Ghost fields, pair connectedness, and scaling: exact results in one-dimensional percolation

The percolation problem is solved exactly in one dimension. The functions obtained bear a strong resemblance to those of the n-vector model on the same lattice. Further, we include a ghost field exactly in all dimensions d, thereby treating the ‘thermodynamics’ of percolation without appealing to the Potts model. In particular, we show for d = 1 that the nature of the singularities near the critical percolation probability, pE = 1, is described by CY, = yp = 1, / 3, = 0, and 6, = CO. We also calculate the pair connectedness and correlation length explicitly, and find 7, = vP= 1, in agreement with the hyperscaling relation dv, = 2-up. Finally, scaling is demonstrated for both the cluster sue distribution and the percolation function analogous to the Gibbs free energy, and the scaling powers are explicitly evaluated; in particular, we find the exponents U = 1 and T = 2.