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 $\mathcal{N}(\phi)=\sum_{i=1}^{n}\phi(\lambda_{i})$ of such matrices, where $\lambda_{i}$ are the eigenvalues of $M_{n}$ and $\phi$ 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.