---
res:
  bibo_abstract:
  - "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 \U0001D532.
    In this paper, we prove the following phase transition\r\nregarding the localization
    length of \U0001D532: 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@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Zhigang
      foaf_name: Bao, Zhigang
      foaf_surname: Bao
  - foaf_Person:
      foaf_givenName: Jaehun
      foaf_name: Lee, Jaehun
      foaf_surname: Lee
      foaf_workInfoHomepage: http://www.librecat.org/personId=96155047-f36a-11ef-b766-8b5ae7cecd49
  - foaf_Person:
      foaf_givenName: Xiaocong
      foaf_name: Xu, Xiaocong
      foaf_surname: Xu
  bibo_doi: 10.1016/j.jfa.2025.111266
  bibo_issue: '4'
  bibo_volume: 290
  dct_date: 2026^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/0022-1236
  - http://id.crossref.org/issn/1096-0783
  dct_language: eng
  dct_publisher: Elsevier@
  dct_title: Phase transition for the bottom singular vector of rectangular random
    matrices@
...