[Paper Review] Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Dang-Zheng Liu, Zhengdong Wang|ArXiv.org|Apr 20, 2009

Random Matrices and Applications13 references3 citations

TL;DR

This paper establishes the weak and almost sure convergence of eigenvalue distributions for random Hermitian Toeplitz and real symmetric Hankel band matrices as the matrix size $N \to \infty$, with bandwidth $b_N \to \infty$ such that $b_N/N \to b \in [0,1]$. The limiting distributions $\gamma_T(b)$ and $\gamma_H(b)$ are universal, symmetric, and their moments are expressed as integrals over pair partitions; notably, $\gamma_T(0)$ is Gaussian and $\gamma_H(0)$ is $|x|e^{-x^2}$, with distinct fourth moments confirming non-Gaussian behavior for $b > 0$. The method relies on trace moment analysis and a novel linear decomposition of Toeplitz matrices using shift operators.

ABSTRACT

Consider real symmetric, complex Hermitian Toeplitz and real symmetric Hankel band matrix models, where the bandwidth $b_{N} a \iy$ but $b_{N}/N o b$, $b\in [0,1]$ as $N o \infty$. We prove that the distributions of eigenvalues converge weakly to universal, symmetric distributions $γ_{_{T}}(b)$ and $γ_{_{H}}(b)$. In the case $b>0$ or $b=0$ but with the addition of $b_{N}\geq C N^{{1/2}+ε_{0}}$ for some positive constants $ε_{0}$ and $C$, we prove almost sure convergence. The even moments of these distributions are the sum of some integrals related to certain pair partitions. In particular, when the bandwidth grows slowly, i.e. $b=0$, $γ_{_{T}}(0)$ is the standard Gaussian distribution and $γ_{_{H}}(0)$ is the distribution $|x| \exp(-x^{2})$. In addition, from the fourth moments we know that the $γ_{_{T}}(b)$'s are different for different $b$'s, the $γ_{_{H}}(b)$'s different for different $b\in [0,{1/2}]$ and the $γ_{_{H}}(b)$'s different for different $b\in [{1/2},1]$.

Motivation & Objective

• To establish the existence and universality of the limit distribution of eigenvalues for random Toeplitz and Hankel band matrices.
• To analyze the dependence of the limiting spectral distribution on the bandwidth ratio $b = \lim b_N/N$.
• To derive explicit formulas for the moments of the limiting distributions using integrals over pair partitions.
• To prove almost sure convergence under mild growth conditions on the bandwidth $b_N$.
• To distinguish the limiting distributions $\gamma_T(b)$ and $\gamma_H(b)$ for different values of $b$ using higher-order moments.

Proposed method

• Modeling Toeplitz and Hankel band matrices as linear combinations of deterministic shift matrices $B$ and $F$ with random coefficients.
• Using the trace moment method to compute $\mathbb{E}[\mathrm{tr}(X_N^k)]$ and analyzing the fourth moment fluctuations to establish convergence.
• Representing the $k$-th moment of the limiting distribution as a sum of integrals over pair partitions $\pi \in \mathcal{P}_2(2k)$.
• Defining the integral $p_\pi(b)$ over $[0,1] \times [-1,1]^k$ involving indicator functions of linear forms in variables $x_l$, capturing bandwidth dependence.
• Applying moment convergence techniques and controlling higher-order cumulants to prove almost sure convergence under $b_N \geq C N^{1/2 + \epsilon_0}$.
• Leveraging the algebraic structure of Toeplitz matrices via the shift operator representation to derive moment formulas.

Experimental results

Research questions

• RQ1 Does the eigenvalue distribution of random Toeplitz band matrices converge weakly as $N \to \infty$ with $b_N/N \to b$?
• RQ2 Are the limiting spectral distributions $\gamma_T(b)$ and $\gamma_H(b)$ universal and symmetric for all $b \in [0,1]$?
• RQ3 How do the fourth moments of $\gamma_T(b)$ and $\gamma_H(b)$ vary with $b$, and what does this imply about their distinctness?
• RQ4 Under what conditions on $b_N$ does the convergence of eigenvalue distributions become almost sure rather than just weak?
• RQ5 Can the limiting distributions be explicitly characterized via integrals over pair partitions of indices?

Key findings

• The limiting eigenvalue distribution $\gamma_T(b)$ is universal and symmetric, with $\gamma_T(0)$ being the standard Gaussian distribution.
• The limiting distribution $\gamma_H(b)$ for Hankel matrices satisfies $\gamma_H(0) = |x| e^{-x^2}$, indicating a non-Gaussian, heavier-tailed shape.
• For $b \in (0,1]$, $\gamma_T(b)$ is not Gaussian, as confirmed by the strictly decreasing fourth moment $m_4(\gamma_T(b))$ on $[0,1]$.
• The fourth moment of $\gamma_H(b)$ strictly decreases on $[0,1/2]$ and strictly increases on $[1/2,1]$, proving that $\gamma_H(b)$'s are distinct for different $b$ in each interval.
• The even moments of $\gamma_T(b)$ and $\gamma_H(b)$ are given by sums of integrals over pair partitions, explicitly computed for $k=2$.
• Almost sure convergence of the eigenvalue distribution is established when $b_N \geq C N^{1/2 + \epsilon_0}$ for some $C>0$ and $\epsilon_0>0$, under moment conditions on the matrix entries.