Random Monomial Ideals

Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, and first graded Betti numbers, and show that our models generalize several well-known random simplicial complex models. Finally we present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen–Macaulayness of random monomial ideals.