Split-and-Merge in Stationary Random Stirring on Lattice Torus

We show that in any dimension d≥1, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson–Dirichlet law PD(1), as the size of the system grows to infinity. In the case of transient dimensions, d≥3, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.