Spectrum of random Toeplitz matrices with band structure.

Kargin, Vladislav

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Perturbed Toeplitz operators and radial determinantal processes

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We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criterion for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.

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The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [ (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting...