---
OA_place: repository
OA_type: green
_id: '22360'
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 \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"
article_number: '111266'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Zhigang
  full_name: Bao, Zhigang
  last_name: Bao
- first_name: Jaehun
  full_name: Lee, Jaehun
  id: 96155047-f36a-11ef-b766-8b5ae7cecd49
  last_name: Lee
- first_name: Xiaocong
  full_name: Xu, Xiaocong
  last_name: Xu
citation:
  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>
  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>
  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.
  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.
  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>.
  short: Z. Bao, J. Lee, X. Xu, Journal of Functional Analysis 290 (2026).
date_created: 2026-07-18T10:32:41Z
date_published: 2026-02-15T00:00:00Z
date_updated: 2026-07-20T11:01:15Z
day: '15'
doi: 10.1016/j.jfa.2025.111266
extern: '1'
external_id:
  arxiv:
  - '2409.01819'
intvolume: '       290'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2409.01819
month: '02'
oa: 1
oa_version: Preprint
publication: Journal of Functional Analysis
publication_identifier:
  eissn:
  - 1096-0783
  issn:
  - 0022-1236
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: Phase transition for the bottom singular vector of rectangular random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 290
year: '2026'
...