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Uniqueness for Gibbs States of Quantum Lattices in Small Mass Regime Sergio Albeverio

Abteilung fur Stochastik, Institut fur Angewandte Mathematik, Universitat Bonn, D 53115 Bonn (Germany); Forschungszentrum BiBoS, Bielefeld (Germany); SFB 237 (Essen{Bochum{Dusseldorf) (Germany); CERFIM, Locarno e-mail albeverio@uni{bonn.de

Yuri Kondratiev

Abteilung fur Stochastik, Institut fur Angewandte Mathematik, Universitat Bonn, D 53115 Bonn (Germany); Forschungszentrum BiBoS, D 33615 Bielefeld (Germany); Institute of Mathematics, Kiev (Ukraine) e-mail [email protected]

Yuri Kozitsky

Institute of Mathematics, Maria Curie-Sklodowska University PL 20-031 Lublin (Poland); Institute for Condensed Matter Physics, Lviv (Ukraine) e-mail [email protected]

Michael Rockner

Fakultet fur Mathematik, Universitat Bielefeld, D 33615 Bielefeld (Germany) e-mail [email protected]

1

Abstract

The model of interacting quantum particles performing anharmonic one{dimensional oscillations around their unstable equilibrium positions, which form the d{ dimensional simple cubic lattice ZZ d , is considered. For this model, it is proved that for every xed value of temperature ?1 , there exists a positive m ( ) such that for the values of the physical mass of the particle m 2 (0; m ( )), the set of tempered Gibbs states consists of exactly one element.

1 Introduction In this paper we consider the following quantum lattice model. To each point of the d{dimensional cubic lattice ZZ d there is attached a quantum particle with the physical mass m which has an unstable equilibrium position at this point. Such particles perform one{dimensional oscillations around their equilibrium positions and interact via an attractive potential. Similar objects have been studied for many years as quite realistic models of a crystalline substance undergoing structural phase transitions (see e.g. [10]). In our case the phase transition is connected with the appearance of macroscopic displacements of the particles (a long{range order) if the dimension d, the mass m, temperature ?1 , and the parameters of the potential energy satisfy certain conditions. These phenomena were studied mathematically in various papers, see e.g. [8], [13], [20], [23]. The essential problem in this context is to understand the role of quantum eects in the phase transitions in such models. As it was justi ed on the physical level [25] and observed experimentally (see [28] and Ch.2.5.4.3 of book [10]), the quantum eects may suppress the long{range ordering. This was proved in paper [29]. Later on it was shown in [3], [4] that not only the long{range order but also any critical anomaly of the displacements of particles from the equilibrium positions are suppressed if the model is "strongly quantum", which may occur in particular if the particle's mass is small. Another important problem of the mathematical theory of models which exhibit such phenomena is the description of their Gibbs states. The uniqueness for such a state implies the absence of all critical anomalies and all the more of the long{ range ordering. Therefore, one may expect that the "strong quantumness" of the model should imply the uniqueness of its Gibbs state. So far, the mentioned uniqueness for the model considered in this work was proved to occur under conditions which do not contain the physical mass of the particle and thus are irrelevant to the "quantumness" of the model. 2

It was done in paper [1] on the base of the logarithmic Sobolev inequalities. An alternative approach to the construction of the Gibbs states for such models lies in the use of cluster expansions. We note here the papers [7], [22] where models of this type with "light" particles were studied by means of dierent versions of cluster expansions. In particular, in paper [7] the existence of temperature Gibbs states was shown by a proof of the convergence of cluster expansions at xed temperature and for small masses. In paper [22] for small values of the particle's mass, the convergence of corresponding cluster expansions was proved for all values of temperature including zero. As a result, the existence of temperature and ground states was proved and a number of the properties of these states have been described. It should be pointed out that for systems with unbounded oscillations, as in the case considered in this work, it is impossible to recover the uniqueness of the state from the convergences of the cluster expansion. However, this is possible in the case of quantum particles which move on compact manifolds. Here the uniqueness of the Euclidean Gibbs measure was shown [6] by using the convergence of the cluster expansions. In this paper we present the proof of the uniqueness for the Gibbs states in the mentioned model for small values of the physical mass of the particle. We follow paper [1] and apply Dobrushin's technique up to the stage where the logarithmic Sobolev inequalities were used to estimate the variances of certain moments, which was crucial in satisfying Dobrushin's uniqueness criterion. Instead, we reduce this problem to the study of spectral properties of the one{particle Hamiltonian which describes the individual oscillations. This study is performed by means of a version of the lattice approximation technique known in the Euclidean quantum eld theory [26]. A similar approach has already been used in our works [3], [4]. It enables us to involve the physical mass into the consideration, so that it appears explicitly in the uniqueness conditions. Unfortunately, it is impossible (at least so far) to remove the temperature dependence from these conditions, in contrast to the proof for the suppression of critical uctuations in papers [3], [4].

