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Volumn 28, Issue 3, 2000, Pages 947-1017

On the Wulff crystal in the Ising model

Author keywords

FK percolation; Ising model; Large deviations; Phase separation; Wulff crystal

Indexed keywords


EID: 0034214575     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1019160324     Document Type: Article
Times cited : (86)

References (71)
  • 1
    • 0001020411 scopus 로고
    • Percolation of the minority spins in high-dimensional ising models
    • 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.
    • (1987) J. Statist. Phys. , vol.49 , pp. 859-865
    • Aizenman, M.1    Bricmont, J.2    Lebowitz, J.L.3
  • 2
    • 0002190594 scopus 로고
    • On a sharp transition from area law to perimeter law in a system of random surfaces
    • AIZENMAN, M., CHAYES, J. T., CHAYES, L., FRÖHLICH, 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.
    • (1983) Comm. Math. Phys. , vol.92 , pp. 19-69
    • Aizenman, M.1    Chayes, J.T.2    Chayes, L.3    Fröhlich, J.4    Russo, L.5
  • 4
    • 0000514175 scopus 로고
    • Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation
    • 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.
    • (1992) Probab. Theory Related Fields , vol.91 , pp. 507-532
    • Alexander, K.S.1
  • 5
    • 0002005827 scopus 로고
    • The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional bernoulli percolation
    • 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.
    • (1990) Comm. Math. Phys. , vol.131 , pp. 1-50
    • Alexander, K.S.1    Chayes, J.T.2    Chayes, L.3
  • 7
    • 0039120595 scopus 로고    scopus 로고
    • Constrained minima of nonlocal free energy functionals
    • BELLETTINI, G., CASSANDRO, M. and PRESUTTI, E. (1996). Constrained minima of nonlocal free energy functionals. J. Statist. Phys. 84 1337-1349.
    • (1996) J. Statist. Phys. , vol.84 , pp. 1337-1349
    • Bellettini, G.1    Cassandro, M.2    Presutti, E.3
  • 10
    • 0001684131 scopus 로고    scopus 로고
    • A general form of the covering principle and relative differentiation of additive functions
    • 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.
    • (1946) Proc. Cambridge Philos. Soc. , vol.41 , pp. 103-110
    • Besicovitch, A.S.1
  • 11
    • 0001684131 scopus 로고    scopus 로고
    • 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.
    • II. Proc. Cambridge Philos. Soc. , vol.42 , pp. 1-10
  • 12
    • 0033473098 scopus 로고    scopus 로고
    • The Wulff construction in three and more dimensions
    • BODINEAU, T. (1999). The Wulff construction in three and more dimensions. Comm. Math. Phys. 207 197-229.
    • (1999) Comm. Math. Phys. , vol.207 , pp. 197-229
    • Bodineau, T.1
  • 13
    • 0003152974 scopus 로고
    • Misura e integrazione sugli insiemi dimensionalmente orientati I, II
    • 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.
    • (1952) Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. , vol.12 , Issue.8 , pp. 3-11
    • Caccioppoli, R.1
  • 14
    • 0002351557 scopus 로고
    • Misura e integrazione sulle varietà parametriche I, II, III
    • CACCIOPPOLI, R. (1952). Misura e integrazione sulle varietà parametriche I, II, III. Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. (8) XII 219-227, 365-373, 629-634.
    • (1952) Rend. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. , vol.12 , Issue.8 , pp. 219-227
    • Caccioppoli, R.1
  • 15
    • 0003310674 scopus 로고    scopus 로고
    • Large deviations for three-dimensional supercritical percolation
    • CERF, R. (1998). Large deviations for three-dimensional supercritical percolation. Astérisque 267 (2000).
    • (1998) Astérisque , pp. 267
    • Cerf, R.1
  • 17
    • 0030491030 scopus 로고    scopus 로고
    • On the 2D stochastic ising model in the phase coexistence region near the critical point
