[{"year":"2017","status":"public","publication_identifier":{"issn":["0010-3640"]},"doi":"10.1002/cpa.21639","abstract":[{"text":"Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.","lang":"eng"}],"issue":"9","day":"01","type":"journal_article","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"month":"09","citation":{"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.","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.","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","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","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","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.","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."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","oa_version":"Submitted Version","date_updated":"2025-09-10T10:58:02Z","fulldoi":"https://doi.org/10.1002/cpa.21639","external_id":{"isi":["000405752100002"],"arxiv":["1512.03703"]},"publication_status":"published","department":[{"_id":"LaEr"}],"_id":"721","page":"1672 - 1705","scopus_import":"1","intvolume":" 70","publication":"Communications on Pure and Applied Mathematics","date_created":"2018-12-11T11:48:08Z","oa":1,"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"isi":1,"publisher":"Wiley","date_published":"2017-09-01T00:00:00Z","corr_author":"1","arxiv":1,"language":[{"iso":"eng"}],"author":[{"full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","last_name":"Ajanki","first_name":"Oskari H"},{"orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger","first_name":"Torben H"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"}],"publist_id":"6959","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","volume":70,"quality_controlled":"1"},{"isi":1,"publisher":"Academic Press","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"oa":1,"date_created":"2018-12-11T11:48:13Z","publication":"Advances in Mathematics","intvolume":" 319","scopus_import":"1","page":"251 - 291","_id":"733","quality_controlled":"1","volume":319,"title":"Convergence rate for spectral distribution of addition of random matrices","author":[{"first_name":"Zhigang","last_name":"Bao","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László"},{"last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231"}],"publist_id":"6935","language":[{"iso":"eng"}],"arxiv":1,"date_published":"2017-10-15T00:00:00Z","corr_author":"1","citation":{"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.","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.","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.","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291.","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","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"},"month":"10","main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"ec_funded":1,"type":"journal_article","day":"15","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."}],"doi":"10.1016/j.aim.2017.08.028","status":"public","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","year":"2017","department":[{"_id":"LaEr"}],"publication_status":"published","external_id":{"isi":["000412150400010"],"arxiv":["1606.03076"]},"fulldoi":"https://doi.org/10.1016/j.aim.2017.08.028","date_updated":"2025-06-04T10:13:45Z","article_processing_charge":"No","oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"article_processing_charge":"No","oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-07-06T13:35:08Z","external_id":{"arxiv":["1602.02312"],"isi":["000409382300005"]},"fulldoi":"https://doi.org/10.4310/ATMP.2017.v21.n3.a5","department":[{"_id":"LaEr"}],"publication_status":"published","publication_identifier":{"issn":["1095-0761"]},"doi":"10.4310/ATMP.2017.v21.n3.a5","status":"public","year":"2017","issue":"3","abstract":[{"text":"We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.","lang":"eng"}],"ec_funded":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.02312"}],"type":"journal_article","day":"25","citation":{"ista":"Bourgade P, Erdös L, Yau H, Yin J. 2017. Universality for a class of random band matrices. Advances in Theoretical and Mathematical Physics. 21(3), 739–800.","ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Universality for a class of random band matrices,” Advances in Theoretical and Mathematical Physics, vol. 21, no. 3. International Press of Boston, pp. 739–800, 2017.","apa":"Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2017). Universality for a class of random band matrices. Advances in Theoretical and Mathematical Physics. International Press of Boston. https://doi.org/10.4310/ATMP.2017.v21.n3.a5","ama":"Bourgade P, Erdös L, Yau H, Yin J. Universality for a class of random band matrices. Advances in Theoretical and Mathematical Physics. 2017;21(3):739-800. doi:10.4310/ATMP.2017.v21.n3.a5","chicago":"Bourgade, Paul, László Erdös, Horng Yau, and Jun Yin. “Universality for a Class of Random Band Matrices.” Advances in Theoretical and Mathematical Physics. International Press of Boston, 2017. https://doi.org/10.4310/ATMP.2017.v21.n3.a5.","mla":"Bourgade, Paul, et al. “Universality for a Class of Random Band Matrices.” Advances in Theoretical and Mathematical Physics, vol. 21, no. 3, International Press of Boston, 2017, pp. 739–800, doi:10.4310/ATMP.2017.v21.n3.a5.","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Advances in Theoretical and Mathematical Physics 21 (2017) 739–800."