2 Temperature loop formalism for Gibbs states As in [1], [3] we consider a translation invariant system of interacting quantum particles performing one{dimensional oscillations, described by the fol-

3

lowing formal Hamiltonian

H=

X

k

Hk + J4

X

(xj ? xk )2 ;



(2.1)

where the indices j , k run over the d{dimensional lattice ZZ d  IRd with the Euclidean distance j : j. The second sum in (2.1) is taken over the nearest neighbours, that is restricted to the pairs with jj ? kj = 1. It describes ferroelectric (J > 0) interactions of the particles whereas the rst term is a sum of one{particle Hamiltonians d2 + 1 x2 + V (x ); (2.2) Hk = ? 21m dx k 2 2 k where m > 0 is the physical mass of the particle and (2.3) V (x) = P2p (x) ? 21 ax2 p X P2p (x) = bl x2l ; p  2: l=2

Here we suppose: a > 0, bp > 0, and bl  0 for all l  p. For d  2, the model has a critical point and long{range order behaviour if its parameters satisfy certain conditions (see e.g. [8], [20]). Each Hamiltonian Hk is an essentially self{adjoint lower semi{bounded operator on the complex Hilbert space L2 (IR; dx) (see e.g. [9]). We will study temperature Euclidean Gibbs states of the system, i.e. Gibbs states constructed using a probability measure on a loop space, see [2], [19], [16] for this approach. Thus the Gibbs state at temperature ?1 is described by the Euclidean (Gibbs) measure on the space of periodic trajectories (loops) taking values in IRZZ d . Consider the space of continuous periodic functions

C (I ) def = f! 2 C ([0; ] ! IR) j !(0) = !( )g and let L2 (I ) be the closure of C (I ) in the Hilbert space of real{valued functions on [0; ] which are square integrable with respect to Lebesgue measure. Here I is the interval [0; ] with the ends identi ed (i.e. a circle of length ). The temperature loop space (TLS) is

def = C (I )ZZ d = f! = (!k )k2ZZ d j !k 2 C (I )g: 4

(2.4)

For ! 2 and   ZZ d , we set ! = (!k )k2 . We shall also consider

; def = f! = (!k )k2 j !k 2 C (I )g;

(2.5)

de ned for all subsets   ZZ d . The spaces ;, are equipped with the product topology and with the {algebra B( ;) generated by the cylinder sets f! j (!k )k2
2 B
;
2 L g; where L is the set of nite subsets of ZZ d , B
is a direct product of Borel subsets fBk  C (I ) j k 2
g. Consider again L2 (I )def = H equipped with the usual inner product (:; :) and norm k:k . De ne on H a strictly positive trace class operator S = (?m
+ 1)?1 , where
stands for the Laplace{ Beltrami operator on the circle I , and m is the physical mass of the particle. In this setting one can de ne a Gaussian measure on H which has zero mean and S as a covariance operator. This measure is uniquely determined by its Fourier transform Z (2.6) exp(i('; !) ) (d!) = exp(? 12 ('; S ') ); ' 2 H : H Actually, the set of continuous loops is a set of full measure (i.e. (C (I )) = 1), and the measure is canonically generated by the oscillator bridge process of length [27]. By means of this Gaussian measure one can de ne Z (2.7)  (d!) = 1 expf? V (!( ))d g (d!)

X I as a probability measure on C (I ). Here X is a normalizing constant and V is given by eq.(2.3). The measure  has all moments, that is

Z

C (I )

k!klC (I )  (d!) < 1; 8l 2 IN:

(2.8)

For every  2 and each nite subset   ZZ d , we de ne the Gibbs measure in , subject to a chosen  , as the probability measure Y (2.9)  ; (d! j ) def = Z 1 ( ) expf? ; (! j )g  (d!k ); ; k2 5

on the loop space ;. Here  ; (!8 j )

Z def = J

< X

(!j ( ) ? !k ( ))2 + 2

4 I :j;k2

X

j 2;k2c

9 =

(2.10)

(!j ( ) ? k ( ))2 ; d;

is the pair interaction in , prescribed by the Hamiltonian (2.1), under the external boundary condition = ZZ d n : c 2 C (I )c ; c def All sums in (2.10) are taken under the condition jj ? kj = 1, and Z

Y Z ; ( ) def = expf? ;(! j )g  (d!k );

; k2

(2.11)

is the nite volume partition function subject to the external boundary condition c . For B 2 B( ) and ! 2 , let 11B (!) take values 1 resp. 0 if ! belongs resp. does not belong to B . Then one can introduce a family of probability kernels f ; j  2 Lg between ( ; B( )) and ( ; B( )) by

 ;(B j ) def =

Z

;

11B (!  c ) ; (d! j ):

(2.12)

These kernels satisfy the consistency conditions which are set as follows (see e.g. [15]). For arbitrary pair 0  , each B 2 B( ), and  2 , =  ; ;0 (B j ) def

Z

 (d!j ) ;0 (B j!)

; =  ; (B j ):

(2.13)

De nition 2.1 A probability measure  on ( ; B( )) is said to be an Euclidean Gibbs state of the lattice model (2.1), (2.2), (2.3) at temperature ?1 if it satis es the Dobrushin{Lanford{Ruelle (DLR) equilibrium equations:   ; =  ; that is Z  (d!) ; (B j!) =  (B ); (2.14)



for all  2 L and B 2 B( ).

6

In order to exclude the states with no physical relevance we impose some a priori conditions restricting the growth of the sequences of moments (see [1], [17]). To this end we introduce

S 0(ZZ d) def = ind plim l2 (ZZ d); 2IN ?p where

l?2 p def =

8 < :

x 2 IR

ZZ d

j

9 =

X

k2ZZ d

(1 + jkj)?p x2k < 1 ; :

(2.15)

We then have the following: De nition 2.2 The class of tempered Gibbs measures G for the model considered consists of the Euclidean Gibbs states de ned above which are such that their moment sequences obey the condition (< k!k k > )k2ZZ d 2 S 0 (ZZ d );

(2.16)

where k : k is the norm in the Hilbert space H = L2 (I ). Here and further on we write Z

< f >= fd

(2.17)

if the integral makes sense. For the existence problem for Euclidean Gibbs measures we refer to [1] and the references therein. Here we only remark that the class G is actually nonempty. Our main aim in this paper is to re ne the uniqueness condition obtained in [1] to involve the quantum nature of the model.

