Tessellations of random maps of arbitrary genus

Understand

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.