# Annales de l’Institut Henri Poincaré D

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**Volume 1, Issue 2, 2014, pp. 139–183**

**DOI: 10.4171/AIHPD/5**

Published online: 2014-07-03

Planar maps, circle patterns and 2D gravity

François David^{[1]}and Bertrand Eynard

^{[2]}(1) CEA, Gif-Sur-Yvette, France

(2) 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 $\mathbb{H}_3$; (3) a discretized version (involving finite difference complex derivative operators $\nabla,\bar\nabla$) 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.

*Keywords: *Circle pattern, Random maps, Conformal invariance, Kähler geometry, 2D gravity, topological gravity

David François, Eynard Bertrand: Planar maps, circle patterns and 2D gravity. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 1 (2014), 139-183. doi: 10.4171/AIHPD/5