Universality for general Wigner-type matrices

Ajanki, Oskari and Erdős, László and Krüger, Torben (2016) Universality for general Wigner-type matrices. Probability Theory and Related Fields. ISSN 0178-8051

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Official URL: http://dx.doi.org/10.1007/s00440-016-0740-2

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.

Item Type: Article
DOI: 10.1007/s00440-016-0740-2
Uncontrolled Keywords: Eigenvector delocalization, Rigidity, Anisotropic local law, Local spectral statistics
Subjects: 500 Science > 510 Mathematics
500 Science > 530 Physics
Research Group: Erdös Group
SWORD Depositor: Sword Import User
Depositing User: Sword Import User
Date Deposited: 09 Nov 2016 15:23
Last Modified: 28 Nov 2017 10:17
URI: https://repository.ist.ac.at/id/eprint/657

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