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 $M\subset S=k[x_1,\ldots ,x_n]$ 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.