Central limit theorem for the free energy of the random field Ising model
aa r X i v : . [ m a t h - ph ] M a r CENTRAL LIMIT THEOREM FOR THE FREE ENERGYOF THE RANDOM FIELD ISING MODEL
SOURAV CHATTERJEE
Abstract.
A central limit theorem is proved for the free energy of therandom field Ising model with all plus or all minus boundary condition,at any temperature (including zero temperature) and any dimension.This solves a problem posed by Wehr and Aizenman in 1990. The proofuses a variant of Stein’s method. Introduction
Take any d ≥
1. Let Λ be any finite subset of Z d . Let ∂ Λ be the setof all x ∈ Z d \ Λ that are adjacent to some point of Λ. Let Σ = {− , } Λ ,Γ = {− , } ∂ Λ and Φ = R Λ . Given σ ∈ Σ, γ ∈ Γ and φ ∈ Φ, define H γ,φ ( σ ) := − X x,y ∈ Λ ,x ∼ y σ x σ y − X x ∈ Λ , y ∈ ∂ Λ ,x ∼ y σ x γ y − X x ∈ Λ φ x σ x , where x ∼ y means that x and y are neighbors. Take any β ∈ [0 , ∞ ].The Ising model on Λ with boundary condition γ , inverse temperature β ,and external field φ , is the probability measure on Σ with probability massfunction proportional to e − βH γ,φ ( σ ) . When β = ∞ , this is simply the uniformprobability measure on the configurations that minimize the energy.Now suppose that ( φ x ) x ∈ Λ are i.i.d. random variables instead of fixedconstants. Then the probability measure defined above becomes a randomprobability measure. This is known as the random field Ising model (RFIM).We will refer to the law of φ x as the random field distribution.The random field Ising model was introduced by Imry and Ma [16] in1975 as a simple example of a disordered system. Imry and Ma predictedthat the model does not have an ordered phase in dimensions one and two,but does exhibit a phase transition in dimensions three and higher. Theexistence of the phase transition in dimension three was partially proved byImbrie [14, 15], who showed that there are two macroscopic ground statesin the 3D RFIM. The phase transition at nonzero temperature was finallyestablished by Bricmont and Kupiainen [6, 7] in 1987, settling the Imry–Ma conjecture in d ≥
3. A few years later, Aizenman and Wehr [3, 4]proved the non-existence of an ordered phase in d ≤
2. The proof of the
Mathematics Subject Classification.
Key words and phrases.
Random field Ising model, central limit theorem, free energy.Research partially supported by NSF grant DMS-1608249.
Imry–Ma conjecture is regarded as a notable success story of mathematicalphysics, because there was considerable debate within the theoretical physicscommunity about the validity of the conjecture. See Bovier [5, Chapter 7]for more details.Another important paper on the random field Ising model is the 1990paper of Wehr and Aizenman [20] on the fluctuations of the free energy ofthe RFIM and related models. The free energy of the RFIM on Λ at inversetemperature β and boundary condition γ is defined as F ( γ, φ, β ) := − β log X σ ∈ Σ e − βH γ,φ ( σ ) . When β = ∞ , the free energy is simply the ground state energy: F ( γ, φ, ∞ ) = min σ ∈ Σ H γ,φ ( σ ) . Wehr and Aizenman [20] proved that under mild conditions on the randomfield distribution, the variance of F is upper and lower bounded by constantmultiples of the size of Λ. In the same paper, Wehr and Aizenman posed theproblem of proving a central limit theorem for F as the size of Λ tends toinfinity. The main result of this paper is a solution of this question for theRFIM with plus or minus boundary condition. The plus boundary conditionis the boundary condition γ where γ ( x ) = +1 for all x ∈ ∂ Λ. Similarly, theminus boundary condition has γ ( x ) = − x ∈ ∂ Λ. When d ≤
2, theresult holds for any boundary condition.Results about fluctuations of the free energy have a number of applica-tions. As stated in [20], bounds on fluctuations of the free energy wereinstrumental in the proof of rounding effects of the quenched randomnesson first-order phase transitions in low-dimensional systems. Another ap-plication in a different model, also discussed in [20], is an inequality forcharacteristic exponents of the model of directed polymers in a random en-vironment. Central limit theorems give the most precise information aboutfluctuations, and they are also mathematically interesting in their own right.Central limit theorems for the free energy have been proved for disorderedsystems with mean-field interactions such as the Sherrington–Kirkpatrickmodel of spin glasses [1, 12, 13]. But as far as I am aware, no such resultswere available for disordered systems on lattices prior to this paper.The main result has two parts, corresponding to the cases β < ∞ and β = ∞ . The β < ∞ case is the following. Theorem 1.1.
