Collisions of random walks in unimodular random graphs: applications to the random geometric graph and long-range percolation

Abstract

We study collision properties of simple random walks in unimodular random rooted graphs. This work continues the study initiated in~\cite{HutchcroftPeres2015}: under recurrence and an integrability condition on the root, two independent random walks collide infinitely often a.s. We prove that, under transience and an integrability assumption involving the Green function on the root, two independent random walks collide only finitely often a.s. We apply these results to several random graphs with unbounded degree: the Gilbert graph, the Delaunay graph, the Gabriel graph; and the long-range percolation model. We use these collision properties to characterize stationary measures of the voter model on these graphs.