  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, first graded
Betti numbers, and present several experimentally-backed conjectures about
regularity, projective dimension, strong genericity, and Cohen-Macaulayness of
random monomial ideals.
