Local Limit Theorems for Random Walks in a Random Environment on a Strip

2019/10/28 by Dmitry Dolgopyat, Dolgopyat, Dmitry, Ilya Goldsheid +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1910.12961

openalex publication_date 2019/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper consists of two parts. In the first part we review recent work on limit theorems for random walks in random environment (RWRE) on a strip with jumps to the nearest layers. In the second part, we prove the quenched Local Limit Theorem (LLT) for the position of the walk in the transient diffusive regime. This fills an important gap in the literature. We then obtain two corollaries of the quenched LLT. The first one is the annealed version of the LLT on a strip. The second one is the proof of the fact that the distribution of the environment viewed from the particle (EVFP) has a limit for a. e. environment. In the case of the random walk with jumps to nearest neighbours in dimension one, the latter result is a theorem of Lally \citeL. Since the strip model incorporates the walks with bounded jumps on a one-dimensional lattice, the second corollary also solves the long standing problem of extending Lalley's result to this case.

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