Recurrence of planar graph limits

TL;DR

This paper proves that certain limits of finite planar graphs are recurrent, including the UIPT and UIPQ, and establishes bounds on random walk avoidance probabilities in bounded degree planar graphs.

Contribution

It demonstrates recurrence for distributional limits of finite planar graphs with exponential tail degree distributions, resolving longstanding conjectures.

Findings

Distributional limits of finite planar graphs are almost surely recurrent.

UIPT and UIPQ are almost surely recurrent.

Probability bounds for random walk avoidance in bounded degree planar graphs.

Abstract

We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, finite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C/log T.