On the Uniqueness of Gibbs States in the Pirogov-Sinai Theory?

Commun. Math. Phys. 189, 311 – 321 (1997) Communications in Mathematical Physics c Springer-Verlag 1997

J.L. Lebowitz1, A.E. Mazel2

1 Department of Mathematics and Physics, Rutgers University, New Brunswick, NJ 08903, USA 2 International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow 113556, Russia

Received: 4 June 1996 / Accepted: 30 October 1996

Dedicated to the memory of Roland Dobrushin

Abstract: We prove that, for low-temperature systems considered in the Pirogov-Sinai theory, uniqueness in the class of translation-periodic Gibbs states implies global uniqueness, i.e. the absence of any non-periodic Gibbs state. The approach to this inﬁnite volume state is exponentially fast.

1. Introduction

The problem of uniqueness of Gibbs states was one of R.L.Dobrushin’s favorite subjects in which he obtained many classical results. In particular when two or more translation-periodic states coexist, it is natural to ask whether there might also exist other, non translation-periodic, Gibbs states, which approach asymptotically, in different spatial directions, the translation periodic ones. The affirmative answer to this question was given by R.L.Dobrushin with his famous construction of such states for the Ising model, using boundary conditions, in three and higher dimensions [D]. Here we consider the opposite± situation: we will prove that in the regions of the low-temperature phase diagram where there is a unique translation-periodic Gibbs state one actually has global uniqueness of the limit Gibbs state. Moreover we show that, uniformly in boundary conditions, the finite volume probability of any local event tends to its infinite volume limit value exponentially fast in the diameter of the domain. The first results concerning this problem in the framework of the Pirogov-Sinai theory [PS] were obtained by R.L.Dobrushin and E.A.Pecherski in [DP]. The Pirogov- Sinai theory describes the low-temperature phase diagram of a wide class of spin lattice models, i.e. it determines all their translation-periodic limit Gibbs states [PS, Z]. The results of [DP], corrected and extended in [Sh], imply that, for any values of parameters at which the model has a unique ground state, the Gibbs state is unique for sufficiently small temperatures. But the closer these parameters are to the points with non unique

? Work supported by NSF grant DMR 92-13424 312 J.L. Lebowitz, A.E. Mazel

ground state the smaller the temperature for which uniqueness of the Gibbs state is given by this method. Independently, an alternative method leading to similar results was developed in [M1,2]. The main difficulty in establishing the results of this type is due to the necessity of having sufficiently detailed knowledge of the partition function in a finite domain with an arbitrary boundary condition. This usually requires a detailed analysis of the geometry of the so called boundary layer produced by such a boundary condition (see [M1,2, Sh]). Here we develop a new simplified approach to the problem. The simplification is achieved by transforming questions concerning the finite volume Gibbs measure with arbitrary boundary conditions into questions concerning the distribution with a stable (in the sense of [Z]) boundary condition. The latter can be easily investigated by means of the polymer expansion constructed for it in the Pirogov-Sinai theory. This also allows the extension of the uniqueness results from systems with a unique ground state to the case with several ground states but unique stable ground state (see [Z]). Since the publication of the paper [PS] about twenty years ago the Pirogov-Sinai theory was extended in different directions. For a good exposition of the initial theory we refer the reader to [Si] and [Sl]. Some of the generalizations can be found in [BKL, BS, DS, DZ, HKZ] and [P]. Below we present our results in the standard settings of [PS and Z]: a finite spin space with a translation-periodic finite potential of finite range,a finite degeneracy of the ground state and a stability of the ground states expressed via the so called Peierls or Gertzik-Pirogov-Sinai condition. The extension to other cases is straightforward. Our method also works for unbounded spins, see [LM].

2. Models and Results

The models are defined on some lattice, which for the sake of simplicity we take to d be the d-dimensional (d 2) cubic lattice Z . The spin variable σx associated with the lattice site x takes values≥ from the finite set S = 1, 2,..., S . The energy of the d { | |} configuration σ SZ is given by the formal Hamiltonian ∈

H0(σ)= U0(σA). (1) A Zd,diam A r ⊂ X ≤ A d A Here σA S is a configuration in A Z and the potential U0(σA): S R, ∈ ⊂ d 7→ satisfies U0(σA)=U0(σA+y) for any y belonging to some subgroup of Z of finite index and the sum is extended over subsets A of Zd with a diameter not exceeding r. d Accordingly for a finite domain V Z , with the boundary conditionσ ¯ c given on its ⊂ V complement V c = Zd V , the conditional Hamiltonian is \

