Random surfaces - Tome (2005) no. 304

Random surfaces

Sheffield, Scott

Astérisque, no. 304 (2005) , 181 p.

MR Zbl | 3 citations dans Numdam

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     language = {en},
     url = {http://www.numdam.org/item/AST_2005__304__R1_0/}
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Sheffield, Scott. Random surfaces. Astérisque, no. 304 (2005), 181 p. http://numdam.org/item/AST_2005__304__R1_0/

Bibliographie
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[1] R. A. Adams - Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975, Pure and Applied Mathematics, Vol. 65. | MR | Zbl

[2] R. K. Ahuja, T. L. Magnanti & J. B. Orlin - Network flows, Prentice Hall Inc., Englewood Cliffs, NJ, 1993, Theory, algorithms, and applications. | MR | Zbl

[3] T. Baker & L. Chayes - On the unicity of discontinuous transitions in the two-dimensional Potts and Ashkin-Teller models, J. Statist. Phys. 93 (1998), no. 1-2, p. 1-15. | MR | Zbl | DOI

[4] R. J. Baxter - Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982. | MR | Zbl

[5] G. Ben Arous & J.-D. Deuschel - The construction of the $(d + 1)$ -dimensional Gaussian droplet, Comm. Math. Phys. 179 (1996), no. 2, p. 467-488. | MR | Zbl | DOI

[6] D. Bertacchi & G. Giacomin - Enhanced interface repulsion from quenched hard-wall randomness, Probab. Theory Relat. Fields 124 (2002), no. 4, p. 487-516. | MR | Zbl | DOI

[7] T. Bodineau, G. Giacomin & Y. Velenik - On entropic reduction of fluctuations, J. Statist. Phys. 102 (2001), no. 5-6, p. 1439-1445. | MR | Zbl | DOI

[8] E. Bolthausen - Random walk representations and entropic repulsion for gradient models, in Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, R. Neth. Acad. Arts Sci.., Amsterdam, 2000, p. 55-83. | MR | Zbl

[9] E. Bolthausen, J.-D. Deuschel & G. Giacomin - Entropic repulsion and the maximum of the two-dimensional harmonic crystal, Ann. Probab. 29 (2001), no. 4, p. 1670-1692. | MR | Zbl

[10] J. Bricmont, A. El Mellouki & J. Fröhlich - Random surfaces in statistical mechanics: roughening, rounding, wetting,... , J. Statist. Phys. 42 (1986), no. 5-6, p. 743-798. | MR | DOI

[11] R. M. Burton & M. Keane - Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989), no. 3, p. 501-505. | MR | Zbl | DOI

[12] L. Chayes - Percolation and ferromagnetism on $𝐙^{2}$ : the $q$ -state Potts cases, Stochastic Process. Appl. 65 (1996), no. 2, p. 209-216. | MR | Zbl | DOI

[13] L. Chayes, D. Mckellar & B. Winn - Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on $ℤ^{2}$ , J. Phys. A 31 (1998), no. 45, p. 9055-9063. | MR | Zbl | DOI

[14] A. Cianchi - Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 3, p. 575-608. | MR | Zbl | EuDML | Numdam

[15] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations 22 (1997), no. 9-10, p. 1629-1646. | MR | Zbl | DOI

[16] A. Cianchi, Some results in the theory of Orlicz spaces and applications to variational problems, in Nonlinear analysis, function spaces and applications, Vol. 6 (Prague, 1998), Acad. Sci. Czech Repub., Prague, 1999, p. 50-92. | MR | Zbl | EuDML

[17] A. Cianchi, A fully anisotropic Sobolev inequality, Pacific J. Math. 196 (2000), no. 2, p. 283-295. | MR | Zbl | DOI

[18] H. Cohn, N. Elkies & J. Propp - Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), no. 1, p. 117-166. | MR | Zbl

[19] H. Cohn, R. Kenyon & J. Propp - A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001), no. 2, p. 297-346 (electronic). | MR | Zbl | DOI

