[{"month":"05","language":[{"iso":"eng"}],"year":"2019","publication":"Random Matrices: Theory and Application","oa":1,"date_updated":"2019-08-02T12:39:21Z","external_id":{"arxiv":["1806.08751"]},"oa_version":"Preprint","author":[{"last_name":"Cipolloni","first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1806.08751"}],"_id":"6488","publication_status":"epub_ahead","type":"journal_article","date_published":"2019-05-09T00:00:00Z","title":"Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","publication_identifier":{"eissn":["20103271"],"issn":["20103263"]},"department":[{"_id":"LaEr"}],"abstract":[{"lang":"eng","text":"We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix W˜ and its minor W. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of W˜ and W. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish."}],"day":"09","citation":{"chicago":"Cipolloni, Giorgio, and László Erdös. “Fluctuations for Differences of Linear Eigenvalue Statistics for Sample Covariance Matrices.” *Random Matrices: Theory and Application*, 2019. https://doi.org/10.1142/S2010326320500069.","apa":"Cipolloni, G., & Erdös, L. (2019). Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices. *Random Matrices: Theory and Application*. https://doi.org/10.1142/S2010326320500069","ieee":"G. Cipolloni and L. Erdös, “Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices,” *Random Matrices: Theory and Application*, 2019.","mla":"Cipolloni, Giorgio, and László Erdös. “Fluctuations for Differences of Linear Eigenvalue Statistics for Sample Covariance Matrices.” *Random Matrices: Theory and Application*, World Scientific Publishing, 2019, doi:10.1142/S2010326320500069.","ista":"Cipolloni G, Erdös L. 2019. Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices. Random Matrices: Theory and Application.","short":"G. Cipolloni, L. Erdös, Random Matrices: Theory and Application (2019).","ama":"Cipolloni G, Erdös L. Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices. *Random Matrices: Theory and Application*. 2019. doi:10.1142/S2010326320500069"},"status":"public","doi":"10.1142/S2010326320500069","publisher":"World Scientific Publishing","date_created":"2019-05-26T21:59:14Z"},{"oa_version":"Preprint","external_id":{"arxiv":["1612.05920"]},"publication":"Annals of Probability","year":"2019","volume":47,"day":"01","issue":"3","department":[{"_id":"LaEr"}],"date_created":"2019-06-02T21:59:13Z","page":"1270-1334","publication_status":"published","main_file_link":[{"url":"https://arxiv.org/abs/1612.05920","open_access":"1"}],"_id":"6511","publication_identifier":{"issn":["00911798"]},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","title":"Local single ring theorem on optimal scale","type":"journal_article","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","first_name":"Zhigang","last_name":"Bao","orcid":"0000-0003-3036-1475"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","last_name":"Schnelli"}],"language":[{"iso":"eng"}],"month":"05","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"date_updated":"2019-08-02T12:39:21Z","abstract":[{"text":"Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).","lang":"eng"}],"publisher":"Project Euclid","status":"public","doi":"10.1214/18-AOP1284","citation":{"chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Single Ring Theorem on Optimal Scale.” *Annals of Probability* 47, no. 3 (2019): 1270–1334. https://doi.org/10.1214/18-AOP1284.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2019). Local single ring theorem on optimal scale. *Annals of Probability*, *47*(3), 1270–1334. https://doi.org/10.1214/18-AOP1284","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local single ring theorem on optimal scale,” *Annals of Probability*, vol. 47, no. 3, pp. 1270–1334, 2019.","mla":"Bao, Zhigang, et al. “Local Single Ring Theorem on Optimal Scale.” *Annals of Probability*, vol. 47, no. 3, Project Euclid, 2019, pp. 1270–334, doi:10.1214/18-AOP1284.","ista":"Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale. Annals of Probability. 47(3), 1270–1334.","short":"Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334.","ama":"Bao Z, Erdös L, Schnelli K. Local single ring theorem on optimal scale. *Annals of Probability*. 2019;47(3):1270-1334. doi:10.1214/18-AOP1284"},"quality_controlled":"1","intvolume":" 47","date_published":"2019-05-01T00:00:00Z"},{"intvolume":" 55","quality_controlled":"1","date_published":"2019-05-01T00:00:00Z","citation":{"mla":"Alt, Johannes, et al. “Location of the Spectrum of Kronecker Random Matrices.” *Annales de l’institut Henri Poincare*, vol. 55, no. 2, 2019, pp. 661–96, doi:10.1214/18-AIHP894.","ieee":"J. Alt, L. Erdös, T. H. Krüger, and Y. Nemish, “Location of the spectrum of Kronecker random matrices,” *Annales de l’institut Henri Poincare*, vol. 55, no. 2, pp. 661–696, 2019.","ama":"Alt J, Erdös L, Krüger TH, Nemish Y. Location of the spectrum of Kronecker random matrices. *Annales de l’institut Henri Poincare*. 2019;55(2):661-696. doi:10.1214/18-AIHP894","ista":"Alt J, Erdös L, Krüger TH, Nemish Y. 2019. Location of the spectrum of Kronecker random matrices. Annales de l’institut Henri Poincare. 55(2), 661–696.","short":"J. Alt, L. Erdös, T.H. Krüger, Y. Nemish, Annales de l’institut Henri Poincare 55 (2019) 661–696.","apa":"Alt, J., Erdös, L., Krüger, T. H., & Nemish, Y. (2019). Location of the spectrum of Kronecker random matrices. *Annales de l’institut Henri Poincare*, *55*(2), 661–696. https://doi.org/10.1214/18-AIHP894","chicago":"Alt, Johannes, László Erdös, Torben H Krüger, and Yuriy Nemish. “Location of the Spectrum of Kronecker Random Matrices.” *Annales de l’institut Henri Poincare* 55, no. 2 (2019): 661–96. https://doi.org/10.1214/18-AIHP894."