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Statistical properties of structured random matrices
Eugene Bogomolny1, Olivier Giraud1
1Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France.
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 low-complexity random matrices is of the intermediate type, characterized by: (i) level repulsion at short distances, (ii) an exponential decrease in the nearest-neighbor distributions at long distances, (iii) a nontrivial value of the spectral compressibility, and (iv) the existence of nontrivial 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.
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