On Fluctuations of Eigenvalues of Random Band Matrices

Abstract

We consider the fluctuations of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form $\mathcal {M}_{ij}=b^{-1/2}u^{1/2}(|i-j|/b)\tilde{w}_{ij}$ with i.i.d. $\tilde{w}_{ij}$ possessing the $(4+\varepsilon )$th moment, where the function u has a finite support $[-C^*,C^*]$, so that M has only $2C_*b+1$ nonzero diagonals. The parameter b (called the bandwidth) is assumed to grow with n in a way such that $b/n\rightarrow 0$. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers (Jana et al., arXiv:1412.2445; Li et al. Random Matrices 2:04, 2013), where CLT was proven under the assumption $n>>b>>n^{1/2}$. Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.