Gibbs Fields: Uniqueness and Decay of Correlations. Revisiting Dobrushin and Pechersky
Diana Conache, Yuri Kondratiev, Yuri Kozitsky, Tanja Pasurek
aa r X i v : . [ m a t h . P R ] J a n GIBBS FIELDS: UNIQUENESS AND DECAY OFCORRELATIONS. REVISITING DOBRUSHIN ANDPECHERSKY
DIANA CONACHE, YURI KONDRATIEV, YURI KOZITSKY,AND TANJA PASUREK
Abstract.
We give a detailed and refined proof of the Dobrushin-Pechersky uniqueness criterion extended to the case of Gibbs fields ongeneral graphs and single-spin spaces, which in particular need not be lo-cally compact. The exponential decay of correlations under the unique-ness condition has also been established. Introduction
A random field on a countable set L is a collection of random variables– spins , indexed by ℓ ∈ L . Each variable is defined on some probabilityspace and takes values in the corresponding single-spin space Ξ ℓ . Typically,it is assumed that each Ξ ℓ is a copy of a Polish space Ξ. In a ‘canonicalversion’, the probability space is (Ξ L , B (Ξ L ) , µ ), where µ is a probabilitymeasure on the Borel σ -field B (Ξ L ). Then also µ is referred to as randomfield. A particular case of such a field is the product measure of some single-spin probability measures σ ℓ . Gibbs random fields with pair interactionsare constructed as ‘perturbations’ of the product measure ⊗ ℓ ∈ L σ ℓ by the‘densities’ exp (cid:16)X W ℓℓ ′ ( ξ ℓ , ξ ℓ ′ ) (cid:17) where W ℓℓ ′ : Ξ × Ξ → R are measurable functions – interaction potentials ,whereas the sum is taken over the set E ⊂ L × L such that W ℓℓ ′ = 0 for( ℓ, ℓ ′ ) ∈ E . The latter condition defines the underlying graph G = ( L , E ).For bounded potentials, the perturbed measures usually exist. Moreover,there is only one such measure if the potentials are small enough and theunderlying graph is enough ‘regular’. If the potentials are unbounded, boththe existence and uniqueness issues turn into serious problems of the theory.Starting from the first results in constructing Gibbs fields with ‘unboundedspins’ [14], attempts to elaborating tools for proving their uniqueness werebeing undertaken [5, 8, 17]. However, except for the results of [17] obtainedfor the potentials and single-spin measures of a special type, and also for This work was financially supported by the DFG through SFB 701: Spektrale Struk-turen und Topologische Methoden in der Mathematik and by the European Commissionunder the project STREVCOMS PIRSES-2013-612669. D. Conache also thanks the sup-port of the IRTG (IGK) 1132 “Stochastics and Real World Models”, Universit¨at Bielefeld. methods applicable to ‘attractive’ potentials, see [11, 18, 22], there is onlyone work presenting a kind of general approach to this problem. This workis due to R. L. Dobrushin and E. A. Pechersky [8], which was first publishedin Russian and later on translated to English. In spite of its great impor-tance, the work remains almost unknown (it has been cited only few times)presumably for the following reasons: (i) the English translation in [8] wasmade with numerous typos and errors of mathematical nature, whereas theRussian version is inaccessible for the most of the readers; (ii) most of theproofs in [8] are very involved and complex, and essential parts of themare only sketched or even missing. In the present publication, we give arefined and complete proof of the Dobrushin-Pechersky result extended inthe following directions: (a) we do not employ the compactness argumentscrucially used in [8]; (b) we settle (in Proposition 2.3 below) the measura-bility issues not even discussed in [8]; (c) instead of the cubic lattice Z d weconsider general graphs as underlying sets of the Gibbs fields. The refine-ment consists, among others, in explicitly calculating the threshold value of K in (2.14) and the constants in the basic estimates in Lemma 3.7. Due to(a), as the single-spin spaces Ξ one can consider just standard Borel spaces,e.g., infinite dimensional spaces which are not locally compact, see [11, 18].Due to (c), one can apply the criterion to varios models employing graphs asunderlying sets. One can also apply the criterion to the equilibrium statesof continuum particle systems, see [22, Chapter 4] and Section 2.3 below.The structure of this paper is as follows. In Section 2, we give necessarypreliminaries and formulate the results in Theorems 2.6 and 2.7. Section3 contains the proof of these theorems based on the estimates obtained inLemmas 3.7 and 3.8, respectively, as well as on a number of other factsproved thein. In Section 4, we perform detailed constructions yielding theproof of the mentioned lemmas.2. Setup and the Result
Notations and preliminaries.
The underlying set for the spin con-figurations of our model is a countable simple connected graph ( L , E ). For avertex ℓ ∈ L , by ∂ℓ we denote the neighborhood of ℓ , i.e., the set of verticesadjacent to ℓ . The vertex degree ∆ ℓ is then the cardinality of ∂ℓ . The onlyassumption regarding the graph is thatsup ℓ ∈ L ∆ ℓ =: ∆ < ∞ , (2.1)i.e., the vertex degree is globally bounded. A given V ⊂ L is said to be an independent set of vertices if ∀ ℓ ∈ V ∂l ∩ V = ∅ . (2.2) IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 3
The chromatic number χ ∈ N is the smallest number such that L = χ − [ j =0 V j , V j − independent , j = 0 , . . . , χ − . (2.3)Obviously, χ ≤ ∆ + 1. However, by Brook’s theorem, see, e.g., [16], for ourgraph we have that χ ≤ ∆ .For a measuarble space ( E, E ), by P ( E ) we denote the set of all proba-bility measures on E . All measurable spaces we deal with in this article arestandard Borel spaces. The prototype example is a Polish space endowedwith the corresponding Borel σ -field. For σ ∈ P ( E ) and a suitable function f : E → R , we write σ ( f ) = Z E f dσ. For our model, the single-spin spaces (Ξ ℓ , B (Ξ ℓ )), ℓ ∈ L , are copies of astandard Borel space (Ξ , B (Ξ)). Then the configuration space X = Ξ L equipped with the product σ -field B ( X ) = B (Ξ L ) is also a standard Borelspace. Likewise, for a nonempty D ⊂ L , Ξ D is the product of Ξ ℓ , ℓ ∈ D .Its elements are denoted by x D = ( x ℓ ) ℓ ∈ D , whereas the elements of X arewritten simply as x = ( x ℓ ) ℓ ∈ L . For y, z ∈ X , by y D × z D c we denote theconfiguration x ∈ X such that x D = y D and x D c = z D c , D c := L \ D . For D ( L , we denote F D = B (Ξ D c ) and write F ℓ if D = { ℓ } . Definition 2.1.
Given ℓ ∈ L , let π ℓ := { π xℓ : x ∈ X } ⊂ P (Ξ ℓ ) be such thatthe map X ∋ x π xℓ ( A ) ∈ R is F ℓ -measurable for each A ∈ B (Ξ ℓ ). A family π = { π ℓ } ℓ ∈ L of the maps of this kind is said to be a one-site specification. Definition 2.2.
