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Universality for Diagonal Eigenvector Overlaps of non-Hermitian Random Matrices

Mohammed Osman

TL;DR

The paper proves universality for the joint distribution of eigenvalues and their diagonal eigenvector overlaps for non-Hermitian Wigner matrices in both bulk and edge, covering real and complex cases. The authors develop a Gauss-divisible framework, leveraging a partial Schur decomposition, resolvent analyses, and a dynamic characteristic flow to connect non-Gaussian ensembles to Gaussian references, then remove the Gaussian component via Green’s function comparison. Central technical advances include two-resolvent local laws, sharp bounds on singular-vector overlaps at the edge, and robust control of the least nonzero singular value, enabling precise asymptotics for the overlap densities ρ_{β,bulk} and ρ_{β,edge}. The results extend the universality paradigm for eigenvector overlaps beyond the complex Ginibre and related integrable ensembles, with potential implications for perturbation theory and spectral stability in non-Hermitian random systems.

Abstract

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the proof we obtain a bound for the least non-zero singular value of $X-z$ when $z$ is an edge eigenvalue and a bound for the inner product between left and right singular vectors of $X-z$ when $|z|=1+O(N^{-1/2})$.

Universality for Diagonal Eigenvector Overlaps of non-Hermitian Random Matrices

TL;DR

The paper proves universality for the joint distribution of eigenvalues and their diagonal eigenvector overlaps for non-Hermitian Wigner matrices in both bulk and edge, covering real and complex cases. The authors develop a Gauss-divisible framework, leveraging a partial Schur decomposition, resolvent analyses, and a dynamic characteristic flow to connect non-Gaussian ensembles to Gaussian references, then remove the Gaussian component via Green’s function comparison. Central technical advances include two-resolvent local laws, sharp bounds on singular-vector overlaps at the edge, and robust control of the least nonzero singular value, enabling precise asymptotics for the overlap densities ρ_{β,bulk} and ρ_{β,edge}. The results extend the universality paradigm for eigenvector overlaps beyond the complex Ginibre and related integrable ensembles, with potential implications for perturbation theory and spectral stability in non-Hermitian random systems.

Abstract

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the proof we obtain a bound for the least non-zero singular value of when is an edge eigenvalue and a bound for the inner product between left and right singular vectors of when .
Paper Structure (19 sections, 45 theorems, 485 equations, 1 figure)

This paper contains 19 sections, 45 theorems, 485 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Main Results
  4. Notation
  5. Partial Schur Decomposition
  6. Proof of Theorem \ref{['thm1']} for Gauss-divisible Matrices
  7. Proof of Theorem \ref{['thm2']} for Gauss-divisible Matrices
  8. Complex matrices:
  9. Real matrices:
  10. Overlap of Singular Vectors
  11. Characteristic Flow
  12. Removal of Gaussian Component
  13. Comparison Lemmas and Proofs of Theorems \ref{['thm1']} and \ref{['thm2']}
  14. Least Non-Zero Singular Value
  15. Overlap
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ be a non-Hermitian Wigner matrix with real ($\beta=1$) or complex ($\beta=2$) entries. Let $f\in C^{2}(\mathbb{F}_{\beta})$ and $g\in C^{5}([0,\infty])$ have compact support and, for $z_{0}\in\mathbb{C}$, define Let $\mathrm{d}m(z,s)$ be the Lebesgue measure on $\mathbb{F}_{\beta}\times[0,\infty]$. Then there is a $\tau>0$ such that:

Figures (1)

  • Figure 1: Clockwise from the top left: the overlap $|\mathbf{w}_{n}^{*}F\mathbf{w}_{m}|$ against the singular value $|\lambda_{n}|$ for different values of $m=1,10,100,1000$. The data is obtained from diagonalising $X-1$ where $X\sim Gin_{2}(1000)$.

Theorems & Definitions (88)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 78 more