Phase transition for the bottom singular vector of rectangular random matrices

In this paper, we consider the rectangular random matrix $X=(x_{ij})\in \mathbb{R}^{N\times n}$ whose entries are iid with tail $\mathbb{P}(|x_{ij}|>t)\sim t^{-\alpha}$ for some $\alpha>0$. We consider the regime $N(n)/n\to \mathsf{a}>1$ 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 $\mathfrak{u}$. In this paper, we prove the following phase transition regarding the localization length of $\mathfrak{u}$: when $\alpha<2$ the localization length is $O(n/\log n)$; when $\alpha>2$ 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 $\alpha$ goes across $2$ is intrinsically different from the one for the top singular vector when $\alpha$ goes across $4$.