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Limit of normalized quadrangulations: the Brownian map. (English) Zbl 1117.60038

The authors built the so-called Brownian map or “continuum random map”, as \(n\to\infty\) of suitably scaled (with scaling \(n^{1/4}\)) random quadrangulations chosen equally likely among the pointed quadrangulations with \(n\) faces. A metric space of pointed abstract maps containing the discrete pointed quadrangulations is constructed and the convergence is weak in this space, the limit being continuous and compact. This limit is described with the help of the Brownian snake with lifetime process the normalized Brownian excursion. First, the authors study a model of rooted quadrangulations which converges to the Brownian map. A model of rooted quadrangulations with random edge lengths is also shown to converge to the Brownian map. By using the canonical surjection from the set of rooted quadrangulations with \(n\) faces onto the set of pointed quadrangulations with \(n\) faces one obtains the first announced result. The convergence of the radius and of the profile of rooted and pointed quadrangulations are also studied.

MSC:

60F99 Limit theorems in probability theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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