},"doi":"10.1214/18-AIHP894","status":"public","abstract":[{"text":"For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.","lang":"eng"}],"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"date_updated":"2019-08-13T13:03:29Z","language":[{"iso":"eng"}],"month":"05","author":[{"first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes","last_name":"Alt"},{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Krüger","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"},{"orcid":"0000-0002-7327-856X","last_name":"Nemish","first_name":"Yuriy","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","full_name":"Nemish, Yuriy"}],"title":"Location of the spectrum of Kronecker random matrices","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["02460203"]},"type":"journal_article","_id":"6240","main_file_link":[{"url":"https://arxiv.org/abs/1706.08343","open_access":"1"}],"publication_status":"published","related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"date_created":"2019-04-08T14:05:04Z","page":"661-696","day":"01","issue":"2","department":[{"_id":"LaEr"}],"year":"2019","publication":"Annales de l'institut Henri Poincare","volume":55,"oa_version":"Preprint","external_id":{"arxiv":["1706.08343"]}},{"file":[{"content_type":"application/pdf","date_updated":"2019-09-17T14:24:13Z","file_name":"2019_Forum_Erdoes.pdf","open_access":1,"file_size":1520344,"access_level":"open_access","date_created":"2019-09-17T14:24:13Z","success":1,"creator":"dernst","relation":"main_file","file_id":"6883"}],"type":"journal_article","publication_identifier":{"eissn":["20505094"]},"article_processing_charge":"No","title":"Random matrices with slow correlation decay","article_number":"e8","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","related_material":{"record":[{"relation":"dissertation_contains","id":"6179","status":"public"}]},"publication_status":"published","_id":"6182","date_created":"2019-03-28T09:05:23Z","article_type":"original","department":[{"_id":"LaEr"}],"day":"26","volume":7,"ddc":["510"],"publication":"Forum of Mathematics, Sigma","year":"2019","accept":"1","external_id":{"arxiv":["1705.10661"]},"oa_version":"Published Version","date_published":"2019-03-26T00:00:00Z","quality_controlled":"1","intvolume":" 7","doi":"10.1017/fms.2019.2","status":"public","citation":{"chicago":"Erdös, László, Torben H Krüger, and Dominik J Schröder. “Random Matrices with Slow Correlation Decay.” *Forum of Mathematics, Sigma* 7 (2019): e8. https://doi.org/10.1017/fms.2019.2.","apa":"Erdös, L., Krüger, T. H., & Schröder, D. J. (2019). Random matrices with slow correlation decay. *Forum of Mathematics, Sigma*, *7*, e8. https://doi.org/10.1017/fms.2019.2","ieee":"L. Erdös, T. H. Krüger, and D. J. Schröder, “Random matrices with slow correlation decay,” *Forum of Mathematics, Sigma*, vol. 7, p. e8, 2019.","mla":"Erdös, László, et al. “Random Matrices with Slow Correlation Decay.” *Forum of Mathematics, Sigma*, vol. 7, Cambridge University Press, 2019, p. e8, doi:10.1017/fms.2019.2.","ista":"Erdös L, Krüger TH, Schröder DJ. 2019. Random matrices with slow correlation decay. Forum of Mathematics, Sigma. 7, e8.","short":"L. Erdös, T.H. Krüger, D.J. Schröder, Forum of Mathematics, Sigma 7 (2019) e8.","ama":"Erdös L, Krüger TH, Schröder DJ. Random matrices with slow correlation decay. *Forum of Mathematics, Sigma*. 2019;7:e8. doi:10.1017/fms.2019.2"},"publisher":"Cambridge University Press","abstract":[{"text":"We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is\r\na systematic diagrammatic control of a multivariate cumulant expansion.","lang":"eng"}],"cc_license":"'https://creativecommons.org/licenses/by/4.0/'","date_updated":"2019-09-17T14:26:55Z","oa":1,"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"month":"03","language":[{"iso":"eng"}],"file_date_updated":"2019-09-17T14:24:13Z","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger"},{"last_name":"Schröder","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}]},{"oa_version":"Published Version","acknowledgement":"The work of the second author was also partially supported by the Hausdorff Center of Mathematics.","external_id":{"arxiv":["1708.01546"]},"publication":"SIAM Journal on Mathematical Analysis","year":"2018","volume":50,"date_created":"2018-12-11T11:45:03Z","page":"3271 - 3290","day":"01","issue":"3","department":[{"_id":"LaEr"}],"title":"Power law decay for systems of randomly coupled differential equations","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","publication_status":"published","_id":"181","main_file_link":[{"url":"https://arxiv.org/abs/1708.01546","open_access":"1"}],"author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger"},{"orcid":"0000-0003-3493-121X","last_name":"Renfrew","first_name":"David T","full_name":"Renfrew, David T","id":"4845BF6A-F248-11E8-B48F-1D18A9856A87"}],"project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"name":"Structured Non-Hermitian Random Matrices","_id":"258F40A4-B435-11E9-9278-68D0E5697425","grant_number":"M02080"}],"date_updated":"2019-08-02T12:37:27Z","oa":1,"language":[{"iso":"eng"}],"month":"01","publisher":"Society for Industrial and Applied Mathematics ","status":"public","doi":"10.1137/17M1143125","citation":{"ista":"Erdös L, Krüger TH, Renfrew DT. 2018. Power law decay for systems of randomly coupled differential equations. SIAM Journal on Mathematical Analysis. 50(3), 3271–3290.","short":"L. Erdös, T.H. Krüger, D.T. Renfrew, SIAM Journal on Mathematical Analysis 50 (2018) 3271–3290.","ama":"Erdös L, Krüger TH, Renfrew DT. Power law decay for systems of randomly coupled differential equations. *SIAM Journal on Mathematical Analysis*. 2018;50(3):3271-3290. doi:10.1137/17M1143125","ieee":"L. Erdös, T. H. Krüger, and D. T. Renfrew, “Power law decay for systems of randomly coupled differential equations,” *SIAM Journal on Mathematical Analysis*, vol. 50, no. 3, pp. 3271–3290, 2018.","mla":"Erdös, László, et al. “Power Law Decay for Systems of Randomly Coupled Differential Equations.” *SIAM Journal on Mathematical Analysis*, vol. 50, no. 3, Society for Industrial and Applied Mathematics , 2018, pp. 3271–90, doi:10.1137/17M1143125.","chicago":"Erdös, László, Torben H Krüger, and David T Renfrew. “Power Law Decay for Systems of Randomly Coupled Differential Equations.” *SIAM Journal on Mathematical Analysis* 50, no. 3 (2018): 3271–90. https://doi.org/10.1137/17M1143125.","apa":"Erdös, L., Krüger, T. H., & Renfrew, D. T. (2018). Power law decay for systems of randomly coupled differential equations. *SIAM Journal on Mathematical Analysis*, *50*(3), 3271–3290. https://doi.org/10.1137/17M1143125"},"abstract":[{"lang":"eng","text":"We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2."