Author(s):

Ajanki, Oskari H; Erdős, László; Krüger, Torben

Article Title: 
Universality for general Wignertype matrices

Affiliation 
IST Austria 
Abstract: 
We consider the local eigenvalue distribution of large selfadjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmEhij2 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.

Keywords: 
Eigenvector delocalization; Rigidity; Anisotropic local law; Local spectral statistics

Journal Title:

Probability Theory and Related Fields

Volume: 
169

Issue 
34

ISSN:

14322064

Publisher:

Springer

Date Published:

20171201

Start Page: 
667

End Page:

727

Copyright Statement: 
CC BY 4.0

URL: 

DOI: 
10.1007/s0044001607402

Notes: 
Open access funding provided by Institute of Science and Technology (IST Austria). Oskari H. Ajanki: Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFBTR 12 Grant of the German Research Council. László Erdős: Partially supported by ERC Advanced Grant RANMAT No. 338804. Torben Krüger: Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFBTR 12 Grant of the German Research Council.

Open access: 
yes (OA journal) 