The topological structure of scaling limits of large planar maps

By Jean-François Le Gall. 2006

225 citations recorded by the source corpus.

We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2a2c83e8be2dee6ecb5108f7070a1efa2c5e9b95

How researchers see this paper

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

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Rescaled bipartite planar maps converge to the Brownian map (2013, background): “The proof of this bound is easily adapted from [9], Lemma 3.”
Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere (2006, methodology): “where the last lower bound is over all choices of the integer p ≥ 1 and of the finite sequence α0, α1, . . . , αp in Te, such that α0 = α and αp = β. We set α ≈ β if and only if D ∗ (α, β) = 0. By Theorem 3.4 in [ L2 ], this is also equivalent to the condition D ◦ (α, β) = 0, for every α, β ∈ Te, al”
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees (2008, methodology): “Beyond strong enumerative results, these methods enabled to finely describe the metric properties of large random planar maps (for example [CS04, BDFG03, Mie07, BG08]), opening the way to the probabilistic study of a possible limiting continuum object, the Brownian planar map, related to the continu”
Random geometry on the sphere (2014, background): “1 below), which was derived in the series of papers [29, 30, 31].”
Liouville quantum gravity and KPZ (2008, background): “This appears to be even more difficult, although the recent progress on topological scaling limits of discrete planar maps by Le Gall [ LG07 ] and others may prove helpful in the special case κ = 8/3, as well as the recent study of geodesics in large planar maps and in the Brownian map [LG08] (see a”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “Le Gall [14] obtained a compactness result, namely that the laws of (nQn, n ≥ 1), form a relatively compact family of probability distributions on M.”
On the sphericity of scaling limits of random planar quadrangulations (2007, methodology): “Our last ingredient is a slight rewriting of the estimates of Le Gall [10] on distances in quadrangulations in terms of encoding processes.”
Scaling limits of random trees and planar maps (2011, background): “The points (i) and (iii) are due to Chassaing and Schaeffer [12], and (ii) is due to Le Gall [30], although these references state these properties in a slightly different context, namely, in the case where v∗ is the root vertex rather than a uniformly chosen vertex.”
Recurrence of planar graph limits (2012, background): “For the case of random p-angulations this research has recently culminated in the work of Le Gall [21] who established this for p = 3 and all even p and independently Miermont [27] for the case p = 4.”
An invariance principle for random planar maps (2006, unknown): “It is likely that methods similar to that of Le Gall (2006) would allow, starting from Theorem 1, to obtain a similar result for Boltzmann distributions on the set −→ M of rooted maps, where this time, the base vertex is taken to be the source of the root edge.”
Gaussian multiplicative chaos and applications: A review (2013, background): “The prototype of such p-angulations is the Brownian map studied in (see [105, 106, 107, 114]) and corresponds to the pure gravity case c = 0.”
Scaling limits of random planar maps with a unique large face (2012, background): “Le Gall showed that the scaling limit of uniform 2p-angulations (all faces of degree 2p) exists along a suitable subsequence and he furthermore showed that its topology is independent of the subsequence and proved that its Hausdorff dimension equals 4 [39].”
Scaling limits of random planar maps with large faces (2011, methodology): “(To be precise, [19] uses a slightly different version of the BDG bijection, with nonnegative labels, but is straightforward to verify that the argument of the proof of Lemma 3.”
Geodesics in large planar maps and in the Brownian map (2008, background): “The following theorem restates the main result of [23] in a form convenient for the present work.”
The Brownian Plane (2012, result): “1 Topology and Hausdorff dimension of the Brownian plane The following proposition is analogous to a result of [13].”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, methodology): “Our approach is adapted from Le Gall [27] for the first assertion, and from Le Gall and Miermont [30] for the Hausdorff dimension.”