Geodesic
The two-point function of bicolored planar maps
Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 4, pp. 335-412

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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.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/21
Classification : 05-XX
Keywords: Planar maps, two-point functions, continued fractions, hard dimers 
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     title = {The two-point function of bicolored planar maps},
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Fusy, Éric; Guitter, Emmanuel. The two-point function of bicolored planar maps. Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 4, pp. 335-412. doi: 10.4171/aihpd/21

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