Cohen--Lenstra heuristics for torsion in homology of random complexes

2017/10/16 by Kahle, Matthew, Lutz, Frank, Newman, Andrew +1 · 3 citations
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.05683

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 \citeAL,LP,LP3 , 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 \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 \citeKalai. We give experimental evidence that in this model also, the torsion is Cohen--Lenstra distributed in the limit.

Cited by

• Using dense graph limit theory to count cocycles of random simplicial complexes
• The fluctuations of the mod p rank of triangular matrices
• The homology torsion growth of determinantal hypertrees

Related

• On the Vanishing of Homology in Random Čech Complexes
• Homomorphisms from the torus
• On torsion in eulerian magnitude homology of Erdos-Renyi random graphs
• Torsion primes in loop space homology
• Homological connectivity of random k-dimensional complexes