H0(σ σ¯ c )= U0(σ ), (2) V | V A A V= ,diam A r ∩ 6 ∅X ≤ c where σA = σA V +¯σA Vc for A V = , i.e. the spin at site x is equal to σx for ∩ ∩ c ∩ 6 ∅ x A V andσ ¯ x for x A V . ∈A ground∩ state of (1)∈ is a configuration∩ σ in Zd whose energy cannot be lowered by changing σ in some local region. We assume that (1) has a finite number of translation- periodic (i.e. invariant under the action of some subgroup of Zd of finite index) ground states. By a standard trick of partitioning the lattice into disjoint cubes Q(y) centered Uniqueness of Gibbs States 313

at y qZd with an appropriate q and enlarging the spin space from S to SQ one can transform∈ the model above into a model on qZd with a translation-invariant potential and only translation-invariant or non-periodic ground states. Hence, without loss of generality, we assume translation-invariance instead of translation-periodicity and we permute the spin so that the ground states of the model will be σ(1),...,σ(m) with (k) d σx = k for any x Z . Taking q>rone obtains a model with nearest neighbor and next nearest neighbor∈ (diagonal) interaction, i.e. the potential is not vanishing only on d lattice cubes Q1 of linear size 1, containing 2 sites. Given a configuration σ in Zd we say that site x is in the kth phase if this configura- (k) tion coincides with σ inside the lattice cube Q2(x) of linear size 2 centered at x. Every connected component of sites not in one of the phases is called a contour of the configuration σ. It is clear that for σ = σ + σ(k) contours are connected subsets of V which V V c we denote byγ ˜1(σ),...,γ˜l(σ). The important observation is that the excess energy of a configuration σ with respect to the energy of the ground state σ(k) is concentrated along the contours of σ. More precisely,

(k) (k) (k) H0(σ σ ) H0(σ σ )= H0(˜γi(σ)), (3) V | V c − V | V c i X where (k) H0(˜γi(σ)) = U0(σQ ) U0(σ ) (4) 1 − Q1 Q1: Q1 γ˜ (σ) X⊆ i d and the sum is taken over the unit lattice cubes Q1, containing 2 sites. The Peierls condition is H0(˜γ (σ)) τ γ˜ (σ) , (5) i ≥ | i | where τ>0 is an absolute constant and γ˜ (σ) denotes the number of sites inγ ˜ (σ). | i | i Consider now a family of Hamiltonians Hn(σ)= Un(σA), n = A Zd,diam A r ⊂ X ≤ 1,...,m 1 satisfying the same conditions as H0 with the same or smaller set of − translation-periodic ground states. For λ =(λ1,...,λm 1) belonging to a neighbor- m 1 − hood of the origin in R − deﬁne a perturbed formal Hamiltonian

m 1 − H = H0 + λnHn. (6) =1 Xn Here λnHn play the role of generalized magnetic ﬁelds removing the degeneracy of the ground state. The ﬁnite volume Gibbs distribution is

exp ( ¯ ) βH σV σV c ( )= − | (7) µV ,σ¯ c σV , V h Ξ(V σ¯ c ) i | V where β>0 is the inverse temperature and µ ,σ¯ (σ ) is the probability of the event V V c V that the conﬁguration in is , given ¯ . Here the conditional Hamiltonian is V σV σV c

m 1 − H(σ σ¯ c )= U(σ ),U()= U ( )(8) V | V Q1 · n · Q1:Q1 V= n=0 X∩ 6 ∅ X and the partition function is 314 J.L. Lebowitz, A.E. Mazel

Ξ(V σ¯ c )= exp βH(σ σ¯ c ) . (9) | V − V | V σ XV h i The notion of a stable ground state was introduced in [Z] (see also the next section) and it is crucial for the Pirogov-Sinai theory because of the following theorem. Theorem [PS, Z]. Consider a Hamiltonian H of the form (6) satisfying all the conditions (k) above. Then for β large enough, β β0(λ), every stable ground state σ generates a translation-invariant Gibbs state ≥

(k) µ ( ) = lim µ ,σ(k) ( ). (10) · V Zd V V c · → These Gibbs states are different for different k and they are the only translation-periodic Gibbs states of the system. An obvious corollary of the above theorem is

