Quenched invariance principle for random walk in time-dependent balanced random environment

Deuschel, Jean–Dominique; Guo, Xiaoqin; Ramı́rez, Alejandro F.

doi:10.1214/16-aihp807

Cited by 22 publications

(30 citation statements)

References 25 publications

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“…typically they tend to stick together. This, and other properties, such as large deviations, have been studied recently in mathematics [5][6][7][8]. On the other hand, there has been much work of the problem of directed polymers (DP), i.e.…”

Section: A Overviewmentioning

confidence: 99%

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Thiery¹,

Doussal²

2016

J. Phys. A: Math. Theor.

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We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin (BC) in [1]. We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time t is performed both (i) in a diffusive vicinity, x ∼ t 1/2 , of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, x ∼ t, of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as t 1/3 and to be distributed according to the Tracy-Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as x ∼ t 3/4 . Our results are confronted to extensive numerical simulations of the model. arXiv:1605.07538v1 [cond-mat.dis-nn]

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Section: A Overviewmentioning

confidence: 99%

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Thiery¹,

Doussal²

2016

J. Phys. A: Math. Theor.

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“…With those assumptions, our model falls into the class of balanced dynamic random environments. For this class of models an invariance principle was proved in [9]. In this paper we give an entirely different proof of the invariance principle for this particular model.…”

Section: Introductionmentioning

confidence: 90%

“…Since random walks in balanced environments are martingales, the key to proving an invariance principle is in proving that the quadratic variation grows linearly. In all previous proofs of invariance principles for random walks in (static or dynamic) environments this was accomplished by proving the existence of an invariant measure for the environment viewed from the particle that was absolutely continuous with respect to the initial measure on environments (see e.g., [22,13,6,9]). In this paper, however, we are able to prove the linear growth of the quadratic variation without any reference to the existence of invariant measures for the environment viewed from the particle.…”

Section: Introductionmentioning

confidence: 99%

Variable speed symmetric random walk driven by the simple symmetric exclusion process

Menezes

Peterson

Xie

2022

Electron. J. Probab.

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We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.

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“…In many papers concerned with the study of the RWRE, establishing the existence of the invariant density has been a crucial (if not the major) part of the work (see [2,6,33,17,18,30,7,23,5,31,29,8] and references therein). As a fundamental problem of the theory of RWRE, this question is interesting and important in its own right.…”

Section: Introductionmentioning

confidence: 99%

Invariant measure for random walks on ergodic environments on a strip

Dolgopyat¹,

Goldsheid²

2019

Ann. Probab.

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Environment viewed from the particle is a powerful method of analyzing random walks (RW) in random environment (RE). It is well known that in this setting the environment process is a Markov chain on the set of environments. We study the fundamental question of existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments for RW on a strip. We first describe all positive sub-exponentially growing solutions of the corresponding invariant density equation in the deterministic setting and then derive necessary and sufficient conditions for the existence of the density when the environment is ergodic in both the transient and the recurrent regimes. We also provide applications of our analysis to the question of positive and null recurrence, the study of the Green functions and to random walks on orbits of a dynamical system. MSC 2010 subject classifications: Primary 60K37; secondary 60J05, 82C44 Keywords and phrases: RWRE, random walks on a strip, invariant measure 1 imsart-aop ver.

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Quenched invariance principle for random walk in time-dependent balanced random environment

Cited by 22 publications

References 25 publications

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

Variable speed symmetric random walk driven by the simple symmetric exclusion process

Invariant measure for random walks on ergodic environments on a strip

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