## Electronic Journal of Probability

### Some families of increasing planar maps

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 56, 1624-1671.

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

https://projecteuclid.org/euclid.ejp/1464819129

Digital Object Identifier
doi:10.1214/EJP.v13-563

Mathematical Reviews number (MathSciNet)
MR2438817

Zentralblatt MATH identifier
1192.60019

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F1

Rights

#### Citation

Albenque, Marie; Marckert, Jean-Francois. Some families of increasing planar maps. Electron. J. Probab. 13 (2008), paper no. 56, 1624--1671. doi:10.1214/EJP.v13-563. https://projecteuclid.org/euclid.ejp/1464819129

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