The Brownian loop measure on Riemann surfaces and applications to length spectra

The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, I will recall the basic facts about surfaces with constant curvature and Brownian motion on them. Then, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. The talk is mainly based on a joint work with Yuhao Xue (IHES).