Shift-invariance for FK-DLR states of a 2D quantum bose-gas
aa r X i v : . [ m a t h - ph ] A p r Shift-invariance for FK-DLR statesof a 2D quantum bose-gas
Y. Suhov , M. Kelbert , I. Stuhl August 27, 2018 Statistical Laboratory, DPMMS, University of Cambridge, UK;Department of Statistics/IME, University of S˜ao Paulo, Brazil;IITP, RAS, Moscow, RussiaE-mail: [email protected] Department of Mathematics, Swansea University, UK;Department of Statistics/IME, University of S˜ao Paulo, BrazilE-mail: [email protected] University of Debrecen, Hungary;Department of Mathematics/IME, University of S˜ao Paulo, BrazilE-mail: [email protected]
Abstract
This paper continues the work [14] and focuses on infinite-volumebosonic states for a quantum system (a quantum gas) in a plane R .We work under similar assumptions upon the form of local Hamil-tonians and the type of the (pair) interaction potential as in [14].The result of the paper is that any infinite-volume FK-DLR func-tional corresponding to the Hamiltonians is shift-invariant, regardlessof whether this functional is unique or not. Keywords: bosonic quantum system in a plane R , FK-DLRstates and functionals, FK-DLR probability measures, shift-invariance . Introduction: FK-DLR states of quantum systems This work continues [14] (and earlier works [5], [6] and [8]). The referenceto [14] are marked by the Roman number I : Eqn (1.1.19. I ), Theorem 1.1. I ,Section 2.3. I and so on. In this paper we provide a justification of the notionof an FK-DLR (Feynman–Kac–Dobrushin–Lanford–Ruelle) state of a quan-tum system in an infinite volume (more generally, an FK-DLR functional ofthe quasi-local C ∗ -algebra). The result of the present paper is that, for aquantum Bose-gas on a plane R , any FK-DLR state is shift-invariant. Thisline of results takes its origin in [1], [9]; we want to stress that a particularimpact upon the present work was made by Refs [10]–[12] (more credit willbe given in due course).We follow the background used in Sects 1. I – 3. I , in a specific situa-tion where the dimension d = 2. Accordingly, Λ = Λ L and Λ standfor squares [ − L, L ] × ⊂ R and [ − L , L ] × ⊂ R , or – more generally –[ − L + c , L + c ] × [ − L + c , L + c ], where c = ( c , c ) ∈ R and Λ ⊃ Λ .As in [14], we denote by z and β the standard thermodynamical variablesof the fugacity and the inverse temperature. The notions of a quantum n -particle Hamiltonian H n, Λ and the Gibbs state ϕ z,β, Λ in Λ are introducedas in Sects 1.1. I (see Eqns (1.1.1. I ) – (1.1.25. I )). We also follow the condi-tions upon the two-body potential V : [ r , ∞ ) → R imposed in [14]. (Here r ∈ (0 , ∞ ) is the hard-core diameter, and we formally set V ( r ) = + ∞ for0 ≤ r < r , conforming with the hard-core condition.) Moreover, we use thecorresponding notation: cf. Eqns (1.1.3. I )–(1.1.5. I ), (1.1.19. I ) and (1.2.9. I ).For the reader’s convenience, we reproduce these conditions (and assume thatthey are valid throughout the paper): V ( r ) = 0 for r ≥ R where R ∈ ( r , ∞ ) , (1 . − V = min (cid:2) V ( r ) : r ≤ r ≤ R (cid:3) , (1 . V = 0 for V ≥ V (1) = max (cid:2) | V ′ ( r ) | : r ≤ r ≤ R (cid:3) , V (2) = max (cid:2) | V ′′ ( r ) | : r ≤ r ≤ R (cid:3) , (1 . ρ := z exp (4 βV R / r ) < . (1 . z > β > { ϕ z,β, Λ } is compact and has limiting pointsas Λ ր R . Moreover, the family of Gibbs states { ϕ z,β, Λ | x (Λ c ) } is compactwhere ϕ z,β, Λ | x (Λ c ) is the Gibbs state in an external potential field generatedby a ‘classical’ configuration x (Λ c ) ⊂ Λ c satisfying (1.1.20. I ). see Theorem1.1. The limiting points for families { ϕ z,β, Λ } and { ϕ z,β, Λ | x (Λ c ) } yield statesof the quasi-local C ∗ -algebra B ; see (1.2.5. I ). Such states describe possible‘thermodynamic phases’ of the quantum Bose-gas in an infinite volume. Atheory proposed in [14] goes a step further: we establish that any such limit-point state ϕ has a particular structure where the operators R Λ yielding the(limiting) density matrices are constructed via an FK representation.More precisely, the integral kernels F Λ ( x , y ) determining the densitymatrices R Λ are written as integrals over spaces of so-called path and loopconfigurations; cf. Sect 2. I . An important rˆole in these formulas is playedby a probability measure (or probability measures) µ on W ∗ ( R ), the spaceof loop configurations (LCs) in the plane R . In a natural sense, the corre-spondence between a functional and a measure is one-to-one. Such a mea-sure µ was called an FK-DLR probability measure (PM) and emerged asa limiting point for the family of similar measures in finite volumes Λ asΛ ր R . The set of FK-DLR PMs is denoted by K = K ( z, β ); in a prob-abilistic terminology these measures are examples of random marked pointprocesses (RMPPs) with marks represented by loops. Accordingly, a class ofstates F + = F + ( z, β ) was introduced, called FK-DLR states, together withits enlargement, F = F ( z, β ) ⊃ F + , giving a class of FK-DLR functionals on B . See Definitions 2.4. I –2.7. I . We stated a result, Theorem 1.2. I , and itsgeneralization, Theorem 2.2. I , claiming that any functional from class F isshift-invariant. For reader’s convenience, we repeat here the statements ofthe latter.The results of this paper are summarised in the following two theorems. The Fock spaces H (Λ ) and H ( S ( s )Λ ) (cf. (1.1.12. I ) ) are related througha pair of mutually inverse shift isomorphisms of Fock spaces U Λ ( s ) : H (Λ ) → H ( S ( s )Λ ) and U Λ ( − s ) : H ( S ( s )Λ ) → H (Λ ) . With the shift isometry S ( s ) : R → R : S ( s ) y = y + s, y ∈ R , nd for the image S ( s )Λ of Λ : S ( s )Λ = [ b + s − L , b + s + L ] × [ b + s − L , b + s + L ] . The isomorphisms U Λ ( s ) and U Λ ( − s ) are given by (cid:0) U Λ ( s ) φ n (cid:1) ( x n ) = φ n ( S ( − s ) x n ) , (cid:0) U Λ ( − s ) φ n (cid:1) ( x n ) = φ n ( S ( s ) x n ) , x n ∈ (Λ ) n , where φ n ∈ L sym2 ((Λ ) n ) , n = 0 , , . . . . Cf. Eqns (1.2.6. I )–(1.2.8. I ). Theorem 1.1. (cf. Theorem 2.2. I ) Assuming conditions (1.1)–(1.4) , let µ be a probability measure from K ( z, β ) . Then the corresponding FK-DLRfunctional ϕ µ ∈ F ( z, β ) is shift-invariant: for any square Λ ⊂ R , vector s ∈ R and operator A ∈ B (Λ ) , ϕ µ ( U S ( s )Λ ( − s ) AU Λ ) = ϕ µ ( A ) . In terms of the corresponding infinite-volume reduced density matrices R Λ : R S ( s )Λ = U Λ ( s ) R Λ U S ( s )Λ ( − s ) . In view of formulas (2.3.5. I )–(2.3.7. I ) relating an FK-DLR functional ϕ to an FK-DLR PM µ , it suffices to verify Theorem 1.2.
Any FK-DLR PM µ is translation invariant: for any s = ( s , s ) ∈ R , square Λ = [ − L , L ] × and event D ∈ W ∗ ( R ) localisedin Λ (i.e., belonging to a sigma-algebra W (Λ ) ; cf Definition I ), µ ( S ( s ) D ) = µ ( D ) . Here S ( s ) D stands for the shifted event localised in the shifted square S ( s )Λ = [ − L + s , s + L ] × [ − L + s , s + L ] .
2. Proof of Theorem 1.2: a tuned-shift argument
In what follows we use the terminology and the system of notation fromSect 1. I and 2. I . The proof of Theorem 1.2 is based on a modification of4n argument developed in [10]–[12]. We want to stress that the paper [12]treating some classes of (Gibbsian) RMPPs does not cover our situationbecause a number of the assumptions used in [12] are (unfortunately) notfulfilled here. Specifically, the condition (2.2) from [12] does not hold in oursituation, as well as conditions specifying what is called a bpsi-function onp. 704 of [12]. The aforementioned modification demands that we use (andinspect) the construction from [11] for classical configurations (CCs) arisingas t -sections of LCs at a given time point t ∈ [0 , β ].Because the argument in the proof does not depend on the direction ofthe vector s , we will assume that s = ( s ,
0) lies along the horizontal axis.Also, due to the group property, we can assume that s ∈ (0 , / Theorem 2.1.
