Counting Coloured Planar Maps: Differential Equations

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 $${q=1, 2}$$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.

[1]  William T. Tutte.  On the Enumeration of Four-Colored Maps , 1969 .

[2]  Gilles Schaeffer,et al.  The degree distribution in bipartite planar maps: applications to the Ising model , 2002 .

[3]  Geoffrey Grimmett.  The Random-Cluster Model , 2002, math/0205237.

[4]  J. Bouttier,et al.  Blocked edges on Eulerian maps and mobiles: application to spanning trees, hard particles and the Ising model , 2007, math/0702097.

[5]  C. Itzykson,et al.  Quantum field theory techniques in graphical enumeration , 1980 .

[6]  I. Goulden,et al.  Combinatorial Enumeration , 2004 .

[7]  Vladimir Kazakov,et al.  The ising model on a random planar lattice: The structure of the phase transition and the exact critical exponents , 1987 .

[8]  D. Shlyakhtenko,et al.  Loop Models, Random Matrices and Planar Algebras , 2010, 1012.0619.

[9]  P. Di Francesco,et al.  2D gravity and random matrices , 1993 .

[10]  Mireille Bousquet-Mélou,et al.  Spanning forests in regular planar maps , 2013, J. Comb. Theory, Ser. A.

[11]  W. T. Tutte.  On the enumeration of planar maps , 1968 .

[12]  W. T. Tutte.  Chromatic solutions. II , 1982 .

[13]  Linxiao Chen.  Basic properties of the infinite critical-FK random map , 2015, 1502.01013.

[14]  Robin Wilson,et al.  Modern Graph Theory , 2013 .

[15]  Doron Zeilberger.  The Umbral Transfer-Matrix Method , III : Counting Animals , 2001 .

[16]  Philippe Di Francesco,et al.  Planar Maps as Labeled Mobiles , 2004, Electron. J. Comb..

[17]  Mireille Bousquet-Mélou,et al.  Counting colored planar maps: Algebraicity results , 2009, J. Comb. Theory, Ser. B.

[18]  S. Sheffield.  Quantum gravity and inventory accumulation , 2011, 1108.2241.

[19]  Vladimir Kazakov,et al.  Ising model on a dynamical planar random lattice: Exact solution , 1986 .

[20]  I. Goulden,et al.  The KP hierarchy, branched covers, and triangulations , 2008, 0803.3980.

[21]  W. T. Tutte.  Chromatic Sums for Rooted Planar Triangulations, V: Special Equations , 1974, Canadian Journal of Mathematics - Journal Canadien de Mathematiques.

[22]  J. Mickelsson.  The Kp Hierarchy , 1989 .

[23]  G. Parisi,et al.  Planar diagrams , 1978 .

[24]  Nicolas Bonichon,et al.  Baxter permutations and plane bipolar orientations , 2008, Electron. Notes Discret. Math..

[25]  F. Y. Wu.  The Potts model , 1982 .

[26]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[27]  Duplantier,et al.  Conformal spectra of polymers on a random surface. , 1988, Physical review letters.

[28]  Mireille Bousquet-Mélou,et al.  Planar maps and algebraic series: polynomial equations with one catalytic variable , 2004 .

[29]  J. Bouttier,et al.  Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model , 2012, 1207.4878.

[30]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[31]  D. Jackson,et al.  A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus , 1990 .

[32]  L. Lipshitz,et al.  Power series solutions of algebraic differential equations , 1984 .

[33]  Olivier Devillers,et al.  Optimal succinct representations of planar maps , 2006, SCG '06.

[34]  Alin Bostan,et al.  Non-D-finite excursions in the quarter plane , 2012, J. Comb. Theory A.

[35]  R. Stanley,et al.  Enumerative Combinatorics: Index , 1999 .

[36]  Mireille Bousquet-Mélou,et al.  Polynomial equations with one catalytic variable, algebraic series and map enumeration , 2006, J. Comb. Theory, Ser. B.

[37]  B. Nienhuis.  Loop models , 2022 .

[38]  Marni Mishna,et al.  Walks with small steps in the quarter plane , 2008, 0810.4387.

[39]  Doron Zeilberger,et al.  The Umbral Transfer-Matrix Method. I. Foundations , 2000, J. Comb. Theory, Ser. A.

[40]  R. Mullin,et al.  On the Enumeration of Tree-Rooted Maps , 1967, Canadian Journal of Mathematics.

[41]  Kilian Raschel,et al.  On the functions counting walks with small steps in the quarter plane , 2012 .

[42]  W. T. Tutte.  Chromatic sums revisited , 1995 .

[43]  Omer Giménez,et al.  Asymptotic enumeration and limit laws of planar graphs , 2005, math/0501269.

[44]  P. Flajolet,et al.  Analytic Combinatorics: RANDOM STRUCTURES , 2009 .

[45]  D. Welsh,et al.  The Potts model and the Tutte polynomial , 2000 .

[46]  W. T. Tutte.  Chromatic sums for rooted planar triangulations: the cases $lambda =1$ and $lambda =2$ , 1973 .

[47]  Mireille Bousquet-Mélou,et al.  Counting planar maps, coloured or uncoloured , 2011, 2004.08792.

[48]  Gilles Schaeer,et al.  Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees , 1997 .

[49]  D. Jackson.  Counting cycles in permutations by group characters, with an application to a topological problem , 1987 .

[50]  W. G. Brown.  On the existence of square roots in certain rings of power series , 1965 .

[51]  G. Bonnet,et al.  The Potts-q random matrix model: loop equations, critical exponents, and rational case , 1999 .

[52]  W. T. Tutte.  On a pair of functional equations of combinatorial interest , 1978 .

[53]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[54]  H. Duminil-Copin,et al.  The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 , 2010, Probability Theory and Related Fields.