},"month":"08","date_published":"2017-08-25T00:00:00Z","language":[{"iso":"eng"}],"arxiv":1,"publist_id":"7337","author":[{"full_name":"Bourgade, Paul","last_name":"Bourgade","first_name":"Paul"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös"},{"last_name":"Yau","first_name":"Horng","full_name":"Yau, Horng"},{"full_name":"Yin, Jun","last_name":"Yin","first_name":"Jun"}],"title":"Universality for a class of random band matrices","quality_controlled":"1","volume":21,"publication":"Advances in Theoretical and Mathematical Physics","intvolume":" 21","scopus_import":"1","page":"739 - 800","das_tickbox":"1","_id":"483","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"date_created":"2018-12-11T11:46:43Z","oa":1,"publisher":"International Press of Boston","isi":1},{"doi":"10.1214/16-ECP38","acknowledgement":"Partially supported by the IST Austria Excellence Scholarship.","status":"public","year":"2017","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"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].","lang":"eng"}],"ec_funded":1,"type":"journal_article","day":"02","file_date_updated":"2018-12-12T10:18:10Z","citation":{"short":"L. Erdös, D.J. Schröder, Electronic Communications in Probability 21 (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.","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.","ista":"Erdös L, Schröder DJ. 2017. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 21, 86.","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","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of functions of Wigner matrices,” Electronic Communications in Probability, vol. 21. Institute of Mathematical Statistics, 2017."},"month":"01","article_processing_charge":"No","oa_version":"Published Version","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-07-29T13:46:19Z","article_number":"86","external_id":{"isi":["000396604900037"]},"fulldoi":"https://doi.org/10.1214/16-ECP38","department":[{"_id":"LaEr"}],"publication_status":"published","publication":"Electronic Communications in Probability","scopus_import":"1","intvolume":" 21","ddc":["510"],"_id":"1144","has_accepted_license":"1","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"date_created":"2018-12-11T11:50:23Z","oa":1,"publisher":"Institute of Mathematical Statistics","isi":1,"pubrep_id":"747","date_published":"2017-01-02T00:00:00Z","file":[{"file_name":"IST-2017-747-v1+1_euclid.ecp.1483347665.pdf","creator":"system","content_type":"application/pdf","date_created":"2018-12-12T10:18:10Z","date_updated":"2018-12-12T10:18:10Z","file_size":440770,"relation":"main_file","file_id":"5329","access_level":"open_access"}],"related_material":{"record":[{"id":"6179","relation":"dissertation_contains","status":"public"}]},"language":[{"iso":"eng"}],"publist_id":"6214","author":[{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","last_name":"Schröder","first_name":"Dominik J"}],"title":"Fluctuations of functions of Wigner matrices","quality_controlled":"1","volume":21},{"date_published":"2016-12-15T00:00:00Z","arxiv":1,"language":[{"iso":"eng"}],"author":[{"full_name":"Lee, Ji","first_name":"Ji","last_name":"Lee"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","publist_id":"6201","volume":26,"quality_controlled":"1","_id":"1157","page":"3786 - 3839","scopus_import":"1","intvolume":" 26","publication":"Annals of Applied Probability","oa":1,"date_created":"2018-12-11T11:50:27Z","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"isi":1,"publisher":"Institute of Mathematical Statistics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","oa_version":"Preprint","date_updated":"2025-09-22T09:55:43Z","fulldoi":"https://doi.org/10.1214/16-AAP1193","external_id":{"arxiv":["1409.4979"],"isi":["000391240100016"]},"publication_status":"published","department":[{"_id":"LaEr"}],"year":"2016","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.","status":"public","doi":"10.1214/16-AAP1193","abstract":[{"lang":"eng","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."}],"issue":"6","type":"journal_article","day":"15","main_file_link":[{"url":"https://arxiv.org/abs/1409.4979","open_access":"1"}],"ec_funded":1,"month":"12","citation":{"short":"J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.","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.","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.","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.","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","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."}},{"publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1214/15-AOP1023","external_id":{"isi":["000376180700016"],"arxiv":["1405.6634"]},"date_updated":"2025-09-22T09:33:43Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Preprint","article_processing_charge":"No","month":"01","citation":{"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.","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.","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","short":"J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.","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.","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."},"day":"01","type":"journal_article","main_file_link":[{"url":"https://arxiv.org/abs/1405.6634","open_access":"1"}],"ec_funded":1,"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."