## Marked Gibbs measures via cluster expansion (1998)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Kondratiev98markedgibbs,

author = {Yuri G. Kondratiev and Tobias Kuna and José L. Silva},

title = {Marked Gibbs measures via cluster expansion},

year = {1998}

}

### OpenURL

### Abstract

We give a sufficiently detailed account on the construction of marked Gibbs measures in the high temperature and low fugacity regime. This is proved for a wide class of underlying spaces and potentials such that stability and integrability conditions are satisfied. That is, for state space we take a locally compact separable metric space X and a separable metric space S for the mark space. This framework allowed us to cover several models of classical and quantum statistical physics. Furthermore, we also show how to extend the construction for more general spaces as e.g., separable standard Borel spaces. The construction of the marked Gibbs measures is based on the method of cluster expansion.

### Citations

319 |
Statistical mechanics: rigorous results
- Ruelle
- 1969
(Show Context)
Citation Context ...se of general underlying and marked space. The results of this paper (which we will give an account below) are based on the so-called cluster expansion method, see e.g., [MM91], [Pen63], [Rue64], and =-=[Rue69]-=-, and we follow closely the scheme of V. A. Malyshev and R. A. Minlos (cf. [MM91, Chap. 3 and 4]), which the authors realized for the con guration space overRd . Let us explain this more precisely. Le... |

75 | Measure Theory - Cohn - 1997 |

60 | Superstable interactions in classical statistical mechanics Commun - Ruelle - 1970 |

49 | Analysis and geometry on configuration spaces - Kondratiev, Röckner - 1998 |

37 |
Some applications of functional integration in statistical mechanics
- Ginibre
- 1970
(Show Context)
Citation Context ...the path space representation of the states in quantum statistical mechanics for Maxwell-Boltzmann statistics. A beautiful description of this connection for the standard density matrices is given in =-=[Gin71]-=-. Ginibre also considers the cases of the Bose-Einstein and FermiDirac statistics. Ginibre does not use any concept of Euclidean Gibbs measure in his considerations, rather he introduce special versio... |

31 |
Measure Theory,” Birkhäuser
- Cohn
- 1980
(Show Context)
Citation Context ...v for projective limits. Thus the construction works as well forX andS separable standard Borel spaces. To this end we recall the de nition and properties of separable standard Borel space, see e.g., =-=[Coh93]-=-, [Geo88] and [Par67]. De nition 5.7 Let (X; F) and (X 0 ; F 0 ) be two measurable spaces. 1. The spaces (X; F) and (X 0 ; F 0 ) are called isomorphic i there exists a measurable bijective mappingf :X... |

24 | Analysis and geometry on configuration spaces: The Gibbsian case. Preprint: Universität - Kondratiev, Röckner - 1997 |

23 | Stochastic analysis on configuration spaces: basic ideas and recent results arXiv:math/9803162v1 [math.PR - Röckner - 1998 |

22 | Phase transition in continuum Potts models
- Georgii, Haggstrom
- 1996
(Show Context)
Citation Context ...nd regularity of ) either 0, or is superstable and lower regular in the sense of [Rue70]. (A4) the positive part + of satis es Z fxjjxj r1g +(x)dx<1: This model is known as continuum Potts model, cf. =-=[GH96]-=-. 19sExample 3.8 Let L (Rd ) be the Banach space of all continuous functions s : [0; ] ! Rd 1 withs(0) =s( ) and = kBT ,kB denotes the Boltzmann constant andT the temperature. On L (Rd ) we consider t... |

16 | Diffeomorphism groups and current algebras: configuration space analysis in quantum theory - Kondratiev, Röckner - 1999 |

15 |
A duality formula on the Poisson space and some applications
- Nualart, Vives
- 1993
(Show Context)
Citation Context ... and (D! 0 )(!) =0otherwise. ! (! 0 ); (D! 0 )(!) := (! [!0 ); if ! \ ! 0 = ;; (2.15) Remark 2.10 Let us mention that the operatorD is related with the Poissonian gradient r P (see e.g., [KSS97b] and =-=[NV95]-=-) by (r P )(!; ^x) =(Df^xg )(!) , (!): 13sFinally, we state some properties of the operatorD, which can be easily checked using the De nition 2.9. Proposition 2.11 Let ; 1; 2 2 A,!2 fin, ^x; ^y 2XS, w... |

