## 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.

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Shcherbina, M. On Fluctuations of Eigenvalues of Random Band Matrices.
*J Stat Phys* **161**, 73–90 (2015). https://doi.org/10.1007/s10955-015-1324-8

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DOI: https://doi.org/10.1007/s10955-015-1324-8