@article{22360,
  abstract     = {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},
  author       = {Bao, Zhigang and Lee, Jaehun and Xu, Xiaocong},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {4},
  publisher    = {Elsevier},
  title        = {{Phase transition for the bottom singular vector of rectangular random matrices}},
  doi          = {10.1016/j.jfa.2025.111266},
  volume       = {290},
  year         = {2026},
}