[{"day":"01","article_processing_charge":"No","scopus_import":"1","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"date_published":"2023-02-01T00:00:00Z","publication":"The Annals of Applied Probability","citation":{"chicago":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1826.","mla":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp. 677–725, doi:10.1214/22-aap1826.","short":"K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725.","ista":"Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1), 677–725.","apa":"Schnelli, K., & Xu, Y. (2023). Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1826","ieee":"K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices,” The Annals of Applied Probability, vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023.","ama":"Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 2023;33(1):677-725. doi:10.1214/22-aap1826"},"article_type":"original","page":"677-725","abstract":[{"text":"We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.","lang":"eng"}],"issue":"1","type":"journal_article","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"14775","status":"public","title":"Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices","intvolume":" 33","month":"02","publication_identifier":{"issn":["1050-5164"]},"doi":"10.1214/22-aap1826","language":[{"iso":"eng"}],"external_id":{"arxiv":["2108.02728"],"isi":["000946432400021"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2108.02728"}],"isi":1,"quality_controlled":"1","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020"}],"ec_funded":1,"author":[{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli"},{"first_name":"Yuanyuan","last_name":"Xu","id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","orcid":"0000-0003-1559-1205","full_name":"Xu, Yuanyuan"}],"date_updated":"2024-01-10T13:31:46Z","date_created":"2024-01-10T09:23:31Z","volume":33,"year":"2023","acknowledgement":"K. Schnelli was supported by the Swedish Research Council Grants VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331.","publication_status":"published","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}]},{"doi":"10.1007/s00220-022-04377-y","language":[{"iso":"eng"}],"external_id":{"isi":["000782737200001"],"arxiv":["2102.04330"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"isi":1,"quality_controlled":"1","publication_identifier":{"issn":["0010-3616"],"eissn":["1432-0916"]},"month":"07","author":[{"full_name":"Schnelli, Kevin","last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"},{"id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","first_name":"Yuanyuan","last_name":"Xu","full_name":"Xu, Yuanyuan"}],"volume":393,"date_created":"2022-04-24T22:01:44Z","date_updated":"2023-08-03T06:34:24Z","acknowledgement":"Kevin Schnelli is supported in parts by the Swedish Research Council Grant VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Yuanyuan Xu is supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331.","year":"2022","department":[{"_id":"LaEr"}],"publisher":"Springer Nature","publication_status":"published","ec_funded":1,"file_date_updated":"2022-08-05T06:01:13Z","date_published":"2022-07-01T00:00:00Z","citation":{"chicago":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical Physics. Springer Nature, 2022. https://doi.org/10.1007/s00220-022-04377-y.","mla":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical Physics, vol. 393, Springer Nature, 2022, pp. 839–907, doi:10.1007/s00220-022-04377-y.","short":"K. Schnelli, Y. Xu, Communications in Mathematical Physics 393 (2022) 839–907.","ista":"Schnelli K, Xu Y. 2022. Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. 393, 839–907.","ieee":"K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices,” Communications in Mathematical Physics, vol. 393. Springer Nature, pp. 839–907, 2022.","apa":"Schnelli, K., & Xu, Y. (2022). Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-022-04377-y","ama":"Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. 2022;393:839-907. doi:10.1007/s00220-022-04377-y"},"publication":"Communications in Mathematical Physics","page":"839-907","article_type":"original","article_processing_charge":"No","has_accepted_license":"1","day":"01","scopus_import":"1","oa_version":"Published Version","file":[{"access_level":"open_access","file_name":"2022_CommunMathPhys_Schnelli.pdf","creator":"dernst","file_size":1141462,"content_type":"application/pdf","file_id":"11726","relation":"main_file","success":1,"checksum":"bee0278c5efa9a33d9a2dc8d354a6c51","date_created":"2022-08-05T06:01:13Z","date_updated":"2022-08-05T06:01:13Z"}],"_id":"11332","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","intvolume":" 393","status":"public","ddc":["510"],"title":"Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices","abstract":[{"text":"We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.","lang":"eng"}],"type":"journal_article"},{"acknowledgement":"The first author is