Scaling limits of random planar maps with a unique large face
By Svante Janson, Sigurdur Orn Stef'ansson. 2012
73 citations recorded by the source corpus.
We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph metric resealed by n(-1/2), are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5a76f604c3eb32e72d54261ad475e0993ca73671
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Rescaled bipartite planar maps converge to the Brownian map (2013, methodology): “This new definition is somehow related to a bijection of Janson and Stefánsson [6] between one-type and two-type trees.”
- Scaling Limits of Random Graphs from Subcritical Classes (2014, background): “see e.g. Haas and Miermont [25] and references therein, and planar maps, see e.g. Albenque and Marckert [2], Bettinelli [9], Caraceni [12], Curien, Haas and Kortchemski [14] and Janson and Stefansson [28]. Although the aforementioned papers identify the CRT as the universal limiting object in variou”