Gibbs Measure - Markov Property

Markov Property

An example of the Markov property of the Gibbs measure can be seen in the Ising model. Here, the probability of a given spin being in state s is, in principle, dependent on all other spins in the model; thus one writes

for this probability. However, the interactions in the Ising model are nearest-neighbor interactions, and thus, one actually has

$P(\sigma_k = s|\sigma_j,\, j\ne k) = P(\sigma_k = s|\sigma_j,\, j\isin N_k)$

where is the set of nearest neighbors of site . That is, the probability at site depends only on the nearest neighbors. This last equation is in the form of a Markov-type statistical independence. Measures with this property are sometimes called Markov random fields. More strongly, the converse is also true: any probability distribution having the Markov property can be represented with the Gibbs measure, given an appropriate energy function; this is the Hammersley–Clifford theorem.

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