## 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.

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