[{"status":"public","author":[{"first_name":"Federico","last_name":"Cornalba","full_name":"Cornalba, Federico"},{"first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","full_name":"Fischer, Julian L","last_name":"Fischer","orcid":"0000-0002-0479-558X"},{"orcid":"0009-0008-1310-7946","first_name":"Jonas","last_name":"Ingmanns","full_name":"Ingmanns, Jonas","id":"71523d30-15b2-11ec-abd3-f80aa909d6b0"},{"first_name":"Claudia","last_name":"Raithel","full_name":"Raithel, Claudia"}],"corr_author":"1","year":"2026","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":54,"date_updated":"2026-05-21T07:21:25Z","language":[{"iso":"eng"}],"keyword":["Weakly interacting particle systems","fluctuating hydrodynamics","Dean-Kawasaki equation","stochastic PDEs","numerical approximation"],"license":"https://creativecommons.org/licenses/by/4.0/","PlanS_conform":"1","page":"155-215","external_id":{"arxiv":["2303.00429"]},"ec_funded":1,"file_date_updated":"2026-05-21T07:11:27Z","arxiv":1,"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"ddc":["510"],"oa_version":"Published Version","issue":"1","title":"Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation","article_processing_charge":"Yes (in subscription journal)","publisher":"Institute of Mathematical Statistics","date_created":"2026-05-20T08:25:25Z","OA_place":"publisher","date_published":"2026-01-01T00:00:00Z","article_type":"original","APC_amount":"1352,08 EUR","department":[{"_id":"JuFi"}],"has_accepted_license":"1","file":[{"checksum":"3e60c0e25a1c96342029a7d2b031505f","file_name":"2026_AnnalsProbability_Cornalba.pdf","file_id":"21906","success":1,"content_type":"application/pdf","file_size":865745,"date_updated":"2026-05-21T07:11:27Z","relation":"main_file","creator":"dernst","access_level":"open_access","date_created":"2026-05-21T07:11:27Z"}],"scopus_import":"1","publication_identifier":{"issn":["0091-1798"],"eissn":["2168-894X"]},"doi":"10.1214/25-aop1763","citation":{"ista":"Cornalba F, Fischer JL, Ingmanns J, Raithel C. 2026. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. The Annals of Probability. 54(1), 155–215.","short":"F. Cornalba, J.L. Fischer, J. Ingmanns, C. Raithel, The Annals of Probability 54 (2026) 155–215.","mla":"Cornalba, Federico, et al. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” <i>The Annals of Probability</i>, vol. 54, no. 1, Institute of Mathematical Statistics, 2026, pp. 155–215, doi:<a href=\"https://doi.org/10.1214/25-aop1763\">10.1214/25-aop1763</a>.","chicago":"Cornalba, Federico, Julian L Fischer, Jonas Ingmanns, and Claudia Raithel. “Density Fluctuations in Weakly Interacting Particle Systems via the Dean–Kawasaki Equation.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2026. <a href=\"https://doi.org/10.1214/25-aop1763\">https://doi.org/10.1214/25-aop1763</a>.","ama":"Cornalba F, Fischer JL, Ingmanns J, Raithel C. Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. <i>The Annals of Probability</i>. 2026;54(1):155-215. doi:<a href=\"https://doi.org/10.1214/25-aop1763\">10.1214/25-aop1763</a>","apa":"Cornalba, F., Fischer, J. L., Ingmanns, J., &#38; Raithel, C. (2026). Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/25-aop1763\">https://doi.org/10.1214/25-aop1763</a>","ieee":"F. Cornalba, J. L. Fischer, J. Ingmanns, and C. Raithel, “Density fluctuations in weakly interacting particle systems via the Dean–Kawasaki equation,” <i>The Annals of Probability</i>, vol. 54, no. 1. Institute of Mathematical Statistics, pp. 155–215, 2026."