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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.
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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 remarkable argument of Eugen Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie-Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.
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 theoretically and numerically. A simple formula describing the velocity distribution (and hence the distribution of the Wigner time delay) is derived, which is capable to explain the algebraic tail of the time delay distribution observed recently in microwave experiments. A dependence of the eigenphases on other external parameters is also discussed. We show that in the semiclassical limit (large number of channels) the curvature distribution of $S$--matrix eigenphases is the same as that corresponding to the curvature distribution of the underlying Hamiltonian and is given by the generalized Cauchy distribution.
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 hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Greens function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jaco
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 generalization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
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