The Annals of Probability

Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model

Olle Häggström, Johan Jonasson, and Russell Lyons

Full-text: Open access

Abstract

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q \geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value and examples of planar regular graphs with regular dual where $p_ \mathrm{c}^{\mathrm{free}} (q) > p_ \mathrm{u}^{\mathrm{wired}} (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove nonrobust phase transition for the Potts model on nonamenable regular graphs.

Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 443-473.

Dates
First available in Project Euclid: 29 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1020107775

Digital Object Identifier
doi:10.1214/aop/1020107775

Mathematical Reviews number (MathSciNet)
MR1894115

Zentralblatt MATH identifier
1025.60044

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Keywords
Percolation Ising model Potts medel planar graph planar dual nonamenable graph robust phase transition

Citation

Häggström, Olle; Jonasson, Johan; Lyons, Russell. Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model. Ann. Probab. 30 (2002), no. 1, 443--473. doi:10.1214/aop/1020107775. https://projecteuclid.org/euclid.aop/1020107775


Export citation

References

  • [1] 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.
  • [2] BABSON, E. and BENJAMINI, I. (1999). Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc. 127 589-597.
  • [3] BAUES, O. and PEYERIMHOFF, N. (2001). Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25 141-159.
  • [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.
  • [6] BENJAMINI, I., LYONS, R., PERES, Y. and SCHRAMM, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29-66.
  • [7] BENJAMINI, I., LYONS, R., PERES, Y. and SCHRAMM, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347-1356.
  • [8] BISKUP, M., BORGS, C., CHAYES, J. T. and KOTECKÝ, R. (2000). Gibbs states of graphical representations in the Potts model with external fields. J. Math. Phys. 41 1170-1210.
  • [9] BLEHER, P.M., RUIZ, J. and ZAGREBNOV, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473-482.
  • [10] BORGS, C. and CHAYES, J. T. (1996). The covariance matrix of the Potts model: a random cluster analysis. J. Statist. Phys. 82 1235-1297.
  • [11] BURTON, R. M. and KEANE, M. S. (1989) Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505.
  • [12] CHABOUD, T. and KENYON, C. (1986). Planar Cayley graphs with regular dual. Internat. J. Algebra Comput. 6 553-561.
  • [13] CHAYES, J. T., CHAYES, L. and SCHONMANN, R. H. (1987). Exponential decay of connectivities in the two-dimensional Ising model. J. Statist. Phys. 49 433-445.
  • [14] CHAYES, J. T., CHAYES, L., SETHNA, J. P. and THOULESS, D. J. (1986). A mean field spin glass with short-range interactions. Comm. Math. Phys. 106 41-89.
  • [15] 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.
  • [16] VAN ENTER, A. (2000). A remark on the notion of robust phase transitions. J. Statist. Phys. 98 1409-1417.
  • [17] EVANS, W., KENYON, C., PERES, Y. and SCHULMAN, L. (2000). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410-433.
  • [18] FLOYD, W. J. and PLOTNICK, S. P. (1987). Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88 1-29.
  • [19] FORTUIN, C. M. and KASTELEYN, P. W. (1972). On the random-cluster model I. Physica 57 536-564.
  • [20] GANDOLFI, A., KEANE, M. S. and NEWMAN, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 511-527.
  • [21] GEORGII, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
  • [22] GEORGII, H.-O., HÄGGSTRÖM, O. and MAES, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena (C. Domb and J. L. Lebowitz, eds.) 1-142. Academic, London.
  • [23] GRIGORCHUK, R. and DE LA HARPE, P. (1997). On problems related to growth, entropy, and spectrum in group theory. J. Dynam. Control Systems 3 51-89.
  • [24] GRIMMETT, G. R. (1995). The stochastic random-cluster process, and the uniqueness of random-cluster measures. Ann. Probab. 23 1461-1510.
  • [25] GRIMMETT, G. R. (1995). Comparison and disjoint-occurrence inequalities for random-cluster models. J. Statist. Phys. 78 1311-1324.
  • [26] GRIMMETT, G. R. and NEWMAN, C. M. (1990). Percolation in +1 dimensions. In Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.) 167-190. Clarendon, Oxford.
  • [27] HÄGGSTRÖM, O. (1996). The random-cluster model on a homogeneous tree. Probab. Theory Related Fields 104 231-253.
  • [28] HÄGGSTRÖM, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423-1436.
  • [29] HÄGGSTRÖM, O. (1998). Random-cluster representations in the study of phase transitions. Markov Process. Related. Fields 4 275-321.
  • [30] HÄGGSTRÖM, O., JONASSON, J. and LYONS, R. (2001). Coupling and Bernoullicity in random-cluster and Potts models. Bernoulli. To appear.
  • [31] JONASSON, J. (1999). The random cluster model on a general graph and a phase transition characterization of nonamenability. Stochastic Process Appl. 79 335-354.
  • [32] JONASSON, J. and STEIF, J. E. (1999). Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 549-559.
  • [33] KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • [34] LIGGETT, T. M., SCHONMANN, R. H. and STACEY, A. M. (1997). Domination by product measures. Ann. Probab. 25 71-95.
  • [35] LYONS, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099-1126. (Version with correct proof of tail triviality of random-cluster measures available at front.math.ucdavis.edu/math.PR/9908177.)
  • [36] LYONS, R. and PERES, Y. (2003). Probability on Trees and Networks. Cambridge Univ. Press. To appear.
  • [37] LYONS, R. and SCHRAMM, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809-1836.
  • [38] NEWMAN, C. M. and SCHULMAN, L. S. (1981). Infinite clusters in percolation models. J. Statist. Phys. 26 613-628.
  • [39] PEMANTLE, R. and STEIF, J. E. (1999). Robust phase transition for Heisenberg and other models on general trees. Ann. Probab. 27 876-912.
  • [40] 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.
  • [41] SCHONMANN, R. H. (2001). Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 271-322.
  • [42] SERIES, C. M. and SINA I, Y. G. (1990). Ising models on the Lobachevsky plane. Comm. Math. Phys. 128 63-76.
  • [43] SWENDSEN, R. H. and WANG, J.-S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58 86-88.
  • [44] TROFIMOV, V. I. (1985). Automorphism groups of graphs as topological groups. Math. Notes 38 717-720.
  • [45] WELSH, D. J. A. (1993). Percolation in the random cluster process and the Q-state Potts model. J. Phys. A 26 2471-2483.
  • [46] WU, C. C. (2000). Ising models on hyperbolic graphs II. J. Statist Phys. 100 893-904.
  • BLOOMINGTON, INDIANA 47405 E-MAIL: rdlyons@indiana.edu