[{"citation":{"mla":"Erdös, László, et al. “Power Law Decay for Systems of Randomly Coupled Differential Equations.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 3, Society for Industrial and Applied Mathematics, 2018, pp. 3271–90, doi:<a href=\"https://doi.org/10.1137/17M1143125\">10.1137/17M1143125</a>.","ama":"Erdös L, Krüger TH, Renfrew DT. Power law decay for systems of randomly coupled differential equations. <i>SIAM Journal on Mathematical Analysis</i>. 2018;50(3):3271-3290. doi:<a href=\"https://doi.org/10.1137/17M1143125\">10.1137/17M1143125</a>","ista":"Erdös L, Krüger TH, Renfrew DT. 2018. Power law decay for systems of randomly coupled differential equations. SIAM Journal on Mathematical Analysis. 50(3), 3271–3290.","chicago":"Erdös, László, Torben H Krüger, and David T Renfrew. “Power Law Decay for Systems of Randomly Coupled Differential Equations.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics, 2018. <a href=\"https://doi.org/10.1137/17M1143125\">https://doi.org/10.1137/17M1143125</a>.","ieee":"L. Erdös, T. H. Krüger, and D. T. Renfrew, “Power law decay for systems of randomly coupled differential equations,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 3. Society for Industrial and Applied Mathematics, pp. 3271–3290, 2018.","apa":"Erdös, L., Krüger, T. H., &#38; Renfrew, D. T. (2018). Power law decay for systems of randomly coupled differential equations. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/17M1143125\">https://doi.org/10.1137/17M1143125</a>","short":"L. Erdös, T.H. Krüger, D.T. Renfrew, SIAM Journal on Mathematical Analysis 50 (2018) 3271–3290."},"date_created":"2018-12-11T11:45:03Z","das_tickbox":"1","department":[{"_id":"LaEr"}],"scopus_import":"1","publication":"SIAM Journal on Mathematical Analysis","acknowledgement":"The work of the second author was also partially supported by the Hausdorff Center of Mathematics.","volume":50,"ec_funded":1,"arxiv":1,"quality_controlled":"1","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Torben H","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger"},{"first_name":"David T","full_name":"Renfrew, David T","orcid":"0000-0003-3493-121X","id":"4845BF6A-F248-11E8-B48F-1D18A9856A87","last_name":"Renfrew"}],"publication_status":"published","ddc":["500"],"year":"2018","oa":1,"article_processing_charge":"No","month":"01","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1708.01546"}],"oa_version":"Published Version","issue":"3","intvolume":"        50","date_updated":"2026-07-06T14:02:06Z","page":"3271 - 3290","title":"Power law decay for systems of randomly coupled differential equations","abstract":[{"lang":"eng","text":"We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Society for Industrial and Applied Mathematics","doi":"10.1137/17M1143125","status":"public","_id":"181","type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"_id":"258F40A4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Structured Non-Hermitian Random Matrices","grant_number":"M02080"}],"external_id":{"isi":["000437018500032"],"arxiv":["1708.01546"]},"language":[{"iso":"eng"}],"isi":1,"fulldoi":"https://doi.org/10.1137/17M1143125","publist_id":"7740","day":"01","date_published":"2018-01-01T00:00:00Z"},{"arxiv":1,"quality_controlled":"1","ec_funded":1,"volume":2018,"publication_identifier":{"issn":["1073-7928"]},"publication":"International Mathematics Research Notices","department":[{"_id":"LaEr"}],"scopus_import":"1","date_created":"2018-12-11T11:49:41Z","related_material":{"record":[{"id":"6179","relation":"dissertation_contains","status":"public"}]},"citation":{"mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” <i>International Mathematics Research Notices</i>, vol. 2018, no. 10, Oxford University Press, 2018, pp. 3255–98, doi:<a href=\"https://doi.org/10.1093/imrn/rnw330\">10.1093/imrn/rnw330</a>.","ama":"Erdös L, Schröder DJ. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. <i>International Mathematics Research Notices</i>. 2018;2018(10):3255-3298. doi:<a href=\"https://doi.org/10.1093/imrn/rnw330\">10.1093/imrn/rnw330</a>","ista":"Erdös L, Schröder DJ. 2018. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. International Mathematics Research Notices. 2018(10), 3255–3298.","chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2018. <a href=\"https://doi.org/10.1093/imrn/rnw330\">https://doi.org/10.1093/imrn/rnw330</a>.","short":"L. Erdös, D.J. Schröder, International Mathematics Research Notices 2018 (2018) 3255–3298.","apa":"Erdös, L., &#38; Schröder, D. J. (2018). Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnw330\">https://doi.org/10.1093/imrn/rnw330</a>","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues,” <i>International Mathematics Research Notices</i>, vol. 2018, no. 10. Oxford University Press, pp. 3255–3298, 2018."},"issue":"10","oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.05163"}],"intvolume":"      2018","month":"05","article_processing_charge":"No","year":"2018","oa":1,"publication_status":"published","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"type":"journal_article","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"status":"public","_id":"1012","doi":"10.1093/imrn/rnw330","publisher":"Oxford University Press","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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."}],"title":"Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues","page":"3255-3298","date_updated":"2026-07-29T13:46:19Z","day":"18","date_published":"2018-05-18T00:00:00Z","fulldoi":"https://doi.org/10.1093/imrn/rnw330","publist_id":"6383","isi":1,"language":[{"iso":"eng"}],"external_id":{"arxiv":["1608.05163"],"isi":["000441668300009"]}},{"date_published":"2018-03-03T00:00:00Z","day":"03","external_id":{"isi":["000431721800005"],"arxiv":["1612.07776 "]},"isi":1,"fulldoi":"https://doi.org/10.1214/17-AAP1302","language":[{"iso":"eng"}],"type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"status":"public","_id":"566","publisher":"Institute of Mathematical Statistics","user_id":"9947682f-b9fa-11ee-9c4a-b3ffaafe6614","doi":"10.1214/17-AAP1302","title":"Local inhomogeneous circular law","abstract":[{"text":"We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X. \r\n\r\n","lang":"eng"}],"page":"148-203","date_updated":"2026-08-05T08:14:17Z","month":"03","oa_version":"Preprint","intvolume":"        28","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1612.07776 "}],"issue":"1","article_processing_charge":"No","oa":1,"year":"2018","publication_status":"published","article_type":"original","author":[{"first_name":"Johannes","full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","last_name":"Alt"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger"}],"ec_funded":1,"volume":28,"quality_controlled":"1","arxiv":1,"department":[{"_id":"LaEr"}],"scopus_import":"1","date_created":"2018-12-11T11:47:13Z","das_tickbox":"1","publication":"Annals of Applied Probability","corr_author":"1","citation":{"chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Inhomogeneous Circular Law.