The boundary of a square tiling of a graph coincides with the Poisson boundary

• 成果类型: Article
• 署名作者: Georgakopoulos, Agelos
• 署名单位: University of Warwick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0601-0
• 发表日期: 2016
• 页码: 773-821
• 关键词: harmonic-functions; planar graphs; random-walks; MANIFOLDS; FORMULA

摘要

Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219-1238, 1996), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann's mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299-314, 1993). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions.