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).

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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. Ara˙jo and Jose Soares de Andrade published in Physica Review E, Vol.70, 066150 (2004).

Another interesting application of percolation is the formation of gypsum ,