skip to content
Dominik Schröder

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

Giorgio Cipolloni, László Erdős, Torben Krüger, Dominik Schröder

Pure Appl. Anal.Vol. 1 (2019)


This is the companion result to the previous paper on cusp universality for complex Hermitian matrices. Lacking Brézin-Hikami-type integral formulae we resort to the more robust Dyson Brownian Motion approach.


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.