Random subcomplexes and Betti numbers of random edge ideals
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 ErdösRényi 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 fixing 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 higherdimensional vertex $k$connectivity that may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.12882
 arXiv:
 arXiv:2104.12882
 Bibcode:
 2021arXiv210412882D
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Combinatorics;
 Mathematics  Probability;
 13D02;
 13F55;
 05C80;
 05E45;
 60B99
 EPrint:
 29 pages, 2 figures