[{"date_published":"2018-06-14T00:00:00Z","citation":{"mla":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” *Probability Theory and Related Fields*, vol. 171, no. 1–2, 543–616, Springer, 2018, doi:10.1007/s00440-017-0787-8.","ista":"Lee J, Schnelli K. 2018. Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. 171(1–2), 543–616.","ieee":"J. Lee and K. Schnelli, “Local law and Tracy–Widom limit for sparse random matrices,” *Probability Theory and Related Fields*, vol. 171, no. 1–2. Springer, 2018.","apa":"Lee, J., & Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. *Probability Theory and Related Fields*. Springer. https://doi.org/10.1007/s00440-017-0787-8","chicago":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” *Probability Theory and Related Fields*. Springer, 2018. https://doi.org/10.1007/s00440-017-0787-8.","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).","ama":"Lee J, Schnelli K. Local law and Tracy–Widom limit for sparse random matrices. *Probability Theory and Related Fields*. 2018;171(1-2). doi:10.1007/s00440-017-0787-8"},"oa":1,"volume":171,"year":"2018","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"doi":"10.1007/s00440-017-0787-8","day":"14","type":"journal_article","publisher":"Springer","issue":"1-2","oa_version":"Preprint","date_updated":"2021-01-12T08:09:33Z","month":"06","abstract":[{"lang":"eng","text":"We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1."}],"scopus_import":1,"publication":"Probability Theory and Related Fields","article_number":"543-616","quality_controlled":"1","publication_status":"published","department":[{"_id":"LaEr"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"main_file_link":[{"url":"https://arxiv.org/abs/1605.08767","open_access":"1"}],"title":"Local law and Tracy–Widom limit for sparse random matrices","publist_id":"7017","intvolume":" 171","ec_funded":1,"status":"public","date_created":"2018-12-11T11:47:56Z","external_id":{"arxiv":["1605.08767"]},"_id":"690","author":[{"last_name":"Lee","full_name":"Lee, Jii","first_name":"Jii"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli"}]},{"intvolume":" 15","article_processing_charge":"No","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"quality_controlled":"1","publication_status":"published","publication_identifier":{"issn":["1980-0436"]},"title":"Transition to shocks in TASEP and decoupling of last passage times","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"},{"call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117"}],"page":"1311-1334","publication":"Latin American Journal of Probability and Mathematical Statistics","scopus_import":1,"has_accepted_license":"1","author":[{"last_name":"Nejjar","id":"4BF426E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter","full_name":"Nejjar, Peter"}],"_id":"70","external_id":{"arxiv":["1705.08836"]},"date_created":"2018-12-11T11:44:28Z","status":"public","ec_funded":1,"file":[{"date_created":"2019-02-14T09:44:10Z","file_name":"2018_ALEA_Nejjar.pdf","access_level":"open_access","checksum":"2ded46aa284a836a8cbb34133a64f1cb","date_updated":"2020-07-14T12:47:46Z","file_size":394851,"relation":"main_file","file_id":"5981","content_type":"application/pdf","creator":"kschuh"}],"ddc":["510"],"language":[{"iso":"eng"}],"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","citation":{"short":"P. Nejjar, Latin American Journal of Probability and Mathematical Statistics 15 (2018) 1311–1334.","ama":"Nejjar P. Transition to shocks in TASEP and decoupling of last passage times. *Latin American Journal of Probability and Mathematical Statistics*. 2018;15(2):1311-1334. doi:10.30757/ALEA.v15-49","ista":"Nejjar P. 2018. Transition to shocks in TASEP and decoupling of last passage times. Latin American Journal of Probability and Mathematical Statistics. 15(2), 1311–1334.","mla":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” *Latin American Journal of Probability and Mathematical Statistics*, vol. 15, no. 2, ALEA, 2018, pp. 1311–34, doi:10.30757/ALEA.v15-49.","ieee":"P. Nejjar, “Transition to shocks in TASEP and decoupling of last passage times,” *Latin American Journal of Probability and Mathematical Statistics*, vol. 15, no. 2. ALEA, pp. 1311–1334, 2018.","chicago":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” *Latin American Journal of Probability and Mathematical Statistics*. ALEA, 2018. https://doi.org/10.30757/ALEA.v15-49.","apa":"Nejjar, P. (2018). Transition to shocks in TASEP and decoupling of last passage times. *Latin American Journal of Probability and Mathematical