• Bernoulli
  • Volume 8, Number 3 (2002), 275-294.

Coupling and Bernoullicity in random-cluster and Potts models

Olle Häggström, Johan Jonasson, and Russell Lyons

Full-text: Open access


An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity.

Article information

Bernoulli, Volume 8, Number 3 (2002), 275-294.

First available in Project Euclid: 8 March 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Cayley graph coupling from the past stochastic domination transitive unimodular


Häggström, Olle; Jonasson, Johan; Lyons, Russell. Coupling and Bernoullicity in random-cluster and Potts models. Bernoulli 8 (2002), no. 3, 275--294.

Export citation


  • [1] Adams, S. (1992) Følner independence and the amenable Ising model. Ergodic Theory Dynam. Systems, 12, 633-657.
  • [2] Aizenman, M., Chayes, J.T., Chayes, L. and Newman, C.M. (1988) Discontinuity of the magnetization in one-dimensional 1/(x-y)^2 Ising and Potts models. J. Statist. Phys., 50, 1-40.
  • [3] Alexander, K. (1995) Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phys., 168, 39-55.
  • [4] Benjamini, I. and Schramm, O. (1996) Percolation beyond Zd, many questions and a few answers. Electron Comm. Probab., 1, 71-82.
  • [5] Benjamini, I. and Schramm, O. (2001) Percolation in the hyperbolic plane. J. Amer. Math. Soc., 14, 487-507. Abstract can also be found in the ISI/STMA publication
  • [6] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999) Group-invariant percolation on graphs. Geom. Funct. Anal., 9, 29-66.
  • [7] Bezuidenhout, C., Grimmett, G. and Kesten, H. (1993) Strict inequality for critical values of Potts models and random-cluster processes. Comm. Math. Phys., 158, 1-16.
  • [8] Chayes, J.T., Chayes, L., Sethna, J.P. and Thouless, D.J. (1986) A mean field spin glass with shortrange interactions. Comm. Math. Phys., 106, 41-89.
  • [9] di Liberto, F., Gallavotti, G. and Russo, L. (1973) Markov processes, Bernoulli schemes, and Ising model. Comm. Math. Phys., 33, 259-282.
  • [10] Diaconis, P. and Freedman, D. (1999) Iterated random functions. SIAM Rev., 41, 45-76. Abstract can also be found in the ISI/STMA publication
  • [11] Edwards, R.G. and Sokal, A.D. (1988) Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D, 38, 2009-2012.
  • [12] Fortuin, C.M. (1972). On the random-cluster model. III The simple random-cluster model. Physica, 59, 545-570.
  • [13] Fortuin, C.M. and Kasteleyn, P.W. (1972) On the random-cluster model. I. Introduction and relation to other models. Physica, 57, 536-564.
  • [14] Georgii, H.-O., Häggström, O. and Maes, C. (2001) The random geometry of equilibrium phases. In C. Domb and J.L. Lebowitz (eds), Phase Transitions and Critical Phenomena, Volume 14, pp. 1-142. London: Academic Press.
  • [15] Grimmett, G.R. (1995) The stochastic random-cluster process, and the uniqueness of random-cluster measures. Ann. Probab., 23, 1461-1510.
  • [16] Häggström, O. (1996) The random-cluster model on a homogeneous tree. Probab. Theory Related Fields, 104, 231-253.
  • [17] Häggström, O. and Peres, Y. (1999) Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Theory Related Fields, 113, 273-285.
  • [18] Häggström, O. and Steif, J.E. (2000) Propp-Wilson algorithms and finitary codings for high noise Markov random fields. Combin. Probab. Comput. 9, 425-439.
  • [19] Häggström, O., Peres, Y. and Schonmann, R. (1999) Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness. In M. Bramson and R. Durrett (eds), Perplexing Probability Problems: Papers in Honor of Harry Kesten, pp. 69-90. Boston. Birkhäuser.
  • [20] Häggström, O., Schonmann, R.H. and Steif, J.E. (2000) The Ising model on diluted graphs and strong amenability. Ann. Probab., 28, 1111-1137.
  • [21] Häggström, O., Jonasson, J. and Lyons, R. (2001) Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. To appear.
  • [22] Higuchi, Y. (1991) Level set representation for the Gibbs states of the ferromagnetic Ising model. Probab. Theory Related Fields, 90, 203-221. Abstract can also be found in the ISI/STMA publication
  • [23] Lindvall, T. (1992) Lectures on the Coupling Method. New York: Wiley.
  • [24] Lyons, R. (2000) Phase transitions on non-amenable graphs. J. Math. Phys., 41, 1099-1126.
  • [25] Newman, C.M. and Schulman, L.S. (1981) Infinite clusters in percolation models. J. Statist. Phys., 26, 613-628.
  • [26] Ornstein, D.S. and Weiss, B. (1973) Zd-actions and the Ising model. Unpublished manuscript.
  • [27] Ornstein, D.S. and Weiss, B. (1987) Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math., 48, 1-141.
  • [28] Propp, J.G. and Wilson, D.B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms, 9, 223-252.
  • [29] Salvatori, M. (1992) On the norms of group-invariant transition operators on graphs. J. Theoret. Probab., 5, 563-576. Abstract can also be found in the ISI/STMA publication
  • [30] Schonmann, R.H. (1999) Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields, 113, 287-300. Abstract can also be found in the ISI/STMA publication
  • [31] Soardi, P.M. and Woess, W. (1990) Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z., 205, 471-486.
  • [32] Steif, J.E. (1991) d-convergence to equilibrium and space-time Bernoullicity for spin systems in the M, E case. Ergodic Theory Dynam. Systems, 11, 547-575.
  • [33] Swendsen, R.H. and Wang, J.-S. (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett., 7-575.
  • [34] Thorisson, H. (1988) Backward limits. Ann. Probab., 16, 914-924. Abstract can also be found in the ISI/STMA publication
  • [35] van den Berg, J. and Steif, J.E. (1999) On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab., 27, 1501-1522. Abstract can also be found in the ISI/STMA publication