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May 2016 Boundaries of planar graphs, via circle packings
Omer Angel, Martin T. Barlow, Ori Gurel-Gurevich, Asaf Nachmias
Ann. Probab. 44(3): 1956-1984 (May 2016). DOI: 10.1214/15-AOP1014

Abstract

We provide a geometric representation of the Poisson and Martin boundaries of a transient, bounded degree triangulation of the plane in terms of its circle packing in the unit disc. (This packing is unique up to Möbius transformations.) More precisely, we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk, and that the space of extremal positive harmonic functions, that is, the Martin boundary, is homeomorphic to the unit circle.

All our results hold more generally for any “good”-embedding of planar graphs, that is, an embedding in the unit disc with straight lines such that angles are bounded away from $0$ and $\pi$ uniformly, and lengths of adjacent edges are comparable. Furthermore, we show that in a good embedding of a planar graph the probability that a random walk exits a disc through a sufficiently wide arc is at least a constant, and that Brownian motion on such graphs takes time of order $r^{2}$ to exit a disc of radius $r$. These answer a question recently posed by Chelkak (2014).

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Omer Angel. Martin T. Barlow. Ori Gurel-Gurevich. Asaf Nachmias. "Boundaries of planar graphs, via circle packings." Ann. Probab. 44 (3) 1956 - 1984, May 2016. https://doi.org/10.1214/15-AOP1014

Information

Received: 1 November 2013; Revised: 1 January 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1339.05061
MathSciNet: MR3502598
Digital Object Identifier: 10.1214/15-AOP1014

Subjects:
Primary: 05C81

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • May 2016
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