- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:04:49
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2014
2
Planar maps, circle patterns and 2D gravity
François
David
CEA, Gif-sur-Yvette, FRANCE
Bertrand
Eynard
CEA Saclay, GIF-SUR-YVETTE CEDEX, FRANCE
Circle pattern, Random maps, Conformal invariance, Kähler geometry, 2D gravity, topological 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 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.
Combinatorics
Convex and discrete geometry
Probability theory and stochastic processes
Quantum theory
139
183
10.4171/AIHPD/5
http://www.ems-ph.org/doi/10.4171/AIHPD/5