Exact solution of a one-dimensional continuum percolation model

TL;DR

This paper presents an exact analytical solution for a one-dimensional continuum percolation model, revealing the critical density at which percolation occurs and providing explicit formulas for key quantities.

Contribution

It introduces an exact solution for the mean cluster size and percolation threshold in a 1D continuum percolation system by mapping it onto a generalized Potts model.

Findings

Mean cluster size diverges at close packing density

Percolation threshold occurs at close packing density

Exact formulas for cluster size and percolation probability

Abstract

I consider a one dimensional system of particles which interact through a hard core of diameter $\si$ and can connect to each other if they are closer than a distance $d$. The mean cluster size increases as a function of the density $\rho$ until it diverges at some critical density, the percolation threshold. This system can be mapped onto an off-lattice generalization of the Potts model which I have called the Potts fluid, and in this way, the mean cluster size, pair connectedness and percolation probability can be calculated exactly. The mean cluster size is $S = 2 \exp[ \rho (d -\si)/(1 - \rho \si)] - 1$ and diverges only at the close packing density $\rho_{cp} = 1 / \si $. This is confirmed by the behavior of the percolation probability. These results should help in judging the effectiveness of approximations or simulation methods before they are applied to higher dimensions.