#### DMCA

## Quasilocality of Projected Gibbs Measures through Analyticity Techniques (1995)

Venue: | Helv. Phys. Acta |

Citations: | 10 - 2 self |

### Citations

530 |
D.: Statistical Mechanics: Rigorous Results
- Ruelle
- 1999
(Show Context)
Citation Context ...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... |

518 |
Abstract Harmonic Analysis
- Hewitt, Ross
- 1963
(Show Context)
Citation Context ...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... |

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

114 | Approach to equilibrium of glauber dynamics in the one phase region - i. the attractive case,
- Martinelli, Olivieri
- 1994
(Show Context)
Citation Context ...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... |

95 | The stochastic random-cluster process, and the uniqueness of random-cluster measures
- Grimmett
- 1995
(Show Context)
Citation Context ...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... |

66 | Completely analytical interactions: constructive description.
- Dobrushin, Shlosman
- 1987
(Show Context)
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 ... |

65 | Statistical mechanics of probabilistic cellular automata
- Lebowitz, Maes, et al.
- 1990
(Show Context)
Citation Context ...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... |

60 |
Constructive criterion for the uniqueness of Gibbs field
- Dobrushin, Shlosman
- 1984
(Show Context)
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 ... |

53 |
A Gibbs description of a system of random variables
- Kozlov
- 1974
(Show Context)
Citation Context ...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... |

49 | High temperature expansions and dynamical systems,
- Bricmont, Kupiainen
- 1996
(Show Context)
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 ... |

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

45 |
Large deviations for the 2D Ising model: a lower bound without cluster expansions,
- Ioffe
- 1994
(Show Context)
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 ... |

44 | For 2–D lattice spin systems weak mixing implies strong mixing.”
- Martinelli, Olivieri, et al.
- 1994
(Show Context)
Citation Context ...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... |

38 |
Projections of Gibbs measures may be non-Gibbsian.
- Schonmann
- 1989
(Show Context)
Citation Context ...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... |

38 |
Potentials for almost Markovian random fields.
- Sullivan
- 1973
(Show Context)
Citation Context ...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 ... |

30 |
Complete analyticity for 2D Ising completed
- Schonmann, Shlosman
- 1995
(Show Context)
Citation Context ...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... |

22 | Ill-defined block-spin transformations at arbitrarily high temperatures,
- Enter
- 1996
(Show Context)
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 ... |

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 ...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... |

22 | Semi–infinite Ising model II. The wetting and layering transitions - Fröhlich, Pfister - 1987 |

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 ...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 ... |

19 | On the layering transition of an SOS surface interacting with a wall
- Cesi, Martinelli
- 1996
(Show Context)
Citation Context ...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... |

19 | Absence of renormalization group pathologies near the critical temperature–two examples
- Haller, Kennedy
- 1996
(Show Context)
Citation Context ...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... |

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

18 |
Velde, The (non-)Gibbsian nature of states invariant under stochastic transformations,
- Maes, Vande
- 1994
(Show Context)
Citation Context ...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... |

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

15 |
Vande Velde, Almost sure quasilocality in the random cluster model ,
- Pfister, K
- 1995
(Show Context)
Citation Context ...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... |

13 |
Layering transition in SOS model with external magnetic
- Dinaburg, Mazel
- 1996
(Show Context)
Citation Context ...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... |

13 |
Large and Moderate Deviation in the Ising Model
- Dobrushin, Shlosman
- 1991
(Show Context)
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 ... |

13 |
High-Temperature Analyticity in Classical Lattice Systems,
- Israel
- 1976
(Show Context)
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... |

13 |
Vande Velde. Defining relative energies for the projected Ising measure
- Maes, K
- 1992
(Show Context)
Citation Context ...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... |

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

11 |
On limits of the Gibbsian formalism in thermodynamics,
- Lorinczi
- 1995
(Show Context)
Citation Context |

11 |
A simple stochastic cluster dynamics: rigorous results
- Martinelli, Scoppola
- 1991
(Show Context)
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... |

9 |
Justification of the renormalization-group method, Theo
- Kashapov
- 1980
(Show Context)
Citation Context ... 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... |

9 |
Finite volume mixing conditions for lattice spin systems and exponential approach to equilibrium of Glauber dynamics
- Martinelli, Olivieri
- 1992
(Show Context)
Citation Context ...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... |

9 |
Uniqueness and half-space nonuniqueness of gibbs states in czech models
- Shlosman
- 1986
(Show Context)
Citation Context |

8 |
Global Markov property in quantum field theory and statistical mechanics: a review on results and problems
- Albeverio, Zegarliński
- 1988
(Show Context)
Citation Context ...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... |

8 |
Regularity Properties of Position-Space Renormalization-Group Transformations: Scope and Limitations of Gibbsian Theory
- Enter, Fernández, et al.
- 1993
(Show Context)
Citation Context ...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... |

8 | Renormalization transformations: Source of examples and problems in probability and statistics. Resenhas do Instituto de Matematica e Estatistica da Universidade de Sao Paulo - Enter, Fernández, et al. - 1994 |

7 |
A remark on different norms and analyticity for many–particle interactions
- Enter, Fernández
- 1989
(Show Context)
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 ... |

7 |
Non-quasilocality of projections of Gibbs measures
- Pfister
- 1994
(Show Context)
Citation Context |

7 |
Remarks on the global Markov property
- Goldstein
- 1980
(Show Context)
Citation Context ...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... |

7 |
Strict convexity ("continuity") of the pressure in lattice systems
- Griffiths, Ruelle
- 1971
(Show Context)
Citation Context ...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... |

7 |
Exact large deviation bounds up to T c for the Ising model in two dimensions, Probab. Theory Rel. Fields 102
- Ioffe
- 1995
(Show Context)
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 ... |

7 |
The two species totally asymmetric simple exclusion process
- Speer
- 1994
(Show Context)
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... |

6 |
Wulff Construction. A Global Shape from
- Dobrushin, Kotecký, et al.
- 1992
(Show Context)
Citation Context |

6 |
Position-space renormalization transformations: some proofs and some problems
- Griffiths, Pearce
- 1978
(Show Context)
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... |

4 |
KoteckyÂ , R.: Pathological behaviour of renormalization-group maps at high ®elds and above the transition temperature
- Enter, ndez, et al.
- 1995
(Show Context)
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 ... |

2 |
Completely analytic random fields
- Dobrushin, Shlosman
- 1985
(Show Context)
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 ... |

2 |
D.W.: Analyticity properties of a lattice gas, Phys
- Gallavotti, Miracle-Sole, et al.
- 1967
(Show Context)
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... |