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Communications in Mathematical Physics

, Volume 354, Issue 1, pp 31–84 | Cite as

Counting Coloured Planar Maps: Differential Equations

  • Olivier Bernardi
  • Mireille Bousquet-Mélou
Article

Abstract

We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic, that is, satisfies a non-trivial polynomial differential equation with respect to the edge variable. We give explicitly a differential system that characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. In statistical physics terms, we solve the q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating function was proved to be algebraic for certain values of q, including \({q=1, 2}\) and 3. It is known to be transcendental in general. In contrast, our differential system holds for an indeterminate q. For certain special cases of combinatorial interest (four colours; proper q-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.

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References

  1. 1.
    Beffara V., Duminil-Copin H.: The self-dual point of the two-dimensional random-cluster model is critical for \({q\geq 1}\). Probab. Theory Relat. Fields 153(3-4), 511–542 (2012)CrossRefMATHGoogle Scholar
  2. 2.
    Bernardi O., Bousquet-Mélou M.: Counting colored planar maps: algebraicity results. J. Combin. Theory Ser. B 101(5), 315–377 (2011) arXiv:0909.1695 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bessis D., Itzykson C., Zuber J.B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1(2), 109–157 (1980)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bollobás B.: Modern Graph Theory, Vol 184 of Graduate Texts in Mathematics. Springer, New York (1998)Google Scholar
  5. 5.
    Bonichon, N., Bousquet-Mélou, M., Fusy, É.: Baxter permutations and plane bipolar orientations. Sém. Lothar. Combin., 61A:Art. B61Ah, 29, 2009/11Google Scholar
  6. 6.
    Borot G., Bouttier J., Guitter E.: Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model. J. Phys. A 45, 494017 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bostan A., Raschel K., Salvy B.: Non-D-finite excursions in the quarter plane. J. Combin. Theory Ser. A 121, 45–63 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Boulatov D.V., Kazakov V.A.: The Ising model on a random planar lattice: the structure of the phase transition and the exact critical exponents. Phys. Lett. B 186(3-4), 379–384 (1987)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bousquet-Mélou, M.: Counting planar maps, coloured or uncoloured. In: Surveys in Combinatorics 2011, volume 392 of London Math. Soc. Lecture Note Ser., pp. 1–49. Cambridge Univ. Press, Cambridge (2011)Google Scholar
  10. 10.
    Bousquet-Mélou M., Courtiel J.: Spanning forests in regular planar maps. J. Combin. Theory Ser. A 135, 1–59 (2015) arXiv:1306.4536 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bousquet-Mélou M., Jehanne A.: Polynomial equations with one catalytic variable, algebraic series and map enumeration. J. Combin. Theory Ser. B 96, 623–672 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bousquet-Mélou, M., Mishna, M.: Walks with small steps in the quarter plane. In: Algorithmic Probability and Combinatorics, volume 520 of Contemp. Math., pp. 1–39. Amer. Math. Soc., Providence (2010)Google Scholar
  13. 13.
    Bousquet-Mélou, M., Schaeffer, G.: The degree distribution of bipartite planar maps: applications to the Ising model. In: Eriksson, K., Linusson, S. (eds.) Formal Power Series and Algebraic Combinatorics, pp. 312–323, Vadstena, Sweden, 2003. Long version on arXiv:math/0211070
  14. 14.
    Bouttier, J., Di Francesco, P., Guitter, E.: Planar maps as labeled mobiles. Electron. J. Combin., 11(1):Research Paper 69 (electronic) (2004)Google Scholar
  15. 15.
    Bouttier J., Francesco P., Guitter E.: Blocked edges on Eulerian maps and mobiles: application to spanning trees, hard particles and the Ising model. J. Phys. A 40(27), 7411–7440 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Brézin E., Itzykson C., Parisi G., Zuber J. B.: Planar diagrams. Comm. Math. Phys. 59(1), 35–51 (1978)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Brown W.G.: On the existence of square roots in certain rings of power series. Math. Ann. 158, 82–89 (1965)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Castelli Aleardi, L., Devillers, O., Schaeffer, G.: Optimal succinct representations of planar maps. In: Computational Geometry (SCG’06), pp. 309–318. ACM, New York (2006)Google Scholar
  19. 19.
    Chen, L.: Basic properties of the infinite critical FK random map. arXiv:1502.01013
  20. 20.
    Denef J., Lipshitz L.: Power series solutions of algebraic differential equations. Math. Ann. 267(2), 213–238 (1984)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Francesco P., Ginsparg P., Zinn-Justin J.: 2D gravity and random matrices. Phys. Rep. 254(1-2), 133 (1995)CrossRefGoogle Scholar
  22. 22.
