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...ia Griffiths II inequality [10], that for each temperature the expectations with “+” boundary conditions are equal to those with “−” boundary conditions. This implies uniqueness by FKG-type arguments =-=[8]-=-. The uniqueness of µ̂+ implies that of µ̂+|−Λ because the distributions {π̂+|−ΛV } are only a finite-volume modification of the kernels {π̂+V } [9, Section 7.4]. Observation 3.5 There exists a consta...

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... blocks is broken because it is a one-dimensional order. The proof of this assertion, from which Claim 3.2 follows, can be done in (at least) two different ways: The first one is via Theorem 18.25 of =-=[9]-=-. Indeed, by considering each block as a single-spin space with as many values as block configurations satisfying the constraint of having a majority “−”, we can map our constrained system into an unc...

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... 4.3 Non-Gibbsianness for an Interval of Temperatures Above Tc (d ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Conclusions and Final Comments 19 References 20 1 Introduction In =-=[27, 28]-=- it was shown how various renormalization-group (RG) maps acting on Gibbs measures produce non-Gibbsian measures. In physicists’ language, this means that a “renormalized Hamiltonian” can not be defin...

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... temperatures above the transition temperature. The first example together with the example of blockaveraging [28] show that non-Gibbsianness can appear deep within the region of complete analyticity =-=[5]-=-, contradicting the intuition explained in [25, 1]. On the other hand, the second example, besides being the first proven example of a “hightemperature” pathology, shows that the condition of complete...

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.... The examples presented there were all valid at low temperatures and mostly either in or close to the coexistence region. The underlying mechanism — pointed out first by Griffiths, Pearce and Israel =-=[11, 12, 19]-=- — is the fact that for the constraints imposed by particular choices of block-spin configurations, the resulting system exhibits a first-order phase transition. For this to happen, it was expected th...

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.... The examples presented there were all valid at low temperatures and mostly either in or close to the coexistence region. The underlying mechanism — pointed out first by Griffiths, Pearce and Israel =-=[11, 12, 19]-=- — is the fact that for the constraints imposed by particular choices of block-spin configurations, the resulting system exhibits a first-order phase transition. For this to happen, it was expected th...

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.... The examples presented there were all valid at low temperatures and mostly either in or close to the coexistence region. The underlying mechanism — pointed out first by Griffiths, Pearce and Israel =-=[11, 12, 19]-=- — is the fact that for the constraints imposed by particular choices of block-spin configurations, the resulting system exhibits a first-order phase transition. For this to happen, it was expected th...

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... G1, . . . , G4 of Georgii’s theorem. One can also prove the existence of four low-temperature Gibbs states using the generalization of Pirogov-Sinai theory due to Bricmont, Kuroda and Lebowitz (BKL) =-=[3]-=-. Let us briefly review BKL theory, as we also apply it later for the example of the Potts model. The central objects of the theory are the restricted ensembles which are families or classes of config...

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... 4.3 Non-Gibbsianness for an Interval of Temperatures Above Tc (d ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Conclusions and Final Comments 19 References 20 1 Introduction In =-=[27, 28]-=- it was shown how various renormalization-group (RG) maps acting on Gibbs measures produce non-Gibbsian measures. In physicists’ language, this means that a “renormalized Hamiltonian” can not be defin...

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...tly many iterations. These include sufficiently sparse (or sufficiently often iterated) decimations in nonzero field [25], decimated projections on a hyperplane [23], and majority [20], block-average =-=[1]-=- and decimation [29] transformations in the (low-temperature) vicinity of the critical point of the two-dimensional Ising model. The case of decimated projections [23] has the peculiarity that the Gib...

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...h the specification {π̂+|−ΛV }. Indeed, the uniqueness of µ̂+ (at all temperatures) follows form ferromagnetism and the uniqueness of the ground state: The latter implies, via Griffiths II inequality =-=[10]-=-, that for each temperature the expectations with “+” boundary conditions are equal to those with “−” boundary conditions. This implies uniqueness by FKG-type arguments [8]. The uniqueness of µ̂+ impl...

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...nness, or to restore it after sufficiently many iterations. These include sufficiently sparse (or sufficiently often iterated) decimations in nonzero field [25], decimated projections on a hyperplane =-=[23]-=-, and majority [20], block-average [1] and decimation [29] transformations in the (low-temperature) vicinity of the critical point of the two-dimensional Ising model. The case of decimated projections...

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...here shown, c.q. argued, to preserve Gibbsianness, or to restore it after sufficiently many iterations. These include sufficiently sparse (or sufficiently often iterated) decimations in nonzero field =-=[25]-=-, decimated projections on a hyperplane [23], and majority [20], block-average [1] and decimation [29] transformations in the (low-temperature) vicinity of the critical point of the two-dimensional Is...

