The Annals of Probability

Boundaries of planar graphs, via circle packings

Omer Angel, Martin T. Barlow, Ori Gurel-Gurevich, and Asaf Nachmias

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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).

Article information

Ann. Probab., Volume 44, Number 3 (2016), 1956-1984.

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

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Zentralblatt MATH identifier

Primary: 05C81: Random walks on graphs

Planar graph random walk Poisson boundary Martin boundary circle packing hyperbolic


Angel, Omer; Barlow, Martin T.; Gurel-Gurevich, Ori; Nachmias, Asaf. Boundaries of planar graphs, via circle packings. Ann. Probab. 44 (2016), no. 3, 1956--1984. doi:10.1214/15-AOP1014.

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