The three-point function of planar quadrangulations

TL;DR

This paper calculates the three-point function for random planar quadrangulations, revealing a universal scaling function in the large size limit that connects to 2D quantum gravity, with explicit formulas and various regimes analyzed.

Contribution

It provides explicit expressions for the universal three-point function of planar quadrangulations in different ensembles and regimes, linking discrete models to continuous quantum gravity.

Findings

Universal three-point function converges in the large quadrangulation limit.

Explicit formulas for the three-point function in grand-canonical and canonical ensembles.

Probability laws for geodesic points and their distances are derived.

Abstract

We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete three-point function converges to a simple universal scaling function, which is the continuous three-point function of pure 2D quantum gravity. We give explicit expressions for this universal three-point function both in the grand-canonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the marked vertices are aligned, we also obtain the probability law for the number of geodesic points, namely vertices that lie on a geodesic path between two given vertices, and at prescribed distances from these vertices.