Duke Math. J. 172 (7), 1293-1332, (15 May 2023) DOI: 10.1215/00127094-2022-0060
Asaf Ferber, Matthew Kwan, Ashwin Sah, Mehtaab Sawhney
KEYWORDS: singularity, Random graphs, random matrices, 05C80
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix) due to the presence of “low-degree dependencies” such as isolated vertices and pairs of degree 1 vertices with the same neighborhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants and , an Erdős–Rényi random graph with n vertices and edge probability typically has the property that its k-core (its largest subgraph with minimum degree at least k) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for “extremely sparse” random matrices with density . A key aspect of our proof is a technique to extract high-degree vertices and use them to “boost” the rank, starting from approximate rank bounds obtainable from (nonquantitative) spectral convergence machinery due to Bordenave, Lelarge, and Salez.