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September  2013, 33(9): 4291-4303. doi: 10.3934/dcds.2013.33.4291

Archimedean ice

Kari Eloranta 1,

1. 

Department of Mathematics, Aalto University, P. O. Box 11100, FI-00076 Aalto, Finland

Received  November 2010 Revised  December 2010 Published  March 2013

The striking boundary dependency, the Arctic Circle Phenomenon, exhibited in the Ice model on the square lattice extends to other planar set-ups. This can be shown using a dynamical formulation which we present for the Archimedean lattices. Critical connectivity results guarantee that the Ice configurations can be generated using the simplest and most efficient local actions. Height functions are utilized throughout the analysis. On a hexagon with suitable boundary height the cellular automaton dynamics generates highly nontrivial Ice equilibria in the triangular and Kagomé cases. On the remaining Archimedean lattice for which the Ice model can be defined, the 3.4.6.4. lattice, the long range behavior is shown to be completely different due to strictly positive entropy for all boundary conditions.
Keywords: Archimedean lattice, cellular automaton, vertex rule, measure of maximal entropy, Ice model, spatial phase transition..
Mathematics Subject Classification: Primary: 37A60, 82C20; Secondary: 82C2.
Citation: Kari Eloranta. Archimedean ice. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4291-4303. doi: 10.3934/dcds.2013.33.4291
References:
[1]

R. J. Baxter, "Exactly Solvable Models In Statistical Mechanics," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982.  Google Scholar

[2]

F. Colomo and A. G. Pronko, The arctic circle revisited, in "Integrable Systems and Random Matrices," Contemp. Math., 458, Amer. Math. Soc., Providence, RI, (2008), 361-376. doi: 10.1090/conm/458/08947.  Google Scholar

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J. H. Conway and J. C. Lagarias, Tilings with polyominoes and combinatorial group theory, J. Combin. Theory, Ser. A 53 (1990), 183-208. doi: 10.1016/0097-3165(90)90057-4.  Google Scholar

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K. Eloranta, Diamond ice, J. Stat. Phys. 96 (1999), 1091-1109. doi: 10.1023/A:1004644418182.  Google Scholar

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B. Grünbaum and G. C. Shephard, "Tilings And Patterns," W. H. Freeman and Company, New York, 1987.  Google Scholar

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W. Jockush, J. Propp and P. Shor, "Random domino tilings and the arctic circle Theorem,", arXiv:math.CO/9801068., ().   Google Scholar

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P. Kasteleyn, The statistics of the dimer on a lattice, I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. Google Scholar

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R. Kenyon, An introduction to the dimer model, in "School and Conference on Probability Theory," ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, (2004), 267-304 (electronic).  Google Scholar

[9]

E. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967), 162-172. Google Scholar

[10]

J. Propp, "Lattice structure for orientation of graphs,", arXiv:math.CO/0209005., ().   Google Scholar

[11]

W. P. Thurston, Conway's tiling groups, Am. Math. Monthly (1990) 757-773. doi: 10.2307/2324578.  Google Scholar

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P. Walters, "An Introduction To Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

R. J. Baxter, "Exactly Solvable Models In Statistical Mechanics," Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982.  Google Scholar

[2]

F. Colomo and A. G. Pronko, The arctic circle revisited, in "Integrable Systems and Random Matrices," Contemp. Math., 458, Amer. Math. Soc., Providence, RI, (2008), 361-376. doi: 10.1090/conm/458/08947.  Google Scholar

[3]

J. H. Conway and J. C. Lagarias, Tilings with polyominoes and combinatorial group theory, J. Combin. Theory, Ser. A 53 (1990), 183-208. doi: 10.1016/0097-3165(90)90057-4.  Google Scholar

[4]

K. Eloranta, Diamond ice, J. Stat. Phys. 96 (1999), 1091-1109. doi: 10.1023/A:1004644418182.  Google Scholar

[5]

B. Grünbaum and G. C. Shephard, "Tilings And Patterns," W. H. Freeman and Company, New York, 1987.  Google Scholar

[6]

W. Jockush, J. Propp and P. Shor, "Random domino tilings and the arctic circle Theorem,", arXiv:math.CO/9801068., ().   Google Scholar

[7]

P. Kasteleyn, The statistics of the dimer on a lattice, I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. Google Scholar

[8]

R. Kenyon, An introduction to the dimer model, in "School and Conference on Probability Theory," ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, (2004), 267-304 (electronic).  Google Scholar

[9]

E. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967), 162-172. Google Scholar

[10]

J. Propp, "Lattice structure for orientation of graphs,", arXiv:math.CO/0209005., ().   Google Scholar

[11]

W. P. Thurston, Conway's tiling groups, Am. Math. Monthly (1990) 757-773. doi: 10.2307/2324578.  Google Scholar

[12]

P. Walters, "An Introduction To Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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