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2008 Some families of increasing planar maps
Marie Albenque, Jean-Francois Marckert
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Electron. J. Probab. 13: 1624-1671 (2008). DOI: 10.1214/EJP.v13-563

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 $n^{1/2}$, 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 $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations.

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Marie Albenque. Jean-Francois Marckert. "Some families of increasing planar maps." Electron. J. Probab. 13 1624 - 1671, 2008. https://doi.org/10.1214/EJP.v13-563

Information

Accepted: 19 September 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1192.60019
MathSciNet: MR2438817
Digital Object Identifier: 10.1214/EJP.v13-563

Subjects:
Primary: 60C05
Secondary: 60F1

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