Local times in critical generations of a random walk in random environment on trees

TL;DR

This paper studies the local times of a null-recurrent biased random walk on Galton-Watson trees, showing convergence to a stable process and providing explicit hitting probabilities.

Contribution

It establishes the convergence of local times in critical generations to a stable continuous-state branching process and derives explicit hitting probability formulas.

Findings

Local times converge to a stable process in the critical regime

Explicit probability formulas for reaching critical generations

Analysis in the sub-diffusive regime of biased walks

Abstract

We consider a null-recurrent randomly biased walk $\mathbb{X}$ on a Galton-Watson tree in the (sub)-diffusive regime and we prove that properly renormalized, the local time in a critical generation converges in law towards some function of a stable continuous-state branching process. We also provide an explicit equivalent of the probability that critical generations are reached by the random walk $\mathbb{X}$.