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Planar maps, circle patterns and 2D gravity. (English) Zbl 1297.52007

Summary: 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 Faddeev-Popov determinant in 2d gravity;
(4)
a combination of Chern classes, thus also establishing a link with topological 2d gravity.

MSC:

52C26 Circle packings and discrete conformal geometry
05C10 Planar graphs; geometric and topological aspects of graph theory
32Q15 Kähler manifolds
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Software:

CirclePack
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Full Text: DOI arXiv

References:

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