Take any d ≥ . Let { Λ n } n ≥ be a sequence of finitenonempty subsets of Z d . For each n , consider the RFIM on Λ n with plusboundary condition, at inverse temperature β ∈ (0 , ∞ ) . Suppose that therandom field distribution has finite moment generating function. Let F n bethe free energy of the model. Suppose that | ∂ Λ n | = o ( | Λ n | ) as n → ∞ . Thenthere is a finite positive constant σ , depending only on β , d and the random LT FOR THE RFIM FREE ENERGY 3 field distribution (and not on the sequence { Λ n } n ≥ ), such that lim n →∞ Var( F n ) | Λ n | = σ , and F n − E ( F n ) p | Λ n | D → N (0 , σ ) , where D → denotes convergence in law, and N (0 , σ ) is the normal distributionwith mean zero and variance σ . The same result holds for minus boundarycondition, possibly with a different value of σ . If d ≤ , then the aboveconclusion holds under any arbitrary sequence of boundary conditions. Note that in Theorem 1.1, the only condition that we imposed on the ran-dom field distribution is that it has finite moment generating function. Forthe β = ∞ case, our proof technique requires that (a) the random field dis-tribution is continuous, and (b) it is a push-forward of the standard normaldistribution under a Lipschitz map (with arbitrary Lipschitz constant). Forexample, the normal distribution with any mean and any variance belongsto this class. The uniform distribution on any interval is another example. Theorem 1.2.
Take any d ≥ and let { Λ n } n ≥ be a sequence of finitesubsets of Z d . Suppose, as in Theorem , that | ∂ Λ n | = o ( | Λ n | ) as n → ∞ .Let G n be the ground state energy of the RFIM on Λ n with plus boundarycondition. Suppose that the random field distribution satisfies the conditionsstated above. Then there is a finite positive constant σ , depending only on d and the random field distribution (and not on the sequence { Λ n } n ≥ ), suchthat lim n →∞ Var( G n ) | Λ n | = σ , and G n − E ( G n ) p | Λ n | D → N (0 , σ ) . The same result holds for minus boundary condition, possibly with a differentvalue of σ . If d ≤ , then the above conclusion holds under any arbitrarysequence of boundary conditions. The main tool for proving Theorem 1.1 is a method of normal approxi-mation introduced in [8], where it was developed as an extension of Stein’smethod [18, 19]. A ‘continuous’ version of this method, developed in [9],is our tool for proving Theorem 1.2. The extension to arbitrary boundaryconditions in d ≤ d ≤
2, famously proved by Aizenman and Wehr [4]. Aquantitative version of the Aizenman–Wehr result, such as the ones recentlyproved in [10] and [2], can be used to obtain rates of convergence in Theo-rems 1.1 and 1.2 when d ≤
2. In particular, if the rate from [2] is used, then
SOURAV CHATTERJEE it should be possible to prove a rate of convergence of order n − α for somesmall positive constant α using the methods of this paper.There are several questions that remain open about central limit theoremsfor the RFIM. The foremost is proving (or disproving) central limit theoremsunder arbitrary boundary conditions in d ≥
3. The main technical difficultyis that for arbitrary boundary conditions, it is not clear how to establish aresult like inequality (3.2) of Section 3, which is crucial for the proof.Another problem is to express the limiting variance σ in some kind ofa closed form, instead of just saying that it exists. The problem of gettingany rate of convergence in d ≥ Technique
First, let us briefly review the main result of [8]. Recall that the Wasser-stein distance d W ( µ, ν ) between the two probability measures µ and ν on R is defined to be the supremum of | R f dµ − R f dν | over all Lipschitz f : R → R with Lipschitz constant 1.Let X be a measurable space and suppose that X = ( X , . . . , X n ) is avector of independent X -valued random variables. Let X ′ = ( X ′ , . . . , X ′ n )be an independent copy of X . Let [ n ] = { , . . . , n } , and for each A ⊆ [ n ],define the random vector X A as X Ai = ( X ′ i if i ∈ A,X i if i A. For each i , let ∆ i f ( X ) := f ( X ) − f ( X { i } ) , and for each A ⊆ [ n ] and i A , let∆ i f ( X A ) := f ( X A ) − f ( X A ∪{ i } ) . Let T := 12 n X i =1 X A ⊆ [ n ] \{ i } ∆ i f ( X )∆ i f ( X A ) n (cid:0) n − | A | (cid:1) . The following theorem is the main result of [8].
Theorem 2.1 ([8]) . Let all terms be defined as above, and let W = f ( X ) .Suppose that W has finite second moment, and let σ := Var( W ) . Let µ be LT FOR THE RFIM FREE ENERGY 5 the law of ( W − E ( W )) /σ and ν be the standard normal distribution on thereal line. Then E ( T ) = σ and d W ( µ, ν ) ≤ p Var( E ( T | W )) σ + 12 σ n X i =1 E | ∆ i f ( X ) | . Recall that the Kolmogorov distance between two probability measures µ and ν on the real line is defined as d K ( µ, ν ) := sup t ∈ R | µ (( −∞ , t ]) − ν (( −∞ , t ]) | . The Kolmogorov distance is more commonly used in probability and statis-tics than the Wasserstein distance. The bound on d W ( µ, ν ) in Theorem 2.1can be used to get a bound on d K ( µ, ν ) using the following simple observa-tion made in Chatterjee and Soundararajan [11]: Let ν denote the standardnormal distribution and let µ be any probability measure on R . Then d K ( µ, ν ) ≤ p d W ( µ, ν ) . (2.1)The combination of Theorem 2.1 and inequality (2.1) usually yields a sub-optimal bound for the Kolmogorov distance. There is a recent improvementof Theorem 2.1 by Lachi`eze-Rey and Peccati [17] that gives optimal boundsfor the Kolmogorov distance in many problems.Theorem 2.1 by itself is a bit difficult to directly apply to the problem athand. We will now synthesize a corollary of Theorem 2.1 that will be moreeasily applicable for the random field Ising model. The main idea here is toapproximate the discrete derivative ∆ i f ( X ) by a function that depends ‘ononly a few coordinates’. We will continue to work in the setting introducedabove.For each 1 ≤ i ≤ n , let g i : X n × X → R be a measurable map. For each i and each p ≥
1, let m p,i := k ∆ i f ( X ) k L p = ( E | ∆ i f ( X ) | p ) /p and let ǫ p,i := k ∆ i f ( X ) − g i ( X, X ′ i ) k L p . First, we have the following generalization of Theorem 2.1.
