Competing long-range bonds and site dilution in the one-dimensional bond-percolation problem

• Autor(a) principal: Fulco,U. L.
• Data de Publicação: 2003
• Outros Autores: Silva,L. R. da, Nobre,F. D., Lucena,L. S.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Brazilian Journal of Physics
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332003000300032

The long-range bond-percolation problem, on a linear chain (d = 1), in the presence of diluted sites (with an occupancy probability p s 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 r ij is given by p ij = <img width=32 height=32 id="_x0000_i1026" src="../../img/revistas/bjp/v33n3/a32img01.gif" align=absbottom>, 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 a (a > 0), one may find a crossover between a nonextensive regime and an extensive regime; in particular, the cases a = 0 and a ® ¥ 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 a and p s. Two characteristic values of a were found, which depend on the site-occupancy probability p s, namely, a1(p s) and a2(p s) (a2(p s) > a1(p s) > 0). The parameter P¥ equals unit, "p > 0, for 0 < a < a1(p s) and vanishes, "p < 1, for a > a2(p s). In the interval a1(p s) < a < a2(p s), the parameter P¥ displays a familiar behavior, i.e., 0 for p < p c(a) and finite otherwise. It is shown that both a1(p s) and a2(p s) decrease with the inclusion of diluted sites. For a fixed p s, it is shown that a convenient variable, p* º p*(p, a, N), may be defined in such a way that plots of P¥ versus p* collapse for different sizes and values of a in the nonextensive regime.