(1) Corollary . If there is only a single stable ground state, say σ , then for β β0(λ) there is a unique translation-periodic Gibbs state ≥

(1) µ ( ) = lim µ ,σ(1) ( ). (11) · V Zd V V c · → Our extension of this result is given by Theorem 1. Under conditions of the Corollary the Gibbs state µ(1)( ) is unique. De- noting this unique state by µ( ) then for ﬁnite A V , such that· dist(A, V c) 2 diam + ( ), any· conﬁguration and⊂ any boundary condition ¯ ≥, d A C1 τ, β, λ, d σA σV c one has

c µ ,σ¯ (σA) µ(σA) exp C2(τ, β, λ, d) dist (A, V ) , (12) V V c − ≤ − where C ,C >0. 1 2 Remark 1. In contrast with [DP, Sh and M1,2] the theorem above treats the situation when there are several ground states with only one of them being stable. Moreover, the result is true for all sufficiently low temperatures not depending on how close the parameters are to the points with non unique ground state. Remark 2. If some of the conditions of the Theorem are violated the statement can be wrong. The simplest counterexample can be constructed from the Ising model in d = 3. It is well-known that at low temperatures this model contains precisely two translation-invariant Gibbs states taken into each other by symmetry and infinitely many non translation-invariant Gibbs states, i.e. the Dobrushin± states mentioned earlier. Identifying configurations taken into each other by symmetry one obtains a model with a unique translation-invariant Gibbs state and infinitely± many non translation-invariant ones. The condition of the Theorem which is not true for this factorized model is the finiteness of the potential: the model contains a hard-core constraint. Indeed, consider a dual lattice formed by the centers of the unit cubes of the initial lattice. Then define on the dual lattice the model with the spin taking 128 values represented by the unit cubes of initial lattice with the following properties: (i) The bonds of the cube are labeled by +1 or 1. − (ii) The product of labels along any of 6 plaquettes is 1. Uniqueness of Gibbs States 315

(iii) The energy of the cube is 1/2 times the sum of the labels of the bonds. − With the hard-core condition saying that for any two neighboring cubes the labels of common bonds are the same in each cube one abtains precisely the factorized Ising model above. Another counterexample based on a gauge model can be found in [B]. Remark 3. We conjecture that for Hamiltonians H satisfying our conditions the statement of our theorem is true at all temperatures. This is known to be the case for systems satisfying FKG inequalities [LML]. Furthermore the only examples of nontranslation- invariant states for such H which we are aware of are those of the Dobrushin interface tye [D], [HKZ] which in d 3 “connect” different pure phases. ≥

3. Preliminaries

In this section we assume that the reader is familiar with the Pirogov-Sinai theory and we only list the appropriate notations and quote some necessary results. A contour is a pair γ =(˜γ,σγ˜ ) consisting of the support γ˜ and the configuration σγ˜ in it. The components of the interior of the contour γ are denoted by Intjγ and the exterior of γ is denoted by Extγ. The family γ,Int˜ jγ,Extγ is a partition of d { } d Z . The configuration σγ˜ can be uniquely extended to the configuration σ0 in Z taking c constant values Ij(γ) and E(γ) on the connected components ofγ ˜ . Generally these values are different for different components. The contour γ is said to be from the phase k if E(γ)=k. The energy of the contour γ is

(E(γ)) H(˜γ(σ)) = U(σQ ) U(σ ) . (13) 1 − Q1 Q1: Q1 γ˜ (σ) X⊆ The statistical weight of γ is w(γ)=exp( βH(γ)) (14) − and satisﬁes βτ γ˜ 0 w(γ) e− | |. (15) ≤ ≤ The renormalized statistical weight of the contour is

Ξ(Int∗γ Ij(γ)) W (γ)=w(γ) j | , (16) Ξ(Int γ E(γ)) j j∗ Y | where for any A Zd we denote ⊂ c A∗ = x A x is not adjacent to A . (17) { ∈ | } The contour γ is stable if 1 W (γ) exp βτ γ˜ (18) ≤ − 3 | | and the ground state σ(k) is stable if all contours γ with E(γ)=kare stable. It is known (see [Z]) that at least one of the ground states is stable. Because of (18) for any x Zd, N 1 and β large enough ∈ ≥ 316 J.L. Lebowitz, A.E. Mazel

