The Annals of Applied Probability

Sharp thresholds for the random-cluster and Ising models

Benjamin Graham and Geoffrey Grimmett

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Abstract

A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point psd(q)=√q∕(1+√q), the Ising model with external field, and the colored random-cluster model. The principal technique is an extension of the influence theorem for monotonic probability measures applied to increasing events with no assumption of symmetry.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 1 (2011), 240-265.

Dates
First available in Project Euclid: 17 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1292598033

Digital Object Identifier
doi:10.1214/10-AAP693

Mathematical Reviews number (MathSciNet)
MR2759201

Zentralblatt MATH identifier
1223.60081

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 60E15: Inequalities; stochastic orderings

Keywords
Random-cluster model Potts model Ising model percolation box-crossing influence sharp threshold colored random-cluster model fuzzy Potts model

Citation

Graham, Benjamin; Grimmett, Geoffrey. Sharp thresholds for the random-cluster and Ising models. Ann. Appl. Probab. 21 (2011), no. 1, 240--265. doi:10.1214/10-AAP693. https://projecteuclid.org/euclid.aoap/1292598033


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References

  • [1] Aizenman, M., Barsky, D. J. and Fernández, R. (1987). The phase transition in a general class of Ising-type models is sharp. J. Statist. Phys. 47 343–374.
  • [2] Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional 1∕|xy|2 Ising and Potts models. J. Statist. Phys. 50 1–40.
  • [3] Aizenman, M. and Fernández, R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. J. Statist. Phys. 44 393–454.
  • [4] Alexander, K. S. (1998). On weak mixing in lattice models. Probab. Theory Related Fields 110 441–471.
  • [5] Bálint, A., Camia, F. and Meester, R. (2009). Sharp phase transition and critical behaviour in 2D divide and colour models. Stochastic Process. Appl. 119 937–965.
  • [6] Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics. Academic Press, London.
  • [7] van den Berg, J. (2008). Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation. Ann. Probab. 36 1880–1903.
  • [8] Bezuidenhout, C. E., Grimmett, G. R. and Kesten, H. (1993). Strict inequality for critical values of Potts models and random-cluster processes. Comm. Math. Phys. 158 1–16.
  • [9] Bollobás, B. and Riordan, O. (2006). The critical probability for random Voronoi percolation in the plane is 1∕2. Probab. Theory Related Fields 136 417–468.
  • [10] Bollobás, B. and Riordan, O. (2006). A short proof of the Harris–Kesten theorem. Bull. London Math. Soc. 38 470–484.
  • [11] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y. and Linial, N. (1992). The influence of variables in product spaces. Israel J. Math. 77 55–64.
  • [12] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  • [13] Chayes, L. (1996). Percolation and ferromagnetism on Z2: The q-state Potts cases. Stochastic Process. Appl. 65 209–216.
  • [14] Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 2009–2012.
  • [15] Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993–3002.
  • [16] Graham, B. T. and Grimmett, G. R. (2006). Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 1726–1745.
  • [17] Grimmett, G. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 1461–1510.
  • [18] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [19] Grimmett, G. (2009). The Random-Cluster Model. Springer, Berlin. Corrected reprint available at http://www.statslab.cam.ac.uk/~grg/books/rcm.html.
  • [20] Grimmett, G. (2010). Probability on Graphs. Cambridge Univ. Press, Cambridge. Available at http://www.statslab.cam.ac.uk/~grg/books/pgs.html.
  • [21] Häggström, O. (1999). Positive correlations in the fuzzy Potts model. Ann. Appl. Probab. 9 1149–1159.
  • [22] Häggström, O. (2001). Coloring percolation clusters at random. Stochastic Process. Appl. 96 213–242.
  • [23] Higuchi, Y. (1993). Coexistence of infinite (∗)-clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 1–33.
  • [24] Higuchi, Y. (1993). A sharp transition for the two-dimensional Ising percolation. Probab. Theory Related Fields 97 489–514.
  • [25] Kahn, J., Kalai, G. and Linial, N. (1988). The influence of variables on Boolean functions. In Proceedings of 29th Symposium on the Foundations of Computer Science 68–80. Computer Science Press, Vashington, DC.
  • [26] Kahn, J. and Weininger, N. (2007). Positive association in the fractional fuzzy Potts model. Ann. Probab. 35 2038–2043.
  • [27] Kalai, G. and Safra, S. (2006). Threshold phenomena and influence: Perspectives from mathematics, computer science, and economics. In Computational Complexity and Statistical Physics. St. Fe Inst. Stud. Sci. Complex. 25–60. Oxford Univ. Press, New York.
  • [28] Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals ½. Comm. Math. Phys. 74 41–59.
  • [29] Kotecký, R. and Shlosman, S. B. (1982). First-order phase transitions in large entropy lattice models. Comm. Math. Phys. 83 493–515.
  • [30] Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J. and Shlosman, S. (1991). Interfaces in the Potts model. I. Pirogov–Sinai theory of the Fortuin–Kasteleyn representation. Comm. Math. Phys. 140 81–91.
  • [31] Laanait, L., Messager, A. and Ruiz, J. (1986). Phases coexistence and surface tensions for the Potts model. Comm. Math. Phys. 105 527–545.
  • [32] Russo, L. (1978). A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39–48.
  • [33] Russo, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229–237.
  • [34] Seymour, P. D. and Welsh, D. J. A. (1978). Percolation probabilities on the square lattice. Ann. Discrete Math. 3 227–245.
  • [35] Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians II 1421–1451. Eur. Math. Soc., Zürich.
  • [36] Smirnov, S. (2010). Conformal invariance in random-cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172 1441–1473.
  • [37] Talagrand, M. (1994). On Russo’s approximate zero-one law. Ann. Probab. 22 1576–1587.
  • [38] Welsh, D. J. A. (1993). Percolation in the random cluster process and Q-state Potts model. J. Phys. A 26 2471–2483.