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Eigenvalues of random matrices from compact classical groups in Wasserstein metric

Bence Borda

TL;DR

This work analyzes how eigenvalues of Haar-distributed matrices from compact classical groups diverge from uniformity on the unit circle under the Wasserstein metric $W_2$. It leverages the determinantal point process structure of the nontrivial eigenangles and a Parseval-based representation to obtain exact formulas for $ ext{E}ig[W_2^2ig]$ and $ ext{Var}ig[W_2^2ig]$, followed by precise asymptotics and a limit law for $W_2^2$ after natural centering and scaling. The results are unified across unitary, orthogonal, and symplectic symmetry classes, yielding group-dependent constants and explicit infinite-series limit laws $oldsymbol{ta}_G$, with the unitary case admitting an explicit density for the limit variable. The findings connect $W_2$-distances to arc-count discrepancies and the characteristic polynomial on the unit circle, enriching the probabilistic description of eigenvalue equidistribution in Dyson ensembles.

Abstract

The circular unitary ensemble and its generalizations concern a random matrix from a compact classical group $\mathrm{U}(N)$, $\mathrm{SU}(N)$, $\mathrm{O}(N)$, $\mathrm{SO}(N)$ or $\mathrm{USp}(N)$ distributed according to the Haar measure. The eigenvalues are known to be very evenly distributed on the unit circle. In this paper, we study the distance from the empirical measure of the eigenvalues to uniformity in the quadratic Wasserstein metric $W_2$. After finding the exact value of the expected value and the variance, we deduce a limit law for the square of the Wasserstein distance. We reformulate our results in terms of the $L^2$ average of the number of eigenvalues in circular arcs, and also in terms of the characteristic polynomial of the matrix on the unit circle.

Eigenvalues of random matrices from compact classical groups in Wasserstein metric

TL;DR

This work analyzes how eigenvalues of Haar-distributed matrices from compact classical groups diverge from uniformity on the unit circle under the Wasserstein metric . It leverages the determinantal point process structure of the nontrivial eigenangles and a Parseval-based representation to obtain exact formulas for and , followed by precise asymptotics and a limit law for after natural centering and scaling. The results are unified across unitary, orthogonal, and symplectic symmetry classes, yielding group-dependent constants and explicit infinite-series limit laws , with the unitary case admitting an explicit density for the limit variable. The findings connect -distances to arc-count discrepancies and the characteristic polynomial on the unit circle, enriching the probabilistic description of eigenvalue equidistribution in Dyson ensembles.

Abstract

The circular unitary ensemble and its generalizations concern a random matrix from a compact classical group , , , or distributed according to the Haar measure. The eigenvalues are known to be very evenly distributed on the unit circle. In this paper, we study the distance from the empirical measure of the eigenvalues to uniformity in the quadratic Wasserstein metric . After finding the exact value of the expected value and the variance, we deduce a limit law for the square of the Wasserstein distance. We reformulate our results in terms of the average of the number of eigenvalues in circular arcs, and also in terms of the characteristic polynomial of the matrix on the unit circle.
Paper Structure (11 sections, 6 theorems, 158 equations, 1 figure, 4 tables)

This paper contains 11 sections, 6 theorems, 158 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A reduction step
  4. The empirical spectral measure as a determinantal point process
  5. Exact formulas for the expected value and the variance
  6. Proof for $\mathrm{U}(N)$ and $\mathrm{SU}(N)$
  7. Proof for $\mathrm{SO}(2N+1)$ and $\mathrm{O}(2N+1)$
  8. Proof for $\mathrm{SO}(2N)$
  9. Proof for $\mathrm{O}^- (2N+2)$ and $\mathrm{USp}(2N)$
  10. Asymptotic formulas for the expected value and the variance
  11. Limit laws

Key Result

Theorem 1

Let $G=\mathrm{U}(N)$, $\mathrm{SU}(N)$, $\mathrm{SO}(2N+1)$, $\mathrm{O}(2N+1)$, $\mathrm{SO}(2N)$, $\mathrm{O}^- (2N+2)$ or $\mathrm{USp}(2N)$, and let $A \in G$ be a random matrix distributed according to the normalized Haar measure on $G$. We have with universal implied constants. The number of nontrivial eigenvalues $N_0$ and the constants $c_G$ and $\sigma_G$ are given in Table tablecGsigma

Figures (1)

  • Figure 1: Regions on which $V(k,\ell)$ is estimated. Note that $V(k,\ell)=0$ in the shaded regions.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof : Proof of Theorem \ref{['exacttheorem']} (i)
  • proof : Proof of Theorem \ref{['exacttheorem']} (ii)
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['exacttheorem']} (iii)
  • ...and 4 more