Geometry of large Boltzmann outerplanar maps. Issue 3 (11th December 2018)

• Record Type: Journal Article
• Authors: Stefánsson, Sigurdur Örn; Stufler, Benedikt
• Is Part Of: Random structures & algorithms. Volume 55:Issue 3(2019)
• Page Start: 742
• Page End: 771
• Publication Date: 2018-12-11
• Subjects: enriched trees -- looptrees -- outerplanar maps -- random trees; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20834
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : 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.