Return probabilities on nonunimodular transitive graphs

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 first at time n. It is folklore conjecture that transient, graphs un∕ρn most of order n−3∕2. We prove this for closed, transitive, amenable nonunimodular subgroup automorphisms. also any un are same ratio fn∕un even tends to an explicit constant. give some examples which holds. For G automorphisms, we weaker asymptotic behavior regarding conjecture, i.e., there positive constant c such fn≥uncnc.