On scaling limits of random Halin-like maps

TL;DR

This paper investigates the scaling limits of certain random maps derived from plane trees, showing they converge to well-known continuous objects like the Brownian CRT or stable looptrees under specific conditions.

Contribution

It introduces a new class of Halin-like maps with Boltzmann weights and establishes their scaling limits as known continuous random structures.

Findings

Scaling limits are either the Brownian CRT or the $eta$-stable looptrees.

Maps with all vertices having one marked corner correspond to non-crossing trees.

Maps with vertices having at least one marked corner relate to outerplanar maps.

Abstract

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the $\alpha$-stable looptrees of Curien and Kortchemski.