Exact solution of a one-dimensional continuum percolation model

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.