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We show that in any dimension $d\ge1$, the cycle-length process of stationaryrandom stirring (or, random interchange) on the lattice torus converges to thecanonical Markovian split-and-merge process with the invariant (and reversible)measure given by the Poisson-Dirichlet law $\mathsf{PD(1)}$, as the size of thesystem grows to infinity. In the case of transient dimensions, $d\ge 3$, theproblem is motivated by attempts to understand the onset of long range order inquantum Heisenberg models via random loop representations of the latter.

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