Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 5, 6 pp.
Recurrence of multiply-ended planar triangulations
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  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 $0$.
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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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