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This study derives an approximation for complex spacing ratios in non-Hermitian systems, offering insights into eigenvalue correlations. The formula rapidly converges to the infinite matrix size limit, providing new analytical results for random matrix theory.

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Area of Science:

  • Mathematics
  • Physics
  • Complex Systems

Background:

  • Complex spacing ratios were recently introduced to analyze eigenvalue correlations in non-Hermitian systems.
  • Analytical results for the probability distribution of these ratios in the large system size limit are currently lacking.

Purpose of the Study:

  • To derive an approximation formula for the probability distribution of complex spacing ratios.
  • To provide analytical results for the Ginibre universality class in random matrix theory.
  • To analyze eigenvalue correlations in non-Hermitian systems in the limit of large system size.

Main Methods:

  • Derivation of an approximation formula for the Ginibre universality class.
  • Analysis of the convergence rate of the formula to the infinite matrix size limit.
  • Calculation of moments for the probability distribution in the limit.

Main Results:

  • An approximation formula for complex spacing ratios in the Ginibre universality class was derived.
  • The formula exhibits exponential convergence to the limit of infinite matrix size.
  • Analytical results for the moments of the distribution in this limit were obtained.

Conclusions:

  • The derived approximation offers a significant advancement in understanding eigenvalue correlations in non-Hermitian systems.
  • This work provides the first analytical results for the probability distribution of complex spacing ratios in the large system size limit.
  • The findings are crucial for the study of random matrix theory and complex systems.