[{"abstract":[{"lang":"eng","text":"We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law."}],"publication_status":"published","title":"Universality for random matrix flows with time dependent density","oa_version":"Submitted Version","page":"1606 - 1656","main_file_link":[{"url":"https://arxiv.org/abs/1504.00650","open_access":"1"}],"quality_controlled":"1","intvolume":" 53","volume":53,"publication_identifier":{"issn":["0246-0203"]},"author":[{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli","first_name":"Kevin"}],"date_created":"2018-12-11T11:47:30Z","doi":"10.1214/16-AIHP765","_id":"615","scopus_import":"1","date_published":"2017-11-01T00:00:00Z","oa":1,"citation":{"apa":"Erdös, L., & Schnelli, K. (2017). Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AIHP765","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656.","mla":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:10.1214/16-AIHP765.","chicago":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-AIHP765.","ama":"Erdös L, Schnelli K. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 2017;53(4):1606-1656. doi:10.1214/16-AIHP765","ieee":"L. Erdös and K. Schnelli, “Universality for random matrix flows with time dependent density,” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4. Institute of Mathematical Statistics, pp. 1606–1656, 2017.","ista":"Erdös L, Schnelli K. 2017. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 53(4), 1606–1656."},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"date_updated":"2025-09-11T07:32:50Z","external_id":{"arxiv":["1504.00650"],"isi":["000416376500005"]},"status":"public","article_processing_charge":"No","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","isi":1,"department":[{"_id":"LaEr"}],"ec_funded":1,"type":"journal_article","project":[{"grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"year":"2017","publisher":"Institute of Mathematical Statistics","corr_author":"1","month":"11","issue":"4","day":"01","publist_id":"7189"},{"article_processing_charge":"No","publication":"Communications on Pure and Applied Mathematics","department":[{"_id":"LaEr"}],"isi":1,"external_id":{"isi":["000405752100002"],"arxiv":["1512.03703"]},"status":"public","date_updated":"2025-09-10T10:58:02Z","language":[{"iso":"eng"}],"day":"01","issue":"9","publist_id":"6959","publisher":"Wiley","corr_author":"1","month":"09","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"}],"year":"2017","ec_funded":1,"quality_controlled":"1","page":"1672 - 1705","main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"publication_status":"published","oa_version":"Submitted Version","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","abstract":[{"lang":"eng","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."}],"oa":1,"date_published":"2017-09-01T00:00:00Z","arxiv":1,"citation":{"ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 2017;70(9):1672-1705. doi:10.1002/cpa.21639","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics, vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:10.1002/cpa.21639.","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics. Wiley, 2017. https://doi.org/10.1002/cpa.21639.","apa":"Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.21639","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.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” Communications on Pure and Applied Mathematics, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:48:08Z","doi":"10.1002/cpa.21639","author":[{"id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","full_name":"Ajanki, Oskari H","last_name":"Ajanki","first_name":"Oskari H"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László"}],"_id":"721","scopus_import":"1","publication_identifier":{"issn":["0010-3640"]},"volume":70,"intvolume":" 70"},{"arxiv":1,"citation":{"ama":"Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 2017;319:251-291. doi:10.1016/j.aim.2017.08.028","mla":"Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” Advances in Mathematics, vol. 319, Academic Press, 2017, pp. 251–91, doi:10.1016/j.aim.2017.08.028.","short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2017.08.028.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2017.08.028","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” Advances in Mathematics, vol. 319. Academic Press, pp. 251–291, 2017."},"date_published":"2017-10-15T00:00:00Z","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","last_name":"Schnelli","orcid":"0000-0003-0954-3231","first_name":"Kevin"}],"date_created":"2018-12-11T11:48:13Z","doi":"10.1016/j.aim.2017.08.028","scopus_import":"1","_id":"733","volume":319,"acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","intvolume":" 319","quality_controlled":"1","page":"251 - 291","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1606.03076"}],"publication_status":"published","oa_version":"Submitted Version","title":"Convergence rate for spectral distribution of addition of random matrices","abstract":[{"lang":"eng","text":"Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum."}],"day":"15","publist_id":"6935","publisher":"Academic Press","month":"10","corr_author":"1","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","year":"2017","ec_funded":1,"article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Advances in Mathematics","external_id":{"arxiv":["1606.03076"],"isi":["000412150400010"]},"status":"public","date_updated":"2025-06-04T10:13:45Z","language":[{"iso":"eng"}]},{"external_id":{"isi":["000404011700017"]},"status":"public","article_processing_charge":"No","publication":"Revista Latino-Americana de Probabilidade e Estatística","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"isi":1,"language":[{"iso":"eng"}],"date_updated":"2025-09-18T10:02:36Z","publisher":"Instituto Nacional de Matematica Pura e Aplicada","corr_author":"1","month":"03","day":"23","publist_id":"7376","ec_funded":1,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"type":"journal_article","year":"2017","page":"299 - 325","main_file_link":[{"url":"http://alea.impa.br/articles/v14/14-17.pdf","open_access":"1"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015)."