Percolation seems rather simple. Conider a simple two-dimensional lattice. Each
of the lattice sites we occupy with a probability p and leave the site empty with a
probability (1-p). For low p we do not expect very large clusters. A Cluster are sites
that are nearest neighbours. At large p we expect a cluster that encompasses almost all
sites. Let us denote the number of sites, that participate in the largest cluster divided
by the volume by P¥. For
small p we should find P¥ = 0 and
for large p P¥ = 1. There exists a 0 <
p = pc <>< 1, such that P¥<>
0 for the frist time. At this critical point we have a (geometrical) phase transition:
![pinfty.jpg (2253 Byte)](images/pinfty.jpg)
Here ß is a critical exponent.
Let nl be the number of clusters of size l, i.e. with l sites,
then we can define the susceptibiltiy as
![susceptibility.jpg (2884 Byte)](images/susceptibility.jpg)
which deiverges as we approach the criticial probability
![suscritical.jpg (2022 Byte)](images/suscritical.jpg)
We have introduced yet another exponent. |