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Spectral statistics of random Toeplitz matrices.

Eugene Bogomolny¹

¹LPTMS, University Paris-Saclay, CNRS, 91405 Orsay, France.

Physical Review. E

|November 20, 2020

Summary

This summary is machine-generated.

Spectral statistics of random Toeplitz matrices reveal surprising universality. The semi-Poisson distribution, found in complex matrices, extends to real matrices, indicating broader applicability in random matrix theory and quantum chaos.

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Area of Science:

Mathematical Physics
Random Matrix Theory
Quantum Chaos

Background:

Investigating spectral statistics of Hermitian random Toeplitz matrices.
Understanding eigenvalue distributions in complex and real matrices.

Purpose of the Study:

To numerically explore spectral statistics of Hermitian random Toeplitz matrices.
To determine the applicability of the semi-Poisson distribution.
To investigate the origins of intermediate spectral statistics.

Main Methods:

Numerical investigation of spectral statistics.
Analysis of eigenvalue distributions for complex and real Toeplitz matrices.
Comparison with Poisson and semi-Poisson distributions.

Main Results:

Complex Toeplitz matrices exhibit eigenvalue statistics well-approximated by the semi-Poisson distribution.
This intermediate behavior is linked to slow off-diagonal decay in Fourier-transformed matrices.
Real Toeplitz matrices show Poisson-like full spectrum statistics, but subspectra follow semi-Poisson distributions.

Conclusions:

The semi-Poisson distribution is a more universal statistical descriptor than previously thought.
Intermediate statistics are prevalent in random matrix ensembles.
Findings advance the understanding of spectral properties in quantum systems.