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This study analyzes eigenvalue distributions in random unitary matrices, revealing connections to fermion statistics and random landscape mechanics. Findings offer insights into the Riemann Zeta function

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Area of Science:

  • Mathematics
  • Statistical Physics
  • Number Theory

Background:

  • Random matrix theory (RMT) is crucial for understanding complex quantum systems and number theory.
  • The Circular Unitary Ensemble (CUE) is a fundamental model in RMT, with applications to quantum chaos and the Riemann Zeta function.
  • Analyzing eigenvalue distributions provides insights into the underlying statistical properties of these systems.

Purpose of the Study:

  • To investigate the maximum of eigenvalue number deviations in random unitary matrices from the CUE(β) ensemble.
  • To establish a connection between these deviations and the statistical mechanics of log-correlated random landscapes.
  • To derive the distribution of this maximum for any β > 0, linking it to fermion statistics and extremal statistics.

Main Methods:

  • Utilizing a mapping to the statistical mechanics of log-correlated random landscapes.
  • Employing an extended Fisher-Hartwig conjecture and the freezing duality conjecture for log-correlated fields.
  • Calculating the cumulants of the distribution of the maximum eigenvalue process.

Main Results:

  • The study derives the cumulants for the maximum of eigenvalue number deviations for any β > 0.
  • The results show a combination of standard fermion counting statistics (free for β=2, interacting for β≠2) and extremal statistics of fractional Brownian motion (Hurst index H=0).
  • Specifically, the β=2 case is conjectured to apply to the statistics of the zeroes of the Riemann Zeta function.

Conclusions:

  • The research provides a novel analytical framework for understanding eigenvalue statistics in random unitary matrices.
  • The findings bridge concepts from random matrix theory, statistical mechanics, and number theory.
  • The derived statistics offer new perspectives on the distribution of eigenvalues and their potential applications, particularly concerning the Riemann Zeta function.