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2018. <a href=\"https://doi.org/10.1214/17-AAP1302\">https://doi.org/10.1214/17-AAP1302</a>.","short":"J. Alt, L. Erdös, T.H. Krüger, Annals of Applied Probability 28 (2018) 148–203.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local inhomogeneous circular law,” <i>Annals of Applied Probability</i>, vol. 28, no. 1. Institute of Mathematical Statistics, pp. 148–203, 2018.","apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2018). Local inhomogeneous circular law. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-AAP1302\">https://doi.org/10.1214/17-AAP1302</a>","mla":"Alt, Johannes, et al. “Local Inhomogeneous Circular Law.” <i>Annals of Applied Probability</i>, vol. 28, no. 1, Institute of Mathematical Statistics, 2018, pp. 148–203, doi:<a href=\"https://doi.org/10.1214/17-AAP1302\">10.1214/17-AAP1302</a>.","ama":"Alt J, Erdös L, Krüger TH. Local inhomogeneous circular law. <i>Annals of Applied Probability</i>. 2018;28(1):148-203. doi:<a href=\"https://doi.org/10.1214/17-AAP1302\">10.1214/17-AAP1302</a>","ista":"Alt J, Erdös L, Krüger TH. 2018. Local inhomogeneous circular law. Annals of Applied Probability. 28(1), 148–203."},"related_material":{"record":[{"id":"149","relation":"dissertation_contains","status":"public"}]}},{"department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:44:53Z","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"6184"},{"relation":"part_of_dissertation","status":"public","id":"6183"},{"status":"public","relation":"part_of_dissertation","id":"1010"},{"relation":"part_of_dissertation","status":"public","id":"550"},{"status":"public","relation":"part_of_dissertation","id":"6240"},{"status":"public","relation":"part_of_dissertation","id":"566"},{"status":"public","relation":"part_of_dissertation","id":"1677"}]},"alternative_title":["ISTA Thesis"],"has_accepted_license":"1","corr_author":"1","citation":{"ista":"Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. Institute of Science and Technology Austria.","ama":"Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:TH_1040\">10.15479/AT:ISTA:TH_1040</a>","mla":"Alt, Johannes. <i>Dyson Equation and Eigenvalue Statistics of Random Matrices</i>. Institute of Science and Technology Austria, 2018, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:TH_1040\">10.15479/AT:ISTA:TH_1040</a>.","short":"J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute of Science and Technology Austria, 2018.","ieee":"J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute of Science and Technology Austria, 2018.","apa":"Alt, J. (2018). <i>Dyson equation and eigenvalue statistics of random matrices</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:TH_1040\">https://doi.org/10.15479/AT:ISTA:TH_1040</a>","chicago":"Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.” Institute of Science and Technology Austria, 2018. <a href=\"https://doi.org/10.15479/AT:ISTA:TH_1040\">https://doi.org/10.15479/AT:ISTA:TH_1040</a>."},"OA_place":"publisher","publication_identifier":{"issn":["2663-337X"]},"ec_funded":1,"supervisor":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László","last_name":"Erdös"}],"pubrep_id":"1040","ddc":["515","519"],"publication_status":"published","author":[{"id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes","first_name":"Johannes","last_name":"Alt"}],"oa_version":"Published Version","month":"07","degree_awarded":"PhD","article_processing_charge":"No","oa":1,"year":"2018","file_date_updated":"2020-07-14T12:44:57Z","abstract":[{"lang":"eng","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."}],"title":"Dyson equation and eigenvalue statistics of random matrices","license":"https://creativecommons.org/licenses/by/4.0/","page":"456","date_updated":"2026-08-12T13:58:56Z","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"type":"dissertation","status":"public","_id":"149","doi":"10.15479/AT:ISTA:TH_1040","publisher":"Institute of Science and Technology Austria","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_published":"2018-07-12T00:00:00Z","day":"12","fulldoi":"https://doi.org/10.15479/AT:ISTA:TH_1040","publist_id":"7772","file":[{"date_updated":"2020-07-14T12:44:57Z","checksum":"d4dad55a7513f345706aaaba90cb1bb8","file_size":5801709,"creator":"dernst","relation":"main_file","date_created":"2019-04-08T13:55:20Z","file_id":"6241","access_level":"open_access","content_type":"application/pdf","file_name":"2018_thesis_Alt.pdf"},{"creator":"dernst","date_created":"2019-04-08T13:55:20Z","relation":"source_file","checksum":"d73fcf46300dce74c403f2b491148ab4","date_updated":"2020-07-14T12:44:57Z","file_size":3802059,"file_name":"2018_thesis_Alt_source.zip","content_type":"application/zip","file_id":"6242","access_level":"closed"}],"language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"}},{"ec_funded":1,"publication_identifier":{"issn":["2010-3263"],"eissn":["2010-3271"]},"arxiv":1,"quality_controlled":"1","article_number":"1950009","date_created":"2019-02-13T10:40:54Z","scopus_import":"1","department":[{"_id":"LaEr"}],"publication":"Random Matrices: Theory and Applications","citation":{"mla":"Erdös, László, and Peter Mühlbacher. “Bounds on the Norm of Wigner-Type Random Matrices.” <i>Random Matrices: Theory and Applications</i>, 1950009, World Scientific Publishing, 2018, doi:<a href=\"https://doi.org/10.1142/s2010326319500096\">10.1142/s2010326319500096</a>.","ama":"Erdös L, Mühlbacher P. Bounds on the norm of Wigner-type random matrices. <i>Random Matrices: Theory and Applications</i>. 2018. doi:<a href=\"https://doi.org/10.1142/s2010326319500096\">10.1142/s2010326319500096</a>","ista":"Erdös L, Mühlbacher P. 2018. Bounds on the norm of Wigner-type random matrices. Random Matrices: Theory and Applications., 1950009.","chicago":"Erdös, László, and Peter Mühlbacher. “Bounds on the Norm of Wigner-Type Random Matrices.” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing, 2018. <a href=\"https://doi.org/10.1142/s2010326319500096\">https://doi.org/10.1142/s2010326319500096</a>.","apa":"Erdös, L., &#38; Mühlbacher, P. (2018). Bounds on the norm of Wigner-type random matrices. <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s2010326319500096\">https://doi.org/10.1142/s2010326319500096</a>","ieee":"L. Erdös and P. Mühlbacher, “Bounds on the norm of Wigner-type random matrices,” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing, 2018.","short":"L. Erdös, P. Mühlbacher, Random Matrices: Theory and Applications (2018)."