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Annales de l’Institut Henri Poincaré D


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Volume 1, Issue 3, 2014, pp. 265–306
DOI: 10.4171/AIHPD/8

Published online: 2014-07-22

On the two-point function of general planar maps and hypermaps

Jérémie Bouttier[1], Éric Fusy[2] and Emmanuel Guitter[3]

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

We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number of faces.

Keywords: random geometry, maps, graph distance, two-point function, discrete integrability, bijections, enumerative combinatorics

Bouttier Jérémie, Fusy Éric, Guitter Emmanuel: On the two-point function of general planar maps and hypermaps. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), 265-306. doi: 10.4171/AIHPD/8