Return probabilities on nonunimodular transitive graphs

Abstract

Consider simple random walk (Xn)n≥0 on a transitive graph with spectral radius ρ. Let un=P[Xn=X0] be the n-step return probability and fn be the first return probability at time n. It is a folklore conjecture that on transient, transitive graphs un∕ρn is at most of the order n−3∕2. We prove this conjecture for graphs with a closed, transitive, amenable and nonunimodular subgroup of automorphisms. We also conjecture that for any transient, transitive graph fn and un are of the same order and the ratio fn∕un even tends to an explicit constant. We give some examples for which this conjecture holds. For a graph G with a closed, transitive, nonunimodular subgroup of automorphisms, we prove a weaker asymptotic behavior regarding to this conjecture, i.e., there is a positive constant c such that fn≥uncnc.