Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime

TL;DR

This paper investigates the limiting distributions of Random Walks in Cooling Random Environments (RWCRE) in two specific transient regimes, revealing Gaussian and mixed distributions influenced by the cooling map's properties.

Contribution

It extends the understanding of RWCRE by analyzing the sub-ballistic and non-diffusive Gaussian regimes, identifying new limiting distribution behaviors.

Findings

In the sub-ballistic regime, limiting distributions are Gaussian or mixtures involving Mittag-Leffler variables.

In the Gaussian regime with non-diffusive scaling, distributions are Gaussian with oscillating scaling factors.

The properties of the cooling map critically influence the limiting distribution characteristics.

Abstract

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the environment is frozen between a fixed sequence of times (called the cooling map) where it is resampled. Naturally the limiting distributions for this model depend both on the structure of the cooling sequence and on distribution $\mu$ from which the environments are sampled. Previous results have considered the cases where $\mu$ is such that the corresponding model of random walks in a fixed random environment (RWRE) is either (1) recurrent, (2) has a Gaussian limit with diffusive scaling (the $\kappa > 2$ case), or (3) has positive speed and a stable, non-Gaussian limit (the $\kappa \in (1,2)$ case). In this paper we examine the limiting distributions in two other transient regimes: the sub-ballistic, non-stable regime (i.e., $\kappa \in (0,1)$), and the Gaussian regime with non-diffusive scaling (i.e., $\kappa = 2$). In the first case we show that the limiting distributions are either Gaussian or a mixture of Gaussian and independent sums of Mittag-Leffler random variables, while in the second case the limiting distributions are always Gaussian but with a scaling that differs from the standard deviation by factor (which can oscillate, but which remains confined to some interval $[\beta,1]$) that depends very delicately on the properties of the cooling map.