Monotonicity of Average Return Probabilities for Random Walks in Random Environments

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 > 0 E [ pt(o;G) ] ≤ E [ pt(o;H) ∣∣ o ∈ V(H)] . §