Lingering random walks in random environment on a strip

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.

Cited by in corpus (5)

• Multi-Gaussian modes of diffusion in a quenched random medium
• Behavior of random walk on discrete point processes
• Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards
• Limit theorems for random walks on a strip in subdiffusive regime
• Anomalous diffusion in disordered multi-channel systems