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Level Statistics and Localization Transitions of Lévy Matrices.
E Tarquini1,2,3, G Biroli2, M Tarzia1
1LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, boîte 121, 75252 Paris cédex 05, France.
Physical Review Letters
|January 23, 2016
Summary
This study fully characterizes Lévy random matrices (LMs), revealing distinct eigenvalue statistics in delocalized and localized phases. Finite-size effects create a broad crossover region, mimicking a mixed phase near the transition.
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.
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