[20] J. H. Conway & J. C. Lagarias - Tiling with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A 53 (1990), no. 2, p. 183-208. | MR | Zbl | DOI

[21] A. Dembo & O. Zeitouni - Large deviations techniques and applications, second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. | MR | Zbl

[22] J.-D. Deuschel & G. Giacomin - Entropic repulsion for the free field: pathwise characterization in $d \geq 3$ , Comm. Math. Phys. 206 (1999), no. 2, p. 447-462. | MR | Zbl | DOI

[23] J.-D. Deuschel & G. Giacomin, Entropic repulsion for massless fields, Stochastic Process. Appl. 89 (2000), no. 2, p. 333-354. | MR | Zbl | DOI

[24] J.-D. Deuschel, G. Giacomin & D. Ioffe - Large deviations and concentration properties for $\nabla ϕ$ interface models, Probab. Theory Relat. Fields 117 (2000), no. 1, p. 49-111. | MR | Zbl | DOI

[25] J.-D. Deuschel & D. W. Stroock - Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press Inc., Boston, MA, 1989. | MR | Zbl

[26] R. Dobrushin, R. Kotecký & S. Shlosman - Wulff construction, Translations of Mathematical Monographs, vol. 104, American Mathematical Society, Providence, RI, 1992, A global shape from local interaction, Translated from the Russian by the authors. | MR | Zbl | DOI

[27] M. D. Donsker & S. R. S. Varadhan - Large deviations from a hydrodynamic scaling limit, Comm. Pure Appl. Math. 42 (1989), no. 3, p. 243-270. | MR | Zbl | DOI

[28] B. Durhuus, J. Fröhlich & T. Jónsson - Self-avoiding and planar random surfaces on the lattice, Nuclear Phys. B 225 (1983), no. 2, FS 9, p. 185-203. | MR | DOI

[29] R. G. Edwards & A. D. Sokal - Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm, Phys. Rev. D (3) 38 (1988), no. 6, p. 2009-2012. | MR | DOI

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

[31] C. M. Fortuin, P. W. Kasteleyn & J. Ginibre - Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), p. 89-103. | MR | Zbl | DOI

[32] M. I. Freidlin & A. D. Wentzell - Random perturbations of dynamical Systems, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1998, Translated from the 1979 Russian original by Joseph Szücs. | MR | Zbl

[33] J. Fröhlich, C.-E. Pfister & T. Spencer - On the statistical mechanics of surfaces, in Stochastic processes in quantum theory and statistical physics (Marseille, 1981), Lecture Notes in Phys., vol. 173, Springer, Berlin, 1982, p. 169-199. | MR | Zbl | DOI

[34] J. Fröhlich & B. Zegarliński - The phase transition in the discrete Gaussian chain with $1 / r^{2}$ interaction energy, J. Statist. Phys. 63 (1991), no. 3-4, p. 455-485. | MR | DOI

[35] J. Fröhlich & T. Spencer - The Berežinskiĭ -Kosterlitz-Thouless transition (energy-entropy arguments and renormalization in defect gases), in Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982), Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, p. 29-138. | MR | DOI

[36] T. Funaki - Recent results on the Ginzburg-Landau $\nabla ϕ$ interface model, in Hydrodynamic limits and related topics (Toronto, ON, 1998), Fields Inst. Commun., vol. 27, Amer. Math. Soc., Providence, RI, 2000, p. 71-81. | MR | Zbl

[37] T. Funaki & T. Nishikawa - Large deviations for the Ginzburg-Landau $\nabla ϕ$ interface model, Probab. Theory Relat. Fields 120 (2001), no. 4, p. 535-568. | MR | Zbl | DOI

[38] T. Funaki & H. Spohn - Motion by mean curvature from the Ginzburg-Landau $\nabla ϕ$ interface model, Comm. Math. Phys. 185 (1997), no. 1, p. 1-36. | MR | Zbl | DOI

