[{"doi":"10.1214/17-EJP42","status":"public","publication_status":"published","ddc":["510","539"],"date_updated":"2026-04-08T14:11:36Z","project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"file_date_updated":"2018-12-12T10:13:39Z","file":[{"access_level":"open_access","relation":"main_file","date_created":"2018-12-12T10:13:39Z","file_name":"IST-2017-807-v1+1_euclid.ejp.1488942016.pdf","file_id":"5024","content_type":"application/pdf","creator":"system","file_size":639384,"date_updated":"2018-12-12T10:13:39Z"}],"day":"08","pubrep_id":"807","article_processing_charge":"No","volume":22,"oa_version":"Published Version","oa":1,"has_accepted_license":"1","external_id":{"isi":["000396611900025"],"arxiv":["1606.07353"]},"citation":{"apa":"Alt, J., Erdös, L., &#38; Krüger, T. H. (2017). Local law for random Gram matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-EJP42\">https://doi.org/10.1214/17-EJP42</a>","ama":"Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. <i>Electronic Journal of Probability</i>. 2017;22. doi:<a href=\"https://doi.org/10.1214/17-EJP42\">10.1214/17-EJP42</a>","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/17-EJP42\">https://doi.org/10.1214/17-EJP42</a>.","mla":"Alt, Johannes, et al. “Local Law for Random Gram Matrices.” <i>Electronic Journal of Probability</i>, vol. 22, 25, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/17-EJP42\">10.1214/17-EJP42</a>.","short":"J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017).","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,” <i>Electronic Journal of Probability</i>, vol. 22. Institute of Mathematical Statistics, 2017."},"year":"2017","ec_funded":1,"publication":"Electronic Journal of Probability","month":"03","title":"Local law for random Gram matrices","related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"}]},"publication_identifier":{"issn":["1083-6489"]},"arxiv":1,"type":"journal_article","quality_controlled":"1","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"}],"author":[{"id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","last_name":"Alt","full_name":"Alt, Johannes"},{"last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"full_name":"Krüger, Torben H","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4821-3297","last_name":"Krüger"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"LaEr"}],"publist_id":"6386","publisher":"Institute of Mathematical Statistics","isi":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"25","_id":"1010","intvolume":"        22","date_published":"2017-03-08T00:00:00Z","language":[{"iso":"eng"}],"scopus_import":"1","date_created":"2018-12-11T11:49:40Z"},{"author":[{"full_name":"Nemish, Yuriy","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","first_name":"Yuriy","orcid":"0000-0002-7327-856X","last_name":"Nemish"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"LaEr"}],"abstract":[{"lang":"eng","text":"We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε."}],"quality_controlled":"1","type":"journal_article","publication_identifier":{"issn":["1083-6489"]},"title":"Local law for the product of independent non-Hermitian random matrices with independent entries","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:49:44Z","scopus_import":"1","intvolume":"        22","_id":"1023","article_number":"22","date_published":"2017-02-06T00:00:00Z","isi":1,"publisher":"Institute of Mathematical Statistics","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publist_id":"6370","oa_version":"Published Version","oa":1,"file":[{"file_name":"IST-2017-802-v1+1_euclid.ejp.1487991681.pdf","file_id":"5149","date_created":"2018-12-12T10:15:29Z","relation":"main_file","access_level":"open_access","date_updated":"2018-12-12T10:15:29Z","file_size":742275,"creator":"system","content_type":"application/pdf"}],"file_date_updated":"2018-12-12T10:15:29Z","volume":22,"article_processing_charge":"No","pubrep_id":"802","day":"06","date_updated":"2025-07-10T11:49:47Z","publication_status":"published","status":"public","doi":"10.1214/17-EJP38","ddc":["510"],"publication":"Electronic Journal of Probability","month":"02","citation":{"ieee":"Y. Nemish, “Local law for the product of independent non-Hermitian random matrices with independent entries,” <i>Electronic Journal of Probability</i>, vol. 22. Institute of Mathematical Statistics, 2017.","ama":"Nemish Y. Local law for the product of independent non-Hermitian random matrices with independent entries. <i>Electronic Journal of Probability</i>. 2017;22. doi:<a href=\"https://doi.org/10.1214/17-EJP38\">10.1214/17-EJP38</a>","apa":"Nemish, Y. (2017). Local law for the product of independent non-Hermitian random matrices with independent entries. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/17-EJP38\">https://doi.org/10.1214/17-EJP38</a>","short":"Y. Nemish, Electronic Journal of Probability 22 (2017).","mla":"Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>, vol. 22, 22, Institute of Mathematical Statistics, 2017, doi:<a href=\"https://doi.org/10.1214/17-EJP38\">10.1214/17-EJP38</a>.","ista":"Nemish Y. 2017. Local law for the product of independent non-Hermitian random matrices with independent entries. Electronic Journal of Probability. 22, 22.","chicago":"Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2017. <a href=\"https://doi.org/10.1214/17-EJP38\">https://doi.org/10.1214/17-EJP38</a>."},"year":"2017","external_id":{"isi":["000396611900022"]},"has_accepted_license":"1"},{"date_updated":"2025-09-10T10:58:02Z","doi":"10.1002/cpa.21639","publication_status":"published","status":"public","oa_version":"Submitted Version","oa":1,"project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"day":"01","article_processing_charge":"No","volume":70,"external_id":{"isi":["000405752100002"],"arxiv":["1512.03703"]},"citation":{"apa":"Ajanki, O. H., Krüger, T. H., &#38; Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. Wiley. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>","ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. <i>Communications on Pure and Applied Mathematics</i>. 