We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the large genus limit of the expected systole.In particular, we will compute the large genus limits of these probabilities. For example, we determine which proportion of the Weil-Petersson volume is asymptotically taken up by the ε-thin part of moduli space.New results. Before we state our results, we need to set up some notation. Given X ∈ M g and an interval [a, b] ⊂ R + , let N g, [a,b] (X) denote the number of primitive closed geodesics on X with lengths in the given interval. Note that in our setup N g, [a,b] : M g → N may be interpreted as a random variable.

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Abstract. We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Riemannian metric.In the hyperbolic case we compute the limit of the expected value of the systole when the number of triangles tends to infinity (approximately 2.484). We also determine the asymptotic probability distribution of the number of curves of any finite length. This turns out to be a Poisson distribution.In the Riemannian case we give an upper bound to the limit supremum and a lower bound to the limit infimum of the expected value of the systole depending only on the metric on the triangle. We also show that this upper bound is sharp in the sense that there is a sequence of metrics for which the limit infimum comes arbitrarily close to the upper bound.The main tool we use is random regular graphs. One of the difficulties in the proof of the limits is controlling the probability that short closed curves are separating. To do this, we first prove that the probability that a random cubic graph has a short separating circuit tends to 0 as the number of vertices tends to infinity and show that this holds for circuits of a length up to log 2 of the number of vertices.

Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order o(log log g) with g being the genus of the surface.