  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.