}],"publication_status":"published","oa_version":"Submitted Version","title":"Fluctuations of the competition interface in presence of shocks","author":[{"last_name":"Ferrari","first_name":"Patrik","full_name":"Ferrari, Patrik"},{"full_name":"Nejjar, Peter","id":"4BF426E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter","last_name":"Nejjar"}],"date_created":"2018-12-11T11:46:31Z","doi":"10.30757/ALEA.v14-17","_id":"447","scopus_import":"1","oa":1,"date_published":"2017-03-23T00:00:00Z","citation":{"ista":"Ferrari P, Nejjar P. 2017. Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. 9, 299–325.","ieee":"P. Ferrari and P. Nejjar, “Fluctuations of the competition interface in presence of shocks,” Revista Latino-Americana de Probabilidade e Estatística, vol. 9. Instituto Nacional de Matematica Pura e Aplicada, pp. 299–325, 2017.","ama":"Ferrari P, Nejjar P. Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. 2017;9:299-325. doi:10.30757/ALEA.v14-17","mla":"Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística, vol. 9, Instituto Nacional de Matematica Pura e Aplicada, 2017, pp. 299–325, doi:10.30757/ALEA.v14-17.","chicago":"Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística. Instituto Nacional de Matematica Pura e Aplicada, 2017. https://doi.org/10.30757/ALEA.v14-17.","short":"P. Ferrari, P. Nejjar, Revista Latino-Americana de Probabilidade e Estatística 9 (2017) 299–325.","apa":"Ferrari, P., & Nejjar, P. (2017). Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. Instituto Nacional de Matematica Pura e Aplicada. https://doi.org/10.30757/ALEA.v14-17"},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","intvolume":" 9","volume":9,"article_type":"original"},{"language":[{"iso":"eng"}],"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6179"}]},"file":[{"content_type":"application/pdf","date_updated":"2018-12-12T10:18:10Z","file_size":440770,"relation":"main_file","access_level":"open_access","creator":"system","file_name":"IST-2017-747-v1+1_euclid.ecp.1483347665.pdf","file_id":"5329","date_created":"2018-12-12T10:18:10Z"}],"date_updated":"2026-04-08T13:55:03Z","ddc":["510"],"status":"public","article_number":"86","external_id":{"isi":["000396604900037"]},"isi":1,"has_accepted_license":"1","department":[{"_id":"LaEr"}],"publication":"Electronic Communications in Probability","article_processing_charge":"No","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ec_funded":1,"year":"2017","pubrep_id":"747","project":[{"grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","month":"01","publisher":"Institute of Mathematical Statistics","day":"02","publist_id":"6214","abstract":[{"lang":"eng","text":"We show that matrix elements of functions of N × N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11]."}],"oa_version":"Published Version","title":"Fluctuations of functions of Wigner matrices","publication_status":"published","file_date_updated":"2018-12-12T10:18:10Z","quality_controlled":"1","acknowledgement":"Partially supported by the IST Austria Excellence Scholarship.","intvolume":" 21","volume":21,"scopus_import":"1","_id":"1144","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J","last_name":"Schröder","orcid":"0000-0002-2904-1856","first_name":"Dominik J"}],"doi":"10.1214/16-ECP38","date_created":"2018-12-11T11:50:23Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ista":"Erdös L, Schröder DJ. 2017. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 21, 86.","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of functions of Wigner matrices,” Electronic Communications in Probability, vol. 21. Institute of Mathematical Statistics, 2017.","chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Functions of Wigner Matrices.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-ECP38.","short":"L. Erdös, D.J. Schröder, Electronic Communications in Probability 21 (2017).","mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Functions of Wigner Matrices.” Electronic Communications in Probability, vol. 21, 86, Institute of Mathematical Statistics, 2017, doi:10.1214/16-ECP38.","ama":"Erdös L, Schröder DJ. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 2017;21. doi:10.1214/16-ECP38","apa":"Erdös, L., & Schröder, D. J. (2017). Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/16-ECP38"},"date_published":"2017-01-02T00:00:00Z","oa":1},{"quality_controlled":"1","file_date_updated":"2018-12-12T10:13:39Z","abstract":[{"text":"We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. ","lang":"eng"}],"publication_status":"published","oa_version":"Published Version","title":"Local law for random Gram matrices","author":[{"full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"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"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger"}],"date_created":"2018-12-11T11:49:40Z","doi":"10.1214/17-EJP42","scopus_import":"1","_id":"1010","citation":{"ista":"Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic Journal of Probability. 22, 25.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,” Electronic Journal of Probability, vol. 22. Institute of Mathematical Statistics, 2017.","ama":"Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. Electronic Journal of Probability. 