    • 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.
    • (1996) J. Statist. Phys. , vol.85 , pp. 55-102
    • Cesi, F.1    Guadagni, G.2    Martinelli, F.3    Schonmann, R.4
  • 18
    • 0001020412 scopus 로고
    • Exponential decay of connectivities in the two-dimensional ising model
    • 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.
    • (1987) J. Statist. Phys. , vol.49 , pp. 433-445
    • Chayes, J.T.1    Chayes, L.2    Schonmann, R.H.3
  • 20
    • 0000245505 scopus 로고
    • Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni
    • 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.
    • (1954) Ann. Mat. Pura Appl. , vol.36 , pp. 191-213
    • De Giorgi, E.1
  • 21
    • 0001844148 scopus 로고
    • Nuovi teoremi relativi alle misure (r - 1)-dimensionali in uno spazio ad r dimensioni
    • DE GIORGI, E. (1955). Nuovi teoremi relativi alle misure (r - 1)-dimensionali in uno spazio ad r dimensioni. Ricerche Mat. 4 95-113.
    • (1955) Ricerche Mat. , vol.4 , pp. 95-113
    • De Giorgi, E.1
  • 22
    • 0002194826 scopus 로고
    • Sulla proprieta isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita
    • 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.
    • (1958) Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat. , vol.5 , Issue.8 , pp. 33-44
    • De Giorgi, E.1
  • 24
    • 0030536786 scopus 로고    scopus 로고
    • Surface order large deviations for high-density percolation
    • DEUSCHEL, J.-D. and PISZTORA, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467-482.
    • (1996) Probab. Theory Related Fields , vol.104 , pp. 467-482
    • Deuschel, J.-D.1    Pisztora, A.2
  • 26
    • 0001539326 scopus 로고
    • Über einen geometrischen satz von Wulff für die gleichgewichtsform von kristallen
    • DINGHAS, A. (1944). Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen. Z. Kristallogr. 105 304-314.
    • (1944) Z. Kristallogr. , vol.105 , pp. 304-314
    • Dinghas, A.1
  • 27
    • 0031518868 scopus 로고    scopus 로고
    • Fluctuations of the phase boundary in the 2D ising ferromagnet
    • DOBRUSHIN, R. K. and HRYNIV, O. (1997). Fluctuations of the phase boundary in the 2D Ising ferromagnet. Comm. Math. Phys. 189 395-445.
    • (1997) Comm. Math. Phys. , vol.189 , pp. 395-445
    • Dobrushin, R.K.1    Hryniv, O.2
  • 29
    • 0001568243 scopus 로고
    • Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction
    • (S. Albeverio, ed.) Cambridge Univ. Press
    • 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.
    • (1992) Ideas and Methods in Quantum and Statistical Physics , pp. 461-483
    • Dobrushin, R.L.1    Shlosman, S.B.2
  • 30
    • 0001440863 scopus 로고
    • Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm
    • 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.
    • (1988) Phys. Rev. D , vol.38 , pp. 2009-2012
    • Edwards, R.G.1    Sokal, A.D.2
  • 35
    • 0000982881 scopus 로고
    • Large deviations for the empirical field of a Gibbs measure
    • FÖLLMER, H. and OREY, S. (1988). Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16 961-977.
    • (1988) Ann. Probab. , vol.16 , pp. 961-977
    • Föllmer, H.1    Orey, S.2
  • 36
    • 0002049957 scopus 로고
    • Large deviations and surface entropy for Markov fields
    • FÖLLMER, H. and ORT, M. (1988). Large deviations and surface entropy for Markov fields. Astérisque 157-158, 173-190.
    • (1988) Astérisque , pp. 157-158
    • Föllmer, H.1    Ort, M.2
  • 39
    • 0041855748 scopus 로고
    • On the random cluster model. I
    • FORTUIN, C. M. and KASTELEYN, P. W. (1972). On the random cluster model. I. Physica 57 536-564.
    • (1972) Physica , vol.57 , pp. 536-564
    • Fortuin, C.M.1    Kasteleyn, P.W.2
  • 42
    • 21344436631 scopus 로고
    • The stochastic random-cluster process and the uniqueness of random-cluster measures
