2020/06/30 by E. Bogomolny, Eugene Bogomolny
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Circular law #Compound Poisson distribution #Distribution (mathematics) #Eigenvalues and eigenvectors #Hermitian matrix #Independent and identically distributed random variables #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Physics #Poisson distribution #Poisson regression #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Random Matrices and Applications #Random matrix #Random variable #Statistics #Sum of normally distributed random variables #Toeplitz matrix #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1103/physreve.102.040101
published as Phys. Rev. E 102, 040101 (2020) · 12 pages 4 figures
arxiv created 2020/07/01 · openalex publication_date 2020/10/06 · arxiv updated 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.