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We study inhomogeneous Bernoulli bond percolation on the graph G Z, where G is a connected quasi-transitive graph. The inhomogeneity is introduced through a random region R around the origin axis \0\, where each edge in R is open with probability q and all other edges are open with probability p. When the region R is defined by stacking or overlapping boxes with random radii centered along the origin axis, we derive conditions on the moments of the radii, based on the growth properties of G, so that for any subcritical p and any q<1, the non-percolative phase persists.
Nascimento et al. (Wed,) studied this question.