--- _id: '2758' abstract: - lang: eng text: 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 n ≫ 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 -tempℓ∞-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. author: - first_name: László full_name: László Erdös id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Benjamin full_name: Schlein, Benjamin last_name: Schlein - first_name: Horng full_name: Yau, Horng-Tzer last_name: Yau citation: ama: Erdös L, Schlein B, Yau H. Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics. 2009;287(2):641-655. doi:10.1007/s00220-008-0636-9 apa: Erdös, L., Schlein, B., & Yau, H. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-008-0636-9 chicago: Erdös, László, Benjamin Schlein, and Horng Yau. “Local Semicircle Law and Complete Delocalization for Wigner Random Matrices.” Communications in Mathematical Physics. Springer, 2009. https://doi.org/10.1007/s00220-008-0636-9. ieee: L. Erdös, B. Schlein, and H. Yau, “Local semicircle law and complete delocalization for Wigner random matrices,” Communications in Mathematical Physics, vol. 287, no. 2. Springer, pp. 641–655, 2009. ista: Erdös L, Schlein B, Yau H. 2009. Local semicircle law and complete delocalization for Wigner random matrices. Communications in Mathematical Physics. 287(2), 641–655. mla: Erdös, László, et al. “Local Semicircle Law and Complete Delocalization for Wigner Random Matrices.” Communications in Mathematical Physics, vol. 287, no. 2, Springer, 2009, pp. 641–55, doi:10.1007/s00220-008-0636-9. short: L. Erdös, B. Schlein, H. Yau, Communications in Mathematical Physics 287 (2009) 641–655. date_created: 2018-12-11T11:59:27Z date_published: 2009-04-01T00:00:00Z date_updated: 2021-01-12T06:59:30Z day: '01' doi: 10.1007/s00220-008-0636-9 extern: 1 intvolume: ' 287' issue: '2' month: '04' page: 641 - 655 publication: Communications in Mathematical Physics publication_status: published publisher: Springer publist_id: '4134' quality_controlled: 0 status: public title: Local semicircle law and complete delocalization for Wigner random matrices type: journal_article volume: 287 year: '2009' ...