Cusp Universality for Random Matrices II: The Real Symmetric Case

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 by Erd˝os et al. (2018, 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 by Landon and Yau (2017, arXiv:1712.03881) to the cusp regime using the optimal local law by Erd˝os et al. (2018, arXiv:1809.03971) and the accurate local shape analysis of the density by Ajanki et al. (2015, arXiv:1506.05095) and Alt et al. (2018, arXiv:1804.07752). We also present a novel PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.