Theorem 2.2.
Let all notation be as above, and let µ , ν and σ be as inTheorem . Let S := 12 n X i =1 X A ⊆ [ n ] \{ i } g i ( X, X ′ i ) g i ( X A , X ′ i ) n (cid:0) n − | A | (cid:1) . Then | σ − E ( S ) | ≤ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) , SOURAV CHATTERJEE and d W ( µ, ν ) ≤ σ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) + √ Var Sσ + 12 σ n X i =1 m ,i . Proof.
For simplicity of notation, let Y i := g i ( X, X i ) and Y Ai := g i ( X A , X ′ i ).Note that for any A and any i A , k ∆ i f ( X )∆ i f ( X A ) − Y i Y Ai k L ≤ k (∆ i f ( X ) − Y i )∆ i f ( X A ) k L + k Y i (∆ i f ( X A ) − Y Ai ) k L ≤ k ∆ i f ( X ) − Y i k L k ∆ i f ( X A ) k L + k Y i k L k ∆ i f ( X A ) − Y Ai k L = ǫ ,i ( m ,i + k Y i k L ) ≤ ǫ ,i ( m ,i + k Y i − ∆ i f ( X ) k L + k ∆ i f ( X ) k L )= 2 ǫ ,i m ,i + ǫ ,i . (2.2)Now, for each i , X A ⊆ [ n ] \{ i } n (cid:0) n − | A | (cid:1) = n − X k =0 |{ A : A ⊆ [ n ] \ { i } , | A | = k }| n (cid:0) n − k (cid:1) = 1 . (2.3)Therefore, k T − S k L ≤ n X i =1 X A ⊆ [ n ] \{ i } k ∆ i f ( X )∆ i f ( X A ) − Y i Y Ai k L n (cid:0) n − | A | (cid:1) ≤ n X i =1 X A ⊆ [ n ] \{ i } ǫ ,i m ,i + ǫ ,i n (cid:0) n − | A | (cid:1) = 12 n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) . Consequently, p Var( T ) ≤ k T − E ( S ) k L ≤ k T − S k L + p Var( S ) ≤ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) + p Var( S ) . This bound, together with Theorem 2.1 and the observation thatVar( E ( T | W )) ≤ Var( T ) , gives the second inequality in the statement of Theorem 2.2. For the firstinequality, recall from Theorem 2.1 that E ( T ) = σ . Then retrace the stepsin the derivation of (2.2) starting with the L norm instead of the L norm, LT FOR THE RFIM FREE ENERGY 7 and finally use the identity (2.3), to get | E ( T ) − E ( S ) | ≤ n X i =1 X A ⊆ [ n ] \{ i } E | ∆ i f ( X )∆ i f ( X A ) − Y i Y Ai | n (cid:0) n − | A | (cid:1) ≤ n X i =1 X A ⊆ [ n ] \{ i } ǫ ,i m ,i + ǫ ,i n (cid:0) n − | A | (cid:1) = 12 n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) . This completes the proof of the theorem. (cid:3)
Theorem 2.2 can be useful only when it is easier to understand Var( S )than Var( T ). The following result gives such a criterion. Proposition 2.3.
Let g i and S be as in Theorem . Suppose that for each i , there is a set N i ⊆ [ n ] such that g i ( x, x ′ i ) is a function of only ( x j ) j ∈ N i and x ′ i . Then Var( S ) ≤ X i,j : N i ∩ N j = ∅ ( m ,i + ǫ ,i ) ( m ,j + ǫ ,j ) . Proof.
Let Y i and Y Ai be as in the proof of Theorem 2.2. Note thatVar( S ) = 14 n X i,j =1 X A ⊆ [ n ] \{ i } ,B ⊆ [ n ] \{ j } Cov( Y i Y Ai , Y j Y Bj ) n (cid:0) n − | A | (cid:1)(cid:0) n − | B | (cid:1) . By independence of coordinates, whenever N i ∩ N j = ∅ ,Cov( Y i Y Ai , Y j Y Bj ) = 0 . Moreover, for any i , j , A and B , | Cov( Y i Y Ai , Y j Y Bj ) | ≤ k Y i Y Ai k L k Y j Y Bj k L ≤ k Y i k L k Y j k L ≤ ( m ,i + ǫ ,i ) ( m ,j + ǫ ,j ) . By (2.3), this completes the proof. (cid:3)
Combining Theorem 2.2 and Proposition 2.3, we get the following result.This is our main tool for proving Theorem 1.1.
Theorem 2.4.
