  Spectral statistics of hermitian random Toeplitz matrices with independent
identically distributed elements is investigated numerically. It is found that
the eigenvalue statistics of complex Toeplitz matrices is surprisingly well
approximated by the semi-Poisson distribution belonging to intermediate-type
statistics observed in certain pseudo-integrable billiards. The origin of
intermediate behaviour could be attributed to the fact that Fourier transformed
random Toeplitz matrices have the same slow decay outside the main diagonal as
critical random matrix ensembles. The statistical properties of the full
spectrum of real random Toeplitz matrices with i.i.d. elements are close to the
Poisson distribution but each of their constituted sub-spectra is again well
described by the semi-Poisson distribution. The findings open new perspective
in intermediate statistics.
