[{"abstract":[{"lang":"eng","text":"We consider the Brown measure of the free circular Brownian motion, a+t√x , with an arbitrary initial condition a , i.e. a is a general non-normal operator and x is a circular element ∗ -free from a . We prove that, under a mild assumption on a , the density of the Brown measure has one of the following two types of behavior around each point on the boundary of its support -- either (i) sharp cut, i.e. a jump discontinuity along the boundary, or (ii) quadratic decay at certain critical points on the boundary. Our result is in direct analogy with the previously known phenomenon for the spectral density of free semicircular Brownian motion, whose singularities are either a square-root edge or a cubic cusp. We also provide several examples and counterexamples, one of which shows that our assumption on a is necessary."}],"oa_version":"Published Version","title":"Density of Brown measure of free circular Brownian motion","publication_status":"published","DOAJ_listed":"1","page":"417-453","quality_controlled":"1","file_date_updated":"2025-04-07T11:21:13Z","acknowledgement":"We thank Ping Zhong for pointing out references [15,19] and providing helpful comments. We also thank the anonymous referee for many valuable comments and proposals to streamline the presentation. This work was partially supported by ERC Advanced Grant “RMTBeyond” No. 10102033.","intvolume":" 30","article_type":"original","volume":30,"publication_identifier":{"eissn":["1431-0643"],"issn":["1431-0635"]},"scopus_import":"1","OA_place":"publisher","_id":"19500","date_created":"2025-04-06T22:01:32Z","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"},{"first_name":"Hong Chang","last_name":"Ji","full_name":"Ji, Hong Chang"}],"doi":"10.4171/DM/999","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","arxiv":1,"citation":{"ieee":"L. Erdös and H. C. Ji, “Density of Brown measure of free circular Brownian motion,” Documenta Mathematica, vol. 30, no. 2. EMS Press, pp. 417–453, 2025.","ista":"Erdös L, Ji HC. 2025. Density of Brown measure of free circular Brownian motion. Documenta Mathematica. 30(2), 417–453.","apa":"Erdös, L., & Ji, H. C. (2025). Density of Brown measure of free circular Brownian motion. Documenta Mathematica. EMS Press. https://doi.org/10.4171/DM/999","ama":"Erdös L, Ji HC. Density of Brown measure of free circular Brownian motion. Documenta Mathematica. 2025;30(2):417-453. doi:10.4171/DM/999","chicago":"Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular Brownian Motion.” Documenta Mathematica. EMS Press, 2025. https://doi.org/10.4171/DM/999.","mla":"Erdös, László, and Hong Chang Ji. “Density of Brown Measure of Free Circular Brownian Motion.” Documenta Mathematica, vol. 30, no. 2, EMS Press, 2025, pp. 417–53, doi:10.4171/DM/999.","short":"L. Erdös, H.C. Ji, Documenta Mathematica 30 (2025) 417–453."},"date_published":"2025-03-20T00:00:00Z","oa":1,"language":[{"iso":"eng"}],"ddc":["510"],"file":[{"file_size":1366865,"date_updated":"2025-04-07T11:21:13Z","content_type":"application/pdf","success":1,"relation":"main_file","checksum":"97a02d18c05f2b9f2048747b140e7d43","file_id":"19523","file_name":"2025_DocumentaMathematica_Erdoes.pdf","date_created":"2025-04-07T11:21:13Z","creator":"dernst","access_level":"open_access"}],"date_updated":"2025-09-30T11:28:02Z","status":"public","external_id":{"isi":["001450119900005"],"arxiv":["2307.08626"]},"has_accepted_license":"1","department":[{"_id":"LaEr"}],"isi":1,"publication":"Documenta Mathematica","article_processing_charge":"Yes","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"},"ec_funded":1,"year":"2025","project":[{"grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"type":"journal_article","month":"03","corr_author":"1","publisher":"EMS Press","license":"https://creativecommons.org/licenses/by/4.0/","OA_type":"gold","issue":"2","day":"20"},{"scopus_import":"1","OA_place":"publisher","_id":"19737","doi":"10.1007/s00440-025-01384-7","date_created":"2025-05-25T22:16:59Z","author":[{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio","orcid":"0000-0002-4901-7992","last_name":"Cipolloni","first_name":"Giorgio"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"full_name":"Ji, Hong Chang","last_name":"Ji","first_name":"Hong Chang"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ieee":"G. Cipolloni, L. Erdös, and H. C. Ji, “Non–Hermitian spectral universality at critical points,” Probability Theory and Related Fields. Springer Nature, 2025.","ista":"Cipolloni G, Erdös L, Ji HC. 2025. Non–Hermitian spectral universality at critical points. Probability Theory and Related Fields., 050603.","apa":"Cipolloni, G., Erdös, L., & Ji, H. C. (2025). Non–Hermitian spectral universality at critical points. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-025-01384-7","mla":"Cipolloni, Giorgio, et al. “Non–Hermitian Spectral Universality at Critical Points.” Probability Theory and Related Fields, 050603, Springer Nature, 2025, doi:10.1007/s00440-025-01384-7.","short":"G. Cipolloni, L. Erdös, H.C. Ji, Probability Theory and Related Fields (2025).","chicago":"Cipolloni, Giorgio, László Erdös, and Hong Chang Ji. “Non–Hermitian Spectral Universality at Critical Points.” Probability Theory and Related Fields. Springer Nature, 2025. https://doi.org/10.1007/s00440-025-01384-7.","ama":"Cipolloni G, Erdös L, Ji HC. Non–Hermitian spectral universality at critical points. Probability Theory and Related Fields. 2025. doi:10.1007/s00440-025-01384-7"},"date_published":"2025-01-01T00:00:00Z","oa":1,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). Supported by ERC Advanced Grant “RMTBeyond” No. 101020331.","article_type":"original","publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s00440-025-01384-7"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"For general large non–Hermitian random matrices X and deterministic normal deformations A, we prove that the local eigenvalue statistics of A + X close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non–Hermitian random matrices (for normal deformations), after bulk and sharp edge universalities have been established in recent years."}],"oa_version":"Published Version","title":"Non–Hermitian spectral universality at critical points","publication_status":"epub_ahead","month":"01","corr_author":"1","publisher":"Springer Nature","OA_type":"hybrid","day":"01","ec_funded":1,"year":"2025","type":"journal_article","project":[{"grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"status":"public","article_number":"050603","external_id":{"isi":["001493091900001"]},"isi":1,"department":[{"_id":"LaEr"}],"publication":"Probability Theory and Related Fields","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"date_updated":"2025-09-30T12:41:58Z"},{"author":[{"first_name":"Volodymyr","last_name":"Riabov","full_name":"Riabov, Volodymyr","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b"}],"doi":"10.1214/23-AIHP1438","date_created":"2024-03-20T09:41:04Z","_id":"15128","scopus_import":"1","OA_place":"repository","oa":1,"date_published":"2025-02-01T00:00:00Z","arxiv":1,"citation":{"ista":"Riabov V. 2025. Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 61(1), 129–154.","ieee":"V. Riabov, “Mesoscopic eigenvalue statistics for Wigner-type matrices,” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 61, no. 1. Institute of Mathematical Statistics, pp. 129–154, 2025.","ama":"Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 2025;61(1):129-154. doi:10.1214/23-AIHP1438","chicago":"Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/23-AIHP1438.","mla":"Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 61, no. 1, Institute of Mathematical Statistics, 2025, pp. 129–54, doi:10.1214/23-AIHP1438.","short":"V. Riabov, Annales de l’institut Henri Poincare (B) Probability and Statistics 61 (2025) 129–154.","apa":"Riabov, V. (2025). Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/23-AIHP1438"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":" 61","acknowledgement":"I would like to express my gratitude to László Erdős for suggesting the project and supervising my work. I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions. Furthermore, I am grateful to Guillaume Dubach for translating the abstract into French.\r\nThe author was supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.","publication_identifier":{"issn":["0246-0203"]},"volume":61,"article_type":"original","page":"129-154","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2301.01712","open_access":"1"}],"quality_controlled":"1","abstract":[{"text":"We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel ingredient is an optimal local law for the two-point function $T(z,\\zeta)$ and a general class of related quantities involving two resolvents at nearby spectral parameters.","lang":"eng"},{"lang":"fre","text":"On établit un théorème limite central universel pour les statistiques linéaires mésoscopiques des valeurs propres d’une matrice de type Wigner au milieu du spectre, avec des fonctions de classe \r\n et à support compact. La principale nouveauté de cette approche est qu’elle repose sur une loi locale optimale pour la fonction à deux points $T(z,\\zeta)$ , ainsi que pour une classe plus générale d’observables impliquant deux résolvantes évaluées en des paramètres proches."