The spectral gap of random compact hyperbolic surfaces

The focus of this talk will be the first non-zero eigenvalue of the Laplacian on a compact hyperbolic surface, also called its spectral gap. Similarly to expanders for graphs, surfaces with a large spectral gap are well-connected, and have good mixing properties. In spite of significant effort in the last 35 years, we have failed to construct examples of large surfaces with an optimal spectral gap.
In this talk, based on joint work with Nalini Anantharaman, I will explain how a new probabilistic approach has led to great progress in this question in recent years. Rather than trying to exhibit examples, we now aim at showing that random surfaces have a large spectral gap with probability close to one. This requires us to have a good model to sample random surfaces, and I will briefly introduce two such models.