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CLT for LES of real valued random centrosymmetric matrices

Indrajit Jana, Sunita Rani

TL;DR

This paper studies fluctuations of the eigenvalues of large real centrosymmetric random matrices through linear eigenvalue statistics (LES). It proves a central limit theorem for the centered LES $L_n^{\circ}(f)$ of analytic test functions, with an explicit limiting variance $V_f$ given by a double contour integral; for polynomial $f(z)=\sum_{k=1}^d a_k z^k$, the variance simplifies to $V_f=\sum_{k=1}^d 2k|a_k|^2$. The authors split the proof into a probabilistic demonstration of Gaussian convergence and a detailed combinatorial calculation of $V_f$ via moments $\mathbb{E}[\operatorname{Tr}(M^k)]$ and $\mathbb{E}[\operatorname{Tr}(M^k)\operatorname{Tr}(M^l)]$, using index-chain methods and mergings (intra-, cross-, partial). The key technical contribution is a tight combinatorial framework that isolates the nonzero contributions from index chains and reveals how centrosymmetry shapes LES fluctuations, with results aligning to the circular law for the eigenvalues. The findings provide exact variance formulas that can inform statistical inference and theoretical studies of structured random matrices in physics and engineering.

Abstract

We study the fluctuations of the eigenvalues of real valued large centrosymmetric random matrices via its linear eigenvalue statistic. This is essentially a central limit theorem (CLT) for sums of dependent random variables. The dependence among them leads to behavior that differs from the classical CLT. The main contribution of this article is finding the expression of the variance of the limiting Gaussian distribution. The crux of the proof lies in combinatorial arguments that involve counting overlapping loops in complete undirected weighted graphs with growing degrees.

CLT for LES of real valued random centrosymmetric matrices

TL;DR

This paper studies fluctuations of the eigenvalues of large real centrosymmetric random matrices through linear eigenvalue statistics (LES). It proves a central limit theorem for the centered LES of analytic test functions, with an explicit limiting variance given by a double contour integral; for polynomial , the variance simplifies to . The authors split the proof into a probabilistic demonstration of Gaussian convergence and a detailed combinatorial calculation of via moments and , using index-chain methods and mergings (intra-, cross-, partial). The key technical contribution is a tight combinatorial framework that isolates the nonzero contributions from index chains and reveals how centrosymmetry shapes LES fluctuations, with results aligning to the circular law for the eigenvalues. The findings provide exact variance formulas that can inform statistical inference and theoretical studies of structured random matrices in physics and engineering.

Abstract

We study the fluctuations of the eigenvalues of real valued large centrosymmetric random matrices via its linear eigenvalue statistic. This is essentially a central limit theorem (CLT) for sums of dependent random variables. The dependence among them leads to behavior that differs from the classical CLT. The main contribution of this article is finding the expression of the variance of the limiting Gaussian distribution. The crux of the proof lies in combinatorial arguments that involve counting overlapping loops in complete undirected weighted graphs with growing degrees.

Paper Structure

This paper contains 6 sections, 5 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Calculation of variance
  4. Analyzing expected product
  5. Single Chain Expectation
  6. Double Chain Expectation

Key Result

Theorem 2.3

Let $M$ be a random matrix satisfying Condition cond:matrixcond_real. Let $f$ be a complex analytic function. Then $\operatorname{L}_{n}^{\circ}(f)\stackrel{d}{\to}N(0, V_{f})$, where $\stackrel{d}{\to}$ stands for convergence in distribution, and The above formula boils down to $V_{f}=\sum_{k=1}^{d}2k|a_{k}|^2$ when $f(z)=\sum_{k=0}^{d}a_{k}z^{k}$ is a polynomial.

Figures (3)

  • Figure 1: Symmetry pattern of a centrosymmetric 5$\times$5 matrix.
  • Figure 2: Histogram of $\operatorname{L}_{n}^{\circ}(f)$ for a $4000\times 4000$ random centrosymmetric matrix, averaged over 750 samples, where matrix entries are Gaussian random variables and test function is $f(x)=x^2+4x^5$. The sample variance of $\operatorname{L}_{n}^{\circ}(f)$ is 166.7225, which is approximately same as suggested by the Proposition \ref{['Prop:Var']}.
  • Figure 3: Sequential intra chain merging

Theorems & Definitions (13)

  • Definition 2.1: Centrosymmetric Matrix
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • Definition 3.2: Index chains
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 3.5
  • ...and 3 more