The Annals of Probability

Limit of normalized quadrangulations: The Brownian map

Jean-François Marckert and Abdelkader Mokkadem

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2144-2202.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1171377440

Digital Object Identifier
doi:10.1214/009117906000000557

Mathematical Reviews number (MathSciNet)
MR2294979

Zentralblatt MATH identifier
1117.60038

Subjects
Primary: 60F99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Keywords
Brownian map quadrangulation planar map limit theorem weak convergence trees abstract maps

Citation

Marckert, Jean-François; Mokkadem, Abdelkader. Limit of normalized quadrangulations: The Brownian map. Ann. Probab. 34 (2006), no. 6, 2144--2202. doi:10.1214/009117906000000557. https://projecteuclid.org/euclid.aop/1171377440


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