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== Correctness ==
It can be shown that this algorithm leads to equilibrium configurations. The first way to prove it is using the theory of [[Markov chain|Markov chains]], either noting that the equilibrium (described by [[Boltzmann distribution|Boltzmann]]-Gibbs distribution) maps into itself, or showing that in a single sweep of the lattice there is a non-zero probability of going from any state of the Markov chain to any other; thus the corresponding irreducible ergodic Markov chain has an asymptotic probability distribution satisfying [[detailed balance]].
Alternatively, we can show explicitily that detailed balance is satisfied. Every transition between two Ising configurations must pass through some bond configuration in the percolation representation. Let's fix a particular bond configuration: what matters in comparing the probabilities related to it is the number of factors <math>q=e^{-2\beta J}</math> for each missing bond between neighboring spins with the same value; the probability of going to a certain Ising configuration compatible with a given bond configuration is uniform (say <math>p</math>). So the ratio of the transition probabilities of going from one state to another is
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Although not analitically clear from the original paper, the reason why all the values of z obtained with the SW algorithm are much lower than the exact lower bound for single-spin-flip algortithms (<math>z\geq\gamma/\nu</math>) is that the correlation length divergence is strictly related to the formation of percolation clusters, which are flipped together. In this way the relaxation time is significantly reduced.
The algorithm is not particularly good for simulating [[Geometrical frustration|frustrated systems]].
==References==
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