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Spectral properties of Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel random matrices with independent identically distributed entries are investigated. Combining numerical and analytic arguments it is demonstrated that spectral statistics of all these random matrices is of intermediate type, characterized by (i) level repulsion at small distances, (ii) an exponential decrease of the nearest-neighbor distributions at large distances, (iii) a non-trivial value of the spectral compressibility, and (iv) the existence of non-trivial fractal dimensions of eigenvectors in Fourier space. Our findings show that intermediate-type statistics is more ubiquitous and universal than was considered so far and open a new direction in random matrix theory.
This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a
We discuss the properties of eigenphases of S--matrices in random models simulating classically chaotic scattering. The energy dependence of the eigenphases is investigated and the corresponding velocity and curvature distributions are obtained both
The eigenvalues of the matrix structure $X + X^{(0)}$, where $X$ is a random Gaussian Hermitian matrix and $X^{(0)}$ is non-random or random independent of $X$, are closely related to Dyson Brownian motion. Previous works have shown how an infinite h
In this paper a geometric method based on Grassmann manifolds and matrix Riccati equations to make hermitian matrices diagonal is presented. We call it Riccati Diagonalization.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the genera