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Outliers for deformed inhomogeneous random matrices

Ruohan Geng, Dang-Zheng Liu, Guangyi Zou

Abstract

Inhomogeneous random matrices with non-trivial variance profiles determined by symmetric stochastic matrices and with independent sub-Gaussian entries up to Hermitian symmetry, encompass a wide range of important models, including sparse Wigner matrices and random band matrices. In these models, the maximum entry variance-a natural proxy for sparsity-serves both as a key structural feature and a primary analytical obstacle. In this paper, we consider low-rank additive perturbations of such matrices and establish a sharp BBP phase transition for extreme eigenvalues at the level of the law of large numbers. Furthermore, in the Gaussian setting, we derive the fluctuations of spectral outliers under suitable conditions on the variance profile and perturbation. These fluctuations exhibit strong non-universality, depending on the eigenvectors, sparsity levels, and the underlying geometric structure. Our proof strategies rely on ribbon graph expansions, upper bounds for diagram functions, large-moment estimates, and the enumeration of typical diagrams.

Outliers for deformed inhomogeneous random matrices

Abstract

Inhomogeneous random matrices with non-trivial variance profiles determined by symmetric stochastic matrices and with independent sub-Gaussian entries up to Hermitian symmetry, encompass a wide range of important models, including sparse Wigner matrices and random band matrices. In these models, the maximum entry variance-a natural proxy for sparsity-serves both as a key structural feature and a primary analytical obstacle. In this paper, we consider low-rank additive perturbations of such matrices and establish a sharp BBP phase transition for extreme eigenvalues at the level of the law of large numbers. Furthermore, in the Gaussian setting, we derive the fluctuations of spectral outliers under suitable conditions on the variance profile and perturbation. These fluctuations exhibit strong non-universality, depending on the eigenvectors, sparsity levels, and the underlying geometric structure. Our proof strategies rely on ribbon graph expansions, upper bounds for diagram functions, large-moment estimates, and the enumeration of typical diagrams.
Paper Structure (44 sections, 47 theorems, 337 equations, 9 figures, 1 table)

This paper contains 44 sections, 47 theorems, 337 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Deformed Wigner matrices
  3. Inhomogeneous random matrices
  4. Models and main results
  5. Applications
  6. Method and structure
  7. Ribbon graphs and graph expansions
  8. Ribbon graph expansions
  9. Graph expansions for sub-Gaussian IRM
  10. Trivalent and typical diagrams
  11. Upper bound estimates
  12. Dominating functions for GOE
  13. GOE dominates IRM
  14. Cluster expansion of weighted diagram functions
  15. Quotient graph $G_0$ and quotient labeling $\bar{\eta}$.
  16. ...and 29 more sections

Key Result

Theorem 1.2

With $X_N$ given in Definition def:inhomo, assume that the nontrivial eigenvalues of $A_N$ satisfy where all $a_{j}$ and $a_{-j}$ are independent of $N$. If $(r+1)\sigma^{*}_N \sqrt{\log N} \to 0$ as $N\to \infty$, then for any fixed positive integer $j$, the $j$-th largest eigenvalue of $X_N$ and the $(N+1-j)$-th largest eigenvalue of $X_N$

Figures (9)

  • Figure 1: An example of oriented ribbon graph: $k=12$, $J=\{1,3,4,5,7,8,11,12\}$, and $\pi=(1\ 3)(4\ 12)(5\ 7)(8 \ 11)$. The boundary edges are colored red.
  • Figure 2: An example of nonoriented ribbon graph: $k=12$, $J=\{1,2,3,4,5,8,9,12\}$, and $\pi=(1\ 4)(2\ 3)(5\ 8)(9 \ 12)$. The boundary edges are colored red, and $\vec{e_2},\vec{e_3}$ are glued in an opposite direction.
  • Figure 3: Okounkov's contraction
  • Figure 4: Reduction of a boundary edge
  • Figure 5: One-boundary loop diagram
  • ...and 4 more figures

Theorems & Definitions (108)

  • Definition 1.1: Inhomogeneous symmetric/Hermitian random matrices
  • Theorem 1.2: BBP transition
  • Theorem 1.3: Fluctuations of outliers
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Band matrices
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • ...and 98 more