[{"language":[{"iso":"eng"}],"arxiv":1,"external_id":{"arxiv":["2208.12220"],"isi":["000905776200001"]},"date_published":"2022-12-01T00:00:00Z","fulldoi":"https://doi.org/10.1063/5.0123441","ddc":["510"],"has_accepted_license":"1","article_processing_charge":"No","type":"journal_article","day":"01","author":[{"full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","last_name":"Henheik"},{"last_name":"Wessel","first_name":"Tom","full_name":"Wessel, Tom"}],"publication_identifier":{"issn":["0022-2488"]},"scopus_import":"1","article_number":"121101","corr_author":"1","oa":1,"date_updated":"2026-07-29T13:18:16Z","file":[{"success":1,"date_updated":"2023-01-27T07:10:52Z","file_id":"12410","creator":"dernst","file_name":"2022_JourMathPhysics_Henheik2.pdf","file_size":5251092,"checksum":"213b93750080460718c050e4967cfdb4","relation":"main_file","access_level":"open_access","date_created":"2023-01-27T07:10:52Z","content_type":"application/pdf"}],"isi":1,"related_material":{"record":[{"status":"public","id":"19540","relation":"dissertation_contains"}]},"intvolume":"        63","file_date_updated":"2023-01-27T07:10:52Z","oa_version":"Published Version","publication":"Journal of Mathematical Physics","month":"12","_id":"12184","ec_funded":1,"article_type":"original","department":[{"_id":"LaEr"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2023-01-15T23:00:52Z","title":"On adiabatic theory for extended fermionic lattice systems","abstract":[{"lang":"eng","text":"We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs."}],"acknowledgement":"It is a pleasure to thank Stefan Teufel for numerous interesting discussions, fruitful collaboration, and many helpful comments on an earlier version of the manuscript. J.H. acknowledges partial financial support from the ERC Advanced Grant No. 101020331 “Random\r\nmatrices beyond Wigner-Dyson-Mehta.” T.W. acknowledges financial support from the DFG research unit FOR 5413 “Long-range interacting quantum spin systems out of equilibrium: Experiment, Theory and Mathematics.\" ","publisher":"AIP Publishing","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331"}],"status":"public","issue":"12","quality_controlled":"1","doi":"10.1063/5.0123441","volume":63,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2022","citation":{"chicago":"Henheik, Sven Joscha, and Tom Wessel. “On Adiabatic Theory for Extended Fermionic Lattice Systems.” <i>Journal of Mathematical Physics</i>. AIP Publishing, 2022. <a href=\"https://doi.org/10.1063/5.0123441\">https://doi.org/10.1063/5.0123441</a>.","apa":"Henheik, S. J., &#38; Wessel, T. (2022). On adiabatic theory for extended fermionic lattice systems. <i>Journal of Mathematical Physics</i>. AIP Publishing. <a href=\"https://doi.org/10.1063/5.0123441\">https://doi.org/10.1063/5.0123441</a>","short":"S.J. Henheik, T. Wessel, Journal of Mathematical Physics 63 (2022).","ieee":"S. J. Henheik and T. Wessel, “On adiabatic theory for extended fermionic lattice systems,” <i>Journal of Mathematical Physics</i>, vol. 63, no. 12. AIP Publishing, 2022.","ista":"Henheik SJ, Wessel T. 2022. On adiabatic theory for extended fermionic lattice systems. Journal of Mathematical Physics. 63(12), 121101.","ama":"Henheik SJ, Wessel T. On adiabatic theory for extended fermionic lattice systems. <i>Journal of Mathematical Physics</i>. 2022;63(12). doi:<a href=\"https://doi.org/10.1063/5.0123441\">10.1063/5.0123441</a>","mla":"Henheik, Sven Joscha, and Tom Wessel. “On Adiabatic Theory for Extended Fermionic Lattice Systems.” <i>Journal of Mathematical Physics</i>, vol. 63, no. 12, 121101, AIP Publishing, 2022, doi:<a href=\"https://doi.org/10.1063/5.0123441\">10.1063/5.0123441</a>."},"publication_status":"published"},{"doi":"10.1142/s2010326322500368","publication_status":"published","citation":{"ista":"Reker J. 2022. On the operator norm of a Hermitian random matrix with correlated entries. Random Matrices: Theory and Applications. 11(4), 2250036.","ama":"Reker J. On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. 2022;11(4). doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>","mla":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4, 2250036, World Scientific Publishing, 2022, doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>.","chicago":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing, 2022. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>.","apa":"Reker, J. (2022). On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>","short":"J. Reker, Random Matrices: Theory and Applications 11 (2022).","ieee":"J. Reker, “On the operator norm of a Hermitian random matrix with correlated entries,” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4. World Scientific Publishing, 2022."},"year":"2022","volume":11,"title":"On the operator norm of a Hermitian random matrix with correlated entries","abstract":[{"lang":"eng","text":"We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one."}],"quality_controlled":"1","issue":"4","status":"public","publisher":"World Scientific Publishing","_id":"11135","month":"10","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2022-04-08T07:11:12Z","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"article_type":"original","related_material":{"record":[{"status":"public","id":"17164","relation":"dissertation_contains"}]},"date_updated":"2026-10-02T11:31:29Z","isi":1,"oa":1,"oa_version":"Preprint","publication":"Random Matrices: Theory and Applications","intvolume":"        11","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2103.03906","open_access":"1"}],"publication_identifier":{"eissn":["2010-3271"],"issn":["2010-3263"]},"scopus_import":"1","author":[{"last_name":"Reker","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","first_name":"Jana","full_name":"Reker, Jana"}],"corr_author":"1","article_number":"2250036","type":"journal_article","article_processing_charge":"No","day":"01","fulldoi":"https://doi.org/10.1142/s2010326322500368","keyword":["Discrete Mathematics and Combinatorics","Statistics","Probability and Uncertainty","Statistics and Probability","Algebra and Number Theory"],"language":[{"iso":"eng"}],"date_published":"2022-10-01T00:00:00Z","external_id":{"isi":["000848873800001"],"arxiv":["2103.03906"]},"arxiv":1},{"language":[{"iso":"eng"}],"keyword":["Mathematical Physics","Statistical and Nonlinear Physics"],"date_published":"2022-07-29T00:00:00Z","external_id":{"isi":["000833007200002"]},"fulldoi":"https://doi.org/10.1007/s10955-022-02965-9","ddc":["530"],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","type":"journal_article","day":"29","author":[{"orcid":"0000-0003-1106-327X","last_name":"Henheik","full_name":"Henheik, Sven Joscha","first_name":"Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb"},{"full_name":"Lauritsen, Asbjørn Bækgaard","id":"e1a2682f-dc8d-11ea-abe3-81da9ac728f1","first_name":"Asbjørn Bækgaard","orcid":"0000-0003-4476-2288","last_name":"Lauritsen"}],"scopus_import":"1","publication_identifier":{"issn":["0022-4715"],"eissn":["1572-9613"]},"article_number":"5","corr_author":"1","oa":1,"isi":1,"date_updated":"2026-10-02T11:43:00Z","file":[{"checksum":"b398c4dbf65f71d417981d6e366427e9","file_size":419563,"relation":"main_file","access_level":"open_access","date_created":"2022-08-08T07:36:34Z","content_type":"application/pdf","success":1,"date_updated":"2022-08-08T07:36:34Z","file_id":"11746","creator":"dernst","file_name":"2022_JourStatisticalPhysics_Henheik.pdf"}],"related_material":{"record":[{"id":"19540","status":"public","relation":"dissertation_contains"},{"relation":"dissertation_contains","status":"public","id":"18135"}]},"intvolume":"       189","file_date_updated":"2022-08-08T07:36:34Z","publication":"Journal of Statistical Physics","oa_version":"Published Version","month":"07","ec_funded":1,"_id":"11732","article_type":"original","department":[{"_id":"GradSch"},{"_id":"LaEr"},{"_id":"RoSe"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_created":"2022-08-05T11:36:56Z","title":"The BCS energy gap at high density","abstract":[{"lang":"eng","text":"We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature."}],"acknowledgement":"We are grateful to Robert Seiringer for helpful discussions and many valuable comments\r\non an earlier version of the manuscript. J.H. acknowledges partial financial support by the ERC Advanced Grant “RMTBeyond’ No. 101020331. Open access funding provided by Institute of Science and Technology (IST Austria)","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020"}],"publisher":"Springer Nature","status":"public","quality_controlled":"1","doi":"10.1007/s10955-022-02965-9","year":"2022","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":189,"citation":{"chicago":"Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “The BCS Energy Gap at High Density.” <i>Journal of Statistical Physics</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s10955-022-02965-9\">https://doi.org/10.1007/s10955-022-02965-9</a>.","apa":"Henheik, S. J., &#38; Lauritsen, A. B. (2022). The BCS energy gap at high density. <i>Journal of Statistical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10955-022-02965-9\">https://doi.org/10.1007/s10955-022-02965-9</a>","short":"S.J. Henheik, A.B. Lauritsen, Journal of Statistical Physics 189 (2022).","ieee":"S. J. Henheik and A. B. Lauritsen, “The BCS energy gap at high density,” <i>Journal of Statistical Physics</i>, vol. 189. Springer Nature, 2022.","ista":"Henheik SJ, Lauritsen AB. 2022. The BCS energy gap at high density. Journal of Statistical Physics. 189, 5.","ama":"Henheik SJ, Lauritsen AB. The BCS energy gap at high density. <i>Journal of Statistical Physics</i>. 2022;189. doi:<a href=\"https://doi.org/10.1007/s10955-022-02965-9\">10.1007/s10955-022-02965-9</a>","mla":"Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “The BCS Energy Gap at High Density.” <i>Journal of Statistical Physics</i>, vol. 189, 5, Springer Nature, 2022, doi:<a href=\"https://doi.org/10.1007/s10955-022-02965-9\">10.1007/s10955-022-02965-9</a>."},"publication_status":"published"},{"oa":1,"date_updated":"2025-04-15T08:05:02Z","intvolume":"         2","oa_version":"Preprint","publication":"Probability and Mathematical Physics","month":"05","_id":"15013","ec_funded":1,"article_type":"original","department":[{"_id":"LaEr"}],"date_created":"2024-02-18T23:01:03Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Spectral radius of random matrices with independent entries","abstract":[{"lang":"eng","text":"We consider random n×n matrices X with independent and centered entries and a general variance profile. We show that the spectral radius of X converges with very high probability to the square root of the spectral radius of the variance matrix of X when n tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the spectral edge."}],"acknowledgement":"Partially supported by ERC Starting Grant RandMat No. 715539 and the SwissMap grant of Swiss National Science Foundation. Partially supported by ERC Advanced Grant RanMat No. 338804. Partially supported by the Hausdorff Center for Mathematics in Bonn.","project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"publisher":"Mathematical Sciences Publishers","issue":"2","status":"public","quality_controlled":"1","doi":"10.2140/pmp.2021.2.221","volume":2,"year":"2021","citation":{"short":"J. Alt, L. Erdös, T.H. Krüger, Probability and Mathematical Physics 2 (2021) 221–280.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Spectral radius of random matrices with independent entries,” <i>Probability and Mathematical Physics</i>, vol. 2, no. 2. Mathematical Sciences Publishers, pp. 221–280, 2021.","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Spectral Radius of Random Matrices with Independent Entries.” <i>Probability and Mathematical Physics</i>. Mathematical Sciences Publishers, 2021. <a href=\"https://doi.org/10.2140/pmp.2021.2.221\">https://doi.org/10.2140/pmp.2021.2.221</a>.","apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2021). Spectral radius of random matrices with independent entries. <i>Probability and Mathematical Physics</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/pmp.2021.2.221\">https://doi.org/10.2140/pmp.2021.2.221</a>","ama":"Alt J, Erdös L, Krüger TH. Spectral radius of random matrices with independent entries. <i>Probability and Mathematical Physics</i>. 2021;2(2):221-280. doi:<a href=\"https://doi.org/10.2140/pmp.2021.2.221\">10.2140/pmp.2021.2.221</a>","mla":"Alt, Johannes, et al. “Spectral Radius of Random Matrices with Independent Entries.” <i>Probability and Mathematical Physics</i>, vol. 2, no. 2, Mathematical Sciences Publishers, 2021, pp. 221–80, doi:<a href=\"https://doi.org/10.2140/pmp.2021.2.221\">10.2140/pmp.2021.2.221</a>.","ista":"Alt J, Erdös L, Krüger TH. 2021. Spectral radius of random matrices with independent entries. Probability and Mathematical Physics. 2(2), 221–280."},"publication_status":"published","language":[{"iso":"eng"}],"arxiv":1,"external_id":{"arxiv":["1907.13631"]},"date_published":"2021-05-21T00:00:00Z","fulldoi":"https://doi.org/10.2140/pmp.2021.2.221","article_processing_charge":"No","type":"journal_article","day":"21","author":[{"full_name":"Alt, Johannes","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","last_name":"Alt"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H"}],"scopus_import":"1","publication_identifier":{"issn":["2690-0998"],"eissn":["2690-1005"]},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1907.13631","open_access":"1"}],"corr_author":"1","page":"221-280"},{"oa":1,"date_updated":"2025-09-10T10:13:20Z","isi":1,"publication":"The Annals of Probability","oa_version":"Preprint","intvolume":"        49","ec_funded":1,"_id":"15259","month":"07","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2024-04-03T07:19:42Z","article_type":"original","department":[{"_id":"LaEr"}],"acknowledgement":"The authors would like to thank Gernot Akemann, Benson Au, Paul Bourgade, Jesper Ipsen, Camille Male, Jamie Mingo, Doron Puder, Emily Redelmeier, Roland Speicher, Wojciech Tarnowski and Ofer Zeitouni for useful discussions, comments and references as well as the anonymous referee for a suggestion that greatly improved one of the theorems.\r\nG.D. gratefully acknowledges support from the grants NSF DMS-1812114 of P. Bourgade (PI) and NSF CAREER DMS-1653602 of L.-P. Arguin (PI), as well as the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.","title":"On words of non-Hermitian random matrices","abstract":[{"lang":"eng","text":"We consider words Gi1⋯Gim involving i.i.d. complex Ginibre matrices and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length m is a Fuss–Catalan distribution with parameter \r\nm+1. This generalizes previous results concerning powers of a complex Ginibre matrix and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices.\r\nThese results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically."}],"quality_controlled":"1","project":[{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"}],"publisher":"Institute of Mathematical Statistics","status":"public","issue":"4","doi":"10.1214/20-aop1496","publication_status":"published","citation":{"mla":"Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>, vol. 49, no. 4, Institute of Mathematical Statistics, 2021, pp. 1886–916, doi:<a href=\"https://doi.org/10.1214/20-aop1496\">10.1214/20-aop1496</a>.","ama":"Dubach G, Peled Y. On words of non-Hermitian random matrices. <i>The Annals of Probability</i>. 2021;49(4):1886-1916. doi:<a href=\"https://doi.org/10.1214/20-aop1496\">10.1214/20-aop1496</a>","ista":"Dubach G, Peled Y. 2021. On words of non-Hermitian random matrices. The Annals of Probability. 49(4), 1886–1916.","ieee":"G. Dubach and Y. Peled, “On words of non-Hermitian random matrices,” <i>The Annals of Probability</i>, vol. 49, no. 4. Institute of Mathematical Statistics, pp. 1886–1916, 2021.","short":"G. Dubach, Y. Peled, The Annals of Probability 49 (2021) 1886–1916.","apa":"Dubach, G., &#38; Peled, Y. (2021). On words of non-Hermitian random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/20-aop1496\">https://doi.org/10.1214/20-aop1496</a>","chicago":"Dubach, Guillaume, and Yuval Peled. “On Words of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2021. <a href=\"https://doi.org/10.1214/20-aop1496\">https://doi.org/10.1214/20-aop1496</a>."