Random subcomplexes and Betti numbers of random edge ideals

TL;DR

This paper investigates the homological properties of random edge ideals derived from Erdős-Rényi graphs, focusing on Betti numbers, regularity, and projective dimension as the number of vertices grows, revealing asymptotic behaviors.

Contribution

It introduces new bounds and distribution results for Betti numbers of random edge ideals, extending prior work to asymptotic regimes and higher-dimensional complexes.

Findings

Sharp bounds on Betti table invariants

Distribution patterns of normalized Betti numbers

Asymptotic properties of random clique complexes

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 Erd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$ we consider the `coedge' ideal $I_G$ 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.