Random regular graphs and the systole of a random surface

Abstract

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.

Finite length spectra of random surfaces and their dependence on genus

“…Then we use the non-negligibility of our set of random surfaces to determine the moments of Z N,w . These turn out to be the same as those in the unrestricted case computed in [Pet13], which implies that the limiting distributions are the same.…”

“…The length of short curves of random surfaces was investigated by the author in [Pet13]. It turns out that in the case of ideal hyperbolic triangles the expected value of the systole converges to a constant for N → ∞ (approximately 2.48), both in the compactified and non-compactified case.…”

“…We also note that Theorem A is a conditional version of Theorem B from [Pet13], which can be obtained from Theorem A by choosing the sequence of sets D N = {0, 1, . .…”

“…The second statement is sharp in the sense that for any prescribed m 2 (d) and fixed x ∈ [0, ∞) we can find a Riemannian metric that comes arbitrarily close to the upper bound. The construction of this metric is the same as the one in [Pet13], the idea is to define a metric that forces short curves to trace a fixed path on every triangle.…”

Poisson approximation of the length spectrum of random surfaces

“…Given a positive real number ℓ, the number of closed geodesics of length ℓ defines a random variable Z ℓ, N : Ω N → , where Ω N denotes our underlying probability space of random surfaces glued out of 2N ideal triangles, N ∈ . In [21], the first author proved that given a finite number of such random variables (i.e., take a finite set of positive real numbers and count the geodesics of exactly these lengths), they converge in distribution to independent Poisson random variables, as N → ∞, with explicit parameters. This was proved using the classical method of moments.…”