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On spectral outliers of inhomogeneous symmetric random matrices

Dylan J. Altschuler, Patrick Oliveira Santos, Konstantin Tikhomirov, Pierre Youssef

Abstract

Sharp conditions for the presence of spectral outliers are well understood for Wigner random matrices with iid entries. In the setting of inhomogeneous symmetric random matrices (i.e., matrices with a non-trivial variance profile), the corresponding problem has been considered only recently. Of special interest is the setting of sparse inhomogeneous matrices since sparsity is both a key feature and a technical obstacle in various aspects of random matrix theory. For such matrices, the largest of the variances of the entries has been used in the literature as a natural proxy for sparsity. We contribute sharp conditions in terms of this parameter for an inhomogeneous symmetric matrix with sub-Gaussian entries to have outliers. Our result implies a ``structural'' universality principle: the presence of outliers is only determined by the level of sparsity, rather than the detailed structure of the variance profile.

On spectral outliers of inhomogeneous symmetric random matrices

Abstract

Sharp conditions for the presence of spectral outliers are well understood for Wigner random matrices with iid entries. In the setting of inhomogeneous symmetric random matrices (i.e., matrices with a non-trivial variance profile), the corresponding problem has been considered only recently. Of special interest is the setting of sparse inhomogeneous matrices since sparsity is both a key feature and a technical obstacle in various aspects of random matrix theory. For such matrices, the largest of the variances of the entries has been used in the literature as a natural proxy for sparsity. We contribute sharp conditions in terms of this parameter for an inhomogeneous symmetric matrix with sub-Gaussian entries to have outliers. Our result implies a ``structural'' universality principle: the presence of outliers is only determined by the level of sparsity, rather than the detailed structure of the variance profile.
Paper Structure (8 sections, 6 theorems, 90 equations)

This paper contains 8 sections, 6 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Main results: spectral outliers of inhomogeneous matrices
  3. The regime with no outliers
  4. Outliers in the subcritical regime
  5. The limiting Empirical Spectral Distribution: A Moment Method Approach
  6. The convergence in expectation
  7. Tail bound and almost-sure convergence
  8. The non-universal regime for the limiting ESD

Key Result

Theorem 1.1

Let $W_n$ be an $n\times n$ symmetric matrix whose entries on and above the diagonal are iid copies of a centered random variable $\xi$ with unit variance. Let $\Sigma_n=(\sigma_{ij})_{1\leq i,j\leq n}$ be a symmetric matrix such that $(\sigma_{ij}^2)_{1\leq i,j\leq n}$ is doubly stochastic. Setting

Theorems & Definitions (13)

  • Theorem 1.1: GNT2014
  • Theorem 1.2: Main result
  • Theorem 1.3: A sufficient condition for existence of outliers
  • Remark 2.1: Communicated by Ramon van Handel
  • Remark 2.2: Heavy-tailed distributions
  • Lemma 3.1
  • Remark 3.2
  • proof : Proof of Lemma \ref{['lem: lower bound norm']}
  • Proposition A.1
  • proof
  • ...and 3 more