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# Planar maps, circle patterns and 2d gravity

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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 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.
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https://hal-cea.archives-ouvertes.fr/cea-01509872
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François David, Bertrand Eynard. Planar maps, circle patterns and 2d gravity. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, European Mathematical Society, 2014, 1, pp.139 - 183. ⟨10.4171/AIHPD/5⟩. ⟨cea-01509872⟩

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