Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices

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}$(\phi)-$\mathbb{E}$ $\mathcal{N}$(\phi)]\stackrel{d}{\to} N(0,V(\phi))$for$b_{n}>>\sqrt{n}$,where$V(\phi)$ is given in the Theorem 1.