## The Annals of Probability

### Boundaries of planar graphs, via circle packings

#### 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).

#### Article information

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

Dates
Revised: January 2015
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410036

Digital Object Identifier
doi:10.1214/15-AOP1014

Mathematical Reviews number (MathSciNet)
MR3502598

Zentralblatt MATH identifier
1339.05061

Subjects
Primary: 05C81: Random walks on graphs

#### Citation

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. https://projecteuclid.org/euclid.aop/1463410036

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