Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices

Abstract

In this paper, we study the fluctuation of linear eigenvalue statistics of Random Band Matrices defined by $M_n=\frac{1}{\sqrt{b_n}}{W}_n$, where $W_n$ is a $n\times n$ band Hermitian random matrix of bandwidth $b_n$, i.e., the diagonal elements and only first $b_n$ off diagonal elements are nonzero. Also variances of the matrix elmements are upto a order of constant. We study the linear eigenvalue statistics $\mathcal{N}(\varphi )={\sum }_{i=1}^n\varphi ({\lambda }_i)$ of such matrices, where ${\lambda }_i$ are the eigenvalues of $M_n$ and $\varphi$ is a sufficiently smooth function. We prove that $\sqrt{\frac{b_n}{n}}[\mathcal{N}(\varphi )-\mathbb{E}\mathcal{N}(\varphi )]\stackrel{d}{\to }N(0,V(\varphi ))$ for $b_n>>\sqrt{n}$, where $V(\varphi )$ is given in the Theorem 1.

arXiv commentIn this version we have corrected several typos and slightly changed the Proposition 2

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• Spectrum of random centrosymmetric matrices; CLT and Circular law
• Fluctuation of eigenvalues of symmetric circulant matrices with independent entries
• Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices
• On Fluctuations for Random Band Toeplitz Matrices
• Process convergence of Fluctuations of linear eigenvalue statistics of random circulant matrices