HAL : hal-00646399, version 2
 arXiv : 1111.7227
 Versions disponibles : v1 (30-11-2011) v2 (21-12-2012)
 Scaling Limit of Random Planar Quadrangulations with a Boundary
 (21/12/2012)
 We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(\sigma_n)$ of integers such that $\sigma_n/\sqrt{2n}$ tends to some $\sigma\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2\sigma_n$ half-edges on the boundary. For $\sigma\in (0,\infty)$, we view $q_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov--Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension $4$ with a boundary of Hausdorff dimension $2$ that is homeomorphic to the two-dimensional disc. For $\sigma=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $\sigma=\infty$, the proper scaling becomes $\sigma_n^{-1/2}$ and we obtain a convergence toward Aldous's CRT.
 1 : Institut Élie Cartan Nancy (IECN) CNRS : UMR7502 – Université de Lorraine
 Équipe de recherche : Probabilités et statistiques
 Domaine : Mathématiques/Probabilités
 Mots Clés : random maps – random trees – Brownian snake – scaling limits – regular convergence – Gromov topology – Hausdorff dimension – Brownian CRT – random metric spaces.
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 hal-00646399, version 2 http://hal.archives-ouvertes.fr/hal-00646399 oai:hal.archives-ouvertes.fr:hal-00646399 Contributeur : Jérémie Bettinelli <> Soumis le : Vendredi 21 Décembre 2012, 16:17:23 Dernière modification le : Mercredi 9 Janvier 2013, 16:32:24