Oriented Percolation Model

A -dimensional discrete percolation model on a regular point lattice  is said to be oriented if  is an oriented lattice. One common such model takes place on the so-called north-east oriented lattice  obtained by orienting each edge of an arbitrary (perhaps unoriented) point lattice  in the direction of increasing coordinate-value.

The above figure shows an example of a subset of a 2-dimensional oriented percolation model on the north-east lattice. Here, each edge has been deleted with probability  for some , independently of all other edges.

Oriented percolation models are especially common in several areas of physics including astrophysics, solid state physics, and particle physics. Worth noting is that, while obvious parallels exist between oriented and unoriented percolation models, the proofs of results in the presence of orientation offer differ greatly from those of their unoriented analogues; indeed, the existence of so-called "one-way streets" restricts the degree of spatial freedom possessed by the flowing fluid (Grimmett 1999).

REFERENCES

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.