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

Tessellations of random maps of arbitrary genus