Swendsen–Wang algorithm

Remarks

The chain has the Ising measure as a stationary distribution, is irreducible and aperiodic, so it is a valid Monte Carlo method. The reason is the Edwards–Sokal coupling of spins and open edges. Its marginals are the Ising model and the random cluster model with $q = 2$, and clauses 1 and 3 sample its two conditional laws. Example: on the graph with two vertices and one edge, let $\pi_a$ and $\pi_d$ be the probabilities that the spins agree or disagree. From an agreeing state the spins still agree after a step with probability $p + (1 - p)/2 = (1 + p)/2$; from a disagreeing state, with probability $1/2$. Stationarity requires $\pi_a (1 - p)/2 = \pi_d / 2$, so $\pi_a / \pi_d = 1/(1 - p) = e^{2\beta J}$, which is the Ising ratio $e^{\beta J} / e^{-\beta J}$. Non-example: opening edges between equal spins with probability $1/2$, whatever $\beta$, breaks the check above unless $e^{2\beta J} = 2$, so it does not sample the Ising model. For the $q$-state Potts model with energy $-J \sum \delta(\sigma_u, \sigma_v)$, the same algorithm works with $p = 1 - e^{-\beta J}$ and a uniform choice among the $q$ values.