10.1007/s00440-016-0740-2
Ajanki, Oskari H
Oskari H
Ajanki
Erdös, László
László
Erdös0000-0001-5366-9603
Krüger, Torben H
Torben H
Krüger
Universality for general Wigner-type matrices
Springer
2017
2018-12-11T11:51:27Z
2019-08-02T12:37:00Z
journal_article
https://research-explorer.app.ist.ac.at/record/1337
https://research-explorer.app.ist.ac.at/record/1337.json
01788051
988843 bytes
application/pdf
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.