Monotonicity of average return probabilities for random walks in random environments

Abstract

We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let P be a unimodular probability measure on rooted networks (G,o) with positive weights wG on its edges and with a percolation subgraph H of G with positive weights wH on its edges. Let P(G,o) denote the conditional law of H given (G,o). Assume that α:=P(G,o)[o∈V(H)]>0 is a constant P-a.s. We show that if P-a.s. whenever e∈E(G) is adjacent to o, E(G,o)[wH(e)|e∈E(H)]P(G,o)[e∈E(H)|o∈V(H)]≤wG(e), then ∀t>0E[pt(o;G)]≤E[pt(o;H)|o∈V(H)].