Existence of the Harmonic Measure for Random Walks on Graphs and in Random Environments

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 ? ^d , d≥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 ?². This is proved using results of Barlow (Ann. Probab. 32:3024–3084, 2004) and Barlow and Hambly (Electron. J. Probab. 14(1):1–27, 2009).