Phase coexistence in Ising, Potts and percolation models
Annales de l'I.H.P. Probabilités et statistiques, Volume 37 (2001) no. 6, p. 643-724
@article{AIHPB_2001__37_6_643_0,
     author = {Cerf, Rapha\"el and Pisztora, \'Agoston},
     title = {Phase coexistence in Ising, Potts and percolation models},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Elsevier},
     volume = {37},
     number = {6},
     year = {2001},
     pages = {643-724},
     zbl = {1006.60094},
     mrnumber = {1863274},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2001__37_6_643_0}
}
Cerf, Raphaël; Pisztora, Ágoston. Phase coexistence in Ising, Potts and percolation models. Annales de l'I.H.P. Probabilités et statistiques, Volume 37 (2001) no. 6, pp. 643-724. http://www.numdam.org/item/AIHPB_2001__37_6_643_0/

[1] K.S. Alexander, Cube-root boundary fluctuations for droplets in random cluster models, Preprint, 2000. | MR 1871907

[2] F.J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (165) (1976). | MR 420406 | Zbl 0327.49043

[3] G. Alberti, G. Bellettini, M. Cassandro, E. Presutti, Surface tension in Ising systems with Kac potentials, J. Stat. Phys. 82 (1996) 743-796. | MR 1372427 | Zbl 1042.82539

[4] K.S. Alexander, J.T. Chayes, L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation, Comm. Math. Phys. 131 (1990) 1-50. | MR 1062747 | Zbl 0698.60098

[5] L. Ambrosio, A. Braides, Functionals defined on partitions in sets of finite perimeter I: Integral representation and Γ-convergence, J. Math. Pures et Appl. 69 (1990) 285-305. | Zbl 0676.49028

[6] L. Ambrosio, A. Braides, Functionals defined on partitions in sets of finite perimeter II: Semicontinuity, relaxation and homogenization, J. Math. Pures et Appl. 69 (1990) 307-333. | MR 1070482 | Zbl 0676.49029

[7] L. Ambrosio, M. Novaga, E. Paolini, Some regularity results for minimal crystals, Preprint, 2000. | Numdam | MR 1932945

[8] Assouad P., Quentin de Gromard T., Sur la dérivation des mesures dans Rn, Note (1998).

[9] O. Benois, T. Bodineau, P. Buttà, E. Presutti, On the validity of van der Waals theory of surface tension, Markov Process. Rel. Fields 3 (1997) 175-198. | MR 1468173 | Zbl 0910.60014

[10] O. Benois, T. Bodineau, E. Presutti, Large deviations in the van der Waals limit, Stochastic Process. Appl. 75 (1998) 89-104. | MR 1629026 | Zbl 0927.60080

[11] A.S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Philos. Soc. 41 (1945) 103-110, Part II. Proc. Cambridge Philos. Soc. 42 (1946) 1-10. | Zbl 0063.00352

[12] T. Bodineau, The Wulff construction in three and more dimensions, Comm. Math. Phys. 207 (1) (1999) 197-229. | MR 1724851 | Zbl 1015.82005

[13] T. Bodineau, D. Ioffe, Y. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes, J. Math. Phys. 41 (3) (2000) 1033-1098. | MR 1757951 | Zbl 0977.82013

[14] R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque 267 (2000). | MR 1774341 | Zbl 0962.60002

[15] R. Cerf, A. Pisztora, On the Wulff crystal in the Ising model, Ann. Probab. 28 (3) (2000) 945-1015. | MR 1797302 | Zbl 1034.82006

[16] G. Congedo, I. Tamanini, Optimal partitions with unbounded data, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX Ser., Rend. Lincei., Mat. Appl. 4 (2) (1993) 103-108. | MR 1233397 | Zbl 0776.49027

[17] G. Congedo, I. Tamanini, On the existence of solutions to a problem in multidimensional segmentation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8 (2) (1991) 175-195. | Numdam | MR 1096603 | Zbl 0729.49003

[18] G. Congedo, I. Tamanini, Regularity properties of optimal segmentations, J. Reine Angew. Math. 420 (1991) 61-84. | MR 1124566 | Zbl 0729.49004

[19] E. De Giorgi, Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 (1955) 95-113. | Zbl 0066.29903

[20] E. De Giorgi, F. Colombini, L.C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore di Pisa (1972). | MR 493669 | Zbl 0296.49031

[21] J.-D. Deuschel, A. Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Relat. Fields 104 (1996) 467-482. | MR 1384041 | Zbl 0842.60023

[22] R.L. Dobrushin, R. Kotecký, S.B. Shlosman, Wulff Construction: A Global Shape from Local Interaction, AMS Translations Series, Providence, RI, 1992. | MR 1181197 | Zbl 0917.60103

[23] R.L. Dobrushin, S.B. Shlosman, Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction, in: Albeverio S. (Ed.), Ideas and Methods in Quantum and Statistical Physics, Cambridge University Press, 1992, pp. 461-483. | MR 1190540 | Zbl 0786.52006

