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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.16894 |
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| _version_ | 1866915684939726848 |
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| author | Curien, Nicolas Fleurat, William Twigt, Adrianus |
| author_facet | Curien, Nicolas Fleurat, William Twigt, Adrianus |
| contents | Can we obtain a Brownian CRT of mass $1/2$ from a CRT of mass $1$ by cutting certain branches?
In this paper, we will answer that question in the much more general setting of self-similar Markov trees. Self-similar Markov trees (ssMt) are random decorated trees that encode the genealogy of a system of particles carrying positive labels, and where particles undergo splitting and growth depending on their labels in a self-similar fashion. Introduced and developed in the recent monograph (Bertoin-Curien-Riera, 2024), they provide a broad generalization of Brownian and stable continuum random trees and arise naturally in various models of random geometry such as the Brownian sphere/disk. The law of a ssMt is characterized by its quadruplet $(\mathrm{a}, σ^2, \boldsymbolΛ; α)$, which specifies the features of the underlying growth-fragmentation mechanism, together with the initial decoration $x>0$. In this work, we focus on special cases of ssMt in which the trees started from different initial values $x>0$ can be coupled into a continuous, increasing family of nested subtrees. In the case of the Brownian and stable continuum random trees, this yields surprisingly simple novel dynamics corresponding to the scaling limit of the leaf-growth algorithms of Luczak-Winkler and Caraceni-Stauffer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16894 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Growing Self-Similar Markov Trees Curien, Nicolas Fleurat, William Twigt, Adrianus Probability Can we obtain a Brownian CRT of mass $1/2$ from a CRT of mass $1$ by cutting certain branches? In this paper, we will answer that question in the much more general setting of self-similar Markov trees. Self-similar Markov trees (ssMt) are random decorated trees that encode the genealogy of a system of particles carrying positive labels, and where particles undergo splitting and growth depending on their labels in a self-similar fashion. Introduced and developed in the recent monograph (Bertoin-Curien-Riera, 2024), they provide a broad generalization of Brownian and stable continuum random trees and arise naturally in various models of random geometry such as the Brownian sphere/disk. The law of a ssMt is characterized by its quadruplet $(\mathrm{a}, σ^2, \boldsymbolΛ; α)$, which specifies the features of the underlying growth-fragmentation mechanism, together with the initial decoration $x>0$. In this work, we focus on special cases of ssMt in which the trees started from different initial values $x>0$ can be coupled into a continuous, increasing family of nested subtrees. In the case of the Brownian and stable continuum random trees, this yields surprisingly simple novel dynamics corresponding to the scaling limit of the leaf-growth algorithms of Luczak-Winkler and Caraceni-Stauffer. |
| title | Growing Self-Similar Markov Trees |
| topic | Probability |
| url | https://arxiv.org/abs/2512.16894 |