Statistics*. ALEA. https://doi.org/10.30757/ALEA.v15-49"},"oa":1,"year":"2018","volume":15,"date_published":"2018-10-01T00:00:00Z","date_updated":"2021-01-12T08:11:24Z","abstract":[{"text":"We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.","lang":"eng"}],"month":"10","article_type":"original","issue":"2","file_date_updated":"2020-07-14T12:47:46Z","oa_version":"Published Version","day":"01","publisher":"ALEA","type":"journal_article","doi":"10.30757/ALEA.v15-49"},{"author":[{"last_name":"Alt","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","full_name":"Alt, Johannes"}],"has_accepted_license":"1","alternative_title":["IST Austria Thesis"],"_id":"149","related_material":{"record":[{"relation":"part_of_dissertation","id":"1010","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"1677"},{"status":"public","id":"550","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"566","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"6183"},{"status":"public","id":"6184","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"6240","status":"public"}]},"date_created":"2018-12-11T11:44:53Z","ec_funded":1,"file":[{"date_updated":"2020-07-14T12:44:57Z","content_type":"application/pdf","file_size":5801709,"relation":"main_file","file_id":"6241","creator":"dernst","date_created":"2019-04-08T13:55:20Z","file_name":"2018_thesis_Alt.pdf","checksum":"d4dad55a7513f345706aaaba90cb1bb8","access_level":"open_access"},{"checksum":"d73fcf46300dce74c403f2b491148ab4","access_level":"closed","date_created":"2019-04-08T13:55:20Z","file_name":"2018_thesis_Alt_source.zip","creator":"dernst","date_updated":"2020-07-14T12:44:57Z","file_id":"6242","relation":"source_file","file_size":3802059,"content_type":"application/zip"}],"status":"public","ddc":["515","519"],"publication_status":"published","department":[{"_id":"LaEr"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"page":"456","title":"Dyson equation and eigenvalue statistics of random matrices","publist_id":"7772","date_updated":"2021-01-12T08:06:48Z","supervisor":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"}],"month":"07","abstract":[{"text":"The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.","lang":"eng"}],"oa_version":"Published Version","file_date_updated":"2020-07-14T12:44:57Z","pubrep_id":"1040","day":"12","type":"dissertation","publisher":"IST Austria","doi":"10.15479/AT:ISTA:TH_1040","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"apa":"Alt, J. (2018). *Dyson equation and eigenvalue statistics of random matrices*. IST Austria. https://doi.org/10.15479/AT:ISTA:TH_1040","chicago":"Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.” IST Austria, 2018. https://doi.org/10.15479/AT:ISTA:TH_1040.","mla":"Alt, Johannes. *Dyson Equation and Eigenvalue Statistics of Random Matrices*. IST Austria, 2018, doi:10.15479/AT:ISTA:TH_1040.","ieee":"J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” IST Austria, 2018.","ista":"Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. IST Austria.","ama":"Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:10.15479/AT:ISTA:TH_1040","short":"J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, IST Austria, 2018."},"oa":1,"year":"2018","date_published":"2018-07-12T00:00:00Z"},{"issue":"10","oa_version":"Preprint","date_updated":"2021-01-12T08:06:34Z","month":"05","abstract":[{"lang":"eng","text":"We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor H and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and H. In particular, our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues ofH and H, around their asymptotic shape, the Vershik'Kerov'Logan'Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense."}],"doi":"10.1093/imrn/rnw330","day":"18","publisher":"Oxford University Press","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"date_published":"2018-05-18T00:00:00Z","oa":1,"citation":{"apa":"Erdös, L., & Schröder, D. J. (2018). Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*. Oxford University Press. https://doi.org/10.1093/imrn/rnw330","chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices*. Oxford University Press, 2018. https://doi.org/10.1093/imrn/rnw330.","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues,” *International Mathematics Research Notices*, vol. 2018, no. 10. Oxford University Press, pp. 3255–3298, 2018.","mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices*, vol. 2018, no. 10, Oxford University Press, 2018, pp. 3255–98, doi:10.1093/imrn/rnw330.","ista":"Erdös L, Schröder DJ. 2018. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. International Mathematics Research Notices. 2018(10), 3255–3298.","ama":"Erdös L, Schröder DJ. