{"year":"2026","language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Journal of Functional Analysis","external_id":{"arxiv":["2409.01819"]},"article_number":"111266","abstract":[{"lang":"eng","text":"In this paper, we consider the rectangular random matrix\r\nX =(xij ) ∈ RN×n whose entries are iid with tail P(|xij | >\r\nt) ∼ t−α for some α> 0. We consider the regime N(n)/n →\r\na > 1 as n tends to infinity. Our main interest lies in the right\r\nsingular vector corresponding to the smallest singular value,\r\nwhich we will refer to as the ``bottom singular vector'', denoted\r\nby 𝔲. In this paper, we prove the following phase transition\r\nregarding the localization length of 𝔲: when α< 2 the\r\nlocalization length is O(n/ log n); when α> 2 the localization\r\nlength is of order n. Similar results hold for all right singular\r\nvectors around the smallest singular value. The variational\r\ndefinition of the bottom singular vector suggests that the\r\nmechanism for this localization-delocalization transition when\r\nα goes across 2 is intrinsically different from the one for the\r\ntop singular vector when α goes across 4"}],"status":"public","author":[{"full_name":"Bao, Zhigang","last_name":"Bao","first_name":"Zhigang"},{"first_name":"Jaehun","id":"96155047-f36a-11ef-b766-8b5ae7cecd49","full_name":"Lee, Jaehun","last_name":"Lee"},{"first_name":"Xiaocong","full_name":"Xu, Xiaocong","last_name":"Xu"}],"intvolume":"       290","volume":290,"day":"15","publication_identifier":{"issn":["0022-1236"],"eissn":["1096-0783"]},"article_type":"original","publication_status":"published","type":"journal_article","doi":"10.1016/j.jfa.2025.111266","month":"02","_id":"22360","publisher":"Elsevier","date_published":"2026-02-15T00:00:00Z","oa_version":"Preprint","title":"Phase transition for the bottom singular vector of rectangular random matrices","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2409.01819","open_access":"1"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Bao, Zhigang, et al. “Phase Transition for the Bottom Singular Vector of Rectangular Random Matrices.” <i>Journal of Functional Analysis</i>, vol. 290, no. 4, 111266, Elsevier, 2026, doi:<a href=\"https://doi.org/10.1016/j.jfa.2025.111266\">10.1016/j.jfa.2025.111266</a>.","chicago":"Bao, Zhigang, Jaehun Lee, and Xiaocong Xu. “Phase Transition for the Bottom Singular Vector of Rectangular Random Matrices.” <i>Journal of Functional Analysis</i>. Elsevier, 2026. <a href=\"https://doi.org/10.1016/j.jfa.2025.111266\">https://doi.org/10.1016/j.jfa.2025.111266</a>.","ieee":"Z. Bao, J. Lee, and X. Xu, “Phase transition for the bottom singular vector of rectangular random matrices,” <i>Journal of Functional Analysis</i>, vol. 290, no. 4. Elsevier, 2026.","ama":"Bao Z, Lee J, Xu X. Phase transition for the bottom singular vector of rectangular random matrices. <i>Journal of Functional Analysis</i>. 2026;290(4). doi:<a href=\"https://doi.org/10.1016/j.jfa.2025.111266\">10.1016/j.jfa.2025.111266</a>","ista":"Bao Z, Lee J, Xu X. 2026. Phase transition for the bottom singular vector of rectangular random matrices. Journal of Functional Analysis. 290(4), 111266.","apa":"Bao, Z., Lee, J., &#38; Xu, X. (2026). Phase transition for the bottom singular vector of rectangular random matrices. <i>Journal of Functional Analysis</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jfa.2025.111266\">https://doi.org/10.1016/j.jfa.2025.111266</a>","short":"Z. Bao, J. Lee, X. Xu, Journal of Functional Analysis 290 (2026)."},"OA_place":"repository","date_updated":"2026-07-20T11:01:15Z","extern":"1","date_created":"2026-07-18T10:32:41Z","arxiv":1,"issue":"4","quality_controlled":"1","OA_type":"green"}