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Wigner surmise for mixed symmetry classes in random matrix theory
Sebastian Schierenberg1, Falk Bruckmann, Tilo Wettig
1Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
This study introduces new formulas for nearest-neighbor spacing distributions in mixed random matrix ensembles. These findings generalize the Wigner surmise and offer accurate approximations for large random matrices.
Area of Science:
- * Physics
- * Mathematics
- * Quantum Chaos
Background:
- * Random matrix theory (RMT) is crucial for understanding complex quantum systems.
- * Nearest-neighbor spacing distributions characterize spectral properties.
- * Existing models (e.g., Wigner surmise) apply to pure ensembles.
Purpose of the Study:
- * To develop analytical formulas for spacing distributions in mixed random matrix ensembles.
- * To interpolate between different symmetry classes and between quantum integrability and nonintegrability.
- * To generalize and extend the applicability of the Wigner surmise.
Main Methods:
- * Derivation of analytical formulas for 2x2 and 4x4 mixed matrix ensembles.
- * Numerical verification of formulas for large-dimension random matrices.
- * Analysis of coupling parameter matching based on local eigenvalue density.
Main Results:
- * Analytical formulas for nearest-neighbor spacing distributions in mixed ensembles were derived.
- * Formulas provide accurate approximations for large random matrices.
- * The Wigner surmise is recovered as a limiting case of the mixed ensembles.
Conclusions:
- * The derived formulas successfully generalize the Wigner surmise.
- * Mixed random matrix ensembles offer a more comprehensive model for spectral statistics.
- * Understanding coupling parameter matching is key for accurate spectral predictions.
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