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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.

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

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Publié par | profil-nechor-2012 |

Nombre de visites sur la page | 13 |

Langue | English |

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