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

TL;DR

This paper establishes a relationship between the length spectrum of Riemann surfaces and Brownian loop measures, providing new formulas connecting geometric, probabilistic, and spectral invariants of these surfaces.

Contribution

It introduces a simple identity linking length spectra with Brownian loop measures on Riemann surfaces, extending understanding of geometric and spectral properties.

Findings

Expressed the total mass of Brownian loops in terms of geodesic lengths.

Connected electrical thickness and Velling–Kirillov potential to Brownian loop measures.

Derived formulas involving zeta-regularized determinants of Laplacians.

Abstract

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.