A given µ ∈ P ( X ) is said to be consistent with a one-sitespecification π in a given D ⊆ L if µ ( ·|F ℓ )( x ) = π xℓ for µ -almost all x andeach ℓ ∈ D . By M D ( π ) we denote the set of all µ ∈ P ( X ) consistent with π in D . We say that µ is consistent with π if it is consistent in L , and writejust M ( π ) in this case.Obviously, µ ∈ M ( π ) if and only if it satisfies the following equation µ ( A ) = Z X Z X I A ( x ) π yℓ ( dx ℓ ) Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) µ ( dy ) (2.4)= Z X (cid:18)Z Ξ I A ( ξ × y { ℓ } c ) π yℓ ( dξ ) (cid:19) µ ( dy ) , which holds for every ℓ ∈ L and A ∈ B ( X ). Here, for η ∈ Ξ, δ η ∈ P (Ξ) isthe Dirac measure centered at η and I A stands for the indicator of A .For a standard Borel space ( E, E ), let ( E , E ) be the product space. For σ, ς ∈ P ( E ), let ̺ ∈ P ( E ) be such that ̺ ( A × E ) = σ ( A ) and ̺ ( E × A ) = ς ( A ) for all A ∈ B ( E ). Then we say that ̺ is a coupling of σ and ς . By C ( σ, ς ) we denote the set of all such couplings. DIANA CONACHE, YURI KONDRATIEV, YURI KOZITSKY, AND TANJA PASUREK
For ξ, η ∈ Ξ, we set υ ( ξ, η ) = ( , if ξ = η ;1 , otherwise , which is a measurable function on Ξ since Ξ is a standard Borel space.Then we equip P (Ξ) with the total variation distance d ( σ, ς ) = sup A ∈B (Ξ) | σ ( A ) − ς ( A ) | , (2.5)that, by duality, can also be written in the form d ( σ, ς ) = inf ̺ ∈C ( σ,ς ) Z Ξ υ ( ξ, η ) ̺ ( dξ, dη ) . Proposition 2.3.
For each ℓ ∈ L and ( x, y ) ∈ X , there exists ̺ x,yℓ ∈C ( π xℓ , π yℓ ) such that: (a) for each B ∈ B (Ξ ℓ ) , the map X ∋ ( x, y ) ̺ x,yℓ ( B ) is F ℓ -measurable; (b) the following holds d ( π xℓ , π yℓ ) = Z Ξ υ ( ξ, η ) ̺ x,yℓ ( dξ, dη ) . (2.6) Proof.
Set ( π xℓ ∧ π yℓ )( A ) = min { π xℓ ( A ); π yℓ ( A ) } , A ∈ B (Ξ ℓ ) . In view of the measurability as in Definition 2.1, the map X ∋ ( x, y ) ( π xℓ ∧ π yℓ )( A ) is F ℓ -measurable since, given a ∈ [0 , { ( x, y ) : a ≤ ( π xℓ ∧ π yℓ )( A ) } = { x : a ≤ π xℓ ( A ) } . Then both maps ( x, y ) ( π xℓ − π xℓ ∧ π yℓ )( A ) and ( x, y ) ( π yℓ − π xℓ ∧ π yℓ )( A )are F ℓ -measurable. By (2.5) also ( x, y ) d ( π xℓ , π yℓ ) is measurable in thesame sense.Set D ℓ = { ( ξ, ξ ) : ξ ∈ Ξ ℓ } . Since Ξ ℓ is a standard Borel space, the map ξ ψ ( ξ ) = ( ξ, ξ ) ∈ D ℓ is measurable. Then, for each B ∈ B (Ξ ℓ ), we havethat ψ − ( B ∩ D ℓ ) ∈ B (Ξ ℓ ), which allows us to define ω x,yℓ ∈ P (Ξ ℓ ) by setting ω x,yℓ ( B ) = ( π xℓ ∧ π yℓ ) (cid:0) ψ − ( B ∩ D ℓ ) (cid:1) . The coupling for which (2.6) holds has the form, see [15, Eq. (5.3), page19], ̺ x,yℓ = ω x,yℓ + ( π xℓ − π xℓ ∧ π yℓ ) ⊗ ( π yℓ − π xℓ ∧ π yℓ ) /d ( π xℓ , π yℓ ) . Then the F ℓ -measurability of the maps ( x, y ) ̺ x,yℓ ( A × A ), A , A ∈B (Ξ ℓ ), follows by the arguments given above. This yields the proof of claim(a) as B (Ξ ℓ ) is a product σ -field. (cid:3) Let ̟ be the family of ̟ ℓ = { ̟ x,yℓ : ( x, y ) ∈ X } , ℓ ∈ L , such that each ̟ x,yℓ is in P (Ξ ℓ ) and, for any B ∈ B (Ξ ℓ ), the map ( x, y ) ̟ x,yℓ ( B ) is F ℓ -measurable. Then ̟ is a one-point specification in the sense of Definition IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 5 M ( ̟ ) of ν ∈ P ( X ) consistent with ̟ . Likein (2.4), ν ∈ M ( ̟ ) if and only if it satisfies ν ( B ) = Z X Z X I B ( x, y ) ̟ y, ˜ yℓ ( dx ℓ , d ˜ x ℓ ) (2.7) × Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) δ ˜ y ℓ ′ ( d ˜ x ℓ ′ ) ν ( dy, d ˜ y ) , which holds for all ℓ ∈ L and B ∈ B ( X ). Proposition 2.4.
Suppose that ̟ x, ˜ xℓ ∈ C ( π xℓ , π ˜ xℓ ) for all ℓ ∈ L and x, ˜ x ∈ X .Then each ν ∈ M ( ̟ ) is a coupling of some µ , µ ∈ M ( π ) .Proof. The equality µ ( A ) = ν ( A × X ), A ∈ B ( X ), determines a probabilitymeasure on X . Thus, for A ∈ B ( X ), by (2.7) we get µ ( A ) = Z X Z X I A ( x ) ̟ y, ˜ yℓ ( dx ℓ , d ˜ x ℓ ) Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) δ ˜ y ℓ ′ ( d ˜ x ℓ ′ ) ν ( dy, d ˜ y )= Z X Z X I A ( x ) π yℓ ( dx ℓ ) Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) ν ( dy, d ˜ y )= Z X Z X I A ( x ) π yℓ ( dx ℓ ) Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) Z X ν ( dy, d ˜ y )= Z X Z X I A ( x ) π yℓ ( dx ℓ ) Y ℓ ′ = ℓ δ y ℓ ′ ( dx ℓ ′ ) µ ( dy ) . Therefore, µ solves (2.4) and hence µ ∈ M ( π ). The same is true for thesecond marginal measure µ . (cid:3) The results.
Our main concern is under which conditions imposedon the family π the set M ( π ) contains one element at most. If each π xℓ is independent of x , the unique element of M ( π ) is the product measure ⊗ ℓ ∈ L π ℓ , which readily follows from (2.4). Therefore, one may try to relatethe uniqueness in question to the weak dependence of π xℓ on x , formulatedin terms of the metric defined in (2.6). Thus, let us take x, y ∈ X suchthat x = y off some ℓ ′ ∈ ∂ℓ , and consider d ( π xℓ , π yℓ ). If this quantity werebounded by a certain κ ℓℓ ′ , uniformly in x and y , this bound (Dobrushin’sestimator, cf. [4, pp. 20, 21]) could be used to formulate the celebratedDobrushin uniqueness condition in the formsup ℓ ∈ L X ℓ ′ ∈ ∂ℓ κ ℓℓ ′ =: ¯ κ < . (2.8)However, in a number of applications, especially where Ξ is a noncompacttopological space, the mentioned boundedness does not hold. The way oftreating such cases suggested in [8] may be outlined as follows. Assume that DIANA CONACHE, YURI KONDRATIEV, YURI KOZITSKY, AND TANJA PASUREK there exists a matrix ( κ ℓℓ ′ ) with the property as in (2.8) such that, for each ℓ ∈ L , the following holds d ( π xℓ , π yℓ ) ≤ X ℓ ′ ∈ ∂ℓ κ ℓℓ ′ υ ( x ℓ ′ , y ℓ ′ ) , (2.9)for x and y belonging to the set X ℓ ( h, K ) := { x ∈ X : h ( x ℓ ′ ) ≤ K for all ℓ ′ ∈ ∂ℓ } . (2.10)Here K > h : Ξ → [0 , + ∞ ) is a given measurablefunction. Clearly, if h is bounded, then X ℓ ( h, K ) = X for big enough K ,and hence (2.9) turns into the mentioned Dobrushin condition. Thus, inorder to cover the case of interest we have to take h unbounded and π xℓ -integrable, with an appropriate control of the dependence of π xℓ ( h ) on x .Namely, we shall assume that, for each ℓ ∈ L and x ∈ X , the following holds π xℓ ( h ) ≤ X ℓ ′ ∈ ∂ℓ c ℓℓ ′ h ( x ℓ ′ ) , (2.11)for some matrix c = ( c ℓℓ ′ ), which satisfies( a ) c ℓℓ ′ ≥
0; ( b ) sup ℓ ∈ L X ℓ ′ ∈ ∂ℓ c ℓℓ ′ =: ¯ c < /∆ χ . (2.12)In the original work [8], the first summand on the right-hand side of (2.11)is a constant C >
0, the value of which determines the scale of K , see (2.10).We thus take it as above for the sake of convenience. Definition 2.5.
