Distribution of Transverse Distances in Directed Animals

TL;DR

This paper establishes a relationship between the distribution of transverse distances in directed animals and two-point correlation functions in lattice gases, deriving exact scaling functions and linking local site distributions to generating functions.

Contribution

It introduces a novel connection between directed animal distributions and lattice gas correlation functions, providing exact scaling forms and functions for specific dimensions.

Findings

Derived the exact scaling function for d=1: f(r) = (√π / 2√3) erfc(r/√3)

Showed that the central site distribution relates to the animals' generating function

Established the scaling form for large s: (s/R_s^d) f(|x|/R_s)

Abstract

We relate $\phi(\bf{x},s)$, the average number of sites at a transverse distance $\bf{x}$ 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 $s$, $\phi(\bf{x},s)$ has the scaling form $\frac{s}{R_s^d} f(|\bf{x}|/R_s)$, 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 $f(r) = \frac{\sqrt{\pi}}{2 \sqrt{3}}erfc(r/\sqrt{3})$. We also show that $\phi(\bf{x}=0,s)$ can be determined in terms of the animals number generating function of the directed animals.