Let all notation be as in Theorem . Let N i be as inProposition . Then | σ − E ( S ) | ≤ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) , SOURAV CHATTERJEE and d W ( µ, ν ) ≤ σ n X i =1 (2 ǫ ,i m ,i + ǫ ,i )+ 12 σ (cid:18) X i,j : N i ∩ N j = ∅ ( m ,i + ǫ ,i ) ( m ,j + ǫ ,j ) (cid:19) / + 12 σ n X i =1 m ,i . Theorem 2.4 will be used in Section 3 to prove Theorem 1.1. However,I have not been able to use Theorem 2.4 to prove Theorem 1.2 (the CLTfor the ground state energy). Instead, a ‘continuous version’ of Theorem 2.4will be used to prove Theorem 1.2. This is presented as Theorem 2.6 below.Let f : R n → R be a differentiable function. Let ∂ i f denote the par-tial derivative of f in the i th coordinate, and let ∇ f = ( ∂ f, . . . , ∂ n f ) bethe gradient of f . Let Z = ( Z , . . . , Z n ) be a vector of i.i.d. standard nor-mal random variables. The main ingredient in the proof of Theorem 2.6is the following lemma, which is a slightly modified version of Lemma 5.3from Chatterjee [9]. Recall that the total variation distance between twoprobability measures µ and ν on the real line is defined as d TV ( µ, ν ) := sup A | µ ( A ) − ν ( A ) | , where the supremum is taken over all Borel subsets of R . Lemma 2.5 ([9]) . Let f and Z be as in the above paragraph and let W := f ( Z ) . Assume that k f ( Z ) k L < ∞ and k ∂ i f ( Z ) k L < ∞ for all i . Let σ := Var( W ) . Let Z ′ be an independent copy of Z , and let T := Z √ t ∇ f ( Z ) · ∇ f ( √ tZ + √ − tZ ′ ) dt. Let µ be the law of ( W − E ( W )) /σ and ν be the standard normal distribution.Then E ( T ) = σ and d TV ( µ, ν ) ≤ p Var( T ) σ . The above lemma is the starting point for the method of ‘second orderPoincar´e inequalities’ developed in [9]. For proving the CLT for the groundstate energy of the RFIM, however, I could not construct a proof usingsecond order Poincar´e inequalities. Instead, the above lemma needs to beused in a different way, more along the lines of Theorem 2.4.For each 1 ≤ i ≤ n , let g i : R n → R be a measurable function and let N i be a set of coordinates such that the value of g i ( x , . . . , x n ) is determinedby ( x j ) j ∈ N i . Suppose that k g i ( Z ) k L < ∞ for all i . For each 1 ≤ i ≤ n and p ≥
1, let m p,i := k ∂ i f ( Z ) k L p (2.4) LT FOR THE RFIM FREE ENERGY 9 and ǫ p,i := k ∂ i f ( Z ) − g i ( Z ) k L p . (2.5)Let g : R n → R n be the function whose i th coordinate map is g i . For0 ≤ t ≤
1, let Z t := √ tZ + √ − tZ ′ , and let S := Z √ t g ( Z ) · g ( Z t ) dt. (2.6)The following theorem gives a continuous analog of Theorem 2.4, in thesetting of Lemma 2.5. Theorem 2.6.
Let ǫ p,i and m p,i be defined as above and let all other vari-ables be defined as in Lemma . Then | σ − E ( S ) | ≤ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) , and d TV ( µ, ν ) ≤ σ n X i =1 (2 ǫ ,i m ,i + ǫ ,i )+ 2 σ (cid:18) X i,j : N i ∩ N j = ∅ ( m .i + ǫ ,i ) ( m ,j + ǫ ,j ) (cid:19) / . Proof.
Note that k T − S k L ≤ Z √ t k∇ f ( Z ) · ∇ f ( Z t ) − g ( Z ) · g ( Z t ) k L dt. But for any t , k∇ f ( Z ) · ∇ f ( Z t ) − g ( Z ) · g ( Z t ) k L ≤ k ( ∇ f ( Z ) − g ( Z )) · ∇ f ( Z t ) k L + k g ( Z ) · ( ∇ f ( Z t ) − g ( Z t )) k L ≤ n X i =1 ( k ( ∂ i f ( Z ) − g i ( Z )) ∂ i f ( Z t ) k L + k g i ( Z )( ∂ i f ( Z t ) − g i ( Z t )) k L ) ≤ n X i =1 ( k ∂ i f ( Z ) − g i ( Z ) k L k ∂ i f ( Z t ) k L + k g i ( Z ) k L k ∂ i f ( Z t ) − g i ( Z t ) k L )= n X i =1 ( ǫ ,i m ,i + k g i ( Z ) k L ǫ ,i ) ≤ n X i =1 ( ǫ ,i m ,i + ( m ,i + ǫ ,i ) ǫ ,i ) . Thus, k T − S k L ≤ n X i =1 (2 ǫ ,i m ,i + ǫ ,i ) . (2.7)On the other hand, p Var( T ) ≤ k T − E ( S ) k L ≤ k T − S k L + p Var( S ) . (2.8)By Jensen’s inequality,Var( S ) = E (cid:18)Z √ t ( g ( Z ) · g ( Z t ) − E ( g ( Z ) · g ( Z t ))) dt (cid:19) ≤ Z √ t Var( g ( Z ) · g ( Z t )) dt = Z √ t n X i,j =1 Cov( g i ( Z ) g i ( Z t ) , g j ( Z ) g j ( Z t )) dt. Now note that if N i ∩ N j = ∅ , thenCov( g i ( Z ) g i ( Z t ) , g j ( Z ) g j ( Z t )) = 0 , and for any i and j ,Cov( g i ( Z ) g i ( Z t ) , g j ( Z ) g j ( Z t )) ≤ k g i ( Z ) g i ( Z t ) k L k g j ( Z ) g j ( Z t ) k L ≤ k g i ( Z ) k L k g j ( Z ) k L ≤ ( m ,i + ǫ ,i ) ( m ,j + ǫ ,j ) . This shows thatVar( S ) ≤ X i,j : N i ∩ N j = ∅ ( m .i + ǫ ,i ) ( m ,j + ǫ ,j ) . Combining this with (2.7), (2.8) and Lemma 2.5, we get the desired boundon d TV ( µ, ν ). For the bound on | σ − E ( S ) | , we proceed as in the proofof (2.7) to obtain a bound on k T − S k L , and then use Lemma 2.5 for theidentity σ = E ( T ). (cid:3) Theorem 2.6 will be used to prove Theorem 1.2 in Section 4. In that proof, f will be the ground state energy of the RFIM on a finite set, considered asa function of the random field. However, it is not a differentiable function ofthe random field. To take care of this issue, we need to extend Theorem 2.6to the slightly larger class of functions. Proposition 2.7.
