Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity

TL;DR

This paper introduces a novel reflected random walk on infinite graphs, linking it to uniform spanning forests and applying it to study complex random planar maps in supercritical Liouville quantum gravity.

Contribution

It defines a new type of reflected random walk and connects it to uniform spanning forests, also applying the theory to analyze random planar maps in supercritical LQG.

Findings

Reflected random walk characterizes the free uniform spanning forest.

Wilson's algorithm applies to reflected walk for connected FUSF.

Conjectures on the behavior of stochastic processes on LQG maps.

Abstract

Let $\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and "reflects off of $\infty$" infinitely many times. We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general. We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge in $(1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also make several conjectures regarding the qualitative behavior of stochastic processes on such maps -- including the FUSF and critical percolation.