2017;70(9):1672-1705. doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>. Wiley, 2017. <a href=\"https://doi.org/10.1002/cpa.21639\">https://doi.org/10.1002/cpa.21639</a>.","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.","mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9, Wiley, 2017, pp. 1672–705, doi:<a href=\"https://doi.org/10.1002/cpa.21639\">10.1002/cpa.21639</a>.","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” <i>Communications on Pure and Applied Mathematics</i>, vol. 70, no. 9. Wiley, pp. 1672–1705, 2017."},"year":"2017","month":"09","publication":"Communications on Pure and Applied Mathematics","ec_funded":1,"arxiv":1,"publication_identifier":{"issn":["0010-3640"]},"corr_author":"1","type":"journal_article","main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","author":[{"full_name":"Ajanki, Oskari H","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","last_name":"Ajanki"},{"full_name":"Krüger, Torben H","last_name":"Krüger","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4821-3297"},{"full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","first_name":"László"}],"department":[{"_id":"LaEr"}],"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."}],"quality_controlled":"1","publisher":"Wiley","isi":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"6959","language":[{"iso":"eng"}],"scopus_import":"1","issue":"9","date_created":"2018-12-11T11:48:08Z","page":"1672 - 1705","_id":"721","intvolume":"        70","date_published":"2017-09-01T00:00:00Z"},{"external_id":{"isi":["000412150400010"],"arxiv":["1606.03076"]},"year":"2017","citation":{"apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">https://doi.org/10.1016/j.aim.2017.08.028</a>","ama":"Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. <i>Advances in Mathematics</i>. 2017;319:251-291. doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">10.1016/j.aim.2017.08.028</a>","short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291.","mla":"Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” <i>Advances in Mathematics</i>, vol. 319, Academic Press, 2017, pp. 251–91, doi:<a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">10.1016/j.aim.2017.08.028</a>.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” <i>Advances in Mathematics</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.aim.2017.08.028\">https://doi.org/10.1016/j.aim.2017.08.028</a>.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” <i>Advances in Mathematics</i>, vol. 319. Academic Press, pp. 251–291, 2017."},"ec_funded":1,"publication":"Advances in Mathematics","month":"10","doi":"10.1016/j.aim.2017.08.028","status":"public","publication_status":"published","date_updated":"2025-06-04T10:13:45Z","article_processing_charge":"No","day":"15","volume":319,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"oa":1,"oa_version":"Submitted Version","publist_id":"6935","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Academic Press","isi":1,"date_published":"2017-10-15T00:00:00Z","page":"251 - 291","intvolume":"       319","_id":"733","scopus_import":"1","date_created":"2018-12-11T11:48:13Z","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","language":[{"iso":"eng"}],"title":"Convergence rate for spectral distribution of addition of random matrices","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1606.03076"}],"arxiv":1,"type":"journal_article","corr_author":"1","quality_controlled":"1","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."}],"department":[{"_id":"LaEr"}],"author":[{"full_name":"Bao, Zhigang","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3036-1475","last_name":"Bao"},{"last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"full_name":"Schnelli, Kevin","last_name":"Schnelli","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin"}]},{"quality_controlled":"1","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"}],"department":[{"_id":"LaEr"}],"author":[{"full_name":"Lee, Ji","first_name":"Ji","last_name":"Lee"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli"}],"title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1409.4979"}],"arxiv":1,"type":"journal_article","date_published":"2016-12-15T00:00:00Z","page":"3786 - 3839","intvolume":"        26","_id":"1157","scopus_import":"1","date_created":"2018-12-11T11:50:27Z","issue":"6","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.","language":[{"iso":"eng"}],"publist_id":"6201","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"Institute of Mathematical Statistics","isi":1,"article_processing_charge":"No","day":"15","volume":26,"project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804"}],"oa":1,"oa_version":"Preprint","doi":"10.1214/16-AAP1193","publication_status":"published","status":"public","date_updated":"2025-09-22T09:55:43Z","ec_funded":1,"month":"12","publication":"Annals of Applied Probability","external_id":{"isi":["000391240100016"],"arxiv":["1409.4979"]},"citation":{"ieee":"J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” <i>Annals of Applied Probability</i>, 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. <i>Annals of Applied Probability</i>. 2016;26(6):3786-3839. doi:<a href=\"https://doi.org/10.1214/16-AAP1193\">10.1214/16-AAP1193</a>","apa":"Lee, J., &#38; Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/16-AAP1193\">https://doi.org/10.1214/16-AAP1193</a>","short":"J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.","mla":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:<a href=\"https://doi.org/10.1214/16-AAP1193\">10.1214/16-AAP1193</a>.","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.","chicago":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2016. <a href=\"https://doi.org/10.1214/16-AAP1193\">https://doi.org/10.1214/16-AAP1193</a>."},"year":"2016"},{"month":"01","publication":"Annals of Probability","ec_funded":1,"year":"2016","citation":{"ieee":"J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” <i>Annals of Probability</i>, vol. 44, no. 3. Institute of Mathematical Statistics, pp. 2349–2425, 2016.","apa":"Lee, J., Schnelli, K., Stetler, B., &#38; Yau, H. (2016). Bulk universality for deformed wigner matrices. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/15-AOP1023\">https://doi.org/10.1214/15-AOP1023</a>","ama":"Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner matrices. <i>Annals of Probability</i>. 