For each k , let f k : R n → R be a differentiable function.Suppose that f ( x ) = lim k →∞ f k ( x ) exists almost everywhere. Further, as-sume that for each i , lim k →∞ ∂ i f k ( x ) exists almost everywhere, and call thelimit ∂ i f ( x ) . Lastly, suppose that for some ǫ > , sup k k f k ( Z ) k L ǫ < ∞ and sup i,k k ∂ i f k ( Z ) k L ǫ < ∞ , (2.9) LT FOR THE RFIM FREE ENERGY 11 where Z = ( Z , . . . , Z n ) is a vector of i.i.d. standard normal random vari-ables. Take any g , . . . , g n as in the paragraph preceding the statement ofTheorem , and define m p,i and ǫ p,i as in (2.4) and (2.5) , assuming that k g i ( Z ) k L ǫ < ∞ for each i . Then the conclusions of Theorem hold forthe function f , treating ∂ i f as its derivative in the i th coordinate.Proof. Let W k := f k ( Z ), σ k := Var( W k ), and µ k be the law of ( W k − E ( W k )) /σ k . Let S be defined as in (2.6). Let ν be the standard normaldistribution. Then Theorem 2.6 gives upper bounds on | σ k − E ( S ) | and d TV ( µ k , ν ) in terms of the L and L norms of ∂ i f k ( Z ) and ∂ i f k ( Z ) − g i ( Z ).As k → ∞ , the a.e. convergence of ∂ i f k to ∂ i f and the condition (2.9)ensure that these norms converge to the corresponding norms of ∂ i f ( Z ) and ∂ i f ( Z ) − g i ( Z ). This immediately implies the validity of the first inequalityof Theorem 2.6 for the function f .Next, note that the a.e. convergence of f k to f and the condition (2.9)ensure that ( W k − E ( W k )) /σ k converges almost surely to ( W − E ( W )) /σ as k → ∞ . This implies that µ k converges to µ weakly. By the well-knowncoupling characterization of total variation distance, for each k there existsa probability measure γ k on R whose one-dimensional marginals are µ k and ν , and γ k ( V ) = d TV ( µ k , ν ) , where V := { ( x, y ) ∈ R : x = y } . Since µ k converges weakly to µ , it follows that the sequence { γ k } k ≥ is atight family of probability measures on R . Let { γ k j } j ≥ be a subsequenceconverging to a limit γ . Then γ has marginals µ and ν . Moreover, since V is an open set, d TV ( µ, ν ) ≤ γ ( V ) ≤ lim inf j →∞ γ k j ( V )= lim inf j →∞ d TV ( µ k j , ν ) . This completes the proof of the proposition. (cid:3) Proof of Theorem C will denote any positive constant that depends only on β , d and the random field distribution. The value of C may change fromline to line or even within a line.We will prove the result under the plus boundary condition only, sincethe argument for the minus boundary condition is the same. Fix an inversetemperature β . Let h σ i i Λ ,γ denote the expected value of the spin at site i under the RFIM on Λ with boundary condition γ , at inverse temperature β . By the FKG property of the random field Ising model, it is a standardfact that for any Λ and any i ∈ Λ, h σ i i Λ ,γ is a monotone increasing functionof the boundary condition γ . From this and the Markovian nature of themodel, it follows that h σ i i Λ , + ≥ h σ i i Λ ′ , + whenever i ∈ Λ ⊆ Λ ′ . Take any i ∈ Z d . For each k , let Λ i,k be the cube of side-length 2 k + 1centered at i . Then the above inequality shows that the limit h σ i i + := lim k →∞ h σ i i Λ i,k , + exists. Therefore, if we let δ k := E |h σ i i Λ i,k , + − h σ i i + | , (3.1)then by translation-invariance, δ k depends only on k and not on i , andlim k →∞ δ k = 0 . (Note that the