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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
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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.

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    [1]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]2 D.R. Nelson and L. Peliti, Fluctuations in Membranes with Crystalline and Hexatic Order, J. Physique 48 (1987) 1085. [3]3 M. Paczuski, M. Kardar and D.R. Nelson, Landau Theory of The Crumpling Transition, Phys. Rev. Lett. 60 (1988) 2638. [4]4 F. David and E. Guitter, Crumpling Transition in Elastic Membranes: Renormalization Group Treatment, Europhys. Lett. 5 (1988) 709. [5]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]6 Y. Kantor and M.V. Jarić, Triangular Lattice Foldings—a Transfer Matrix Study, Europhys. Lett. 11 (1990) 157-161. [7]7 P. Di Francesco and E. Guitter, Entropy of Folding of the Triangular Lattice, Europhys. Lett. 26 (1994) 455.
  • M. Bowick, P. Di Francesco, O. Golinelli, and E. Guitter, Three-dimensional folding of the triangular lattice, Nuclear Phys. B 450 (1995), no. 3, 463–494. MR 1351678, DOI
  • [9]9 P. Di Francesco and E. Guitter, Folding Transition of the Triangular Lattice, Phys. Rev. E50 (1994) 4418-4426. [10]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.
  • P. Di Francesco, Folding transitions of the square-diagonal lattice, Nuclear Phys. B 528 (1998), no. 3, 453–468. MR 1643647, DOI
  • [12]12 R.J. Baxter, Colorings of a Hexagonal Lattice, J. Math. Phys. 11 (1970) 784-789, ; and q-Colourings of the Triangular Lattice, J. Phys. A19 Math. Gen. (1986) 2821-2839.
  • P. Di Francesco, Folding the square-diagonal lattice, Nuclear Phys. B 525 (1998), no. 3, 507–548. MR 1639304, DOI
  • [14]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 ; see also the book by P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Singapore (1991) for a review.
  • Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982. MR 690578
  • [16]16 This problem is discussed in the mathematical entertainment section, edited by A. Shen, of the Mathematical Intelligencer, Volume 19 number 4 (1997) 48.
  • P. Di Francesco, P. Ginsparg, and J. Zinn-Justin, $2$D gravity and random matrices, Phys. Rep. 254 (1995), no. 1-2, 133. MR 1320471, DOI
  • P. Di Francesco, B. Eynard, and E. Guitter, Coloring random triangulations, Nuclear Phys. B 516 (1998), no. 3, 543–587. MR 1625175, DOI
  • [19]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.
  • I. Krichever, O. Lipan, P. Wiegmann, and A. Zabrodin, Quantum integrable models and discrete classical Hirota equations, Comm. Math. Phys. 188 (1997), no. 2, 267–304. MR 1471815, DOI
  • C. Itzykson and J. B. Zuber, The planar approximation. II, J. Math. Phys. 21 (1980), no. 3, 411–421. MR 562985, DOI
  • [22]22 Harish-Chandra, Differential operators on a semi-simple Lie algebra, Amer. Jour. of Math 79 (1957) 87 ; 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.
  • W. T. Tutte, A census of planar maps, Canadian J. Math. 15 (1963), 249–271. MR 146823, DOI
  • [24]24 A. Sainte-Laguë, Avec des nombres et des lignes (Récréations Mathématiques), Vuibert, Paris (1937). [25]25 J. Touchard, Contribution à l’étude du problème des timbres poste, Canad. J. Math. 2 (1950) 385-398.
  • W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193–199. MR 221957, DOI
  • V. I. Arnol′d, The branched covering ${\bf C}{\rm P}^2\to S^4$, hyperbolicity and projective topology, Sibirsk. Mat. Zh. 29 (1988), no. 5, 36–47, 237 (Russian); English transl., Siberian Math. J. 29 (1988), no. 5, 717–726 (1989). MR 971226, DOI
  • Ki Hyoung Ko and Lawrence Smolinsky, A combinatorial matrix in $3$-manifold theory, Pacific J. Math. 149 (1991), no. 2, 319–336. MR 1105701
  • [29]29 S. Lando and A. Zvonkin, Plane and projective meanders, Theor. Comp. Sci. 117 (1993) 227-241 ; and Meanders, Selecta Math. Sov. 11 (1992) 117-144.
  • P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding, and arch statistics, Math. Comput. Modelling 26 (1997), no. 8-10, 97–147. Combinatorics and physics (Marseilles, 1995). MR 1492504, DOI
  • P. Di Francesco, O. Golinelli, and E. Guitter, Meanders and the Temperley-Lieb algebra, Comm. Math. Phys. 186 (1997), no. 1, 1–59. MR 1462755, DOI
  • [32]32 R. Bacher, Meander Algebras, prépublication de l’Institut Fourier n$^{o}$ $478$ (1999).
  • P. Di Francesco, O. Golinelli, and E. Guitter, Meanders: a direct enumeration approach, Nuclear Phys. B 482 (1996), no. 3, 497–535. MR 1427435, DOI
  • [34]34 O. Golinelli, A Monte-Carlo study of meanders, preprint cond-mat/9906329, to appear in EPJ B (2000). [35]35 I. Jensen, Enumeration of plane meanders, preprint cond-mat/9910313. [36]36 P. Di Francesco, O. Golinelli and E. Guitter, Meanders: exact asymptotics, preprint cond-mat/9910453, to appear in Nucl. Phys. B (2000). [37]37 P. Di Francesco, Matrix model combinatorics: applications to folding and coloring, M.S.R.I. lecture notes, preprint math-ph/9911002 (1999). [38]38 J. Jacobsen and J. Kondev, Field theory of compact polymers on the square lattice, Nucl. Phys. B 532 [FS], (1998) 635-688 ; Transition from the compact to the dense phase of two-dimensional polymers, J. Stat. Phys. 96, (1999) 21-48. [39]39 P. Di Francesco, E. Guitter and J. Jacobsen, work in progress. [40]40 V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal Structure of 2-D Quantum Gravity, Mod. Phys. Lett. A3 (1988) 819, ; F. David, Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge, Mod. Phys. Lett. A3 (1988) 1651, ; 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.
  • P. Di Francesco, Meander determinants, Comm. Math. Phys. 191 (1998), no. 3, 543–583. MR 1608551, DOI
  • P. Di Francesco, ${\rm SU}(N)$ meander determinants, J. Math. Phys. 38 (1997), no. 11, 5905–5943. MR 1480837, DOI
  • Dietmar Bisch and Vaughan Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), no. 1, 89–157. MR 1437496, DOI
  • P. di Francesco, New integrable lattice models from Fuss-Catalan algebras, Nuclear Phys. B 532 (1998), no. 3, 609–634. MR 1657030, DOI
  • A. B. Zamolodchikov, Tetrahedron equations and the relativistic $S$-matrix of straight-strings in $2+1$-dimensions, Comm. Math. Phys. 79 (1981), no. 4, 489–505. MR 623964

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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 19, 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