Planar maps, circle patterns and 2d gravity

TL;DR

This paper connects random planar triangulations with Delaunay triangulations through circle pattern techniques, revealing their measure as a sum over spanning trees, a Kähler volume form, a discretized Polyakov determinant, and a topological Chern class, linking 2d gravity and hyperbolic geometry.

Contribution

It introduces a novel framework relating random planar maps to complex geometry, hyperbolic tessellations, and topological invariants, providing new insights into 2d gravity models.

Findings

Measure expressed as sum over 3-spanning-trees

Identifies a Kähler metric volume form on Delaunay triangulations

Discretizes Polyakov's conformal Faddeev-Popov determinant

Abstract

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a K\"ahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space; (3) a discretized version (involving finite difference complex derivative operators) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.