2016;44(3):2349-2425. doi:<a href=\"https://doi.org/10.1214/15-AOP1023\">10.1214/15-AOP1023</a>","chicago":"Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2016. <a href=\"https://doi.org/10.1214/15-AOP1023\">https://doi.org/10.1214/15-AOP1023</a>.","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.” <i>Annals of Probability</i>, vol. 44, no. 3, Institute of Mathematical Statistics, 2016, pp. 2349–425, doi:<a href=\"https://doi.org/10.1214/15-AOP1023\">10.1214/15-AOP1023</a>.","ista":"Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425."},"external_id":{"isi":["000376180700016"],"arxiv":["1405.6634"]},"oa":1,"oa_version":"Preprint","volume":44,"day":"01","article_processing_charge":"No","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804"}],"date_updated":"2025-09-22T09:33:43Z","status":"public","publication_status":"published","doi":"10.1214/15-AOP1023","date_created":"2018-12-11T11:50:47Z","issue":"3","scopus_import":"1","language":[{"iso":"eng"}],"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.  ","date_published":"2016-01-01T00:00:00Z","_id":"1219","intvolume":"        44","page":"2349 - 2425","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","isi":1,"publisher":"Institute of Mathematical Statistics","publist_id":"6115","department":[{"_id":"LaEr"}],"author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli"},{"full_name":"Stetler, Ben","last_name":"Stetler","first_name":"Ben"},{"last_name":"Yau","first_name":"Horngtzer","full_name":"Yau, Horngtzer"}],"quality_controlled":"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."}],"main_file_link":[{"url":"https://arxiv.org/abs/1405.6634","open_access":"1"}],"type":"journal_article","arxiv":1,"title":"Bulk universality for deformed wigner matrices"},{"doi":"10.4171/JST/132","publication_status":"published","status":"public","date_updated":"2025-09-22T09:32:55Z","article_processing_charge":"No","day":"01","volume":6,"oa_version":"Preprint","oa":1,"external_id":{"arxiv":["1408.3961"],"isi":["000388627000004"]},"citation":{"ieee":"R. Froese, D. Lee, C. Sadel, W. Spitzer, and G. Stolz, “Localization for transversally periodic random potentials on binary trees,” <i>Journal of Spectral Theory</i>, vol. 6, no. 3. European Mathematical Society, pp. 557–600, 2016.","apa":"Froese, R., Lee, D., Sadel, C., Spitzer, W., &#38; Stolz, G. (2016). Localization for transversally periodic random potentials on binary trees. <i>Journal of Spectral Theory</i>. European Mathematical Society. <a href=\"https://doi.org/10.4171/JST/132\">https://doi.org/10.4171/JST/132</a>","ama":"Froese R, Lee D, Sadel C, Spitzer W, Stolz G. Localization for transversally periodic random potentials on binary trees. <i>Journal of Spectral Theory</i>. 2016;6(3):557-600. doi:<a href=\"https://doi.org/10.4171/JST/132\">10.4171/JST/132</a>","chicago":"Froese, Richard, Darrick Lee, Christian Sadel, Wolfgang Spitzer, and Günter Stolz. “Localization for Transversally Periodic Random Potentials on Binary Trees.” <i>Journal of Spectral Theory</i>. European Mathematical Society, 2016. <a href=\"https://doi.org/10.4171/JST/132\">https://doi.org/10.4171/JST/132</a>.","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.","mla":"Froese, Richard, et al. “Localization for Transversally Periodic Random Potentials on Binary Trees.” <i>Journal of Spectral Theory</i>, vol. 6, no. 3, European Mathematical Society, 2016, pp. 557–600, doi:<a href=\"https://doi.org/10.4171/JST/132\">10.4171/JST/132</a>.","short":"R. Froese, D. Lee, C. Sadel, W. Spitzer, G. Stolz, Journal of Spectral Theory 6 (2016) 557–600."},"year":"2016","month":"01","publication":"Journal of Spectral Theory","title":"Localization for transversally periodic random potentials on binary trees","arxiv":1,"type":"journal_article","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1408.3961"}],"abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","author":[{"full_name":"Froese, Richard","last_name":"Froese","first_name":"Richard"},{"first_name":"Darrick","last_name":"Lee","full_name":"Lee, Darrick"},{"last_name":"Sadel","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","first_name":"Christian","orcid":"0000-0001-8255-3968","full_name":"Sadel, Christian"},{"full_name":"Spitzer, Wolfgang","first_name":"Wolfgang","last_name":"Spitzer"},{"last_name":"Stolz","first_name":"Günter","full_name":"Stolz, Günter"}],"department":[{"_id":"LaEr"}],"publist_id":"6112","publisher":"European Mathematical Society","isi":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"557 - 600","intvolume":"         6","_id":"1223","date_published":"2016-01-01T00:00:00Z","language":[{"iso":"eng"}],"scopus_import":"1","issue":"3","date_created":"2018-12-11T11:50:48Z"},{"isi":1,"publisher":"Springer","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"6067","language":[{"iso":"eng"}],"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.","issue":"3","date_created":"2018-12-11T11:50:59Z","scopus_import":"1","_id":"1257","intvolume":"       343","page":"881 - 919","date_published":"2016-05-01T00:00:00Z","type":"journal_article","corr_author":"1","title":"A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes","author":[{"last_name":"Sadel","first_name":"Christian","orcid":"0000-0001-8255-3968","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","full_name":"Sadel, Christian"},{"full_name":"Virág, Bálint","last_name":"Virág","first_name":"Bálint"}],"department":[{"_id":"LaEr"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"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."}],"quality_controlled":"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,” <i>Communications in Mathematical Physics</i>, 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. <i>Communications in Mathematical Physics</i>. 2016;343(3):881-919. doi:<a href=\"https://doi.org/10.1007/s00220-016-2600-4\">10.1007/s00220-016-2600-4</a>","apa":"Sadel, C., &#38; Virág, B. (2016). A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. <i>Communications in Mathematical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s00220-016-2600-4\">https://doi.org/10.1007/s00220-016-2600-4</a>","short":"C. Sadel, B. Virág, Communications in Mathematical Physics 343 (2016) 881–919.","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.","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.” <i>Communications in Mathematical Physics</i>, vol. 343, no. 3, Springer, 2016, pp. 881–919, doi:<a href=\"https://doi.org/10.1007/s00220-016-2600-4\">10.1007/s00220-016-2600-4</a>.","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.” <i>Communications in Mathematical Physics</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00220-016-2600-4\">https://doi.org/10.1007/s00220-016-2600-4</a>."