[{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2026-03-13T00:00:00Z","year":"2026","oa_version":"Published Version","date_updated":"2026-03-23T13:32:14Z","day":"13","type":"journal_article","publisher":"Wiley","status":"public","doi":"10.1002/cpa.70040","OA_place":"publisher","article_type":"original","OA_type":"hybrid","arxiv":1,"article_processing_charge":"Yes (via OA deal)","citation":{"ieee":"E. L. Giacomelli, C. Hainzl, P. T. Nam, and R. Seiringer, “The Huang–Yang formula for the low-density Fermi gas: Upper bound,” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2026.","mla":"Giacomelli, Emanuela L., et al. “The Huang–Yang Formula for the Low-Density Fermi Gas: Upper Bound.” <i>Communications on Pure and Applied Mathematics</i>, Wiley, 2026, doi:<a href=\"https://doi.org/10.1002/cpa.70040\">10.1002/cpa.70040</a>.","short":"E.L. Giacomelli, C. Hainzl, P.T. Nam, R. Seiringer, Communications on Pure and Applied Mathematics (2026).","chicago":"Giacomelli, Emanuela L., Christian Hainzl, Phan Thành Nam, and Robert Seiringer. “The Huang–Yang Formula for the Low-Density Fermi Gas: Upper Bound.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2026. <a href=\"https://doi.org/10.1002/cpa.70040\">https://doi.org/10.1002/cpa.70040</a>.","ista":"Giacomelli EL, Hainzl C, Nam PT, Seiringer R. 2026. The Huang–Yang formula for the low-density Fermi gas: Upper bound. Communications on Pure and Applied Mathematics.","ama":"Giacomelli EL, Hainzl C, Nam PT, Seiringer R. The Huang–Yang formula for the low-density Fermi gas: Upper bound. <i>Communications on Pure and Applied Mathematics</i>. 2026. doi:<a href=\"https://doi.org/10.1002/cpa.70040\">10.1002/cpa.70040</a>","apa":"Giacomelli, E. L., Hainzl, C., Nam, P. T., &#38; Seiringer, R. (2026). The Huang–Yang formula for the low-density Fermi gas: Upper bound. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.70040\">https://doi.org/10.1002/cpa.70040</a>"},"_id":"21472","title":"The Huang–Yang formula for the low-density Fermi gas: Upper bound","month":"03","publication":"Communications on Pure and Applied Mathematics","scopus_import":"1","author":[{"first_name":"Emanuela L.","full_name":"Giacomelli, Emanuela L.","last_name":"Giacomelli"},{"first_name":"Christian","full_name":"Hainzl, Christian","last_name":"Hainzl"},{"full_name":"Nam, Phan Thành","last_name":"Nam","first_name":"Phan Thành"},{"last_name":"Seiringer","full_name":"Seiringer, Robert","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-6781-0521"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1002/cpa.70040"}],"acknowledgement":"We thank the referees for valuable remarks. This work was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the TRR 352 – Project-ID 470903074. PTN was partially supported by the European Research Council via the ERC Consolidator Grant RAMBAS – Project-Nr. 10104424.\r\nOpen access publishing facilitated by Università degli Studi di Milano, as part of the Wiley - CRUI-CARE agreement.","publication_identifier":{"eissn":["1097-0312"],"issn":["0010-3640"]},"oa":1,"external_id":{"arxiv":["2409.17914"]},"department":[{"_id":"RoSe"}],"date_created":"2026-03-22T23:04:33Z","language":[{"iso":"eng"}],"publication_status":"epub_ahead","abstract":[{"lang":"eng","text":"We study the ground state energy of a gas of spin 1/2 fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density e, with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang–Yang correction term of order e^7/3. Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation."}],"quality_controlled":"1"},{"file":[{"relation":"main_file","access_level":"open_access","success":1,"checksum":"fbcc9cc7bf274f024e4f4afc9c208f96","content_type":"application/pdf","date_updated":"2025-01-09T09:36:41Z","date_created":"2025-01-09T09:36:41Z","file_id":"18803","file_name":"2024_CommPureApplMath_Erdoes.pdf","creator":"dernst","file_size":566963}],"acknowledgement":"László Erdős is partially supported by ERC Advanced Grant “RMTBeyond” No. 101020331. Hong Chang Ji is supported by ERC Advanced Grant “RMTBeyond” No. 101020331.","oa":1,"publication_identifier":{"issn":["0010-3640"],"eissn":["1097-0312"]},"publication":"Communications on Pure and Applied