}],"publication_status":"published","oa_version":"Preprint","title":"Mesoscopic eigenvalue statistics for Wigner-type matrices","publisher":"Institute of Mathematical Statistics","corr_author":"1","month":"02","day":"01","issue":"1","OA_type":"green","ec_funded":1,"type":"journal_article","project":[{"grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"year":"2025","external_id":{"isi":["001427953600004"],"arxiv":["2301.01712"]},"status":"public","article_processing_charge":"No","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","isi":1,"department":[{"_id":"GradSch"},{"_id":"LaEr"}],"language":[{"iso":"eng"}],"date_updated":"2025-05-19T13:54:31Z"},{"article_type":"original","publication_identifier":{"issn":["0377-9017"],"eissn":["1573-0530"]},"volume":116,"acknowledgement":"L.E. and J.H. are supported by the ERC Advanced Grant “RMTBeyond” No. 101020331. Moreover, J.H. acknowledges (partial) financial support by the ERC Consolidator Grant “ProbQuant” (jointly with the Swiss State Secretariat for Education, Research and Innovation). C.V. was (partially) supported by the German Academic Scholarship Foundation and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 352 – Project-ID 470903074. Moreover, C.V. acknowledges (partial) financial support by the ERC Starting Grant “FermiMath\" No. 101040991 and the ERC Consolidator Grant “RAMBAS” No. 10104424, funded by the European Union. Open access funding provided by Institute of Science and Technology (IST Austria).","intvolume":" 116","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ista":"Erdös L, Henheik SJ, Vogel C. 2025. Normal typicality and dynamical typicality for a random block-band matrix model. Letters in Mathematical Physics. 116, 5.","ieee":"L. Erdös, S. J. Henheik, and C. Vogel, “Normal typicality and dynamical typicality for a random block-band matrix model,” Letters in Mathematical Physics, vol. 116. Springer Nature, 2025.","mla":"Erdös, László, et al. “Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model.” Letters in Mathematical Physics, vol. 116, 5, Springer Nature, 2025, doi:10.1007/s11005-025-02037-5.","chicago":"Erdös, László, Sven Joscha Henheik, and Cornelia Vogel. “Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model.” Letters in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s11005-025-02037-5.","short":"L. Erdös, S.J. Henheik, C. Vogel, Letters in Mathematical Physics 116 (2025).","ama":"Erdös L, Henheik SJ, Vogel C. Normal typicality and dynamical typicality for a random block-band matrix model. Letters in Mathematical Physics. 2025;116. doi:10.1007/s11005-025-02037-5","apa":"Erdös, L., Henheik, S. J., & Vogel, C. (2025). Normal typicality and dynamical typicality for a random block-band matrix model. Letters in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s11005-025-02037-5"},"oa":1,"date_published":"2025-12-26T00:00:00Z","OA_place":"publisher","scopus_import":"1","_id":"20925","date_created":"2026-01-04T23:01:33Z","doi":"10.1007/s11005-025-02037-5","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","last_name":"Henheik","orcid":"0000-0003-1106-327X"},{"full_name":"Vogel, Cornelia","id":"1cd0554a-ea28-11f0-9f40-ff76440883cd","first_name":"Cornelia","last_name":"Vogel"}],"oa_version":"Published Version","title":"Normal typicality and dynamical typicality for a random block-band matrix model","publication_status":"epub_ahead","abstract":[{"text":"We prove normal typicality and dynamical typicality for a (centered) random block-band matrix model with block-dependent variances. A key feature of our model is that we achieve intermediate equilibration times, an aspect that has not been proven rigorously in any model before. Our proof builds on recently established concentration estimates for products of resolvents of Wigner type random matrices (Erdős and Riabov in Commun Math Phys 405(12): 282, 2024) and an intricate analysis of the deterministic approximation.","lang":"eng"}],"pmid":1,"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s11005-025-02037-5"}],"year":"2025","type":"journal_article","project":[{"call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"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"},"ec_funded":1,"OA_type":"hybrid","day":"26","month":"12","corr_author":"1","publisher":"Springer Nature","ddc":["510"],"date_updated":"2026-01-05T11:22:25Z","language":[{"iso":"eng"}],"has_accepted_license":"1","department":[{"_id":"LaEr"}],"publication":"Letters in Mathematical Physics","article_processing_charge":"Yes (via OA deal)","status":"public","article_number":"5","external_id":{"pmid":["41459414"]},"PlanS_conform":"1"},{"language":[{"iso":"eng"}],"date_updated":"2026-02-18T08:35:38Z","external_id":{"arxiv":["2404.17512"]},"status":"public","article_processing_charge":"No","department":[{"_id":"LaEr"}],"publication":"The Annals of Probability","ec_funded":1,"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"}],"type":"journal_article","year":"2025","publisher":"Institute of Mathematical Statistics","month":"11","corr_author":"1","OA_type":"green","issue":"6","day":"01","abstract":[{"lang":"eng","text":"For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic."}],"publication_status":"published","oa_version":"Preprint","title":"On the spectral edge of non-Hermitian random matrices","page":"2256-2308","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2404.17512"}],"quality_controlled":"1","acknowledgement":"The authors would like to thank the anonymous referee for providing helpful comments and suggestions. We also thank Joscha Henheik and Volodymyr Riabov for pointing out a gap in an earlier version of the proof of equation (3.18). The first, third, and fourth authors are supported by ERC Advanced Grant “RMTBeyond” No. 101020331.","intvolume":" 53","article_type":"original","volume":53,"publication_identifier":{"eissn":["2168-894X"],"issn":["0091-1798"]},"author":[{"first_name":"Andrew J","last_name":"Campbell","full_name":"Campbell, Andrew J","id":"582b06a9-1f1c-11ee-b076-82ffce00dde4"},{"last_name":"Cipolloni","orcid":"0000-0002-4901-7992","first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Ji","first_name":"Hong Chang","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d","full_name":"Ji, Hong Chang"}],"date_created":"2026-02-17T07:58:20Z","doi":"10.1214/25-aop1761","OA_place":"repository","_id":"21271","arxiv":1,"citation":{"ista":"Campbell AJ, Cipolloni G, Erdös L, Ji HC. 2025. On the spectral edge of non-Hermitian random matrices. The Annals of Probability. 53(6), 2256–2308.","ieee":"A. J. Campbell, G. Cipolloni, L. Erdös, and H. C. Ji, “On the spectral edge of non-Hermitian random matrices,” The Annals of Probability, vol. 53, no. 6. Institute of Mathematical Statistics, pp. 2256–2308, 2025.","mla":"Campbell, Andrew J., et al. “On the Spectral Edge of Non-Hermitian Random Matrices.” The Annals of Probability, vol. 53, no. 6, Institute of Mathematical Statistics, 2025, pp. 2256–308, doi:10.1214/25-aop1761.","short":"A.J. Campbell, G. Cipolloni, L. Erdös, H.C. Ji, The Annals of Probability 53 (2025) 2256–2308.","chicago":"Campbell, Andrew J, Giorgio Cipolloni, László Erdös, and Hong Chang Ji. “On the Spectral Edge of Non-Hermitian Random Matrices.” The Annals of Probability. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/25-aop1761.","ama":"Campbell AJ, Cipolloni G, Erdös L, Ji HC. On the spectral edge of non-Hermitian random matrices. The Annals of Probability. 