Systematic derivation of percolation thresholds in continuum systems

Analytic expressions for the critical percolation density in the continuum are derived using the direct-connectedness expansion method. Demonstrating a systematic application of this method for systems of permeable cubes or spheres, it is concluded that the present expansion is quite practical for the derivation of critical parameters of continuum systems. Correspondingly we also utilize the method to find the percolation threshold of the so-far unstudied continuum systems of oriented hypercubes. The results obtained are in excellent agreement with Monte Carlo simulations.