Scaling limits of random outerplanar maps with independent link-weights

TL;DR

This paper introduces a new bijection and proof technique for analyzing the scaling limits of large outerplanar maps, including cases with edge weights, and provides new bounds and enumeration results.

Contribution

It presents a novel bijection and a simplified proof for the scaling limit of outerplanar maps, extending to first-passage percolation and subclasses like bipartite maps.

Findings

Established scaling limits for outerplanar maps with weights

Derived sharp tail bounds for map diameters

Recovered asymptotic enumeration formulas

Abstract

The scaling limit of large simple outerplanar maps was established by Caraceni using a bijection due to Bonichon, Gavoille and Hanusse. The present paper introduces a new bijection between outerplanar maps and trees decorated with ordered sequences of edge-rooted dissections of polygons. We apply this decomposition in order to provide a new, short proof of the scaling limit that also applies to the general setting of first-passage percolation. We obtain sharp tail-bounds for the diameter and recover the asymptotic enumeration formula for outerplanar maps. Our methods also enable us treat subclasses such as bipartite outerplanar maps.