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Percolation of arbitrary words in \(\{0, 1\}^ \mathbb{N}\). (English) Zbl 0832.60095

Summary: Let \({\mathcal G}\) be a (possibly directed) locally finite graph with countably infinite vertex set \({\mathcal V}\). Let \(\{X(v) : v \in {\mathcal V}\}\) be an i.i.d. family of random variables with \[ P\{X(v) = 1\} = 1 - P \{X(v) = 0\} =p. \] Finally, let \(\xi = (\xi_1, \xi_2, \ldots)\) be a generic element of \(\{0,1\}^\mathbb{N}\); such a \(\xi\) is called a word. We say that the word \(\xi\) is seen from a vertex \(v\) if there exists a self- avoiding path \((v, v_1, v_2, \ldots)\) on \({\mathcal G}\) starting at \(v\) and such that \(X(v_i) = \xi_i\) for \(i \geq 1\). The traditional problem in (site) percolation is whether \(P \{(1,1,1, \dots)\) is seen from \(v\} > 0\). So-called \(AB\)-percolation occurs if \(P\{ (1,0,1,0,1,0, \dots)\) is seen from \(v\} > 0\). Here we investigate (a) whether \(P\){all words are seen from \(v\} > 0\) (for a fixed \(v)\) and (b) whether \(P\){all words are seen from some \(v\} = 1\). We show that both answers are positive if \({\mathcal G} = \mathbb{Z}^d\), or even \(\mathbb{Z}^d_+\) with all edges oriented in the “positive direction”, when \(d\) is sufficiently large. We show that on the oriented \(\mathbb{Z}^3_+\) the answer to (a) is negative, but we do not know the answer to (b) on \(\mathbb{Z}^3_+\). Various graphs \({\mathcal G}\) are constructed (almost all of them trees) for which the set of words \(\xi\) which can be seen from a given \(v\) (or from some \(v)\) is large, even though it is w.p. 1 not the set of all words.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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