Recurrence of multiply-ended planar triangulations

Ori Gurel-Gurevich; Asaf Nachmias; Juan Souto

doi:https://doi.org/10.1214/16-ECP4418

Recurrence of multiply-ended planar triangulations

Ori Gurel-Gurevich, Asaf Nachmias, Juan Souto

The Hebrew University of Jerusalem, Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	A05
Pages (from-to)	1-6
Number of pages	6
Journal	Electronic Communications in Probability
Volume	22
DOIs	https://doi.org/10.1214/16-ECP4418
State	Published - 2017

Keywords

Circle packing
Random walk

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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https://doi.org/10.1214/16-ECP4418

Cite this

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AB - 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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KW - Random walk

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JO - Electronic Communications in Probability

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Recurrence of multiply-ended planar triangulations

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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