Distribution of Transverse Distances in Directed Animals

Abstract

We relate $\ddot{I}•(\mathbf{x},\mathbf{s})$, the average number of sites at a transverse distance $\mathbf{x}$ in the directed animals with $s$ sites in $d$ transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbor exclusion in $d$ dimensions. For large $s$, $\ddot{I}•(\mathbf{x},\mathbf{s})$ has the scaling form $\frac{s}{R_s^d}f(|\mathbf{x}|/{\mathbf{R}}_{\mathbf{s}})$, 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 $f(r)=\frac{\sqrt{\ddot{I}}€}{2\sqrt{3}}erfc(r/\sqrt{3})$. We also show that $\ddot{I}•(\mathbf{x}=0,\mathbf{s})$ can be determined in terms of the animals number generating function of the directed animals.

10 latex pages,1 eps-figure