3 Uniqueness theorem We start this section with the formulation of the uniqueness theorem. Theorem 3.1 For the model considered, let the class of tempered Gibbs measures G be as in De nition 2.2. Then for every , there exists a positive m ( ) such that for all values of the physical mass m 2 (0; m ( )), the class G consists of exactly one element, that is jG j = 1.

Remark 3.1 The value of m depends also on the concrete choice of the function V (2.3) as well as on the value of the interaction intensity J . 7

Following the scheme used in [1] we metrize the space of measures. Let (X ; ) be a complete separable metric space, M be the set of all probability measures on B(X ), and 

M1 def = 2Mj



Z

X

(x; x0 )(dx) < 1 ;

(3.1)

for some x0 2 X . Further, let Lip(X ) denote the set of all Lipschitz functions f : X ! IR, and   j f ( x ) ? f ( y ) j def [f ]Lip = sup (3.2) (x; y) j x; y 2 X ; x 6= y ; Lip1 (X ) def = ff : X ! IR j [f ]Lip  1g : For given 1 , 2 2 M1 , we set

R(1 ; 2 )

def =

sup

 Z

X

f (x)1 (dx) ?

Z

X

(3.3)

(3.4)  f (x)2 (dx) j f 2 Lip1(X ) :

A signi cant role in the proof will be played by Dobrushin's matrix. It is de ned in terms of the measures  ; given by eqs. (2.9) { (2.11) with the one{point subset  = fkg. To simplify notations we set

fjgc = jc ;  ;fkg (B j) = k (B j): Dobrushin's matrix elements Ck;j ; k; j 2 ZZ d are:

(3.5) )

(

Ckj def = sup R(kk(:j?);kk(:j)) j ;  2 ; jc = jc : j j

(3.6)

We will use them to check Dobrushin's uniqueness criterion [11], [12], [15], [21]. Proposition 3.1 (Dobrushin's uniqueness criterion) For the model considered, let the following condition be satis ed 8
 x ; ') = J f< f (:; ') > x ? < f > x < (:; ') > x g = J Cov x (f; (:; ') ): (3.13) 9

Our next aim is to nd an estimate of this derivative which does not depend on x 2 H . We start by using the Schwarz inequality q

j(rx < f >x ; ') j  J Varx f Varx (:; ') ; p

(3.14)

where Z

Var x f =

Z

?

H

f 2 (!) x (d!) Z

H H Z

Var x (:; ') =

H

Z

?

f (!)f (!0 ) x(d!) x (d!0 );

(!; ')2  x(d!) Z

H H

Z

H H

(3.16)

(!; ') (!0 ; ')  x (d!) x (d!0 );

that may also be rewritten as follows Z Z 1 x Var f = 2 (f (!) ? f (!0 ))2  x (d!) x (d!0 ); H H Z Var x (:; ') = 12

(3.15)

(! ? !0 ; ')2  x(d!) x (d!0 ):

(3.17) (3.18)

Suppose now that we have estimated the rst variance by a constant which continuously depends on , on the parameters a, bl of the function (2.3), and on the value of the physical mass m. At the same time let the second variance be bounded by a function of the a, bl , and m, multiplied by the square of the norm of '. It is very important that both estimates would hold uniformly for all x 2 H . Then the mean{value theorem together with the representation of the Dobrushin coecients (3.11) imply that the conditions (3.7), (3.9) may be satis ed provided the product of the mentioned bounds is suciently small. Below we shall implement this idea. It can be observed that the expression (3.18) de nes a bounded positive quadratic form on H . That is Var x (:; ') def = ('; Kx ') (3.19) Z Z 1 k! ? !0k2  x(d!) x(d!0 ):  2 k'k2 H H 10

The operator Kx which appears here has the following integral representation Z (Kx ')( ) = Kx (;  0 )'( 0 )d 0 ;  2 I ; (3.20) I

with the kernel Kx (;  0 ) given by Z Z [!( ) ? !0 ( )][!( 0 ) ? !0 ( 0 )] x (d!) x (d!0 ): (3.21) Kx (;  0 ) = 21 H H Clearly, for every x 2 H , the operator Kx is symmetric and positive, its kernel (3.21) is a continuous function of  ,  0 2 I , and Z Z 1 trace(Kx ) = 2 k! ? !0k2  x(d!) x(d!0 ) < 1: (3.22) H H For a bounded linear operator T : H ! H , let Spp(T ) denote the pure point spectrum of T and let kT k be its operator norm. For a positive compact operator T , one has

kT k = max Spp(T ):

(3.23)

On the other hand, for the norm of a symmetric positive T , one has (see e.g [24], p.216): ) j ' 2 H n f0g g: (3.24) kT k = supf (';k'T' 2 k

Now we modify our notations introducing

P~2p (x) =

p

X

l=2

21?l bl x2l ; p  2

(3.25)

(3.26) V~ (x) = P~2p (x) ? 12 ax2 ; d2 + 1 (2Jd + 1)q2 + V~ (q): H~ = ? 21m dx (3.27) 2 2 The latter operator is de ned in L2 (IR; dx) and the operator q acts as follows (q )(x) def = x (x):

11

(3.28)

Further, along with the measures de ned by (2.7) and (3.10), we introduce (

)

Z ~ (d!) = ~1 exp ? V~ (!( ))d (d!); X ( I ) Z Z X~ = exp ? V~ (!( ))d (d!);

H

I

(3.29)

