Abstract
A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds. If V denotes the number of atoms removed, then the grand canonical Boltzmann weight is q V, 0<q<1. As shown by Cerf and Kenyon, in the limit q→1 a deterministic shape is attained, which has the three facets (100), (010), (001), and a rounded piece interpolating between them. We analyse the step statistics as q→1. In the rounded piece it is given by a determinantal process based on the discrete sine-kernel. Exactly at the facet edge, the steps have more space to meander. Their statistics is again determinantal, but this time based on the Airy-kernel. In particular, the border step is well approximated by the Airy process, which has been obtained previously in the context of growth models. Our results are based on the asymptotic analysis for space-time inhomogeneous transfer matrices.
Similar content being viewed by others
REFERENCES
H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A: Math. Gen. 15:L631-L637 (1982).
R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Comm. Math. Phys 222:147-179 (2001).
H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85:117-166 (1996), arXiv:math.CO/0008243.
H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, N.Y. J. Math. 4:137-165 (1998).
P. L. Ferrari, M. Prähofer, and H. Spohn, Fluctuations of an Atomic Ledge Bordering a Crystalline Facet, (2003), preprint, arXiv:cond-mat/0303162.
T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau ∇ φ interface model, Comm. Math. Phys. 185:1-36 (1997).
J. Gravner, C. A. Tracy, and H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Stat. Phys. 102:1085-1132 (2001).
K. Johansson, Discrete Polynuclear Growth and Determinantal Processes, (2002), preprint, arXiv:math.PR/0206208.
K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields 123:225-280 (2002).
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys. 4:287-293 (1963).
R. Kenyon, The planar dimer model with boundary: a survey, Directions in Mathematical Quasicrystals, CRM Monogr. Ser. 13:307-328 (2000).
R. Kenyon, Dominos and Gaussian free field, Ann. Prob. 29:1128-1137 (2001).
A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16:581-603 (2003).
V. L. Pokrovsky and A. L. Talapov, Ground state, spectrum, and phase diagram of two-dimensional incommensurate crystals, Phys. Rev. Lett. 42:65-68 (1978).
M. Prähofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys. 108:1071-1106 (2002).
M. Reed and B. Simon: Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic Press, New York, 1978).
T. Shirai and Y. Takahashi, Random Point Fields Associated with Certain Fredholm Determinants I: Fermion, Poisson, and Boson Point Processes (2002), preprint, RIMS-1368.
A. Soshnikov, Determinantal random point fields, Russ. Math. Surv. 55:923-976 (2000).
H. Spohn, Tracer dynamics in Dyson's model of interacting Brownian particles, J. Stat. Phys. 47:669-680 (1987).
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159:151-174 (1994).
F. Y. Wu, Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric, Phys. Rev. 168:539-543 (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ferrari, P.L., Spohn, H. Step Fluctuations for a Faceted Crystal. Journal of Statistical Physics 113, 1–46 (2003). https://doi.org/10.1023/A:1025703819894
Issue Date:
DOI: https://doi.org/10.1023/A:1025703819894