The Annals of Probability· 2013Q1
Uniqueness and universality of the Brownian map
- 339citations
- Q1SCImago
- 2013year
Short summary
Rescaled random planar maps (q-angulations with n faces) converge to a universal limit, the Brownian map, for q>=4 (even) and q=3.
AI-generated from the title and abstract; the full text is not read.
Key points
- Rescaled vertex sets of random planar q-angulations (n faces) converge to the Brownian map.
- Convergence is shown for q>=4 (even) and q=3.
- The scaling factor is c_q * n^(-1/4) for the graph distance.
- The Brownian map is a universal limit space.
- The result solves a question by Schramm concerning triangulations.
AI-generated from the title and abstract; the full text is not read.
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 $(\mathbf{m}_{n},c_{q}n^{-1/4}d_{\mathrm{gr}})$ 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.
The authors' abstract, as published at the source. The Annals of Probability, 2013 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics