Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases

arXiv:2610.22658v1 Announce Type: cross Abstract: Consider the bipartite Erd\H{o}s--R\'enyi graph $\mathbb{G}(n,m,d/\sqrt{mn})$ at the critical sparsity $d = b\log(N)$ for some $b > 0$ with $N = n + m$, and constant aspect ratio $\gamma = n/m \geq 1$. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Mar\v{c}enko--Pastur (MP) distribution, is split into two phases: in the bulk, singular vectors are completely delocalized, while near the edges, singular vectors are semilocalized, with their mass concentrating on small disjoint balls, centered at vertices with extreme degrees. As $b$ decreases, we identify sharp thresholds for the emergence of outlier singular values. For explicit $b_{\rm R}^{\star}$ and $b_{\rm L}^{\star}$ as functions of $\gamma$, we prove that when $b > b_{\rm R}^{\star}$, there is no outlier outside the bulk; when $b_{\rm L}^{\star}