2025;53(6):2256-2308. doi:10.1214/25-aop1761","apa":"Campbell, A. J., Cipolloni, G., Erdös, L., & Ji, H. C. (2025). On the spectral edge of non-Hermitian random matrices. The Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/25-aop1761"},"date_published":"2025-11-01T00:00:00Z","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"19547"},{"id":"20575","status":"public","relation":"dissertation_contains"}]},"date_updated":"2026-04-07T12:32:19Z","ddc":["510"],"file":[{"file_name":"2025_CommMathPhysics_Erdoes.pdf","date_created":"2025-09-10T07:48:21Z","checksum":"abd32af7b8ca6dc5b9080823a433986b","file_id":"20336","access_level":"open_access","creator":"dernst","relation":"main_file","content_type":"application/pdf","file_size":1465827,"date_updated":"2025-09-10T07:48:21Z","success":1}],"status":"public","article_number":"253","external_id":{"arxiv":["2410.06813"],"isi":["001565019000005"]},"PlanS_conform":"1","isi":1,"has_accepted_license":"1","department":[{"_id":"LaEr"}],"publication":"Communications in Mathematical Physics","article_processing_charge":"Yes (via OA deal)","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":"2025","type":"journal_article","month":"09","corr_author":"1","publisher":"Springer Nature","OA_type":"hybrid","issue":"10","day":"01","abstract":[{"lang":"eng","text":"For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner–Dyson–Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality (Alt et al. in Ann Probab 48(2):963–1001, 2020), while cusp universality was proven only for Wigner-type matrices with independent entries (Cipolloni et al. in Pure Appl Anal 1:615–707, 2019; Erdős et al. in Commun. Math. Phys. 378:1203–1278, 2018). As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus we also provide a significantly simplified alternative proof of the local eigenvalue universality in the previously studied bulk (Erdős et al. in Forum Math. Sigma 7:E8, 2019) and edge (Alt et al. in Ann Probab 48(2):963–1001, 2020) regimes."}],"oa_version":"Published Version","title":"Cusp universality for correlated random matrices","publication_status":"published","file_date_updated":"2025-09-10T07:48:21Z","quality_controlled":"1","acknowledgement":"We thank Giorgio Cipolloni for many productive discussions and the anonymous referees for several useful suggestions and spotting some typos. Open access funding provided by Institute of Science and Technology (IST Austria).","intvolume":" 406","article_type":"original","publication_identifier":{"eissn":["1432-0916"],"issn":["0010-3616"]},"volume":406,"OA_place":"publisher","scopus_import":"1","_id":"20322","doi":"10.1007/s00220-025-05417-z","date_created":"2025-09-10T05:38:17Z","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","full_name":"Henheik, Sven Joscha","last_name":"Henheik","orcid":"0000-0003-1106-327X","first_name":"Sven Joscha"},{"full_name":"Riabov, Volodymyr","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","first_name":"Volodymyr","last_name":"Riabov"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ama":"Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 2025;406(10). doi:10.1007/s00220-025-05417-z","short":"L. Erdös, S.J. Henheik, V. Riabov, Communications in Mathematical Physics 406 (2025).","mla":"Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics, vol. 406, no. 10, 253, Springer Nature, 2025, doi:10.1007/s00220-025-05417-z.","chicago":"Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s00220-025-05417-z.","apa":"Erdös, L., Henheik, S. J., & Riabov, V. (2025). Cusp universality for correlated random matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-025-05417-z","ista":"Erdös L, Henheik SJ, Riabov V. 2025. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 406(10), 253.","ieee":"L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated random matrices,” Communications in Mathematical Physics, vol. 406, no. 10. Springer Nature, 2025."},"arxiv":1,"date_published":"2025-09-01T00:00:00Z","oa":1},{"ec_funded":1,"acknowledgement":" Supported by the ERC\r\nAdvanced Grant ”RMTBeyond” No. 101020331.","year":"2025","type":"preprint","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331"}],"month":"06","OA_place":"repository","_id":"20576","corr_author":"1","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"},{"id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","full_name":"Riabov, Volodymyr","last_name":"Riabov","first_name":"Volodymyr"}],"doi":"10.48550/ARXIV.2506.06441","date_created":"2025-10-29T19:09:03Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","day":"06","citation":{"ieee":"L. Erdös and V. Riabov, “The zigzag strategy for random band matrices,” arXiv. .","ista":"Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv, 10.48550/ARXIV.2506.06441.","apa":"Erdös, L., & Riabov, V. (n.d.). The zigzag strategy for random band matrices. arXiv. https://doi.org/10.48550/ARXIV.2506.06441","ama":"Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv. doi:10.48550/ARXIV.2506.06441","chicago":"Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2506.06441.","short":"L. Erdös, V. Riabov, ArXiv (n.d.).","mla":"Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” ArXiv, doi:10.48550/ARXIV.2506.06441."},"date_published":"2025-06-06T00:00:00Z","oa":1,"abstract":[{"lang":"eng","text":"We prove that a very general class of $N\\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\\gg \\sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the $\\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically."}],"language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"dissertation_contains","id":"20575","status":"public"}]},"oa_version":"Preprint","title":"The zigzag strategy for random band matrices","date_updated":"2026-04-07T12:32:19Z","publication_status":"draft","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2506.06441","open_access":"1"}],"status":"public","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"publication":"arXiv","article_processing_charge":"No"},{"oa":1,"date_published":"2025-11-03T00:00:00Z","citation":{"ieee":"V. Riabov, “Universality in random matrices with spatial structure,” Institute of Science and Technology Austria, 2025.","ista":"Riabov V. 2025. Universality in random matrices with spatial structure. Institute of Science and Technology Austria.","apa":"Riabov, V. (2025). Universality in random matrices with spatial structure. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-20575","short":"V. Riabov, Universality in Random Matrices with Spatial Structure, Institute of Science and Technology Austria, 2025.","mla":"Riabov, Volodymyr. Universality in Random Matrices with Spatial Structure. Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-20575.","chicago":"Riabov, Volodymyr. “Universality in Random Matrices with Spatial Structure.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-20575.","ama":"Riabov V. Universality in random matrices with spatial structure. 2025. doi:10.15479/AT-ISTA-20575"},"supervisor":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"}],"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2025-10-29T19:12:24Z","author":[{"id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","full_name":"Riabov, Volodymyr","last_name":"Riabov","first_name":"Volodymyr"}],"doi":"10.15479/AT-ISTA-20575","_id":"20575","OA_place":"publisher","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-064-0"]},"alternative_title":["ISTA Thesis"],"acknowledgement":"The work comprising this thesis was supported by the ERC Advanced Grant \"RMTBeyond\"\r\nNo.101020331 awarded to my advisor.","file_date_updated":"2025-10-29T18:54:53Z","page":"436","publication_status":"published","title":"Universality in random matrices with spatial structure","oa_version":"Published Version","abstract":[{"lang":"eng","text":"This thesis deals with eigenvalue and eigenvector universality results for random matrix ensembles equipped with non-trivial spatial structure. We consider both mean-field models with a general variance profile (Wigner-type matrices) and correlation structure (correlated matrices) among the entries, as well as non-mean-field random band matrices with bandwidth W >> N^(1/2).\r\n\r\nTo extract the universal properties of random matrix spectra and eigenvectors, we obtain concentration estimates for their resolvent, the local laws, which generalize the celebrated Wigner semicircle law for a broad class of random matrices to much finer spectral scales. The local laws hold for both a single resolvent as well as for products of multiple resolvents, known as resolvent chains, and express the remarkable approximately-deterministic behavior of these objects down to the microscopic scale.\r\n\r\nOur primary tool for establishing the local laws is the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous random matrices. Our proof method systematically addresses the challenges arising from non-trivial spatial structures and is robust to all types of singularities in the spectrum, as we demonstrate in the correlated setting. Furthermore, we incorporate the analysis of the deterministic resolvent chain approximations into the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit for establishing multi-resolvent local laws.\r\n\r\nUsing these methods, we prove complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis, and Wigner-Dyson universality in the bulk for random band matrices down to the optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality of local eigenvalue statistics at the cups for random matrices with correlated entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices in the bulk of the spectrum.