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Annales de l’Institut Henri Poincaré D

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Volume 2, Issue 4, 2015, pp. 335–412
DOI: 10.4171/AIHPD/21

Published online: 2015-11-14

The two-point function of bicolored planar maps

Éric Fusy[1] and Emmanuel Guitter[2]

(1) École Polytechnique, Palaiseau, France
(2) CEA Saclay, Gif-Sur-Yvette, France

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.

Keywords: Planar maps, two-point functions, continued fractions, hard dimers

Fusy Éric, Guitter Emmanuel: The two-point function of bicolored planar maps. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 2 (2015), 335-412. doi: 10.4171/AIHPD/21