= Cluster algorithm
A <cluster algorithm> for <Monte Carlo method> simulation updates a correlated group of variables together. Near a <critical point>, large correlated regions can make single-variable updates mix slowly; collective moves can change those regions more efficiently without changing the intended equilibrium <probability distribution>.
For a zero-field ferromagnetic <Ising model> with energy $-J\sum_{\{i,j\}}s_is_j$, $J>0$, a standard construction independently activates bonds between equal neighboring spins with <probability> $1-e^{-2\beta J}$. The connected components are then independently assigned spin $+1$ or $-1$ with equal <probability>. Marginalizing these auxiliary bonds gives the original spin distribution, explaining why the collective update preserves it. The improvement depends on the model and update rule; grouping spins arbitrarily would not establish correct sampling.<Nonuniversal critical dynamics in Monte Carlo simulations by Robert H. Swendsen and Jian-Sheng Wang>[1987, cluster construction for spin simulations]{ref}
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