The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. After recalling basic facts about surfaces with constant curvature, their length spectrum, and Brownian motion on them, 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.