},"day":"01","intvolume":"        54","publication_status":"published","fulldoi":"https://doi.org/10.1214/25-aop1763","type":"journal_article","project":[{"name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","call_identifier":"H2020"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"}],"OA_type":"hybrid","_id":"21894","oa":1,"month":"01","quality_controlled":"1","acknowledgement":"All authors gratefully acknowledge funding from the Austrian Science Fund (FWF) through the project F65. CR gratefully acknowledges support from the Austrian Science Fund (FWF), grants P30000, P33010, W1245. FC gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411.","abstract":[{"lang":"eng","text":"The Dean–Kawasaki equation—one of the most fundamental SPDEs of\r\nfluctuating hydrodynamics—has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form, it is highly\r\nsingular and fails to be renormalizable, even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is\r\nnatural to introduce a suitable regularization, for example, by applying a formal spatial discretization or by truncating high-frequency noise: This yields\r\nwell-posed equations that should still precisely approximate the law of the\r\nparticle density fluctuations.\r\nIn the present work, we prove that a regularization in the form of a formal\r\ndiscretization of the Dean–Kawasaki equation indeed accurately describes\r\ndensity fluctuations in systems of weakly interacting diffusing particles: We\r\nshow that, in suitable weak metrics, the law of fluctuations as predicted by\r\nthe discretized Dean–Kawasaki SPDE approximates the law of fluctuations\r\nof the original particle system, up to an error that is of arbitrarily high order in\r\nthe inverse particle number and a discretization error. In particular, the Dean–\r\nKawasaki equation provides a means for efficient and accurate simulations of\r\ndensity fluctuations in weakly interacting particle systems."}],"publication":"The Annals of Probability"},{"external_id":{"arxiv":["2404.17512"]},"oa_version":"Preprint","issue":"6","ec_funded":1,"arxiv":1,"article_processing_charge":"No","title":"On the spectral edge of non-Hermitian random matrices","date_created":"2026-02-17T07:58:20Z","OA_place":"repository","publisher":"Institute of Mathematical Statistics","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","corr_author":"1","author":[{"first_name":"Andrew J","last_name":"Campbell","full_name":"Campbell, Andrew J","id":"582b06a9-1f1c-11ee-b076-82ffce00dde4"},{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","last_name":"Cipolloni","full_name":"Cipolloni, Giorgio","first_name":"Giorgio","orcid":"0000-0002-4901-7992"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603"},{"first_name":"Hong Chang","full_name":"Ji, Hong Chang","last_name":"Ji","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d"}],"volume":53,"language":[{"iso":"eng"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2404.17512","open_access":"1"}],"date_updated":"2026-02-18T08:35:38Z","page":"2256-2308","OA_type":"green","oa":1,"month":"11","_id":"21271","acknowledgement":"The authors would like to thank the anonymous referee for providing helpful comments and suggestions. We also thank Joscha Henheik and Volodymyr Riabov for pointing out a gap in an earlier version of the proof of equation (3.18). The first, third, and fourth authors are supported by ERC Advanced Grant “RMTBeyond” No. 101020331.","quality_controlled":"1","publication":"The Annals of Probability","abstract":[{"text":"For general non-Hermitian large random matrices X and deterministic deformation matrices A, we prove that the local eigenvalue statistics of A+X close to the typical edge points of its spectrum are universal. Furthermore, we show that, under natural assumptions, on A the spectrum of A+X does not have outliers at a distance larger than the natural fluctuation scale of the eigenvalues. As a consequence, the number of eigenvalues in each component of Spec(A+X) is deterministic.","lang":"eng"}],"department":[{"_id":"LaEr"}],"date_published":"2025-11-01T00:00:00Z","article_type":"original","citation":{"mla":"Campbell, Andrew J., et al. “On the Spectral Edge of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>, vol. 53, no. 6, Institute of Mathematical Statistics, 2025, pp. 2256–308, doi:<a href=\"https://doi.org/10.1214/25-aop1761\">10.1214/25-aop1761</a>.","chicago":"Campbell, Andrew J, Giorgio Cipolloni, László Erdös, and Hong Chang Ji. “On the Spectral Edge of Non-Hermitian Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2025. <a href=\"https://doi.org/10.1214/25-aop1761\">https://doi.org/10.1214/25-aop1761</a>.","ista":"Campbell AJ, Cipolloni G, Erdös L, Ji HC. 2025. On the spectral edge of non-Hermitian random matrices. The Annals of Probability. 