Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

TL;DR

This paper proves that eigenvalue statistics near cusp singularities in complex Hermitian Wigner-type matrices are universal, following a Pearcey process, thereby completing the universality classification for such matrices.

Contribution

It establishes the universality of local eigenvalue statistics at cusp points for complex Hermitian Wigner-type matrices, including approximate cusps, and proves an optimal local law at the cusp.

Findings

Eigenvalue statistics at cusps follow a Pearcey process.

Universality holds for both exact and approximate cusps.

Main technical result is an optimal local law at the cusp.

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:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.