⃗.
    
  * If A∈ℱ_x⃗,y⃗, then θ_h⃗(A)∈ℱ_x⃗+(h_1,h_1),y⃗+(h_2,h_2) and ℙ(θ_h⃗(A))=ℙ(A).
    
  * θ_h⃗_1θ_h⃗_2=θ_h⃗_1+h⃗_2 for all h⃗_1,h⃗_2∈ℝ^2, and θ_0⃗=Id. 

We say this two-dimensional noise is a black noise when the only linear random variable is 0, i.e., any F∈ L^2(Ω) satisfying 𝔼[ F| ℱ_R_1∪ R_2] = 𝔼[ F| ℱ_R_1 ] + 𝔼[ F| ℱ_R_2 ] for all rectangles R_1,R_2 as in Item (2), implies that F=0.

There are two well-known examples of two-dimensional black noises arising from simple probabilistic systems. The first is the scaling limit of critical planar percolation, which was conjectured to be black noise by Tsirelson <cit.> and later proved by Schramm and Smirnov <cit.>. The second is the Brownian web, which was proved to be a black noise by Ellis and Feldheim <cit.>.
In the context of this definition, we can state the following conjecture, due to Bálint Virág (in conversation). Define the σ-algebras ℱ_(s,t),(x,y) to be the ones generated by the random variables ∫_a^b dℒ∘π (see <cit.>) as one varies over all s≤ a<b≤ t and all continuous deterministic paths π: [a,b]→ [x,y]. Then one has the following conjecture.

 The directed landscape is a two-dimensional black noise. 


The difficulty of the two-dimensional problem is showing that the noise property holds in the first place, in particular Item (2) in <ref>. If true, this would address a problem posed by Ellis and Feldheim <cit.>. Their work demonstrates that the Brownian web is an example of a two-dimensional black noise, and they ask for further relevant examples of two-dimensional black noise arising from simple probabilistic systems.

There are several key properties of the directed landscape that are crucial to the proof that it is a black noise, and which distinguish it from many prelimiting models in the KPZ universality class that are inherently white noise.

 

  * Scale invariance: Obtaining a rescaled version of the same space-time process at every temporal scale is a crucial ingredient to many parts of the proof. For example, in <ref> we get a spatial mixing rate for the directed landscape of order e^-d k^3 on a time interval [0,1]. This mixing rate immediately translates to a mixing rate of e^-d ϵ^-2 k^3 for on a smaller time interval [0,ϵ].


  * Zero Temperature: Many of our estimates would be weakened or fail for positive temperature analogues like the KPZ equation, as the maximum would get replaced by some convolution operation which is not as local in some sense. 


  * Local Brownianity: Even for many zero-temperature 
conjectural prelimits (such as inviscid stochastic Burgers equations driven by white noise) the local Brownianity of the spatial process fails. Improved spatial regularity of the solution actually hurts many of the estimates one needs to prove the black noise property, as the better regularity “widens” the set of x-values where the spatial height process is close to the argmax. Controlling the size of the latter set of x-values is really the heart of the proof, as quantified in <ref>.
 



 §.§ Notation

Throughout the paper, we use the font 𝔅, ℭ as a way to denote random constants. The letters C,d are used, respectively, to denote deterministic constants that may get larger or smaller from line to line when performing an estimate. Typically, such C,d will be (respectively) a multiplying constant and an exponential decay rate of some quantity. For a metric space X we will let C(X) denote the space of continuous functions X→ℝ, equipped with the topology of uniform convergence on compacts. We often deal with random variables defined on the underlying probability space C(ℝ) or C(ℝ^2) or C(ℝ^4_↑), and these random variables should be understood as being real-valued Borel-measurable functions with respect to the topology of uniform convergence on compact sets (which is always separable and metrizable, hence Polish).




 §.§ Structure of the Paper
 In <ref>, we reduce the problem of demonstrating that the directed landscape is a black noise to demonstrating a specific variance estimate about the Airy sheet (<ref>). <ref> through <ref> are dedicated to obtaining this estimate, which relies on a mixing property of the directed landscape (<ref>), as well as estimates on argmaxes of Airy sheets (<ref>) and estimates on Bessel processes (<ref>). Then in <ref>, we conclude that the directed landscape is a black noise. Finally in <ref>, we prove <ref> from <ref>. 



 §.§ Acknowledgements
 SP thanks Duncan Dauvergne, Jeremy Quastel, and Bálint Virág for discussions and context about this problem during a visit to Toronto. SP and ZH thank Ivan Corwin and Yu Gu for comments on an earlier draft of the paper. ZH was supported by the Fernholz Foundation's Minerva Fellowship Program and Ivan Corwin's grant, NSF DMS-1811143. ZH thanks E. Himwich and W. Himwich for discussions about black noise. 



§ REDUCING THE BLACK NOISE PROBLEM TO AN ESTIMATE ON THE AIRY SHEET

In this section, we reduce the problem of showing that the directed landscape is a black noise to a 
specific variance bound for the Airy sheet. 


    Let (Ω, 𝒢,ℙ) be a standard probability space, and suppose that 𝒢 is generated by some countably infinite collection of random variables {X_n}_n=1^∞ taking values in some Polish space 𝒳. Consider any bounded measurable F:Ω→ℝ. Then for each m∈ℕ there exists a Borel measurable map Ψ_m:𝒳^m→ℝ such that sup_x_1,...,x_m∈𝒳 |Ψ_m(x_1,...,x_m)| ≤F_L^∞(Ω), and moreover the random variables F_m:= Ψ_m(X_1,...,X_m) satisfy F=lim_m→∞ F_m ℙ-almost surely as well as lim_m→∞F_m-F_L^p(Ω)= 0 for 1≤ p<∞.



    We define 𝒢_m:=σ(X_1,...,X_m) and F_m:= 𝔼[F|𝒢_m]. Using approximation by 𝒢_m-measurable simple functions, we see that F_m are of the form Ψ_m(X_1,...,X_m) for some Borel measurable Ψ_m:𝒳^m→ℝ. If needed, we replace Ψ_m by (-F_L^∞(Ω)) ∨Ψ_m ∧F_L^∞(Ω), and this still satisfies the same relation F_m=Ψ_m (X_1,...,X_m) on a set of measure 1. By Doob's martingale convergence theorem, F_m→ F almost surely and in every L^p(Ω).


In his survey, Tsirelson <cit.> gives a few examples in which it is possible to prove something is a black noise, and provides a necessary and sufficient condition for a given noise to be black. We quote the following result from that survey, see <cit.>.

 Let (Ω, (ℱ_s,t)_s<t, ℙ, θ_h) be a noise in the sense of <ref>. The space M of ℱ_0,1-measurable linear random variables (<ref>) is a closed linear subspace of L^2(Ω), and moreover the orthogonal projection of any F∈ L^2(Ω) onto M is given by 
    lim_n→∞∑_k=1^2^n(𝔼[ F| ℱ_(k-1)2^-n,k2^-n] - 𝔼[F]) ,
 where the limit is understood in L^2(Ω).


 A noise (Ω, (ℱ_s,t)_s<t, ℙ, θ_h) is a black noise 
if and only if the following condition holds on a dense linear subspace D⊂ L^2(Ω): 

    ∑_k=1^2^n(𝔼[ F| ℱ_(k-1)2^-n,k2^-n]-𝔼[F]) L^2(Ω)→ 0,        for  all     F∈ D,
 or equivalently, 

    lim_n→∞∑_k=1^2^nVar(𝔼[ F| ℱ_(k-1)2^-n,k2^-n] )= 0,         for  all     F∈ D.


We recall the definition of the Directed Landscape and the associated noise (<ref>) and use Ω to denote the underlying probability space of ℒ. There is a particular class of functions that are most useful to us in the context of the directed landscape. 
We define D⊂ L^2(Ω) to be those functionals of the landscape that are finite linear combinations of “polynomial functions" of the form 
    F=∏_j=1^m ∏_i=1^n_jℒ_s_j,t_j(x_ij,y_ij),
 
where (s_j,t_j) are disjoint intervals of ℝ whose endpoints s_j,t_j are both dyadic numbers (of the form k2^-N with k∈ℤ and N∈ℕ), furthermore x_ij,y_ij∈ℝ are all distinct points in ℝ, and m,n_j∈ℕ. 

    The set D is a dense linear subspace of L^2(Ω) with Ω as in <ref>.

 The space L^2(γ_1)⊗⋯⊗ L^2(γ_m) is isometrically isomorphic as a Hilbert space to L^2(γ_1⊗⋯⊗γ_m) for any Borel probability measures γ_i on a Polish space, under the canonical isomorphism (F_1⊗⋯⊗ F_m) (x_1,...,x_m) ≅ F_1(x_1)⋯ F_m(x_m). Therefore, in order to show that functions of the form (<ref>) are dense in L^2(Ω), it suffices to show that any function of the form ∏_j=1^m φ_j(ℒ_s_j,t_j(x_1,y_1),...,ℒ_s_j,t_j(x_n,y_n)) is in the L^2-closure of D, where m,n∈ℕ and φ_j are bounded measurable functions on ℝ^n. 
Indeed, functionals of the latter form are dense in L^2(Ω) by the following argument: by <ref> one can take n↑∞ to approximate any functional of the form ∏_j=1^m Φ_j (ℒ_s_j,t_j) where m∈ℕ and Φ_j:C(ℝ^2)→ℝ are bounded measurable, and ℒ_s,t:ℝ^2→ℝ denotes the random function given by (x,y)↦ℒ_s,t(x,y). Then in turn by taking linear combinations and using the tensorization property of L^2 one can approximate any functional of the form Ψ(ℒ_s_1,t_1,...,ℒ_s_m,t_m) where m∈ℕ and Ψ: C(ℝ^2)^m→ℝ is bounded measurable, and finally by <ref> one can take m↑∞ and the latter types of functionals can then approximate any general functional F(ℒ).

We therefore focus on showing that any function of the form ∏_j=1^m φ_j(ℒ_s_j,t_j(x_1,y_1),...,ℒ_s_j,t_j(x_n,y_n)) is in the L^2(Ω)-closure of D, where m,n∈ℕ and φ_j are bounded measurable functions on ℝ^n. By independence of the different terms in the product, it suffices to prove that, for m=1, such functions may be approximated in L^2(Ω) by functions of the form (<ref>) with m=1. In turn, it suffices to show that polynomial functions on ℝ^n are dense in L^2(μ) where μ is the probability measure on ℝ^n given by the joint law of (ℒ_s,t(x_1,y_1),...,ℒ_s,t(x_n,y_n)) for some fixed interval (s,t) and fixed x⃗,y⃗∈ℝ^n. Since the marginals here are deterministically shifted Tracy-Widom GUE laws which have a super-exponential tail decay, we conclude that μ must have a finite moment generating function everywhere. Therefore, if f∈ L^2(μ) was orthogonal to all polynomials, then by taking the Taylor expansion of the exponential and interchanging the infinite sum with the integral over ℝ^n, we would have that ∫_ℝ^n f(x) e^ i∑_1^n λ_j x_jμ(dx) = 0 for all λ = (λ_1,...,λ_n)∈ℝ^n. This interchange can be justified by Fubini, using the super-exponential tail decay on μ. By inverting the Fourier transform (which is a linear isomorphism on 𝒮'(ℝ^n)), this implies that the measure f dμ∈𝒮'(ℝ^n) must be the zero measure, i.e., f=0 μ-almost everywhere. 


Due to the independence of the σ-algebras associated to disjoint intervals (property (1) in <ref>), we can ignore the product over j in (<ref>), and thus without loss of generality, we may assume that m=1 in (<ref>). We label the single interval by (s_1,t_1)=(s,t). By shifting and rescaling the landscape, we can assume without loss of generality that (s,t)=(0,1). Therefore, we have reduced the black noise problem to demonstrating that 

    lim_n→∞∑_k=1^2^nVar(𝔼[ ∏_i=1^nℒ_0,1(x_j,y_j) | ℱ^ℒ_(k-1)2^-n,k2^-n])= 0,       for all x⃗,y⃗∈ℝ^n,  and  n∈ℕ.

The entire problem is now formulated in terms of the Airy sheet 𝒮:=ℒ_0,1. In fact, we can simplify things further. 

    In order to show that the directed landscape is a black noise, it suffices to show that for any η∈ (0,1/2) and n∈ℕ and x⃗,y⃗∈ℝ^n,
    
    lim_n→∞∑_η 2^n≤ k ≤ (1-η)2^nVar(𝔼[ ∏_i=1^nℒ_0,1(x_j,y_j) | ℱ^ℒ_(k-1)2^-n,k2^-n])= 0,


 Let F:= ∏_i=1^nℒ_0,1(x_j,y_j), and let G denote the linear random variable obtained by applying the orthogonal projection of F onto the closed linear subspace of L^2(Ω) consisting of linear ℱ^ℒ_0,1 measurable random variables. 

By applying <ref>, we see from (<ref>) that G must be measurable with respect to ℱ^ℒ_0,η∨ℱ^ℒ_1-η,1, in other words, all dependency on ℱ^ℒ_η,1-η vanishes. This is true for all η∈ (0,1/2). Therefore, G is measurable with respect to ℋ:=⋂_η∈ (0,1/2)ℱ^ℒ_0,η∨ℱ^ℒ_1-η,1. We claim that ℋ is a trivial σ-algebra, only containing sets of measure 0 or 1. It is clear that ℋ is contained in ℱ^ℒ_0,1. However, ℋ is also independent of the σ-algebra generated by ⋃_η∈ (0,1/2)ℱ^ℒ_η,1-η. We claim that the σ-algebra generated by ⋃_η∈ (0,1/2)ℱ^ℒ_η,1-η is all of ℱ^ℒ_0,1. To see this, note by the continuity of the directed landscape in all parameters that e.g. the random variable ℒ_0,1(x,y) can be written as the almost sure limit as η→ 0 of ℒ_η,1-η(x,y). Thus ℋ⊂ℱ^ℒ_0,1 is independent of ℱ^ℒ_0,1, proving that it is a trivial σ-algebra.

Since the random variable G is ℋ-measurable, we conclude that it is almost surely constant. But G is an L^2-limit of the sequence of random variables as in <ref>, and therefore (<ref>) holds.


The following corollary is an immediate consequence.


    Fix some ϱ>0. In order to show that the directed landscape is a black noise, it suffices to show that for any η∈ (0,1/2),  n∈ℕ and x⃗,y⃗∈ℝ^n,  there exists some C = C(n,η,x⃗,y⃗)>0 such that for all a,b∈ [η,1-η],
    
    Var(𝔼[ ∏_i=1^nℒ_0,1(x_j,y_j) | ℱ^ℒ_a,b]) ≤ C (b-a)^1+ϱ,



The rest of the paper will focus on proving (<ref>), for a value of ϱ that is slightly smaller than 1/15. This bound may not be sharp. 















§ A STRONG MIXING PROPERTY FOR THE LANDSCAPE UNDER SPATIAL SHIFTS


In this section, we prove a strong mixing property for the directed landscape under spatial shifts (<ref>) that is crucial to the arguments in <ref>. First, we state a theorem from <cit.> that is used in the proof. 
  Fix u=(x,t;y,s)∈ℝ^4_↑. There is almost surely a unique directed geodesic Π_u from (x,s) to (y,t), i.e., a path π:[s,t]→ℝ such that π(s) = x , π(t)=y and for all k∈ℕ and all s=s_0<...<s_k=t, 
    ℒ_s,t(x,y) = ∑_j=1^k ℒ_t_k-1,t_k (π(t_k-1),π(t_k)).
 Its distribution only depends on u through scaling: as random continuous functions from [0,1]→ℝ, 
 
    Π_(x,t;y,s)(s+(t-s)r)d=Π_(0,0;0,1)(r)+x+(y-x)r.
 Moreover, for u=(0,0;0,1), there is a random constant ℭ such that, for all s>0 with t,t+s∈[0,1],
 
    |Π_u(t+s)-Π_u(t)| ≤ℭs^2/3log^1/3(2/s).
 The random constant satisfies 𝔼[a^ℭ^3]<∞ for some a>1. 
    
