[{"abstract":[{"lang":"eng","text":"We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently."}],"title":"Jointly convex quantum Jensen divergences","oa_version":"Preprint","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1712.05324"}],"page":"67-78","quality_controlled":"1","acknowledgement":"The author was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01) and partially supported by the Hungarian National Research, Development and Innovation Office – NKFIH (grant no. K124152)","intvolume":"       576","article_type":"original","volume":576,"scopus_import":"1","_id":"405","author":[{"orcid":"0000-0003-1109-5511","last_name":"Virosztek","first_name":"Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel"}],"date_created":"2018-12-11T11:46:17Z","doi":"10.1016/j.laa.2018.03.002","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"chicago":"Virosztek, Daniel. “Jointly Convex Quantum Jensen Divergences.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2019. <a href=\"https://doi.org/10.1016/j.laa.2018.03.002\">https://doi.org/10.1016/j.laa.2018.03.002</a>.","short":"D. Virosztek, Linear Algebra and Its Applications 576 (2019) 67–78.","mla":"Virosztek, Daniel. “Jointly Convex Quantum Jensen Divergences.” <i>Linear Algebra and Its Applications</i>, vol. 576, Elsevier, 2019, pp. 67–78, doi:<a href=\"https://doi.org/10.1016/j.laa.2018.03.002\">10.1016/j.laa.2018.03.002</a>.","ama":"Virosztek D. Jointly convex quantum Jensen divergences. <i>Linear Algebra and Its Applications</i>. 2019;576:67-78. doi:<a href=\"https://doi.org/10.1016/j.laa.2018.03.002\">10.1016/j.laa.2018.03.002</a>","apa":"Virosztek, D. (2019). Jointly convex quantum Jensen divergences. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2018.03.002\">https://doi.org/10.1016/j.laa.2018.03.002</a>","ista":"Virosztek D. 2019. Jointly convex quantum Jensen divergences. Linear Algebra and Its Applications. 576, 67–78.","ieee":"D. Virosztek, “Jointly convex quantum Jensen divergences,” <i>Linear Algebra and Its Applications</i>, vol. 576. Elsevier, pp. 67–78, 2019."},"arxiv":1,"oa":1,"date_published":"2019-09-01T00:00:00Z","language":[{"iso":"eng"}],"date_updated":"2025-04-15T06:50:00Z","status":"public","external_id":{"arxiv":["1712.05324"],"isi":["000470955300005"]},"department":[{"_id":"LaEr"}],"isi":1,"publication":"Linear Algebra and Its Applications","article_processing_charge":"No","ec_funded":1,"year":"2019","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","grant_number":"291734"}],"type":"journal_article","month":"09","corr_author":"1","publisher":"Elsevier","day":"01","publist_id":"7424"},{"volume":173,"publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"article_type":"original","intvolume":"       173","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).\r\n","oa":1,"date_published":"2019-02-01T00:00:00Z","citation":{"apa":"Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2019). Stability of the matrix Dyson equation and random matrices with correlations. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-018-0835-z\">https://doi.org/10.1007/s00440-018-0835-z</a>","ama":"Ajanki OH, Erdös L, Krüger TH. Stability of the matrix Dyson equation and random matrices with correlations. <i>Probability Theory and Related Fields</i>. 2019;173(1-2):293–373. doi:<a href=\"https://doi.org/10.1007/s00440-018-0835-z\">10.1007/s00440-018-0835-z</a>","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 173 (2019) 293–373.","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” <i>Probability Theory and Related Fields</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s00440-018-0835-z\">https://doi.org/10.1007/s00440-018-0835-z</a>.","mla":"Ajanki, Oskari H., et al. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” <i>Probability Theory and Related Fields</i>, vol. 173, no. 1–2, Springer, 2019, pp. 293–373, doi:<a href=\"https://doi.org/10.1007/s00440-018-0835-z\">10.1007/s00440-018-0835-z</a>.","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Stability of the matrix Dyson equation and random matrices with correlations,” <i>Probability Theory and Related Fields</i>, vol. 173, no. 1–2. Springer, pp. 293–373, 2019.","ista":"Ajanki OH, Erdös L, Krüger TH. 2019. Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields. 173(1–2), 293–373."},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","author":[{"full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H","last_name":"Ajanki"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger","first_name":"Torben H"}],"date_created":"2018-12-11T11:46:25Z","doi":"10.1007/s00440-018-0835-z","_id":"429","scopus_import":"1","publication_status":"published","title":"Stability of the matrix Dyson equation and random matrices with correlations","oa_version":"Published Version","abstract":[{"lang":"eng","text":"We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent."}],"file_date_updated":"2020-07-14T12:46:26Z","quality_controlled":"1","page":"293–373","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"type":"journal_article","year":"2019","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"publist_id":"7394","issue":"1-2","day":"01","publisher":"Springer","corr_author":"1","month":"02","ddc":["510"],"file":[{"file_name":"2018_ProbTheory_Ajanki.pdf","date_created":"2018-12-17T16:12:08Z","checksum":"f9354fa5c71f9edd17132588f0dc7d01","file_id":"5720","access_level":"open_access","creator":"dernst","relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:46:26Z","file_size":1201840}],"date_updated":"2026-04-03T09:46:51Z","language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","publication":"Probability Theory and Related Fields","has_accepted_license":"1","department":[{"_id":"LaEr"}],"isi":1,"external_id":{"isi":["000459396500007"]},"status":"public"},{"status":"public","department":[{"_id":"LaEr"}],"has_accepted_license":"1","article_processing_charge":"No","related_material":{"record":[{"relation":"part_of_dissertation","id":"6184","status":"public"},{"id":"6186","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"6185","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"1012"},{"status":"public","id":"1144","relation":"part_of_dissertation"},{"id":"6182","status":"public","relation":"part_of_dissertation"}]},"language":[{"iso":"eng"}],"ddc":["515","519"],"file":[{"date_created":"2019-03-28T08:53:52Z","checksum":"6926f66f28079a81c4937e3764be00fc","file_name":"2019_Schroeder_Thesis.tar.gz","file_id":"6180","creator":"dernst","access_level":"closed","relation":"source_file","file_size":7104482,"date_updated":"2020-07-14T12:47:21Z","content_type":"application/x-gzip"},{"file_size":4228794,"date_updated":"2020-07-14T12:47:21Z","content_type":"application/pdf","checksum":"7d0ebb8d1207e89768cdd497a5bf80fb","date_created":"2019-03-28T08:53:52Z","file_name":"2019_Schroeder_Thesis.pdf","file_id":"6181","creator":"dernst","access_level":"open_access","relation":"main_file"}],"date_updated":"2026-04-08T13:55:03Z","corr_author":"1","month":"03","publisher":"Institute of Science and Technology Austria","day":"18","degree_awarded":"PhD","ec_funded":1,"year":"2019","project":[{"grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"dissertation","page":"375","file_date_updated":"2020-07-14T12:47:21Z","abstract":[{"lang":"eng","text":"In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.