Competing Long-Range Bonds and Site Dilution in the One-Dimensional Bond-Percolation Problem

The long-range bond-percolation problem, on a linear chain (d = 1), in the presence of diluted sites (with an occupancy probability ps for an active site) is studied by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j, separated by a distance rij is given by pij = p/r ij , where p represents the usual occupancy probability between nearest-neighbor sites. This model allows one to analyse the competition between long-range bonds (which enhance percolation) and diluted sites (which weaken percolation). By varying the parameter α (α ≥ 0), one may find a crossover between a nonextensive regime and an extensive regime; in particular, the cases α = 0 and α →∞ represent, respectively, two well-known limits, namely, the mean-field (infinite-range bonds) and first-neighbor-bond limits. The percolation order parameter, P∞, was investigated numerically for different values of α and ps. Two characteristic values of α were found, which depend on the site-occupancy probability ps, namely, α1(ps) and α2(ps) (α2(ps) > α1(ps) ≥ 0). The parameter P∞ equals unit, ∀p > 0, for 0 ≤ α ≤ α1(ps) and vanishes, ∀p < 1, for α > α2(ps). In the interval α1(ps) < α < α2(ps), the parameter P∞ displays a familiar behavior, i.e., 0 for p ≤ pc(α) and finite otherwise. It is shown that both α1(ps) and α2(ps) decrease with the inclusion of diluted sites. For a fixed ps, it is shown that a convenient variable, p∗ ≡ p∗(p, α, N), may be defined in such a way that plots of P∞ versus p∗ collapse for different sizes and values of α in the nonextensive regime.