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

We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses the Brownian loop measure introduced by Lawler and Werner. In particular, we express the total mass of Brownian loops in a fixed free homotopy class on any Riemann surface in terms of the length of the geodesic representative for the complete constant curvature metric. This expression also allows us to write the electrical thickness of a compact set in $\mathbb C$ separating $0$ and $\infty$, or the Velling--Kirillov K\"ahler potential, in terms of the Brownian loop measure and the zeta-regularized determinant of Laplacian as a renormalization of the Brownian loop measure with respect to the length spectrum.