Percolation in an interactive cluster-growth model

Abstract

Interactions are included in the standard model of site percolation by specifying two numbers, $p_0$ and $p_1$, which are the probabilities of occupying a site on a lattice if none or at least one of the neighboring sites are occupied, respectively. This is expected to be a better description of systems where interactions between neighbors can either enhance or deter site occupation. Monte Carlo methods are used to simulate the irreversible growth of clusters starting from an empty two-dimensional square lattice, and continuing through the percolation threshold. The value of the percolation threshold $p_c$ and other properties of the system depend on the parameter r=$p_1$/$p_0$. For r=1, the system corresponds to random percolation. As r→∞, compact Eden clusters are produced, which link together to form larger, fractal clusters. As r→0, neighboring sites become less easily occupied, resulting in checked ‘‘domains’’ separated by antiphase boundaries. Several quantities are analyzed to determine the critical exponents ν, γ, and β, including the threshold distribution width, the mean cluster size, and the infinite cluster size. For all finite values of r, the data are found to be consistent with the exponents of random percolation.