Geometry of large Boltzmann outerplanar maps

We study the phase diagram of random outerplanar maps sampled according to nonnegative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The Bolt...

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Veröffentlicht in:	Random structures & algorithms 2019-10, Vol.55 (3), p.742-771
Hauptverfasser:	Stefánsson, Sigurdur Örn, Stufler, Benedikt
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Sprache:	eng
Schlagworte:	Apexes enriched trees looptrees outerplanar maps Phase diagrams random trees
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creator	Stefánsson, Sigurdur Örn Stufler, Benedikt
description	We study the phase diagram of random outerplanar maps sampled according to nonnegative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The Boltzmann outerplanar maps are then shown to converge in the Gromov‐Hausdorff sense towards the α‐stable looptree introduced by Curien and Kortchemski (2014), with the parameter α depending on the specific weight‐sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree.
doi_str_mv	10.1002/rsa.20834
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subjects	Apexes enriched trees looptrees outerplanar maps Phase diagrams random trees
title	Geometry of large Boltzmann outerplanar maps
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