| Résume | The spectral gap of a hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. It measures its the connectivity as well as the speed of mixing for various dynamics. I will present results obtained with Nalini Anantharaman, where we prove that random hyperbolic surfaces of large genus have an optimal spectral gap. I will explain how we relate this spectral question to a problem of geodesic counting, and introduce new ideas we have developed to count closed geodesics on a random hyperbolic surface. |
