Size distribution of clusters in site-percolation on random recursive tree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911999608225792 |
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| author | Gu, Chenlin Yuan, Linglong |
| author_facet | Gu, Chenlin Yuan, Linglong |
| contents | We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben-Naim [J. Phys. A48(2015), no.4, 0405001, 15 pp.]. For a random recursive tree of size $n$, let every site have probability ${p \in (0,1)}$ to remain and with probability $(1-p)$ to be removed. As $n\to\infty,$ we show that the proportion of the remaining clusters of size $k$ is of order $k^{-1-\frac{1}{p}}$, resulting in a Yule-Simon distribution; the largest cluster size is of order $n^{p}$, and admits a non-trivial scaling limit. The proofs are based on the embedding of this model in the multi-type branching processes, and a coupling with the bond-percolation on random recursive trees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_12515 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Size distribution of clusters in site-percolation on random recursive tree Gu, Chenlin Yuan, Linglong Probability 60J27, 60J85, 60J80 We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben-Naim [J. Phys. A48(2015), no.4, 0405001, 15 pp.]. For a random recursive tree of size $n$, let every site have probability ${p \in (0,1)}$ to remain and with probability $(1-p)$ to be removed. As $n\to\infty,$ we show that the proportion of the remaining clusters of size $k$ is of order $k^{-1-\frac{1}{p}}$, resulting in a Yule-Simon distribution; the largest cluster size is of order $n^{p}$, and admits a non-trivial scaling limit. The proofs are based on the embedding of this model in the multi-type branching processes, and a coupling with the bond-percolation on random recursive trees. |
| title | Size distribution of clusters in site-percolation on random recursive tree |
| topic | Probability 60J27, 60J85, 60J80 |
| url | https://arxiv.org/abs/2408.12515 |