2017;22. doi:10.1214/17-EJP42","mla":"Alt, Johannes, et al. “Local Law for Random Gram Matrices.” Electronic Journal of Probability, vol. 22, 25, Institute of Mathematical Statistics, 2017, doi:10.1214/17-EJP42.","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-EJP42.","short":"J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017).","apa":"Alt, J., Erdös, L., & Krüger, T. H. (2017). Local law for random Gram matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/17-EJP42"},"arxiv":1,"date_published":"2017-03-08T00:00:00Z","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":" 22","volume":22,"publication_identifier":{"issn":["1083-6489"]},"article_number":"25","external_id":{"isi":["000396611900025"],"arxiv":["1606.07353"]},"status":"public","article_processing_charge":"No","has_accepted_license":"1","isi":1,"department":[{"_id":"LaEr"}],"publication":"Electronic Journal of Probability","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"file":[{"date_updated":"2018-12-12T10:13:39Z","file_size":639384,"content_type":"application/pdf","relation":"main_file","creator":"system","access_level":"open_access","file_id":"5024","date_created":"2018-12-12T10:13:39Z","file_name":"IST-2017-807-v1+1_euclid.ejp.1488942016.pdf"}],"ddc":["510","539"],"date_updated":"2026-04-08T14:11:36Z","publisher":"Institute of Mathematical Statistics","month":"03","day":"08","publist_id":"6386","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"pubrep_id":"807","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"}],"type":"journal_article","year":"2017"},{"date_created":"2018-12-11T11:47:07Z","author":[{"first_name":"Johannes","last_name":"Alt","full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87"}],"doi":"10.1214/17-ECP97","scopus_import":"1","_id":"550","citation":{"ieee":"J. Alt, “Singularities of the density of states of random Gram matrices,” Electronic Communications in Probability, vol. 22. Institute of Mathematical Statistics, 2017.","ista":"Alt J. 2017. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 22, 63.","apa":"Alt, J. (2017). Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/17-ECP97","ama":"Alt J. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 2017;22. doi:10.1214/17-ECP97","short":"J. Alt, Electronic Communications in Probability 22 (2017).","mla":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” Electronic Communications in Probability, vol. 22, 63, Institute of Mathematical Statistics, 2017, doi:10.1214/17-ECP97.","chicago":"Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-ECP97."},"date_published":"2017-11-21T00:00:00Z","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","intvolume":" 22","volume":22,"publication_identifier":{"issn":["1083-589X"]},"quality_controlled":"1","file_date_updated":"2020-07-14T12:47:00Z","abstract":[{"lang":"eng","text":"For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities."}],"publication_status":"published","title":"Singularities of the density of states of random Gram matrices","oa_version":"Published Version","publisher":"Institute of Mathematical Statistics","month":"11","corr_author":"1","publist_id":"7265","day":"21","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"pubrep_id":"926","type":"journal_article","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"year":"2017","article_number":"63","external_id":{"isi":["000416389200001"]},"status":"public","article_processing_charge":"No","department":[{"_id":"LaEr"}],"isi":1,"has_accepted_license":"1","publication":"Electronic Communications in Probability","language":[{"iso":"eng"}],"related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"ddc":["539"],"date_updated":"2026-04-08T14:11:36Z","file":[{"date_updated":"2020-07-14T12:47:00Z","file_size":470876,"content_type":"application/pdf","relation":"main_file","checksum":"0ec05303a0de190de145654237984c79","date_created":"2018-12-12T10:08:04Z","file_id":"4663","file_name":"IST-2018-926-v1+1_euclid.ecp.1511233247.pdf","creator":"system","access_level":"open_access"}]},{"date_created":"2018-12-11T11:51:27Z","doi":"10.1007/s00440-016-0740-2","author":[{"full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H","last_name":"Ajanki"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger","first_name":"Torben H"}],"_id":"1337","scopus_import":"1","oa":1,"date_published":"2017-12-01T00:00:00Z","citation":{"ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type matrices,” Probability Theory and Related Fields, vol. 169, no. 3–4. Springer, pp. 667–727, 2017.","ista":"Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727.","apa":"Ajanki, O. H., Erdös, L., & Krüger, T. H. (2017). Universality for general Wigner-type matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-016-0740-2","ama":"Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 2017;169(3-4):667-727. doi:10.1007/s00440-016-0740-2","mla":"Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields, vol. 169, no. 3–4, Springer, 2017, pp. 667–727, doi:10.1007/s00440-016-0740-2.","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields. Springer, 2017. https://doi.org/10.1007/s00440-016-0740-2.","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727."},"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","intvolume":" 169","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). ","publication_identifier":{"issn":["0178-8051"]},"volume":169,"page":"667 - 727","file_date_updated":"2020-07-14T12:44:44Z","quality_controlled":"1","abstract":[{"text":"We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.","lang":"eng"}],"publication_status":"published","oa_version":"Published Version","title":"Universality for general Wigner-type matrices","publisher":"Springer","corr_author":"1","month":"12","issue":"3-4","day":"01","publist_id":"5930","ec_funded":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"type":"journal_article","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"pubrep_id":"657","year":"2017","external_id":{"isi":["000414358400002"]},"status":"public","article_processing_charge":"Yes (via OA deal)","publication":"Probability Theory and Related Fields","has_accepted_license":"1","isi":1,"department":[{"_id":"LaEr"}],"language":[{"iso":"eng"}],"date_updated":"2026-04-16T09:55:44Z","ddc":["510","530"],"file":[{"content_type":"application/pdf","file_size":988843,"date_updated":"2020-07-14T12:44:44Z","relation":"main_file","checksum":"29f5a72c3f91e408aeb9e78344973803","date_created":"2018-12-12T10:08:25Z","file_name":"IST-2017-657-v1+2_s00440-016-0740-2.pdf","file_id":"4686","creator":"system","access_level":"open_access"}]},{"date_updated":"2026-04-16T09:55:56Z","file":[{"file_id":"4665","checksum":"67afa85ff1e220cbc1f9f477a828513c","file_name":"IST-2016-489-v1+1_s00440-015-0692-y.pdf","date_created":"2018-12-12T10:08:05Z","access_level":"open_access","creator":"system","relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:45:00Z","file_size":1615755}],"ddc":["530"],"language":[{"iso":"eng"}],"department":[{"_id":"LaEr"}],"isi":1,"has_accepted_license":"1","publication":"Probability Theory and Related Fields","article_processing_charge":"Yes (via OA deal)","status":"public","external_id":{"isi":["000398842700004"]},"year":"2017","pubrep_id":"489","type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ec_funded":1,"publist_id":"5644","issue":"3-4","day":"01","month":"04","corr_author":"1","publisher":"Springer","title":"Delocalization for a class of random block band matrices","oa_version":"Published Version","publication_status":"published","abstract":[{"lang":"eng","text":"We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized."}],"quality_controlled":"1","file_date_updated":"2020-07-14T12:45:00Z","page":"673 - 776","article_type":"original","publication_identifier":{"issn":["0178-8051"]},"volume":167,"acknowledgement":"Z. Bao was supported by ERC Advanced Grant RANMAT No. 338804; L. Erdős was partially supported by ERC Advanced Grant RANMAT No. 338804.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). The authors are very grateful to the anonymous referees for careful reading and valuable comments, which helped to improve the organization.","intvolume":" 167","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","citation":{"apa":"Bao, Z., & Erdös, L. (2017). Delocalization for a class of random block band matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-015-0692-y","short":"Z. Bao, L. Erdös, Probability Theory and Related Fields 167 (2017) 673–776.","chicago":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” Probability Theory and Related Fields. Springer, 2017. https://doi.org/10.1007/s00440-015-0692-y.","mla":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” Probability Theory and Related Fields, vol. 167, no. 3–4, Springer, 2017, pp. 673–776, doi:10.1007/s00440-015-0692-y.","ama":"Bao Z, Erdös L. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 2017;167(3-4):673-776. doi:10.1007/s00440-015-0692-y","ieee":"Z. Bao and L. Erdös, “Delocalization for a class of random block band matrices,” Probability Theory and Related Fields, vol. 167, no. 3–4. Springer, pp. 673–776, 2017.","ista":"Bao Z, Erdös L. 2017. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 167(3–4), 673–776."},"oa":1,"date_published":"2017-04-01T00:00:00Z","scopus_import":"1","_id":"1528","doi":"10.1007/s00440-015-0692-y","author":[{"first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"}],"date_created":"2018-12-11T11:52:32Z"},{"article_processing_charge":"No","department":[{"_id":"LaEr"}],"isi":1,"publication":"Annals of Applied Probability","external_id":{"isi":["000391240100016"],"arxiv":["1409.4979"]},"status":"public","date_updated":"2025-09-22T09:55:43Z","language":[{"iso":"eng"}],"issue":"6","day":"15","publist_id":"6201","publisher":"Institute of Mathematical Statistics","month":"12","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"year":"2016","ec_funded":1,"quality_controlled":"1","page":"3786 - 3839","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1409.4979"}],"publication_status":"published","oa_version":"Preprint","title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","abstract":[{"text":"We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay.","lang":"eng"}],"arxiv":1,"citation":{"ama":"Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 2016;26(6):3786-3839. doi:10.1214/16-AAP1193","mla":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:10.1214/16-AAP1193.","chicago":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/16-AAP1193.","short":"J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.","apa":"Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AAP1193","ista":"Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839.","ieee":"J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” Annals of Applied Probability, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016."},"date_published":"2016-12-15T00:00:00Z","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1214/16-AAP1193","date_created":"2018-12-11T11:50:27Z","author":[{"last_name":"Lee","first_name":"Ji","full_name":"Lee, Ji"},{"first_name":"Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli","full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"scopus_import":"1","_id":"1157","volume":26,"acknowledgement":"We thank Horng-Tzer Yau for numerous discussions and remarks. We are grateful to Ben Adlam, Jinho Baik, Zhigang Bao, Paul Bourgade, László Erd ̋os, Iain Johnstone and Antti Knowles for comments. We are also grate-\r\nful to the anonymous referee for carefully reading our manuscript and suggesting several improvements.","intvolume":" 26"},{"oa_version":"Preprint","title":"Bulk universality for deformed wigner matrices","publication_status":"published","abstract":[{"lang":"eng","text":"We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N."}],"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1405.6634"}],"page":"2349 - 2425","volume":44,"acknowledgement":"J.C. was supported in part by National Research Foundation of Korea Grant 2011-0013474 and TJ Park Junior Faculty Fellowship.\r\nK.S. was supported by ERC Advanced Grant RANMAT, No. 338804, and the \"Fund for Math.\"\r\nB.S. was supported by NSF GRFP Fellowship DGE-1144152.\r\nH.Y. was supported in part by NSF Grant DMS-13-07444 and Simons investigator fellowship. We thank Paul Bourgade, László Erd ̋os and Antti Knowles for helpful comments. We are grateful to the Taida Institute for Mathematical\r\nSciences and National Taiwan Universality for their hospitality during part of this\r\nresearch. We thank Thomas Spencer and the Institute for Advanced Study for their\r\nhospitality during the academic year 2013–2014. ","intvolume":" 44","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ama":"Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner matrices. Annals of Probability. 