The Annals of Probability

On the Wulff crystal in the Ising model

Raphaël Cerf and Ágoston Pisztora

Full-text: Open access


We study the phase separation phenomenon in the Ising model in dimensions $d \geq 3$. To this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization $m^*$. We confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase. Moreover, the rescaled droplet is asymptotically close to a definite deterministic shape, the Wulff crystal, which minimizes the surface free energy. In the course of the proof we establish a surface order large deviation principle for the magnetization. Our results are valid for temperatures $T$ below a limit of slab-thresholds $\hat{T}_c$ conjectured to agree with the critical point $T_c$. Moreover, $T$ should be such that there exist only two extremal translation invariant Gibbs states at that temperature, a property which can fail for at most countably many values and which is conjectured to be true for every $T$. The proofs are based on the Fortuin–Kasteleyn representation of the Ising model along with coarse-graining techniques.To handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis. Finally,we propose a heuristic picture that for subcritical temperatures close enough to $T_c$, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.

Article information

Ann. Probab., Volume 28, Number 3 (2000), 947-1017.

First available in Project Euclid: 18 April 2002

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] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60F10: Large deviations

Phase separation Wulff crystal Ising model large deviations FK


Cerf, Raphaël; Pisztora, Ágoston. On the Wulff crystal in the Ising model. Ann. Probab. 28 (2000), no. 3, 947--1017. doi:10.1214/aop/1019160324.