},"year":"2016","external_id":{"isi":["000374659800005"]},"has_accepted_license":"1","month":"05","publication":"Communications in Mathematical Physics","ec_funded":1,"date_updated":"2025-09-22T09:02:32Z","status":"public","publication_status":"published","doi":"10.1007/s00220-016-2600-4","ddc":["510","539"],"oa_version":"Published Version","oa":1,"file":[{"file_name":"IST-2016-703-v1+1_s00220-016-2600-4.pdf","file_id":"5119","date_created":"2018-12-12T10:15:02Z","relation":"main_file","checksum":"4fb2411d9c2f56676123165aad46c828","access_level":"open_access","file_size":800792,"date_updated":"2020-07-14T12:44:42Z","creator":"system","content_type":"application/pdf"}],"project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"file_date_updated":"2020-07-14T12:44:42Z","volume":343,"article_processing_charge":"Yes (via OA deal)","pubrep_id":"703","day":"01"},{"abstract":[{"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.","lang":"eng"}],"department":[{"_id":"LaEr"}],"author":[{"first_name":"Paul","last_name":"Bourgade","full_name":"Bourgade, Paul"},{"orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"Erdös, László"},{"last_name":"Yau","first_name":"Horngtzer","full_name":"Yau, Horngtzer"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"title":"Fixed energy universality for generalized wigner matrices","main_file_link":[{"url":"https://arxiv.org/abs/1407.5606","open_access":"1"}],"corr_author":"1","type":"journal_article","arxiv":1,"date_published":"2016-10-01T00:00:00Z","intvolume":"        69","_id":"1280","page":"1815 - 1881","date_created":"2018-12-11T11:51:07Z","issue":"10","scopus_import":"1","language":[{"iso":"eng"}],"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.","publist_id":"6036","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","isi":1,"publisher":"Wiley-Blackwell","volume":69,"day":"01","article_processing_charge":"No","project":[{"grant_number":"338804","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"oa":1,"oa_version":"Preprint","status":"public","publication_status":"published","doi":"10.1002/cpa.21624","date_updated":"2025-09-22T08:35:52Z","ec_funded":1,"publication":"Communications on Pure and Applied Mathematics","month":"10","year":"2016","citation":{"chicago":"Bourgade, Paul, László Erdös, Horngtzer Yau, and Jun Yin. “Fixed Energy Universality for Generalized Wigner Matrices.” <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell, 2016. <a href=\"https://doi.org/10.1002/cpa.21624\">https://doi.org/10.1002/cpa.21624</a>.","mla":"Bourgade, Paul, et al. “Fixed Energy Universality for Generalized Wigner Matrices.” <i>Communications on Pure and Applied Mathematics</i>, vol. 69, no. 10, Wiley-Blackwell, 2016, pp. 1815–81, doi:<a href=\"https://doi.org/10.1002/cpa.21624\">10.1002/cpa.21624</a>.","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.","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881.","ama":"Bourgade P, Erdös L, Yau H, Yin J. Fixed energy universality for generalized wigner matrices. <i>Communications on Pure and Applied Mathematics</i>. 2016;69(10):1815-1881. doi:<a href=\"https://doi.org/10.1002/cpa.21624\">10.1002/cpa.21624</a>","apa":"Bourgade, P., Erdös, L., Yau, H., &#38; Yin, J. (2016). Fixed energy universality for generalized wigner matrices. <i>Communications on Pure and Applied Mathematics</i>. Wiley-Blackwell. <a href=\"https://doi.org/10.1002/cpa.21624\">https://doi.org/10.1002/cpa.21624</a>","ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Fixed energy universality for generalized wigner matrices,” <i>Communications on Pure and Applied Mathematics</i>, vol. 69, no. 10. Wiley-Blackwell, pp. 1815–1881, 2016."},"external_id":{"isi":["000382932900001"],"arxiv":["1407.5606"]}},{"scopus_import":"1","issue":"3","date_created":"2018-12-11T11:52:00Z","language":[{"iso":"eng"}],"date_published":"2016-08-01T00:00:00Z","page":"672 - 719","intvolume":"       271","_id":"1434","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"Academic Press","isi":1,"publist_id":"5764","department":[{"_id":"LaEr"}],"author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László"},{"last_name":"Schnelli","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","full_name":"Schnelli, Kevin"}],"quality_controlled":"1","abstract":[{"lang":"eng","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."}],"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1508.05905"}],"arxiv":1,"corr_author":"1","type":"journal_article","title":"Local stability of the free additive convolution","month":"08","publication":"Journal of Functional Analysis","ec_funded":1,"external_id":{"arxiv":["1508.05905"],"isi":["000378013400009"]},"year":"2016","citation":{"ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,” <i>Journal of Functional Analysis</i>, vol. 271, no. 3. Academic Press, pp. 672–719, 2016.","ama":"Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution. <i>Journal of Functional Analysis</i>. 2016;271(3):672-719. doi:<a href=\"https://doi.org/10.1016/j.jfa.2016.04.006\">10.1016/j.jfa.2016.04.006</a>","apa":"Bao, Z., Erdös, L., &#38; Schnelli, K. (2016). Local stability of the free additive convolution. <i>Journal of Functional Analysis</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jfa.2016.04.006\">https://doi.org/10.1016/j.jfa.2016.04.006</a>","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the Free Additive Convolution.” <i>Journal of Functional Analysis</i>. Academic Press, 2016. <a href=\"https://doi.org/10.1016/j.jfa.2016.04.006\">https://doi.org/10.1016/j.jfa.2016.04.006</a>.","mla":"Bao, Zhigang, et al. “Local Stability of the Free Additive Convolution.” <i>Journal of Functional Analysis</i>, vol. 271, no. 3, Academic Press, 2016, pp. 672–719, doi:<a href=\"https://doi.org/10.1016/j.jfa.2016.04.006\">10.1016/j.jfa.2016.04.006</a>.","ista":"Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 672–719.","short":"Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719."