}],"issue":"3","year":"2016","status":"public","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. ","doi":"10.1214/15-AOP1023","volume":44,"quality_controlled":"1","title":"Bulk universality for deformed wigner matrices","publist_id":"6115","author":[{"first_name":"Jioon","last_name":"Lee","full_name":"Lee, Jioon"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin"},{"full_name":"Stetler, Ben","first_name":"Ben","last_name":"Stetler"},{"first_name":"Horngtzer","last_name":"Yau","full_name":"Yau, Horngtzer"}],"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2016-01-01T00:00:00Z","publisher":"Institute of Mathematical Statistics","isi":1,"oa":1,"date_created":"2018-12-11T11:50:47Z","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"_id":"1219","page":"2349 - 2425","scopus_import":"1","intvolume":" 44","publication":"Annals of Probability"},{"month":"05","citation":{"short":"C. Sadel, B. Virág, Communications in Mathematical Physics 343 (2016) 881–919.","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.","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","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.","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.","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"},"file_date_updated":"2020-07-14T12:44:42Z","type":"journal_article","day":"01","ec_funded":1,"abstract":[{"lang":"eng","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."}],"issue":"3","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"year":"2016","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.","status":"public","doi":"10.1007/s00220-016-2600-4","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1007/s00220-016-2600-4","external_id":{"isi":["000374659800005"]},"date_updated":"2025-09-22T09:02:32Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","pubrep_id":"703","publisher":"Springer","isi":1,"oa":1,"date_created":"2018-12-11T11:50:59Z","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"}],"has_accepted_license":"1","_id":"1257","page":"881 - 919","ddc":["510","539"],"scopus_import":"1","intvolume":" 343","publication":"Communications in Mathematical Physics","volume":343,"quality_controlled":"1","title":"A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes","author":[{"full_name":"Sadel, Christian","orcid":"0000-0001-8255-3968","last_name":"Sadel","first_name":"Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Virág, Bálint","last_name":"Virág","first_name":"Bálint"}],"publist_id":"6067","language":[{"iso":"eng"}],"file":[{"date_updated":"2020-07-14T12:44:42Z","date_created":"2018-12-12T10:15:02Z","creator":"system","file_name":"IST-2016-703-v1+1_s00220-016-2600-4.pdf","content_type":"application/pdf","access_level":"open_access","checksum":"4fb2411d9c2f56676123165aad46c828","file_size":800792,"relation":"main_file","file_id":"5119"}],"date_published":"2016-05-01T00:00:00Z","corr_author":"1"},{"publisher":"Wiley-Blackwell","isi":1,"scopus_import":"1","intvolume":" 69","publication":"Communications on Pure and Applied Mathematics","_id":"1280","page":"1815 - 1881","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"date_created":"2018-12-11T11:51:07Z","oa":1,"publist_id":"6036","title":"Fixed energy universality for generalized wigner matrices","author":[{"full_name":"Bourgade, Paul","last_name":"Bourgade","first_name":"Paul"},{"last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603"},{"full_name":"Yau, Horngtzer","first_name":"Horngtzer","last_name":"Yau"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"volume":69,"corr_author":"1","date_published":"2016-10-01T00:00:00Z","arxiv":1,"language":[{"iso":"eng"}],"ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/1407.5606","open_access":"1"}],"type":"journal_article","day":"01","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.","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","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","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.","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.","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881."},"month":"10","doi":"10.1002/cpa.21624","year":"2016","status":"public","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 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."}],"issue":"10","external_id":{"arxiv":["1407.5606"],"isi":["000382932900001"]},"fulldoi":"https://doi.org/10.1002/cpa.21624","department":[{"_id":"LaEr"}],"publication_status":"published","article_processing_charge":"No","oa_version":"Preprint","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2025-09-22T08:35:52Z"},{"year":"2016","status":"public","doi":"10.1016/j.jfa.2016.04.006","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"}],"issue":"3","type":"journal_article","day":"01","main_file_link":[{"url":"http://arxiv.org/abs/1508.05905","open_access":"1"}],"ec_funded":1,"month":"08","citation":{"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.","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","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","ista":"Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 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.","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."