On the bottom of the spectrum of random hyperbolic surfaces

Although there are several ways to ''choose a compact hyperbolic surface at random'', putting the Weil-Petersson probability measure on the moduli space of hyperbolic surfaces of a given topology is certainly the most natural.

The work of M. Mirzakhani has made possible the study of this probabilistic model: it is one of the only model of ''random riemannian manifolds'' where some explicit calculations are actually possible. One may thus ask questions about of the geometry and the spectral statistics of the laplacian of a randomly chosen surface – in analogy with what is usually asked for models of random graphs.

I will be interested in the spectral gap of the laplacian for a random compact hyperbolic surface, in the limit of large genus (j.w. Laura Monk).