},"month":"09","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1802.05175"}],"oa_version":"Preprint","oa":1,"year":"2018","article_processing_charge":"No","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László"},{"last_name":"Mühlbacher","full_name":"Mühlbacher, Peter","first_name":"Peter"}],"publication_status":"published","status":"public","_id":"5971","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"World Scientific Publishing","doi":"10.1142/s2010326319500096","title":"Bounds on the norm of Wigner-type random matrices","abstract":[{"lang":"eng","text":"We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation."}],"date_updated":"2026-08-12T14:33:01Z","day":"26","date_published":"2018-09-26T00:00:00Z","external_id":{"arxiv":["1802.05175"],"isi":["000477677200002"]},"language":[{"iso":"eng"}],"isi":1,"fulldoi":"https://doi.org/10.1142/s2010326319500096"},{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"publist_id":"6141","fulldoi":"https://doi.org/10.1007/s00220-016-2805-6","file":[{"content_type":"application/pdf","file_name":"IST-2016-722-v1+1_s00220-016-2805-6.pdf","access_level":"open_access","file_id":"5102","date_created":"2018-12-12T10:14:47Z","relation":"main_file","creator":"system","file_size":1033743,"date_updated":"2020-07-14T12:44:39Z","checksum":"ddff79154c3daf27237de5383b1264a9"}],"isi":1,"language":[{"iso":"eng"}],"external_id":{"isi":["000393696700005"]},"day":"01","date_published":"2017-02-01T00:00:00Z","doi":"10.1007/s00220-016-2805-6","publisher":"Springer","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"_id":"1207","status":"public","page":"947 - 990","date_updated":"2025-07-10T11:50:22Z","file_date_updated":"2020-07-14T12:44:39Z","abstract":[{"lang":"eng","text":"The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix."}],"title":"Local law of addition of random matrices on optimal scale","article_processing_charge":"Yes (via OA deal)","year":"2017","oa":1,"intvolume":"       349","oa_version":"Published Version","issue":"3","month":"02","publication_status":"published","author":[{"last_name":"Bao","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin"}],"pubrep_id":"722","ddc":["530"],"quality_controlled":"1","publication_identifier":{"issn":["0010-3616"]},"ec_funded":1,"volume":349,"has_accepted_license":"1","citation":{"ista":"Bao Z, Erdös L, Schnelli K. 2017. Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. 349(3), 947–990.","mla":"Bao, Zhigang, et al. “Local Law of Addition of Random Matrices on Optimal Scale.” <i>Communications in Mathematical Physics</i>, vol. 349, no. 3, Springer, 2017, pp. 947–90, doi:<a href=\"https://doi.org/10.1007/s00220-016-2805-6\">10.1007/s00220-016-2805-6</a>.","ama":"Bao Z, Erdös L, Schnelli K. Local law of addition of random matrices on optimal scale. <i>Communications in Mathematical Physics</i>. 2017;349(3):947-990. doi:<a href=\"https://doi.org/10.1007/s00220-016-2805-6\">10.1007/s00220-016-2805-6</a>","short":"Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 947–990.","apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2017). Local law of addition of random matrices on optimal scale. <i>Communications in Mathematical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s00220-016-2805-6\">https://doi.org/10.1007/s00220-016-2805-6</a>","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local law of addition of random matrices on optimal scale,” <i>Communications in Mathematical Physics</i>, vol. 349, no. 3. Springer, pp. 947–990, 2017.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Law of Addition of Random Matrices on Optimal Scale.” <i>Communications in Mathematical Physics</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00220-016-2805-6\">https://doi.org/10.1007/s00220-016-2805-6</a>."},"publication":"Communications in Mathematical Physics","department":[{"_id":"LaEr"}],"scopus_import":"1","date_created":"2018-12-11T11:50:43Z"},{"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publisher":"Springer","doi":"10.1007/s00440-016-0740-2","_id":"1337","status":"public","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"type":"journal_article","date_updated":"2026-04-16T09:55:44Z","page":"667 - 727","title":"Universality for general Wigner-type matrices","abstract":[{"text":"We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.","lang":"eng"}],"file_date_updated":"2020-07-14T12:44:44Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"external_id":{"isi":["000414358400002"]},"language":[{"iso":"eng"}],"file":[{"creator":"system","relation":"main_file","date_created":"2018-12-12T10:08:25Z","checksum":"29f5a72c3f91e408aeb9e78344973803","date_updated":"2020-07-14T12:44:44Z","file_size":988843,"file_name":"IST-2017-657-v1+2_s00440-016-0740-2.pdf","content_type":"application/pdf","file_id":"4686","access_level":"open_access"}],"fulldoi":"https://doi.org/10.1007/s00440-016-0740-2","publist_id":"5930","isi":1,"day":"01","date_published":"2017-12-01T00:00:00Z","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).  ","publication_identifier":{"issn":["0178-8051"]},"volume":169,"ec_funded":1,"quality_controlled":"1","citation":{"ista":"Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727.","ama":"Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices. <i>Probability Theory and Related Fields</i>. 2017;169(3-4):667-727. doi:<a href=\"https://doi.org/10.1007/s00440-016-0740-2\">10.1007/s00440-016-0740-2</a>","mla":"Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.” <i>Probability Theory and Related Fields</i>, vol. 169, no. 3–4, Springer, 2017, pp. 667–727, doi:<a href=\"https://doi.org/10.1007/s00440-016-0740-2\">10.1007/s00440-016-0740-2</a>.","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727.","apa":"Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2017). Universality for general Wigner-type matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-016-0740-2\">https://doi.org/10.1007/s00440-016-0740-2</a>","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type matrices,” <i>Probability Theory and Related Fields</i>, vol. 169, no. 3–4. Springer, pp. 667–727, 2017.","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for General Wigner-Type Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00440-016-0740-2\">https://doi.org/10.1007/s00440-016-0740-2</a>."},"corr_author":"1","has_accepted_license":"1","date_created":"2018-12-11T11:51:27Z","scopus_import":"1","department":[{"_id":"LaEr"}],"publication":"Probability Theory and Related Fields","year":"2017","oa":1,"article_processing_charge":"Yes (via OA deal)","month":"12","issue":"3-4","intvolume":"       169","oa_version":"Published Version","author":[{"last_name":"Ajanki","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","full_name":"Ajanki, Oskari H","first_name":"Oskari H"},{"last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-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"}],"publication_status":"published","ddc":["510","530"],"pubrep_id":"657"},{"publication":"Probability Theory and Related Fields","date_created":"2018-12-11T11:52:32Z","department":[{"_id":"LaEr"}],"scopus_import":"1","has_accepted_license":"1","citation":{"chicago":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00440-015-0692-y\">https://doi.org/10.1007/s00440-015-0692-y</a>.","short":"Z. Bao, L. Erdös, Probability Theory and Related Fields 167 (2017) 673–776.","ieee":"Z. Bao and L. Erdös, “Delocalization for a class of random block band matrices,” <i>Probability Theory and Related Fields</i>, vol. 167, no. 3–4. Springer, pp. 673–776, 2017.","apa":"Bao, Z., &#38; Erdös, L. (2017). Delocalization for a class of random block band matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-015-0692-y\">https://doi.org/10.1007/s00440-015-0692-y</a>","mla":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” <i>Probability Theory and Related Fields</i>, vol. 167, no. 3–4, Springer, 2017, pp. 673–776, doi:<a href=\"https://doi.org/10.1007/s00440-015-0692-y\">10.1007/s00440-015-0692-y</a>.","ama":"Bao Z, Erdös L. Delocalization for a class of random block band matrices. <i>Probability Theory and Related Fields</i>. 