EXACT SOLUTION OF A ONE-DIMENSIONAL CONTINUUM PERCOLATION MODEL

I consider a one-dimensional system of particles that interact through a hard core of diameter sigma and can connect to each other if they a re closer than a distance d. The mean cluster size increases as a func tion of the density rho until it diverges at some critical density, th e percolation threshold. This system can be mapped onto an off-lattice generalization of the Potts model, which I have called the Potts flui d, and in this way, the mean cluster size, pair connectedness, and per colation probability can be calculated exactly. The mean cluster size is S=2 exp[rho(d-sigma)/(1-rho sigma)]-1 and diverges only at tile clo se-packing density rho(CP)=1/sigma. This is confirmed by the behavior of the percolation probability. These results should help in judging t he effectiveness of approximations or simulation methods before they a re applied to higher dimensions.