Mathematics","file_date_updated":"2025-01-09T09:36:41Z","scopus_import":"1","author":[{"orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"Erdös, László"},{"last_name":"Ji","full_name":"Ji, Hong Chang","first_name":"Hong Chang","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d"}],"isi":1,"intvolume":"        77","ec_funded":1,"volume":77,"quality_controlled":"1","license":"https://creativecommons.org/licenses/by-nc-nd/4.0/","abstract":[{"lang":"eng","text":"We consider N×N non-Hermitian random matrices of the form X+A, where A is a general deterministic matrix and N−−√X consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by N1+o(1) and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1); both results are optimal up to the factor No(1). The latter result complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549) and improves the N-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819, arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower tail estimate for the small singular values of X+A−z, is of independent interest."}],"publication_status":"published","date_created":"2024-05-12T22:01:02Z","department":[{"_id":"LaEr"}],"external_id":{"arxiv":["2301.04981"],"isi":["001217139900001"]},"corr_author":"1","language":[{"iso":"eng"}],"status":"public","tmp":{"name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","image":"/images/cc_by_nc_nd.png","short":"CC BY-NC-ND (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode"},"issue":"9","day":"01","type":"journal_article","publisher":"Wiley","has_accepted_license":"1","date_published":"2024-09-01T00:00:00Z","project":[{"grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020"}],"date_updated":"2025-09-08T07:25:47Z","oa_version":"Published Version","year":"2024","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"3785-3840","_id":"15378","month":"09","title":"Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices","article_processing_charge":"Yes (via OA deal)","citation":{"apa":"Erdös, L., &#38; Ji, H. C. (2024). Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.22201\">https://doi.org/10.1002/cpa.22201</a>","ama":"Erdös L, Ji HC. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. 2024;77(9):3785-3840. doi:<a href=\"https://doi.org/10.1002/cpa.22201\">10.1002/cpa.22201</a>","ista":"Erdös L, Ji HC. 2024. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 77(9), 3785–3840.","chicago":"Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the Eigenvalue Condition Number of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2024. <a href=\"https://doi.org/10.1002/cpa.22201\">https://doi.org/10.1002/cpa.22201</a>.","short":"L. Erdös, H.C. Ji, Communications on Pure and Applied Mathematics 77 (2024) 3785–3840.","mla":"Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the Eigenvalue Condition Number of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>, vol. 77, no. 9, Wiley, 2024, pp. 3785–840, doi:<a href=\"https://doi.org/10.1002/cpa.22201\">10.1002/cpa.22201</a>.","ieee":"L. Erdös and H. C. Ji, “Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices,” <i>Communications on Pure and Applied Mathematics</i>, vol. 77, no. 9. Wiley, pp. 3785–3840, 2024."},"ddc":["510"],"arxiv":1,"article_type":"original","OA_type":"hybrid","OA_place":"publisher","doi":"10.1002/cpa.22201"},{"arxiv":1,"ddc":["510"],"doi":"10.1002/cpa.22028","article_type":"original","_id":"10405","title":"Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices","month":"05","article_processing_charge":"Yes (via OA deal)","citation":{"ista":"Cipolloni G, Erdös L, Schröder DJ. 2023. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 76(5), 946–1034.","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2023. <a href=\"https://doi.org/10.1002/cpa.22028\">https://doi.org/10.1002/cpa.22028</a>.","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2023). Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.22028\">https://doi.org/10.1002/cpa.22028</a>","ama":"Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices. <i>Communications on Pure and Applied Mathematics</i>. 2023;76(5):946-1034. doi:<a href=\"https://doi.org/10.1002/cpa.22028\">10.1002/cpa.22028</a>","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,” <i>Communications on Pure and Applied Mathematics</i>, vol. 76, no. 5. Wiley, pp. 946–1034, 2023.