supported in part by Hong Kong RGC Grant GRF 16301519 and NSFC 11871425. The second author is supported in part by ERC Advanced Grant RANMAT 338804. The third author is supported in part by Swedish Research Council Grant VR-2017-05195 and the Knut and Alice Wallenberg Foundation","year":"2021","department":[{"_id":"LaEr"}],"publisher":"Cambridge University Press","publication_status":"published","author":[{"last_name":"Bao","first_name":"Zhigang","orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"full_name":"Erdös, László","first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603"},{"full_name":"Schnelli, Kevin","first_name":"Kevin","last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231"}],"volume":9,"date_created":"2021-06-13T22:01:33Z","date_updated":"2023-08-08T14:03:40Z","article_number":"e44","ec_funded":1,"file_date_updated":"2021-06-15T14:40:45Z","oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000654960800001"],"arxiv":["2008.07061"]},"project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"isi":1,"quality_controlled":"1","doi":"10.1017/fms.2021.38","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["20505094"]},"month":"05","_id":"9550","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","intvolume":" 9","status":"public","ddc":["510"],"title":"Equipartition principle for Wigner matrices","file":[{"creator":"cziletti","content_type":"application/pdf","file_size":483458,"access_level":"open_access","file_name":"2021_ForumMath_Bao.pdf","success":1,"checksum":"47c986578de132200d41e6d391905519","date_updated":"2021-06-15T14:40:45Z","date_created":"2021-06-15T14:40:45Z","file_id":"9555","relation":"main_file"}],"oa_version":"Published Version","type":"journal_article","abstract":[{"text":"We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. ","lang":"eng"}],"citation":{"ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Equipartition principle for Wigner matrices,” Forum of Mathematics, Sigma, vol. 9. Cambridge University Press, 2021.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2021). Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2021.38","ista":"Bao Z, Erdös L, Schnelli K. 2021. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 9, e44.","ama":"Bao Z, Erdös L, Schnelli K. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 2021;9. doi:10.1017/fms.2021.38","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Equipartition Principle for Wigner Matrices.” Forum of Mathematics, Sigma. Cambridge University Press, 2021. https://doi.org/10.1017/fms.2021.38.","short":"Z. Bao, L. Erdös, K. Schnelli, Forum of Mathematics, Sigma 9 (2021).","mla":"Bao, Zhigang, et al. “Equipartition Principle for Wigner Matrices.” Forum of Mathematics, Sigma, vol. 9, e44, Cambridge University Press, 2021, doi:10.1017/fms.2021.38."},"publication":"Forum of Mathematics, Sigma","article_type":"original","date_published":"2021-05-27T00:00:00Z","scopus_import":"1","article_processing_charge":"No","has_accepted_license":"1","day":"27"},{"publication_identifier":{"eissn":["15658538"],"issn":["00217670"]},"month":"11","oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1804.11199","open_access":"1"}],"external_id":{"isi":["000611879400008"],"arxiv":["1804.11199"]},"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"isi":1,"quality_controlled":"1","doi":"10.1007/s11854-020-0135-2","language":[{"iso":"eng"}],"ec_funded":1,"year":"2020","acknowledgement":"Supported in part by Hong Kong RGC Grant ECS 26301517.\r\nSupported in part by ERC Advanced Grant RANMAT No. 338804.\r\nSupported in part by the Knut and Alice Wallenberg Foundation and the Swedish Research Council Grant VR-2017-05195.","department":[{"_id":"LaEr"}],"publisher":"Springer Nature","publication_status":"published","author":[{"orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao","first_name":"Zhigang","full_name":"Bao, Zhigang"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","first_name":"László","last_name":"Erdös","full_name":"Erdös, László"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"volume":142,"date_updated":"2023-08-24T11:16:03Z","date_created":"2021-02-07T23:01:15Z","scopus_import":"1","article_processing_charge":"No","day":"01","citation":{"ieee":"Z. Bao, L. Erdös, and K. Schnelli, “On the support of the free additive convolution,” Journal d’Analyse Mathematique, vol. 142. Springer Nature, pp. 323–348, 2020.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2020). On the support of the free additive convolution. Journal d’Analyse Mathematique. Springer Nature. https://doi.org/10.1007/s11854-020-0135-2","ista":"Bao Z, Erdös L, Schnelli K. 2020. On the support of the free additive convolution. Journal d’Analyse Mathematique. 142, 323–348.","ama":"Bao Z, Erdös L, Schnelli K. On the support of the free additive convolution. Journal d’Analyse Mathematique. 