C3N W (γ) e− , (19) ≤ γ:(Extγ)c x, γ˜ N X3 | |≥ where C3 = C3(τ,β,d) is positive and monotone increasing in τ and β. In particular

W (γ) C4, (20) ≤ γ:(Extγ)c x X 3

C3 (k) where C4 = e− . For the stable ground state σ the corresponding partition function can be represented as

βH(σ(k) σ(k) ) Ξ(V k)=e− V | V c W (γ ), (21) | i [γ ]s V, E([γ ]s)=k i i ∈ X i Y where the sum is taken over all collections of contours [γi] such thatγ ˜i are disjoint, E(γi)=kfor all i andγ ˜i V for all i. Representation (21) and⊆ estimate (18) allow to write an absolutely convergent polymer expansion

log Ξ(V k)= βH(σ(k) σ(k))+ W(π(k)), (22) | − V | V c π(k) V X∈ where the sum is taken over so called polymers π(k) of the phase k belonging to the (k) domain V . By definition a polymer π =(γi) is a collection of, not necessarily different, contours γ of the phase k such that γ˜ is connected. The statistical weight W (π(k)) i ∪i i is uniquely defined via W (γi) and satisfies the estimate (see [Se])

1 W (π(k)) exp βτ 6d γ˜ (23) | |≤ − 3 − | i| " i # X implying (k) C3N W (π ) e− . (24) | |≤ (k) c π =(γi): i(Extγi) x, γ˜ i N ∪ X3 i | |≥ (k) Denote by µ ( γi , ext) the probabilityP of the event that all contours of the collec- V { } tion γ are external ones inside V . By the construction { i} (k) µ ( γi , ext) W (γi). (25) V { } ≤ i Y From the polymer expansion (22) and estimate (24) it is not hard to conclude that for γ with dist( γ˜ ,Vc) γ˜ { i} ∪i i ≥|∪i i| (k) (k) µ ( γi , ext) µ ( γi , ext) V { } − { } (26) (k) c µ ( γi , ext) i γ˜i exp C 5 dist ( iγ˜i,V ) , ≤ { } |∪ | − ∪ where C5 = C5(τ,β,d) is positive and monotone increasing in τ and β. For any A V , dist(A, V c) diamA, and any σ estimate (26) implies in a standard way that ∈ ≥ A (k) (k) c µ (σA) µ (σA) exp C6 dist (A, V ) , (27) V − ≤ −

Uniqueness of Gibbs States 317

where again C6 = C6(τ,β,d) is positive and monotone increasing in τ and β. From now on we suppose that σ(1) is the only stable ground state of H and denote by (1) µ ( ) and µ ¯ c ( ) the Gibbs distribution with the stable boundary condition σ c and V · V,σV · V an arbitrary boundary conditionσ ¯ V c respectively. Clearly µ ,σ¯ c (σA) depends only on V V σ¯ ∂V , where ∂V = x V c : x is adjacent to V , (28) { ∈ } e and we freely use the notation µV,σ¯ ∂V (σA). For a domain V ﬁx a collection γi of all external contours in V touching ∂V and suppose that the number of the points{ at} which γ˜ touches ∂V is not greater than L. Consider a smaller domain V 0 = Int∗γ with ∪i i ∪i∪j j i (Ij(γi)) e the boundary condition σδV0 = i j σ∂Int γ .Given γi V and M> i γ˜i j∗ i { } ∈ | | denote by γ e,M the event that the number of unstable (see Def. 2 of Sect. 3.2 in [2]) E{ i} P P P sites in V 0 is not less than M γ˜ . According to the Theorem of Sect. 3.2 in [Z], − i | i|

P C7(τ,β,λ,d)(M γ˜ i )+C8(τ,β,λ,d) γ˜i µ e e− − i | | i | |. (29) V 0,σ0 γi ,M ∂V 0 E{ } ≤ P P The positive constants C7 andC8 tend to 0 as β or (β,λ) approaches the manifold (1) →∞ e on which σ is not the only stable ground state. For different γi the events γ e,M { } E{ i} are disjoint and for their union V,M,L = γ e V γ e,M one has the estimate E ∪{ i} ∈ E{ i} e µ = µ e µ γ V V,M,L V 0,σ0 γi ,M V i E ∂V 0 E{ } { } γ e V { iX} ∈ C7(M γ˜ i )+C8 γ˜ i e− − i | | | | W (γ ) ≤ i γ e V i { iX} ∈ P Y 1 C7M ( τβ C7 C8) γ˜i e− e− 3 − − | | ≤ γ e V i { iX} ∈ Y C7M L e− (1 + C4) ≤ C7M+C4L e− . (30) ≤ Finally observe that for any A V and any σ ⊆ A C9L(¯σ∂V ) C9L(¯σ∂V ) µ (σA)e− µ ,σ¯ (σA) µ (σA)e , (31) V ≤ V V c ≤ V where L(¯σ ) is the number of sites x ∂V withσ ¯ = 1 and ∂V ∈ x 6 d C9 =2 βmax U(σQ1 ) . (32) σQ1 | |

