Planar maps, circle patterns and 2D gravity

  • François David

    CEA, Gif-Sur-Yvette, France
  • Bertrand Eynard

    CEA Saclay, Gif-Sur-Yvette, France


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. 139–183

DOI 10.4171/AIHPD/5