Vol. 1, No. 4, 2019

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

Giorgio Cipolloni, László Erdős, Torben Krüger and Dominik Schröder

Vol. 1 (2019), No. 4, 615–707
DOI: 10.2140/paa.2019.1.615

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 Erdős 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 Erdős 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.

cusp universality, Dyson Brownian motion, local law
Mathematical Subject Classification 2010
Primary: 60B20, 15B52
Received: 28 January 2019
Revised: 17 June 2019
Accepted: 22 July 2019
Published: 12 October 2019
Giorgio Cipolloni
IST Austria
László Erdős
IST Austria
Torben Krüger
University of Bonn
Dominik Schröder
IST Austria