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Communications on Pure and Applied Mathematics 76 (2023) 946–1034.","mla":"Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” <i>Communications on Pure and Applied Mathematics</i>, vol. 76, no. 5, Wiley, 2023, pp. 946–1034, doi:<a href=\"https://doi.org/10.1002/cpa.22028\">10.1002/cpa.22028</a>."},"date_published":"2023-05-01T00:00:00Z","has_accepted_license":"1","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"},{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","grant_number":"665385","call_identifier":"H2020","name":"International IST Doctoral Program"}],"oa_version":"Published Version","year":"2023","date_updated":"2025-03-31T16:00:54Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"946-1034","status":"public","tmp":{"name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","image":"/images/cc_by_nc_nd.png","short":"CC BY-NC-ND (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode"},"day":"01","type":"journal_article","publisher":"Wiley","issue":"5","publication_status":"published","external_id":{"isi":["000724652500001"],"arxiv":["1912.04100"]},"department":[{"_id":"LaEr"}],"date_created":"2021-12-05T23:01:41Z","language":[{"iso":"eng"}],"corr_author":"1","volume":76,"quality_controlled":"1","abstract":[{"lang":"eng","text":"We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. "}],"file_date_updated":"2023-10-04T09:21:48Z","scopus_import":"1","publication":"Communications on Pure and Applied Mathematics","author":[{"last_name":"Cipolloni","full_name":"Cipolloni, Giorgio","first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4901-7992"},{"full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603"},{"orcid":"0000-0002-2904-1856","last_name":"Schröder","full_name":"Schröder, Dominik J","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"ec_funded":1,"intvolume":"        76","isi":1,"file":[{"relation":"main_file","access_level":"open_access","success":1,"content_type":"application/pdf","checksum":"8346bc2642afb4ccb7f38979f41df5d9","date_updated":"2023-10-04T09:21:48Z","file_id":"14388","date_created":"2023-10-04T09:21:48Z","file_name":"2023_CommPureMathematics_Cipolloni.pdf","creator":"dernst","file_size":803440}],"acknowledgement":"L.E. would like to thank Nathanaël Berestycki and D.S.would like to thank Nina Holden for valuable discussions on the Gaussian freefield.G.C. and L.E. are partially supported by ERC Advanced Grant No. 338804.G.C. received funding from the European Union’s Horizon 2020 research and in-novation programme under the Marie Skłodowska-Curie Grant Agreement No.665385. D.S. is supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.","oa":1,"publication_identifier":{"eissn":["1097-0312"],"issn":["0010-3640"]}},{"license":"https://creativecommons.org/licenses/by/4.0/","abstract":[{"text":"We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem.","lang":"eng"}],"volume":74,"quality_controlled":"1","date_created":"2020-10-04T22:01:37Z","external_id":{"isi":["000572991500001"]},"department":[{"_id":"RoSe"}],"language":[{"iso":"eng"}],"publication_status":"published","acknowledgement":"Partial support through National Science Foundation GrantDMS-1363432 (R.L.F.) and the European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation programme (grant agreementNo 694227; R.S.), is acknowledged. Open access funding enabled and organizedby Projekt DEAL.","file":[{"creator":"dernst","file_name":"2021_CommPureApplMath_Frank.pdf","date_created":"2021-03-11T10:03:30Z","file_id":"9236","file_size":334987,"content_type":"application/pdf","checksum":"5f665ffa6e6dd958aec5c3040cbcfa84","date_updated":"2021-03-11T10:03:30Z","success":1,"access_level":"open_access","relation":"main_file"}],"oa":1,"publication_identifier":{"eissn":["1097-0312"],"issn":["0010-3640"]},"scopus_import":"1","author":[{"first_name":"Rupert","full_name":"Frank, Rupert","last_name":"Frank"},{"orcid":"0000-0002-6781-0521","first_name":"Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","full_name":"Seiringer, Robert","last_name":"Seiringer"}],"publication":"Communications on Pure and Applied Mathematics","file_date_updated":"2021-03-11T10:03:30Z","intvolume":"        