Lingering random walks in random environment on a strip

TL;DR

This paper studies recurrent random walks in random environments on a strip, establishing conditions under which they exhibit a specific log tb2 asymptotic behavior, and classifying their asymptotics based on the environment's support.

Contribution

It provides a complete classification of asymptotic behaviors of recurrent random walks in i.i.d. environments on a strip, including explicit conditions and algebraic descriptions.

Findings

RW exhibits (log tb2) behavior outside an algebraic hyperplane.

If the environment's support lies on the hyperplane, the RW is a martingale with CLT-governed behavior.

The approach extends to one-dimensional walks with bounded jumps, offering a full classification.

Abstract

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the "(log t)-squared" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.