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*. 2018;2018(10):3255-3298. doi:10.1093/imrn/rnw330","short":"L. Erdös, D.J. Schröder, International Mathematics Research Notices 2018 (2018) 3255–3298."},"year":"2018","volume":2018,"_id":"1012","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","orcid":"0000-0001-5366-9603"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","first_name":"Dominik J"}],"status":"public","ec_funded":1,"external_id":{"arxiv":["1608.05163"]},"related_material":{"record":[{"id":"6179","relation":"dissertation_contains","status":"public"}]},"date_created":"2018-12-11T11:49:41Z","department":[{"_id":"LaEr"}],"publication_identifier":{"issn":["10737928"]},"quality_controlled":"1","publication_status":"published","publist_id":"6383","title":"Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"main_file_link":[{"url":"https://arxiv.org/abs/1608.05163","open_access":"1"}],"page":"3255-3298","intvolume":" 2018","scopus_import":1,"publication":"International Mathematics Research Notices"},{"oa_version":"Submitted Version","issue":"9","month":"09","abstract":[{"lang":"eng","text":"Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur."}],"date_updated":"2021-01-12T08:12:24Z","doi":"10.1002/cpa.21639","publisher":"Wiley-Blackwell","type":"journal_article","day":"01","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"date_published":"2017-09-01T00:00:00Z","year":"2017","volume":70,"citation":{"ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. *Communications on Pure and Applied Mathematics*. 2017;70(9):1672-1705. doi:10.1002/cpa.21639","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” *Communications on Pure and Applied Mathematics*. Wiley-Blackwell, 2017. https://doi.org/10.1002/cpa.21639.","apa":"Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. *Communications on Pure and Applied Mathematics*. Wiley-Blackwell. https://doi.org/10.1002/cpa.21639","mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” *Communications on Pure and Applied Mathematics*, vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:10.1002/cpa.21639.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” *Communications on Pure and Applied Mathematics*, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017.","ista":"Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 70(9), 1672–1705."},"oa":1,"_id":"721","author":[{"last_name":"Ajanki","full_name":"Ajanki, Oskari H","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"}],"status":"public","ec_funded":1,"date_created":"2018-12-11T11:48:08Z","publist_id":"6959","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1512.03703"}],"page":"1672 - 1705","department":[{"_id":"LaEr"}],"publication_identifier":{"issn":["00103640"]},"publication_status":"published","quality_controlled":"1","intvolume":" 70","scopus_import":1,"publication":"Communications on Pure and Applied Mathematics"},{"scopus_import":1,"publication":"Advances in Mathematics","publist_id":"6935","title":"Convergence rate for spectral distribution of addition of random matrices","project":[{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1606.03076"}],"page":"251 - 291","department":[{"_id":"LaEr"}],"publication_status":"published","quality_controlled":"1","intvolume":" 319","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","status":"public","ec_funded":1,"date_created":"2018-12-11T11:48:13Z","_id":"733","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli"}],"date_published":"2017-10-15T00:00:00Z","year":"2017","volume":319,"citation":{"chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics*. Academic Press, 2017. https://doi.org/10.1016/j.aim.2017.08.028.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. *Advances in Mathematics*. Academic Press. https://doi.org/10.1016/j.aim.2017.08.028","mla":"Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics*, vol. 319, Academic Press, 2017, pp. 251–91, doi:10.1016/j.aim.2017.08.028.","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” *Advances in Mathematics*, vol. 319. Academic Press, pp. 251–291, 2017.","ama":"Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. *Advances in Mathematics*. 2017;319:251-291. doi:10.1016/j.aim.2017.08.028","short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291."