Percolation, to quote a visitor
who gave a colloquium many years ago
at the Technion,
``is so simple that you could even explain it to your wife.''
(The audience reponse indicated some surprise and it was pointed out to a very
embarassed visitor that the only member of the audience active in percolation
theory was in fact someone's wife. Despite this, when referreing said visitors
recent submission to PRL I recommended publication.)
Here is an explanation suitable for husbands and other non-physicists:
If a container is filled with metal balls an electric current can pass thru
it. If the container were filled with glass beads, no current would pass thru.
What percentage of metal beads is needed so that electricity can pass thru???
Instead of beads of metal and glass we could study alloys made of
metallic and insulating atoms, and ask what percentage of the atoms
must be metallic in order that the alloy will conduct electricity.
The change from insulating to conducting state that occurs as the percentage
of metal balls is increased is called the percolation phase transition.
This is a phase transition very much like the phase transition
between magnetic and nonmagnetic states in a ferromagnet or
Ising model. In a magnet the phase transition occurs at Tc:
below this temperature the system is ferromagnetic, above paramagnetic.
The percolation transition occurs at the percolation threshold
below this there is no connection, above it the system is
Lets think about a one dimensional string of beads. How many must be
metallic for a current to pass thru them? i.e. what is pc
for this system.
I have a special relationship to percolation theory because I selected it
as a topic for my own undergraduate project more years ago than I care to
remember, and later edited a book on
"Percolation Structures and Processes". See an extract here .
Percolation can be studied by analytic (probablistic methods) but a case as
simple as the site problem on the square lattice cannot be solved with these methods. Thus in general simulation or exact enumeration (series expansions) is needed.
However, there are certain cases about which some things can be said exactly,
so lets start here.