Publication : t13/189

Planar maps, circle patterns and 2d gravity

David F. (CEA, IPhT (Institut de Physique Théorique), F-91191 Gif-sur-Yvette, France)
Eynard B. (CEA, IPhT (Institut de Physique Théorique), F-91191 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 K\"ahler 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.
Année de publication : 2014
Revue : Ann. Inst. Henri Poincaré Comb. Phys. Interact 1 139-183 (2014)
DOI : 10.4171/AIHPD/5
Preprint : arXiv:1307.3123
Lien :
PACS : MSC: Primary 52C26, 05C10; Secondary 2Q15, 60G55, 81T40.
Keywords : Circle pattern, random maps, conformal invariance, Kähler geometry, 2D gravity, topological gravity.
Langue : Anglais

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  • article-triangulations-1.pdf
  • H3triangle-2.png


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