Limit of normalized quadrangulations: The Brownian map

Abstract

Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ 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.Comment: Published at http://dx.doi.org/10.1214/009117906000000557 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The topological structure of scaling limits of large planar maps

“…Other applications in the spirit of the present work can be found in the recent papers [12], [25] and [26] that were mentioned earlier. Note in particular that the random metric space (T e / ≈, D * ) that is discussed above is essentially equivalent to the Brownian map of [26], although the presentation there is different. See also [5], [6], [11] and [19] for various results about random infinite planar triangulations and quadrangulations and their asymptotic properties.…”

“…In view of the correspondence between maps and mobiles, it seems plausible that scaling limits of large bipartite planar maps can be described in terms of continuous random trees. This idea already appeared in the pioneering work of Chassaing and Schaeffer [12], and was then developed by Marckert and Mokkadem [26], who defined and studied the so-called Brownian map. It was argued in [26] that the Brownian map is in some weak sense the limit of rescaled uniformly distributed random quadrangulations of the plane (see also Marckert and Miermont [25] for recent work along the same lines).…”

“…This idea already appeared in the pioneering work of Chassaing and Schaeffer [12], and was then developed by Marckert and Mokkadem [26], who defined and studied the so-called Brownian map. It was argued in [26] that the Brownian map is in some weak sense the limit of rescaled uniformly distributed random quadrangulations of the plane (see also Marckert and Miermont [25] for recent work along the same lines). The point of view of the present paper is however different from the one in [26] or in [25].…”

“…It was argued in [26] that the Brownian map is in some weak sense the limit of rescaled uniformly distributed random quadrangulations of the plane (see also Marckert and Miermont [25] for recent work along the same lines). The point of view of the present paper is however different from the one in [26] or in [25]. For every given planar map M, we equip the set m of its vertices with the graph distance, and our aim is to study the resulting compact metric space when the number of faces of the map tends to infinity.…”

Volume growth and heat kernel estimates for the continuum random tree

“…It is known that, for any fixed s, the process W (s) has the same distribution as W , (see [26], Proposition 4.9). From this fact, an elementary application of Fubini's theorem allows it to be deduced that, for any measurable A ⊆ U (1) , However, if we observe that the real tree associated with W (s) has exactly the same structure as T , apart from the fact that it is rooted at σ s instead of ρ, and recall that µ is the image of ℓ under the map s → σ s , then we are able to obtain to conclude from (11) that, heuristically, any property of the random real tree T that holds when the root is assumed to be ρ will also hold when we consider the root to be σ, for µ-a.e.…”