Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes

Abstract

ABSTRACTWe study torsion in homology of the random d-complex Y ∼ Yd(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 Hd − 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 in Hd(Y) first appear with high probability.Our main study is the limiting distribution on the q-part of the torsion subgroup of Hd − 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 |Aut(H)|.We also study the torsion in homology of the uniform random -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 n2 [Kalai 83]. We give experimental evidence that in this model also, the torsion is Cohen–Lenstra distributed in the limit.