Percolation of words on the hypercubic lattice with one-dimensional long-range interactions

We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter 0 or 1 according to Bernoulli r.v.’s with parameter p. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on Zd−1×Z, d⩾3, where each edge parallel to Zd−1 has length one and is open with probability ϵ, while edges of length n parallel to Z are open with probability pn. We prove that if the sum of pn diverges, then for any ϵ and p, there is a K such that all words are seen from the origin with probability close to 1, even if all connections with length larger than K are suppressed.