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...a major role. In that description phase transitions occur at limit points of the zeroes of the finite volume partition functions in the complex plane when the thermodynamic limit is taken (see, e.g., =-=[47]-=-). The study of analyticity involves thus the introduction of complex interactions and complex measures. A stronger version of analyticity, called complete analyticity, has been introduced by Dobrushi...

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...ubset of the Banach algebra of continuous functions on the spectrum. The range is, however, dense in the sup-norm topology, and the maximal ideal space for A q is \Omega\Gamma For details we refer to =-=[27]-=-. 8 We are interested to see whether at high enough temperatures an interaction with certain regularity properties exists for the subsystem consisting of thesspin variables, after the oe spin variable...

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...king explicit the temperature in the densities. Any measure consistent with the above specification \Pi \Phi is called a Gibbs measure. For further details of the theory of Gibbs measures we refer to =-=[20, 15]. \Pi is s-=-aid to be a uniformly nonnull specification with respect to the reference measuresif there is an " ? 0 such that for every E 2 Fs, (E) ? 0 impliess(!; E)s", for every ! 2 \Omega ;s2 P f (Z d...

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...ime the infinite-volume Gibbs measure is unique. Layering phase transitions are long range order phenomena localized in `shells' of sites of negligible volume compared to the size of the whole system =-=[5, 50, 4, 44, 42, 43]-=-. Since layering transitions have after all no effect on the bulk phase diagram, this phenomenon suggests a weaker notion of uniqueness in the bulk. If the Basuev phenomenon would occur, then it presu...

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...nd Sokal [14, 15, 16]. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models =-=[25, 46]-=-, projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific con...

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... states of which the strongest available is an analyticity property uniform in all volumes, called complete analyticity. Complete analyticity has been introduced and studied by Dobrushin and Shlosman =-=[6, 7, 8]-=-. A weaker version of it, that might be termed restricted complete analyticity, is a property uniform only in sufficiently regular volumes [41] (see below). 2 In this paper we want to investigate the ...

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...om nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata =-=[35, 40]-=-, and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statistical mechanics, which is the question whe...

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... states of which the strongest available is an analyticity property uniform in all volumes, called complete analyticity. Complete analyticity has been introduced and studied by Dobrushin and Shlosman =-=[6, 7, 8]-=-. A weaker version of it, that might be termed restricted complete analyticity, is a property uniform only in sufficiently regular volumes [41] (see below). 2 In this paper we want to investigate the ...

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...hat \Pi is a Gibbs specification with respect to it. 2. \Pi is quasilocal, and uniformly nonnull with respect to the reference measure. Two versions of the theorem go back to Sullivan [52] and Kozlov =-=[34]-=-; for a discussion see [15]. We will consider below projections of Gibbs measures obtained by summing over spins living on a selected infinite sublattice L ae Z d . (We also require L c to be infinite...

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... (G "bZ 2 ) ja fs(oe;s)j = jjf jj q For a discussion on the relation between possible norms and uniqueness properties (uniqueness, exponential decay of correlations, analyticity) in general, see =-=also [12, 3]-=-. Let us replace e v k by the complex function e z k , such that je z k j = e v k . Then the condition it has to satisfy is jj e z k \Gamma 1 e z k \Gamma 1 + q jj q ! 1 (3.25) Consider the case jj e ...

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...oblem originally dates back to the late `70s when Griffiths and Pearce, and Israel noticed that in certain renormalization examples one cannot take for granted the existence of an effective potential =-=[22, 23, 31]-=-. Later a comprehensive investigation of these so called renormalization-pathologies has been carried out by van Enter, Fern'andez and Sokal [14, 15, 16]. In parallel with these studies a number of ot...

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...ll temperatures above the critical point at any field the proof of these lemmas was given in [44]. The gap in the proof for the rest of the uniqueness region was closed in [49], relying on results in =-=[10, 9, 28, 29]-=-. Notice that the strong mixing behaviour above actually coincides with complete analyticity restricted to large squares. Proof of the theorem: By the results of [44, 49], the strong mixing condition ...

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...ider now the case d = 2 with the same notations as above. Theorem 4.3 For every h 6= 0,sh is a Gibbs measure for some absolutely summable interaction, at any temperature. First we need two lemmas. In =-=[44]-=- Gibbs measures for the two dimensional Ising model are characterized by mixing conditions (see also [41]). These properties give detailed information about the way local fluctuations are decoupled in...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...tion \Phi such that \Pi is a Gibbs specification with respect to it. 2. \Pi is quasilocal, and uniformly nonnull with respect to the reference measure. Two versions of the theorem go back to Sullivan =-=[52]-=- and Kozlov [34]; for a discussion see [15]. We will consider below projections of Gibbs measures obtained by summing over spins living on a selected infinite sublattice L ae Z d . (We also require L ...

