The Annals of Probability

Uniqueness and universality of the Brownian map

Jean-François Le Gall

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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 $\mathbf{m}_{n}$ be the vertex set of $M_{n}$, which is equipped with the graph distance $d_{\mathrm{gr}}$. Both when $q\geq4$ is an even integer and when $q=3$, there exists a positive constant $c_{q}$ such that the rescaled metric spaces $(\mathbf{m}_{n},c_{q}n^{-1/4}d_{\mathrm{gr}})$ 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.

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Ann. Probab., Volume 41, Number 4 (2013), 2880-2960.

First available in Project Euclid: 3 July 2013

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F17: Functional limit theorems; invariance principles
Secondary: 05C80: Random graphs [See also 60B20]

Brownian map planar map graph distance triangulation scaling limit Gromov–Hausdorff convergence geodesic


Le Gall, Jean-François. Uniqueness and universality of the Brownian map. Ann. Probab. 41 (2013), no. 4, 2880--2960. doi:10.1214/12-AOP792.

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