\r\n\r\nFinally, this thesis also contains other applications of the multi-resolvent local laws to spatially-inhomogeneous random matrices, obtained prior to the development of the Zigzag strategy. In particular, we provide a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type matrices in all spectral regimes, including the novel cusps, and rigorously establish the prethermalization phenomenon for deformed Wigner matrices.\r\n\r\nThe main body of this thesis consists of seven research papers (listed on page xi), each presented in a separate chapter with its own introduction and all relevant context, suitable to be read independently. We ask the reader’s indulgence for the repetitions in the historical overviews and other minor redundancies that remain among the chapters as a result. The overall Introduction, preceding the chapters, provides a condensed, informal summary of the main ideas and concepts at the core of these works.\r\n"}],"degree_awarded":"PhD","day":"3","publisher":"Institute of Science and Technology Austria","corr_author":"1","month":"11","project":[{"grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"type":"dissertation","year":"2025","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":"No","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"has_accepted_license":"1","status":"public","file":[{"date_updated":"2025-10-29T18:53:59Z","file_size":7536583,"content_type":"application/pdf","success":1,"date_created":"2025-10-29T18:53:59Z","file_id":"20577","checksum":"6a0487b2b66bb35d44b394756d44b8b4","file_name":"riabov_thesis-pdfa.pdf","access_level":"open_access","creator":"vriabov","relation":"main_file"},{"date_updated":"2025-10-29T18:54:53Z","file_size":17841612,"content_type":"application/x-zip-compressed","relation":"source_file","date_created":"2025-10-29T18:54:53Z","checksum":"224efda6bf9864d296a1e5e0124c1e8f","file_name":"manuscript.zip","file_id":"20578","creator":"vriabov","access_level":"closed"}],"ddc":["515","519"],"date_updated":"2026-04-07T12:32:20Z","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"20322"},{"relation":"part_of_dissertation","id":"18764","status":"public"},{"relation":"part_of_dissertation","id":"13317","status":"public"},{"relation":"part_of_dissertation","id":"19368","status":"deleted"},{"relation":"part_of_dissertation","id":"18554","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"20576"},{"relation":"part_of_dissertation","id":"17174","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"19547"},{"relation":"part_of_dissertation","id":"19598","status":"public"}]},"language":[{"iso":"eng"}]},{"volume":193,"publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"article_type":"original","intvolume":" 193","acknowledgement":"I would like to express my gratitude to László Erdős for his careful guidance and supervision of my work. I am also thankful to Jana Reker and Joscha Henheik for many helpful discussions. Open access funding provided by Institute of Science and Technology (IST Austria).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_published":"2025-12-01T00:00:00Z","arxiv":1,"citation":{"apa":"Riabov, V. (2025). Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-025-01373-w","short":"V. Riabov, Probability Theory and Related Fields 193 (2025) 1183–1237.","mla":"Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields, vol. 193, Springer Nature, 2025, pp. 1183–237, doi:10.1007/s00440-025-01373-w.","chicago":"Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields. Springer Nature, 2025. https://doi.org/10.1007/s00440-025-01373-w.","ama":"Riabov V. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 2025;193:1183-1237. doi:10.1007/s00440-025-01373-w","ieee":"V. Riabov, “Linear Eigenvalue statistics at the cusp,” Probability Theory and Related Fields, vol. 193. Springer Nature, pp. 1183–1237, 2025.","ista":"Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 193, 1183–1237."},"_id":"19598","scopus_import":"1","OA_place":"publisher","author":[{"last_name":"Riabov","first_name":"Volodymyr","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","full_name":"Riabov, Volodymyr"}],"doi":"10.1007/s00440-025-01373-w","date_created":"2025-04-20T22:01:28Z","title":"Linear Eigenvalue statistics at the cusp","oa_version":"Published Version","publication_status":"published","abstract":[{"lang":"eng","text":"We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear eigenvalue statistics at the cusp-like singularities were not studied in any ensemble. Our analysis covers not only the exact cusps but the entire transitionary regime from the square-root singularity at a regular edge through the sharp cusp to the bulk. We identify a new one-parameter family of functionals that govern the limiting bias and variance, continuously interpolating between the previously known formulas in the bulk and at a regular edge. Since cusps are the only possible singularities besides the regular edges, our result gives a complete description of the linear eigenvalue statistics in all regimes."}],"file_date_updated":"2025-12-30T13:10:05Z","quality_controlled":"1","page":"1183-1237","year":"2025","type":"journal_article","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"},"day":"01","OA_type":"hybrid","corr_author":"1","month":"12","publisher":"Springer Nature","date_updated":"2026-04-07T12:32:19Z","file":[{"relation":"main_file","checksum":"700229b280725c0d6aad0d71362cce5f","date_created":"2025-12-30T13:10:05Z","file_id":"20916","file_name":"2025_ProbTheoryRelatFields_Riabov.pdf","access_level":"open_access","creator":"dernst","content_type":"application/pdf","file_size":919213,"date_updated":"2025-12-30T13:10:05Z","success":1}],"ddc":["510"],"related_material":{"record":[{"id":"20575","status":"public","relation":"dissertation_contains"}]},"language":[{"iso":"eng"}],"publication":"Probability Theory and Related Fields","isi":1,"has_accepted_license":"1","department":[{"_id":"LaEr"}],"article_processing_charge":"Yes (via OA deal)","status":"public","PlanS_conform":"1","external_id":{"isi":["001466997300001"],"arxiv":["2307.07432"]}},{"ddc":["510"],"date_updated":"2026-04-07T12:37:10Z","file":[{"relation":"main_file","file_name":"2025_ErgodicTheory_Henheik.pdf","file_id":"18828","date_created":"2025-01-13T08:51:40Z","checksum":"650fe115d998fe0ac3a8d0c7519447c8","creator":"dernst","access_level":"open_access","content_type":"application/pdf","date_updated":"2025-01-13T08:51:40Z","file_size":659100,"success":1}],"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"19540"}]},"language":[{"iso":"eng"}],"article_processing_charge":"Yes (via OA deal)","publication":"Ergodic Theory and Dynamical Systems","department":[{"_id":"LaEr"}],"has_accepted_license":"1","isi":1,"external_id":{"isi":["001308182000001"]},"status":"public","project":[{"grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d"},{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","grant_number":"885707","call_identifier":"H2020","name":"Spectral rigidity and integrability for billiards and geodesic flows"}],"type":"journal_article","year":"2025","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"},"issue":"2","day":"01","OA_type":"hybrid","publisher":"Cambridge University Press","corr_author":"1","month":"02","publication_status":"published","oa_version":"Published Version","title":"Deformational rigidity of integrable metrics on the torus","abstract":[{"text":"It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.","lang":"eng"}],"quality_controlled":"1","file_date_updated":"2025-01-13T08:51:40Z","page":"467-503","publication_identifier":{"eissn":["1469-4417"],"issn":["0143-3857"]},"volume":45,"article_type":"original","intvolume":" 45","acknowledgement":"I am very grateful to Vadim Kaloshin for suggesting the topic, his guidance during this project, and many helpful comments on an earlier version of the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr Riabov for interesting discussions. Partial financial support by the ERC Advanced Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received funding from the European Research Council (ERC) ERC Grant No. 885707.","oa":1,"date_published":"2025-02-01T00:00:00Z","citation":{"apa":"Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. Cambridge University Press. https://doi.org/10.1017/etds.2024.48","chicago":"Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” Ergodic Theory and Dynamical Systems. Cambridge University Press, 2025. https://doi.org/10.1017/etds.2024.48.","mla":"Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” Ergodic Theory and Dynamical Systems, vol. 45, no. 2, Cambridge University Press, 2025, pp. 467–503, doi:10.1017/etds.2024.48.","short":"S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503.","ama":"Henheik SJ. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 2025;45(2):467-503. doi:10.1017/etds.2024.48","ieee":"S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,” Ergodic Theory and Dynamical Systems, vol. 45, no. 2. Cambridge University Press, pp. 467–503, 2025.","ista":"Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 45(2), 467–503."