Let µ be an FK-DLR PM, Λ be a square [ − L , L ] × and an event D ⊂ W ∗ r ( R ) be given, localized in Λ : D ∈ W (Λ ) . Then µ ( S ( s ) D ) + µ ( S ( − s ) D ) − µ ( D ) ≥ . (2 . We introduce the functionals K and L for path andloop configurations : K (Ω ∗ ) = X ω ∗ ∈ Ω ∗ k ( ω ∗ ) , K (Ω ∗ ) = X ω ∗ ∈ Ω ∗ k ( ω ∗ ) , L (Ω ∗ ) = Y ω ∗ ∈ Ω ∗ k ( ω ∗ ) . For a given (large) L we introduce the squareΛ = [ − L, L ] × [ − L, L ] ⊃ Λ (2 . µ ( S ( ± s ) D ) and µ ( D ) as integrals of conditional expec- In short, the paper [12] employs an approach based on sup-norm conditions whereasthe situation under consideration in this paper requires the use of integral-type norms. Acrucial fact is that a Jacobian emerging in the course of the construction has the form(3.23) suitable for our purposes. W (Λ c ): Z W ∗ r ( R ) µ (d Ω ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W r (Λ c ) (cid:17) × Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∈ S ( ± s ) D (cid:17) z K (Ω ∗ Λ ) L (Ω ∗ Λ ) exp (cid:2) − h (Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) (2 . µ ( D ) is recovered at s = 0, with S (0) = Id.)Furthermore, again as in [10]–[12], we employ maps T ± L = T ± L,L ( s ) : W ∗ ( R ) → W ∗ r ( R ). These are applied to the concatenated loop configura-tion (LC) Ω ∗ Λ ∨ Ω ∗ Λ c in the expressions from Eqn (4.3), in the the correspondingcase of shift S ( ± s ). Important properties of maps T ± L are:(i) The maps (Ω ∗ Λ , Ω ∗ Λ c ) T ± L (Ω ∗ Λ , Ω ∗ Λ c ) are one-to-one, and a numberof ‘nice’ properties hold true when the LC Ω ∗ Λ ∨ Ω ∗ Λ c lies in a ‘good’ set G L ⊂ W ∗ r ( R ). (Viz., for Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L the loops from Ω ∗ Λ ∩ W ∗ r (Λ ) willnot interact with loops from Ω ∗ Λ c .) The set G L carries asymptotically a fullmeasure as L → ∞ . See below.(ii) For a ‘good’ LC Ω ∗ = Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L over R , the ‘external’ part Ω ∗ Λ c is preserved under T ± n . In other words, the maps are non-trivial only on thepart Ω ∗ Λ (although the way Ω ∗ Λ is transformed depends upon Ω ∗ Λ c (and on Ω ∗ Λ , of course)). For that reason we will often address T ± L as a ‘tuned’ shift Ω ∗ e Ω ∗ = ( T ± L Ω ∗ Λ ) ∨ Ω ∗ Λ c or, dealing with a pair (Ω ∗ Λ , Ω ∗ Λ c ) ∈ W ∗ (Λ , Λ c ),Ω ∗ Λ e Ω ∗ Λ = T ± L Ω ∗ Λ ∈ W ∗ r (Λ) . (2 . ♯ Ω ∗ Λ = ♯ e Ω ∗ Λ andtransforms a loop ω ∗ ∈ Ω ∗ Λ as ω ∗ e ω ∗ where k ( e ω ∗ ) = k ( ω ∗ ). Consequently,functionals K and L are preserved: K ( e Ω ∗ ) = K (Ω ∗ ) and L ( e Ω ∗ ) = L (Ω ∗ ).Next, for all t ∈ [0 , k ( ω ∗ ) β ], point e ω ∗ ( t ) ∈ R is obtained as a ‘tuned shift’ e ω ∗ ( t ) = ω ∗ ( t ) ± s R ± L h ω ∗ ; t ; { Ω ∗ Λ } ( t ) ∪ { Ω ∗ Λ c } ( t ) i ; (2 . R ± L consists of a loop ω ∗ ∈W ∗ r , a time point t ∈ [0 , k ( ω ∗ ) β ] and the t -section { Ω ∗ Λ } ( t ) ∪ { Ω ∗ Λ c } ( t ) = The symbol used in [10]–[12] is T instead of T . The idea of using maps T ± L goes backto [1] and [9]. Ω ∗ Λ ∨ Ω ∗ Λ c } ( t ) ∈ C r ( R ) of an LC Ω ∗ Λ ∨ Ω ∗ Λ c . Here, as in [14], C ( R ) stands forthe collection of finite or countable (unordered) subsets x ⊂ R (includingthe empty set) and C r ( R ) ⊂ C ( R ) for the collection of subsets x withmin h | x − x ′ | : x, x ′ ∈ x , x = x ′ i ≥ r . (iv) For simplicity, let us omit henceforth the symbols ± whenever pos-sible. The value w L h ω ∗ ; t ; { Ω ∗ Λ } ( t ) ∪ { Ω ∗ Λ c } ( t ) i is a non-negative number.Moreover, when Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L then for ω ∗ ∈ Ω ∗ Λ ∩ W ∗ r (Λ ) and 0 ≤ t ≤ k ( ω ∗ ) β , w L h ω ∗ ; t ; { Ω ∗ Λ } ( t ) ∪ { Ω ∗ Λ c } ( t ) i ≡ , ≤ t ≤ k ( ω ∗ ) β. Consequently, in accordance with (2.5), for ω ∗ ∈ W ∗ r ( x ) with x ∈ Λ and t ∈ [0 , k ( ω ∗ ) β ] the point e ω ∗ ( t ) = ω ∗ ( t ) + s . Therefore, the loops ω ∗ fromΩ ∗ = Ω ∗ Λ ∩ W ∗ r (Λ ) are shifted intact by the amount s under the map (2.4).Consequently, the integral energy h (Ω ∗ ) is not changed under tuned shifts.(v) The set S ( s )( D ∩ G L ) will have a µ -measure close to that of S ( s ) D ;moreover, the probability µ ( S ( s )( D ∩ G L )) will be written in the form µ ( S ( ± s )( D ∩ G L )) = Z W ∗ r ( R ) µ (d Ω ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W r (Λ c ) (cid:17) × Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L ∩ D (cid:17) z K (Ω ∗ Λ ) L (Ω ∗ Λ ) × J ± L (Ω ∗ Λ ∨ Ω ∗ Λ c ) exp (cid:2) − h ( T ± L ( s )Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) (2 . J ± L = J ± L,s gives the Jacobian of transformation T ± L ( s ). Byvirtue of properties above (cf. particularly (i) and (iv)), the impact of tT L upon the energy h ( tT L Ω ∗ Λ | Ω ∗ Λ c ) will be felt through the LC Ω ∗ Λ \ Λ = Ω ∗ Λ ∩W ∗ r (Λ \ Λ ) only. (More precisely, through a LC Ω ∗ Λ \ Λ R ( L ) where Λ R ( L ) =[ − R ( L ) , R ( L )] × and R ( L ) ր ∞ with L . See Eqn (3.2) below.) Essentially,the same remains true about the Jacobian J L (Ω ∗ Λ ∨ Ω ∗ Λ c ).(vi) In fact, a detailed analysis shows that second-order incremental ex-pressions h J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) i / (2 . h h ( T + L ( s )Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L ( s )Ω ∗ Λ | Ω ∗ Λ c ) − h (Ω ∗ Λ | Ω ∗ Λ c ) i (2 . Theorem 2.2.
