On fluctuations of eigenvalues of random band matrices

TL;DR

This paper proves a central limit theorem for linear eigenvalue statistics of random band matrices with growing bandwidth, extending previous results and providing a versatile method applicable to various classical random matrix models.

Contribution

It generalizes CLT results for eigenvalue statistics of band matrices without restrictive bandwidth growth conditions and introduces a method applicable to many random matrix models.

Findings

Proved CLT for eigenvalue statistics with minimal bandwidth assumptions.

Extended results to classical models like deformed Wigner and covariance matrices.

Developed a versatile method for proving CLT in diverse random matrix contexts.

Abstract

We consider the fluctuation of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form $\mathcal{M}_{ij}=b^{-1/2}u^{1/2}(|i-j|)\tilde w_{ij}$ with i.i.d. $w_{ij}$ possessing the $(4+\varepsilon)$th moment, where the function $u$ has a finite support $[-C^*,C^*]$, so that $M$ has only $2C_*b+1$ nonzero diagonals. The parameter $b$ (called the bandwidth) is assumed to grow with $n$ in a way that $b/n\to 0$. Without any additional assumptions on the growth of $b$ we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers [8] and [11], where CLT was proven under the assumption $n>>b>>n^{1/2}$. Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales, etc.