Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Folding and coloring problems in mathematics and physics

Author: P. Di Francesco
Journal: Bull. Amer. Math. Soc. 37 (2000), 251-307
MSC (2000): Primary 82-02; Secondary 82B20, 82B41, 83C27, 05A15, 05A16, 05C15, 05C30, 05C80, 05E99, 03D20, 16G99
Published electronically: April 10, 2000
MathSciNet review: 1754642
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We review various folding problems arising in the physics of membranes and polymers. These are (1) the phantom folding of tethered membranes, i.e. the two-dimensional lattice folding; (2) the phantom folding of fluid membranes, i.e. the folding of tessellations of arbitrary genus; (3) the self-avoiding folding of polymers, i.e. the meander problem. All three problems are found to be related to coloring problems and possess one kind of underlying integrable structure, in different guises. Many mathematical results follow from taking advantage of this fact.

References [Enhancements On Off] (What's this?)

  • [1] Y. Kantor and D.R. Nelson, Crumpling Transition in Polymerized Membranes, Phys. Rev. Lett. 58 (1987) 2774 and Phase Transitions in Flexible Polymeric Surfaces, Phys. Rev. A 36 (1987) 4020.
  • [2] D.R. Nelson and L. Peliti, Fluctuations in Membranes with Crystalline and Hexatic Order, J. Physique 48 (1987) 1085.
  • [3] M. Paczuski, M. Kardar and D.R. Nelson, Landau Theory of The Crumpling Transition, Phys. Rev. Lett. 60 (1988) 2638.
  • [4] F. David and E. Guitter, Crumpling Transition in Elastic Membranes: Renormalization Group Treatment, Europhys. Lett. 5 (1988) 709.
  • [5] M. Baig, D. Espriu and J. Wheater, Phase Transitions in Random Surfaces, Nucl. Phys. B314 (1989) 587; R. Renken and J. Kogut, Scaling Behavior at the Crumpling Transition, Nucl. Phys. B342 (1990) 753; R. Harnish and J. Wheater, The Crumpling Transition of Crystalline Random Surfaces, Nucl. Phys. B350 (1991) 861; J. Wheater and P. Stephenson, On the Crumpling Transition in Crystalline Random Surfaces, Phys. Lett. B302 (1993) 447.
  • [6] Y. Kantor and M.V. Jaric, Triangular Lattice Foldings--a Transfer Matrix Study, Europhys. Lett. 11 (1990) 157-161.
  • [7] P. Di Francesco and E. Guitter, Entropy of Folding of the Triangular Lattice, Europhys. Lett. 26 (1994) 455.
  • [8] M. Bowick, P. Di Francesco, O. Golinelli and E. Guitter, 3D folding of the triangular lattice, Nucl. Phys. B450[FS] (1995) 463-494. MR 97a:82014
  • [9] P. Di Francesco and E. Guitter, Folding Transition of the Triangular Lattice, Phys. Rev. E50 (1994) 4418-4426.
  • [10] P. Di Francesco, E. Guitter and S. Mori, Folding of the triangular lattice with quenched bending rigidity, Phys. Rev. E 55 No. 1 (1997) 237-251.
  • [11] P. Di Francesco, Folding Transitions of the Square-Diagonal Lattice, Nucl. Phys. B528 (1998) 453-468. MR 99h:82029
  • [12] R.J. Baxter, Colorings of a Hexagonal Lattice, J. Math. Phys. 11 (1970) 784-789, MR 42:1457; and q-Colourings of the Triangular Lattice, J. Phys. A19 Math. Gen. (1986) 2821-2839. MR 87k:82123
  • [13] P. Di Francesco, Folding the Square-Diagonal Lattice, Nucl. Phys. B525[FS] (1998) 507-548. MR 99k:82033
  • [14] H. Temperley and 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. Roy. Soc. A322 (1971) 251-280 MR 58:16425; see also the book by P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Singapore (1991) for a review. MR 92m:82030
  • [15] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982). MR 86i:82002a
  • [16] This problem is discussed in the mathematical entertainment section, edited by A. Shen, of the Mathematical Intelligencer, Volume 19 number 4 (1997) 48.
  • [17] P. Di Francesco, P. Ginsparg and J. Zinn-Justin, 2D Gravity and Random Matrices, Physics Reports 254 (1995) 1-131. MR 96c:81191
  • [18] P. Di Francesco, B. Eynard and E. Guitter, Coloring Random Triangulations, Nucl. Phys. B516[FS] (1998) 543-587. MR 2000c:82039
  • [19] B. Eynard and C. Kristjansen, An iterative solution of the 3-color problem on a random lattice, Nucl. Phys. B516[FS] (1998) 592-542. CMP 98:13
  • [20] I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Comm. Math. Phys. 188 (1997) 267. MR 99c:58076
  • [21] C. Itzykson and J.-B. Zuber, The planar approximation II, J. Math. Phys. 21 (1980) 411. MR 81a:81068