~x (d!) = ~1 exp ?Jdk!k2 + J (!; x) ~ (d!); (3.30) Zx where the partition function Z~x may be written as in (3.10) but with ~ instead of  . Having these new measures we introduce the operator K~ x by n

o

means of eq.(3.20) with the kernel Z Z 1 0 ~ [!( ) ? !0 ( )][!( 0 ) ? !0 ( 0 )]~ x (d!)~ x (d!0 ): (3.31) Kx (;  ) = 2 H H

Its trace is nite and can be calculated just as in (3.22) with the measure ~ instead of  . The construction of the bounds mentioned above is based upon the following lemmas, which are proved in Section 4 below. Lemma 3.1 For every x 2 H and all ' 2 H ('; Kx ') 

K~ 0

k'k2 :



(3.32)

Lemma 3.2 For every x 2 H , trace(Kx )  trace(K~ 0 ):

(3.33)

It is known (see [9], p.57) that (the closure of) H~ given by (3.27) is a self{ adjoint operator with a discrete spectrum, all its eigenvalues E~l , l 2 IN are simple. We set (3.34)
~ def = inf fjE~l ? E~l0 j l; l0 2 IN; l 6= lg: Lemma 3.3 For the spectral radius of the operator K~ 0, the following estimate holds (3.35) maxSpp(K~ 0 )  1~ 2 : m
12

Lemma 3.4 There exists a positive constant 0 (independent of m) such that for suciently small positive values of the physical mass m, the following estimate holds 1   m pp?+11 : (3.36) m
~ 2 0

To estimate the variance of the Lipschitz function f given by eq.(3.17) one may use the logarithmic Sobolev inequality as it was done in [1]. In the case where the function V de ned by (2.3) is V (x) = ? 21 ax2 + P4 (x) = ? 21 ax2 + b2 x4 ; this estimate looks as follows. Lemma 3.5 For all x 2 H and arbitrary f 2 Lip1(H ), 0

Var x f  2Jd +e 1 + a=4 ;

(3.37)

where the exponent 0 is de ned by the parameters of the function V 25 a2 : 0 = 288 b2 In the case of V given by eq. (2.3) with p > 2 one may use the following estimate of the variance of f , which is linear in . Lemma 3.6 There exists a positive constant 0 (independent of m and ) such that for all x 2 H , arbitrary f 2 Lip1 (H ), and for suciently small m, the following estimate holds Var x f  0 m?1=(p+1) : (3.38)

Proof of Theorem 3.1. We start with estimating of the variance Varx (:; ') given by eqs. (3.16), (3.18), (3.19). By means of Lemma 3.1 and (3.23) one obtains

Var x (:; ') = ('; Kx ') 

K~ 0

k'k2 = maxSpp(K~ 0 )k'k2 ; and further by means of Lemma 3.3, Lemma 3.4 p?1

Var x (:; ')  0 m p+1 k'k2 ; 13

(3.39)

that holds for suciently small positive m. In the case p > 2 one may use the estimate (3.38) and choose m so small that both latter esimates hold. This yields for the distance (3.4) p?2

R( x ;  y )  kx ? yk J 0 0 m 2(p+1) ; p

which means in view of (3.11), (3.9) that the condition of Dobrushin's criterion (3.7) is satis ed provided p?2 (3.40) m p+1 < (2Jd)12   : 0 0 Therefore, in the case p > 2 the upper bound for m is p+1

? p?2 (3.41) m ( ) = p+1 : [(2dJ )2 0 0 ] p?2 For p = 2, we use the estimate (3.37), choose m so small that (3.39) also

holds, and obtain

R( x;  y )  kx ? yk



Je 0 =2 m 61

s

0

2Jd + 1 + a=4 ;

which implies in turn that Dobrushin's criterion is satis ed provided  3 2 Jd + 1 + a= 4 def ? 3  0 m < m( ) = e ; (2Jd)2 0

(3.42)

4 Proof of Lemmas The proof of the rst two lemmas is performed by means of a version of the lattice approximation technique known in Euclidean quantum eld theory (see [26], Chapter VIII). We start with the following heuristic representation of the Gaussian measure de ned by (2.6) (see e.g. eq. (5.2) in [1]): (

)

Z Z Y 1 1 1 2

(d!) = C exp ? 2 m !_ ( )d ? 2 !2 ( )d d!( ); (4.1) I I  2I

14

where !_ ( ) def = d!=d . Recall that this measure is supported on C (I )  H . Let us consider the following base fek ( ) j k 2 Kg of H : s

s

ek ( ) = 2 cos k; k > 0; ek ( ) = ? 2 sin k; k < 0; e0 ( ) = p1 ; (4.2) where K def = fk = 2  j  2 ZZ g: (4.3) Then



!( ) =

X

k2K

!^ k ek ( ); !^ k =

Z

I

!( )ek ( )d;

(4.4)

and the measure (4.1) may be written as the following product

(d!) =

Y

k2K

s

k (d!^ k ) =

k (d!^ k );

(4.5)

mk2 + 1 expf? 1 (mk2 + 1)^!2 gd!^ : k k 2 2

Sometimes it will be convenient to use the following version of the Fourier transformation (4.4): X p (4.6) !( ) = !~ p1 eik ; i = ?1; k2K

k



with

!~ k = p1 (^!k ? i!^ ?k ); k > 0; !~ k = p1 (^!?k + i!^ k); k < 0; !~ 0 = !^ 0: (4.7)

2 2 Let C (I ) be the adjoint space of C (I ). The space of continuous cylinder functions on C (I ) is de ned as follows

