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Inhomogeneous symmetric random matrix: eigenvalues and eigenvectors

Zeyan Song, Hanchao Wang

TL;DR

This work analyzes the spectral properties of inhomogeneous symmetric random matrices with independent, non-identically distributed subgaussian entries. It develops a distance-based framework built around log-RLCD and high-dimensional Littlewood–Offord techniques to translate spectral questions into small-ball probabilities, yielding sharp tail bounds for eigenvalue gaps, tight bounds on k-th smallest singular values, and no-gap delocalization for eigenvectors. Key contributions include a Distance Theorem for random linear spaces, sharp eigen-gap tail bounds with optimal exponents, a quantitative bound for singular values of principal submatrices, and improved delocalization results that advance understanding beyond prior symmetric-matrix results. Collectively, these results extend spectral analysis to non-iid subgaussian symmetric matrices and provide robust tools for controlling eigenvalues, singular values, and eigenvectors in high-dimensional random settings.

Abstract

Let A be an n-by-n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and eigenvectors. Our first result gives an upper tail bound for eigenvalue gaps. We show that for k <= n / log n, 1 <= i <= n - k, and epsilon > 0, P( lambda_{i+k} - lambda_i <= epsilon * n^{-1/2} ) <= (C * epsilon)^{(k^2 + k) / 2} + exp(-c * n). This bound is optimal when k is fixed. Our second result provides a quantitative estimate on the small singular values of A. Combining a recent theorem of Han Yi (arXiv:2506.01155), we prove that for c * log n <= k <= sqrt(n) and epsilon >= 0, P( sigma_{n-k+1}(A) <= k * epsilon * n^{-1/2} ) <= (C * epsilon)^{c * k^2} + exp(-c * k * n). Finally, using a distance-based analytical framework for eigenvalue gaps, we obtain quantitative bounds for singular values of principal submatrices and for delocalization of eigenvectors. In particular, we establish a no-gap delocalization estimate for eigenvectors of A, improving a result of Rudelson and Vershynin (Geom. Funct. Anal. 26 (2016), 1716-1776).

Inhomogeneous symmetric random matrix: eigenvalues and eigenvectors

TL;DR

This work analyzes the spectral properties of inhomogeneous symmetric random matrices with independent, non-identically distributed subgaussian entries. It develops a distance-based framework built around log-RLCD and high-dimensional Littlewood–Offord techniques to translate spectral questions into small-ball probabilities, yielding sharp tail bounds for eigenvalue gaps, tight bounds on k-th smallest singular values, and no-gap delocalization for eigenvectors. Key contributions include a Distance Theorem for random linear spaces, sharp eigen-gap tail bounds with optimal exponents, a quantitative bound for singular values of principal submatrices, and improved delocalization results that advance understanding beyond prior symmetric-matrix results. Collectively, these results extend spectral analysis to non-iid subgaussian symmetric matrices and provide robust tools for controlling eigenvalues, singular values, and eigenvectors in high-dimensional random settings.

Abstract

Let A be an n-by-n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and eigenvectors. Our first result gives an upper tail bound for eigenvalue gaps. We show that for k <= n / log n, 1 <= i <= n - k, and epsilon > 0, P( lambda_{i+k} - lambda_i <= epsilon * n^{-1/2} ) <= (C * epsilon)^{(k^2 + k) / 2} + exp(-c * n). This bound is optimal when k is fixed. Our second result provides a quantitative estimate on the small singular values of A. Combining a recent theorem of Han Yi (arXiv:2506.01155), we prove that for c * log n <= k <= sqrt(n) and epsilon >= 0, P( sigma_{n-k+1}(A) <= k * epsilon * n^{-1/2} ) <= (C * epsilon)^{c * k^2} + exp(-c * k * n). Finally, using a distance-based analytical framework for eigenvalue gaps, we obtain quantitative bounds for singular values of principal submatrices and for delocalization of eigenvectors. In particular, we establish a no-gap delocalization estimate for eigenvectors of A, improving a result of Rudelson and Vershynin (Geom. Funct. Anal. 26 (2016), 1716-1776).

Paper Structure

This paper contains 39 sections, 52 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Quantitative estimate for the singular values
  3. Delocalization for inhomogeneous symmetric matrices
  4. Distance theorem for random linear space
  5. Notation and proof sketch
  6. Notation
  7. Proof sketch
  8. Distance theorem
  9. Tail bound for the gaps of the eigenvalue
  10. No-gap delocaliaztion of the eigenvectors
  11. Quantitative estimate for the singular values
  12. Preliminaries
  13. Randomized logarithmic least common denominator
  14. Decomposition and net of the sphere
  15. Approximating for unstructure vectors
  16. ...and 24 more sections

Key Result

Theorem 1.1

For all $T \ge 1$, there exist $C_{Theorem A}$ and $c_{Theorem A}$ depending only on $T$ such that the following holds. Let $n \in \mathbb{N}^{+}$ with $n \ge C_{Theorem A }$ and $k \in \mathbb{N}^{+}$ with $1 \le k \le c_{Theorem A}n/\log{n}$. Let $A$ be an $n \times n$ symmetric random matrix with

Theorems & Definitions (94)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6: Theorem 1.5 in Rudelson_gafa
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 84 more