2020;142:323-348. doi:10.1007/s11854-020-0135-2","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “On the Support of the Free Additive Convolution.” Journal d’Analyse Mathematique. Springer Nature, 2020. https://doi.org/10.1007/s11854-020-0135-2.","short":"Z. Bao, L. Erdös, K. Schnelli, Journal d’Analyse Mathematique 142 (2020) 323–348.","mla":"Bao, Zhigang, et al. “On the Support of the Free Additive Convolution.” Journal d’Analyse Mathematique, vol. 142, Springer Nature, 2020, pp. 323–48, doi:10.1007/s11854-020-0135-2."},"publication":"Journal d'Analyse Mathematique","page":"323-348","article_type":"original","date_published":"2020-11-01T00:00:00Z","type":"journal_article","abstract":[{"text":"We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].","lang":"eng"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"9104","intvolume":" 142","title":"On the support of the free additive convolution","status":"public","oa_version":"Preprint"},{"scopus_import":"1","day":"01","article_processing_charge":"No","page":"1270-1334","publication":"Annals of Probability","citation":{"chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability. Institute of Mathematical Statistics, 2019. https://doi.org/10.1214/18-AOP1284.","mla":"Bao, Zhigang, et al. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability, vol. 47, no. 3, Institute of Mathematical Statistics, 2019, pp. 1270–334, doi:10.1214/18-AOP1284.","short":"Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334.","ista":"Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale. Annals of Probability. 47(3), 1270–1334.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2019). Local single ring theorem on optimal scale. Annals of Probability. Institute of Mathematical Statistics. 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. Institute of Mathematical Statistics, pp. 1270–1334, 2019.","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"},"date_published":"2019-05-01T00:00:00Z","type":"journal_article","abstract":[{"lang":"eng","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)."}],"issue":"3","status":"public","title":"Local single ring theorem on optimal scale","intvolume":" 47","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"6511","oa_version":"Preprint","month":"05","publication_identifier":{"issn":["00911798"]},"isi":1,"quality_controlled":"1","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"external_id":{"isi":["000466616100003"],"arxiv":["1612.05920"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1612.05920","open_access":"1"}],"language":[{"iso":"eng"}],"doi":"10.1214/18-AOP1284","ec_funded":1,"publication_status":"published","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}],"year":"2019","date_created":"2019-06-02T21:59:13Z","date_updated":"2023-08-28T09:32:29Z","volume":47,"author":[{"full_name":"Bao, Zhigang","first_name":"Zhigang","last_name":"Bao","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3036-1475"},{"last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"first_name":"Kevin","last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin"}]},{"project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1605.08767","open_access":"1"}],"external_id":{"arxiv":["1605.08767"]},"oa":1,"language":[{"iso":"eng"}],"doi":"10.1007/s00440-017-0787-8","month":"06","department":[{"_id":"LaEr"}],"publisher":"Springer","publication_status":"published","year":"2018","volume":171,"date_created":"2018-12-11T11:47:56Z","date_updated":"2021-01-12T08:09:33Z","author":[{"last_name":"Lee","first_name":"Jii","full_name":"Lee, Jii"},{"last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"article_number":"543-616","publist_id":"7017","ec_funded":1,"citation":{"ieee":"J. Lee and K. Schnelli, “Local law and Tracy–Widom limit for sparse random matrices,” Probability Theory and Related Fields, vol. 171, no. 1–2. Springer, 2018.","apa":"Lee, J., & Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-017-0787-8","ista":"Lee J, Schnelli K. 2018. Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. 171(1–2), 543–616.","ama":"Lee J, Schnelli K. Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. 2018;171(1-2). doi:10.1007/s00440-017-0787-8","chicago":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” Probability Theory and Related Fields. Springer, 2018. https://doi.org/10.1007/s00440-017-0787-8.","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).","mla":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” Probability Theory and Related Fields, vol. 171, no. 1–2, 543–616, Springer, 2018, doi:10.1007/s00440-017-0787-8."},"publication":"Probability Theory and Related Fields","date_published":"2018-06-14T00:00:00Z","scopus_import":1,"day":"14","intvolume":" 171","status":"public","title":"Local law and Tracy–Widom limit for sparse random matrices","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"690","oa_version":"Preprint","type":"journal_article","issue":"1-2","abstract":[{"text":"We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.","lang":"eng"}]},{"oa_version":"Submitted Version","title":"Universality for random matrix flows with time dependent density","status":"public","intvolume":" 53","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","_id":"615","abstract":[{"text":"We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law.","lang":"eng"}],"issue":"4","type":"journal_article","date_published":"2017-11-01T00:00:00Z","page":"1606 - 1656","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","citation":{"mla":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:10.1214/16-AIHP765.","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656.","chicago":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-AIHP765.","ama":"Erdös L, Schnelli K. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 2017;53(4):1606-1656. doi:10.1214/16-AIHP765","ista":"Erdös L, Schnelli K. 2017. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 53(4), 1606–1656.","apa":"Erdös, L., & Schnelli, K. (2017). Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AIHP765","ieee":"L. Erdös and K. Schnelli, “Universality for random matrix flows with time dependent density,” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4. Institute of Mathematical Statistics, pp. 1606–1656, 2017."},"day":"01","scopus_import":1,"date_created":"2018-12-11T11:47:30Z","date_updated":"2021-01-12T08:06:22Z","volume":53,"author":[{"full_name":"Erdös, László","last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Schnelli, Kevin","last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"publication_status":"published","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}],"year":"2017","publist_id":"7189","ec_funded":1,"language":[{"iso":"eng"}],"doi":"10.1214/16-AIHP765","quality_controlled":"1","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1504.00650"}],"month":"11","publication_identifier":{"issn":["02460203"]}},{"ec_funded":1,"publist_id":"6141","file_date_updated":"2020-07-14T12:44:39Z","author":[{"last_name":"Bao","first_name":"Zhigang","orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"volume":349,"date_updated":"2023-09-20T11:16:57Z","date_created":"2018-12-11T11:50:43Z","year":"2017","department":[{"_id":"LaEr"}],"publisher":"Springer","publication_status":"published","publication_identifier":{"issn":["00103616"]},"month":"02","doi":"10.1007/s00220-016-2805-6","language":[{"iso":"eng"}],"oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"isi":["000393696700005"]},"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"isi":1,"quality_controlled":"1","issue":"3","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"}],"type":"journal_article","pubrep_id":"722","oa_version":"Published Version","file":[{"checksum":"ddff79154c3daf27237de5383b1264a9","date_created":"2018-12-12T10:14:47Z","date_updated":"2020-07-14T12:44:39Z","relation":"main_file","file_id":"5102","file_size":1033743,"content_type":"application/pdf","creator":"system","access_level":"open_access","file_name":"IST-2016-722-v1+1_s00220-016-2805-6.pdf"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1207","intvolume":" 349","title":"Local law of addition of random matrices on optimal scale","ddc":["530"],"status":"public","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","day":"01","scopus_import":"1","date_published":"2017-02-01T00:00:00Z","citation":{"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.","short":"Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 947–990.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Law of Addition of Random Matrices on Optimal Scale.” Communications in Mathematical Physics. Springer, 2017. 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","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.","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. Springer, pp. 947–990, 2017.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-016-2805-6"},"publication":"Communications in Mathematical Physics","page":"947 - 990"},{"citation":{"short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.","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.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2017.08.028.","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","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” Advances in Mathematics, vol. 319. Academic Press, pp. 251–291, 2017.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2017.08.028","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291."},"publication":"Advances in Mathematics","page":"251 - 291","date_published":"2017-10-15T00:00:00Z","scopus_import":"1","article_processing_charge":"No","day":"15","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"733","intvolume":" 319","title":"Convergence rate for spectral distribution of addition of random matrices","status":"public","oa_version":"Submitted Version","type":"journal_article","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."