FC = fu : C (I ) ! IR j (91:; : : : ; n 2 C (I ) ); (9un 2 C (IRn )) u(!) = un (1 (!); : : : ; n (!))g: Given a collection of f1 ; : : : ; l g = fj g, j 2 I , l 2 IN , let j 2 C (I ) ; j =

1; : : : ; l be de ned as follows j (!) = !(j ). Let also

FC (1; : : : ; l ) def = fu 2 FCj(9ul 2 C (IRl )) u(!) = ul (!(1 ); : : : ; !(l ))g:

(4.8)

15

Furthermore, suppose that for chosen fj g, all numbers j = are rational. Such  form a dense subset Q  I . Then there exist sequences fN (k) ; k 2 IN g, fn(jk) ; k 2 IN g, j = 1; : : : ; l each of which tends to in nity, such that

j = (n(jk) =N (k) ) ; 8k 2 IN; and 8j = 1; : : : ; l: (4.9) To simplify our notations we drop the symbol (k) assuming that N and nj

tend to in nity in such a way that (4.9) holds. Clearly, we may also suppose that all such N are even. The set of values of N satisfying these conditions depends on the choice of fj g, we denote it by N (1 ; : : : ; l ). If the collection fj 2 Q j j = 1; : : : ; lg is xed, we write simply N . For N = 2L; L 2 IN , we introduce (4.10) K def = fk = 2  j ? L + 1    L;  2 ZZ g; N

and



(N ) (d!) =

Y

k2KN

k (d!^ k )

Y

KnKN

(^!k )d!^ k ;

(4.11)

where  is the Dirac {function. Proposition 4.1 The sequence of measures f (N ) j N = 2L; L 2 IN g weakly converges to the measure (4.5). For every N = 2L; L 2 IN and each x 2 H , the functions

F (!) def = exp F~ (!) def = exp

(

Z

?Jdk!k2 + J (!; x) ?

Z



(

)

?Jdk!k2 + J (!; x) ?

are (N ) {integrable. The proof is evident. We denote Z MN def = Fd (N ) ; C (I ) and  x (d!) def =

M~ N def =

I

Z

C (I )

I

V (!( ))d ; (4.12) V~ (!( ))d ; (4.13)

~ (N ) ; Fd

1 (N ) MN F (!) (d!); ~Nx (d!) def = ~1 F~ (!) (N ) (d!); MN N

16

)

(4.14) (4.15)

The next assertion follows directly from the above one and from the expressions (2.7), (3.10), (3.29), and (3.30). Proposition 4.2 Given a choice of 1 ; : : : ; l , let u 2 FC (1 ; : : : ; l ) be { integrable. Then for arbitrary x 2 H and every N 2 N (1 ; : : : ; l ), this function is (N ) {integrable and the following convergences hold for every

x 2 H

Z

Z

C (I )

C (I )

u(!)Nx (d!) ! u(!)~Nx (d!) !

Z

Z

C (I ) C (I )

u(!) x (d!); u(!)~ x (d!):

Let us consider the integral of the type of Z

C (I )

u(!)F (!) (N ) (d!);

with u satisfying the conditions of the latter assertion. Every !( ) in the arguments of u and F appears as (i) (ii) (iii)

!(j ) = !( nNj ); Z

I

(4.16)

x( )!( )d;

Z

I

!2r ( )d; r 2 IN:

On the other hand, each such !( ) has the representation (4.4) in which !^k = 0 for all k 2 K n KN because all k with such k are degenerate at zero (see (4.11)). Then

s

X !^ k k =N (nj ); !(j ) = !^ k ek ( nNj ) = N k2KN k2KN X

(4.17)

where for q 2 QN def = fq = 2=N j ? L + 1    L;  2 ZZ g, and for n = 0; 1; : : : ; N ? 1, q (n) are de ned as follows r

r

q (n) = N2 cos qn; q > 0; q (n) = ? N2 sin qn; q < 0; 0 (n) = p1 : N (4.18) 17

Note that for k varying in KN , q = k=N runs over the set QN . By means of (4.4), the second expressions in (4.16) may be written in the following form Z

I

x( )!( )d =

X

k2KN

x^k !^ k ; x^k =

Z

x( )ek ( )d:

I

(4.19)

As for the third term in (4.16), it is more convenient to transform it by means of (4.6), (4.7) Z

I

!2r ( )d = =

X

k1 ;:::;k2r 2KN X

k1 ;:::;k2r 2KN

!~ k1 : : : !~ k2r ?r

Z

I

exp(i(k1 + : : : + k2r ) )d

!~ k1 : : : !~ k2r ?r+1 (k1 + : : : + k2r ):

(4.20)

Here (0) = 1 and (k) = 0 for k 6= 0. Now we introduce s

n N !( N ); n = 0; 1; : : : ; N ? 1; s^q = !^ k ; s~q = !~ k k = N q; q 2 QN ;

sn = for which one has

sn = s^q =

s~q eiqn ; s^q q (n) = p1 N q2QN q2QN X

X

NX ?1 n=0

snq (n); s~q = p1

NX ?1

N n=0

sne?iqn :

(4.21)

(4.22) (4.23)

Then (4.20) may be rewritten as follows Z

I

!2r ( )d

(4.24)

NX ?1 X = N r 1r?1 (k1 + : : : + k2r ) sn1 : : : sn2r n1 ;:::;n2r =0 k1 ;:::k2r 2KN

exp(? i N (k1 n1 + : : : + k2r n2r ))

L NX ?1 X = N r 1r?1 sn1 : : : sn2r n1 ;:::;n2r =0 1 ;:::2r?1 ?L+1

18

2 ( (n ? n )) : : : exp(? i2 ( (n ? n )) exp(? iN 1 1 2r N 2r?1 2r?1 2r

NX ?1 ?1 2r?1 NX = NNr r?1 s2nr = N n=0 n=0

s

Ns n

2r

!

