UNICELLULAR MAPS VS. HYPERBOLIC SURFACES IN LARGE GENUS: SIMPLE CLOSED CURVES

• 成果类型: Article
• 署名作者: Janson, Svante; Louf, Baptiste
• 署名单位: Uppsala University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1601
• 发表日期: 2023
• 页码: 899-929
• 关键词: rooted maps; geodesics

摘要

We study uniformly random maps with a single face, genus g, and size n, as n, g -infinity with g = o(n), in continuation of several previous works on the geometric properties of high genus maps. We calculate the number of short simple cycles, and we show convergence of their lengths (after a well-chosen rescaling of the graph distance) to a Poisson process, which happens to be exactly the same as the limit law obtained by Mirzakhani and Petri (Comment. Math. Helv. 94 (2019) 869-889) when they studied simple closed geodesics on random hyperbolic surfaces under the Weil-Petersson measure as g - infinity.This leads us to conjecture that these two models are somehow the same in the limit, which would allow to translate problems on hyperbolic surfaces in terms of random trees, thanks to a powerful bijection of Chapuy, Feray and Fusy (J. Combin. Theory Ser. A 2013 (120) 2064-2092).