Let h , K , κ , and c be as in (2.8) – (2.12). Then byΠ( h, K, κ, c ) we denote the set of one-site specifications π for which bothestimates (2.9), (2.10) and (2.11) hold true for each ℓ ∈ L .Given µ ∈ M ( π ), the integrability assumed in (2.11) does not yet implythat h is µ -integrable. For π satisfying (2.11), by M ( π, h ) we denote thesubset of M ( π ) consisting of those measures for which the following holds µ ( h ) := sup ℓ ∈ L Z X h ( x ℓ ) µ ( dx ) < ∞ . (2.13)In a similar way, we introduce the set M D ( π, h ) for a given D ⊂ L , cf.Definition 2.2.From now on we fix the graph, the function h , and the matrices c and κ .Thereafter, we set K ∗ = max (cid:26) ∆ χ +1 ¯ c (1 − ¯ κ ) ; 2 ∆ χ +1 (2 ∆ χ − + 1 − ¯ c∆ χ )(1 − ¯ κ ) (1 − ¯ c∆ χ ) (cid:27) . (2.14) Theorem 2.6.
For each
K > K ∗ and π ∈ Π( h, K, κ, c ) , the set M ( π, h ) contains at most one element. An important characteristic of the states µ ∈ M ( π ) is the decay of corre-lations. Fix two distinct vertices ℓ , ℓ ∈ L and consider bounded functions IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 7 f, g : X → R + , such that f is B (Ξ ℓ )-measurable and g is B (Ξ ℓ )-measurable.Set Cov µ ( f ; g ) = µ ( f g ) − µ ( f ) µ ( g ) , and let δ denote the path distance on the underlying graph. Theorem 2.7.
Let π and K be as in Theorem 2.6, and M ( π, h ) be nonemptyand hence contain a single state µ . Let also f and g be as just described and k · k ∞ denote the sup-norm on X . Then there exist positive C K and α K ,dependent on K only, such that | Cov µ ( f ; g ) | ≤ C K k f k ∞ k g k ∞ exp [ − α K δ ( ℓ , ℓ )] . (2.15)2.3. Comments and applications.
Let us make some comments to theabove results. For further comments related to the proof of these results seethe end of Section 3. • According to [20, Section 8], the elements of M ( π ) as in Definition2.2 are one-site Gibbs states. In [9, Theorem 1.33, page 23] and [20,Section 8], there are given conditions under which the elements of M ( π ) are ‘usual’ Gibbs states, e.g., in the sense of [9, Definition1.23, page 16]. This, in particular, holds if π is a subset of the set ofall local kernels Π D defined for all finite D ⊂ L , which determine thestates. In this case, Theorem 2.6 yields the existence and uniquenessof the usual states, see [22]. • The condition in (2.13) is usually satisfied for tempered measures,i.e., for those elements of M ( π ) which are supported on tempered configurations, cf., e.g., [14]. • As mentioned above, we do not require that h be compact in thesense of [8]. This our extension gets important if one deals withsingle-spin spaces which are not locally compact, e.g., with spaces ofH¨older continuous functions as in [1, 11, 18]. • In contrast to [8, Theorem 1], in (2.14) we give an explicit expressionfor the threshold value K ∗ , which depends only on the parametersof the underlying graph and on the norms ¯ c and ¯ κ . • The novelty of Theorem 2.7 consists in the following. The decayof correlations under the uniqueness condition was proven only forcompact single-spin spaces, see [12], where the classical Dobrushincriterion can be applied. For ‘unbounded spins’, the correspondingresults are usually obtained by cluster expansions, see, e.g., [21],where the correlations are shown to decay due to ‘weak enough’interactions’ and no information on the number of states is available. • The parameters C K and α K in (2.15) are also given explicitly, see(3.24) below.Now we turn to briefly outlining possible applications of Theorems 2.6and 2.7. A more detailed discussion of this issue can be found in [22], seealso the related parts of [18]. Further results in these directions will bepublished in forthcoming articles. DIANA CONACHE, YURI KONDRATIEV, YURI KOZITSKY, AND TANJA PASUREK
By means of Theorems 2.6 and 2.7 the uniqueness of equilibrium statesand the decay of correlations can be established in the following models: • Systems of classical N -dimensional anharmonic oscillators describedby the energy functional H ( x ) = X ℓ ∈ L V ( ξ ℓ ) + X ( ℓ,ℓ ′ ) ∈ E W ℓℓ ′ ( ξ ℓ , ξ ℓ ′ ) , ξ ℓ ∈ R N , N ∈ N • Systems of quantum N -dimensional anharmonic oscillators describedby the Hamiltonian H = X ℓ ∈ L H ℓ + X ( ℓ,ℓ ′ ) ∈ E W ℓℓ ′ ( q ℓ , q ℓ ′ ) , where q ℓ = ( q (1) ℓ , . . . , q ( N ) ℓ ) is the position operator and H ℓ is the one-particle Hamiltonian defined on the corresponding physical Hilbertspace. States of such models are constructed in a path integral ap-proach as probability measures on the products of continuous peri-odic functions, which are not locally compact, see [1, 11, 18]. • Systems of interacting particles in the continuum (e.g. R d ), includ-ing the Lebowitz-Mazel-Presutti model [13], and systems of ‘parti-cles’ lying on the cone of discrete measures introduced in [10]. Notethat to continuum systems the original version [8] of the Dobrushin-Pechersky criterion was used in [3, 19].3. The Proof of Theorems 2.6 and 2.7
The ingredients of the proof.
First we introduce the notion of lo-cality . By writing D ⋐ L we mean that D is a nonempty finite subset of L . For such D , elements of B (Ξ D ) ⊂ B ( X ) are called local sets. A function f : X → R is called local if it is B (Ξ D )-measurable for some D ⋐ L . Likewise, B ∈ B ( X ) is local if B ∈ B ((Ξ × Ξ) D ) for such D . Locality of functions f : X → R is defined in the same way. Lemma 3.1.
Given a one-site specification π and µ , µ ∈ M ( π ) , supposethere exists ν ∗ ∈ C ( µ , µ ) such that Z X υ ( x ℓ , y ℓ ) ν ∗ ( dx, dy ) = 0 , (3.1) holding for all ℓ ∈ L . Then µ = µ .Proof. Local sets A ⊂ X are measure defining, that is, µ , µ ∈ P ( X )coincide if they coincide on local sets. For A ∈ B (Ξ D ) and the indicator I A ,we have | I A ( x ) − I A ( y ) | ≤ X ℓ ∈ D υ ( x ℓ , y ℓ ) , and then | µ ( A ) − µ ( A ) | ≤ X ℓ ∈ D Z X υ ( x ℓ , y ℓ ) ν ∗ ( dx, dy ) = 0 , IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 9 which yields the proof. (cid:3)
The proof of Theorem 2.6 will be done by showing that, for each µ , µ ∈M ( π, h ), the set C ( µ , µ ) contains a certain ν ∗ such that (3.1) holds. Thiscoupling ν ∗ will be obtained by taking the limit in the topology of localsetwise convergence, cf. [9], which we introduce as follows. Definition 3.2.
A net { ν α } α ∈ I ⊂ P ( X ) is said to be convergent to a ν ∗ ∈P ( X ) in the topology of local setwise convergence ( L -topology, for short),if ν α ( B ) → ν ∗ ( B ) for all local B ∈ B ( X ). Or, equivalently, ν α ( f ) → ν ∗ ( f )for all bounded local functions. The same definition applies also to nets { µ α } α ∈ I ⊂ P ( X ).Note that the L -topology is Hausdorff, but not metrizable if Ξ is not acompact topological space. Lemma 3.3.
