Quenched limit theorems for nearest neighbour random walks in 1D random environment

TL;DR

This paper studies one-dimensional random walks in random environments, showing that trap passage times become asymptotically independent exponential variables, leading to quenched limit theorems in the subdiffusive regime.

Contribution

It introduces a new asymptotic independence result for trap passage times, enabling quenched limit theorems in the subdiffusive regime of 1D random walks.

Findings

Trap passage times are asymptotically independent exponential variables.

Parameters of these variables form an asymptotic Poisson process.

Weak quenched limit theorems are established in the subdiffusive regime.

Abstract

It is well known that random walks in one dimensional random environment can exhibit subdiffusive behavior due to presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.