[39] T. Funaki & S. Olla - Fluctuations for $\nabla ϕ$ interface model on a wall, Stochastic Process. Appl. 94 (2001), no. 1, p. 1-27. | MR | Zbl | DOI

[40] A. Gandolfi, M. Keane & L. Russo - On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation, Ann. Probab. 16 (1988), no. 3, p. 1147-1157. | MR | Zbl | DOI

[41] R. J. Gardner - The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, p. 355-405 (electronic). | MR | Zbl | DOI

[42] H.-O. Georgii - Gibbs measures and phase transitions, de Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. | MR | Zbl

[43] H.-O. Georgii & Y. Higuchi - Percolation and number of phases in the twodimensional Ising model, J. Math. Phys. 41 (2000), no. 3, p. 1153-1169, Probabilistic techniques in equilibrium and nonequilibrium statistical physics. | MR | Zbl | DOI

[44] G. Giacomin - Anharmonic lattices, random walks and random interfaces, Recent research developments in statistical physics I (2000), p. 97-118.

[45] G. Giacomin, Aspects of statistical mechanics of random surfaces, 2001, Preliminary (and partial) notes of the lectures given at IHP.

[46] G. Giacomin, Limit theorems for random interface models of Ginzburg-Landau $\nabla ϕ$ type, in Stochastic partial differential equations and applications (Trento, 2002), Lecture Notes in Pure and Appl. Math., vol. 227, Dekker, New York, 2002, p. 235-253. | MR | Zbl

[47] G. Giacomin, On stochastic domination in the Brascamp-Lieb framework, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 3, p. 507 514. | MR | Zbl | DOI

[48] G. Giacomin, S. Olla & H. Spohn - Equilibrium fluctuations for $\nabla ϕ$ interface model, Ann. Probab. 29 (2001), no. 3, p. 1138-1172. | MR | Zbl

[49] H. H. Goldstine - A history of the calculus of variations from, the 17th through the 19th century, Studies in the History of Mathematics and Physical Sciences, vol. 5, Springer-Verlag, New York, 1980. | MR | Zbl | DOI

[50] G. Grimmett - Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. | MR | Zbl

[51] M. Z. Guo, G. C. Papanicolaou & S. R. S. Varadhan - Nonlinear diffusion limit for a System with nearest neighbor interactions, Comm. Math. Phys. 118 (1988), no. 1, p. 31-59. | MR | Zbl | DOI

[52] B. Helffer & J. Sjöstrand - On the correlation for Kac-like models in the convex case, J. Statist. Phys. 74 (1994), no. 1-2, p. 349-409. | MR | Zbl | DOI

[53] J. Hershberger & J. Snoeyink - Computing minimum length paths of a given homotopy class, Comput. Geom. 4 (1994), no. 2, p. 63-97. | MR | Zbl | DOI

[54] M. Huber - A bounding chain for Swendsen-Wang, Random Structures Algorithms 22 (2003), no. 1, p. 43-59. | MR | Zbl | DOI

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

[56] P. Kastelyn - The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1965), p. 1209-1225. | Zbl

[57] J. H. B. Kemperman - On the $F K G$ -inequality for measures on a partially ordered space, Nederl. Akad. Wetensch. Proc. Ser. A 80=Indag. Math. 39 (1977), no. 4, p. 313-331. | MR | Zbl | DOI

[58] R. Kenyon - Local statistics of lattice dimers, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 5, p. 591-618. | MR | Zbl | EuDML | Numdam | DOI

[59] R. Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, p. 759-795. | MR | Zbl | DOI

[60] R. Kenyon, Long-range properties of spanning trees, J. Math. Phys. 41 (2000), no. 3, p. 1338-1363, Probabilistic techniques in equilibrium and nonequilibrium statistical physics. | MR | Zbl | DOI

[61] R. Kenyon, The planar dimer model with boundary: a survey, in Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, p. 307-328. | MR | Zbl | DOI