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","oa_version":"Preprint","date_updated":"2025-09-18T14:00:03Z","fulldoi":"https://doi.org/10.1016/j.jfa.2016.04.006","external_id":{"arxiv":["1508.05905"],"isi":["000378013400009"]},"publication_status":"published","department":[{"_id":"LaEr"}],"_id":"1434","page":"672 - 719","intvolume":" 271","scopus_import":"1","publication":"Journal of Functional Analysis","oa":1,"date_created":"2018-12-11T11:52:00Z","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"isi":1,"publisher":"Academic Press","corr_author":"1","date_published":"2016-08-01T00:00:00Z","arxiv":1,"language":[{"iso":"eng"}],"author":[{"orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao","first_name":"Zhigang"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231"}],"title":"Local stability of the free additive convolution","publist_id":"5764","volume":271,"quality_controlled":"1"},{"publication":"Journal of Statistical Physics","intvolume":" 163","scopus_import":"1","ddc":["510"],"page":"280 - 302","_id":"1489","has_accepted_license":"1","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"date_created":"2018-12-11T11:52:19Z","oa":1,"isi":1,"publisher":"Springer","pubrep_id":"516","corr_author":"1","date_published":"2016-04-01T00:00:00Z","file":[{"relation":"main_file","file_id":"4869","file_size":660602,"checksum":"7139598dcb1cafbe6866bd2bfd732b33","access_level":"open_access","content_type":"application/pdf","creator":"system","file_name":"IST-2016-516-v1+1_s10955-016-1479-y.pdf","date_updated":"2020-07-14T12:44:57Z","date_created":"2018-12-12T10:11:16Z"}],"language":[{"iso":"eng"}],"author":[{"full_name":"Ajanki, Oskari H","last_name":"Ajanki","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger","first_name":"Torben H","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H"}],"title":"Local spectral statistics of Gaussian matrices with correlated entries","publist_id":"5698","quality_controlled":"1","volume":163,"doi":"10.1007/s10955-016-1479-y","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.","status":"public","year":"2016","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"issue":"2","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"}],"ec_funded":1,"day":"01","type":"journal_article","file_date_updated":"2020-07-14T12:44:57Z","citation":{"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.","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.","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","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","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.","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016) 280–302."},"month":"04","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2025-09-18T11:17:50Z","external_id":{"isi":["000373132700003"]},"fulldoi":"https://doi.org/10.1007/s10955-016-1479-y","department":[{"_id":"LaEr"}],"publication_status":"published"},{"date_updated":"2025-09-18T11:00:43Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Preprint","article_processing_charge":"No","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1007/s00023-015-0456-3","external_id":{"isi":["000377994000003"],"arxiv":["1501.04287"]},"issue":"7","abstract":[{"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 rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete adjacency operator (Laplacian) on ℤd and λ a random potential. ","lang":"eng"}],"status":"public","year":"2016","doi":"10.1007/s00023-015-0456-3","month":"07","citation":{"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.","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.","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.","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","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","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."},"type":"journal_article","day":"01","ec_funded":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1501.04287"}],"language":[{"iso":"eng"}],"arxiv":1,"corr_author":"1","date_published":"2016-07-01T00:00:00Z","quality_controlled":"1","volume":17,"author":[{"full_name":"Sadel, Christian","orcid":"0000-0001-8255-3968","first_name":"Christian","last_name":"Sadel","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87"}],"title":"Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel","publist_id":"5558","date_created":"2018-12-11T11:53:00Z","oa":1,"project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"page":"1631 - 1675","_id":"1608","publication":"Annales Henri Poincare","intvolume":" 17","scopus_import":"1","publisher":"Birkhäuser","isi":1},{"department":[{"_id":"LaEr"}],"publication_status":"published","external_id":{"arxiv":["1310.7057"],"isi":["000373163300006"]},"fulldoi":"https://doi.org/10.1007/s00440-014-0610-8","date_updated":"2025-09-18T10:46:46Z","article_processing_charge":"No","oa_version":"Preprint","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.","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.","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.","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","ista":"Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.","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","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."