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 an $n\times n$ band Hermitian random matrix of bandwidth $b_{n}$; i.e., the diagonal elements and only the first $b_{n}$ off-diagonal elements are nonzero. We study the linear eigenvalue statistics ${\cal N}(\phi)=\sum_{i=1}^{n}\phi(\l_{i})$ of such matrices, where $\l_{i}$ are the eigenvalues of $M_{n}$ and $\phi$ is a sufficiently smooth function. We prove that $\sqrt{\frac{b_{n}}{n}}[{\cal N}(\phi)-\E {\cal N}(\phi)]\stackrel{d}{\to} N(0,V(\phi))$ for $b_{n}\gg\sqrt{n}$, where $V(\phi)$ is given in Theorem 1.