Can the root cluster remain largest forever in random recursive tree percolation?
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917553710825472 |
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| author | Zheng, Yushu |
| author_facet | Zheng, Yushu |
| contents | We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01881 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Can the root cluster remain largest forever in random recursive tree percolation? Zheng, Yushu Probability 60C05, 05C80, 60K35 We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$. |
| title | Can the root cluster remain largest forever in random recursive tree percolation? |
| topic | Probability 60C05, 05C80, 60K35 |
| url | https://arxiv.org/abs/2606.01881 |