BOUNDARIES OF PLANAR GRAPHS, VIA CIRCLE PACKINGS

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 t...

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Bibliographic Details
Published in:The Annals of probability 2016-05, Vol.44 (3), p.1956-1984
Main Authors: Angel, Omer, Barlow, Martin T., Gurel-Gurevich, Ori, Nachmias, Asaf
Format: Article
Language:English
Subjects:
Brownian motion
Circles
Geometry
Graph representations
Graphs
Harmonic functions
Line segments
Markov chains
Mathematical functions
Mathematical theorems
Planar graphs
Poisson distribution
Random walk
Random walk theory
Stochastic models
Studies
Triangulation
Vertices
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Summary: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 π 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 r2 to exit a disc of radius r. These answer a question recently posed by Chelkak (2014).
ISSN:0091-1798
2168-894X
DOI:10.1214/15-AOP1014