Volume (2000) no. 267

Large deviations for three dimensional supercritical percolation

Cerf, Raphaël

Astérisque, no. 267 (2000), 183 p.

@book{AST_2000__267__R1_0,
     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},
     zbl = {0962.60002},
     mrnumber = {1774341},
     language = {en},
     url = {http://www.numdam.org/item/AST_2000__267__R1_0}
}

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

References

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. | Article | MR 728447 | Zbl 0529.60099

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. | Article | MR 1151807 | Zbl 0739.60089

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. | Article | MR 1062747 | Zbl 0698.60098

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. | Article | MR 1216574 | Zbl 0812.49032

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. | Article | MR 1372427 | Zbl 1042.82539

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. | Article | MR 1412081 | Zbl 1081.82502

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. | Numdam | MR 1632984 | Zbl 0904.35041

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 1468173 | Zbl 0910.60014

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 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. | Article | MR 12325 | Zbl 0063.00352

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. | Article | MR 14414 | Zbl 0063.00353

13. T. Bodineau, The Wulff construction in three and more dimensions., preprint (1999). | Article | MR 1724851 | Zbl 1015.82005

14. Y. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, 1988. | MR 936419 | Zbl 0633.53002

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 47756 | Zbl 0048.03704

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 49991 | Zbl 0048.03705

17. R. Cerf, Large deviations for i.i.d. random compact sets, Proc. Amer. Math. Soc. 127 (1999), 2431-2436. | Article | MR 1487361 | Zbl 0934.60017

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. | MR 1777542 | Zbl 0974.60089

19. R. Cerf and A. Pisztora, On the Wulff crystal in the Ising model, preprint (1999). | MR 1797302 | Zbl 1034.82006

20. L. Cesari, Surface area, Princeton University Press, 1956. | MR 74500 | Zbl 0073.04101

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. | Article | MR 1413237 | Zbl 0937.82004

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 1233397 | Zbl 0776.49027

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. | Article | Numdam | MR 1096603 | Zbl 0729.49003

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. | Article | MR 62214 | Zbl 0055.28504

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 74499 | Zbl 0066.29903

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 98331 | Zbl 0116.07901

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 493669 | Zbl 0296.49031

28. A. Dembo and O. Zeitouni, Large deviations techniques and applications, Jones and Bartlett publishers, 1993. | MR 1202429 | Zbl 1177.60035 | Zbl 0793.60030

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. | Article | MR 1384041 | Zbl 0842.60023

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 12454 | Zbl 0028.43001

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. | Article | MR 1181197 | Zbl 0917.60103

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 1190540 | Zbl 0786.52006

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 1158660 | Zbl 0804.28001

34. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics 85, Cambridge Univ. Press, 1985. | MR 867284 | Zbl 0587.28004

35. H. Federer, Geometric measure theory, Springer-Verlag, 1969. | MR 257325 | Zbl 0874.49001 | Zbl 0176.00801

36. I. Fonseca, Lower semicontinuity of surface energies, Proc. Roy. Soc. Edinb. Sect. A 120 no.1/2 (1992), 99-115. | Article | MR 1149987 | Zbl 0757.49013

37. I. Fonseca, The Wulff theorem revisited, Proc. Roy. Soc. Lond. Ser. A 432 no.1884 (1991), 125-145. | Article | MR 1116536 | Zbl 0725.49017

38. I. Fonseca and L. Giovanni, Bulk and contact energies: nucleation and relaxation, SIAM J. Math. Anal. 30 no.1 (1999), 190-219. | Article | MR 1656999 | Zbl 0924.49010

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. | Article | MR 1130601 | Zbl 0752.49019

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 0722.49017

41. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, 1984. | Article | MR 775682 | Zbl 0545.49018

42. G. R. Grimmett, Percolation, Springer-Verlag, 1989 | MR 995460 | Zbl 0691.60089

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. | Article | MR 1068308 | Zbl 0711.60100

44. B. L. Gurevich and G. E. Shilov, Integral, measure and derivative: a unified approach, Prentice-Hall, 1966. | MR 194571 | Zbl 0138.27501

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. | Article | MR 720406 | Zbl 0541.92006

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. | Article | MR 1602044 | Zbl 0897.60038

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. | Article | MR 1257822 | Zbl 0946.82502

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. | Article | MR 1339736 | Zbl 0830.60018

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. | Article | MR 1660207 | Zbl 0929.60076

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. | Article | MR 1055419 | Zbl 0705.60092

51. S. Lang, Differential manifolds, 2nd ed., Springer-Verlag, 1985. | MR 772023 | Zbl 0551.58001

52. U. Massari and M. Miranda, Minimal surfaces of codimension one, Mathematics Studies 91, North-Holland, 1984 | MR 795963 | Zbl 0565.49030

U. Massari and M. Miranda, Minimal surfaces of codimension one, Notas de Matematica 95, North-Holland, 1984. | MR 795963 | Zbl 0565.49030

53. U. Massari and I. Tamanini, Regularity properties of optimal segmentations, J. Reine Angew. Math. 420 (1991), 61-84. | MR 1124566 | Zbl 0729.49004

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 1333890 | Zbl 0819.28004

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. | Article | MR 1171142 | Zbl 0900.82029

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. | Article | Zbl 1106.82303

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. | Article | MR 997568 | Zbl 0691.49036

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 1149430

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. | Article | MR 1483597 | Zbl 0904.60022

60. A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 no.4 (1996), 427-466. | Article | MR 1384040 | Zbl 0842.60022

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 803213 | Zbl 0593.46026

62. T. Quentin De Gromard, Strong approximation in $B V (Ω)$ , preprint (2000). | MR 803213 | Zbl 0593.46026

63. R. T. Rockafellar, Convex analysis, Princeton Univ. Press, New Jersey, 1970. | Article | MR 274683

64. A. Rosenfeld, Three-dimensional digital topology, Inform. and Control 50 no.2 (1981), 119-127. | Article | MR 665508 | Zbl 0484.68054

65. W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987. | MR 924157 | Zbl 0925.00005

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. | Article | MR 1392417 | Zbl 1081.82547

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. | Article | MR 1334405 | Zbl 0821.60097

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. | Article | MR 1627669 | Zbl 0915.60090

69. Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 no.4 (1978), 568-588. | Article | MR 493671 | Zbl 0392.49022

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 420407 | Zbl 0317.49053

71. Jean E. Taylor, Unique structure of solutions to a class of nonelliptic variational problems, Proc. Sympos. Pure Math. 27 (1975), 419-427. | Article | MR 388225 | Zbl 0317.49054

72. A. Visintin, Models of phase transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, 1996. | MR 1423808 | Zbl 0882.35004

73. A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 no.2 (1991), 175-201. | Article | MR 1111612 | Zbl 0736.49030

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. | Article | Zbl 0168.07402

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. | JFM 67.0164.01 | MR 4289 | Zbl 0025.03603

77. W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation., Graduate Texts in Mathematics 120, Springer-Verlag, 1989. | Article | MR 1014685 | Zbl 0692.46022