Cohen--Lenstra heuristics for torsion in homology of random complexes

Abstract

We study torsion in homology of the random $d$-complex $Y \sim Y_d(n,p)$ experimentally. Our experiments suggest that there is almost always a moment in the process where there is an enormous burst of torsion in homology $H_{d-1}(Y)$. This moment seems to coincide with the phase transition studied in \cite{AL,LP,LP3} , where cycles in $H_d(Y)$ first appear with high probability. Our main study is the limiting distribution on the $q$-part of the torsion subgroup of $H_{d-1}(Y)$ for small primes $q$. We find strong evidence for a limiting Cohen--Lenstra distribution, where the probability that the $q$-part is isomorphic to a given $q$-group $H$ is inversely proportional to the order of the automorphism group $|\mbox{Aut}(H)|$. We also study the torsion in homology of the uniform random $\Q$-acyclic $2$-complex. This model is analogous to a uniform spanning tree on a complete graph, but more complicated topologically since Kalai showed that the expected order of the torsion group is exponentially large in $n^2$ \cite{Kalai}. We give experimental evidence that in this model also, the torsion is Cohen--Lenstra distributed in the limit.