Spanning forests in regular planar maps

We address the enumeration ofp-valent planar maps equipped with a spanning forest, with a weightz per face and a weightu per component of the forest. Equivalently, we count regular maps equipped with a spanning tree, with a weightz per face and a weight := u + 1 per internally active edge, in the sense of Tutte. This enumeration problem corresponds to the limitq! 0 of theq-state Potts model on the (dual)p-angulations. Our approach is purely combinatorial. The generating function, denoted by F(z;u), is expressed in terms of a pair of series defined by an implicit system involving doubly hypergeometric functions. We derive from this system that F(z;u) is differentially algebraic, that is, satisfies a differential equation (inz) with polynomial coefficients inz and u. This has recently been proved for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. Foru 1, we study the singularities ofF(z;u) and the corresponding asymptotic behaviour of itsn th coefficient. Foru > 0, we find the standard asymptotic behaviour of planar maps, with a subexponential factorn 5=2 . Atu = 0 we witness a phase transition with a factor n 3 . When u2 ( 1;0), we obtain an extremely unusual behaviour in n 3 =(logn) 2 . To our knowledge, this is a new "universality class" of planar maps.

[1]  Edward A. Bender,et al.  The asymptotic number of rooted maps on a surface , 1986, J. Comb. Theory, Ser. A.

[2]  Jean-Marc DAUL.  Q-states Potts model on a random planar lattice , 1995 .

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

[4]  J. Bouttier,et al.  A recursive approach to the O(n) model on random maps via nested loops , 2011, 1106.0153.

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

[6]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

[7]  Ovidiu Costin,et al.  Infinite-order phase transition in a classical spin system , 1990 .

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

[9]  Jean-Franccois Marckert,et al.  Invariance principles for random bipartite planar maps , 2005, math/0504110.

[10]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

[11]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[12]  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.

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

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

[15]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[16]  Julien Courtiel,et al.  Spanning forests in regular planar maps (conference version) , 2016 .

[17]  W. T. Tutte.  A Census of Planar Maps , 1963, Canadian Journal of Mathematics.

[18]  Michael Drmota,et al.  Systems of functional equations , 1997, Random Struct. Algorithms.

[19]  L. Lipshitz,et al.  D-finite power series , 1989 .

[20]  Stanley Burris,et al.  Characteristic Points of Recursive Systems , 2009, Electron. J. Comb..

[21]  Mireille Bousquet-Mélou,et al.  Counting Coloured Planar Maps: Differential Equations , 2015, 1507.02391.

[22]  P. Zinn-Justin.  The Dilute Potts Model on Random Surfaces , 1999 .

[23]  Haipeng Luo,et al.  Upper and lower bounds for Fv(4,4;5) , 2010, Electron. J. Comb..

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

[25]  Philippe Flajolet,et al.  Random maps, coalescing saddles, singularity analysis, and Airy phenomena , 2001, Random Struct. Algorithms.

[26]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[27]  Éric Fusy,et al.  A bijection for triangulations, quadrangulations, pentagulations, etc , 2010, J. Comb. Theory, Ser. A.

[28]  Omer Angel.  Growth and percolation on the uniform infinite planar triangulation , 2002 .

[29]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[30]  Julien Courtiel,et al.  Combinatoire du polynôme de Tutte et des cartes planaires. (Combinatorics of the Tutte polynomial and planar maps) , 2014, 1411.0737.

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

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

[33]  R. Tennant.  Algebra , 1941, Nature.

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

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

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

[37]  Criel Merino López.  Chip firing and the tutte polynomial , 1997 .

[38]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

[39]  Grégory Miermont,et al.  Scaling limits of random planar maps with large faces , 2011 .

[40]  Olivier Bernardi.  A Characterization of the Tutte Polynomial via Combinatorial Embeddings , 2006 .

[41]  Béla Bollobás,et al.  The phase transition in the uniformly grown random graph has infinite order , 2005 .

[42]  J. Bouttier,et al.  More on the O(n) model on random maps via nested loops: loops with bending energy , 2012, 1202.5521.

[43]  Maurizio Serva,et al.  Exact solutions and infinite-order phase transitions for a general class of Ising models on the regularized Apollonian network , 2014 .

[44]  Rodney Baxter.  Dichromatic Polynomials and Potts Models Summed Over Rooted Maps , 2000 .

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

[46]  J. Bouttier,et al.  Planar Maps and Continued Fractions , 2010, 1007.0419.

[47]  Jürg Fröhlich,et al.  The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and the Coulomb gas , 1981 .

[48]  Sergio Caracciolo,et al.  Spanning Forests on Random Planar Lattices , 2009, 0903.4432.

[49]  Gilles Schaeffer,et al.  A Bijection for Rooted Maps on Orientable Surfaces , 2007, SIAM J. Discret. Math..

[50]  P. Francesco,et al.  Census of planar maps: From the one-matrix model solution to a combinatorial proof , 2002, cond-mat/0207682.

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

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

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

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

[55]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

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

[57]  Ronald C. Mullin.  On the average activity of a spanning tree of a rooted map , 1967 .

[58]  Robert Cori,et al.  The sand-pile model and Tutte polynomials , 2003, Adv. Appl. Math..

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