• DOI:10.2140/paa.2019.1.615
• Corpus ID: 119319888

Cusp universality for random matrices, II: The real symmetric case

@article{Cipolloni2018CuspUF,
  title={Cusp universality for random matrices, II: The real symmetric case},
  author={Giorgio Cipolloni and L'aszl'o ErdHos and Torben Kruger and Dominik Schr{\"o}der},
  journal={Pure and Applied Analysis},
  year={2018}
}

• Giorgio Cipolloni, L'aszl'o ErdHos, +1 author Dominik Schröder
• Published 9 November 2018
• Mathematics
• Pure and Applied Analysis

We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.