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## Dynamics of Gibbs Models

Suppose that consists of finite common states. For example, as in Ising model. Then we can construct the transition probability at each by

Site update. is a reversible Markov chain with the stationary distribution  since

.

However, it is not irreducible. Now take a fixed ordering of . Define , call it systematic site update. Then is an irreducible transition probability with the stationary distribution . Alternatively we can introduce a distribution over , and define , which is called random site update.