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arXiv · 2503.19705

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

Daniel Amankwah·Jakob Björnberg·Sigurdur Örn Stefánsson·Benedikt Stufler·Joonas Turunen
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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).

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Daniel Amankwah, Jakob Björnberg, Sigurdur Örn Stefánsson, Benedikt Stufler, Joonas Turunen. 2025-03-25. The scaling limit of random 2-connected series-parallel maps. https://arxiv.org/abs/2503.19705

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