Phase transition for the bottom singular vector of rectangular random matrices

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