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In the Monte Carlo simulation of both Lattice field-theories and of models of Statistical Mechanics, identities verified by exact mean-values such as Schwinger-Dyson equations, Guerra relations, Callen identities, etc., provide well known and sensitive tests of thermalization bias as well as checks of pseudo random number generators.
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We shall need to recall the main features of the Swendsen-Wang dynamics, as formulated in Ref. [ 25 ] . The configuration space is expanded: to the original spins { 𝒮 x → } \{{\cal S}_{\vec{x}}\} in the lattice sites, we add bond variables b x → y → = 0 , 1 b_{\vec{x}\vec{y}}\!=\!0,1 in the lattice bonds joining nearest-neighbors sites x → \vec{x} and y → \vec{y} (the marginal probability for the spins is the Boltzmann weight of the original Ising model). The b = 1 b\!=\!1 bonds are said to be occupied. Two lattice sites connected by a chain of occupied bonds are said to belong to the same cluster
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