Monte Carlo results for continuum percolation in low and high dimensions

We report on Monte Carlo simulations of continuum percolation thresholds, by implementing highly efficient algorithms for very large samples. Our work, which includes percolation of hyperspheres, hypercubes, and boxes, in various dimensions, sizes, and shapes, has confirmed the expected dependence of the threshold on Vex, the total excluded volume, and on Bc, the average number of bonds per site. We have further confirmed that Vex = Bc, and that Bc is dependent on the objects shape, for which we offer a possible explanation. In particular we find that, counterintuitively, one can have Bc <1, as we have found for hyperspheres of dimension ≥12. From our many results for differently sized hyperspheres, we were also able to derive the correlation length exponent ν solely from the behavior of the thresholds using finite-size scaling.