Fluctuations of the Horton-Strahler number of stable Galton-Watson trees
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914028035506176 |
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| author | Khanfir, Robin |
| author_facet | Khanfir, Robin |
| contents | The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned to have size $n$ (including the Catalan trees), which are the finite-dimensional marginals of stable Lévy trees. While these random variables are known to grow as a multiple of $\ln n$ in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton-Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, i.e. the stable Lévy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fluctuations of the Horton-Strahler number of stable Galton-Watson trees Khanfir, Robin Probability 60C05, 60J80, 60F05, 60F17, 05C05, 60D05 The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned to have size $n$ (including the Catalan trees), which are the finite-dimensional marginals of stable Lévy trees. While these random variables are known to grow as a multiple of $\ln n$ in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton-Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, i.e. the stable Lévy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number. |
| title | Fluctuations of the Horton-Strahler number of stable Galton-Watson trees |
| topic | Probability 60C05, 60J80, 60F05, 60F17, 05C05, 60D05 |
| url | https://arxiv.org/abs/2401.13771 |