Universality for general Wigner-type matrices Journal Article


Author(s): Ajanki, Oskari H; Erdős, László; Krüger, Torben
Article Title: Universality for general Wigner-type matrices
Affiliation IST Austria
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.
Keywords: Eigenvector delocalization; Rigidity; Anisotropic local law; Local spectral statistics
Journal Title: Probability Theory and Related Fields
ISSN: 1432-2064
Publisher: Springer  
Date Published: 2016-09-19
Start Page: Epub ahead of print
Copyright Statement: CC BY 4.0
URL:
DOI: 10.1007/s00440-016-0740-2
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 SFB-TR 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 SFB-TR 12 Grant of the German Research Council.
Open access: yes (OA journal)
IST Austria Authors
  1. László Erdős
    102 Erdős
  2. Oskari Ajanki
    3 Ajanki