Central Limit Theorems for Cavity and Local Fields of the Sherrington-Kirkpatrick Model

One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature regime of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient approach for obtaining the limiting distributions, the nature of this method restricts the derivation of optimal and general results. In this paper, our study based on Gaussian interpolation obtains the CLT for the cavity field. With the help of this result, we conclude the CLT for local fields. In both cases, more refined moment estimates are given.


The Sherrington-Kirkpatrick Model and TAP Equations
Let N be a positive integer. Consider the space of configurations Σ N = {−1, +1} N . The elements σ = (σ 1 , . . . , σ N ) ∈ Σ N are called spin configurations and σ i 's are called spins. Suppose that g = {g ij } 1≤i<j≤N are i.i.d. standard Gaussian random variables with g ij = g ji , which is called the disorder. For a given realization of g, we define the Hamiltonian H N , with inverse temperature β > 0 and external field h ∈ R, by for σ ∈ Σ N and we then define the Gibbs measure G N on Σ N by where Z N is the normalizing constant, called the partition function. The model we just defined here is the so-called Sherrington-Kirkpatrick (SK) model [2] and our study in this paper will concentrate on the high temperature region only. We use σ 1 , σ 2 , etc. to denote configurations chosen independently from the Gibbs measure (with the same given disorder). These are also called replicas in physics. Given a function f on Σ n N = (Σ N ) n , the quenched average of f on the product space (Σ n N , G ⊗n N ) is defined as f = σ 1 ,...,σ n f (σ 1 , . . . , σ n )G N ( σ 1 ) · · · G N ({σ n }).
One of the main approaches to studying the SK model in the high temperature phase involves the overlap of the configurations σ 1 , σ 2 , It turns out that in the limit, as N tends to infinity, this quantity will converge a.s. to a constant q, which is the unique solution to where z is a standard Gaussian random variable. More precisely, concluding from Theorem 1.4.1 in Talagrand [4], for fixed β 0 < 1/2, we have for every β ≤ β 0 and h, where K is a constant depending on k and β 0 only. By using replicas, many quantities that will be used in our study can be controlled through (3) (see Section 1.10 of [4] for details): Letσ i j = σ i j − σ j . If we define then, for fixed β 0 < 1/2 and k ∈ N, there exists some K depending on β 0 and k only such that for any β ≤ β 0 and h, max 1≤i,j≤N,i =j Since each spin takes only two values, the Gibbs measure can be completely determined by the quenched averages σ 1 , . . . , σ N . This observation provides us another main approach to studying SK model in the high temperature regime via the Thouless-Anderson-Palmer (TAP) system of equations as outlined in [5]: Here ≈ means that two quantities are approximately equal with high probability. The first rigorous proof of the validity of TAP equations in the high temperature phase based on the remarkable formulation of the cavity method was established by Theorem 2.4.20 in Talagrand's book [3]. Later in the forthcoming edition [4] of [3], Corollary 1.7.8 implies that all N equations hold simultaneously with high probability. In view of the cavity method, the idea is to reduce the N-spin system to a smaller system by removing a spin. More precisely the procedure can be described as follows. Recall formula (1) for the Hamiltonian H N on Σ N with inverse temperature β and external field h. Let 1 ≤ i ≤ N be a fixed site and write where H N −1 is the Hamiltonian for the (N − 1)-spin system defined by for ρ = (σ 1 , . . . , σ i−1 , σ i+1 , . . . , σ N ) ∈ Σ N −1 and β − = (N − 1)/Nβ, and l i is defined as Here, for convenience, the dependence of H N −1 on the i-th spin remains implicit and we use · − to denote the quenched average induced by H N −1 . Then the following fundamental identity holds Under · − , we call l i the cavity field (slightly different from the cavity field defined in Section 1.2.1) and under · , we call l i the local field. Therefore, to prove the TAP equations, we are led to the important study of the limit laws for cavity and local fields.
The key observation to establish the limit law for the cavity field is that from the definition of l i , {g ij } j≤N,j =i is independent of the randomness of · − , which motivates our approach by using Gaussian interpolation on the cavity field. Consequently, we then deduce the limit law for the local field. In both cases, the quantitative results for the moment estimates are given and will be stated in the following section.

Main Results
In the rest of the paper φ µ,σ 2 stands for the Gaussian density with mean µ and variance σ 2 . Suppose U : R d → R is continuous. We say that U is of moderate growth if lim x →∞ U(x) exp (−a x 2 ) = 0 for every a > 0.