}],"quality_controlled":"1","intvolume":" 50","date_published":"2018-01-01T00:00:00Z","publist_id":"7740"},{"month":"03","language":[{"iso":"eng"}],"date_updated":"2019-09-17T12:52:57Z","oa":1,"author":[{"full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","last_name":"Alt"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"last_name":"Krüger","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"}],"date_published":"2018-03-03T00:00:00Z","quality_controlled":"1","intvolume":" 28","abstract":[{"text":"We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X. \r\n\r\n","lang":"eng"}],"status":"public","doi":"10.1214/17-AAP1302","citation":{"chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Inhomogeneous Circular Law.” *Annals Applied Probability * 28, no. 1 (2018): 148–203. https://doi.org/10.1214/17-AAP1302.","apa":"Alt, J., Erdös, L., & Krüger, T. H. (2018). Local inhomogeneous circular law. *Annals Applied Probability *, *28*(1), 148–203. https://doi.org/10.1214/17-AAP1302","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local inhomogeneous circular law,” *Annals Applied Probability *, vol. 28, no. 1, pp. 148–203, 2018.","mla":"Alt, Johannes, et al. “Local Inhomogeneous Circular Law.” *Annals Applied Probability *, vol. 28, no. 1, Institute of Mathematical Statistics, 2018, pp. 148–203, doi:10.1214/17-AAP1302.","short":"J. Alt, L. Erdös, T.H. Krüger, Annals Applied Probability 28 (2018) 148–203.","ista":"Alt J, Erdös L, Krüger TH. 2018. Local inhomogeneous circular law. Annals Applied Probability . 28(1), 148–203.","ama":"Alt J, Erdös L, Krüger TH. Local inhomogeneous circular law. *Annals Applied Probability *. 2018;28(1):148-203. doi:10.1214/17-AAP1302"},"publisher":"Institute of Mathematical Statistics","volume":28,"publication":"Annals Applied Probability ","year":"2018","external_id":{"arxiv":["1612.07776 "]},"oa_version":"Preprint","publication_status":"published","related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1612.07776 "}],"_id":"566","type":"journal_article","article_processing_charge":"No","title":"Local inhomogeneous circular law","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"1","department":[{"_id":"LaEr"}],"day":"03","page":"148-203","date_created":"2018-12-11T11:47:13Z"},{"oa":1,"date_updated":"2019-09-17T12:55:37Z","month":"09","language":[{"iso":"eng"}],"year":"2018","publication":"arXiv","external_id":{"arxiv":["1809.03971"]},"oa_version":"Preprint","author":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","first_name":"Torben H","last_name":"Krüger"},{"last_name":"Schröder","first_name":"Dominik J","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2018-09-11T00:00:00Z","type":"preprint","title":"Cusp universality for random matrices I: Local law and the complex hermitian case","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","_id":"6185","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1809.03971"}],"related_material":{"record":[{"status":"public","id":"6179","relation":"dissertation_contains"}]},"publication_status":"submitted","citation":{"ista":"Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. arXiv.","short":"L. Erdös, T.H. Krüger, D.J. Schröder, ArXiv (n.d.).","ama":"Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices I: Local law and the complex hermitian case. *arXiv*.","ieee":"L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices I: Local law and the complex hermitian case,” *arXiv*. .","mla":"Erdös, László, et al. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” *ArXiv*.","chicago":"Erdös, László, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case.” *ArXiv*, n.d.","apa":"Erdös, L., Krüger, T. H., & Schröder, D. J. (n.d.). Cusp universality for random matrices I: Local law and the complex hermitian case. *ArXiv*."},"page":"50","status":"public","date_created":"2019-03-28T10:21:15Z","department":[{"_id":"LaEr"}],"abstract":[{"lang":"eng","text":"For complex Wigner-type matrices, i.e. Hermitian random matrices with\r\nindependent, not necessarily identically distributed entries above the\r\ndiagonal, we show that at any cusp singularity of the limiting eigenvalue\r\ndistribution the local eigenvalue statistics are universal and form a Pearcey\r\nprocess. Since the density of states typically exhibits only square root or\r\ncubic root cusp singularities, our work complements previous results on the\r\nbulk and edge universality and it thus completes the resolution of the\r\nWigner-Dyson-Mehta universality conjecture for the last remaining universality\r\ntype in the complex Hermitian class. Our analysis holds not only for exact\r\ncusps, but approximate cusps as well, where an extended Pearcey process\r\nemerges. As a main technical ingredient we prove an optimal local law at the\r\ncusp for both symmetry classes. This result is also used in the companion paper\r\n[arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type\r\nmatrices is proven."}],"day":"11"},{"department":[{"_id":"LaEr"}],"abstract":[{"text":"We prove that the local eigenvalue statistics of real symmetric Wigner-type\r\nmatrices near the cusp points of the eigenvalue density are universal. Together\r\nwith the companion paper [arXiv:1809.03971], which proves the same result for\r\nthe complex Hermitian symmetry class, this completes the last remaining case of\r\nthe Wigner-Dyson-Mehta universality conjecture after bulk and edge\r\nuniversalities have been established in the last years. We extend the recent\r\nDyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp\r\nregime using the optimal local law from [arXiv:1809.03971] and the accurate\r\nlocal shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752].