If one consideres the 1 in a
configuration x in X={0,1} ^{ G }
where G= Z^{ d } as 'conducting cells' and
0 as 'isolating cells' one has a percolation problem.
Can one
march to infinity on the conducting 1's to infinity,
if one is allowed in each step to go from one cell to a one
of the 3^{ d 1 neighboring cells?
The set of cells which can be reached from a 'conducting cell' are
called a 'cluster'.
The classical percolation problem is when x is choosen randomly
in X with respect to the product measure of the measure
p(1)=p, p(0) = 1p on {0,1}. In this case, there exists a critical
value p * such that for
p smaller than p * , all clusters are
finite and for p bigger than p * , there
exists at least one infinite cluster.
The percolation problem is also interesting in less random situations.
The general problem is that one has an ergodic
Z d action on a probability space (Y,m)
and a random variable f:Y > R which takes values 0 or 1. If the
Z d action is generated by
T j and one writes
T n =
T 1 n1
T 2 n2
...
T d nd ,
where n=(n1, ... , nd) one has for almost all y in Y
a configuration x in X given by
x n = f(T n )
An example is almost periodic percolation.
With this, one means that the closure of
all translates of x form a uniquely ergodic subshift.
One gets such configurations
with the probability space given by the circle with the Lebesgue
measure and letting
T j to be ergodic translations.
Choose f to be the characteristic function of an interval I with length p.
The animated picture on this page shows such configurations in the
case d=2, where the length p of interval goes from 0.9 to 0.1
} 
The Mathematica code to
generate such configutations is
IsInI[x_,a_,b_]:=If[a<=x<=b,1,0];f[n_,m_]:=Mod[N[Sqrt[2]*n+Sqrt[3]*m],1]
Percolation[c_]:=ListDensityPlot[Table[
IsInI[f[i,j],0,c],{i,100},{j,100}], Mesh>False,
Frame>True,FrameTicks>False,Axes>False];