    Duplantier B., Kostov I.: Conformal spectra of polymers on a random surface. Phys. Rev. Lett. 61(13), 1433–1437 (1988)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Eynard B., Bonnet G.: The Potts-q random matrix model: loop equations, critical exponents, and rational case. Phys. Lett. B 463(2-4), 273–279 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  25. 25.
    Fortuin C. M., Kasteleyn P. W.: On the random cluster model: I. Introduction and relation to other models. Physica 57, 536–564 (1972)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Giménez O., Noy M.: Asymptotic enumeration and limit laws of planar graphs. J. Am. Math. Soc 22(2), 309–329 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Goulden, I. P., Jackson, D. M.: Combinatorial enumeration. Wiley, New York (1983). Wiley-Interscience Series in Discrete MathematicsGoogle Scholar
  28. 28.
    Goulden I. P., Jackson D. M.: The KP hierarchy, branched covers, and triangulations. Adv. Math. 219(3), 932–951 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Guionnet A., Jones V. F. R., Shlyakhtenko D., Zinn-Justin P.: Loop models, random matrices and planar algebras. Comm. Math. Phys. 316(1), 45–97 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jackson D. M.: Counting cycles in permutations by group characters, with application to a topological problem. Trans. Am. Math. Soc. 299(2), 785–801 (1987)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jackson D. M., Visentin T. I.: A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus. Trans. Am. Math. Soc. 322(1), 343–363 (1990)MathSciNetMATHGoogle Scholar
  32. 32.
    Kazakov V. A.: Ising model on a dynamical planar random lattice: exact solution. Phys. Lett. A 119(3), 140–144 (1986)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Kurkova I., Raschel K.: On the functions counting walks with small steps in the quarter plane. Publ. Math. Inst. Hautes études Sci. 116, 69–114 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mullin R. C.: On the enumeration of tree-rooted maps. Can. J. Math. 19, 174–183 (1967)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Odlyzko, A. M., Richmond, L. B.: A differential equation arising in chromatic sum theory. In: Proceedings of the Fourteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1983), volume 40, pp. 263–275 (1983)Google Scholar
  36. 36.
    Schaeffer, G.: Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees. Electron. J. Combin., 4(1):Research Paper 20 (electronic) (1997)Google Scholar
  37. 37.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 90), pp. 138–148, San Francisco, CA (1990)Google Scholar
  38. 38.
    Sheffield S.: Quantum gravity and inventory accumulation. Ann. PNBA B 44(6), 3804–3848 (2016) arXiv:1108.2241 MathSciNetMATHGoogle Scholar
  39. 39.
    Stanley R. P.: Enumerative Combinatorics 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  40. 40.
    Tutte W. T.: On the enumeration of planar maps. Bull. Am. Math. Soc 74, 64–74 (1968)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Tutte W. T.: On the enumeration of four-colored maps. SIAM J. Appl. Math. 17, 454–460 (1969)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Tutte W. T.: Chromatic sums for rooted planar triangulations. II. The case \({\lambda =\tau +1}\). Can. J. Math 25, 657–671 (1973)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Tutte W. T.: Chromatic sums for rooted planar triangulations. III. The case \({\lambda =3}\). Can. J. Math. 25, 780–790 (1973)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Tutte W. T.: Chromatic sums for rooted planar triangulations. IV. The case \({\lambda =\infty }\). Can. J. Math. 25, 929–940 (1973)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Tutte W.T.: Chromatic sums for rooted planar triangulations: the cases \({\lambda =1}\) and \({\lambda =2}\). Can. J. Math. 25, 426–447 (1973)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Tutte W. T.: Chromatic sums for rooted planar triangulations. V. Special equations. Can. J. Math. 26, 893–907 (1974)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Tutte W. T.: On a pair of functional equations of combinatorial interest. Aequ. Math. 17(2-3), 121–140 (1978)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Tutte W. T.: Chromatic solutions. Can. J. Math. 34(3), 741–758 (1982)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Tutte W. T.: Chromatic solutions. II. Can. J. Math. 34(4), 952–960 (1982)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Tutte, W. T.: Map-colourings and differential equations. In: Progress in Graph Theory (Waterloo, ON, 1982), pp. 477–485. Academic Press, Toronto (1984)Google Scholar
  51. 51.
    Tutte W. T.: Chromatic sums revisited. Aequ. Math. 50(1-2), 95–134 (1995)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Welsh D.J.A., Merino C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41(3), 1127–1152 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Wu F. Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Zeilberger D.: The umbral transfer-matrix method: I. Foundations. J. Comb. Theory, Ser. A 91, 451–463 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.CNRS, LaBRI, Université de BordeauxTalenceFrance

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