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

TL;DR

This paper proves that the cycle-length process in stationary random stirring on a lattice torus converges to a split-and-merge process governed by the Poisson-Dirichlet law as the system size increases, with implications for quantum Heisenberg models.

Contribution

It establishes the convergence of the cycle-length process to a canonical split-and-merge process with Poisson-Dirichlet invariant measure in any dimension.

Findings

Cycle-length process converges to split-and-merge process

Invariant measure is Poisson-Dirichlet law PD(1)

Results apply to dimensions d≥1 and relate to quantum Heisenberg models

Abstract

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.