Cusp universality for random matrices I: Local law and the complex hermitian case
submitted
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Torben H
Krüger
author 3020C786-F248-11E8-B48F-1D18A9856A87
Dominik J
Schröder
author 408ED176-F248-11E8-B48F-1D18A9856A87
LaEr
department
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.
2018
eng
arXiv
1809.03971
50
/record/6179
L. Erdös, T.H. Krüger, D.J. Schröder, ArXiv (n.d.).
Erdös, László, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” <i>ArXiv</i>, n.d.
Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. <i>arXiv</i>.
Erdös, L., Krüger, T. H., & Schröder, D. J. (n.d.). Cusp universality for random matrices I: Local law and the complex hermitian case. <i>ArXiv</i>.
Erdös, László, et al. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” <i>ArXiv</i>.
L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices I: Local law and the complex hermitian case,” <i>arXiv</i>. .
Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. arXiv.
61852019-03-28T10:21:15Z2020-01-17T09:49:13Z