},"volume":49,"year":"2021","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"language":[{"iso":"eng"}],"date_published":"2021-07-01T00:00:00Z","external_id":{"isi":["000681349000008"],"arxiv":["1904.04312"]},"arxiv":1,"fulldoi":"https://doi.org/10.1214/20-aop1496","type":"journal_article","article_processing_charge":"No","day":"01","publication_identifier":{"issn":["0091-1798"]},"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1904.04312"}],"author":[{"full_name":"Dubach, Guillaume","first_name":"Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","orcid":"0000-0001-6892-8137","last_name":"Dubach"},{"last_name":"Peled","full_name":"Peled, Yuval","first_name":"Yuval"}],"corr_author":"1","page":"1886-1916"},{"fulldoi":"https://doi.org/10.1007/s00220-021-04239-z","ddc":["510"],"has_accepted_license":"1","arxiv":1,"external_id":{"isi":["000712232700001"],"arxiv":["2012.13215"]},"date_published":"2021-10-29T00:00:00Z","language":[{"iso":"eng"}],"page":"1005–1048","corr_author":"1","author":[{"orcid":"0000-0002-4901-7992","last_name":"Cipolloni","full_name":"Cipolloni, Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","first_name":"Giorgio"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","first_name":"Dominik J"}],"scopus_import":"1","publication_identifier":{"issn":["0010-3616"],"eissn":["1432-0916"]},"day":"29","article_processing_charge":"Yes (via OA deal)","type":"journal_article","article_type":"original","department":[{"_id":"LaEr"}],"date_created":"2021-11-07T23:01:25Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","month":"10","_id":"10221","intvolume":"       388","file_date_updated":"2022-02-02T10:19:55Z","oa_version":"Published Version","publication":"Communications in Mathematical Physics","oa":1,"date_updated":"2025-04-15T06:53:08Z","file":[{"relation":"main_file","checksum":"a2c7b6f5d23b5453cd70d1261272283b","file_size":841426,"date_created":"2022-02-02T10:19:55Z","content_type":"application/pdf","access_level":"open_access","success":1,"date_updated":"2022-02-02T10:19:55Z","file_id":"10715","file_name":"2021_CommunMathPhys_Cipolloni.pdf","creator":"cchlebak"}],"isi":1,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":388,"year":"2021","citation":{"apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2021). Eigenstate thermalization hypothesis for Wigner matrices. <i>Communications in Mathematical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00220-021-04239-z\">https://doi.org/10.1007/s00220-021-04239-z</a>","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Eigenstate Thermalization Hypothesis for Wigner Matrices.” <i>Communications in Mathematical Physics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00220-021-04239-z\">https://doi.org/10.1007/s00220-021-04239-z</a>.","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Eigenstate thermalization hypothesis for Wigner matrices,” <i>Communications in Mathematical Physics</i>, vol. 388, no. 2. Springer Nature, pp. 1005–1048, 2021.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Communications in Mathematical Physics 388 (2021) 1005–1048.","ista":"Cipolloni G, Erdös L, Schröder DJ. 2021. Eigenstate thermalization hypothesis for Wigner matrices. Communications in Mathematical Physics. 388(2), 1005–1048.","mla":"Cipolloni, Giorgio, et al. “Eigenstate Thermalization Hypothesis for Wigner Matrices.” <i>Communications in Mathematical Physics</i>, vol. 388, no. 2, Springer Nature, 2021, pp. 1005–1048, doi:<a href=\"https://doi.org/10.1007/s00220-021-04239-z\">10.1007/s00220-021-04239-z</a>.","ama":"Cipolloni G, Erdös L, Schröder DJ. Eigenstate thermalization hypothesis for Wigner matrices. <i>Communications in Mathematical Physics</i>. 2021;388(2):1005–1048. doi:<a href=\"https://doi.org/10.1007/s00220-021-04239-z\">10.1007/s00220-021-04239-z</a>"},"publication_status":"published","doi":"10.1007/s00220-021-04239-z","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"publisher":"Springer Nature","issue":"2","status":"public","quality_controlled":"1","abstract":[{"lang":"eng","text":"We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020)."}],"title":"Eigenstate thermalization hypothesis for Wigner matrices","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria)."},{"language":[{"iso":"eng"}],"date_published":"2021-09-28T00:00:00Z","has_accepted_license":"1","fulldoi":"https://doi.org/10.1214/21-EJP686","ddc":["519"],"type":"journal_article","article_processing_charge":"No","day":"28","publication_identifier":{"eissn":["1083-6489"]},"scopus_import":"1","author":[{"full_name":"Dubach, Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","first_name":"Guillaume","orcid":"0000-0001-6892-8137","last_name":"Dubach"}],"article_number":"124","oa":1,"file":[{"relation":"main_file","file_size":735940,"checksum":"1c975afb31460277ce4d22b93538e5f9","content_type":"application/pdf","date_created":"2021-11-15T10:10:17Z","access_level":"open_access","file_id":"10288","date_updated":"2021-11-15T10:10:17Z","success":1,"file_name":"2021_ElecJournalProb_Dubach.pdf","creator":"cchlebak"}],"date_updated":"2025-04-14T07:43:47Z","oa_version":"Published Version","publication":"Electronic Journal of Probability","intvolume":"        26","file_date_updated":"2021-11-15T10:10:17Z","_id":"10285","ec_funded":1,"month":"09","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_created":"2021-11-14T23:01:25Z","article_type":"original","department":[{"_id":"LaEr"}],"acknowledgement":"We acknowledge partial support from the grants NSF DMS-1812114 of P. Bourgade (PI) and NSF CAREER DMS-1653602 of L.-P. Arguin (PI). This project has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. We would like to thank Paul Bourgade and László Erdős for many helpful comments.","title":"On eigenvector statistics in the spherical and truncated unitary ensembles","abstract":[{"lang":"eng","text":"We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur."}],"quality_controlled":"1","project":[{"name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"publisher":"Institute of Mathematical Statistics","status":"public","doi":"10.1214/21-EJP686","citation":{"chicago":"Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated Unitary Ensembles.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2021. <a href=\"https://doi.org/10.1214/21-EJP686\">https://doi.org/10.1214/21-EJP686</a>.","apa":"Dubach, G. (2021). On eigenvector statistics in the spherical and truncated unitary ensembles. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/21-EJP686\">https://doi.org/10.1214/21-EJP686</a>","short":"G. Dubach, Electronic Journal of Probability 26 (2021).","ieee":"G. Dubach, “On eigenvector statistics in the spherical and truncated unitary ensembles,” <i>Electronic Journal of Probability</i>, vol. 26. Institute of Mathematical Statistics, 2021.","ista":"Dubach G. 2021. On eigenvector statistics in the spherical and truncated unitary ensembles. Electronic Journal of Probability. 26, 124.","ama":"Dubach G. On eigenvector statistics in the spherical and truncated unitary ensembles. <i>Electronic Journal of Probability</i>. 2021;26. doi:<a href=\"https://doi.org/10.1214/21-EJP686\">10.1214/21-EJP686</a>","mla":"Dubach, Guillaume. “On Eigenvector Statistics in the Spherical and Truncated Unitary Ensembles.” <i>Electronic Journal of Probability</i>, vol. 26, 124, Institute of Mathematical Statistics, 2021, doi:<a href=\"https://doi.org/10.1214/21-EJP686\">10.1214/21-EJP686</a>."},"publication_status":"published","year":"2021","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":26},{"department":[{"_id":"LaEr"}],"article_type":"original","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2020-09-11T08:35:50Z","month":"01","_id":"8373","ec_funded":1,"intvolume":"       609","publication":"Linear Algebra and its Applications","oa_version":"Preprint","isi":1,"date_updated":"2025-04-14T07:50:40Z","oa":1,"year":"2021","volume":609,"citation":{"apa":"Pitrik, J., &#38; Virosztek, D. (2021). A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>","chicago":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>.","ieee":"J. Pitrik and D. Virosztek, “A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means,” <i>Linear Algebra and its Applications</i>, vol. 609. Elsevier, pp. 203–217, 2021.","short":"J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217.","ista":"Pitrik J, Virosztek D. 2021. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear Algebra and its Applications. 609, 203–217.","mla":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>, vol. 609, Elsevier, 2021, pp. 203–17, doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>.","ama":"Pitrik J, Virosztek D. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and its Applications</i>. 2021;609:203-217. doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>"},"publication_status":"published","doi":"10.1016/j.laa.2020.09.007","status":"public","publisher":"Elsevier","project":[{"_id":"26A455A6-B435-11E9-9278-68D0E5697425","grant_number":"846294","call_identifier":"H2020","name":"Geometric study of Wasserstein spaces and free probability"},{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"quality_controlled":"1","abstract":[{"text":"It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.","lang":"eng"}],"title":"A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means","acknowledgement":"The authors are grateful to Milán Mosonyi for fruitful discussions on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum Information Theory, No. 96 141, and by Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01), by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","fulldoi":"https://doi.org/10.1016/j.laa.2020.09.007","arxiv":1,"date_published":"2021-01-15T00:00:00Z","external_id":{"isi":["000581730500011"],"arxiv":["2002.11678"]},"language":[{"iso":"eng"}],"keyword":["Kubo-Ando mean","weighted multivariate mean","barycenter"],"page":"203-217","author":[{"last_name":"Pitrik","first_name":"József","full_name":"Pitrik, József"},{"orcid":"0000-0003-1109-5511","last_name":"Virosztek","full_name":"Virosztek, Daniel","first_name":"Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87"}],"main_file_link":[{"url":"https://arxiv.org/abs/2002.11678","open_access":"1"}],"publication_identifier":{"issn":["0024-3795"]},"scopus_import":"1","day":"15","article_processing_charge":"No","type":"journal_article"},{"corr_author":"1","author":[{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","first_name":"Giorgio","full_name":"Cipolloni, Giorgio","orcid":"0000-0002-4901-7992","last_name":"Cipolloni"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"scopus_import":"1","day":"01","article_processing_charge":"Yes (via OA deal)","type":"journal_article","ddc":["510"],"fulldoi":"https://doi.org/10.1007/s00440-020-01003-7","has_accepted_license":"1","arxiv":1,"external_id":{"arxiv":["1908.00969"],"isi":["000572724600002"]},"date_published":"2021-02-01T00:00:00Z","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2021","publication_status":"published","citation":{"ama":"Cipolloni G, Erdös L, Schröder DJ. Edge universality for non-Hermitian random matrices. <i>Probability Theory and Related Fields</i>. 2021. doi:<a href=\"https://doi.org/10.1007/s00440-020-01003-7\">10.1007/s00440-020-01003-7</a>","mla":"Cipolloni, Giorgio, et al. “Edge Universality for Non-Hermitian Random Matrices.” <i>Probability Theory and Related Fields</i>, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s00440-020-01003-7\">10.1007/s00440-020-01003-7</a>.","ista":"Cipolloni G, Erdös L, Schröder DJ. 2021. Edge universality for non-Hermitian random matrices. Probability Theory and Related Fields.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields (2021).","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Edge universality for non-Hermitian random matrices,” <i>Probability Theory and Related Fields</i>. Springer Nature, 2021.","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Edge Universality for Non-Hermitian Random Matrices.” <i>Probability Theory and Related Fields</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00440-020-01003-7\">https://doi.org/10.1007/s00440-020-01003-7</a>.","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2021). Edge universality for non-Hermitian random matrices. <i>Probability Theory and Related Fields</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00440-020-01003-7\">https://doi.org/10.1007/s00440-020-01003-7</a>"},"doi":"10.1007/s00440-020-01003-7","status":"public","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"name":"International IST Doctoral Program","call_identifier":"H2020","grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425"}],"publisher":"Springer Nature","quality_controlled":"1","title":"Edge universality for non-Hermitian random matrices","abstract":[{"text":"We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.","lang":"eng"}],"department":[{"_id":"LaEr"}],"article_type":"original","date_created":"2020-10-04T22:01:37Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","month":"02","ec_funded":1,"_id":"8601","file_date_updated":"2020-10-05T14:53:40Z","oa_version":"Published Version","publication":"Probability Theory and Related Fields","isi":1,"date_updated":"2026-04-02T14:03:52Z","file":[{"relation":"main_file","file_size":497032,"checksum":"611ae28d6055e1e298d53a57beb05ef4","date_created":"2020-10-05T14:53:40Z","content_type":"application/pdf","access_level":"open_access","success":1,"file_id":"8612","date_updated":"2020-10-05T14:53:40Z","creator":"dernst","file_name":"2020_ProbTheory_Cipolloni.pdf"}],"oa":1},{"language":[{"iso":"eng"}],"alternative_title":["ISTA Thesis"],"date_published":"2021-01-25T00:00:00Z","has_accepted_license":"1","degree_awarded":"PhD","ddc":["510"],"fulldoi":"https://doi.org/10.15479/AT:ISTA:9022","type":"dissertation","article_processing_charge":"No","day":"25","publication_identifier":{"issn":["2663-337X"]},"author":[{"first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio","orcid":"0000-0002-4901-7992","last_name":"Cipolloni"}],"corr_author":"1","page":"380","file":[{"date_created":"2021-01-25T14:19:03Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","file_size":4127796,"checksum":"5a93658a5f19478372523ee232887e2b","creator":"gcipollo","file_name":"thesis.pdf","success":1,"date_updated":"2021-01-25T14:19:03Z","file_id":"9043"},{"access_level":"closed","date_created":"2021-01-25T14:19:10Z","content_type":"application/zip","checksum":"e8270eddfe6a988e92a53c88d1d19b8c","file_size":12775206,"relation":"source_file","creator":"gcipollo","file_name":"Thesis_files.zip","date_updated":"2021-01-25T14:19:10Z","file_id":"9044"}],"date_updated":"2026-04-08T06:59:33Z","oa":1,"oa_version":"Published Version","file_date_updated":"2021-01-25T14:19:10Z","ec_funded":1,"_id":"9022","month":"01","date_created":"2021-01-21T18:16:54Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"acknowledgement":"I gratefully acknowledge the financial support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385 and my advisor’s ERC Advanced Grant No. 338804.","title":"Fluctuations in the spectrum of random matrices","abstract":[{"lang":"eng","text":"In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.\r\nIn the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time\r\n(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.\r\nIn the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result\r\nimproves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices."}],"supervisor":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"}],"status":"public","publisher":"Institute of Science and Technology Austria","project":[{"name":"International IST Doctoral Program","call_identifier":"H2020","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","grant_number":"665385"},{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"doi":"10.15479/AT:ISTA:9022","publication_status":"published","citation":{"mla":"Cipolloni, Giorgio. <i>Fluctuations in the Spectrum of Random Matrices</i>. Institute of Science and Technology Austria, 2021, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9022\">10.15479/AT:ISTA:9022</a>.","ama":"Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:9022\">10.15479/AT:ISTA:9022</a>","ista":"Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. Institute of Science and Technology Austria.","ieee":"G. Cipolloni, “Fluctuations in the spectrum of random matrices,” Institute of Science and Technology Austria, 2021.","short":"G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, Institute of Science and Technology Austria, 2021.","apa":"Cipolloni, G. (2021). <i>Fluctuations in the spectrum of random matrices</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:9022\">https://doi.org/10.15479/AT:ISTA:9022</a>","chicago":"Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.” Institute of Science and Technology Austria, 2021. <a href=\"https://doi.org/10.15479/AT:ISTA:9022\">https://doi.org/10.15479/AT:ISTA:9022</a>."},"OA_place":"publisher","year":"2021"},{"quality_controlled":"1","publisher":"Elsevier","project":[{"call_identifier":"H2020","grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425","name":"Geometric study of Wasserstein spaces and free probability"}],"status":"public","issue":"3","acknowledgement":"D. Virosztek was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","title":"The metric property of the quantum Jensen-Shannon divergence","abstract":[{"text":"In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.","lang":"eng"}],"publication_status":"published","citation":{"ama":"Virosztek D. The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. 2021;380(3). doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>","mla":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>, vol. 380, no. 3, 107595, Elsevier, 2021, doi:<a href=\"https://doi.org/10.1016/j.aim.2021.107595\">10.1016/j.aim.2021.107595</a>.","ista":"Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics. 380(3), 107595.","short":"D. Virosztek, Advances in Mathematics 380 (2021).","ieee":"D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,” <i>Advances in Mathematics</i>, vol. 380, no. 3. Elsevier, 2021.","chicago":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” <i>Advances in Mathematics</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>.","apa":"Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2021.107595\">https://doi.org/10.1016/j.aim.2021.107595</a>"},"year":"2021","volume":380,"doi":"10.1016/j.aim.2021.107595","oa_version":"Preprint","publication":"Advances in Mathematics","intvolume":"       380","oa":1,"isi":1,"date_updated":"2025-04-14T07:50:40Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","date_created":"2021-01-22T17:55:17Z","article_type":"original","department":[{"_id":"LaEr"}],"ec_funded":1,"_id":"9036","month":"03","day":"26","type":"journal_article","article_processing_charge":"No","article_number":"107595","scopus_import":"1","publication_identifier":{"issn":["0001-8708"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.10447"}],"author":[{"orcid":"0000-0003-1109-5511","last_name":"Virosztek","full_name":"Virosztek, Daniel","first_name":"Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87"}],"external_id":{"isi":["000619676100035"],"arxiv":["1910.10447"]},"date_published":"2021-03-26T00:00:00Z","arxiv":1,"keyword":["General Mathematics"],"language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1016/j.aim.2021.107595"},{"article_number":"2103.04817","citation":{"ista":"Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv, 2103.04817.","mla":"Arguin, Louis-Pierre, et al. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” <i>ArXiv</i>, 2103.04817, doi:<a href=\"https://doi.org/10.48550/arXiv.2103.04817\">10.48550/arXiv.2103.04817</a>.","ama":"Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2103.04817\">10.48550/arXiv.2103.04817</a>","apa":"Arguin, L.-P., Dubach, G., &#38; Hartung, L. (n.d.). Maxima of a random model of the Riemann zeta function over intervals of varying length. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2103.04817\">https://doi.org/10.48550/arXiv.2103.04817</a>","chicago":"Arguin, Louis-Pierre, Guillaume Dubach, and Lisa Hartung. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2103.04817\">https://doi.org/10.48550/arXiv.2103.04817</a>.","ieee":"L.-P. Arguin, G. Dubach, and L. Hartung, “Maxima of a random model of the Riemann zeta function over intervals of varying length,” <i>arXiv</i>. .","short":"L.-P. Arguin, G. Dubach, L. Hartung, ArXiv (n.d.)."},"publication_status":"submitted","year":"2021","doi":"10.48550/arXiv.2103.04817","main_file_link":[{"url":"https://arxiv.org/abs/2103.04817","open_access":"1"}],"author":[{"last_name":"Arguin","first_name":"Louis-Pierre","full_name":"Arguin, Louis-Pierre"},{"first_name":"Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","full_name":"Dubach, Guillaume","orcid":"0000-0001-6892-8137","last_name":"Dubach"},{"last_name":"Hartung","first_name":"Lisa","full_name":"Hartung, Lisa"}],"project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"day":"08","status":"public","acknowledgement":"The research of L.-P. A. is supported in part by the grant NSF CAREER DMS-1653602. G. D. gratefully acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. The research of L. H. is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 233630050 -TRR 146, Project-ID 443891315 within SPP 2265 and Project-ID 446173099.","type":"preprint","article_processing_charge":"No","title":"Maxima of a random model of the Riemann zeta function over intervals of varying length","abstract":[{"lang":"eng","text":"We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate.\r\nIt is shown that the deterministic level of the maximum interpolates smoothly between the ones\r\nof log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from\r\n3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of\r\nlog-correlated variables with time-dependent variance and rate occur. A key ingredient of the\r\nproof is a precise upper tail tightness estimate for the maximum of the model on intervals of\r\nsize one, that includes a Gaussian correction. This correction is expected to be present for the\r\nRiemann zeta function and pertains to the question of the correct order of the maximum of\r\nthe zeta function in large intervals."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2021-03-09T11:08:15Z","department":[{"_id":"LaEr"}],"_id":"9230","ec_funded":1,"fulldoi":"https://doi.org/10.48550/arXiv.2103.04817","month":"03","external_id":{"arxiv":["2103.04817"]},"date_published":"2021-03-08T00:00:00Z","oa_version":"Preprint","publication":"arXiv","arxiv":1,"oa":1,"date_updated":"2025-04-14T07:43:51Z","language":[{"iso":"eng"}]},{"fulldoi":"https://doi.org/10.48550/arXiv.2103.11389","external_id":{"arxiv":["2103.11389"]},"date_published":"2021-03-21T00:00:00Z","arxiv":1,"language":[{"iso":"eng"}],"corr_author":"1","article_number":"2103.11389","main_file_link":[{"url":"https://arxiv.org/abs/2103.11389","open_access":"1"}],"author":[{"full_name":"Dubach, Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","first_name":"Guillaume","last_name":"Dubach","orcid":"0000-0001-6892-8137"},{"full_name":"Mühlböck, Fabian","id":"6395C5F6-89DF-11E9-9C97-6BDFE5697425","first_name":"Fabian","last_name":"Mühlböck","orcid":"0000-0003-1548-0177"}],"day":"21","type":"preprint","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2021-03-23T05:38:48Z","department":[{"_id":"LaEr"},{"_id":"ToHe"}],"ec_funded":1,"_id":"9281","month":"03","oa_version":"Preprint","publication":"arXiv","related_material":{"record":[{"relation":"other","status":"public","id":"9946"}]},"date_updated":"2025-04-15T06:26:12Z","oa":1,"publication_status":"submitted","citation":{"short":"G. Dubach, F. Mühlböck, ArXiv (n.d.).","ieee":"G. Dubach and F. Mühlböck, “Formal verification of Zagier’s one-sentence proof,” <i>arXiv</i>. .","chicago":"Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence Proof.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2103.11389\">https://doi.org/10.48550/arXiv.2103.11389</a>.","apa":"Dubach, G., &#38; Mühlböck, F. (n.d.). Formal verification of Zagier’s one-sentence proof. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2103.11389\">https://doi.org/10.48550/arXiv.2103.11389</a>","ama":"Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2103.11389\">10.48550/arXiv.2103.11389</a>","mla":"Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence Proof.” <i>ArXiv</i>, 2103.11389, doi:<a href=\"https://doi.org/10.48550/arXiv.2103.11389\">10.48550/arXiv.2103.11389</a>.","ista":"Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. arXiv, 2103.11389."