2017;167(3-4):673-776. doi:<a href=\"https://doi.org/10.1007/s00440-015-0692-y\">10.1007/s00440-015-0692-y</a>","ista":"Bao Z, Erdös L. 2017. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 167(3–4), 673–776."},"corr_author":"1","quality_controlled":"1","ec_funded":1,"volume":167,"publication_identifier":{"issn":["0178-8051"]},"acknowledgement":"Z. Bao was supported by ERC Advanced Grant RANMAT No. 338804; L. Erdős was partially supported by ERC Advanced Grant RANMAT No. 338804.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). The authors are very grateful to the anonymous referees for careful reading and valuable comments, which helped to improve the organization.","pubrep_id":"489","ddc":["530"],"author":[{"last_name":"Bao","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László","last_name":"Erdös"}],"publication_status":"published","article_type":"original","issue":"3-4","intvolume":"       167","oa_version":"Published Version","month":"04","oa":1,"year":"2017","article_processing_charge":"Yes (via OA deal)","abstract":[{"text":"We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.","lang":"eng"}],"file_date_updated":"2020-07-14T12:45:00Z","title":"Delocalization for a class of random block band matrices","date_updated":"2026-04-16T09:55:56Z","page":"673 - 776","_id":"1528","status":"public","type":"journal_article","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"doi":"10.1007/s00440-015-0692-y","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publisher":"Springer","day":"01","date_published":"2017-04-01T00:00:00Z","language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1007/s00440-015-0692-y","file":[{"access_level":"open_access","file_id":"4665","file_name":"IST-2016-489-v1+1_s00440-015-0692-y.pdf","content_type":"application/pdf","file_size":1615755,"checksum":"67afa85ff1e220cbc1f9f477a828513c","date_updated":"2020-07-14T12:45:00Z","date_created":"2018-12-12T10:08:05Z","relation":"main_file","creator":"system"}],"isi":1,"publist_id":"5644","external_id":{"isi":["000398842700004"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"}},{"external_id":{"isi":["000404011700017"]},"fulldoi":"https://doi.org/10.30757/ALEA.v14-17","publist_id":"7376","isi":1,"language":[{"iso":"eng"}],"day":"23","date_published":"2017-03-23T00:00:00Z","page":"299 - 325","date_updated":"2025-09-18T10:02:36Z","title":"Fluctuations of the competition interface in presence of shocks","abstract":[{"text":"We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).","lang":"eng"}],"publisher":"Instituto Nacional de Matematica Pura e Aplicada","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.30757/ALEA.v14-17","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"type":"journal_article","status":"public","_id":"447","publication_status":"published","article_type":"original","author":[{"first_name":"Patrik","full_name":"Ferrari, Patrik","last_name":"Ferrari"},{"last_name":"Nejjar","full_name":"Nejjar, Peter","first_name":"Peter","id":"4BF426E2-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","oa":1,"year":"2017","month":"03","oa_version":"Submitted Version","main_file_link":[{"open_access":"1","url":"http://alea.impa.br/articles/v14/14-17.pdf"}],"intvolume":"         9","corr_author":"1","citation":{"ista":"Ferrari P, Nejjar P. 2017. Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. 9, 299–325.","mla":"Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” <i>Revista Latino-Americana de Probabilidade e Estatística</i>, vol. 9, Instituto Nacional de Matematica Pura e Aplicada, 2017, pp. 299–325, doi:<a href=\"https://doi.org/10.30757/ALEA.v14-17\">10.30757/ALEA.v14-17</a>.","ama":"Ferrari P, Nejjar P. Fluctuations of the competition interface in presence of shocks. <i>Revista Latino-Americana de Probabilidade e Estatística</i>. 2017;9:299-325. doi:<a href=\"https://doi.org/10.30757/ALEA.v14-17\">10.30757/ALEA.v14-17</a>","short":"P. Ferrari, P. Nejjar, Revista Latino-Americana de Probabilidade e Estatística 9 (2017) 299–325.","apa":"Ferrari, P., &#38; Nejjar, P. (2017). Fluctuations of the competition interface in presence of shocks. <i>Revista Latino-Americana de Probabilidade e Estatística</i>. Instituto Nacional de Matematica Pura e Aplicada. <a href=\"https://doi.org/10.30757/ALEA.v14-17\">https://doi.org/10.30757/ALEA.v14-17</a>","ieee":"P. Ferrari and P. Nejjar, “Fluctuations of the competition interface in presence of shocks,” <i>Revista Latino-Americana de Probabilidade e Estatística</i>, vol. 9. Instituto Nacional de Matematica Pura e Aplicada, pp. 299–325, 2017.","chicago":"Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” <i>Revista Latino-Americana de Probabilidade e Estatística</i>. Instituto Nacional de Matematica Pura e Aplicada, 2017. <a href=\"https://doi.org/10.30757/ALEA.v14-17\">https://doi.org/10.30757/ALEA.v14-17</a>."},"scopus_import":"1","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"date_created":"2018-12-11T11:46:31Z","publication":"Revista Latino-Americana de Probabilidade e Estatística","ec_funded":1,"volume":9,"quality_controlled":"1"},{"quality_controlled":"1","ec_funded":1,"volume":22,"publication_identifier":{"issn":["1083-589X"]},"article_number":"63","publication":"Electronic Communications in Probability","date_created":"2018-12-11T11:47:07Z","scopus_import":"1","department":[{"_id":"LaEr"}],"has_accepted_license":"1","related_material":{"record":[{"id":"149","relation":"dissertation_contains","status":"public"}]},"citation":{"chicago":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/17-ECP97\">https://doi.org/10.1214/17-ECP97</a>.","ieee":"J. Alt, “Singularities of the density of states of random Gram matrices,” <i>Electronic Communications in Probability</i>, vol. 22. Institute of Mathematical Statistics, 2017.","apa":"Alt, J. (2017). Singularities of the density of states of random Gram matrices. <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-ECP97\">https://doi.org/10.1214/17-ECP97</a>","short":"J. Alt, Electronic Communications in Probability 22 (2017).","mla":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” <i>Electronic Communications in Probability</i>, vol. 22, 63, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/17-ECP97\">10.1214/17-ECP97</a>.","ama":"Alt J. Singularities of the density of states of random Gram matrices. <i>Electronic Communications in Probability</i>. 