74","isi":1,"ec_funded":1,"article_processing_charge":"No","citation":{"ieee":"R. Frank and R. Seiringer, “Quantum corrections to the Pekar asymptotics of a strongly coupled polaron,” <i>Communications on Pure and Applied Mathematics</i>, vol. 74, no. 3. Wiley, pp. 544–588, 2021.","mla":"Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar Asymptotics of a Strongly Coupled Polaron.” <i>Communications on Pure and Applied Mathematics</i>, vol. 74, no. 3, Wiley, 2021, pp. 544–88, doi:<a href=\"https://doi.org/10.1002/cpa.21944\">10.1002/cpa.21944</a>.","short":"R. Frank, R. Seiringer, Communications on Pure and Applied Mathematics 74 (2021) 544–588.","chicago":"Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar Asymptotics of a Strongly Coupled Polaron.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2021. <a href=\"https://doi.org/10.1002/cpa.21944\">https://doi.org/10.1002/cpa.21944</a>.","ista":"Frank R, Seiringer R. 2021. Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. Communications on Pure and Applied Mathematics. 74(3), 544–588.","ama":"Frank R, Seiringer R. Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. <i>Communications on Pure and Applied Mathematics</i>. 2021;74(3):544-588. doi:<a href=\"https://doi.org/10.1002/cpa.21944\">10.1002/cpa.21944</a>","apa":"Frank, R., &#38; Seiringer, R. (2021). Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.21944\">https://doi.org/10.1002/cpa.21944</a>"},"_id":"8603","month":"03","title":"Quantum corrections to the Pekar asymptotics of a strongly coupled polaron","article_type":"original","doi":"10.1002/cpa.21944","ddc":["510"],"issue":"3","day":"01","type":"journal_article","publisher":"Wiley","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"544-588","has_accepted_license":"1","date_published":"2021-03-01T00:00:00Z","project":[{"grant_number":"694227","_id":"25C6DC12-B435-11E9-9278-68D0E5697425","name":"Analysis of quantum many-body systems","call_identifier":"H2020"}],"date_updated":"2025-07-10T11:57:12Z","year":"2021","oa_version":"Published Version"},{"day":"01","type":"journal_article","publisher":"Wiley","issue":"9","status":"public","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"1672 - 1705","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"date_published":"2017-09-01T00:00:00Z","year":"2017","oa_version":"Submitted Version","date_updated":"2025-09-10T10:58:02Z","article_processing_charge":"No","citation":{"apa":"Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>","ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. 2017;70(9):1672-1705. doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>","ista":"Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 70(9), 1672–1705.","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2017. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>.","mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>.","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017."},"_id":"721","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","month":"09","publist_id":"6959","doi":"10.1002/cpa.21639","arxiv":1,"main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"oa":1,"publication_identifier":{"issn":["0010-3640"]},"publication":"Communications on Pure and Applied Mathematics","scopus_import":"1","author":[{"full_name":"Ajanki, Oskari H","last_name":"Ajanki","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","full_name":"Krüger, Torben H"},{"full_name":"Erdös, László","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603"}],"ec_funded":1,"isi":1,"intvolume":"        70","abstract":[{"text":"Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.","lang":"eng"}],"volume":70,"quality_controlled":"1","department":[{"_id":"LaEr"}],"external_id":{"isi":["000405752100002"],"arxiv":["1512.03703"]},"date_created":"2018-12-11T11:48:08Z","corr_author":"1","language":[{"iso":"eng"}],"publication_status":"published"},{"month":"05","quality_controlled":"1","title":"Arnol′d diffusion in a pendulum lattice","_id":"8500","volume":67,"citation":{"ieee":"V. Kaloshin, M. Levi, and M. Saprykina, “Arnol′d diffusion in a pendulum lattice,” <i>Communications on Pure and Applied Mathematics</i>, vol. 67, no. 5. Wiley, pp. 748–775, 2014.","mla":"Kaloshin, Vadim, et al. “Arnol′d Diffusion in a Pendulum Lattice.” <i>Communications on Pure and Applied Mathematics</i>, vol. 67, no. 5, Wiley, 2014, pp. 748–75, doi:<a href=\"https://doi.org/10.1002/cpa.21509\">10.1002/cpa.21509</a>.","short":"V. Kaloshin, M. Levi, M. Saprykina, Communications on Pure and Applied Mathematics 67 (2014) 748–775.","chicago":"Kaloshin, Vadim, Mark Levi, and Maria Saprykina. “Arnol′d Diffusion in a Pendulum Lattice.