A Theorem about Random Fields
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چکیده
Averintsev [1] and Spitzer [2] proved that the class of Markov fields is identical to the class of Gibbs ensembles when the domain is a finite subset of the cubic lattice and each site may be in either of two given states. Hammersley and Clifford [3] proved the same result for the more general case when the domain is the set of sites of an arbitrary finite graph and the number of possible states for each site is finite. In order to show this, they extended the notion of a Gibbs ensemble to embrace more complex interactions than occur on the cubic lattice. Their method was circuitous and showed merely the existence of a potential function for a Markov field with little indication of its form. In [4], Preston gives a more direct approach to the two-state problem and presents an explicit formula for the potential. We show here that the equivalence of Markov fields and Gibbs ensembles follows immediately from a very simple application of the Mobius inversion theorem of [5] which allows us to construct a natural expression for the potential function of a Markov field. We confine our attention to the set of sites of an arbitrary finite graph and allow each site to be in any one of a countable set of states. The two-state solution of Preston emerges as a corollary.
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تاریخ انتشار 1973