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...s in a non-zero field, and all temperatures above the critical point at any field the proof of these lemmas was given in [44]. The gap in the proof for the rest of the uniqueness region was closed in =-=[49]-=-, relying on results in [10, 9, 28, 29]. Notice that the strong mixing behaviour above actually coincides with complete analyticity restricted to large squares. Proof of the theorem: By the results of...

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...obvious `safe regions', since failure of Gibbsianness occurs even deep within the uniqueness region as recent examples show (in particular, above the critical temperature or at large magnetic fields) =-=[11, 13]-=-. Results for regions close to the critical point are few and far between. In [33, 2] it is suggested that the critical temperature at which a phase transition can occur in the constrained system for ...

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...ranted the existence of an effective potential [22, 23, 31]. Later a comprehensive investigation of these so called renormalization-pathologies has been carried out by van Enter, Fern'andez and Sokal =-=[14, 15, 16]-=-. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections...

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...ueness region as recent examples show (in particular, above the critical temperature or at large magnetic fields) [11, 13]. Results for regions close to the critical point are few and far between. In =-=[33, 2]-=- it is suggested that the critical temperature at which a phase transition can occur in the constrained system for certain majority-rule schemes and block averaging transformations, is strictly lower ...

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...ime the infinite-volume Gibbs measure is unique. Layering phase transitions are long range order phenomena localized in `shells' of sites of negligible volume compared to the size of the whole system =-=[5, 50, 4, 44, 42, 43]-=-. Since layering transitions have after all no effect on the bulk phase diagram, this phenomenon suggests a weaker notion of uniqueness in the bulk. If the Basuev phenomenon would occur, then it presu...

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...tical point one may have a behaviour devoid of pathologies at least in certain cases, but they fall short of being rigorous proofs. Furthermore there are two recent more precise results available. In =-=[26]-=- the absence of pathologies near the critical point has been shown for decimation on a rectangular lattice and some Kadanoff transformations on a triangular lattice, both applied to the Ising model. I...

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...ueness region as recent examples show (in particular, above the critical temperature or at large magnetic fields) [11, 13]. Results for regions close to the critical point are few and far between. In =-=[33, 2]-=- it is suggested that the critical temperature at which a phase transition can occur in the constrained system for certain majority-rule schemes and block averaging transformations, is strictly lower ...

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...om nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata =-=[35, 40]-=-, and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statistical mechanics, which is the question whe...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...nd Sokal [14, 15, 16]. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models =-=[25, 46]-=-, projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific con...

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...ime the infinite-volume Gibbs measure is unique. Layering phase transitions are long range order phenomena localized in `shells' of sites of negligible volume compared to the size of the whole system =-=[5, 50, 4, 44, 42, 43]-=-. Since layering transitions have after all no effect on the bulk phase diagram, this phenomenon suggests a weaker notion of uniqueness in the bulk. If the Basuev phenomenon would occur, then it presu...

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...ll temperatures above the critical point at any field the proof of these lemmas was given in [44]. The gap in the proof for the rest of the uniqueness region was closed in [49], relying on results in =-=[10, 9, 28, 29]-=-. Notice that the strong mixing behaviour above actually coincides with complete analyticity restricted to large squares. Proof of the theorem: By the results of [44, 49], the strong mixing condition ...

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...at sufficiently high temperatures for the qstate Potts model (qs2). As a consequence we obtain complete analyticity also for its decimations on the sublattice bZ d . We use the technique developed in =-=[30, 19]-=-. The key observation is as follows: By using Gelfand's theory, a (sufficiently large) subset of the space of continuous observables can be related to a Banach algebra of functions on the configuratio...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...ut by van Enter, Fern'andez and Sokal [14, 15, 16]. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics =-=[51, 45]-=-, percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata [35, 40], and others, showing that the measure r...

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... conditional probabilities which are essentially non-quasilocal at this configuration. 2. in the uniqueness region where the potential for the image measure is an analytic function of the temperature =-=[31, 32]-=-. Surprisingly enough, however, there are no obvious `safe regions', since failure of Gibbsianness occurs even deep within the uniqueness region as recent examples show (in particular, above the criti...

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...en introduced and studied by Dobrushin and Shlosman [6, 7, 8]. A weaker version of it, that might be termed restricted complete analyticity, is a property uniform only in sufficiently regular volumes =-=[41]-=- (see below). 2 In this paper we want to investigate the behaviour of states at high temperatures or in the presence of a magnetic field in two examples in which non-Gibbsianness occurs at certain val...

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...ime the infinite-volume Gibbs measure is unique. Layering phase transitions are long range order phenomena localized in `shells' of sites of negligible volume compared to the size of the whole system =-=[5, 50, 4, 44, 42, 43]-=-. Since layering transitions have after all no effect on the bulk phase diagram, this phenomenon suggests a weaker notion of uniqueness in the bulk. If the Basuev phenomenon would occur, then it presu...

