Cohen–Lenstra Heuristics for Torsion in Homology of Random Complexes

ABSTRACT

We study torsion in homology of the random d-complex Y ∼ 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 [Aronshtam and Linial 13, Linial and Peled 16, Linial and Peled 17], 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 |Aut(H)|.

We also study the torsion in homology of the uniform random $\mathbb{Q}$-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]. We give experimental evidence that in this model also, the torsion is Cohen–Lenstra distributed in the limit.

KEYWORDS:

• random simplicial complex
• Q-acyclic complex
• hypertree
• phase transition

Acknowledgments

M. K. is grateful to Sam Payne and Yuval Peled for helpful and inspiring conversations. All four authors are grateful to ICERM for visits during the special semester program Topology in Motion in the fall of 2016, where much of this work was completed. F.H.L. is also thankful to HIM Bonn for a stay for the Special Hausdorff Program Applied and Computational Algebraic Topology in 2017.

Additional information

Funding

M. K. was partially supported by NSF-DMS #1352386; F. H. L. was partially supported by the DFG Coll. Research Center TRR 109, “Discretization in Geometry and Dynamics”; A. N. and K. P. were partially supported by NSF-DMS #1547357.