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 Kähler 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.
Cite this article
François David, Bertrand Eynard, Planar maps, circle patterns and 2D gravity. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 2, pp. 139–183DOI 10.4171/AIHPD/5