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

TL;DR

This paper introduces a new derivation of the distance-dependent two-point function for random planar triangulations, utilizing recursive relations and classical Tutte results to explicitly solve the problem.

Contribution

It provides a novel derivation method for the two-point function using recursive boundary relations and classical enumeration results, connecting old and new combinatorial techniques.

Findings

Explicit recursive relation for slice generating functions

Connection to Tutte's classical enumeration results

Constructive solution to the two-point function

Abstract

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.