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Area of Science:

  • Mathematics
  • Probability Theory
  • Statistical Mechanics

Background:

  • Spectral properties of random matrices are crucial in various scientific fields.
  • Understanding the behavior of large random matrices requires advanced analytical techniques.

Purpose of the Study:

  • To investigate the cumulant approach for analyzing spectral properties of large random matrices.
  • To detail the joint cumulants of high traces for large unitary random matrices.

Main Methods:

  • Utilizing the cumulant approach to analyze spectral properties.
  • Focusing on joint cumulants of high traces of unitary random matrices.

Main Results:

  • Detailed analysis of joint cumulants for large unitary random matrices.
  • Proof of Gaussian fluctuation for pair-counting statistics with non-smooth test functions.

Conclusions:

  • The cumulant approach provides powerful tools for studying random matrix spectral properties.
  • The findings contribute to the understanding of statistical fluctuations in large random matrix ensembles.