# Cusp universality for random matrices I: local law and the complex Hermitian case

László Erdős, Torben Krüger, Dominik Schröder

Comm. Math. Phys.Vol. 378 (2020)

## Summary

Hermitian random matrices can only feature three universal eigenvalue statistics: Sine in the bulk, Airy at square-root edges and Pearcey at cubic-root cusps. In this work we complete the universality program for those matrices by proving Pearcey-kernel universality at all cusp points.## Abstract

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper arXiv:1811.0455 where the cusp universality for real symmetric Wigner-type matrices is proven.