### Citations

573 | Quantum complexity theory - Bernstein, Vazirani - 1997 |

237 | Correlation inequalities on some partially ordered sets
- Fortuin, Kasteleyn, et al.
- 1971
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Citation Context ...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... |

222 |
Gibbs measures and phase transitions, Walter de Gruyter
- Georgii
- 1988
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Citation Context ... 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... |

175 | Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory
- Enter, Fernández, et al.
- 1993
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Citation Context ... 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... |

66 | Completely analytical interactions: constructive description.
- Dobrushin, Shlosman
- 1987
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Citation Context ... 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... |

47 |
A.: Mathematical Properties of Position-Space RenormalizationGroup Transformations
- Griffiths, Pearce
- 1979
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Citation Context .... 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... |

30 |
Position-space renormalization-group transformations: Some proofs and some problems,
- Griffiths, Pearce
- 1978
(Show Context)
Citation Context .... 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... |

29 |
Banach algebras and Kadanoff transformations, in Random Fields (Esztergom,
- Israel
- 1979
(Show Context)
Citation Context .... 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... |

23 |
First order phase transitions in lattice and continuous systems
- Bricmont, Kuroda, et al.
- 1985
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Citation Context ... 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... |

22 | The low-temperature behavior of disordered magnets, - Chayes, Chayes, et al. - 1985 |

22 |
Renormalization transformations in the vicinity of first-order phase transitions: What can and cannot go wrong,
- Enter, Fernandez, et al.
- 1991
(Show Context)
Citation Context ... 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... |

20 | Some numerical results on the block spin transformation for the 2d Ising model at the critical point
- Benfatto, Marinari, et al.
- 1995
(Show Context)
Citation Context ...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... |

20 |
Results and Theorems
- Griffiths
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Citation Context ...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... |

19 |
Vande Velde. A note on the projection of Gibbs measures
- Lörinczi, K
- 1994
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Citation Context ...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... |

19 | Some remarks on pathologies of renormalization-group transformations,
- Martinelli, Olivieri
- 1993
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Citation Context ...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... |

18 |
Some rigorous results on majority rule renormalization group transformations near the critical point
- Kennedy
- 1993
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Citation Context ...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... |

15 |
Some remarks on almost Gibbs states
- Lörinczi, Winnink
- 1993
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Citation Context ...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... |

13 |
Singular renormalization group transformations and first order phase transitions (I
- Hasenfratz, Hasenfratz
- 1988
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Citation Context ...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 ... |

11 |
Some results on the projected two-dimensional Ising model,
- Lorinczi
- 1994
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Citation Context ...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... |

10 | Low temperature series for renormalized operators: the ferromagnetic square Ising model
- Salas
- 1995
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Citation Context ...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... |

9 |
On the structure of low-temperature phases in three dimensional spin models with random impurities: A general Pirogov-Sinai approach
- Zahradnik
- 1992
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Citation Context ...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... |

7 |
Non-quasilocality of projections of Gibbs measures
- Pfister
- 1994
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Citation Context ... 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... |

6 |
On the relation between finite range potentials and subshifts of finite type’, Probab. Theory Related Fields 101
- Häggström
- 1995
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Citation Context ...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 ... |

5 |
Classification of Gibbs’ states on Smale spaces and onedimensional lattice systems
- Haydn
- 1994
(Show Context)
Citation Context ...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... |

5 |
Vande Velde. Private communication
- unknown authors
(Show Context)
Citation Context ... 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... |

3 |
First-order transitions in large-entropy lattice models
- Kotecký, Shlosman
- 1982
(Show Context)
Citation Context ...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... |

2 |
Gibbs states and subshifts of finite type. University of Goteborg preprint
- Haggstrom
- 1993
(Show Context)
Citation Context ...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 ... |

2 |
On phase transitions for subshifts of finite type. University of Goteborg preprint
- Haggstrom
- 1993
(Show Context)
Citation Context ...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 ... |