Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

By Jean-François Le Gall, F. Paulin. 2006

158 citations recorded by the source corpus.

Abstract.We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/342ee31ba7eaa1dfaea3a7b7752a87cc8d9e4f16

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

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.”
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 limits of random planar maps with a unique large face (2012, background): “We consider rooted and pointed planar maps as is done in [43] which is different from the original case [16] where the maps were pointed but not rooted.”
Geodesics in large planar maps and in the Brownian map (2008, background): “For instance the non-existence of small “bottlenecks” in large 2p-angulations was derived in [24] as a consequence of the convergence (1) and the fact that (m∞, D) is homeomorphic to S(2).”
On the sphericity of scaling limits of random planar quadrangulations (2007, methodology): “• To be perfectly accurate, Le Gall & Paulin showed the same result for uniform 2 k - angulations (maps with degree-2 k faces) with n faces.”
Some families of increasing planar maps (2007, methodology): “Then Le Gall & Paulin [30] show that this limiting space is topologically a sphere.”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “This last step, however, is not mandatory in order to identify the topology of the limit: Le Gall and Paulin [LGP08], and later Miermont [Mie08], showed that any possible limit is homeomorphic to the two-dimensional sphere.”
Uniqueness and universality of the Brownian map (2011, background): “concerned with canonical embeddings of the Brownian map: It is known [18] that the space (m∞, D ∗) is a.”
Scaling Limits for Random Quadrangulations of Positive Genus (2010, background): “He also proved, together with Paulin [19], that every Brownian map is almost surely homeomorphic to the two-dimensional sphere.”
The topological structure of scaling limits of large planar maps (2006, background): “They are derived in greater detail in the subsequent paper [24] where they play an important role (see Lemma 3.”
Scaling limits of random trees and planar maps (2011, background): “This result was first obtained by Le Gall and Paulin [34], by arguing directly on the quotient space M = T”
The Brownian Plane (2012, methodology): “The sphericity of the Brownian map was already applied in [17] to prove the non-existence of small bottlenecks in large uniform quadrangulations.”