The scaling limit of random 2-connected series-parallel maps

TL;DR

This work analyzes the scaling limit of weighted random rooted, two-connected series-parallel maps with $n$ edges. By exploiting a bijection to leaf-labeled plane trees and a Bienaymé–Galton–Watson/simply generated-trees framework, the authors model the maps via weighted random trees and use a Markov-additive geodesic comparison to relate map distances to tree distances. Under a critical, light-tailed weight condition, the rescaled metric spaces $(\mathsf{SP}^\mu_n, c_\mu n^{-1/2} d_{\mathsf{SP}_n})$ converge in distribution to a constant multiple of Aldous' continuum random tree $\mathcal{T}_{\mathrm{e}}$ in the Gromov–Hausdorff sense, with optimal diameter tail bounds. The results extend the CRT universality from simpler tree-like models to the nuanced class of 2-connected SP maps and connect to broader themes in the study of random planar structures and their potential links to Liouville quantum gravity.

Abstract

A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series-parallel maps with $n$ edges and show that under some integrability conditions on these weights, the maps with distances rescaled by a factor $n^{-1/2}$ converge to a constant multiple of Aldous' continuum random tree (CRT) in the Gromov--Hausdorff sense. The proof relies on a bijection between a set of trees with $n$ leaves and a set of series-parallel maps with $n$ edges, which enables one to compare geodesics in the maps and in the corresponding trees via a Markov chain argument introduced by Curien, Haas and Kortchemski (2015).

Figures (7)

• Figure 1: A simulation of uniform 2-connected series-parallel maps with different numbers of edges. The endpoints of the root edge are represented by $\bullet$ and $\circ$.
• Figure 2: Left: An example of a rooted, two-connected SP-map. The directed root edge is indicated by an arrow. Right: Its corresponding labeled plane tree where $\circ$ denotes the label $P$ and $\bullet$ denotes the label $S$.
• Figure 3: The root edge is removed from a rooted two-connected SP-map. The resulting map is either two-connected (left) or it is not (right).
• Figure 4: The removal of the first child of the root. Here, the vertices $\circ$ correspond to label $P$ and the vertices $\bullet$ to label $S$.
• Figure 5: Definition of $\varphi^\ast$ from $\varphi$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Lemma 3.1
• proof
• Corollary 3.2