Theory of continuum percolation. III. Low-density expansion

We use a mapping between the continuum percolation model and the Potts fluid (a system of interacting [Formula Presented]-state spins which are free to move in the continuum) to derive the low-density expansion of the pair connectedness and the mean cluster size. We prove that given an adequate identification of functions, the result is equivalent to the density expansion derived from a completely different point of view by Coniglio, DeAngelis, and Forlani [J. Phys. A 10, 1123 (1977)] to describe physical clustering in a gas. We then apply our expansion to a system of hypercubes with a hard core interaction. The calculated critical density is within approximately 5% of the results of simulations, and is thus much more precise than previous theoretical results which were based on integral equations. We suggest that this is because integral equations smooth out overly the partition function (i.e., they describe predominantly its analytical part), while our method targets instead the part which describes the phase transition (i.e., the singular part).