Geometry of large Boltzmann outerplanar maps

Abstract

We study the phase diagram of random outerplanar maps sampled according to non-negative 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 $\alpha$-stable looptree introduced by Curien and Kortchemski (2014), with the parameter $\alpha$ 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.