On Fluctuations for Random Band Toeplitz Matrices

Yiting Li; Xin Sun

paper

On Fluctuations for Random Band Toeplitz Matrices

arXiv:1412.5232

Abstract

In this paper we study two one-parameter families of random band Toeplitz matrices: \[ A_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}δ_{|i-j|\le[b_nt]}\Big)_{i,j=1}^n \quad\text{and}\quad B_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}(t)δ_{|i-j|\le b_n}\Big)_{i,j=1}^n \] where 1. $a_{0} = 0$ , ${a_{1}, a_{2}, ..}$ in $A_{n} (t)$ are independent random variables and $a_{- i} = a_{i}$ 2. $a_{0} (t) = 0$ , ${a_{1} (t), a_{2} (t), ...}$ in $B_{n} (t)$ are independent copies of the standard Brownian motion at time $t$ and $a_{- i} (t) = a_{i} (t)$ . As $t$ varies, the empirical measures $μ (A_{n} (t))$ and $μ (B_{n} (t))$ are measure valued stochastic processes. The purpose of this paper is to study the fluctuations of $μ (A_{n} (t))$ and $μ (B_{n} (t))$ as $n$ goes to $\infty$ . Given a monomial $f (x) = x^{p}$ with $p \geq 2$ , the corresponding rescaled fluctuations of $μ (A_{n} (t))$ and $μ (B_{n} (t))$ are \[\sqrt{b_n}\Big(\int f(x)dμ(A_n(t))-E[\int f(x)dμ(A_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(A_n(t)^p)-E[\text{tr}(A_n(t)^p)]\Big), \quad(1)\] \[\sqrt{b_n}\Big(\int f(x)dμ(B_n(t))-E[\int f(x)dμ(B_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(B_n(t)^p)-E[\text{tr}(B_n(t)^p)]\Big) \quad(2)\] respectively. We will prove that (1) and (2) converge to centered Gaussian families ${Z_{p} (t)}$ and ${W_{p} (t)}$ respectively. The covariance structure $E [Z_{p} (t_{1}) Z_{q} (t_{2})]$ and $E [W_{p} (t_{1}) W_{q} (t_{2})]$ are obtained for all $p, q \geq 2$ ; $t_{1}, t_{2} \geq 0,$ and are both homogeneous polynomials of $t_{1}$ and $t_{2}$ for fixed $p, q$ . In particular, $Z_{2} (t)$ is the Brownian motion and $Z_{3} (t)$ is the same as $W_{2} (t)$ up to a constant. The main method of this paper is the moment method.

25 pages. To appear in Random Matrices: Theory and Applications