## 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

### Keywords:

percolation of words; truncation question
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### References:

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