Planar maps, circle patterns and 2d gravity

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 Kahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space H3; (3) a discretized version (involving finite dierence complex derivative operators '; ') of Polyakov's