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Invariance principles for spatial multitype Galton–Watson trees

Grégory Miermont — 2008

Annales de l'I.H.P. Probabilités et statistiques

We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the...

Tessellations of random maps of arbitrary genus

Grégory Miermont — 2009

Annales scientifiques de l'École Normale Supérieure

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 one to encode such structures by 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...

The brownian cactus I. Scaling limits of discrete cactuses

Nicolas CurienJean-François Le GallGrégory Miermont — 2013

Annales de l'I.H.P. Probabilités et statistiques

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space E , one can associate an -tree called the continuous cactus of E . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian cactus...

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