Abstract
We study the surface tension and the phenomenon of phase coexistence for the Ising model on \({\mathbb{Z}^d\,(d\, \geqslant\, 2)}\) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations: upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value τ q of surface tension to maximal flows (first passage times if d = 2). For a broad class of distributions of the couplings we show that the inequality \({\tau^a\, \leqslant\, \tau^q}\) –where τ a is the surface tension under the averaged Gibbs measure – is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models.
Similar content being viewed by others
References
Aizenman M., Chayes J.T., Chayes L., Newman C.M.: The phase boundary in dilute and random Ising and Potts ferromagnets. J. Phys. A 20(5), L313–L318 (1987)
Aizenman M., Chayes J.T., Chayes L., Newman C.M.: Discontinuity of the magnetization in one-dimensional 1/| x−y|2 Ising and Potts models. J. Stat. Phys. 50(1-2), 1–40 (1988)
Ambrosio L., Braides A.: Functionals defined on partitions in sets of finite perimeter. II. Semicontinuity, relaxation and homogenization. J. Math. Pures Appl. (9) 69(3), 307–333 (1990)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. New York: The Clarendon Press oxford University Press, 2000
Bodineau T.: The Wulff construction in three and more dimensions. Commun. Math. Phys. 207(1), 197–229 (1999)
Bodineau T., Ioffe D., Velenik Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41(3), 1033–1098 (2000)
Bollobás, B.: Graph theory. Volume 63 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1979
Carmona P., Guerra F., Hu Y., Menjane O.: Strong disorder for a certain class of directed polymers in a random environment. J. Theoret. Probab. 19(1), 134–151 (2006)
Cerf, R.: Large deviations for three dimensional supercritical percolation. Astérisque 267, vi+177 (2000)
Cerf, R.: The Wulff crystal in Ising and percolation models. Volume 1878 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 2006
Cerf R., Pisztora A.: On the Wulff crystal in the Ising model. Ann. Probab. 28(3), 947–1017 (2000)
Chafaï, D.: Inégalités de Poincaré et de Gross pour les mesures de Bernoulli, de Poisson, et de Gauss. Unpublished, available at http://hal.archives-ouvertes.fr/ccsd-00012428, 2005
Chayes J.T., Chayes L.: Bulk transport properties and exponent inequalities for random resistor and flow networks. Commun. Math. Phys. 105(1), 133–152 (1986)
Chayes J.T., Chayes L., Fröhlich J.: The low-temperature behavior of disordered magnets. Commun. Math. Phys. 100(3), 399–437 (1985)
Chayes L., Machta J., Redner O.: Graphical representations for Ising systems in external fields. J. Stat. Phys. 93(1-2), 17–32 (1998)
Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. In: Stochastic analysis on large scale interacting systems, Volume 39 of Adv. Stud. Pure Math., Tokyo: Math. Soc. Japan, 2004, pp. 115–142
Comets F., Vargas V.: Majorizing multiplicative cascades for directed polymers in random media. ALEA Lat. Am. J. Probab. Math. Stat. 2, 267–277 (2006) (electronic)
Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Volume 38 of Applications of Mathematics (New York), second edition, New York: Springer-Verlag, 1998
Deuschel J.-D., Pisztora A.: Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104(4), 467–482 (1996)
Dobrushin, R., Kotecký, R., Shlosman, S.: Wulff construction: A global shape from local interaction, Volume 104 of Translations of Mathematical Monographs. Providence, RI: Amer. Math. Soc., 1992
Durrett R., Liggett T.M.: The shape of the limit set in Richardson’s growth model. Ann. Probab. 9(2), 186–193 (1981)
Edwards R.G., Sokal A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38(6), 2009–2012 (1988)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. New York: Springer-Verlag, 1969
Fonseca I.: The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432(1884), 125–145 (1991)
Fonseca I., Müller S.: A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119(1-2), 125–136 (1991)
Grimmett G., Marstrand J.M.: The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430(1879), 439–457 (1990)
Huse D.A., Henley C.L.H.: Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Let. 54(25), 2708–2711 (1985)
Ioffe D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74(1-2), 411–432 (1994)
Ioffe D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Related Fields 102(3), 313–330 (1995)
Ioffe D., Schonmann R.H.: Dobrushin-Kotecký-Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199(1), 117–167 (1998)
Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)
Kesten, H.: Aspects of first passage percolation. In: École d’été de probabilités de Saint-Flour, XIV—1984, Volume 1180 of Lecture Notes in Math., Berlin: Springer, 1986, pp. 125–264
Kesten H.: Surfaces with minimal random weights and maximal flows: a higher-dimensional version of first-passage percolation. Illinois J. Math. 31(1), 99–166 (1987)
Kingman J.F.C.: The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30, 499–510 (1968)
Ledoux, M.: The concentration of measure phenomenon. Volume 89 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 2001
Messager A., Miracle-Solé S., Ruiz J.: Convexity properties of the surface tension and equilibrium crystals. J. Stat. Phys. 67(3-4), 449–470 (1992)
Pfister C.-E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64(7), 953–1054 (1991)
Pisztora A.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104(4), 427–466 (1996)
Rossignol, R., Théret, M.: Lower large deviations for maximal flows through a box in first passage percolation. http://arXiv.org/abs/0801.0967v1[math.PR], 2008
Taylor J.E.: Crystalline variational problems. Bull. Amer. Math. Soc. 84(4), 568–588 (1978)
Théret M.: On the small maximal flows in first passage percolation. Annales de la faculté des sciences de Toulouse 17(1), 207–219 (2008)
Théret M.: Upper large deviations for the maximal flow in first passage percolation. Stoch. Proc. Appl. 117(9), 1208–1233 (2007)
Wouts, M.: Glauber dynamics in the dilute Ising model below T c . In preparation
Wouts M.: A coarse graining for the Fortuin-Kasteleyn measure in random media. Stoch. Procs. Appl. 118(11), 1929–1972 (2008)
Wouts, M.: The dilute Ising model : phase coexistence at equilibrium & dynamics in the region of phase transition. Ph.D. thesis, Université Paris 7 - Paris Diderot, available at http://tel.archives-ouvertes.fr/tel-00272899, 2007
Zhang Y.: Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys. 98(3-4), 799–811 (2000)
Zhang, Y.: Limit theorems for maximum flows on a lattice. http://arXiv.org/abs/0710.4589[math.PR], 2007
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Toninelli
Rights and permissions
About this article
Cite this article
Wouts, M. Surface Tension in the Dilute Ising Model. The Wulff Construction. Commun. Math. Phys. 289, 157–204 (2009). https://doi.org/10.1007/s00220-009-0782-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0782-8