2017;22. doi:<a href=\"https://doi.org/10.1214/17-ECP97\">10.1214/17-ECP97</a>","ista":"Alt J. 2017. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 22, 63."},"corr_author":"1","oa_version":"Published Version","intvolume":"        22","month":"11","year":"2017","oa":1,"article_processing_charge":"No","pubrep_id":"926","ddc":["539"],"author":[{"last_name":"Alt","first_name":"Johannes","full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87"}],"publication_status":"published","status":"public","_id":"550","type":"journal_article","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"doi":"10.1214/17-ECP97","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"Institute of Mathematical Statistics","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."}],"file_date_updated":"2020-07-14T12:47:00Z","title":"Singularities of the density of states of random Gram matrices","date_updated":"2026-04-08T14:11:36Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"date_published":"2017-11-21T00:00:00Z","day":"21","language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1214/17-ECP97","publist_id":"7265","isi":1,"file":[{"creator":"system","relation":"main_file","date_created":"2018-12-12T10:08:04Z","checksum":"0ec05303a0de190de145654237984c79","date_updated":"2020-07-14T12:47:00Z","file_size":470876,"file_name":"IST-2018-926-v1+1_euclid.ecp.1511233247.pdf","content_type":"application/pdf","file_id":"4663","access_level":"open_access"}],"external_id":{"isi":["000416389200001"]}},{"publist_id":"7247","fulldoi":"https://doi.org/10.1090/cln/028","language":[{"iso":"eng"}],"date_published":"2017-01-01T00:00:00Z","day":"01","page":"226","date_updated":"2025-04-15T08:05:02Z","title":"A Dynamical Approach to Random Matrix Theory","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"}],"publisher":"American Mathematical Society","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1090/cln/028","type":"book","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"status":"public","_id":"567","publication_status":"published","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"full_name":"Yau, Horng","first_name":"Horng","last_name":"Yau"}],"article_processing_charge":"No","year":"2017","month":"01","intvolume":"        28","oa_version":"None","corr_author":"1","citation":{"apa":"Erdös, L., &#38; Yau, H. (2017). <i>A Dynamical Approach to Random Matrix Theory</i> (Vol. 28). American Mathematical Society. <a href=\"https://doi.org/10.1090/cln/028\">https://doi.org/10.1090/cln/028</a>","ieee":"L. Erdös and H. Yau, <i>A Dynamical Approach to Random Matrix Theory</i>, vol. 28. American Mathematical Society, 2017.","short":"L. Erdös, H. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 2017.","chicago":"Erdös, László, and Horng Yau. <i>A Dynamical Approach to Random Matrix Theory</i>. Vol. 28. Courant Lecture Notes. American Mathematical Society, 2017. <a href=\"https://doi.org/10.1090/cln/028\">https://doi.org/10.1090/cln/028</a>.","ista":"Erdös L, Yau H. 2017. A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 226p.","mla":"Erdös, László, and Horng Yau. <i>A Dynamical Approach to Random Matrix Theory</i>. Vol. 28, American Mathematical Society, 2017, doi:<a href=\"https://doi.org/10.1090/cln/028\">10.1090/cln/028</a>.","ama":"Erdös L, Yau H. <i>A Dynamical Approach to Random Matrix Theory</i>. Vol 28. American Mathematical Society; 2017. doi:<a href=\"https://doi.org/10.1090/cln/028\">10.1090/cln/028</a>"},"alternative_title":["Courant Lecture Notes"],"department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:47:13Z","series_title":"Courant Lecture Notes","publication_identifier":{"isbn":["9-781-4704-3648-3"],"eisbn":["978-1-4704-4194-4"]},"ec_funded":1,"volume":28,"quality_controlled":"1"},{"scopus_import":"1","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:47:30Z","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","corr_author":"1","citation":{"ama":"Erdös L, Schnelli K. Universality for random matrix flows with time dependent density. <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. 2017;53(4):1606-1656. doi:<a href=\"https://doi.org/10.1214/16-AIHP765\">10.1214/16-AIHP765</a>","mla":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:<a href=\"https://doi.org/10.1214/16-AIHP765\">10.1214/16-AIHP765</a>.","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.","chicago":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/16-AIHP765\">https://doi.org/10.1214/16-AIHP765</a>.","apa":"Erdös, L., &#38; Schnelli, K. (2017). Universality for random matrix flows with time dependent density. <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/16-AIHP765\">https://doi.org/10.1214/16-AIHP765</a>","ieee":"L. Erdös and K. Schnelli, “Universality for random matrix flows with time dependent density,” <i>Annales de l’institut Henri Poincare (B) Probability and Statistics</i>, vol. 53, no. 4. Institute of Mathematical Statistics, pp. 1606–1656, 2017.","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656."},"publication_identifier":{"issn":["0246-0203"]},"ec_funded":1,"volume":53,"arxiv":1,"quality_controlled":"1","publication_status":"published","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"last_name":"Schnelli","first_name":"Kevin","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"month":"11","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1504.00650"}],"oa_version":"Submitted Version","issue":"4","intvolume":"        53","article_processing_charge":"No","year":"2017","oa":1,"title":"Universality for random matrix flows with time dependent density","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"}],"date_updated":"2025-09-11T07:32:50Z","page":"1606 - 1656","type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"status":"public","_id":"615","publisher":"Institute of Mathematical Statistics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1214/16-AIHP765","date_published":"2017-11-01T00:00:00Z","day":"01","external_id":{"isi":["000416376500005"],"arxiv":["1504.00650"]},"fulldoi":"https://doi.org/10.1214/16-AIHP765","publist_id":"7189","isi":1,"language":[{"iso":"eng"}]},{"type":"journal_article","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"_id":"1010","status":"public","doi":"10.1214/17-EJP42","publisher":"Institute of Mathematical Statistics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file_date_updated":"2018-12-12T10:13:39Z","abstract":[{"text":"We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. ","lang":"eng"}],"title":"Local law for random Gram matrices","date_updated":"2026-04-08T14:11:36Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"day":"08","date_published":"2017-03-08T00:00:00Z","publist_id":"6386","fulldoi":"https://doi.org/10.1214/17-EJP42","file":[{"file_name":"IST-2017-807-v1+1_euclid.ejp.1488942016.pdf","content_type":"application/pdf","access_level":"open_access","file_id":"5024","date_created":"2018-12-12T10:13:39Z","relation":"main_file","creator":"system","file_size":639384,"date_updated":"2018-12-12T10:13:39Z"}],"isi":1,"language":[{"iso":"eng"}],"external_id":{"isi":["000396611900025"],"arxiv":["1606.07353"]},"arxiv":1,"quality_controlled":"1","ec_funded":1,"publication_identifier":{"issn":["1083-6489"]},"volume":22,"article_number":"25","publication":"Electronic Journal of Probability","scopus_import":"1","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:49:40Z","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"149"}]},"has_accepted_license":"1","citation":{"ista":"Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic Journal of Probability. 