[24] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. | MR 1158660 | Zbl 0804.28001

[25] K.J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge Univ. Press, 1985. | MR 867284 | Zbl 0587.28004

[26] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. | MR 257325 | Zbl 0874.49001

[27] C.M. Fortuin, P.W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica 57 (1972) 536-564. | MR 359655

[28] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. | MR 775682 | Zbl 0545.49018

[29] G.R. Grimmett, Percolation, Grundlehren der Mathematischen Wissenschaften, 321, Springer-Verlag, Berlin, 1999. | MR 1707339 | Zbl 0926.60004

[30] G.R. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Probab. 23 (1995) 1461-1510. | MR 1379156 | Zbl 0852.60105

[31] G.R. Grimmett, J.M. Marstrand, The supercritical phase of percolation is well behaved, Proc. R. Soc. Lond. Ser. A 430 (1990) 439-457. | MR 1068308 | Zbl 0711.60100

[32] J. Hass, M. Hutchings, R. Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc. 1 (3) (1995) 98-102, (electronic). | MR 1369639 | Zbl 0864.53007

[33] D. Ioffe, Large deviations for the 2D Ising model: a lower bound without cluster expansions, J. Stat. Phys. 74 (1993) 411-432. | MR 1257822 | Zbl 0946.82502

[34] D. Ioffe, Exact large deviation bounds up to Tc for the Ising model in two dimensions, Probab. Theory Relat. Fields 102 (1995) 313-330. | MR 1339736 | Zbl 0830.60018

[35] D. Ioffe, R. Schonmann, Dobrushin-Kotecký-Shlosman Theorem up to the critical temperature, Comm. Math. Phys. 199 (1998) 117-167. | MR 1660207 | Zbl 0929.60076

[36] H. Kesten, Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab. 18 (1990) 537-555. | MR 1055419 | Zbl 0705.60092

[37] S. Lang, Differential Manifolds, Springer-Verlag, 1985. | MR 772023 | Zbl 0551.58001

[38] G. Lawlor, F. Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math. 166 (1) (1994) 55-82. | MR 1306034 | Zbl 0830.49028

[39] J.L. Lebowitz, A.E. Mazel, Yu.M. Suhov, An Ising interface between two walls: competition between two tendencies, Rev. Math. Phys. 8 (5) (1996) 669-687. | MR 1405769 | Zbl 0858.60099

[40] G.P. Leonardi, Optimal subdivisions of n-dimensional domains. Ph.D. Thesis, Università di Trento, 1998.

[41] U. Massari, M. Miranda, Minimal Surfaces of Codimension One, North-Holland Mathematics Studies 91, Notas de Matematica, 95, North-Holland, 1984. | MR 795963 | Zbl 0565.49030

[42] U. Massari, L. Pepe, Sull'approssimazione degli aperti lipschitziani di Rn con varietà differenziabili, Bollettino U.M.I. (4) 10 (1974) 532-544. | MR 365318 | Zbl 0316.49031

[43] A. Messager, S. Miracle-Solé, J. Ruiz, Convexity properties of the surface tension and equilibrium crystals, J. Stat. Phys. 67 (3/4) (1992) 449-469. | MR 1171142 | Zbl 0900.82029

[44] S. Miracle-Solé, Surface tension, step free energy, and facets in the equilibrium crystal, J. Stat. Phys. 79 (1/2) (1995) 183-214. | Zbl 1106.82303

[45] C.M. Newman, Topics in Disordered Systems, Lectures in Mathematics, ETH Zürich, Birkhäuser, 1997. | MR 1480664 | Zbl 0897.60093

[46] C.E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta 64 (1991) 953-1054. | MR 1149430

[47] C.E. Pfister, Y. Velenik, Interface, surface tension and reentrant pinning transition in the 2D Ising model, Comm. Math. Phys. 204 (2) (1999) 269-312. | MR 1704276 | Zbl 0937.82016

[48] C.E. Pfister, Y. Velenik, Large deviations and continuum limit in the 2D Ising model, Probab. Theory Related Fields 109 (1997) 435-506. | MR 1483597 | Zbl 0904.60022

[49] A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (1996) 427-466. | MR 1384040 | Zbl 0842.60022

[50] T. Quentin De Gromard, Approximation forte dans BV(Ω), C. R. Acad. Sci. Paris, Ser. I 301 (1985) 261-264. | Zbl 0593.46026

[51] R.H. Schonmann, Second order large deviation estimates for ferromagnetic systems in the phase coexistence region, Comm. Math. Phys. 112 (1987) 409-422. | MR 908546

[52] Y. Velenik, Ph.D. Thesis, EPFL, 1997.

[53] A.I. Vol'Pert, The spaces BV and quasilinear equations, Math. USSR Sb. 2 (1967) 225-267. | MR 216338 | Zbl 0168.07402

[54] W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, 1989. | MR 1014685 | Zbl 0692.46022