The three-point function of planar quadrangulations

Understand

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.