Recurrence of multiply-ended planar triangulations

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 $0$.