Geometry of large Boltzmann outerplanar maps

TL;DR

This paper analyzes the geometric phase transition of large Boltzmann outerplanar maps, showing their convergence to stable looptrees and describing the shape transition from circle to Brownian tree.

Contribution

It establishes the convergence of weighted outerplanar maps to $oldsymbol{ ext{α}}$-stable looptrees and characterizes the geometric transition depending on face weights.

Findings

Maps converge to $ ext{α}$-stable looptrees as vertices grow large

The shape transitions from a circle to a Brownian tree

The convergence depends on the specific weight-sequence

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.