Favorite sites of randomly biased walks on a supercritical Galton-Watson tree

TL;DR

This paper studies the favorite sites of biased random walks on supercritical Galton-Watson trees, revealing how the set of favorite sites behaves depending on a key parameter, with detailed probabilistic analysis and tail estimates.

Contribution

It introduces a comprehensive analysis of favorite sites for null-recurrent biased walks on Galton-Watson trees, identifying different regimes based on a parameter and providing new probabilistic insights.

Findings

Favorite sites are bounded for  < , tight at =2, and oscillate for <.

Complete characterization of the cardinality of favorite sites for <.

Utilizes tail estimates and change of measure techniques for multi-type Galton-Watson trees.

Abstract

Erd\H{o}s and R\'ev\'esz initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter $\kappa \in (1, \infty]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $\kappa \in (2, \infty]$, tight in the case $\kappa=2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $\kappa \in (1, 2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $\kappa \in (2, \infty]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.