Electronic Communications in Probability

Large cycles in random permutations related to the Heisenberg model

Jakob Björnberg

Full-text: Open access


We study the weighted version of the interchange process where a permutation receives weight $\theta^{\#\mathrm{cycles}}$.  For $\theta=2$ this is Tóth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $\theta>1$, that large cycles appear at "low temperature".

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 55, 11 pp.

Accepted: 27 July 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Interchange process Heisenberg model

This work is licensed under a Creative Commons Attribution 3.0 License.


Björnberg, Jakob. Large cycles in random permutations related to the Heisenberg model. Electron. Commun. Probab. 20 (2015), paper no. 55, 11 pp. doi:10.1214/ECP.v20-4328. https://projecteuclid.org/euclid.ecp/1465320982

Export citation


  • Alon, Gil; Kozma, Gady. The probability of long cycles in interchange processes. Duke Math. J. 162 (2013), no. 9, 1567–1585.
  • Angel, Omer. Random infinite permutations and the cyclic time random walk. Discrete random walks (Paris, 2003), 9–16 (electronic), Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.
  • Berestycki, Nathanaël. Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles. Electron. J. Probab. 16 (2011), no. 5, 152–173.
  • Berestycki, Nathanaël; Durrett, Rick. A phase transition in the random transposition random walk. Probab. Theory Related Fields 136 (2006), no. 2, 203–233.
  • J. E. Björnberg and D. Ueltschi, Decay of transverse correlations in quantum heisenberg models, Journal of Mathematical Physics 56 (2015), no. 4.
  • Bollobás, B.; Grimmett, G.; Janson, S. The random-cluster model on the complete graph. Probab. Theory Related Fields 104 (1996), no. 3, 283–317.
  • Goldschmidt, Christina; Ueltschi, Daniel; Windridge, Peter. Quantum Heisenberg models and their probabilistic representations. Entropy and the quantum II, 177–224, Contemp. Math., 552, Amer. Math. Soc., Providence, RI, 2011.
  • Hammond, Alan. Infinite cycles in the random stirring model on trees. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), no. 1, 85–104.
  • Hammond, Alan. Sharp phase transition in the random stirring model on trees. Probab. Theory Related Fields 161 (2015), no. 3-4, 429–448.
  • Penrose, O. Bose-Einstein condensation in an exactly soluble system of interacting particles. J. Statist. Phys. 63 (1991), no. 3-4, 761–781.
  • Schramm, Oded. Compositions of random transpositions [ ]. Selected works of Oded Schramm. Volume 1, 2, 571–593, Sel. Works Probab. Stat., Springer, New York, 2011.
  • Tóth, Bálint. Phase transition in an interacting Bose system. An application of the theory of Ventselʹ and FreÄ­dlin. J. Statist. Phys. 61 (1990), no. 3-4, 749–764.
  • Tóth, Bálint. Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet. Lett. Math. Phys. 28 (1993), no. 1, 75–84.