22, 25.","mla":"Alt, Johannes, et al. “Local Law for Random Gram Matrices.” <i>Electronic Journal of Probability</i>, vol. 22, 25, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/17-EJP42\">10.1214/17-EJP42</a>.","ama":"Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. <i>Electronic Journal of Probability</i>. 2017;22. doi:<a href=\"https://doi.org/10.1214/17-EJP42\">10.1214/17-EJP42</a>","apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2017). Local law for random Gram matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-EJP42\">https://doi.org/10.1214/17-EJP42</a>","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,” <i>Electronic Journal of Probability</i>, vol. 22. Institute of Mathematical Statistics, 2017.","short":"J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017).","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/17-EJP42\">https://doi.org/10.1214/17-EJP42</a>."},"oa_version":"Published Version","intvolume":"        22","month":"03","article_processing_charge":"No","oa":1,"year":"2017","pubrep_id":"807","ddc":["510","539"],"publication_status":"published","author":[{"last_name":"Alt","full_name":"Alt, Johannes","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87"}]},{"type":"journal_article","status":"public","_id":"1023","publisher":"Institute of Mathematical Statistics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.1214/17-EJP38","title":"Local law for the product of independent non-Hermitian random matrices with independent entries","file_date_updated":"2018-12-12T10:15:29Z","abstract":[{"text":"We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.","lang":"eng"}],"date_updated":"2025-07-10T11:49:47Z","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"day":"06","date_published":"2017-02-06T00:00:00Z","external_id":{"isi":["000396611900022"]},"fulldoi":"https://doi.org/10.1214/17-EJP38","isi":1,"file":[{"access_level":"open_access","file_id":"5149","content_type":"application/pdf","file_name":"IST-2017-802-v1+1_euclid.ejp.1487991681.pdf","file_size":742275,"date_updated":"2018-12-12T10:15:29Z","relation":"main_file","date_created":"2018-12-12T10:15:29Z","creator":"system"}],"publist_id":"6370","language":[{"iso":"eng"}],"publication_identifier":{"issn":["1083-6489"]},"volume":22,"quality_controlled":"1","article_number":"22","scopus_import":"1","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:49:44Z","publication":"Electronic Journal of Probability","citation":{"ista":"Nemish Y. 2017. Local law for the product of independent non-Hermitian random matrices with independent entries. Electronic Journal of Probability. 22, 22.","ama":"Nemish Y. Local law for the product of independent non-Hermitian random matrices with independent entries. <i>Electronic Journal of Probability</i>. 2017;22. doi:<a href=\"https://doi.org/10.1214/17-EJP38\">10.1214/17-EJP38</a>","mla":"Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>, vol. 22, 22, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/17-EJP38\">10.1214/17-EJP38</a>.","ieee":"Y. Nemish, “Local law for the product of independent non-Hermitian random matrices with independent entries,” <i>Electronic Journal of Probability</i>, vol. 22. Institute of Mathematical Statistics, 2017.","apa":"Nemish, Y. (2017). Local law for the product of independent non-Hermitian random matrices with independent entries. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-EJP38\">https://doi.org/10.1214/17-EJP38</a>","short":"Y. Nemish, Electronic Journal of Probability 22 (2017).","chicago":"Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/17-EJP38\">https://doi.org/10.1214/17-EJP38</a>."},"has_accepted_license":"1","month":"02","oa_version":"Published Version","intvolume":"        22","article_processing_charge":"No","year":"2017","oa":1,"ddc":["510"],"pubrep_id":"802","publication_status":"published","author":[{"last_name":"Nemish","first_name":"Yuriy","full_name":"Nemish, Yuriy","orcid":"0000-0002-7327-856X","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87"}]},{"title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","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":"2025-09-10T10:58:02Z","page":"1672 - 1705","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"type":"journal_article","_id":"721","status":"public","publisher":"Wiley","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1002/cpa.21639","date_published":"2017-09-01T00:00:00Z","day":"01","external_id":{"isi":["000405752100002"],"arxiv":["1512.03703"]},"fulldoi":"https://doi.org/10.1002/cpa.21639","isi":1,"publist_id":"6959","language":[{"iso":"eng"}],"scopus_import":"1","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:48:08Z","publication":"Communications on Pure and Applied Mathematics","corr_author":"1","citation":{"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.","ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. 2017;70(9):1672-1705. doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>","mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017.","apa":"Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>","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.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2017. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>."