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Zheng-Dong Wang

Publications and source records attributed to Zheng-Dong Wang.

6 recordsLinked to original sources
arXiv2011

A note on eigenvalues of random block Toeplitz matrices with slowly growing bandwidth

This paper can be thought of as a remark of \cite{llw}, where the authors studied the eigenvalue distribution $μ_{X_N}$ of random block Toeplitz band matrices with given block order $m$. In this note we will give explicit density functions of $\lim\limits_{N\to\infty}μ_{X_N}$ when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of $m\times m$ Gaussian unitary ensemble (GUE for short). The series $\{\lim\limits_{N\to\infty}μ_{X_N}|m\in\mathbb{N}\}$ can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks.

math.PR↗
arXiv2010

Fluctuation of Eigenvalues for Random Toeplitz and Related Matrices

Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random variables such that \be \mathbb{E}[a_{j}]=0, \ \ \mathbb{E}[|a_{j}|^{2}]=1 \ \ \textrm{for}\,\ \ j=0,1,2,...,\ee (homogeneity of 4-th moments) \be{κ=\mathbb{E}[|a_{j}|^{4}],}\ee \noindent and further (uniform boundedness)\be\sup\limits_{j\geq 0} \mathbb{E}[|a_{j}|^{k}]=C_{k}<\iy\ \ \ \textrm{for} \ \ \ k\geq 3.\ee Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree $\geq 2$. Without the assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hankel matrices and product of several Toeplitz matrices in a flavor of free probability theory etc. Since Toeplitz matrices are quite different from the Wigner and Wishart matrices, our results enrich this topic.

math.PR↗
arXiv2010

Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulae of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit $γ_{_T}^{(m)}$ $(γ_{_H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where $m$ is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that $γ_{_T}^{(m)}$ $(γ_{_H}^{(m)})$ converges weakly to the semicircle law as $m\ra\iy$.

math.PR↗
arXiv2009

Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Consider real symmetric, complex Hermitian Toeplitz and real symmetric Hankel band matrix models, where the bandwidth $b_{N}\ra \iy$ but $b_{N}/N \to b$, $b\in [0,1]$ as $N\to \infty$. 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 $b\in [{1/2},1]$.

math.PR↗
arXiv2008

Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).

math-ph↗