The most visited sites of biased random walks on trees

Yueyun Hu,Zhan Shi

doi:10.1214/ejp.v20-4051

Abstract

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton-Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erdős and Révész for simple symmetric random walk on the line.

Highlights

We consider a biased random walk (Xn) on a supercritical Galton–Watson tree T, rooted at ∅
∅, which is considered as the parent of such that ω(∅, ∅) + x: ← x =∅ ω(∅, x) = 1
We proved in [16] that it was the case for Sinai’s one-dimensional random walk in random environment

Summary

Introduction

We consider a (randomly) biased random walk (Xn) on a supercritical Galton–Watson tree T, rooted at ∅. For the symmetric Bernoulli random walk on Z, Erdos and Révész [11] conjectured: (a) tightness for the family of most visited sites, and (b) the cardinality of the set of most visited sites being eventually bounded by 2. The present paper is devoted to studying both questions for biased walks on trees; our answer is as follows. Concerning the tightness question for most visited sites, biased walks on trees behave very differently from recurrent one-dimensional nearest-neighbour random walks (whether the environment is random or deterministic). To the best of our knowledge, this is the first non-trivial example of null recurrent Markov chain whose most visited sites are tight. We give a precise statement of the main result of this paper, Theorem 2.1

Statement of results

Findings

A Appendix