absolute value in (3.1) is unnecessary, since the randomvariable inside is nonnegative. But we keep it anyway, to emphasize the pointthat h σ i i Λ i,k , + ≈ h σ i i + with high probability when k is large.) Moreover,given any k and Λ such that Λ i,k ⊆ Λ, h σ i i + ≤ h σ i i Λ , + ≤ h σ i i Λ i,k , + . Consequently, E |h σ i i Λ , + − h σ i i Λ i,k , + | ≤ δ k . (3.2)Now take any nonempty set Λ ⊆ Z d . Fix β and let F be the free energyof the RFIM on Λ with plus boundary condition, at inverse temperature β .Consider F as a function of the random field ( φ i ) i ∈ Λ , and let ∆ i F be thechange in the value of F when φ i is replaced by an independent copy φ ′ i , asin Theorem 2.1. Let α i := β ( φ ′ i − φ i ) . Then note that ∆ i F = − β log h e α i σ i i Λ , + = − β log h cosh α i + σ i sinh α i i Λ , + = − β log(cosh α i + h σ i i Λ , + sinh α i ) . In particular, k ∆ i F k L ≤ k α i k L β ≤ C. (3.3)Now fix some k ≥
1. For each i ∈ Λ, let N i := Λ i,k ∩ Λ . Let g i := − β log(cosh α i + h σ i i N i , + sinh α i ) . Clearly, k g i k L ≤ C. (3.4) LT FOR THE RFIM FREE ENERGY 13
For any x ∈ [ − , α i + x sinh α i lies between the numbers e − α i and e α i . The derivative of the logarithm function in this interval isbounded above by e | α i | . Therefore, for any x, y ∈ [ − , | log(cosh α i + x sinh α i ) − log(cosh α i + y sinh α i ) | ≤ e | α i | | x − y | . Thus, | ∆ i F − g i | ≤ e | α i | |h σ i i Λ , + − h σ i i N i , + | , and so k ∆ i F − g i k L ≤ k e | α i | k L kh σ i i Λ , + − h σ i i N i , + k L ≤ C ( E |h σ i i Λ , + − h σ i i N i , + | ) / . Let Λ ′ be the set of all i ∈ Λ that are at a distance at least k from theboundary of Λ. Then for each i ∈ Λ ′ , N i = Λ i,k , and therefore by (3.2), k ∆ i F − g i k L ≤ Cδ / k . (3.5)On the other hand, if i Λ ′ , then by (3.3) and (3.4), k ∆ i F − g i k L ≤ C. (3.6)For each i ∈ Λ, note that the number of j such that N i ∩ N j = ∅ is boundedby Ck d . Also, clearly, | Λ \ Λ ′ | ≤ Ck d | ∂ Λ | . (3.7)Finally, from [20], we know thatVar( F ) ≥ C | Λ | . (3.8)We now have all the estimates required for using Theorem 2.4. Let m p,i := k ∆ i F k L p and ǫ p,i := k ∆ i F − g i k L p . By the estimates obtained above, X i ∈ Λ (2 ǫ ,i m ,i + ǫ ,i ) ≤ C | Λ ′ | δ / k + C | Λ \ Λ ′ |≤ C | Λ | δ / k + Ck d | ∂ Λ | . Next, note that X i,j : N i ∩ N j = ∅ ( m ,i + ǫ ,i ) ( m ,j + ǫ ,j ) ≤ Ck d | Λ | . Finally, X i ∈ Λ m ,i ≤ C | Λ | . Let µ denote the law of ( F − E ( F )) / p Var( F ) and let ν denote the standardnormal distribution. Plugging the above bounds into Theorem 2.4, andusing the lower bound (3.8), we get d W ( µ, ν ) ≤ Cδ / k + Ck d | ∂ Λ || Λ | + Ck d/ p | Λ | . Let F n and Λ n be as in the statement of the theorem. Let µ n be the law of( F n − E ( F n )) / p Var( F n ). Since | ∂ Λ n | = o ( | Λ n | ) as n → ∞ , the above boundshows that lim sup n →∞ d W ( µ n , ν ) ≤ Cδ / k . However, k is arbitrary, and δ k → k → ∞ . This shows that µ n convergesto ν in the Wasserstein metric.To complete the proof of Theorem 1.1, it only remains to show that theratio Var( F n ) / | Λ n | tends to a finite nonzero limit. For this, we will use thefirst inequality of Theorem 2.4 and the following simple lemma. Lemma 3.1.
For any integers m ≥ l ≥ and n ≥ , n X k =0 (cid:0) nk (cid:1)(cid:0) n + mk + l (cid:1) = n + m + 1( m + 1) (cid:0) ml (cid:1) . Proof.
