Skip to main content Accessibility help
×
Home
  • Print publication year: 1998
  • Online publication date: December 2009

Appendix H - Directed percolation

Summary

Directed percolation is a special case of standard percolation (for a review on percolation theory, see Kinzel [1983], Stauffer and Aharony [1992]) in which only certain directions are allowed. To describe standard percolation, let us imagine a lattice where a given fraction p of the sites are ‘occupied’ and the rest are empty. Now define a cluster as a group of neighbouring occupied sites. All sites within a cluster are thus connected to each other by one unbroken chain of nearest-neighbour bonds connecting only occupied sites. Percolation theory deals with the number and properties of these clusters as the fraction p of occupied sites is varied. The occupied sites are assumed to be randomly distributed over the entire lattice and p is therefore the probability that a given site is occupied. (Note that the variant described here is called site-percolation: we started out with a lattice where the sites are either occupied or empty. One could as well introduce bond-percolation, where a fraction of the bonds are blocked.)

As the number p is increased, the clusters will, on average, grow in size and as p approaches a critical value, pc, there will be a finite probability that one connected cluster will extend over the entire lattice. We call the value pc the percolation threshold and to understand the statistical behaviour around this point it is necessary to take into account the strong critical fluctuations, which give rise to scaling laws with universal critical exponents.