On the Riemann surface type of Random Planar Maps

TL;DR

This paper proves that certain random Riemann surfaces derived from infinite planar maps are parabolic, meaning Brownian motion on them is recurrent, by analyzing their distributional limits.

Contribution

It establishes the parabolic nature of Riemann surfaces from specific infinite planar maps, extending previous work on distributional limits of random triangulations.

Findings

Both the Angel-Schramm and Sheffield infinite constructions are parabolic.

Brownian motion on these surfaces is recurrent.

Results follow from general theorems on limits of random triangulations.

Abstract

We show that the (random) Riemann surfaces of the Angel-Schramm Uniform Infinite Planar Triangulation and of Sheffield's infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random unbiased disc triangulations, following work of Benjamini and Schramm.