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CLT for LES of correlated Non-Hermitian Random Matrices

Indrajit Jana, Sunita Rani

TL;DR

The paper proves a central limit theorem for linear eigenvalue statistics of two correlated non-Hermitian random matrices, showing joint convergence to a bivariate Gaussian with variance depending on inter-matrix correlations and fourth-order cumulants. The authors build a resolvent-based framework, leveraging Girko's formula, a local law for products of resolvents, and stability operators to reduce LES fluctuations to Gaussian fluctuations of resolvent traces, carefully controlling error terms via cumulant expansions. They extend classical results for independent or single-matrix models (Rider-Silverstein; Cipolloni-Erdős-Schröder) to the correlated two-matrix setting and obtain explicit covariance structures; in the analytic case, the variance simplifies to a familiar form. The results also yield the limiting LES for random centrosymmetric matrices, underscoring the method’s reach across structured ensembles and highlighting the role of higher-order cumulants in non-Hermitian spectral fluctuations.

Abstract

We consider two $n\times n$ non-Hermitian random matrices such that the $ij$th entry of one matrix is correlated with the $ij$th entry of the other matrix. However, the entries of any particular matrix are i.i.d. random variables. We study the asymptotic behavior of the combined spectrum, and the limit of the linear eigenvalue statistic defined on the combined spectrum. We show that if the random variables are centered with variance $1/n$ and having finite moments, then the centered \textit{Linear Eigenvalue Statistics} (LESs) converge jointly to a bivariate Gaussian distribution. We assumed that the test function used in the LES belongs to Sobolev $H^{2+δ}$ space. The variance of the limiting Gaussian distribution depends on correlation structure of the matrix entries and the fourth order mixed cumulants of the matrix entries. This generalizes the previous results by Rider, Silverstein (2006), Cipolloni, Erdős, Schröder (2023). In particular, we obtain the limiting LES of random centrosymmetric matrices.

CLT for LES of correlated Non-Hermitian Random Matrices

TL;DR

The paper proves a central limit theorem for linear eigenvalue statistics of two correlated non-Hermitian random matrices, showing joint convergence to a bivariate Gaussian with variance depending on inter-matrix correlations and fourth-order cumulants. The authors build a resolvent-based framework, leveraging Girko's formula, a local law for products of resolvents, and stability operators to reduce LES fluctuations to Gaussian fluctuations of resolvent traces, carefully controlling error terms via cumulant expansions. They extend classical results for independent or single-matrix models (Rider-Silverstein; Cipolloni-Erdős-Schröder) to the correlated two-matrix setting and obtain explicit covariance structures; in the analytic case, the variance simplifies to a familiar form. The results also yield the limiting LES for random centrosymmetric matrices, underscoring the method’s reach across structured ensembles and highlighting the role of higher-order cumulants in non-Hermitian spectral fluctuations.

Abstract

We consider two non-Hermitian random matrices such that the th entry of one matrix is correlated with the th entry of the other matrix. However, the entries of any particular matrix are i.i.d. random variables. We study the asymptotic behavior of the combined spectrum, and the limit of the linear eigenvalue statistic defined on the combined spectrum. We show that if the random variables are centered with variance and having finite moments, then the centered \textit{Linear Eigenvalue Statistics} (LESs) converge jointly to a bivariate Gaussian distribution. We assumed that the test function used in the LES belongs to Sobolev space. The variance of the limiting Gaussian distribution depends on correlation structure of the matrix entries and the fourth order mixed cumulants of the matrix entries. This generalizes the previous results by Rider, Silverstein (2006), Cipolloni, Erdős, Schröder (2023). In particular, we obtain the limiting LES of random centrosymmetric matrices.

Paper Structure

This paper contains 20 sections, 13 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Main result
  4. Primary reductions
  5. Local law for product of resolvents
  6. Approximation of $G_{1}^{(t)}BG_{2}^{(t)}$ via stability operators
  7. Central limit theorem for linear eigenvalue statistic
  8. Central limit theorem for resolvents
  9. Evaluation of $I_{3}^{(1)}$
  10. Evaluation of $T_{1}^{(1)}, T_{2}^{(1)}, T_{3}^{(1)}, T_{4}^{(1)}$
  11. Reduced form of $I^{(1)}+I^{(2)}$
  12. Evaluation of $S_{i}^{(t)}$ terms in \ref{['eq:I^(1)+I^(2)Noataions']}
  13. Evaluation of $S_{2}^{(t)}$s
  14. Evaluation of $S_{i}^{(t)}$s, when $i=1, 3, 4, 5, 6$
  15. Evaluation of $T_{5}^{(t)}$ terms in \ref{['eq:I^(1)+I^(2)Noataions']}
  16. ...and 5 more sections

Key Result

Theorem 3.2

Let $X^{(1)}$ and $X^{(2)}$ be two $n\times n$ random matrices satisfying Condition Condtion::. Let $f\in H^{2+\delta}(\Omega)$ be a test function, where $\delta > 0$ and $\overline{\mathbf{D}}\subset \Omega\subset\mathbb{C}.$ Then the centered linear statistics $\mathcal{L}_n{(f)}$, defined in eq:L The above notations are defined as follows; $m_i^{(t)}=(m^{z_i})^{(t)}(\iota\eta_i)$ is the soluti

Theorems & Definitions (19)

  • Definition 2.1: Stochastic domination
  • Theorem 3.2: Central Limit Theorem for linear eigenvalue statistics
  • Theorem 3.3: CLT for resolvents
  • Theorem 4.1: Optimal local law for $G^{(t)}$,10.1214/21-EJP591, Theorem 3.1
  • Theorem 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 5.1: cipolloni2023central, Lemma 4.3, 4.4
  • Corollary 5.2
  • Lemma 5.3
  • ...and 9 more