Unicellular maps vs. hyperbolic surfaces in large genus: simple closed curves. (English) Zbl 1516.05045

Summary: We study uniformly random maps with a single face, genus \(g\), and size \(n\), as \(n,g \to \infty\) 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 M. Mirzakhani and B. Petri [Comment. Math. Helv. 94, No. 4, 869–889 (2019; Zbl 1472.57028)] when they studied simple closed geodesics on random hyperbolic surfaces under the Weil-Petersson measure as \(g \to \infty\).
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 G. Chapuy et al. [J. Comb. Theory, Ser. A 120, No. 8, 2064–2092 (2013; Zbl 1278.05081)].


05C10 Planar graphs; geometric and topological aspects of graph theory
05C80 Random graphs (graph-theoretic aspects)
60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
60B05 Probability measures on topological spaces
60B10 Convergence of probability measures
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI arXiv


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