Quasi-lattice approximation of statistical systems with strong superstable interactions. Correlation functions
aa r X i v : . [ m a t h - ph ] J a n Quasi-latti e approximation of statisti al systems withstrong superstable intera tions. Correlation fun tionsA. L. Rebenko , M. V. Terty hnyi Institute of Mathemati s, Ukrainian National A ademy of S ien es, Kyiv, Ukrainerebenkovolia able. om ; rebenkoimath.kiev.ua Fa ulty of me hani s and mathemati s, Kyiv Shev henko university, Kyiv, Ukrainemt4ukr.net Abstra tA ontinuous in(cid:28)nite system of point parti les intera ting via two-body strongsuperstable potential is onsidered in the framework of lassi al statisti al me hani s.We de(cid:28)ne some kind of approximation of main quantities, whi h des ribe ma ros op-i al and mi ros opi al hara teristi s of systems, su h as grand partition fun tionand orrelation fun tions. The pressure of an approximated system onverge to thepressure of the initial system if the parameter of approximation a → for any valuesof an inverse temperature β > and a hemi al a tivity z . The same result is truefor the family of orrelation fun tions in the region of small z.Keywords :Strong superstable potential, quasi-latti e approximation, orrela-tion fun tionsMathemati s Subje t Classi(cid:28) ation : 82B05; 82B211 Introdu tionThe main a hievements of mathemati al physi s in resear h of riti al phenomena are onne ted (cid:28)rst of all with studying in(cid:28)nite latti e systems. But one an see totally anothersituation on erning ontinuous systems. The mathemati al results have been obtained inthe majority of ases only for the small values of parameters β = kT ( where T is aemperature) and a hemi al a tivity z . The resear h of ontinuous systems in the area of riti al values of these parameters is restri ted to some arti(cid:28) ial models like the Widom-Rowlinson model [25℄ or with (cid:28)eld theory of type Hamiltonian [9℄, and the methods ofinvestigation are opied from latti e systems(see, e.g., [22℄, [10℄, using Peierls' argument,[2℄, using Pirogov-Sinai theory or [3℄,[4℄, using random luster expansion). Another typeof arguments was invented by Gruber and Gri(cid:30)ths [5℄ and used in [19℄,[6℄ to prove theexisten e of orientational ordering transitions in the ontinuous-spin models of ferro(cid:29)uid.Some important hara teristi s of riti al phenomena an be also des ribed by usinglatti e approximation of ontinuous systems. It was espe ially su essful to apply lat-ti e approximation to resear h of the models of quantum (cid:28)eld theory (see, e.g., [23℄ andreferen es therein). Substantial progress was also rea hed in studying models of latti e-gas([20℄). But the main disadvantage of the last example is that it does not ontain theparameter that ensures the transition to the lassi al ontinuous gas.On the other hand the main mathemati al problems in the resear h of in(cid:28)nite ontinu-ous systems appear be ause it is ne essary to take into a ount all possible on(cid:28)gurationsof parti les, even if the probability of their o urren e is rather small. One of possible waysto solve this problem is to introdu e hard- ore potentials. It helps to avoid mathemati aldi(cid:30) ulties whi h is onne ted with an a umulation of many number of parti les in thesmall volume , but at the same time it leads to some new problems, that is onne ted withinterpretation of physi al results and appli ation of some mathemati al methods.In the present arti le we propose some intermediate approximation of several main quan-tities, whi h des ribe ma ros opi al and mi ros opi al hara teristi s of systems, su h asgrand partition fun tion and