Computational studies of bond-site percolation.

Abstract

Percolation theory enters in various areas of research including critical phenomena and phase transitions. Bond-site percolation is a generalization of pure percolation motivated by the fact that bond-site is close to many physical realities. This work relies on a numerical study of percolation in lattices. A lattice is a regular pattern of sites also known as nodes or vertices connected by bonds also known as links or edges. Sites may be occupied or unoccupied, where the concentration ps is the fraction of occupied sites. The quantity pb is the fraction of open bonds. A cluster is a set of occupied sites connected by opened bonds. The bond-site percolation problem is formulated as follows: we consider an infinite lattice whose sites and bonds are at random or correlated and either allowed or forbidden with probabilities ps and pb that any site and any bond are occupied and open respectively. If those probabilities are small, there appears a sprinkling of isolated clusters each consisting of occupied sites connected by open bonds surrounded by numerous unoccupied sites. As the probabilities increase, reaching critical values above which there is an infinitely large cluster, then percolation is taking place. This means that one can cross the entire lattice by going successively from one occupied site connected by a opened bond to a neighbouring occupied site. The sudden onset of a spanning cluster happens at particular values of ps and pb, called the critical concentrations. Quantities related to cluster configuration (mean cluster and correlation length) and individual cluster structure (size and gyration radius of clusters ) are determined and compared for different models. In our studies, the Monte Carlo approach is applied while some authors used series expansion and renormalization group methods. The contribution of this work is the application of models in which the probability of opening a bond depends on the occupancy of sites. Compared with models in which probabilities of opening bonds are uncorrelated with the occupancy of sites, in the suppressed bond-site percolation, the higher site occupancy is needed to reach percolation. The approach of suppressed bond-site percolation is extended by considering direction of percolation along bonds (directed suppressed bond-site percolation). Fundamental results for models of suppressed bond-site percolation and directed suppressed bond-site percolation are the numerical determination of phase boundary between the percolating and non-percolating regions. Also, it appears that the spanning cluster around critical concentration is independent on models. This is an intrinsic property of a system.