Almost sure convergence for stochastically biased random walks on trees

TL;DR

This paper proves that a biased random walk on a supercritical Galton--Watson tree exhibits almost sure convergence of its maximal displacement scaled by (log n)^3, under certain conditions, revealing slow movement behavior in a random environment.

Contribution

It establishes almost sure convergence results for the maximal displacement of a biased random walk in a random tree environment, extending understanding of slow movement phenomena.

Findings

Maximal displacement scaled by (log n)^3 converges almost surely.

Convergence holds under general integrability assumptions.

Results apply to the recurrent case of the random walk.

Abstract

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant.