## Spectrum of random Toeplitz matrices with band structure.(English)Zbl 1188.15036

Summary: The author considers the eigenvalues of symmetric Toeplitz matrices with independent random entries and band structure. We assume that the entries of the matrices have zero mean and a uniformly bounded 4th moment, and we study the limit of the eigenvalue distribution when both the size of the matrix and the width of the band with non-zero entries grow to infinity. It is shown that if the bandwidth/size ratio converges to zero, then the limit of the eigenvalue distributions is Gaussian. If the ratio converges to a positive limit, then the distributions converge to a non-Gaussian distribution, which depends only on the limit ratio. A formula for the fourth moment of this distribution is derived.

### MSC:

 15B52 Random matrices (algebraic aspects) 15B05 Toeplitz, Cauchy, and related matrices 15A18 Eigenvalues, singular values, and eigenvectors
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