Cohen–Lenstra conjecture for torsion in random 2-trees

Let $\mathcal{T}^2(n)$ be the set of $2$-trees on $n$ vertices, and let $X$ be drawn uniformly from $\mathcal{T}^2(n)$. For a fixed prime $q$, let the Sylow $q$-subgroup of $H_1(X)$ be the subgroup consisting of elements of $q$-power order. Cohen–Lenstra conjecture for random 2-trees. For a fixed prime $q$, the Sylow $q$-subgroup of $H_1(X)$ is asymptotically distributed according to the Cohen–Lenstra distribution, which assigns probability

$$\frac{\prod_{k=1}^{\infty}(1-q^{-k})}{|\operatorname{Aut}(G)|}$$

to any finite abelian $q$-group $G$. The conjecture extends the proposed distribution for the torsion burst in the Linial–Meshulam model; the paper's evidence is experimental, and rapid mixing of the sampling Markov chain is noted as an open issue.