[{"status":"public","type":"journal_article","_id":"2776","title":"Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues","publist_id":"4114","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Antti","full_name":"Knowles, Antti","last_name":"Knowles"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng-Tzer"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"extern":1,"date_updated":"2021-01-12T06:59:37Z","citation":{"mla":"Erdös, László, et al. “Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues.” Communications in Mathematical Physics, vol. 314, no. 3, Springer, 2012, pp. 587–640, doi:10.1007/s00220-012-1527-7.","apa":"Erdös, L., Knowles, A., Yau, H., & Yin, J. (2012). Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-012-1527-7","ama":"Erdös L, Knowles A, Yau H, Yin J. Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. 2012;314(3):587-640. doi:10.1007/s00220-012-1527-7","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Communications in Mathematical Physics 314 (2012) 587–640.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues,” Communications in Mathematical Physics, vol. 314, no. 3. Springer, pp. 587–640, 2012.","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues.” Communications in Mathematical Physics. Springer, 2012. https://doi.org/10.1007/s00220-012-1527-7.","ista":"Erdös L, Knowles A, Yau H, Yin J. 2012. Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. 314(3), 587–640."},"month":"09","intvolume":" 314","quality_controlled":0,"publisher":"Springer","abstract":[{"lang":"eng","text":"We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments."}],"issue":"3","doi":"10.1007/s00220-012-1527-7","date_published":"2012-09-01T00:00:00Z","volume":314,"date_created":"2018-12-11T11:59:32Z","page":"587 - 640","day":"01","publication":"Communications in Mathematical Physics","publication_status":"published","year":"2012"},{"_id":"2774","status":"public","type":"journal_article","extern":1,"citation":{"ista":"Erdös L, Fournais S, Solovej J. 2012. Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. 312(3), 847–882.","chicago":"Erdös, László, Søren Fournais, and Jan Solovej. “Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field.” Communications in Mathematical Physics. Springer, 2012. https://doi.org/10.1007/s00220-012-1468-1.","short":"L. Erdös, S. Fournais, J. Solovej, Communications in Mathematical Physics 312 (2012) 847–882.","ieee":"L. Erdös, S. Fournais, and J. Solovej, “Scott correction for large atoms and molecules in a self-generated magnetic field,” Communications in Mathematical Physics, vol. 312, no. 3. Springer, pp. 847–882, 2012.","ama":"Erdös L, Fournais S, Solovej J. Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. 2012;312(3):847-882. doi:10.1007/s00220-012-1468-1","apa":"Erdös, L., Fournais, S., & Solovej, J. (2012). Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-012-1468-1","mla":"Erdös, László, et al. “Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field.” Communications in Mathematical Physics, vol. 312, no. 3, Springer, 2012, pp. 847–82, doi:10.1007/s00220-012-1468-1."},"date_updated":"2021-01-12T06:59:36Z","title":"Scott correction for large atoms and molecules in a self-generated magnetic field","author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"László Erdös","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"last_name":"Fournais","full_name":"Fournais, Søren","first_name":"Søren"},{"first_name":"Jan","full_name":"Solovej, Jan P","last_name":"Solovej"}],"publist_id":"4116","abstract":[{"lang":"eng","text":"We consider a large neutral molecule with total nuclear charge Z in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Zα 2 ≤ κ 0 for a sufficiently small κ 0, where α denotes the fine structure constant. We show that, in the simultaneous limit Z → ∞, α → 0 such that κ = Zα 2 is fixed, the ground state energy of the system is given by a two term expansion c 1Z 7/3 + c 2(κ) Z 2 + o(Z 2). The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion."}],"intvolume":" 312","month":"06","quality_controlled":0,"publisher":"Springer","publication":"Communications in Mathematical Physics","day":"01","year":"2012","publication_status":"published","date_created":"2018-12-11T11:59:31Z","date_published":"2012-06-01T00:00:00Z","volume":312,"issue":"3","doi":"10.1007/s00220-012-1468-1","page":"847 - 882"},{"title":"A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices","publist_id":"4117","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"László Erdös","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"full_name":"Yau, Horng-Tzer","last_name":"Yau","first_name":"Horng"}],"extern":1,"citation":{"ista":"Erdös L, Yau H. 2012. A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electronic Journal of Probability. 17.","chicago":"Erdös, László, and Horng Yau. “A Comment on the Wigner-Dyson-Mehta Bulk Universality Conjecture for Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2012. https://doi.org/10.1214/EJP.v17-1779.","ama":"Erdös L, Yau H. A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electronic Journal of Probability. 2012;17. doi:10.1214/EJP.v17-1779","apa":"Erdös, L., & Yau, H. (2012). A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v17-1779","short":"L. Erdös, H. Yau, Electronic Journal of Probability 17 (2012).","ieee":"L. Erdös and H. Yau, “A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices,” Electronic Journal of Probability, vol. 17. Institute of Mathematical Statistics, 2012.","mla":"Erdös, László, and Horng Yau. “A Comment on the Wigner-Dyson-Mehta Bulk Universality Conjecture for Wigner Matrices.” Electronic Journal of Probability, vol. 17, Institute of Mathematical Statistics, 2012, doi:10.1214/EJP.v17-1779."