},"publication_identifier":{"issn":["0010-3640"]},"volume":70,"ec_funded":1,"arxiv":1,"quality_controlled":"1","publication_status":"published","author":[{"full_name":"Ajanki, Oskari H","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","last_name":"Ajanki"},{"full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"}],"month":"09","intvolume":"        70","main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"oa_version":"Submitted Version","issue":"9","article_processing_charge":"No","year":"2017","oa":1},{"quality_controlled":"1","arxiv":1,"volume":319,"ec_funded":1,"acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","publication":"Advances in Mathematics","department":[{"_id":"LaEr"}],"scopus_import":"1","date_created":"2018-12-11T11:48:13Z","corr_author":"1","citation":{"chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” <i>Advances in Mathematics</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">https://doi.org/10.1016/j.aim.2017.08.028</a>.","apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">https://doi.org/10.1016/j.aim.2017.08.028</a>","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” <i>Advances in Mathematics</i>, vol. 319. Academic Press, pp. 251–291, 2017.","short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.","mla":"Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” <i>Advances in Mathematics</i>, vol. 319, Academic Press, 2017, pp. 251–91, doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">10.1016/j.aim.2017.08.028</a>.","ama":"Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. <i>Advances in Mathematics</i>. 2017;319:251-291. doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">10.1016/j.aim.2017.08.028</a>","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291."},"oa_version":"Submitted Version","intvolume":"       319","main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"month":"10","article_processing_charge":"No","oa":1,"year":"2017","publication_status":"published","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","first_name":"Zhigang","last_name":"Bao"},{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"last_name":"Schnelli","first_name":"Kevin","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"type":"journal_article","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"_id":"733","status":"public","doi":"10.1016/j.aim.2017.08.028","publisher":"Academic Press","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","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."}],"title":"Convergence rate for spectral distribution of addition of random matrices","date_updated":"2025-06-04T10:13:45Z","page":"251 - 291","date_published":"2017-10-15T00:00:00Z","day":"15","publist_id":"6935","fulldoi":"https://doi.org/10.1016/j.aim.2017.08.028","isi":1,"language":[{"iso":"eng"}],"external_id":{"arxiv":["1606.03076"],"isi":["000412150400010"]}},{"article_processing_charge":"No","year":"2017","oa":1,"month":"08","issue":"3","main_file_link":[{"url":"https://arxiv.org/abs/1602.02312","open_access":"1"}],"oa_version":"Submitted Version","intvolume":"        21","publication_status":"published","author":[{"last_name":"Bourgade","first_name":"Paul","full_name":"Bourgade, Paul"},{"last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Yau","full_name":"Yau, Horng","first_name":"Horng"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"volume":21,"publication_identifier":{"issn":["1095-0761"]},"ec_funded":1,"arxiv":1,"quality_controlled":"1","citation":{"ama":"Bourgade P, Erdös L, Yau H, Yin J. Universality for a class of random band matrices. <i>Advances in Theoretical and Mathematical Physics</i>. 2017;21(3):739-800. doi:<a href=\"https://doi.org/10.4310/ATMP.2017.v21.n3.a5\">10.4310/ATMP.2017.v21.n3.a5</a>","mla":"Bourgade, Paul, et al. “Universality for a Class of Random Band Matrices.” <i>Advances in Theoretical and Mathematical Physics</i>, vol. 21, no. 3, International Press of Boston, 2017, pp. 739–800, doi:<a href=\"https://doi.org/10.4310/ATMP.2017.v21.n3.a5\">10.4310/ATMP.2017.v21.n3.a5</a>.","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.","chicago":"Bourgade, Paul, László Erdös, Horng Yau, and Jun Yin. “Universality for a Class of Random Band Matrices.” <i>Advances in Theoretical and Mathematical Physics</i>. International Press of Boston, 2017. <a href=\"https://doi.org/10.4310/ATMP.2017.v21.n3.a5\">https://doi.org/10.4310/ATMP.2017.v21.n3.a5</a>.","ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Universality for a class of random band matrices,” <i>Advances in Theoretical and Mathematical Physics</i>, vol. 21, no. 3. International Press of Boston, pp. 739–800, 2017.","apa":"Bourgade, P., Erdös, L., Yau, H., &#38; Yin, J. (2017). Universality for a class of random band matrices. <i>Advances in Theoretical and Mathematical Physics</i>. International Press of Boston. <a href=\"https://doi.org/10.4310/ATMP.2017.v21.n3.a5\">https://doi.org/10.4310/ATMP.2017.v21.n3.a5</a>","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Advances in Theoretical and Mathematical Physics 21 (2017) 739–800."},"department":[{"_id":"LaEr"}],"scopus_import":"1","das_tickbox":"1","date_created":"2018-12-11T11:46:43Z","publication":"Advances in Theoretical and Mathematical Physics","external_id":{"arxiv":["1602.02312"],"isi":["000409382300005"]},"fulldoi":"https://doi.org/10.4310/ATMP.2017.v21.n3.a5","publist_id":"7337","isi":1,"language":[{"iso":"eng"}],"day":"25","date_published":"2017-08-25T00:00:00Z","publisher":"International Press of Boston","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.4310/ATMP.2017.v21.n3.a5","project":[{"grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","_id":"483","status":"public","page":"739 - 800","date_updated":"2026-07-06T13:35:08Z","title":"Universality for a class of random band matrices","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"}]},{"article_processing_charge":"No","oa":1,"year":"2017","month":"01","intvolume":"        21","oa_version":"Published Version","publication_status":"published","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","first_name":"László","last_name":"Erdös"},{"id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J","orcid":"0000-0002-2904-1856","first_name":"Dominik J","last_name":"Schröder"}],"ddc":["510"],"pubrep_id":"747","acknowledgement":"Partially supported by the IST Austria Excellence Scholarship.","article_number":"86","volume":21,"ec_funded":1,"quality_controlled":"1","citation":{"chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Functions of Wigner Matrices.” <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/16-ECP38\">https://doi.org/10.1214/16-ECP38</a>.","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of functions of Wigner matrices,” <i>Electronic Communications in Probability</i>, vol. 21. Institute of Mathematical Statistics, 2017.","apa":"Erdös, L., &#38; Schröder, D. J. (2017). Fluctuations of functions of Wigner matrices. <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/16-ECP38\">https://doi.org/10.1214/16-ECP38</a>","short":"L. Erdös, D.J. Schröder, Electronic Communications in Probability 21 (2017).","mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Functions of Wigner Matrices.” <i>Electronic Communications in Probability</i>, vol. 21, 86, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/16-ECP38\">10.1214/16-ECP38</a>.","ama":"Erdös L, Schröder DJ. Fluctuations of functions of Wigner matrices. <i>Electronic Communications in Probability</i>. 2017;21. doi:<a href=\"https://doi.org/10.1214/16-ECP38\">10.1214/16-ECP38</a>","ista":"Erdös L, Schröder DJ. 2017. