Random walk in cooling random environment: ergodic limits and concentration inequalities

TL;DR

This paper establishes strong laws and large deviation principles for a one-dimensional random walk in a cooling dynamic environment, using multi-layer decompositions and concentration inequalities to analyze ergodic limits and fluctuations.

Contribution

It extends previous work by proving strong laws and quenched large deviations in the cooling regime, introducing new concentration inequalities and ergodic theorems for sums of random variables.

Findings

Strong law of large numbers in the cooling regime

Quenched large deviation principle established

Concentration inequality for static environment displacement

Abstract

In previous work by Avena and den Hollander, a model of a one-dimensional random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a growing sequence of deterministic times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over increasing periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moments bounds and concentration estimates. Along the way we derive two results of independent interest, namely, a concentration inequality for the cumulants of the displacement in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the speed of divergence of the increments of the resampling times.