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Brownian Loops and the Selberg Zeta Function

Roman Lemonde, Jian Wang

TL;DR

This work builds a bridge between stochastic Brownian loop measures and spectral geometry on hyperbolic surfaces. By introducing Brownian motion with killing and computing the loop mass in essential homotopy classes, the authors relate the total loop mass to the Selberg zeta function via $-\log Z_X\left(\tfrac12+\sqrt{\tfrac14+\kappa}\right)$, providing a probabilistic interpretation of zeta values on $(\delta,\infty)$. They further connect these loop masses to renormalized heat traces and to regularized determinants of the Laplacian: in the compact case, through $\det_\zeta\Delta$ and Sarnak-type identities, and in the infinite-area case, via the 0-trace determinant $\det_0(\Delta_X+s(s-1))$ alongside a 0-trace–loop-mass correspondence. The results unify stochastic and spectral viewpoints, showing that loop masses encode resonances and regularization data, and offer probabilistic interpretations of determinant formulas in both finite and infinite-area hyperbolic surfaces.

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.

Brownian Loops and the Selberg Zeta Function

TL;DR

This work builds a bridge between stochastic Brownian loop measures and spectral geometry on hyperbolic surfaces. By introducing Brownian motion with killing and computing the loop mass in essential homotopy classes, the authors relate the total loop mass to the Selberg zeta function via , providing a probabilistic interpretation of zeta values on . They further connect these loop masses to renormalized heat traces and to regularized determinants of the Laplacian: in the compact case, through and Sarnak-type identities, and in the infinite-area case, via the 0-trace determinant alongside a 0-trace–loop-mass correspondence. The results unify stochastic and spectral viewpoints, showing that loop masses encode resonances and regularization data, and offer probabilistic interpretations of determinant formulas in both finite and infinite-area hyperbolic surfaces.

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.
Paper Structure (10 sections, 11 theorems, 63 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Brownian loop measure.
  4. Selberg zeta functions
  5. The Brownian Loop Measure in Homotopy Classes
  6. Trace and determinant
  7. Brownian loop measure and regularized heat trace
  8. Brownian loop measure and regularized determinant of the Laplacian
  9. The Compact Case
  10. The infinite-area case

Key Result

Theorem 1

Let $X$ be as above. Then for any $\kappa\geq-\frac{1}{4}$ such that $\frac{1}{2}+\sqrt{\frac{1}{4}+\kappa}>\delta$, where for each $\gamma\in \mathcal{P}_X$, $m\in \mathbb N^*$, $\mathcal{C}_X(\gamma^m)$ denotes the set of all oriented closed curves on $X$ that are freely homotopic to $\gamma^m$.

Figures (1)

  • Figure 1: Illustration of a closed geodesic (yellow) and a Brownian loop (blue) of the same homotopic type on the hyperbolic cylinder. Both the geodesic and the Brownian loop are lifted to the universal cover $\mathbb{H}$. The light green region is a fundamental domain of the hyperbolic cylinder.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Proposition 2.2: lawler
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • ...and 11 more