Swendsen–Wang algorithm: Difference between revisions

Revision as of 04:19, 7 May 2019

Revision as of 08:45, 4 August 2019

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It can be shown that this algorithm leads to equilibrium configurations. The first way to prove it is using the theory of [[Markov chain]]s, 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 explicitilyexplicitly 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

<math>\frac{P_{\lbrace\sigma\rbrace\rightarrow\lbrace\sigma'\rbrace}}{P_{\lbrace\sigma'\rbrace\rightarrow\lbrace\sigma\rbrace}}=\frac{Pr\left(\lbrace\sigma'\rbrace|B.C.\right)Pr\left(B.C.|\lbrace\sigma\rbrace\right)}{Pr\left(\lbrace\sigma\rbrace|B.C.\right)Pr\left(B.C.|\lbrace\sigma'\rbrace\right)}=\frac{p\cdot \exp\left[-2\beta\sum\limits_{<l,m>}\delta_{\sigma_l,\sigma_m}J_{lm}\right]}{p\cdot \exp\left[-2\beta\sum\limits_{<l,m>}\delta_{\sigma'_l,\sigma'_m}J_{lm}\right]}