Given µ , µ ∈ P ( X ) , let { ν α } α ∈ I ⊂ C ( µ , µ ) be convergentto a certain ν ∈ P ( X ) in the L -topology. Then ν ∈ C ( µ , µ ) . The proof of this lemma is rather obvious. The coupling in question ν ∗ will be constructed within a step-by-step procedure based on the mapping( R ℓ ν )( f ) = Z X (cid:18)Z Ξ f ( ξ × x { ℓ } c , η × y { ℓ } c ) ̺ x,yℓ ( dξ, dη ) (cid:19) ν ( dx, dy ) , (3.2)where ℓ ∈ L , ̺ x,yℓ is as in (2.6), and f : X → R is a function such that both ν ( f ) and the integral on the right-hand side of (3.2) exist. Lemma 3.4.
For each ℓ ∈ L , the mapping (3.2) has the following properties:(a) if ν ∈ C ( µ , µ ) for some µ , µ ∈ M ( π ) , then also R ℓ ν ∈ C ( µ , µ ) ; (b)if f is F ℓ ( X ) -measurable and ν -integrable, then ( R ℓ ν )( f ) = ν ( f ) .Proof. Claim (a) is true since ̺ x,yℓ ∈ C ( π xℓ , π yℓ ) for all x, y ∈ X . Claim (b)follows by the fact that the considered f in (3.2) is independent of ξ and η ,and that ̺ x,yℓ is a probability measure. (cid:3) Given ℓ ∈ L , we set Y ℓ = { ( x , x ) ∈ X : υ ( x ℓ , x ℓ ) ≤ X ℓ ′ ∈ ∂ℓ υ ( x ℓ ′ , x ℓ ′ ) } . Lemma 3.5.
For each ν ∈ P ( X ) and ℓ ∈ L , it follows that ( R ℓ ν )( Y ℓ ) = 1 .Proof. If ( x , x ) is in Y cℓ , then υ ( x ℓ , x ℓ ) = 1 and υ ( x ℓ ′ , x ℓ ′ ) = 0 for all ℓ ′ ∈ ∂ℓ , which follows by the fact that υ takes values in { , } . This meansthat x ℓ = x ℓ and x ℓ ′ = x ℓ ′ for all ℓ ′ ∈ ∂ℓ . For such ( x , x ), the definitionof π implies that π x ℓ = π x ℓ , and hence Z Ξ υ ( ξ, η ) ̺ x ,x ℓ ( dξ, dη ) = d ( π x ℓ , π x ℓ ) = 0 , which by (3.2) yields ( R ℓ ν )( Y cℓ ) = 0. (cid:3) The proof of Theorem 2.6 will be done by showing that, for each µ , µ ∈M ( π, h ), there exists ν ∗ ∈ C ( µ , µ ), for wich (3.1) holds. To this end weconstruct a sequence { ˆ ν n } n ∈ N ⊂ C ( µ , µ ) such that γ (ˆ ν n ) := sup ℓ ∈ L Z X υ ( x ℓ , x ℓ )ˆ ν n ( dx , dx ) → , n → + ∞ . (3.3)This sequence will be obtained by a procedure based on the mapping (3.2)and the estimates which we derive in the next subsection. The proof ofTheorem 2.7 will be obtained as a byproduct.3.2. The main estimates.
In the sequel, we use the following functionsindexed by ℓ ∈ L I ℓ ( x , x ) = υ ( x ℓ , x ℓ ) , H iℓ ( x , x ) = h ( x iℓ ) , i = 1 , . (3.4)By claim (b) of Lemma 3.4, we have that( R ℓ ν )( I ℓ ) = ν ( I ℓ ) , ( R ℓ ν )( I ℓ H iℓ ) = ν ( I ℓ H iℓ ) for ℓ = ℓ , ℓ = ℓ , whenever H iℓ is ν -integrable. We recall that ̺ x,yℓ in (3.2) is a coupling of π xℓ and π yℓ , for which (2.9) and (2.11) hold true. Lemma 3.6.
Let ν ∈ P ( X ) be such that the integrals on both sides of (3.2)exist for f = H iℓ , ℓ ∈ L and i = 1 , . Then the following estimates hold ( R ℓ ν )( I ℓ ) ≤ X ℓ ′ ∈ ∂ℓ κ ℓℓ ′ ν ( I ℓ ′ ) + K − X i =1 , X ℓ ,ℓ ∈ ∂ℓ ν ( I ℓ H iℓ ) , (3.5)( R ℓ ν )( I ℓ H iℓ ) ≤ ν ( I ℓ ) + X ℓ ∈ ∂ℓ c ℓℓ ν ( I ℓ H iℓ ) , (3.6)( R ℓ ν )( I ℓ H iℓ ) ≤ X ℓ ∈ ∂ℓ ν ( I ℓ H iℓ ) , ℓ = ℓ, (3.7)( R ℓ ν )( I ℓ H iℓ ) ≤ X ℓ ∈ ∂ℓ ν ( I ℓ ) + X ℓ ,ℓ ∈ ∂ℓ c ℓℓ ν ( I ℓ H iℓ ) . (3.8) Proof.
The proof of (3.7) readily follows by Lemma 3.5. Let us prove (3.5).By (2.6) and (3.2), we have( R ℓ ν )( I ℓ ) = Z X d ( π x ℓ , π x ℓ ) ν ( dx , dx )= Z X ℓ ( x ) ℓ ( x ) d ( π x ℓ , π x ℓ ) ν ( dx , dx )+ Z X (cid:2) − ℓ ( x ) ℓ ( x ) (cid:3) d ( π x ℓ , π x ℓ ) ν ( dx , dx ) , IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 11 where ℓ is the indicator of the set defined in (2.10). By (2.9), we have Z X ℓ ( x ) ℓ ( x ) d ( π x ℓ , π x ℓ ) ν ( dx , dx ) ≤ X ℓ ′ ∈ ∂ℓ κ ℓℓ ′ ν ( I ℓ ′ ) , which yields the first term of the right-hand side of (3.5). By (2.10), wehave (cid:2) − ℓ ( x ) ℓ ( x ) (cid:3) ≤ X i =1 , X ℓ ∈ ∂ℓ (cid:2) − I h ≤ K ( x iℓ ) (cid:3) , where I h ≤ K is the indicator of { ξ ∈ Ξ : h ( ξ ) ≤ K } . Then the second termof the right-hand side of (3.5) cannot exceed the following X i =1 , X ℓ ∈ ∂ℓ Z X (cid:2) − I h ≤ K ( x iℓ ) (cid:3) d ( π x ℓ , π x ℓ ) ν ( dx , dx ) ≤ K − X i =1 , X ℓ ∈ ∂ℓ Z X h ( x iℓ ) d ( π x ℓ , π x ℓ ) ν ( dx , dx ) ≤ K − X i =1 , X ℓ ,ℓ ∈ ∂ℓ ν ( I ℓ H iℓ ) . The latter line has been obtained by (3.7).Let us prove now (3.6). By (3.2) and the fact that ̺ x,yℓ ∈ C ( π xℓ , π yℓ ), wehave ( R ℓ ν )( I ℓ H iℓ ) = Z X (cid:18)Z Ξ h ( ξ ) π x i ( dξ ) (cid:19) υ ( x ℓ , x ℓ ) ν ( dx , dx ) ≤ RHS(3 . , where we have used (2.11). To prove (3.8) we employ Lemma 3.5, by whichwe get LHS(3 . ≤ X ℓ ∈ ∂ℓ ( R ℓ ν )( I ℓ H iℓ ) ≤ RHS(3 . , where the latter estimate follows by (3.6). (cid:3) From the lemma just proven it follows that along with the parameter γ ( ν )defined in (3.3) one has to control also the following λ ( ν ) = max i =1 , sup ℓ,ℓ ′ ∈ L ν ( I ℓ H iℓ ′ ) , (3.9)where ν ∈ C ( µ , µ ), µ , µ ∈ M h ( π ), and π ∈ Π( h, K, κ, c ), see Definition2.5. The proof of Theorem 2.6.