}],"main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"oa":1,"external_id":{"isi":["000412150400010"]},"project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","isi":1,"doi":"10.1016/j.aim.2017.08.028","language":[{"iso":"eng"}],"month":"10","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","year":"2017","publisher":"Academic Press","department":[{"_id":"LaEr"}],"publication_status":"published","author":[{"first_name":"Zhigang","last_name":"Bao","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang"},{"last_name":"Erdös","first_name":"László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"volume":319,"date_updated":"2023-09-28T11:30:42Z","date_created":"2018-12-11T11:48:13Z","ec_funded":1,"publist_id":"6935"},{"scopus_import":1,"day":"15","publication":"Annals of Applied Probability","citation":{"ama":"Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 2016;26(6):3786-3839. doi:10.1214/16-AAP1193","ista":"Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839.","ieee":"J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” Annals of Applied Probability, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016.","apa":"Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AAP1193","mla":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:10.1214/16-AAP1193.","short":"J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.","chicago":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/16-AAP1193."},"page":"3786 - 3839","date_published":"2016-12-15T00:00:00Z","type":"journal_article","abstract":[{"lang":"eng","text":"We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay."}],"issue":"6","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","_id":"1157","title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","status":"public","intvolume":" 26","oa_version":"Preprint","month":"12","oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1409.4979"}],"quality_controlled":"1","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"doi":"10.1214/16-AAP1193","language":[{"iso":"eng"}],"publist_id":"6201","ec_funded":1,"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.","year":"2016","publication_status":"published","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}],"author":[{"last_name":"Lee","first_name":"Ji","full_name":"Lee, Ji"},{"last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"date_created":"2018-12-11T11:50:27Z","date_updated":"2021-01-12T06:48:43Z","volume":26},{"abstract":[{"lang":"eng","text":"We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N."}],"issue":"3","type":"journal_article","oa_version":"Preprint","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","_id":"1219","status":"public","title":"Bulk universality for deformed wigner matrices","intvolume":" 44","day":"01","scopus_import":1,"date_published":"2016-01-01T00:00:00Z","publication":"Annals of Probability","citation":{"chicago":"Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/15-AOP1023.","short":"J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.","mla":"Lee, Jioon, et al. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability, vol. 44, no. 3, Institute of Mathematical Statistics, 2016, pp. 2349–425, doi:10.1214/15-AOP1023.","apa":"Lee, J., Schnelli, K., Stetler, B., & Yau, H. (2016). Bulk universality for deformed wigner matrices. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOP1023","ieee":"J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” Annals of Probability, vol. 44, no. 3. Institute of Mathematical Statistics, pp. 2349–2425, 2016.","ista":"Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425.","ama":"Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner matrices. Annals of Probability. 2016;44(3):2349-2425. doi:10.1214/15-AOP1023"},"page":"2349 - 2425","ec_funded":1,"publist_id":"6115","author":[{"first_name":"Jioon","last_name":"Lee","full_name":"Lee, Jioon"},{"full_name":"Schnelli, Kevin","last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Ben","last_name":"Stetler","full_name":"Stetler, Ben"},{"full_name":"Yau, Horngtzer","last_name":"Yau","first_name":"Horngtzer"}],"date_updated":"2021-01-12T06:49:10Z","date_created":"2018-12-11T11:50:47Z","volume":44,"year":"2016","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. ","publication_status":"published","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}],"month":"01","doi":"10.1214/15-AOP1023","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1405.6634"}],"oa":1,"quality_controlled":"1","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}]},{"oa_version":"Preprint","title":"Local stability of the free additive convolution","status":"public","intvolume":" 271","_id":"1434","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.","lang":"eng"}],"issue":"3","type":"journal_article","date_published":"2016-08-01T00:00:00Z","page":"672 - 719","publication":"Journal of Functional Analysis","citation":{"ama":"Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution. Journal of Functional Analysis. 