Integrals of general geometric random variables on the moduli space of hyperbolic surfaces

Abstract

In this article we provide an integration formula enabling us to integrate random variables defined on the moduli space of hyperbolic surfaces which involve the lengths of closed geodesics of any prescribed topological type, at fixed genus and number of boundaries. We then compute the general form of length functions of closed loops in Fenchel-Nielsen coordinates. Using this expression together with our integration formula, we express the expectations of geometric random variables as integrals over R with almost polynomial density. As a consequence, given an arbitrary closed loop $\gamma$, we obtain an improvement of Mirzakhani's asymptotic equivalent of the Weil-Petersson expectation E[N$\gamma$(a)], when a $\to$ $\infty$, of the number of geodesics with same topological type as $\gamma$ of length at most a. We finally provide, for a loop $\gamma$ filling a one-holed torus or a four-holed sphere, an interpretation of the density function associated with $\gamma$ as the period of the Hamiltonian flow of the length function of $\gamma$.