},"month":"02","main_file_link":[{"url":"http://arxiv.org/abs/1310.7057","open_access":"1"}],"ec_funded":1,"day":"01","type":"journal_article","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. "}],"issue":"1-2","doi":"10.1007/s00440-014-0610-8","year":"2016","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.","status":"public","volume":164,"quality_controlled":"1","publist_id":"5215","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231"}],"title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","arxiv":1,"language":[{"iso":"eng"}],"corr_author":"1","date_published":"2016-02-01T00:00:00Z","isi":1,"publisher":"Springer","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"oa":1,"date_created":"2018-12-11T11:54:31Z","scopus_import":"1","intvolume":" 164","publication":"Probability Theory and Related Fields","_id":"1881","page":"165 - 241"},{"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2016-01-01T00:00:00Z","volume":6,"quality_controlled":"1","publist_id":"6112","title":"Localization for transversally periodic random potentials on binary trees","author":[{"full_name":"Froese, Richard","first_name":"Richard","last_name":"Froese"},{"full_name":"Lee, Darrick","last_name":"Lee","first_name":"Darrick"},{"full_name":"Sadel, Christian","orcid":"0000-0001-8255-3968","last_name":"Sadel","first_name":"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"}],"date_created":"2018-12-11T11:50:48Z","oa":1,"_id":"1223","das_tickbox":"1","page":"557 - 600","scopus_import":"1","intvolume":" 6","publication":"Journal of Spectral Theory","publisher":"EMS Press","isi":1,"date_updated":"2026-07-06T11:55:04Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","article_processing_charge":"No","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.4171/JST/132","external_id":{"arxiv":["1408.3961"],"isi":["000388627000004"]},"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."}],"issue":"3","year":"2016","status":"public","doi":"10.4171/JST/132","month":"01","citation":{"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. EMS Press, 2016. https://doi.org/10.4171/JST/132.","mla":"Froese, Richard, et al. “Localization for Transversally Periodic Random Potentials on Binary Trees.” Journal of Spectral Theory, vol. 6, no. 3, EMS Press, 2016, pp. 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.","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","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. EMS Press. https://doi.org/10.4171/JST/132","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. EMS Press, 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."},"day":"01","type":"journal_article","main_file_link":[{"url":"https://arxiv.org/abs/1408.3961","open_access":"1"}]},{"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2015-02-01T00:00:00Z","volume":333,"quality_controlled":"1","title":"The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case","publist_id":"4818","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"full_name":"Knowles, Antti","first_name":"Antti","last_name":"Knowles"}],"oa":1,"date_created":"2018-12-11T11:56:05Z","_id":"2166","page":"1365 - 1416","intvolume":" 333","scopus_import":"1","publication":"Communications in Mathematical Physics","isi":1,"publisher":"Springer","date_updated":"2025-09-23T13:39:37Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","oa_version":"Preprint","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1007/s00220-014-2119-5","external_id":{"arxiv":["1309.5106"],"isi":["000348303100008"]},"abstract":[{"lang":"eng","text":"We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107, 2014). "}],"issue":"3","year":"2015","status":"public","doi":"10.1007/s00220-014-2119-5","month":"02","citation":{"ista":"Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 333(3), 1365–1416.","apa":"Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2119-5","ama":"Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 2015;333(3):1365-1416. doi:10.1007/s00220-014-2119-5","ieee":"L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case,” Communications in Mathematical Physics, vol. 333, no. 3. Springer, pp. 1365–1416, 2015.","short":"L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416.","mla":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics, vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:10.1007/s00220-014-2119-5.","chicago":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics. Springer, 2015. https://doi.org/10.1007/s00220-014-2119-5."},"day":"01","type":"journal_article","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1309.5106"}]},{"abstract":[{"lang":"eng","text":"This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed."}],"issue":"1","year":"2015","status":"public","acknowledgement":"B.Z. was supported in part by NSFC Grant 11071213, ZJNSF Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11, and by a Grant R-155-000-131-112 at the National University of Singapore\r\n","doi":"10.1214/14-AOS1281","month":"02","citation":{"apa":"Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/14-AOS1281","ista":"Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 43(1), 382–421.","ama":"Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 2015;43(1):382-421. doi:10.1214/14-AOS1281","ieee":"Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” Annals of Statistics, vol. 43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015.","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/14-AOS1281.","mla":"Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281."