The Annals of Probability

On the 2D Ising Wulff crystal near criticality

R. Cerf and R. J. Messikh

Full-text: Open access


We study the behavior of the two-dimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal.

Article information

Ann. Probab., Volume 38, Number 1 (2010), 102-149.

First available in Project Euclid: 25 January 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Large deviations criticality phase coexistence Wulff shape Ising model


Cerf, R.; Messikh, R. J. On the 2D Ising Wulff crystal near criticality. Ann. Probab. 38 (2010), no. 1, 102--149. doi:10.1214/08-AOP449.

Export citation


  • [1] Alexander, K. S. (1998). On weak mixing in lattice models. Probab. Theory Related Fields 110 441–471.
  • [2] Alexander, K. S. (2004). Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32 441–487.
  • [3] Alexander, K. S. (2001). Cube-root boundary fluctuations for droplets in random cluster models. Comm. Math. Phys. 224 733–781.
  • [4] Alexander, K., Chayes, J. T. and Chayes, L. (1990). The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys. 131 1–50.
  • [5] Barouch, E., Mc Coy, B. M., Tracy, C. A. and Wu, T. T. (1976). Spin–spin correlation functions for the two dimensional Ising model: Exact theory in the scaling region. Phys. Rev. B 13 316–375.
  • [6] Bodineau, T. (1999). The Wulff construction in three and more dimensions. Comm. Math. Phys. 207 197–229.
  • [7] Bodineau, T. (2005). Slab percolation for the Ising model. Probab. Theory Related Fields 132 83–118.
  • [8] Bricmont, J., Lebowitz, J. L. and Pfister, C. E. (1981). On the local structure of the phase separation line in the two-dimensional Ising system. J. Stat. Phys. 26 313–332.
  • [9] Cerf, R. (2000). Large deviations for three dimensional supercritical percolation. Astérisque 267 vi+177.
  • [10] Cerf, R. (2004). The Wulff Crystal in Ising and Percolation Models. Ecole d’été de probabilités, Saint Flour.
  • [11] Cerf, R. and Messikh, R. J. (2006). The 2d-Ising Model Near Criticality: A FK-percolation Analysis.
  • [12] Cerf, R. and Pisztora, Á. (2000). On the Wulff crystal in the Ising model. Ann. Probab. 28 947–1017.
  • [13] Cerf, R. and Pisztora, Á. (2001). Phase coexistence in Ising, Potts and percolation models. Ann. Inst. H. Poincaré Probab. Statist. 37 643–724.
  • [14] Cheng, H. and Wu, T. T. (1967). Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. III. Phys. Rev. 164 719–735.
  • [15] Couronné, O. and Messikh, R. J. (2004). Surface order large deviations for 2D FK-percolation and Potts models. Stochastic Process. Appl. 113 81–99.
  • [16] De Giorgi, E. (1958). Sulla proprietà isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 33–44.
  • [17] Deuschel, J.-D. and Pisztora, Á. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467–482.
  • [18] Dobrushin, R., Kotecký, R. and Shlosman, S. (1992). Wulff Construction. Translations of Mathematical Monographs 104. Amer. Math. Soc., Providence, RI.
  • [19] 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.
  • [20] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
  • [21] Grimmett, G. (1997). Percolation and disordered systems. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1665 153–300. Springer, Berlin.
  • [22] Grimmett, G. (1994). The random-cluster model. In Probability, Statistics and Optimisation 49–63. Wiley, Chichester.
  • [23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [24] Ioffe, D. (1994). Large deviations for the 2D Ising model: A lower bound without cluster expansions. J. Stat. Phys. 74 411–432.
  • [25] Ioffe, D. (1995). Exact large deviation bounds up to Tc for the Ising model in two dimensions. Probab. Theory Related Fields 102 313–330.
  • [26] Ioffe, D. and Schonmann, R. H. (1998). Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199 117–167.
  • [27] Kasteleyn, P. W. (1967). Graph theory and crystal physics. In Graph Theory and Theoretical Physics 43–110. Academic Press, London.
  • [28] Laanait, L., Messager, A. and Ruiz, J. (1986). Phases coexistence and surface tensions for the Potts model. Comm. Math. Phys. 105 527–545.
  • [29] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71–95.
  • [30] Mc Coy, B. M. and Wu, T. T. (1973). The Two Dimensional Ising Model. Harvard Univ. Press, Cambridge, MA.
  • [31] Messikh, R. J. (2009). On the surface tension of the 2d Ising model near criticality. Preprint.
  • [32] Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65 117–149.
  • [33] Pfister, C.-E. (1991). Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64 953–1054.
  • [34] Pfister, C.-E. and Velenik, Y. (1997). Large deviations and continuum limit in the 2D Ising model. Probab. Theory Related Fields 109 435–506.
  • [35] Pisztora, Á. (1996). Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427–466.
  • [36] Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120.
  • [37] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
  • [38] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.