 

With <ref> in hand, we prove that there exists a probability space on which we can couple three copies of the directed landscape in such a way that two of them are independent, and the third agrees with the other two sufficiently far from the origin. 
 There exist three copies of the directed landscape ℒ^i=(ℒ^i_s,t)_0≤ s<t≤ 1 with i=0,1,2, all coupled onto the same probability space (Ω,ℱ,ℙ_triple) so that:

    
  * ℒ^1, ℒ^2 are independent under ℙ_triple. 
    
  * There exists a positive random variable X:Ω→ (0,∞) such that ℒ_s,t^0(x,y) = ℒ_s,t^1(x,y) for all x,y>X and 0≤ s<t≤ 1, and furthermore ℒ_s,t^0(x,y) = ℒ_s,t^2(x,y) for all x,y<-X and 0≤ s<t≤ 1.
    
  * ℙ_triple(X>u)≤ Ce^-du^3 where C,d>0 do not depend on u>0.


The proof is inspired by <cit.>, which proves a coupling result for copies of the directed landscape via approximation by a last passage model. 
 
    We consider two fields of independent, identically distributed exponential random variables ω^i = {ω^i_t,x}_t≥ 0,x∈ℤ for i=1,2, independent of one another. We define a third field of independent, identically distributed variables ω^0_t,x by ω^0_t,x := ω^1_t,x if x≥ 0 and ω^0_t,x := ω^2_t,x if x< 0.

    For i=0,1,2 and x,y∈ℤ and 0≤ m<n∈ℤ, we consider the fields of last passage times 
    L_m,n^i(x,y):= max{ W^i(π): π:{m,...,n}→ℤ ; π(m)=x,π(n)=y}
 where π denotes a nearest neighbor path π:{m,...,n}→ℤ and its length is defined by 
    W^i(π):=  ∑_i=m^n ω^i_i,π(i).

    For x,y∈ε^2/3ℤ and s,t∈εℤ, we then define a rescaled last passage time 
    ℒ^i,ε_s,t (x,y):= ε^1/3(L_ε^-1s,ε^-1t^i(ε^-2/3x,ε^-2/3y) - ε^-1(t-s)).

    <cit.> shows that the marginal laws of each ℒ^i,ε_s,t (x,y) individually converge in law as ε→ 0 to the directed landscape in the uniform-on-compact topology. Here, the compact set can be taken to be any compact subset of {(s,t,x,y): 0≤ s<t≤ 1, (x,y)∈ℝ^2}, see <ref>. 
    
    Furthermore, if π_opt^i,m,n,x,y denotes the
    maximizing path, satisfying L^i_m,n(x,y) = W^i(π_opt^i,m,n,x,y), then 
    <cit.> implies that the paths 
    Π^i,s,t,x,y;ε(r):= ε^2/3π_opt^i,ε^-1s,ε^-1t,ε^-2/3x,ε^-2/3y(ε^-1 r),
 converge in law as ε→ 0, jointly in all variables (s,t,x,y,r), with respect to the uniform-on-compact topology on { (s,t,x,y,r):0≤ s<t≤ 1, s≤ r≤ t, (x,y)∈ℤ^2 }, to geodesic paths Π^i,s,t,x,y(r) of the corresponding directed landscapes ℒ^i, for i=0,1,2. This convergence occurs jointly with the height profiles ℒ^i,ε_s,t (x,y).

    Now let (ℒ^0,ℒ^1,ℒ^2) be any joint limit point of (ℒ^0,ε,ℒ^1,ε,ℒ^2,ε) as ε→ 0. We define the random variable 
    X:= inf{x∈ℕ: inf_r∈ [0,1]Π^0,0,1,x,x(r)>0 , sup_r∈ [0,1]Π^0,0,1,-x,-x (r)<0},
 which tracks the smallest positive integer value of x for which both of the directed geodesic paths from x→ x and from -x→-x do not cross 0 in ℒ^0. We verify that ℒ^1,ℒ^2 are independent and that ℒ_s,t^0(x,y) = ℒ_s,t^1(x,y) for all x,y>X and 0≤ s<t≤ 1 by noting that the analogous facts hold trivially in the prelimit. Furthermore, ℒ_s,t^0(x,y) = ℒ_s,t^2(x,y) for all x,y<-X and 0≤ s<t≤ 1, again because the analogous fact holds in the prelimit.

    It remains to verify the tail bound in Item (3) of the lemma statement. The moment bound of <ref> implies that for any fixed x∈ℝ the probability of any geodesic path from x→ x varying more than a distance u>0 from its starting point decays like e^-du^3 for some d>0. This directly implies Item (3) through a union bound.


We use <ref> to prove a strong mixing property for the landscape under diagonal shifts. 

 The directed landscape ℒ=(ℒ_s,t)_0≤ s<t≤ 1 is α-mixing under spatial shifts, with mixing rate α(k)≤ Ce^-dk^3 for some universal constants C,d>0. More precisely, we have the following stronger estimate. For -∞≤ a<b≤∞ let 𝒢_a,b denote the σ-algebra generated by the random variables ℒ_s,t(x,y) where 0≤ s<t≤ 1 and x,y ∈ (a,b). Then 
    |(F,G)| ≤  Ce^-dk^3𝔼[F^4]^1/4𝔼[G^4]^1/4,
 where C,d>0 are independent of k and F,G ∈ L^4(Ω) that are respectively measurable with respect to 𝒢_-∞,0 and 𝒢_k,∞. 


The α-mixing property is a very strong form of mixing that is uniform over all observables (see 
the survey by Bradley <cit.> for further explanation). Roughly speaking, the result says that under translation of the landscaped by k spatial units, the decay of correlation is upper bounded by e^-d k^3, where d>0 is bounded away from zero uniformly over all observables and k>0. Moreover, the remarkably fast cubic-exponential rate of decay  means that we can treat the field as virtually independent on well-separated scales. 

 By translation invariance, it suffices to prove the claim with the assumption “F measurable with respect to 𝒢_-∞,0" replaced by the statement “F measurable with respect to 𝒢_-∞,-k" (this replacement is just for notational convenience). Therefore, we will consider F,G that are respectively measurable with respect to 𝒢_-∞,-k and 𝒢_k,∞. 

We assume that ℒ^0,ℒ^1,ℒ^2 are the same couplings and X is the same random variable as in <ref>. We will use 𝔼_triple[·] to denote the expectation with respect to the measure on the coupled space from <ref>. Consider the measurable functions F,G as above and write F_i:=F(ℒ^i),G_i:=G(ℒ^i) for i=0,1,2. Then by independence, 𝔼_triple[F_2G_1] =𝔼_triple[F_2]𝔼_triple[G_1], thus 

    |(F,G)|    = |𝔼_triple[ F_0G_0-F_2G_1]| 
       ≤ |𝔼_triple[ (F_0-F_2+F_2) (G_0-G_1+G_1) -F_2G_1 ]| 
       = |𝔼_triple[(F_0-F_2)(G_0-G_1)] +𝔼_triple[(F_0-F_2)G_1] + 𝔼_triple[F_2(G_0-G_1)]| 
       ≤𝔼_triple[(F_0-F_2)^2]^1/2𝔼_triple[(G_0-G_1)^2]^1/2 + 𝔼_triple[(F_0-F_2)^2]^1/2𝔼_triple[G_1^2]^1/2
             + 𝔼_triple[F_2^2]^1/2𝔼_triple[(G_0-G_1)^2]^1/2.

Now recall that F,G are, respectively, measurable with respect to 𝒢_-∞,-k and 𝒢_k,∞. Therefore, F_0=F_2 and G_0=G_1 on the event that X ≤ k. By <ref> we thus have that 
    𝔼_triple[(F_0-F_2)^2]^1/2   =  𝔼_triple[(F_0-F_2)^21_{X>k}]^1/2
       ≤ 2 𝔼[F^4]^1/4ℙ_triple(X>k)^1/2≤ 2𝔼[F^4]^1/4Ce^-1/2 d k^3.
 We used Cauchy-Schwarz and (F_0-F_2)^2 ≤ 2(F_0^2+F_2^2) in the first inequality. Likewise, one may prove that 𝔼_triple[(G_0-G_1)^2]^1/2≤ 2𝔼[G^4]^1/4 Ce^-1/2 dk^3. On the other hand, it is clear from Jensen that 𝔼_triple[F_2^2]^1/2≤𝔼_triple[F_2^4]^1/4 = 𝔼[F^4]^1/4 and likewise 𝔼_triple[G_1^2]^1/2≤𝔼_triple[G_1^4]^1/4 = 𝔼[G^4]^1/4. 


The choice of L^4 is not optimal in this proof, and applying Hölder's inequality rather than Cauchy-Schwarz could improve the assumptions to F,G∈ L^p(Ω) with a bound 𝔼[|F|^p]^1/p𝔼[|G|^p]^1/p for any p>2. It is natural to ask what the strongest or optimal form of mixing is for spatial shifts in the landscape, such as β-mixing or ϕ-mixing, but we will not pursue this question. <ref> and <ref> provide a flavor of how to prove mixing estimates, but we actually need a stronger version of these results. 


    Fix T>0, and consider any compact disjoint intervals I_1,J_1,...,I_N,J_N⊂ℝ. Define the σ-fields 
    ℱ_I:= σ( {ℒ_s,t(x,y): 0≤ s<t≤ T, x and y  are in the same interval  I_r  for some  r≤ N}

    
    ℱ_J:= σ( {ℒ_s,t(x,y): 0≤ s<t≤ T,  x and y  are in the same interval  J_r  for some  r≤ N}.

    Define the positive real number 
    D:= inf{|x-y| : x∈⋃_1^N I_r,   y∈⋃_1^N J_r}.

    Then we have the bound 
    |(F,G)| ≤  CNe^-dT^-2D^3𝔼[F^4]^1/4𝔼[G^4]^1/4.

    Here C,d>0 are universal constants that are independent of N,T,{I_r,J_r}_r=1^N,D and F,G ∈ L^4(Ω) that are respectively measurable with respect to ℱ_I and ℱ_J.


We remark that not all I_r need to be strictly to the left of all J_r or vice versa: they may very well be mixed amongst each other in a nontrivial way.


Without loss of generality, as in <ref>, we may assume that T=1. By scale invariance of the directed landscape <ref> (4), all of the intervals could be replaced by their T^-2/3-rescalings. This changes the value of D to T^-2/3D, which respects the estimate. In the case that T=1, the proof uses an explicit coupling construction that is very similar to the proofs of <ref> and <ref>. We now sketch the details of this coupling construction.

Without loss of generality, we assume that the intervals are ordered I_1,J_1,I_2,J_2,...,I_N,J_N so that the supremum of I_1 is less than or equal to the infimum of J_1, the supremum of J_1 is less than the infimum of I_2, and so on. This does not lose any generality because we can always combine consecutive instances of the I_r or J_r without changing the value of D.

We relabel I_1,J_1,I_2,J_2,...,I_N,J_N as U_1,...,U_2N and define a_r to be the midpoint of the segment in the complement of ⋃_i≤ 2N U_i that is between U_r and U_r+1. We define a fattening of I_r and J_r, by setting I̅_r:= (a_2r-1,a_2r) and J̅_r:= (a_2r,a_2r+1) for r=1,...,N. In this notation, we say that a_2N+1:=∞. We also define  b_r,c_r to be those points in the segment in the complement of ⋃_i≤ 2N U_i that is between U_r and U_r+1 that are (respectively) exactly one-third and two-thirds of the way across this segment.

If L^1,L^2∈ C(ℝ_↑^4) and I⊂ℝ, 
then we will say that L^1,L^2 agree on I×[0,T] if L^1_a,b(x,y)=L^2_a,b(x,y) for all 0≤ a<b≤ T and all a,b∈ I. Just as we did in <ref>, we construct three copies ℒ^0,ℒ^1,ℒ^2 of the directed landscape, so that ℒ^1, ℒ^2 are independent, and furthermore, with high probability, ℒ^0 agrees with ℒ^1 on I_r × [0,T] and ℒ^0 agrees with ℒ^2 on J_r × [0,T] for all 1≤ r≤ N. We have the following construction based on the same argument as <ref>. 

Consider fields of independent, identically distributed exponential random variables ω^i = {ω^i(t,x)}_t≥ 0,x∈ℤ for i=1,2, independent of one another. Define a third field of independent, identically distributed variables ω_ε^0={ω_ε^0(t,x)}_t≥ 0,x∈ℤ which agrees with ω^1 on ℤ_≥ 0×⋃ ((ε^-2/3I̅_r)∩ℤ) and agrees with ω^2 on ℤ_≥ 0×⋃ ((ε^-2/3J̅_r)∩ℤ). Just as we did in <ref>, we take a joint limit point of the last passage times for all three fields (together with all of the geodesic paths associated to the finite collection of endpoints given by b_1,c_1,...,b_N,c_N) as ε→ 0. We will obtain three copies of the directed landscape ℒ^0,ℒ^1, ℒ^2 coupled onto the same space (together with the respective geodesics).



Consider the event E in which all of the 4N geodesic paths from the points b_r to b_r and c_r to c_r  in ℒ^0 do not cross any of the values a_r, and also do not cross the endpoints of the intervals I_r,J_r (which in particular implies that they do not intersect). By the same argument as in <ref>, the values ℒ^0_s,t(x,y) will agree with either of ℒ^1 or ℒ^2 as long as x,y lie strictly in between any given consecutive geodesics Π_r and Π_r+1. See <ref> for a geometric depiction of this event. 

We apply a union bound over all 4N of the geodesics, together with an application of <ref>, to show that the event E occurs with probability greater than 1-C(4N)e^-dD^3, where C,d do not depend on N,D or the choice of intervals. Consequently, by repeating the exact argument as that in <ref> (replacing the event {X>k} appearing there with the event E here) we obtain the required bound. 


Specializing <ref> to the case T=1 and projecting the result to the two-time marginal (s,t)=(0,1), we obtain the following corollary about strong mixing of the Airy sheet under diagonal shifts.


    Consider the Airy sheet 𝒮 = ℒ_0,1. Fix T>0, and consider any compact disjoint intervals I_1,...,I_N,J_1,...,J_N⊂ℝ^2. Define the σ-fields 
    
    𝒢_I     := σ( {𝒮(x,y):  (x,y)∈ I_r× I_r  for some  r≤ N}),
        
    𝒢_J     := σ( {𝒮(x,y):   (x,y)∈ J_r× J_r  for some  r≤ N}).

    Define 
    D:= inf{|x-y| : x∈⋃_1^N I_r,   y∈⋃_1^N J_r}.

    Then we have the bound 
    |(F,G)| ≤  CNe^-dD^3𝔼[F^4]^1/4𝔼[G^4]^1/4.

    Here C,d>0 are universal constants that are independent of N,{I_r,J_r}_r=1^N,D and F,G ∈ L^4(Ω) that are respectively measurable with respect to 𝒢_I and 𝒢_J.



