Some families of increasing planar maps

Niveau: Supérieur
Some families of increasing planar maps Marie Albenque Jean-Franc¸ois Marckert LIAFA, CNRS UMR 7089 CNRS, LaBRI, UMR 5800 Universite Paris Diderot - Paris 7 Universite Bordeaux 1 75205 Paris Cedex 13 351 cours de la Liberation 33405 Talence cedex Abstract Stack-triangulations appear as natural objects when one wants to define some families of increasing triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with 2n faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by n1/2, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by (6/11) logn converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations. 1 Introduction Consider a rooted triangulation of the plane. Choose a finite triangular face ABC and add inside a new vertex O and the three edges AO, BO and CO. Starting at time 1 from a single rooted triangle, after k such evolutions, a triangulation with 2k+2 faces is obtained. The set of triangulations ?2k with 2k faces that can be reached by this growing procedure is not the set of all rooted triangulations with 2k faces.

• maps having


• bipartite map


• triangulation


• rooted bipartite


• gromov-hausdorff topology


• apollonian space-filling


• maps