Uniqueness and universality of the Brownian map

Details Uniqueness and universality of the Brownian map Uniqueness and universality of the Brownian map

2011-05-24

arxiv.org/abs/1105.4842v1

Mathematics

Probability

70 pages

Scientific paper

We consider a random planar map M(n) which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m(n) be the vertex set of M(n), which is equipped with the graph distance d_{gr}. Both when q=3 and when q is an even integer greater than 3, there exists a positive constant c_q such that the rescaled metric spaces (m(n),c_q n^{-1/4} d_{gr}) converge in distribution in the Gromov-Hausdorff sense, towards a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

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