Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge
Abstract
This is a continuation of our earlier paper (Tao and Vu, <ExternalRef> <RefSource>http://arxiv.org/abs/0908.1982v4[math.PR]</RefSource> <RefTarget Address="http://arxiv.org/abs/0908.1982v4[math.PR]" TargetType="URL"/> </ExternalRef>, 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (<ExternalRef> <RefSource>http://arxiv.org/abs/0908.1982v4[math.PR]</RefSource> <RefTarget Address="http://arxiv.org/abs/0908.1982v4[math.PR]" TargetType="URL"/> </ExternalRef>, 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697-733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- September 2010
- DOI:
- 10.1007/s00220-010-1044-5
- arXiv:
- arXiv:0908.1982
- Bibcode:
- 2010CMaPh.298..549T
- Keywords:
-
- Mathematics - Probability;
- 15A52
- E-Print:
- 24 pages, no figures, to appear, Comm. Math. Phys. One new reference added