The topological structure of scaling limits of large planar maps
type: Item
instance of: scholarly article
title: The topological structure of scaling limits of large planar maps
zbMATH Open document ID: 1132.60013
published in: Inventiones Mathematicae
publication date: 2007-10-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0607567
review text: The purpose of this work is to investigate continuous limits of rescaled planar maps. Let \(p\) be a fixed integer \((p\geq2)\) and consider for every integer \(n\geq2\) 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 \(n\to\infty\) towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. The author proves 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. Finally, the last section contains the calculation of the Hausdorff dimension of the limiting metric space.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 5199720
zbMATH Keywords: scaling limit
zbMATH Keywords: planar map
zbMATH Keywords: uniform distribution
zbMATH Keywords: Gromov--Hausdorff distance
author: J.-F. Le Gall
reviewed by: N. G. Gamkrelidze
MaRDI profile type: Publication
OpenAlex ID: W2026147416
arXiv ID: math/0607567
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DOI: 10.1007/S00222-007-0059-9
description: scientific article