We define an inhomogeneous percolation model on “ladder graphs” obtained as direct products of an arbitrary graph $$G = (V,E)$$ G = ( V , E ) and the set of integers $${\mathbb {Z}}$$ Z . (Vertices are thought of as having a “vertical” component indexed by an integer.) We make two natural choices for the set of edges, producing an unoriented graph $${\mathbb {G}}$$ G and an oriented graph $$\vec {{\mathbb {G}}}$$ G → . These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite “column” are open with probability q and all other edges are open with probability p . For all fixed q one can define the critical percolation threshold $$p_\mathrm{c}(q)$$ p c ( q ) . We show that this function is continuous in (0, 1).

中文翻译：

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梯形图上的非均匀渗透

我们在作为任意图 $$G = (V,E)$$ G = ( V , E ) 和整数集 $${\mathbb {Z} }$$ Z 。（顶点被认为有一个由整数索引的“垂直”分量。）我们对边集做出两个自然选择，生成一个无向图 $${\mathbb {G}}$$ G 和一个有向图 $ $\vec {{\mathbb {G}}}$$ G → . 这些图被赋予了渗透配置，其中，固定的无限“列”内的边独立地以概率 q 开放，所有其他边以概率 p 开放。对于所有固定的 q，可以定义临界渗透阈值 $$p_\mathrm{c}(q)$$ pc ( q ) 。我们证明这个函数在 (0, 1) 上是连续的。