The scaling limit of random 2-connected series-parallel maps
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916662189490176 |
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| author | Amankwah, Daniel Björnberg, Jakob Stefánsson, Sigurdur Örn Stufler, Benedikt Turunen, Joonas |
| author_facet | Amankwah, Daniel Björnberg, Jakob Stefánsson, Sigurdur Örn Stufler, Benedikt Turunen, Joonas |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The scaling limit of random 2-connected series-parallel maps Amankwah, Daniel Björnberg, Jakob Stefánsson, Sigurdur Örn Stufler, Benedikt Turunen, Joonas Probability 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). |
| title | The scaling limit of random 2-connected series-parallel maps |
| topic | Probability |
| url | https://arxiv.org/abs/2503.19705 |