On Fluctuations for Random Band Toeplitz Matrices

TL;DR

This paper analyzes the fluctuations of empirical measures of two families of random band Toeplitz matrices, showing they converge to Gaussian processes with explicitly derived covariance structures, using the moment method.

Contribution

It introduces a detailed study of the Gaussian fluctuations of spectral measures for random band Toeplitz matrices with both independent entries and Brownian motion entries, including covariance structures.

Findings

Fluctuations converge to centered Gaussian families.

Covariance structures are explicitly derived.

Special cases include Brownian motion and quadratic fluctuations.

Abstract

In this paper we study two one-parameter families of random band Toeplitz matrices: \[ A_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}\delta_{|i-j|\le[b_nt]}\Big)_{i,j=1}^n \quad\text{and}\quad B_n(t)=\frac{1}{\sqrt{b_n}}\Big(a_{i-j}(t)\delta_{|i-j|\le b_n}\Big)_{i,j=1}^n \] 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 $\mu(A_n(t))$ and $\mu(B_n(t))$ are measure valued stochastic processes. The purpose of this paper is to study the fluctuations of $\mu(A_n(t))$ and $\mu(B_n(t))$ as $n$ goes to $\infty$. Given a monomial $f(x)=x^p$ with $p\ge2$, the corresponding rescaled fluctuations of $\mu(A_n(t))$ and $\mu(B_n(t))$ are \[\sqrt{b_n}\Big(\int f(x)d\mu(A_n(t))-E[\int f(x)d\mu(A_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(A_n(t)^p)-E[\text{tr}(A_n(t)^p)]\Big), \quad(1)\] \[\sqrt{b_n}\Big(\int f(x)d\mu(B_n(t))-E[\int f(x)d\mu(B_n(t))]\Big)=\frac{\sqrt{b_n}}{n}\Big(\text{tr}(B_n(t)^p)-E[\text{tr}(B_n(t)^p)]\Big) \quad(2)\] 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 $p,q\ge 2$; $t_1,t_2\ge 0,$ 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.