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<p>We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben–Naim (<i>J. Phys. A.</i>, 2015). For a random recursive tree of size <i>n</i>, let every site have probability <i>p</i> ∈ (0, 1) to remain and with probability (1 − <i>p</i>) to be removed. As <i>n</i> → ∞, we show that the proportion of the remaining clusters of size <i>k</i> is of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k^{-1-{1\over{p}}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>k</mi> <mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </math></EquationSource> </InlineEquation>, resulting in a Yule–Simon distribution; the largest cluster size is of order <i>n</i><sup><i>p</i></sup>, 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.</p>

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Size Distribution of Clusters in Site-percolation on Random Recursive Tree

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Chenlin Gu,
  • Linglong Yuan

摘要

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. A., 2015). For a random recursive tree of size n, let every site have probability p ∈ (0, 1) to remain and with probability (1 − p) to be removed. As n → ∞, we show that the proportion of the remaining clusters of size k is of order \(k^{-1-{1\over{p}}}\) k − 1 − 1 p , resulting in a Yule–Simon distribution; the largest cluster size is of order np, 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.