Local limits of uniform triangulations in high genus

• 成果类型: Article
• 署名作者: Budzinski, Thomas; Louf, Baptiste
• 署名单位: University of British Columbia; Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00986-3
• 发表日期: 2021
• 页码: 1-47
• 关键词: scaling limit; MAPS; geodesics

摘要

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in Curien (Probab Theory Relat Fields 165(3):509-540, 2016). The proof relies on a combinatorial argument and the Goulden-Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the planeTis weakly Markovian in the sense that the probability to observe a finite triangulationtaround the root only depends on the perimeter and volume oft, thenTis a mixture of PSHT.