10 | Generalized Appell systems
- Kondratiev, Silva, et al.
- 1997
(Show Context)
Citation Context ...or!;! 0 2 fin and (D! 0 )(!) =0otherwise. ! (! 0 ); (D! 0 )(!) := (! [!0 ); if ! \ ! 0 = ;; (2.15) Remark 2.10 Let us mention that the operatorD is related with the Poissonian gradient r P (see e.g., =-=[KSS97b]-=- and [NV95]) by (r P )(!; ^x) =(Df^xg )(!) , (!): 13sFinally, we state some properties of the operatorD, which can be easily checked using the De nition 2.9. Proposition 2.11 Let ; 1; 2 2 A,!2 fin, ^x... |

10 |
Theory of Graphs, volume 38
- Ore
- 1962
(Show Context)
Citation Context ...sure z on B( ) such that z = zp ,1 . The measure z is called marked Poisson measure. 2.3 Basic concepts in graph theory Now we are going to introduce some standard concepts of graph theory, see e.g., =-=[Ore67]-=- for more details. LetX be a non empty set. A partition ofX is a family of non empty subsets (Xi)i2I ofX; called parts, such thatXi \Xj S = ; fori6=jand iXi =X. The set of all partitions ofX where all... |

8 |
Cluster property of the correlation functions of classical gases
- Ruelle
(Show Context)
Citation Context ...res in the case of general underlying and marked space. The results of this paper (which we will give an account below) are based on the so-called cluster expansion method, see e.g., [MM91], [Pen63], =-=[Rue64]-=-, and [Rue69], and we follow closely the scheme of V. A. Malyshev and R. A. Minlos (cf. [MM91, Chap. 3 and 4]), which the authors realized for the con guration space overRd . Let us explain this more ... |

7 |
Probability measures on metric spaces, Probability and
- Parthasarathy
- 1967
(Show Context)
Citation Context ...ts. Thus the construction works as well forX andS separable standard Borel spaces. To this end we recall the de nition and properties of separable standard Borel space, see e.g., [Coh93], [Geo88] and =-=[Par67]-=-. De nition 5.7 Let (X; F) and (X 0 ; F 0 ) be two measurable spaces. 1. The spaces (X; F) and (X 0 ; F 0 ) are called isomorphic i there exists a measurable bijective mappingf :X !X 0 such that its i... |

7 | measures on the configuration space and unitary representations of the group of diffeomorphisms - Shimomura, Poisson - 1994 |

6 |
Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit
- Aizenman, Goldstein, et al.
- 1978
(Show Context)
Citation Context ... de ned, see Section 3. Let us mention that such measures are called states in statistical physics of continuous systems and in probability theory they are known as marked point random elds, cf. e.g. =-=[AGL78]-=-, [GZ93], [Kin93], and [MM91]. The marked Poisson measures are constructed in Subsection 2.2. Finally, in Subsection 2.3 (resp. Subsection 2.4) we introduce some facts from graph theory (resp. *-calcu... |

6 |
Random Fields, volume 534
- Preston
- 1976
(Show Context)
Citation Context ...! \ ! 0 = ; as E (! [! 0 )=E (!)+E (! 0 )+W (!;! 0 ): (3.3) Now we can de ne grand canonical marked Gibbs measures. 15sDe nition 3.1 For any 2Bc(X) the marked speci cation for any! 2 ,F 2B( ) by (see =-=[Pre76]-=-) ; (!;F):= 1f ~ Z ; <1g (!)[ ~Z ; (!)] ,1 Z ; is de ned 1F (!Xn [! 0 ) (3.4) exp[,E (!Xn [! 0 )] z (d! 0 ); (3.5) where> 0 is the inverse temperature. ~Z ; ~Z ; (!) := Z is called partition function:... |

5 |
Properties of marked Gibbs measures in high temperature regime
- Kuna
- 1998
(Show Context)
Citation Context ... that the limit measure ful ls the DLR equation, see Subsection 5.2, Theorem 5.6, and hence it is a Gibbs measure. Let us mention that using further consequences of the cluster expansion developed in =-=[Kun98]-=- and the general results from [KK98] it is possible to show that the limit measure is a Gibbs measure foramuch wider class of potentials. We would like to emphasize that the above results (specially t... |

5 | Convergence of fugacity expansion for fluid and lattice gases - Penrose - 1963 |

