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Article Dans Une Revue Annals of Probability Année : 2011

Scaling limits of random planar maps with large faces

Résumé

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index $\alpha\in(1,2)$. When the number $n$ of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index $\alpha$. In particular, the profile of distances in the map, rescaled by the factor $n^{−1/2\alpha}$, converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as $n\to\infty$, at least along suitable subsequences, towards a limiting random compact metric space whose Hausdorff dimension is equal to $2\alpha$.
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Dates et versions

hal-00405123 , version 1 (18-07-2009)

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Jean-François Le Gall, Grégory Miermont. Scaling limits of random planar maps with large faces. Annals of Probability, 2011, 39 (1), pp.1--69. ⟨hal-00405123⟩
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