article published yes Zhigang Bao author Jaehun Lee author 96155047-f36a-11ef-b766-8b5ae7cecd49 Xiaocong Xu author In this paper, we consider the rectangular random matrix X =(xij ) ∈ RN×n whose entries are iid with tail P(|xij | > t) ∼ t−α for some α> 0. We consider the regime N(n)/n → a > 1 as n tends to infinity. Our main interest lies in the right singular vector corresponding to the smallest singular value, which we will refer to as the ``bottom singular vector'', denoted by 𝔲. In this paper, we prove the following phase transition regarding the localization length of 𝔲: when α< 2 the localization length is O(n/ log n); when α> 2 the localization length is of order n. Similar results hold for all right singular vectors around the smallest singular value. The variational definition of the bottom singular vector suggests that the mechanism for this localization-delocalization transition when α goes across 2 is intrinsically different from the one for the top singular vector when α goes across 4 Elsevier2026 eng 0022-1236 1096-0783 2409.0181910.1016/j.jfa.2025.111266 2904 yes 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. 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> 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> 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>. 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. Z. Bao, J. Lee, X. Xu, Journal of Functional Analysis 290 (2026). 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>. 223602026-07-18T10:32:41Z2026-07-20T11:01:15Z