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Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus

Laura Monk, Rares Stan

TL;DR

The paper proves that the Dirac spectrum of typical high-genus hyperbolic surfaces with a nontrivial spin structure converges, in a probabilistic sense, to the Dirac spectrum of the hyperbolic plane. It achieves this via a Dirac-adapted Selberg trace formula and a carefully crafted family of test functions $h_t$, yielding a precise density limit $\frac{1}{4\pi}\int_a^b \lambda\coth(\pi\lambda)\,d\lambda$ with explicit remainder $O((b+1)/\sqrt{\log g})$. A uniform Weyl law and multiplicity bounds follow, valid uniformly for typical surfaces under the Weil--Petersson measure, while pathological examples show the Dirac spectrum can misbehave on certain non-typical surfaces. The results illuminate how Benjamini--Schramm convergence to the plane governs Dirac spectral statistics, with implications for quantum chaos and spectral geometry on random hyperbolic surfaces.

Abstract

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus $g$ with $o(\sqrt{g})$ cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

Spectral convergence of the Dirac operator on typical hyperbolic surfaces of high genus

TL;DR

The paper proves that the Dirac spectrum of typical high-genus hyperbolic surfaces with a nontrivial spin structure converges, in a probabilistic sense, to the Dirac spectrum of the hyperbolic plane. It achieves this via a Dirac-adapted Selberg trace formula and a carefully crafted family of test functions , yielding a precise density limit with explicit remainder . A uniform Weyl law and multiplicity bounds follow, valid uniformly for typical surfaces under the Weil--Petersson measure, while pathological examples show the Dirac spectrum can misbehave on certain non-typical surfaces. The results illuminate how Benjamini--Schramm convergence to the plane governs Dirac spectral statistics, with implications for quantum chaos and spectral geometry on random hyperbolic surfaces.

Abstract

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil-Petersson surfaces of large genus with cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.
Paper Structure (25 sections, 18 theorems, 84 equations)

This paper contains 25 sections, 18 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Setting and motivation
  3. Spectral convergence of the Dirac operator
  4. Upper bounds and pathological surfaces
  5. Applications
  6. Acknowledgements
  7. Preliminaries
  8. Spin structures and the Dirac operator
  9. The spectrum of Dirac operators
  10. Cusps and nontrivial spin structures
  11. Multiplicities and counting functions
  12. Pathological examples of Dirac spectra
  13. The Selberg trace formula for Dirac operators
  14. Random hyperbolic surfaces
  15. Plan of the proof and first estimates
  16. ...and 10 more sections

Key Result

Theorem 1

Let $(k(g))_{g \geq 2}$ be a sequence of non-negative integers such that $k(g) = o(\sqrt{g})$ as $g \rightarrow + \infty$. There exists a constant $C>0$ such that, for any $0\leq a \leq b$, any $g \geq 2$, any $X \in \mathcal{A}_{g,k(g)}$, and any nontrivial spin structure $\varepsilon$ on $X$, we h where the remainder satisfies:

Theorems & Definitions (36)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • proof
  • ...and 26 more