Brownian Loops and the Selberg Zeta Function
Probability
2026-01-21 v1 Geometric Topology
Spectral Theory
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.
Keywords
Cite
@article{arxiv.2601.13086,
title = {Brownian Loops and the Selberg Zeta Function},
author = {Roman Lemonde and Jian Wang},
journal= {arXiv preprint arXiv:2601.13086},
year = {2026}
}
Comments
19 pages, 1 figure