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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2410.22152 |
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- 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.