Georgia Institute of Technology College of Sciences

The bi-adjacency matrix of an Erdős–Rényi random bipartite graph with bounded aspect ratio is a rectangular random matrix with Bernoulli entries. Depending on the sparsity parameter $p$, its spectral behavior may either resemble that of a classical Wishart matrix or depart from this universal regime. In this talk, we study the extreme singular values at the critical density $np=c\log n$. We present the first quantitative characterization of the emergence of outlier singular values outside the Marčenko–Pastur law and determine their precise locations as functions of the largest and smallest degree vertices in the underlying random graph, which can be seen as an analogue of the Bai–Yin theorem in the sparse setting. These results uncover a clear mechanism by which combinatorial structures in sparse graphs generate spectral outliers. Joint work with Ioana Dumitriu, Haixiao Wang and Zhichao Wang.