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This study proves universality of local eigenvalue statistics at cusp singularities for random matrices, completing the Wigner-Dyson-Mehta conjecture. The findings apply to a broad class of matrices, including those with correlated entries.

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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.