Cusp Universality for Correlated Random Matrices

Area of Science:

• Random Matrix Theory
• Mathematical Physics
• Spectral Statistics

Background:

• The Wigner-Dyson-Mehta universality conjecture posits universal local eigenvalue statistics in random matrices.
• Previous work established universality in the bulk and edge spectral regimes for general random matrices.
• Cusp universality was previously shown only for specific types of random matrices with independent entries.

Purpose of the Study:

• To prove local eigenvalue statistics universality at cusp singularities for correlated real symmetric and complex Hermitian random matrices.
• To complete the proof of the Wigner-Dyson-Mehta universality conjecture across all spectral regimes.
• To establish universality for a more general class of random matrices than previously studied.

Main Methods:

• Development of an optimal local law at the cusp singularity using the "Zigzag strategy".
• The "Zigzag strategy" combines the characteristic flow method with a Green function comparison argument.
• The local law is proven uniformly across the entire spectrum.

Main Results:

• Universality of local eigenvalue statistics is proven for random matrices at cusp singularities.
• This result extends universality to a broader class of random matrices, including those with correlated entries.
• A simplified proof for bulk and edge universality is also provided.

Conclusions:

• The study successfully proves cusp universality, completing the Wigner-Dyson-Mehta conjecture for a wide range of random matrices.
• The "Zigzag strategy" offers a powerful new technique for analyzing random matrix properties.
• The findings have significant implications for understanding spectral properties in quantum systems and statistical physics.