Percolation in negative field and lattice animals.

Abstract

We study in detail percolation in a negative ‘‘ghost’’ field, and show that the percolation model crosses over, in the presence of a negative field h, to the lattice-animal model, as predicted by the field theory. This was done by exact solutions in one dimension and on a Cayley tree, and series expansions in general dimension. We confirm the scaling picture near the percolation threshold, and study the extended scaling ansatz for all values of h in terms of the nonlinear scaling field gh. Estimates for gh are obtained as a function of h in all dimensions. We also show how information on percolation clusters in all concentrations up to the percolation threshold may be extracted by studying the critical behavior of the generalized susceptibilities χk(p,h) near their critical point pc(h) as a function of h, and obtain data on the cluster distribution function and on the ratio of perimeter bonds to cluster bonds, for large clusters for all 0≤p≤pc. The crossover function is studied in one dimension, mean-field theory and the e expansion.