\r\nIn the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.\r\nIn the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure."}],"title":"From Dyson to Pearcey: Universal statistics in random matrix theory","oa_version":"Published Version","publication_status":"published","_id":"6179","OA_place":"publisher","doi":"10.15479/AT:ISTA:th6179","date_created":"2019-03-28T08:58:59Z","author":[{"orcid":"0000-0002-2904-1856","last_name":"Schröder","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa":1,"date_published":"2019-03-18T00:00:00Z","supervisor":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"}],"citation":{"apa":"Schröder, D. J. (2019). <i>From Dyson to Pearcey: Universal statistics in random matrix theory</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:th6179\">https://doi.org/10.15479/AT:ISTA:th6179</a>","ama":"Schröder DJ. From Dyson to Pearcey: Universal statistics in random matrix theory. 2019. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th6179\">10.15479/AT:ISTA:th6179</a>","chicago":"Schröder, Dominik J. “From Dyson to Pearcey: Universal Statistics in Random Matrix Theory.” Institute of Science and Technology Austria, 2019. <a href=\"https://doi.org/10.15479/AT:ISTA:th6179\">https://doi.org/10.15479/AT:ISTA:th6179</a>.","mla":"Schröder, Dominik J. <i>From Dyson to Pearcey: Universal Statistics in Random Matrix Theory</i>. Institute of Science and Technology Austria, 2019, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:th6179\">10.15479/AT:ISTA:th6179</a>.","short":"D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix Theory, Institute of Science and Technology Austria, 2019.","ieee":"D. J. Schröder, “From Dyson to Pearcey: Universal statistics in random matrix theory,” Institute of Science and Technology Austria, 2019.","ista":"Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random matrix theory. Institute of Science and Technology Austria."},"publication_identifier":{"issn":["2663-337X"]},"alternative_title":["ISTA Thesis"]},{"related_material":{"record":[{"relation":"dissertation_contains","id":"6179","status":"public"}]},"language":[{"iso":"eng"}],"date_updated":"2026-04-08T13:55:03Z","ddc":["510"],"file":[{"date_created":"2019-09-17T14:24:13Z","file_id":"6883","file_name":"2019_Forum_Erdoes.pdf","checksum":"933a472568221c73b2c3ce8c87bf6d15","access_level":"open_access","creator":"dernst","relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:47:22Z","file_size":1520344}],"external_id":{"arxiv":["1705.10661"],"isi":["000488847100001"]},"article_number":"e8","status":"public","article_processing_charge":"No","publication":"Forum of Mathematics, Sigma","has_accepted_license":"1","isi":1,"department":[{"_id":"LaEr"}],"ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"type":"journal_article","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"year":"2019","publisher":"Cambridge University Press","corr_author":"1","month":"03","day":"26","abstract":[{"text":"We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is\r\na systematic diagrammatic control of a multivariate cumulant expansion.","lang":"eng"}],"publication_status":"published","oa_version":"Published Version","title":"Random matrices with slow correlation decay","quality_controlled":"1","file_date_updated":"2020-07-14T12:47:22Z","intvolume":"         7","publication_identifier":{"eissn":["2050-5094"]},"volume":7,"article_type":"original","author":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger"},{"full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","first_name":"Dominik J","last_name":"Schröder","orcid":"0000-0002-2904-1856"}],"date_created":"2019-03-28T09:05:23Z","doi":"10.1017/fms.2019.2","_id":"6182","scopus_import":"1","oa":1,"date_published":"2019-03-26T00:00:00Z","arxiv":1,"citation":{"apa":"Erdös, L., Krüger, T. H., &#38; Schröder, D. J. (2019). Random matrices with slow correlation decay. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2019.2\">https://doi.org/10.1017/fms.2019.2</a>","mla":"Erdös, László, et al. “Random Matrices with Slow Correlation Decay.” <i>Forum of Mathematics, Sigma</i>, vol. 7, e8, Cambridge University Press, 2019, doi:<a href=\"https://doi.org/10.1017/fms.2019.2\">10.1017/fms.2019.2</a>.","chicago":"Erdös, László, Torben H Krüger, and Dominik J Schröder. “Random Matrices with Slow Correlation Decay.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2019. <a href=\"https://doi.org/10.1017/fms.2019.2\">https://doi.org/10.1017/fms.2019.2</a>.","short":"L. Erdös, T.H. Krüger, D.J. Schröder, Forum of Mathematics, Sigma 7 (2019).","ama":"Erdös L, Krüger TH, Schröder DJ. Random matrices with slow correlation decay. <i>Forum of Mathematics, Sigma</i>. 2019;7. doi:<a href=\"https://doi.org/10.1017/fms.2019.2\">10.1017/fms.2019.2</a>","ieee":"L. Erdös, T. H. Krüger, and D. J. Schröder, “Random matrices with slow correlation decay,” <i>Forum of Mathematics, Sigma</i>, vol. 7. Cambridge University Press, 2019.","ista":"Erdös L, Krüger TH, Schröder DJ. 2019. Random matrices with slow correlation decay. Forum of Mathematics, Sigma. 7, e8."