On fluctuations of eigenvalues of random band matrices

Abstract

We consider the fluctuation of linear eigenvalue statistics of random band $n\times n$ matrices whose entries have the form ${\mathcal{M}}_{ij}=b^{-1/2}{u}^{1/2}(|i-j|){\tilde{w}}_{ij}$ with i.i.d. $w_{ij}$ possessing the $(4+\epsilon )$th moment, where the function $u$ has a finite support $[-C^{\ast },C^{\ast }]$, so that $M$ has only $2C_{\ast }b+1$ nonzero diagonals. The parameter $b$ (called the bandwidth) is assumed to grow with $n$ in a way that $b/n\to 0$. 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 [8] and [11], 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.

15 pages

References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Gaussian fluctuations for linear spectral statistics of large random covariance matrices
• Products of independent elliptic random matrices

Cited by in corpus (11)

• Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices
• Fluctuations of eigenvalues of patterned random matrices
• CLT for non-Hermitian random band matrices with variance profiles
• Fluctuation of linear eigenvalue statistics of reverse circulant matrices with independent entries
• On fluctuations of global and mesoscopic linear eigenvalue statistics of generalized Wigner matrices
• Quantitative CLT for linear eigenvalue statistics of Wigner matrices
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
• Spectrum of random centrosymmetric matrices; CLT and Circular law
• Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices
• Central Limit Theorem for Linear Eigenvalue Statistics for Submatrices of Wigner Random Matrices
• Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices