[{"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","publication_identifier":{"issn":["00911798"]},"quality_controlled":"1","type":"journal_article","year":"2019","publication_status":"published","external_id":{"arxiv":["1612.05920"]},"issue":"3","intvolume":" 47","citation":{"ista":"Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale. Annals of Probability. 47(3), 1270–1334.","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.","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","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","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."},"oa":1,"date_published":"2019-05-01T00:00:00Z","month":"05","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"volume":47,"publisher":"Project Euclid","page":"1270-1334","_id":"6511","department":[{"_id":"LaEr"}],"doi":"10.1214/18-AOP1284","title":"Local single ring theorem on optimal scale","main_file_link":[{"url":"https://arxiv.org/abs/1612.05920","open_access":"1"}],"date_created":"2019-06-02T21:59:13Z","day":"01","date_updated":"2019-08-02T12:39:21Z","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Schnelli, Kevin","last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"language":[{"iso":"eng"}],"status":"public","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)."}],"publication":"Annals of Probability"},{"language":[{"iso":"eng"}],"status":"public","abstract":[{"lang":"eng","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."}],"publication":"Probability Theory and Related Fields","date_created":"2018-12-11T11:47:56Z","day":"14","date_updated":"2019-08-02T12:39:27Z","author":[{"last_name":"Lee","first_name":"Jii","full_name":"Lee, Jii"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin"}],"title":"Local law and Tracy–Widom limit for sparse random matrices","main_file_link":[{"url":"https://arxiv.org/abs/1605.08767","open_access":"1"}],"publist_id":"7017","volume":171,"publisher":"Springer","_id":"690","department":[{"_id":"LaEr"}],"doi":"10.1007/s00440-017-0787-8","oa":1,"date_published":"2018-06-14T00:00:00Z","month":"06","project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"publication_status":"published","external_id":{"arxiv":["1605.08767"]},"intvolume":" 171","issue":"1-2","citation":{"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","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.","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).","chicago":"Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” *Probability Theory and Related Fields* 171, no. 1–2 (2018). https://doi.org/10.1007/s00440-017-0787-8.","apa":"Lee, J., & Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. *Probability Theory and Related Fields*, *171*(1–2). 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).","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, 2018."},"oa_version":"Preprint","type":"journal_article","quality_controlled":"1","year":"2018","article_number":"543-616","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"language":[{"iso":"eng"}],"status":"public","abstract":[{"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.","lang":"eng"}],"publication":"Advances in Mathematics","date_created":"2018-12-11T11:48:13Z","day":"15","date_updated":"2019-08-02T12:39:29Z","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao","first_name":"Zhigang","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","full_name":"Erdös, László"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"title":"Convergence rate for spectral distribution of addition of random matrices","main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"publist_id":"6935","volume":319,"publisher":"Academic Press","page":"251 - 291","_id":"733","department":[{"_id":"LaEr"}],"doi":"10.1016/j.aim.2017.08.028","date_published":"2017-10-15T00:00:00Z","oa":1,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"month":"10","publication_status":"published","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","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.","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","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.","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","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.","ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291."},"intvolume":" 319","oa_version":"Submitted Version","type":"journal_article","quality_controlled":"1","year":"2017","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","file":[{"file_name":"IST-2016-722-v1+1_s00220-016-2805-6.pdf","date_updated":"2018-12-12T10:14:47Z","relation":"main_file","date_created":"2018-12-12T10:14:47Z","creator":"system","content_type":"application/pdf","file_id":"5102","open_access":1,"access_level":"open_access","file_size":1033743}],"year":"2017","type":"journal_article","quality_controlled":"1","accept":"1","publication_identifier":{"issn":["00103616"]},"oa_version":"Published Version","issue":"3","intvolume":" 349","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.","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.","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","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.","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."},"publication_status":"published","date_published":"2017-02-01T00:00:00Z","file_date_updated":"2018-12-12T10:14:47Z","oa":1,"month":"02","project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"_id":"1207","doi":"10.1007/s00220-016-2805-6","department":[{"_id":"LaEr"}],"page":"947 - 990","ddc":["530"],"volume":349,"publisher":"Springer","publist_id":"6141","title":"Local law of addition of random matrices on optimal scale","date_updated":"2019-08-02T12:36:55Z","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang"},{"full_name":"Erdös, László","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli"}],"pubrep_id":"722","date_created":"2018-12-11T11:50:43Z","cc_license":"'https://creativecommons.org/licenses/by/4.0/'","day":"01","publication":"Communications in Mathematical Physics","status":"public","abstract":[{"lang":"eng","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."