orrelation fun tions. The main idea is in the following: wesplit the spa e R d into noninterse ting hyper ubes with a volume a d and de(cid:28)ne approx-imated grand partition fun tion and the family of approximated orrelation fun tions insu h a way, that they take into a ount only su h on(cid:28)gurations of parti les in R d , whenthere is not more than one parti le in ea h ube.It was shown in this work, that for the potentials whi h have non integrable singularityin the neighborhood of the origin(strong superstable potentials) the pressure of the ap-proximated system onverge to the pressure of the initial system if a → for any valuesof an inverse temperature β > and a hemi al a tivity z . The same result is true for thefamily of orrelation fun tions in the region of small z.2 Notations and main results2.1 Con(cid:28)guration spa eLet R d be a d -dimensional Eu lidean spa e. The set of positions { x i } i ∈ N of identi alparti les is onsidered to be a lo ally (cid:28)nite subset in R d and the set of all su h subsets reates the on(cid:28)guration spa e: Γ = Γ R d := (cid:8) γ ⊂ R d (cid:12)(cid:12) | γ ∩ Λ | < ∞ , for all Λ ∈ B c ( R d ) (cid:9) , where | A | denotes the ardinality of the set A and B c ( R d ) denote the systems of all boundedBorel sets in R d . We also need to de(cid:28)ne the spa e of (cid:28)nite on(cid:28)gurations Γ : Γ = G n ∈ N Γ ( n ) , Γ ( n ) := { η ⊂ R d | | η | = n } , N = N ∪ { } . For every Λ ∈ B c ( R d ) one an de(cid:28)ne a mapping N Λ : Γ → N of the form N Λ ( γ ) := | γ ∩ Λ | = | γ Λ | . The Borel σ -algebra B (Γ) is equal to σ ( N Λ (cid:12)(cid:12) Λ ∈ B c ( R d ) ) and additionally one may intro-du e the following (cid:28)ltration B Λ (Γ) := σ ( N Λ ′ (cid:12)(cid:12) Λ ′ ∈ B c ( R d ) , Λ ′ ⊂ Λ ) , see [11℄, [12℄, [1℄ for details.We need also to de(cid:28)ne Γ Λ := { η ∈ Γ | η ⊂ Λ } . By B (Γ Λ ) we denote the orresponding σ -algebra on Γ Λ . For the given intensity measure σ (in this ontext σ is Lebesgue measure on B ( R d ) ) and any n ∈ N the produ t measure σ ⊗ n an be onsidered as a measure on ^ ( R d ) n = (cid:8) ( x , . . . , x n ) ∈ ( R d ) n (cid:12)(cid:12) x k = x l if k = l (cid:9) and hen e as a measure σ ( n ) on Γ ( n ) through the map sym n : ^ ( R d ) n ∋ ( x , ..., x n )
7→ { x , ..., x n } ∈ Γ ( n ) . De(cid:28)ne the Lebesgue-Poisson measure λ zσ on B (Γ ) by the formula: λ zσ := X n ≥ z n n ! σ ( n ) . (2.1)3he restri tion of λ zσ to B (Γ Λ ) we also denote by λ zσ . For more detailed stru ture ofthe on(cid:28)guration spa es Γ , Γ , Γ Λ see [1℄.As in [16℄ de(cid:28)ne two additional on(cid:28)guration spa es: a spa e of dilute on(cid:28)gurationsand a spa e of dense on(cid:28)gurations.Let a > be arbitrary. Following [21℄ for ea h r ∈ Z d we de(cid:28)ne an elementary ubewith an edge a and a enter r ∆ a ( r ) := { x ∈ R d | a ( r i − / ≤ x i < a ( r i + 1 / } . (2.2)We will write ∆ instead of ∆ a ( r ) , if a ube ∆ is onsidered to be arbitrary and there isno reason to emphasize that it is entered at the on rete point r ∈ Z d . Let ∆ a be thepartition of R d into ubes ∆ a ( r ) . Without loss of generality onsider only that Λ ∈ B c ( R d ) whi h is union of ubes ∆ a ( r ) . Then for any X j Λ whi h is a union of ubes ∆ ∈ ∆ a de(cid:28)ne Γ dilX := { γ ∈ Γ X | N ∆ ( γ ) = 0 ∨ for all ∆ ⊂ X } (2.3)and Γ denX := { γ ∈ Γ X | N ∆ ( γ ) ≥ for all ∆ ⊂ X } . (2.4)2.2 De(cid:28)nition of the systemThe energy of any on(cid:28)guration γ ∈ Γ Λ or γ ∈ Γ is de(cid:28)ned by the following formula: U φ ( γ ) = U ( γ ) := X { x,y }⊂ γ φ ( | x − y | ) , (2.5)where {· , ·} means sum over all possible di(cid:27)erent ouples of parti les from