Tessellations of random maps of arbitrary genus

Details Tessellations of random maps of arbitrary genus Tessellations of random maps of arbitrary genus

2007-12-21

arxiv.org/abs/0712.3688v2

Mathematics

Probability

58pp, 6 figures. One figure added, minor corrections

Scientific paper

We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic.

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