kk-independent percolation on trees

Consider the class of kk-independent bond or site percolations with parameter pp on a tree TT. We derive tight bounds on pp for both almost sure percolation and almost sure nonpercolation. The bounds are continuous functions of kk and the branching number of TT. This extends previous results by Lyons for the independent case (k=0k=0) and by Balister & Bollobás for 11-independent bond percolations. Central to our argumentation are moment method bounds à la Lyons supplemented by explicit percolation models à la Balister & Bollobás. An indispensable tool is the minimality and explicit construction of Shearer’s measure on the kk-fuzz of ZZ.