Recurrence of multiply-ended planar triangulations

TL;DR

This paper establishes a criterion linking the recurrence of bounded degree planar triangulations to the polar nature of accumulation points in circle packings, extending previous results to more general embeddings.

Contribution

It generalizes a theorem by He and Schramm by relating recurrence to accumulation point polarity in broader classes of embeddings.

Findings

Recurrence is characterized by the polarity of accumulation points.

The result applies to circle packings and straight-line embeddings with bounded angles.

It extends previous work to graphs with multiple ends and general embeddings.

Abstract

In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability 1). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from 0.