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

Dmitri Ioffe, Balint A Toth*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
33 Downloads (Pure)


We show that in any dimension 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 PD(1), as the size of the system grows to infinity. In the case of transient dimensions, 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.
Original languageEnglish
Pages (from-to)630–653
Number of pages24
JournalJournal of Statistical Physics
Publication statusPublished - 1 Feb 2020


  • random interchange process
  • split-and-merge
  • Poisson-Dirichlet distribution
  • quantum Heisenberg model


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