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We present a new derivation of the distance-dependent two-point function ofrandom planar triangulations. As it is well-known, this function is intimatelyrelated to the generating functions of so-called slices, which are pieces oftriangulation having boundaries made of shortest paths of prescribed length. Weshow that the slice generating functions are fully determined by a directrecursive relation on their boundary length. Remarkably, the kernel of thisrecursion is some quantity introduced and computed by Tutte a long time ago inthe context of a global enumeration of planar triangulations. We may thus relyon these old results to solve our new recursion relation explicitly in aconstructive way.

The distance-dependent two-point function of triangulations: a new derivation from old results