Excluded volume and its relation to the onset of percolation

The general relationship between the percolation threshold of systems of various objects and the excluded volume associated with these objects is discussed. In particular, we derive the average excluded area and the average excluded volume associated with two- and three-dimensional randomly oriented objects. The results yield predictions for the dependencies, of the percolation critical concentration of various kinds of "sticks," on the stick aspect ratio and the anisotropy of the stick orientation distribution. Comparison of the present results with available Monte Carlo data shows that the percolation threshold of the sticks is described by the above dependencies. On the other hand, the numerical values of the excluded area and the excluded volume are not dimensional invariants as suggested in the literature, but rather depend on the randomness of the stick orientations. The usefulness of the present results for percolation-threshold problems in the continuum is discussed. In particular, it is shown that the excluded area and the excluded volume give the number of bonds per object $B_c$ when the objects are all the same size. In the case where there is a distribution of object sizes, the proper average of the excluded area or volume is a dimensional invariant while $B_c$ is not.