Random walks in cooling random environments

TL;DR

This paper introduces a new model of one-dimensional random walks in dynamic environments with resampling times that grow in different regimes, analyzing their probabilistic behavior and potential phase transitions.

Contribution

It develops a novel interpolation between static and fully resampled environments by resampling along deterministic times, and establishes laws of large numbers and CLTs for these regimes.

Findings

Proves weak laws of large numbers for all regimes.

Establishes central limit theorems in the linear and polynomial regimes.

Suggests a crossover phenomenon in the scaling behavior for polynomial and exponential regimes.

Abstract

We propose a model of a one-dimensional random walk in dynamic random environment that interpolates between two classical settings: (I) the random environment is sampled at time zero only; (II) the random environment is resampled at every unit of time. In our model the random environment is resampled along an increasing sequence of deterministic times. We consider the annealed version of the model, and look at three growth regimes for the resampling times: (R1) linear; (R2) polynomial; (R3) exponential. We prove weak laws of large numbers and central limit theorems. We list some open problems and conjecture the presence of a crossover for the scaling behaviour in regimes (R2) and (R3).