Directed Percolation - Lattice Models of Directed Percolation

Lattice Models of Directed Percolation

One of the simplest realizations of DP is bond directed percolation. This model is a directed variant of ordinary (isotropic) percolation and can be introduced as follows. The figure shows a tilted square lattice with bonds connecting neighboring sites. The bonds are permeable (open) with probability and impermeable (closed) otherwise. The sites and bonds may be interpreted as holes and randomly distributed channels of a porous medium.

The difference between ordinary and directed percolation is illustrated to the right. In isotropic percolation a spreading agent (e.g. water) introduced at a particular site percolates along open bonds, generating a cluster of wet sites. Contrarily, in directed percolation the spreading agent can pass open bonds only along a preferred direction in space, as indicated by the arrow. The resulting red cluster is directed in space.

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