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

TL;DR

This work analyzes RWCRE in two challenging transient regimes where the underlying RWRE exhibits distinct limiting behavior: the sub-ballistic regime with $\kappa\in(0,1)$ and the Gaussian critical regime with $\kappa=2$. For $\kappa\in(0,1)$, the authors prove a replacement-type limit showing subsequential limits are mixtures of independent Mittag-Leffler variables and Gaussians, parameterized by a limit vector $\lambda_*\in\ell^2$, with pure or mixed limits arising in special cases; this leverages variance asymptotics and $L^p$ convergence results for the RWRE. In the $\kappa=2$ regime, the limiting distribution remains Gaussian but with a non-trivial and delicately cooling-map-dependent scaling $\beta_n$, and they provide an explicit formula for $\beta_n$ in terms of truncated variances; they show that $L^2$ convergence fails for the RWRE and that the proper RWCRE scaling can vary and even oscillate with $n$. The paper also develops sharp tail estimates for the RWRE in the $\kappa=2$ case and uses a sophisticated decomposition into large and small cooling intervals to prove the Gaussian limits for RWCRE under general cooling maps, including several illustrative examples of polynomial, exponential, and oscillatory cooling. Together, these results complete the limiting-distribution picture for one-dimensional RWCRE in the sub-ballistic and Gaussian-critical regimes and highlight the crucial role of the cooling map in shaping asymptotics. $

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 $μ$ from which the environments are sampled. Previous results have considered the cases where $μ$ 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 $κ> 2$ case), or (3) has positive speed and a stable, non-Gaussian limit (the $κ\in (1,2)$ case). In this paper we examine the limiting distributions in two other transient regimes: the sub-ballistic, non-stable regime (i.e., $κ\in (0,1)$), and the Gaussian regime with non-diffusive scaling (i.e., $κ= 2$). 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 $[β,1]$) that depends very delicately on the properties of the cooling map.

Theorems & Definitions (60)

• Definition 1.1: $\kappa$-regular measures
• Remark 1.2
• Theorem 1.3: KKS75
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10: RWRE $L^p$ Convergence $\kappa\in(0,1)$