Statistical properties of structured random matrices

TL;DR

This paper investigates the spectral properties of structured Hermitian random matrices, revealing that their statistics are of an intermediate type with unique features, expanding the understanding of universality in random matrix theory.

Contribution

It demonstrates that spectral statistics of structured Hermitian random matrices are of an intermediate type, a finding that broadens the scope of universality in random matrix theory.

Findings

Spectral statistics exhibit level repulsion at small distances.

Nearest-neighbor distributions decay exponentially at large distances.

Eigenvectors in Fourier space have non-trivial fractal dimensions.

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.