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...1; 8 j 2 , has been used. Note that a global specification does exist since for the Ising potential the global Markov property is known to hold in the whole uniqueness regime, in arbitrary dimensions =-=[21, 1]-=-. The `globalization' cf. (4.1) is made in such a way thatsh is consistent with ~ \Gamma h . We construct the specification \Pi h = f h V g for the projection given on(\Omega ; F ) by using the global...

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...ranted the existence of an effective potential [22, 23, 31]. Later a comprehensive investigation of these so called renormalization-pathologies has been carried out by van Enter, Fern'andez and Sokal =-=[14, 15, 16]-=-. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections...

Citation Context

... (G "bZ 2 ) ja fs(oe;s)j = jjf jj q For a discussion on the relation between possible norms and uniqueness properties (uniqueness, exponential decay of correlations, analyticity) in general, see =-=also [12, 3]-=-. Let us replace e v k by the complex function e z k , such that je z k j = e v k . Then the condition it has to satisfy is jj e z k \Gamma 1 e z k \Gamma 1 + q jj q ! 1 (3.25) Consider the case jj e ...

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...xamples emerged from also other sources, such as models from nonequilibrium statistical mechanics [51, 45], percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices =-=[48, 39, 36, 37, 38, 17]-=-, probabilistic cellular automata [35, 40], and others, showing that the measure relevant in their specific context was not a Gibbs measure. These examples are related to the inverse problem of statis...

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...1; 8 j 2 , has been used. Note that a global specification does exist since for the Ising potential the global Markov property is known to hold in the whole uniqueness regime, in arbitrary dimensions =-=[21, 1]-=-. The `globalization' cf. (4.1) is made in such a way thatsh is consistent with ~ \Gamma h . We construct the specification \Pi h = f h V g for the projection given on(\Omega ; F ) by using the global...

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...blem is better understood than the existence aspect: indeed, within a large class of potentials it is true that if there does exist such an interaction, then it is unique up to `physical equivalence' =-=[24]-=-. In many interesting cases this inverse problem comes up for a measure which is the image of a Gibbs measure under a transformation (e.g., renormalization transformations, lower dimensional projectio...

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...ll temperatures above the critical point at any field the proof of these lemmas was given in [44]. The gap in the proof for the rest of the uniqueness region was closed in [49], relying on results in =-=[10, 9, 28, 29]-=-. Notice that the strong mixing behaviour above actually coincides with complete analyticity restricted to large squares. Proof of the theorem: By the results of [44, 49], the strong mixing condition ...

Citation Context

...ut by van Enter, Fern'andez and Sokal [14, 15, 16]. In parallel with these studies a number of other examples emerged from also other sources, such as models from nonequilibrium statistical mechanics =-=[51, 45]-=-, percolation models [25, 46], projections of Gibbs measures to lower dimensional sublattices [48, 39, 36, 37, 38, 17], probabilistic cellular automata [35, 40], and others, showing that the measure r...

Citation Context

...ll temperatures above the critical point at any field the proof of these lemmas was given in [44]. The gap in the proof for the rest of the uniqueness region was closed in [49], relying on results in =-=[10, 9, 28, 29]-=-. Notice that the strong mixing behaviour above actually coincides with complete analyticity restricted to large squares. Proof of the theorem: By the results of [44, 49], the strong mixing condition ...

Citation Context

...oblem originally dates back to the late `70s when Griffiths and Pearce, and Israel noticed that in certain renormalization examples one cannot take for granted the existence of an effective potential =-=[22, 23, 31]-=-. Later a comprehensive investigation of these so called renormalization-pathologies has been carried out by van Enter, Fern'andez and Sokal [14, 15, 16]. In parallel with these studies a number of ot...

Citation Context

...obvious `safe regions', since failure of Gibbsianness occurs even deep within the uniqueness region as recent examples show (in particular, above the critical temperature or at large magnetic fields) =-=[11, 13]-=-. Results for regions close to the critical point are few and far between. In [33, 2] it is suggested that the critical temperature at which a phase transition can occur in the constrained system for ...

Citation Context

... states of which the strongest available is an analyticity property uniform in all volumes, called complete analyticity. Complete analyticity has been introduced and studied by Dobrushin and Shlosman =-=[6, 7, 8]-=-. A weaker version of it, that might be termed restricted complete analyticity, is a property uniform only in sufficiently regular volumes [41] (see below). 2 In this paper we want to investigate the ...

Citation Context

...at sufficiently high temperatures for the qstate Potts model (qs2). As a consequence we obtain complete analyticity also for its decimations on the sublattice bZ d . We use the technique developed in =-=[30, 19]-=-. The key observation is as follows: By using Gelfand's theory, a (sufficiently large) subset of the space of continuous observables can be related to a Banach algebra of functions on the configuratio...