},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd"},{"day":"12","issue":"4","publisher":"MSP","month":"10","type":"journal_article","project":[{"grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"},{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385"}],"year":"2019","ec_funded":1,"article_processing_charge":"No","department":[{"_id":"LaEr"}],"publication":"Pure and Applied Analysis ","external_id":{"arxiv":["1811.04055"]},"status":"public","date_updated":"2026-04-08T13:55:02Z","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"6179","relation":"dissertation_contains"}]},"arxiv":1,"citation":{"apa":"Cipolloni, G., Erdös, L., Krüger, T. H., &#38; Schröder, D. J. (2019). Cusp universality for random matrices, II: The real symmetric case. <i>Pure and Applied Analysis </i>. MSP. <a href=\"https://doi.org/10.2140/paa.2019.1.615\">https://doi.org/10.2140/paa.2019.1.615</a>","short":"G. Cipolloni, L. Erdös, T.H. Krüger, D.J. Schröder, Pure and Applied Analysis  1 (2019) 615–707.","chicago":"Cipolloni, Giorgio, László Erdös, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices, II: The Real Symmetric Case.” <i>Pure and Applied Analysis </i>. MSP, 2019. <a href=\"https://doi.org/10.2140/paa.2019.1.615\">https://doi.org/10.2140/paa.2019.1.615</a>.","mla":"Cipolloni, Giorgio, et al. “Cusp Universality for Random Matrices, II: The Real Symmetric Case.” <i>Pure and Applied Analysis </i>, vol. 1, no. 4, MSP, 2019, pp. 615–707, doi:<a href=\"https://doi.org/10.2140/paa.2019.1.615\">10.2140/paa.2019.1.615</a>.","ama":"Cipolloni G, Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices, II: The real symmetric case. <i>Pure and Applied Analysis </i>. 2019;1(4):615–707. doi:<a href=\"https://doi.org/10.2140/paa.2019.1.615\">10.2140/paa.2019.1.615</a>","ieee":"G. Cipolloni, L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices, II: The real symmetric case,” <i>Pure and Applied Analysis </i>, vol. 1, no. 4. MSP, pp. 615–707, 2019.","ista":"Cipolloni G, Erdös L, Krüger TH, Schröder DJ. 2019. Cusp universality for random matrices, II: The real symmetric case. Pure and Applied Analysis . 1(4), 615–707."},"oa":1,"date_published":"2019-10-12T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2019-03-28T10:21:17Z","author":[{"last_name":"Cipolloni","orcid":"0000-0002-4901-7992","first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio"},{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297"},{"first_name":"Dominik J","last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"doi":"10.2140/paa.2019.1.615","scopus_import":"1","_id":"6186","article_type":"original","volume":1,"publication_identifier":{"eissn":["2578-5885"],"issn":["2578-5893"]},"intvolume":"         1","quality_controlled":"1","page":"615–707","main_file_link":[{"url":"https://arxiv.org/abs/1811.04055","open_access":"1"}],"publication_status":"published","oa_version":"Preprint","title":"Cusp universality for random matrices, II: The real symmetric case","abstract":[{"lang":"eng","text":"We prove that the local eigenvalue statistics of real symmetric Wigner-type\r\nmatrices near the cusp points of the eigenvalue density are universal. Together\r\nwith the companion paper [arXiv:1809.03971], which proves the same result for\r\nthe complex Hermitian symmetry class, this completes the last remaining case of\r\nthe Wigner-Dyson-Mehta universality conjecture after bulk and edge\r\nuniversalities have been established in the last years. We extend the recent\r\nDyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp\r\nregime using the optimal local law from [arXiv:1809.03971] and the accurate\r\nlocal shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752].\r\nWe also present a PDE-based method to improve the estimate on eigenvalue\r\nrigidity via the maximum principle of the heat flow related to the Dyson\r\nBrownian motion."}]},{"intvolume":"        55","volume":55,"publication_identifier":{"issn":["0246-0203"]},"_id":"6240","scopus_import":"1","author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297"},{"full_name":"Nemish, Yuriy","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","first_name":"Yuriy","orcid":"0000-0002-7327-856X","last_name":"Nemish"}],"date_created":"2019-04-08T14:05:04Z","doi":"10.1214/18-AIHP894","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2019-05-01T00:00:00Z","oa":1,"arxiv":1,"citation":{"chicago":"Alt, Johannes, László Erdös, Torben H Krüger, and Yuriy Nemish. “Location of the Spectrum of Kronecker Random Matrices.” <i>Annales de l’institut Henri Poincare</i>. Institut Henri Poincaré, 2019. <a href=\"https://doi.org/10.1214/18-AIHP894\">https://doi.org/10.1214/18-AIHP894</a>.","mla":"Alt, Johannes, et al. “Location of the Spectrum of Kronecker Random Matrices.” <i>Annales de l’institut Henri Poincare</i>, vol. 55, no. 2, Institut Henri Poincaré, 2019, pp. 661–96, doi:<a href=\"https://doi.org/10.1214/18-AIHP894\">10.1214/18-AIHP894</a>.","short":"J. Alt, L. Erdös, T.H. Krüger, Y. Nemish, Annales de l’institut Henri Poincare 55 (2019) 661–696.","ama":"Alt J, Erdös L, Krüger TH, Nemish Y. Location of the spectrum of Kronecker random matrices. <i>Annales de l’institut Henri Poincare</i>. 2019;55(2):661-696. doi:<a href=\"https://doi.org/10.1214/18-AIHP894\">10.1214/18-AIHP894</a>","apa":"Alt, J., Erdös, L., Krüger, T. H., &#38; Nemish, Y. (2019). Location of the spectrum of Kronecker random matrices. <i>Annales de l’institut Henri Poincare</i>. Institut Henri Poincaré. <a href=\"https://doi.org/10.1214/18-AIHP894\">https://doi.org/10.1214/18-AIHP894</a>","ista":"Alt J, Erdös L, Krüger TH, Nemish Y. 2019. Location of the spectrum of Kronecker random matrices. Annales de l’institut Henri Poincare. 55(2), 661–696.","ieee":"J. Alt, L. Erdös, T. H. Krüger, and Y. Nemish, “Location of the spectrum of Kronecker random matrices,” <i>Annales de l’institut Henri Poincare</i>, vol. 55, no. 2. Institut Henri Poincaré, pp. 661–696, 2019."},"abstract":[{"text":"For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.","lang":"eng"}],"oa_version":"Preprint","title":"Location of the spectrum of Kronecker random matrices","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1706.08343"}],"page":"661-696","quality_controlled":"1","ec_funded":1,"year":"2019","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"}],"type":"journal_article","month":"05","publisher":"Institut Henri Poincaré","issue":"2","day":"01","related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"}]},"language":[{"iso":"eng"}],"date_updated":"2026-04-08T14:11:36Z","status":"public","external_id":{"isi":["000467793600003"],"arxiv":["1706.08343"]},"publication":"Annales de l'institut Henri Poincare","department":[{"_id":"LaEr"}],"isi":1,"article_processing_charge":"No"},{"year":"2018","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"type":"journal_article","ec_funded":1,"issue":"1-2","day":"04","publist_id":"7615","month":"06","publisher":"Springer Nature","date_updated":"2025-04-15T06:50:21Z","language":[{"iso":"eng"}],"publication":"Acta Scientiarum Mathematicarum","department":[{"_id":"LaEr"}],"article_processing_charge":"No","status":"public","external_id":{"arxiv":["1802.03305"]},"publication_identifier":{"issn":["0001-6969"],"eissn":["2064-8316"]},"volume":84,"article_type":"original","intvolume":"        84","acknowledgement":"The author was supported by the ISTFELLOW program of the Institute of Science and Technol- ogy Austria (project code IC1027FELL01) and partially supported by the Hungarian National Research, Development and Innovation Office, NKFIH (grant no. K124152).