K-independent percolation on trees

TL;DR

This paper establishes precise bounds on the percolation threshold p for k-independent percolations on infinite trees, extending prior results and utilizing moment methods and Shearer's measure.

Contribution

It provides tight bounds on percolation thresholds for k-independent models on trees, generalizing previous work for independent and 1-independent cases.

Findings

Derived tight bounds on p for a.s. percolation and nonpercolation

Extended Lyons' and Bollob extasciitildeas & Balister's results to k-independent cases

Utilized moment bounds and Shearer's measure in proofs

Abstract

Consider the class of k-independent bond, respectively site, percolations with parameter p on an infinite tree T. We derive tight bounds on p for both a.s. percolation and a.s. 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 Bollob\`as & Balister for 1-independent bond percolations. Central to our argumentation are moment method bounds \`a la Lyons supplemented by explicit percolation models \`a la Bollob\`as & Balister. An indispensable tool is the minimality and explicit construction of Shearer's measure on the k-fuzz of Z.