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

Abstract

We consider the randomly biased random walk on trees in the slow movement regime as in Hu and Shi (2016), whose potential is given by a branching random walk in the boundary case. We study the heavy range up to the nth return to the root, i.e., the number of edges visited more than kn times. For kn=nθ with θ∈(0,1), we obtain the convergence in probability of the rescaled heavy range, which improves one result of Andreoletti and Diel (2020).