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_published":"2018-06-04T00:00:00Z","arxiv":1,"citation":{"ieee":"D. Virosztek, “Maps on probability measures preserving certain distances - a survey and some new results,” <i>Acta Scientiarum Mathematicarum</i>, vol. 84, no. 1–2. Springer Nature, pp. 65–80, 2018.","ista":"Virosztek D. 2018. Maps on probability measures preserving certain distances - a survey and some new results. Acta Scientiarum Mathematicarum. 84(1–2), 65–80.","apa":"Virosztek, D. (2018). Maps on probability measures preserving certain distances - a survey and some new results. <i>Acta Scientiarum Mathematicarum</i>. Springer Nature. <a href=\"https://doi.org/10.14232/actasm-018-753-y\">https://doi.org/10.14232/actasm-018-753-y</a>","mla":"Virosztek, Daniel. “Maps on Probability Measures Preserving Certain Distances - a Survey and Some New Results.” <i>Acta Scientiarum Mathematicarum</i>, vol. 84, no. 1–2, Springer Nature, 2018, pp. 65–80, doi:<a href=\"https://doi.org/10.14232/actasm-018-753-y\">10.14232/actasm-018-753-y</a>.","short":"D. Virosztek, Acta Scientiarum Mathematicarum 84 (2018) 65–80.","chicago":"Virosztek, Daniel. “Maps on Probability Measures Preserving Certain Distances - a Survey and Some New Results.” <i>Acta Scientiarum Mathematicarum</i>. Springer Nature, 2018. <a href=\"https://doi.org/10.14232/actasm-018-753-y\">https://doi.org/10.14232/actasm-018-753-y</a>.","ama":"Virosztek D. Maps on probability measures preserving certain distances - a survey and some new results. <i>Acta Scientiarum Mathematicarum</i>. 2018;84(1-2):65-80. doi:<a href=\"https://doi.org/10.14232/actasm-018-753-y\">10.14232/actasm-018-753-y</a>"},"_id":"284","scopus_import":"1","author":[{"last_name":"Virosztek","orcid":"0000-0003-1109-5511","first_name":"Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel"}],"date_created":"2018-12-11T11:45:36Z","doi":"10.14232/actasm-018-753-y","title":"Maps on probability measures preserving certain distances - a survey and some new results","oa_version":"Preprint","publication_status":"published","abstract":[{"text":"Borel probability measures living on metric spaces are fundamental\r\nmathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric W_p for arbitrary p &gt;= 1, and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.","lang":"eng"}],"quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1802.03305","open_access":"1"}],"page":"65 - 80"},{"publisher":"Society for Industrial and Applied Mathematics ","month":"01","publist_id":"7740","issue":"3","day":"01","ec_funded":1,"project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FWF","name":"Structured Non-Hermitian Random Matrices","grant_number":"M02080","_id":"258F40A4-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","year":"2018","external_id":{"arxiv":["1708.01546"],"isi":["000437018500032"]},"status":"public","article_processing_charge":"No","department":[{"_id":"LaEr"}],"isi":1,"publication":"SIAM Journal on Mathematical Analysis","language":[{"iso":"eng"}],"date_updated":"2025-04-15T08:05:02Z","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297"},{"last_name":"Renfrew","orcid":"0000-0003-3493-121X","first_name":"David T","id":"4845BF6A-F248-11E8-B48F-1D18A9856A87","full_name":"Renfrew, David T"}],"doi":"10.1137/17M1143125","date_created":"2018-12-11T11:45:03Z","scopus_import":"1","_id":"181","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>.","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>.","short":"L. Erdös, T.H. Krüger, D.T. Renfrew, SIAM Journal on Mathematical Analysis 50 (2018) 3271–3290.","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>","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>","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.","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."},"arxiv":1,"oa":1,"date_published":"2018-01-01T00:00:00Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","acknowledgement":"The work of the second author was also partially supported by the Hausdorff Center of Mathematics.","intvolume":"        50","volume":50,"page":"3271 - 3290","main_file_link":[{"url":"https://arxiv.org/abs/1708.01546","open_access":"1"}],"quality_controlled":"1","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."}],"publication_status":"published","title":"Power law decay for systems of randomly coupled differential equations","oa_version":"Published Version"},{"arxiv":1,"citation":{"ista":"Betea D, Bouttier J, Nejjar P, Vuletic M. 2018. The free boundary Schur process and applications I. Annales Henri Poincare. 19(12), 3663–3742.","ieee":"D. Betea, J. Bouttier, P. Nejjar, and M. Vuletic, “The free boundary Schur process and applications I,” <i>Annales Henri Poincare</i>, vol. 19, no. 12. Springer Nature, pp. 3663–3742, 2018.","short":"D. Betea, J. Bouttier, P. Nejjar, M. Vuletic, Annales Henri Poincare 19 (2018) 3663–3742.","mla":"Betea, Dan, et al. “The Free Boundary Schur Process and Applications I.” <i>Annales Henri Poincare</i>, vol. 19, no. 12, Springer Nature, 2018, pp. 3663–742, doi:<a href=\"https://doi.org/10.1007/s00023-018-0723-1\">10.1007/s00023-018-0723-1</a>.","chicago":"Betea, Dan, Jeremie Bouttier, Peter Nejjar, and Mirjana Vuletic. “The Free Boundary Schur Process and Applications I.” <i>Annales Henri Poincare</i>. Springer Nature, 2018. <a href=\"https://doi.org/10.1007/s00023-018-0723-1\">https://doi.org/10.1007/s00023-018-0723-1</a>.","ama":"Betea D, Bouttier J, Nejjar P, Vuletic M. The free boundary Schur process and applications I. <i>Annales Henri Poincare</i>. 