For any δ > there exists L ∗ = L ∗ ( δ ) > such that for L ≥ L ∗ (A) µ ( G L ) = Z W ∗ r ( R ) µ (dΩ ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W ∗ r (Λ c ) (cid:17) × Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L (cid:17) × z K (Ω ∗ Λ ) L (Ω ∗ Λ ) exp (cid:2) − h (Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) ≥ − δ. (2 . The probabilities µ ( S ( ± s )( D ∩ G L )) are represented in the form (2.6)with the following properties: ∀ Ω ∗ Λ ∈ W ∗ r (Λ) , Ω ∗ Λ c ∈ W ∗ r (Λ c ) with Ω ∗ Λ ∨ Ω ∗ Λ c ∈G L ; (Ca) h J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) i / ≥ − δ ; (Cb) h ( T + L ( s )Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L ( s )Ω ∗ Λ | Ω ∗ Λ c ) − h (Ω ∗ Λ | Ω ∗ Λ c ) ≤ δ . The proof of Theorem 2.2 is carried on in the next sections.
Remark.
It is the pair of inequalities (Ca), (Cb) (together with thedefinition of the ‘good’ set G L ) where one crucially uses the fact that thephysical dimension of the system equals 2.We now show how to deduce the statement of Theorem 2.1 from that ofTheorem 2.2. Owing to Theorem 2.2 (A), (B), we can write:the LHS of (2.1) + 3 δ ≥ µ ( S ( s )( D ∩ G L )) + µ ( S ( − s )( D ∩ G L )) − µ ( D ∩ G L )= Z W ∗ r ( R ) µ (d Ω ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W r (Λ c ) (cid:17) × Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L ∩ D (cid:17) z K (Ω ∗ Λ ) L (Ω ∗ Λ ) × n J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) exp (cid:2) − h ( T + L Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) + J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) exp (cid:2) − h ( T − L Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) − (cid:2) − h (Ω ∗ Λ | Ω ∗ Λ c ) (cid:3)o . (2 . Z W ∗ r ( R ) µ (d Ω ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W r (Λ c ) (cid:17) Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L ∩ D (cid:17) × z K (Ω ∗ Λ ) L (Ω ∗ Λ ) (cid:16)h J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) i / × exp n − (cid:2) h ( T + L Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L Ω ∗ Λ | Ω ∗ Λ c ) (cid:3)(cid:14) o − exp (cid:2) − h (Ω ∗ Λ | Ω ∗ Λ c ) (cid:3)(cid:17) . (2 . − δ ) e − δ/ − Z W ∗ r ( R ) µ (d Ω ∗ Λ c ) (cid:16) Ω ∗ Λ c ∈ W r (Λ c ) (cid:17) × Z W ∗ r (Λ) dΩ ∗ Λ (cid:16) Ω ∗ Λ ∨ Ω ∗ Λ c ∈ G L ∩ D (cid:17) × z K (Ω ∗ Λ ) L (Ω ∗ Λ ) exp (cid:2) − h (Ω ∗ Λ | Ω ∗ Λ c ) (cid:3) = 2[(1 − δ ) e − δ/ − µ ( G L ∩ D ) ≥ − δ ) e − δ/ − − δ ) . (2 . δ can be made arbitrarily small, we obtain the inequality (2.2).