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-09-22T22:01:43Z","author":[{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik"}],"doi":"10.1017/etds.2024.48","_id":"18112","OA_place":"publisher","scopus_import":"1"},{"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"},"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"}],"type":"journal_article","year":"2025","publisher":"Springer Nature","corr_author":"1","month":"01","day":"30","OA_type":"hybrid","related_material":{"record":[{"status":"public","id":"19540","relation":"dissertation_contains"}]},"language":[{"iso":"eng"}],"ddc":["510"],"date_updated":"2026-04-07T12:37:10Z","file":[{"content_type":"application/pdf","date_updated":"2025-02-05T07:01:40Z","file_size":828335,"success":1,"relation":"main_file","checksum":"ee07edf5f85a6f2651926b2f8760af74","date_created":"2025-02-05T07:01:40Z","file_name":"2025_LettersMathPhysics_Erdoes.pdf","file_id":"19004","access_level":"open_access","creator":"dernst"}],"external_id":{"pmid":["39896265"],"isi":["001409618800002"],"arxiv":["2410.08108"]},"article_number":"14","status":"public","article_processing_charge":"Yes (via OA deal)","publication":"Letters in Mathematical Physics","department":[{"_id":"LaEr"}],"has_accepted_license":"1","isi":1,"intvolume":" 115","acknowledgement":"We thank Giorgio Cipolloni for helpful discussions in a closely related joint project. Open access funding provided by Institute of Science and Technology (IST Austria). All authors were supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.","volume":115,"publication_identifier":{"issn":["1573-0530"]},"article_type":"original","date_created":"2025-02-05T06:48:29Z","doi":"10.1007/s11005-025-01904-5","author":[{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","last_name":"Henheik","orcid":"0000-0003-1106-327X"},{"orcid":"0000-0003-1491-4623","last_name":"Kolupaiev","first_name":"Oleksii","id":"149b70d4-896a-11ed-bdf8-8c63fd44ca61","full_name":"Kolupaiev, Oleksii"}],"_id":"19001","scopus_import":"1","OA_place":"publisher","oa":1,"date_published":"2025-01-30T00:00:00Z","citation":{"ista":"Erdös L, Henheik SJ, Kolupaiev O. 2025. Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. 115, 14.","ieee":"L. Erdös, S. J. Henheik, and O. Kolupaiev, “Loschmidt echo for deformed Wigner matrices,” Letters in Mathematical Physics, vol. 115. Springer Nature, 2025.","mla":"Erdös, László, et al. “Loschmidt Echo for Deformed Wigner Matrices.” Letters in Mathematical Physics, vol. 115, 14, Springer Nature, 2025, doi:10.1007/s11005-025-01904-5.","short":"L. Erdös, S.J. Henheik, O. Kolupaiev, Letters in Mathematical Physics 115 (2025).","chicago":"Erdös, László, Sven Joscha Henheik, and Oleksii Kolupaiev. “Loschmidt Echo for Deformed Wigner Matrices.” Letters in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s11005-025-01904-5.","ama":"Erdös L, Henheik SJ, Kolupaiev O. Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. 2025;115. doi:10.1007/s11005-025-01904-5","apa":"Erdös, L., Henheik, S. J., & Kolupaiev, O. (2025). Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s11005-025-01904-5"},"arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"We consider two Hamiltonians that are close to each other, H1≈H2, and analyze the time-decay of the corresponding Loschmidt echo M(t):=|⟨ψ0,eitH2e−itH1ψ0⟩|2 that expresses the effect of an imperfect time reversal on the initial state ψ0. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such H1 and H2.","lang":"eng"}],"pmid":1,"publication_status":"published","title":"Loschmidt echo for deformed Wigner matrices","oa_version":"Published Version","quality_controlled":"1","file_date_updated":"2025-02-05T07:01:40Z"},{"OA_place":"publisher","scopus_import":"1","_id":"18764","author":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","last_name":"Henheik","orcid":"0000-0003-1106-327X"},{"full_name":"Reker, Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","first_name":"Jana","last_name":"Reker"},{"full_name":"Riabov, Volodymyr","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","first_name":"Volodymyr","last_name":"Riabov"}],"doi":"10.1007/s00023-024-01518-y","date_created":"2025-01-05T23:01:59Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","arxiv":1,"citation":{"ista":"Erdös L, Henheik SJ, Reker J, Riabov V. 2025. Prethermalization for deformed Wigner matrices. Annales Henri Poincare. 26, 1991–2033.","ieee":"L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed Wigner matrices,” Annales Henri Poincare, vol. 26. Springer Nature, pp. 1991–2033, 2025.","mla":"Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” Annales Henri Poincare, vol. 26, Springer Nature, 2025, pp. 1991–2033, doi:10.1007/s00023-024-01518-y.","short":"L. Erdös, S.J. Henheik, J. Reker, V. Riabov, Annales Henri Poincare 26 (2025) 1991–2033.","chicago":"Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization for Deformed Wigner Matrices.” Annales Henri Poincare. Springer Nature, 2025. https://doi.org/10.1007/s00023-024-01518-y.","ama":"Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner matrices. Annales Henri Poincare. 2025;26:1991-2033. doi:10.1007/s00023-024-01518-y","apa":"Erdös, L., Henheik, S. J., Reker, J., & Riabov, V. (2025). Prethermalization for deformed Wigner matrices. Annales Henri Poincare. Springer Nature. https://doi.org/10.1007/s00023-024-01518-y"},"date_published":"2025-06-01T00:00:00Z","oa":1,"acknowledgement":"All authors were supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nJ.R. was additionally supported by the ERC Advanced Grant “LDRaM” No. 884584.\r\nWe thank Peter Reimann and Lennart Dabelow for helpful comments. Open access funding provided by Institute of Science and Technology (IST Austria).","intvolume":" 26","article_type":"original","publication_identifier":{"issn":["1424-0637"]},"volume":26,"page":"1991-2033","file_date_updated":"2025-06-25T05:38:34Z","quality_controlled":"1","abstract":[{"lang":"eng","text":"We prove that a class of weakly perturbed Hamiltonians of the form H_λ= H_0 + λW, with W being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by H_λ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order λ^{-2}. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix H_λ."}],"oa_version":"Published Version","title":"Prethermalization for deformed Wigner matrices","publication_status":"published","month":"06","corr_author":"1","publisher":"Springer Nature","OA_type":"hybrid","day":"01","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"},"ec_funded":1,"year":"2025","project":[{"call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"type":"journal_article","status":"public","external_id":{"isi":["001385326500001"],"arxiv":["2310.06677"]},"isi":1,"has_accepted_license":"1","department":[{"_id":"LaEr"}],"publication":"Annales Henri Poincare","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17174"},{"id":"20575","status":"public","relation":"dissertation_contains"},{"relation":"dissertation_contains","status":"public","id":"19540"}]},"date_updated":"2026-04-07T12:37:11Z","ddc":["510"],"file":[{"content_type":"application/pdf","file_size":977773,"date_updated":"2025-06-25T05:38:34Z","success":1,"file_id":"19895","file_name":"2025_AnnalesHenriPoincare_Erdoes.pdf","checksum":"49e6a934db540206f7eaa0c798553ded","date_created":"2025-06-25T05:38:34Z","access_level":"open_access","creator":"dernst","relation":"main_file"}]},{"title":"How a space-time singularity helps remove the ultraviolet divergence problem","oa_version":"Preprint","publication_status":"draft","abstract":[{"lang":"eng","text":"Particle creation terms in quantum Hamiltonians are usually ultraviolet\r\ndivergent and thus mathematically ill defined. A rather novel way of solving\r\nthis problem is based on imposing so-called interior-boundary conditions on the\r\nwave function. Previous papers showed that this approach works in the\r\nnon-relativistic regime, but particle creation is mostly relevant in the\r\nrelativistic case after all. In flat relativistic space-time (that is,\r\nneglecting gravity), the approach was previously found to work only for certain\r\nsomewhat artificial cases. Here, as a way of taking gravity into account, we\r\nconsider curved space-time, specifically the super-critical\r\nReissner-Nordstr\\\"om space-time, which features a naked timelike singularity.\r\nWe find that the interior-boundary approach works fully in this setting; in\r\nparticular, we prove rigorously the existence of well-defined, self-adjoint\r\nHamiltonians with particle creation at the singularity, based on\r\ninterior-boundary conditions. We also non-rigorously analyze the asymptotic\r\nbehavior of the Bohmian trajectories and construct the corresponding Bohm-Bell\r\nprocess of particle creation, motion, and annihilation. The upshot is that in\r\nquantum physics, a naked space-time singularity need not lead to a breakdown of\r\nphysical laws, but on the contrary allows for boundary conditions governing\r\nwhat comes out of the singularity and thereby removing the ultraviolet\r\ndivergence."