Spectral statistics of Toeplitz matrices

TL;DR

This paper explores the spectral statistics of hermitian and real random Toeplitz matrices, revealing semi-Poisson and Poisson distributions and suggesting a connection to intermediate-type statistics seen in certain physical systems.

Contribution

It provides the first numerical investigation linking Toeplitz matrix spectral statistics to intermediate distributions like semi-Poisson, expanding understanding of their spectral behavior.

Findings

Eigenvalue statistics of complex Toeplitz matrices follow semi-Poisson distribution.

Real Toeplitz matrices' spectra are close to Poisson, with sub-spectra following semi-Poisson.

Fourier transformed Toeplitz matrices exhibit slow decay similar to critical random matrix ensembles.

Abstract

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.