Large deviations for three dimensional supercritical percolation

Cerf, Raphaël

Astérisque, no. 267 (2000) , 183 p.

MR Zbl | 4 citations dans Numdam

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     author = {Cerf, Rapha\"el},
     title = {Large deviations for three dimensional supercritical percolation},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {267},
     year = {2000},
     mrnumber = {1774341},
     zbl = {0962.60002},
     language = {en},
     url = {http://www.numdam.org./item/AST_2000__267__R1_0/}
}

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Cerf, Raphaël. Large deviations for three dimensional supercritical percolation. Astérisque, no. 267 (2000), 183 p. http://numdam.org/item/AST_2000__267__R1_0/

Bibliographie
Cité par

1. M. Aizenman, J. T. Chayes, L. Chayes, J. Fröhlich and L. Russo, On a sharp transition from area law to perimeter law in a system of random surfaces, Commun. Math. Phys. 92 no.1 (1983), 19-69. | MR | Zbl | DOI

2. K. S. Alexander, Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation, Probab. Theory Related Fields 91 no.3/4 (1992), 507-532. | MR | Zbl | DOI

3. K. S. Alexander, J. T. Chayes and L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation, Commun. Math. Phys. 131 no.1 (1990), 1-50. | MR | Zbl | DOI

4. F. Almgren, Questions and answers about area-minimizing surfaces and geometric measure theory, Differential geometry: partial differential equations on manifolds. R. Greene (ed.) et al., Proc. Symp. Pure Math. 54 (1993), 29-53. | MR | Zbl | DOI

5. G. Alberti, G. Bellettini, M. Cassandro and E. Presutti, Surface tension in Ising systems with Kac potentials, J. Statist. Phys. 82 no.3/4 (1996), 743-796. | MR | Zbl | DOI

6. P. Assouad and T. Quentin De Gromard, Sur la derivation des mesures dans $ℝ^{n}$ , Note (1998).

7. G. Bellettini, M. Cassandro and E. Presutti, Constrained minima of nonlocal free energy functionals, J. Statist. Phys. 84 no.5/6 (1996), 1337-1349. | MR | Zbl | DOI

8. G. Bellettini, M. Novaga, Comparison results between minimal barriers and viscosity solutions for geometric evolutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 no.1 (1998), 97-131. | MR | Zbl | EuDML | Numdam

9. O. Benois, T. Bodineau, P. Buttà and E. Presutti, On the validity of van der Waals theory of surface tension, Markov Process. Related Fields 3 no.2 (1997), 175-198. | MR | Zbl

10. O. Benois, T. Bodineau and E. Presutti, Large deviations in the van der Waals limit, Stochastic Process. Appl. 75 no.1 (1998), 89-104. | MR | Zbl

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. | MR | Zbl | DOI

12. A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions. II., Proc. Cambridge Philos. Soc. 42 (1946), 1-10. | MR | Zbl | DOI

13. T. Bodineau, The Wulff construction in three and more dimensions., preprint (1999). | MR | Zbl | DOI

14. Y. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, 1988. | MR | Zbl

15. R. Caccioppoli, Misura e integrazione sugli insiemi dimensionalmente orientati I, II, Rend. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Ser. VIII Vol. XII fasc. 1, 2 (gennaio-febbraio 1952), 3-11, 137-146. | MR | Zbl

16. R. Caccioppoli, Misura e integrazione sulle varietà parametriche I, II, III, Rend. Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., Ser. VIII Vol. XII fasc. 3 , 4 , 6 (marzo-aprile-giugno 1952), 219-227, 365-373, 629-634. | MR | Zbl

17. R. Cerf, Large deviations for i.i.d. random compact sets, Proc. Amer. Math. Soc. 127 (1999), 2431-2436. | MR | Zbl | DOI

18. R. Cerf, Large deviations of the finite cluster shape for two-dimensional percolation in the Hausdorff and $L^{1}$ topology, J. Theor. Probab. 12 no.4 (1999), 1137-1163. | Zbl | MR

19. R. Cerf and A. Pisztora, On the Wulff crystal in the Ising model, preprint (1999). | MR | Zbl

20. L. Cesari, Surface area, Princeton University Press, 1956. | MR | Zbl

21. F. Cesi, G. Guadagni, F. Martinelli and R. H. Schonmann, On the $2 D$ stochastic Ising model in the phase coexistence region near the critical point, J. Statist. Phys. 85 no.1/2 (1996), 55-102. | MR | Zbl | DOI