    
    



   




§ USEFUL ESTIMATES ON AIRY SHEETS AND ARGMAXES


We begin this section by establishing some useful notation. 

For f,g∈ C(ℝ) and L∈ C(ℝ^2) we define 

    f∘ L∘ g:= sup_x,y∈ℝ f(x)+L(x,y)+g(y).
 In this section, we prove a bound 
(<ref>) on the covariance of f∘ℒ_0,ϵ∘ g and u∘ℒ_0,ϵ∘ v for deterministic functions f,g,u,v all lying within a certain restricted domain. This estimate, and the additional estimates on the covariance of products of functions of this nature (<ref> and <ref>), are critical inputs to the proof of the estimate (<ref>). 
Most of this section is devoted to setting up bounds which are used to prove <ref>, <ref>, and <ref>. The following lemma from <cit.> is useful in our arguments.








    

    
    
    

    
    
    

[Lemma 5.3, <cit.>]  There exists a deterministic c>0 and a random ℭ>0 so that 
    |𝒮(x,y)+(x-y)^2| ≤ℭ+clog^2/3 (2+|x|+|y|).

Furthermore 𝔼[a^ℭ^3/2]<∞ for some a>1.

We use this lemma to prove a helpful property of the point (x,y) which maximizes f∘ℒ_0,ϵ∘ g. 
  Let f,g: ℝ→ℝ be deterministic continuous functions satisfying -2 x^2 -B ≤ f(x),g(x) ≤ -1/2 x^2+B for some B>0. We define
the random variable 

    (X_ϵ,Y_ϵ) := argmax_(x,y)∈ℝ^2  f(x) + ℒ_0,ϵ(x,y)+g(y).

We define z_max:= argmax_z∈ℝ f(z)+g(z), and assume that this point is unique. Then for M^2>1+100B, 
    ℙ( X_ϵ^2+Y_ϵ^2> M^2) ≤ C e^-dϵ^-1/2M^3
 where C,d>0 are universal constants not depending on f,g,M,ϵ, B.



    By the scale invariance of the directed landscape, we can replace ℒ_0,ϵ(x,y) by the process 
    ϵ^1/3ℒ_0,1 (ϵ^-2/3 x,ϵ^-2/3y) = ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y).
 In the notation of <ref> we have 
    |ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+ ϵ^-1 (x-y)^2|≤ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)),
 where c is deterministic and ℭ is a non-negative random variable satisfying ℙ( ℭ>u) ≤ Ce^-du^3/2 for u ≥ 0. From now on, we use the notation 𝒮^ϵ(x,y):=ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y). In particular,
    
    (f+g)(z_max) - ℭϵ^1/3 - cϵ^1/3log^2/3(2+2ϵ^-2/3|z_max|)    ≤ f(z_max) + 𝒮^ϵ(z_max,z_max) +g(z_max) 
       ≤ f(X_ϵ) + 𝒮^ϵ(X_ϵ,Y_ϵ) +g(Y_ϵ)


    By the assumptions of the proposition, on the event X_ϵ^2+Y_ϵ^2>M^2 we have that 
    f(X_ϵ)+g(Y_ϵ) ≤ 2B - 1/2(X_ϵ^2+Y_ϵ^2)≤ 2B-1/2 M^2.
 However,  <ref> also implies that 
    𝒮^ϵ(X_ϵ,Y_ϵ) ≤ℭϵ^1/3 + cϵ^1/3log^2/3 (ϵ^-2/3(2+ |X_ϵ|+|Y_ϵ|)).

     From the conditions imposed on f,g by the statement of the proposition, it is clear that z_max∈[-2√(B),2√(B)]. Combining all of this with (<ref>) we obtain the following bound on ℭ:
     
    2ℭϵ^1/3   ≥ (f+g)(z_max) - 2B+1/2 M^2 +cϵ^1/3(- log^2/3 (2+ ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) - log^2/3 (2+2ϵ^-2/3|z_max|)) 
       ≥ -z_max^2 -4B +1/2 M^2 +cϵ^1/3(- log^2/3 (2+ ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) - log^2/3 (2+2ϵ^-2/3|z_max|)) 
       ≥ -8B +1/2M^2 +cϵ^1/3(- log^2/3 (2+ ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) - log^2/3 (2+2ϵ^-2/3B)).

     The deterministic constant c may grow from line to line. Therefore, for sufficiently small ϵ>0 (below some threshold that is deterministic and independent of B≥ 2), c ϵ^1/3log^2/3 (2+2ϵ^-2/3B) ≤ B. 
     
     Under the additional assumption that X_ϵ^2+Y_ϵ^2 ≤ (2M)^2, we find that c ϵ^1/3log^2/3 (2+ ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) ≤1/4 M^2 for sufficiently small ϵ>0 (below some threshold that is deterministic and independent of M≥ 2). Therefore, the last expression admits a lower bound by -9B +1/4 M^2. We have shown that 
    ℙ( M^2 ≤ X_ϵ^2 +Y_ϵ^2 ≤ (2M)^2) ≤ℙ( 2ℭϵ^1/3≥ -9B +1/4M^2),
 for ϵ≤ϵ_0 with ϵ_0 deterministic and independent of B,M. Summing over dyadically sized concentric annuli, we obtain 
     
    ℙ(X_ϵ^2 +Y_ϵ^2 >M^2)    = ∑_k=0^∞ℙ( 4^k M^2 < X_ϵ^2 +Y_ϵ^2 ≤ 4^k+1M^2 ) 
       ≤∑_k=0^∞ℙ( 2ℭϵ^1/3≥ -9B + 4^k-1 M^2) 
       ≤∑_k=0^∞ Ce^-dϵ^-1/2(4^k-1M^2 - 9B)^3/2
       ≤∑_k=0^∞ Ce^-dϵ^-1/2 8^k-2 (M^2-36B)^3/2.

         
      In the last line we used the bound 9B≤ 9· 4^k B. We can obtain a factor of 8^-k in front of the exponential terms, through the bounds e^-u≤ 2u^-1 e^-u/2≤ 2u^-1 e^-u/8^k. This allows us to bound the infinite sum by a constant and obtain a bound only in terms of the first summand. In summary, the final line of the equation above can be bounded above by a quantity of the form 
    Cϵ^1/2(M^2-36B)^-3/2 e^-d ϵ^-1/2(M^2-36B)^3/2,
 where the value of d can be reduced if desired. If M^2 is larger than 1+100B, then (M^2-36B)^-3/2 can be bounded above by 1. Furthermore, M^2>1+100B>72B implies that M^2-36B >1/2 M^2, and therefore e^-d ϵ^-1/2(M^2-36B)^3/2≤ e^- d ϵ^-1/2 M^3, where once again, we can make c larger and d smaller if desired.

We use <ref> to obtain the following bound on |X_ϵ-Y_ϵ|. 

    Let f,g: ℝ→ℝ be deterministic continuous functions satisfying -2 x^2 -B ≤ f(x),g(x) ≤ -1/2 x^2+B for some B>0. Define
the random variable 

    (X_ϵ,Y_ϵ) := argmax_(x,y)∈ℝ^2  f(x) + ℒ_0,ϵ(x,y)+g(y).

We define z_max:= argmax_z∈ℝ f(z)+g(z), and assume that this point is unique. Then 
    ℙ(|X_ϵ-Y_ϵ|>ϵ^1/2-δ)≤ Ce^-d ϵ^-1/2(ϵ^-2δ - 108 B)^3/2 + Ce^-d  log^6/5(1/ϵ),

where C,d>0 are universal constants independent of f,g,B, ϵ∈ (0,1].


Intuitively, the bound ϵ^1/2-δ should not be optimal, as one would expect based on the scaling properties of the landscape that the typical value of |X_ϵ-Y_ϵ| should be of order ϵ^2/3. Without additional regularity assumptions on f or g, this is simply not true. However, with additional Hölder regularity assumptions we will later show that this is indeed true, see <ref> below.


    As in previous arguments, we replace ℒ_0,ϵ(x,y) by ϵ^1/3ℒ_0,1 (ϵ^-2/3 x,ϵ^-2/3y) = ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y). From <ref>, we have the bound 
    |ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+ ϵ^-1 (x-y)^2|≤ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)),
 where c is deterministic and ℭ is a non-negative random variable satisfying ℙ( ℭ>u) ≤ Ce^-du^3/2 for u ≥ 0. 
    
    We estimate the probability of the event K_ϵ:={|X_ϵ-Y_ϵ|>ϵ^1/2-δ}. On one hand, on the event K_ϵ, ϵ^-1(X_ϵ - Y_ϵ)^2 > ϵ^-2δ. From the conditions on f,g imposed by the statement of the proposition, it is also clear that |z_max| ≤√(2) B. Therefore, on K_ϵ we find that 
    -ϵ^-2δ   +2B ≥ -ϵ^-2δ +sup_x f(x) + sup_y g(y)
       ≥ f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) 
       ≥[f(X_ϵ) +ϵ^1/3𝒮 (ϵ^-2/3 X_ϵ,ϵ^-2/3Y_ϵ)+g(Y_ϵ)] - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|))
       ≥[f(z_max)+ϵ^1/3𝒮(ϵ^-2/3z_max,ϵ^-2/3 z_max)+g(z_max)] - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) 
       ≥ f(z_max)+g(z_max) -(ℭϵ^1/3 + cϵ^1/3log^2/3 (2 +2ϵ^-2/3 |z_max|)) 
        
            - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) 
       ≥ (-B - 1/2z_max^2)+(-B - 1/2z_max^2) -2ℭϵ^1/3
            - cϵ^1/3( log^2/3 (2+2ϵ^2/3|z_max|) + log^2/3 ( 2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|) ) ) 
       ≥ -4B -2ℭϵ^1/3  -cϵ^1/3( log^2/3 (2+2ϵ^-2/3B)  + log^2/3 ( 2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|) )).
 Since c is deterministic, for sufficiently small deterministic ϵ>0 (below some threshold independent of B≥ 2 say) we have an elementary bound cϵ^1/3log^2/3 (2+2ϵ^-2/3B) ≤ 2B. We define the event 
    H_ϵ:={|X_ϵ|+|Y_ϵ| ≤ 1+log^2/5(1/ϵ)+100B}.
 By <ref> we find that 1-ℙ(H_ϵ) ≤ Ce^-d log^6/5(1/ϵ). On the event H_ϵ we likewise have that cϵ^1/3log^2/3 ( 2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|) )≤ 100B, again assuming that ϵ is below some sufficiently small positive deterministic threshold. On K_ϵ∩ H_ϵ we find that 2ℭϵ^1/3≥ϵ^-2δ -108B . Therefore, we obtain 
    
    ℙ(K_ϵ)    ≤ℙ(H_ϵ∩ K_ϵ)+ℙ(H_ϵ^c) 
       ≤ℙ( 2ℭϵ^1/3≥ϵ^-2δ -108B) + Ce^-d log^6/5(1/ϵ)
       ≤ Ce^-d ϵ^-1/2(ϵ^-2δ - 108 B)^3/2 + Ce^-d  log^6/5(1/ϵ).

    This completes the proof.


We introduce several quantities which will be important to our argument.
 Let f,g: ℝ→ℝ be continuous functions decaying parabolically at infinity, that is, -Ax^2 -B ≤ f(x),g(x) ≤ -Cx^2-D for some A>C>0 and B,D∈ℝ. Assume that f,g achieve their maximum values at unique input values z_max^f, z_max^g. 
For ϵ,M>0 and α∈ (0,1), we define

    𝒮(f;ϵ)   := {z∈ℝ: f(z) ≥ f(z_max^f)-ϵ}
    Γ(f,g;ϵ)   := inf{ |z-z'|: |f(z)-f(z_max^f)|<ϵ,   |g(z')-g(z_max^g)|<ϵ}.
    
    ℌ𝔬𝔩(g,α;M)   := sup_s,t∈ [-M,M]
    s≠ t|g(t)-g(s)|/|t-s|^α.

We define N_ϵ(f) to be the smallest number of intervals of length ϵ^2 required to cover 𝒮(f;ϵ). 


    



    


 Let f,g: ℝ→ℝ be deterministic functions satisfying -2x^2 -B ≤ f(x),g(x) ≤ -1/2x^2+B for some B≥ 1. Define
the random variable 

    (X_ϵ,Y_ϵ) := argmax_(x,y)∈ℝ^2  f(x) + ℒ_0,ϵ(x,y)+g(y).

We also denote z_max:= argmax_z∈ℝ f(z)+g(z), which we assume is unique. Fix δ∈ (0,1/3]. Then 

    ℙ(|X_ϵ-Y_ϵ|>ϵ^2/3-δ)   ≤ Ce^-d  log^6/5(1/ϵ)+1_{ B≥ dϵ^-1/3} +1_{ℌ𝔬𝔩(g, 1/2(1-3/10δ),   1+100B+log^2/5(1/ϵ)) >ϵ^-δ/10}

where C,d>0 may depend on δ but are independent of f,g,B and ϵ∈ (0,1].


Here and in later propositions, we do not optimize the constants such as 3/10,2/5,6/5,100. The current values suffice for arguments in <ref>.



    By scale invariance of the directed landscape we can replace ℒ_0,ϵ(x,y) by ϵ^1/3ℒ_0,1 (ϵ^-2/3 x,ϵ^-2/3y) = ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y), where 𝒮 is the Airy sheet as usual. In the notation of <ref> we have 
    |ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+ ϵ^-1 (x-y)^2|≤ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)),
 where c is deterministic and ℭ is a non-negative random variable satisfying ℙ( ℭ>u) ≤ Ce^-du^3/2 for u ≥ 0. 
    
    We choose v:= 3δ/10-6δ> 3/10δ and define an inductive sequence of exponents κ_n for n=0,1,2,3,... as follows: 
    κ_0=1/3 and κ_n+1 := 1-v/2 +1/2(1/2 -v)κ_n.

    As n→∞, κ_n↑2/3-1/2δ. This convergence necessarily occurs exponentially quickly by standard iteration theory for linear recurrences. 
    For n=0,1,2,3,... define the events 
    K^(n)_ϵ:= { |X_ϵ-Y_ϵ| ≤ϵ^κ_n}.

    We will now prove that there exist universal constants C,d>0 so for all n≥ 1 and ϵ∈ (0,1], 
    
    ℙ(  K^(n-1)_ϵ\ K_ϵ^(n) ) ≤ 1_{ B≥ dϵ^-1} + 1_{ℌ𝔬𝔩(g;1/2-v/2; 1+100B+log^2/5(1/ϵ))> ϵ^-δκ_n-1/2} + Ce^-dlog^6/5(1/ϵ).