the on(cid:28)gu-ration γ , φ ( | x − y | ) - pair intera tion potential. De(cid:28)ne also intera tion energy between on(cid:28)gurations η, γ ∈ Γ by: W ( η ; γ ) := X x ∈ ηy ∈ γ φ ( | x − y | ) . (2.6)We introdu e 3 kinds of intera tions, whi h will be used in this arti le:De(cid:28)nition 1. Intera tion is alled:a) stable (S), if there exists B >0 su h that: U ( γ ) ≥ − B | γ | , for any γ ∈ Γ ; (2.7)4) superstable (SS), if there exist A ( a ) > , B ( a ) ≥ and a > su h that: U ( γ ) ≥ A ( a ) X ∆ ∈ ∆ a : | γ ∆ |≥ | γ ∆ | − B ( a ) | γ | , for any γ ∈ Γ ; (2.8) ) strong superstable (SSS), if there exist A ( a ) > , B ( a ) ≥ , m ≥ and a > su hthat: U ( γ ) ≥ A ( a ) X ∆ ∈ ∆ a : | γ ∆ |≥ | γ ∆ | m − B ( a ) | γ | , for any γ ∈ Γ (2.9)for any a ≤ a .In the above onditions onstants A ( a ) , B ( a ) depend on ∆ a and onsequently on a . Ina ordan e with these de(cid:28)nitions there is a problem to des ribe the ne essary onditionson 2-body potential, whi h ensure stability, superstability or strong superstability of anin(cid:28)nite statisti al system. For the latest review and some new results on this problem see[18℄ and [24℄ for many-body ase.(A): Assumption on the intera tion potential. In this arti le we onsider ageneral type of potentials φ , whi h are ontinuous on R + \ { } and for whi h there exist r > , R > r , ϕ > , ϕ > , and ε > su h that: φ ( | x | ) ≡ − φ − ( | x | ) ≥ − ϕ | x | d + ε for | x | ≥ R, ; (2.10) φ ( | x | ) ≡ φ + ( | x | ) ≥ ϕ | x | s , s ≥ d for | x | ≤ r , (2.11)where φ + ( | x | ) := max { , φ ( | x | ) } , φ − ( | x | ) := − min { , φ ( | x | ) } . (2.12)Note that in the Eq. (2.9) the onstant a ≤ r . For the intera tion potentials whi h satisfythe assumption (A) de(cid:28)ne two important hara teristi s (for any ∆ ∈ ∆ a with a < r ): υ ( a ) := X ∆ ′ ∈ ∆ a sup x ∈ ∆ sup y ∈ ∆ ′ φ − ( | x − y | ); (2.13) b ( a ) := inf { x,y }⊂ ∆ φ + ( | x − y | ) . (2.14)Due to the translation invarian e of the 2-body potential υ and b do not depend on theposition of ∆ . The following statement is true.5roposition 2.1. Let potential φ satisfy the assumption (A). Then the intera tion isstrong superstable and the energy U satis(cid:28)es the inequality (2.9) with some < a < a andif s > d then m = 2 , A = A ( a ) = b − υ > , B = B ( a ) = υ , (2.15)Proof. For any γ ∈ Γ and any a > U ( γ ) = X { x,y }⊂ γ φ ( | x − y | ) = X ∆ ∈ ∆ a : | γ ∆ |≥ X { x,y }⊂ γ ∆ φ ( | x − y | ) + X { ∆ , ∆ ′ }⊂ ∆ a X x ∈ γ ∆ y ∈ γ ∆ ′ φ ( | x − y | ) ≥ X ∆ ∈ ∆ a : | γ ∆ |≥ | γ ∆ | ( | γ ∆ | − b − X { ∆ , ∆ ′ }⊂ ∆ a : | γ ∆ |≥ , | γ ∆ ′ |≥ | γ ∆ || γ ∆ ′ | sup x ∈ γ ∆ sup y ∈ γ ∆ ′ φ − ( | x − y | ) − υ | γ | ≥ X ∆ ∈ ∆ a : | γ ∆ |≥ | γ ∆ | (cid:18) b − υ (cid:19) − υ | γ | . We use the de(cid:28)nitions (2.12)(cid:21)(2.14) and the inequality: | γ ∆ || γ ∆ ′ | ≤
12 ( | γ ∆ | + | γ ∆ ′ | ) In the ase s = d the following statement is true (see [18℄ for details): for any su(cid:30) ientlysmall ε > there exists a onstant B = B ( ε, a ) su h that the following inequality holds: U ( γ ) ≥ X ∆ ∈ ∆ a , | γ ∆ |≥ (cid:16) C d log | γ ∆ | − v − ε log | γ ∆ | (cid:17) | γ ∆ | − B | γ | , (2.16)where (see [7℄) C d = 1 a d π d d Γ (cid:0) d (cid:1) ϕ , (2.17) Γ( · ) is a lassi al gamma-fun tion.The system of parti les is strong superstable (SSS) be ause for any ε > one an (cid:28)ndsu h numbers N ≥ and B = B ( N ; ε, a ) that for any | γ ∆ | > N C d log | γ ∆ | > v . (2.18)It follows from (2.16) - (2.18) that if s = d we an put A ( a ) = K s ( ε ) υ , B ( a ) = L s ( ε ) υ + M s ( ε ) , K s ( ε ) , L s ( ε ) , M s ( ε ) do not depend on the parameter a .In the sequel we will use the estimates (2.15) of the onstants