We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always be n, where n is the number of variables in the polynomial ring. As a consequence we prove that Cohen.Macaulayness is a rare property. We characterize when a random monomial ideal is generic/strongly generic, and when it is Scarf.i.e., when the algebraic Scarf complex of M S = k[x1, . . . , xn] gives a minimal free resolution of S/M. It turns out, outside of a very specific ratio of model parameters, random monomial ideals are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.

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.