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This is a continuation of our earlier paper on the universality of theeigenvalues of Wigner random matrices. The main new results of this paper arean extension of the results in that paper from the bulk of the spectrum up tothe edge. In particular, we prove a variant of the universality results ofSoshnikov for the largest eigenvalues, assuming moment conditions rather thansymmetry conditions. The main new technical observation is that there is asignificant bias in the Cauchy interlacing law near the edge of the spectrumwhich allows one to continue ensuring the delocalization of eigenvectors.

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