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

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