4. Proof of Theorem

We are now ready to prove the theorem. Given σV denote by (σV ) the union of the connected components of the set x V : σ =1 adjacent to x δV :σ =1. { ∈ x 6 } { ∈ x 6 } This set is called the boundary layer of σV . Take an integer N>0 and suppose that V contains a cube Q6N with sides of length 6N centered at the origin. From now on all cubes are assumed to be centered at the origin. Let Q ,N0 6Nbe the maximal N 0 ≥ cube contained in V . Denote ∂0V = ∂QN0 ∂V . First we consider boundary conditions (1) ∩ (1) σ¯ which coincide with σ on ∂V ∂0V and differ from σ on ∂0V by at most √N ∂V \ lattice sites. Introduce the event 0 = σ : (σ ) Q4 = . By construction for E { V V ∩ N 6 ∅} 318 J.L. Lebowitz, A.E. Mazel

σ 0 every (σ ) touches ∂V at some site x ∂V withσ ¯ = 1 and there exists at V ∈E i V ∈ x 6 least one component i intersecting Q4N . Without loss of generality we suppose that it is ( ). This leads to the estimate 1 σV C9√N µ ,σ¯ ( 0) e µ ( 0) V V c E ≤ V E C9√N C3N √N e √Ne− (1+C4) ≤ C10N e− . (33) ≤ In the first inequality of (33) we used (31) reducing the problem to the calculation for the stable boundary condition σ(1). The second inequality comes in a standard way from the cluster expansion for ( ). Indeed, in the domain with the stable boundary µV V (1) · condition σ every component i contains an external contour γi such that E(γi)=1, V c γ˜ and (Ext(γ ))c. One may simply say that γ is the external boundary of i ⊆ i i ⊆ i i i and clearlyγ ˜i touches ∂V .If1 intersects Q4N thenγ ˜1 intersects or encloses Q4N . The number of possibilities to choose the site x ∂V, σ¯ = 1 at whichγ ˜1 touches ∂V ∈ x 6 does not exceed √N which produces the factor √N in the estimate. The next factor estimates the sum of the statistical weights of all possible γ1 touching this site. It is based on (19) and takes into account the fact that the diameter ofγ ˜1, and hence γ˜1 ,is | | not less than N. The constant C4 (see (20)) estimates the sum of the statistical weights √N of all possible γi touching a given lattice site and (1+C4) estimates the statistical weight of all possibilities to choose γi, i =1 . The whole estimate uses (25) and the fact that µ ( γ , ext) is the upper bound{ for6 the} sum of µ -probabilities of boundary V { i} V layers = i having γi as the boundary of i. The third inequality in (33) is trivial { } 2 2 for C10 =0.5C3 and N 4(C9 +1+C4) C3− . c ≥ c Denote by 0 the complement of 0.Ifσ 0 then (V (σ ))∗ Q4N 2 (see E E V ∈E \ V ⊇ c− (17) for the definition of ( )∗). It is not hard to see that the configuration σ equals · V ∈E0 1 on the boundary of (V (σ ))∗.NowfixA Q2 and σ . In view of (27) one has \ V ⊂ N A

