Some families of increasing planar maps

TL;DR

This paper studies the asymptotic properties of increasing planar maps, specifically stack-triangulations and quadrangulations, under different probabilistic models, revealing convergence to infinite maps, continuum trees, and logarithmic distance scaling.

Contribution

It introduces new asymptotic results for increasing families of planar maps, including convergence to infinite maps and continuum trees under various distributions.

Findings

Uniform distribution converges to an infinite map in local topology.

Rescaled maps converge to the continuum random tree in Gromov-Hausdorff topology.

Distances between random points scale with (6/11)log n, converging to 1 in probability.

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.