Some families of increasing planar maps

By Marie Albenque, Jean-François Marckert. 2007

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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.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/753f3372bce7e44154021af501f311d5f2f88973

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In Computational Theory and Mathematics

In Mathematical Physics

In Signal Processing

Svante Janson

Citations in the corpus

Scaling limits of random planar maps with a unique large face (2012, background): “Other examples of planar maps converging to Aldous’ continuum tree are stack triangulations [1] and random dissections of polygons [20].”