Spectral statistics of hermitian random Toeplitz matrices with independent identically distributed elements is investigated numerically. It is found that 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 are close to the Poisson distribution but each of their constituent sub-spectra 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 that was considered before.

中文翻译：

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Toeplitz矩阵的光谱统计

数值研究了具有独立相同分布元素的埃尔米特随机Toeplitz矩阵的频谱统计。已经发现，复杂的Toeplitz矩阵的特征值统计令人惊讶地通过半Poisson分布得到了很好的近似，该半Poisson分布属于在某些拟可积分台球中观察到的中间类型统计。中间行为的起源可归因于以下事实：傅立叶变换后的随机Toeplitz矩阵在主对角线上具有与临界随机矩阵集合相同的缓慢衰减。真正的随机Toeplitz矩阵的全谱统计特性都接近于Poisson分布，但它们的每个构成子谱再次由Semi-Poisson分布很好地描述。