[62] R. Kenyon, Dominos and the Gaussian free field, Ann. Probab. 29 (2001), no. 3, p. 1128-1137. | MR | Zbl | DOI

[63] R. Kenyon, A. Okounkov & S. Sheffield - Dimers and Amoebae, http://arxiv.org/abs/math-ph/0311005, to appear in Ann. Math. | MR | Zbl

[64] R. W. Kenyon & D. B. Wilson - Critical resonance in the non-intersecting lattice path model, Probab. Theory Relat. Fields 130 (2004), no. 3, p. 289-318. | MR | Zbl | DOI

[65] J. L. Lebowitz & A. E. Mazel - On the uniqueness of Gibbs states in the Pirogov-Sinai theory, Comm. Math. Phys. 189 (1997), no. 2, p. 311-321. | MR | Zbl | DOI

[66] J. L. Lebowitz & A. Martin-Löf - On the uniqueness of the equilibrium state for Ising spin Systems, Comm. Math. Phys. 25 (1972), p. 276-282. | MR | DOI

[67] M. Luby, D. Randall & A. Sinclair - Markov chain algorithms for planar lattice structures, SIAM J. Comput. 31 (2001), no. 1, p. 167-192 (electronic). | MR | Zbl | DOI

[68] W. S. Massey - A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. | MR | Zbl

[69] V. G. Maz'Ja - Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985, Translated from the Russian by T. O. Shaposhnikova. | MR

[70] V. G. Maz'Ya & S. V. Poborchi - Differentiable functions on bad domains, World Scientific Publishing Co. Inc., River Edge, NJ, 1997. | MR | Zbl

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

[72] A. Naddaf & T. Spencer - On homogenization and scaling limit of some gradient perturbations of a massless free field, Comm. Math. Phys. 183 (1997), no. 1, p. 55-84. | MR | Zbl

[73] A. Pisztora - Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Relat. Fields 104 (1996), no. 4, p. 427-466. | MR | Zbl

[74] J. Propp - Generating random elements of finite distributive lattices, Electron. J. Combin. 4 (1997), no. 2, p. Research Paper 15, approx. 12 pp. (electronic), The Wilf Festschrift (Philadelphia, PA, 1996). | MR | Zbl | EuDML

[75] J. G. Propp & D. B. Wilson - Exact sampling with coupled Markov chains and applications to statistical mechanics, in Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995), vol. 9, 1996, p. 223-252. | MR | Zbl

[76] M. Renardy & R. C. Rogers - An introduction to partial differential equations, second ed., Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004. | MR | Zbl

[77] J. Rougemont - Evaporation of droplets in the two-dimensional Ginzburg-Landau equation, Phys. D 140 (2000), no. 3-4, p. 267-282. | MR | Zbl

[78] W. Rudin - Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987. | MR | Zbl

[79] S. Sheffield - Ribbon tilings and multidimensional height functions, Trans. Amer. Math. Soc. 354 (2002), no. 12, p. 4789-4813 (electronic). | MR | Zbl

[80] S. Sheffield, Gaussian free fields for mathematicians, arXiv: math. PR/0312099 (2004). | MR | Zbl

[81] S. Sheffield & O. Schramm - Contour lines of the $2 D$ Gaussian free field, In preparation (2006).

[82] A. J. De Souza - An extension operator for Orlicz-Sobolev spaces, An. Acad. Brasil. Ciênc. 53 (1981), no. 1, p. 9-12. | MR | Zbl

[83] H. Spohn - Interface motion in models with stochastic dynamics, J. Statist. Phys. 71 (1993), no. 5-6, p. 1081-1132. | MR | Zbl

[84] R. H. Swendsen & J.-S. Wang - Nonuniversal critical dynamics in Monte Carlo simulations, Physical Review Letters 58 (1987), p. 86-88.

[85] G. Wulff - Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Kristallflächen, Zeit. Krystall. Min. 34 (1901), p. 449-530.