Annales de l’Institut Henri Poincaré D
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Published online: 2017-05-23
The distance-dependent two-point function of triangulations: a new derivation from old resultsEmmanuel Guitter (1) CEA Saclay, Gif-sur-Yvette, France
We present a new derivation of the distance-dependent two-point function of random planar triangulations. As it is well-known, this function is intimately related to the generating functions of so-called slices, which are pieces of triangulation having boundaries made of shortest paths of prescribed length. We show that the slice generating functions are fully determined by a direct recursive relation on their boundary length. Remarkably, the kernel of this recursion is some quantity introduced and computed by Tutte a long time ago in the context of a global enumeration of planar triangulations. We may thus rely on these old results to solve our new recursion relation explicitly in a constructive way.
Keywords: Planar maps, triangulations, two-point function, hull
Guitter Emmanuel: The distance-dependent two-point function of triangulations: a new derivation from old results. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), 177-211. doi: 10.4171/AIHPD/38