}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2409.00677"}],"acknowledgement":"JH gratefully acknowledges partial financial support by the ERC Advanced\r\nGrant “RMTBeyond” No. 101020331.","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","oa":1,"date_published":"2025-02-28T00:00:00Z","arxiv":1,"citation":{"apa":"Henheik, S. J., Poudyal, B., & Tumulka, R. (n.d.). How a space-time singularity helps remove the ultraviolet divergence problem. arXiv. https://doi.org/10.48550/arXiv.2409.00677","ama":"Henheik SJ, Poudyal B, Tumulka R. How a space-time singularity helps remove the ultraviolet divergence problem. arXiv. doi:10.48550/arXiv.2409.00677","mla":"Henheik, Sven Joscha, et al. “How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem.” ArXiv, doi:10.48550/arXiv.2409.00677.","chicago":"Henheik, Sven Joscha, Bipul Poudyal, and Roderich Tumulka. “How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2409.00677.","short":"S.J. Henheik, B. Poudyal, R. Tumulka, ArXiv (n.d.).","ieee":"S. J. Henheik, B. Poudyal, and R. Tumulka, “How a space-time singularity helps remove the ultraviolet divergence problem,” arXiv. .","ista":"Henheik SJ, Poudyal B, Tumulka R. How a space-time singularity helps remove the ultraviolet divergence problem. arXiv, 10.48550/arXiv.2409.00677."},"_id":"19552","OA_place":"repository","doi":"10.48550/arXiv.2409.00677","author":[{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik"},{"full_name":"Poudyal, Bipul","first_name":"Bipul","last_name":"Poudyal"},{"full_name":"Tumulka, Roderich","last_name":"Tumulka","first_name":"Roderich"}],"date_created":"2025-04-11T12:07:25Z","date_updated":"2026-04-07T12:37:11Z","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"19540"}]},"language":[{"iso":"eng"}],"publication":"arXiv","department":[{"_id":"LaEr"}],"article_processing_charge":"No","status":"public","external_id":{"arxiv":["2409.00677"]},"year":"2025","type":"preprint","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"}],"ec_funded":1,"day":"28","corr_author":"1","month":"02"},{"corr_author":"1","month":"01","day":"30","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"},"ec_funded":1,"year":"2025","type":"preprint","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"}],"status":"public","external_id":{"arxiv":["2410.10718"]},"publication":"arXiv","department":[{"_id":"LaEr"}],"article_processing_charge":"No","related_material":{"record":[{"id":"19540","status":"public","relation":"dissertation_contains"}]},"language":[{"iso":"eng"}],"date_updated":"2026-04-07T12:37:11Z","_id":"19546","OA_place":"repository","author":[{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio","orcid":"0000-0002-4901-7992","last_name":"Cipolloni","first_name":"Giorgio"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik","full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb"},{"full_name":"Kolupaiev, Oleksii","id":"149b70d4-896a-11ed-bdf8-8c63fd44ca61","first_name":"Oleksii","last_name":"Kolupaiev","orcid":"0000-0003-1491-4623"}],"doi":"10.48550/arXiv.2410.10718","date_created":"2025-04-11T08:34:49Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","oa":1,"date_published":"2025-01-30T00:00:00Z","citation":{"ama":"Cipolloni G, Erdös L, Henheik SJ, Kolupaiev O. Eigenvector decorrelation for random matrices. arXiv. doi:10.48550/arXiv.2410.10718","short":"G. Cipolloni, L. Erdös, S.J. Henheik, O. Kolupaiev, ArXiv (n.d.).","mla":"Cipolloni, Giorgio, et al. “Eigenvector Decorrelation for Random Matrices.” ArXiv, doi:10.48550/arXiv.2410.10718.","chicago":"Cipolloni, Giorgio, László Erdös, Sven Joscha Henheik, and Oleksii Kolupaiev. “Eigenvector Decorrelation for Random Matrices.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2410.10718.","apa":"Cipolloni, G., Erdös, L., Henheik, S. J., & Kolupaiev, O. (n.d.). Eigenvector decorrelation for random matrices. arXiv. https://doi.org/10.48550/arXiv.2410.10718","ista":"Cipolloni G, Erdös L, Henheik SJ, Kolupaiev O. Eigenvector decorrelation for random matrices. arXiv, 10.48550/arXiv.2410.10718.","ieee":"G. Cipolloni, L. Erdös, S. J. Henheik, and O. Kolupaiev, “Eigenvector decorrelation for random matrices,” arXiv. ."},"arxiv":1,"acknowledgement":"Supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2410.10718","open_access":"1"}],"abstract":[{"text":"We study the sensitivity of the eigenvectors of random matrices, showing that\r\neven small perturbations make the eigenvectors almost orthogonal. More\r\nprecisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show\r\nthat their bulk eigenvectors become asymptotically orthogonal as soon as\r\n$\\mathrm{Tr}(D_1-D_2)^2\\gg 1$, or their respective energies are separated on a\r\nscale much bigger than the local eigenvalue spacing. Furthermore, we show that\r\nquadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic\r\nmatrix $A\\in\\mathbf{C}^{N\\times N}$ in a specific subspace of codimension one\r\nare of size $N^{-1/2}$. This proves a generalization of the Eigenstate\r\nThermalization Hypothesis to eigenvectors belonging to two different spectral\r\nfamilies.","lang":"eng"}],"title":"Eigenvector decorrelation for random matrices","oa_version":"Preprint","publication_status":"draft"},{"page":"720","file_date_updated":"2025-04-23T14:11:05Z","abstract":[{"lang":"eng","text":"This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. \r\n\t\r\nIn Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices.\tIn order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. \r\n\t\r\nIn Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. \r\n\t\r\nIn Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. \r\n\t\r\nAdditionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs). "}],"publication_status":"published","title":"Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems","oa_version":"Published Version","date_created":"2025-04-10T21:21:18Z","doi":"10.15479/AT-ISTA-19540","author":[{"last_name":"Henheik","orcid":"0000-0003-1106-327X","first_name":"Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","full_name":"Henheik, Sven Joscha"}],"OA_place":"publisher","_id":"19540","citation":{"ieee":"S. J. Henheik, “Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems,” Institute of Science and Technology Austria, 2025.","ista":"Henheik SJ. 2025. Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. Institute of Science and Technology Austria.","apa":"Henheik, S. J. (2025). Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-19540","ama":"Henheik SJ. Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. 2025. doi:10.15479/AT-ISTA-19540","chicago":"Henheik, Sven Joscha. “Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-19540.","mla":"Henheik, Sven Joscha. Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems. Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-19540.","short":"S.J. Henheik, Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems, Institute of Science and Technology Austria, 2025."},"supervisor":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"}],"date_published":"2025-04-10T00:00:00Z","oa":1,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","alternative_title":["ISTA Thesis"],"publication_identifier":{"isbn":["978-3-99078-057-2"],"issn":["2663-337X"]},"status":"public","article_processing_charge":"No","has_accepted_license":"1","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"language":[{"iso":"eng"}],"related_material":{"record":[{"id":"14343","status":"public","relation":"part_of_dissertation"},{"id":"18656","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"13317"},{"relation":"part_of_dissertation","id":"11732","status":"public"},{"status":"public","id":"12184","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"14421","status":"public"},{"relation":"part_of_dissertation","id":"10623","status":"public"},{"relation":"part_of_dissertation","id":"18112","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"19001"},{"id":"10642","status":"public","relation":"part_of_dissertation"},{"status":"public","id":"19545","relation":"part_of_dissertation"},{"status":"public","id":"19546","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"19550"},{"relation":"part_of_dissertation","id":"19551","status":"public"},{"relation":"part_of_dissertation","id":"19552","status":"public"},{"relation":"part_of_dissertation","status":"public","id":"14542"},{"status":"public","id":"17049","relation":"part_of_dissertation"},{"id":"18764","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"19547"},{"relation":"part_of_dissertation","id":"19548","status":"public"}]},"file":[{"date_updated":"2025-04-10T21:14:18Z","file_size":4107587,"content_type":"application/zip","date_created":"2025-04-10T21:14:18Z","checksum":"b8477ae5578436c72c3bb4193ad34ac5","file_name":"Henheik-Thesis_source_final.zip","file_id":"19542","access_level":"closed","creator":"shenheik","relation":"source_file"},{"success":1,"content_type":"application/pdf","date_updated":"2025-04-11T13:16:05Z","file_size":9999492,"relation":"main_file","creator":"shenheik","access_level":"open_access","checksum":"e9fc0ea12ec46c9f71110c33217c4140","date_created":"2025-04-11T13:16:05Z","file_id":"19553","file_name":"Henheik-Thesis-pdfa_FINAL.pdf"},{"content_type":"application/pdf","date_updated":"2025-04-23T14:10:27Z","file_size":13276442,"relation":"other","file_id":"19615","date_created":"2025-04-23T14:10:27Z","checksum":"f94580f86c785e7108eb116cd189e225","file_name":"Henheik-Thesis-Volume1_print.pdf","creator":"cchlebak","access_level":"closed"},{"relation":"other","creator":"cchlebak","access_level":"closed","date_created":"2025-04-23T14:11:05Z","file_name":"Henheik-Thesis-Volume2_print.pdf","file_id":"19616","checksum":"b927ead3c78020ffb32918911deedb74","content_type":"application/pdf","date_updated":"2025-04-23T14:11:05Z","file_size":7628767}],"ddc":["519"],"date_updated":"2026-04-07T12:37:12Z","publisher":"Institute