The proof is based on constructinga sequence with the property (3.3). Given µ , µ ∈ M ( π, h ) with π ∈ Π( h, K, κ, c ), we take an arbitrary ν ∈ C ( µ , µ ) and construct ν ∈ C ( µ , µ )by applying the mapping defined in (3.2) to the initial ν with ℓ runningover the set L . Each time we use the estimates derived in Lemma 3.6. Thenthe first two elements of the sequence in question are set ˆ ν = ν and ˆ ν = ν .Afterwards, we produce ˆ ν from ˆ ν , etc.Recall that the underlying graph is supposed to have the property definedin (2.1) and χ ≤ ∆ is its chromatic number. Set A = 2 ∆ χ +1 − ¯ κ . (3.10)Then, for K > K ∗ , see (2.14), the following holds K − < ¯ c (1 − ¯ κ )4 ∆ χ +1 , AK − < ¯ c/ . (3.11) Lemma 3.7.
For
K > K ∗ , take π ∈ Π( h, K, κ, c ) and µ , µ ∈ M ( π, h ) .Then for each ν ∈ C ( µ , µ ) there exists ν ∈ C ( µ , µ ) for which the follow-ing estimates hold γ ( ν ) ≤ (cid:2) ¯ κ + AK − (cid:3) γ ( ν ) + 2 AK − λ ( ν ) , (3.12) λ ( ν ) ≤ ∆ χ − γ ( ν ) + ¯ c∆ χ λ ( ν ) . (3.13)The proof of the lemma will be given in the subsequent parts of the paper. Proof Theorem 2.6:
As already mentioned, we let ˆ ν ∈ C ( µ , µ ) and ˆ ν ∈C ( µ , µ ) be the measures on the left-hand sides and right-hand sides of(3.12) and (3.13), respectively. Then we apply to ˆ ν the same reconstructionprocedure and obtain ˆ ν ∈ C ( µ , µ ), for which both estimates (3.12), (3.13)hold with ˆ ν on the right-hand sides. We repeat this due times and obtainˆ ν n ∈ C ( µ , µ ) such that γ (ˆ ν n ) λ (ˆ ν n ) ! ≤ [ M ( K )] n γ ( ν ) λ ( ν ) ! , (3.14)where M ( K ) is the matrix defined by the right-hand sides of (3.12) and(3.13). Its spectral radius is r K = 12 h ¯ κ + AK − + ¯ c∆ χ + p (¯ κ + AK − − ¯ c∆ χ ) + 8 ∆ χ AK − i . (3.15)For K > K ∗ , see (2.14), we have r K <
1, which by (3.14) yields (3.3) andthereby completes the proof.3.4.
The proof of Theorem 2.7.
The proof of this theorem is based onthe version of the estimates in Lemma 3.7 obtained in a subset D ⊂ L . Forsuch D , we define ∂ D = { ℓ ′ ∈ D c : ∂ℓ ′ ∩ D = ∅} , IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 13 which is the external boundary of D . For ν ∈ P ( X ) such that all H iℓ , i = 1 , ℓ ∈ D ∪ ∂ D are ν -integrable, see (3.4), we set, cf. (3.3) and (3.9), γ D ( ν ) = sup ℓ ∈ D ν ( I ℓ ) , λ D ( ν ) = max i =1 , sup ℓ ,ℓ ∈ D ν ( I ℓ H iℓ ) . (3.16)Next, for ℓ as in (2.15) and N = δ ( ℓ , ℓ ), we set D = { ℓ } , D k = D k − ∪ ∂ D k − , k = 1 , . . . , N − . Let µ x ( · ) denote the conditional measure µ ( ·|F D N − )( x ). For brevity, we saythat ν x ∈ P ( X ) is F D N − -measurable if the maps x ν x ( B ) are F D N − -measurable for all B ∈ B ( X ). Clearly, ν x = µ x ⊗ µ possesses this property.The version of Lemma 3.7 which we need is the following statement. Lemma 3.8.
Let π , K , and µ be as in Theorem 2.7 and ν x = µ x ⊗ µ . Thenthere exist ν x , . . . , ν xN − ∈ C ( µ x , µ ) , all F D N − -measurable, such that for theparameters defined in (3.16) the following estimates hold γ D N − s − ( ν xs ) λ D N − s − ( ν xs ) ! ≤ M ( K ) γ D N − s ( ν xs − ) λ D N − s ( ν xs − ) ! , (3.17) for all s = 1 , . . . , N − and µ -almost all x ∈ X .Proof of Theorem 2.7: Since g is F D N − -measurable, we have Z X f ( x ) g ( x ) µ ( dx ) = Z X g ( x ) (cid:18)Z X f ( y ) µ x ( dy ) (cid:19) µ ( dx ) , which yields Cov µ ( f ; g ) = Z X g ( x )Φ( x ) µ ( dx ) , (3.18)where Φ( x ) = Z X ( f ( y ) − f ( z )) µ x ( dy ) µ ( dz ) . (3.19)For each ν xs , s = 0 , . . . , N −
1, as in Lemma 3.8, we then haveΦ( x ) = Z X ( f ( y ) − f ( z )) ν xs ( dy, dz ) , (3.20)and hence | Φ( x ) | ≤ k f k ∞ ν xN − ( I ℓ ) = 2 k f k ∞ γ D ( ν xN − ) . (3.21)Note that the function defined in (3.19), (3.20) is related to the quantitywhich characterizes mixing in state µ , cf. [12, Proposition 2.5].Let v s and v s − denote the column vector on the left-hand and right-handsides of (3.17), respectively. Set ξ = ∆ χ − r K − ¯ c∆ χ = r K − ¯ κ − AK − AK − > , and let T be the 2 × T = ξ and T = 1. Then thematrix f M ( K ) := T M ( K ) T − , (3.22)cf. [2, Corollary 2.9.4, page 102], is positive and such that both its rows sumup to r K . Set ˜ v s = T v s and let ˜ v is , i = 1 ,
2, be the entries of ˜ v s . By (3.17)we then get k ˜ v s k := max { ˜ v s ; ˜ v s } ≤ k f M ( K ) kk ˜ v s − k = r K max { ˜ v s − ; ˜ v s − } , which yields γ D ( ν xN − ) ≤ r N − K max { γ D N − ( ν x ); ξ − λ D N − ( ν x ) } . (3.23)Applying this estimate in (3.21) and then in (3.18) we arrive at (2.15) with,cf. (3.15) and (2.13), α K = − log r K , C K = 2 r − K max { ξ − µ ( h ) } . (3.24)Let us make now further comments on the above results and their proof. • The mapping in (3.2), which is the main reconstruction tool, seeSection 4 below, was first introduced in another seminal paper byR. L. Dobrushin [7]. In a rather general context, it was used in [6].The main feature of this mapping, which was not pointed out in [8],is the measurability of the coupling ̺ x,yℓ in ( x, y ) ∈ X . A similarproperty of the couplings in Lemma 3.8 was crucial for the proof ofTheorem 2.7. • We avoid using ‘compactness’ of h , and hence the related topologicalproperties of the single-spin space Ξ, by employing the L -topology,see Definition 3.2. • In contrast