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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 $\mathcal{N}(ϕ)=\sum_{i=1}^{n}ϕ(λ_{i})$ of such matrices, where $λ_{i}$ are the eigenvalues of $M_{n}$ and $ϕ$ is a sufficiently smooth function. We prove that $\sqrt{\frac{b_{n}}{n}}[\mathcal{N}(ϕ)-\mathbb{E} \mathcal{N}(ϕ)]\stackrel{d}{\to} N(0,V(ϕ))$ for $b_{n}>>\sqrt{n}$, where $V(ϕ)$ is given in the Theorem 1.