# Universality for general Wigner-type matrices

O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727.

IST-2017-657-v1+2_s00440-016-0740-2.pdf 988.84 KB

Journal Article | Published | English
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Department
Abstract
We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.
Publishing Year
Date Published
2017-12-01
Journal Title
Probability Theory and Related Fields
Acknowledgement
Open access funding provided by Institute of Science and Technology (IST Austria).
Volume
169
Issue
3-4
Page
667 - 727
ISSN
IST-REx-ID

### Cite this

Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 2017;169(3-4):667-727. doi:10.1007/s00440-016-0740-2
Ajanki, O. H., Erdös, L., & Krüger, T. H. (2017). Universality for general Wigner-type matrices. Probability Theory and Related Fields, 169(3–4), 667–727. https://doi.org/10.1007/s00440-016-0740-2
Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields 169, no. 3–4 (2017): 667–727. https://doi.org/10.1007/s00440-016-0740-2.
O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type matrices,” Probability Theory and Related Fields, vol. 169, no. 3–4, pp. 667–727, 2017.
Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727.
Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields, vol. 169, no. 3–4, Springer, 2017, pp. 667–727, doi:10.1007/s00440-016-0740-2.
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