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Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble.

Bertrand Lacroix-A-Chez-Toine1, Jeyson Andrés Monroy Garzón2, Christopher Sebastian Hidalgo Calva2

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Summary
This summary is machine-generated.

We analyzed complex random matrices and discovered a new intermediate regime for eigenvalue fluctuations. This finding bridges typical and large deviation behaviors, offering universal insights into random matrix theory.

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Area of Science:

  • Mathematics
  • Physics
  • Statistics

Background:

  • The Ginibre ensemble is a standard model for complex random matrices.
  • Understanding eigenvalue distributions is crucial in various scientific fields.

Purpose of the Study:

  • To compute the full distribution and cumulants of eigenvalues within a disk for the Ginibre ensemble.
  • To investigate the fluctuation regimes of eigenvalue counts in the large N limit.

Main Methods:

  • Exact computation for finite N.
  • Asymptotic analysis in the large N limit.
  • Importance sampling Monte Carlo simulations for validation.

Main Results:

  • Identified three fluctuation regimes for eigenvalue counts: Gaussian, intermediate (O(sqrt[N])), and large deviation.
  • The intermediate regime, previously overlooked, ensures smooth transitions between typical and large deviation behaviors.
  • Demonstrated that all centered cumulants are of order O(sqrt[N]) and controlled by the intermediate regime.
  • Derived an explicit and universal intermediate deviation function.

Conclusions:

  • The study reveals a novel intermediate regime in random matrix eigenvalue fluctuations.
  • This regime is universal and crucial for understanding the full spectrum of behaviors.
  • The findings have implications for fields utilizing random matrix theory.