:

Return to (4.19) and rewrite it as follows Z

=

x( )!( )d =

I NX ?1 n=0

X

k2KN

x^k !^ k

snhn(x); hn (x) def =

X

k2KN

(4.25)

x^k  k=N (n):

(4.26)

At last (4.17) now takes the form s

!(j ) = !( nNj ) = N snj :

(4.27)

In these new variables fsn jn = 0; : : : ; N ? 1g the measures (N ) , Nx , and ~Nx given by (4.11), (4.14), and (4.15) respectively, may be rede ned as the following probability measures on IRN :

(N ) (ds) = 1 exp CN

Nh (ds) = M1 (

N

exp ? 2Jd2? a

(

NX ?1 NX ?1 ? m2 ( N )2 [sn+1 ? sn]2 ? 21 s2n n=0 n=0

)

NY ?1 n=0

dsn (4.28)

NX ?1 n=0

s2n + J

NX ?1 n=0

snhn(x) ?

NX ?1

s

)

P ( N s ) (N ) (ds); 2p n n=0 N

(4.29) ~Nh (ds) = ~1 M N s ( ) NX ?1 NX ?1 NX ?1 2 Jd ? a N exp ? 2 s2n + J snhn(x) ? N P~2p ( sn) (N ) (ds): n=0

n=0

n=0

Therewith, Proposition 4.2 may be reformulated as follows. 19

Proposition 4.3 [Lattice approximation] Let the collection f1; : : : ; l g, j 2 Q , l 2 IN , be xed. Then for every {integrable function u 2 FC (1 ; : : : ; l ), for every x 2 H , and for arbitrary sequence N 2 N (1 ; : : : ; l ), N ! 1, the following convergences hold s

s

ul ( N sn1 ; : : : ; N snl )Nh (ds) ?! N IR s s Z N N s )~ h (ds) ?! u ( s ; : : : ; n l 1 nl N IRN Z

Z

C (I ) Z

C (I )

u(!) x (d!); (4.30) u(!)~ x (d!);

where for given x 2 H , hn = hn (x) is de ned by (4.25).

Our main tools for the proof of the lemmas formulated in the preceding section are given in the following two statements. Lemma 4.1 [FKG inequality] For every x 2 H , and for every ;  0 2 I

Kx (;  0 )  0:

(4.31)

Proof. Let us rst choose ;  0 2 Q , so that FC (;  0 ), N (;  0 ), and hn are de ned. Then by Proposition 4.3, the following convergence holds

N (< s s 0 > h ? < s > h < s 0 > h ) ?! K (;  0 ); x l N l  N l l N where l = (= )N and l0 = ( 0 = )N . But the FKG inequality

(4.32)

< slsl0 >Nh  < sl >Nh < sl0 >Nh ; holds since Nh , with arbitrary real hn , is a general ferromagnet (see Theorems VIII.16, VIII.17 p. 280{283, book [26]). Thus the inequality (4.31) holds on the dense subset Q  I . From the continuity of the kernel Kx (;  0 ) and (4.32) the proof of (4.31) follows.

Lemma 4.2 [GKS inequality] For every x 2 H , and for every ;  0 2 I Kx (;  0 )  K~ 0 (;  0 ): (4.33) Proof. As in the previous lemma we start with ;  0 2 Q . Then for the

kernels Kx , K~ 0 , one has the representations (3.21), (3.31). For chosen ;  0 , 20

we construct their lattice approximations

< sl sl0 >Nh ? < sl >Nh < sl0 >Nh

= 21 (s ? sl0 )(s0l ? s0l0 )(Nh Nh )(ds; ds0 ) 2N l IR Z Z sn p? sn0 s0n p? s0n0 = M12 2 2 IR2N N NX ?1 NX ?1 expf? 2Jd2? a (s2n + s0n 2 ) + J (sn + s0n )hn (x) n=0 n=0 Z Z

NX ?1

s

(4.34)

s

[P ( N s ) + P ( N s0 )]g( (N ) (N ) )(ds; ds0 ); ? 2p n 2p n n=0 N < sl sl0 >~N0 ? < sl >~N0 < sl0 >~N0 =< sl sl0 >~N0

(4.35)

Z NX ?1 NX ?1 1 1 2 0 = ~ slsl expf? 2 (2Jd ? a) sn ? N P~2p ( N sn)g (N ) (ds); N MN IR n=0 n=0 s

For the polynomials P2p and P~2p de ned by (2.3), (3.25), one has P2p ( xp+ y ) + P2p ( xp? y ) = P~2p (x) + P~2p (y) + 2p (xjy); x; y 2 IR; (4.36) 2 2 where pX ?1 x2k d2k P~ pX ?1 def 2 p 2p (xjy) = k (y)x2k ; (4.37) 2k (y) = (2 k )! dy k=1 k=1

Since bk  0, all coecients k (y) are nonnegative for every y 2 IR. To separate the variables in the integral in (4.34) we use the following orthogonal transformation of IRN  IRN (4.38) n = p1 (sn ? s0n ); n = p1 (sn + s0n); 2 2 sn = p1 (n + n ); s0n = p1 (n ? n ): 2 2

For the Gaussian measure (N ) (N ) , the orthogonality implies

(N ) (N ) (ds; ds0 ) = (N ) (N ) (d; d); 21

which, together with (4.36), yields in (4.34)

