2008/03/31 by László Erdős, Laszlo Erdos, Benjamin Schlein +2 · 8 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #math-ph #math.MP #math.PR #msc:15A52 #msc:82B44
paper · pdf · doi:10.1007/s00220-008-0636-9
14 pages, LateX file. An appendix by J. Bourgain was added. Final version, to appear in Comm. Math. Phys
arxiv created 2008/09/23 · openalex publication_date 2008/09/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We consider N× N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales η≫ N-1 (log N)8. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ^∞-norm of the corresponding eigenvectors is of order O(N-1/2), modulo logarithmic corrections. The upper bound O(N-1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.