    • GRIMMETT, G. R. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 1461-1510.
    • (1995) Ann. Probab. , vol.23 , pp. 1461-1510
    • Grimmett, G.R.1
  • 43
    • 0001626028 scopus 로고
    • The supercritical phase of percolation is well behaved
    • 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.
    • (1990) Proc. Roy. Soc. Lond. Ser. A , vol.430 , pp. 439-457
    • Grimmett, G.R.1    Marstrand, J.M.2
  • 44
    • 0031632711 scopus 로고    scopus 로고
    • On local behaviour of the phase separation line in the 2D ising model
    • HRYNIV, O. (1998). On local behaviour of the phase separation line in the 2D Ising model. Probab. Theory Related Fields 110 91-107.
    • (1998) Probab. Theory Related Fields , vol.110 , pp. 91-107
    • Hryniv, O.1
  • 45
    • 21344496131 scopus 로고
    • Large deviations for the 2D ising model: A lower bound without cluster expansions
    • IOFFE, D. (1993). Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Statist. Phys. 74 411-432.
    • (1993) J. Statist. Phys. , vol.74 , pp. 411-432
    • Ioffe, D.1
  • 46
    • 21844490341 scopus 로고
    • c for the ising model in two dimensions
    • c for the Ising model in two dimensions. Probab. Theory Related Fields 102 313-330.
    • (1995) Probab. Theory Related Fields , vol.102 , pp. 313-330
    • Ioffe, D.1
  • 47
    • 0032270185 scopus 로고    scopus 로고
    • Dobrushin-Kotecký-Shlosman theorem up to the critical temperature
    • IOFFE, D. and SCHONMANN, R. (1998). Dobrushin-Kotecký-Shlosman Theorem up to the critical temperature. Comm. Math. Phys. 199 117-167.
    • (1998) Comm. Math. Phys. , vol.199 , pp. 117-167
    • Ioffe, D.1    Schonmann, R.2
  • 48
    • 0002023496 scopus 로고
    • The probability of a large finite cluster in supercritical bernoulli percolation
    • KESTEN, H. and ZHANG, Y. (1990). The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 537-555.
    • (1990) Ann. Probab. , vol.18 , pp. 537-555
    • Kesten, H.1    Zhang, Y.2
  • 49
    • 0001446038 scopus 로고
    • Coexistence of phases in ising ferromagnets
    • LEBOWITZ, J. L. (1977). Coexistence of phases in Ising ferromagnets. J. Statist. Phys. 16 463-476.
    • (1977) J. Statist. Phys. , vol.16 , pp. 463-476
    • Lebowitz, J.L.1
  • 50
    • 0001742454 scopus 로고
    • On the uniqueness of the equilibrium state for ising spin systems
    • LEBOWITZ, J. L. and MARTIN-LÖF, A. (1972). On the uniqueness of the equilibrium state for Ising spin systems. Comm. Math. Phys. 25 276-282.
    • (1972) Comm. Math. Phys. , vol.25 , pp. 276-282
    • Lebowitz, J.L.1    Martin-Löf, A.2
  • 54
    • 0000565923 scopus 로고
    • Convexity properties of the surface tension and equilibrium crystals
    • MESSAGER, A., MIRACLE-SOLE, S. and RUIZ, J. (1992). Convexity properties of the surface tension and equilibrium crystals. J. Statist. Phys. 67 449-469.
    • (1992) J. Statist. Phys. , vol.67 , pp. 449-469
    • Messager, A.1    Miracle-Sole, S.2    Ruiz, J.3
  • 55
    • 21844509991 scopus 로고
    • Surface tension, step free energy, and facets in the equilibrium crystal
    • MIRACLE-SOLÉ, S. (1995). Surface tension, step free energy, and facets in the equilibrium crystal. J. Statist. Phys. 79 183-214.
    • (1995) J. Statist. Phys. , vol.79 , pp. 183-214
    • Miracle-Solé, S.1
  • 56
    • 0003383184 scopus 로고
    • Disordered ising systems and random cluster representations
    • G. Grimmett, ed. Kluwer, Dordrecht
    • NEWMAN, C. M. (1993). Disordered Ising systems and random cluster representations. In Probability and Phase Transition (G. Grimmett, ed.) 247-260. Kluwer, Dordrecht.
    • (1993) Probability and Phase Transition , pp. 247-260
    • Newman, C.M.1
  • 57
    • 0000652715 scopus 로고
    • Large deviations for Gibbs random fields
