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A note on the eigenvalue rigidity of hyperbolic surfaces in the random cover model

Elena Kim, Zhongkai Tao

Abstract

Let $X$ be a compact connected orientable hyperbolic surface and $X_n$ be a degree $n$ random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on $X_n$ converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. Our proof relies on the Selberg trace formula and a variant of the polynomial method.

A note on the eigenvalue rigidity of hyperbolic surfaces in the random cover model

Abstract

Let be a compact connected orientable hyperbolic surface and be a degree random cover. We show that, with high probability, the distribution of eigenvalues of the Laplacian on converges to the spectral measure of the hyperbolic plane with polynomially decaying error. This is analogous to the eigenvalue rigidity property for graphs of Huang--Yau [arXiv:2102.00963] and improves the logarithmic bound of Monk [arXiv:2002.00869]. Our proof relies on the Selberg trace formula and a variant of the polynomial method.
Paper Structure (12 sections, 7 theorems, 73 equations)

This paper contains 12 sections, 7 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Previous work
  3. Proof idea
  4. Preliminaries
  5. Surface group
  6. Selberg Trace Formula
  7. Proof of Theorem \ref{['thm:cover']}
  8. The proof of Theorem \ref{['thm:cover']}
  9. Proof of Proposition \ref{['prop:expectation_squared']}
  10. Uniform bound
  11. Polynomial expansion
  12. Markov brothers' inequality

Key Result

Theorem 1

Let $X$ be a compact connected orientable hyperbolic surface of genus $g\geq 2$. For any $\epsilon>0$, there exist $\alpha=\alpha(g,\epsilon)>0$ and $C=C(X,\epsilon)>0$ such that the following is true. Let $X_n$ be a degree $n$ cover of $X$ taken uniformly at random and let $\lambda_j(X_n)$ be the $ where $\lambda_j\geq 1/4$ is defined by Moreover, we have the following Weyl law for $N_{X_n}(\Lam

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Theorem \ref{['thm:cover']}
  • Lemma 3.3
  • proof
  • ...and 3 more