of Science and Technology Austria","month":"04","corr_author":"1","degree_awarded":"PhD","day":"10","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"},"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331"}],"type":"dissertation","year":"2025"},{"language":[{"iso":"eng"}],"related_material":{"record":[{"id":"19540","status":"public","relation":"dissertation_contains"}]},"file":[{"relation":"main_file","file_id":"19549","date_created":"2025-04-11T09:13:31Z","checksum":"f49e06e8dba819f7ad52a202e287ebca","file_name":"Henheik_JSpectralTheory_2025.pdf","creator":"cchlebak","access_level":"open_access","date_updated":"2025-04-11T09:13:31Z","file_size":779158,"content_type":"application/pdf","success":1}],"ddc":["500"],"date_updated":"2026-04-07T12:37:11Z","external_id":{"arxiv":["2312.11310"],"isi":["001438931600009"]},"status":"public","article_processing_charge":"No","has_accepted_license":"1","department":[{"_id":"LaEr"},{"_id":"RoSe"}],"isi":1,"publication":"Journal of Spectral Theory","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":[{"grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d"},{"name":"Mathematical Challenges in BCS Theory of Superconductivity","grant_number":"I06427","_id":"bda63fe5-d553-11ed-ba76-a16e3d2f256b"}],"year":"2025","publisher":"EMS Press","month":"01","corr_author":"1","OA_type":"gold","issue":"1","day":"09","abstract":[{"text":"We consider the BCS energy gap „.T / (essentially given by „.T / \u0019 .T; p\u0016/,\r\nthe BCS order parameter) at all temperatures 0 \u0014 T \u0014 Tc up to the critical one, Tc, and show\r\nthat, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola\r\n[Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these\r\nrestrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors\r\nwith non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of\r\ntemperatures close to Tc.\r\n\r\n\r\n .","lang":"eng"}],"DOAJ_listed":"1","publication_status":"published","title":"Universal behavior of the BCS energy gap","oa_version":"Published Version","page":"305–352","quality_controlled":"1","file_date_updated":"2025-04-11T09:13:31Z","acknowledgement":"We thank Andreas Deuchert, Christian Hainzl, Edwin Langmann, Marius Lemm, Robert Seiringer, and Jan Philip Solovej for helpful discussions,\r\nand Edwin Langmann and Robert Seiringer for valuable comments on an earlier version of the manuscript.\r\nFunding. Joscha Henheik gratefully acknowledges partial financial support by the\r\nERC Advanced Grant “RMTBeyond” No. 101020331. Asbjørn Bækgaard Lauritsen\r\ngratefully acknowledges partial financial support by the Austrian Science Fund (FWF)\r\nthrough grant DOI 10.55776/I6427 (as part of the SFB/TRR 352).\r\n","intvolume":" 15","article_type":"original","volume":15,"publication_identifier":{"eissn":["1664-0403"]},"date_created":"2025-04-11T09:19:28Z","author":[{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik"},{"id":"e1a2682f-dc8d-11ea-abe3-81da9ac728f1","full_name":"Lauritsen, Asbjørn Bækgaard","orcid":"0000-0003-4476-2288","last_name":"Lauritsen","first_name":"Asbjørn Bækgaard"}],"doi":"10.4171/JST/540","scopus_import":"1","OA_place":"publisher","_id":"19548","arxiv":1,"citation":{"ieee":"S. J. Henheik and A. B. Lauritsen, “Universal behavior of the BCS energy gap,” Journal of Spectral Theory, vol. 15, no. 1. EMS Press, pp. 305–352, 2025.","ista":"Henheik SJ, Lauritsen AB. 2025. Universal behavior of the BCS energy gap. Journal of Spectral Theory. 15(1), 305–352.","apa":"Henheik, S. J., & Lauritsen, A. B. (2025). Universal behavior of the BCS energy gap. Journal of Spectral Theory. EMS Press. https://doi.org/10.4171/JST/540","ama":"Henheik SJ, Lauritsen AB. Universal behavior of the BCS energy gap. Journal of Spectral Theory. 2025;15(1):305–352. doi:10.4171/JST/540","mla":"Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior of the BCS Energy Gap.” Journal of Spectral Theory, vol. 15, no. 1, EMS Press, 2025, pp. 305–352, doi:10.4171/JST/540.","short":"S.J. Henheik, A.B. Lauritsen, Journal of Spectral Theory 15 (2025) 305–352.","chicago":"Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior of the BCS Energy Gap.” Journal of Spectral Theory. EMS Press, 2025. https://doi.org/10.4171/JST/540."},"oa":1,"date_published":"2025-01-09T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"doi":"10.1007/s10959-023-01275-4","date_created":"2023-08-06T22:01:13Z","author":[{"full_name":"Campbell, Andrew J","id":"582b06a9-1f1c-11ee-b076-82ffce00dde4","first_name":"Andrew J","last_name":"Campbell"},{"first_name":"Sean","last_name":"O’Rourke","full_name":"O’Rourke, Sean"}],"_id":"13975","scopus_import":"1","oa":1,"date_published":"2024-03-01T00:00:00Z","citation":{"apa":"Campbell, A. J., & O’Rourke, S. (2024). Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. Springer Nature. https://doi.org/10.1007/s10959-023-01275-4","short":"A.J. Campbell, S. O’Rourke, Journal of Theoretical Probability 37 (2024) 933–973.","chicago":"Campbell, Andrew J, and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” Journal of Theoretical Probability. Springer Nature, 2024. https://doi.org/10.1007/s10959-023-01275-4.","mla":"Campbell, Andrew J., and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” Journal of Theoretical Probability, vol. 37, Springer Nature, 2024, pp. 933–73, doi:10.1007/s10959-023-01275-4.","ama":"Campbell AJ, O’Rourke S. Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. 2024;37:933-973. doi:10.1007/s10959-023-01275-4","ieee":"A. J. Campbell and S. O’Rourke, “Spectrum of Lévy–Khintchine random laplacian matrices,” Journal of Theoretical Probability, vol. 37. Springer Nature, pp. 933–973, 2024.","ista":"Campbell AJ, O’Rourke S. 2024. Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. 37, 933–973."},"arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":" 37","acknowledgement":"The first author thanks Yizhe Zhu for pointing out reference [30]. We thank David Renfrew for comments on an earlier draft. We thank the anonymous referee for a careful reading and helpful comments.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).","publication_identifier":{"eissn":["1572-9230"],"issn":["0894-9840"]},"volume":37,"article_type":"original","page":"933-973","file_date_updated":"2024-07-22T09:41:21Z","quality_controlled":"1","abstract":[{"text":"We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn where An\r\n is a real symmetric random matrix and Dn is a diagonal matrix whose entries are equal to the corresponding row sums of An. If An is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of Ln is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices An with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of Ln converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which Ln converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure.","lang":"eng"}],"publication_status":"published","title":"Spectrum of Lévy–Khintchine random laplacian matrices","oa_version":"Published Version","publisher":"Springer Nature","corr_author":"1","month":"03","day":"01","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","year":"2024","external_id":{"isi":["001038341000001"],"arxiv":["2210.07927"]},"status":"public","article_processing_charge":"Yes (via OA deal)","publication":"Journal of Theoretical Probability","isi":1,"department":[{"_id":"LaEr"}],"has_accepted_license":"1","language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","file_size":555070,"date_updated":"2024-07-22T09:41:21Z","success":1,"relation":"main_file","file_name":"2024_JourTheorProbab_Campbell.pdf","checksum":"f7793d313104c70422140c5e6494c779","file_id":"17300","date_created":"2024-07-22T09:41:21Z","access_level":"open_access","creator":"dernst"}],"ddc":["510"],"date_updated":"2024-07-22T09:41:42Z"},{"day":"01","publisher":"Springer Nature","month":"04","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020"}],"type":"journal_article","year":"2024","ec_funded":1,"article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Probability Theory and Related Fields","external_id":{"arxiv":["2210.12060"],"isi":["001118972500001"]},"status":"public","date_updated":"2025-08-05T13:28:15Z","language":[{"iso":"eng"}],"citation":{"chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields. Springer Nature, 2024. https://doi.org/10.1007/s00440-023-01229-1.