2016;44(3):2349-2425. doi:10.1214/15-AOP1023","chicago":"Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/15-AOP1023.","short":"J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.","mla":"Lee, Jioon, et al. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability, vol. 44, no. 3, Institute of Mathematical Statistics, 2016, pp. 2349–425, doi:10.1214/15-AOP1023.","apa":"Lee, J., Schnelli, K., Stetler, B., & Yau, H. (2016). Bulk universality for deformed wigner matrices. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOP1023","ista":"Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425.","ieee":"J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” Annals of Probability, vol. 44, no. 3. Institute of Mathematical Statistics, pp. 2349–2425, 2016."},"arxiv":1,"date_published":"2016-01-01T00:00:00Z","oa":1,"scopus_import":"1","_id":"1219","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"first_name":"Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli","full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Stetler","first_name":"Ben","full_name":"Stetler, Ben"},{"full_name":"Yau, Horngtzer","last_name":"Yau","first_name":"Horngtzer"}],"date_created":"2018-12-11T11:50:47Z","doi":"10.1214/15-AOP1023","date_updated":"2025-09-22T09:33:43Z","language":[{"iso":"eng"}],"department":[{"_id":"LaEr"}],"isi":1,"publication":"Annals of Probability","article_processing_charge":"No","status":"public","external_id":{"isi":["000376180700016"],"arxiv":["1405.6634"]},"year":"2016","type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"ec_funded":1,"publist_id":"6115","issue":"3","day":"01","month":"01","publisher":"Institute of Mathematical Statistics"},{"abstract":[{"lang":"eng","text":"We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, Qr, and a random transversally periodic potential, κQt, with coupling constant κ. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large κ. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing κ. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder."}],"publication_status":"published","title":"Localization for transversally periodic random potentials on binary trees","oa_version":"Preprint","page":"557 - 600","main_file_link":[{"url":"https://arxiv.org/abs/1408.3961","open_access":"1"}],"quality_controlled":"1","intvolume":" 6","volume":6,"doi":"10.4171/JST/132","date_created":"2018-12-11T11:50:48Z","author":[{"full_name":"Froese, Richard","last_name":"Froese","first_name":"Richard"},{"full_name":"Lee, Darrick","last_name":"Lee","first_name":"Darrick"},{"first_name":"Christian","last_name":"Sadel","orcid":"0000-0001-8255-3968","full_name":"Sadel, Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Spitzer","first_name":"Wolfgang","full_name":"Spitzer, Wolfgang"},{"full_name":"Stolz, Günter","last_name":"Stolz","first_name":"Günter"}],"scopus_import":"1","_id":"1223","citation":{"ieee":"R. Froese, D. Lee, C. Sadel, W. Spitzer, and G. Stolz, “Localization for transversally periodic random potentials on binary trees,” Journal of Spectral Theory, vol. 6, no. 3. European Mathematical Society, pp. 557–600, 2016.","ista":"Froese R, Lee D, Sadel C, Spitzer W, Stolz G. 2016. Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. 6(3), 557–600.","apa":"Froese, R., Lee, D., Sadel, C., Spitzer, W., & Stolz, G. (2016). Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. European Mathematical Society. https://doi.org/10.4171/JST/132","ama":"Froese R, Lee D, Sadel C, Spitzer W, Stolz G. Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. 2016;6(3):557-600. doi:10.4171/JST/132","short":"R. Froese, D. Lee, C. Sadel, W. Spitzer, G. Stolz, Journal of Spectral Theory 6 (2016) 557–600.","mla":"Froese, Richard, et al. “Localization for Transversally Periodic Random Potentials on Binary Trees.” Journal of Spectral Theory, vol. 6, no. 3, European Mathematical Society, 2016, pp. 557–600, doi:10.4171/JST/132.","chicago":"Froese, Richard, Darrick Lee, Christian Sadel, Wolfgang Spitzer, and Günter Stolz. “Localization for Transversally Periodic Random Potentials on Binary Trees.” Journal of Spectral Theory. European Mathematical Society, 2016. https://doi.org/10.4171/JST/132."},"arxiv":1,"oa":1,"date_published":"2016-01-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"date_updated":"2025-09-22T09:32:55Z","external_id":{"isi":["000388627000004"],"arxiv":["1408.3961"]},"status":"public","article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Journal of Spectral Theory","type":"journal_article","year":"2016","publisher":"European Mathematical Society","month":"01","issue":"3","day":"01","publist_id":"6112"},{"abstract":[{"text":"We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.","lang":"eng"}],"publication_status":"published","title":"Local stability of the free additive convolution","oa_version":"Preprint","page":"672 - 719","main_file_link":[{"url":"http://arxiv.org/abs/1508.05905","open_access":"1"}],"quality_controlled":"1","intvolume":" 271","volume":271,"date_created":"2018-12-11T11:52:00Z","doi":"10.1016/j.jfa.2016.04.006","author":[{"first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Schnelli","orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"scopus_import":"1","_id":"1434","arxiv":1,"citation":{"ista":"Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 672–719.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,” Journal of Functional Analysis, vol. 271, no. 3. Academic Press, pp. 672–719, 2016.","ama":"Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution. Journal of Functional Analysis. 2016;271(3):672-719. doi:10.1016/j.jfa.2016.04.006","mla":"Bao, Zhigang, et al. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis, vol. 271, no. 3, Academic Press, 2016, pp. 672–719, doi:10.1016/j.jfa.2016.04.006.","short":"Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis. Academic Press, 2016. https://doi.org/10.1016/j.jfa.2016.04.006.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2016). Local stability of the free additive convolution. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2016.04.006"},"oa":1,"date_published":"2016-08-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"date_updated":"2025-09-18T14:00:03Z","external_id":{"arxiv":["1508.05905"],"isi":["000378013400009"]},"status":"public","article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Journal of Functional Analysis","ec_funded":1,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"type":"journal_article","year":"2016","publisher":"Academic Press","month":"08","corr_author":"1","day":"01","publist_id":"5764","issue":"3"},{"year":"2016","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"type":"journal_article","pubrep_id":"703","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ec_funded":1,"day":"01","publist_id":"6067","issue":"3","corr_author":"1","month":"05","publisher":"Springer","date_updated":"2025-09-22T09:02:32Z","ddc":["510","539"],"file":[{"file_size":800792,"date_updated":"2020-07-14T12:44:42Z","content_type":"application/pdf","date_created":"2018-12-12T10:15:02Z","file_id":"5119","file_name":"IST-2016-703-v1+1_s00220-016-2600-4.pdf","checksum":"4fb2411d9c2f56676123165aad46c828","creator":"system","access_level":"open_access","relation":"main_file"}],"language":[{"iso":"eng"}],"publication":"Communications in Mathematical Physics","isi":1,"department":[{"_id":"LaEr"}],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","status":"public","external_id":{"isi":["000374659800005"]},"volume":343,"intvolume":" 343","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). The work of C. Sadel was supported by NSERC Discovery Grant 92997-2010 RGPIN and by the People Programme (Marie Curie Actions) of the EU 7th Framework Programme FP7/2007-2013, REA Grant 291734.","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2016-05-01T00:00:00Z","oa":1,"citation":{"ieee":"C. Sadel and B. Virág, “A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes,” Communications in Mathematical Physics, vol. 343, no. 3. Springer, pp. 881–919, 2016.","ista":"Sadel C, Virág B. 2016. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. 343(3), 881–919.","apa":"Sadel, C., & Virág, B. (2016). A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-016-2600-4","ama":"Sadel C, Virág B. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. 2016;343(3):881-919. doi:10.1007/s00220-016-2600-4","chicago":"Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications in Mathematical Physics. Springer, 2016. https://doi.org/10.1007/s00220-016-2600-4.","mla":"Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications in Mathematical Physics, vol. 343, no. 3, Springer, 2016, pp. 881–919, doi:10.1007/s00220-016-2600-4.","short":"C. Sadel, B. Virág, Communications in Mathematical Physics 343 (2016) 881–919."},"_id":"1257","scopus_import":"1","doi":"10.1007/s00220-016-2600-4","author":[{"full_name":"Sadel, Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","first_name":"Christian","orcid":"0000-0001-8255-3968","last_name":"Sadel"},{"last_name":"Virág","first_name":"Bálint","full_name":"Virág, Bálint"}],"date_created":"2018-12-11T11:50:59Z","title":"A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes","oa_version":"Published Version","publication_status":"published","abstract":[{"text":"We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix (Formula presented.). Focusing on the eigenvalues of (Formula presented.) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required (Formula presented.) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.","lang":"eng"}],"quality_controlled":"1","file_date_updated":"2020-07-14T12:44:42Z","page":"881 - 919"},{"article_processing_charge":"No","publication":"Communications on Pure and Applied Mathematics","isi":1,"department":[{"_id":"LaEr"}],"external_id":{"arxiv":["1407.5606"],"isi":["000382932900001"]},"status":"public","date_updated":"2025-09-22T08:35:52Z","language":[{"iso":"eng"}],"day":"01","publist_id":"6036","issue":"10","publisher":"Wiley-Blackwell","corr_author":"1","month":"10","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"year":"2016","ec_funded":1,"page":"1815 - 1881","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1407.5606"}],"publication_status":"published","title":"Fixed energy universality for generalized wigner matrices","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics."}],"oa":1,"date_published":"2016-10-01T00:00:00Z","arxiv":1,"citation":{"ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Fixed energy universality for generalized wigner matrices,” Communications on Pure and Applied Mathematics, vol. 69, no. 10. Wiley-Blackwell, pp. 1815–1881, 2016.","ista":"Bourgade P, Erdös L, Yau H, Yin J. 2016. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 69(10), 1815–1881.","apa":"Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2016). Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. Wiley-Blackwell. https://doi.org/10.1002/cpa.21624","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881.","mla":"Bourgade, Paul, et al. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics, vol. 69, no. 10, Wiley-Blackwell, 2016, pp. 1815–81, doi:10.1002/cpa.21624.","chicago":"Bourgade, Paul, László Erdös, Horngtzer Yau, and Jun Yin. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics. Wiley-Blackwell, 2016. https://doi.org/10.1002/cpa.21624.","ama":"Bourgade P, Erdös L, Yau H, Yin J. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 2016;69(10):1815-1881. doi:10.1002/cpa.21624"},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","author":[{"full_name":"Bourgade, Paul","last_name":"Bourgade","first_name":"Paul"},{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Yau","first_name":"Horngtzer","full_name":"Yau, Horngtzer"},{"last_name":"Yin","first_name":"Jun","full_name":"Yin, Jun"}],"doi":"10.1002/cpa.21624","date_created":"2018-12-11T11:51:07Z","_id":"1280","scopus_import":"1","volume":69,"intvolume":" 69","acknowledgement":"The work of P.B. was partially supported by National Sci-\r\nence Foundation Grant DMS-1208859. The work of L.E. was partially supported\r\nby ERC Advanced Grant RANMAT 338804. The work of H.