Return probabilities on nonunimodular transitive graphs

Consider simple random walk $(S_n)_{ngeq0}$ on a transitive graph with spectral radius $rho$. Let $u_n=mathbb{P}[S_n=S_0]$ be the $n$-step return probability. It is a folklore conjecture that on transient, transitive graphs $u_n/rho^n$ 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 study the first return probability $f_n$. For a graph $G$ with a closed, transitive, nonunimodular subgroup of automorphisms, we show that there is a positive constant $c$ such that $f_ngeq frac{u_n}{cn^c}$. We also make some conjectures related to $f_n$ and $u_n$ for transient, transitive graphs.