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.

arXiv comment62 pages. Updated version with additional references

References in corpus (6)

• Universal singularity at the closure of a gap in a random matrix theory
• Level Spacing of Random Matrices in an External Source
• Edge Universality of Beta Ensembles
• The Pearcey Process
• Airy processes with wanderers and new universality classes
• Propagation of singular behavior for Gaussian perturbations of random matrices

Cited by in corpus (8)

• Edge Universality for non-Hermitian Random Matrices
• Correlated Random Matrices: Band Rigidity and Edge Universality
• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
• Large deviations for the largest eigenvalue of matrices with variance profiles
• Critical Behavior of Non-Intersecting Brownian Motions
• Pearcey universality at cusps of polygonal lozenge tiling
• Quenched universality for deformed Wigner matrices
• Non-Hermitian spectral universality at critical points