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In two dimensions, the first fact ("no percolation in the critical phase") is proved for many lattices, using duality. Substantial progress has been made on two-dimensional percolation through the conjecture of Oded Schramm that the scaling limit of a large cluster may be described in terms of a Schramm–Loewner evolution.
The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice.
In two dimensional square lattice percolation is defined as follows. A site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 1 – p; the corresponding problem is called site percolation, see Fig. 2. Percolation typically exhibits universality.
Directed percolation (DP) refers to percolation in which the fluid can flow only in one direction along bonds—such as only in the downward direction on a square lattice rotated by 45 degrees. This system is referred to as "1 + 1 dimensional DP" where the two dimensions are thought of as space and time.
Interpreting the preferred direction as a temporal degree of freedom, directed percolation can be regarded as a stochastic process that evolves in time. In a minimal, two-parameter model [1] that includes bond and site DP as special cases, a one-dimensional chain of sites evolves in discrete time , which can be viewed as a second dimension, and all sites are updated in parallel.
The RC model is a generalization of percolation, where each cluster is weighted by a factor of . Given a configuration ω {\displaystyle \omega } , we let C ( ω ) {\displaystyle C(\omega )} be the number of open clusters, or alternatively the number of connected components formed by the open bonds.
Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents ...
One of the simplest examples is Bernoulli percolation in a two dimensional square lattice. Sites are randomly occupied with probability . A cluster is defined as a collection of nearest neighbouring occupied sites. For small values of the occupied sites form only small local clusters.