Phase transitions for percolation of words in one dimension

In this paper, we investigate models of percolation of words on $\mathbb{Z}_+$ with long-range connections. The underlying graph is oriented and constructed according to a sequence of non-negative ranges. In the first model, the sequence of ranges is considered random and, in the second model, deterministic and non-decreasing. In both cases, we establish a phase transition for the occurrence of percolation of all words simultaneously from the origin.