A cohesive account on the ergodic behaviours and scaling limits of Random Walks in Cooling Random Environments

TL;DR

This paper surveys recent results and introduces new insights into the ergodic behaviors and scaling limits of Random Walks in Cooling Random Environments, highlighting structural mechanisms relevant to disordered systems with resetting.

Contribution

It provides a unified overview of RWCRE in 1d and introduces conceptual frameworks applicable to broader disordered systems with perturbations.

Findings

Recurrence criteria for RWCRE in 1d

Law of large numbers and large deviations established

Identification of structural mechanisms like ergodic limits

Abstract

Transport in disordered media is a central theme in probability and statistical physics, where randomness in the underlying medium produces phenomena such as localization, anomalous scaling, and slow relaxation. A paradigmatic model for transport in disordered media is that of Random Walks in Random Environments (RWRE), which has been extensively studied since the 1970's and is by now well understood in one dimension. More recently, several works have explored perturbations of models of transport in disordered media aimed at interpolating between static disorder and fully homogenized dynamics. Random walks in cooling random environments (RWCRE), introduced in this context, constitute a key example: the environment is dynamically resampled at prescribed times and kept fixed in between, giving rise to a delicate ``quasi-ergodic'' structure in time allowing to interpolate between homogeneous random walks and classical RWRE. The purpose of this paper is twofold. A first goal is to offer an original survey on the main results for RWCRE in 1d: recurrence criteria, law of large numbers, large deviations, and fluctuation phenomena across different resampling regimes. Those results have been derived in a series of recent works and are complemented here with a number of new statements aiming at presenting a unified phenomenological picture. As a second goal, we try to extract a coherent conceptual picture that highlights structural mechanisms -- such as ergodic limits, persistence under perturbations and replacement principles -- that extend beyond the specific setting of RWCRE and are relevant for a broader class of disordered systems that can be perturbed by introducing independent resetting in the same fashion.