A ( a ) and B ( a ) , be ausethe proof of the main results is the same for both ases. (cid:4) Proposition 2.2. It follows from the Proposition (2.1) that for the potentials whi h satisfythe onditions (2.10)- (2.12) the inequality (2.7) holds with B = d − s d sd φ s ϕ d ! s − d , (2.19)where the onstant φ is very lose to R R d φ − ( | x | ) dx .Proof. We an put a = a m in su h a way that b ( a m ) = 2 υ ( a m ) . From the de(cid:28)nitions(2.13), (2.14) it is lear that b ( a m ) ≥ ϕ d s a sm and υ ( a m ) = a dm φ as lim a → a d υ ( a ) = R R d φ − ( | x | ) dx . As a result a m ≥ (cid:0) ϕ φ (cid:1) s − d d s s − d ) . (2.20)The estimate (2.19) of the onstant B dire tly follows from (2.20) and (2.15). The end ofproof. (cid:4) Remark 2.1. It is important to stress that the onstant B in (2.19) does not depend onthe partition ∆ a and depends only on the potential φ and dimension of the spa e.Remark 2.2. Indeed, for the potentials whi h satisfy the assumption (A) the inequality(2.9) holds with m = 1 + s/d (see [18℄). But for our purpose it is su(cid:30) ient to apply (2.9)with (2.15) and (2.7) with (2.19).2.3 Partition fun tions, orresponding pressure and orrelationfun tionsThe main hara teristi s of Gibbs states are orrelation fun tions. A family of (cid:28)nite volume orrelation fun tions with empty boundary onditions for the grand anoni al ensemble is7e(cid:28)ned by the following formula: ρ Λ ( η ; z, β ) := z | η | Z Λ ( z, β ) Z Γ Λ e − βU ( η ∪ γ ) λ zσ ( dγ ) , η ∈ Γ Λ , (2.21)where Z Λ ( z, β ) := Z Γ Λ e − βU ( γ ) λ zσ ( dγ ) (2.22)is the grand partition fun tion whi h plays the role of normalizing onstant in the de(cid:28)nitionof the Gibbs measure. Besides it has independent important physi al meaning for thede(cid:28)nition of the thermodynami fun tion(cid:21)pressure: p ( z, β ) = lim | Λ |→∞ p Λ ( z, β ) = 1 β lim | Λ |→∞ | Λ | log Z Λ ( z, β ) , (2.23)The existen e of this limit for the above de(cid:28)ned system of parti les is well-known result(see, e.g., [21℄).To de(cid:28)ne above mentioned approximation let us introdu e the following family of or-relation fun tions: ρ ( − )Λ ( η ; z, β, a ) := z | η | Z ( − )Λ ( z, β, a ) Z Γ Λ e − βU ( η ∪ γ ) Y ∆ ∈ ∆ a ∩ Λ χ ∆ − ( η ∪ γ ) λ zσ ( dγ ) , η ∈ Γ Λ , (2.24) Z ( − )Λ ( z, β, a ) := Z Γ dil Λ e − βU ( γ ) λ zσ ( dγ ) = Z Γ Λ e − βU ( γ ) Y ∆ ∈ ∆ a ∩ Λ χ ∆ − ( γ ) λ zσ ( dγ ) . (2.25)where we introdu ed B ∆ (Γ Λ ) -measurable fun tion χ ∆ − by the formula: χ ∆ − ( γ ) = ( , for γ with N ∆ ( γ ) = | γ ∆ | = 0 ∨ , , otherwise . (2.26)Remark 2.3. By de(cid:28)nition ρ ( − )Λ ( η ; z, β ; a ) = 0 for η Γ ( dil )Λ One an de(cid:28)ne the orresponding pressure: p ( − ) ( z, β, a ) = lim | Λ |→∞ p ( − )Λ ( z, β, a ) = 1 β lim | Λ |→∞ | Λ | log Z ( − )Λ ( z, β, a ) . (2.27)Remark 2.4. The main point of this approximation onsists that in expressions for thebasi hara teristi s of the system integration is arried out not over all spa e of on(cid:28)g-urations Γ Λ , but only over those on(cid:28)gurations whi h ontain for the given partition ∆ a not more than one parti le in ea h ube ∆ ∈ ∆ a . That fa t is surprising as for an in(cid:28)nitesystem the set of su h on(cid:28)gurations in Γ is the set of measure zero with respe t to thePoisson measure and the Gibbs measure. Nevertheless, as we shall see in following se -tion, the basi hara teristi s of the approximated system ( even in a thermodynami limit Λ ր R d ) an be somehow lose to the orresponding hara teristi s of the initial system.8.4 Main resultsWe prove the results for the in(cid:28)nite volume hara teristi s, so let us de(cid:28)ne the sequen eof bounded Lebesgue measurable