Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes
Matthew Kahle, Frank H. Lutz, Andrew Newman, Kyle Parsons · Experimental Mathematics · 2018
We 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 [Aronshtam and Linial 13] L. Aronshtam and N. Linial. “When does top Homology in a Random Simplicial Complex Vanish?” Rand. Struct. Algo. 46 (2013), 26–35.[Crossref], [Web of Science ®] , [Google Scholar], Linial and Peled 16 [Linial and Peled 16] N. Linial and Y. Peled. “On the Phase Transition in Random Simplicial Complexes.” Annals of Mathematics 184 (2016), 745–773.[Crossref], [Web of Science ®] , [Google Scholar], Linial and Peled 17 [Linial and Peled 17] N. Linial and Y. Peled. Random simplicial complexes: Around the phase transition, pp. 543–570, Springer International Publishing, Cham, 2017. [Google Scholar]], 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 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 n2 [Kalai 83 [Kalai 83] G. Kalai. “Enumeration of Q-acyclic Simplicial Complexes.” Israel Journal of Mathematics 45 (1983), 337–351.[Crossref], [Web of Science ®] , [Google Scholar]]. We give experimental evidence that in this model also, the torsion is Cohen–Lenstra distributed in the limit.