Jean-François Marckert

Limit of normalized quadrangulations: The Brownian map

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.

Invariance principles for random bipartite planar maps

It is conjectured in the Physics literature that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing & Schaeffer, have shown that the radius of a random quadrangulation with

The rotation correspondence is asymptotically a dilatation

n

The CRT is the scaling limit of unordered binary trees

faces converges in distribution once rescaled by

The height and width of simple trees

n^{1/4}

Asymptotics of trees with a prescribed degree sequence and applications

to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, di Francesco \&\ Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution: the radius of such maps, conditioned to have

Noncrossing trees are almost conditioned Galton--Watson trees

n

Quasi-optimal energy-efficient leader election algorithms in radio networks

faces (or

Many Empty Triangles have a Common Edge

n

One more approach to the convergence of the empirical process to the Brownian bridge

vertices) and under a criticality assumption, converges in distribution once rescaled by