I am trying to find a research problem in commutative algebra that involves probabilistic studies. Just like recent trends in algebraic geometry to study "average" behaviours by introducing measures on the space of polynomials, I want to know if there are important topics at the intersection of randomness and commutative algebra as well? I know of one recent work that appears quite interesting - https://arxiv.org/pdf/1701.07130.pdf. I'd be grateful for some more references.
For instance, I want to see if one can study the average betti number of "random graded modules"; a sort of a probabilistic version of the Eisenbud-Horrocks problem. Does this even make sense? I have no idea how I'd define a random module though.
Pardon me for the unclear tone of the question. I am miles behind even beginner level.