Maximal local time of randomly biased random walks on a Galton-Watson tree

TL;DR

This paper investigates the maximum local time of a recurrent random walk on a Galton-Watson tree in a random environment, revealing its asymptotic behavior during multiple excursions.

Contribution

It introduces a novel analysis of the maximal local time using the structure of a multi-type Galton-Watson tree derived from the walk's edge local times.

Findings

Asymptotic behavior of the maximal local time during n excursions.

Characterization of the maximal type of the associated Galton-Watson tree.

Results apply to diffusive and sub-diffusive regimes.

Abstract

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.