Tessellations of random maps of arbitrary genus

Grégory Miermont (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Random maps; scaling limits; random snakes; asymptotic enumeration; geodesics

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Grégory Miermont. Tessellations of random maps of arbitrary genus. Annales Scientifiques de l'École Normale Supérieure, 2009, 42 (5), pp.725--781. ⟨hal-00200685v2⟩