},"type":"journal_article","day":"01","main_file_link":[{"url":"https://arxiv.org/abs/1304.5690","open_access":"1"}],"date_updated":"2025-09-29T11:02:34Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","oa_version":"Preprint","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1214/14-AOS1281","external_id":{"arxiv":["1304.5690"],"isi":["000349738500014"]},"date_created":"2018-12-11T11:52:25Z","oa":1,"_id":"1505","page":"382 - 421","intvolume":" 43","publication":"Annals of Statistics","publisher":"Institute of Mathematical Statistics","isi":1,"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2015-02-01T00:00:00Z","volume":43,"quality_controlled":"1","publist_id":"5672","author":[{"first_name":"Zhigang","last_name":"Bao","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475"},{"last_name":"Pan","first_name":"Guangming","full_name":"Pan, Guangming"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}],"title":"Universality for the largest eigenvalue of sample covariance matrices with general population"},{"abstract":[{"text":"Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1).","lang":"eng"}],"issue":"3","year":"2015","status":"public","doi":"10.3150/14-BEJ615","month":"08","citation":{"short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","mla":"Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” Bernoulli, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:10.3150/14-BEJ615.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” Bernoulli. Bernoulli Society for Mathematical Statistics and Probability, 2015. https://doi.org/10.3150/14-BEJ615.","ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. Bernoulli. 2015;21(3):1600-1628. doi:10.3150/14-BEJ615","ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628.","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). The logarithmic law of random determinant. Bernoulli. Bernoulli Society for Mathematical Statistics and Probability. https://doi.org/10.3150/14-BEJ615","ieee":"Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” Bernoulli, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics and Probability, pp. 1600–1628, 2015."},"day":"01","type":"journal_article","main_file_link":[{"url":"http://arxiv.org/abs/1208.5823","open_access":"1"}],"date_updated":"2025-09-23T13:59:56Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Preprint","article_processing_charge":"No","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.3150/14-BEJ615","external_id":{"arxiv":["1208.5823"],"isi":["000356993100012"]},"date_created":"2018-12-11T11:52:25Z","oa":1,"_id":"1506","page":"1600 - 1628","intvolume":" 21","publication":"Bernoulli","publisher":"Bernoulli Society for Mathematical Statistics and Probability","isi":1,"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2015-08-01T00:00:00Z","volume":21,"quality_controlled":"1","title":"The logarithmic law of random determinant","author":[{"full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Guangming","last_name":"Pan","full_name":"Pan, Guangming"},{"full_name":"Zhou, Wang","last_name":"Zhou","first_name":"Wang"}],"publist_id":"5671"},{"department":[{"_id":"LaEr"}],"publication_status":"published","external_id":{"isi":["000354943600029"]},"fulldoi":"https://doi.org/10.1109/TIT.2015.2421894","date_updated":"2025-09-23T07:57:31Z","oa_version":"None","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"short":"Z. Bao, G. Pan, W. Zhou, IEEE Transactions on Information Theory 61 (2015) 3413–3426.","mla":"Bao, Zhigang, et al. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory, vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:10.1109/TIT.2015.2421894.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory. IEEE, 2015. https://doi.org/10.1109/TIT.2015.2421894.","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. IEEE. https://doi.org/10.1109/TIT.2015.2421894","ama":"Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 2015;61(6):3413-3426. doi:10.1109/TIT.2015.2421894","ieee":"Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices,” IEEE Transactions on Information Theory, vol. 61, no. 6. IEEE, pp. 3413–3426, 2015.","ista":"Bao Z, Pan G, Zhou W. 2015. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 61(6), 3413–3426."},"month":"06","type":"journal_article","day":"01","abstract":[{"lang":"eng","text":"In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004)."}],"issue":"6","doi":"10.1109/TIT.2015.2421894","year":"2015","acknowledgement":"G. Pan was supported by MOE Tier 2 under Grant 2014-T2-2-060 and in part by Tier 1 under Grant RG25/14 through the Nanyang Technological University, Singapore. W. Zhou was supported by the National University of Singapore, Singapore, under Grant R-155-000-131-112.