to the estimates obtained in [8, Lemma 5], our estimatein (3.13) is independent of K . The only constant in (3.12) is given ex-plicitly in (3.10). This allowed us to calculate explicitly the spectralradius (3.15), which was then used to obtain the decay parameter α K , see (3.24). • The proof of Lemma 3.8 was performed in the spirit of the proof ofProposition 2.5 of [12]. Our Φ( x ) in (3.19), (3.20) can be used toprove a kind of mixing in state µ . However, here we cannot esti-mate this function uniformly in x , and hence employ its measurableestimate (3.21) which is then integrated in (3.18). • The transformation used in (3.22) allowed us to find explicitly theoperator norm of M ( K ) equal to its spectral radius r K . This thenwas used to find in (3.23) the exact rate of the decay of correlationsin µ . IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 15 Proof of Lemmas 3.7 and 3.8
For the partition (2.3) of the set of vertices L , which has the property(2.2), we set U j = j [ i =0 V i , W j = L \ U j , j = 0 , . . . , χ − . (4.1)The measure ν in (3.12), (3.13) will be obtained in the course of consecutivereconstructions with ℓ ∈ V j . The first step is4.1. Reconstruction over V . Let { ℓ , ℓ , . . . , } be any numbering of theelements of V . Set V ( n )0 = { ℓ , . . . , ℓ n } , ν ( n )0 = R ℓ n R ℓ n − · · · R ℓ ν , n ∈ N . (4.2)Our first task is to estimate ν ( n )0 ( I ℓ ). By claim (b) of Lemma 3.4 we havethat ν ( n )0 ( I ℓ ) = ν ( I ℓ ) , for ℓ / ∈ V ( n )0 . (4.3)For k ≤ n , by (2.2) and claim (b) of Lemma 3.4, and then by (3.5) and(4.3), we have ν ( n )0 ( I ℓ k ) = ν ( k )0 ( I ℓ k ) ≤ X ℓ ∈ ∂ℓ k κ ℓ k ℓ ν ( I ℓ ) + K − X i =1 , X ℓ,ℓ ′ ∈ ∂ℓ k ν ( I ℓ H iℓ ′ ) ≤ ¯ κγ ( ν ) + 2 ∆ K − λ ( ν ) , (4.4)see also (2.8), (3.3), and (3.9).Next we turn to estimating ν ( n )0 ( I ℓ H iℓ ′ ). As in (4.3) we have ν ( n )0 ( I ℓ H iℓ ′ ) = ν ( I ℓ H iℓ ′ ) for ℓ, ℓ ′ / ∈ V ( n )0 . For k < m ≤ n , by claim (b) of Lemma 3.4, and then by (3.6), (3.5), (4.4),and (3.7), we have ν ( n )0 ( I ℓ k H iℓ m ) = ν ( m )0 ( I ℓ k H iℓ m ) ≤ ν ( k )0 ( I ℓ k ) + X ℓ ∈ ∂ℓ m c ℓ m ℓ ν ( k )0 ( I ℓ k H iℓ ) ≤ ¯ κγ ( ν ) + 2 ∆ K − λ ( ν ) + X ℓ ∈ ∂ℓ m c ℓ m ℓ X ℓ ′ ∈ ∂ℓ k ν ( I ℓ ′ H iℓ ) ≤ ¯ κγ ( ν ) + (cid:2) ∆ ¯ c + 2 ∆ K − (cid:3) λ ( ν ) . For k ≤ n , by (3.8) we have ν ( n )0 ( I ℓ k H iℓ k ) = ν ( k )0 ( I ℓ k H iℓ k ) ≤ X ℓ ∈ ∂ℓ k ν ( I ℓ ) + X ℓ,ℓ ′ ∈ ∂ℓ k c ℓ k ℓ ′ ν ( I ℓ H iℓ ′ ) ≤ ∆γ ( ν ) + ∆ ¯ cλ ( ν ) . Next, for m < k ≤ n , by (3.7) and (3.6) we have ν ( n )0 ( I ℓ k H iℓ m ) = ν ( k )0 ( I ℓ k H iℓ m ) ≤ X ℓ ∈ ∂ℓ k ν ( m )0 ( I ℓ H iℓ m ) ≤ X ℓ ∈ ∂ℓ k ν ( I ℓ ) + X ℓ ′ ∈ ∂ℓ m c ℓ m ℓ ′ ν ( I ℓ H iℓ ′ ) ≤ ∆γ ( ν ) + ∆ ¯ cλ ( ν ) . Now we consider the case where k ≤ n and ℓ / ∈ V ( n )0 . Then by (3.7) we have ν ( n )0 ( I ℓ k H iℓ ) = ν ( k )0 ( I ℓ k H iℓ ) ≤ X ℓ ′ ∈ ∂ℓ k ν ( I ℓ ′ H iℓ ) ≤ ∆λ ( ν ) . For k ≤ n and ℓ / ∈ V ( n )0 , we also have by (3.6) that ν ( n )0 ( I ℓ H iℓ k ) = ν ( k )0 ( I ℓ H iℓ k ) ≤ ν ( I ℓ ) + X ℓ ′ ∈ ∂ℓ k c ℓ k ℓ ′ ν ( I ℓ H iℓ ′ ) ≤ γ ( ν ) + ¯ cλ ( ν ) . (4.5)Now let us consider the sequence { ν ( n )0 } n ∈ N defined in (4.2). By claim (b)of Lemma 3.4 it stabilizes on local sets B ∈ B ( X ), and hence is convergentin the L -topology. Let ν be its limit. By Lemma 3.3 we have that ν ∈C ( µ , µ ). At the same time, by (4.1), (4.3), and (4.4) it follows that ν ( I ℓ ) ≤ ( ¯ κγ ( ν ) + 2 ∆ K − λ ( ν ) , for ℓ ∈ V ; γ ( ν ) , for ℓ ∈ W . (4.6)Similarly, by (4.4) – (4.5) we obtain ν ( I ℓ H iℓ ′ ) ≤ ∆γ ( ν ) + (cid:2) ∆ ¯ c + 2 ∆ K − (cid:3) λ ( ν ) , ℓ, ℓ ′ ∈ V ; ∆λ ( ν ) , ℓ ∈ V , ℓ ′ ∈ W ; γ ( ν ) + ¯ cλ ( ν ) , ℓ ∈ W , ℓ ′ ∈ V ; λ ( ν ) , ℓ, ℓ ′ ∈ W . (4.7)These estimates complete the reconstruction over V .4.2. Reconstruction over V j : Proof of Lemma 3.7. Here we assumethat ν j satisfies the following estimates, cf. (4.6), where A is as in (3.10): ν j ( I ℓ ) ≤ ( (cid:2) ¯ κ + AK − (cid:3) γ ( ν ) + 2 AK − λ ( ν ) , for ℓ ∈ U j − ; γ ( ν ) , for ℓ ∈ W j − . (4.8) IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 17
And also, cf. (4.7), ν j ( I ℓ H iℓ ′ ) ≤ ∆ j γ ( ν ) + ¯ c∆ j +1 λ ( ν ) , ℓ, ℓ ′ ∈ U j − ; ∆ j λ ( ν ) , ℓ ∈ U j − , ℓ ′ ∈ W j − ; jγ ( ν ) + ¯ cλ ( ν ) , ℓ ∈ W j − , ℓ ′ ∈ V j − ; λ ( ν ) , ℓ, ℓ ′ ∈ W j − . (4.9)Since W ∆ − = ∅ , see (4.1), for j = ∆ − ∆ < A , see (3.10). Also(4.7) agrees with (4.9), which follows from the fact that¯ c∆ + 2 ∆ K − < ¯ c∆ + AK − ≤ ¯ c∆ + ¯ c/ < ¯ c∆ ≤ ¯ c∆ j +1 , j = 1 , . . . χ − , see (3.10) and (3.11).Thus, our aim now is to prove that the estimates as in (4.8) and (4.9)hold also for j + 1. Note that the last lines in these estimates follow by claim(b) of Lemma 3.4. As above, we enumerate V j = { ℓ , ℓ , · · · } and set ν ( n ) j = R ℓ n R ℓ n − · · · R ℓ ν j . For k ≤ n , by (3.5) we have, cf. (4.4), ν ( n ) j ( I ℓ k ) = ν ( k ) j ( I ℓ k ) ≤ X ℓ ∈ ∂ℓ k ∩ U j − κ ℓ k ℓ ν j ( I ℓ ) + X ℓ ∈ ∂ℓ k ∩ W j κ ℓ k ℓ ν j ( I ℓ )+ K − X i =1 , X ℓ,ℓ ′ ∈ ∂ℓ k ∩ U j − ν j ( I ℓ H iℓ ′ )+ K − X i =1 , X ℓ ∈ ∂ℓ k ∩ U j − X ℓ ′ ∈ ∂ℓ k ∩ W j ν j ( I ℓ H iℓ ′ )+ K − X i =1 , X ℓ ∈ ∂ℓ k ∩ W j X ℓ ′ ∈ ∂ℓ k ∩ U j − ν j ( I ℓ H iℓ ′ )+ K − X i =1 , X ℓ,ℓ ′ ∈ ∂ℓ k ∩ W j ν j ( I ℓ H iℓ ′ ) . Now we use the assumptions in (4.8) and (4.9) and obtain herefrom ν ( n ) j ( I ℓ k ) ≤ h ¯ κ + K − (cid:16) ¯ κA + 2 ∆ j ∆ j + 2 j∆ j e ∆ j (cid:17)i γ ( ν ) (4.10)+ 2 K − h ¯ κA + ¯ c∆ j +1 ∆ j + ∆ j ∆ j e ∆ j + ¯ c∆ j e ∆ j + e ∆ j i λ ( ν ) , where ∆ j := | ∂ℓ k ∩ U j − | , e ∆ j := | ∂ℓ k ∩ W j | . To prove that ¯ κA + 2 ∆ j ∆ j + 2 j∆ j e ∆ j ≤ A see the first line in (4.8), we use (3.10), take into account that ∆ ≥ j ≤ ∆ j , j = 1 , , . . . χ −
