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Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics

Yotam Maoz

TL;DR

The paper investigates smooth linear spectral statistics of twisted Laplacians on random $n$-covers of a fixed compact hyperbolic surface, establishing a central limit theorem in a double limit where $n\to\infty$ first and then $L\to\infty$, with fluctuations matching GOE/GUE predictions. It develops a detailed trace-formula framework, analyzes the oscillatory and diagonal contributions, and exploits recent results on the joint distribution and independence of the fixed-point variables $F_{n}(\gamma)$ to prove Gaussian limits for all moments. It further proves a probabilistic convergence for the centered energy variance when averaging over the height parameter $\alpha$, showing convergence to the GOE/GUE variance in the regime $L=o(T)$. The methods connect random covering spaces to random-matrix-type statistics through precise geodesic counting and partition-based moment bounds, and they relate to analogous Weil–Petersson results for moduli-space random surfaces. Overall, the work strengthens the bridge between spectral statistics on random hyperbolic surfaces and universal random-matrix behavior, with implications for energy-window fluctuations and variance predictions in geometric quantum chaos.

Abstract

We study smooth linear spectral statistics of twisted Laplacians on random $n$-covers of a fixed compact hyperbolic surface $X$. We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree $n$ covers of $X$. The second is the energy variance of a typical surface. In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the energy variance of a typical random $n$-cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let $n$, the covering degree, go to $\infty$ then let $L\to \infty$ where $1/L$ is the window length.

Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics

TL;DR

The paper investigates smooth linear spectral statistics of twisted Laplacians on random -covers of a fixed compact hyperbolic surface, establishing a central limit theorem in a double limit where first and then , with fluctuations matching GOE/GUE predictions. It develops a detailed trace-formula framework, analyzes the oscillatory and diagonal contributions, and exploits recent results on the joint distribution and independence of the fixed-point variables to prove Gaussian limits for all moments. It further proves a probabilistic convergence for the centered energy variance when averaging over the height parameter , showing convergence to the GOE/GUE variance in the regime . The methods connect random covering spaces to random-matrix-type statistics through precise geodesic counting and partition-based moment bounds, and they relate to analogous Weil–Petersson results for moduli-space random surfaces. Overall, the work strengthens the bridge between spectral statistics on random hyperbolic surfaces and universal random-matrix behavior, with implications for energy-window fluctuations and variance predictions in geometric quantum chaos.

Abstract

We study smooth linear spectral statistics of twisted Laplacians on random -covers of a fixed compact hyperbolic surface . We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree covers of . The second is the energy variance of a typical surface. In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the energy variance of a typical random -cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let , the covering degree, go to then let where is the window length.
Paper Structure (30 sections, 22 theorems, 305 equations)

This paper contains 30 sections, 22 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Statistics for a Fixed Height and a Random Surface
  3. The Centered Energy Variance of a Random Surface
  4. Related Results
  5. Overview of the Paper
  6. Background
  7. Covers of $X$
  8. The Trace Formula
  9. The Variables $F_{n}(\gamma)$
  10. Counting Bounds
  11. Proof of Theorem \ref{['main thm 1']}
  12. Outline of the Proof
  13. Odd $k$
  14. Even $k$
  15. Proof of Theorem \ref{['B bound']}
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $X$ and $\chi$ as before, in addition fix $\alpha\in\mathbb{R}$, then: where $\Sigma_{\text{GOE}}^{2}(\psi)$ is the "smoothed" number variance of random matrices for the $\text{GOE}$ model in the large dimension limit and is given by: and $\Sigma_{\text{GUE}}^{2}(\psi)=\frac{1}{2}\Sigma_{\text{GOE}}^{2}(\psi)$.

Theorems & Definitions (31)

  • Theorem 1.2: Naud 2022
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.6
  • Theorem 2.1: Twisted trace formula
  • Theorem 2.2: Corollary 1.7 in PZ
  • Theorem 2.3: Theorem 1.8 in surfaces
  • Corollary 2.4: Corollary 1.9 in surfaces
  • Corollary 2.5: Corollary 1.10 in surfaces
  • Lemma 2.6: Proposition 3.1 in Naud
  • ...and 21 more