Geometrically constrained statistical systems on regular and random lattices: From folding to meanders

Physics Reports

Volumn 415, Issue 1, 2005, Pages 1-88

  Di Francesco, Philippe ^a   Guitter, E ^a  

^a CEA SACLAY   (France)

Author keywords

Coloring; Eulerian gravity; Folding; Fully packed loops; Hamiltonian cycles; Meanders

Indexed keywords

EID: 22144433325     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2005.05.001     Document Type: Review

References (109)

1. J. Buchner, T. Kiefhaber (Eds.), Protein Folding Handbook, Wiley, New York, 2005.
2. D. Nelson, T. Piran, S. Weinberg (Eds.), Statistical Mechanics of Membranes and Surfaces, second ed., World Scientific Publishing, Singapore, 2004.
3. A. Lobkovsky S. Gentges H. Li D. Morse T. Witten Scaling properties of stretching ridges in a crumpled elastic sheet Science 270 1995 1482-1485
4. P. Di Francesco E. Guitter Entropy of folding of the triangular lattice Europhys. Lett. 26 1994 455-460 arXiv:condmat/9402058.
5. P. Di Francesco E. Guitter Folding transition of the triangular lattice Phys. Rev. E 50 1994 4418-4426 arXiv:condmat/9406041.
6. M. Bowick P. Di Francesco O. Golinelli E. Guitter Three-dimensional folding of the triangular lattice Nucl. Phys. B 450 FS 1995 463-494 arXiv:cond-mat/9502063.
7. P. Di Francesco E. Guitter S. Mori Folding of the triangular lattice with quenched random bending rigidity Phys. Rev. E 55 1997 237-251arXiv:cond-mat/9607077.
8. M. Bowick O. Golinelli E. Guitter S. Mori Geometrical folding transitions of the triangular lattice in the face-centered cubic lattice Nucl. Phys. B 495 FS 1997 583-607 arXiv:cond-mat/9611105.
9. P. Di Francesco B. Eynard E. Guitter Coloring random triangulations Nucl. Phys. B 516 FS 1998 543-587 arXiv:condmat/9711050.
10. B. Eynard E. Guitter C. Kristjansen Hamiltonian cycles on a random three-coordinate lattice Nucl. Phys. B 528 1998 523-532 arXiv:cond-mat/ 9801281.
11. E. Guitter C. Kristjansen J. Nielsen Hamiltonian cycles on random Eulerian triangulations Nucl. Phys. B 546 1999 731-750 arXiv:cond-mat/9811289.
12. P. Di Francesco E. Guitter C. Kristjansen Fully packed O (n = 1) model on random Eulerian triangulations Nucl. Phys. B 549 1999 657-667 arXiv:cond-mat/9902082
13. P. Di Francesco O. Golinelli E. Guitter Meander, folding and arch statistics Math. Comput. Modelling 26 1997 97-147 arXiv:hep-th/9506030.
14. P. Di Francesco O. Golinelli E. Guitter Meanders and the Temperley-Lieb algebra Commun. Math. Phys. 186 1997 1-59
15. P. Di Francesco O. Golinelli E. Guitter Meanders: A direct enumeration approach Nucl. Phys. B 482 FS 1996 497-535 59, arXiv:hep-th/9602025.
16. P. Di Francesco O. Golinelli E. Guitter Meanders: Exact asymptotics Nucl. Phys. B 570 2000 699-712 arXiv:condmat/9910453.
17. P. Di Francesco E. Guitter J. Jacobsen Exact meander asymptotics: A numerical check Nucl. Phys. B 580 2000 757-795 arXiv:cond-mat/0003008.
18. P. Di Francesco Folding and coloring problems in mathematics and physics Bull. Am. Math. Soc. 37 3 2000 251-307
19. P. Di Francesco Matrix model combinatorics: Applications to folding and coloring In: P. Bleher A. Eds. Its Random Matrices and their Applications vol. 40 2001 Cambridge University Press, MSRI Publications Cambridge 111-170 arXiv:math-ph/9911002.
20. P. Di Francesco, Geometrically constrained statistical models on fixed and random lattices: From hard squares to meanders, arXiv:cond-mat/ 0211591.
21. D.R. Nelson L. Peliti Fluctuations in membranes with crystalline and hexatic order J. Phys. 48 1987 1085-1092
22. Y. Kantor D.R. Nelson Crumpling transition in polymerized membranes Phys. Rev. Lett. 58 1987 2774-2777