    
    To prove this, notice 
    
    f(X_ϵ)     -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) 
       ≥ 
        f(X_ϵ) +ϵ^1/3𝒮 (ϵ^-2/3 X_ϵ,ϵ^-2/3Y_ϵ)+g(Y_ϵ) - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|))
       ≥ f(z_max)+ϵ^1/3𝒮(ϵ^-2/3z_max,ϵ^-2/3 z_max)+g(z_max) - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|)) 
       ≥ f(z_max)+g(z_max) -(ℭϵ^1/3 + cϵ^1/3log^2/3 (2 +2ϵ^-2/3 |z_max|)) 
        
             - ℭϵ^1/3- c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|))
  From the conditions on f,g imposed by the statement of the proposition, it follows that |z_max| ≤√(2) B. Since c is deterministic, for sufficiently small deterministic ϵ>0, which does not depend on the choice of B≥ 1, we have an elementary bound cϵ^1/3log^2/3 (2+2ϵ^-2/3B) ≤ Cϵ^1/3 -2/3v B^v. This follows from the fact that log^2/3(2+u) ≤ Cu^v for all u≥ 1 where C=C(v) does not depend on u.
    We define H_ϵ to be the event 
    H_ϵ:={|X_ϵ|+|Y_ϵ| ≤ 1+log^2/5(1/ϵ)+100B}.
 By <ref>, we conclude 1-ℙ(H_ϵ) ≤ Ce^-d log^6/5(1/ϵ). Likewise, on the event H_ϵ we have cϵ^1/3log^2/3 ( 2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|) )≤ C ϵ^1/3-2/3v B^v. Here we have assumed again that ϵ is below some sufficiently small positive deterministic threshold. Consequently, on the event H_ϵ, we obtain the following bound
    
    f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) ≥ f(z_max)+g(z_max) -2ℭϵ^1/3 -Cϵ^1/3-2/3v B^v.

    On the event K^(n-1)_ϵ∩ H_ϵ, 
    |g(X_ϵ)-g(Y_ϵ)| ≤ C ϵ^-v κ_n-1 /2 |X_ϵ-Y_ϵ|^1/2-v/2≤ϵ^(1/2 - v)κ_n-1 as long as we have ℌ𝔬𝔩(g;1/2-v/2; 1+100B+log^2/5(1/ϵ))≤ϵ^-vκ_n-1/2. When we combine this bound with (<ref>), on the event H_ϵ∩ K^(n-1)_ϵ, we obtain the bound
    
    f(z_max) -ϵ^-1(X_ϵ-Y_ϵ)^2 +g(z_max)    ≥ f(X_ϵ) -ϵ^-1(X_ϵ-Y_ϵ)^2 +g(X_ϵ) 
       ≥ f(X_ϵ) -ϵ^-1(X_ϵ-Y_ϵ)^2 +g(Y_ϵ) - |g(X_ϵ)-g(Y_ϵ)| 
       ≥ f(z_max)+g(z_max) -2ℭϵ^1/3 -Cϵ^1/3-2/3v B^v -  ϵ^(1/2 -v)κ_n-1
       ≥ f(z_max)+g(z_max) -2ℭϵ^1/3 -Cϵ^(1/2 - v)κ_n-1B^v.

    In the final line, we use the fact that B≥ 1 and that (1/2-v)κ_n is always less than 1/3-2/3v. As long as ℌ𝔬𝔩(g;1/2-v/2; 1+100B+log^2/5(1/ϵ))≤ϵ^-vκ_n-1/2, we have by the recursive definition of the κ_n that 1+(1/2 - v)κ_n-1 = 2κ_n+v and therefore the above expression immediately gives 
    (X_ϵ-Y_ϵ)^2 ≤ 2ℭϵ^4/3 +Cϵ^1+(1/2 - v)κ_n-1 B^v=2ℭϵ^4/3 +Cϵ^2κ_n+v B^v,
 for some large enough deterministic constant C=C(v) that does not depend on B,f,g,ϵ. The right-hand side of the previous expression is less than ϵ^2κ_n as long as ℭ≤1/4ϵ^-2v (because 2κ_n≤4/3-2v for all n) and B≤1/2ϵ^-1 C^-1/v, which proves (<ref>) once we use the facts that ℙ(H_ϵ^c) ≤ Ce^-dlog^6/5(1/ϵ) and ℙ(ℭ > 1/2ϵ^-2v) ≤ Ce^-dϵ^-3v≤ Ce^-dlog^6/5(1/ϵ).

    We use (<ref>) to prove the bound in the proposition. By using <ref> and the fact that κ_0 is chosen to be 1/3, 
    ℙ( K^(0)_ϵ) ≤ Ce^-d ϵ^-1/2(ϵ^-1/3 - 108 B)^3/2≤ Ce^-dϵ^-5/6 + 1_{ 200B≥ϵ^-1/3}.

    
    Using this bound and then repeatedly summing over (<ref>) up to a large enough value of n such that κ_n reaches the desired threshold will give the desired bound (this is a finite sum). This is because v>3/10δ for δ∈ (0,1/3], as explained earlier. 
   
    
    
    
    
    
    
        
        
    
    
    

    


 Let f,g: ℝ→ℝ be deterministic functions satisfying -2x^2 -B ≤ f(x),g(x) ≤ -1/2x^2+B for some B≥ 1. Define the random variable 

    (X_ϵ,Y_ϵ) := argmax_(x,y)∈ℝ^2  f(x) + ℒ_0,ϵ(x,y)+g(y).

We also denote z_max:= argmax_z∈ ℝ f(z)+g(z), which we assume is unique. Fix δ∈ (0,1/3]. Then

    ℙ(X_ϵ∉𝒮(f+g,ϵ^1/3- δ))    ≤  Ce^-d  log^6/5(1/ϵ) +1_{ B≥ dϵ^-1/3} +1_{ℌ𝔬𝔩(g, 1/2(1-3/20δ),   1+100B+log^2/5(1/ϵ)) >ϵ^-δ/20},
    
    ℙ(Y_ϵ∉𝒮(f+g,ϵ^1/3- δ))    ≤  Ce^-d  log^6/5(1/ϵ) +1_{ B≥ dϵ^-1/3} +1_{ℌ𝔬𝔩(f, 1/2(1-3/20δ),   1+100B+log^2/5(1/ϵ)) >ϵ^-δ/20}

where C,d>0 may depend on δ but are independent of f,g,B and ϵ∈ (0,1]. 



The bound on Y_ϵ follows from the bound on X_ϵ by a symmetry argument. Therefore, we focus on the first bound.
By the scale invariance of the directed landscape, we replace ℒ_0,ϵ(x,y) by ϵ^1/3ℒ_0,1 (ϵ^-2/3 x,ϵ^-2/3y) = ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y). In the notation of <ref> we have 
    |ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+ ϵ^-1 (x-y)^2|≤ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)),
 where c is deterministic and ℭ is a non-negative random variable satisfying ℙ( ℭ>u) ≤ Ce^-du^3/2 for u ≥ 0. 
We fix an arbitrary δ_1>0 (the value of which will be specified at a later point). We define the event 
    H_ϵ:={|X_ϵ|+|Y_ϵ| ≤ 1+log^2/5(1/ϵ)+100B}.
 We recall on H_ϵ, from (<ref>) for ϵ below a sufficiently small positive deterministic threshold
    
    f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) ≥ f(z_max)+g(z_max) -2ℭϵ^1/3 -Cϵ^1/3-2/3δ_1 B^δ_1.

    We also define the event 
    F_ϵ:= { |X_ϵ - Y_ϵ| ≤ϵ^2/3-δ_1}.
 By <ref>, F_ϵ has probability greater than 1-Ce^-dlog^6/5(1/ϵ) as long as B≤ dϵ^-1/3 and 
    ℌ𝔬𝔩(g,1/2(1-δ_2),1+100B+log^2/5(1/ϵ)) ≤ϵ^- δ_2/3,
 where δ_2:=3/10δ_1. From the expression above, if ℌ𝔬𝔩(g,1/2(1-δ_2), 1+100B+log^2/5(1/ϵ)) ≤ϵ^- δ_2/3, then on the event F_ϵ∩ H_ϵ we have the bound
    
    f(X_ϵ) + g(X_ϵ)    > f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(X_ϵ) 
       ≥ f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) - |g(Y_ϵ)-g(X_ϵ)| 
       ≥ f(X_ϵ) -ϵ^-1(X_ϵ - Y_ϵ)^2+g(Y_ϵ) - ϵ^-δ_2/3 |X_ϵ-Y_ϵ|^1/2-δ_2
       ≥ f(z_max)+g(z_max) -2ℭϵ^1/3 -Cϵ^1/3-2/3δ_1 B^δ_1 - ϵ^1/3 -δ_2 - 1/2δ_1 +δ_1δ_2.

    With δ as in the proposition statement, we make a specific choice of δ_1 so that 2/3δ_1<1/2δ and δ_2+1/2δ_1 -δ_1δ_2<1/2δ. For precision, take δ_1:=1/2δ. From the previous expression, it follows that in order to have X_ϵ∉𝒮(f+g,ϵ^1/3- δ), for ϵ<3^-2/δ it must be true that ℭ> 1/6ϵ^-δ or B>1/6ϵ^-2/3 C^-1/δ_1. Therefore, 
    ℙ(X_ϵ∉𝒮(f+g,ϵ^1/3- δ) ≤ℙ(ℭ> 1/6ϵ^-δ) + 1_{B>1/6ϵ^-2/3 C^-1/δ_1} + ℙ(H_ϵ^c) + ℙ(F_ϵ^c).
 To finish the proof, we apply the tail bounds on ℭ and the upper bounds on the latter two terms, all of which have already been discussed. 
    



 
  
    
    
        
        
    
    
    

Using the preceding bounds, we can prove the following estimate on the directed landscape. 
 Let f,g: ℝ→ℝ be deterministic functions satisfying -2x^2 -B ≤ f(x),g(x) ≤ -1/2x^2+B for some B≥ 1. Then for any δ>0 and p>1,

    𝔼[ |f∘ℒ_0,ϵ∘ g - max(f+g)|^p]^1/p
    ≤ CB^2 δϵ^1/3-δ +CB(Ce^-d  log^6/5(1/ϵ) +1_{ B≥ dϵ^-1/3} +1_{ℌ𝔬𝔩(g, 1/2(1-3/40δ),   1+100B+log^2/5(1/ϵ)) >ϵ^-δ/40}).
 Here C,d may depend on p but are uniform over all f,g,B, and ϵ∈(0,1].


 As before, by scale invariance of the directed landscape, we use the substitution, ℒ_0,ϵ(x,y) by ϵ^1/3ℒ_0,1 (ϵ^-2/3 x,ϵ^-2/3y) = ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y). In the notation of <ref> we have 

    |ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+ ϵ^-1 (x-y)^2|≤ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)),
 where c is deterministic and ℭ is a non-negative random variable satisfying ℙ( ℭ>u) ≤ Ce^-du^3/2 for u ≥ 0. Let X_ϵ,Y_ϵ,z_max be as in <ref> and <ref>. We note that  
    
    |f(X_ϵ)    +ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y)+g(Y_ϵ) - (f(z_max)+g(z_max))| 
       ≤ |f(X_ϵ) + g(X_ϵ) - (f(z_max)+g(z_max))| + |ϵ^1/3𝒮 (ϵ^-2/3 X_ϵ,ϵ^-2/3Y_ϵ)+ ϵ^-1 (X_ϵ-Y_ϵ)^2| 
            + |g(Y_ϵ)-g(X_ϵ)| +ϵ^-1 (X_ϵ-Y_ϵ)^2.
 We can now define the event 
    J_ϵ:= { X_ϵ∈𝒮(f+g,ϵ^1/3- δ)}∩{|X_ϵ-Y_ϵ| ≤ϵ^2/3-δ}∩{|X_ϵ|+|Y_ϵ| ≤ 1+100B +log^2/5(1/ϵ)}.
 
    Furthermore, under the assumption that ℌ𝔬𝔩(g,1/2-δ, 1+100B +log^2/5(1/ϵ))<ϵ^-δ, the last expression is bounded (term-by-term) by
    
    ϵ^1/3-δ +( ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|X_ϵ|+|Y_ϵ|))) + ϵ^-δ/3|X_ϵ-Y_ϵ|^1/2-δ +ϵ^-1 (X_ϵ-Y_ϵ)^2 
    ≤ϵ^1/3-δ +(ℭϵ^1/3 +ϵ^1/3-2/3δB^δ) + ϵ^(2/3-δ)(1/2-δ) + ϵ^1/3-2δ.

    For the second term we used the following argument: since c is deterministic, for sufficiently small deterministic ϵ>0 which does not depend on the choice of B≥ 1, we have an elementary bound cϵ^1/3log^2/3 (2+2ϵ^-2/3B) ≤ Cϵ^1/3 -2/3δB^δ. This bound follows immediately from the fact that log^2/3(2+u) ≤ Cu^δ for all u≥ 1, where C=C(δ) does not depend on u. Note that all powers of ϵ in the last expression are at worst 1/3-2δ. Since ℭ is the only random term in the last expression, these bounds imply that 
    
    𝔼[ |f∘ℒ_0,ϵ∘ g - max(f+g)|^p· 1_J_ϵ]^1/p≤ CB^δϵ^1/3-2δ.
 
    Replacing δ by 2δ we obtain a bound of the same form as in the proposition statement. It remains to bound what occurs outside J_ϵ. Using <ref> and <ref> and <ref>, and the elementary inequality (a+b)^1/2≤ a^1/2+b^1/2, we obtain 
    ℙ(J_ϵ^c)^1/2≤ Ce^-d  log^6/5(1/ϵ) +1_{ B≥ dϵ^-1/3} +1_{ℌ𝔬𝔩(g, 1/2(1-3/20δ),   1+100B+log^2/5(1/ϵ)) >ϵ^-δ/20}.
 By Cauchy-Schwarz and Minkowski, we furthermore have 
    
    𝔼[ |f∘ℒ_0,ϵ∘ g - max(f+g)|^p· 1_J_ϵ^c]^1/p≤( 𝔼[|f∘ℒ_0,ϵ∘ g|^2p]^1/2p + max(f+g)) ·ℙ(J_ϵ^c)^1/2.
 Clearly, max(f+g)≤ 2B. We will also show that 𝔼[|f∘ℒ_0,ϵ∘ g|^2p]^1/2p is bounded above by B times some absolute constant. As before, we write ℒ_0,ϵ as 𝒮^ϵ(x,y):=ϵ^1/3𝒮 (ϵ^-2/3 x,ϵ^-2/3y). We apply (<ref>) to obtain that 
    f(x)+𝒮^ϵ(x,y)+g(y)    ≤ 2B-1/2(x^2+y^2) +ℭϵ^1/3+ c ϵ^1/3log^2/3(2+ϵ^-2/3(|x|+|y|)) 
       ≤ 2B - 1/2(x^2+y^2) +ℭϵ^1/3 +2+|x|+|y|,

    where the last bound is valid for sufficiently small ϵ>0 below some deterministic threshold. Let m denote the maximum value achieved by the function (x,y)↦ 2+|x|+|y|- 1/2(x^2+y^2) over all of ℝ^2. Then in order for the last expression to exceed 2B+m+u, it would be necessary for the constant ℭ to exceed u. Therefore,
    
    ℙ( f∘ℒ_0,ϵ∘ g> 2B+m+u) ≤ Ce^-du^3/2,
 which implies that  𝔼[(f∘ℒ_0,ϵ∘ g)_+^2p]^1/2p (where u_+:=max{u,0}) is bounded above by B multiplied by an absolute constant. To bound the negative part, we note by similar arguments that the probability that f∘ℒ_0,ϵ∘ g<-2B yields a large value of ℭ. One can show this by considering what happens at (z_max,z_max). This yields an analogous bound on 𝔼[(f∘ℒ_0,ϵ∘ g)_-^2p]^1/2p (where u_- = max{0,-u}).
    


We now come to the main result of this section, which we obtain by applying the mixing results of <ref> with the estimates proved in this section. We begin by stating a covariance bound that will be crucial to understanding the mechanisms behind the black noise property. 


