Random matrices: Universality of local eigenvalue statistics up to the edge

Abstract

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov 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.

24 pages, no figures, to appear, Comm. Math. Phys. One new reference added

References in corpus (7)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Universality for mathematical and physical systems
• Universality of Random Matrices and Local Relaxation Flow
• Wegner estimate and level repulsion for Wigner random matrices
• Bulk Universality for Wigner Matrices
• Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation

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