Spectral statistics of random Toeplitz matrices.

Abstract

The spectral statistics of Hermitian random Toeplitz matrices with independent and identically distributed elements are investigated numerically. It is found that eigenvalue statistics of complex Toeplitz matrices are surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudointegrable billiards. The origin of intermediate behavior 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 are close to the Poisson distribution, but each of their constituent subspectra is again well described by the semi-Poisson distribution. The findings indicate that intermediate statistics in general and the semi-Poisson distribution in particular are more universal than considered before.