Planar stochastic hyperbolic triangulations

• 成果类型: Article
• 署名作者: Curien, Nicolas
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0638-4
• 发表日期: 2016
• 页码: 509-540
• 关键词: brownian map; quadrangulation; percolation; GROWTH; limit

摘要

Pursuing the approach of Angel and Ray (Ann Probab, 2015) we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices naturally generalize Angel and Schramm's uniform infinite planar triangulation (UIPT) and are hyperbolic in flavor. We prove that they exhibit a sharp exponential volume growth, are non-Liouville, and that the simple random walk on them has positive speed almost surely. We conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size. An artistic representation of a random (3-connected) triangulation of the plane with hyperbolic flavor.