1) and obtain2 ∆ j ∆ j + 2 j∆ j e ∆ j ≤ ∆ j ∆ j (cid:16) ∆ j + e ∆ j ( j/∆ j ) (cid:17) ≤ ∆ j +2 ≤ A (1 − ¯ κ ) , where we have taken into account that j + 2 ≤ χ + 1, see (3.10). To provethat the coefficient at λ ( ν ) in (4.10) agrees with that in (4.8) we use thefollowing estimates¯ c∆ j +1 ∆ j + ∆ j ∆ j e ∆ j + ¯ c∆ j e ∆ j + e ∆ j = ¯ c∆ j +1 ∆ j (cid:16) ∆ j + e ∆ j ∆ − j (cid:17) + ∆ j e ∆ j (cid:16) ∆ j + e ∆ j ∆ − ( j +1) (cid:17) ≤ ∆ + ∆ j +2 ≤ ∆ j +2 ≤ A (1 − ¯ κ ) . For ℓ ∈ U j − , ν ( n ) j ( I ℓ ) = ν j ( I ℓ ) and hence obeys the first line of (4.8). For ℓ ∈ W j , again ν ( n ) j ( I ℓ ) = ν j ( I ℓ ) and hence obeys the second line of (4.8).Here we also used that ¯ c < /∆ χ and j + 1 ≤ χ , see (2.12). Thus, (4.8) with j + 1 holds true.Now we turn to estimating ν ( n ) j ( I ℓ H iℓ ′ ). In the situation where ℓ, ℓ ′ ∈ U j − ∪ W j , we have that ν ( n ) j ( I ℓ H iℓ ′ ) = ν j ( I ℓ H iℓ ′ ) and hence obeys (4.9). Letus consider first the cases where only one vertex of ℓ, ℓ ′ lies in V j .For ℓ ′ ∈ U j − and k ≤ n , by (3.7) and the first and third lines in (4.9) weobtain ν ( n ) j ( I ℓ k H iℓ ′ ) = ν ( k ) j ( I ℓ k H iℓ ′ ) ≤ X ℓ ∈ ∂ℓ k ∩ U j − ν j ( I ℓ H iℓ ′ ) + X ℓ ∈ ∂ℓ k ∩ W j ν j ( I ℓ H iℓ ′ ) ≤ h ∆ j ∆ j + j e ∆ j i γ ( ν ) + h ¯ c∆ j +1 ∆ j + ¯ c e ∆ j i λ ( ν ) ≤ ∆ j +1 γ ( ν ) + ¯ c∆ j +2 λ ( ν ) , which yields the first line in (4.9) with j + 1.For ℓ ′ ∈ W j and k ≤ n , by (3.7) and the second and fourth lines in (4.9)it follows that ν ( n ) j ( I ℓ k H iℓ ′ ) = ν ( k ) j ( I ℓ k H iℓ ′ ) ≤ X ℓ ∈ ∂ℓ k ∩ U j − ν j ( I ℓ H iℓ ′ ) + X ℓ ∈ ∂ℓ k ∩ W j ν j ( I ℓ H iℓ ′ ) ≤ (cid:16) ∆ j ∆ j + e ∆ j (cid:17) λ ( ν ) ≤ ∆ j +1 λ ( ν ) , which agrees with the second line in (4.9). IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 19
For ℓ ∈ U j − and k ≤ n , by (3.6) and the first and second lines in (4.9)we get ν ( n ) j ( I ℓ H iℓ k ) = ν ( k ) j ( I ℓ H iℓ k ) ≤ ν j ( I ℓ ) + X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ ν j ( I ℓ H iℓ ′ )+ X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ν j ( I ℓ H iℓ ′ ) ≤ (cid:2) ¯ κ + AK − (cid:3) γ ( ν )+2 AK − λ ( ν ) + (cid:2) ∆ j γ ( ν ) + ¯ c∆ j +1 λ ( ν ) (cid:3) X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + ∆ j λ ( ν ) X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ . In order for this to agree with the first line in (4.9), it is enough that thefollowing holds¯ κ + AK − + ∆ j X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ ≤ ∆ j +1 , (4.11)2 AK − + ¯ c∆ j +1 X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + ∆ j X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ≤ ¯ c∆ j +2 . Recall that we assume ∆ ≥
2. By (3.11) and (2.12) we get that the left-handside of the first line in (4.11) does not exceed¯ κ + ¯ c/ ∆ − < < ∆ j +1 , for j = 1 , . . . , χ − . Likewise, the left-hand side of the second line in (4.11) does not exceed¯ c + ¯ c + ¯ c∆ j ≤ ¯ c (2 + ∆ j ) < ¯ c∆ j +2 for j = 1 , . . . , χ − . For ℓ ∈ W j and k ≤ n , by (3.6) and the third and fourth lines in (4.9) weget ν ( n ) j ( I ℓ H iℓ k ) = ν ( k ) j ( I ℓ H iℓ k ) ≤ ν j ( I ℓ )+ X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ ν j ( I ℓ H iℓ ′ ) + X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ν j ( I ℓ H iℓ ′ ) (4.12) ≤ γ ( ν ) + [ jγ ( ν ) + ¯ cλ ( ν )] X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + λ ( ν ) X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ≤ (1 + j ¯ c ) γ ( ν ) + ¯ c X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ λ ( ν ) , which clearly agrees with the third line in (4.9). Now we consider the cases where both ℓ, ℓ ′ lie in V j . For k < m ≤ n , byfirst (3.6) and (3.7), and then by (3.5), we have ν ( n ) j ( I ℓ k H iℓ m ) = ν ( m ) j ( I ℓ k H iℓ m ) ≤ ν ( k ) j ( I ℓ k ) + X ℓ ′ ∈ ∂ℓ m c ℓ m ℓ ′ ν ( k ) j ( I ℓ k H iℓ ′ ) ≤ X ℓ ∈ ∂ℓ k κ ℓ k ℓ ν j ( I ℓ ) + K − X s =1 , X ℓ,ℓ ′ ∈ ∂ℓ k ν j ( I ℓ H sℓ ′ )+ X ℓ ′ ∈ ∂ℓ m c ℓ m ℓ ′ X ℓ ∈ ∂ℓ k ν j ( I ℓ H iℓ ′ ) . (4.13)The next step is to split the sums in (4.13) as it has been done in, e.g.,(4.12), and then use (4.8) and (4.9). By doing so we get ν ( n ) j ( I ℓ k H iℓ m ) ≤ (cid:2) (¯ κ + AK − ) γ ( ν ) + 2 AK − λ ( ν ) (cid:3) X ℓ ∈ ∂ℓ k ∩ U j − κ ℓ k ℓ + γ ( ν ) X ℓ ∈ ∂ℓ k ∩ W j κ ℓ k ℓ + 2 K − ∆ j (cid:2) ∆ j γ ( ν ) + ¯ c∆ j +1 λ ( ν ) (cid:3) +2 K − ∆ j e ∆ j (cid:2) ∆ j λ ( ν ) + jγ ( ν ) + ¯ cλ ( ν ) (cid:3) + 2 K − e ∆ j λ ( ν )+ ∆ j (cid:2) ∆ j γ ( ν ) + ¯ c∆ j +1 λ ( ν ) (cid:3) X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + ∆ j ∆ j λ ( ν ) X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ + e ∆ j ( jγ ( ν ) + ¯ cλ ( ν )) X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + e ∆ j λ ( ν ) X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ . IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 21
