Average Behavior of Minimal Free Resolutions of Monomial Ideals

We describe the typical homological properties of monomial ideals defined by random generating sets. We show that, under mild assumptions, random monomial ideals (RMI’s) will almost always have 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. We give a rigorous proof that Cohen-Macaulayness is a “rare” property. We characterize when an RMI is generic/strongly generic, and when it “is Scarf”—in other words, when the algebraic Scarf complex of $M\subset S=k[x_1,\dots ,x_n]$ gives a minimal free resolution of $S\mathrm{/}M$. As a result we see that, outside of a very specific ratio of model parameters, RMI’s are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers.