LST for d-Dimensional Lattices

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN- N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

We first generalize the nomenclature. Consider a Euclidean d-dimensional lattice L, with translation group Gp. A frame, F, of L is defined to be a finite subset of (not necessarily contiguous) sites of L that is closed under (i) intersection, (ii) union, (iii) difference and (iv) operations g Gp. A block, Bp, is a specific assignment 

The d-dimensional analogue of the operator tt, mapping JV-block probabilities to (N + l)-block probabilities, is an operator, Fj, defining the Bayesian extension of Pp to Ppi along dimension j. Here, Ppi = where is an extension of 

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