regions of Λ l ⊂ R d : Λ ⊂ Λ ⊂ . . . ⊂ Λ n ⊂ . . . , ∪ l Λ l = R d . (2.28)We onsider only su h Λ l ∈ B c ( R d ) whi h is union of ubes ∆ a ( r ) de(cid:28)ned by (2.2).Theorem 2.1. Let the intera tion potential φ ( | x | ) satisfy the assumptions (A). Then thelimits p ( z, β ) = 1 β lim l →∞ | Λ l | log Z Λ l ( z, β ) , (2.29) p ( − ) ( z, β, a ) = 1 β lim l →∞ | Λ l | log Z ( − )Λ l ( z, β, a ) (2.30)are (cid:28)nite and for any ε > there exists a = a ( z, ε ) > su h that: | p ( z, β ) − p ( − ) ( z, β, a ) | < ε (2.31)holds for all positive z, β and a ∈ (0 , a ( z, ε )) .The proof of the limit (2.29) is well known result [21℄. The proof of (2.30) and (2.31)one an (cid:28)nd in [17℄. But for the ompleteness of the presentation we give a sket h of theproof in the next se tion. A similar result is true for the orrelation fun tions in the (cid:28)xedvolume Λ :Theorem 2.2. Let the intera tion potential φ ( | x | ) satisfy the assumptions (A). Then forany ε > , any (cid:28)xed Λ and any on(cid:28)guration η ∈ Γ there exists a = a ( z, β, ε ) > su hthat: | ρ Λ ( η ; z, β ) − ρ ( − )Λ ( η ; z, β, a ) | < ε. (2.32)To formulate a similar result for the limit orrelation fun tions in the in(cid:28)nite volumenote that for any on(cid:28)guration η ∈ Γ and any sequen e (2.28), su h that η ⊂ Λ thereexists subsequen e (Λ ′ k ) of (Λ l ) , su h that lim k →∞ ρ Λ ′ k ( η ; z, β ) = ρ ( η ; z, β ) < ∞ (2.33)for all positive z, β uniformly on B c (Γ ) . This result follows from the uniform bounds ofthe family { ρ Λ : Λ ∈ B c ( R d ) } . (see [21℄, [16℄, [14℄).9t is also lear that the same uniform bounds hold for the family of { ρ ( − )Λ : Λ ∈ B c ( R d ) } .So, there exists subsequen e ( Λ ′′ m ) of the sequen e ( Λ ′ k ) su h that one an de(cid:28)ne ρ ( − ) ( η ; z, β, a ) = lim m →∞ ρ ( − )Λ ′′ m ( η ; z, β, a ) < ∞ . (2.34)In the ase of small values of a hemi al a tivity z there exists the unique limit ρ ( η ; z, β ) that is a solution of Kirkwood-Salzburg(KS) equations in the spa e E ξ (see [20℄). In thenext hapter we will show, that a similar equations an be easily written for the fun tions ρ ( − ) ( η ; z, β ) that is a unique solution of these equations for su(cid:30) iently small values ofparameters z or β .Theorem 2.3. Let the intera tion potential φ ( | x | ) satisfy the assumptions (A). Then forany ε > , su(cid:30) iently small z and any on(cid:28)guration η ∈ Γ there exists a = a ( z, β, ε ) > su h that: | ρ ( η ; z, β ) − ρ ( − ) ( η ; z, β, a ) | < ε. (2.35)holds for all a ∈ (0 , a ( z, ε )) .Corollary 2.1. The inequalities (2.31), (2.32), (2.35) ensure the existen e of limits: lim a → p ( − ) ( z, β, a ) = p ( z, β ) (2.36)for any positive z, β > , η ∈ Γ ; lim a → ρ ( − )Λ ( η ; z, β, a ) = ρ Λ ( η ; z, β ) . (2.37)for any positive z, β > , η ∈ Γ , any (cid:28)xed Λ and lim a → ρ ( − ) ( η ; z, β, a ) = ρ ( η ; z, β ) . (2.38)for small positive z, any β > and η ∈ Γ . ∆ ∈ ∆ a as χ ∆+ ( γ ) := 1 − χ ∆ − ( γ ) . Then we use the following partition of the unity for any γ ∈ Γ Λ :10 = Y ∆ ⊂ Λ (cid:2) χ ∆ − ( γ ) + χ ∆+ ( γ ) (cid:3) = N Λ X n =0 X { ∆ ,..., ∆ n }⊂ Λ n Y i =1 χ ∆ i + ( γ ) Y ∆ ⊂ Λ \∪ ni =1 ∆ i χ ∆ − ( γ )= X X ⊆ Λ e χ X + ( γ ) e χ Λ \ X − ( γ ) , (3.1)where N Λ = | Λ | /a d (here the symbol | · | means Lebesgue measure of the set Λ ) is thenumber of ubes ∆ in the volume Λ , and e χ X ± ( γ ) := Y ∆ ⊂ X χ ∆ ± ( γ ) . (3.2)Inserting (3.1) into (2.22) we obtain: Z Λ ( z, β ) = X X ⊆ Λ Z Γ Λ e − βU ( γ ) e χ X + ( γ ) e χ Λ \ X − ( γ ) λ zσ ( dγ ) . (3.3)It is obvious, that the (cid:28)rst term in (3.3) (at X = ∅ ) oin ides with Z ( − )Λ ( z, β, a ) (see (2.25)).Using in(cid:28)nite