3. Definition of transformations T ± L As was said earlier, the maps Ω ∗ T ± L Ω ∗ Λ ∨ Ω ∗ Λ c are determined bytransforming the t -sections { T ± L Ω ∗ Λ } ( t ) of the LC Ω ∗ Λ , for each t ∈ [0 , β ].Denoting by T ± L = T ± L ( ± s ) the map acting on CCs from C r (Λ), we can write: { T ± L Ω ∗ Λ ∨ Ω ∗ Λ c } ( t ) = { T ± L Ω ∗ Λ } ( t ) ∨ Ω ∗ Λ c ( t )= (cid:0) T ± L [ { Ω ∗ Λ } ( t )] (cid:1) ∨ Ω ∗ Λ c ( t ) . (3 . t -section { Ω ∗ Λ } ( t ) istransformed depends on { Ω ∗ Λ c } ( t ), although { Ω ∗ Λ c } ( t ) itself is not movingwhen Ω ∗ ∈ G L .More precisely, set: R ( L ) = (cid:0) log log L (cid:1) / , Λ R ( L ) = [ − R ( L ) , R ( L )] × , (3 . − L, L ] × where L = L − L / . (3 . , Λ R ( L ) , Λ and Λ satisfiesΛ ⊂ Λ R ( L ) ⊂ Λ ⊂ Λ . The transformed CC T ± L [ { Ω ∗ Λ } ( t )] ∈ C r (Λ) is formed by points e ω ∗± L ( lβ + t )obtained, as a result of shifts in the (positive) horizontal direction, from thepoints ω ∗ ( lβ + t ) where t ∈ [0 , β ], l = 0 , . . . , k ( ω ∗ ) − ω ∗ ∈ Ω ∗ Λ : e ω ∗± L ( lβ + t ) = ω ∗ ( lβ + t ) ± p L ( ω ∗ ( lβ + t )) s. (3 . p L ( ω ∗ ( lβ + t )) ≥ { Ω ∗ Λ } ( t ) and { Ω ∗ Λ c } ( t ) and are constructed recursively; cf. [11]. When ω ∗ ( lβ + t ) ∈ Λ \ Λ,we have that p L ( ω ∗ ( lβ + t )) = 0 and e ω ∗± L ( lβ + t ) = ω ∗ ( lβ + t ) . In other words, a loop ω ∗ ∈ Ω ∗ is affected only at points ω ∗ ( t ) lying in Λ.In the course of construction of values p L ( ω ∗ ( lβ + t )), we employ thefunction u ∈ [0 , ∞ ) τ L ( u ) determined as follows: τ L ( u ) = , ≤ u ≤ R ( L ) , − Q ( u − R ( L )) Q ( L − R ( L )) , R ( L ) ≤ u ≤ L, , u ≥ L, (3 . Q ( u ) = Z u q ( v )d v, with q ( v ) = 11 ∨ v | log v | . (3 . p ( ω ∗ ( lβ + t )) = p L ( ω ∗ ( lβ + t )) are related to results of aseries of minimizations, over points ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) ∩ Λ, of subse-quently introduced functions e t ( j ) ( · ; t ) = e t ( j ) L ( · ; t ). Here j runs from 0 to ♯ (cid:0) { Ω ∗ } ( t ) ∩ Λ (cid:1) and the functions are y ∈ R e t ( j ) ( y ; t ) ∈ [0 , , ≤ j ≤ X ω ∗ ∈ Ω ∗ X ≤ l
0) where s ∈ [0 , / y ∈ R e m (1) ( y ; t ) indicates by howmuch a particle at point y could be moved when we take into account theparticles placed at points ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) ∩ Λ c (which do not move)and the particles placed at points ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) ∩ Λ (which aremoved by p ). Consequently, e t (2) ( y ; t ) indicates how much a movement byquantity e t (0) ( y ; t ) should be reduced in presence of hard-core particles atpoints ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) ∩ Λ c and ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) ∩ Λ.Next, we minimise the function e t (2) ( · ; t ) over the t -section( { Ω ∗ } ( t ) \ { Ω } ( t )) ∩ Λ and, like before, set: p = min he t (2) L ( y ; t ) : y ∈ ( { Ω ∗ } ( t ) \ { Ω } ( t )) ∩ Λ i ,P κ +1 = arg min he t (2) ( y ; t ) : y ∈ ( { Ω ∗ } ( t ) \ { Ω } ( t )) ∩ Λ i . (3 . { Ω ∗ } ( t ) \ { Ω } ( t )) ∩ Λ, we list all these points: P κ +1 , . . . , P κ + κ (in anyorder). As earlier, the value p is assigned to each of those points as p ( P j ): p ( ω ∗ ( lβ + t )) = p , if ω ∗ ∈ Ω ∗ , ≤ l < k ( ω ∗ ) ,ω ∗ ( lβ + t ) ∈ Λ and e t (2) ( ω ∗ ( lβ + t ); t ) = p . And so on: this procedure is iterated until we exhaust all points in { Ω ∗ } ( t ) ∩ Λ. (Recall, their number and their positions vary with t ∈ [0 , β ].)At the end, we obtain a resulting function e t = e t L ( · ; t ): y ∈ R e t ( y ) where e t ( y ) = e t (0) ( y ) ∧ e m ( y ) (3 . e m ( y ) = e m L ( y ; t ) = ^ ω ∗ ( lβ + t ) ∈ { Ω ∗ } ( t ) m ω ∗ ( lβ + t ) , p ( ω ∗ ( lβ + t )) × s ( y ) . (3 . p ( ω ∗ ( lβ + t )) = 0 when ω ∗ ( lβ + t ) ∈ Λ c . Observe that e t L ( y ; t ) = 0 for y ∈ Λ c and e t L ( y ; t ) = 1 for y ∈ Λ R ( L ) . (3 . J ± L (Ω ∗ Λ ∨ Ω ∗ Λ c ) of the transform T ± L turns out to be of theform: J ± L (Ω ∗ Λ ∨ Ω ∗ Λ c ) = exp " Z β d t X ω ∗ ∈ Ω ∗ Λ ∨ Ω ∗ Λ c × X ≤ l Consider the events L (1) L = { Ω ∗ ∈ W ∗ r ( R ) : α R \ Λ ( ω ∗ ) = 1 ∀ ω ∗ ∈ Ω ∗ with x ( ω ∗ ) ∈ Λ c } (3 . . nd L (2) L = { Ω ∗ ∈ W ∗ r ( R ) : α Λ R ( L ) ( ω ∗ ) = 1 ∀ ω ∗ ∈ Ω ∗ with x ( ω ∗ ) ∈ Λ } (3 . . In other words, (a) for Ω ∗ ∈ L (1) L , every loop ω ∗ from Ω ∗ which startsat a point x ( ω ∗ ) outside square Λ does not reach square Λ , while (b) for Ω ∗ ∈ L (2) L , every loop ω ∗ from Ω ∗ Λ (which starts in Λ ) does not leave square Λ R ( L ) . Then, under condition (1.4) , lim L →∞ µ L ( L (1) L ) = lim L →∞ µ L ( L (2) L ) = 1 , (3 . ∀ µ ∈ K ( z, β ) . Proof of Lemma L →∞ µ L ( L (1) L ) = 1. At first we write µ ( W ∗ r \ L (1) L )= µ ( ∃ at least one loop ω ∗ with x ( ω ∗ ) ∈ Λ c reaching Λ) ≤ Z µ (d Ω ∗ ) X ω ∗ ∈ Ω ∗ Λc (cid:16) ω ∗ ( t ) ∈ Λ for some t ∈ [0 , k ( ω ∗ ) β ] (cid:17) . By virtue of the Campbell theorem, the last integral equals Z d ω ∗ ρ ( ω ∗ ) (cid:16) x ( ω ∗ ) ∈ Λ c but ω ∗ ( t ) ∈ Λ for some t ∈ [0 , k ( ω ∗ ) β ] (cid:17) . By the Ruelle bound (cf. Eqns (2.3.18. I )–(2.3.20. I )) this does not exceed Z Λ c d x Z W ∗ ( x ) P x (d ω ∗ ) ρ k ( ω ∗ ) k ( ω ∗ ) (cid:16) ω ∗ ( t ) ∈ Λ for some t ∈ [0 , k ( ω ∗ ) β ] (cid:17) . Next, we observe that the loop ω ∗ with the endpoint x = ( x , x ) ∈ Λ c (i.e., with max | x j | m ≥ L ) can reach Λ only if at least one of its one-dimensional components (i.e., a scalar Brownian bridge with the endpoint x j , j = 1 or 2) deviates from its origin by at least ( | x j | − L ) + L / . Therefore,the last displayed expression is upper-bounded by2 × X k ≥ ρ k k √ πβk Z ∞ L / d x exp (cid:2) − x / (2 kβ ) (cid:3) ≤ X k ≥ ρ k √ π k exp [ − L / / (2 kβ )] L / / √ kβ + p L / / ( kβ ) + 4 /π . (3 . B ( t ) withendpoints and y ≥ ∀ A > yP βk , y n sup [ B ( t ) : 0 ≤ t ≤ βk ] ≥ A o = 1 √ πβk e − (2 A − y ) / (2 βk ) (3 . . ∀ A ∈ (0 , ∞ ), e − A / A + √ A + 2 ≤ Z ∞ A e − t / d t ≤ e − A / A + p A + /π . (3 . . L → ∞ . Thiscompletes the proof of Lemma 3.1.In what follows we will assume that an LC Ω ∗ lies in L L . Together with(3.22) this will imply that the loops ω ∗ ∈ Ω ∗ with x ( ω ∗ ) ∈ Λ c remainsunaffected by transformations T ± ( s ). 4. Estimates for the Jacobians To guarantee assertions (A) and (Ca) of Theorem 2.2 we need to securethat the good set G L carries a large measure and contains only those LCs Ω ∗ ∈ W ∗ ( R ) for which the expression J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) can beappropriately controlled. To this end, consider a random variable Σ J ( Ω ∗ ) =Σ J L ( Ω ∗ ) given by the RHS of (3.24):Σ J ( Ω ∗ ) : Ω ∗ Z β d t X x ∈{ Ω ∗ } ( t ) h(cid:0) ∂ e t L (cid:1) ( x ; t ) i = X ω ∗ ∈ Ω ∗ Z β d t X ≤ l The mean value of Σ (1) is assessed as follows: Z W ∗ r ( R ) µ (d Ω ∗ )Σ (1) L ( Ω ∗ ) ≤ C γ ( L ) (4 . where C ∈ (0 , ∞ ) is a constant and the quantity γ ( L ) is defined as follows: γ ( L ) := Z Λ q ( | x | m − R ( L ) − ǫ − a/ Q ( L − R ( L ) − ǫ − a/ d x, with lim L →∞ γ ( L ) = 0 . (4 . roof of Proposition Z W ∗ r ( R ) µ (d Ω ∗ )Σ (1) L ( Ω ∗ )= Z W ∗ d ω ∗ ρ ( ω ∗ ) Z k ( ω ∗ ) β τ L ( | ω ∗ ( t ) | m )d t . (4 . Z W ∗ d ω ∗ ρ k ( ω ∗ ) k ( ω ∗ ) Z k ( ω ∗ ) β τ L ( | ω ∗ ( t ) | m )d t . (4 . x ( ω ∗ ) ∈ Λ R ( L ) , we estimate Z L ( | ω ∗ ( t ) | m ) ≤ Q ( L − R ( L ) − ǫ − a/ ; (4 . Z W ∗ d ω ∗ ( x ( ω ∗ ) ∈ Λ R ( L ) ) ρ k ( ω ∗ ) k ( ω ∗ ) Z k ( ω ∗ ) β τ L ( | ω ∗ ( t ) | m )d t does not exceed (2 R ( L )) [ Q ( L − R ( L ) − ǫ − a/ X k ≥ ( kβ ) ρ k (2 πkβ ) k = (cid:0) log log L (cid:1) / π [ Q ( L − R ( L ) − ǫ − a/ X k ≥ ρ k k< ρ (cid:0) log log L (cid:1) / π (1 − ρ ) [ Q ( L − R ( L ) − ǫ − a/ . (4 . ω ∗ reaches Λ R ( L ) . For given x Λ R ( L ) and ω ∗ ∈ W ∗ ( x ) this can occur wheneither (i) k ( ω ∗ ) is large – say, k ( ω ∗ ) > (cid:2) | x | m − R ( L ) (cid:3)(cid:14) k ( ω ∗ ) ≤ (cid:2) | x | m − R ( L ) (cid:3)(cid:14) ω ∗ deviates from x , in the max-distance, by at least | x | m − R ( L ). Then thecorresponding part of expression (4.14) Z W ∗ d ω ∗ ( x ( ω ∗ ) Λ R ( L ) ) × ( ω ∗ ( t ) ∈ Λ R ( L ) for some t ∈ [0 , k ( ω ∗ ) β ]) × ρ k ( ω ∗ ) k ( ω ∗ ) Z k ( ω ∗ ) β τ L ( | ω ∗ ( t ) | m )d t 20s upper-bounded by Z R d x (cid:26) X k ≥| x | m / ( kβ ) ρ k (2 πkβ ) k + X ≤ k ≤| x | m / ( kβ ) ρ k k Z W kβ (0) P kβ (d ω ∗ ) × (cid:16) max (cid:2) | ω ∗ ( t ) | m : 0 ≤ t ≤ kβ (cid:3) > | x | m (cid:17)(cid:27) . (4 . X k ≥| x | m / ρ k πk ≤ ρ | x | m / π (1 − ρ ) , and its contribution into the integral Z R d x does not exceed a constant. Toestimate the second sum, one can use the inequalities (3.27.1,2). This yields: X ≤ k ≤| x | / ( kβ ) ρ k k Z W kβ (0) P kβ (d ω ∗ ) × (cid:16) max (cid:2) | ω ∗ ( t ) | m : 0 ≤ t ≤ kβ (cid:3) > | x | m (cid:17) ≤ | x | m + p | x | + 4 /π e −| x | m /β π (1 − ρ ) . (4 . Z R d x also does not exceed aconstant.More generally, for a given r > R ( L ) we consider the contribution into(4.14) from loops ω ∗ with x ( ω ∗ ) Λ r such that | ω ∗ ( t ) | m = r for some t ∈ [0 , k ( ω ∗ ) β ]. Repeating the above argument, we conclude that this con-tribution again is less than or equal to a constant times τ L ( r ). Note that allconstants can be made uniform; this implies that(4 . ≤ C [ Q ( L − R ( L ) − ǫ − a/ × (cid:20)(cid:0) log log L (cid:1) / + Z LR ( L ) [ q ( r − R ( L ) − ǫ − a/ d r (cid:21) . (4 . C γ ( L )) goes to 0as L → ∞ . This finishes the proof of Proposition 4.1.21t is instructive to note that the relation (4.9) does not require a smallnessfor ǫ .We now pass to random variable Σ (2) L = Σ (2 , L + Σ (2 , L . Proposition 4.2. For ǫ small enough, lim L →∞ Z W ∗ r ( R ) µ (d Ω ∗ )Σ (2) L ( Ω ∗ ) = 0 . (4 . Proof of Proposition C ) timesthe sum I , + I , . Here the term I , = I , L is specified as follows: I , = Z β d t Z d ω ∗ X ≤ l,l ′′ 5. Estimates for the change in the energy.Concluding remarks In this section we assess the expressionexp h h ( T + L ( s )Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L ( s )Ω ∗ Λ | Ω ∗ Λ c ) − h (Ω ∗ Λ | Ω ∗ Λ c ) i (cf. (2.8)). The argument is based on the same idea as in Sect 8.6 of [11](again we partially borrow the system of notation from there). In the courseof the argument we will produce a further (and final) specification of the set G L ⊂ W ∗ r ( R ) of good LCs. Namely, given Ω ∗ ∈ W ∗ r ( R ), we set, as before, Ω ∗ Λ = { ω ∗ ∈ Ω ∗ : x ( ω ∗ ) ∈ Λ } , Ω ∗ Λ c = { ω ∗ ∈ Ω ∗ : x ( ω ∗ ) ∈ Λ c } . Then write h ( T + L ( s ) Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L ( s ) Ω ∗ Λ | Ω ∗ Λ c ) − h ( Ω ∗ Λ | Ω ∗ Λ c )= Z β d t n E [ { T + ( s ) Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )]+ E [ { T − ( s ) Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )] − E [ { Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )] o . (5 . E [ { T ± ( s ) Ω ∗ } Λ ( t ) |{ Ω ∗ Λ c } ( t )] is defined as the sum12 X x,x ′ ∈{ Ω ∗ Λ } ( t ) V (cid:0) | x ± s e t ( x ) − x ′ ∓ s e t ( x ′ ) | (cid:1) + X