By the well-known formula for the beta integral, Z x k + l (1 − x ) n + m − k − l dx = ( k + l )!( n + m − k − l )!( n + m + 1)!= 1( n + m + 1) (cid:0) n + mk + l (cid:1) . Thus, again by the beta integral formula, n X k =0 (cid:0) nk (cid:1)(cid:0) n + mk + l (cid:1) = Z ( n + m + 1) n X k =0 (cid:18) nk (cid:19) x k + l (1 − x ) n + m − k − l dx = Z ( n + m + 1) x l (1 − x ) m − l dx = ( n + m + 1) l !( m − l )!( m + 1)! . This completes the proof of the lemma. (cid:3)
We will now show that under the conditions of Theorem 1.1, Var( F n ) / | Λ n | tends to a finite nonzero limit. Fix k ≥ N i and g i be as before.Consider g i as a function of ( φ j ) j ∈ Λ . For each A ⊆ Λ such that i A , let g Ai be the value of g i after replacing φ j with an independent copy φ ′ j for each j ∈ A . Then the quantity S of Theorem 2.4 is simply12 X i ∈ Λ X A ⊆ Λ \{ i } g i g Ai | Λ | (cid:0) | Λ |− | A | (cid:1) . LT FOR THE RFIM FREE ENERGY 15
This can be rewritten as12 X i ∈ Λ X A ⊆ N i \{ i } X A ⊆ Λ \ N i g i g A ∪ A i | Λ | (cid:0) | Λ |− | A | + | A | (cid:1) . But for any i , A and A as in the above display, the definition of g i impliesthat g A ∪ A i = g A i . Thus, S = X i ∈ Λ S i , where S i := 12 X A ⊆ N i \{ i } g i g A i (cid:18) X A ⊆ Λ \ N i | Λ | (cid:0) | Λ |− | A | + | A | (cid:1) (cid:19) . (3.9)Let p (Λ , i, A ) denote the term within the brackets in the above display.Note that by (2.3), X A ⊆ N i \{ i } p (Λ , i, A ) = X A ⊆ Λ \{ i } | Λ | (cid:0) | Λ |− | A | (cid:1) = 1 . Consequently, for any i ∈ Λ, E | S i | ≤ X A ⊆ N i \{ i } p (Λ , i, A ) E | g i g A i | ≤ C. (3.10)On the other hand, it is not difficult to see from the expression (3.9) and thedefinitions of g i , N i and Λ ′ that E ( S i ) is the same for all i ∈ Λ ′ . Withoutloss of generality, suppose that the origin 0 is in Λ ′ . Then by the precedingremark, E ( S ) = | Λ ′ | E ( S ) + X i ∈ Λ \ Λ ′ E ( S i ) . By (3.10) and (3.7), this gives | E ( S ) − | Λ | E ( S ) | ≤ Ck d | ∂ Λ | . (3.11) On the other hand, by Lemma 3.1, for any A ⊆ N , p (Λ , , A ) = 1 | Λ | X A ⊆ Λ \ N (cid:0) | Λ |− | A | + | A | (cid:1) = 1 | Λ | | Λ \ N | X k =0 X A ⊆ Λ \ N , | A | = k (cid:0) | Λ |− | A | + k (cid:1) = 1 | Λ | | Λ \ N | X k =0 (cid:0) | Λ \ N | k (cid:1)(cid:0) | Λ |− | A | + k (cid:1) = 1 | N | (cid:0) | N |− | A | (cid:1) . This shows that when 0 ∈ Λ ′ , E ( S ) depends only on k , β , d and the randomfield distribution, and not on Λ.On the other hand, by the first inequality of Theorem 2.4, | Var( F ) − E ( S ) | ≤ X i ∈ Λ (2 ǫ ,i m ,i + ǫ ,i ) , (3.12)where m ,i = k ∆ i F k L and ǫ ,i = k ∆ i F − g i k L , as before. Proceeding as inthe proof of (3.3), we get m ,i ≤ C for all i . Similarly, proceeding as in theproofs of (3.5) and (3.6), we get that for any i ∈ Λ, ǫ ,i ≤ C , and for i ∈ Λ ′ , ǫ ,i ≤ Cδ / k , where δ k is defined as in (3.1). By (3.7) and (3.12), this gives | Var( F ) − E ( S ) | ≤ Cδ / k | Λ | + Ck d | ∂ Λ | . (3.13)Now let F n and Λ n be as in the statement of Theorem 1.1. By (3.11) and(3.13), it follows that for each k , there is some number a k depending only on k , β , d and the random field distribution, and not on the sequence { Λ n } n ≥ ,such that lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) Var( F n ) | Λ n | − a k (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ / k . Since δ k → k → ∞ , this shows that { a k } k ≥ is a Cauchy sequence.Let a be the limit of this sequence. Then a depends only on β , d andthe random field distribution, and Var( F n ) / | Λ n | converges to a as n → ∞ .This completes the proof of Theorem 1.1, except for the last assertion about d ≤
2. When d ≤
2, the famous uniqueness result of Aizenman and Wehr[4] for the infinite volume Gibbs state implies thatlim k →∞ h σ i i Λ i,k , + = lim k →∞ h σ i i Λ i,k , − . LT FOR THE RFIM FREE ENERGY 17
This, together with FKG, implies that instead of (3.2) we have the strongerestimate E (sup γ |h σ i i Λ ,γ − h σ i i Λ i,k , + | ) ≤ E |h σ i i Λ i,k , − − h σ i i Λ i,k , + | → k → ∞ . The rest of the proof goes through as before.4.
