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

Abstract

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