x ∈{ Ω ∗ Λ } ( t ) x ′ ∈{ Ω ∗ Λc } ( t ) V (cid:0) | x ± s e t ( x ) − x ′ ∓ s e t ( x ′ ) | (cid:1) (5 . E [ { Ω ∗ } ( t ) | Ω ∗ Λ c ( t )] is obtained by omitting the terms containing theshift-vector s . Cf. (2.3.11. I )–(2.3.12. I ). Recall, our aim is to guarantee thaton the good set G L , the absolute values of the variables Σ ( i ) L ( Ω ∗ ) are small.Two straightforward bounds turn out to be helpful: (cid:12)(cid:12)(cid:12) E [ { T + ( s ) Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )] + E [ { T − ( s ) Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )] − E [ { Ω ∗ Λ } ( t ) |{ Ω ∗ Λ c } ( t )] (cid:12)(cid:12)(cid:12) ≤ V (2) s × ( X x,x ′ ∈{ Ω ∗ Λ } ( t ) + X x ∈{ Ω ∗ Λ } ( t ) x ′ ∈{ Ω ∗ Λc } ( t ) ) | e t ( x ) − e t ( x ′ ) | ( | x − x ′ | ≤ R ) (5 . | e t ( x ) − e t ( x ′ ) | ≤ | e t ( x ) − τ L ( | x | m ) | + | τ L ( | x | m ) − τ L ( | x ′ | m ) | + | τ L ( | x ′ | m ) − e t ( x ′ ) | . (5 . (cid:12)(cid:12) h ( T + L ( s ) Ω ∗ Λ | Ω ∗ Λ c ) + h ( T − L ( s ) Ω ∗ Λ | Ω ∗ Λ c ) − h ( Ω ∗ Λ | Ω ∗ Λ c ) (cid:12)(cid:12) ≤ V (2) s Z β d t ( X x,x ′ ∈{ Ω ∗ Λ } ( t ) + X x ∈{ Ω ∗ Λ } ( t ) x ′ ∈{ Ω ∗ Λc } ( t ) ) ( | x − x ′ | ≤ R ) × h | e t ( x ) − τ L ( | x | m ) | + | τ L ( | x | m ) − τ L ( | x ′ | m ) | i =: Σ (3) L ( Ω ∗ ) + Σ (4) L ( Ω ∗ ) (5 . (3) L and Σ (4) L emerge when we expand the sum of squares inthe parentheses.As above, we will try to make sure that the expected values of variablesΣ (3) L and Σ (4) L vanish as L → ∞ : Lemma 5.1. lim L →∞ Z µ (d Ω ∗ )Σ (3) L ( Ω ∗ ) = lim L →∞ Z µ (d Ω ∗ )Σ (4) L ( Ω ∗ ) = 0 . (5 . roof of Lemma (4) L . It is instructive to expandΣ (4) L ( Ω ∗ ) = Σ (4 , L ( Ω ∗ ) + Σ (4 , L ( Ω ∗ ) . Here Σ (4 , L ( Ω ∗ ) gives a single-loop contribution to Σ (4) L ( Ω ∗ ) while Σ (4 , L ( Ω ∗ )yields a contribution from pairs of loops:Σ (4 , L ( Ω ∗ ) = X ω ∗ ∈ Ω ∗ Λ Z β d t ( X ≤ l Given Ω ∗ ∈ G L , the transformations T ± L ( s ) : Ω ∗ e Ω ∗ ∈ W ∗ r ( R ) possess the following properties: (i) The maps T ± ( s ) are measurable and − . (ii) e Ω ∗ Λ c = Ω ∗ Λ c and e Ω ∗ Λ = S ( s ) Ω ∗ Λ . Moreover, there exists a − correspondence between the loops e ω ∗ ∈ e Ω ∗ Λ and ω ∗ ∈ Ω ∗ Λ such that e ω ∗ isobtained as a deformation of ω ∗ via tuned shifts of t -sections, in the mannerdescribed in Section . In particular, k ( e ω ∗ ) = k ( ω ∗ ) . (iii) The equality (2.6) holds true, where the expression h J + L (Ω ∗ Λ ∨ Ω ∗ Λ c ) J − L (Ω ∗ Λ ∨ Ω ∗ Λ c ) i / is close to uniformly in Ω ∗ for L large. (iv) The quantity (2.8) is close to uniformly in Ω ∗ when L is largeenough. The assertion of Theorem 2.1 then follows.29 cknowledgments This work has been conducted under Grant 2011/20133-0 provided by theFAPESP, Grant 2011.5.764.35.0 provided by The Reitoria of the Universidadede S˜ao Paulo, Grants 2012/04372-7 and 11/51845-5 provided by the FAPESP.The authors express their gratitude to NUMEC and IME, Universidade deS˜ao Paulo, Brazil, for the warm hospitality. References [1] J. Fr¨ohlich and C. Pfister. On the absence of spontaneous symmetrybreaking and of crystalline ordering in two-dimensional systems. Com-mun. Math. Phys. , , 1981, 277–298[2] H.-O. Georgii, Gibbs Measures and Phase Transitions. Walter deGruyter, Berlin, 1988[3] K. Itˆo, H.P. McKean. Diffusion Processes and Their Sample Paths.Springer, Berlin, 1996[4] S. Karlin, H.M. Taylor. A Second Course in Stochastic Processes. Aca-demic Press, New York, 1981[5] M. Kelbert, Y. Suhov. A quantum MerminWagner theorem for quan-tum rotators on two-dimensional graphs. Journ. Math. 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