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Dynamics of Gibbs Models

Suppose that $ \Lambda$ consists of finite common states. For example, $ \Lambda = \{-1,+1\}$ as in Ising model. Then we can construct the transition probability at each $ v\in V$ by

$\displaystyle \mathbf{P}_v(x,y) := \begin{cases}
\dfrac{\pi(y)}{\sum_{\lambda\...
...eq v} = (y_w)_{ w\neq v}$; }  [0.5in]
0 & \mbox{ otherwise. }
\end{cases}
$

Site update. $ \mathbf{P}_v$ is a reversible Markov chain with the stationary distribution $ \pi$ since

$ \pi(\mathbf{x}) \mathbf{P}_v(\mathbf{x},\mathbf{y})
= \pi(\mathbf{x}) \dfrac{\pi(\mathbf{y})}
{\sum_{\lambda\in\Lambda} \pi((\mathbf{x}; x_v \gets \lambda))}$ $ = \pi(\mathbf{y}) \dfrac{\pi(\mathbf{x})}
{\sum_{\lambda\in\Lambda} \pi((\mat...
...y}; y_v \gets \lambda))}
= \pi(\mathbf{y}) \mathbf{P}_v(\mathbf{y},\mathbf{x})$.

However, it is not irreducible. Now take a fixed ordering $ v_1,\ldots,v_n$ of $ V$. Define $ \mathbf{P} := \mathbf{P}_{v_1} \cdots \mathbf{P}_{v_n}$, call it systematic site update. Then $ \mathbf{P}$ is an irreducible transition probability with the stationary distribution $ \pi$. Alternatively we can introduce a distribution $ \rho(v)$ over $ V$, and define $ \mathbf{P} = \sum_{v\in V} \rho(v) \mathbf{P}_{v}$, which is called random site update.


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