-T. Y. was partially\r\nsupported by National Science Foundation Grant DMS-1307444 and a Simons In-\r\nvestigator award. The work of J.Y. was partially supported by National Science\r\nFoundation Grant DMS-1207961. The major part of this research was conducted\r\nwhen all authors were visiting IAS and were also supported by National Science\r\nFoundation Grant DMS-1128255."},{"abstract":[{"lang":"eng","text":"We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues. "}],"publication_status":"published","title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","oa_version":"Preprint","page":"165 - 241","main_file_link":[{"url":"http://arxiv.org/abs/1310.7057","open_access":"1"}],"quality_controlled":"1","intvolume":" 164","acknowledgement":"Most of the presented work was obtained while Kevin Schnelli was staying at the IAS with the support of\r\nThe Fund For Math.","volume":164,"doi":"10.1007/s00440-014-0610-8","date_created":"2018-12-11T11:54:31Z","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","last_name":"Schnelli","orcid":"0000-0003-0954-3231","first_name":"Kevin"}],"_id":"1881","scopus_import":"1","oa":1,"date_published":"2016-02-01T00:00:00Z","citation":{"apa":"Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-014-0610-8","ama":"Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 2016;164(1-2):165-241. doi:10.1007/s00440-014-0610-8","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.","mla":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8.","chicago":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields. Springer, 2016. https://doi.org/10.1007/s00440-014-0610-8.","ieee":"J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” Probability Theory and Related Fields, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.","ista":"Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241."},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"date_updated":"2025-09-18T10:46:46Z","external_id":{"arxiv":["1310.7057"],"isi":["000373163300006"]},"status":"public","article_processing_charge":"No","publication":"Probability Theory and Related Fields","department":[{"_id":"LaEr"}],"isi":1,"ec_funded":1,"type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"year":"2016","publisher":"Springer","corr_author":"1","month":"02","issue":"1-2","publist_id":"5215","day":"01"},{"page":"280 - 302","quality_controlled":"1","file_date_updated":"2020-07-14T12:44:57Z","abstract":[{"text":"We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries. ","lang":"eng"}],"oa_version":"Published Version","title":"Local spectral statistics of Gaussian matrices with correlated entries","publication_status":"published","scopus_import":"1","_id":"1489","doi":"10.1007/s10955-016-1479-y","author":[{"first_name":"Oskari H","last_name":"Ajanki","full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"}],"date_created":"2018-12-11T11:52:19Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ama":"Ajanki OH, Erdös L, Krüger TH. Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. 2016;163(2):280-302. doi:10.1007/s10955-016-1479-y","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016) 280–302.","mla":"Ajanki, Oskari H., et al. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” Journal of Statistical Physics, vol. 163, no. 2, Springer, 2016, pp. 280–302, doi:10.1007/s10955-016-1479-y.","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” Journal of Statistical Physics. Springer, 2016. https://doi.org/10.1007/s10955-016-1479-y.","apa":"Ajanki, O. H., Erdös, L., & Krüger, T. H. (2016). Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-016-1479-y","ista":"Ajanki OH, Erdös L, Krüger TH. 2016. Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. 163(2), 280–302.","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local spectral statistics of Gaussian matrices with correlated entries,” Journal of Statistical Physics, vol. 163, no. 2. Springer, pp. 280–302, 2016."},"date_published":"2016-04-01T00:00:00Z","oa":1,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). Oskari H. Ajanki was Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12 Grant of the German Research Council. László Erdős was Partially supported by ERC Advanced Grant RANMAT No. 338804. Torben Krüger was Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12 Grant of the German Research Council.","intvolume":" 163","volume":163,"status":"public","external_id":{"isi":["000373132700003"]},"department":[{"_id":"LaEr"}],"isi":1,"has_accepted_license":"1","publication":"Journal of Statistical Physics","article_processing_charge":"Yes (via OA deal)","language":[{"iso":"eng"}],"date_updated":"2025-09-18T11:17:50Z","ddc":["510"],"file":[{"relation":"main_file","date_created":"2018-12-12T10:11:16Z","file_name":"IST-2016-516-v1+1_s10955-016-1479-y.pdf","checksum":"7139598dcb1cafbe6866bd2bfd732b33","file_id":"4869","creator":"system","access_level":"open_access","date_updated":"2020-07-14T12:44:57Z","file_size":660602,"content_type":"application/pdf"}],"month":"04","corr_author":"1","publisher":"Springer","day":"01","issue":"2","publist_id":"5698","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ec_funded":1,"year":"2016","pubrep_id":"516","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"type":"journal_article"},{"date_updated":"2025-09-18T11:00:43Z","language":[{"iso":"eng"}],"article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Annales Henri Poincare","external_id":{"isi":["000377994000003"],"arxiv":["1501.04287"]},"status":"public","project":[{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","year":"2016","ec_funded":1,"day":"01","publist_id":"5558","issue":"7","publisher":"Birkhäuser","month":"07","corr_author":"1","publication_status":"published","title":"Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At d dimensional growth for d>2 this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d dimensional growth with d<2 one has pure point spectrum in this energy region. At exactly uniform 2 dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum (d≤2) to absolutely continuous spectrum (d≥3) for random operators of the type rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete adjacency operator (Laplacian) on ℤd and λ a random potential. "}],"quality_controlled":"1","page":"1631 - 1675","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1501.04287"}],"volume":17,"intvolume":" 17","citation":{"mla":"Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel.” Annales Henri Poincare, vol. 17, no. 7, Birkhäuser, 2016, pp. 1631–75, doi:10.1007/s00023-015-0456-3.","short":"C. Sadel, Annales Henri Poincare 17 (2016) 1631–1675.","chicago":"Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel.” Annales Henri Poincare. Birkhäuser, 2016. https://doi.org/10.1007/s00023-015-0456-3.","ama":"Sadel C. Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. 2016;17(7):1631-1675. doi:10.1007/s00023-015-0456-3","apa":"Sadel, C. (2016). Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. Birkhäuser. https://doi.org/10.1007/s00023-015-0456-3","ista":"Sadel C. 2016. Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. 17(7), 1631–1675.","ieee":"C. Sadel, “Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel,” Annales Henri Poincare, vol. 17, no. 7. Birkhäuser, pp. 1631–1675, 2016."},"arxiv":1,"oa":1,"date_published":"2016-07-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:53:00Z","doi":"10.1007/s00023-015-0456-3","author":[{"id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","full_name":"Sadel, Christian","orcid":"0000-0001-8255-3968","last_name":"Sadel","first_name":"Christian"}],"scopus_import":"1","_id":"1608"},{"department":[{"_id":"LaEr"}],"isi":1,"has_accepted_license":"1","publication":"Nature Communications","article_processing_charge":"No","status":"public","article_number":"6977","external_id":{"isi":["000353705000003"]},"date_updated":"2025-09-23T08:44:01Z","file":[{"content_type":"application/pdf","file_size":1151501,"date_updated":"2020-07-14T12:45:17Z","relation":"main_file","access_level":"open_access","creator":"system","file_id":"5245","file_name":"IST-2016-451-v1+1_ncomms7977.pdf","date_created":"2018-12-12T10:16:54Z","checksum":"c4cffb5c8b245e658a34eac71a03e7cc"}],"ddc":["530"],"language":[{"iso":"eng"}],"publist_id":"5282","day":"24","month":"04","publisher":"Nature Publishing Group","year":"2015","pubrep_id":"451","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"quality_controlled":"1","file_date_updated":"2020-07-14T12:45:17Z","title":"Evolutionary games of condensates in coupled birth-death processes","oa_version":"Published Version","publication_status":"published","abstract":[{"text":"Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.","lang":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"apa":"Knebel, J., Weber, M., Krüger, T. H., & Frey, E. (2015). Evolutionary games of condensates in coupled birth-death processes. Nature Communications. Nature Publishing Group. https://doi.org/10.1038/ncomms7977","mla":"Knebel, Johannes, et al. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” Nature Communications, vol. 6, 6977, Nature Publishing Group, 2015, doi:10.1038/ncomms7977.","short":"J. Knebel, M. Weber, T.H. Krüger, E. Frey, Nature Communications 6 (2015).","chicago":"Knebel, Johannes, Markus Weber, Torben H Krüger, and Erwin Frey. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” Nature Communications. Nature Publishing Group, 2015. https://doi.org/10.1038/ncomms7977.","ama":"Knebel J, Weber M, Krüger TH, Frey E. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 2015;6. doi:10.1038/ncomms7977","ieee":"J. Knebel, M. Weber, T. H. Krüger, and E. Frey, “Evolutionary games of condensates in coupled birth-death processes,” Nature Communications, vol. 6. Nature Publishing Group, 2015.","ista":"Knebel J, Weber M, Krüger TH, Frey E. 2015. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 6, 6977."},"oa":1,"date_published":"2015-04-24T00:00:00Z","scopus_import":"1","_id":"1824","doi":"10.1038/ncomms7977","date_created":"2018-12-11T11:54:13Z","author":[{"full_name":"Knebel, Johannes","last_name":"Knebel","first_name":"Johannes"},{"full_name":"Weber, Markus","last_name":"Weber","first_name":"Markus"},{"last_name":"Krüger","orcid":"0000-0002-4821-3297","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"},{"first_name":"Erwin","last_name":"Frey","full_name":"Frey, Erwin"}],"volume":6,"intvolume":" 6"},{"language":[{"iso":"eng"}],"date_updated":"2025-10-22T10:23:38Z","status":"public","external_id":{"isi":["000349364100002"],"arxiv":["1309.5107"]},"publication":"Annales Henri Poincare","isi":1,"department":[{"_id":"LaEr"}],"article_processing_charge":"No","ec_funded":1,"year":"2015","type":"journal_article","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"month":"03","publisher":"Springer","day":"01","issue":"3","publist_id":"5233","abstract":[{"text":"The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas.\r\n","lang":"eng"}],"title":"The Altshuler-Shklovskii formulas for random band matrices II: The general case","oa_version":"Preprint","publication_status":"published","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1309.5107"}],"page":"709 - 799","intvolume":" 16","volume":16,"_id":"1864","scopus_import":"1","date_created":"2018-12-11T11:54:26Z","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László"},{"full_name":"Knowles, Antti","first_name":"Antti","last_name":"Knowles"}],"doi":"10.1007/s00023-014-0333-5","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"date_published":"2015-03-01T00:00:00Z","citation":{"chicago":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare. Springer, 2015. https://doi.org/10.1007/s00023-014-0333-5.","mla":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:10.1007/s00023-014-0333-5.","short":"L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.","ama":"Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 2015;16(3):709-799. doi:10.1007/s00023-014-0333-5","apa":"Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. Springer. https://doi.org/10.1007/s00023-014-0333-5","ista":"Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.","ieee":"L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices II: The general case,” Annales Henri Poincare, vol. 16, no. 3. Springer, pp. 709–799, 2015."},"arxiv":1}]