  • [22] Harish-Chandra, Differential operators on a semi-simple Lie algebra, Amer. Jour. of Math 79 (1957) 87 MR 18:809d; J. Duistermaat and G. Heckman, On the variation of cohomology of the symplectic form of the reduced phase space, Inv. Math. 69 (1982) 259-268. MR 84h:58051a
  • [23] W. Tutte, A Census of Planar Maps, Canad. Jour. of Math. 15 (1963) 249. MR 26:4343
  • [24] A. Sainte-Laguë, Avec des nombres et des lignes (Récréations Mathématiques), Vuibert, Paris (1937).
  • [25] J. Touchard, Contribution à l'étude du problème des timbres poste, Canad. J. Math. 2 (1950) 385-398. MR 12:312i
  • [26] W. Lunnon, A map-folding problem, Math. of Comp. 22 (1968) 193-199. MR 36:5009
  • [27] V. Arnold, The branched covering of $CP_{2} \to S_{4}$, hyperbolicity and projective topology, Siberian Math. Jour. 29 (1988) 717-726. MR 90a:57037
  • [28] K.H. Ko, L. Smolinsky, A combinatorial matrix in $3$-manifold theory, Pacific. J. Math 149 (1991) 319-336. MR 92d:57008
  • [29] S. Lando and A. Zvonkin, Plane and projective meanders, Theor. Comp. Sci. 117 (1993) 227-241 MR 94i:05004; and Meanders, Selecta Math. Sov. 11 (1992) 117-144. MR 93k:05013
  • [30] P. Di Francesco, O. Golinelli and E. Guitter, Meander, Folding and Arch Statistics, Math. Comput. Modelling, Vol. 26, No.8-10 (1997) 97-147. MR 99f:82029
  • [31] P. Di Francesco, O. Golinelli and E. Guitter, Meanders and the Temperley-Lieb Algebra, Commun. Math. Phys. 186 (1997), 1-59. MR 99f:82028
  • [32] R. Bacher, Meander Algebras, prépublication de l'Institut Fourier n$^{o}$ $478$ (1999).
  • [33] P. Di Francesco, O. Golinelli and E. Guitter, Meanders: a direct enumeration approach, Nucl. Phys. B482[FS] (1996), 497-535. MR 97j:82074
  • [34] O. Golinelli, A Monte-Carlo study of meanders, preprint cond-mat/9906329, to appear in EPJ B (2000).
  • [35] I. Jensen, Enumeration of plane meanders, preprint cond-mat/9910313.
  • [36] P. Di Francesco, O. Golinelli and E. Guitter, Meanders: exact asymptotics, preprint cond-mat/9910453, to appear in Nucl. Phys. B (2000).
  • [37] P. Di Francesco, Matrix model combinatorics: applications to folding and coloring, M.S.R.I. lecture notes, preprint math-ph/9911002 (1999).
  • [38] J. Jacobsen and J. Kondev, Field theory of compact polymers on the square lattice, Nucl. Phys. B 532 [FS], (1998) 635-688 MR 99f:82027; Transition from the compact to the dense phase of two-dimensional polymers, J. Stat. Phys. 96, (1999) 21-48.
  • [39] P. Di Francesco, E. Guitter and J. Jacobsen, work in progress.
  • [40] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2-D Quantum Gravity, Mod. Phys. Lett. A3 (1988) 819, MR 89i:83039; F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge, Mod. Phys. Lett. A3 (1988) 1651, MR 89k:81138; J. Distler and H. Kawai, Conformal Field Theory and 2-D Quantum Gravity or Who's Afraid of Joseph Liouville?, Nucl. Phys. B321 (1989) 509. MR 90g:81226
  • [41] P. Di Francesco, Meander Determinants, Commun. Math. Phys. 191 (1998) 543-583. MR 99e:05007
  • [42] P. Di Francesco, SU(N) Meander Determinants, Jour. Math. Phys. 38 (1997) 5905-5943. MR 99k:05019
  • [43] D. Bisch and V. Jones, Algebras associated to intermediate subfactors, Inv. Math. 128 (1997) 89-157. MR 99c:46072
  • [44] P. Di Francesco, New Integrable Lattice Models from Fuss-Catalan Algebras, Nucl. Phys. B532 (1998) 609-634. MR 99k:82020
  • [45] A.B. Zamolodchikov, Tetrahedron Equations and the Relativistic $S$-Matrix of Straight Strings in $(2+1)$-Dimensions, Commun. Math. Phys. 79 (1981) 489-505. MR 82k:81081

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 82-02, 82B20, 82B41, 83C27, 05A15, 05A16, 05C15, 05C30, 05C80, 05E99, 03D20, 16G99

Retrieve articles in all journals with MSC (2000): 82-02, 82B20, 82B41, 83C27, 05A15, 05A16, 05C15, 05C30, 05C80, 05E99, 03D20, 16G99

Additional Information

P. Di Francesco
Affiliation: Service de Physique Théorique, C.E.A. Saclay, F-91191 Gif sur Yvette, France

Received by editor(s): November 1, 1998
Received by editor(s) in revised form: February 10, 2000
Published electronically: April 10, 2000
Additional Notes: Work partially supported by NSF grant PHY-9722060.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society