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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics
  • Random Matrix Theory

Background:

  • Lévy or heavy-tailed random matrices (LMs) are crucial in various physical systems.
  • Understanding their spectral properties, especially localization transitions, is an ongoing challenge.

Purpose of the Study:

  • To provide a comprehensive theoretical and numerical study of Lévy random matrices.
  • To determine the localization transition and phase diagram of LMs.
  • To analyze eigenvalue statistics and finite-size effects in LMs.

Main Methods:

  • Analysis of the self-consistent equation for the resolvent's diagonal elements.
  • Application of supersymmetric field theory and Dyson Brownian motion.
  • Numerical simulations to confirm theoretical predictions.

Main Results:

  • Established the equation for the localization transition and mapped the phase diagram.
  • Showed eigenvalue statistics match the Gaussian orthogonal ensemble in the delocalized phase and are Poisson-like in the localized phase.
  • Observed a wide crossover region due to a rapidly diverging characteristic scale for finite-size effects.

Conclusions:

  • A complete theory of Lévy matrices is now established, integrating analytical and numerical findings.
  • The study clarifies the behavior of LMs across delocalized, localized, and crossover regimes.
  • Findings are crucial for understanding complex systems exhibiting heavy-tailed random matrix behavior.