AI Uncovers Semi-Poisson Statistics in Random Toeplitz Matrices

A recent study delves into the spectral statistics of random Toeplitz matrices, revealing that complex Hermitian variants exhibit intermediate statistics closely aligned with the semi-Poisson distribution. This finding challenges traditional classifications in random matrix theory, where Gaussian orthogonal ensembles and Poisson distributions dominate. The research employs numerical diagonalization of matrices up to 1000x1000 dimensions, analyzing eigenvalue spacing and correlation functions. For complex Toeplitz matrices with independent, identically distributed Gaussian elements, the nearest-neighbor distribution and two-point formfactor match semi-Poisson predictions, as shown in figures comparing computational results to theoretical curves. In contrast, real symmetric Toeplitz matrices display Poisson-like statistics in their full spectrum, but splitting into symmetric and skew-symmetric sub-spectra uncovers semi-Poisson behavior in each. Eigenfunction analysis in momentum space further supports this, with fractal dimensions indicating multifractality, such as D1 ≈ 0.6 and D2 ≈ 0.2, suggesting non-trivial scaling properties. The study links these results to critical random matrix ensembles, where slow decay of off-diagonal elements in Fourier-transformed matrices drives intermediate statistics. This work highlights the broader relevance of semi-Poisson distributions in fields like quantum physics and signal processing, urging further exploration into the universality of such statistical patterns.