Acta Mathematica· 2013Q1
The Brownian map is the scaling limit of uniform random plane quadrangulations
- 318citations
- Q1SCImago
- 2013year
Short summary
Uniform random quadrangulations of the sphere with n faces, when scaled by n−1/4 and viewed with the Gromov–Hausdorff topology, converge to the Brownian map as n approaches infinity.
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Key points
- Uniform random quadrangulations of the sphere converge to the Brownian map in the Gromov–Hausdorff topology.
- The convergence is established for quadrangulations scaled by a factor of n−1/4.
- The proof utilizes the concept of geodesic stars, which are specific configurations of geodesics within the maps.
- This finding confirms a conjecture by Le Gall regarding the limiting behavior of random plane maps.
AI-generated from the title and abstract; the full text is not read.
Abstract
We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.
The authors' abstract, as published at the source. Acta Mathematica, 2013 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics