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

Yilin Wang, Yuhao Xue

TL;DR

This work connects stochastic geometry and spectral theory on Riemann surfaces by expressing geodesic length data through the Brownian loop measure μ_X. By exploiting conformal invariance and a precise analysis of loops in fixed homotopy classes (hyperbolic and flat cases), the authors derive exact identities relating the length spectra of a surface and its punctured or bounded variants. They further show how these identities yield explicit computations (e.g., for annuli) and relate the Brownian loop mass to the zeta-regularized determinant of the Laplacian via a length-spectrum renormalization, illuminating deep links between probabilistic loop measures, trace formulas, and spectral invariants. The results provide new tools to study length spectra, with potential connections to classical identities and Teichmüller theory through the appearance of Kähler potentials and determinant formulas.

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 Kähler 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.

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

TL;DR

This work connects stochastic geometry and spectral theory on Riemann surfaces by expressing geodesic length data through the Brownian loop measure μ_X. By exploiting conformal invariance and a precise analysis of loops in fixed homotopy classes (hyperbolic and flat cases), the authors derive exact identities relating the length spectra of a surface and its punctured or bounded variants. They further show how these identities yield explicit computations (e.g., for annuli) and relate the Brownian loop mass to the zeta-regularized determinant of the Laplacian via a length-spectrum renormalization, illuminating deep links between probabilistic loop measures, trace formulas, and spectral invariants. The results provide new tools to study length spectra, with potential connections to classical identities and Teichmüller theory through the appearance of Kähler potentials and determinant formulas.

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 separating and , or the Velling--Kirillov Kähler 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.
Paper Structure (13 sections, 21 theorems, 85 equations, 2 figures)

This paper contains 13 sections, 21 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Identity on surface length spectra
  3. Explicit computations of Brownian loop measure
  4. Zeta-regularized determinant of Laplacian
  5. Brownian loop measure on Riemannian surfaces
  6. Brownian loop measure in a homotopy class
  7. On hyperbolic surfaces
  8. On flat tori
  9. Consequences
  10. Puncturing the surface
  11. On surfaces with boundary
  12. Zeta-regularized determinant of Laplacian
  13. Basics on Brownian motion on Riemannian manifolds

Key Result

Lemma 1.1

If $(X,g)$ admits a complete hyperbolic metric that is conformal to $g$, the total mass of the Brownian loop measure in an essential homotopy class of closed curves on $(X,g)$ equals where $\gamma$ is the unique hyperbolic geodesic in the homotopy class and $\ell_\gamma (X)$ is the hyperbolic length of $\gamma$.

Figures (2)

  • Figure 1: Illustration of the annulus $\mathbb{A}_r = \left \langle \sigma \right \rangle\backslash\mathbb{H}$ with the complete hyperbolic metric induced by $\mathbb{H}$. The blue curve is the unique simple closed geodesic in $\mathbb A_r$.
  • Figure 2: Illustration of the annuli in the proof of Corollary \ref{['cor:touching_annulus']}.

Theorems & Definitions (37)

  • Lemma 1.1: See Lemma \ref{['lem:blm-hyp']}
  • Lemma 1.2: See Lemma \ref{['lem:blm-flat']}
  • Theorem 1.3: See Theorem \ref{['thm-id-hyp']}
  • Remark 1.4
  • Corollary 1.5: See Corollary \ref{['cor:annulus']}
  • Corollary 1.6: See Corollary \ref{['cor:thick']}
  • Corollary 1.7
  • Corollary 1.8: See Corollary \ref{['cor:boundary_total']}
  • Proposition 1.9: See Proposition \ref{['prop:det_loop']}
  • Remark 2.1
  • ...and 27 more