Unimodular hyperbolic triangulations: circle packing and random walk

• 成果类型: Article
• 署名作者: Angel, Omer; Hutchcroft, Tom; Nachmias, Asaf; Ray, Gourab
• 署名单位: University of British Columbia; Tel Aviv University; University of Cambridge
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0653-9
• 发表日期: 2016
• 页码: 229-268
• 关键词: harmonic-functions; planar graphs; percolation; CONVERGENCE; MAPS

摘要

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons [1]. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem [19]. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.