Length spectra of random metric map of large genus: a Teichmüller approach.

In this talk, after a brief historical review, I will explain how to sample a random hyperbolic surface. After that, we will focus on the length spectrum of a random surface. More precisely, we will investigate short closed geodesics on a random hyperbolic surface of large genus. It turns out that the lengths of these geodesics are distributed in exactly the same way as those of the short cycles in a large random graph. This is a joint work with Simon Barazer and Alessandro Giacchetto.