\r\nWe also present a PDE-based method to improve the estimate on eigenvalue\r\nrigidity via the maximum principle of the heat flow related to the Dyson\r\nBrownian motion.","lang":"eng"}],"day":"09","page":"60","citation":{"apa":"Cipolloni, G., Erdös, L., Krüger, T. H., & Schröder, D. J. (n.d.). Cusp universality for random matrices II: The real symmetric case. *ArXiv*.","chicago":"Cipolloni, Giorgio, László Erdös, Torben H Krüger, and Dominik J Schröder. “Cusp Universality for Random Matrices II: The Real Symmetric Case.” *ArXiv*, n.d.","ama":"Cipolloni G, Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices II: The real symmetric case. *arXiv*.","ista":"Cipolloni G, Erdös L, Krüger TH, Schröder DJ. Cusp universality for random matrices II: The real symmetric case. arXiv.","short":"G. Cipolloni, L. Erdös, T.H. Krüger, D.J. Schröder, ArXiv (n.d.).","mla":"Cipolloni, Giorgio, et al. “Cusp Universality for Random Matrices II: The Real Symmetric Case.” *ArXiv*.","ieee":"G. Cipolloni, L. Erdös, T. H. Krüger, and D. J. Schröder, “Cusp universality for random matrices II: The real symmetric case,” *arXiv*. ."},"status":"public","date_created":"2019-03-28T10:21:17Z","main_file_link":[{"url":"https://arxiv.org/abs/1811.04055","open_access":"1"}],"_id":"6186","publication_status":"submitted","related_material":{"record":[{"status":"public","id":"6179","relation":"dissertation_contains"}]},"date_published":"2018-11-09T00:00:00Z","type":"preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Cusp universality for random matrices II: The real symmetric case","article_processing_charge":"No","external_id":{"arxiv":["1811.04055"]},"oa_version":"Preprint","author":[{"full_name":"Cipolloni, Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","first_name":"Giorgio","last_name":"Cipolloni"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","first_name":"Torben H","last_name":"Krüger"},{"last_name":"Schröder","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}],"month":"11","language":[{"iso":"eng"}],"year":"2018","publication":"arXiv","date_updated":"2019-09-17T13:17:51Z","oa":1},{"publication":"Probability Theory and Related Fields","ddc":["510"],"year":"2018","oa_version":"Published Version","accept":"1","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).\r\n","publication_status":"epub_ahead","_id":"429","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Stability of the matrix Dyson equation and random matrices with correlations","type":"journal_article","file":[{"file_name":"2018_ProbTheory_Ajanki.pdf","open_access":1,"file_size":1201840,"content_type":"application/pdf","date_updated":"2018-12-17T16:12:08Z","creator":"dernst","relation":"main_file","file_id":"5720","access_level":"open_access","success":1,"date_created":"2018-12-17T16:12:08Z"}],"day":"17","department":[{"_id":"LaEr"}],"date_created":"2018-12-11T11:46:25Z","language":[{"iso":"eng"}],"month":"02","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"_id":"BFDF9788-01D1-11E9-AC17-EBD7A21D5664","name":"IST Austria Open Access Fund"}],"oa":1,"cc_license":"'https://creativecommons.org/licenses/by/4.0/'","date_updated":"2019-08-02T12:38:26Z","author":[{"last_name":"Ajanki","first_name":"Oskari H","full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"last_name":"Krüger","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","first_name":"Torben H"}],"file_date_updated":"2018-12-17T16:12:08Z","publist_id":"7394","quality_controlled":"1","date_published":"2018-02-17T00:00:00Z","abstract":[{"text":"We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.","lang":"eng"}],"publisher":"Springer","status":"public","doi":"10.1007/s00440-018-0835-z","citation":{"apa":"Ajanki, O. H., Erdös, L., & Krüger, T. H. (2018). Stability of the matrix Dyson equation and random matrices with correlations. *Probability Theory and Related Fields*. https://doi.org/10.1007/s00440-018-0835-z","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” *Probability Theory and Related Fields*, 2018. https://doi.org/10.1007/s00440-018-0835-z.","mla":"Ajanki, Oskari H., et al. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” *Probability Theory and Related Fields*, Springer, 2018, doi:10.1007/s00440-018-0835-z.","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Stability of the matrix Dyson equation and random matrices with correlations,” *Probability Theory and Related Fields*, 2018.","ama":"Ajanki OH, Erdös L, Krüger TH. Stability of the matrix Dyson equation and random matrices with correlations. *Probability Theory and Related Fields*. 2018. doi:10.1007/s00440-018-0835-z","ista":"Ajanki OH, Erdös L, Krüger TH. 2018. Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields.","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields (2018)."}},{"oa_version":"Preprint","external_id":{"arxiv":["1608.05163"]},"year":"2018","publication":"International Mathematics Research Notices","volume":2018,"day":"18","department":[{"_id":"LaEr"}],"issue":"10","date_created":"2018-12-11T11:49:41Z","page":"3255-3298","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.05163"}],"_id":"1012","publication_status":"published","related_material":{"record":[{"status":"public","id":"6179","relation":"dissertation_contains"}]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues","article_processing_charge":"No","publication_identifier":{"issn":["10737928"]},"type":"journal_article","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Schröder","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J","first_name":"Dominik J"}],"language":[{"iso":"eng"}],"month":"05","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"date_updated":"2019-09-17T11:33:35Z","abstract":[{"lang":"eng","text":"We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor H and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and H. In particular, our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues ofH and H, around their asymptotic shape, the Vershik'Kerov'Logan'Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense."}],"publisher":"Oxford University Press","citation":{"chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices* 2018, no. 10 (2018): 3255–98. https://doi.org/10.1093/imrn/rnw330.","apa":"Erdös, L., & Schröder, D. J. (2018). Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*, *2018*(10), 3255–3298. https://doi.org/10.1093/imrn/rnw330","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues,” *International Mathematics Research Notices*, vol. 2018, no. 10, pp. 3255–3298, 2018.","mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices*, vol. 2018, no. 10, Oxford University Press, 2018, pp. 3255–98, doi:10.1093/imrn/rnw330.","short":"L. Erdös, D.J. Schröder, International Mathematics Research Notices 2018 (2018) 3255–3298.","ista":"Erdös L, Schröder DJ. 2018. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. International Mathematics Research Notices. 2018(10), 3255–3298.","ama":"Erdös L, Schröder DJ. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*. 2018;2018(10):3255-3298. doi:10.1093/imrn/rnw330"},"doi":"10.1093/imrn/rnw330","status":"public","publist_id":"6383","intvolume":" 2018","quality_controlled":"1","date_published":"2018-05-18T00:00:00Z"},{"date_created":"2019-02-13T10:40:54Z","publisher":"World Scientific Publishing","status":"public","doi":"10.1142/s2010326319500096","citation":{"apa":"Erdös, L., & Mühlbacher, P. (2018). Bounds on the norm of Wigner-type random matrices. *Random Matrices: Theory and Applications*. https://doi.org/10.1142/s2010326319500096","chicago":"Erdös, László, and Peter Mühlbacher. “Bounds on the Norm of Wigner-Type Random Matrices.” *Random Matrices: Theory and Applications*, 2018. https://doi.org/10.1142/s2010326319500096.","mla":"Erdös, László, and Peter Mühlbacher. “Bounds on the Norm of Wigner-Type Random Matrices.” *Random Matrices: Theory and Applications*, 1950009, World Scientific Publishing, 2018, doi:10.1142/s2010326319500096.","ieee":"L. Erdös and P. Mühlbacher, “Bounds on the norm of Wigner-type random matrices,” *Random matrices: Theory and applications*, 2018.","ama":"Erdös L, Mühlbacher P. Bounds on the norm of Wigner-type random matrices. *Random matrices: Theory and applications*. 2018. doi:10.1142/s2010326319500096","short":"L. Erdös, P. Mühlbacher, Random Matrices: Theory and Applications (2018).","ista":"Erdös L, Mühlbacher P. 2018. Bounds on the norm of Wigner-type random matrices. Random matrices: Theory and applications."},"day":"26","abstract":[{"lang":"eng","text":"We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation."}],"department":[{"_id":"LaEr"}],"quality_controlled":"1","publication_identifier":{"issn":["2010-3263","2010-3271"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"1950009","title":"Bounds on the norm of Wigner-type random matrices","date_published":"2018-09-26T00:00:00Z","type":"journal_article","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1802.05175"}],"_id":"5971","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"full_name":"Mühlbacher, Peter","first_name":"Peter","last_name":"Mühlbacher"}],"oa_version":"Preprint","external_id":{"arxiv":["1802.05175"]},"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"date_updated":"2019-08-02T12:39:06Z","oa":1,"publication":"Random matrices: Theory and applications","year":"2018","language":[{"iso":"eng"}],"month":"09"},{"department":[{"_id":"LaEr"}],"day":"20","abstract":[{"lang":"eng","text":"We study the unique solution $m$ of the Dyson equation \\[ -m(z)^{-1} = z - a\r\n+ S[m(z)] \\] on a von Neumann algebra $\\mathcal{A}$ with the constraint\r\n$\\mathrm{Im}\\,m\\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is\r\na self-adjoint element of $\\mathcal{A}$ and $S$ is a positivity-preserving\r\nlinear operator on $\\mathcal{A}$. We show that $m$ is the Stieltjes transform\r\nof a compactly supported $\\mathcal{A}$-valued measure on $\\mathbb{R}$. Under\r\nsuitable assumptions, we establish that this measure has a uniformly\r\n$1/3$-H\\\"{o}lder continuous density with respect to the Lebesgue measure, which\r\nis supported on finitely many intervals, called bands. In fact, the density is\r\nanalytic inside the bands with a square-root growth at the edges and internal\r\ncubic root cusps whenever the gap between two bands vanishes. The shape of\r\nthese singularities is universal and no other singularity may occur. We give a\r\nprecise asymptotic description of $m$ near the singular points. These\r\nasymptotics generalize the analysis at the regular edges given in the companion\r\npaper on the Tracy-Widom universality for the edge eigenvalue statistics for\r\ncorrelated random matrices [arXiv:1804.07744] and they play a key role in the\r\nproof of the Pearcey universality at the cusp for Wigner-type matrices\r\n[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band\r\nmass formula from [arXiv:1804.07744] to the von Neumann algebra setting by\r\nshowing that the spectral mass of the bands is topologically rigid under\r\ndeformations and we conclude that these masses are quantized in some important\r\ncases."}],"status":"public","page":"72","citation":{"short":"J. Alt, L. Erdös, T.H. Krüger, ArXiv:1804.07752 (n.d.).","ista":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv:1804.07752.","ama":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. *arXiv:180407752*.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” *arXiv:1804.07752*. .","mla":"Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” *ArXiv:1804.07752*.","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” *ArXiv:1804.07752*, n.d.","apa":"Alt, J., Erdös, L., & Krüger, T. H. (n.d.). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. *ArXiv:1804.07752*."