}],"language":[{"iso":"eng"}]},{"year":"2017","oa_version":"Submitted Version","publication_identifier":{"issn":["02460203"]},"quality_controlled":"1","type":"journal_article","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","month":"11","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"date_published":"2017-11-01T00:00:00Z","issue":"4","intvolume":" 53","citation":{"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","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656.","mla":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability and Statistics*, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:10.1214/16-AIHP765.","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*, *53*(4), 1606–1656. https://doi.org/10.1214/16-AIHP765","chicago":"Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability and Statistics* 53, no. 4 (2017): 1606–56. https://doi.org/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.","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, pp. 1606–1656, 2017."},"publication_status":"published","publist_id":"7189","main_file_link":[{"url":"https://arxiv.org/abs/1504.00650","open_access":"1"}],"title":"Universality for random matrix flows with time dependent density","page":"1606 - 1656","department":[{"_id":"LaEr"}],"doi":"10.1214/16-AIHP765","_id":"615","publisher":"Institute of Mathematical Statistics","volume":53,"publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law."}],"status":"public","author":[{"orcid":"0000-0001-5366-9603","first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin","full_name":"Schnelli, Kevin"}],"date_updated":"2019-08-02T12:39:11Z","day":"01","date_created":"2018-12-11T11:47:30Z"},{"title":"Local stability of the free additive convolution","publist_id":"5764","main_file_link":[{"url":"http://arxiv.org/abs/1508.05905","open_access":"1"}],"volume":271,"publisher":"Academic Press","page":"672 - 719","_id":"1434","doi":"10.1016/j.jfa.2016.04.006","department":[{"_id":"LaEr"}],"language":[{"iso":"eng"}],"status":"public","abstract":[{"text":"We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.","lang":"eng"}],"publication":"Journal of Functional Analysis","date_created":"2018-12-11T11:52:00Z","day":"01","date_updated":"2019-08-02T12:37:08Z","author":[{"full_name":"Bao, Zhigang","last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin"}],"oa_version":"Preprint","type":"journal_article","quality_controlled":"1","year":"2016","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"date_published":"2016-08-01T00:00:00Z","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"month":"08","publication_status":"published","citation":{"apa":"Bao, Z., Erdös, L., & Schnelli, K. (2016). Local stability of the free additive convolution. *Journal of Functional Analysis*, *271*(3), 672–719. https://doi.org/10.1016/j.jfa.2016.04.006","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the Free Additive Convolution.” *Journal of Functional Analysis* 271, no. 3 (2016): 672–719. https://doi.org/10.1016/j.jfa.2016.04.006.","ama":"Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution. *Journal of Functional Analysis*. 2016;271(3):672-719. doi:10.1016/j.jfa.2016.04.006","short":"Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719.","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.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,” *Journal of Functional Analysis*, vol. 271, no. 3, pp. 672–719, 2016.","ista":"Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 672–719."},"issue":"3","intvolume":" 271"},{"doi":"10.1214/15-AOP1023","department":[{"_id":"LaEr"}],"_id":"1219","page":"2349 - 2425","publisher":"Institute of Mathematical Statistics","volume":44,"publist_id":"6115","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1405.6634"}],"title":"Bulk universality for deformed wigner matrices","author":[{"full_name":"Lee, Jioon","last_name":"Lee","first_name":"Jioon"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli"},{"full_name":"Stetler, Ben","last_name":"Stetler","first_name":"Ben"},{"last_name":"Yau","first_name":"Horngtzer","full_name":"Yau, Horngtzer"}],"date_updated":"2019-08-02T12:36:56Z","day":"01","date_created":"2018-12-11T11:50:47Z","publication":"Annals of Probability","abstract":[{"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.","lang":"eng"}],"status":"public","language":[{"iso":"eng"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","year":"2016","type":"journal_article","quality_controlled":"1","oa_version":"Preprint","citation":{"chicago":"Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” *Annals of Probability* 44, no. 3 (2016): 2349–2425. https://doi.org/10.1214/15-AOP1023.","apa":"Lee, J., Schnelli, K., Stetler, B., & Yau, H. (2016). Bulk universality for deformed wigner matrices. *Annals of Probability*, *44*(3), 2349–2425. https://doi.org/10.1214/15-AOP1023","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","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.","short":"J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.","ista":"Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425.","ieee":"J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” *Annals of Probability*, vol. 44, no. 3, pp. 2349–2425, 2016."