< slsl0 >Nh ? < sl >Nh < sl0 >Nh

= M12 N +

  0 expf? 12 (2Jd ? a) 2N l l

Z Z

IR

NX ?1

p

2J

n=0

NX ?1

n hn (x) ? s

NX ?1 n=0

NX ?1 n=0

(4.39) (n2 + n 2 )

S2p (n jn) s

[P~ ( N  ) + P~ ( N  )]g( (N ) (N ) )(d; d); ? 2p n 2p n n=0 N For # 2 [0; 1], we set Z ll0 (#) = Y (1#)

? where

NX ?1

s

1 0 expf? (2Jd ? a)   l l N 2

IR

NX ?1

NX ?1 n=0

n2

P~ ( N  ) ? #  ( j )g (N ) (d); 2p n 2p n n n=0 N n=0

(4.40)

NX ?1 NX ?1 1 2 Y (#) = N expf? 2 (2Jd ? a) n ? N P~2p ( N n) IR n=0 n=0 s

Z

? #

NX ?1 n=0

2p (n jn )g (N ) (d );

(4.41)

and use them in (4.39)

< slsl0 >Nh ? < sl >Nh < sl0 >Nh

NX ?1 = M12 2N ll0 (1)Y (1) expf? 12 (2Jd ? a) n 2 N IR n=0

(4.42)

Z

p

NX ?1

NX ?1

s

P~2p ( N n )g (N ) (d): n=0 n=0 Both functions ll0 (#), Y (#) are continuous on [0; 1] and dierentiable on (0; 1). Furthermore, ?1 ?1 pX @  0 (#) = ? 1 [Z NX 2k @# ll Y (#) IRN ) n=0 k=1 k (n )n l l0 +

2J

nhn (x) ?

22

NX ?1 NX ?1 N ?1 ~2p ( N n) ? # X 2p (n jn )g (N ) (d ) expf? 12 (2Jd ? a) n2 ? P n=0 n=0 N n=0 Z ?1 NX ?1 NX ?1 pX k (n)n2k expf? 21 (2Jd ? a) n2 ? < l l0 ># N IR n=0 k=1 n=0 s

?1 P~ ( N  ) ? # NX 2p (n jn )g (N ) (d )]: ? 2p n N n=0 n=0 NX ?1

s

In what follows

@  0 (#) = @# ll

(4.43)

?1 NX ?1 pX

k (n ) < n2k l l0 ># ? < n2k >#< l l0 ># ? Y (1#) n=0 k=1 



where the expectations < : ># are taken with respect to the measure NX ?1 # (ds) def = Y (1#) expf? 12 (2Jd ? a) n2 n=0 s NX ?1 N ? 1 ~2p ( N n ) ? # X 2p (n jn )g (N ) (d ); ? P n=0 N n=0

(4.44)

with the Gaussian measure (N ) (d ) given by (4.28). For every  2 IRN and # 2 [0; 1], the measure # (ds) is an even ferromagnet, thus by Theorem VIII.14 (see pp. 273, 274 of [26]), its moments obey the GKS inequality, in particular

< n2k l l0 >#  < n2k >#< l l0 >#;

which means that the right{hand side of (4.43) is nonpositive, that yields (4.45) 0  ll0 (#)  ll0 (0); 8# 2 [0; 1] Now one may employ the latter in (4.42) and obtain (4.46) < slsl0 >Nh ? < sl >Nh < sl0 >Nh Z NX ?1 0 (0) 1  ll  M 2 2N Y (1) expf? 2 (2Jd ? a) n2 N IR n=0 +

p

2J

NX ?1 n=0

n hn (x) ?

NX ?1

s

P~2p ( N n)g (N ) (d) n=0 23

Z Z ?1 1 (2Jd ? a) NX ll0 (0) = M (n2 + n 2 ) exp f? 2 2N 2 IR N n=0

NX ?1

p

2J

+

n=0

NX ?1

n hn (x) ? s

NX ?1 n=0

2p (n jn ) s

[P~ ( N  ) + P~ ( N  )]g( (N ) (N ) )(d; d): ? 2p n 2p n n=0 N Returning in the latter integral to the initial variables by means of (4.38) one obtains

< sl sl0 >Nh ? < sl >Nh < sl0 >Nh

ll0 (0)  M 2 N

Z

?1 N ?1 1 (2Jd ? a) NX 2 + s0 2 ) + J X (sn + s0 )hn (x) ( s exp f? n n n 2N 2

IR

NX ?1

(4.47)

s

n=0

s

n=0

[P ( N s ) + P ( N s0 )]g( (N ) (N ) )(ds; ds0 ) 2p n 2p n n=0 N Z Z = ll0 (0) N N Nh (ds)Nh (ds0 ) = ll0 (0):

?