23. Y. Kantor D.R. Nelson Phase transitions in flexible polymeric surfaces Phys. Rev. A 36 1987 4020-4032
24. M. Paczuski M. Kardar D.R. Nelson Landau theory of the crumpling transition Phys. Rev. Lett. 60 1988 2638-2640
25. F. David E. Guitter Crumpling transition in elastic membranes: renormalization group treatment Europhys. Lett. 5 1988 709-713
26. P. Di Francesco Folding the square-diagonal lattice Nucl. Phys. B 525 1998 507-548 arXiv:cond-mat/9803051
27. P. Di Francesco Folding transitions of the square-diagonal lattice Nucl. Phys. B 528 1998 609-634arXiv:condmat/9804073.
28. Y. Kantor M.V. Jarić Triangular lattice foldings: A transfer matrix study Europhys. Lett. 11 1990 157-161
29. R.J. Baxter Colorings of a hexagonal lattice J. Math. Phys. 11 1970 784-789
30. E.N.M. Cirillo G. Gonnella A. Pelizzola Folding transitions of the triangular lattice with defects Phys. Rev. E 53 1996 1479-1484 arXiv:hep-th/9507161.
31. E.N.M. Cirillo G. Gonnella A. Pelizzola Folding transition of the triangular lattice in a discrete three-dimensional space Phys. Rev. E 53 1996 3253-3256 arXiv:hep-th/9512069
32. H.W.J. Blöte B. Nienhuis Fully-packed loop model on the honeycomb lattice Phys. Rev. Lett. 72 1994 1372-1375
33. R.J. Baxter Exactly Solved Models in Statistical Mechanics 1984 Academic Press London
34. For an introduction to the Coulomb gas formalism, see B. Nienhuis, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, vol. 11, Academic Press, New York, 1987.
35. E.F. Shender V.B. Cherepanov P.C.W. Holdsworth A.J. Berlinsky Kagome antiferromagnet with defects: Satisfaction, frustration, and spin folding in a random spin system Phys. Rev. Lett. 70 1993 3812-3815 arXiv:cond-mat/9303031.
36. J. Kondev, J. de Gier, B. Nienhuis, Operator spectrum and exact exponents of the fully-packed loop model, J. Phys. A 29: Math. Gen. (1996) 6489-6504, arXiv:cond-mat/9603170.
37. For a general introduction to conformal field theory, see P. Di Francesco, P. Mathieu, D. Sénéchal. Conformal Field Theory, Graduate Texts in Contemporary Physics, first ed., 1996, pp. 1-890; second ed., 1999, Springer, Berlin, pp. 1-890.
38. P.A. Mac Mahon, Combinatory Analysis, vol. 2, Cambridge University Press, Cambridge, 1916, reprinted by Chelsea, New York, 1960.
39. M.T. Batchelor J. Suzuki C.M. Yung Exact results for Hamiltonian walks from the solution of the fully-packed loop model on the honeycomb lattice Phys. Rev. Lett. 73 1994 2646-2649 arXiv:cond-mat/9308083.
40. V. Kazakov Bilocal regularization of models of random surfaces Phys. Lett. B 150 1985 282-284
41. F. David Planar diagrams, two-dimensional lattice gravity and surface models Nucl. Phys. B 257 1985 45-58
42. J. Ambjorn B. Durhuus J. Fröhlich Diseases of triangulated random surface models and possible cures Nucl. Phys. B 257 1985 433-449
43. V. Kazakov I. Kostov A. Migdal Critical properties of randomly triangulated planar random surfaces Phys. Lett. B 157 1985 295-300
44. E. Brézin C. Itzykson G. Parisi J.-B. Zuber Planar diagrams Commun. Math. Phys. 59 1978 35-51
45. P. Di Francesco P. Ginsparg J. Zinn-Justin 2D gravity and random matrices Phys. Rep. 254 1995 1-131 arXiv:hepth/9306153.
46. B. Eynard, Random Matrices, Saclay Lecture Notes, 2000, http://www-spht.cea.fr/lectures_notes.shtml.
47. V.G. Knizhnik A.M. Polyakov A.B. Zamolodchikov Fractal structure of 2D quantum gravity Mod. Phys. Lett. A 3 1988 819-826
48. F. David Conformal field theories coupled to 2D gravity in the conformal gauge Mod. Phys. Lett. A 3 1988 1651-1656
49. J. Distler H. Kawai Conformal field theory and 2D quantum gravity Nucl. Phys. B 321 1989 509-527
