# Percolation

## Percolation is the simplest model for random media.
It is typically defined on a lattice by independently
assigning to each site one or zero with probability p
or 1-p respectively. It is in fact the q->1 limit of the
Potts model .
At a certain threshold value pc
one infinite, spanning cluster appears which at pc is
a "volatile fractal". This we showed with
Gene Stanley
in a paper published in Physical
Review Letters Vol. 53, p.1121 (1984).

##
Percolation clusters can also been grown from one single seed.
This gives rise to models for the spreading of epidemics.
With Gene Stanley we investigated the structure of such
epidemic clusters in a generalized way as presented in our
paper published in Z. Phys.B Vol.60, p.165-170 (1985)

##
The part of the infinite cluster which would carry a current
if a potential is applied on both sides is called the
"backbone" and at pc it has the structure of a necklace made out
of blobs of different sizes. The size distribution follows
a powerlaw. This was shown using a new numerical technique
that we called "burning" with Dani Hong and Gene Stanley in
a
paper published in J. Phys. A Vol.17, L261 (1984).

## Part of the backbone is the minimum path, i.e. the shortest
path on the infinite cluster connecting two end points. At the
threshold this path is fractal and it dimension can be calculated
with high precision using the "burning" method as done with
Gene Stanley in a
paper published in J. Phys. A Vol.21, L829-L833 (2003).

##
While only the largest cluster precurses the infinite one
(for d < 6) also the second largest cluster etc are maximal
at the threshold. With Alla Margolina and Dietrich Stauffer
we studied the scaling between the size of the second largest
cluster and the system size as
published in Physics Letters Vol. 93A, p.73-75 (1982).

## In d dimension in fact one has d-1 percolation transitions,
namely the percolation of hyperplanes of dimensions less
than d. With
Janos Kertesz we derived relations between the
critical densities and the correlation length exponents
of these different transitions using duality as discussed in a
paper published in J. Phys. A Vol.18, L1109-L1112 (1985).

## If each site of a cluster can be connected to a common seed
such that the connecting path does never bend back one talks of
"directed percolation". This is simpler than usual percolation since
one direction can be identified with a time. But one has more exponents,
namely those in the direction of time and those perpendicular to it.
By modifying the occupation probabilities at the border with respect to
those of the bulk one can have a cross over to compact clusters
as shown with J.F.F. Mendez and R. Dickman
in Phys. Rev. E Vol.54, R3071-R3074 (1996).

## Directed percolation is in the same universality class as the
quantum Hamiltonian of the Reggeon field theory. By solving for the
largest eigenvalues of its transfer matrix we obtained with
Malte Henkel the low lying states of the spectrum as presented
in J. Phys. A Vol.23, 3719-3727 (1990). The self similarity
of the spectrum seems to indicate for a picture involving interacting
pseudo particles (excitations).

## The most recent paper on percolation concerns invasion percolation,
namely when the invasion takes place several times. In this multiple
invasion one observes a crossover in the fractal dimension to the one
of the optimal path which is about 1.22 in two dimensions as discussed
in a paper with Ascanio D. Araújo
and Jose Soares de Andrade published in Physica Review E, Vol.70, 066150
(2004).

## Another interesting application of percolation is the
formation of gypsum ,