Large deviations for random walks in random environment on a Galton-Watson tree

TL;DR

This paper establishes a large deviation principle for the hitting times of a random walk in a random environment on a Galton-Watson tree, analyzing both quenched and annealed cases and exploring polynomial regimes in subexponential scenarios.

Contribution

It introduces the first large deviation principles for hitting times in this setting and investigates polynomial regimes in subexponential cases, extending previous one-dimensional results.

Findings

Large deviation principles for u and annealed hitting times

Identification of polynomial regimes in subexponential cases

Dependence on tail distribution of first regeneration time

Abstract

Consider a random walk in random environment on a supercritical Galton--Watson tree, and let $\tau_n$ be the hitting time of generation $n$. The paper presents a large deviation principle for $\tau_n/n$, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time.