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Spectral independence of almost fully correlated random matrices

Oleksii Kolupaiev

TL;DR

This work demonstrates that two highly correlated random matrices can exhibit asymptotically independent local eigenvalue fluctuations in the bulk, provided a decorrelation parameter satisfies $\alpha \gg N^{-1}$. The authors develop a robust framework combining bulk local laws, Dyson Brownian Motion universality, and a novel Green Function Theorem to handle degenerate correlation structures, along with a linear filtering mechanism to model general correlations. The main result shows the joint $m$- and $n$-point eigenvalue statistics factorize up to $N^{-c(m,n)}$ after rescaling near fixed bulk energies, and this independence persists even under a broad class of linear filters. The threshold $\alpha \sim N^{-1}$ is proven optimal, highlighting a sharp boundary between independence and strong coupling in the microscopic spectrum of correlated random matrices.

Abstract

We study the joint spectral properties of two coupled random matrices $H^{(1)}$ and $H^{(2)}$, which are either real symmetric or complex Hermitian. The entries of these matrices exhibit polynomially decaying correlations, both within each matrix and between them. Surprisingly, we find that under extremely weak decorrelation condition, permitting $H^{(1)}$ and $H^{(2)}$ to be almost fully correlated, the fluctuations of their individual eigenvalues in the bulk of the spectrum are still asymptotically independent. Furthermore, we demonstrate that this decorrelation condition is optimal.

Spectral independence of almost fully correlated random matrices

TL;DR

This work demonstrates that two highly correlated random matrices can exhibit asymptotically independent local eigenvalue fluctuations in the bulk, provided a decorrelation parameter satisfies . The authors develop a robust framework combining bulk local laws, Dyson Brownian Motion universality, and a novel Green Function Theorem to handle degenerate correlation structures, along with a linear filtering mechanism to model general correlations. The main result shows the joint - and -point eigenvalue statistics factorize up to after rescaling near fixed bulk energies, and this independence persists even under a broad class of linear filters. The threshold is proven optimal, highlighting a sharp boundary between independence and strong coupling in the microscopic spectrum of correlated random matrices.

Abstract

We study the joint spectral properties of two coupled random matrices and , which are either real symmetric or complex Hermitian. The entries of these matrices exhibit polynomially decaying correlations, both within each matrix and between them. Surprisingly, we find that under extremely weak decorrelation condition, permitting and to be almost fully correlated, the fluctuations of their individual eigenvalues in the bulk of the spectrum are still asymptotically independent. Furthermore, we demonstrate that this decorrelation condition is optimal.

Paper Structure

This paper contains 15 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Set-up and main results
  3. Model
  4. A pair of correlated Wigner-type matrices
  5. General correlated case
  6. Eigenvalue correlation functions
  7. Main result
  8. Proof of Theorem \ref{['theo:main']}
  9. Gaussian divisible case: Proof of Proposition \ref{['prop:DBM']}
  10. GFT: proof of Proposition \ref{['prop:p_comparison']}
  11. Proofs of additional technical results
  12. Optimality of Assumption \ref{['ass:W-type']}(iii): justification of Example \ref{['ex:optimal']}
  13. Local law: proof of Proposition \ref{['prop:loc_law']}
  14. Properties of the GFT flow: proof of Lemma \ref{['lemma:match']}
  15. Proof of Lemma \ref{['lem:cond_rho']}

Key Result

Theorem 2.4

Assume one of the following two sets of conditions: Fix a (small) $\kappa>0$ and take $E^{(j)}\in\mathbf{B}^{(j)}_\kappa$ for $j=1,2$. Then for any $m, n\in\mathbf{N}$ and a compactly supported smooth test function $F\in C_c^{\infty}(\mathbf{R}^{m+n})$ it holds that where $c(m,n)>0$ depends only on $m$ and $n$, and the implicit constant in $\mathcal{O}$ depends on $F$.

Theorems & Definitions (11)

  • Example 2.3
  • Theorem 2.4: Independence of local statistics
  • Example 2.5: Optimality of $N^{-1}$-threshold
  • Proposition 3.1: Local law in the bulk of the spectrum
  • Proposition 3.2
  • Lemma 3.3: Properties of the GFT flow
  • Proposition 3.4: GFT
  • Lemma 4.1
  • proof : Proof of Proposition \ref{['prop:DBM']}
  • Proposition 5.1: Green function comparison
  • ...and 1 more