2018;19(12):3663-3742. doi:<a href=\"https://doi.org/10.1007/s00023-018-0723-1\">10.1007/s00023-018-0723-1</a>","apa":"Betea, D., Bouttier, J., Nejjar, P., &#38; Vuletic, M. (2018). The free boundary Schur process and applications I. <i>Annales Henri Poincare</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00023-018-0723-1\">https://doi.org/10.1007/s00023-018-0723-1</a>"},"date_published":"2018-11-13T00:00:00Z","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1007/s00023-018-0723-1","date_created":"2018-12-11T11:47:09Z","author":[{"first_name":"Dan","last_name":"Betea","full_name":"Betea, Dan"},{"full_name":"Bouttier, Jeremie","last_name":"Bouttier","first_name":"Jeremie"},{"last_name":"Nejjar","first_name":"Peter","id":"4BF426E2-F248-11E8-B48F-1D18A9856A87","full_name":"Nejjar, Peter"},{"full_name":"Vuletic, Mirjana","last_name":"Vuletic","first_name":"Mirjana"}],"scopus_import":"1","_id":"556","article_type":"original","publication_identifier":{"issn":["1424-0637"]},"volume":19,"intvolume":"        19","quality_controlled":"1","file_date_updated":"2020-07-14T12:47:03Z","page":"3663-3742","publication_status":"published","oa_version":"Published Version","title":"The free boundary Schur process and applications I","abstract":[{"lang":"eng","text":"We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions."}],"issue":"12","publist_id":"7258","day":"13","publisher":"Springer Nature","month":"11","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"},{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics"}],"type":"journal_article","year":"2018","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"article_processing_charge":"Yes (via OA deal)","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"has_accepted_license":"1","isi":1,"publication":"Annales Henri Poincare","external_id":{"isi":["000450487900003"],"arxiv":["1704.05809"]},"status":"public","file":[{"relation":"main_file","creator":"dernst","access_level":"open_access","date_created":"2019-01-21T15:18:55Z","file_name":"2018_Annales_Betea.pdf","file_id":"5866","checksum":"0c38abe73569b7166b7487ad5d23cc68","date_updated":"2020-07-14T12:47:03Z","file_size":3084674,"content_type":"application/pdf"}],"ddc":["500"],"date_updated":"2025-09-18T07:34:29Z","language":[{"iso":"eng"}]},{"day":"26","publisher":"World Scientific Publishing","month":"09","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"year":"2018","ec_funded":1,"article_processing_charge":"No","publication":"Random matrices: Theory and applications","department":[{"_id":"LaEr"}],"isi":1,"external_id":{"isi":["000477677200002"],"arxiv":["1802.05175"]},"article_number":"1950009","status":"public","date_updated":"2025-04-15T08:05:02Z","language":[{"iso":"eng"}],"date_published":"2018-09-26T00:00:00Z","oa":1,"arxiv":1,"citation":{"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>","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>","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>.","short":"L. Erdös, P. Mühlbacher, Random Matrices: Theory and Applications (2018).","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>.","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.","ista":"Erdös L, Mühlbacher P. 2018. Bounds on the norm of Wigner-type random matrices. Random matrices: Theory and applications., 1950009."},"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Mühlbacher, Peter","first_name":"Peter","last_name":"Mühlbacher"}],"date_created":"2019-02-13T10:40:54Z","doi":"10.1142/s2010326319500096","_id":"5971","scopus_import":"1","publication_identifier":{"issn":["2010-3263"],"eissn":["2010-3271"]},"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1802.05175"}],"publication_status":"published","oa_version":"Preprint","title":"Bounds on the norm of Wigner-type random matrices","abstract":[{"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.","lang":"eng"}]},{"volume":171,"intvolume":"       171","oa":1,"date_published":"2018-06-14T00:00:00Z","arxiv":1,"citation":{"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,” <i>Probability Theory and Related Fields</i>, vol. 171, no. 1–2. Springer, 2018.","chicago":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s00440-017-0787-8\">https://doi.org/10.1007/s00440-017-0787-8</a>.","mla":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” <i>Probability Theory and Related Fields</i>, vol. 171, no. 1–2, 543–616, Springer, 2018, doi:<a href=\"https://doi.org/10.1007/s00440-017-0787-8\">10.1007/s00440-017-0787-8</a>.","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. <i>Probability Theory and Related Fields</i>. 2018;171(1-2). doi:<a href=\"https://doi.org/10.1007/s00440-017-0787-8\">10.1007/s00440-017-0787-8</a>","apa":"Lee, J., &#38; Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-017-0787-8\">https://doi.org/10.1007/s00440-017-0787-8</a>"},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1007/s00440-017-0787-8","author":[{"full_name":"Lee, Jii","first_name":"Jii","last_name":"Lee"},{"orcid":"0000-0003-0954-3231","last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"date_created":"2018-12-11T11:47:56Z","_id":"690","scopus_import":"1","publication_status":"published","title":"Local law and Tracy–Widom limit for sparse random matrices","oa_version":"Preprint","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."}],"quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1605.08767","open_access":"1"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"type":"journal_article","year":"2018","ec_funded":1,"issue":"1-2","day":"14","publist_id":"7017","publisher":"Springer","month":"06","date_updated":"2025-09-10T14:00:58Z","language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Probability Theory and Related Fields","isi":1,"department":[{"_id":"LaEr"}],"external_id":{"arxiv":["1605.08767"],"isi":["000432129600012"]},"article_number":"543-616","status":"public"},{"volume":15,"publication_identifier":{"issn":["1980-0436"]},"article_type":"original","intvolume":"        15","date_published":"2018-10-01T00:00:00Z","oa":1,"arxiv":1,"citation":{"ama":"Nejjar P. Transition to shocks in TASEP and decoupling of last passage times. <i>Latin American Journal of Probability and Mathematical Statistics</i>. 2018;15(2):1311-1334. doi:<a href=\"https://doi.org/10.30757/ALEA.v15-49\">10.30757/ALEA.v15-49</a>","short":"P. Nejjar, Latin American Journal of Probability and Mathematical Statistics 15 (2018) 1311–1334.","chicago":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” <i>Latin American Journal of Probability and Mathematical Statistics</i>. Instituto Nacional de Matematica Pura e Aplicada, 2018. <a href=\"https://doi.org/10.30757/ALEA.v15-49\">https://doi.org/10.30757/ALEA.v15-49</a>.","mla":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” <i>Latin American Journal of Probability and Mathematical Statistics</i>, vol. 15, no. 2, Instituto Nacional de Matematica Pura e Aplicada, 2018, pp. 1311–34, doi:<a href=\"https://doi.org/10.30757/ALEA.v15-49\">10.30757/ALEA.v15-49</a>.","apa":"Nejjar, P. (2018). Transition to shocks in TASEP and decoupling of last passage times. <i>Latin American Journal of Probability and Mathematical Statistics</i>. Instituto Nacional de Matematica Pura e Aplicada. <a href=\"https://doi.org/10.30757/ALEA.v15-49\">https://doi.org/10.30757/ALEA.v15-49</a>","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.","ieee":"P. Nejjar, “Transition to shocks in TASEP and decoupling of last passage times,” <i>Latin American Journal of Probability and Mathematical Statistics</i>, vol. 15, no. 2. Instituto Nacional de Matematica Pura e Aplicada, pp. 1311–1334, 2018."