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Uniqueness and universality of the Brownian map
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abstract
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 $(\mathbf{m}_n,c_qn^{-1/4}d_\mathrm{gr})$ 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.
Forward citations
Cited by 2 Pith papers
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Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity
A random walk that reflects off the boundary at infinity yields new algorithmic constructions of the free uniform spanning forest and a conjectural embedding framework for supercritical Liouville quantum gravity.
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Random surfaces and Liouville quantum gravity
An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.
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