Continuum asymptotics for tree growth models with uniform backward dynamics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915863404216320 |
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| author | Geldbach, David |
| author_facet | Geldbach, David |
| contents | We study (plane) tree-valued Markov chains $(T_n,n \geq 1)$ with uniform backward dynamics and show that they can be obtained by sampling from a real tree. As non--plane trees, every such Markov chain is represented by a weighted real tree. We equip this real tree with a planar order as well as some extra functions for the full representation theorem. We also show that under an inhomogeneous rescaling after trimming leaves $(T_n, n\geq 1)$ converges to a random real tree in the Gromov--Prokhorov metric. This makes use of a special class of real trees, interval partition trees, which were introduced by Forman (2020). Moreover, this generalises and sheds some new light on work by Evans, Grübel and Wakolbinger (2017) on the binary special case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04336 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuum asymptotics for tree growth models with uniform backward dynamics Geldbach, David Probability 60G09, 54E70, 60C05 We study (plane) tree-valued Markov chains $(T_n,n \geq 1)$ with uniform backward dynamics and show that they can be obtained by sampling from a real tree. As non--plane trees, every such Markov chain is represented by a weighted real tree. We equip this real tree with a planar order as well as some extra functions for the full representation theorem. We also show that under an inhomogeneous rescaling after trimming leaves $(T_n, n\geq 1)$ converges to a random real tree in the Gromov--Prokhorov metric. This makes use of a special class of real trees, interval partition trees, which were introduced by Forman (2020). Moreover, this generalises and sheds some new light on work by Evans, Grübel and Wakolbinger (2017) on the binary special case. |
| title | Continuum asymptotics for tree growth models with uniform backward dynamics |
| topic | Probability 60G09, 54E70, 60C05 |
| url | https://arxiv.org/abs/2309.04336 |