Quenched Central Limit Theorem in a Corner Growth Setting

We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths.


Introduction and main results
A number of well-known probabilistic models derive their underlying complexity from a variant of the following simple setup.Put independent and identically distributed weights at each vertex of the two-dimensional integer lattice.Given a lattice point in the first quadrant, consider all paths in the lattice from the origin to this point that only move up or to the right at each step.Each such path has a random energy given by the sum of the weights along the path, and so the collection of random energies indexed by the up-right paths exhibits a complicated dependence structure.This dependence is at the heart of the difficulty in understanding such models as the totally asymmetric simple exclusion process (TASEP), the infinite tandem queue, the random directed polymer, or the corner growth model.
For example in the corner growth model, or directed nearest neighbor last-passage percolation on the 2d lattice, the fundamental issue is to understand the distribution of the maximum-energy path to a given point.This maximal energy represents the last passage time to the point, or time at which it joins the growing corner shape.In directed polymer models one assigns a Boltzmann weight to each path according to its random energy, and one is concerned with the sum of all polymer weights, or partition function, which normalizes the polymer weights into probabilities.(See e.g.[Cor12] for a description of the model as well as its relationship to equivalent models).Both are difficult models because of the high degree of dependence in the joint distribution of the collection of path energies.A more detailed discussion of these models as they relate to our result appears further below.
In this paper we derive a result about the joint distribution of the path energies which to our knowledge has not been observed previously.Namely, we show that conditional on the environment of weights, the empirical distribution of the family of path energies is approximately Gaussian.More precisely, for almost every environment, the energy of an up-right path selected uniformly at random is asymptotically normally distributed as the mesh gets small.This is the content of our main theorem, which is proved for generally distributed weights with nonzero variance and under further moment assumptions (in fact, in our setting the weights need not even be identically distributed).We now introduce some notation, describe our results, and then discuss connections to the corner growth and directed random polymer models.
1.1.Up-right paths in a random environment.We work inside the positive quadrant Z 2