},"oa":1,"oa_version":"Preprint","day":"01","article_processing_charge":"No","volume":271,"project":[{"grant_number":"338804","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"date_updated":"2025-09-18T14:00:03Z","doi":"10.1016/j.jfa.2016.04.006","publication_status":"published","status":"public"},{"language":[{"iso":"eng"}],"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.","issue":"2","date_created":"2018-12-11T11:52:19Z","scopus_import":"1","_id":"1489","intvolume":"       163","page":"280 - 302","date_published":"2016-04-01T00:00:00Z","isi":1,"publisher":"Springer","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"5698","author":[{"full_name":"Ajanki, Oskari H","last_name":"Ajanki","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H"},{"full_name":"Erdös, László","last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Krüger, Torben H","last_name":"Krüger","first_name":"Torben H","orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"department":[{"_id":"LaEr"}],"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"}],"quality_controlled":"1","type":"journal_article","corr_author":"1","title":"Local spectral statistics of Gaussian matrices with correlated entries","month":"04","publication":"Journal of Statistical Physics","ec_funded":1,"citation":{"chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” <i>Journal of Statistical Physics</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s10955-016-1479-y\">https://doi.org/10.1007/s10955-016-1479-y</a>.","mla":"Ajanki, Oskari H., et al. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” <i>Journal of Statistical Physics</i>, vol. 163, no. 2, Springer, 2016, pp. 280–302, doi:<a href=\"https://doi.org/10.1007/s10955-016-1479-y\">10.1007/s10955-016-1479-y</a>.","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.","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016) 280–302.","apa":"Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2016). Local spectral statistics of Gaussian matrices with correlated entries. <i>Journal of Statistical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s10955-016-1479-y\">https://doi.org/10.1007/s10955-016-1479-y</a>","ama":"Ajanki OH, Erdös L, Krüger TH. Local spectral statistics of Gaussian matrices with correlated entries. <i>Journal of Statistical Physics</i>. 2016;163(2):280-302. doi:<a href=\"https://doi.org/10.1007/s10955-016-1479-y\">10.1007/s10955-016-1479-y</a>","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local spectral statistics of Gaussian matrices with correlated entries,” <i>Journal of Statistical Physics</i>, vol. 163, no. 2. Springer, pp. 280–302, 2016."},"year":"2016","external_id":{"isi":["000373132700003"]},"has_accepted_license":"1","oa_version":"Published Version","oa":1,"file_date_updated":"2020-07-14T12:44:57Z","file":[{"content_type":"application/pdf","creator":"system","date_updated":"2020-07-14T12:44:57Z","file_size":660602,"checksum":"7139598dcb1cafbe6866bd2bfd732b33","access_level":"open_access","relation":"main_file","date_created":"2018-12-12T10:11:16Z","file_name":"IST-2016-516-v1+1_s10955-016-1479-y.pdf","file_id":"4869"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"volume":163,"day":"01","pubrep_id":"516","article_processing_charge":"Yes (via OA deal)","date_updated":"2025-09-18T11:17:50Z","status":"public","publication_status":"published","doi":"10.1007/s10955-016-1479-y","ddc":["510"]},{"publist_id":"5558","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","isi":1,"publisher":"Birkhäuser","date_published":"2016-07-01T00:00:00Z","intvolume":"        17","_id":"1608","page":"1631 - 1675","issue":"7","date_created":"2018-12-11T11:53:00Z","scopus_import":"1","language":[{"iso":"eng"}],"title":"Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1501.04287"}],"type":"journal_article","corr_author":"1","arxiv":1,"quality_controlled":"1","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&gt;2 this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d dimensional growth with d&lt;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. "}],"department":[{"_id":"LaEr"}],"author":[{"full_name":"Sadel, Christian","last_name":"Sadel","first_name":"Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-8255-3968"}],"citation":{"ieee":"C. Sadel, “Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel,” <i>Annales Henri Poincare</i>, vol. 17, no. 7. Birkhäuser, pp. 1631–1675, 2016.","ama":"Sadel C. Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. <i>Annales Henri Poincare</i>. 2016;17(7):1631-1675. doi:<a href=\"https://doi.org/10.1007/s00023-015-0456-3\">10.1007/s00023-015-0456-3</a>","apa":"Sadel, C. (2016). Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. <i>Annales Henri Poincare</i>. Birkhäuser. <a href=\"https://doi.org/10.1007/s00023-015-0456-3\">https://doi.org/10.1007/s00023-015-0456-3</a>","mla":"Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel.” <i>Annales Henri Poincare</i>, vol. 17, no. 7, Birkhäuser, 2016, pp. 1631–75, doi:<a href=\"https://doi.org/10.1007/s00023-015-0456-3\">10.1007/s00023-015-0456-3</a>.","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.","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.” <i>Annales Henri Poincare</i>. Birkhäuser, 2016. <a href=\"https://doi.org/10.1007/s00023-015-0456-3\">https://doi.org/10.1007/s00023-015-0456-3</a>."},"year":"2016","external_id":{"arxiv":["1501.04287"],"isi":["000377994000003"]},"ec_funded":1,"publication":"Annales Henri Poincare","month":"07","status":"public","publication_status":"published","doi":"10.1007/s00023-015-0456-3","date_updated":"2025-09-18T11:00:43Z","volume":17,"day":"01","article_processing_charge":"No","project":[{"name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"291734"}],"oa":1,"oa_version":"Preprint"},{"oa_version":"Preprint","oa":1,"project":[{"grant_number":"338804","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"day":"01","article_processing_charge":"No","volume":164,"date_updated":"2025-09-18T10:46:46Z","doi":"10.1007/s00440-014-0610-8","status":"public","publication_status":"published","publication":"Probability Theory and Related Fields","month":"02","ec_funded":1,"external_id":{"isi":["000373163300006"],"arxiv":["1310.7057"]},"year":"2016","citation":{"ieee":"J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.","mla":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:<a href=\"https://doi.org/10.1007/s00440-014-0610-8\">10.1007/s00440-014-0610-8</a>.","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.","ista":"Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.","chicago":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00440-014-0610-8\">https://doi.org/10.1007/s00440-014-0610-8</a>.","ama":"Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. 