  We show that in any dimension $d\ge1$, 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 $\mathsf{PD(1)}$, as the size of the
system grows to infinity. In the case of transient dimensions, $d\ge 3$, 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.
