论文标题
在有限及时图上存在渗透阈值
Existence of a percolation threshold on finite transitive graphs
论文作者
论文摘要
令$（g_n）$为一系列有限连接的顶点传递图，并倾向于无穷大。我们说，一系列参数$（p_n）$是一个渗透阈值，如果对于每一个$ \ varepsilon> 0 $，比例$ \ weft \ lett \ lvert \ lvert k_1 \ rvert \ rvert \ rvert \ rvert $ rvert $ rvert $ rvert $ thetices intercolation Percolation Percolation Percolation Percolation Percolation $ \ \ Mathbb {p} _p^g $ selplies as splate canteries n splate can \ lim_ {n \ to \ infty} \ mathbb {p} _ {（1+ \ varepsilon）p_n}^{g_n}^{g_n} \ left（\ left \ lest \ lvert k_1 \ rvert \ rvert \ rvert \ rvert \ rvert \ rvert \ rvert \ rvert \ geq al \ lim_ {n \ to \ infty} \ mathbb {p} _ {（1- \ varepsilon）p_n}^{g_n} \ left（\ left \ left \ lvert k_1 \ rvert k_1 \ rvert \ rert \ rerver \ rvert \ rvert \ geq al \ geqα\ right） \ end {split} \]我们证明$（g_n）$在且仅当$（g_n）$不包含特定无限的密集图病理子序列时才具有渗透阈值。我们的论点使用了Vanneuville通过耦合[van22]的相变的新证明，以及我们最近与Hutchcroft关于巨型群集的独特性[EH21]的作品。
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].
