Cusp Universality for Random Matrices II: The Real Symmetric Case
Abstract
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.
Cite
@article{arxiv.1811.04055,
title = {Cusp Universality for Random Matrices II: The Real Symmetric Case},
author = {Giorgio Cipolloni and László Erdős and Torben Krüger and Dominik Schröder},
journal= {arXiv preprint arXiv:1811.04055},
year = {2019}
}
Comments
62 pages. Updated version with additional references