On the sphericity of scaling limits of random planar quadrangulations

By Grégory Miermont. 2007

77 citations recorded by the source corpus.

We give a new proof of a theorem by Le Gall and Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4e05c5ca29cd1de66f191ec0bfb677dc0ec9d603

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Scaling Limits for Random Quadrangulations of Positive Genus (2010, background): “Miermont [22] later gave a variant proof of this fact.”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “Then Le Gall & Paulin identified this topology with the topology of the 2-dimensional sphere [18, 24].”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “As we consider in this work surfaces with a boundary, the notion of 1-regularity used by Miermont in [Mie08] is no longer sufficient: we will also need here the notion of 0-regularity, which we will expose in Section 5.1.”
Scaling limits of random trees and planar maps (2011, methodology): “We follow the approach of [39], which consists in establishing the absence of “bottlenecks” in the Brownian map before proving via a theorem of Whyburn that this space is homeomorphic to the sphere.”
Random geometry on the sphere (2014, background): “Note that the Brownian map (m∞, D) is almost surely homeomorphic to the sphere S(2) [35] (see also [39]), even though its Hausdorff dimension is equal to 4.”