Rescaled bipartite planar maps converge to the Brownian map

By C. Abraham. 2013

70 citations recorded by the source corpus.

For every integer $n\geq 1$, we consider a random planar map $\mathcal{M}_n$ which is uniformly distributed over the class of all rooted bipartite planar maps with $n$ edges. We prove that the vertex set of $\mathcal{M}_n$ equipped with the graph distance rescaled by the factor $(2n)^{-1/4}$ converges in distribution, in the Gromov-Hausdorff sense, to the Brownian map. This complements several recent results giving the convergence of various classes of random planar maps to the Brownian map.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b50b72b0ff788fcc44edeb21b30a333e76cd9ebd

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Random geometry on the sphere (2014, background): “Similarly, analogs of (1) hold for general planar maps with a fixed number of edges [11] or for bipartite planar maps with a fixed number of edges [1].”
The hull process of the Brownian plane (2014, background): “A fundamental object is the random compact metric space known as the Brownian map, which has been proved to be the universal scaling limit of several important classes of random planar maps conditioned to have a large size (see in particular [1, 3, 6, 23, 30]).”
Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps (2014, methodology): “5 The non-pointed case We follow here the method given in [1], Section 6.”
The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection (2013, methodology): “Let us also mention that, in parallel to our work, Céline 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.”