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 21, 86."},"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6179"}]},"has_accepted_license":"1","scopus_import":"1","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:50:23Z","publication":"Electronic Communications in Probability","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"external_id":{"isi":["000396604900037"]},"isi":1,"fulldoi":"https://doi.org/10.1214/16-ECP38","file":[{"creator":"system","date_created":"2018-12-12T10:18:10Z","relation":"main_file","date_updated":"2018-12-12T10:18:10Z","file_size":440770,"file_name":"IST-2017-747-v1+1_euclid.ecp.1483347665.pdf","content_type":"application/pdf","file_id":"5329","access_level":"open_access"}],"publist_id":"6214","language":[{"iso":"eng"}],"date_published":"2017-01-02T00:00:00Z","day":"02","publisher":"Institute of Mathematical Statistics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1214/16-ECP38","type":"journal_article","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"_id":"1144","status":"public","date_updated":"2026-07-29T13:46:19Z","title":"Fluctuations of functions of Wigner matrices","file_date_updated":"2018-12-12T10:18:10Z","abstract":[{"lang":"eng","text":"We show that matrix elements of functions of N × N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11]."}]},{"abstract":[{"lang":"eng","text":"We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay."}],"title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","date_updated":"2025-09-22T09:55:43Z","page":"3786 - 3839","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","status":"public","_id":"1157","doi":"10.1214/16-AAP1193","publisher":"Institute of Mathematical Statistics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","day":"15","date_published":"2016-12-15T00:00:00Z","publist_id":"6201","fulldoi":"https://doi.org/10.1214/16-AAP1193","isi":1,"language":[{"iso":"eng"}],"external_id":{"arxiv":["1409.4979"],"isi":["000391240100016"]},"publication":"Annals of Applied Probability","department":[{"_id":"LaEr"}],"scopus_import":"1","date_created":"2018-12-11T11:50:27Z","citation":{"ista":"Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839.","ama":"Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. 2016;26(6):3786-3839. doi:<a href=\"https://doi.org/10.1214/16-AAP1193\">10.1214/16-AAP1193</a>","mla":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:<a href=\"https://doi.org/10.1214/16-AAP1193\">10.1214/16-AAP1193</a>.","short":"J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.","ieee":"J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” <i>Annals of Applied Probability</i>, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016.","apa":"Lee, J., &#38; Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/16-AAP1193\">https://doi.org/10.1214/16-AAP1193</a>","chicago":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2016. <a href=\"https://doi.org/10.1214/16-AAP1193\">https://doi.org/10.1214/16-AAP1193</a>."},"quality_controlled":"1","arxiv":1,"volume":26,"ec_funded":1,"acknowledgement":"We thank Horng-Tzer Yau for numerous discussions and remarks. We are grateful to Ben Adlam, Jinho Baik, Zhigang Bao, Paul Bourgade, László Erd ̋os, Iain Johnstone and Antti Knowles for comments. We are also grate-\r\nful to the anonymous referee for carefully reading our manuscript and suggesting several improvements.","publication_status":"published","author":[{"full_name":"Lee, Ji","first_name":"Ji","last_name":"Lee"},{"last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","first_name":"Kevin"}],"issue":"6","oa_version":"Preprint","intvolume":"        26","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1409.4979"}],"month":"12","article_processing_charge":"No","oa":1,"year":"2016"},{"author":[{"last_name":"Lee","full_name":"Lee, Jioon","first_name":"Jioon"},{"last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin"},{"full_name":"Stetler, Ben","first_name":"Ben","last_name":"Stetler"},{"full_name":"Yau, Horngtzer","first_name":"Horngtzer","last_name":"Yau"}],"publication_status":"published","month":"01","intvolume":"        44","main_file_link":[{"url":"https://arxiv.org/abs/1405.6634","open_access":"1"}],"issue":"3","oa_version":"Preprint","year":"2016","oa":1,"article_processing_charge":"No","date_created":"2018-12-11T11:50:47Z","department":[{"_id":"LaEr"}],"scopus_import":"1","publication":"Annals of Probability","citation":{"apa":"Lee, J., Schnelli, K., Stetler, B., &#38; Yau, H. (2016). Bulk universality for deformed wigner matrices. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/15-AOP1023\">https://doi.org/10.1214/15-AOP1023</a>","ieee":"J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” <i>Annals of Probability</i>, vol. 44, no. 3. Institute of Mathematical Statistics, pp. 2349–2425, 2016.","short":"J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.","chicago":"Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2016. <a href=\"https://doi.org/10.1214/15-AOP1023\">https://doi.org/10.1214/15-AOP1023</a>.","ista":"Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425.","ama":"Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner matrices. <i>Annals of Probability</i>. 2016;44(3):2349-2425. doi:<a href=\"https://doi.org/10.1214/15-AOP1023\">10.1214/15-AOP1023</a>","mla":"Lee, Jioon, et al. “Bulk Universality for Deformed Wigner Matrices.” <i>Annals of Probability</i>, vol. 44, no. 3, Institute of Mathematical Statistics, 2016, pp. 2349–425, doi:<a href=\"https://doi.org/10.1214/15-AOP1023\">10.1214/15-AOP1023</a>."},"ec_funded":1,"volume":44,"arxiv":1,"quality_controlled":"1","acknowledgement":"J.C. was supported in part by National Research Foundation of Korea Grant 2011-0013474 and TJ Park Junior Faculty Fellowship.\r\nK.S. was supported by ERC Advanced Grant RANMAT, No. 338804, and the \"Fund for Math.\"\r\nB.S. was supported by NSF GRFP Fellowship DGE-1144152.\r\nH.Y. was supported in part by NSF Grant DMS-13-07444 and Simons investigator fellowship. We thank Paul Bourgade, László Erd ̋os and Antti Knowles for helpful comments. We are grateful to the Taida Institute for Mathematical\r\nSciences and National Taiwan Universality for their hospitality during part of this\r\nresearch. We thank Thomas Spencer and the Institute for Advanced Study for their\r\nhospitality during the academic year 2013–2014.  ","day":"01","date_published":"2016-01-01T00:00:00Z","external_id":{"isi":["000376180700016"],"arxiv":["1405.6634"]},"language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1214/15-AOP1023","publist_id":"6115","isi":1,"title":"Bulk universality for deformed wigner matrices","abstract":[{"text":"We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.","lang":"eng"}],"date_updated":"2025-09-22T09:33:43Z","page":"2349 - 2425","status":"public","_id":"1219","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"Institute of Mathematical Statistics","doi":"10.1214/15-AOP1023"}]