},"date_created":"2019-03-28T09:20:06Z","publication_status":"submitted","related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"}]},"main_file_link":[{"url":"https://arxiv.org/abs/1804.07752","open_access":"1"}],"_id":"6183","type":"preprint","date_published":"2018-04-20T00:00:00Z","title":"The Dyson equation with linear self-energy: Spectral bands, edges and cusps","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["1804.07752"]},"author":[{"full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","last_name":"Alt"},{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"last_name":"Krüger","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H"}],"oa_version":"Preprint","month":"04","publication":"arXiv:1804.07752","year":"2018","language":[{"iso":"eng"}],"date_updated":"2019-08-13T13:03:29Z"},{"date_updated":"2019-08-13T13:46:45Z","publication":"arXiv:1804.07744","year":"2018","language":[{"iso":"eng"}],"month":"04","author":[{"last_name":"Alt","full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes"},{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"last_name":"Krüger","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H"},{"last_name":"Schröder","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}],"oa_version":"Preprint","external_id":{"arxiv":["1804.07744"]},"title":"Correlated random matrices: Band rigidity and edge universality","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_published":"2018-04-20T00:00:00Z","type":"preprint","publication_status":"submitted","related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"},{"relation":"dissertation_contains","status":"public","id":"6179"}]},"main_file_link":[{"url":"https://arxiv.org/abs/1804.07744","open_access":"1"}],"_id":"6184","date_created":"2019-03-28T09:20:08Z","status":"public","page":"26","citation":{"ieee":"J. Alt, L. Erdös, T. H. Krüger, and D. J. Schröder, “Correlated random matrices: Band rigidity and edge universality,” *arXiv:1804.07744*. .","mla":"Alt, Johannes, et al. “Correlated Random Matrices: Band Rigidity and Edge Universality.” *ArXiv:1804.07744*.","ista":"Alt J, Erdös L, Krüger TH, Schröder DJ. Correlated random matrices: Band rigidity and edge universality. arXiv:1804.07744.","short":"J. Alt, L. Erdös, T.H. Krüger, D.J. Schröder, ArXiv:1804.07744 (n.d.).","ama":"Alt J, Erdös L, Krüger TH, Schröder DJ. Correlated random matrices: Band rigidity and edge universality. *arXiv:180407744*.","chicago":"Alt, Johannes, László Erdös, Torben H Krüger, and Dominik J Schröder. “Correlated Random Matrices: Band Rigidity and Edge Universality.” *ArXiv:1804.07744*, n.d.","apa":"Alt, J., Erdös, L., Krüger, T. H., & Schröder, D. J. (n.d.). Correlated random matrices: Band rigidity and edge universality. *ArXiv:1804.07744*."},"day":"20","abstract":[{"lang":"eng","text":"We prove edge universality for a general class of correlated real symmetric\r\nor complex Hermitian Wigner matrices with arbitrary expectation. Our theorem\r\nalso applies to internal edges of the self-consistent density of states. In\r\nparticular, we establish a strong form of band rigidity which excludes\r\nmismatches between location and label of eigenvalues close to internal edges in\r\nthese general models."}],"department":[{"_id":"LaEr"}]},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Universality for general Wigner-type matrices","publication_identifier":{"issn":["01788051"]},"file":[{"creator":"system","relation":"main_file","file_id":"4686","access_level":"open_access","date_created":"2018-12-12T10:08:25Z","file_name":"IST-2017-657-v1+2_s00440-016-0740-2.pdf","file_size":988843,"open_access":1,"content_type":"application/pdf","date_updated":"2018-12-12T10:08:25Z"}],"type":"journal_article","_id":"1337","publication_status":"published","date_created":"2018-12-11T11:51:27Z","page":"667 - 727","day":"01","issue":"3-4","department":[{"_id":"LaEr"}],"year":"2017","publication":"Probability Theory and Related Fields","ddc":["510","530"],"volume":169,"oa_version":"Published Version","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). ","accept":"1","intvolume":" 169","quality_controlled":"1","date_published":"2017-12-01T00:00:00Z","publist_id":"5930","publisher":"Springer","citation":{"mla":"Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.” *Probability Theory and Related Fields*, vol. 169, no. 3–4, Springer, 2017, pp. 667–727, doi:10.1007/s00440-016-0740-2.","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type matrices,” *Probability Theory and Related Fields*, vol. 169, no. 3–4, pp. 667–727, 2017.","ama":"Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices. *Probability Theory and Related Fields*. 2017;169(3-4):667-727. doi:10.1007/s00440-016-0740-2","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727.","ista":"Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727.","apa":"Ajanki, O. H., Erdös, L., & Krüger, T. H. (2017). Universality for general Wigner-type matrices. *Probability Theory and Related Fields*, *169*(3–4), 667–727. https://doi.org/10.1007/s00440-016-0740-2","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for General Wigner-Type Matrices.” *Probability Theory and Related Fields* 169, no. 3–4 (2017): 667–727. https://doi.org/10.1007/s00440-016-0740-2."},"status":"public","doi":"10.1007/s00440-016-0740-2","abstract":[{"text":"We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.","lang":"eng"}],"pubrep_id":"657","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"},{"_id":"BFDF9788-01D1-11E9-AC17-EBD7A21D5664","name":"IST Austria Open Access Fund"}],"date_updated":"2019-08-02T12:37:00Z","cc_license":"'https://creativecommons.org/licenses/by/4.0/'","oa":1,"language":[{"iso":"eng"}],"month":"12","file_date_updated":"2018-12-12T10:08:25Z","author":[{"first_name":"Oskari H","full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","last_name":"Ajanki"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","last_name":"Krüger"}]},{"publist_id":"6386","date_published":"2017-03-08T00:00:00Z","quality_controlled":"1","intvolume":" 22","pubrep_id":"807","abstract":[{"lang":"eng","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∗. "}],"doi":"10.1214/17-EJP42","status":"public","citation":{"ista":"Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic Journal of Probability. 22, 25.","short":"J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017) 25.","ama":"Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. *Electronic Journal of Probability*. 