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Harmonic Functions On Planar And Almost Planar Graphs And Manifolds, Via Circle Packings (0)
BibTeX
@MISC{Packings_harmonicfunctions,
author = {Via Circle Packings and Itai Benjamini and Oded Schramm},
title = {Harmonic Functions On Planar And Almost Planar Graphs And Manifolds, Via Circle Packings},
year = {}
}
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Abstract
. The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If P is a bounded circle packing in R 2 whose contacts graph is a bounded valence triangulation of a disk, then, with probability 1 , the simple random walk on P converges to a limit point. Moreover, in this situation any continuous function on the limit set of P extends to a continuous harmonic function on the closure of the contacts graph of P ; that is, this Dirichlet problem is solvable. We define the notions of almost planar graphs and manifolds, and show that under the assumptions of transience and bounded local geometry these possess non constant, harmonic, bounded, Dirichlet functions. Let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph. 1. Introduction In this paper we study several aspects of harmonic functions on planar graphs and generalizations. Our main results a...
Keyphrases
planar graph harmonic function almost planar graph via circle packing dirichlet function dirichlet problem bounded valence transient planar graph local geometry limit set several aspect non constant limit point continuous function simple random walk main result continuous harmonic function contact graph bounded valence triangulation bounded circle packing posse non