2016;271(3):672-719. doi:10.1016/j.jfa.2016.04.006","ista":"Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 672–719.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,” Journal of Functional Analysis, vol. 271, no. 3. Academic Press, pp. 672–719, 2016.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2016). Local stability of the free additive convolution. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2016.04.006","mla":"Bao, Zhigang, et al. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis, vol. 271, no. 3, Academic Press, 2016, pp. 672–719, doi:10.1016/j.jfa.2016.04.006.","short":"Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis. Academic Press, 2016. https://doi.org/10.1016/j.jfa.2016.04.006."},"day":"01","scopus_import":1,"date_updated":"2021-01-12T06:50:42Z","date_created":"2018-12-11T11:52:00Z","volume":271,"author":[{"full_name":"Bao, Zhigang","last_name":"Bao","first_name":"Zhigang","orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603"},{"first_name":"Kevin","last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin"}],"publication_status":"published","department":[{"_id":"LaEr"}],"publisher":"Academic Press","year":"2016","ec_funded":1,"publist_id":"5764","language":[{"iso":"eng"}],"doi":"10.1016/j.jfa.2016.04.006","quality_controlled":"1","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"main_file_link":[{"url":"http://arxiv.org/abs/1508.05905","open_access":"1"}],"month":"08"},{"doi":"10.1007/s00440-014-0610-8","language":[{"iso":"eng"}],"main_file_link":[{"url":"http://arxiv.org/abs/1310.7057","open_access":"1"}],"oa":1,"quality_controlled":"1","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"month":"02","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin"}],"date_created":"2018-12-11T11:54:31Z","date_updated":"2021-01-12T06:53:49Z","volume":164,"acknowledgement":"Most of the presented work was obtained while Kevin Schnelli was staying at the IAS with the support of\r\nThe Fund For Math.","year":"2016","publication_status":"published","department":[{"_id":"LaEr"}],"publisher":"Springer","publist_id":"5215","ec_funded":1,"date_published":"2016-02-01T00:00:00Z","publication":"Probability Theory and Related Fields","citation":{"chicago":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields. Springer, 2016. https://doi.org/10.1007/s00440-014-0610-8.","mla":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8.","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.","apa":"Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-014-0610-8","ieee":"J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” Probability Theory and Related Fields, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.","ama":"Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 2016;164(1-2):165-241. doi:10.1007/s00440-014-0610-8"},"page":"165 - 241","day":"01","scopus_import":1,"oa_version":"Preprint","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","_id":"1881","title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","status":"public","intvolume":" 164","abstract":[{"lang":"eng","text":"We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues. "}],"issue":"1-2","type":"journal_article"},{"scopus_import":1,"month":"09","day":"01","quality_controlled":"1","publication":"Reviews in Mathematical Physics","oa":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.8015"}],"citation":{"ista":"Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8), 1550018.","ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” Reviews in Mathematical Physics, vol. 27, no. 8. World Scientific Publishing, 2015.","apa":"Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. World Scientific Publishing. https://doi.org/10.1142/S0129055X1550018X","ama":"Lee J, Schnelli K. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 2015;27(8). doi:10.1142/S0129055X1550018X","chicago":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics. World Scientific Publishing, 2015. https://doi.org/10.1142/S0129055X1550018X.","mla":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:10.1142/S0129055X1550018X.","short":"J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015)."},"language":[{"iso":"eng"}],"date_published":"2015-09-01T00:00:00Z","doi":"10.1142/S0129055X1550018X","article_number":"1550018","type":"journal_article","abstract":[{"lang":"eng","text":"We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix."}],"issue":"8","publist_id":"5475","title":"Edge universality for deformed Wigner matrices","publication_status":"published","status":"public","intvolume":" 27","publisher":"World Scientific Publishing","department":[{"_id":"LaEr"}],"_id":"1674","year":"2015","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T06:52:26Z","date_created":"2018-12-11T11:53:24Z","oa_version":"Preprint","volume":27,"author":[{"last_name":"Lee","first_name":"Jioon","full_name":"Lee, Jioon"},{"orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin","full_name":"Schnelli, Kevin"}]}]