2016;164(1-2):165-241. doi:<a href=\"https://doi.org/10.1007/s00440-014-0610-8\">10.1007/s00440-014-0610-8</a>","apa":"Lee, J., &#38; Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-014-0610-8\">https://doi.org/10.1007/s00440-014-0610-8</a>"},"author":[{"full_name":"Lee, Jioon","last_name":"Lee","first_name":"Jioon"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"department":[{"_id":"LaEr"}],"quality_controlled":"1","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&gt;λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ&gt;λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ&gt;λ+, while they are completely delocalized for λ&lt;λ+. Similar results hold for the lowest eigenvalues. "}],"arxiv":1,"corr_author":"1","type":"journal_article","main_file_link":[{"url":"http://arxiv.org/abs/1310.7057","open_access":"1"}],"title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","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.","language":[{"iso":"eng"}],"scopus_import":"1","issue":"1-2","date_created":"2018-12-11T11:54:31Z","page":"165 - 241","_id":"1881","intvolume":"       164","date_published":"2016-02-01T00:00:00Z","publisher":"Springer","isi":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"5215"},{"quality_controlled":"1","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). "}],"author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László"},{"first_name":"Antti","last_name":"Knowles","full_name":"Knowles, Antti"}],"department":[{"_id":"LaEr"}],"title":"The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case","type":"journal_article","arxiv":1,"main_file_link":[{"url":"http://arxiv.org/abs/1309.5106","open_access":"1"}],"intvolume":"       333","_id":"2166","page":"1365 - 1416","date_published":"2015-02-01T00:00:00Z","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:56:05Z","issue":"3","scopus_import":"1","publist_id":"4818","isi":1,"publisher":"Springer","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","volume":333,"day":"01","article_processing_charge":"No","oa_version":"Preprint","oa":1,"publication_status":"published","status":"public","doi":"10.1007/s00220-014-2119-5","date_updated":"2025-09-23T13:39:37Z","month":"02","publication":"Communications in Mathematical Physics","citation":{"chicago":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” <i>Communications in Mathematical Physics</i>. Springer, 2015. <a href=\"https://doi.org/10.1007/s00220-014-2119-5\">https://doi.org/10.1007/s00220-014-2119-5</a>.","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.” <i>Communications in Mathematical Physics</i>, vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:<a href=\"https://doi.org/10.1007/s00220-014-2119-5\">10.1007/s00220-014-2119-5</a>.","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.","ama":"Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. <i>Communications in Mathematical Physics</i>. 2015;333(3):1365-1416. doi:<a href=\"https://doi.org/10.1007/s00220-014-2119-5\">10.1007/s00220-014-2119-5</a>","apa":"Erdös, L., &#38; Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. <i>Communications in Mathematical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s00220-014-2119-5\">https://doi.org/10.1007/s00220-014-2119-5</a>","ieee":"L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case,” <i>Communications in Mathematical Physics</i>, vol. 333, no. 3. Springer, pp. 1365–1416, 2015."},"year":"2015","external_id":{"arxiv":["1309.5106"],"isi":["000348303100008"]}},{"citation":{"ama":"Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. <i>Annals of Statistics</i>. 2015;43(1):382-421. doi:<a href=\"https://doi.org/10.1214/14-AOS1281\">10.1214/14-AOS1281</a>","apa":"Bao, Z., Pan, G., &#38; Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. <i>Annals of Statistics</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/14-AOS1281\">https://doi.org/10.1214/14-AOS1281</a>","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.","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","mla":"Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” <i>Annals of Statistics</i>, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:<a href=\"https://doi.org/10.1214/14-AOS1281\">10.1214/14-AOS1281</a>.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” <i>Annals of Statistics</i>. Institute of Mathematical Statistics, 2015. <a href=\"https://doi.org/10.1214/14-AOS1281\">https://doi.org/10.1214/14-AOS1281</a>.","ieee":"Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” <i>Annals of Statistics</i>, vol. 43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015."},"year":"2015","external_id":{"isi":["000349738500014"],"arxiv":["1304.5690"]},"month":"02","publication":"Annals of Statistics","status":"public","publication_status":"published","doi":"10.1214/14-AOS1281","date_updated":"2025-09-29T11:02:34Z","volume":43,"day":"01","article_processing_charge":"No","oa":1,"oa_version":"Preprint","publist_id":"5672","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","isi":1,"publisher":"Institute of Mathematical Statistics","date_published":"2015-02-01T00:00:00Z","_id":"1505","intvolume":"        43","page":"382 - 421","issue":"1","date_created":"2018-12-11T11:52:25Z","language":[{"iso":"eng"}],"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","title":"Universality for the largest eigenvalue of sample covariance matrices with general population","main_file_link":[{"url":"https://arxiv.org/abs/1304.5690","open_access":"1"}],"type":"journal_article","arxiv":1,"abstract":[{"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 (&amp;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 &amp;Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.","lang":"eng"}],"quality_controlled":"1","department":[{"_id":"LaEr"}],"author":[{"full_name":"Bao, Zhigang","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3036-1475","last_name":"Bao"},{"full_name":"Pan, Guangming","last_name":"Pan","first_name":"Guangming"},{"full_name":"Zhou, Wang","last_name":"Zhou","first_name":"Wang"}]},{"author":[{"orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","last_name":"Bao","full_name":"Bao, Zhigang"},{"first_name":"Guangming","last_name":"Pan","full_name":"Pan, Guangming"},{"full_name":"Zhou, Wang","last_name":"Zhou","first_name":"Wang"}],"department":[{"_id":"LaEr"}],"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 &lt;∞, 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"}],"quality_controlled":"1","arxiv":1,"type":"journal_article","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1208.5823"}],"title":"The logarithmic law of random determinant","language":[{"iso":"eng"}],"issue":"3","date_created":"2018-12-11T11:52:25Z","page":"1600 - 1628","_id":"1506","intvolume":"        21","date_published":"2015-08-01T00:00:00Z","publisher":"Bernoulli Society for Mathematical Statistics and Probability","isi":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"5671","oa_version":"Preprint","oa":1,"article_processing_charge":"No","day":"01","volume":21,"date_updated":"2025-09-23T13:59:56Z","doi":"10.3150/14-BEJ615","publication_status":"published","status":"public","month":"08","publication":"Bernoulli","external_id":{"isi":["000356993100012"],"arxiv":["1208.5823"]},"citation":{"ieee":"Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” <i>Bernoulli</i>, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics and Probability, pp. 1600–1628, 2015.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability, 2015. <a href=\"https://doi.org/10.3150/14-BEJ615\">https://doi.org/10.3150/14-BEJ615</a>.","ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628.","short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","mla":"Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” <i>Bernoulli</i>, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:<a href=\"https://doi.org/10.3150/14-BEJ615\">10.3150/14-BEJ615</a>.","ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. <i>Bernoulli</i>. 2015;21(3):1600-1628. doi:<a href=\"https://doi.org/10.3150/14-BEJ615\">10.3150/14-BEJ615</a>","apa":"Bao, Z., Pan, G., &#38; Zhou, W. (2015). The logarithmic law of random determinant. <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability. <a href=\"https://doi.org/10.3150/14-BEJ615\">https://doi.org/10.3150/14-BEJ615</a>"},"year":"2015"},{"month":"08","publication":"Journal of the European Mathematical Society","citation":{"mla":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” <i>Journal of the European Mathematical Society</i>, vol. 17, no. 8, European Mathematical Society, 2015, pp. 1927–2036, doi:<a href=\"https://doi.org/10.4171/JEMS/548\">10.4171/JEMS/548</a>.","ista":"Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 17(8), 1927–2036.","short":"L. Erdös, H. Yau, Journal of the European Mathematical Society 17 (2015) 1927–2036.","chicago":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” <i>Journal of the European Mathematical Society</i>. European Mathematical Society, 2015. <a href=\"https://doi.org/10.4171/JEMS/548\">https://doi.org/10.4171/JEMS/548</a>.","ama":"Erdös L, Yau H. Gap universality of generalized Wigner and β ensembles. <i>Journal of the European Mathematical Society</i>. 2015;17(8):1927-2036. doi:<a href=\"https://doi.org/10.4171/JEMS/548\">10.4171/JEMS/548</a>","apa":"Erdös, L., &#38; Yau, H. (2015). Gap universality of generalized Wigner and β ensembles. <i>Journal of the European Mathematical Society</i>. European Mathematical Society. <a href=\"https://doi.org/10.4171/JEMS/548\">https://doi.org/10.4171/JEMS/548</a>","ieee":"L. Erdös and H. Yau, “Gap universality of generalized Wigner and β ensembles,” <i>Journal of the European Mathematical Society</i>, vol. 17, no. 8. European Mathematical Society, pp. 1927–2036, 2015."},"year":"2015","external_id":{"arxiv":["1211.3786"],"isi":["000360822900003"]},"oa_version":"Preprint","oa":1,"volume":17,"day":"01","article_processing_charge":"No","date_updated":"2025-09-23T09:08:38Z","status":"public","publication_status":"published","doi":"10.4171/JEMS/548","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:52:26Z","issue":"8","scopus_import":"1","intvolume":"        17","_id":"1508","page":"1927 - 2036","date_published":"2015-08-01T00:00:00Z","isi":1,"publisher":"European Mathematical Society","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"5669","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László"},{"full_name":"Yau, Horng","last_name":"Yau","first_name":"Horng"}],"department":[{"_id":"LaEr"}],"abstract":[{"lang":"eng","text":"We consider generalized Wigner ensembles and general β-ensembles with analytic potentials for any β ≥ 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any C4(ℝ) potential."}],"quality_controlled":"1","type":"journal_article","arxiv":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1211.3786"}],"title":"Gap universality of generalized Wigner and β ensembles"},{"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)."}],"quality_controlled":"1","department":[{"_id":"LaEr"}],"author":[{"full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3036-1475","first_name":"Zhigang","last_name":"Bao"},{"full_name":"Pan, Guangming","last_name":"Pan","first_name":"Guangming"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}],"title":"Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices","type":"journal_article","date_published":"2015-06-01T00:00:00Z","page":"3413 - 3426","_id":"1585","intvolume":"        61","scopus_import":"1","issue":"6","date_created":"2018-12-11T11:52:52Z","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","language":[{"iso":"eng"}],"publist_id":"5586","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publisher":"IEEE","isi":1,"day":"01","article_processing_charge":"No","volume":61,"oa_version":"None","doi":"10.1109/TIT.2015.2421894","status":"public","publication_status":"published","date_updated":"2025-09-23T07:57:31Z","month":"06","publication":"IEEE Transactions on Information Theory","external_id":{"isi":["000354943600029"]},"citation":{"ieee":"Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices,” <i>IEEE Transactions on Information Theory</i>, vol. 61, no. 6. IEEE, pp. 3413–3426, 2015.","ama":"Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. <i>IEEE Transactions on Information Theory</i>. 