A probability measure on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm , denote by the associated random graph model. Let denote the collection of 1‐ipms on G for which each edge is included in with probability at least p. For , Balister and Bollobás asked for the value of the least p_⋆ such that for all p > p_⋆ and all , almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p_⋆. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f_1, G(p), the infimum over all of the probability that is connected. We determine f_1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.

中文翻译：

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1个独立随机图的长路径和连通性

一个概率测度的边缘集合的曲线图的子集G ^是一个1个非依赖性 概率测度（1-IPM）上ģ如果事件判断由边缘集，是在离开图中的距离至少为1 ģ是独立的。给定1-ipm ，由关联的随机图模型表示。让表示的1-IPMS集合上ģ针对每个边缘被包括在与至少概率p。为，Balister和Bollobás要求最少的值p _⋆使得对于所有p  >  p _⋆和所有，几乎可以肯定包含一个无限分量。在本文中，我们显着改善了p _previous的先前下界。我们还确定了直线和梯形格子上长路径出现的1个独立临界概率。最后，对于有限图G，我们研究f _{1，  G}（p），即所有连通概率的最小值。当G是一条路径，一个完整的图和一个长度为5的循环时_，我们精确地确定f _{1，  G}（p）。