Some families of increasing planar maps

Abstract

Stack-triangulations appear as natural objects when one wants to define some increasing families of 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.

The continuum random tree is the scaling limit of unlabeled unrooted trees

“…The CRT plays a central role in the study of the geometric shape of large discrete structures. It arises as scaling limit for a variety of models [5,[7][8][9]15,18,19,30,36]. Although scaling limits describe asymptotic global properties, they do not contain information on local properties, such as the limiting degree distribution of a randomly chosen vertex in a graph.…”

Some Properties of Random Apollonian Networks

“…A counterexample is shown in Figure 5 where the height of the random ternary tree can be made arbitrarily large but the diameter is 2. Albenque and Marckert proved in [2] that if v, u are two i.i.d. uniformly random internal vertices, i.e., v, u ≥ 4, then the distance d(u, v) tends to 6 11 log n with probability 1 as the number of vertices n of the RAN grows to infinity.…”

“…CCF-1013110. We would like to thank Luc Devroye and Alexis Darrasse for pointing out references [12] and [2,30] respectively.…”

Ancestors and descendants in evolving k‐tree models

“…The model of ordered increasing k-trees has been introduced by the authors in [21], where also a study of some parameters such as, e.g., the degree of the nodes and the local clustering coefficient, has been sketched briefly. We want to mention further that the special instance d = 1 of d-ary increasing k-trees leads to network models, which have been introduced previously and that are known as random Apollonian networks, see, e.g., [1].…”