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Geometry of large Boltzmann outerplanar maps
Author(s) -
Stefánsson Sigurdur Örn,
Stufler Benedikt
Publication year - 2019
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20834
Subject(s) - mathematics , hausdorff space , infinity , boundary (topology) , boltzmann constant , sequence (biology) , geometry , combinatorics , mathematical analysis , physics , genetics , biology , thermodynamics
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.