Percolation is a logarithmic CFT. Among other things, this means it is not unitary.

This page focuses on “standard percolation” — see related theories below for alternative definitions that yield different CFTs.

Percolation can be defined as either bond or site percolation, and on any geometry of lattice. In both cases, by “lattice” we mean an (infinite) undirected graph.

Because percolation is not a unitary CFT, there is no Hamiltonian lattice theory that flows to percolation in the IR. All lattices in this section are spacetime lattices.

Usually percolation is defined on a square lattice. Many other possibilities exist, with the same critical behavior.

The most straightforward computational method is to instantiate a random lattice of some finite size and compute cluster sizes. This gives a single sample, and the procedure can be repeated as much as desired to improve the statistics. Note that although this is a Monte Carlo method, there is no (non-trivial) Markov chain. In other words, there is no “thermalization” or “mixing time” needed in the simulation.

Percolation in one dimension is, unsurprisingly, exactly solvable. Clearly no critical cluster can exist unless $p=1$, so that's the critical point. Working out the average cluster size at arbitrary $p$ can be an exercise.

For bond percolation on a 2-dimensional square lattice, $p_c = 0.5$ is known exactly.

In six or more dimensions critical percolation is described by mean-field theory.

The Bethe lattice can be thought of as the $d\rightarrow\infty$ lattice. It is the infinite tree in which each vertex has the same number ($z$ — called the coordination number) of neighbors. Note that $z=1$ is invalid, and $z=2$ is just a one-dimensional lattice.

As with many statistical models, percolation can be exactly solved on a Bethe lattice: see for instance these notes. The critical exponent is $\beta = 1$; the percolation threshold is $p_c = (z-1)^{-1}$.

• Directed percolation

• Wikipedia