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Cohen-Lenstra heuristics for torsion in homology of random complexes. (English) Zbl 1460.05206

Summary: 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 L. Aronshtam and N. Linial [ Random Struct. Algorithms 46, No. 1, 26–35 (2015; Zbl 1326.55019)], and N. Linial and Y. Peled [Ann. Math. (2) 184, No. 3, 745–773 (2016; Zbl 1348.05193); in: A journey through discrete mathematics. A tribute to Jiří Matoušek. Cham: Springer. 543–570 (2017; Zbl 1380.05205)], 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 \(|\operatorname{Aut}(H)|\).
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 G. Kalai [Isr. J. Math. 45, 337–351 (1983; Zbl 0535.57011)] showed that the expected order of the torsion group is exponentially large in \(n^2\). We give experimental evidence that in this model also, the torsion is Cohen-Lenstra distributed in the limit.

MSC:

05E45 Combinatorial aspects of simplicial complexes
05C80 Random graphs (graph-theoretic aspects)
60C05 Combinatorial probability
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