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...e it after sufficiently many iterations. These include sufficiently sparse (or sufficiently often iterated) decimations in nonzero field [25], decimated projections on a hyperplane [23], and majority =-=[20]-=-, block-average [1] and decimation [29] transformations in the (low-temperature) vicinity of the critical point of the two-dimensional Ising model. The case of decimated projections [23] has the pecul...

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...esults of [7]. The same is true for the case of block averaging in a field (analyzed in [28, p. 1014]). This raises the possibility of restoring a weak form of Gibbsianness defined only almost-surely =-=[1, 22, 24, 6, 17]-=-. REFERENCES 20 For the high-temperature pathologies of the decimated Potts models, we expect them to disappear if the decimation transformation is repeated sufficiently many times. Alternatively, for...

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...omplete analyticity may be violated above the transition temperature, answering a question posed by Roland Dobrushin. We mention that Griffiths and Pearce [11, 12], and also Hasenfratz and Hasenfratz =-=[16]-=-, presented arguments suggesting the existence of “peculiarities” for majorityrule transformations at some precisely tuned (high) values of the magnetic field. Our discussion shows that the situation ...

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...esults of [7]. The same is true for the case of block averaging in a field (analyzed in [28, p. 1014]). This raises the possibility of restoring a weak form of Gibbsianness defined only almost-surely =-=[1, 22, 24, 6, 17]-=-. REFERENCES 20 For the high-temperature pathologies of the decimated Potts models, we expect them to disappear if the decimation transformation is repeated sufficiently many times. Alternatively, for...

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...es of this non-Gibbsianness on computational schemes (renormalization-group calculations, image-processing algorithms) which assume the existence of a renormalized Hamiltonian in the usual sense (see =-=[26]-=- for a pioneer study in this direction). 2 BASIC SET-UP 3 2 Basic Set-up We consider finite-spin systems in ZZd, that is a space of the form Ω = (Ω0) Z d —the configuration space — where Ω0 — the sing...

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...ular temperatures) can be overcome by choosing the decimated spins in a random fashion, for instance 2 with probability f and 1 otherwise. By using a random version of Pirogov-Sinai due to Zahradńık =-=[30]-=- we can then prove the analogue of Theorem 4.2 for a whole interval of temperatures above Tc. Zahradńık’s proof of the existence of coexisting phases for random systems only applies for small disorde...

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... majority-rule acting on the Ising model in a strong field, this set of pathological configurations is of measure zero with respect to the (unique) Ising Gibbs state. This follows from the results of =-=[7]-=-. The same is true for the case of block averaging in a field (analyzed in [28, p. 1014]). This raises the possibility of restoring a weak form of Gibbsianness defined only almost-surely [1, 22, 24, 6...

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...ng model [20, 1, 29], though highly suggestive, are not conclusive because they involve only (judiciously) selected block-spin configurations. Of related interest are the transformations presented in =-=[13, 15, 14]-=- which are “anti-pathological” in the sense that they can produce Gibbs measures out of non-Gibbsian ones. In this paper we present two new examples of non-Gibbsianness that show the ubiquity of this ...

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...esults of [7]. The same is true for the case of block averaging in a field (analyzed in [28, p. 1014]). This raises the possibility of restoring a weak form of Gibbsianness defined only almost-surely =-=[1, 22, 24, 6, 17]-=-. REFERENCES 20 For the high-temperature pathologies of the decimated Potts models, we expect them to disappear if the decimation transformation is repeated sufficiently many times. Alternatively, for...

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... interval around the transition temperature Tc includes (a whole subinterval of) temperatures where the decimation transformation produces non-Gibbsianness. This is to be contrasted with some results =-=[20, 1, 29]-=- suggesting an opposite conclusion for neighborhoods of the critical temperature of the Ising model. Although the arguments presented in these works are not completely rigorous — they are based on num...

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...ent to) the Ising model. On the other hand for large q very different properties emerge, in particular it is known that for q sufficiently high the Potts model exhibits a first-order phase transition =-=[21, 3]-=- with critical inverse temperature βc = 1 2d ln q + O(1/q) . (4.2) Our results apply to models with q sufficiently high, and we find it useful to present them in three steps of increasing technical co...

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...ng model [20, 1, 29], though highly suggestive, are not conclusive because they involve only (judiciously) selected block-spin configurations. Of related interest are the transformations presented in =-=[13, 15, 14]-=- which are “anti-pathological” in the sense that they can produce Gibbs measures out of non-Gibbsian ones. In this paper we present two new examples of non-Gibbsianness that show the ubiquity of this ...

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...ng model [20, 1, 29], though highly suggestive, are not conclusive because they involve only (judiciously) selected block-spin configurations. Of related interest are the transformations presented in =-=[13, 15, 14]-=- which are “anti-pathological” in the sense that they can produce Gibbs measures out of non-Gibbsian ones. In this paper we present two new examples of non-Gibbsianness that show the ubiquity of this ...