Dimension Drop for Transient Random Walks on Galton-Watson Trees in Random Environments

TL;DR

This paper demonstrates the dimension drop phenomenon for harmonic measures of transient random walks in random environments on infinite Galton-Watson trees, using ergodic theory and regeneration times.

Contribution

It provides an explicit construction of the invariant measure for the forward environment, extending previous work to include the dimension drop phenomenon.

Findings

Dimension drop holds for harmonic measure on Galton-Watson trees

Explicit invariant measure construction for the environment seen by the walk

Application of ergodic theory techniques to random walks in random environments

Abstract

We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite Galton-Watson tree without leaves. We use regeneration times and ergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peres in 1996 to give an explicit construction of the invariant measure for the forward environment seen by the particule at exit times which is absolutely continuous with respect to the joint law of the tree and the path of the random walk.