Average Behavior of Minimal Free Resolutions of Monomial Ideals

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 subset of S = k[x(1),..., x(n)] 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.