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On the Spectral Form Factor for Random Matrices
Giorgio Cipolloni1, László Erdős2, Dominik Schröder3
1Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544 USA.
Communications in Mathematical Physics
|July 3, 2023
Summary
This study rigorously proves the spectral form factor (SFF) prediction for disordered quantum systems and random matrices. Our findings extend universality to larger spectral scales, matching physics predictions across various regimes.
Area of Science:
- Quantum Physics
- Mathematical Physics
- Statistical Mechanics
Background:
- The spectral form factor (SFF) is crucial for testing universality in disordered quantum systems.
- Previous mathematical results for SFF were limited to only two exactly solvable models.
Purpose of the Study:
- To rigorously prove the physics prediction of SFF for a broad class of random matrices.
- To extend the understanding of SFF universality beyond existing mathematical frameworks.
Main Methods:
- Utilized multi-resolvent local laws, a robust mathematical method.
- Analyzed Wigner matrices and the monoparametric ensemble.
Main Results:
- Proved the SFF prediction up to an intermediate time scale for a large class of random matrices.
- Demonstrated that SFF universality can be triggered by a single random parameter in the monoparametric ensemble.
- Showcased that the derived formulas accurately predict SFF in the 'slope-dip-ramp' regime.
Conclusions:
- The study provides rigorous mathematical validation for physics predictions concerning SFF.
- Established universality of SFF for a wider range of random matrices, including the monoparametric ensemble.
- The findings significantly advance the mathematical understanding of disordered quantum systems.
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