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For disordered systems, the time average in Eq. 9 is followed by a disorder average, which strongly diminishes the influence of the particular Monte Carlo dynamics.
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For disordered systems, the time average in Eq. 9 is followed by a disorder average, which strongly diminishes the influence of the particular Monte Carlo dynamics.
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22
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66749108119
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RC8161
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S. S. Lavenberg and P. D. Welch, IBM Corporation, Yorktown Heights, New York Research Report No. RC8161, 1980 (unpublished).
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66749150478
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note
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We shall need to recall the main features of the Swendsen-Wang dynamics, as formulated in Ref. The configuration space is expanded; to the original spins { S x } in the lattice sites, we add bond variables b x y =0,1 in the lattice bonds joining nearest-neighbor sites x and y (the marginal probability for the spins is the Boltzmann weight of the original Ising model). The 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. The clusters partition the lattice into connected components. A crucial role is played by the conditional probabilities. That of the spins, given the bonds is: all the spins in a cluster take the same value, the spin values in different clusters are statistically independent and equal ±1 with 50% probability. For that of the bonds, given the spins, different bonds are statistically independent, and a given bond can be one only if it connects spins of equal sign and, in that case, with probability 1- e-2κ. The two Schrödinger-type operators, Pspin and Pbond belong to the heat-bath category; Pspin (Pbond) leave the bonds (spins) unchanged and choose the new spin (bond) configuration according to the conditional probability of the spins (bonds), given the bonds (spins). As all heat-bath dynamics, both Pbond and Pspin verify detailed balance. Their product PSW = Pspin Pbond verifies only the softer balance condition, Eq. 10.
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28
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4243225907
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10.1103/PhysRevLett.63.827
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X. J. Li and A. D. Sokal, Phys. Rev. Lett. 63, 827 (1989). 10.1103/PhysRevLett.63.827
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Li, X.J.1
Sokal, A.D.2
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66749188324
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Combining ηCt ηSt =0 with Eqs. 3 8 9 10, we recover the well-known cluster estimator for U4: U4 = 3 (□c nc2) 2 -2 □c nc4 □ □c nc2 □ 2.
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Combining ηCt ηSt □ =0 with Eqs. 3 8 9 10, we recover the well-known cluster estimator for U4: U4 = 3 (□c nc2) 2 -2 □c nc4 □ □c nc2 □ 2.
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30
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66749132838
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For finite L, gC >0 while the large- L limit for U4 is 3.
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For finite L, gC >0 while the large- L limit for U4 is 3.
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31
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66749121902
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For instance, M2 □ = Pspin M2 □ = C□. A telegraphic proof of CC BSW (0) =0 is also obtained by combining C M2 □ = Pspin C M2 □ with Pspin C M2 =C Pspin M2 = C2 and with Eq. 18.
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For instance, M2 □ = Pspin M2 □ = C□. A telegraphic proof of CC BSW (0) =0 is also obtained by combining C M2 □ = Pspin C M2 □ with Pspin C M2 =C Pspin M2 = C2 and with Eq. 18.
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