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....any forbidden pattern. Shifts of nite type are those which can be de ned by a nite set of forbidden patterns. This property is a conjugacy invariant, see for instance [15] or [13] Many natural examples of two dimensional shifts of nite type arise from lattice systems in statistical mechanics [1]. The dynamic of multi dimensional shifts is much more complex than the one of one dimensional shifts. For instance the entropy of a shift, which is a conjugacy invariant that gives the complexity of the allowed patterns (i.e. patterns contained in some con guration of the shift) is easily ....

....of nite type X = XF where F is the following set of patterns: x x x with x 2 A. The con gurations of this bidimensional shift are the three colorings of a square lattice. Two adjacent cells have a di erent color. It turns out that the exact value of the entropy of this shift is known (see [1]) and equal to h(X) 3 2 log 4 : 2: Example 2.14 We now give an example of a two dimensional shift of nite type X for which the exact value of its entropy of allowed blocks is not known. Let A be the alphabet f0; 1g. We de ne the shift of nite type X = XF where F is the following set of ....

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R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989.

.... N chessboard have been extensively studied (e.g. for kings, in [18] 30] The capacity calculations in [4] were formulated in terms of counting independent sets of vertices in graphs. The capacities are also closely related to gases, lattices, and Ising model entropies in statistical mechanics [2]. In addition to run length constraints, other types of constraints can be used to model two dimensional channels for certain applications [1] 8] 9] 10] 12] 13] 23] 24] 25] 27] 28] For example, run length constraints along diagonals in both directions (northwest southeast and ....

.... as checkerboard constraints [29] Two dimensional (d; 1) constraints are examples of checkerboard constraints, in which case the set S is the union of the intervals [ d; d] on both the horizontal and vertical axes in the plane (i.e. a shape) Likewise the hexagonal grid constraint studied in [2] is a checkerboard constraint. It was noted in [29] For example, in two dimensional optical recording systems bits may be stored on media in the form of dark or bright patterns. As the storage disk is read, these patterns pass through various lenses and other image forming devices, thus ....

R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

....27, 1997 1 Introduction Partitions and compositions of integers are, besides their intrinsic interests, usually used as theoretical models for evolutionary processes in different contexts: statistical mechanics, theory of quantum strings, population biology, nonparametric statistics, etc. cf. [1, 4, 8, 10, 12, 30, 49, 54]. Also parameters in partitions often have natural interpretations in terms of characters in symmetric groups; cf. 15, 47] Thus properties (statistical, algebraic, analytic, of these objects received constant attention in the literature. In many situations, the notion of degree of ....

R. J. Baxter. Exactly solved models in statistical mechanics . Academic Press, London, (1982).

....cic , Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109 TIT.2003.810642 I. INTRODUCTION P ROBABILITY distributions defined by graphs arise in a variety of fields, including coding theory, e.g. 5] 6] artificial intelligence, e.g. 1] 7] statistical physics [8], as well as image processing and computer vision, e.g. 9] Given a graphical model, one important problem is computing marginal distributions of variables at each node of the graph. For acyclic graphs (i.e. trees) standard and highly efficient algorithms exist for this task. In contrast, ....

....on which basis one would like to draw inferences about . For example, in the context of error correcting codes (e.g. 2] the collection represents the bits received from the noisy channel, whereas the vector represents the transmitted codeword. Similarly, in image processing or computer vision [8], the vector represents noisy observations of image pixels or features. One standard inference problem, and that of central interest in this paper, is the computation of the marginal distributions for each node . This task, which in this paper will be called optimal estimation or inference, is ....

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R. J. Baxter, Exactly Solved Models in Statistical Mechanics.New York: Academic, 1982.

....27, 1997 1 Introduction Partitions and compositions of integers are, besides their intrinsic interests, usually used as theoretical models for evolutionary processes in di#erent contexts: statistical mechanics, theory of quantum strings, population biology, nonparametric statistics, etc. cf. [1, 4, 8, 10, 12, 30, 49, 54]. Also parameters in partitions often have natural interpretations in terms of characters in symmetric groups; cf. 15, 47] Thus properties (statistical, algebraic, analytic, of these objects received constant attention in the literature. In many situations, the notion of degree of ....

R. J. Baxter. Exactly solved models in statistical mechanics . Academic Press, London, (1982).

.... algorithm for decoding lowdensity parity check codes [6] the turbo decoding algorithm [7] 8] Pearl s belief propagation algorithm for inference on Bayesian networks [9] the Kalman filter for signal processing [10] 11] and the transfer matrix approach in statistical mechanics [12]. MERL Cambridge Research Lab, 201 Broadway, 8th Floor, Cambridge MA 02139. yedidia merl.com Electrical Engineering and Computer Science, MIT Artificial Intelligence Laboratory, NE43a, Cambridge MA 02139. wtf ai.mit.edu School of Computer Science and Engineering, The Hebrew University of ....

R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

....z : The start matrix for is equal to b a b z : Then (W ) c a b z c z b z c a z z c z z z : 15 Example 3 A famous example where the entropy is exactly computed is the following one. Take A = f0; 1; 2g, and H = V = f00; 11; 22g. The value of the entropy is equal to 2 log 3 , see [2]. The associated substitution given by our method is = H;V a z b c z z c z b z b z z c a z c a z z c z b z a z z a b z z z z z z z z z z z : The start matrix for is equal to z b c a z c a b z : As in Section 4, we have Theorem 7 Let m be the dominant eigenvalue ....

R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, 1989.

.... proof based on the notion of heaps of pieces [3, 4, 22] However, this combinatorial method has not (yet) been extended to animals in three dimensions, for which the very difficult solution of the corresponding gas model, called the hard hexagon model, remains the unique enumeration technique [2]. Proposition 1.1 [3, 4, 11, 12, 14] The area generating function for square lattice directed animals is S 0 (t) 1 1 t (a) b) c) Figure 4: Three dimensional oriented lattices. Looking upon animals as heaps of pieces shows that the area generating function for directed animals on ....

....function of the model. Then the density is ff 1 2 where 1;2 = 1 ff Sigma Delta) 2 with Delta = 1 Gamma ff) 4fffi and i = i = ff. Moreover, ff i 1 1 Sigma (ff Gamma 1) Delta 14 Proof. This calculation is very classical in statistical physics (see [2] Chap. 2 for instance) We differentiate the partition function with respect to ff to obtain: NZ Z ff C The right hand side of this identity is the density of the model, thus we only need to compute Z. Let V be the function defined by V (0; 0) V (0; 1) 1, V (1; 0) fffi and V (1; 1) ....

R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press (1982).

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R. J. Baxter, Exactly solved models in statistical mechanics (Academic, London, 1982).

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R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London--New York, 1982).

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R. J. Baxter, editor. Exactly Solved Models in Statistical Mechanics. Academic Press, New York, 1982.

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Baxter, R. J. Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1982.

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R. J. Baxter, Exactly solved models in statistical mechanics. Academic Press, London, 1982. xii+486 pp.

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R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, New York, 1982.

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R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

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R.J. BAXTER, Exactly solved models in statistical mechanics, Academic Press, London, 1982.

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R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, 1989.

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R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

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R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, London and New York, 1982.

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R.J. BAXTER, Exactly solved models in statistical mechanics, Academic Press, London, 1982.

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R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, London and New York, 1982.

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R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).

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R. J. Baxter. Exactly Solved Models in Statistical Mechanics. Academic Press, 1982.

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R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, (1982)

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R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982.

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