22. G. Congedo and I. Tamanini, Optimal partitions with unbounded data, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX Ser., Rend. Lincei., Mat. Appl. 4 no.2 (1993), 103-108. | MR | Zbl | EuDML

23. G. Congedo and I. Tamanini, On the existence of solutions to a problem in multidimensional segmentation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8 no.2 (1991), 175-195. | MR | Zbl | EuDML | Numdam | DOI

24. E. De Giorgi, Su una teoria generale della misura $(r - 1)$ -dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. IV Ser. 36 (1954), 191-213. | MR | Zbl | DOI

25. E. De Giorgi, Nuovi teoremi relativi alle misure $(r - 1)$ -dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 (1955), 95-113. | MR | Zbl

26. E. De Giorgi, Sulla proprieta isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. VIII Ser. 5, (1958), 33-44. | MR | Zbl

27. E. De Giorgi, F. Colombini and L. C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore di Pisa, 1972. | MR | Zbl

28. A. Dembo and O. Zeitouni, Large deviations techniques and applications, Jones and Bartlett publishers, 1993. | MR | Zbl

29. J. D. Deuschel and A. Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Related Fields 104 no.4 (1996), 467-482. | MR | Zbl | DOI

30. A. Dinghas, Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen, Z. Kristallogr., Mineral. Petrogr. 105 Abt.A (1944), 304-314. | MR | Zbl

31. R. L. Dobrushin, R. Kotecký and S. B. Shlosman, Wulff construction: a global shape from local interaction, AMS translations series, Providence (Rhode Island), 1992. | MR | Zbl | DOI

32. R. L. Dobrushin and S. B. Shlosman, Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction, Ideas and methods in quantum and statistical physics 2 (1992), S. Albeverio (ed.), Cambridge University Press, 461-483. | MR | Zbl

33. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, Boca Raton: CRC Press, 1992. | MR | Zbl

34. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics 85, Cambridge Univ. Press, 1985. | MR | Zbl

35. H. Federer, Geometric measure theory, Springer-Verlag, 1969. | MR | Zbl

36. I. Fonseca, Lower semicontinuity of surface energies, Proc. Roy. Soc. Edinb. Sect. A 120 no.1/2 (1992), 99-115. | MR | Zbl | DOI

37. I. Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. Lond. Ser. A 432 no.1884 (1991), 125-145. | MR | Zbl | DOI

38. I. Fonseca and L. Giovanni, Bulk and contact energies: nucleation and relaxation, SIAM J. Math. Anal. 30 no.1 (1999), 190-219. | MR | Zbl | DOI

39. I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinb. Sect. A 119 no.1/2 (1991), 125-136. | MR | Zbl | DOI

40. C. Giacomelli and I. Tamanini, Un tipo di approssimazione "dall'interno" degli insiemi di perimetro finito, Rend. Mat. Acc. Lincei s.9 v.1 (1990), 181-187. | Zbl | EuDML

41. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, 1984. | MR | Zbl | DOI

42. G. R. Grimmett, Percolation, Springer-Verlag, 1989 | MR | Zbl

43. G. R. Grimmett and J. M. Marstrand, The supercritical phase of percolation is well behaved, Proc. Roy. Soc. Lond., Ser. A 430 no.1879 (1990), 439-457. | MR | Zbl | DOI

44. B. L. Gurevich and G. E. Shilov, Integral, measure and derivative: a unified approach, Prentice-Hall, 1966. | MR | Zbl

45. G. T. Herman and D. Webster, Surfaces of organs in discrete three-dimensional space, Mathematical aspects of computerized tomography, Oberwolfach, Lecture Notes in Med. Inform. 8 (1981), Springer, 204-224. | MR | Zbl | DOI

46. O. Hryniv, On local behaviour of the phase separation line in the $2 D$ Ising model, Probab. Theory Related Fields 110 no.1 (1998), 91-107. | MR | Zbl | DOI

47. D. Ioffe, Large deviations for the $2 D$ Ising model: a lower bound without cluster expansions, J. Statist. Phys. 74 no.1/2 (1994), 411-432. | Zbl | MR | DOI

48. D. Ioffe, Exact large deviations bounds up to $T_{c}$ for the Ising model in two dimensions, Probab. Theory Related Fields 102 no.3 (1995), 313-330. | MR | Zbl | DOI

