Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes

We study torsion in homology of the randomd-complexY similar to 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 homologyH(d- 1)(Y). This moment seems to coincide with the phase transition studied in [Aronshtam and Linial 13,Linial and Peled 16,Linial and Peled 17], where cycles inH(d)(Y) first appear with high probability. Our main study is the limiting distribution on theq-part of the torsion subgroup ofH(d- 1)(Y) for small primesq. We find strong evidence for a limiting Cohen-Lenstra distribution, where the probability that theq-part is isomorphic to a givenq-groupHis inversely proportional to the order of the automorphism group |Aut(H)|. We also study the torsion in homology of the uniform randomQ-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 inn(2)[Kalai 83]. We give experimental evidence that in this model also, the torsion is Cohen-Lenstra distributed in the limit.