},"year":"2021","doi":"10.48550/arXiv.2103.11389","status":"public","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"title":"Formal verification of Zagier's one-sentence proof","abstract":[{"lang":"eng","text":"We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics."}]},{"project":[{"call_identifier":"H2020","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","grant_number":"665385","name":"International IST Doctoral Program"}],"publisher":"Institute of Mathematical Statistics","status":"public","quality_controlled":"1","abstract":[{"text":"We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.","lang":"eng"}],"title":"Fluctuation around the circular law for random matrices with real entries","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":26,"year":"2021","citation":{"mla":"Cipolloni, Giorgio, et al. “Fluctuation around the Circular Law for Random Matrices with Real Entries.” <i>Electronic Journal of Probability</i>, vol. 26, 24, Institute of Mathematical Statistics, 2021, doi:<a href=\"https://doi.org/10.1214/21-EJP591\">10.1214/21-EJP591</a>.","ama":"Cipolloni G, Erdös L, Schröder DJ. Fluctuation around the circular law for random matrices with real entries. <i>Electronic Journal of Probability</i>. 2021;26. doi:<a href=\"https://doi.org/10.1214/21-EJP591\">10.1214/21-EJP591</a>","ista":"Cipolloni G, Erdös L, Schröder DJ. 2021. Fluctuation around the circular law for random matrices with real entries. Electronic Journal of Probability. 26, 24.","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Fluctuation around the circular law for random matrices with real entries,” <i>Electronic Journal of Probability</i>, vol. 26. Institute of Mathematical Statistics, 2021.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Electronic Journal of Probability 26 (2021).","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2021). Fluctuation around the circular law for random matrices with real entries. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/21-EJP591\">https://doi.org/10.1214/21-EJP591</a>","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Fluctuation around the Circular Law for Random Matrices with Real Entries.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2021. <a href=\"https://doi.org/10.1214/21-EJP591\">https://doi.org/10.1214/21-EJP591</a>."},"publication_status":"published","doi":"10.1214/21-EJP591","intvolume":"        26","file_date_updated":"2021-05-25T13:24:19Z","oa_version":"Published Version","publication":"Electronic Journal of Probability","oa":1,"isi":1,"file":[{"relation":"main_file","checksum":"864ab003ad4cffea783f65aa8c2ba69f","file_size":865148,"content_type":"application/pdf","date_created":"2021-05-25T13:24:19Z","access_level":"open_access","file_id":"9423","date_updated":"2021-05-25T13:24:19Z","success":1,"file_name":"2021_EJP_Cipolloni.pdf","creator":"kschuh"}],"date_updated":"2026-04-02T14:00:37Z","department":[{"_id":"LaEr"}],"date_created":"2021-05-23T22:01:44Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","month":"03","ec_funded":1,"_id":"9412","day":"23","article_processing_charge":"No","type":"journal_article","article_number":"24","author":[{"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","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}],"publication_identifier":{"eissn":["1083-6489"]},"scopus_import":"1","arxiv":1,"external_id":{"arxiv":["2002.02438"],"isi":["000641855600001"]},"date_published":"2021-03-23T00:00:00Z","language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1214/21-EJP591","ddc":["510"],"has_accepted_license":"1"},{"language":[{"iso":"eng"}],"external_id":{"arxiv":["2008.07061"],"isi":["000654960800001"]},"date_published":"2021-05-27T00:00:00Z","arxiv":1,"has_accepted_license":"1","fulldoi":"https://doi.org/10.1017/fms.2021.38","ddc":["510"],"type":"journal_article","article_processing_charge":"No","day":"27","scopus_import":"1","publication_identifier":{"eissn":["2050-5094"]},"author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Schnelli","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","full_name":"Schnelli, Kevin"}],"article_number":"e44","oa":1,"isi":1,"file":[{"file_name":"2021_ForumMath_Bao.pdf","creator":"cziletti","success":1,"file_id":"9555","date_updated":"2021-06-15T14:40:45Z","access_level":"open_access","date_created":"2021-06-15T14:40:45Z","content_type":"application/pdf","file_size":483458,"checksum":"47c986578de132200d41e6d391905519","relation":"main_file"}],"date_updated":"2026-04-07T08:36:39Z","oa_version":"Published Version","publication":"Forum of Mathematics, Sigma","intvolume":"         9","file_date_updated":"2021-06-15T14:40:45Z","ec_funded":1,"_id":"9550","month":"05","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","date_created":"2021-06-13T22:01:33Z","article_type":"original","department":[{"_id":"LaEr"}],"acknowledgement":"The first author is supported in part by Hong Kong RGC Grant GRF 16301519 and NSFC 11871425. The second author is supported in part by ERC Advanced Grant RANMAT 338804. The third author is supported in part by Swedish Research Council Grant VR-2017-05195 and the Knut and Alice Wallenberg Foundation","title":"Equipartition principle for Wigner matrices","abstract":[{"text":"We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. ","lang":"eng"}],"quality_controlled":"1","project":[{"call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"publisher":"Cambridge University Press","status":"public","doi":"10.1017/fms.2021.38","publication_status":"published","citation":{"ista":"Bao Z, Erdös L, Schnelli K. 2021. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 9, e44.","mla":"Bao, Zhigang, et al. “Equipartition Principle for Wigner Matrices.” <i>Forum of Mathematics, Sigma</i>, vol. 9, e44, Cambridge University Press, 2021, doi:<a href=\"https://doi.org/10.1017/fms.2021.38\">10.1017/fms.2021.38</a>.","ama":"Bao Z, Erdös L, Schnelli K. Equipartition principle for Wigner matrices. <i>Forum of Mathematics, Sigma</i>. 2021;9. doi:<a href=\"https://doi.org/10.1017/fms.2021.38\">10.1017/fms.2021.38</a>","apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2021). Equipartition principle for Wigner matrices. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fms.2021.38\">https://doi.org/10.1017/fms.2021.38</a>","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Equipartition Principle for Wigner Matrices.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2021. <a href=\"https://doi.org/10.1017/fms.2021.38\">https://doi.org/10.1017/fms.2021.38</a>.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Equipartition principle for Wigner matrices,” <i>Forum of Mathematics, Sigma</i>, vol. 9. Cambridge University Press, 2021.","short":"Z. Bao, L. Erdös, K. Schnelli, Forum of Mathematics, Sigma 9 (2021)."},"year":"2021","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":9},{"day":"01","article_processing_charge":"Yes (in subscription journal)","type":"journal_article","page":"4205–4269","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","full_name":"Krüger, Torben H"},{"last_name":"Nemish","orcid":"0000-0002-7327-856X","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","first_name":"Yuriy","full_name":"Nemish, Yuriy"}],"scopus_import":"1","publication_identifier":{"issn":["1424-0637"],"eissn":["1424-0661"]},"arxiv":1,"external_id":{"arxiv":["1911.05112"],"isi":["000681531500001"]},"date_published":"2021-12-01T00:00:00Z","language":[{"iso":"eng"}],"fulldoi":"https://doi.org/10.1007/s00023-021-01085-6","ddc":["510"],"has_accepted_license":"1","project":[{"call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"publisher":"Springer Nature","status":"public","quality_controlled":"1","abstract":[{"text":"In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.","lang":"eng"}],"title":"Scattering in quantum dots via noncommutative rational functions","acknowledgement":"The authors are very grateful to Yan Fyodorov for discussions on the physical background and for providing references, and to the anonymous referee for numerous valuable remarks.","volume":22,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2021","publication_status":"published","citation":{"ista":"Erdös L, Krüger TH, Nemish Y. 2021. Scattering in quantum dots via noncommutative rational functions. Annales Henri Poincaré . 