≥1
of the two-dimensional integer lattice.By convention, coordinates (i, j) ∈ Z 2 ≥1 refer to squares, see Figure 1.A (fixed) environment w is an assignment of a real number w ij ∈ R to every square of Z 2 ≥1 .We call the w ij weights.Fix integers M, N ≥ 1 and consider the rectangle 1 ≤ i ≤ M , 1 ≤ j ≤ N inside the quadrant.Denote it by M,N .The environment w restricted to M,N is a vector in R M N .
An up-right path σ from (1, 1) to (M, N ) is a collection To each up-right path σ we associate a vector Here and below 1 A is the indicator of A.
For any up-right path σ = {(i k , j k )} define its energy with respect to an environment w as where •, • is the standard inner product in R M N .Now suppose both the environment and the path are chosen at random, independently of each other.Assume the random environment w consists of independent random variables w ij defined on a probability space (Ω w , F w , P w ) with for some p (to be specified later) and K > 0.Here E w is expectation with respect to P w .Assume that σ is a random up-right path chosen (according to some distribution) from all paths inside a given subset Σ M,N ⊆ M,N .A simple example is when Σ M,N = M,N and σ is chosen uniformly from all M +N −2 M −1 possible paths.In any case, σ is a random path defined on some (Ω σ , F σ , P σ ), and expectation with respect to P σ will be denoted E σ .
We thus think of the energy of a path Y σ , w in (1.1) as a random variable defined on (Ω w × Ω σ , F w × F σ , P w × P σ ) which depends on both the randomness in the environment and in the path (the path is independent from the environment).The goal of this paper is to show that for certain natural distributions of σ and P w -almost every environment, the (quenched) random variable Y σ , w depending on the random path σ is asymptotically Gaussian as M, N → ∞.A precise formulation is given next.1.2.Quenched central limit theorems.Let the environment w = {w ij } consist of independent random variables satisfying (1.2).Let M = ⌊ξN ⌋, where ξ > 0 is fixed.
Theorem 1.1.If p > 12 and σ is chosen uniformly from all up-right paths in the rectangle M,N , then P w -almost surely, where the convergence in distribution to the standard normal is with respect to the marginal P σ .
In other words, for large N and almost every environment w, the empirical distribution of the family of path energies will be approximately Gaussian.The assumption p > 12 may seem unexpected.As will be seen in the proofs, it arises from a combination of the moment assumptions needed for our concentration inequality and our path counting estimates.
Corollary 1.2.The statement of Theorem 1.1 remains valid (still with p > 12) when σ is chosen uniformly from all up-right paths passing through each of the points 2, (a).
Theorem 1.1 and Corollary 1.2 are proven in Section 3.1.To illustrate that our approach can yield similar results for other path families, as long as suitable path counting arguments are available, we will also prove the analogous result for the family of paths avoiding a hole of fixed proportion in the center of M,N .For simplicity in Theorem 1.3 below we assume that M = N (though a suitably modified statement can be established for M = ⌊ξN ⌋ as well).Let B = ⌊βN ⌋, where β ∈ (0, 1) is fixed, and define the subset Σ N,N = {(i, j) ∈ N,N : max (|i − N/2|, |j − N/2|) ≥ B/2}; see Figure 2, (b).
Theorem 1.3.If p > 12, and σ be chosen uniformly from the set of up-right paths that remain in Σ N,N , then P w -almost surely, This theorem is proven in Section 3.2.
Remark 1.4.In (1.2) we assumed that the environment random variables w ij have mean 0 and variance 1.By shifting and scaling the energies Y σ , w one readily sees that all our results formulated above can be extended to independent environments with arbitrary constant mean and nonzero variance.1.3.Relation to other models.We now briefly compare our setting with other models based on directed up-right paths in a random environment.Consider first the directed polymer models introduced in statistical physics in [HH85]; see also e.g.[IS88], [Sep12].In the lattice setting, the directed polymer partition function is defined as (we continue to assume that M = N ) where the outer sum is taken over all up-right paths σ inside N,N , and β > 0 is the inverse temperature.The polymer weight of a path σ is defined as The study of the asymptotic behavior of Z N and Q N as N → ∞ has received a lot of attention in the past 30 years.Of particular interest are the asymptotic fluctuations of the free energy log Z N (w).These fluctuations are expected to grow as N 1/3 under mild assumptions.However, this scaling behavior is currently known only for a number of integrable cases (that is for special choices of the distribution of w ij leading to exact formulae for the Laplace transform of Z N ); see [OO15], [BC14], [BCF14].In integrable cases, the fluctuations themselves are governed by one of the Tracy-Widom distributions [TW94], which is characteristic for the Kardar-Parisi-Zhang universality [Cor12].Study of the asymptotic fluctuations of log Z N (w) when the integrability is not known presents a major open problem in the field.
Passing to the zero temperature limit β → ∞ turns the free energy log Z N (w) into the last passage percolation time: where the maximum is taken over all up-right paths inside N,N .Assume that the environment variables w ij are nonnegative.This does not significantly restrict generality since if the distribution of w ij is bounded from below, one can achieve nonnegativity by adding a fixed constant to all the w ij .We can then interpret the nonnegative w ij as random waiting times in the corner growth model so that (1.5) becomes the time at which the growing interface covers (N, N ).For further details on corner growth we refer to [Joh00], [Sep09], [Cor12], [BDS16], [Sep17].Asymptotic fluctuations of G N (w) in integrable cases (when the w ij 's have exponential or geometric distribution) have been shown to converge on scale N 1/3 to the Tracy-Widom distribution [Joh00].Again, the problem of asymptotic fluctuations in the corner growth model with other distributions of w ij (for which exact formulae are not known to exist) is open.
Our results (in particular, (1.3)) mean that that the path energies Y σ , w asymptotically behave as (2N − 1)Ew ij + ζ √ 2N − 1 (with ζ Gaussian).Let us compare this with the order of G N (w) known exactly for special distributions of w ij from, e.g., [Joh00].(A similar comparison may be performed in the polymer case, but we omit it.)When w ij are geometric, P( 1−q , which is smaller (however, the difference goes to zero as q ց 0).Similarly, for w ij exponential with mean 1, the last passage time behaves as EG N (w) ∼ 4N , and typical values of Y σ , w are of order (2N − 1)Ew ij ∼ 2N .
Therefore, while our results indicate that the asymptotic quenched behavior of the typical values of the path energies Y σ , w is universal (i.e., does not depend on the distribution of the w ij under mild assumptions), this conclusion does not extend to extreme values of Y σ , w responsible for the asymptotics of the last passage time G N (w).
In Section 2 we employ Talagrand's concentration inequality to establish a quenched central limit theorem (Theorem 2.2) modulo an estimate on the distribution of the up-right path σ.
In Section 3 we obtain the needed combinatorial estimates for natural ensembles of up-right paths described in Section 1.2 above, and complete the proofs of Theorem 1.1, Theorem 1.3, and Corollary 1.2.
Acknowledgements.We appreciate helpful discussions with Alexander Drewitz, Elizaveta Rebrova, Timo Seppäläinen, and Roman Vershynin.Part of this work was completed when MM and LP participated at a workshop "Analytic Tools in Probability and Applications" at IMA in 2015, and we are grateful to the Institute and the workshop's organizers for hospitality and support.MM is partially supported by grant #315593 from the Simons Foundation.LP is partially supported by the NSF grant DMS-1664617.

