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Combinatorial maps and hyperbolic surfaces in high genus

By Baptiste Louf

Appears in collection : Probability and Geometry in, on and of non-Euclidian spaces / Probabilités et géométrie dans, sur et des espaces non-euclidiens

In this talk, we consider two models of random geometry of surfaces in high genus: combinatorial maps and hyperbolic surfaces. The geometric properties of random hyperbolic surfaces under the Weil–Petersson measure as the genus tends to infinity have been studied for around 15 years now, building notably on enumerative results of Mirzakhani. On the other hand, random combinatorial maps were first studied in the planar/finite genus case, and then in the high genus regime starting 10 years ago with unicellular (i.e. one-faced) maps. In a joint work with Svante Janson, we noticed some numerical coincidence regarding the counting of short closed curves in unicellular maps/hyperbolic surfaces in high genus (comparing it to results of Mirzakhani and Petri). This leads us to conjecture some similarities between the two models in the limit, and raises several other open questions.

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Citation data

  • DOI 10.24350/CIRM.V.20099303
  • Cite this video Louf, Baptiste (05/10/2023). Combinatorial maps and hyperbolic surfaces in high genus. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20099303
  • URL https://dx.doi.org/10.24350/CIRM.V.20099303


  • JANSON, Svante et LOUF, Baptiste. Unicellular maps vs. hyperbolic surfaces in large genus: Simple closed curves. The Annals of Probability, 2023, vol. 51, no 3, p. 899-929. - http://dx.doi.org/10.1214/22-AOP1601

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