2017;22:25. doi:10.1214/17-EJP42","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,” *Electronic Journal of Probability*, vol. 22, p. 25, 2017.","mla":"Alt, Johannes, et al. “Local Law for Random Gram Matrices.” *Electronic Journal of Probability*, vol. 22, Institute of Mathematical Statistics, 2017, p. 25, doi:10.1214/17-EJP42.","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” *Electronic Journal of Probability* 22 (2017): 25. https://doi.org/10.1214/17-EJP42.","apa":"Alt, J., Erdös, L., & Krüger, T. H. (2017). Local law for random Gram matrices. *Electronic Journal of Probability*, *22*, 25. https://doi.org/10.1214/17-EJP42"},"publisher":"Institute of Mathematical Statistics","month":"03","language":[{"iso":"eng"}],"date_updated":"2019-08-13T13:03:29Z","cc_license":"'https://creativecommons.org/licenses/by/4.0/'","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"author":[{"last_name":"Alt","first_name":"Johannes","full_name":"Alt, Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"last_name":"Krüger","first_name":"Torben H","full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87"}],"file_date_updated":"2018-12-12T10:13:39Z","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"149"}]},"publication_status":"published","_id":"1010","type":"journal_article","file":[{"date_created":"2018-12-12T10:13:39Z","access_level":"open_access","creator":"system","file_id":"5024","relation":"main_file","content_type":"application/pdf","date_updated":"2018-12-12T10:13:39Z","file_size":639384,"open_access":1,"file_name":"IST-2017-807-v1+1_euclid.ejp.1488942016.pdf"}],"publication_identifier":{"issn":["10836489"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_number":"25","title":"Local law for random Gram matrices","department":[{"_id":"LaEr"}],"day":"08","date_created":"2018-12-11T11:49:40Z","volume":22,"ddc":["510","539"],"publication":"Electronic Journal of Probability","year":"2017","accept":"1","external_id":{"arxiv":["1606.07353"]},"oa_version":"Published Version"},{"oa_version":"Published Version","accept":"1","year":"2017","publication":"Probability Theory and Related Fields","ddc":["530"],"volume":167,"date_created":"2018-12-11T11:52:32Z","page":"673 - 776","day":"01","department":[{"_id":"LaEr"}],"issue":"3-4","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Delocalization for a class of random block band matrices","publication_identifier":{"issn":["01788051"]},"type":"journal_article","file":[{"content_type":"application/pdf","date_updated":"2018-12-12T10:08:05Z","file_name":"IST-2016-489-v1+1_s00440-015-0692-y.pdf","open_access":1,"file_size":1615755,"access_level":"open_access","date_created":"2018-12-12T10:08:05Z","creator":"system","relation":"main_file","file_id":"4665"}],"_id":"1528","publication_status":"published","file_date_updated":"2018-12-12T10:08:05Z","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","first_name":"Zhigang","last_name":"Bao","orcid":"0000-0003-3036-1475"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"oa":1,"date_updated":"2019-08-02T12:37:13Z","cc_license":"'https://creativecommons.org/licenses/by/4.0/'","language":[{"iso":"eng"}],"month":"04","publisher":"Springer","citation":{"short":"Z. Bao, L. Erdös, Probability Theory and Related Fields 167 (2017) 673–776.","ista":"Bao Z, Erdös L. 2017. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 167(3–4), 673–776.","ama":"Bao Z, Erdös L. Delocalization for a class of random block band matrices. *Probability Theory and Related Fields*. 2017;167(3-4):673-776. doi:10.1007/s00440-015-0692-y","ieee":"Z. Bao and L. Erdös, “Delocalization for a class of random block band matrices,” *Probability Theory and Related Fields*, vol. 167, no. 3–4, pp. 673–776, 2017.","mla":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” *Probability Theory and Related Fields*, vol. 167, no. 3–4, Springer, 2017, pp. 673–776, doi:10.1007/s00440-015-0692-y.","chicago":"Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” *Probability Theory and Related Fields* 167, no. 3–4 (2017): 673–776. https://doi.org/10.1007/s00440-015-0692-y.","apa":"Bao, Z., & Erdös, L. (2017). Delocalization for a class of random block band matrices. *Probability Theory and Related Fields*, *167*(3–4), 673–776. https://doi.org/10.1007/s00440-015-0692-y"},"doi":"10.1007/s00440-015-0692-y","status":"public","abstract":[{"text":"We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.","lang":"eng"}],"pubrep_id":"489","intvolume":" 167","quality_controlled":"1","date_published":"2017-04-01T00:00:00Z","publist_id":"5644"},{"publisher":"American Mathematical Society","date_created":"2018-12-11T11:47:13Z","citation":{"apa":"Erdös, L., & Yau, H. (2017). *A dynamical approach to random matrix theory* (Vol. 28). American Mathematical Society.","chicago":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28. Courant Lecture Notes. American Mathematical Society, 2017.","mla":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28, American Mathematical Society, 2017.","ieee":"L. Erdös and H. Yau, *A dynamical approach to random matrix theory*, vol. 28. American Mathematical Society, 2017.","ama":"Erdös L, Yau H. *A Dynamical Approach to Random Matrix Theory*. Vol 28. American Mathematical Society; 2017.","short":"L. Erdös, H. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 2017.","ista":"Erdös L, Yau H. 2017. A dynamical approach to random matrix theory, American Mathematical Society, 226p."},"page":"226","status":"public","abstract":[{"text":"This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.\r\n","lang":"eng"}],"day":"01","department":[{"_id":"LaEr"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"A dynamical approach to random matrix theory","intvolume":" 28","publication_identifier":{"isbn":["9781470436483"],"eisbn":["978-1-4704-4194-4"]},"quality_controlled":"1","date_published":"2017-01-01T00:00:00Z","type":"book","_id":"567","publication_status":"published","publist_id":"7247","alternative_title":["Courant Lecture Notes"],"oa_version":"None","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Horng","full_name":"Yau, Horng","last_name":"Yau"}],"project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"series_title":"Courant Lecture Notes","date_updated":"2019-08-02T12:39:03Z","year":"2017","language":[{"iso":"eng"}],"month":"01","volume":28},{"date_updated":"2019-08-02T12:39:28Z","oa":1,"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"month":"09","language":[{"iso":"eng"}],"author":[{"last_name":"Ajanki","first_name":"Oskari H","full_name":"Ajanki, Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger"},{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"}],"date_published":"2017-09-01T00:00:00Z","intvolume":" 70","quality_controlled":"1","publist_id":"6959","citation":{"ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. *Communications on Pure and Applied Mathematics*. 