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We consider a random planar map M₍ which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m₍ be the vertex set of M₍, which is equipped with the graph distance d₆ₑ. Both when q4 is an even integer and when q=3, there exists a positive constant cₐ such that the rescaled metric spaces (m₍, cₐn^-1/4d₆ₑ) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.
Jean‐François Le Gall (Mon,) studied this question.
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