Limit of normalized quadrangulations: The Brownian map

By Jean-François Marckert, Abdelkader Mokkadem. 2004

148 citations recorded by the source corpus.

Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of the Brownian map is defined. The weak convergences hold in these metric spaces.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b41dacc5f8a3c31254169efb2bcc5088a8d2185b

How researchers see this paper

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

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Recurrence of planar graph limits (2012, background): “The existence of such a limit was first suggested by Chassaing and Schaefer [13], Le Gall [20], Marckert and Mokkadem [25] who named it the Brownian map.”
Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “scaled by n−1/4, and let µ∗ n denote the law of Xn. Problem 4.1. Show that the weak limit limn→∞ µ∗n with respect to the GromovHausdorﬀ metric exists. Determine the properties of the limit. Note that [68] provesconvergence of samples from µn, but the metric used there on the samples from µn is very ”
Scaling Limit of Random Planar Quadrangulations with a Boundary (2011, background): “The problem was first addressed by Marckert and Mokkadem [33], who constructed a candidate limiting space called the Brownian map, and showed the convergence toward it in another sense.”
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”
Local structure of random quadrangulations (2005, background): “f the same order 1/R2 and it’s generating function has the main singularity of the same order 3/2. There should also exist certain natural mapping of branching process structure into the brownian map [14]. Finally, we want to note the following statement in [1]: ”A boundary of l links will have the ”
Some families of increasing planar maps (2007, background): “In Marckert & Mokkadem [37], quadrangulations are shown to be obtained as the gluing of two trees, thanks to the Schaeﬀer’s bijection (see [40, 13, 37]) between quadrangulations and well labeled trees.”
Geodesics in large planar maps and in the Brownian map (2008, methodology): “Marckert and Mokkadem [ 28 ] considered the case of quadrangulations and proved a weak form of the convergence (1).”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “The main goal of this work is to prove these conjectures, namely that n−1/4Qn converges in distribution as n!∞ to the conjectured limit of [23] and [15].”
Scaling Limits for Random Quadrangulations of Positive Genus (2010, background): “An important step toward this result has been made by Le Gall [17] who showed the tightness of the laws of these metric spaces, and that every possible limiting space— commonly called Brownian map, in reference to Marckert and Mokkadem’s terminology—is in fact almost surely of Hausdorff dimension 4.”
Conditioned Brownian trees (2005, unknown): “Roughly speaking, N0 is obtained by the same construction as above, but instead of considering a normalized Brownian excursion, we now let e be distributed according to the (infinite) Itô measure of Brownian excursions.”