2017;70(9):1672-1705. doi:10.1002/cpa.21639","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","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.” *Communications on Pure and Applied Mathematics*, vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi: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, pp. 1672–1705, 2017.","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*, *70*(9), 1672–1705. https://doi.org/10.1002/cpa.21639","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” *Communications on Pure and Applied Mathematics* 70, no. 9 (2017): 1672–1705. https://doi.org/10.1002/cpa.21639."},"status":"public","doi":"10.1002/cpa.21639","publisher":"Wiley-Blackwell","abstract":[{"text":"Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.","lang":"eng"}],"volume":70,"year":"2017","publication":"Communications on Pure and Applied Mathematics","oa_version":"Submitted Version","type":"journal_article","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["00103640"]},"_id":"721","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1512.03703"}],"publication_status":"published","page":"1672 - 1705","date_created":"2018-12-11T11:48:08Z","issue":"9","department":[{"_id":"LaEr"}],"day":"01"},{"year":"2017","publication":"Advances in Mathematics","volume":319,"oa_version":"Submitted Version","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","title":"Convergence rate for spectral distribution of addition of random matrices","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","_id":"733","main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"publication_status":"published","date_created":"2018-12-11T11:48:13Z","page":"251 - 291","day":"15","department":[{"_id":"LaEr"}],"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"date_updated":"2019-08-02T12:39:29Z","oa":1,"language":[{"iso":"eng"}],"month":"10","author":[{"orcid":"0000-0003-3036-1475","last_name":"Bao","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","first_name":"Kevin"}],"intvolume":" 319","quality_controlled":"1","date_published":"2017-10-15T00:00:00Z","publist_id":"6935","publisher":"Academic Press","citation":{"apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. *Advances in Mathematics*, *319*, 251–291. https://doi.org/10.1016/j.aim.2017.08.028","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics* 319 (2017): 251–91. 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.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” *Advances in Mathematics*, vol. 319, pp. 251–291, 2017.","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","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."},"status":"public","doi":"10.1016/j.aim.2017.08.028","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."}]},{"oa_version":"Published Version","accept":"1","year":"2017","ddc":["530"],"publication":"Communications in Mathematical Physics","volume":349,"day":"01","department":[{"_id":"LaEr"}],"issue":"3","date_created":"2018-12-11T11:50:43Z","page":"947 - 990","_id":"1207","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Local law of addition of random matrices on optimal scale","publication_identifier":{"issn":["00103616"]},"file":[{"file_name":"IST-2016-722-v1+1_s00220-016-2805-6.pdf","open_access":1,"file_size":1033743,"date_updated":"2018-12-12T10:14:47Z","content_type":"application/pdf","relation":"main_file","file_id":"5102","creator":"system","access_level":"open_access","date_created":"2018-12-12T10:14:47Z"}],"type":"journal_article","author":[{"orcid":"0000-0003-3036-1475","last_name":"Bao","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","first_name":"László"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","first_name":"Kevin","last_name":"Schnelli"}],"file_date_updated":"2018-12-12T10:14:47Z","language":[{"iso":"eng"}],"month":"02","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"cc_license":"'https://creativecommons.org/licenses/by/4.0/'","date_updated":"2019-08-02T12:36:55Z","oa":1,"abstract":[{"text":"The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.","lang":"eng"}],"pubrep_id":"722","publisher":"Springer","citation":{"apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Local law of addition of random matrices on optimal scale. *Communications in Mathematical Physics*, *349*(3), 947–990. https://doi.org/10.1007/s00220-016-2805-6","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Law of Addition of Random Matrices on Optimal Scale.” *Communications in Mathematical Physics* 349, no. 3 (2017): 947–90. https://doi.org/10.1007/s00220-016-2805-6.","ama":"Bao Z, Erdös L, Schnelli K. Local law of addition of random matrices on optimal scale. *Communications in Mathematical Physics*. 2017;349(3):947-990. doi:10.1007/s00220-016-2805-6","short":"Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 947–990.","ista":"Bao Z, Erdös L, Schnelli K. 2017. Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. 349(3), 947–990.","mla":"Bao, Zhigang, et al. “Local Law of Addition of Random Matrices on Optimal Scale.” *Communications in Mathematical Physics*, vol. 349, no. 3, Springer, 2017, pp. 947–90, doi:10.1007/s00220-016-2805-6.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local law of addition of random matrices on optimal scale,” *Communications in Mathematical Physics*, vol. 349, no. 3, pp. 947–990, 2017."},"doi":"10.1007/s00220-016-2805-6","status":"public","publist_id":"6141","intvolume":" 349","quality_controlled":"1","date_published":"2017-02-01T00:00:00Z"}]