22, 4205–4269.","ama":"Erdös L, Krüger TH, Nemish Y. Scattering in quantum dots via noncommutative rational functions. <i>Annales Henri Poincaré </i>. 2021;22:4205–4269. doi:<a href=\"https://doi.org/10.1007/s00023-021-01085-6\">10.1007/s00023-021-01085-6</a>","mla":"Erdös, László, et al. “Scattering in Quantum Dots via Noncommutative Rational Functions.” <i>Annales Henri Poincaré </i>, vol. 22, Springer Nature, 2021, pp. 4205–4269, doi:<a href=\"https://doi.org/10.1007/s00023-021-01085-6\">10.1007/s00023-021-01085-6</a>.","chicago":"Erdös, László, Torben H Krüger, and Yuriy Nemish. “Scattering in Quantum Dots via Noncommutative Rational Functions.” <i>Annales Henri Poincaré </i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00023-021-01085-6\">https://doi.org/10.1007/s00023-021-01085-6</a>.","apa":"Erdös, L., Krüger, T. H., &#38; Nemish, Y. (2021). Scattering in quantum dots via noncommutative rational functions. <i>Annales Henri Poincaré </i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00023-021-01085-6\">https://doi.org/10.1007/s00023-021-01085-6</a>","short":"L. Erdös, T.H. Krüger, Y. Nemish, Annales Henri Poincaré  22 (2021) 4205–4269.","ieee":"L. Erdös, T. H. Krüger, and Y. Nemish, “Scattering in quantum dots via noncommutative rational functions,” <i>Annales Henri Poincaré </i>, vol. 22. Springer Nature, pp. 4205–4269, 2021."},"doi":"10.1007/s00023-021-01085-6","intvolume":"        22","file_date_updated":"2022-05-12T12:50:27Z","oa_version":"Published Version","publication":"Annales Henri Poincaré ","oa":1,"isi":1,"file":[{"file_id":"11365","date_updated":"2022-05-12T12:50:27Z","success":1,"creator":"dernst","file_name":"2021_AnnHenriPoincare_Erdoes.pdf","relation":"main_file","checksum":"8d6bac0e2b0a28539608b0538a8e3b38","file_size":1162454,"content_type":"application/pdf","date_created":"2022-05-12T12:50:27Z","access_level":"open_access"}],"date_updated":"2025-04-15T08:04:59Z","article_type":"original","department":[{"_id":"LaEr"}],"date_created":"2021-08-15T22:01:29Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","month":"12","ec_funded":1,"_id":"9912"},{"fulldoi":"https://doi.org/10.1016/j.jfa.2020.108639","external_id":{"isi":["000559623200009"],"arxiv":["1708.01597"]},"date_published":"2020-10-15T00:00:00Z","arxiv":1,"keyword":["Analysis"],"language":[{"iso":"eng"}],"article_number":"108639","corr_author":"1","publication_identifier":{"issn":["0022-1236"]},"scopus_import":"1","main_file_link":[{"url":"https://arxiv.org/abs/1708.01597","open_access":"1"}],"author":[{"first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"full_name":"Schnelli, Kevin","first_name":"Kevin","last_name":"Schnelli"}],"day":"15","type":"journal_article","article_processing_charge":"No","date_created":"2022-03-18T10:18:59Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_type":"original","department":[{"_id":"LaEr"}],"ec_funded":1,"_id":"10862","month":"10","oa_version":"Preprint","publication":"Journal of Functional Analysis","intvolume":"       279","oa":1,"isi":1,"date_updated":"2025-04-15T08:05:01Z","publication_status":"published","citation":{"apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2020). Spectral rigidity for addition of random matrices at the regular edge. <i>Journal of Functional Analysis</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jfa.2020.108639\">https://doi.org/10.1016/j.jfa.2020.108639</a>","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Spectral Rigidity for Addition of Random Matrices at the Regular Edge.” <i>Journal of Functional Analysis</i>. Elsevier, 2020. <a href=\"https://doi.org/10.1016/j.jfa.2020.108639\">https://doi.org/10.1016/j.jfa.2020.108639</a>.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Spectral rigidity for addition of random matrices at the regular edge,” <i>Journal of Functional Analysis</i>, vol. 279, no. 7. Elsevier, 2020.","short":"Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 279 (2020).","ista":"Bao Z, Erdös L, Schnelli K. 2020. Spectral rigidity for addition of random matrices at the regular edge. Journal of Functional Analysis. 279(7), 108639.","mla":"Bao, Zhigang, et al. “Spectral Rigidity for Addition of Random Matrices at the Regular Edge.” <i>Journal of Functional Analysis</i>, vol. 279, no. 7, 108639, Elsevier, 2020, doi:<a href=\"https://doi.org/10.1016/j.jfa.2020.108639\">10.1016/j.jfa.2020.108639</a>.","ama":"Bao Z, Erdös L, Schnelli K. Spectral rigidity for addition of random matrices at the regular edge. <i>Journal of Functional Analysis</i>. 2020;279(7). doi:<a href=\"https://doi.org/10.1016/j.jfa.2020.108639\">10.1016/j.jfa.2020.108639</a>"},"year":"2020","volume":279,"doi":"10.1016/j.jfa.2020.108639","quality_controlled":"1","project":[{"call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"publisher":"Elsevier","issue":"7","status":"public","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804.","title":"Spectral rigidity for addition of random matrices at the regular edge","abstract":[{"text":"We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4], [5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.","lang":"eng"}]},{"citation":{"apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2020). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>Documenta Mathematica</i>. EMS Press. <a href=\"https://doi.org/10.4171/dm/780\">https://doi.org/10.4171/dm/780</a>","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>Documenta Mathematica</i>. EMS Press, 2020. <a href=\"https://doi.org/10.4171/dm/780\">https://doi.org/10.4171/dm/780</a>.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” <i>Documenta Mathematica</i>, vol. 25. EMS Press, pp. 1421–1539, 2020.","short":"J. Alt, L. Erdös, T.H. Krüger, Documenta Mathematica 25 (2020) 1421–1539.","ista":"Alt J, Erdös L, Krüger TH. 2020. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. Documenta Mathematica. 25, 1421–1539.","mla":"Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>Documenta Mathematica</i>, vol. 25, EMS Press, 2020, pp. 1421–539, doi:<a href=\"https://doi.org/10.4171/dm/780\">10.4171/dm/780</a>.","ama":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>Documenta Mathematica</i>. 2020;25:1421-1539. doi:<a href=\"https://doi.org/10.4171/dm/780\">10.4171/dm/780</a>"},"publication_status":"published","volume":25,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2020","doi":"10.4171/dm/780","quality_controlled":"1","status":"public","publisher":"EMS Press","title":"The Dyson equation with linear self-energy: Spectral bands, edges and cusps","abstract":[{"lang":"eng","text":"We study the unique solution m of the Dyson equation \\( -m(z)^{-1} = z\\1 - a + S[m(z)] \\) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving linear operator on A. We show that m is the Stieltjes transform of a compactly supported A-valued measure on R. Under suitable assumptions, we establish that this measure has a uniformly 1/3-Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of m near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-12-18T10:37:43Z","department":[{"_id":"LaEr"}],"article_type":"original","_id":"14694","month":"09","oa_version":"Published Version","publication":"Documenta Mathematica","file_date_updated":"2023-12-18T10:42:32Z","intvolume":"        25","related_material":{"record":[{"status":"public","id":"6183","relation":"earlier_version"}]},"file":[{"content_type":"application/pdf","date_created":"2023-12-18T10:42:32Z","access_level":"open_access","relation":"main_file","checksum":"12aacc1d63b852ff9a51c1f6b218d4a6","file_size":1374708,"creator":"dernst","file_name":"2020_DocumentaMathematica_Alt.pdf","file_id":"14695","date_updated":"2023-12-18T10:42:32Z","success":1}],"date_updated":"2025-04-15T08:05:00Z","oa":1,"corr_author":"1","page":"1421-1539","publication_identifier":{"issn":["1431-0635"],"eissn":["1431-0643"]},"author":[{"id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","full_name":"Alt, Johannes","last_name":"Alt"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"orcid":"0000-0002-4821-3297","last_name":"Krüger","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","full_name":"Krüger, Torben H"}],"day":"01","type":"journal_article","article_processing_charge":"Yes","has_accepted_license":"1","ddc":["510"],"fulldoi":"https://doi.org/10.4171/dm/780","external_id":{"arxiv":["1804.07752"]},"date_published":"2020-09-01T00:00:00Z","arxiv":1,"keyword":["General Mathematics"],"language":[{"iso":"eng"}]},{"doi":"10.2140/pmp.2020.1.101","volume":1,"year":"2020","publication_status":"published","citation":{"ama":"Cipolloni G, Erdös L, Schröder DJ. Optimal lower bound on the least singular value of the shifted Ginibre ensemble. <i>Probability and Mathematical Physics</i>. 