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Outlier eigenvalues and eigenvectors of generalized Wigner matrices with finite-rank perturbations

Bishakh Bhattacharya, Arijit Chakrabarty, Rajat Subhra Hazra

TL;DR

We analyze a generalized Wigner matrix $W_N$ perturbed by a finite-rank deterministic matrix, focusing on outlier eigenvalues and the associated eigenvectors. Under mild assumptions on the variance profile and perturbation growth, we establish a central limit theorem for the top $k$ outliers, with a joint Gaussian limit whose covariance is given by quadratic forms in the planted directions. We also prove Gaussian fluctuations for the alignment of the outlier eigenvectors with the perturbation directions, alongside delocalization bounds, and we develop a functional central limit theorem for the eigenvector process viewed as a function on $[0,1]$ within a Sobolev space framework. A martingale CLT for quadratic forms of the Wigner matrix is proved and used to control the key fluctuation terms. Together, these results provide a detailed description of edge behavior and eigenvector dynamics in dense, inhomogeneous random matrices with finite-rank perturbations, with implications for signal extraction in noisy high-dimensional systems.

Abstract

A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. We prove central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for non-aligned components. Furthermore, we discuss the convergence of the eigenvector process in a Sobolev space framework.

Outlier eigenvalues and eigenvectors of generalized Wigner matrices with finite-rank perturbations

TL;DR

We analyze a generalized Wigner matrix perturbed by a finite-rank deterministic matrix, focusing on outlier eigenvalues and the associated eigenvectors. Under mild assumptions on the variance profile and perturbation growth, we establish a central limit theorem for the top outliers, with a joint Gaussian limit whose covariance is given by quadratic forms in the planted directions. We also prove Gaussian fluctuations for the alignment of the outlier eigenvectors with the perturbation directions, alongside delocalization bounds, and we develop a functional central limit theorem for the eigenvector process viewed as a function on within a Sobolev space framework. A martingale CLT for quadratic forms of the Wigner matrix is proved and used to control the key fluctuation terms. Together, these results provide a detailed description of edge behavior and eigenvector dynamics in dense, inhomogeneous random matrices with finite-rank perturbations, with implications for signal extraction in noisy high-dimensional systems.

Abstract

A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. We prove central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for non-aligned components. Furthermore, we discuss the convergence of the eigenvector process in a Sobolev space framework.
Paper Structure (28 sections, 25 theorems, 417 equations)

This paper contains 28 sections, 25 theorems, 417 equations.

Table of Contents

  1. Introduction
  2. Outline:
  3. The Setup and the Main Results
  4. Setup and Assumptions:
  5. Notation:
  6. Main Results.
  7. Eigenvalue Fluctuations – Proofs of Main Results
  8. Proof of Theorem \ref{['Th.EigenValueCLT']}: a sketch
  9. Step 1: Resolvent expansion.
  10. Step 2: Truncation (Lemma \ref{['Le.S1']}).
  11. Step 3: Replacing higher-order terms (Lemma \ref{['Le.S2']}).
  12. Step 4: Deterministic approximation (Lemma \ref{['Le.S3']}).
  13. Step 5: Bounding the difference (Lemma \ref{['Le.S4']}).
  14. Step 6: Extracting the main fluctuation (Lemma \ref{['Le.S5']}).
  15. Step 7: Final expression.
  16. ...and 13 more sections

Key Result

Theorem 2.1

( Central Limit Theorem for Eigenvalues) Under Assumptions assump:A1 and assump:A2, for all $i$ such that $1 \leq i \leq k$, we have the expansion As a consequence, the vector of fluctuations of the top $k$ eigenvalues converges in distribution to a multivariate Gaussian: where $(G_1,\ldots,G_k)$ is a centered Gaussian $\mathbb R^k$-valued random vector with covariance for all $1 \le i,j \le k$

Theorems & Definitions (60)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3: Gaussian fluctuation of eigenvector alignment
  • Theorem 2.4: Eigenvector delocalization
  • Theorem 2.5
  • Remark 2.2: Identification of the Limit in the Homogeneous Case
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 50 more