On the scaling limit of random planar maps with large faces

DOI10.1214/10-AOP549zbMATH Open1204.05088arXiv0907.3262OpenAlexW2064871074MaRDI QIDQ624656

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Publication date: 9 February 2011

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Abstract: 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 $alphain(1,2)$. 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 $n^{-1/2alpha}$, 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 $noinfty$, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to $2alpha$.

Full work available at URL: https://arxiv.org/abs/0907.3262

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