The clusters shown in the figure are the result
of a diffusion and aggregation process on
a square lattice. In the beginning
one has a random distribution of individual particles at a
given density that perform a Brownian motion. When two particles
touch they aggregate irreversibly. The such formed clusters
continue diffusing and aggregating. This model is called
cluster-cluster aggregation at finite density and describes
for instance the formation of soot or of colloidal suspensions.
With Max Kolb we found that using the model of cluster-cluster
aggregation one obtains a type of sol-gel transition with
a critical behaviour different from the usual percolation
as shown in the
paper published in J.Phys.A, Vol.18, L435-L441 (1985).
In a further
paper published in J.Phys.A, Vol.19, L1027-L1031 (1986)
we were able to present the scaling relations pertaining
to the different regimes defined by different initial
densities. Finally in the
paper published in Physical Review Letters, Vol. 59,
p.454-457 (1987) we showed that only at low densities
the clusters are volume fractals while at higher
densities one obtains clusters that are compact but have
a fractal surface with a new dimension.
A short review about the scaling laws and relations
to mean-field can be found in the
proceedings of the Cargese School "On Growth and Forms"
(Martinus Nijhoff Publishers, Dordrecht, 1986), pp. 222-226.