Limit Law for the Cavity Field
Suppose g 1 , . . . , g N are i.i.d. standard Gaussian r.v.s independent of the randomness of {g ij } i<j≤N . Define the cavity field l by The name "cavity field" is due to the important role of l played in the cavity method as we have already seen in Section 1.1. Note that the quenched average of l is The limit law of the centered distribution l−r under the Gibbs measure was firstly studied by Talagrand and it is approximately a centered Gaussian distribution with variance 1−q.
The exact quantitative result is given by Theorem 1.7.11 in Talagrand [4].
Theorem 1. [4, Theorem 1.7.11] Let β 0 < 1/2 and k ∈ N. Suppose that U is an infinitely differentiable function defined on R and the derivatives of all orders of U are of moderate growth. Then for any β ≤ β 0 and h, we have where K is a constant depending on k, U, and β 0 only.
Note that here and the Theorem 2 and 3 below, we put strong condition on U for convenience. In fact, as we can see, from the proofs of Theorem 1, 2, and 3, this strong condition can be easily released.
By applying Theorem 1, Talagrand proved the TAP equations, see Theorem 2.4.20 in [3] and Theorem 1.7.7 in [4]. However, his argument relies heavily on the special property of the exponential function and it seems impossible to deduce the limit law for local fields from Theorem 1. To overcome this difficulty, it would be very helpful if we have good quantitative results for the limit law of l, which is also one of the research problems proposed by Talagrand ([4], Research Problem 1.7.12). In our study we prove that the limit of l is still concentrative and under the Gibbs measure, l is approximately Gaussian with mean r and variance 1 − q. Our main quantitative result is stated as follows.
Theorem 2. Let β 0 < 1/2 and k ∈ N. Suppose that U is an infinitely differentiable function defined on R and the derivatives of all orders of U are of moderate growth. Then for any β ≤ β 0 and h, we have where K is a constant depending on k, U, and β 0 only.
In [3], p. 87 and also [4], Section 1.5, Talagrand gave intuitive arguments to support that the limit law of the cavity field is approximately a Gaussian distribution with center r and variance 1 − q. Indeed, he proved that when k = 1, the left-hand side of (11) should be small without any error bound. Later on Chatterjee [1] based on Stein's method obtained the first quantitative result that when k = 1 and U is a bounded measurable function U, the left-hand side of (11) has an error bound c(β 0 ) U ∞ / √ N, where c(β 0 ) is a constant depending on β 0 only. Hence, Theorem 2 justifies Talagrand's conjecture and also improves Chatterjee's error bound if suitable smoothness on U is assumed. Informatively, as we will see, Theorem 2 is the bridge to study the limit law for local fields, which is the main advantage that Theorem 1 does not have.

Limit Law for the Local Fields
Recall that the local field l i at site i is defined by formula (7). For 1 ≤ i ≤ N, suppose that ν i is a random measure, whose density is the mixture of two Gaussian densities Then we prove that the local field l i under the Gibbs measure is close to ν i in the following sense: Theorem 3. Let β 0 < 1/2 and k ∈ N. Suppose that U is an infinitely differentiable function defined on R and the derivatives of all orders of U are of moderate growth. Then for any 1 ≤ i ≤ N, β ≤ β 0 , and h, where K is a constant depending on β 0 , k, and U only.
Again, by applying Stein's method, Chatterjee [1] proved the first quantitative result regarding the limit law for the local fields that when k = 1 and U is a bounded measurable function U, the left-hand side of (13) has an error bound c(β 0 ) U ∞ / √ N, where c(β 0 ) is a constant depending on β 0 only. Thus, Theorem 3 improves this error bound if we require some smoothness on U. Recall formula (8). If we put U(x) = tanh(βx + h) and modify the proof for Theorem 3 slightly, then we obtain the same quantitative result for the TAP equations in Talagrand [4, Theorem 1.7.7] : Corollary 1. Let β 0 < 1/2 and k ∈ N. Then for each 1 ≤ i ≤ N, any β ≤ β 0 , and h, we have where K is a constant depending on β 0 , and k only.
Proof. Let us notice a useful formula from [1], equation (9), which is stated as follows.
Suppose that X is a random variable, whose density is the mixture of two Gaussian densities: Then a straightforward computation yields Hence, if we apply Theorem 3 and use (14) with a = β and b = h, then we get the announced result. Note that the constant K still does not depend on h, which can be verified by going through the proof for Theorem 3 and using the uniform boundedness of U(x) = tanh(βx + h).

Proofs
This section is devoted to proving Theorems 2 and 3. In the following proofs, the constants K, without mentioning specifically, are always assumed to satisfy the requirements of the corresponding announced statements. Note that we use E ζ to stand for the expectation with respect to the randomness of ζ when ζ is a random variable.