    Consider functions f,g,u,v each lying in between -B-2x^2 and B-1/2x^2. Recall Γ and N_ϵ as defined in (<ref>). Then for any δ>0, 
    |(f∘ℒ_0,ϵ∘ g, u ∘ℒ_0,ϵ∘ v)|     ≤ CB^2δmin{ϵ^2/3 - δ, (N_ϵ^1/3-δ(f+g)+N_ϵ^1/3-δ(u+v))e^-c ϵ^-2Γ(f+g,u+v;ϵ^1/3-δ)^3}
        +CB^2 ·( 𝔈_ϵ(f;B)+𝔈_ϵ(g;B)+𝔈_ϵ(u;B)+𝔈_ϵ(v;B)).

    Here C,d are universal constants uniform over all f,g,u,v, ϵ,B and  
    𝔈_ϵ(f,B):= e^-d  log^6/5(1/ϵ) +1_{ B≥ dϵ^-1/3} + 1_{ℌ𝔬𝔩(f, 1/2(1-3/40δ),   1+100B+log^2/5(1/ϵ))>ϵ^-δ/40}.



By Cauchy-Schwartz,

    |(f∘ℒ_0,ϵ∘ g, u ∘ℒ_0,ϵ∘ v)|   = |(f∘ℒ_0,ϵ∘ g - max(f+g), u ∘ℒ_0,ϵ∘ v - max(u+v))| 
       ≤𝔼[(f∘ℒ_0,ϵ∘ g - max(f+g))^2 ]^1/2𝔼[(u∘ℒ_0,ϵ∘ v - max(u+v))^2 ]^1/2 .

    Using <ref>, we get that the last expression is bounded above by CB^2 δϵ^2/3-δ +CBe^-dlog^6/5(1/ϵ) as long as B≤ dϵ^-1/3 and ℌ𝔬𝔩(g, 1/2(1-3/40δ),   1+100B+log^2/5(1/ϵ))≤ϵ^-δ/40 and ℌ𝔬𝔩(v, 1/2(1-3/40δ),   1+100B+log^2/5(1/ϵ))≤ϵ^-δ/40. 
    
    We define the event J_ϵ^f,g:= { X_ϵ^f,g,Y_ϵ^f,g∈𝒮(f+g,ϵ^1/3- δ)}∩{|X_ϵ^f,g-Y_ϵ^f,g| ≤ϵ^2/3-δ}∩{|X_ϵ^f,g|+|Y_ϵ^f,g| ≤ 1+100B +log^2/5(1/ϵ)}, where X_ϵ^f,g,Y_ϵ^f,g are the argmaxes as defined in <ref> and <ref> respectively. 
    We define the event J^u,v_ϵ in the same way, but with f,g replaced by u,v. Finally, we define J_ϵ^f,g,u,v:= J^f,g_ϵ∩ J^u,v_ϵ. We condition on the event J_ϵ^f,g,u,v, and this allows us to use <ref> to obtain a bound of the form (N_ϵ^1/3-δ(f+g)+N_ϵ^1/3-δ(u+v))e^-c ϵ^-2Γ(f+g,u+v;ϵ^1/3-δ)^3. 
    Indeed by setting T:=ϵ in <ref>, we see that the value of D in that theorem corresponds to Γ(f+g,u+v,ϵ^1/3-δ), and furthermore N_ϵ^1/3-δ(f+g)+N_ϵ^1/3-δ(u+v) gives an upper bound for the value of N which appears in that theorem. 
    Consequently, we apply <ref> to obtain the bound 
    |Cov((f∘ℒ_0,ϵ∘ g)· 1_J_ϵ^f,g,u,v  , (u ∘ℒ_0,ϵ∘ v)· 1_J_ϵ^f,g,u,v)|  ≤(N_ϵ^1/3-δ(f+g)+N_ϵ^1/3-δ(u+v))e^-c ϵ^-2Γ(f+g,u+v;ϵ^1/3-δ)^3,

    because on J_ϵ^f,g,u,v the maximum on all of ℝ^2 in the definitions of f∘ℒ_0,ϵ∘ g,u∘ℒ_0,ϵ∘ v can be replaced by a maximum on some set of the form which appears in <ref>. It remains to control what happens on the complement of J_ϵ^f,g,u,v. By applying Cauchy-Schwarz twice we have 
    
    |Cov   (f∘ℒ_0,ϵ∘ g , u ∘ℒ_0,ϵ∘ v)| ≤|Cov((f∘ℒ_0,ϵ∘ g)· 1_J_ϵ^f,g,u,v  , (u ∘ℒ_0,ϵ∘ v)· 1_J_ϵ^f,g,u,v)|  
            + 2𝔼[(f∘ℒ_0,ϵ∘ g - max(f+g))^4 ]^1/4𝔼[(u∘ℒ_0,ϵ∘ v - max(u+v))^4 ]^1/4· (1-ℙ(J_ϵ^f,g,u,v))^1/2.
 Using arguments similar to (<ref>), it follows that (1-ℙ(J_ϵ^f,g,u,v))^1/2 yields an upper bound consisting of terms of the form appearing in 𝔈_ϵ. The upper bounds for all other terms have already been discussed.



    Consider functions f,g,u,v,u̅, v̅ each lying in between -B-2x^2 and B-1/2x^2. Define Γ(ϵ):=min{Γ(f+g,u+v;ϵ^1/3-δ_1),Γ(f+g,u̅+v̅;ϵ^1/3-δ_1)}. Then for any δ>0 
    
    |Cov(f∘   ℒ_0,ϵ∘ g  ,   (u ∘ℒ_0,ϵ∘ v-max(u+v)) ((u̅∘ℒ_0,ϵ∘v̅-max(u̅+v̅)))| 
       ≤  CB^2δmin{ϵ ^1 - δ, (N_ϵ^1/3-δ(f+g)+N_ϵ^1/3-δ(u+v)+N_ϵ^1/3-δ(u̅+v̅))e^-c ϵ^-2Γ(ϵ)^3}
                 +CB^3 ·( 𝔈_ϵ(f;B)+𝔈_ϵ(g;B)+𝔈_ϵ(u;B)+𝔈_ϵ(v;B) + 𝔈_ϵ(u̅;B)+𝔈_ϵ(v̅;B)).
 
    Here C,d are universal constants uniform over all f,g,u,v, ϵ,B, and 𝔈_ϵ was defined in <ref>.



The proof is very similar to that of the previous theorem. Note, by applying Cauchy-Schwartz twice, that

    |Cov(f∘ℒ_0,ϵ∘ g, (u ∘ℒ_0,ϵ∘ v-max(u+v)) ((u̅∘ℒ_0,ϵ∘v̅-max(u̅+v̅)))| 
    ≤𝔼[(f∘ℒ_0,ϵ∘ g - max(f+g))^2 ]^1/2𝔼[(u∘ℒ_0,ϵ∘ v - max(u+v))^4 ]^1/4𝔼[(u̅∘ℒ_0,ϵ∘v̅ - max(u̅ +v̅))^4 ]^1/4 .

    Using <ref>, we get that the last expression is bounded above by CB^2δϵ^1-δ +CBe^-dlog^6/5(1/ϵ) as long as B≤ dϵ^-1/3 and ℌ𝔬𝔩(g, 1/2(1- δ_1),   1+100B+log^2/5(1/ϵ))≤ϵ^-δ_1/3 and ℌ𝔬𝔩(v, 1/2(1- δ_1),   1+100B+log^2/5(1/ϵ))≤ϵ^-δ_1/3, where δ_1=3/40δ.

    Next, we proceed exactly as in the previous theorem. We condition on the event J_ϵ^f,g,u,v, u̅,v̅, which allows us to use <ref> to obtain a bound of the form 
    (N_ϵ^1/3-δ_1(f+g)+N_ϵ^1/3-δ_1(u+v)+N_ϵ^1/3-δ_1(u̅+v̅))e^-c ϵ^-2Γ(ϵ)^3.
 Indeed by setting T:=ϵ in that theorem, the value of D corresponds to Γ(ϵ), and moreover N_ϵ^1/3-δ_1(f+g)+N_ϵ^1/3-δ_1(u+v)+N_ϵ^1/3-δ_1(u̅+v̅) gives an upper bound for the value of N appearing in that theorem. It remains to control what occurs on the complement of J_ϵ^f,g,u,v, u̅,v̅. This can be done completely analogously to the proof of Theorem <ref>. As we observed there, this will only yield error terms of the form 𝔈_ϵ.



Consider functions f_i,g_i,u_j,v_j (1≤ i≤ k,1≤ j≤ℓ) each lying in between -B-2x^2 and B-1/2x^2. 
Then for any δ_1>0 there exists δ_2>0 so that 
    |Cov(∏_i=1^k (f_i∘ℒ_0,ϵ∘ g_i-max(f_i+g_i)), ∏_j=1^ℓ (u_j ∘ℒ_0,ϵ∘ v_j-max(u_j+v_j)))| 
    ≤  CB^δ_2ϵ^1/3(k+ℓ)-δ_1 +CB^k+ℓ·( ∑_i 𝔈_ϵ(f_i;B)+ 𝔈_ϵ(g_i;B)+∑_j 𝔈_ϵ(u_j;B) +  𝔈_ϵ(v_j;B)).

    Here C,d are universal constants uniform over all f_i,g_i,u_i,v_i, ϵ,B, and 𝔈_ϵ is as defined in <ref>.



    We apply Cauchy-Schwartz to the covariance, and then apply Hölder's inequality to obtain
    
    |Cov   (∏_i=1^k (f_i∘ℒ_0,ϵ∘ g_i-max(f_i+g_i)), ∏_j=1^ℓ (u_j ∘ℒ_0,ϵ∘ v_j-max(u_j+v_j)))| 
       ≤𝔼[∏_i=1^k (f_i∘ℒ_0,ϵ∘ g_i-max(f_i+g_i))^2 ]^1/2𝔼[ ∏_j=1^ℓ (u_j ∘ℒ_0,ϵ∘ v_j-max(u_j+v_j))^2]^1/2
       ≤∏_i=1^k𝔼[ (f_i∘ℒ_0,ϵ∘ g_i-max(f_i+g_i))^2k]^1/(2k)∏_j=1^ℓ𝔼[ (u_j ∘ℒ_0,ϵ∘ v_j-max(u_j+v_j))^2k]^1/(2k)
        .

    We use <ref> to obtain the claim (note here that unlike the previous two theorems, one does not need <ref> as we are simply using the unrefined bound without leveraging the mixing property, as this will be sufficient later).



    In all propositions above, the constants 1/2 and 2 in the parabolic decay rates of f,g could have been replaced by A^-1 and A for arbitrary A>1. This would affect the constants such as 100 and 108 as well as the constants C,d. While we cannot control the behavior of these constants as A↑∞, there is never an issue for a fixed finite A, no matter how large. This will be important later due to the nature of <ref>, because we will want f,g to be “typical" sample paths of parabolic Airy_2 processes of scales within the interval [η,1] for some fixed but very small η>0. In this case A can be as large as Cη^-1 for some universal C>0.

    Another remark is that the constant Ce^-dlog^6/5(1/ϵ) can be improved to Ce^-dlog^3M(1/ϵ) for any M>0, wherever it appears. However, the cost is that we then need to replace all instances of {ℌ𝔬𝔩(f, 1/2(1-δ̃),   1+100B+log^2/5(1/ϵ))>ϵ^-δ̃/3} by the respective interval {ℌ𝔬𝔩(f, 1/2(1-δ̃),   1+100B+log^M(1/ϵ))>ϵ^-δ̃/3}, where δ̃ denotes the various instances of 3/10δ,3/20δ, 3/40δ and other convenient choices of multiplying constants that we made in the propositions above. For the purposes of this paper, the value of M can be taken to be any value larger than 1/3, however we chose M=2/5 in order to be precise.




§ ESTIMATES ON 3-DIMENSIONAL BESSEL PROCESSES



 Let R be a three-dimensional Bessel process, and let p∈ℕ. Then there exists C_p>0 such that we have a uniform bound over all s_1≤ ...≤ s_p and ϵ∈ (0,1]

    ℙ( inf_t∈ [s_i,s_i+1] R(t) ≤ϵ , ∀ 1≤ i ≤ p) ds_1⋯ ds_p≤ C_p ϵ^p ∏_i=0^p-1min{1,(s_i+1-s_i)^-3/2(1+log_+^3/2(s_i+1-s_i))},
where s_0:=0. 


 We will prove the bound first in the case that ϵ=1. We interpret the Bessel process as the Euclidean norm of a three-dimensional standard Brownian motion, that is, R(t) = |B⃗(t)|. It is a helpful fact that if B⃗=(B_1,B_2,B_3), then through a union bound and well-known formulas for the distribution of the supremum of a Brownian motion on [0,1], 

    ℙ( sup_t∈[0,1]|B⃗(t)| >a) ≤ 3ℙ (sup_t∈ [0,1] B_1(t) >a/3) ≤ 6 e^-a^2 /18.
 
It is also helpful to note the following fact 

    ℙ( |B⃗(1)| >b,  inf_t∈ [0,1] |B⃗(t)| ≤ 1 ||B⃗(0)|=a) ≤ Ce^-1/36(a^2+ b^2),

where C>0 does not depend on a,b>0. This follows from the strong Markov property. Indeed, define the stopping time τ as the first time t that |B⃗(t)| = 1 and the Brownian motion W⃗(t):= B⃗(t+τ)-B⃗(τ). The event {|B⃗(1)| >b,  inf_t∈ [0,1] |B⃗(t)| ≤ 1} is contained in the event {sup_t∈ [0,1] |W⃗(t)| > b-1}, and the second event is moreover independent of the ℱ^B_τ-measurable event {inf_t∈ [0,1] |B⃗(t)|<1}={τ≤ 1}. Therefore, by conditioning on the latter event, we see that the left side of (<ref>) is bounded above by 
    ℙ(sup_t∈ [0,1] |W⃗(t)| > b-1)·ℙ(inf_t∈ [0,1] |B⃗(t)|<1|B⃗(0)=a).
 By (<ref>) this is bounded above by 36 e^-1/18((a-1)^2+(b-1)^2). Now use (b-1)^2 ≥1/2 b^2 - 1^2 and (a-1)^2 ≥1/2a^2-1^2 and we finally arrive at (<ref>) with C= 36 e^1/9. 


We return to proving the lemma in the case ϵ=1. Without loss of generality, we will assume that s_i+1-s_i>3. In order for inf_t∈ [s_i,s_i+1] R(t) ≤ 1 for all 1≤ i ≤ p+1, at least one of the following three things must happen for each i =1,...,p:


  * sup_t∈ [s_i,s_i+1] |B(t)-B(s_i+1)| > 9√(log(s_i-s_i-1)).

  * |B(s_i+1)-B(s_i+1)| ≤ 18√(log(s_i-s_i-1)).

  * sup_t∈ [s_i+1,s_i+1+1] |B(t)-B(s_i+1)| > 9√(log(s_i-s_i-1)).

We partition [s_1,s_p) into 2p disjoint intervals of the form [0,s_1),[s_1,s_1+1),[s_1+1,s_2), [s_2,s_2+1),[s_2+1,s_3),..., [s_p,s_p+1), which we denote by I_1,J_1,I_2,J_2,I_3,..., J_p. For a given realization of the Brownian motion B⃗, we say that an interval I_i or J_i is active if one of the conditions above holds on that interval. We have shown that among any three consecutive sub-intervals in the partition I_1,J_1,I_2,J_2,I_3,..., J_p, at least one must be active. Some intervals may furthermore be doubly active in the sense that the conditions within the above conditions may be satisfied for two distinct values of i. Furthermore, at least one of the first two sub-intervals I_1,J_1 must be active.
Consequently, by counting the doubly active intervals twice, there must be at least p sub-intervals from I_1,J_1,I_2,J_2,I_3,..., J_p that are active, each corresponding to a unique value of i in the list above. 
Notice that e^-(9√(log(s_i+1-s_i)))^2/18≤ 1∧ (s_i+1-s_i)^-2≤ 1∧ (s_i+1-s_i)^-3/2, and the standard 3-dimensional Gaussian tail bound tells us that 

    ℙ(|B(s_i+1)-B(s_i+1)| ≤ 18√(log(s_i+1-s_i))) ≤max{ 1,18^3· (s_i+1-s_i)^-3/2 (1+log^3/2_+(s_i+1-s_i))}.

