Phase coexistence in Ising, Potts and percolation models

Raphaël Cerf; Ágoston Pisztora

Annales de l'I.H.P. Probabilités et statistiques (2001)

  • Volume: 37, Issue: 6, page 643-724
  • ISSN: 0246-0203

How to cite

top

Cerf, Raphaël, and Pisztora, Ágoston. "Phase coexistence in Ising, Potts and percolation models." Annales de l'I.H.P. Probabilités et statistiques 37.6 (2001): 643-724. <http://eudml.org/doc/77703>.

@article{Cerf2001,
author = {Cerf, Raphaël, Pisztora, Ágoston},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
language = {eng},
number = {6},
pages = {643-724},
publisher = {Elsevier},
title = {Phase coexistence in Ising, Potts and percolation models},
url = {http://eudml.org/doc/77703},
volume = {37},
year = {2001},
}

TY - JOUR
AU - Cerf, Raphaël
AU - Pisztora, Ágoston
TI - Phase coexistence in Ising, Potts and percolation models
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2001
PB - Elsevier
VL - 37
IS - 6
SP - 643
EP - 724
LA - eng
UR - http://eudml.org/doc/77703
ER -

References

top
  1. [1] K.S. Alexander, Cube-root boundary fluctuations for droplets in random cluster models, Preprint, 2000. MR1871907
  2. [2] F.J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc.4 (165) (1976). Zbl0327.49043MR420406
  3. [3] G. Alberti, G. Bellettini, M. Cassandro, E. Presutti, Surface tension in Ising systems with Kac potentials, J. Stat. Phys.82 (1996) 743-796. Zbl1042.82539MR1372427
  4. [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. Zbl0698.60098MR1062747
  5. [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. Zbl0676.49028
  6. [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. Zbl0676.49029MR1070482
  7. [7] L. Ambrosio, M. Novaga, E. Paolini, Some regularity results for minimal crystals, Preprint, 2000. Zbl1066.49021MR1932945
  8. [8] Assouad P., Quentin de Gromard T., Sur la dérivation des mesures dans Rn, Note (1998). 
  9. [9] O. Benois, T. Bodineau, P. Buttà, E. Presutti, On the validity of van der Waals theory of surface tension, Markov Process. Rel. Fields3 (1997) 175-198. Zbl0910.60014MR1468173
  10. [10] O. Benois, T. Bodineau, E. Presutti, Large deviations in the van der Waals limit, Stochastic Process. Appl.75 (1998) 89-104. Zbl0927.60080MR1629026
  11. [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. Zbl0063.00352
  12. [12] T. Bodineau, The Wulff construction in three and more dimensions, Comm. Math. Phys.207 (1) (1999) 197-229. Zbl1015.82005MR1724851
  13. [13] T. Bodineau, D. Ioffe, Y. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes, J. Math. Phys.41 (3) (2000) 1033-1098. Zbl0977.82013MR1757951
  14. [14] R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque267 (2000). Zbl0962.60002MR1774341
  15. [15] R. Cerf, A. Pisztora, On the Wulff crystal in the Ising model, Ann. Probab.28 (3) (2000) 945-1015. Zbl1034.82006MR1797302
  16. [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. Zbl0776.49027MR1233397
  17. [17] G. Congedo, I. Tamanini, On the existence of solutions to a problem in multidimensional segmentation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire8 (2) (1991) 175-195. Zbl0729.49003MR1096603
  18. [18] G. Congedo, I. Tamanini, Regularity properties of optimal segmentations, J. Reine Angew. Math.420 (1991) 61-84. Zbl0729.49004MR1124566
  19. [19] E. De Giorgi, Nuovi teoremi relativi alle misure (r−1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat.4 (1955) 95-113. Zbl0066.29903
  20. [20] E. De Giorgi, F. Colombini, L.C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore di Pisa (1972). Zbl0296.49031MR493669
  21. [21] J.-D. Deuschel, A. Pisztora, Surface order large deviations for high-density percolation, Probab. Theory Relat. Fields104 (1996) 467-482. Zbl0842.60023MR1384041
  22. [22] R.L. Dobrushin, R. Kotecký, S.B. Shlosman, Wulff Construction: A Global Shape from Local Interaction, AMS Translations Series, Providence, RI, 1992. Zbl0917.60103MR1181197
