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 neighbor 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 animals number generating function of the directed animals.