Critical site percolation and cutsets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917965774979072 |
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| author | Li, Zhongyang |
| author_facet | Li, Zhongyang |
| contents | In 2003, Kahn conjectured a characterization of the critical percolation probability $p_c$
in terms of vertex cut sets (\cite{JK03}). Later, Lyons and Peres (2016) conjectured a similar characterization of $p_c$
, but in terms of edge cut sets (\cite{LP16}). Both conjectures were subsequently proven by Tang (\cite{pt23}) for bond percolation and site percolation on bounded-degree graphs. Tang further conjectured that Kahn's vertex-cut characterization for $p_c^{site}$
and the Lyons-Peres edge-cut characterization for $p_c^{site}$ would hold for site percolation on any infinite, connected, locally finite graph.
In this paper, we establish Kahn's vertex-cut characterization for $p_c^{site}$ by adapting arguments from \cite{jmh57a, DCT15}. Additionally, we disprove the Lyons-Peres edge-cut characterization for $p_c^{site}$ by constructing a counterexample. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical site percolation and cutsets Li, Zhongyang Probability In 2003, Kahn conjectured a characterization of the critical percolation probability $p_c$ in terms of vertex cut sets (\cite{JK03}). Later, Lyons and Peres (2016) conjectured a similar characterization of $p_c$ , but in terms of edge cut sets (\cite{LP16}). Both conjectures were subsequently proven by Tang (\cite{pt23}) for bond percolation and site percolation on bounded-degree graphs. Tang further conjectured that Kahn's vertex-cut characterization for $p_c^{site}$ and the Lyons-Peres edge-cut characterization for $p_c^{site}$ would hold for site percolation on any infinite, connected, locally finite graph. In this paper, we establish Kahn's vertex-cut characterization for $p_c^{site}$ by adapting arguments from \cite{jmh57a, DCT15}. Additionally, we disprove the Lyons-Peres edge-cut characterization for $p_c^{site}$ by constructing a counterexample. |
| title | Critical site percolation and cutsets |
| topic | Probability |
| url | https://arxiv.org/abs/2410.22152 |