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 & Bollobs for 1-independent bond percolations. Central to our argumentation are moment method bounds la Lyons supplemented by explicit percolation models la Balister & Bollobs. An indispensable tool is the minimality and explicit construction of Shearer's measure on the k-fuzz of Z. © 2011 Elsevier B.V. All rights reserved.