Theory of continuum percolation I. General formalism

TL;DR

This paper develops a comprehensive theoretical framework for continuum percolation by introducing the Potts fluid model, linking statistical mechanics methods to percolation phenomena in continuous systems.

Contribution

It introduces the Potts fluid model as a new formalism, enabling the application of statistical mechanics to continuum percolation problems.

Findings

Potts magnetization relates to percolation probability

Susceptibility corresponds to mean cluster size

Correlation functions are linked to pair-connectedness

Abstract

The theoretical basis of continuum percolation has changed greatly since its beginning as little more than an analogy with lattice systems. Nevertheless, there is yet no comprehensive theory of this field. A basis for such a theory is provided here with the introduction of the Potts fluid, a system of interacting $s$-state spins which are free to move in the continuum. In the $s \to 1$ limit, the Potts magnetization, susceptibility and correlation functions are directly related to the percolation probability, the mean cluster size and the pair-connectedness, respectively. Through the Hamiltonian formulation of the Potts fluid, the standard methods of statistical mechanics can therefore be used in the continuum percolation problem.