Large deviations for random walk in a space--time product environment

TL;DR

This paper investigates large deviation principles for random walks in space-time environments, establishing convergence of the environment process and equivalence of averaged and quenched deviations near typical velocities.

Contribution

It introduces a framework for analyzing the environment Markov chain conditioned on the walk's velocity, proving convergence to a stationary process and equivalence of large deviations in higher dimensions.

Findings

Convergence of the environment process to a stationary measure under conditioning.

Equivalence of averaged and quenched large deviations near typical velocities in dimensions d≥3.

Identification of the limiting process as a Doob h-transform of the original kernel.

Abstract

We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space--time product environment $\omega\in\Omega$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}\omega)_{n\geq0}$ where $T$ denotes the shift on $\Omega$. Conditioned on the particle having asymptotic mean velocity equal to any given $\xi$, we show that the empirical process of the environment Markov chain converges to a stationary process $\mu_{\xi}^{\infty}$ under the averaged measure. When $d\geq3$ and $\xi$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity $\xi$, the empirical process of the environment Markov chain converges to $\mu_{\xi}^{\infty}$ under the quenched measure as well. In this case, we show that $\mu_{\xi}^{\infty}$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob $h$-transform.