The Brownian map is the scaling limit of uniform random plane quadrangulations
type: Item
instance of: scholarly article
title: The Brownian map is the scaling limit of uniform random plane quadrangulations
zbMATH Open document ID: 1278.60124
author: Grégory Miermont
published in: Acta Mathematica
publication date: 2013-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1104.1606
review text: The central object of study in this work is quadrangulation. A map is called quadrangulation if all faces have degree four, where the degree of a face is the number of edges that are incident to it (an edge which is incident to only one face has to be counted twice in the computation of the degree of this face). The author proves that uniform random quadrangulations of the sphere with \(n\) faces, endowed with the usual graph distance and renormalized by \(n-1/4\), converge as \(n\to\infty\) in distribution for the Gromov-Hausdorff topology to a limiting metric space. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 6199563
zbMATH Keywords: Brownian map
zbMATH Keywords: quadrangulations
zbMATH Keywords: scaling limit
reviewed by: N. G. Gamkrelidze
MaRDI profile type: Publication
OpenAlex ID: W2095362975
arXiv ID: 1104.1606
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DOI: 10.1007/S11511-013-0096-8
description: scientific article; zbMATH DE number 6199563