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Determinants of Laplacians on converging hyperbolic surfaces

Renan Gross, Guy Lachman, Asaf Nachmias

TL;DR

This work analyzes the asymptotics of the log determinant of the Laplacian on sequences of compact hyperbolic surfaces under local (BS) convergence to a random rooted limit. The authors decompose the zeta-regularized determinant into a volume term and heat-kernel remainders, and establish uniform control of small-time and large-time contributions using collar geometry, heat-kernel bounds, and Vitali convergence under the condition of uniformly integrable short geodesics. They prove that if short geodesics are uniformly integrable, the normalized log determinant converges to a constant $E_\mu$ depending only on the limit law $\mu$, with a complementary limsup inequality when this condition fails; they also relate $E_\mu$ to the derivative of a zeta-type function for the limit. The results extend previous work by Naud beyond the hyperbolic plane and connect to concepts like Lyons’ tree entropy and $L^2$-invariants, clarifying how local geometric constraints govern spectral determinants in large-volume random hyperbolic geometry.

Abstract

Let $S_k$ be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of $S_k$ converges to a constant depending only the law of the limiting surface.

Determinants of Laplacians on converging hyperbolic surfaces

TL;DR

This work analyzes the asymptotics of the log determinant of the Laplacian on sequences of compact hyperbolic surfaces under local (BS) convergence to a random rooted limit. The authors decompose the zeta-regularized determinant into a volume term and heat-kernel remainders, and establish uniform control of small-time and large-time contributions using collar geometry, heat-kernel bounds, and Vitali convergence under the condition of uniformly integrable short geodesics. They prove that if short geodesics are uniformly integrable, the normalized log determinant converges to a constant depending only on the limit law , with a complementary limsup inequality when this condition fails; they also relate to the derivative of a zeta-type function for the limit. The results extend previous work by Naud beyond the hyperbolic plane and connect to concepts like Lyons’ tree entropy and -invariants, clarifying how local geometric constraints govern spectral determinants in large-volume random hyperbolic geometry.

Abstract

Let be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of converges to a constant depending only the law of the limiting surface.
Paper Structure (6 sections, 9 theorems, 76 equations, 3 figures)

This paper contains 6 sections, 9 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous work and remarks
  3. Injectivity radius and heat kernel bounds
  4. Proofs
  5. The case $\mathbf{r(x)^2 \geq t}$
  6. The case $\mathbf{r(x)^2 \leq t}$

Key Result

Theorem 1

There exists a universal constant $E_{\mathbb{H}} > 0$ so that the following holds. Let ${S}_k$ be a sequence of compact hyperbolic surfaces of volume tending to $\infty$ that locally converges to a random rooted surface $(\mathbf{{S}}_\infty, \mathbf{x}_0)$ with distribution $\mu$. Define $E_{\mu}\ where $p_{t}^{{S}}(x,y)$ is the heat-kernel of the Laplacian on a Riemannian surface ${S}$.

Figures (3)

  • Figure 1: In addition to the hyperbolic plane, a sequence of hyperbolic surfaces can converge to various infinite structures, such as a thick grid, Jacob's ladder, or a Cantor tree. Image taken from angel_hutchcroft_nachmias_ray_unimodular_random_maps.
  • Figure 2: When $\tilde{x}$ is far from the boundary of the the collar, the group elements which move it the least are the collar translations.
  • Figure 3: For large $t$, many copies of $B(x, r(x))$ can fit in $B(x,\sqrt{t})$.

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • Lemma 4
  • Theorem 5: gross_lachman_nachmias_sharp_lower_bounds, Theorem 2 and Corollary 9
  • Theorem 6: Bounding the heat kernel by volume of balls; Corollary 3.1 in li_yau_main with $\alpha=3/2$ and $\varepsilon=1/2$
  • Remark 7
  • Remark 8
  • Lemma 9: Collar lemma
  • Proposition 10: Injectivity radius estimate
  • ...and 7 more