A variational principle for domino tilings

Journal of the American Mathematical Society

Volumn 14, Issue 2, 2001, Pages 297-346

  Cohn, Henry ^a,b   Kenyon, Richard ^c   Propp, James ^d  

^a HARVARD UNIVERSITY   (United States)

^b MICROSOFT RESEARCH   (United States)

^c UNIVERSITÉ PARIS SUD   (France)

^d UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Dimer model; Dominos; Matchings; Random tiling; Variational principle

Indexed keywords

EID: 0035580737     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-00-00355-6     Document Type: Article

References (50)

1. H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A 15 (1982), L631-L637. MR 83k:82046
2. H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A 15 (1982), L631-L637. MR 83k:82046
3. R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab. 21 (1993), 1329-1371.
4. MR 94m:60019
5. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), 117-166. MR 97k:52026
6. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), 117-166. MR 97k:52026
7. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998), 137-165. MR 99j:60011
8. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. Math. 4 (1998), 137-165. MR 99j:60011
9. N. Destainville, R. Mosseri, and F. Bailly, Configurational entropy of co-dimension one tilings and directed membranes, J. of Statistical Physics 87 (1997), 697-754. MR 98i:52029
10. N. Destainville, R. Mosseri, and F. Bailly, Configurational entropy of co-dimension one tilings and directed membranes, J. of Statistical Physics 87 (1997), 697-754. MR 98i:52029
11. N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating sign matrices and domino tilings, J. Algebraic Combin. 1 (1992), 111-132 and 219-234. MR 94f:52035; MR 94f:52036
12. N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating sign matrices and domino tilings, J. Algebraic Combin. 1 (1992), 111-132 and 219-234. MR 94f:52035; MR 94f:52036
13. N. Elkies, G. Kuperberg, M. Larsen, and J. Propp, Alternating sign matrices and domino tilings, J. Algebraic Combin. 1 (1992), 111-132 and 219-234. MR 94f:52035; MR 94f:52036
14. H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. MR 41:1976
15. H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. MR 41:1976
16. J.-C. Fournier, Pavage des figures planes sans trous par des dominos: fondement graphique de l'algorithme de Thurston et parallélisation, Compte Rendus de L'Acad. des Sci., Série I 320 (1995), 107-112. MR 95k:52031
17. J.-C. Fournier, Pavage des figures planes sans trous par des dominos: fondement graphique de l'algorithme de Thurston et parallélisation, Compte Rendus de L'Acad. des Sci., Série I 320 (1995), 107-112. MR 95k:52031
18. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1977. MR 57:13109
19. D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1977. MR 57:13109
20. D. Gupta and J. Propp, work in preparation.
21. M. Höffe, Zufallsparkettierungen und Dimermodelle, Thesis, Institut für Theoretische Physik der Eberhard-Karls-Universität, Tübingen, September 1997.
22. M. T. Jaekel and J. M. Maillard, Inversion relations and disorder solutions on Potts models, J. Phys. A 17 (1984), 2079-2094. MR 86f:82054
23. M. T. Jaekel and J. M. Maillard, Inversion relations and disorder solutions on Potts models, J. Phys. A 17 (1984), 2079-2094. MR 86f:82054
24. W. Jockusch, J. Propp, and P. Shor, Random domino tilings and the arctic circle theorem, preprint, 1995.
25. P. W. Kasteleyn, The statistics of dimers on a lattice, I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225.
26. P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys. 4 (1963), 287-293. MR 27:3394
27. P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys. 4 (1963), 287-293. MR 27:3394
28. C. Kenyon and E. Remila, Perfect matchings on the triangular lattice, Discrete Math. 152 (1996), 191-210. MR 97a:05171
29. C. Kenyon and E. Remila, Perfect matchings on the triangular lattice, Discrete Math. 152 (1996), 191-210. MR 97a:05171
30. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. Henri Poincaré, Probabilités, 33 (1997), 591-618. MR 99b:82039
31. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. Henri Poincaré, Probabilités, 33 (1997), 591-618. MR 99b:82039
32. R. Kenyon, The planar dimer model with boundary: a survey, CRM Proceedings and Lecture Notes, to appear.
33. L. S. Levitov, Equivalence of the dimer resonating-valence-bond problem to the quantum roughening problem, Phys. Rev. Lett. 64 (1990), 92-94. MR 90k:82078
34. L. S. Levitov, Equivalence of the dimer resonating-valence-bond problem to the quantum roughening problem, Phys. Rev. Lett. 64 (1990), 92-94. MR 90k:82078
35. 4-symmetry on the checkerboard Potts model, J. Phys. A 18 (1984), 833-846. MR 86e:82049
36. 4-symmetry on the checkerboard Potts model, J. Phys. A 18 (1984), 833-846. MR 86e:82049
37. J. Milnor, Volumes of hyperbolic three-manifolds, Chapter 7 in lecture notes of W. P. Thurston, The Geometry and Topology of Three-manifolds, Princeton, 1978.
38. J. Propp, Lattice structure for orientations of graphs, preprint, 1993.
39. J. Propp, Boundary-dependent local behavior for 2-D dimer models, Internat. J. Modern Phys. B 11 (1997), 183-187. MR 98a:82025
40. J. Propp, Boundary-dependent local behavior for 2-D dimer models, Internat. J. Modern Phys. B 11 (1997), 183-187. MR 98a:82025
41. J. G. Propp and D. B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9 (1996), 223-252. MR 99k:60176
42. J. G. Propp and D. B. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms 9 (1996), 223-252. MR 99k:60176
43. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987. MR 88k:00002
44. W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, New York, 1987. MR 88k:00002
45. N. Saldanha, C. Tomei, M. Casarin, Jr., and D. Romualdo, Spaces of domino tilings, Discrete Comput. Geom. 14 (1995), 207-233. MR 96e:52050
46. N. Saldanha, C. Tomei, M. Casarin, Jr., and D. Romualdo, Spaces of domino tilings, Discrete Comput. Geom. 14 (1995), 207-233. MR 96e:52050
47. H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics - an exact result, Phil. Mag. 6 (1961), 1061-1063. MR 24:B2436
48. H. N. V. Temperley and M. E. Fisher, Dimer problem in statistical mechanics - an exact result, Phil. Mag. 6 (1961), 1061-1063. MR 24:B2436
49. W. P. Thurston, Conway's tiling groups, Amer. Math. Monthly 97 (1990), 757-773. MR 91k:52028
50. W. P. Thurston, Conway's tiling groups, Amer. Math. Monthly 97 (1990), 757-773. MR 91k:52028