Random permutations and the Heisenberg model

We discuss probabilistic representations of certain quantum spin systems, including the ferromagnetic Heisenberg model, in terms of random permutations. The cycle structure of the random permutations is connected with the correlation structure in the spin-system, and it is expected that this cycle structure converges to a distribution known as Poisson--Dirichlet, in the limit of large systems. This problem is still open but we present some partial progress.