By applying a union bound over all possible choices of p-tuples of active and doubly active sub-intervals from I_1,J_1,I_2,J_2,I_3,..., J_p, and applying the independence of increments of the 3-dimensional Brownian motion, we obtain the bound in the lemma. This is a consequence of (<ref>) for doubly active intervals and (<ref>) or (<ref>) for active ones. Each of the bounds for the increments [s_i+1,s_i] will be used exactly once. This proves the claim when ϵ=1.

For arbitrary ϵ>0, we need to explain why we obtain a factor ϵ^p. We begin by explaining this factor in the case that p=1. If we condition on reaching the value 1 inside the interval [s_i,s_i+1], the probability of reaching ϵ inside that interval is bounded by the probability of reaching ϵ ever again after time s_i. Using the Markov property, and the optional stopping theorem for the martingale R^-1 (stopping when this martingale hits ϵ^-1), we show that the latter probability is exactly ϵ. Therefore, for the p=1 case we have the bound 

    ℙ( inf_t∈ [s_i,s_i+1] R(t)<ϵ)    = ℙ( inf_t∈ [s_i,s_i+1] R(t)<ϵ|inf_t∈ [s_i,s_i+1] R(t)<1)ℙ( inf_t∈ [s_i,s_i+1] R(t)<1) 
       ≤ϵ  ·  s_1^-3/2 (1+log_+^3/2 s_1) .
 While this illustrates the bound in the case p=1, the same argument extends in a straightforward way to 
general values of p.



    Let R be a standard two-sided 3-dimensional Bessel process. Let N_ϵ(R) denote the smallest number of intervals of length ϵ^2 needed to cover the random set {t∈ℝ: R(t)≤ϵ}. Then sup_ϵ∈ (0,1]𝔼[N_ϵ(R)^p]<∞ for all p≥ 1.



This random variable is bounded above by 
    ϵ^-2∫_ℝ 1_{inf_t∈ [s,s+ϵ^2]R(t) ≤ϵ} ds .
 By scale invariance of the Bessel process, we know that the above random variable has the same law as 
    ∫_ℝ 1_{inf_t∈ [s,s+1] R(t) ≤ 1} ds.
 Let p be an even integer, and take the p^th moment, then apply Fubini to interchange the integral and expectation. We obtain that 
    𝔼[N_ϵ(R)^p]   ≤∫_ℝ^pℙ( inf_t∈ [s_i,s_i+1] R(t) ≤ 1, ∀ 1≤ i ≤ p) ds_1⋯ ds_p 
       =2^p  p! ·∫_{0<s_1<...<s_p}ℙ( inf_t∈ [s_i,s_i+1] R(t) ≤ 1, ∀ 1≤ i ≤ p) ds_1⋯ ds_p

By <ref> the integrand is bounded above by 

    C_p∏_i=0^p-1min{1,(s_i+1-s_i)^-3/2(1+log_+^3/2(s_i+1-s_i))},
 where s_0:=0. The fact that previous integral is finite follow from this bound. 



























    We define a Bessel process pinned at t∈ℝ to be the process R(∙ -t) where R is a two-sided 3-dimensional Bessel process, centered at zero. We will denote by ℙ_Bes3^t_1,t_2 to be the law on the canonical space C(ℝ)^2, of two independent Bessel processes R_1, R_2 pinned at t_1,t_2 respectively. 


  Fix δ∈ (0,1/2) and 0<γ̅<γ<1. Then there exists C>0 so that for all ϵ∈ (0,1] we have the estimate 

    sup_|t_1-t_2| ≥ϵ^-γ   ℙ_Bes3^t_1,t_2(  there   exist   s_1,s_2∈ℝ  such   that    R_1(s_1) + R_2(s_2) ≤ϵ^- δ  and    |s_1-s_2|≤ 2ϵ^-γ̅) 
       ≤ Cϵ^3/2γ -δ(2+γ-γ̅) -1/2γ̅.




For the black noise result, we will see that this bound is most useful when γ is slightly smaller than 1/3 and δ,γ̅ are very close to 0. In this case, note that the right side can be made arbitrarily close to ϵ^1/2. In practice, we will only use this estimate with γ=4/15 and γ̅=δ close to 0. 

    The proof will proceed using a simple union bound. If such values of s_1,s_2∈ℝ exist as specified above, then there exists some n∈ℤ such that all of the following simultaneously occur: |s_1-n| ≤ϵ^-γ̅, |s_s-n| ≤ϵ^-γ̅, R_1(s_1) ≤ϵ^-δ, R_2(s_2) ≤ϵ^-δ. Applying this union bound over all n∈ℤ, we find that the desired probability is upper bounded by 
        
    ∑_n∈ℤℙ_Bes3^t_1,t_2(  there exist  s_1,s_2∈ℝ such that  |s_1-ϵ^-γ̅n| ≤ϵ^-γ̅,|s_2-ϵ^-γ̅n| ≤ϵ^-γ̅,R_1(s_1) ≤ϵ^-δ, R_2(s_2) ≤ϵ^-δ).

    Without any loss of generality we can replace (t_1,t_2) by (0,t_2-t_1) and assume that T:=t_2-t_1>0, thus by assumption T>ϵ^-γ. We can estimate the probability in the summand, using the scaling invariance of the Bessel process to rewrite 
    
    ℙ_Bes3^0,T(  there exist  s_1,s_2∈ℝ such that |s_1-ϵ^-γ̅n| ≤ϵ^-γ̅,|s_2-ϵ^-γ̅n| ≤ϵ^-γ̅,R_1(s_1) ≤ϵ^-δ, R_2(s_2) ≤ϵ^-δ) 
       = ℙ_Bes3^0,ϵ^γ̅T(  there exist  s_1,s_2∈ℝ such that |s_1-n| ≤ 1,|s_2-n| ≤ 1,R_1(s_1) ≤ϵ^1/2γ̅-δ, R_2(s_2) ≤ϵ^1/2γ̅-δ) 
       = ∏_i=1^2 ℙ_Bes3^0,ϵ^γ̅T(  there exists  s_i∈ [n-1,n+1]  such that R_i(s_i) ≤ϵ^1/2γ̅-δ) 
       ≤ C [ ϵ^1/2γ̅-δ |n|^-3/2+δ][ϵ^1/2γ̅-δ |ϵ^γ̅ T-n|^-3/2+δ] 
       =C ϵ^γ̅-2δ |n|^-3/2+δ|ϵ^γ̅ T-n|^-3/2+δ.

    In the third line, we used the independence of R_1 and R_2, and <ref> with p=1 in the fourth line (we replaced the logarithmic correction by a weaker bound of size |n|^δ for simplicity). 
    Now notice that uniformly over N∈ℕ we have by symmetry 
    ∑_n=1^N 1/n^3/2-δ(N-n)^3/2-δ≤ 2∑_1≤ n≤ N/21/n^3/2-δ(N-n)^3/2-δ≤ 2(2/N)^3/2-δ∑_1≤ n≤ N/2n^-3/2+δ≤2^5/2ζ(32-δ)/N^3/2 -δ.

    Using this bound with N=ϵ^γ̅T≥ϵ^γ̅-γ, one finds that the desired probability is upper bounded by Cϵ^γ̅-2δ (ϵ^γ̅T)^-3/2+δ≤ Cϵ^γ̅-2δ (ϵ^γ̅-γ)^-3/2+δ = Cϵ^1/2 (3γ-γ̅) -δ(2+γ-γ̅). 




§ PROOF OF THE MAIN THEOREM


    Consider a standard Brownian motion B on [0,1], and let T denote its unique argmax. Fix α∈ (0,1). Define the process U:=(B(T)-B(T+t))_t∈ [-α T, α(1- T)]. Consider a three-dimensional Bessel process R independent of T and B, and define the process V:= (R(t))_t∈ [-α T, α(1- T)]. Then U is absolutely continuous with respect to V, and the Radon-Nikodym derivative is deterministically bounded by C(1-α)^-1/2 for some universal constant C>0. 


In the statement of <ref>, we can think of a continuous function on a random closed interval I⊂ [-1,1] as an element of C[-1,1] that is constant away from I. The last statement is sharp: if α=1 then the Radon-Nikodym is only in L^3-δ as opposed to L^∞, as discussed in <cit.>.


    Conditioned on T, the process (B(T)-B(T+t))_t∈ [- T, 1- T] is a two-sided Brownian meander, as discussed in <cit.>. The left side is a meander of length T and the right side is a meander of length 1-T. When we condition on T, these two meanders are independent of one another. Therefore, it suffices to show that for a Brownian meander X of length 1, the Radon-Nikodym derivative (X(t))_t∈ [0,α] is absolutely continuous with respect to a Bessel process on the same interval (R(t))_t∈ [0,α].

    In the proof of <cit.>, the authors observe that if α=1, then the Radon-Nikodym derivative is simply given by √(π/2)· R(1)^-1. Let ℱ^R_t denote the filtration of the Bessel process. Then for general α∈ [0,1] the Radon-Nikodym derivative of X with respect to R on the interval [0,α] is simply given by 
    √(π/2)·𝔼[ R(1)^-1 | ℱ^R_α] = √(π/2)· f_α(R(α)),

    where the expectation is taken with respect to a 3-dimensional Bessel Process and f_α:[0,∞)→ (0,∞) is the unique function satisfying 
    
    f_α(|x⃗|) = (2π(1-α))^-3/2∫_ℝ^3 e^-|y⃗|^2/2(1-α)|x⃗-y⃗|^-1dy⃗.

    In other words, x⃗↦ f_α(|x⃗|) is the radial function on ℝ^3 given by convolving the three-dimensional heat kernel with the harmonic function x⃗↦ |x⃗|^-1. The equality above follows from the Markov property and the fact that the Bessel process R can be realized as the Euclidean norm of a three-dimensional Brownian motion. A direct computation shows that f_α_L^∞(ℝ^3)≤ C(1-α)^-1/2 for a universal constant C. 



An Airy process of scale s>0 centered at y∈ℝ is the process 
    𝒜_y^s(x):= s^1/3𝒫_1(s^-2/3(x-y)),       x∈ℝ
 where 𝒫_1 is the  parabolic Airy_2 process; the top curve of the line ensemble in <ref>.



     Fix η∈ (0,1) and let 𝒜_x^t and 𝒜̃_̃ỹ^̃s̃ denote two independent Airy processes of scales t,s ∈ (η,1] centered at x,y∈ℝ respectively. We also define the notation 𝒱:=𝒜_x^t+𝒜̃_̃ỹ^̃s̃. Let I=[a,b] be a compact interval. Then (𝒱(t+a) - V(a))_t∈ [0,b-a] is absolutely continuous with respect to a Brownian motion of diffusion rate 4. Furthermore, for all p≥ 1, the Radon-Nikodym derivative has a finite p^th moment which is bounded above by Ce^Cp^2 (a^2+b^2) for some universal constant C=C(p,η,x,y)>0 not depending on p,a,b,s,t such that 0≤ b-a≤ 10. 


The choice b-a≤ 10 is arbitrary, but we will never use this bound for intervals larger than length 10.


    We reduce the claim to the case where a=-b. Since the Airy_2 process is stationary without the addition of the parabola, and since the difference of two quadratic functions is linear, translating the interval I to be centered at the origin amounts to adding a linear drift term of the form mt+b to the process 𝒱(t), for some m whose absolute value can be bounded above by C(|a|+|b|). 

    The Airy process on [-a,a] with |a|≤ 10 is absolutely continuous with respect to a standard Brownian motion on that interval, and moreover the Radon-Nikodym derivative is deterministically bounded due to <cit.>. 
    
    Brownian motion with a linear drift of slope m on [0,b-a] is absolutely continuous with respect to Brownian motion with no drift on that same interval, with Radon-Nikodym derivative given by e^± mB_b-a - m^2/2(b-a)^2. The p^th moment of this random variable is simply e^(p^2-p)m^2(b-a)^2. Combining this with the result of the previous paragraph finishes the proof.


    




(The crucial bound) 
Fix η>0 and recall Γ from (<ref>). Let 𝒜_x^t, 𝒜̃_̃ỹ^̃s̃, 𝒜̅_̅u̅^̅a̅,𝒜̂_̂v̂^̂b̂ denote four independent Airy processes of scale t,s,a,b respectively, centered at fixed real numbers x,y,u,v respectively. For any δ>0, there exists a constant C=C(η, x,y,u,v,δ)>0 independent of δ∈ (0,1/3), such that we have the following bound uniformly over ϵ∈ (0,1]: 
    sup_η≤ s,t,a,b≤ 1ℙ( Γ(𝒜_x^t+𝒜̃_̃ỹ^̃s̃, 𝒜̅_̅u̅^̅a̅+𝒜̂_̂v̂^̂b̂, ϵ^1/3-δ) ≤ϵ^2/3-δ) ≤ Cϵ^2/5-4δ.



We are not sure whether ϵ^2/5 is the optimal bound, however we are unable to improve it through the methods of this paper. The crucial input to proving that the directed landscape is a black noise is the fact that 2/5-δ is larger than 1/3 for sufficiently small δ. The bound in <ref> is sufficient because it will give us a bound almost as small as ϵ^2/3·ϵ^2/5 = ϵ^16/15 in (<ref>). This eventually implies a bound in (<ref>) with ϱ smaller than 1/15. 


In this proof, we leverage the cube-exponential tail decay of the argmaxes of the Airy processes, and combine this with the results of <ref> and <ref> to write everything in terms of Bessel processes, where we can apply the bound from <ref>. We make the following definitions:

    
  * Denote 𝒱_1:= 𝒜_x^t+𝒜̃_̃ỹ^̃s̃ and 𝒱_2:= 𝒜̅_̅u̅^̅a̅+𝒜̂_̂v̂^̂b̂ henceforth.
    
    
  * Let G_1 denote the argmax on ℝ of 𝒱_1 and let G_2 denote the argmax on ℝ of 𝒱_2. 

    
  * For k∈ℤ, let E^(1)_k denote the event that G_1∈ [k,k+5] and let E^(2)_k denote the event that G_2 ∈ [k,k+5]. Note that E^(i)_k are independent for distinct i. 

    
  * For k_1,k_2,ℓ∈ℤ let F_k_1,k_2,ℓ denote the event that G_1∈ [k_1,k_1+5] , G_2 ∈ [k_2,k_2+5], and furthermore the maximum values of both 𝒱_1,𝒱_2 on the interval [ℓ,ℓ+5] are within 1 of their absolute maximum values on all of ℝ.

    
  * For i=1,2 and k ∈ℤ let H^(i)_k denote the event that there exist two points of distance at least k from each other such that the values of 𝒱_i at both points are within 1 of max_z∈ℝ𝒱_i(z). Note that H^(i)_k are independent for distinct i, and 
    F_k_1,k_2,ℓ⊂ H^(1)_k_1-ℓ∩ H^(2)_k_2-ℓ.


