BOUNDARIES OF PLANAR GRAPHS, VIA CIRCLE PACKINGS

• 成果类型: Article
• 署名作者: Angel, Omer; Barlow, Martin T.; Gurel-Gurevich, Ori; Nachmias, Asaf
• 署名单位: University of British Columbia; Hebrew University of Jerusalem; Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1014
• 发表日期: 2016
• 页码: 1956-1984
• 关键词: harmonic-functions; harnack principle; martin boundary; domains; CONVERGENCE; INEQUALITY

摘要

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 Mobius 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).