50. This problem is discussed in the mathematical entertainments section, A. Shen (Ed.), Math. Intell. 19(4) (1997) 48.
51. W. Tutte A census of planar maps Can. J. Math. 15 1963 249-271
52. P. Di Francesco Rectangular matrix models and combinatorics of colored graphs Nucl. Phys. B 648 2003 461-496 arXiv:cond-mat/0208037.
53. G. Schaeffer Bijective census and random generation of Eulerian planar maps Electron. J. Combin. 4 1997 R20
54. G. Schaeffer Conjugaison d'arbres et cartes combinatoires aléatoires, Ph.D. Thesis, Université Bordeaux I, 1998.
55. J. Bouttier P. Di Francesco E. Guitter Census of planar maps: From the one-matrix model solution to a combinatorial proof Nucl. Phys. B 645 2002 477-499 arXiv:cond-mat/0207682.
56. J. Bouttier P. Di Francesco E. Guitter Counting colored random triangulations Nucl. Phys. B 641 FS 2002 519-532 arXiv:cond-mat/0206452.
57. M. Bousquet-Mélou G. Schaeffer Enumeration of planar constellations Adv. Appl. Math. 24 2000 337-368
58. B. Eynard C. Kristjansen An iterative solution of the three-color problem on a random lattice Nucl. Phys. B 516 1998 529-542 arXiv:cond-mat/9710199.
59. I. Kostov Exact solution of the three-color problem on a random lattice Phys. Lett. B 549 2002 245-252 arXiv:hepth/0005190
60. I. Kostov O (n) vector model on a planar random lattice: Spectrum of anomalous dimensions Mod. Phys. Lett. A 4 1989 217-226
61. B. Eynard C. Kristjansen More on the exact solution of the O (n) model on a random lattice and an investigation of the case n > 2 Nucl. Phys. B 466 1996 463-487 arXiv:hep-th/9512052.
62. B. Duplantier I. Kostov Conformal spectra of polymers on a random surface Phys. Rev. Lett. 61 1988 1433-1437
63. For a combinatorial interpretation of this formula, see R. Cori, S. Dulucq, G. Viennot Eds. Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory A 43 (1986) 1-22.
64. V. Kazakov P. Zinn-Justin Two-matrix model with ABAB interaction Nucl. Phys. B 546 1999 647-668 arXiv:hepth/9808043.
65. E. Lucas Théorie des Nombres I 1891 Gauthier-Villars Paris reedited by Blanchard, Paris, 1961.
66. A. Sainte Laguë, Avec des Nombres et des Lignes: Récréations Mathématiques, Vuibert, Paris, 1937, p. 147.
67. J. Touchard Contributions à l'étude du problème des timbres poste Can. J. Math. 2 1950 385-398
68. J. Koehler Folding a strip of stamps J. Combin. Theory 5 1968 135-152
69. W. Lunnon A map-folding problem Math. Comput. 22 1968 193-199
70. 4, hyperbolicity and projective topology Siberian Math. J. 29 1988 717-726
71. S. Lando A. Zvonkin Plane and projective meanders Theor. Comput. Sci. 117 1993 227-241
72. S. Lando A. Zvonkin Meanders Selecta Math. Sov. 11 1992 117-144
73. K.H. Ko L. Smolinsky A combinatorial matrix in 3-manifold theory Pacific. J. Math. 149 1991 319-336
74. K. Hoffman K. Mehlhorn P. Rosenstiehl R. Tarjan Sorting Jordan sequences in linear time using level-linked search trees Inform. Control 68 1986 170-184
75. A. Phillips, The Visual Mind: Art and Mathematics, MIT Press, Cambridge, MA, p. 65, 1993; see also http://www.math.sunysb.edu/~tony/mazes/.
76. L. Chekhov C. Kristjansen Hermitian matrix model with plaquette interaction Nucl.Phys. B 479 FS 1996 683-696 arXiv:hep-th/9605013.
77. Y. Makeenko Strings matrix models and meanders Nucl. Phys. Proc. Suppl. 49 1996 226-237 arXiv:hep-th/9512211
78. G. Semenoff R. Szabo Fermionic matrix models Int. J. Mod. Phys. A 12 1997 2135-2292 arXiv:hep-th/9605140
79. H. Temperley E. Lieb Relations between the percolation and coloring problems and other graph-theoretical problems associated with regular planar lattices: Some exact results for the percolation problem Proc. R. Soc. A 322 1971 251-280
80. see also the book by P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Singapore, 1991, for a review.
81. P. Di Francesco Meander determinants Commun. Math. Phys. 191 1998 543-583 arXiv:hep-th/9612026.
82. P. Di Francesco SU (N) meander determinants J. Math. Phys. 38 1997 5905-5943 arXiv:hep-th/9702181.
83. J. Jacobsen J. Kondev Field theory of compact polymers on the square lattice Nucl. Phys. B 532 FS 1998 635-688 arXiv:cond-mat/9804048.
84. D. Dei Cont B. Nienhuis The packing of two species of polygons on the square lattice J. Phys. A: Math. Gen. 37 2004 3085-3100 arXiv:cond-mat/0311244
85. 2 model, arXiv:condmat/0412018.
86. 2 model J. Phys. A: Math. Gen. 37 2004 7213-7225 arXiv:math-ph/0402008.
87. J.L. Jacobsen J. Kondev Transition from the compact to the dense phase of two-dimensional polymers J. Stat. Phys. 96 1999 21-48 48, arXiv:cond-mat/9811085
88. I. Jensen, Enumerations of plane meanders, arXiv:cond-mat/9910313, 1999. Talk presented at Stat Phys-Taipei, 1999.
89. I. Jensen A. Guttmann Critical exponents of plane meanders J. Phys. A 33 2000 L187-L192 arXiv:cond-mat/0004321
90. I. Jensen A transfer matrix approach to the enumeration of plane meanders J. Phys. A 33 2000 5953-5963 arXiv:condmat/0008178
91. O. Golinelli A Monte-Carlo study of meanders Eur. Phys. J. B 14 2000 145-155 arXiv:cond-mat/9906329.
92. See for instance D. Poulalhon, G. Schaeffer, A note on bipartite Eulerian planar maps, http://www.loria.fr/~schaeffe/
93. D. Gaunt M. Fisher Hard-sphere lattice gases, I: Plane-square lattice J. Chem. Phys. 43 1965 2840-2863
94. L. Runnels L. Combs J. Salvant Exact finite methods of lattice statistics, II: Honeycomb-lattice gas of hard molecules J. Chem. Phys. 47 1967 4015-4020
95. R.J. Baxter I.G. Enting S.K. Tsang Hard square lattice gas J. Stat. Phys. 22 1980 465-489
96. R.J. Baxter Hard hexagons: Exact solution J. Phys. A 13 1980 L61-L70
97. R.J. Baxter S.K. Tsang Entropy of hard hexagons J. Phys. A 13 1980 1023-1030
98. see also R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1987.
99. R.J. Baxter, Planar lattice gases with nearest-neighbour exclusion, Ann. Combin. (3) (1999) 191-203, arXiv:cond-mat/9811264.
100. J. Bouttier, P. Di Francesco, E. Guitter, Critical and tricritical hard objects on bicolourable random lattices: Exact solutions, J. Phys. A 35: Math. Gen. (2002) 3821-3854, arXiv:cond-mat/0201213.
101. G. Schaeffer P. Zinn-Justin On the asymptotic number of plane curves and alternating knots Exp. Math. 13 4 2005 483-494 arXiv:math-ph/0304034.
102. D. Bisch V. Jones Algebras associated to intermediate subfactors Invent. Math. 128 1997 89-157
103. W. Lunnon Multi-dimensional strip folding Comput. J. 14 1971 75-79
104. P. Chassaing G. Schaeffer Random planar lattices and integrated superBrownian excursion Probab. Theory Related Fields 128 2 2004 161-212 arXiv:math.CO/0205226.
105. B. Duplantier Random walks and quantum gravity in two dimensions Phys. Rev. Lett. 81 1998 5489-5492
106. B. Duplantier Harmonic measure exponents for two-dimensional percolation Phys. Rev. Lett. 82 1999 3940-3943 arXiv:cond-mat/9901008.
107. G. Lawler O. Schramm W. Werner Values of Brownian intersection exponents I: Half-plane exponents, II: Plane exponents, III: Two-sided exponents Acta Math. 187 2001 237-273, 275-308
108. G. Lawler,O. Schramm,W.Werner, Ann. Inst. H. Poincaré PR 38 (2002) 109-123, arXiv:math.PR/9911084, 0003156, 0005294
109. J. Bouttier P. Di Francesco E. Guitter Planar maps as labeled mobiles Elec. J. Combin. 11 1 2004 R69 arXiv:math.CO/0405099.