Heavy range of the randomly biased walk on Galton-Watson trees in the slow movement regime

TL;DR

This paper investigates the heavy range of a randomly biased walk on Galton-Watson trees in the slow movement regime, establishing convergence results for the number of edges visited frequently, thus advancing understanding of such stochastic processes.

Contribution

It provides new convergence results for the heavy range of biased random walks on trees, extending previous work and covering the case where edges are visited more than a power of the number of returns.

Findings

Convergence in probability of the rescaled heavy range for $k_n=n^ heta$.

Improvement over previous results by [AD20].

Analysis in the boundary case of the branching random walk potential.

Abstract

We consider the randomly biased random walk on trees in the slow movement regime as in [HS16], whose potential is given by a branching random walk in the boundary case. We study the heavy range up to the $n$-th return to the root, i.e., the number of edges visited more than $k_n$ times. For $k_n=n^\theta$ with $\theta\in(0,1)$, we obtain the convergence in probability of the rescaled heavy range, which improves one result of [AD20].