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-独立随机图中的长路径和连通性


概率测度 如果由在G中图形距离至少为 1 的边集确定的事件是独立的，则图G的边集的子集上的 是G上的 1独立概率测度 (1-ipm)。给定 1-ipm ，表示为 相关的随机图模型。让 表示 1‐ipms 的集合 在G上，每条边都包含在 概率至少为p 。为了 ，Balister 和 Bollobás 要求最小p _⋆的值，使得对于所有p > p _⋆并且所有 , 几乎可以肯定包含无限分量。在本文中，我们显着改进了p _⋆之前的下限。我们还确定了直线格子和阶梯格子上出现长路径的 1 独立临界概率。最后，对于有限图G，我们研究f _{1, G} ( p )，即所有的下确界 的概率 已连接。当G是一条路径、一个完全图且长度至多为 5 的环时，我们准确地确定f _{1, G} ( p )。