Recurrence of multiply-ended planar triangulations

Ori Gurel-Gurevich; Asaf Nachmias; Juan Souto

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arXiv · 1506.00221

Recurrence of multiply-ended planar triangulations

Ori Gurel-Gurevich·Asaf Nachmias·Juan Souto

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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.

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Ori Gurel-Gurevich, Asaf Nachmias, Juan Souto. 2015-05-31. Recurrence of multiply-ended planar triangulations. https://arxiv.org/abs/1506.00221

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