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

Abstract

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, a ghost field is included exactly in all dimensions d, thereby treating the 'thermodynamics' of percolation without appealing to the Potts model. In particular, it is shown that for d=1 that the nature of the singularities near the critical percolation probability, p_c=1, is described by alpha _p= gamma _p=1, beta _p=0, and delta _p= infinity . The pair connectedness and correlation length are calculated explicitly, and eta _p= nu _p=1, in agreement with the hyperscaling relation d nu _p=2- alpha _p. Finally, scaling is demonstrated for both the cluster size distribution and the percolation function analogous to the Gibbs free energy, and the scaling powers are explicitly evaluated; in particular, the exponents sigma =1 and tau =2, are found.