Using dense graph limit theory to count cocycles of random simplicial complexes

Abstract

We develop a limit theory for $1$-cochains of complete graphs with coefficients from a finite abelian group. We prove an analogue of the large deviation principle of Chatterjee and Varadhan for random cochains. We use these new tools to prove results about the homology of random $2$-dimensional simplicial complexes. More specifically, we prove that if $T_n$ is a random $2$-dimensional determinantal hypertree on $n$ vertices and $p$ is any prime, then \[\frac{\dim H_1(T_n,\mathbb{F}_p)}{n^2}\] converges to zero in probability. The same result holds for random $1$-out 2-complexes.