2020;1(1):101-146. doi:<a href=\"https://doi.org/10.2140/pmp.2020.1.101\">10.2140/pmp.2020.1.101</a>","mla":"Cipolloni, Giorgio, et al. “Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble.” <i>Probability and Mathematical Physics</i>, vol. 1, no. 1, Mathematical Sciences Publishers, 2020, pp. 101–46, doi:<a href=\"https://doi.org/10.2140/pmp.2020.1.101\">10.2140/pmp.2020.1.101</a>.","ista":"Cipolloni G, Erdös L, Schröder DJ. 2020. Optimal lower bound on the least singular value of the shifted Ginibre ensemble. Probability and Mathematical Physics. 1(1), 101–146.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Probability and Mathematical Physics 1 (2020) 101–146.","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Optimal lower bound on the least singular value of the shifted Ginibre ensemble,” <i>Probability and Mathematical Physics</i>, vol. 1, no. 1. Mathematical Sciences Publishers, pp. 101–146, 2020.","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble.” <i>Probability and Mathematical Physics</i>. Mathematical Sciences Publishers, 2020. <a href=\"https://doi.org/10.2140/pmp.2020.1.101\">https://doi.org/10.2140/pmp.2020.1.101</a>.","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2020). Optimal lower bound on the least singular value of the shifted Ginibre ensemble. <i>Probability and Mathematical Physics</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/pmp.2020.1.101\">https://doi.org/10.2140/pmp.2020.1.101</a>"},"abstract":[{"lang":"eng","text":"We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant z∈C. We prove an optimal lower tail estimate on this singular value in the critical regime where z is around the spectral edge, thus improving the classical bound of Sankar, Spielman and Teng (SIAM J. Matrix Anal. Appl. 28:2 (2006), 446–476) for the particular shift-perturbation in the edge regime. Lacking Brézin–Hikami formulas in the real case, we rely on the superbosonization formula (Comm. Math. Phys. 283:2 (2008), 343–395)."}],"title":"Optimal lower bound on the least singular value of the shifted Ginibre ensemble","acknowledgement":"Partially supported by ERC Advanced Grant No. 338804. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 66538","publisher":"Mathematical Sciences Publishers","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"},{"call_identifier":"H2020","grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425","name":"International IST Doctoral Program"}],"status":"public","issue":"1","quality_controlled":"1","month":"11","_id":"15063","ec_funded":1,"article_type":"original","department":[{"_id":"LaEr"}],"date_created":"2024-03-04T10:27:57Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_updated":"2025-07-10T11:51:06Z","intvolume":"         1","oa_version":"Preprint","publication":"Probability and Mathematical Physics","author":[{"last_name":"Cipolloni","orcid":"0000-0002-4901-7992","first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László"},{"last_name":"Schröder","orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"publication_identifier":{"issn":["2690-0998"]},"scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1908.01653","open_access":"1"}],"page":"101-146","corr_author":"1","article_processing_charge":"No","type":"journal_article","day":"16","fulldoi":"https://doi.org/10.2140/pmp.2020.1.101","language":[{"iso":"eng"}],"keyword":["General Medicine"],"arxiv":1,"date_published":"2020-11-16T00:00:00Z","external_id":{"arxiv":["1908.01653"]}},{"author":[{"last_name":"Pitrik","full_name":"Pitrik, Jozsef","first_name":"Jozsef"},{"full_name":"Virosztek, Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","first_name":"Daniel","last_name":"Virosztek","orcid":"0000-0003-1109-5511"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1903.10455"}],"scopus_import":"1","publication_identifier":{"eissn":["1573-0530"],"issn":["0377-9017"]},"page":"2039-2052","corr_author":"1","article_processing_charge":"No","type":"journal_article","day":"01","fulldoi":"https://doi.org/10.1007/s11005-020-01282-0","language":[{"iso":"eng"}],"arxiv":1,"date_published":"2020-08-01T00:00:00Z","external_id":{"arxiv":["1903.10455"],"isi":["000551556000002"]},"doi":"10.1007/s11005-020-01282-0","year":"2020","volume":110,"publication_status":"published","citation":{"chicago":"Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” <i>Letters in Mathematical Physics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s11005-020-01282-0\">https://doi.org/10.1007/s11005-020-01282-0</a>.","apa":"Pitrik, J., &#38; Virosztek, D. (2020). Quantum Hellinger distances revisited. <i>Letters in Mathematical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11005-020-01282-0\">https://doi.org/10.1007/s11005-020-01282-0</a>","short":"J. Pitrik, D. Virosztek, Letters in Mathematical Physics 110 (2020) 2039–2052.","ieee":"J. Pitrik and D. Virosztek, “Quantum Hellinger distances revisited,” <i>Letters in Mathematical Physics</i>, vol. 110, no. 8. Springer Nature, pp. 2039–2052, 2020.","ista":"Pitrik J, Virosztek D. 2020. Quantum Hellinger distances revisited. Letters in Mathematical Physics. 110(8), 2039–2052.","ama":"Pitrik J, Virosztek D. Quantum Hellinger distances revisited. <i>Letters in Mathematical Physics</i>. 2020;110(8):2039-2052. doi:<a href=\"https://doi.org/10.1007/s11005-020-01282-0\">10.1007/s11005-020-01282-0</a>","mla":"Pitrik, Jozsef, and Daniel Virosztek. “Quantum Hellinger Distances Revisited.” <i>Letters in Mathematical Physics</i>, vol. 110, no. 8, Springer Nature, 2020, pp. 2039–52, doi:<a href=\"https://doi.org/10.1007/s11005-020-01282-0\">10.1007/s11005-020-01282-0</a>."},"abstract":[{"lang":"eng","text":"This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. "}],"title":"Quantum Hellinger distances revisited","acknowledgement":"J. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum\r\nInformation Theory, No. 96 141, and by the Hungarian National Research, Development and Innovation\r\nOffice (NKFIH) via Grants Nos. K119442, K124152 and KH129601. D. Virosztek was supported by the\r\nISTFELLOW program of the Institute of Science and Technology Austria (Project Code IC1027FELL01),\r\nby the European Union’s Horizon 2020 research and innovation program under the Marie\r\nSklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National\r\nResearch, Development and Innovation Office (NKFIH) via Grants Nos. K124152 and KH129601.\r\nWe are grateful to Milán Mosonyi for drawing our attention to Ref.’s [6,14,15,17,\r\n20,21], for comments on earlier versions of this paper, and for several discussions on the topic. We are\r\nalso grateful to Miklós Pálfia for several discussions; to László Erdös for his essential suggestions on the\r\nstructure and highlights of this paper, and for his comments on earlier versions; and to the anonymous\r\nreferee for his/her valuable comments and suggestions.","status":"public","issue":"8","publisher":"Springer Nature","project":[{"grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"Geometric study of Wasserstein spaces and free probability"},{"call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","name":"International IST Postdoc Fellowship Programme"}],"quality_controlled":"1","month":"08","ec_funded":1,"_id":"7618","department":[{"_id":"LaEr"}],"article_type":"original","date_created":"2020-03-25T15:57:48Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","isi":1,"date_updated":"2025-10-09T08:23:15Z","oa":1,"intvolume":"       110","oa_version":"Preprint","publication":"Letters in Mathematical Physics"}]