Proof of Theorem C will denote any positive constant that depends only on d and the random field distribution. The value of C may change from lineto line or even within a line. As before, we will only present the proof forthe plus boundary condition, since the argument for the minus boundarycondition is the same.Fix a finite nonempty set Λ ⊆ Z d and consider the RFIM on Λ withplus boundary condition. By the assumed condition on the random fielddistribution, the random field φ i at a site i ∈ Λ can be expressed as u ( Z i ),where ( Z i ) i ∈ Λ are i.i.d. standard normal random variables and u is a Lips-chitz map. Moreover, since the random field distribution is continuous, theground state is unique with probability one. Let ˆ σ denote the ground stateand let G denote the energy of the ground state. Let F β denote the freeenergy at inverse temperature β and let h σ i i β denote the expected value of σ i at inverse temperature β (on Λ, under plus boundary condition). Thenit is not hard to show that G = lim β →∞ F β . Moreover, by the uniqueness of the ground state, it follows easily that almostsurely, ˆ σ i = lim β →∞ h σ i i β . (4.1)Let ∂ i F β be the derivative of F β with respect to Z i . Then ∂ i F β = − u ′ ( Z i ) h σ i i β . Thus, with probability one,lim β →∞ ∂ i F β = − u ′ ( Z i )ˆ σ i . Call the above limit ∂ i G . It is now easy to see from Proposition 2.7 thatTheorem 2.6 may be applied to the function G , treating ∂ i G as its partialderivative with respect to Z i .For each i ∈ Z d , let ˆ σ ki be the ground state value of the spin at site i inthe RFIM on a box of side-length 2 k + 1 centered at i with plus boundarycondition. By (4.1) and the FKG property of the RFIM, it follows (similarlyas in the proof of Theorem 1.1) that ˆ σ ki ≥ ˆ σ k +1 i for all k . Letˆ σ ∞ i := lim k →∞ ˆ σ ki . In particular, if we let δ k := E | ˆ σ ki − ˆ σ ∞ i | , (4.2)then lim k →∞ δ k = 0 . Now fix some k . If Λ ′ is defined as in the proof of Theorem 1.1, then for any i ∈ Λ ′ , E | ˆ σ i − ˆ σ ki | ≤ δ k . Let g i := − u ′ ( Z i )ˆ σ ki . Then the above inequality shows that when i ∈ Λ ′ , ǫ ,i := k ∂ i G − g i k L ≤ Cδ / k . When i ∈ Λ \ Λ ′ , we trivially have ǫ ,i ≤ C . Also, clearly, m ,i := k ∂ i G k L ≤ C. Let N i be as in the proof of Theorem 1.1. From [20], we know that σ := Var( G ) ≥ C | Λ | . Armed with these estimates, we may now proceed as in the proof of Theo-rem 1.1, and using Theorem 2.6 instead of Theorem 2.4, we get d TV ( µ, ν ) ≤ Cδ / k + Ck d | ∂ Λ || Λ | + Ck d/ p | Λ | , where µ is the law of ( G − E ( G )) /σ , and ν is the standard normal distribu-tion.Let G n and Λ n be as in the statement of Theorem 1.2. Let µ n be the lawof ( G n − E ( G n )) / p Var( G n ). Since | ∂ Λ n | = o ( | Λ n | ) as n → ∞ , the abovebound shows that lim sup n →∞ d W ( µ n , ν ) ≤ Cδ / k . However, k is arbitrary, and δ k → k → ∞ . This shows that µ n convergesto ν in the Wasserstein metric.To complete the proof of Theorem 1.2, it only remains to show that theratio Var( G n ) / | Λ n | tends to a finite nonzero limit. As before, fix k ≥ N i and g i be as above. Consider g i as a function of ( Z j ) j ∈ Λ . For each i ,let Z ′ i be an independent copy of Z i , and for each 0 ≤ t ≤
1, let Z ti := √ tZ i + √ − tZ ′ i . Let g ti be the value of g i after replacing each Z j by Z tj . Then the quantity S of Theorem 2.6 is simply Z √ t X i ∈ Λ g i g ti dt. This can be rewritten as S = X i ∈ Λ S i , LT FOR THE RFIM FREE ENERGY 19 where S i := Z √ t g i g ti dt. By the definitions of g i , N i and Λ ′ , it follows that E ( S i ) is the same for all i ∈ Λ ′ . Without loss of generality, suppose that the origin 0 is in Λ ′ . Thus, E ( S ) = | Λ ′ | E ( S ) + X i ∈ Λ \ Λ ′ E ( S i ) . As in the proof of Theorem 1.1, this gives | E ( S ) − | Λ | E ( S ) | ≤ Ck d | ∂ Λ | . (4.3)Moreover, it is clear that when 0 ∈ Λ ′ , E ( S ) depends only on k , d and therandom field distribution, and not on Λ. On the other hand, by the firstinequality of Theorem 2.6, | Var( G ) − E ( S ) | ≤ X i ∈ Λ (2 ǫ ,i m ,i + ǫ ,i ) , where m ,i = k ∂ i G k L and ǫ ,i = k ∂ i G − g i k L . Proceeding as in the proofof Theorem 1.1, this gives | Var( G ) − E ( S ) | ≤ Cδ / k | Λ | + Ck d | ∂ Λ | , (4.4)where δ k is now defined as in (4.2).Let G n and Λ n be as in the statement of Theorem 1.2. By (4.3) and (4.4),it follows that for each k , there is some number a k depending only on k , d and the random field distribution, and not on the sequence { Λ n } n ≥ , suchthat lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) Var( G n ) | Λ n | − a k (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ / k . It is now easy to complete proof as in the last part of the proof of Theo-rem 1.1. The case d ≤ Acknowledgments
I thank Persi Diaconis for a number of useful comments, and Nguyen TienDung for pointing out some omissions in the first draft. I also thank theanonymous referees for several useful suggestions.
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