Cusp universality for random matrices I: Local law and the complex hermitian case

L. Erdös, T.H. Krüger, D.J. Schröder, ArXiv:1809.03971 (n.d.).

Preprint | Submitted | English
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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.
Publishing Year
Date Published
2018-09-11
Journal Title
arXiv:1809.03971
Page
50
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Cite this

Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. arXiv:180903971.
Erdös, L., Krüger, T. H., & Schröder, D. J. (n.d.). Cusp universality for random matrices I: Local law and the complex hermitian case. ArXiv:1809.03971.
Erdös, László, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” ArXiv:1809.03971, n.d.
L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices I: Local law and the complex hermitian case,” arXiv:1809.03971. .
Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. arXiv:1809.03971.
Erdös, László, et al. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” ArXiv:1809.03971.

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