The Scaling Limit of Random Outerplanar Maps

TL;DR

This paper proves that large random outerplanar maps, when scaled appropriately, converge to a scaled version of Aldous' Brownian tree, revealing their universal limiting shape.

Contribution

It establishes the scaling limit of random outerplanar maps as a scaled Brownian tree using a specific bijection, extending understanding of their geometric structure.

Findings

Random outerplanar maps converge to a scaled Brownian tree.

The convergence is in the Gromov-Hausdorff sense.

The scaling factor is rac{7 \u221a{2}}{9}.

Abstract

A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with $n$ vertices suitably rescaled by a factor $1/ \sqrt{n}$ converge in the Gromov-Hausdorff sense to $\displaystyle{\frac{7 \sqrt{2}}{9}}$ times Aldous' Brownian tree. The proof uses the bijection of Bonichon, Gavoille and Hanusse.