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

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...

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Bibliographic Details
Published in:Communications in mathematical physics 2020, Vol.378 (2), p.1203-1278
Main Authors: Erdős, László, Krüger, Torben, Schröder, Dominik
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Cusps
Eigenvalues
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Matrix methods
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Singularity (mathematics)
Symmetry
Theoretical
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Summary: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 the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055 ) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631 ), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969 ).
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-019-03657-4