May 2023 Unicellular maps vs. hyperbolic surfaces in large genus: Simple closed curves
Svante Janson, Baptiste Louf
Author Affiliations +
Ann. Probab. 51(3): 899-929 (May 2023). DOI: 10.1214/22-AOP1601


We study uniformly random maps with a single face, genus g, and size n, as n,g 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.

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, Féray and Fusy (J. Combin. Theory Ser. A 2013 (120) 2064–2092).

Funding Statement

Supported by the Knut and Alice Wallenberg Foundation.


We are grateful to Bram Petri and Stephan Wagner for enlightening discussions. We also thank Thomas Budzinski, Nicolas Curien, and Yunhui Wu for comments on the first version of this paper.


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Svante Janson. Baptiste Louf. "Unicellular maps vs. hyperbolic surfaces in large genus: Simple closed curves." Ann. Probab. 51 (3) 899 - 929, May 2023.


Received: 1 December 2021; Revised: 1 June 2022; Published: May 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583058
zbMATH: 1516.05045
Digital Object Identifier: 10.1214/22-AOP1601

Primary: 05C10 , 05C80 , 60C05 , 60D05

Keywords: hyperbolic surfaces , Large genus , Random maps

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 3 • May 2023
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