Gaussian concentration and quenched Central Limit Theorems
In this section we focus on general concentration estimates, and establish our quenched CLT modulo combinatorial estimates which are postponed until Section 3.
2.1.General concentration lemma.Let us work in a more general setting.Suppose Σ is a set with n elements equipped with independent weights {w a | a ∈ Σ} satisfying conditions (1.2) for p ≥ 3.For a fixed R > 0 we define the truncations (2.1) Let σ be a random subset of Σ having almost surely m elements.As before, P w and P σ stand for the marginal probability measures corresponding to {w a } and σ, respectively, and similarly for expectations.Let Y σ a := 1 a∈σ , and let Y σ denote the corresponding random vector in R n .Define Next, let γ ∼ N(0, 1) denote the standard Gaussian measure on R, and let µ w and µ (R) w be the quenched distributions (conditioned on w) of the random variables (2.3) respectively.In particular, for a test function f : R → R we have Lemma 2.1.Under the above assumptions, there exist absolute constants C, c > 0 (not depending on parameters of the model) and a constant κ > 0 depending only on K and p in (1.2) such that for any s, t, R > 0 and any convex 1-Lipschitz1 function f : R → R we have Proof.We have The terms (2.5) and (2.7) on the right will be estimated by elementary methods, (2.6) via Talagrand's concentration inequality for independent bounded random variables, and (2.8) via an appropriate version of the central limit theorem.Note that the terms (2.7) and (2.8) are deterministic.
We start with the last term (2.8).As a consequence of results of Esseen [Ess58], we have a bound on Wasserstein-1 distance that is independent of the path σ; see for example [Gol07, Proposition 2.2] for a statement of this bound that depends only on finite third moments (as we assume here).Note also that the L 1 -distance of distribution functions as used in [Gol07] is equivalent to the Wassertein-1 distance in our setting since we use 1-Lipschitz test functions.For all σ (recall that σ has m elements a.s.) we have where κ > 0 depends only on the third absolute moment of w a and can therefore be bounded in terms of the constants p and K from (1.2).Since (2.9) holds for all σ, an application of Fubini's theorem implies that (2.10) (In particular, this implies that the expectation E w f dµ w in (2.8) is finite.)Let us turn to (2.6).Consider the function F : R Σ → R defined by Since f is convex, F is an average of convex functions on R Σ , and so is also convex.Let us now estimate the Lipschitz constant of F .We have for w, w ′ ∈ R Σ : (2.11) Here the third line follows because Y σ has nonnegative components, |w − w ′ | denotes componentwise absolute value, and the last line follows from the Cauchy-Schwarz inequality.Thus, we see that F has Lipschitz constant at most L/ √ m.
Using this and applying Talagrand's inequality for bounded independent random variables [Tal95] w we obtain for any t > 0 and for some absolute constants C, c > 0, (2.12) It remains to consider terms (2.5) and (2.7).From the Lipschitz estimate (2.11) we have .
2.2.Quenched central limit theorem.Our aim now is to apply the general Lemma 2.1 to obtain the quenched central limit theorem in the corner growth setting described in Section 1.
Recall that we choose N (the vertical dimension of the rectangle in Figure 1) to be the main parameter going to infinity, and we let M = ⌊ξN ⌋ for some ξ > 0. The parameters in Lemma 2.1 are instantiated as follows: Here we assume that for each N , Σ M,N ⊆ M,N is a given subset and that σ is chosen according to some distribution such that σ ∈ Σ M,N almost surely (it replaces the set Σ in Lemma 2.1).We further assume that as N → ∞, for some 0 < η ≤ 2 and 0 < λ ≤ η/2.3For the specific subsets Σ M,N considered in this paper, the parameter L(N ) will be estimated separately in Section 3 below.Note that the constants C, c, K, p, κ in Lemma 2.1 are independent of N .
Theorem 2.2.Under the above assumptions, if λ < 1 2 and p > 6 1−2λ then P w -almost surely, 1 where the convergence in distribution to the standard normal is with respect to the marginal P σ .
Proof.To get the desired P w -almost sure convergence in distribution, we will choose the parameters R, s, t in Lemma 2.1 depending on N and apply the Borel-Cantelli lemma. 4That is, from the left side of (2.4) we see that we must have lim (2.17)Moreover, to use Borel-Cantelli the right side of (2.4) must be summable, which is equivalent to for some absolute constant N 0 .Let R(N ) ∼ N ρ for some ρ > 0. Then for (2.17) and (2.18) to hold under our assumption (2.16) it is necessary and sufficient that The latter inequality holds if p > 6 1−2λ , which completes the proof.

