Statistical properties of structured random matrices

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.