The Annals of Probability
- Ann. Probab.
- Volume 34, Number 6 (2006), 2144-2202.
Limit of normalized quadrangulations: The Brownian map
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.
Ann. Probab., Volume 34, Number 6 (2006), 2144-2202.
First available in Project Euclid: 13 February 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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