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Percolation of words on \(\mathbb Z^{d}\) with long-range connections. (English) Zbl 1231.60118

Summary: Consider an independent site percolation model on \(\mathbb Z^{d}\), with parameter \(p \in \) (0, 1), where all long-range connections in the axis directions are allowed. In this work, we show that, given any parameter \(p\), there exists an integer \(K(p)\) such that all binary sequences (words) \(\xi \in\{0, 1\}^{\mathbb N}\) can be seen simultaneously, almost surely, even if all connections with length larger than \(K(p)\) are suppressed. We also show some results concerning how \(K(p)\) should scale with \(p\) as \(p\) goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
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