Brownian Loops and the Selberg Zeta Function

TL;DR

This paper explores the relationship between Brownian loop measures on hyperbolic surfaces and the Selberg zeta function, offering new insights into spectral geometry and probabilistic interpretations of Laplacian determinants.

Contribution

It introduces a method to compute Brownian loop measures with killing in homotopy classes and links these measures to the Selberg zeta function for geometrically finite surfaces.

Findings

Computed the mass of Brownian loops with killing in homotopy classes.

Established a connection between total loop mass and the Selberg zeta function.

Provided a probabilistic interpretation of regularized Laplacian determinants.

Abstract

We study the Brownian loop measure on hyperbolic surfaces for Brownian motion with a constant killing rate. We compute the mass of Brownian loops with killing in a free homotopy class and then relate the total mass of loops in all essential homotopy classes to the Selberg zeta function when the surface is geometrically finite. As an application, we provide a probabilistic interpretation of different notions of regularized determinants of Laplacian, in both the compact and infinite-area cases.