Existence of the harmonic measure for random walks on graphs and in random environments

TL;DR

This paper establishes conditions under which harmonic measures from infinity exist for transient random walks on weighted graphs, including models on  lattices and supercritical percolation clusters, expanding understanding of harmonic measures in random environments.

Contribution

It provides a new sufficient condition for the existence of harmonic measure from infinity in random walks on weighted graphs, verified for specific models like the random conductance model and supercritical clusters.

Findings

Harmonic measure exists for certain random walks on weighted graphs.

The condition is verified for the random conductance model on  lattices.

Harmonic measure also exists for random walks on supercritical clusters of .

Abstract

We give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs. In particular, this condition is verified by the random conductance model on $\Z^d$, $d\geq 3$, when the conductances are i.i.d. and the bonds with positive conductance percolate. The harmonic measure from infinity also exists for random walks on supercritical clusters of $\Z^2$. This is proved using results of Barlow (2004).