Orateur
Description
Maps come with different shapes, such as trees or triangulations with many more edges. Many classes of maps have been enumerated (2-connected maps, trees, quadrangulations...), notably by Tutte, and a phenomenon of universality has been demonstrated: for the majority of them, the number of elements of size n in the class has an asymptotic of the form k 1/(r^n n^(5/2)), for a certain k and a certain r. Nevertheless, there are classes of degenerate'' maps whose behaviour is similar to that of trees, and whose number of elements of size n has an asymptotic of the form k/(r^n n^(3/2)), as for example outerplanar maps. This dichotomy of behaviour is not only observed for enumeration, but also for metrics. Indeed, in the
tree'' case, the distance between two random vertices is in n^(1/2), against n^{1/4} for uniform planar maps of size $n$. This work focuses on what happens between these two very different regimes. We highlight a model depending on a parameter u>0 which exhibits the expected behaviours, and a transition between the two: depending on the position of $u$ with respect to u_C, the behaviour is that of one or the other universality class. More precisely, we observe a subcritical'' regime where the scale limit of the maps is the Brownian map, a
supercritical'' regime where it is the Brownian tree and finally a critical regime where it is the 3/2-stable tree. The results are obtained using a robust method, which can be used to study a variety of similar models