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2014. <a href=\"https://doi.org/10.1002/cpa.21509\">https://doi.org/10.1002/cpa.21509</a>.","ista":"Kaloshin V, Levi M, Saprykina M. 2014. Arnol′d diffusion in a pendulum lattice. Communications on Pure and Applied Mathematics. 67(5), 748–775.","ama":"Kaloshin V, Levi M, Saprykina M. Arnol′d diffusion in a pendulum lattice. <i>Communications on Pure and Applied Mathematics</i>. 2014;67(5):748-775. doi:<a href=\"https://doi.org/10.1002/cpa.21509\">10.1002/cpa.21509</a>","apa":"Kaloshin, V., Levi, M., &#38; Saprykina, M. (2014). Arnol′d diffusion in a pendulum lattice. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.21509\">https://doi.org/10.1002/cpa.21509</a>"},"article_processing_charge":"No","abstract":[{"lang":"eng","text":"The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way."}],"publication_status":"published","article_type":"original","language":[{"iso":"eng"}],"doi":"10.1002/cpa.21509","date_created":"2020-09-18T10:47:01Z","publication_identifier":{"issn":["0010-3640"]},"status":"public","issue":"5","type":"journal_article","day":"01","publisher":"Wiley","date_updated":"2022-08-25T13:58:13Z","oa_version":"None","year":"2014","date_published":"2014-05-01T00:00:00Z","page":"748-775","intvolume":"        67","extern":"1","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","author":[{"orcid":"0000-0002-6051-2628","full_name":"Kaloshin, Vadim","last_name":"Kaloshin","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425"},{"first_name":"Mark","full_name":"Levi, Mark","last_name":"Levi"},{"first_name":"Maria","full_name":"Saprykina, Maria","last_name":"Saprykina"}],"keyword":["Applied Mathematics","General Mathematics"],"publication":"Communications on Pure and Applied Mathematics"},{"day":"01","type":"journal_article","publisher":"Wiley","issue":"9","status":"public","publication_identifier":{"issn":["0010-3640","1097-0312"]},"author":[{"last_name":"Dolgopyat","full_name":"Dolgopyat, Dmitry","first_name":"Dmitry"},{"orcid":"0000-0002-6051-2628","first_name":"Vadim","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","full_name":"Kaloshin, Vadim","last_name":"Kaloshin"},{"last_name":"Koralov","full_name":"Koralov, Leonid","first_name":"Leonid"}],"publication":"Communications on Pure and Applied Mathematics","keyword":["Applied Mathematics","General Mathematics"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","extern":"1","intvolume":"        57","page":"1127-1158","date_published":"2004-09-01T00:00:00Z","oa_version":"None","year":"2004","date_updated":"2021-01-12T08:19:50Z","article_processing_charge":"No","abstract":[{"lang":"eng","text":"We consider the evolution of a connected set on the plane carried by a space periodic incompressible stochastic flow. While for almost every realization of the stochastic flow at time t most of the particles are at a distance of order equation image away from the origin, there is a measure zero set of points that escape to infinity at the linear rate. We study the set of points visited by the original set by time t and show that such a set, when scaled down by the factor of t, has a limiting nonrandom shape."}],"citation":{"ieee":"D. Dolgopyat, V. Kaloshin, and L. Koralov, “A limit shape theorem for periodic stochastic dispersion,” <i>Communications on Pure and Applied Mathematics</i>, vol. 57, no. 9. Wiley, pp. 1127–1158, 2004.","mla":"Dolgopyat, Dmitry, et al. “A Limit Shape Theorem for Periodic Stochastic Dispersion.” <i>Communications on Pure and Applied Mathematics</i>, vol. 57, no. 9, Wiley, 2004, pp. 1127–58, doi:<a href=\"https://doi.org/10.1002/cpa.20032\">10.1002/cpa.20032</a>.","short":"D. Dolgopyat, V. Kaloshin, L. Koralov, Communications on Pure and Applied Mathematics 57 (2004) 1127–1158.","chicago":"Dolgopyat, Dmitry, Vadim Kaloshin, and Leonid Koralov. “A Limit Shape Theorem for Periodic Stochastic Dispersion.