Path counting
Our goal in this section is to obtain estimates of L(N ) of the form (2.16) with λ < 1 2 for the concrete families of up-right paths described in Section 1.2.These estimates lead to quenched central limit theorems.Similar estimates have appeared in the context of random polymers before, for example, see [AKQ14, Appendix A]. ) and so λ = 1 2 , which is not quite good enough for our purpose.We will show however that one can in fact take λ = 1 4 using a better estimate of M k .Recall that M = ⌊ξN ⌋, ξ > 0.
Lemma 3.2.Let σ be chosen uniformly from all possible paths in M,N .There exists C > 0 such that for all N large enough, M +N k=2 M k ≤ C √ N .
Proof.Fix k = 2, . . ., M + N and a = (i, j) ∈ (k) M,N , that is, i + j = k.Then which is the hypergeometric distribution.Indeed, the numerator counts pairs of paths from (1, 1) to (i, j) and from (i, j) to (M, N ), and the denominator counts all possible paths.Let us denote the probability (3.3) by p

Figure 2 .
Figure 2. Subsets of the rectangle from which the up-right path σ is chosen uniformly in (a) Corollary 1.2, (b) Theorem 1.3.

3. 1 .
All possible paths -proofs of Theorem 1.1 and Corollary 1.2.We start with the case when σ is chosen uniformly at random from the set of all possible up-right paths in the rectangle M,N , so Σ M,N = M,N .Let us slice the rectangle as follows::= {a = (i, j) : 1 ≤ i ≤ M, 1 ≤ j ≤ N, i + j = k} .denoted M k := max a∈ (k) M,N P σ (a ∈ σ).Remark 3.1.The estimate (3.2) holds for any distribution of the up-right path σ.Moreover, since the maximum probability over (k) M,N can always be bounded by 1, we have the trivial estimate L(N ) 2 ≤ M + N − 1.In our regime (M proportional to N ) this estimate leads to L(N ) = O(N 1 2 (k) i .By looking at ratios p (k) i /p (k) i+1 and comparing this to 1 one can .19) and that s(N ) and t(N ) tend to zero sufficiently slowly as negative powers of N .Setting ρ = 1 2 −λ−ε for small enough ε > 0, one can check that condition (2.19) (together with our assumptions η ≤ 2 and λ ≤ η/2 coming from (2.2) and (2.15)) is equivalent to 0