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Outlier eigenvalues for full rank deformed single ring random matrices

Ching-Wei Ho, Zhi Yin, Ping Zhong

TL;DR

The paper analyzes outlier eigenvalues of the full-rank deformed single ring model $A_n + U_n\,\Sigma_n\, V_n$ where $U_n,V_n$ are Haar unitary and $A_n$, $\Sigma_n$ converge in $*$-moments to $a$ and $\Sigma$ in a $W^*$-probability space. Building on Brown measure theory and free probability, the authors derive a two-domain framework outside the Brown measure support: an outer domain $\Theta_{\rm out}$ where outliers may persist and a inner domain $\Theta_{\rm in}$ where outliers do not occur, with precise stability results for full-rank perturbations and finite-rank perturbations respectively. The main methodology combines subordination techniques for free convolutions, resolvent and polynomial convergence arguments, and Weingarten calculus for Haar matrices to compare the perturbed and unperturbed models, culminating in a Rouche-type argument that tracks zeros of determinant-like quantities. The results extend earlier work by Benaych-Georges and Rochet to full-rank deformations and provide a rigorous link between the Brown measure of $a+T$ (with $T$ R-diagonal) and the spectral behavior of large deformed matrices, including explicit domain descriptions and stability criteria. Overall, the work advances the understanding of how low- and full-rank perturbations interact with non-Hermitian random matrix limits governed by Brown measures, with implications for spectral stability in complex systems. The techniques have potential applications in physics-inspired random matrix models and in operator-algebraic descriptions of large complex systems where non-self-adjoint perturbations are relevant.

Abstract

Let $A_n$ be an $n \times n$ deterministic matrix and $Σ_n$ be a deterministic non-negative matrix such that $A_n$ and $Σ_n$ converge in $*$-moments to operators $a$ and $Σ$ respectively in some $W^*$-probability space. We consider the full rank deformed model $A_n + U_n Σ_n V_n,$ where $U_n$ and $V_n$ are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of $A_n + U_nΣ_n V_n$ in two domains that are outside the support of the Brown measure of $a +u Σ$. We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When $A_n$ has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).

Outlier eigenvalues for full rank deformed single ring random matrices

TL;DR

The paper analyzes outlier eigenvalues of the full-rank deformed single ring model where are Haar unitary and , converge in -moments to and in a -probability space. Building on Brown measure theory and free probability, the authors derive a two-domain framework outside the Brown measure support: an outer domain where outliers may persist and a inner domain where outliers do not occur, with precise stability results for full-rank perturbations and finite-rank perturbations respectively. The main methodology combines subordination techniques for free convolutions, resolvent and polynomial convergence arguments, and Weingarten calculus for Haar matrices to compare the perturbed and unperturbed models, culminating in a Rouche-type argument that tracks zeros of determinant-like quantities. The results extend earlier work by Benaych-Georges and Rochet to full-rank deformations and provide a rigorous link between the Brown measure of (with R-diagonal) and the spectral behavior of large deformed matrices, including explicit domain descriptions and stability criteria. Overall, the work advances the understanding of how low- and full-rank perturbations interact with non-Hermitian random matrix limits governed by Brown measures, with implications for spectral stability in complex systems. The techniques have potential applications in physics-inspired random matrix models and in operator-algebraic descriptions of large complex systems where non-self-adjoint perturbations are relevant.

Abstract

Let be an deterministic matrix and be a deterministic non-negative matrix such that and converge in -moments to operators and respectively in some -probability space. We consider the full rank deformed model where and are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of in two domains that are outside the support of the Brown measure of . We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).

Paper Structure

This paper contains 18 sections, 42 theorems, 260 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Deformed single ring model
  3. The Brown measure and convergence of eigenvalue distributions
  4. Main results
  5. Outline of the proofs
  6. Auxiliary results in free probability theory
  7. Basic notions of free probability
  8. Support of the Brown measure of $a+T$
  9. Properties of the support via subordination functions
  10. No outliers; proof of Theorem \ref{['thm:no-outlier']}
  11. Proof of Proposition \ref{['prop:no-outlier-1']} and Corollary \ref{['cor:no-outlier-1']}
  12. Proof of Proposition \ref{['prop:no-outlier-2']}
  13. Outliers in the outer domain; proof of Theorem \ref{['thm:outlier']}
  14. Spectral radius of $R_n^{'}(z) Y_n$
  15. Convergence of a random matrix polynomial
  16. ...and 3 more sections

Key Result

Theorem 1.1

BercoviciZhong2022 Suppose that $a$ and $T$ are freely independent in $(\mathcal{A}, \tau)$, and $T$ is R-diagonal. Then the support of the Brown measure $\mu_{a+ T}$ is contained in the closure of $\Omega(T,a)$. Moreover, the density formula of $\mu_{a+T}$ can be expressed in terms of the subordina

Figures (1)

  • Figure 1: The matrix $A_n^{"}$ has eigenvalues (marked red) at $0.75 + 0.25i$, $0.65+0.25i$, $-1.5+1.5i$, $-2 + 1.2 i$, $-1 - i,$ and $-1-0.5i$. The outliers of $A_n + U_n\Sigma_n V_n$ are near $-1.5+1.5i$, $-2 + 1.2 i$, $-1 - i$ and $-1-0.5i$.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 72 more