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November 2006 Limit of normalized quadrangulations: The Brownian map
Jean-François Marckert, Abdelkader Mokkadem
Ann. Probab. 34(6): 2144-2202 (November 2006). DOI: 10.1214/009117906000000557


Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.


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Jean-François Marckert. Abdelkader Mokkadem. "Limit of normalized quadrangulations: The Brownian map." Ann. Probab. 34 (6) 2144 - 2202, November 2006.


Published: November 2006
First available in Project Euclid: 13 February 2007

zbMATH: 1117.60038
MathSciNet: MR2294979
Digital Object Identifier: 10.1214/009117906000000557

Primary: 60F99 , 60K35
Secondary: 60C05 , 60F05

Keywords: abstract maps , Brownian map , limit theorem , Planar map , quadrangulation , trees , weak convergence

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 6 • November 2006
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