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

Kargin, Vladislav

Electron. Commun. Probab. 14, 412-423 (2009).

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

Keywords:

random Toeplitz matrices; eigenvalues; eigenvalue distribution; non-Gaussian distribution