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Universality laws for random matrices via exchangeable counterparts

Joel A. Tropp

Abstract

Recently, Brailovskaya & van Handel (GAFA, 2024) established a suite of nonasymptotic universality laws which demonstrate that the spectral statistics of an independent sum of random matrices mirror the spectral statistics of a Gaussian random matrix with the same first- and second-order moments. This paper develops a more elementary proof of their main results by means of a new implementation of the method of exchangeable counterparts.

Universality laws for random matrices via exchangeable counterparts

Abstract

Recently, Brailovskaya & van Handel (GAFA, 2024) established a suite of nonasymptotic universality laws which demonstrate that the spectral statistics of an independent sum of random matrices mirror the spectral statistics of a Gaussian random matrix with the same first- and second-order moments. This paper develops a more elementary proof of their main results by means of a new implementation of the method of exchangeable counterparts.
Paper Structure (61 sections, 27 theorems, 196 equations)

This paper contains 61 sections, 27 theorems, 196 equations.

Table of Contents

  1. Main results and related work
  2. Independent sums
  3. The Gaussian proxy
  4. Random matrix statistics
  5. Monomial moments
  6. Mean spectral distribution
  7. Spectral support
  8. Universality: Proof strategies
  9. Interpolation
  10. Cumulant expansions
  11. Exchangeable counterparts
  12. Other related work
  13. Roadmap
  14. Notation
  15. Covariance identities: Scalar setting
  16. ...and 46 more sections

Key Result

Theorem I

Let $\bm{X}$ be an independent sum eqn:indep-sum-intro of random self-adjoint matrices, and let $\bm{Z}$ be the matching Gaussian model eqn:gauss-intro. For each natural number $p \in \mathbb{N}$, Furthermore, when $p L^2(\bm{X}) \leq \sigma^2(\bm{X})$, The statistics $\sigma^2(\bm{X})$ and $L(\bm{X})$ are defined in sec:statistics-intro.

Theorems & Definitions (56)

  • Theorem I: Monomial moments: Universality
  • Theorem II: Cauchy transform: Universality
  • Corollary 1.1: Spectral functions: Universality
  • Theorem III: Resolvent norm: Universality
  • Corollary 1.2: Spectrum: Universality
  • proof : Proof sketch
  • Definition 2.2: Exchangeable family
  • Proposition 2.3: Independent sum: Linear regression property
  • proof
  • Proposition 2.4: Covariance identity: Independent sum
  • ...and 46 more