-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 & Bollobás for 1 -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 k -fuzz of Z .