\r\n","status":"public","volume":61,"quality_controlled":"1","author":[{"last_name":"Bao","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475"},{"full_name":"Pan, Guangming","first_name":"Guangming","last_name":"Pan"},{"full_name":"Zhou, Wang","first_name":"Wang","last_name":"Zhou"}],"publist_id":"5586","title":"Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices","language":[{"iso":"eng"}],"date_published":"2015-06-01T00:00:00Z","isi":1,"publisher":"IEEE","date_created":"2018-12-11T11:52:52Z","intvolume":" 61","scopus_import":"1","publication":"IEEE Transactions on Information Theory","_id":"1585","page":"3413 - 3426"},{"volume":27,"quality_controlled":"1","title":"Edge universality for deformed Wigner matrices","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"5475","arxiv":1,"language":[{"iso":"eng"}],"date_published":"2015-09-01T00:00:00Z","isi":1,"publisher":"World Scientific Publishing","oa":1,"date_created":"2018-12-11T11:53:24Z","_id":"1674","intvolume":" 27","scopus_import":"1","publication":"Reviews in Mathematical Physics","publication_status":"published","department":[{"_id":"LaEr"}],"fulldoi":"https://doi.org/10.1142/S0129055X1550018X","external_id":{"isi":["000362566600001"],"arxiv":["1407.8015"]},"article_number":"1550018","date_updated":"2025-09-23T08:32:19Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa_version":"Preprint","article_processing_charge":"No","month":"09","citation":{"chicago":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics. World Scientific Publishing, 2015. https://doi.org/10.1142/S0129055X1550018X.","mla":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:10.1142/S0129055X1550018X.","short":"J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015).","ista":"Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8), 1550018.","ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” Reviews in Mathematical Physics, vol. 27, no. 8. World Scientific Publishing, 2015.","apa":"Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. World Scientific Publishing. https://doi.org/10.1142/S0129055X1550018X","ama":"Lee J, Schnelli K. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 2015;27(8). doi:10.1142/S0129055X1550018X"},"type":"journal_article","day":"01","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.8015"}],"abstract":[{"lang":"eng","text":"We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V 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 of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix."}],"issue":"8","year":"2015","status":"public","doi":"10.1142/S0129055X1550018X"},{"date_updated":"2025-10-22T10:23:38Z","article_processing_charge":"No","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"LaEr"}],"publication_status":"published","external_id":{"isi":["000349364100002"],"arxiv":["1309.5107"]},"fulldoi":"https://doi.org/10.1007/s00023-014-0333-5","issue":"3","abstract":[{"lang":"eng","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"}],"doi":"10.1007/s00023-014-0333-5","status":"public","year":"2015","citation":{"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.","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","short":"L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.","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."},"month":"03","ec_funded":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1309.5107"}],"day":"01","type":"journal_article","language":[{"iso":"eng"}],"arxiv":1,"date_published":"2015-03-01T00:00:00Z","volume":16,"author":[{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös"},{"full_name":"Knowles, Antti","first_name":"Antti","last_name":"Knowles"}],"title":"The Altshuler-Shklovskii formulas for random band matrices II: The general case","publist_id":"5233","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"oa":1,"date_created":"2018-12-11T11:54:26Z","publication":"Annales Henri Poincare","scopus_import":"1","intvolume":" 16","page":"709 - 799","_id":"1864","isi":1,"publisher":"Springer"},{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","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."}],"doi":"10.1038/ncomms7977","status":"public","year":"2015","citation":{"ista":"Knebel J, Weber M, Krüger TH, Frey E. 2015. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 6, 6977.","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.","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","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","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.","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)."},"month":"04","day":"24","type":"journal_article","file_date_updated":"2020-07-14T12:45:17Z","date_updated":"2025-09-23T08:44:01Z","article_number":"6977","oa_version":"Published Version","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","department":[{"_id":"LaEr"}],"publication_status":"published","external_id":{"isi":["000353705000003"]},"fulldoi":"https://doi.org/10.1038/ncomms7977","has_accepted_license":"1","oa":1,"date_created":"2018-12-11T11:54:13Z","publication":"Nature Communications","intvolume":" 6","scopus_import":"1","ddc":["530"],"_id":"1824","pubrep_id":"451","publisher":"Nature Publishing Group","isi":1,"language":[{"iso":"eng"}],"date_published":"2015-04-24T00:00:00Z","file":[{"file_name":"IST-2016-451-v1+1_ncomms7977.pdf","creator":"system","content_type":"application/pdf","date_updated":"2020-07-14T12:45:17Z","date_created":"2018-12-12T10:16:54Z","file_size":1151501,"relation":"main_file","file_id":"5245","access_level":"open_access","checksum":"c4cffb5c8b245e658a34eac71a03e7cc"}],"quality_controlled":"1","volume":6,"publist_id":"5282","title":"Evolutionary games of condensates in coupled birth-death processes","author":[{"last_name":"Knebel","first_name":"Johannes","full_name":"Knebel, Johannes"},{"full_name":"Weber, Markus","last_name":"Weber","first_name":"Markus"},{"full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297","last_name":"Krüger","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Frey, Erwin","last_name":"Frey","first_name":"Erwin"}]}]