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The topological structure of scaling limits of large planar maps

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We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map M n which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of M n , equipped with the graph distance rescaled by the factor n -1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.

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  1. Ecole normale supérieure de Paris, DMA-ENS, 45 rue d’Ulm, 75005, Paris, France

    Jean-François Le Gall

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  1. Jean-François Le Gall
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Le Gall, JF. The topological structure of scaling limits of large planar maps. Invent. math. 169, 621–670 (2007). https://doi.org/10.1007/s00222-007-0059-9

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  • DOI: https://doi.org/10.1007/s00222-007-0059-9

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