The two-point function of bicolored planar maps

Fusy, Éric; Guitter, Emmanuel

Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 4, pp. 335-412.

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 coe�fficients 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.

Classification : 05-XX
Mots-clés : Planar maps, two-point functions, continued fractions, hard dimers