3 |
Gibbs Random Fields: Cluster Expansions. Mathematics and its applications
- Malyshev, Minlos
- 1991
(Show Context)
Citation Context ...uch kind of measures in the case of general underlying and marked space. The results of this paper (which we will give an account below) are based on the so-called cluster expansion method, see e.g., =-=[MM91]-=-, [Pen63], [Rue64], and [Rue69], and we follow closely the scheme of V. A. Malyshev and R. A. Minlos (cf. [MM91, Chap. 3 and 4]), which the authors realized for the con guration space overRd . Let us ... |

3 |
Canonical and microcanonical Gibbs states
- Preston
- 1979
(Show Context)
Citation Context ..., , ; (!;F) := 1 fZ ; <1g (!) Z ; (!) Z 1F (!Xn [! 0 )e ,E (! Xn [! 0 ) (d! 0 ); (cf. De nition 3.1 in Section 3). It is well-known that is a speci cation in the sense of [Pre76, Section 6] (see also =-=[Pre79]-=- and [Pre80]) for the given pair potential . Shortly speaking, a marked Gibbs measure is de ned as a probability measure which has as conditional expectation the speci cation ; . The aforementioned li... |

3 | Differential geometry on compound Poisson space. Methods of Functional Analysis and Topology - Kondratiev, Silva, et al. - 1998 |

2 |
Phase transition and Martin boundary
- Föllmer
- 1975
(Show Context)
Citation Context ...lle (DLR) equations. Let Ggc(; ) denote the set of all such probability measures . ; Remark 3.2 1. It is well-known that f g 2Bc(X) is a fBXn (,)g 2Bc(X)speci cation in the following sense (see e.g., =-=[Fol75]-=-, [Pre76], [Pre79]), 0 for all; 2Bc(X): (S1) (S2) (S3) (S4) ; 0 ; (!; ) 2f0; 1g for all! 2 . ; (;Y) is BXn ( )-measurable for allY 2B( ). ; (;Y \Y 0 )= 1Y 0 = ( ; 0 ; 0 ; ; (;Y) for allY 2B( ),Y 0 2BX... |

2 |
measures on the con guration space and unitary representations of the group of di eomorphisms
- Shimomura, \Poisson
- 1994
(Show Context)
Citation Context ...llows from Theorem 5.11 that for any 2 IX and anyn 2N the set (S) n is a separable standard Borel space. Therefore, by the same argument (^S) n =Sn is also a separable standard Borel space, see e.g., =-=[Shi94]-=-. 36sNow taking into account the isomorphism (cf. 2.4) between (^S) n =Sn and (n) the same holds for (n) , Hence is also a separable standard Borel space as well as (n) X by Theorem 5.11, (1). Finally... |

1 |
Analysis and geometry on con guration spaces: The Gibbsian case
- Kondratiev, Rockner
- 1997
(Show Context)
Citation Context ... framework is on the one hand related to the examples in statistical physics we would like to cover, see Examples 3.5 - 3.8 below and also Subsection 5.4. On the other hand, in recent papers [AKR98], =-=[AKR97a]-=- (see also lecture notes [Roc98]) the authors put special emphasis in the construction of di erential geometry on the simple con guration space ,X over a manifoldX, i.e., ,X := fX jj \Kj<1for any comp... |

1 |
Di eomorphism groups and current algebras: con guration spaces analysis in quantum theory
- Kondratiev, Rockner
- 1997
(Show Context)
Citation Context ...a manifoldX, i.e., ,X := fX jj \Kj<1for any compactKXg; (cf. (2.1)) via a lifting of the geometry from the underlying manifoldX (see as well [KSS97a] for an extension for compound Poisson spaces). In =-=[AKR97b]-=- the authors applied the aforementioned di erential geometry to construct representations of current algebras and hence non-relativistic quantum eld theories. This provides a scheme of canonical quant... |

1 |
Analysis and geometry on con guration spaces
- Kondratiev, Rockner
- 1998
(Show Context)
Citation Context ...s general framework is on the one hand related to the examples in statistical physics we would like to cover, see Examples 3.5 - 3.8 below and also Subsection 5.4. On the other hand, in recent papers =-=[AKR98]-=-, [AKR97a] (see also lecture notes [Roc98]) the authors put special emphasis in the construction of di erential geometry on the simple con guration space ,X over a manifoldX, i.e., ,X := fX jj \Kj<1fo... |

