Geodesics filling a pair of pants on a random hyperbolic surface

The moduli space of a hyperbolic surface is a probability space when equipped with the Weil-Petersson measure. In this presentation, we will see some results of Mirzakhani on the integration of random variables defined on the moduli space, whose definition involves lengths of simple closed geodesics. We will then try to compute the expression of the length of a non-simple geodesic in Fenchel-Nielsen coordinates and see how this will help us to obtain an integration formula for random variables involving geodesics filling a pair of pants. As a consequence of this new integration formula, we obtain an improvement of the asymptotic expansion of the expectation of the number of closed geodesics filling a pair of pants with given topological type, as their length tends to the infinity.