Boundaries of planar graphs, via circle packings

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

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


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).

Original languageAmerican English
Pages (from-to)1956-1984
Number of pages29
JournalAnnals of Probability
Issue number3
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2016.


  • Circle packing
  • Hyperbolic
  • Martin boundary
  • Planar graph
  • Poisson boundary
  • Random walk


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