c C6N µ ,σ¯ (σA ) µ(σA) e− . (34) V V c |E0 − ≤

This gives us

µ ,σ¯ (σA) µ(σA) µ ,σ¯ (σA 0) µ(σA) µ ,σ¯ ( 0) V V c − ≤ V V c |E − V V c E c c + µ ,σ¯ (σA ) µ(σA) µ ,σ¯ ( ) V V c |E0 − V V c E0 C10N C6N e− + e− ≤ C11N e− , (35) ≤ where C11 =0.5 min(C10,C6) and N log 2/C11. ≥ To extend (35) to the wider class of boundary conditions we suppose that V Q8 ⊇ N and QN 0 ,N0 8Nis the maximal cube contained in V . Now we consider boundary conditionsσ ¯ ≥which coincide with σ(1) on ∂V ∂ V and differ from σ(1) on ∂ V by ∂V 0 0 2 \ (i) at most (√N) lattice sites. Denote 1i = Q8N 2i+2 Q8N 2i and let (σ )bea − \ − V union of the connected components of the set x V Q8N 2i : σx =1 adjacent to x ∂V :¯σ=1 ,i=1,...,N. Introduce disjoint{ ∈ events\ − 6 } { ∈ x6 } Uniqueness of Gibbs States 319

(1) 1 = σ : (σ ) 11 < √N , E { V | V ∩ | } (1) (i 1) i = σ : (σ ) 11 √N,..., − (σ ) 1i 1 √N, E { V | V ∩ |≥ | V ∩ − |≥ (i)(σ ) 1 < √N | V ∩ i| } and

N c = . (36) Ec Ei ÿ =1 ! [i If σ then the boundary contour (σ ) contains at least N√N sites. Hence (30) V ∈Ec V implies the following estimate for the probability of c 2 E C9(√N) µ ,σ¯ ( c) e µ ( c) V V c E ≤ V E 2 2 C9(√N) C7N√N+C4(√N) e e− ≤ C12N√N e− , (37) ≤ 2 2 where C12 =0.5C7 and N 4(C9 + C4) C7− . ≥ (i) For σ iconsider the volume Vi =(V (σ ))∗ Q8N 2i with the boundary V ∈E \ V ∪ − conditionσ ¯ c + σV V . By construction the number of sites x ∂Vi with σx = 1 is less V \ i ∈ 6 than √N and one can apply (35) to obtain the bound

C11N µ ,σ¯ (σA i) µ(σA) e− . (38) V V c |E − ≤

Joining (37) and (38) we conclude

µ ,σ¯ c (σA) µ(σA) = µ ,σ¯ c ( i) µ ,σ¯ c (σA i) µ(σA) V V − V V E V V |E − i=1 X

+ µ ,σ¯ c ( c) µ ,σ¯ c (σA c) µ(σA) V V E V V |E − C11N C12N√N e− + e− ≤ C11N 2e− , (39) ≤ 2 where N (C11/C12) . Expression (39) is a version of (35) which is weaker by the factor 2 in≥ the RHS but is applicable to the wider class of boundary conditions containing (√N)2 unstable sites instead of √N for (35). The argument leading from (35) to (39) can be iterated several times. The ﬁrst iteration treats the following situation. Suppose that V Q10 and Q ,N0 10N ⊇ N N 0 ≥ is the maximal cube contained in V . Consider boundary conditionsσ ¯ ∂V which coincide (1) (1) 3 with σ on ∂V ∂0V and differ from σ on ∂0V by at most (√N) lattice sites. Then the analogue of (39)\ is

C11N µ ,σ¯ (σA) µ(σA) 3e− . (40) V V c − ≤

320 J.L. Lebowitz, A.E. Mazel

Similarly after 2d iterations one obtains that for any V Q2 with the boundary ⊇ dN conditionσ ¯ containing not more than (√N)2d unstable sites ∂V

C11N µ ,σ¯ (σA) µ(σA) 2de− . (41) V V c − ≤

2 2 2 2 2 Now set C1 =2dmax 4( C9 +1+C4) C3− , log 2/C11, 4(C9 + C4) C7− , (C11/C12) c and for any A and σA consider a cube Q2L with L C1 and dist(A, Q ) (1 1/d)L. ≥ 2L ≥ − Taking N = L/d and C2 = C11/2d for V = Q2L one obtains (12) from (41). For any V Q2 with ∂V ∂Q2 = we have ⊇ L ∩ L 6 ∅

µ ,σ¯ (σA) µ(σA) µQ ,σ (σA) µ(σA) µ ,σ¯ (σ∂Q ) V V c − ≤ 2L ∂Q2L − V V c 2L σ X∂Q2L c exp C2 dist (A, V ) , (42) ≤ − which ﬁnishes the proof of the Theorem.

Acknowledgement. We are greatly indebted to F.Cesi, F.Martinelli and S.Shlosman for many very helpful discussions.

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Communicated by Ya. G. Sinai