},"date_updated":"2021-01-12T06:59:36Z","status":"public","type":"journal_article","_id":"2773","date_published":"2012-04-10T00:00:00Z","doi":"10.1214/EJP.v17-1779","volume":17,"date_created":"2018-12-11T11:59:31Z","day":"10","publication":"Electronic Journal of Probability","year":"2012","publication_status":"published","month":"04","intvolume":" 17","quality_controlled":0,"publisher":"Institute of Mathematical Statistics","abstract":[{"lang":"eng","text":"Recently we proved [3, 4, 6, 7, 9, 10, 11] that the eigenvalue correlation functions of a general class of random matrices converge, weakly with respect to the energy, to the corresponding ones of Gaussian matrices. Tao and Vu [15] gave a proof that for the special case of Hermitian Wigner matrices the convergence can be strengthened to vague convergence at any fixed energy in the bulk. In this article we show that this theorem is an immediate corollary of our earlier results. Indeed, a more general form of this theorem also follows directly from our work [2]."}]},{"abstract":[{"text":"We consider a magnetic Schrödinger operator in two dimensions. The magnetic field is given as the sum of a large and constant magnetic field and a random magnetic field. Moreover, we allow for an additional deterministic potential as well as a magnetic field which are both periodic. We show that the spectrum of this operator is contained in broadened bands around the Landau levels and that the edges of these bands consist of pure point spectrum with exponentially decaying eigenfunctions. The proof is based on a recent Wegner estimate obtained in Erdos and Hasler (Commun. Math. Phys., preprint, arXiv:1012.5185) and a multiscale analysis.","lang":"eng"}],"month":"03","intvolume":" 146","quality_controlled":0,"publisher":"Springer","day":"01","publication":"Journal of Statistical Physics","year":"2012","publication_status":"published","volume":146,"date_published":"2012-03-01T00:00:00Z","issue":"5","doi":"10.1007/s10955-012-0445-6","date_created":"2018-12-11T11:59:31Z","page":"900 - 923","_id":"2771","status":"public","type":"journal_article","extern":1,"date_updated":"2021-01-12T06:59:35Z","citation":{"ieee":"L. Erdös and D. Hasler, “Anderson localization at band edges for random magnetic fields,” Journal of Statistical Physics, vol. 146, no. 5. Springer, pp. 900–923, 2012.","short":"L. Erdös, D. Hasler, Journal of Statistical Physics 146 (2012) 900–923.","apa":"Erdös, L., & Hasler, D. (2012). Anderson localization at band edges for random magnetic fields. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-012-0445-6","ama":"Erdös L, Hasler D. Anderson localization at band edges for random magnetic fields. Journal of Statistical Physics. 2012;146(5):900-923. doi:10.1007/s10955-012-0445-6","mla":"Erdös, László, and David Hasler. “Anderson Localization at Band Edges for Random Magnetic Fields.” Journal of Statistical Physics, vol. 146, no. 5, Springer, 2012, pp. 900–23, doi:10.1007/s10955-012-0445-6.","ista":"Erdös L, Hasler D. 2012. Anderson localization at band edges for random magnetic fields. Journal of Statistical Physics. 146(5), 900–923.","chicago":"Erdös, László, and David Hasler. “Anderson Localization at Band Edges for Random Magnetic Fields.” Journal of Statistical Physics. Springer, 2012. https://doi.org/10.1007/s10955-012-0445-6."},"title":"Anderson localization at band edges for random magnetic fields","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"David","last_name":"Hasler","full_name":"Hasler, David G"}],"publist_id":"4119"},{"author":[{"last_name":"Bourgade","full_name":"Bourgade, Paul","first_name":"Paul"},{"last_name":"Erdös","full_name":"László Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Yau","full_name":"Yau, Horng-Tzer","first_name":"Horng"}],"publist_id":"4112","title":"Bulk universality of general β-ensembles with non-convex potential","citation":{"chicago":"Bourgade, Paul, László Erdös, and Horng Yau. “Bulk Universality of General β-Ensembles with Non-Convex Potential.” Journal of Mathematical Physics. American Institute of Physics, 2012. https://doi.org/10.1063/1.4751478.","ista":"Bourgade P, Erdös L, Yau H. 2012. Bulk universality of general β-ensembles with non-convex potential. Journal of Mathematical Physics. 53(9).","mla":"Bourgade, Paul, et al. “Bulk Universality of General β-Ensembles with Non-Convex Potential.” Journal of Mathematical Physics, vol. 53, no. 9, American Institute of Physics, 2012, doi:10.1063/1.4751478.","ama":"Bourgade P, Erdös L, Yau H. Bulk universality of general β-ensembles with non-convex potential. Journal of Mathematical Physics. 2012;53(9). doi:10.1063/1.4751478","apa":"Bourgade, P., Erdös, L., & Yau, H. (2012). Bulk universality of general β-ensembles with non-convex potential. Journal of Mathematical Physics. American Institute of Physics. https://doi.org/10.1063/1.4751478","short":"P. Bourgade, L. Erdös, H. Yau, Journal of Mathematical Physics 53 (2012).","ieee":"P. Bourgade, L. Erdös, and H. Yau, “Bulk universality of general β-ensembles with non-convex potential,” Journal of Mathematical Physics, vol. 53, no. 9. American Institute of Physics, 2012."},"date_updated":"2021-01-12T06:59:38Z","extern":1,"type":"journal_article","status":"public","_id":"2778","date_created":"2018-12-11T11:59:33Z","volume":53,"doi":"10.1063/1.4751478","issue":"9","date_published":"2012-09-28T00:00:00Z","publication_status":"published","year":"2012","publication":"Journal of Mathematical Physics","day":"28","quality_controlled":0,"publisher":"American Institute of Physics","intvolume":" 53","month":"09","abstract":[{"text":"We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β > 0. This removes the convexity assumption appeared in the earlier work [P. Bourgade, L. Erdös, and H.-T. Yau, Universality of general β-ensembles, preprint arXiv:0907.5605 (2011)]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a "convexified measure" so that the local statistics are preserved under this convexification.","lang":"eng"}]}]