The two-point function of bicolored planar maps

Eric Fusy (1) , Emmanuel Guitter (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customary, the maps are enumerated with arbitrary degree-dependent face weights, but the novelty here is that we also introduce color-dependent vertex weights. Explicit expressions are given for vertex-bicolored maps with bounded face degrees in the form of ratios of determinants of fixed size. Our approach is based on a slice decomposition of maps which relates the distance-dependent two-point function to the coefficients of the continued fraction expansions of some distance-independent map generating functions. Special attention is paid to the case of vertex-bicolored quadrangulations and hexangulations, whose two-point functions are also obtained in a more direct way involving equivalences with hard dimer statistics. A few consequences of our results, as well as some extension to vertex-tricolored maps, are also discussed.

Mots clés: Planar maps; two-point functions; continued fractions; hard dimers

Domaines: Combinatoire [math.CO]

Eric Fusy, Emmanuel Guitter. The two-point function of bicolored planar maps. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, 2015, 2 (4), pp.335-412. ⟨10.4171/AIHPD/21⟩. ⟨hal-01188333⟩