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Let's consider a typical ferromagnetic Ising model with only nearest-neighbour interaction.
* Starting from a given configuration of spins, we associate to each pair of nearest neighbours on sites <math>n,m</math> a random variable <math>b_{n,m}\in \lbrace 0,1\rbrace</math> which is interpreted in the following way: if <math>b_{n,m}=0</math> there is no link between the sites <math>n</math> and <math>m</math>; if <math>b_{n,m}=1</math>, <math>\sigma_n</math> and <math>\sigma_m</math> are connected. These values are assigned according to the following (conditional) probability distribution:
<math>P\left[b_{n,m}=0|\sigma_n\neq\sigma_m\right]=1</math>;
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Since the first term contains a restriction on the spin values whereas there is no restriction in the second term, the weighting factors (properly normalized) can be interpreted as probabilities of forming/not forming a link between the sites: <math>P_{<n,m>\;link}=1-e^{-2\beta J_{nm}}.</math>
The process can be easily adapted to antiferromagnetic spin systems, as it is sufficient to eliminate <math>Z_{n,m}^{same}</math> in favor of <math>Z_{n,m}^{diff}</math>
* After assigning the bond variables, we identify the same-spin clusters formed by connected sites and make an inversion of all the variables in the cluster with probability 1/2. At the following time step we have a new starting Ising configuration, which will produce a new clustering and a new collective spin-flip.
== Correctness ==
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