Equivalence of the two-dimensional directed-site animal problem to Baxter's hard-square lattice-gas model

Abstract

The directed-site animal problem on the square and triangular lattices is shown to be equivalent to Baxter's hard-square lattice-gas model with anisotropic next-nearest-neighbor interactions at a disorder point. The exact solution of the latter is used to determine the animal numbers as functions of their size in the two cases. Expressions in closed form are proposed for the number of animals on an infinite strip in terms of width and animal size, and for the average transverse extent of animals of a given size.