49. D. Ioffe and R. H. Schonmann, Dobrushin-Kotecký-Shlosman Theorem up to the critical temperature, Commun. Math. Phys. 199 no.1 (1998), 117-167. | MR | Zbl | DOI

50. H. Kesten and Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab. 18 no.2 (1990), 537-555. | MR | Zbl | DOI

51. S. Lang, Differential manifolds, 2nd ed., Springer-Verlag, 1985. | MR | Zbl

52. U. Massari and M. Miranda, Minimal surfaces of codimension one, Mathematics Studies 91, North-Holland, 1984 | MR | Zbl

U. Massari and M. Miranda, Minimal surfaces of codimension one, Notas de Matematica 95, North-Holland, 1984. | MR | Zbl

53. U. Massari and I. Tamanini, Regularity properties of optimal segmentations, J. Reine Angew. Math. 420 (1991), 61-84. | MR | Zbl | EuDML

54. P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability., Cambridge Studies in Advanced Math. 44, Cambridge Univ. Press, 1995. | MR | Zbl

55. A. Messager, S. Miracle-Solé and J. Ruiz, Convexity properties of the surface tension and equilibrium crystals, J. Statist. Phys. 67 no.3/4 (1992), 449-470. | MR | Zbl | DOI

56. S. Miracle-Solé, Surface tension, step free energy, and facets in the equilibrium crystal, J. Statist. Phys. 79 no.1/2 (1995), 183-214. | Zbl | DOI

57. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math. 42 no.5 (1989), 577-685. | MR | Zbl | DOI

58. C. E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta 64 no.7 (1991), 953-1054. | MR

59. C. E. Pfister and Y. Velenik, Large deviations and continuum limit in the $2 D$ Ising model, Probab. Theory Related Fields 109 no.4 (1997), 435-506. | MR | Zbl | DOI

60. A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 no.4 (1996), 427-466. | MR | Zbl | DOI

61. T. Quentin De Gromard, Approximation forte dans $B V (Ω)$ , C. R. Acad. Sci. Paris, Ser. I 301 no.6 (1985), 261-264. | MR | Zbl

62. T. Quentin De Gromard, Strong approximation in $B V (Ω)$ , preprint (2000). | Zbl | MR

63. R. T. Rockafellar, Convex analysis, Princeton Univ. Press, New Jersey, 1970. | MR | DOI

64. A. Rosenfeld, Three-dimensional digital topology, Inform. and Control 50 no.2 (1981), 119-127. | MR | Zbl | DOI

65. W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987. | MR | Zbl

66. R. H. Schonmann and S. B. Shlosman, Constrained variational problem with applications to the Ising model, J. Statist. Phys. 83 no.5/6 (1996), 867-905. | MR | Zbl | DOI

67. R. H. Schonmann and S. B. Shlosman, Complete analyticity for $2 D$ Ising model completed, Commun. Math. Phys. 170 no.2 (1995), 453-482. | MR | Zbl | DOI

68. R. H. Schonmann and S. B. Shlosman, Wulff droplets and the metastable relaxation of kinetic Ising models, Commun. Math. Phys. 194 no. 2 (1998), 389-462. | MR | Zbl | DOI

69. Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 no.4 (1978), 568-588. | MR | Zbl | DOI

70. Jean E. Taylor, Existence and structure of solutions to a class of nonelliptic variational problems, Symposia Mathematica 14 no.4 (1974), 499-508. | MR | Zbl

71. Jean E. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Proc. Sympos. Pure Math. 27 (1975), 419-427. | MR | Zbl | DOI

72. A. Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, 1996. | MR | Zbl

73. A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 no.2 (1991), 175-201. | MR | Zbl | DOI

74. A. I. Vol'Pert, The spaces $B V$ and quasilinear equations, Math. USSR-Sbornik 2 no.2 (1967), 225-267, translation of Mat. Sbornik, Tom 73 (115), (1967), no.2. | Zbl | DOI

75. G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflächen, Z. Krystallogr. 34 (1901), 449-530.

76. B. Younovitch, Sur la dérivation des fonctions absolument additives d'ensemble., C. R. (Doklady) Acad. Sci. URSS, n. Ser. 30 (1941), 112-114. | MR | Zbl | JFM

77. W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation., Graduate Texts in Mathematics 120, Springer-Verlag, 1989. | MR | Zbl | DOI