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:44:28Z","doi":"10.30757/ALEA.v15-49","author":[{"id":"4BF426E2-F248-11E8-B48F-1D18A9856A87","full_name":"Nejjar, Peter","last_name":"Nejjar","first_name":"Peter"}],"_id":"70","scopus_import":"1","publication_status":"published","title":"Transition to shocks in TASEP and decoupling of last passage times","oa_version":"Published Version","abstract":[{"lang":"eng","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."}],"file_date_updated":"2020-07-14T12:47:46Z","quality_controlled":"1","page":"1311-1334","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"},{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","year":"2018","ec_funded":1,"issue":"2","day":"01","publisher":"Instituto Nacional de Matematica Pura e Aplicada","month":"10","date_updated":"2025-04-14T07:27:49Z","file":[{"date_updated":"2020-07-14T12:47:46Z","file_size":394851,"content_type":"application/pdf","relation":"main_file","file_id":"5981","date_created":"2019-02-14T09:44:10Z","file_name":"2018_ALEA_Nejjar.pdf","checksum":"2ded46aa284a836a8cbb34133a64f1cb","access_level":"open_access","creator":"kschuh"}],"ddc":["510"],"language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Latin American Journal of Probability and Mathematical Statistics","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"has_accepted_license":"1","isi":1,"external_id":{"isi":["000460475800022"],"arxiv":["1705.08836"]},"status":"public"},{"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","oa_version":"Preprint","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.05163"}],"page":"3255-3298","quality_controlled":"1","intvolume":"      2018","volume":2018,"publication_identifier":{"issn":["1073-7928"]},"scopus_import":"1","_id":"1012","date_created":"2018-12-11T11:49:41Z","author":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","first_name":"Dominik J","last_name":"Schröder","orcid":"0000-0002-2904-1856"}],"doi":"10.1093/imrn/rnw330","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"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>","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>.","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>","ista":"Erdös L, Schröder DJ. 2018. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. International Mathematics Research Notices. 2018(10), 3255–3298.","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."},"arxiv":1,"oa":1,"date_published":"2018-05-18T00:00:00Z","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"dissertation_contains","id":"6179","status":"public"}]},"date_updated":"2026-04-08T13:55:03Z","status":"public","external_id":{"arxiv":["1608.05163"],"isi":["000441668300009"]},"department":[{"_id":"LaEr"}],"isi":1,"publication":"International Mathematics Research Notices","article_processing_charge":"No","ec_funded":1,"year":"2018","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"}],"month":"05","publisher":"Oxford University Press","issue":"10","day":"18","publist_id":"6383"},{"publisher":"Institute of Science and Technology Austria","month":"07","corr_author":"1","degree_awarded":"PhD","publist_id":"7772","day":"12","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"pubrep_id":"1040","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"dissertation","year":"2018","status":"public","article_processing_charge":"No","department":[{"_id":"LaEr"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"6240"},{"relation":"part_of_dissertation","id":"6184","status":"public"},{"status":"public","id":"566","relation":"part_of_dissertation"},{"id":"6183","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"1010","relation":"part_of_dissertation"},{"id":"550","status":"public","relation":"part_of_dissertation"},{"id":"1677","status":"public","relation":"part_of_dissertation"}]},"ddc":["515","519"],"file":[{"file_size":5801709,"date_updated":"2020-07-14T12:44:57Z","content_type":"application/pdf","date_created":"2019-04-08T13:55:20Z","file_name":"2018_thesis_Alt.pdf","file_id":"6241","checksum":"d4dad55a7513f345706aaaba90cb1bb8","creator":"dernst","access_level":"open_access","relation":"main_file"},{"relation":"source_file","creator":"dernst","access_level":"closed","file_name":"2018_thesis_Alt_source.zip","checksum":"d73fcf46300dce74c403f2b491148ab4","file_id":"6242","date_created":"2019-04-08T13:55:20Z","file_size":3802059,"date_updated":"2020-07-14T12:44:57Z","content_type":"application/zip"}],"date_updated":"2026-04-08T14:11:37Z","author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"}],"date_created":"2018-12-11T11:44:53Z","doi":"10.15479/AT:ISTA:TH_1040","OA_place":"publisher","_id":"149","supervisor":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"}],"citation":{"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>","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>.","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>.","short":"J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute of Science and Technology Austria, 2018.","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>","ieee":"J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute of Science and Technology Austria, 2018.","ista":"Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. Institute of Science and Technology Austria."},"date_published":"2018-07-12T00:00:00Z","oa":1,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","alternative_title":["ISTA Thesis"],"publication_identifier":{"issn":["2663-337X"]},"page":"456","file_date_updated":"2020-07-14T12:44:57Z","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"}],"publication_status":"published","title":"Dyson equation and eigenvalue statistics of random matrices","oa_version":"Published Version"},{"article_type":"original","volume":28,"intvolume":"        28","arxiv":1,"citation":{"ama":"Alt J, Erdös L, Krüger TH. Local inhomogeneous circular law. <i>Annals Applied Probability </i>. 