    • OLLA, S. (1988). Large deviations for Gibbs random fields. Probab. Theory Related Fields 77 395-409.
    • (1988) Probab. Theory Related Fields , vol.77 , pp. 395-409
    • Olla, S.1
  • 58
    • 0000981771 scopus 로고
    • Large deviations and phase separation in the two-dimensional ising model
    • PFISTER, C. E. (1991). Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64 953-1054.
    • (1991) Helv. Phys. Acta , vol.64 , pp. 953-1054
    • Pfister, C.E.1
  • 59
    • 0031505092 scopus 로고    scopus 로고
    • Large deviations and continuum limit in the 2D ising model
    • PFISTER, C. E. and VELENIK, Y. (1997). Large deviations and continuum limit in the 2D Ising model. Probab. Theory Related Fields 109 435-506.
    • (1997) Probab. Theory Related Fields , vol.109 , pp. 435-506
    • Pfister, C.E.1    Velenik, Y.2
  • 60
    • 0042416901 scopus 로고    scopus 로고
    • Surface order large deviations for ising, potts and percolation models
    • PISZTORA, Á. (1996). Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104 427-466.
    • (1996) Probab. Theory Related Fields , vol.104 , pp. 427-466
    • Pisztora, Á.1
  • 63
    • 0000798606 scopus 로고
    • Second order large deviation estimates for ferromagnetic systems in the phase coexistence region
    • SCHONMANN, R. H. (1987). Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Comm. Math. Phys. 112 409-422.
    • (1987) Comm. Math. Phys. , vol.112 , pp. 409-422
    • Schonmann, R.H.1
  • 64
    • 0030543461 scopus 로고    scopus 로고
    • Constrained variational problem with applications to the ising model
    • SCHONMANN, R. and SHLOSMAN, S. B. (1996). Constrained variational problem with applications to the Ising model. J. Statist. Phys. 83 867-905.
    • (1996) J. Statist. Phys. , vol.83 , pp. 867-905
    • Schonmann, R.1    Shlosman, S.B.2
  • 65
    • 34249762051 scopus 로고    scopus 로고
    • Complete analyticity for the 2D ising model completed
    • SCHONMANN, R. and SHLOSMAN, S. B. (1996). Complete analyticity for the 2D Ising model completed. Comm. Math. Phys. 170 453-482.
    • (1996) Comm. Math. Phys. , vol.170 , pp. 453-482
    • Schonmann, R.1    Shlosman, S.B.2
  • 66
    • 0032098595 scopus 로고    scopus 로고
    • Wulff droplets and the metastable relaxation of kinetic ising models
    • SCHONMANN, R. and SHLOSMAN, S. B. (1998). Wulff droplets and the metastable relaxation of kinetic Ising models. Comm. Math. Phys. 194 389-462.
    • (1998) Comm. Math. Phys. , vol.194 , pp. 389-462
    • Schonmann, R.1    Shlosman, S.B.2
  • 67
    • 0001123141 scopus 로고
    • Existence and structure of solutions to a class of nonelliptic variational problems
    • TAYLOR, J. E. (1974). Existence and structure of solutions to a class of nonelliptic variational problems. Sympos. Math. 14 499-508.
    • (1974) Sympos. Math. , vol.14 , pp. 499-508
    • Taylor, J.E.1
  • 68
    • 0000500627 scopus 로고
    • Unique structure of solutions to a class of nonelliptic variational problems
    • TAYLOR, J. E. (1975). Unique structure of solutions to a class of nonelliptic variational problems. Proc. Symp. Pure Math. 27 419-427.
    • (1975) Proc. Symp. Pure Math. , vol.27 , pp. 419-427
    • Taylor, J.E.1
  • 69
    • 84966261517 scopus 로고
    • Crystalline variational problems
    • TAYLOR, J. E. (1978). Crystalline variational problems. Bull. Amer. Math. Soc. 84 568-588.
    • (1978) Bull. Amer. Math. Soc. , vol.84 , pp. 568-588
    • Taylor, J.E.1
  • 70
    • 0001421454 scopus 로고
    • Zur frage der geschwindigkeit des wachstums und der auflösung der kristallflächen
    • WULFF, G. (1901). Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Z. Kristallogr. 34 449-530.
    • (1901) Z. Kristallogr. , vol.34 , pp. 449-530
    • Wulff, G.1


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