Random regular graphs and the systole of a random surface

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 goes 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 for the number of vertices going to infinity and show that this holds for circuits of a length up to $\log_2$ of the number of vertices.