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Spectral statistics of random matrices, including Toeplitz and Hankel types, exhibit intermediate behavior. This finding reveals a more universal and widespread nature of these spectral properties in random matrix theory.

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Area of Science:

  • Mathematics
  • Physics
  • Spectral Theory
  • Random Matrix Theory

Background:

  • Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel random matrices are fundamental in various scientific domains.
  • Understanding their spectral properties is crucial for advancing theoretical frameworks.

Purpose of the Study:

  • To investigate the spectral properties of specific low-complexity random matrices.
  • To characterize the nature of their spectral statistics and eigenvector behavior.

Main Methods:

  • Combined numerical simulations and analytical arguments.
  • Analysis of spectral statistics, including level repulsion and nearest-neighbor distributions.
  • Investigation of spectral compressibility and eigenvector fractal dimensions.

Main Results:

  • Demonstrated that spectral statistics for these matrices are of the intermediate type.
  • Identified key characteristics: level repulsion, exponential decay in distributions, nontrivial spectral compressibility, and fractal eigenvector dimensions.
  • Established universality of intermediate-type statistics.

Conclusions:

  • Intermediate-type spectral statistics are more ubiquitous than previously understood.
  • These findings open new avenues in the field of random matrix theory.
  • The study highlights the universal nature of spectral properties in complex systems.