},"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"doi":"10.1016/j.aim.2017.08.028","publisher":"Academic Press","type":"journal_article","day":"15","oa_version":"Submitted Version","month":"10","abstract":[{"text":"Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.","lang":"eng"}],"date_updated":"2021-01-12T08:13:07Z"},{"quality_controlled":"1","publication_status":"published","publication_identifier":{"issn":["10950761"]},"department":[{"_id":"LaEr"}],"project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"page":"739 - 800","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.02312"}],"title":"Universality for a class of random band matrices","publist_id":"7337","intvolume":" 21","scopus_import":1,"publication":"Advances in Theoretical and Mathematical Physics","_id":"483","author":[{"first_name":"Paul","full_name":"Bourgade, Paul","last_name":"Bourgade"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"last_name":"Yau","full_name":"Yau, Horng","first_name":"Horng"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"ec_funded":1,"status":"public","date_created":"2018-12-11T11:46:43Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"date_published":"2017-08-25T00:00:00Z","oa":1,"citation":{"chicago":"Bourgade, Paul, László Erdös, Horng Yau, and Jun Yin. “Universality for a Class of Random Band Matrices.” *Advances in Theoretical and Mathematical Physics*. International Press, 2017. https://doi.org/10.4310/ATMP.2017.v21.n3.a5.","apa":"Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2017). Universality for a class of random band matrices. *Advances in Theoretical and Mathematical Physics*. International Press. https://doi.org/10.4310/ATMP.2017.v21.n3.a5","ista":"Bourgade P, Erdös L, Yau H, Yin J. 2017. Universality for a class of random band matrices. Advances in Theoretical and Mathematical Physics. 21(3), 739–800.","ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Universality for a class of random band matrices,” *Advances in Theoretical and Mathematical Physics*, vol. 21, no. 3. International Press, pp. 739–800, 2017.","mla":"Bourgade, Paul, et al. “Universality for a Class of Random Band Matrices.” *Advances in Theoretical and Mathematical Physics*, vol. 21, no. 3, International Press, 2017, pp. 739–800, doi:10.4310/ATMP.2017.v21.n3.a5.","ama":"Bourgade P, Erdös L, Yau H, Yin J. Universality for a class of random band matrices. *Advances in Theoretical and Mathematical Physics*. 2017;21(3):739-800. doi:10.4310/ATMP.2017.v21.n3.a5","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Advances in Theoretical and Mathematical Physics 21 (2017) 739–800."},"volume":21,"year":"2017","oa_version":"Submitted Version","issue":"3","date_updated":"2021-01-12T08:00:57Z","abstract":[{"text":"We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.","lang":"eng"}],"month":"08","doi":"10.4310/ATMP.2017.v21.n3.a5","day":"25","type":"journal_article","publisher":"International Press"},{"article_number":"63","publication":"Electronic Communications in Probability","scopus_import":1,"intvolume":" 22","title":"Singularities of the density of states of random Gram matrices","publist_id":"7265","project":[{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"department":[{"_id":"LaEr"}],"publication_status":"published","publication_identifier":{"issn":["1083589X"]},"quality_controlled":"1","related_material":{"record":[{"relation":"dissertation_contains","id":"149","status":"public"}]},"date_created":"2018-12-11T11:47:07Z","ddc":["539"],"status":"public","ec_funded":1,"file":[{"date_created":"2018-12-12T10:08:04Z","file_name":"IST-2018-926-v1+1_euclid.ecp.1511233247.pdf","checksum":"0ec05303a0de190de145654237984c79","access_level":"open_access","date_updated":"2020-07-14T12:47:00Z","relation":"main_file","content_type":"application/pdf","file_id":"4663","file_size":470876,"creator":"system"}],"has_accepted_license":"1","author":[{"last_name":"Alt","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","full_name":"Alt, Johannes"}],"_id":"550","year":"2017","volume":22,"citation":{"short":"J. Alt, Electronic Communications in Probability 22 (2017).","ama":"Alt J. Singularities of the density of states of random Gram matrices. *Electronic Communications in Probability*. 2017;22. doi:10.1214/17-ECP97","ista":"Alt J. 2017. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 22, 63.","mla":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” *Electronic Communications in Probability*, vol. 22, 63, Institute of Mathematical Statistics, 2017, doi:10.1214/17-ECP97.","ieee":"J. Alt, “Singularities of the density of states of random Gram matrices,” *Electronic Communications in Probability*, vol. 22. Institute of Mathematical Statistics, 2017.","chicago":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” *Electronic Communications in Probability*. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-ECP97.","apa":"Alt, J. (2017). Singularities of the density of states of random Gram matrices. *Electronic Communications in Probability*. Institute of Mathematical Statistics. https://doi.org/10.1214/17-ECP97"},"oa":1,"date_published":"2017-11-21T00:00:00Z","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Institute of Mathematical Statistics","type":"journal_article","day":"21","doi":"10.1214/17-ECP97","month":"11","abstract":[{"lang":"eng","text":"For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities."}],"date_updated":"2021-01-12T08:02:34Z","file_date_updated":"2020-07-14T12:47:00Z","oa_version":"Published Version","pubrep_id":"926"},{"intvolume":" 28","language":[{"iso":"eng"}],"publication_status":"published","quality_controlled":"1","publication_identifier":{"eisbn":["978-1-4704-4194-4"],"isbn":["9781470436483"]},"department":[{"_id":"LaEr"}],"project":[{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"page":"226","title":"A dynamical approach to random matrix theory","publist_id":"7247","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"apa":"Erdös, L., & Yau, H. (2017). *A dynamical approach to random matrix theory* (Vol. 28). American Mathematical Society.","chicago":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28. Courant Lecture Notes. American Mathematical Society, 2017.","ieee":"L. Erdös and H. Yau, *A dynamical approach to random matrix theory*, vol. 28. American Mathematical Society, 2017.","mla":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28, American Mathematical Society, 2017.","ista":"Erdös L, Yau H. 2017. A dynamical approach to random matrix theory, American Mathematical Society, 226p.","ama":"Erdös L, Yau H. *A Dynamical Approach to Random Matrix Theory*. Vol 28. American Mathematical Society; 2017.","short":"L. Erdös, H. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 2017."},"volume":28,"year":"2017","date_published":"2017-01-01T00:00:00Z","date_updated":"2020-01-16T12:37:45Z","author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"last_name":"Yau","full_name":"Yau, Horng","first_name":"Horng"}],"abstract":[{"lang":"eng","text":"This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.\r\n"}],"month":"01","alternative_title":["Courant Lecture Notes"],"_id":"567","oa_version":"None","date_created":"2018-12-11T11:47:13Z","day":"01","type":"book","publisher":"American Mathematical Society","ec_funded":1,"series_title":"Courant Lecture Notes","status":"public"},{"language":[{"iso":"eng"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","year":"2017","volume":53,"citation":{"apa":"Erdös, L., & Schnelli, K. (2017). Universality for random matrix flows with time dependent density. *Annales de l’institut Henri Poincare (B) Probability and Statistics*. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AIHP765","chicago":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability and Statistics*. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-AIHP765.","ieee":"L. Erdös and K. Schnelli, “Universality for random matrix flows with time dependent density,” *Annales de l’institut Henri Poincare (B) Probability and Statistics*, vol. 53, no. 4. Institute of Mathematical Statistics, pp. 1606–1656, 2017.","ista":"Erdös L, Schnelli K. 2017. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 53(4), 1606–1656.","mla":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability and Statistics*, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:10.1214/16-AIHP765.","ama":"Erdös L, Schnelli K. Universality for random matrix flows with time dependent density. *Annales de l’institut Henri Poincare (B) Probability and Statistics*. 2017;53(4):1606-1656. doi:10.1214/16-AIHP765","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656."},"oa":1,"date_published":"2017-11-01T00:00:00Z","abstract":[{"text":"We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law.","lang":"eng"}],"month":"11","date_updated":"2021-01-12T08:06:22Z","oa_version":"Submitted Version","issue":"4","publisher":"Institute of Mathematical Statistics","type":"journal_article","day":"01","doi":"10.1214/16-AIHP765","intvolume":" 53","title":"Universality for random matrix flows with time dependent density","publist_id":"7189","page":"1606 - 1656","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1504.00650"}],"project":[{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"department":[{"_id":"LaEr"}],"publication_status":"published","publication_identifier":{"issn":["02460203"]},"quality_controlled":"1","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","scopus_import":1,"author":[{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli"}],"_id":"615","date_created":"2018-12-11T11:47:30Z","status":"public","ec_funded":1}]