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Abstract
We prove that a uniform rooted plane map with n edges converges in distribution after asuitable normalization to the Brownian map for the Gromov–Hausdorff topology. A recent bijection due to Ambjørn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.
Highlights
The topic of limits of random maps has met an increasing interest over the last two decades, as it is recognized that such objects provide natural model of discrete and continuous 2-dimensional geometries [ADJ97, AS03]
Recall that a plane map is a cellular embedding of a finite graph into the sphere, considered up to orientation-preserving homeomorphisms
We see a map m as a metric space by endowing the set V (m) of its vertices with its natural graph metric dm: the graph distance between two vertices is the minimal number of edges of a path linking them
Summary
The topic of limits of random maps has met an increasing interest over the last two decades, as it is recognized that such objects provide natural model of discrete and continuous 2-dimensional geometries [ADJ97, AS03]. We choose at random a map of “size” n in a given class and look at the limit as n → ∞ in the sense of the Gromov–Hausdorff topology [Gro99] of the corresponding metric space, once rescaled by the proper factor This question first arose in [CS04], focusing on the class of plane quadrangulations, that is, maps whose faces are of degree 4, and where the size is defined as the number of faces. The key to our study is to use a combination of the Cori–Vauquelin–Schaeffer bijection, together with a recent bijection due to Ambjørn and Budd [AB13], that allows to couple directly a uniform (pointed) map with n edges and a uniform quadrangulation with n faces, while preserving distances asymptotically This allows to transfer known results from uniform quadrangulations to uniform maps, in a way that is comparatively easier than a method based on the Bouttier–Di Francesco–Guitter bijection. Let us mention that, in parallel to our work, Celine Abraham [Abr13] has obtained a similar result to ours for uniform bipartite maps, by using an approach based on the Bouttier–Di Francesco–Guitter bijection
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