Random Subcomplexes and Betti Numbers of Random Edge Ideals

Abstract

Abstract We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb {K}}[x_1, \dots x_n]$, utilizing methods from the Erd̋s–Rényi model of random graphs. Here, for a graph $G \sim G(n, p),$ we consider the coedge ideal $I_G$ generated by monomials corresponding to the missing edges of $G$ and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve setting the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions, we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way, we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.