Integral equations for some correlated percolation models

The properties of the continuum Potts model are used to derive integral equations for the properties of correlated percolation. Specifically, extended Born–Greeen–Yvon (BGY) equations are derived for the two-point connectedness function in continuum percolation systems. Two different types of correlations among the percolating elements are considered: those due to a two-body potential and those due to impenetrable inclusions occupying a fraction of the system volume. In both cases, the superposition approximation is derived and solved numerically. The variation of the percolation threshold with increasing correlation is calculated and discussed. Finally, higher order corrections to the superposition approximation are obtained by analyzing the relevant Mayer series.