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields 188 (2024) 1131–1182.","mla":"Cipolloni, Giorgio, et al. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields, vol. 188, Springer Nature, 2024, pp. 1131–82, doi:10.1007/s00440-023-01229-1.","ama":"Cipolloni G, Erdös L, Schröder DJ. Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. 2024;188:1131-1182. doi:10.1007/s00440-023-01229-1","apa":"Cipolloni, G., Erdös, L., & Schröder, D. J. (2024). Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-023-01229-1","ista":"Cipolloni G, Erdös L, Schröder DJ. 2024. Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. 188, 1131–1182.","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Mesoscopic central limit theorem for non-Hermitian random matrices,” Probability Theory and Related Fields, vol. 188. Springer Nature, pp. 1131–1182, 2024."},"arxiv":1,"oa":1,"date_published":"2024-04-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio","last_name":"Cipolloni","orcid":"0000-0002-4901-7992","first_name":"Giorgio"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J","orcid":"0000-0002-2904-1856","last_name":"Schröder","first_name":"Dominik J"}],"date_created":"2023-10-08T22:01:17Z","doi":"10.1007/s00440-023-01229-1","scopus_import":"1","_id":"14408","article_type":"original","publication_identifier":{"eissn":["1432-2064"],"issn":["0178-8051"]},"volume":188,"acknowledgement":"The authors are grateful to Joscha Henheik for his help with the formulas in Appendix B.\r\nLászló Erdős supported by ERC Advanced Grant “RMTBeyond” No. 101020331. Dominik Schröder supported by the SNSF Ambizione Grant PZ00P2 209089.","intvolume":" 188","quality_controlled":"1","page":"1131-1182","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2210.12060"}],"publication_status":"published","title":"Mesoscopic central limit theorem for non-Hermitian random matrices","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We prove that the mesoscopic linear statistics ∑if(na(σi−z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H20-functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/24-EJP1247","short":"J. Reker, Electronic Journal of Probability 29 (2024).","chicago":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2024. https://doi.org/10.1214/24-EJP1247.","mla":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” Electronic Journal of Probability, vol. 29, 191, Institute of Mathematical Statistics, 2024, doi:10.1214/24-EJP1247.","ama":"Reker J. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 2024;29. doi:10.1214/24-EJP1247","ieee":"J. Reker, “Multi-point functional central limit theorem for Wigner matrices,” Electronic Journal of Probability, vol. 29. Institute of Mathematical Statistics, 2024.","ista":"Reker J. 2024. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 29, 191."},"oa":1,"date_published":"2024-12-20T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1214/24-EJP1247","date_created":"2025-01-05T23:01:58Z","author":[{"id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","full_name":"Reker, Jana","last_name":"Reker","first_name":"Jana"}],"OA_place":"publisher","scopus_import":"1","_id":"18762","article_type":"original","volume":29,"publication_identifier":{"eissn":["1083-6489"]},"acknowledgement":"I am very grateful to László Erdős for suggesting the topic and many valuable discussions during my work on the project. I would also like to thank the two anonymous referees for their careful reading of the manuscript and detailed feedback.\r\nPartially supported by ERC Advanced Grants “RMTBeyond” No. 101020331 and “LDRaM” No. 884584.","intvolume":" 29","file_date_updated":"2025-01-08T08:44:03Z","quality_controlled":"1","DOAJ_listed":"1","publication_status":"published","title":"Multi-point functional central limit theorem for Wigner matrices","oa_version":"Published Version","abstract":[{"text":"Consider the random variable $\\mathrm{Tr}( f_1(W)A_1\\dots f_k(W)A_k)$ where $W$ is an $N\\times N$ Hermitian Wigner matrix, $k\\in\\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\\dots, f_k$ as well as bounded deterministic matrices $A_1,\\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\\dots,f_k$ and the number of traceless matrices among $A_1,\\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\\geq2$. As an application, we consider the fluctuation of $\\mathrm{Tr}(\\mathrm{e}^{\\mathrm{i} tW}A_1\\mathrm{e}^{-\\mathrm{i} tW}A_2)$ around its thermal value $\\mathrm{Tr}(A_1)\\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.","lang":"eng"}],"OA_type":"gold","day":"20","publisher":"Institute of Mathematical Statistics","month":"12","corr_author":"1","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"type":"journal_article","year":"2024","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","has_accepted_license":"1","department":[{"_id":"LaEr"}],"isi":1,"publication":"Electronic Journal of Probability","article_number":"191","external_id":{"arxiv":["2307.11028"],"isi":["001381599200001"]},"status":"public","ddc":["510"],"file":[{"date_created":"2025-01-08T08:44:03Z","file_name":"2024_ElectrJournProbability_Reker.pdf","file_id":"18773","checksum":"67178feaa8630a332599d3037a3fe70e","access_level":"open_access","creator":"dernst","relation":"main_file","content_type":"application/pdf","file_size":812428,"date_updated":"2025-01-08T08:44:03Z","success":1}],"date_updated":"2025-09-09T11:59:15Z","language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"earlier_version","id":"17173","status":"public"}]}},{"month":"02","corr_author":"1","publisher":"Institute of Mathematical Statistics","day":"01","issue":"1B","ec_funded":1,"year":"2024","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","grant_number":"101020331"}],"type":"journal_article","status":"public","external_id":{"arxiv":["2208.12206"],"isi":["001163006100021"]},"isi":1,"department":[{"_id":"LaEr"}],"publication":"Annals of Applied Probability","article_processing_charge":"No","language":[{"iso":"eng"}],"date_updated":"2025-09-04T12:08:11Z","scopus_import":"1","_id":"15025","doi":"10.1214/23-AAP2000","author":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0003-2625-495X","last_name":"McKenna","first_name":"Benjamin","id":"b0cc634c-d549-11ee-96c8-87338c7ad808","full_name":"McKenna, Benjamin"}],"date_created":"2024-02-25T23:00:56Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ama":"Erdös L, McKenna B. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. 2024;34(1B):1623-1662. doi:10.1214/23-AAP2000","short":"L. Erdös, B. McKenna, Annals of Applied Probability 34 (2024) 1623–1662.","mla":"Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” Annals of Applied Probability, vol. 34, no. 1B, Institute of Mathematical Statistics, 2024, pp. 1623–62, doi:10.1214/23-AAP2000.","chicago":"Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” Annals of Applied Probability. Institute of Mathematical Statistics, 2024. https://doi.org/10.1214/23-AAP2000.","apa":"Erdös, L., & McKenna, B. (2024). Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/23-AAP2000","ista":"Erdös L, McKenna B. 2024. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. 34(1B), 1623–1662.","ieee":"L. Erdös and B. McKenna, “Extremal statistics of quadratic forms of GOE/GUE eigenvectors,” Annals of Applied Probability, vol. 34, no. 1B. Institute of Mathematical Statistics, pp. 1623–1662, 2024."},"arxiv":1,"oa":1,"date_published":"2024-02-01T00:00:00Z","acknowledgement":"The first author was supported by the ERC Advanced Grant “RMTBeyond” No. 101020331. The second author was supported by Fulbright Austria and the Austrian Marshall Plan Foundation.","intvolume":" 34","article_type":"original","publication_identifier":{"issn":["1050-5164"]},"volume":34,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2208.12206","open_access":"1"}],"page":"1623-1662","quality_controlled":"1","abstract":[{"text":"We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than √N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble.","lang":"eng"}],"title":"Extremal statistics of quadratic forms of GOE/GUE eigenvectors","oa_version":"Preprint","publication_status":"published"}]