53(6), 2256–2308.","short":"A.J. Campbell, G. Cipolloni, L. Erdös, H.C. Ji, The Annals of Probability 53 (2025) 2256–2308.","apa":"Campbell, A. J., Cipolloni, G., Erdös, L., &#38; Ji, H. C. (2025). On the spectral edge of non-Hermitian random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/25-aop1761\">https://doi.org/10.1214/25-aop1761</a>","ieee":"A. J. Campbell, G. Cipolloni, L. Erdös, and H. C. Ji, “On the spectral edge of non-Hermitian random matrices,” <i>The Annals of Probability</i>, vol. 53, no. 6. Institute of Mathematical Statistics, pp. 2256–2308, 2025.","ama":"Campbell AJ, Cipolloni G, Erdös L, Ji HC. On the spectral edge of non-Hermitian random matrices. <i>The Annals of Probability</i>. 2025;53(6):2256-2308. doi:<a href=\"https://doi.org/10.1214/25-aop1761\">10.1214/25-aop1761</a>"},"doi":"10.1214/25-aop1761","publication_identifier":{"eissn":["2168-894X"],"issn":["0091-1798"]},"publication_status":"published","fulldoi":"https://doi.org/10.1214/25-aop1761","type":"journal_article","day":"01","intvolume":"        53","project":[{"grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d"}]},{"language":[{"iso":"eng"}],"main_file_link":[{"url":" https://doi.org/10.48550/arXiv.1811.11598","open_access":"1"}],"date_updated":"2025-04-14T07:27:47Z","page":"591-648","year":"2022","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","author":[{"id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","full_name":"Dello Schiavo, Lorenzo","last_name":"Dello Schiavo","first_name":"Lorenzo","orcid":"0000-0002-9881-6870"}],"corr_author":"1","volume":50,"isi":1,"article_processing_charge":"No","title":"The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold","date_created":"2022-05-08T22:01:44Z","publisher":"Institute of Mathematical Statistics","external_id":{"arxiv":["1811.11598"],"isi":["000773518500005"]},"oa_version":"Preprint","issue":"2","ec_funded":1,"arxiv":1,"fulldoi":"https://doi.org/10.1214/21-AOP1541","publication_status":"published","type":"journal_article","day":"01","intvolume":"        50","project":[{"name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","grant_number":"716117","call_identifier":"H2020"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"department":[{"_id":"JaMa"}],"date_published":"2022-03-01T00:00:00Z","article_type":"original","doi":"10.1214/21-AOP1541","publication_identifier":{"eissn":["2168-894X"],"issn":["0091-1798"]},"citation":{"mla":"Dello Schiavo, Lorenzo. “The Dirichlet–Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold.” <i>Annals of Probability</i>, vol. 50, no. 2, Institute of Mathematical Statistics, 2022, pp. 591–648, doi:<a href=\"https://doi.org/10.1214/21-AOP1541\">10.1214/21-AOP1541</a>.","chicago":"Dello Schiavo, Lorenzo. “The Dirichlet–Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2022. <a href=\"https://doi.org/10.1214/21-AOP1541\">https://doi.org/10.1214/21-AOP1541</a>.","ista":"Dello Schiavo L. 2022. The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold. Annals of Probability. 50(2), 591–648.","short":"L. Dello Schiavo, Annals of Probability 50 (2022) 591–648.","ieee":"L. Dello Schiavo, “The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold,” <i>Annals of Probability</i>, vol. 50, no. 2. Institute of Mathematical Statistics, pp. 591–648, 2022.","apa":"Dello Schiavo, L. (2022). The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/21-AOP1541\">https://doi.org/10.1214/21-AOP1541</a>","ama":"Dello Schiavo L. The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold. <i>Annals of Probability</i>. 