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\(k\)-independent percolation on trees. (English) Zbl 1245.82026

Summary: Consider the class of \(k\)-independent bond or site percolations with parameter \(p\) on a tree \(\mathbb T\). 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 \(\mathbb T\). 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 \(\mathbb Z\).

MSC:

82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C05 Trees
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References:

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