    
  * For k∈ℤ and i=1,2, let G_i^(k) denote the argmax on [k,k+5] of 𝒱_i and M_k^(i) := 𝒱_i(G^(k)_i) denote the maximum value of 𝒱_i on [k,k+5].

    
  * For ℓ∈ℤ let K_ℓ^ϵ denote the event that there are points z_1∈ [4/5ℓ+1/5 G_1^(ℓ),4/5 (ℓ+5) +1/5 G_1^(ℓ)]⊂ [ℓ,ℓ+5] and z_2∈ [4/5ℓ+1/5 G_2^(ℓ),4/5 (ℓ+5) +1/5 G_2^(ℓ)]⊂ [ℓ,ℓ+5] such that |z_1-z_2| ≤ϵ^2/5, and moreover 𝒱_i(z_i)> M^(i)_ℓ -ϵ^1/3-δ for i=1,2.


We use a union bound for the probability that Γ(𝒜_x^t+𝒜̃_̃ỹ^̃s̃, 𝒜̅_̅u̅^̅a̅+𝒜̂_̂v̂^̂b̂, ϵ^1/3-δ) ≤ϵ^2/3-δ. The basis for this union bound is the following inclusion: 

    {Γ(𝒜_x^t+𝒜̃_̃ỹ^̃s̃, 𝒜̅_̅u̅^̅a̅+𝒜̂_̂v̂^̂b̂, ϵ^1/3-δ) ≤ϵ^2/3-δ}   ⊂⋃_k_1,k_2,ℓ∈ℤ E_k_1^(1)∩ E_k_2^(2)∩ F_k_1,k_2,ℓ∩ K^ϵ_ℓ
       ⊂⋃_k_1,k_2,ℓ∈ℤ E_k_1^(1)∩ E_k_2^(2)∩ H^(1)_k_1-ℓ∩ H^(2)_k_2-ℓ∩ K^ϵ_ℓ.

In this equation, we note that E_k_1^(1)∩ E_k_2^(2)∩ F_k_1,k_2,ℓ=F_k_1,k_2,ℓ. The purpose of writing the inclusion in this form will be clear in Step 4. The reason for this inclusion is that intervals of the form I_ℓ:= ⋂_i=1,2[4/5ℓ+1/5 G_i^(ℓ),4/5 (ℓ+4) +1/5 G_i^(ℓ)] cover all of ℝ as ℓ varies through all of ℤ, and successive intervals I_ℓ,I_ℓ+1 overlap in an interval of length at least 1 (this is because each of the two intervals comprising I_ℓ has length 4, and therefore I_ℓ itself has length at least 3). Consequently, if z_1,z_2∈ℝ such that |z_1-z_2|<ϵ^2/5≤ 1 then z_1,z_2∈ I_ℓ for some ℓ. The union bound yields an infinite sum over (k_1,k_2,ℓ) ∈ℤ^3, and then Hölder's inequality yields the desired bound. We break this argument into four steps, obtaining individual bounds on the events E_k_1^(1) , E_k_2^(2) , H^(1)_k_1-ℓ, H^(2)_k_2-ℓ , K^ϵ_ℓ.




Step 1. Using the fact that G_1,G_2 can be interpreted as points of geodesic locations in the directed landscape, we use <ref> to obtain ℙ( E^(i)_k) ≤ Ce^-d|k|^3 for some C,d>0 independent of k∈ℤ and i=1,2.   




Step 2. We claim that ℙ(H^(i)_k) ≤ Ce^-d|k|^3. This is almost immediate from <ref>, since the parabolic Airy_2 process embeds into the Airy sheet by setting (x,y)=(t,0). Either the maximum of 𝒱_i is smaller than 1-k (which has a probability less than e^-c|k|^3 since the maximum of each 𝒱_i is Tracy-Widom GUE distributed with a deterministic shift which depends on x,y, which we ignore), or else there exist two points of distance greater than k from each other such that the value of 𝒱_1 at both points is bigger than -k. In order for the latter event to occur, the constant ℭ in <ref> would have to be of order k^2, which has a probability less than Ce^-d|k|^3.




Step 3. To bound ℙ(K_ℓ^ϵ), we first note that the event K_ℓ^ϵ only depends on the behavior of the processes 𝒱_i restricted to the interval [ℓ,ℓ+5], which is the entire point of the union decomposition above. Using <ref>, and using Hölder's inequality to get rid of the Radon-Nikodym derivative, we obtain ℙ(K_ℓ^ϵ) ≤ Ce^Cq^2ℓ^2·ℙ_BM^⊗ 2 ( K_ℓ^ϵ)^1/p. The latter probability is taken with respect to two independent Brownian motions on [0,5]. Under these Brownian motions, the argmaxes G_1^(ℓ) and G_2^(ℓ) have arcsine laws, and we can verify that ℙ_BM^⊗ 2( |G_1^(ℓ) - G_2^(ℓ)|<ϵ^2/5) <C ϵ^2/5. Therefore, 
    ℙ_BM^⊗ 2(K_ℓ^ϵ∩{|G_1^(ℓ) - G_2^(ℓ)|< ϵ^2/5})^1/p < ϵ^p^-12/5.
 It only remains to estimate ℙ_BM^⊗ 2( K_ℓ^ϵ∩{|G_1^(ℓ) - G_2^(ℓ)|≥ϵ^2/5}). To do this, we apply the result of <ref> (with α =4/5, since the interval [4/5ℓ+1/5 G_i^(ℓ),4/5 (ℓ+5) +1/5 G_i^(ℓ)]⊂ [ℓ,ℓ+5] has length 4) to rewrite the probability in terms of Bessel processes, at the extra cost of a deterministic factor. Rescaling these Bessel processes by ϵ^-2/3 in time and by ϵ^1/3 in space puts us in the setting of <ref>, with γ = 2/3 - 2/5 = 4/15. We apply that proposition with γ̅=δ, where δ there is chosen the same as δ here. 
Applying <ref> yields a bound of ϵ^2/5-3δ, since ϵ^2/5 = (ϵ^4/15)^3/2 and the remaining negative exponents in that proposition can be bounded above by 3δ. In summary, 
    ℙ(K_ℓ^ϵ)≤ Ce^Cℓ^2ϵ^p^-1(2/5-3δ),
 
where C is uniform over ℓ,ϵ.




Step 4. Finally, we apply the union bound in (<ref>), then use Hölder's inequality with some other conjugate exponents p',q' (possibly different from the exponents p,q from earlier) and thanks to the results of Steps 1-3 we obtain 

    ℙ(Γ(𝒜_x^t+𝒜̃_̃ỹ^̃s̃, 𝒜̅_̅u̅^̅a̅+𝒜̂_̂v̂^̂b̂, ϵ^1/3-δ) ≤ϵ^2/3-δ)    ≤∑_k_1,k_2,ℓ∈ℤℙ(E_k_1^(1)∩ E_k_2^(2)∩ H^(1)_k_1-ℓ∩ H^(2)_k_2-ℓ∩ K^ϵ_ℓ) 
       ≤∑_k_1,k_2,ℓ∈ℤℙ(E_k_1^(1)∩ E_k_2^(2))^1/2q'ℙ(H^(1)_k_1-ℓ∩ H^(2)_k_2-ℓ)^1/2q'ℙ (K^ϵ_ℓ)^1/p'
       ≤ϵ^1/pp' (2/5-3δ)∑_k_1,k_2,ℓ∈ℤ e^-c/2q'(|k_1|^3 + |k_2|^3+|ℓ-k_1|^3 + |ℓ-k_2|^3)· e^Cq^2/p'ℓ^2.

For the term e^Cq^2/p'ℓ^2, we use the bound ℓ^2 = 1/2ℓ^2 +1/2ℓ^2 ≤ k_1^2 +(ℓ-k_1)^2 + k_2^2 + (ℓ-k_2)^2. We perform the sums in the expression above, first over ℓ and then over (k_1,k_2). The cube-exponential decay overwhelms the square-exponential growth terms in each case, which shows that the infinite sum converges to a finite value for any (fixed) choice of p,p'. We conclude the proof by taking p,p' very close to 1 (and thus q,q' large) so that 1/pp' (2/5-3δ) = 2/5-4δ.

















   Fix η∈ (0,1). Let 𝒜_x^t and 𝒜̃_̃ỹ^̃s̃ denote two independent Airy processes of scale t,s centered at x,y∈ℝ respectively, and denote 𝒱:=𝒜_x^t+𝒜̃_̃ỹ^̃s̃. As in <ref>, let N_ϵ(𝒱) denote the smallest number of intervals of length ϵ^2 needed to cover the random set 𝒮(𝒱,ϵ). Then uniformly over all scales s,t∈ [η,1], sup_ϵ∈ (0,1]𝔼[N_ϵ(𝒱)^p]<∞ for all p≥ 1.




Let N^k_ϵ(𝒱) denote the smallest number of intervals contained in [k,k+2] that are needed to cover the random set 𝒮(𝒱,ϵ)∩ [k,k+2]. We have a trivial bound N_ϵ(𝒱)≤∑_k∈ℤ N^k_ϵ(𝒱) and Minkowski's inequality implies 

    sup_ϵ∈ (0,1]𝔼[N_ϵ(𝒱)^p]^1/p≤∑_k∈ℤsup_ϵ∈ (0,1]𝔼[N^k_ϵ(𝒱)^p]^1/p.

Let E_k denote the event that N^k_ϵ(𝒱) is nonzero, that is, the event that 𝒮(𝒱,ϵ)∩ [k,k+2] is nonempty. Then either the maximum of 𝒱 on all ℝ is smaller than 1-k (which has a probability less than e^-d|k|^3 since we know that the maximum of 𝒱 is Tracy-Widom GUE distributed plus a deterministic shift depending only on x,y which we are ignoring) or else there exists a point in [k,k+2] such that the value of 𝒱 at this point is bigger than -k (since ϵ≤ 1 by assumption). But the constant ℭ in <ref> would have to be of order k^2 for the latter event to occur, which has a probability less than Ce^-d|k|^3. Thus ℙ(E_k) ≤ Ce^-d|k|^3 for some C,d>0.

We can bound N_ϵ^k(𝒱) by the number M_ϵ^k(𝒱) of intervals of size ϵ^2 contained in [k-1,k+3] needed to cover the set of points z∈ [k,k+2] such that 𝒱(z) is within ϵ of its maximum value on [k-1,k+3]. We apply <ref> and <ref> with α=3/4, and see that up to a deterministic factor of size Ce^Ck^2 and possibly enlarging p due to Hölder's inequality, the process 𝒱 can be replaced by a standard two-sided Bessel process. By this observation and <ref> we conclude that 𝔼[ M_ϵ^k(𝒱)^p]≤ Ce^Ck^2 for some absolute constants C,d independent of k,ϵ but possibly depending on p. We find that 
    𝔼[N^k_ϵ(𝒱)^p] =𝔼[N^k_ϵ(𝒱)^p1_E_k] ≤𝔼[M^k_ϵ(𝒱)^2p]^1/2pℙ(1_E_k)^1/2≤ Ce^-d|k|^3· Ce^Ck^2,
 where C,d are uniform over all k,ϵ. We sum over k∈ℤ and use (<ref>) to obtain the desired bound.



    Let 𝒮 be the Airy sheet, and for a function f:ℝ^2→ℝ let [f]_α,M :=sup_s⃗≠t⃗∈ [-M,M]^2 |t⃗-s⃗|^-α |f(t⃗)-f(s⃗)| denote the α-Hölder semi-norm of f on [-M,M]^2. For any α<1/2 and δ∈ (0,1/2), there exists C,d>0 such that uniformly over M>0 and u>0, 
    ℙ( [𝒮]_α,M >u) ≤ CM^10e^-dM^-δu^3/2.




    This follows from <cit.>, by taking τ = 0 and observing that log^1/2(bξ^-1) ≤ C(δ)b^δξ^-δ for arbitrary δ,ξ>0.


  We fix η>0 and recall the error terms 𝔈_ϵ(f,B) from <ref>. We consider two independent Airy processes 𝒜^s_x,𝒜^t_y of scales s,t ∈ [η,1] and centered at x,y∈ℝ respectively, and let 𝒱:= 𝒜^s_x+𝒜^t_y. We define the random variable 
    𝔅:= inf{ B>0 : -B-2η^-1 u^2 ≤𝒱(u) ≤ B-2η u^2,  for all u∈ℝ}.
 Fix p≥ 1. Then 𝔼[𝔅^p]<∞, and there exist C,d>0 depending only on η,p (and not on s,t,ϵ) such that 
    
    𝔼[ 𝔈_ϵ(𝒱,𝔅)^p] ≤ Ce^-dlog^6/5(1/ϵ).




    By <ref>, the random variable 𝔅 is almost surely finite, and in fact satisfies ℙ(𝔅>u) ≤ Ce^-du^3/2 for some C,d>0 that may depend on η and x_i,y_i (but not on s,t). We see that 𝔼[ 1_{𝔅>dϵ^-1/3}] ≤ Ce^-d ϵ^-1 where the constants C,d may have decreased or increased (respectively).

    It remains to bound the Hölder semi-norm term in the expression for 𝔈_ϵ. We set δ̃:=3/40δ, and note that the probability of 100 𝔅> log^4/5(1/ϵ) decays like Ce^-dlog^6/5(1/ϵ). Consequently, we just need to show that 
    
    ℙ( ℌ𝔬𝔩(𝒱, 1/2(1-δ̃), 1+ 2log^4/5(1/ϵ) )>ϵ^-δ̃/3) ≤ Ce^-dlog^6/5(1/ϵ).

    We use <ref> to see that the left side is bounded above by Clog^40/5(1/ϵ) · e^-d log^-4δ/5(1/ϵ)·ϵ^-δ̃/2. This is far smaller than the required bound. 


Now we are finally in a position to prove (<ref>).


    The estimate (<ref>) holds for any ϱ<1/15, consequently the directed landscape is a black noise.


 Using the metric composition property of the directed landscape, we can write the conditional expectation inside the variance of (<ref>) for 0≤ a<b≤ 1 as 

    𝔼[ ∏_i=1^n ℒ_0,1(x_j,y_j) | ℱ^ℒ_a,b]     = 𝔼[ ∏_i=1^n f_i^a ∘ℒ_a,b∘ g_i^b|ℱ_a,b^ℒ] 
    
        = ∫_C(ℝ)^n∫_C(ℝ)^n∏_i=1^n (u_i ∘ℒ_a,b∘ v_i) ν_a(du⃗)μ_b(dv⃗),
 where f_i^t:= f_i∘_ℓℒ_0,t and g_i^t:= ℒ_t,1∘_r g_i, where the left and right metric compositions are defined as 

    (f ∘_ℓ L)(y):= sup_x∈ℝ f(x)+L(x,y),        (L∘_r g)(x) = sup_y∈ℝ L(x,y)+g(y),

and where ν_a and μ_b are respectively the laws of (f_1^a,...,f_n^a) and (g^b_1,...,g^b_n). Fix η,δ∈(0,1/2). From the last expression we deduce that to prove the theorem, it suffices to show that 
    ∫_(C(ℝ)^n)^4Cov( ∏_i=1^n (f_i ∘ℒ_0,b-a∘ g_i), ∏_j=1^n (u_j ∘ℒ_0,b-a∘ v_j))ν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗) ≤ C(b-a)^16/15-δ.

where η≤ a<b≤ 1-η and C is allowed to depend upon η,δ∈ (0,1/2) and on x_j,y_j but not on a,b. 

