Geodesics in large planar maps and in the Brownian map

By Jean-François Le Gall. 2008

137 citations recorded by the source corpus.

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3b5f1c9bcbaae9d0241e6836d77c73d27632e654

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Random geometry on the sphere (2014, background): “1 below), which was derived in the series of papers [29, 30, 31].”
Scaling limits of random trees and planar maps (2011, background): “2At this point, it should be noted that [31, 34, 32] consider another version of Schaeffer’s bijection, where no distinguished vertex v∗ has to be considered.”
Uniqueness and universality of the Brownian map (2011, background): “The present work can be viewed as a continuation and in a sense a conclusion to our preceding papers [15] and [16].”
The Brownian Plane (2012, background): “By a simple argument (see the beginning of Section 3 in [14]), there must exist two vertices b and b′ such that b ∈ T1, b′ ∈ T∞\T1 and ω(T ) = Π(b) = Π(b′), so that in particular b ≈ b′.”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “3) and bounding the second term using [28], Lemma 6.”
A view from infinity of the uniform infinite planar quadrangulation (2012, result): “These two properties are reminiscent of the work of Le Gall on geodesics in the Brownian Map [26].”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, methodology): “A formulation of the main result of [15] is that this path projects into (S,D) as a geodesic γ from ρ to p(s), and that any D-geodesic from ρ can be described in this way.”