Folding and coloring problems in mathematics and physics
HTML articles powered by AMS MathViewer
- by P. Di Francesco PDF
- Bull. Amer. Math. Soc. 37 (2000), 251-307 Request permission
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
-
[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 10.1016/0550-3213(95)00290-9 [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 10.1016/S0550-3213(98)00431-3
- Julian Bonder, Über die Darstellung gewisser, in der Theorie der Flügelschwingungen auftretender Integrale durch Zylinderfunktionen, Z. Angew. Math. Mech. 19 (1939), 251–252 (German). MR 42, DOI 10.1002/zamm.19390190409
- P. Di Francesco, Folding the square-diagonal lattice, Nuclear Phys. B 525 (1998), no. 3, 507–548. MR 1639304, DOI 10.1016/S0550-3213(98)00320-4
- Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
- 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 10.1016/0370-1573(94)00084-G
- P. Di Francesco, B. Eynard, and E. Guitter, Coloring random triangulations, Nuclear Phys. B 516 (1998), no. 3, 543–587. MR 1625175, DOI 10.1016/S0550-3213(98)00037-6 [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 10.1007/s002200050165
- C. Itzykson and J. B. Zuber, The planar approximation. II, J. Math. Phys. 21 (1980), no. 3, 411–421. MR 562985, DOI 10.1063/1.524438
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- W. T. Tutte, A census of planar maps, Canadian J. Math. 15 (1963), 249–271. MR 146823, DOI 10.4153/CJM-1963-029-x [24]24 A. Sainte-Laguë, Avec des nombres et des lignes (Récréations Mathématiques), Vuibert, Paris (1937).
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193–199. MR 221957, DOI 10.1090/S0025-5718-1968-0221957-8
- V. I. Arnol′d, The branched covering $\textbf {C}\textrm {P}^2\to S^4$, 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 10.1007/BF00970265
- Ki Hyoung Ko and Lawrence Smolinsky, A combinatorial matrix in $3$-manifold theory, Pacific J. Math. 149 (1991), no. 2, 319–336. MR 1105701, DOI 10.2140/pjm.1991.149.319
- Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
- 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 10.1016/S0895-7177(97)00202-1
- 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 10.1007/BF02885671 [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 10.1016/S0550-3213(96)00505-6 [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).
- Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99 [39]39 P. Di Francesco, E. Guitter and J. Jacobsen, work in progress.
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
- P. Di Francesco, Meander determinants, Comm. Math. Phys. 191 (1998), no. 3, 543–583. MR 1608551, DOI 10.1007/s002200050277
- P. Di Francesco, $\textrm {SU}(N)$ meander determinants, J. Math. Phys. 38 (1997), no. 11, 5905–5943. MR 1480837, DOI 10.1063/1.532173
- Dietmar Bisch and Vaughan Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), no. 1, 89–157. MR 1437496, DOI 10.1007/s002220050137
- P. di Francesco, New integrable lattice models from Fuss-Catalan algebras, Nuclear Phys. B 532 (1998), no. 3, 609–634. MR 1657030, DOI 10.1016/S0550-3213(98)00603-8
- 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, DOI 10.1016/0550-3213(89)90409-4
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
- Email: philippe@spht.saclay.cea.fr
- 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.
- © Copyright 2000 American Mathematical Society
- 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
- DOI: https://doi.org/10.1090/S0273-0979-00-00870-3
- MathSciNet review: 1754642