2015;61(6):3413-3426. doi:<a href=\"https://doi.org/10.1109/TIT.2015.2421894\">10.1109/TIT.2015.2421894</a>","apa":"Bao, Z., Pan, G., &#38; Zhou, W. (2015). Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. <i>IEEE Transactions on Information Theory</i>. IEEE. <a href=\"https://doi.org/10.1109/TIT.2015.2421894\">https://doi.org/10.1109/TIT.2015.2421894</a>","mla":"Bao, Zhigang, et al. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” <i>IEEE Transactions on Information Theory</i>, vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:<a href=\"https://doi.org/10.1109/TIT.2015.2421894\">10.1109/TIT.2015.2421894</a>.","short":"Z. Bao, G. Pan, W. Zhou, IEEE Transactions on Information Theory 61 (2015) 3413–3426.","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.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” <i>IEEE Transactions on Information Theory</i>. IEEE, 2015. <a href=\"https://doi.org/10.1109/TIT.2015.2421894\">https://doi.org/10.1109/TIT.2015.2421894</a>."},"year":"2015"},{"author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"last_name":"Schnelli","orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"department":[{"_id":"LaEr"}],"quality_controlled":"1","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."}],"type":"journal_article","arxiv":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.8015"}],"title":"Edge universality for deformed Wigner matrices","language":[{"iso":"eng"}],"issue":"8","date_created":"2018-12-11T11:53:24Z","scopus_import":"1","_id":"1674","intvolume":"        27","article_number":"1550018","date_published":"2015-09-01T00:00:00Z","isi":1,"publisher":"World Scientific Publishing","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publist_id":"5475","oa_version":"Preprint","oa":1,"volume":27,"article_processing_charge":"No","day":"01","date_updated":"2025-09-23T08:32:19Z","status":"public","publication_status":"published","doi":"10.1142/S0129055X1550018X","publication":"Reviews in Mathematical Physics","month":"09","citation":{"ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” <i>Reviews in Mathematical Physics</i>, vol. 27, no. 8. World Scientific Publishing, 2015.","ama":"Lee J, Schnelli K. Edge universality for deformed Wigner matrices. <i>Reviews in Mathematical Physics</i>. 2015;27(8). doi:<a href=\"https://doi.org/10.1142/S0129055X1550018X\">10.1142/S0129055X1550018X</a>","apa":"Lee, J., &#38; Schnelli, K. (2015). Edge universality for deformed Wigner matrices. <i>Reviews in Mathematical Physics</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/S0129055X1550018X\">https://doi.org/10.1142/S0129055X1550018X</a>","short":"J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015).","mla":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” <i>Reviews in Mathematical Physics</i>, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:<a href=\"https://doi.org/10.1142/S0129055X1550018X\">10.1142/S0129055X1550018X</a>.","ista":"Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8), 1550018.","chicago":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” <i>Reviews in Mathematical Physics</i>. World Scientific Publishing, 2015. <a href=\"https://doi.org/10.1142/S0129055X1550018X\">https://doi.org/10.1142/S0129055X1550018X</a>."},"year":"2015","external_id":{"isi":["000362566600001"],"arxiv":["1407.8015"]}},{"publist_id":"5472","isi":1,"publisher":"American Institute of Physics","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"1677","intvolume":"        56","article_number":"103301","date_published":"2015-10-09T00:00:00Z","language":[{"iso":"eng"}],"issue":"10","date_created":"2018-12-11T11:53:25Z","scopus_import":"1","title":"The local semicircle law for random matrices with a fourfold symmetry","related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"}]},"type":"journal_article","corr_author":"1","arxiv":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1506.04683"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale."}],"author":[{"full_name":"Alt, Johannes","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","last_name":"Alt"}],"department":[{"_id":"LaEr"}],"year":"2015","citation":{"ieee":"J. Alt, “The local semicircle law for random matrices with a fourfold symmetry,” <i>Journal of Mathematical Physics</i>, vol. 56, no. 10. American Institute of Physics, 2015.","ama":"Alt J. The local semicircle law for random matrices with a fourfold symmetry. <i>Journal of Mathematical Physics</i>. 2015;56(10). doi:<a href=\"https://doi.org/10.1063/1.4932606\">10.1063/1.4932606</a>","apa":"Alt, J. (2015). The local semicircle law for random matrices with a fourfold symmetry. <i>Journal of Mathematical Physics</i>. American Institute of Physics. <a href=\"https://doi.org/10.1063/1.4932606\">https://doi.org/10.1063/1.4932606</a>","ista":"Alt J. 2015. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 56(10), 103301.","short":"J. Alt, Journal of Mathematical Physics 56 (2015).","mla":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” <i>Journal of Mathematical Physics</i>, vol. 56, no. 10, 103301, American Institute of Physics, 2015, doi:<a href=\"https://doi.org/10.1063/1.4932606\">10.1063/1.4932606</a>.","chicago":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” <i>Journal of Mathematical Physics</i>. American Institute of Physics, 2015. <a href=\"https://doi.org/10.1063/1.4932606\">https://doi.org/10.1063/1.4932606</a>."},"external_id":{"arxiv":["1506.04683"],"isi":["000364237000026"]},"ec_funded":1,"publication":"Journal of Mathematical Physics","month":"10","status":"public","publication_status":"published","doi":"10.1063/1.4932606","date_updated":"2026-04-08T14:11:36Z","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804"}],"volume":56,"article_processing_charge":"No","day":"09","oa_version":"Preprint","oa":1}]