Escape regimes of biased random walks on Galton-Watson trees

TL;DR

This paper investigates the behavior of biased random walks on Galton-Watson trees, revealing new trapping phenomena and extending understanding of walk regimes in trees with different offspring distributions, especially with infinite variance.

Contribution

It extends previous results to include offspring laws with infinite variance and uncovers new trapping phenomena in subcritical trees, contrasting with supercritical cases.

Findings

Infinite variance offspring laws lead to new trapping phenomena.

Subcritical trees always exhibit sub-ballistic behavior.

Supercritical trees can have ballistic phases, unlike subcritical ones.

Abstract

We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.