Distribution of transverse distances in directed animals

Abstract

We relate φ(x, s), the average number of sites at a transverse distance x in the directed animals with s sites in d transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbour exclusion in d dimensions. For large s, φ(x, s) has the scaling form s/R^d_s f(|x|/R_s), where R_s is the root-mean square radius of gyration of animals of s sites. We determine the exact scaling function for d = 1 to be f(r) = √π/2√3 erfc(r/√3). We also show that φ(x = 0, s) can be determined in terms of the animal number generating function of the directed animals.