2018;28(1):148-203. doi:<a href=\"https://doi.org/10.1214/17-AAP1302\">10.1214/17-AAP1302</a>","mla":"Alt, Johannes, et al. “Local Inhomogeneous Circular Law.” <i>Annals 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>.","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Inhomogeneous Circular Law.” <i>Annals 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 Applied Probability  28 (2018) 148–203.","apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2018). Local inhomogeneous circular law. <i>Annals Applied Probability </i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-AAP1302\">https://doi.org/10.1214/17-AAP1302</a>","ista":"Alt J, Erdös L, Krüger TH. 2018. Local inhomogeneous circular law. Annals Applied Probability . 28(1), 148–203.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local inhomogeneous circular law,” <i>Annals Applied Probability </i>, vol. 28, no. 1. Institute of Mathematical Statistics, pp. 148–203, 2018."},"oa":1,"date_published":"2018-03-03T00:00:00Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"},{"first_name":"Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87"}],"doi":"10.1214/17-AAP1302","date_created":"2018-12-11T11:47:13Z","scopus_import":"1","_id":"566","publication_status":"published","title":"Local inhomogeneous circular law","oa_version":"Preprint","abstract":[{"lang":"eng","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"}],"quality_controlled":"1","page":"148-203","main_file_link":[{"url":"https://arxiv.org/abs/1612.07776 ","open_access":"1"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"type":"journal_article","year":"2018","ec_funded":1,"issue":"1","day":"03","publisher":"Institute of Mathematical Statistics","month":"03","corr_author":"1","date_updated":"2026-04-08T14:11:36Z","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"article_processing_charge":"No","department":[{"_id":"LaEr"}],"isi":1,"publication":"Annals Applied Probability ","external_id":{"isi":["000431721800005"],"arxiv":["1612.07776 "]},"status":"public"},{"type":"preprint","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"year":"2018","acknowledgement":"Partially funded by ERC Advanced Grant RANMAT No. 338804.\r\nPartially supported by the Hausdorff Center for Mathematics.\r\n","ec_funded":1,"date_published":"2018-04-20T00:00:00Z","oa":1,"citation":{"ama":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and  cusps. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.1804.07752\">10.48550/arXiv.1804.07752</a>","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and  Cusps.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.1804.07752\">https://doi.org/10.48550/arXiv.1804.07752</a>.","mla":"Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and  Cusps.” <i>ArXiv</i>, 1804.07752, doi:<a href=\"https://doi.org/10.48550/arXiv.1804.07752\">10.48550/arXiv.1804.07752</a>.","short":"J. Alt, L. Erdös, T.H. Krüger, ArXiv (n.d.).","apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (n.d.). The Dyson equation with linear self-energy: Spectral bands, edges and  cusps. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.1804.07752\">https://doi.org/10.48550/arXiv.1804.07752</a>","ista":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and  cusps. arXiv, 1804.07752.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and  cusps,” <i>arXiv</i>. ."},"arxiv":1,"day":"20","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2019-03-28T09:20:06Z","author":[{"id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes","last_name":"Alt","first_name":"Johannes"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297"}],"doi":"10.48550/arXiv.1804.07752","_id":"6183","month":"04","date_updated":"2026-04-08T14:11:36Z","publication_status":"draft","oa_version":"Preprint","title":"The Dyson equation with linear self-energy: Spectral bands, edges and  cusps","related_material":{"record":[{"relation":"later_version","status":"public","id":"14694"},{"relation":"dissertation_contains","id":"149","status":"public"}]},"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We study the unique solution $m$ of the Dyson equation \\[ -m(z)^{-1} = z - a\r\n+ S[m(z)] \\] on a von Neumann algebra $\\mathcal{A}$ with the constraint\r\n$\\mathrm{Im}\\,m\\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is\r\na self-adjoint element of $\\mathcal{A}$ and $S$ is a positivity-preserving\r\nlinear operator on $\\mathcal{A}$. We show that $m$ is the Stieltjes transform\r\nof a compactly supported $\\mathcal{A}$-valued measure on $\\mathbb{R}$. Under\r\nsuitable assumptions, we establish that this measure has a uniformly\r\n$1/3$-H\\\"{o}lder continuous density with respect to the Lebesgue measure, which\r\nis supported on finitely many intervals, called bands. In fact, the density is\r\nanalytic inside the bands with a square-root growth at the edges and internal\r\ncubic root cusps whenever the gap between two bands vanishes. The shape of\r\nthese singularities is universal and no other singularity may occur. We give a\r\nprecise asymptotic description of $m$ near the singular points. These\r\nasymptotics generalize the analysis at the regular edges given in the companion\r\npaper on the Tracy-Widom universality for the edge eigenvalue statistics for\r\ncorrelated random matrices [arXiv:1804.07744] and they play a key role in the\r\nproof of the Pearcey universality at the cusp for Wigner-type matrices\r\n[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band\r\nmass formula from [arXiv:1804.07744] to the von Neumann algebra setting by\r\nshowing that the spectral mass of the bands is topologically rigid under\r\ndeformations and we conclude that these masses are quantized in some important\r\ncases."}],"article_processing_charge":"No","publication":"arXiv","department":[{"_id":"LaEr"}],"external_id":{"arxiv":["1804.07752"]},"article_number":"1804.07752","main_file_link":[{"url":"https://arxiv.org/abs/1804.07752","open_access":"1"}],"status":"public"},{"file_date_updated":"2020-07-14T12:44:39Z","quality_controlled":"1","page":"947 - 990","publication_status":"published","title":"Local law of addition of random matrices on optimal scale","oa_version":"Published Version","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."