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

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 ${\mathbb{Z}}^d$, $d\ge 3$, 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 ${\mathbb{Z}}^2$. This is proved using results of Barlow (2004).

arXiv comment25 p. and 2 figures