Return probabilities on nonunimodular transitive graphs

Consider simple random walk $(X_n)_{n\geq0}$ on a transitive graph with spectral radius $\rho$. Let $u_n=\mathbb{P}[X_n=X_0]$ be the $n$-step return probability and $f_n$ be the first return probability at time $n$. 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 conjecture that for any transient, transitive graph $f_n$ and $u_n$ are of the same order and the ratio $f_n/u_n$ 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 $f_n\geq \frac{u_n}{cn^c}$.