Swendsen-Wang algorithm ($p=1-e^{-2\beta J}$)

The Swendsen-Wang algorithm is a cluster Markov chain Monte Carlo method that updates connected groups of variables together. For the ferromagnetic Ising model with no external field, let $s_i\in \{-1,+1\}$ and $H(s)=-J{\sum }_{\{i,j\}\in E}{s}_i{s}_j$, where $J>0$ and $\beta$ is inverse temperature. One step places a bond on each equal-spin edge independently with probability $p=1-e^{-2\beta J}$, leaves unequal-spin edges unbonded, then assigns each connected bond cluster an independent uniformly chosen sign.^[ref]

The bond and spin updates are conditional draws from a joint spin-bond distribution. Summing out the bonds recovers the Boltzmann distribution, so alternating the two conditional draws preserves the intended equilibrium law.^[ref]

For two connected vertices initially carrying equal spins, the next spins agree with probability $p+(1-p)/2=(1+p)/2$: a retained bond forces agreement, while an absent bond permits independent signs. Unlike a single-spin update, one step can reverse a large correlated region. This often reduces critical slowing down, although efficiency depends on the model and the observable. The Ising formula above should not be applied unchanged to a frustrated interaction or an external field.