Theory of continuum percolation III. Low density expansion

TL;DR

This paper develops a low density expansion for continuum percolation using a mapping to the Potts fluid, providing more accurate critical density estimates for hypercubes than previous integral equation methods.

Contribution

It introduces a novel low density expansion approach for continuum percolation based on a mapping to the Potts fluid, improving critical density predictions.

Findings

Critical density within 5% of simulations

More precise than integral equation methods

Validates the phase transition focus of the approach

Abstract

We use a previously introduced mapping between the continuum percolation model and the Potts fluid (a system of interacting s-states 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 et al. [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).