Return probabilities on nonunimodular transitive graphs

TL;DR

This paper investigates return probabilities of simple random walks on nonunimodular transitive graphs, proving a conjecture about their decay rate and proposing a related conjecture about the relationship between first return and return probabilities.

Contribution

It proves the folklore conjecture on the decay rate of return probabilities for certain nonunimodular transitive graphs and introduces a new conjecture relating first return and return probabilities.

Findings

Proves the decay rate of return probabilities is at most order n^{-3/2} for specific graphs.

Shows the ratio of first return to return probabilities tends to a constant in some cases.

Establishes a lower bound for the first return probability in terms of the return probability.

Abstract

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