Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite connected subgraphs of $G$ and consider a model of Bernoulli bond percolation on $\mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $\mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}\left(q\right)$ is a continuous function in $\left(0,1\right)$, provided that the graphs in $\mathcal{C}$ are "well-spaced" in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szab\'o and Valesin.

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关于梯形图非均匀渗透的注记

设 $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ 为无限连通图 $G=(V,E)$ 的笛卡尔积和整数集 $\mathbb{Z}$。我们选择 $G$ 的有限连通子图的集合 $\mathcal{C}$ 并考虑在 $\mathbb{G}$ 上的伯努利键渗流模型，该模型为每个边分配开放概率 $q$，其投影到$G$ 位于 $\mathcal{C}$ 的某个子图中，并且每个其他边的概率为 $p$。我们证明临界渗透阈值 $p_{c}\left(q\right)$ 是 $\left(0,1\right)$ 中的连续函数，前提是 $\mathcal{C}$ 中的图是$G$ 中的“良好间隔”及其顶点集具有统一有界的基数。这概括了最近由 Szab\'o 和 Valesin 得出的结果。