In order for this to agree with the first line in (4.9), it is enough that thefollowing two estimate hold(¯ κ + AK − ) X ℓ ∈ ∂ℓ k ∩ U j − κ ℓ k ℓ + X ℓ ∈ ∂ℓ k ∩ W j κ ℓ k ℓ + 2 K − ∆ j ∆ j (4.14)+2 K − j∆ j e ∆ j + ∆ j ∆ j X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + e ∆ j X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ ≤ ∆ j +1 AK − X ℓ ∈ ∂ℓ k ∩ U j − κ ℓ k ℓ + 2 K − ∆ j ¯ c∆ j +1 + 2 K − ∆ j e ∆ j ∆ j (4.15)+2 K − ¯ c∆ j e ∆ j + 2 K − e ∆ j + ¯ c∆ j ∆ j +1 X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + ∆ j ∆ j X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ + ¯ c e ∆ j X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + e ∆ j X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ ≤ ¯ c∆ j +2 . Taking into account that ¯ κ < c/ K − ∆ j ∆ j (cid:16) ∆ j + e ∆ j ( j/∆ j ) (cid:17) + ¯ c∆ j +1 ≤ c/ c/ c∆ j +1 < ∆ χ < ∆ j +1 . To prove (4.15) we use (3.11), (2.12), the inequality ∆ j e ∆ j ≤ ∆ /
4, andperform the following calculationsLHS(4 . ≤ AK − ¯ κ + 12 K − ∆ j +2 + 2 K − (cid:16) ∆ j + ¯ c∆ j e ∆ j + e ∆ j (cid:17) + ∆ j X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + ∆ j ∆ j X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ +¯ c e ∆ j X ℓ ′ ∈ ∂ℓ m ∩ U j − c ℓ m ℓ ′ + e ∆ j X ℓ ′ ∈ ∂ℓ m ∩ W j c ℓ m ℓ ′ ≤ ¯ c + ¯ c∆ j +2 ∆ χ +1 + ¯ c∆ ∆ χ +1 + ¯ c∆ j +1 + ¯ c∆ < ¯ c∆ j +2 , which holds even for j = 1, χ = 2, and ∆ = 2. Next, for k ≤ n , by (3.8) we have ν ( n ) j ( I ℓ k H iℓ k ) = ν ( k ) j ( I ℓ k H iℓ k ) ≤ X ℓ ∈ ∂ℓ k ν j ( I ℓ ) + X ℓ,ℓ ′ ∈ ∂ℓ k c ℓ k ℓ ′ ν j ( I ℓ H iℓ ′ ) (4.16)As above, we split the sums in (4.16) and then use (4.8) and (4.9), andobtain ν ( n ) j ( I ℓ k H iℓ k ) ≤ ∆ j (cid:2) ¯ κ + AK − (cid:3) γ ( ν ) + ∆ j AK − λ ( ν )+ e ∆ j γ ( ν ) + ∆ j (cid:2) ∆ j γ ( ν ) + ¯ c∆ j +1 λ ( ν ) (cid:3) X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + ∆ j ∆ j λ ( ν ) X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ + e ∆ j ( jγ ( ν ) + ¯ cλ ( ν )) X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + e ∆ j λ ( ν ) X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ . (4.17)In order for this to agree with the first line in (4.9), it is sufficient that thefollowing two inequalities hold ∆ j (cid:2) ¯ κ + AK − (cid:3) + e ∆ j + (cid:16) ∆ j ∆ j + j e ∆ j (cid:17) X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ ≤ ∆ j +1 , (4.18)2 AK − ∆ j + ¯ c∆ j +1 ∆ j X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + ∆ j ∆ j X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ (4.19)+¯ c f ∆ j X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + e ∆ j X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ≤ ¯ c∆ j +2 . By means of (3.11) we getLHS(4 . ≤ ∆ + ∆AK − + ¯ c∆ j +1 < ∆ + 12 ∆ χ − + 1 < ∆ j +1 . Similarly,LHS(4 . ≤ ¯ c∆ j + ∆ j X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + ∆ j ∆ j X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ +¯ c e ∆ j X ℓ ′ ∈ ∂ℓ k ∩ U j − c ℓ k ℓ ′ + e ∆ j X ℓ ′ ∈ ∂ℓ k ∩ W j c ℓ k ℓ ′ ≤ ¯ c∆ + ¯ c∆ j ∆ j + ¯ c e ∆ j < ¯ c∆ + ¯ c∆ j +1 ≤ ¯ c∆ j +2 . IBBS FIELDS: UNIQUENESS AND DECAY OF CORRELATIONS 23
Now we consider the case where m < k ≤ n . By (3.7), and then by (3.6),we have ν ( n ) j ( I ℓ k H iℓ m ) = ν ( k ) j ( I ℓ k H iℓ m ) ≤ X ℓ ∈ ∂ℓ k ν ( m ) j ( I ℓ H iℓ m ) (4.20) ≤ X ℓ ∈ ∂ℓ k ν j ( I ℓ ) + X ℓ ∈ ∂ℓ k X ℓ ′ ∈ ∂ℓ m c ℓ m ℓ ′ ν j ( I ℓ H iℓ ′ ) . Again we split the sums in (4.20) and then use (4.8) and (4.9), and obtainthat ν ( n ) j ( I ℓ k H iℓ m ) ≤ RHS(4 . . Thus, we have that (4.9) with j + 1 holds also in this case. The proof iscomplete.4.3. The proof of Lemma 3.8.
Assume that we have given ν xs − ∈ C ( µ x , µ )with the properties in question. Then we split D N − s − into independent sub-sets by taking intersections with the sets V j , as in (2.3). Let ℓ , . . . , ℓ m be anumbering of D N − s − ∩ V . Set˜ ν x = ν xs − and ˜ ν xk = R ℓ k ˜ ν xk − , k = 1 , . . . , m, where R ℓ is defined in (3.2). Thus, ˜ ν xm is F D N − -measurable, and ˜ ν xm ( I ℓ ) and˜ ν xm ( I ℓ H iℓ ′ ), ℓ, ℓ ′ ∈ D N − s − , satisfy the inequalities in (4.6) and (4.7), respec-tively, in which the right-hand sides contain γ D N − s ( ν xs − ) and λ D N − s ( ν xs − ).Then we perform the reconstruction over the remaining independent subsetsof D N − s − and obtain an element of C ( µ x , µ ), which we denote by ν xs . Its F D N − -measurability is then guarateed by construction, and the parameters γ D N − s − ( ν xs ) and λ D N − s − ( ν xs ) satisfy the first-line inequalities in (4.8) and(4.9), respectively, and hence (3.17) with γ D N − s ( ν xs − ) and λ D N − s ( ν xs − ) onthe right-hand side. The F D N − -measurability of ν x is straightforward. References [1] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky, and M. R¨ockner,
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Fakut¨at f¨ur Mathematik, Universit¨at Bielefeld, Bielefeld D-33615, Ger-many
E-mail address : [email protected] Fakut¨at f¨ur Mathematik, Universit¨at Bielefeld, Bielefeld D-33615, Ger-many
E-mail address : [email protected] Instytut Matematyki, Uniwersytet Marii Curie-Sk lodowskiej, 20-031 Lublin,Poland
E-mail address : [email protected] Fakut¨at f¨ur Mathematik, Universit¨at Bielefeld, Bielefeld D-33615, Ger-many
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