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 a(n 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.

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

• Directed percolation