1 |
Gibbs Measures and Phase Transitions, volume XIV. de Gruyter
- Georgii
- 1988
(Show Context)
Citation Context ...jective limits. Thus the construction works as well forX andS separable standard Borel spaces. To this end we recall the de nition and properties of separable standard Borel space, see e.g., [Coh93], =-=[Geo88]-=- and [Par67]. De nition 5.7 Let (X; F) and (X 0 ; F 0 ) be two measurable spaces. 1. The spaces (X; F) and (X 0 ; F 0 ) are called isomorphic i there exists a measurable bijective mappingf :X !X 0 suc... |

1 |
Large deviations and the maximuum entropy principle for marked point random fields
- Georgii, Zessin
- 1993
(Show Context)
Citation Context ...see Section 3. Let us mention that such measures are called states in statistical physics of continuous systems and in probability theory they are known as marked point random elds, cf. e.g. [AGL78], =-=[GZ93]-=-, [Kin93], and [MM91]. The marked Poisson measures are constructed in Subsection 2.2. Finally, in Subsection 2.3 (resp. Subsection 2.4) we introduce some facts from graph theory (resp. *-calculus) whi... |

1 |
Correlation functional for marked Gibbs measures and Ruelle bound
- Kondratiev, Kuna, et al.
- 1993
(Show Context)
Citation Context ...R equation, see Subsection 5.2, Theorem 5.6, and hence it is a Gibbs measure. Let us mention that using further consequences of the cluster expansion developed in [Kun98] and the general results from =-=[KK98]-=- it is possible to show that the limit measure is a Gibbs measure foramuch wider class of potentials. We would like to emphasize that the above results (specially the one of Theorem 5.3) are strongly ... |

1 | Euclidean Gibbs states for quantum continuous systems with Boltzmann statistics via cluster expansion. Methods of Functional Analysis and Topology - Kondratiev, Lytvynov, et al. - 1997 |

1 |
Di erential geometry on compound Poisson space. Methods of Functional Analysis and Topology
- Kondratiev, Silva, et al.
- 1997
(Show Context)
Citation Context ...erential geometry on the simple con guration space ,X over a manifoldX, i.e., ,X := fX jj \Kj<1for any compactKXg; (cf. (2.1)) via a lifting of the geometry from the underlying manifoldX (see as well =-=[KSS97a]-=- for an extension for compound Poisson spaces). In [AKR97b] the authors applied the aforementioned di erential geometry to construct representations of current algebras and hence non-relativistic quan... |

1 |
Convergence of fugacity expansions for uids and lattice gases
- Penrose
- 1963
(Show Context)
Citation Context ... of measures in the case of general underlying and marked space. The results of this paper (which we will give an account below) are based on the so-called cluster expansion method, see e.g., [MM91], =-=[Pen63]-=-, [Rue64], and [Rue69], and we follow closely the scheme of V. A. Malyshev and R. A. Minlos (cf. [MM91, Chap. 3 and 4]), which the authors realized for the con guration space overRd . Let us explain t... |

1 |
Speci cations and their Gibbs states
- Preston
- 1980
(Show Context)
Citation Context ...:= 1 fZ ; <1g (!) Z ; (!) Z 1F (!Xn [! 0 )e ,E (! Xn [! 0 ) (d! 0 ); (cf. De nition 3.1 in Section 3). It is well-known that is a speci cation in the sense of [Pre76, Section 6] (see also [Pre79] and =-=[Pre80]-=-) for the given pair potential . Shortly speaking, a marked Gibbs measure is de ned as a probability measure which has as conditional expectation the speci cation ; . The aforementioned limit measure ... |

1 |
Stochastic analysis on con guration spaces: Basic ideas and recent results
- Rockner
- 1998
(Show Context)
Citation Context ...ated to the examples in statistical physics we would like to cover, see Examples 3.5 - 3.8 below and also Subsection 5.4. On the other hand, in recent papers [AKR98], [AKR97a] (see also lecture notes =-=[Roc98]-=-) the authors put special emphasis in the construction of di erential geometry on the simple con guration space ,X over a manifoldX, i.e., ,X := fX jj \Kj<1for any compactKXg; (cf. (2.1)) via a liftin... |

1 | Orientational ordering transition in a continuous-spin ferrofluid - Romano, Zagrebnov - 1998 |

1 | Specifications and their Gibbs states - Preston - 1980 |

1 | Theory of Graphs, volume 38 of American Mathematical Society Colloquium Publications - Ore - 1967 |

1 | Random Fields, volume 534 of Lectures Notes in Math - Preston - 1976 |