IR

IR

Comparing (4.40), (4.44), and (4.29) one gets ll0 (0) =< l l0 >0 =< l l0 >0 ; and 0 = ~N0 ; that yields in (4.47)

< sl sl0 >Nh ? < sl >Nh < sl0 >Nh  < l l0 >0 =< sl sl0 >~N0 ;

(4.48)

where (4.35) has also been used. Now passing to the limit N ! 1 one obtains from (4.32) and (4.48) the proof of the stated inequality (4.33) on the dense subset Q  I . To extend the proof to I it is enough to employ the continuity of the kernels of Kx and K~ 0 . Now we may prove the lemmas stated in the preceding section. Proof of Lemma 3.1. Applying the FKG inequality (4.31), the GKS inequality (4.33), and the representation of the norm of K~ 0 (3.24) one obtains ('; Kx ') = j('; Kx ') j 



Z

Z

I I

Z

Z

I I

Kx (;  0 )j'( )jj'( 0 )jdd 0

K~ 0(;  0 )j'( )jj'( 0 )jdd 0 

K~ 0

kj'jk2 =

K~ 0

k'k2 : 24





Proof of Lemma 3.2. For the kernels of Kx and K~ 0 , one has trace(Kx ) =

Z

I

Kx (;  )d; trace(K~ 0 ) =

Z

I

K~ 0 (;  )d;

which gives (3.33) by means of the GKS inequality (4.33). A statement similar to Lemma 3.3 has already been proved earlier (see e.g. Lemma 2.3 of paper [3]), but for the completeness, we repeat it here. Proof of Lemma 3.3. For the case x = 0, one has in (3.31)

K~ 0 (;  0 ) =

Z

!( )!( 0 )~ 0 (d!):

H

(4.49)

The periodicity of the loops !( ) implies that the kernel of K~ 0 is also periodic K~ 0 (;  0 ) = K~ 0 ( + #;  0 + #); 8# 2 [0; ] (4.50) where the addition is modulo . The latter yields in turn Z Z n o maxS (K~ ) = K~ (0;  )d = 1 trace qe? H~ qe?( ? )H~ d; (4.51) pp

where

0

I

0

Z

I

Z = trace(e? H~ );

(4.52) and the operators H~ , q are de ned by (3.27) and (3.28). We recall that the eigenvalues of H~ , E~l are simple. We denote by l the corresponding eigenfunction and set qll0 = ( l ; q l0 ) . For symmetry reasons qll = 0, then (4.51) may be written as follows ~ ~ 0 e? E~l0 ? e? E~l ) X ; (qll0 )2 (El ? El )( maxSpp(K~ 0 ) = Z1 (E~l ? E~l0 )2 l;l02IN;l6=l0 The case of zero denominator is excluded, thus this denominator may be estimated by means of (3.34), which yields X maxSpp(K~ 0 )  Z1 ~12 (qll0 )2 (E~l ? E~l0 )(e? E~l0 ? e? E~l )
l;l02IN ~ q]e? H~ = 1 ; = Z1 ~12 trace[q; [H; m
~ 2
where [:; :] means commutator. 25

Proof of Lemma 3.4. Let us return to the expressions (3.25){(3.27). For some positive , consider the unitary operator U on H , which acts as

follows

(U )(x) = 1=2 (x): Then we have on a core for d=dx, resp. q

d )U ?1 = ?1 ( d ); U ( dx  dx

U qU?1 = q:

(4.53)

Now we put  = m?1=(2p+2) and conclude that the operators H~ and H~ (m) p = T0 + m1=(p+1) T1 H~ (m) = m? p+1 T; T def

(4.54)

are unitary equivalent. Here

d2 + 21?p b q2p ; T0 = ? 12 dx p 2 T1 = 21 (2Jd + 1 ? a)m p+1 q2 + p?2

pX ?1 l=2

21?l m

p?l?1 p+1

bl q2l :

Let
and
0 be de ned by the expression (3.34) but with the eigenvalues of the operators T and T0 respectively. Then p
~ = m? p+1
;

(4.55)

It can be observed (see e.g [18]) that the operator T is a perturbation of T0 , which is analytic at the point m = 0. This implies the existence of a certain constant c0 such that
 c 0
0 ; (4.56) for suciently small positive values of m. These arguments yield the estimate (3.36) with 0 = 1=(c0
0 )2 . Proof of Lemma 3.5. The estimate of the type of (3.37) has been proved in paper [1] on the base of the logarithmic Sobolev inequality. Its realization in the form of (3.37) is obtained for the choice of V~ (3.25), (3.26) with p = 2. For more details one may see section 6 of [1]. Proof of Lemma 3.6. Let us return to the de nition of the variance of f 2 Lip1(H ) (3.17). By means of (3.2), (3.3), (3.22), (3.33), (4.49), (4.50) 26

one obtains Var x f

(4.57) Z Z

2 1

! ? ! 0  x (d! ) x (d! 0 ) = trace(K )  2 x H H Z ~ = < q2 >;  trace(K0 ) = K~ 0 (;  )d = trace(q2 e? H~ ) def

Z

H

where Z is given by (4.52). For maxSpp(K~ 0 ), we have the estimate (3.35). On the other hand, the same quantity may be expressed in terms of the Duhamel two{point function [14] and hence be estimated from below. This estimate comes from Bogolyubov's inequalities, it was obtained in [14] and for our case it reads   (4.58) < q2 > f 4m < q2 >  maxSpp(K~ 0 ); where the function f was de ned in [14] implicitly by the relation f (xtanhx) = 1 tanhx; x 2 (0; +1); and was estimated there as

x

1 (1 ? e?x )  f (x):

(4.59)

x

Applying this estimate together with (3.35) in (4.58) one gets p

m p+1 ; 
~ c0
0

2m < q2 > (1 ? e? =4m )  1

where we have used (4.55), (4.56). It follows from the latter that one may nd a constant 0 such that 1

< q2 > 0 m? p+1 ; holds for suciently small positive values of m. Applying this in (4.57) one obtains (3.38)

Acknowledgement Yuri Kozitsky is grateful for kind hospitality ex-

tended to him at Bonn and Bochum Universities where his part of this work has mainly been performed. His research was supported in part by the Fundamental Research Fund of Ukraine under the grant 2.4/173 that is also acknowledged. 27

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