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Largest gaps between bulk eigenvalues of unitary-invariant random Hermitian matrices

Christophe Charlier

TL;DR

The authors analyze largest gaps between bulk eigenvalues of unitary-invariant random Hermitian matrices with a general potential $V$ and establish a universal extreme-value limit. By deriving sharp bulk kernel asymptotics via Riemann–Hilbert analysis and linking gap events to CUE gap probabilities, they show that the rescaled gaps $\tau_k^{(n)}$ converge to a gamma-Gumbel distribution shifted by an explicit $c_{V,I}$ that depends on the interval $I$ and the potential $V$. The limit involves the equilibrium density $\rho$, the vanishing order $q$, and explicit constants determined by the minimizers of $\rho$ on $I$. The results extend Feng and Wei beyond Gaussian potentials and yield a precise, multi-faceted description of extreme bulk gaps, with specializations to GUE, LUE, and JUE, and connections to Poisson point processes for gap counts. The work advances the understanding of extreme-value statistics in random matrix theory under broad, analytically tractable conditions.

Abstract

We study $n\times n$ random Hermitian matrix ensembles that are invariant under unitary conjugation. Let $I$ be a finite union of intervals lying in the bulk, and let $m_{k}^{(n)}$ be the $k$-th largest gap between consecutive eigenvalues lying in $I$. We prove that the rescaled gap $\smash{τ_{k}^{(n)}}$, which is defined by \begin{align*} m_{k}^{(n)} = \frac{1}{2π\inf_{I}ρ} \bigg( \frac{\sqrt{32 \log n}}{n} + \frac{3q-8}{2q} \frac{ \log(2\log n)}{n \sqrt{2\log n}} + \frac{4τ_{k}^{(n)}}{n \sqrt{2\log n}} \bigg), \end{align*} converges in distribution as $n\to +\infty$ to a gamma-Gumbel random variable that is shifted by an explicit constant $c_{V,I}$ depending only on $I$ and on the potential $V$. Here $ρ$ is the density of the equilibrium measure and $q\in \N_{>0}$ is the highest order at which $ρ(x)$ approaches $\inf_{I}ρ$ with $x\in I$; for example, if $ρ(x)=1/(π\sqrt{x(1-x)})$, then $q=2$ if $\frac{1}{2}\in \overline{I}$ and $q=1$ otherwise. This work extends a result of Feng and Wei beyond the Gaussian potential.

Largest gaps between bulk eigenvalues of unitary-invariant random Hermitian matrices

TL;DR

The authors analyze largest gaps between bulk eigenvalues of unitary-invariant random Hermitian matrices with a general potential and establish a universal extreme-value limit. By deriving sharp bulk kernel asymptotics via Riemann–Hilbert analysis and linking gap events to CUE gap probabilities, they show that the rescaled gaps converge to a gamma-Gumbel distribution shifted by an explicit that depends on the interval and the potential . The limit involves the equilibrium density , the vanishing order , and explicit constants determined by the minimizers of on . The results extend Feng and Wei beyond Gaussian potentials and yield a precise, multi-faceted description of extreme bulk gaps, with specializations to GUE, LUE, and JUE, and connections to Poisson point processes for gap counts. The work advances the understanding of extreme-value statistics in random matrix theory under broad, analytically tractable conditions.

Abstract

We study random Hermitian matrix ensembles that are invariant under unitary conjugation. Let be a finite union of intervals lying in the bulk, and let be the -th largest gap between consecutive eigenvalues lying in . We prove that the rescaled gap , which is defined by \begin{align*} m_{k}^{(n)} = \frac{1}{2π\inf_{I}ρ} \bigg( \frac{\sqrt{32 \log n}}{n} + \frac{3q-8}{2q} \frac{ \log(2\log n)}{n \sqrt{2\log n}} + \frac{4τ_{k}^{(n)}}{n \sqrt{2\log n}} \bigg), \end{align*} converges in distribution as to a gamma-Gumbel random variable that is shifted by an explicit constant depending only on and on the potential . Here is the density of the equilibrium measure and is the highest order at which approaches with ; for example, if , then if and otherwise. This work extends a result of Feng and Wei beyond the Gaussian potential.
Paper Structure (10 sections, 11 theorems, 193 equations)

This paper contains 10 sections, 11 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Outline.
  3. Estimates on the kernel
  4. Gap probabilities of \ref{['def of eig distribution']}
  5. Gap probabilities of the CUE
  6. Proof of Theorem \ref{['thm:main']}
  7. Upper bound \ref{['limsup']}
  8. Lower bound \ref{['liminf']}
  9. A characterization of $\Sigma_{k}(a_{1},\ldots,a_{k})$
  10. Acknowledgements.

Key Result

Theorem 1.2

Suppose $V:\mathbb{R}\to \mathbb{R}\cup\{+\infty\}$ is a potential satisfying Assumptions ass:V, and let $I$ be a finite union of intervals such that $\overline{I}$ has pairwise distinct endpoints and satisfies $\overline{I} \subset \mathcal{S}^{\circ}$. Let $\lambda_{1}<\dots<\lambda_{n}$ be sample where $q$ is as in def of q. For any fixed $x \in \mathbb{R}$, the random variables $\#\{\tau_{k}^{

Theorems & Definitions (25)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 3.1
  • ...and 15 more