• Corpus ID: 233407515

Random subcomplexes and Betti numbers of random edge ideals

@inproceedings{Dochtermann2021RandomSA,
  title={Random subcomplexes and Betti numbers of random edge ideals},
  author={Anton Dochtermann and Andrew Newman},
  year={2021}
}

• A. Dochtermann, Andrew Newman
• Published 26 April 2021
• Mathematics

We study homological properties of random quadratic monomial ideals in a polynomial ring R = K[x1, . . . xn], utilizing methods from the Erdős–Rényi model of random graphs. Here for a graph G ∼ G(n,p) we consider the ‘coedge’ ideal IG corresponding to the missing edges of G, and study Betti numbers of R/IG as n tends to infinity. Our main results involve fixing the edge probability p = p(n) so that asymptotically almost surely the Krull dimension of R/IG is fixed. Under these conditions we establish various properties regarding the Betti table of R/IG, 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.