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

Abstract

We show that in any dimension $d\ge 1$, 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 $\mathsf{PD}(1)$, 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.

We dedicate this paper to Joel Lebowitz on the occasion of his 90th birthday with deep respect for his scientific and moral accomplishment