divisible property of the Lebesgue-Poisson measure (see for example (2.5) in[15℄) one dedu e that: Z Λ ( z, β ) = Z ( − )Λ ( z, β, a ) X ∅6 = X ⊆ Λ Z Γ X e ρ ( − )Λ \ X ( γ ; a ) e χ X + ( γ ) λ zσ ( dγ ) := Z ( − )Λ ( z, β, a ) Z (+)Λ ( z, β, a ) , (3.4)where e ρ ( − )Λ \ X ( γ X ; a ) = e − βU ( γ X ) Z ( − )Λ ( z, β, a ) Z Γ Λ \ X e − βW ( γ X | γ ′ ) − βU ( γ ′ ) e χ Λ \ X − ( γ ′ ) λ zσ ( dγ ′ ) , (3.5)We, also, de(cid:28)ne p (+) ( z, β, a ) in the same way as in (2.30) p (+) ( z, β, a ) = lim l →∞ p (+)Λ l ( z, β, a ) = 1 β lim l →∞ | Λ l | log Z (+)Λ l ( z, β, a ) (3.6)Consequently, in order to prove the Theorem 2.1 we have to estimate the value of p (+) ( z, β, a ) . Using Proposition 2.1 (Eqs. (2.9), (2.15)) one an obtain: e − βU ( γ X ) ≤ Y ∆ ∈ ∆ a ∩ X e − βA ( a ) | γ ∆ | + βB ( a ) | γ ∆ | , A ( a ) = b − υ , B ( a ) = υ . (3.7)Taking into a ount assumption (A)(Eqs. (2.10)) and (2.13) we obtain: e − βW ( γ X | γ ′ ) ≤ Y ∆ ∈ ∆ a ∩ X e βυ | γ ∆ | . (3.8)11sing in(cid:28)nite divisible property of the measure λ zσ and using (3.7), (3.8) we have: Z Γ X ρ ( − )Λ \ X ( γ X ; a ) e χ X + ( γ ) λ zσ ( dγ ) ≤ Z ( − )Λ \ X ( z, β, a ) Z ( − )Λ ( z, β, a ) × Y ∆ ∈ ∆ a ∩ X Z Γ ∆ e − βA | γ ∆ | + βB | γ ∆ | + βυ | γ ∆ | χ ∆+ ( γ ∆ ) λ zσ ( dγ ∆ ) . As a result, using de(cid:28)nition of Lebesgue-Poisson measure ( see (2.1)) one an obtain thefollowing estimate: Z Γ ∆ e − β A | γ ∆ | + β ( B + υ ) | γ ∆ | χ ∆+ ( γ ∆ ) λ zσ ( dγ ∆ ) = ∞ X n =2 ( a d z ) n n ! e − β ( b − υ ) n + β υ n ≤ ǫ ( a ) , (3.9)with ǫ ( a ) = 12 z a d e − β ( b − υ ) exp { za d e − β ( b − υ ) } . (3.10)Now from the de(cid:28)nition of N Λ , Z (+)Λ ( z, β, a ) (see (3.4)) and above estimates we have: log Z (+)Λ ( z, β, a ) ≤ log X ∅6 = X ⊆ Λ ǫ ( a ) N X = log " N Λ X k =1 N Λ ! k !( N Λ − k )! ǫ ( a ) k = log [1 + ǫ ( a )] N Λ = | Λ | a d log [1 + ǫ ( a )] . (3.11)As a result p (+) ( z, β ; a ) ≤ βa d log [1 + ǫ ( a )] . It is important for the proof of the theorem to (cid:28)nd out the asymptoti behavior of ǫ ( a ) at a → . It follows from the Eq.(3.10) and the orresponding behavior of b and υ (see(2.10)-(2.14)). As a result we have: ǫ ( a ) ∼ a d e − as , s ≥ d. (3.12)So, lim a → p (+) ( z, β ; a ) = 0 . The end of the proof. (cid:4)
12 Proof of Theorem 2.2Inserting (3.1) with an argument η ∪ γ into (2.21) we obtain: ρ Λ ( η ; z, β ) = z | η | Z Λ ( z, β ) X X ⊆ Λ Z Γ Λ e − βU ( η ∪ γ ) e χ X + ( η ∪ γ ) e χ Λ \ X − ( η ∪ γ ) λ zσ ( dγ ) . (4.1)Extra ting the (cid:28)rst term at X = ∅ and using the de(cid:28)nitions (2.24),(2.25) we an rewrite(4.1) in the following form: ρ Λ ( η ; z, β ) = Z ( − )Λ ( z, β, a ) Z Λ ( z, β ) ρ ( − )Λ ( η ; z, β, a ) + R Λ ( η ; z, β, a ) , (4.2)where R Λ ( η ; z, β, a ) = z | η | Z Λ ( z, β ) X ∅6 = X ⊆ Λ Z Γ Λ e − βU ( η ∪ γ ) e χ X + ( η ∪ γ ) e χ Λ \ X − ( η ∪ γ ) λ zσ ( dγ ) . (4.3)The proof of the Theorem 2.2 is based on two te hni al lemmas.Lemma 4.1. Let the intera tion potential φ ( | x | ) satisfy the assumptions (A). Then forany (cid:28)xed volume Λ ∈ B c ( R d ) and any on(cid:28)guration η ∈ Γ the following holds:lim a → R Λ ( η ; z, β, a ) = 0 . (4.4)Proof. See Appendix. (cid:4) Lemma 4.2. Let the intera tion potential φ ( | x | ) satisfy the assumptions (A). Then forany (cid:28)xed volume Λ ∈ B c ( R d ) the following holds:lim a → Z ( − )Λ ( z, β, a ) Z Λ ( z, β ) = 1 . (4.5)Proof. It follows from the estimates (3.11), thatlim a → Z ( − )Λ ( z, β, a ) Z Λ ( z, β ) ≥ . From the other hand in a ordan e with the de(cid:28)nitions (2.22), (2.25) it is lear that Z ( − )Λ ( z, β, a ) Z Λ ( z, β ) ≤ . As a result we have lim a → Z ( − )Λ ( z, β, a ) Z Λ ( z, β ) = 1 .
