{"title":"Extremal eigenvalues and eigenvectors of deformed Wigner matrices","oa":1,"intvolume":"       164","citation":{"apa":"Lee, J., &#38; Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. Springer. <a href=\"https://doi.org/10.1007/s00440-014-0610-8\">https://doi.org/10.1007/s00440-014-0610-8</a>","ieee":"J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2. Springer, pp. 165–241, 2016.","ama":"Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. <i>Probability Theory and Related Fields</i>. 2016;164(1-2):165-241. doi:<a href=\"https://doi.org/10.1007/s00440-014-0610-8\">10.1007/s00440-014-0610-8</a>","chicago":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00440-014-0610-8\">https://doi.org/10.1007/s00440-014-0610-8</a>.","short":"J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.","mla":"Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” <i>Probability Theory and Related Fields</i>, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:<a href=\"https://doi.org/10.1007/s00440-014-0610-8\">10.1007/s00440-014-0610-8</a>.","ista":"Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241."},"doi":"10.1007/s00440-014-0610-8","ec_funded":1,"date_published":"2016-02-01T00:00:00Z","quality_controlled":"1","publist_id":"5215","abstract":[{"text":"We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ&gt;λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ&gt;λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ&gt;λ+, while they are completely delocalized for λ&lt;λ+. Similar results hold for the lowest eigenvalues. ","lang":"eng"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","type":"journal_article","publisher":"Springer","day":"01","main_file_link":[{"url":"http://arxiv.org/abs/1310.7057","open_access":"1"}],"issue":"1-2","language":[{"iso":"eng"}],"volume":164,"year":"2016","date_created":"2018-12-11T11:54:31Z","date_updated":"2021-01-12T06:53:49Z","month":"02","publication_status":"published","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"oa_version":"Preprint","scopus_import":1,"_id":"1881","status":"public","department":[{"_id":"LaEr"}],"publication":"Probability Theory and Related Fields","author":[{"first_name":"Jioon","full_name":"Lee, Jioon","last_name":"Lee"},{"first_name":"Kevin","full_name":"Schnelli, Kevin","last_name":"Schnelli","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231"}],"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.","page":"165 - 241"}