2022;50(2):591-648. doi:<a href=\"https://doi.org/10.1214/21-AOP1541\">10.1214/21-AOP1541</a>"},"scopus_import":"1","acknowledgement":"Research supported by the Sonderforschungsbereich 1060 and the Hausdorff Center for Mathematics. The author gratefully acknowledges funding of his current position at IST Austria by the Austrian Science Fund (FWF) grant F65 and by the European Research Council (ERC, Grant agreement No. 716117, awarded to Prof. Dr. Jan Maas).","quality_controlled":"1","publication":"Annals of Probability","abstract":[{"text":"We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.","lang":"eng"}],"oa":1,"month":"03","_id":"11354"},{"oa":1,"month":"05","_id":"11418","publication":"Annals of Probability","abstract":[{"lang":"eng","text":"We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N\r\nWigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048)."}],"acknowledgement":"L.E. would like to thank Zhigang Bao for many illuminating discussions in an early stage of this research. The authors are also grateful to Paul Bourgade for his comments on the manuscript and the anonymous referee for several useful suggestions.","quality_controlled":"1","citation":{"apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Normal fluctuation in quantum ergodicity for Wigner matrices. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/21-AOP1552\">https://doi.org/10.1214/21-AOP1552</a>","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Normal fluctuation in quantum ergodicity for Wigner matrices,” <i>Annals of Probability</i>, vol. 50, no. 3. Institute of Mathematical Statistics, pp. 984–1012, 2022.","ama":"Cipolloni G, Erdös L, Schröder DJ. Normal fluctuation in quantum ergodicity for Wigner matrices. <i>Annals of Probability</i>. 2022;50(3):984-1012. doi:<a href=\"https://doi.org/10.1214/21-AOP1552\">10.1214/21-AOP1552</a>","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Normal Fluctuation in Quantum Ergodicity for Wigner Matrices.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2022. <a href=\"https://doi.org/10.1214/21-AOP1552\">https://doi.org/10.1214/21-AOP1552</a>.","mla":"Cipolloni, Giorgio, et al. “Normal Fluctuation in Quantum Ergodicity for Wigner Matrices.” <i>Annals of Probability</i>, vol. 50, no. 3, Institute of Mathematical Statistics, 2022, pp. 984–1012, doi:<a href=\"https://doi.org/10.1214/21-AOP1552\">10.1214/21-AOP1552</a>.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Annals of Probability 50 (2022) 984–1012.","ista":"Cipolloni G, Erdös L, Schröder DJ. 2022. Normal fluctuation in quantum ergodicity for Wigner matrices. Annals of Probability. 50(3), 984–1012."},"publication_identifier":{"issn":["0091-1798"],"eissn":["2168-894X"]},"doi":"10.1214/21-AOP1552","scopus_import":"1","department":[{"_id":"LaEr"}],"date_published":"2022-05-01T00:00:00Z","article_type":"original","fulldoi":"https://doi.org/10.1214/21-AOP1552","publication_status":"published","type":"journal_article","day":"01","intvolume":"        50","oa_version":"Preprint","issue":"3","arxiv":1,"external_id":{"arxiv":["2103.06730"],"isi":["000793963400005"]},"date_created":"2022-05-29T22:01:53Z","publisher":"Institute of Mathematical Statistics","article_processing_charge":"No","title":"Normal fluctuation in quantum ergodicity for Wigner matrices","volume":50,"isi":1,"year":"2022","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","author":[{"orcid":"0000-0002-4901-7992","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio","last_name":"Cipolloni","first_name":"Giorgio"},{"orcid":"0000-0001-5366-9603","first_name":"László","full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-2904-1856","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J","last_name":"Schröder","first_name":"Dominik J"}],"page":"984-1012","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/2103.06730","open_access":"1"}],"date_updated":"2023-08-03T07:16:53Z"}]