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

Consider the bipartite Erdős--Rényi 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 $γ= n/m \geq 1$. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Marčenko