Nonconcentration of return times

TL;DR

This paper investigates the distribution of first return times for simple random walks on infinite recurrent graphs, revealing heavy tails and nonconcentration properties, and explores implications for collision behaviors in graph products.

Contribution

It establishes universal bounds on return time distributions and constructs examples demonstrating these bounds are tight, also analyzing collision properties in graph products.

Findings

Return time distribution is heavy tailed and nonconcentrated.

Constructed recurrent graph examples attain the bounds.

In certain graph products, random walks collide infinitely often.

Abstract

We show that the distribution of the first return time $\tau$ to the origin, v, of a simple random walk on an infinite recurrent graph is heavy tailed and nonconcentrated. More precisely, if $d_v$ is the degree of v, then for any $t\geq1$ we have \[\mathbf{P}_v(\tau\ge t)\ge\frac{c}{d_v\sqrt{t}}\] and \[\mathbf{P}_v(\tau=t\mid\tau\geq t)\leq\frac{C\log(d_vt)}{t}\] for some universal constants $c>0$ and $C<\infty$. The first bound is attained for all t when the underlying graph is $\mathbb{Z}$, and as for the second bound, we construct an example of a recurrent graph G for which it is attained for infinitely many t's. Furthermore, we show that in the comb product of that graph G with $\mathbb{Z}$, two independent random walks collide infinitely many times almost surely. This answers negatively a question of Krishnapur and Peres [Electron. Commun. Probab. 9 (2004) 72-81] who asked whether every comb product of two infinite recurrent graphs has the finite collision property.