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

Daniel Boivin (1) , Clément Rau (2)

1 LM - Laboratoire de mathématiques de Brest
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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).

Mots clés: Harmonic measure; supercritical percolation clusters; effective conductance; Harnack inequality; random conductance model

Domaines: Probabilités [math.PR]

Daniel Boivin, Clément Rau. Existence of the harmonic measure for random walks on graphs and in random environments. 2011. ⟨hal-00643780v2⟩