},"issue":"3","intvolume":" 44","acknowledgement":"J.C. was supported in part by National Research Foundation of Korea Grant 2011-0013474 and TJ Park Junior Faculty Fellowship.\r\nK.S. was supported by ERC Advanced Grant RANMAT, No. 338804, and the \"Fund for Math.\"\r\nB.S. was supported by NSF GRFP Fellowship DGE-1144152.\r\nH.Y. was supported in part by NSF Grant DMS-13-07444 and Simons investigator fellowship. We thank Paul Bourgade, László Erd ̋os and Antti Knowles for helpful comments. We are grateful to the Taida Institute for Mathematical\r\nSciences and National Taiwan Universality for their hospitality during part of this\r\nresearch. We thank Thomas Spencer and the Institute for Advanced Study for their\r\nhospitality during the academic year 2013–2014. ","publication_status":"published","month":"01","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"date_published":"2016-01-01T00:00:00Z"},{"title":"Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1409.4979"}],"publist_id":"6201","volume":26,"publisher":"Institute of Mathematical Statistics","page":"3786 - 3839","_id":"1157","department":[{"_id":"LaEr"}],"doi":"10.1214/16-AAP1193","language":[{"iso":"eng"}],"status":"public","abstract":[{"text":"We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay.","lang":"eng"}],"publication":"Annals of Applied Probability","date_created":"2018-12-11T11:50:27Z","day":"15","date_updated":"2019-08-02T12:36:53Z","author":[{"last_name":"Lee","first_name":"Ji","full_name":"Lee, Ji"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"oa_version":"Preprint","quality_controlled":"1","type":"journal_article","year":"2016","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"date_published":"2016-12-15T00:00:00Z","month":"12","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"publication_status":"published","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.","issue":"6","intvolume":" 26","citation":{"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, pp. 3786–3839, 2016.","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.","ama":"Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. *Annals of Applied Probability*. 2016;26(6):3786-3839. doi:10.1214/16-AAP1193","mla":"Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” *Annals of Applied Probability*, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:10.1214/16-AAP1193.","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* 26, no. 6 (2016): 3786–3839. https://doi.org/10.1214/16-AAP1193.","apa":"Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. *Annals of Applied Probability*, *26*(6), 3786–3839. https://doi.org/10.1214/16-AAP1193"}},{"status":"public","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. "}],"language":[{"iso":"eng"}],"publication":"Probability Theory and Related Fields","date_created":"2018-12-11T11:54:31Z","day":"01","date_updated":"2019-08-02T12:37:29Z","author":[{"full_name":"Lee, Jioon","last_name":"Lee","first_name":"Jioon"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","last_name":"Schnelli"}],"title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","publist_id":"5215","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1310.7057"}],"volume":164,"publisher":"Springer","_id":"1881","department":[{"_id":"LaEr"}],"doi":"10.1007/s00440-014-0610-8","page":"165 - 241","oa":1,"date_published":"2016-02-01T00:00:00Z","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems"}],"month":"02","publication_status":"published","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.","citation":{"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","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.","chicago":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” *Probability Theory and Related Fields* 164, no. 1–2 (2016): 165–241. https://doi.org/10.1007/s00440-014-0610-8.","apa":"Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. *Probability Theory and Related Fields*, *164*(1–2), 165–241. 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, pp. 165–241, 2016.","ista":"Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241."},"issue":"1-2","intvolume":" 164","quality_controlled":"1","type":"journal_article","oa_version":"Preprint","year":"2016","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87"},{"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."}],"status":"public","language":[{"iso":"eng"}],"month":"09","oa":1,"date_published":"2015-09-01T00:00:00Z","publication":"Reviews in Mathematical Physics","publication_status":"published","day":"01","date_created":"2018-12-11T11:53:24Z","intvolume":" 27","citation":{"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).","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* 27, no. 8 (2015). https://doi.org/10.1142/S0129055X1550018X.","apa":"Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices. *Reviews in Mathematical Physics*, *27*(8). https://doi.org/10.1142/S0129055X1550018X","ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” *Reviews in Mathematical Physics*, vol. 27, no. 8, 2015.","ista":"Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8)."},"author":[{"last_name":"Lee","first_name":"Jioon","full_name":"Lee, Jioon"},{"full_name":"Schnelli, Kevin","last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87"}],"issue":"8","date_updated":"2019-01-24T13:04:02Z","quality_controlled":"1","type":"journal_article","title":"Edge universality for deformed Wigner matrices","oa_version":"Preprint","main_file_link":[{"url":"http://arxiv.org/abs/1407.8015","open_access":"1"}],"publist_id":"5475","article_number":"1550018","year":"2015","publisher":"World Scientific Publishing","volume":27,"doi":"10.1142/S0129055X1550018X","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","department":[{"_id":"LaEr"}],"_id":"1674"}]