Unicellular maps vs hyperbolic surfaces in large genus: simple closed curves

TL;DR

This paper investigates the properties of large genus random maps with a single face, showing that the distribution of short simple cycles converges to a Poisson process, similar to hyperbolic surfaces, suggesting a deep connection between these models.

Contribution

It establishes a link between random maps and hyperbolic surfaces by analyzing cycle distributions and conjecturing their equivalence in the large genus limit.

Findings

Number of short simple cycles converges to a Poisson process.

Cycle length distribution matches that of hyperbolic surfaces.

Suggests a potential equivalence between random maps and hyperbolic geometry models.

Abstract

We study uniformly random maps with a single face, genus $g$, and size $n$, as $n,g\rightarrow \infty$ 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 Mirzakhani and Petri (2019) when they studied simple closed geodesics on random hyperbolic surfaces under the Weil-Petersson measure as $g\rightarrow \infty$. 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 Chapuy, F\'eray and Fusy (2013).