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Spectrum of random centrosymmetric matrices; CLT and Circular law

Indrajit Jana, Sunita Rani

TL;DR

This work analyzes linear eigenvalue statistics and the spectral distribution of random centrosymmetric matrices with i.i.d. entries. It establishes a central limit theorem for analytic test functions, giving an explicit variance in terms of the test function's derivative and a circular law for the limiting spectrum under proper scaling. The authors leverage a block-diagonal reduction via a counter-identity, resolvent techniques, and a martingale CLT framework to derive tightness, finite-dimensional convergence, and the covariance structure of the limiting Gaussian process. They also perform a detailed combinatorial variance analysis using index chains to show the LES fluctuations depend only on the even-power terms, connecting the variance to $\sum_{k=1}^d 2k a_k^2$ for polynomials $P_d$. Overall, the results deepen understanding of spectral fluctuations in structured non-Hermitian random matrices and extend circular law phenomena to centrosymmetric ensembles.

Abstract

We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we also argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law.

Spectrum of random centrosymmetric matrices; CLT and Circular law

TL;DR

This work analyzes linear eigenvalue statistics and the spectral distribution of random centrosymmetric matrices with i.i.d. entries. It establishes a central limit theorem for analytic test functions, giving an explicit variance in terms of the test function's derivative and a circular law for the limiting spectrum under proper scaling. The authors leverage a block-diagonal reduction via a counter-identity, resolvent techniques, and a martingale CLT framework to derive tightness, finite-dimensional convergence, and the covariance structure of the limiting Gaussian process. They also perform a detailed combinatorial variance analysis using index chains to show the LES fluctuations depend only on the even-power terms, connecting the variance to for polynomials . Overall, the results deepen understanding of spectral fluctuations in structured non-Hermitian random matrices and extend circular law phenomena to centrosymmetric ensembles.

Abstract

We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we also argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law.
Paper Structure (15 sections, 15 theorems, 96 equations, 5 figures)

This paper contains 15 sections, 15 theorems, 96 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and layout
  3. Main result
  4. Truncation and reduction
  5. Circular Law for Random Centrosymmetric Matrix
  6. Proof of the central limit theorem
  7. Proof of tightness
  8. Proof of \ref{['eq: first equivalent cond of MCLT']}
  9. Proof of \ref{['eq: second equivalent cond of MCLT']}
  10. Proof of \ref{['eq: third equivalent cond of MCLT']}
  11. Calculation of variance
  12. Analyzing expected product
  13. Single Chain Expectation
  14. Double chain Expectation
  15. Covariance kernel of $\operatorname{Tr} \hat{\mathcal{R}}_{z}(M)$

Key Result

Theorem 3.4

Suppose $M$ is a centrosymmetric random matrix satisfying Condition cond:matrixcond. Let $f:\mathbb{C}\to\mathbb{C}$ be an analytic function. Then $\operatorname{L}_{f}^{\circ}(M)\stackrel{d}{\to}N(0, \sigma_{f}^{2})$, where In particular, if $f$ is a polynomial given by $P_{d}(z)=\sum_{k=0}^{d}a_{k}z^{k}$, then $\sigma_{p_d}^2=\sum_{k=1}^{d}2ka_{k}^2$.

Figures (5)

  • Figure 1: Symmetry pattern of a centrosymmetric 5$\times$5 matrix.
  • Figure 2: Scatter plot of real and imaginary parts of the eigenvalues of a $5000\times 5000$ random centrosymmetric matrix having standard Gaussian entries.
  • Figure 3: Histogram of $\operatorname{L}^\circ(P_{d})/\sqrt{n}$ of a $4000\times 4000$ random centrosymmetric matrix sampled 1000 times. The entries of the matrix are drawn from the Gaussian distribution, where $P_{d}(x)=2x^3+x^4$.
  • Figure 4: Sequential cross chain merging
  • Figure 5: Non sequential cross chain merging

Theorems & Definitions (28)

  • Definition 3.1: Centrosymmetric Matrix
  • Definition 3.2: Poincaré inequality
  • Theorem 3.4
  • Definition 4.1: Counter-identity Matrix
  • Theorem 4.2
  • Theorem 5.1: Circular Law, tao2008random
  • Theorem 5.2: Circular Law for Random Centrosymmetric Matrix
  • proof
  • Proposition 6.1
  • Lemma 6.2
  • ...and 18 more