the Fortuin-Kasteleyn random-cluster representation
/ Fortuin: FOR-town; Kasteleyn: KAS-tuh-line /
The random-cluster model is the master object that unifies percolation and the Ising/Potts spin models under one roof. It is a single two-parameter family of dependent edge-percolation measures, and tuning one parameter recovers ordinary percolation, the Ising model, or the q-state Potts model. The Fortuin-Kasteleyn representation is the precise dictionary translating spin questions into connectivity (percolation) questions, and it is the reason percolation methods power the rigorous theory of phase transitions in magnets.
The random-cluster measure phi_{p,q} on edge configurations omega in {0,1}^E assigns to omega a weight proportional to (product over edges of p^(omega(e)) (1-p)^(1-omega(e))) times q^(k(omega)), where k(omega) is the NUMBER OF OPEN CLUSTERS in omega. The extra factor q^(k(omega)) is what introduces dependence: unlike Bernoulli percolation, opening an edge can merge two clusters and lower k, so for q > 1 the model rewards connectivity and edges are positively correlated (it satisfies the FKG lattice condition). When q = 1 the factor disappears and phi_{p,1} is exactly Bernoulli(p) percolation. The Edwards-Sokal coupling links it to spins: sample edges from phi_{p,q} with p = 1 - e^(-beta), then colour each open cluster independently and uniformly with one of q colours; the resulting site colours are exactly distributed as the q-state Potts model at inverse temperature beta (q = 2 gives Ising). Conversely, spin correlations equal connection probabilities: in Ising, E[sigma_x sigma_y] = phi(x is connected to y), so the magnetic order-disorder transition IS the percolation transition of the random-cluster model.
This equivalence is enormously powerful: it lets one prove magnetic phase transitions, exponential decay of spin correlations, and continuity of the Ising magnetization using FKG, BK, and Russo-type arguments imported from percolation. It is how Smirnov's conformal-invariance program reaches the Ising model and how Duminil-Copin and collaborators proved continuity of the phase transition for 1 <= q <= 4 and its discontinuity (first-order) for q > 4 in two dimensions. Two honest caveats: q^(k(omega)) makes the model NON-product and genuinely dependent, so the clean independence-based arguments of Bernoulli percolation must be replaced by their FKG-monotonicity analogues; and the representation requires q to be a positive integer for the spin coupling, though the random-cluster measure itself makes sense for any real q >= 1 (and the q -> 0, p -> 0 limit connects to spanning trees and forests).
To compute the Ising two-point function E[sigma_0 sigma_x] at inverse temperature beta, set p = 1 - e^(-2beta) and sample the q = 2 random-cluster model; then E[sigma_0 sigma_x] = phi_{p,2}(0 is connected to x). Spin correlations decaying exponentially is the SAME statement as cluster connections decaying exponentially — the magnetic and the percolative pictures are one.
Edwards-Sokal: colour FK clusters at random and you get Potts; spin correlations equal connection probabilities.
The factor q^(k(omega)) makes the random-cluster model dependent (NOT a product measure) for q != 1, so Bernoulli-percolation arguments relying on independence must be replaced by FKG/monotonicity versions. The spin coupling needs q to be a positive integer, though the measure itself extends to real q >= 1.