k-independent percolation on trees

Consider the class of k-independent bond or site percolations with parameter p on a tree T. We derive tight bounds on p for both almost sure percolation and almost sure nonpercolation. The bounds are continuous functions of k and the branching number of T. This extends previous results by Lyons for the independent case (k = 0) and by Balister & Bollobas for 1-independent bond percolations. Central to our argumentation are moment method bounds a la Lyons supplemented by explicit percolation models a la Balister & Bollobas. An indispensable tool is the mini mality and explicit construction of Shearer's measure on the k-fuzz of Z. (C) 2011 Published by Elsevier B.V.

考虑在树\(T\)上具有参数\(p\)的\(k\)-独立键渗流或位点渗流类别。我们推导出了几乎必然渗流和几乎必然不渗流时\(p\)的紧界。这些界是\(k\)以及\(T\)的分支数的连续函数。这扩展了莱昂斯（Lyons）对于独立情形（\(k = 0\)）以及巴利斯特（Balister）和博洛巴斯（Bollobas）对于\(1\)-独立键渗流的先前结果。我们论证的核心是类似于莱昂斯的矩方法界，并由类似于巴利斯特和博洛巴斯的显式渗流模型作为补充。一个不可或缺的工具是在\(Z\)的\(k\)-模糊上的希勒（Shearer）测度的极小性和显式构造。（\(C\)）2011由爱思唯尔（Elsevier）B.V.出版