Export citation


  • [1] Aizenman, M., Bricmont, J. and Lebowitz, J. L. (1987). Percolation of the minority spins in high-dimensional Ising models. J. Statist. Phys. 49 859-865.
  • [2] Aizenman, M., Chayes, J. T., Chayes, L., Fr ¨ohlich, J. and Russo, L. (1983). On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92 19-69.
  • [3] Alberti, G., Bellettini, G., Cassandro, M. and Presutti, E. (1996). Surface tension in Ising systems with Kac potentials. J. Statist. Phys. 82 743-796.
  • [4] Alexander, K. S. (1992). Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation. Probab. Theory Related Fields 91 507-532.
  • [5] Alexander, K. S., 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.
  • [6] Assouad, P. and Quentin de Gromard, T. (1998). Sur la d´erivation des mesures dans n. Note.
  • [7] Bellettini, G., Cassandro, M. and Presutti, E. (1996). Constrained minima of nonlocal free energy functionals. J. Statist. Phys. 84 1337-1349.
  • [8] Benois, O., Bodineau, T., Butt a, P. and Presutti, E. (1997). On the validity of van der Waals theory of surface tension. Markov Process. Related Fields 3 175-198.
  • [9] Benois, O., Bodineau, T. and Presutti, E. (1998). Large deviations in the van der Waals limit. Stochastic Process. Appl. 75 89-104.
  • [10] Besicovitch, A. S. (1946). A general form of the covering principle and relative differentiation of additive functions. Proc. Cambridge Philos. Soc. 41 103-110; II. Proc. Cambridge Philos. Soc. 42 1-10.
  • [11] Bodineau, T. (1999). The Wulff construction in three and more dimensions. Comm. Math. Phys. 207 197-229.
  • [12] Caccioppoli, R. (1952). Misura e integrazione sugli insiemi dimensionalmente orientati I, II. Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. (8) XII 3-11, 137-146.
  • [13] Caccioppoli, R. (1952). Misura e integrazione sulle variet a parametriche I, II, III. Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. (8) XII 219-227, 365-373, 629-634.
  • [14] Cerf, R. (1998). Large deviations for three-dimensional supercritical percolation. Ast´erisque
  • 267 (2000).
  • [15] Cerf, R. and Pisztora, ´A. (2000). Phase coexistence in Ising, Potts and percolation models. Preprint.
  • [16] Cesi, F., Guadagni, G., Martinelli, F. and Schonmann, R. (1996). On the 2D stochastic Ising model in the phase coexistence region near the critical point. J. Statist. Phys. 85 55-102.
  • [17] 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.
  • [18] Comets, F. (1986). Grandes d´eviations pour des champs de Gibbs sur d. C. R. Acad. Sci. S´er. I 303 511-513.
  • [19] De Giorgi, E. (1954). Su una teoria generale della misura r-1 -dimensionale in uno spazio ad r dimensioni. Ann. Mat. Pura Appl. 36 191-213.
  • [20] De Giorgi, E. (1955). Nuovi teoremi relativi alle misure r 1 -dimensionali in uno spazio ad r dimensioni. Ricerche Mat. 4 95-113.
  • [21] De Giorgi, E. (1958). Sulla proprieta isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. (8) 5 33-44.
  • [22] De Giorgi, E., Colombini, F. and Piccinini, L. C. (1972). Frontiere orientate di misura minima e questioni collegate. Scuola Normale Superiore di Pisa.
  • [23] Deuschel, J.-D. and Pisztora, ´A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467-482.
  • [24] Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, New York.
  • [25] Dinghas, A. (1944). ¨Uber einen geometrischen Satz von Wulff f ¨ur die Gleichgewichtsform von Kristallen.Kristallogr. 105 304-314.
  • [26] Dobrushin, R. K. and Hryniv, O. (1997). Fluctuations of the phase boundary in the 2D Ising ferromagnet. Comm. Math. Phys. 189 395-445.
  • [27] Dobrushin, R. L., Koteck´y, R. and Shlosman, S. B. (1992). Wulff construction: a global shape from local interaction. Amer. Math. Soc. Transl. Ser. 2.
  • [28] Dobrushin, R. L. and Shlosman, S. B. (1992). Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction. In Ideas and Methods in Quantum and Statistical Physics (S. Albeverio, ed.) 461-483. Cambridge Univ. Press.
  • [29] 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.
  • [30] Ellis, R. S. (1986). Entropy, Large Deviations and Statistical Mechanics. Springer, New York.
  • [31] Evans, L. C. and Gariepy, R. F. (1992). Measure theory and fine properties of functions. CRC Press, Boca Raton, FL.
  • [32] Falconer, K. J. (1985). The Geometry of Fractal Sets. Cambridge Univ. Press.
  • [33] Federer, H. (1969). Geometric Measure Theory. Springer, New York.
  • [34] F ¨ollmer, H. and Orey, S. (1988). Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16 961-977.
  • [35] F ¨ollmer, H. and Ort, M. (1988). Large deviations and surface entropy for Markov fields. Ast´erisque 157-158, 173-190.
  • [36] Fonseca, I. (1991). The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 125-145.
  • [37] Fonseca, I. and M ¨uller, S. (1991). A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sec. A 119 125-136.
  • [38] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random cluster model. I. Physica 57 536-564.
  • [39] Giusti, E. (1984). Minimal surfaces and functions of bounded variation. Birkh¨auser, Boston.
  • [40] Grimmett, G. R. (1989). Percolation. Springer, New York.
  • [41] Grimmett, G. R. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 1461-1510.
  • [42] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. Lond. Ser. A 430 439-457.
  • [43] Hryniv, O. (1998). On local behaviour of the phase separation line in the 2D Ising model. Probab. Theory Related Fields 110 91-107.
  • [44] Ioffe, D. (1993). Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Statist. Phys. 74 411-432.
  • [45] Ioffe, D. (1995). Exact large deviation bounds up to Tc for the Ising model in two dimensions. Probab. Theory Related Fields 102 313-330.
  • [46] Ioffe, D. and Schonmann, R. (1998). Dobrushin-Koteck´y-Shlosman Theorem up to the critical temperature. Comm. Math. Phys. 199 117-167.
  • [47] Kesten, H. and Zhang, Y. (1990). The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 537-555.
  • [48] Lebowitz, J. L. (1977). Coexistence of phases in Ising ferromagnets. J. Statist. Phys. 16 463-476.
  • [49] Lebowitz, J. L. and Martin-L ¨of, A. (1972). On the uniqueness of the equilibrium state for Ising spin systems. Comm. Math. Phys. 25 276-282.
  • [50] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71-95.
  • [51] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Univ. Press.
  • [52] Massari, U. and Miranda, M. (1994). Minimal Surfaces of Codimension One. North-Holland, Amsterdam.
  • [53] Messager, A., Miracle-Sol´e, S. and Ruiz, J. (1992). Convexity properties of the surface tension and equilibrium crystals. J. Statist. Phys. 67 449-469.
  • [54] Miracle-Sol´e, S. (1995). Surface tension, step free energy, and facets in the equilibrium crystal. J. Statist. Phys. 79 183-214.
  • [55] Newman, C. M. (1993). Disordered Ising systems and random cluster representations. In Probability and Phase Transition (G. Grimmett, ed.) 247-260. Kluwer, Dordrecht.
  • [56] Olla, S. (1988). Large deviations for Gibbs random fields. Probab. Theory Related Fields 77 395-409.
  • [57] Pfister, C. E. (1991). Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64 953-1054.
  • [58] Pfister, C. E. and Velenik, Y. (1997). Large deviations and continuum limit in the 2D Ising model. Probab. Theory Related Fields 109 435-506.
  • [59] Pisztora, ´A. (1996). Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427-466.
  • [60] Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • [61] Schneider, R. (1993). Convex bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
  • [62] Schonmann, R. H. (1987). Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Comm. Math. Phys. 112 409-422.
  • [63] Schonmann, R. and Shlosman, S. B. (1996). Constrained variational problem with applications to the Ising model. J. Statist. Phys. 83 867-905.
  • [64] Schonmann, R. and Shlosman, S. B. (1996). Complete analyticity for the 2D Ising model completed. Comm. Math. Phys. 170 453-482.
  • [65] Schonmann, R. and Shlosman, S. B. (1998). Wulff droplets and the metastable relaxation of kinetic Ising models. Comm. Math. Phys. 194 389-462.
  • [66] Taylor, J. E. (1974). Existence and structure of solutions to a class of nonelliptic variational problems. Sympos. Math. 14 499-508.
  • [67] Taylor, J. E. (1975). Unique structure of solutions to a class of nonelliptic variational problems. Proc. Symp. Pure Math. 27 419-427.
  • [68] Taylor, J. E. (1978). Crystalline variational problems. Bull. Amer. Math. Soc. 84 568-588.
  • [69] Wulff, G. (1901). Zur Frage der Geschwindigkeit des Wachstums und der Aufl¨osung der Kristallfl¨achen.Kristallogr. 34 449-530.
  • [70] Ziemer, W. P. (1989). Weakly differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Springer, New York.
  • CNRS, Universit´e Paris Sud Math´ematique, B atiment 425 91405 Orsay Cedex France Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, Pennsylvania 15213-3890