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2004. <a href=\"https://doi.org/10.1002/cpa.20032\">https://doi.org/10.1002/cpa.20032</a>.","ista":"Dolgopyat D, Kaloshin V, Koralov L. 2004. A limit shape theorem for periodic stochastic dispersion. Communications on Pure and Applied Mathematics. 57(9), 1127–1158.","ama":"Dolgopyat D, Kaloshin V, Koralov L. A limit shape theorem for periodic stochastic dispersion. <i>Communications on Pure and Applied Mathematics</i>. 2004;57(9):1127-1158. doi:<a href=\"https://doi.org/10.1002/cpa.20032\">10.1002/cpa.20032</a>","apa":"Dolgopyat, D., Kaloshin, V., &#38; Koralov, L. (2004). A limit shape theorem for periodic stochastic dispersion. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.20032\">https://doi.org/10.1002/cpa.20032</a>"},"_id":"8517","volume":57,"quality_controlled":"1","title":"A limit shape theorem for periodic stochastic dispersion","month":"09","date_created":"2020-09-18T10:49:12Z","doi":"10.1002/cpa.20032","language":[{"iso":"eng"}],"article_type":"original","publication_status":"published"},{"article_type":"original","publist_id":"4161","doi":"10.1002/(SICI)1097-0312(200006)53:6&lt;667::AID-CPA1&gt;3.0.CO;2-5","arxiv":1,"article_processing_charge":"No","citation":{"ista":"Erdös L, Yau H. 2000. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Communications on Pure and Applied Mathematics. 53(6), 667–735.","chicago":"Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation.” <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell, 2000. <a href=\"https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5\">https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5</a>.","apa":"Erdös, L., &#38; Yau, H. (2000). Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell. <a href=\"https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5\">https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5</a>","ama":"Erdös L, Yau H. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. <i>Communications on Pure and Applied Mathematics</i>. 2000;53(6):667-735. doi:<a href=\"https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5\">10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5</a>","ieee":"L. Erdös and H. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,” <i>Communications on Pure and Applied Mathematics</i>, vol. 53, no. 6. Wiley-Blackwell, pp. 667–735, 2000.","short":"L. Erdös, H. Yau, Communications on Pure and Applied Mathematics 53 (2000) 667–735.","mla":"Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation.” <i>Communications on Pure and Applied Mathematics</i>, vol. 53, no. 6, Wiley-Blackwell, 2000, pp. 667–735, doi:<a href=\"https://doi.org/10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5\">10.1002/(SICI)1097-0312(200006)53:6&#38;lt;667::AID-CPA1&#38;gt;3.0.CO;2-5</a>."},"_id":"2731","month":"06","title":"Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","page":"667 - 735","extern":"1","date_published":"2000-06-01T00:00:00Z","date_updated":"2023-05-03T09:09:04Z","year":"2000","oa_version":"Preprint","issue":"6","type":"journal_article","publisher":"Wiley-Blackwell","day":"01","status":"public","date_created":"2018-12-11T11:59:18Z","external_id":{"arxiv":["math-ph/9901020"]},"language":[{"iso":"eng"}],"publication_status":"published","abstract":[{"lang":"eng","text":"We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section."}],"volume":53,"quality_controlled":"1","scopus_import":"1","author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"first_name":"Horng","full_name":"Yau, Horng","last_name":"Yau"}],"publication":"Communications on Pure and Applied Mathematics","intvolume":"        53","acknowledgement":"Partially supported by U.S. National Science Foundation grants DMS-9403462, 9703752. We would like to thank H. Spohn for his several comments and discussions on this project. Part of this work was done during the time when L. E. visited the Erwin Schrödinger Institute in Vienna and when both authors visited the Center of Theoretical Sciences in Taiwan. We thank them for the hospitality and the support of this work.","publication_identifier":{"issn":["0010-3640"]}}]