Existence of a percolation threshold on finite transitive graphs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916156008300544 |
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| author | Easo, Philip |
| author_facet | Easo, Philip |
| contents | Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1 \right\rVert$ of vertices contained in the largest cluster under bond percolation $\mathbb{P}_p^G$ satisfies both \[ \begin{split}
\lim_{n \to \infty} \mathbb{P}_{(1+\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 1 \quad \text{for some $α> 0$, and}
\lim_{n \to \infty} \mathbb{P}_{(1-\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 0 \quad \text{for all $α> 0$}. \end{split}\] We prove that $(G_n)$ has a percolation threshold if and only if $(G_n)$ does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville's new proof of the sharpness of the phase transition for infinite graphs via couplings [Van22] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [EH21]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_09501 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Existence of a percolation threshold on finite transitive graphs Easo, Philip Probability Mathematical Physics Combinatorics Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1 \right\rVert$ of vertices contained in the largest cluster under bond percolation $\mathbb{P}_p^G$ satisfies both \[ \begin{split} \lim_{n \to \infty} \mathbb{P}_{(1+\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 1 \quad \text{for some $α> 0$, and} \lim_{n \to \infty} \mathbb{P}_{(1-\varepsilon)p_n}^{G_n} \left( \left\lVert K_1 \right\rVert \geq α\right) &= 0 \quad \text{for all $α> 0$}. \end{split}\] We prove that $(G_n)$ has a percolation threshold if and only if $(G_n)$ does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville's new proof of the sharpness of the phase transition for infinite graphs via couplings [Van22] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [EH21]. |
| title | Existence of a percolation threshold on finite transitive graphs |
| topic | Probability Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2208.09501 |