13 Proof of Theorem 2.3Using de(cid:28)nitions (2.1), (2.26) we
an rewrite the de(cid:28)nition (2.24) for the family of
orre-lation fun
tions ρ ( − )Λ ( · ; z, β, a ) in the following form: ρ ( − )Λ ( η ; z, β, a ) = z | η | Z ( − )Λ ( z, β, a ) e − βU ( η ) Y ∆ ∈ Λ η χ ∆ − ( η ∆ ) (cid:20) N Λ \ Λ η X k =1 z k X { ∆ ,..., ∆ k }⊂ (Λ \ Λ η ) ∩ ∆ a Z ∆ · · · Z ∆ k e − βW ( η ; { y ,...,y k } ) e − βU ( { y , ··· y k } ) dy . . . dy k (cid:21) , (5.1)where Λ η is a union of
ubes of ∆ a whi
h
ontain points from the
on(cid:28)guration η (and inthe sequel we will use su
h a notation) and summation is taken over all possible sets of
ubes from ∆ a that belong to the area Λ \ Λ η . We prove the theorem using KS equationsfor the fun
tions ρ ( η ; z, β ) and ρ ( − ) ( η ; z, β, a ) . Remind that KS equations for the fun
tions ρ ( η ; z, β ) an be written in the form of one operator equation(see [20℄) ρ = z e Kρ + zδ, (5.2)where operator e K a
ts on an arbitrary fun
tion ϕ a
ording with the rule ( e Kϕ )( { x } ) = ∞ X k =1 k ! Z R d · · · Z R d k Y i =1 (cid:0) e − βφ ( | y i − x | ) − (cid:1) × ϕ ( { y , . . . , y k } ) dy · · · dy k , if | η | = 1 ( η = { x } ); (5.3) ( e Kϕ )( η ) = X x ∈ η e π ( x ; η \ { x } ) e − βW ( x ; η \{ x } ) (cid:2) ϕ ( η \ { x } ) + ∞ X k =1 k ! Z R d · · · Z R d × k Y i =1 (cid:0) e − βφ ( | y i − x | ) − (cid:1) ϕ ( η \ { x } ∪ { y , . . . , y k } ) dy · · · dy k (cid:3) , if | η | ≥ , (5.4)where e π ( x ; η \ { x } ) = π W ( x ; η \ { x } ) P y ∈ η π W ( y ; η \ { y } ) , π W ( x ; η \ { x } ) = if W ( x ; η \ { x } ) ≥ − B, otherwise , (5.5) ρ := { ρ ( η ; z, β ) } η ∈ Γ , (5.6) δ ( η ) = 1 if | η | = 1 and δ ( η ) = 0 otherwise. 14emark 5.1. Operator e K = Π K in the Ruelle's notation [20℄ and (5.4), (5.5) are exa
trealization of the operator Π Operator e K is bounded operator in Bana
h spa
e of measurable bounded fun
tions E ξ ( ξ > with the norm || ϕ || ξ = sup η ∈ Γ | ϕ ( η ) | ξ −| η | (5.7)The solution of the equation (5.2)
an be represented in the form of
onvergent in E ξ (andpoint
onvergent for any (cid:28)xed η ∈ Γ ) series ρ ( η ; z, β ) = ∞ X n =0 z n +1 ( e K n δ )( η ; z, β ) , (5.8)if | z | ≤ e − βB − C ( β ) − , C ( β ) = Z R d (cid:12)(cid:12) e − βφ ( | x | ) − (cid:12)(cid:12) dx (5.9)and intera
tion satis(cid:28)es the
onditions (2.7), (2.10), (2.11).One
an write similar equations for the fun
tions ρ ( − )Λ ( η ; z, β, a ) . It
an be easily donein the way like it was shown in [13℄ for the
ase of latti
e gas. Let us pro
eed with severalnew notations that
orrespond the notations in the spa
e of
on(cid:28)gurations in the latti
egas system (see [13℄). De(cid:28)ne the spa
e C = C ∆ a of
on(cid:28)gurations of
ubes from ∆ a . Let s = { ∆ η , . . . , ∆ | η | η } be the (cid:28)nite
on(cid:28)guration of | η | ubes from ∆ a with all points fromthe
on(cid:28)guration η ∈ Γ and s ′ = s \ { ∆ η } . Let us denote by C (cid:28)n ∆ a a spa
e of all (cid:28)nite
on(cid:28)gurations of
ubes from C (see also [13℄) and c = { ∆ , . . . , ∆ k } ∈ C (cid:28)n ∆ a be any (cid:28)nite
on(cid:28)guration of k ubes from ∆ a , k = 0 , . . . , | γ | ; ( if k = 0 c = ∅ ) .For te
hni
al reason we also introdu
e new potential ˆ φ ( x, y ) = φ ( | x − y | ) + φ or ∆ a ( x, y ) , where φ or ∆ a ( x, y ) = + ∞ if x, y ∈ ∆ ∈ ∆ a , if x ∈ ∆ , y ∈ ∆ ′ and ∆ = ∆ ′ . (5.10)As in the de(cid:28)nition (5.1) all points of the
on(cid:28)gurations η, γ are situated in di(cid:27)erent
ubeswe
an put the potential ˆ φ instead of the potential φ in the de(cid:28)nitions (2.24), (2.25). Letus de(cid:28)ne also a potential ˆ φ (∆ , ∆ ′ ) as the family of potentials ˆ φ ( x, y ) : ˆ φ (∆ , ∆ ′ ) = n ˆ φ ( x, y ) (cid:12)(cid:12) x ∈ ∆ , y ∈ ∆ ′ o , (5.11) ˆ φ (∆ , ∆) = + ∞ for any ∆ ∈ ∆ a . c = { ∆ , ..., ∆ k } , s = { ∆ η , ..., ∆ mη } , m = | η | the fun
