Electronic Communications in Probability

Recurrence of multiply-ended planar triangulations

Ori Gurel-Gurevich, Asaf Nachmias, and Juan Souto

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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 $0$.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 5, 6 pp.

Received: 12 July 2015
Accepted: 17 May 2016
First available in Project Euclid: 6 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C81: Random walks on graphs 52C26: Circle packings and discrete conformal geometry

random walk circle packing

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Gurel-Gurevich, Ori; Nachmias, Asaf; Souto, Juan. Recurrence of multiply-ended planar triangulations. Electron. Commun. Probab. 22 (2017), paper no. 5, 6 pp. doi:10.1214/16-ECP4418. https://projecteuclid.org/euclid.ecp/1483671681

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