Cusp Universality for Correlated Random Matrices

TL;DR

This paper proves that the local eigenvalue statistics at cusp singularities are universal for a broad class of correlated random matrices, completing the Wigner-Dyson-Mehta universality conjecture across all spectral regimes.

Contribution

It establishes cusp universality for correlated matrices, extending previous results limited to Wigner-type matrices with independent entries, using an optimal local law via the Zigzag strategy.

Findings

Proves cusp universality for correlated matrices

Establishes an optimal local law at the cusp

Re-establishes universality in bulk and edge regimes

Abstract

For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality [arXiv:1804.07744], while cusp universality was proven only for Wigner-type matrices with independent entries [arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus also re-establishing universality of the local eigenvalue statistics in the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744] regimes.