tions U ( c ) , W ( s ; c ) , ρ ( − )Λ ( s ; z, β, a ) , ρ ( − ) ( s ; z, β, a ) are the families (see (5.10)) of the
orresponding U ( γ ) , W ( η ; γ ) , ρ ( − )Λ ( η ; z, β, a ) , ρ ( − ) ( η ; z, β, a ) with γ = { γ ∆ , ..., γ ∆ k } , η = { η ∆ η , ..., η ∆ mη } and at a → every
ube shrinks in the
orresponding point so that c → γ , s → η .Con(cid:28)guration η ∈ Γ in de(cid:28)nition of the fun
tion ρ ( η ; z, β ) is (cid:28)xed and
oordinates of
ubes ∆ η , . . . , ∆ | η | η in R d hange, but Lebesgue measure of Λ η tends to zero (mess Λ η ( a ) → . The energy U ( γ ) of the
on(cid:28)guration γ ∈ Γ X , X ⊆ Λ in these notations is U ( c ) = X ≤ i
1) (1 + ǫ ( a )) | Λ \ Λ η | ad . (6.12)It follows from (6.4), (6.7), (6.12) that: R Λ ( η ; z, β, a ) ≤ ( ze β υ ) | η | (1 + ǫ ( a )) | Λ \ Λ η | ad − (cid:18) ǫ ( a ) | Λ \ Λ η | a d + (2 | η | − × (1 + ǫ ( a )) e − β ( b − υ ) e za d | η | (cid:19) → , if a → . (6.13)The lemma is proven. (cid:4) Proof of the lemma (5.1)We have to prove that for any ε > there exists a ε that for any a < a ε the followingestimate holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X { ∆ ,..., ∆ k }⊂ ∆ a \ Λ η Z ∆ dx · · · Z ∆ k dx k F − ( x , . . . , x k ; a ) − k ! Z ( R d ) k F ( x , . . . , x k ) dx · · · dx k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. (6.14)21rom the integrability onditions of the fun tions F − , F one an obtain that for any ε > there exists bounded Λ ε ⊂ R d , su h that (cid:12)(cid:12)(cid:12)(cid:12) X { ∆ ,..., ∆ k }⊂ ∆ a \ Λ η Z ∆ dx · · · Z ∆ k dx k F − ( x , . . . , x k ; a ) − X { ∆ ,..., ∆ k }⊂ (∆ a \ Λ η ) ∩ Λ ε Z ∆ dx · · · Z ∆ k dx k F − ( x , . . . , x k ; a ) (cid:12)(cid:12)(cid:12)(cid:12) < ε , (6.15)and (cid:12)(cid:12)(cid:12)(cid:12) k ! Z ( R d ) k F ( x , . . . , x k ) dx · · · dx k − k ! Z Λ kε F ( x , . . . , x k ) dx · · · dx k (cid:12)(cid:12)(cid:12)(cid:12) < ε . (6.16)Using (6.14)(cid:21) (6.16) it is easy to noti e that the proof of the lemma an be redu ed toveri(cid:28) ation the fa t that for any ε > there exists a ε = f ( ε ) > that for any a < a ε thefollowing estimate is true: R = (cid:12)(cid:12)(cid:12)(cid:12) X { ∆ ,..., ∆ k }⊂ (∆ a \ Λ η ) ∩ Λ ε Z ∆ dx · · · Z ∆ k dx k F − ( x , . . . , x k ; a ) − k ! Z Λ kε F ( x , . . . , x k ) dx · · · dx k (cid:12)(cid:12)(cid:12)(cid:12) < ε . (6.17)Dividing ea h integral over Λ ε into the sum of integrals over ∆ ∈ ∆ a ∩ Λ ε one an arrangetwo terms in (6.17) into three ones to get estimate R ≤ R + R + R ,R = k − X j =1 X { k ,...,k j } ,k + ··· + k j = k k ! · · · k j ! ′ X π ∈ P j X { ∆ ,..., ∆ j }⊂ ∆ a ∩ Λ ε Z ∆ dx · · · Z ∆ dx k π (1) · · · Z ∆ j dx k − k π ( j ) +1 · · · Z ∆ j | F ( x , . . . , x k ) | dx k , (6.18) R = X { ∆ ,..., ∆ k }⊂ (∆ a \ Λ η ) ∩ Λ ε Z ∆ dx · · · Z ∆ k dx k | F − ( x , . . . , x k ; a ) − F ( x , . . . , x k ) | , (6.19) R = X { ∆ ,..., ∆ k }⊂ ∆ a ∩ Λ ε , { ∆ ,..., ∆ k }∩ Λ η = ∅ Z ∆ dx · · · Z ∆ k dx k | F ( x , . . . , x k ) | , (6.20)22here P j is a set of all permutations of numbers { , . . . , j } , but the sum P ′ π ∈ P j meansthat we onsider only di(cid:27)erent permutations of numbers { k , . . . , k j } (for example if k i = k j the permutation of numbers k i , k j is onsidered only on e). Then for R we have: R < k − X j =1 j ! X { k ,...,k j } ,k + ··· + k j = k k ! · · · k j ! ′ X π ∈ P j X ∆ ⊂ ∆ a ∩ Λ ε ,..., ∆ j ⊂ ∆ a ∩ Λ ε Z ∆ dx · · · Z ∆ dx k π (1) · · · Z ∆ j dx k − k π ( j ) +1 · · · Z ∆ j | F ( x , . . . , x k ) | dx k < k − X j =1 a dk j ! X { k ,...,k j } ,k + ··· + k j = k k ! · · · k j ! ′ X π ∈ P j X ∆ ⊂ ∆ a ∩ Λ ε ,..., ∆ j ⊂ ∆ a ∩ Λ ε sup { x ,...,x k }∈ ( R d ) k | F ( x , . . . , x k ) | < k − X j =1 a d ( k − j ) j ! | Λ ε | j X { k ,...,k j } ,k + ··· + k j = k k ! · · · k j ! ′ X π ∈ P j sup { x ,...,x k }∈ ( R d ) k | F ( x , . . . , x k ) | → if a → . (6.21)For R : R < k ! 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