}],"citation":{"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>.","short":"Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 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>","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>","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.","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."},"oa":1,"date_published":"2017-02-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli","orcid":"0000-0003-0954-3231"}],"date_created":"2018-12-11T11:50:43Z","doi":"10.1007/s00220-016-2805-6","scopus_import":"1","_id":"1207","publication_identifier":{"issn":["0010-3616"]},"volume":349,"intvolume":"       349","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"LaEr"}],"has_accepted_license":"1","isi":1,"publication":"Communications in Mathematical Physics","external_id":{"isi":["000393696700005"]},"status":"public","date_updated":"2025-07-10T11:50:22Z","file":[{"content_type":"application/pdf","file_size":1033743,"date_updated":"2020-07-14T12:44:39Z","creator":"system","access_level":"open_access","file_name":"IST-2016-722-v1+1_s00220-016-2805-6.pdf","checksum":"ddff79154c3daf27237de5383b1264a9","file_id":"5102","date_created":"2018-12-12T10:14:47Z","relation":"main_file"}],"ddc":["530"],"language":[{"iso":"eng"}],"publist_id":"6141","issue":"3","day":"01","publisher":"Springer","month":"02","pubrep_id":"722","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"type":"journal_article","year":"2017","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"}},{"intvolume":"        22","volume":22,"publication_identifier":{"issn":["1083-6489"]},"scopus_import":"1","_id":"1023","author":[{"id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","full_name":"Nemish, Yuriy","last_name":"Nemish","orcid":"0000-0002-7327-856X","first_name":"Yuriy"}],"date_created":"2018-12-11T11:49:44Z","doi":"10.1214/17-EJP38","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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.","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.","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>.","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>.","short":"Y. Nemish, Electronic Journal of Probability 22 (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>"},"date_published":"2017-02-06T00:00:00Z","oa":1,"abstract":[{"lang":"eng","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."}],"title":"Local law for the product of independent non-Hermitian random matrices with independent entries","oa_version":"Published Version","publication_status":"published","file_date_updated":"2018-12-12T10:15:29Z","quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"year":"2017","pubrep_id":"802","type":"journal_article","month":"02","publisher":"Institute of Mathematical Statistics","publist_id":"6370","day":"06","language":[{"iso":"eng"}],"date_updated":"2025-07-10T11:49:47Z","ddc":["510"],"file":[{"relation":"main_file","access_level":"open_access","creator":"system","date_created":"2018-12-12T10:15:29Z","file_id":"5149","file_name":"IST-2017-802-v1+1_euclid.ejp.1487991681.pdf","content_type":"application/pdf","file_size":742275,"date_updated":"2018-12-12T10:15:29Z"}],"status":"public","article_number":"22","external_id":{"isi":["000396611900022"]},"department":[{"_id":"LaEr"}],"has_accepted_license":"1","isi":1,"publication":"Electronic Journal of Probability","article_processing_charge":"No"},{"title":"Universality for a class of random band matrices","oa_version":"Submitted Version","publication_status":"published","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"}],"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.02312"}],"page":"739 - 800","publication_identifier":{"issn":["1095-0761"]},"volume":21,"intvolume":"        21","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"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, 2017. <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.","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, 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>.","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>","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. <a href=\"https://doi.org/10.4310/ATMP.2017.v21.n3.a5\">https://doi.org/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.","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, pp. 739–800, 2017."},"arxiv":1,"oa":1,"date_published":"2017-08-25T00:00:00Z","scopus_import":"1","_id":"483","author":[{"last_name":"Bourgade","first_name":"Paul","full_name":"Bourgade, Paul"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"full_name":"Yau, Horng","last_name":"Yau","first_name":"Horng"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"date_created":"2018-12-11T11:46:43Z","doi":"10.4310/ATMP.2017.v21.n3.a5","date_updated":"2025-09-18T09:52:57Z","language":[{"iso":"eng"}],"department":[{"_id":"LaEr"}],"isi":1,"publication":"Advances in Theoretical and Mathematical Physics","article_processing_charge":"No","status":"public","external_id":{"isi":["000409382300005"],"arxiv":["1602.02312"]},"year":"2017","type":"journal_article","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"ec_funded":1,"day":"25","issue":"3","publist_id":"7337","month":"08","publisher":"International Press"},{"department":[{"_id":"LaEr"}],"article_processing_charge":"No","status":"public","date_updated":"2025-04-15T08:05:02Z","series_title":"Courant Lecture Notes","language":[{"iso":"eng"}],"publist_id":"7247","day":"01","corr_author":"1","month":"01","publisher":"American Mathematical Society","year":"2017","type":"book","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"ec_funded":1,"quality_controlled":"1","page":"226","title":"A Dynamical Approach to Random Matrix Theory","oa_version":"None","publication_status":"published","abstract":[{"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","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2017-01-01T00:00:00Z","citation":{"ista":"Erdös L, Yau H. 2017. A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 226p.","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.","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>.","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>.","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>","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>"},"_id":"567","date_created":"2018-12-11T11:47:13Z","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"}],"doi":"10.1090/cln/028","publication_identifier":{"isbn":["9-781-4704-3648-3"],"eisbn":["978-1-4704-4194-4"]},"volume":28,"alternative_title":["Courant Lecture Notes"],"intvolume":"        28"}]