Invariant properties of the percolation thresholds in the soft-core–hard-core transition

We report the first Monte Carlo study of the average critical number of bonds per site $B_c$ for three-dimensional continuum systems. The results show that for a system of spheres the dependence of $B_c$ on the soft-shell to hard-core ratio can be mapped onto the site-percolation–bond-percolation transition in lattices. For a system of very long cylinders, $B_c$ appears to be a dimensional invariant. It is predicted that for any continuum system the above dependence will be equal or weaker than that of a system of spheres.