  23. [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. Zbl0786.52006MR1190540
  24. [24] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992. Zbl0804.28001MR1158660
  25. [25] K.J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge Univ. Press, 1985. Zbl0587.28004MR867284
  26. [26] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. Zbl0874.49001MR257325
  27. [27] C.M. Fortuin, P.W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica57 (1972) 536-564. MR359655
  28. [28] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, 1984. Zbl0545.49018MR775682
  29. [29] G.R. Grimmett, Percolation, Grundlehren der Mathematischen Wissenschaften, 321, Springer-Verlag, Berlin, 1999. Zbl0926.60004MR1707339
  30. [30] G.R. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Probab.23 (1995) 1461-1510. Zbl0852.60105MR1379156
  31. [31] G.R. Grimmett, J.M. Marstrand, The supercritical phase of percolation is well behaved, Proc. R. Soc. Lond. Ser. A430 (1990) 439-457. Zbl0711.60100MR1068308
  32. [32] J. Hass, M. Hutchings, R. Schlafly, The double bubble conjecture, Electron. Res. Announc. Amer. Math. Soc.1 (3) (1995) 98-102, (electronic). Zbl0864.53007MR1369639
  33. [33] D. Ioffe, Large deviations for the 2D Ising model: a lower bound without cluster expansions, J. Stat. Phys.74 (1993) 411-432. Zbl0946.82502MR1257822
  34. [34] D. Ioffe, Exact large deviation bounds up to Tc for the Ising model in two dimensions, Probab. Theory Relat. Fields102 (1995) 313-330. Zbl0830.60018MR1339736
  35. [35] D. Ioffe, R. Schonmann, Dobrushin–Kotecký–Shlosman Theorem up to the critical temperature, Comm. Math. Phys.199 (1998) 117-167. Zbl0929.60076MR1660207
  36. [36] H. Kesten, Y. Zhang, The probability of a large finite cluster in supercritical Bernoulli percolation, Ann. Probab.18 (1990) 537-555. Zbl0705.60092MR1055419
  37. [37] S. Lang, Differential Manifolds, Springer-Verlag, 1985. Zbl0551.58001MR772023
  38. [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. Zbl0830.49028MR1306034
  39. [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. Zbl0858.60099MR1405769
  40. [40] G.P. Leonardi, Optimal subdivisions of n-dimensional domains. Ph.D. Thesis, Università di Trento, 1998. 
  41. [41] U. Massari, M. Miranda, Minimal Surfaces of Codimension One, North-Holland Mathematics Studies 91, Notas de Matematica, 95, North-Holland, 1984. Zbl0565.49030MR795963
  42. [42] U. Massari, L. Pepe, Sull'approssimazione degli aperti lipschitziani di Rn con varietà differenziabili, Bollettino U.M.I. (4)10 (1974) 532-544. Zbl0316.49031MR365318
  43. [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. Zbl0900.82029MR1171142
  44. [44] S. Miracle-Solé, Surface tension, step free energy, and facets in the equilibrium crystal, J. Stat. Phys.79 (1/2) (1995) 183-214. Zbl1106.82303
  45. [45] C.M. Newman, Topics in Disordered Systems, Lectures in Mathematics, ETH Zürich, Birkhäuser, 1997. Zbl0897.60093MR1480664
  46. [46] C.E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta64 (1991) 953-1054. MR1149430
  47. [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. Zbl0937.82016MR1704276
  48. [48] C.E. Pfister, Y. Velenik, Large deviations and continuum limit in the 2D Ising model, Probab. Theory Related Fields109 (1997) 435-506. Zbl0904.60022MR1483597
  49. [49] A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields104 (1996) 427-466. Zbl0842.60022MR1384040
  50. [50] T. Quentin de Gromard, Approximation forte dans BV(Ω), C. R. Acad. Sci. Paris, Ser. I301 (1985) 261-264. Zbl0593.46026
  51. [51] R.H. Schonmann, Second order large deviation estimates for ferromagnetic systems in the phase coexistence region, Comm. Math. Phys.112 (1987) 409-422. MR908546
  52. [52] Y. Velenik, Ph.D. Thesis, EPFL, 1997. 
  53. [53] A.I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR Sb.2 (1967) 225-267. Zbl0168.07402MR216338
  54. [54] W.P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, 1989. Zbl0692.46022MR1014685

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.