  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 $f_n\geq \frac{u_n}{cn^c}$.
