Statistical properties of structured random matrices

Abstract

Spectral properties of Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel random matrices with independent identically distributed entries are investigated. Combining numerical and analytic arguments it is demonstrated that spectral statistics of all these random matrices is of intermediate type, characterized by (i) level repulsion at small distances, (ii) an exponential decrease of the nearest-neighbor distributions at large distances, (iii) a non-trivial value of the spectral compressibility, and (iv) the existence of non-trivial fractal dimensions of eigenvectors in Fourier space. Our findings show that intermediate-type statistics is more ubiquitous and universal than was considered so far and open a new direction in random matrix theory.

arXiv comment34 pages, 7 figures

References in corpus (11)

• Anderson Transitions
• Exact relations between multifractal exponents at the Anderson transition
• Eigenvalue Spacing Distribution for the Ensemble of Real Symmetric Toeplitz Matrices
• Joint probability densities of level spacing ratios in random matrices
• Short-range plasma model for intermediate spectral statistics
• Eigenfunction entropy and spectral compressibility for critical random matrix ensembles
• Intermediate statistics in quantum maps
• Perturbation approach to multifractal dimensions for certain critical random matrix ensembles
• Spectral properties of quantized barrier billiards
• Higher-order level spacings in random matrix theory based on Wigner's conjecture
• Spectral statistics of Toeplitz matrices

Cited by in corpus (5)

• Symmetry Violation of Quantum Multifractality: Gaussian fluctuations versus Algebraic Localization
• Barrier billiard and random matrices
• Unitary matrix integrals, symmetric polynomials, and long-range random walks
• Random matrices associated with general barrier billiards
• Universal Relation between Spectral and Wavefunction Properties at Criticality