×

3D crystal: how flat its flat facets are? (English) Zbl 1076.74016

Summary: We investigate the hypothesis that the (random) crystal of the \((-)\)-phase inside the \((+)\)-phase of the 3D canonical Ising model has flat facets. We argue that it might need to be weakened, due to the possibility of formation of an extra monolayer on a facet. We then prove this weaker hypothesis for the solid-on-solid model.

MSC:

74E15 Crystalline structure
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05A16 Asymptotic enumeration
05A17 Combinatorial aspects of partitions of integers
74E35 Random structure in solid mechanics
82D25 Statistical mechanics of crystals
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Van Beijeren, H.: Interface sharpness in the Ising system. Commun. Math. Phys. 40, 1-6 (1975) · doi:10.1007/BF01614092
[2] Van Beijeren, H., Nolden, I.: In: W. Schommers, P. Blankenhagen, (eds.), Structure and Dynamics of Surfaces II, Berlin: Springer, 1987
[3] Bodineau, T.: The Wulff construction in three and more dimensions. Commun. Math. Phys. 207, 197-229 (1999) · Zbl 1015.82005 · doi:10.1007/s002200050724
[4] Bricmont, J., Fontaine, J.-R., Lebowitz, J.L.: Surface tension, percolation, and roughening. J. Stat. Phys. 29(2), 193-203 (1982) · doi:10.1007/BF01020782
[5] Bricmont, J., Fröhlich, J., El Mellouki, A.: Random surfaces in statistical mechanics: roughening, rounding, wetting, ... . J. Stat. Phys. 42(5-6), 743-798 (1986)
[6] Bodineau, T., Ioffe, D.: Stability of interfaces and stochastic dynamics in the regime of partial wetting. Ann. Henri Poincaré, 5, 871-914 (2004) · Zbl 1106.82022 · doi:10.1007/s00023-004-0184-6
[7] Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41, 1033-1098 (2000) · Zbl 0969.00035 · doi:10.1063/1.533180
[8] Cerf, R., Pisztora, A.: On the Wulff crystal in the Ising model. Ann. Probab. 28(3), 947-1017 (2000) · Zbl 1034.82006
[9] Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Commun. Math. Phys. 222(1), 147-179 (2001) · Zbl 1013.82010 · doi:10.1007/s002200100505
[10] Dobrushin, R.L.: The Gibbs state that describes the coexistence of phases for a three-dimensional Ising model. (In Russian) Teor. Verojatnost. i Primenen. 17, 619-639 (1972)
[11] Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction: a global shape from local interaction. AMS translations series, Providence, RI: Am. Math. Soc., 1992 · Zbl 0917.60103
[12] Ferrari, P.L., Prähofer, M., Spohn, H.: Fluctuations of an Atomic Ledge Bordering a Crystalline Facet. Phys. Rev. E 69, 035102 (2004) · doi:10.1103/PhysRevE.69.035102
[13] Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1-46 (2003) · Zbl 1116.82331 · doi:10.1023/A:1025703819894
[14] Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Commun. Math. Phys. 81(4), 527-602 (1981) · doi:10.1007/BF01208273
[15] Ioffe, D., Schonmann, R.: Dobrushin-Kotecký-Shlosman theory up to the critical temperature. Commun. Math. Phys. 199, 117-167 (1998) · Zbl 0929.60076 · doi:10.1007/s002200050497
[16] Maes, Ch., Redig, F., Shlosman, S., van Moffaert, A.: Percolation, Path Large Deviations and Weakly Gibbs States. Commun. Math. Phys. 209, 517-545 (2000) · Zbl 0945.60098 · doi:10.1007/s002200050029
[17] Miracle-Sole, S.: Surface tension, step free energy and facets in the equilibrium crystal. J. Stat. Phys. 79, 183-214 (1995) · Zbl 1106.82303 · doi:10.1007/BF02179386
[18] Miracle-Sole, S.: Facet shapes in a Wulff crystal. In: Mathematical results in statistical mechanics (Marseilles, 1998), River Edge, NJ: World Sci. Publishing, 1999, pp. 83-101 · Zbl 1055.82512
[19] Rottman, C., Wortis, M.: Statistical Mechanics of Equilibrium Crystal Shapes: Interfacial Phase Diagrams and Phase Transitions. Phys. Rep. 103, 59-79 (1984) · doi:10.1016/0370-1573(84)90066-8
[20] Schonmann, R.H., Shlosman, S.: Constrained variational problem with applications to the Ising model. J. Stat. Phys. 83, 867-905 (1996) · Zbl 1081.82547 · doi:10.1007/BF02179548
[21] Schonmann, R.H., Shlosman, S.: Complete analyticity for 2D Ising completed. Commun. Math. Phys. 170, 453-482 (1995) · Zbl 0821.60097 · doi:10.1007/BF02108338
[22] S. Shlosman: The Wulff construction in statistical mechanics and in combinatorics. Russ. Math. Surv. 56(4), 709-738 (2001) · Zbl 1035.82013
[23] Shlosman, S.: Zero temperature Ising crystal. In preparation
[24] Vershik, A., Kerov, S.: Asymptotic of the largest and typical dimensions of irreducible representations of a symmetric group. Funct. Anal. Appl. 19, 21-31 (1985) · Zbl 0592.20015 · doi:10.1007/BF01086021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.