In this proof, we abbreviate [n]:={1,...,n}. By taking the Taylor expansion of the function (x_1,...,x_n)↦ x_1⋯ x_n around some fixed value (a_1,...,a_n), we rewrite it as ∑_S⊂ [n] a_S (x-a)_S^c, where x_S=x_i_1⋯ x_i_r if S={i_1,...,i_r}. We apply this fact with x_i:=f_i∘ℒ_0,ϵ∘ g_i, and a_i:=max(f_i+g_i), which allows us to rewrite the products in (<ref>) as linear combinations of quantities of the form max(f+g)_S· (f∘ℒ_0,ϵ∘ g - max(f+g))_S^c, whereas before S is a subset of [n] and the subscript denotes the product of the respective quantity over all indices which lie in S. Consequently, to prove (<ref>) it suffices to show that for all subsets S,T⊂[n], 
    ∫_(C(ℝ)^n)^4Cov( (f∘ℒ_0,ϵ∘ g     - max(f+g))_S^c, (u∘ℒ_0,ϵ∘ v - max(u+v))_T^c) 
       ·max(f+g)_Smax(u+v)_T ν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗) ≤ Cϵ^16/15-δ.

Under ν_a^⊗ 2⊗μ_b^⊗ 2 the f_i,g_i,u_i,v_i are all distributed as independent parabolic Airy_2 processes of different scales (these scales are always in [η,1-η]), perhaps recentered at some values depending on x_i,y_i (which are fixed). Consequently, the individual maxima of f_i+g_i and u_i+v_i each have Tracy-Widom GUE laws (perhaps rescaled and recentered). Therefore, 
 
    sup_η≤ a<b≤ 1-δmax_S,T⊂[n]∫_(C(ℝ)^n)^4|max(f+g)_Smax(u+v)_T |^qν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗)<∞.


By Hölder's inequality (choosing p very close to 1 and thus q large), to prove (<ref>) it suffices to show that for all S,T⊂[n] one has 
 
    [∫_(C(ℝ)^n)^4|Cov( (f∘ℒ_0,ϵ   ∘ g - max(f+g))_S^c, 
       (u∘ℒ_0,ϵ∘ v - max(u+v))_T^c)|^pν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗)]^1/p≤ Cϵ^16/15-δ.

Let #S denote the cardinality of the set S. If # (S^c) + #(T^c) ≥ 4, then we may apply <ref> to obtain a bound of size ϵ^4/3-δ plus some number of “error terms" which by <ref> have a super-polynomial rate of decay as ϵ→ 0. This achieves the goal in (<ref>). 

By a symmetry argument, what remains is to prove (<ref>) in the case when (#(S^c),#(T^c)) = (1,1) or (#(S^c),#(T^c)) = (1,2).
We consider the case when (#(S^c),#(T^c)) = (1,1). In this case, the covariance in the integrand reduces to (f_1∘ℒ_0,ϵ∘ g_1, u_1∘ℒ_0,ϵ∘ v_1). We apply <ref> and <ref> (with k+ℓ=2) and we obtain (modulo some “error terms" which by <ref> have a super-polynomial rate of decay as ϵ→ 0) an upper bound given by  
    [∫_(C(ℝ)^n)^4   𝔅^2δmin{ϵ ^2/3 - δ, 
       (N_ϵ^1/3-δ(f_1+g_1) + N_ϵ^1/3-δ(u_1+v_1))· e^-c ϵ^-2Γ(f_1+g_1,u_1+v_1;ϵ^1/3-δ)^3}ν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗)]^1/p .

By <ref> and another application of Hölder's inequality with p very close to 1, we can ignore the factor of 𝔅^2 δ, since 𝔼[𝔅^2δ q]<∞ for the conjugate exponent q. We use the elementary bound min{a,b}≤ a^θ b^1-θ for θ∈ [0,1], and see that the last expression is bounded above by

    [∫_(C(ℝ)^n)^4ϵ^(1-θ)(2/3 - δ)    ·(N_ϵ^1/3-δ(f_1+g_1)+N_ϵ^1/3-δ(u_1+v_1))^θ
       · e^-c θϵ^-2Γ(f_1+g_1,u_1+v_1;ϵ^1/3-δ)^3ν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗)]^1/p 
         ,

for all θ∈ [0,1]. We split the integral into two parts according to Γ(f_1+g_1,u_1+v_1,ϵ^1/3-δ)<ϵ^2/3-δ or Γ(u_1+v_1,f_1+g_1,ϵ^1/3-δ)≥ϵ^2/3-δ. 
On the latter event we obtain an exponential rate of decay of order e^-cϵ^-3δ, which decays to 0 super-polynomially fast as ϵ→ 0. In the case that Γ(f_1+g_1,u_1+v_1,ϵ^1/3-δ)<ϵ^2/3-δ, we apply <ref> and Hölder's inequality (with p close to 1 and q large) to argue that we can disregard the term (N_ϵ^1/3-δ(f_1+g_1)+N_ϵ^1/3-δ(u_1+v_1))^θ at the cost of making the value of p slightly larger. Then we apply <ref> to see that the integral in (<ref>) will decay as ϵ^(1-θ)(2/3 - δ) ·ϵ^2/5-4δ. In this setting, we can make the exponent on ϵ as close to 16/15 as desired by taking θ,δ close to 0 and p close to 1.

Next, we consider the case when (#(S^c),#(T^c)) = (1,2). The argument in this case is very similar, we apply <ref> instead of <ref>, and we still apply <ref>, in this case with k+ℓ=3. In (<ref>) this gives (modulo the error terms) a bound of size 
    [∫_(C(ℝ)^n)^4ϵ^(1-θ)(1 - δ) ·(N_ϵ^1/3-δ(f_1+g_1)+N_ϵ^1/3-δ(u_1   +v_1)+ N_ϵ^1/3-δ(u_2+ v_2))^θ
       · e^-c θϵ^-2Γ(ϵ)^3ν_a^⊗ 2(df⃗,du⃗)μ_b^⊗ 2(dg⃗,dv⃗)]^1/p,

where Γ(ϵ):=min{Γ(f_1+g_1,u_1+v_1;ϵ^1/3-δ),Γ(f_1+g_1,u_2+v_2;ϵ^1/3-δ)}. 
We split the integral into two parts according to Γ(ϵ)<ϵ^2/3-δ or Γ(ϵ)≥ϵ^2/3-δ. The latter of these terms yields a super-polynomially fast rate of decay to zero in (<ref>), whereas the former can be bounded by <ref> and <ref>, noting from <ref> and a union bound that 
    ℙ(Γ(ϵ)<ϵ^2/3-δ) ≤ℙ(Γ(f_1+g_1,u_1+v_1;ϵ^1/3-δ)<ϵ^2/3-δ) +ℙ(Γ(f_1+g_1,u_2+v_2;ϵ^1/3-δ) <ϵ^2/3-δ) ≤ Cϵ^2/5-4δ.
 This will yield an expression at most ϵ^(1-θ)(1 - δ) ·ϵ^2/5-4δ in (<ref>). By choosing δ,θ close to 0, the exponent on ϵ can be made as close to 7/5 as desired by taking θ close to 0, which exceeds the desired threshold of ϵ^16/15-δ. 




§ INDEPENDENCE FROM GAUSSIAN WHITE NOISE
 

In this section, we prove <ref>. 


Assume that we have a probability space equipped with a noise, (Ω, (ℱ_s,t)_s<t , P, θ), and that ℒ,ξ are (respectively) a directed landscape and a standard white noise which are adapted to (and together generate) the filtration ℱ_s,t, 
    ℱ_s,t := ℱ^ξ_s,t∨ℱ^ℒ_s,t,
 where ℱ_s,t^ξ is the filtration generated by the white noise and ℱ_s,t^ℒ is the filtration generated by the directed landscape. Our goal is to show that ℱ^ξ,ℱ^ℒ are independent of each other. We will break the proof down into five steps. 




Step 1. For h∈ L^2(ℝ^2), we define the pairing (h,ξ) := ∫_ℝ^2 h(t,x)ξ(dt,dx). We claim that these are linear random variables on this probability space, and that we have E[(h,ξ)|ℱ_s,t] = (h1_[s,t]×ℝ,ξ). This would be obvious if ℱ^ξ_s,t was the full σ-algebra, but needs some verification in the present context, where ℱ_s,t may contain strictly more information than ℱ^ξ_s,t.

Note that the restriction ξ|_[s,t] is ℱ_s,t-measurable by assumption, therefore the definition of a noise (<ref>) implies that it is independent of ℱ_t,u and ℱ_a,s for any a<s<t<u. From this fact, we can prove the claim by writing the almost sure identity 
    (h,ξ) = (h1_(-∞,s]×ℝ,ξ) + (h1_[s,t]×ℝ,ξ) +(h1_[t,∞)×ℝ],ξ) ,
 and then noting that the three terms on the right are respectively ℱ_-∞,s-measurable, ℱ_s,t-measurable, and ℱ_t,∞-measurable. By taking the conditional expectation given ℱ_s,t, and using independence of these three σ-algebras, the first and third term vanish while the second term is unaffected by the conditional expectation.




Step 2. We argue that (h,ξ)^ℒ:=E[(h,ξ)|ℱ^ℒ] is a linear random variable on the restricted noise (Ω, (ℱ_s,t^ℒ)_s<t , P, θ) for any h∈ L^2(ℝ^2). The black noise property of ℒ proved in <ref> will then imply that E[(h,ξ)|ℱ^ℒ]=0 for any h∈ L^2(ℝ^2). 

To prove this, it suffices to show that E[(h,ξ)^ℒ |ℱ^ℒ_s,t]= (h1_[s,t]×ℝ,ξ)^ℒ. This is because h↦ (h,ξ)^ℒ is clearly a linear map from L^2(ℝ^2) → L^2(Ω, ℱ^ℒ,P). By the tower property of conditional expectation, for s<t we have

    𝔼[ 𝔼[(h,ξ)|ℱ^ℒ] | ℱ^ℒ_s,t]    = 𝔼[(h,ξ) | ℱ^ℒ_s,t] 
       = 𝔼[ 𝔼[(h,ξ)|ℱ_s,t] | ℱ^ℒ_s,t] 
       = 𝔼[ 𝔼[ (h1_[s,t]×ℝ ,ξ) ]|ℱ^ℒ_s,t] = 𝔼[ 𝔼[(h1_[s,t]×ℝ,ξ)|ℱ^ℒ] | ℱ^ℒ_s,t].

We used the result of Step 1 in the third equality. This proves the desired claim.




Step 3. Let h_s_j,t_j∈ L^2(ℝ^2) be supported on (s_j,t_j) ×ℝ, where (s_j,t_j) are disjoint intervals. Using Step 2 and the noise property, 
    𝔼[ ∏_j=1^n F_s_j,t_j(ℒ) · (h_s_j,t_j,ξ) ] = ∏_j=1^n 𝔼[ F_s_j,t_j(ℒ)· (h_s_j,t_j,ξ)] = 0,
 (we use the result of Step 2 in the last equality). The F_s_j,t_j:C(ℝ^4_↑) →ℝ are arbitrary bounded functionals which are ℱ_s_j,t_j^ℒ (1≤ j≤ n)-measurable. 




Step 4. As before, let h_s_j,t_j∈ L^2(ℝ^2) be supported on (s_j,t_j) ×ℝ, where (s_j,t_j) are disjoint intervals. Using standard stochastic calculus tools for the Brownian motion, we will show that linear combinations of random variables of the form c+∏_j=1^n(h_s_j,t_j,ξ) are actually dense in L^2(Ω,ℱ^ξ,P), as one varies over all c∈ℝ, n∈ℕ and disjoint intervals (s_j,t_j).

There are many ways to prove the desired claim, but we will take the following approach. First choose an orthonormal basis {e_j}_j≥ 1 of L^2(ℝ) and define the independent, identically distributed Brownian motions B^j(t):= (1_[0,t]⊗ e_j, ξ). Note that ℱ^ξ_s,t is precisely the σ-algebra generated by (B^j(u)-B^j(s))_u∈ [s,t],j∈ℕ. Fix some natural numbers ℓ and p_1<...<p_ℓ. Let t^n_j = j2^-n, and notice that by standard construction of stochastic integrals we have (as n→∞) the convergence

    ∑_σ∈ S_p_ℓ∑_s2^n ≤ k_1<k_2<...<k_p_ℓ≤ t2^n
    k_i∈ℤ∏_j=1^ℓ   ∏_r=p_j-1^p_j-1( B^j(t^n_k_σ(r)+1)-B^j(t^n_k_σ(r))  ) 
    L^2(Ω)⟶   ∑_σ∈ S_p_ℓ∫_s ≤ s_1<s_2<...<s_p_ℓ≤ t∏_j=1^ℓ∏_r=p_j-1^p_j-1 dB^j(s_k_σ(r)+1) 
        =∏_j=1^ℓ H_p_j-p_j-1((t-s)^-1/2(B^j(t)-B^j(s))),
 
where H_p(x) is the p^th standard Hermite polynomial and S_k is the symmetric group of permutations of {1,...,k}. But for fixed s<t, the random variables of the form of the right side form an orthonormal set whose closed linear span in L^2(Ω) (as one varies over natural numbers ℓ and 0= p_0<...<p_ℓ,) includes all random variables of the form φ(B^1(t)-B^1(s),..., B^n(t)-B^n(s)) where n∈ℕ and φ:ℝ^n→ℝ is bounded measurable, 
by e.g. the standard theory of Gaussian chaos <cit.>. Consequently, the linear span E in L^2(Ω) of random variables of the form c+∏_j=1^n(h_s_j,t_j,ξ) contains any random variable of the form ∏_j=1^m φ_j(B^1(t_j)-B^1(s_j),..., B^n(t_j)-B^n(s_j)), where m,n∈ℕ and φ_j:ℝ^n→ℝ are bounded measurable and (s_j,t_j) are disjoint intervals. Therefore, by the tensorization property of L^2 spaces (explained in the proof of <ref>) the linear span E contains all random variables of the form Ψ( (B^i(t_j)-B^i(s_j))_1≤ j≤ m, 1≤ i≤ n), where m,n∈ℕ and Ψ:ℝ^m× n→ℝ is bounded measurable and (s_j,t_j) are disjoint intervals with dyadic endpoints. This proves the claim, as random variables of the latter form can be used to approximate arbitrary functionals G(ξ), by letting m,n↑∞ and using e.g. the result of <ref>.
 





Step 5. Using the results of Steps 3 and 4, we apply a density argument that E[F(ℒ)G(ξ)] = E[F(ℒ)]E[G(ξ)] for all bounded measurable F,G on the respective canonical spaces, completing the proof.

By the result of Step 4, it suffices to prove the claim when G(ξ) = c+∏_j=1^r(h_s_j,t_j,ξ), where c∈ℝ and (s_j,t_j) are disjoint intervals with endpoints taking values in {k2^-n:k∈ℤ} for some n∈ℕ. 

In turn, it suffices to prove the claim when F is a finite linear combination of functionals of the form ∏_j=0^r+1 F_s_j,t_j(ℒ) where F_s_j,t_j:C(ℝ^4_↑) are ℱ_s_j,t_j^ℒ-measurable and the extreme endpoints are defined by s_0:=-∞, t_0:=s_1, s_r+1:=t_n, t_r+1:=+∞. 
Indeed, the finite linear span of such functionals is dense in L^2(Ω, ℱ^ℒ,P) by the tensorization property of L^2 (see proof of <ref>). 
But this is immediate from the result of Step 3.



alpha