Proof of Theorem 2
Using replicas, we set for 1 ≤ i ≤ 2k, Suppose that ξ, ξ 1 , . . . , ξ 2k are i.i.d. Gaussian r.v.s with mean zero and variance 1 − q and they are independent of {g j } j≤N and {g ij } i<j≤N . Recalling definitions (9), (10), and (15), we consider the Gaussian interpolations, Suppose that U is a real-valued function defined on R and is of moderate growth. Define and Notice that ψ(0) = 0 and ψ(1) is equal to the left-hand side of (11). Now, the main idea used to prove Theorem 2 is to control the derivatives of ψ up to some optimal order by using Gaussian integration by parts. In order to justify the application of this calculus result, Lemma 1 is needed. It says that mild composition and integration of functions of moderate growth are still of moderate growth. On the other hand, to differentiate ψ, we need the differentiability of V , which is the reason why we establish Lemma 2 below.
Then all of the functions defined above are of moderate growth.
Proof. It is easy to see that U 0 , U 1 , and U 2 are of moderate growth. For U 3 , since as |x| → ∞, for all a > 0, it follows that U 3 is also of moderate growth. The function U 4 is of moderate growth since lim sup for all M > 0 and also the fact that U is of moderate growth. Since Thus, Since U is of moderate growth, U(y) exp − by 2 2σ 2 can be regarded as a bounded function in y. On the other hand, since is finite and independent of x, by taking b to be small enough and letting |x| tend to infinity, we conclude, from (19), that U 5 is of moderate growth. This completes the proof. Proof. For any x, x ′ , y ∈ R, by the mean value theorem, we can find some z(x, x ′ , y) between x and x ′ so that U(x + y) − U(x ′ + y) = U ′ (z(x, x ′ , y) + y)(x − x ′ ). Since U ′ is of moderate growth, for any M 1 , M 2 > 0, By the continuity of and by the dominated convergence theorem, ψ ′ (x) = E ξ U ′ (ξ + x).
In view of Theorems 1, 2, and 3, the constants K are independent of N and β. The main reason is because, when conditioning on the randomness of {g ij } i<j≤N , the cavity field and its quenched average are centered Gaussian distributions, whose variances are bounded above by some constants, which are independent of {g ij } i<j≤N , N, and β. Therefore, we still have good control on the moment estimates of the cavity field and its quenched average. This observation will be used repeatedly and for convenience, we formulate it as Lemma 3. ii) For N ∈ N, suppose that is a family of random variables such that |X N j,β | ≤ K 1 for 1 ≤ j ≤ N, and β ∈ J.
iii) Let f 1 , f 2 : N × I × J → R be measurable functions such that iv) Let U : R → R be a continuous function. Suppose that there are some A > 0 and 0 < a < min {(8K 2 Then there is a constant K > 0 such that where E 0 means the expectation with respect to g N j : j ≤ N, N ∈ N and z.
Proof. From the given conditions, we obtain Now integrating this inequality, we find the left-hand side of (20) is then bounded above by which is a finite constant independent of N and β since a satisfies iv).
In the sequel, we use E 0 to denote the expectation with respect to the randomness of {g j } j≤N and {ξ i } i≤2k . Recall formulas (10) and (16) for r and u i , respectively. The following lemma, as an application of Gaussian integration by parts, is our main equation to control the derivatives of all orders of ψ.
Lemma 4. Let k ∈ N. Suppose that V 1 , V 2 , . . . , V 2k : R 2 → R are twice continuously differentiable functions and their first and second order partial derivatives are of moderate growth. Define Then ϕ is differentiable on (0, 1) and Here, ∂ a+b V i ∂ a x∂ b y means that we differentiate V i with respect to the first variable a times and with respect to the second variable b times.
Proof. To prove the differentiability of ϕ, it suffices to prove, with the help of the mean value theorem and the dominated convergence theorem, that for 0 < δ < 1/2 and 1 ≤ i ≤ 2k, for some constant K. Note that Since ∂V i ∂x and V j are of moderate growth, for any a > 0, there exists some A > 0 such that Set z, g N j = ((1 − q) −1 ξ i , g j ) and also let X N j,β , f 1 (N, t, β), f 2 (N, t, β), U(x) be any one of the following vectors Then by choosing a small enough and applying Lemma 3, there exists a constant K independent of β and N such that Therefore, from (23) and Cauchy-Schwarz inequality, and by Hölder's inequality, (22) holds.
Remark 1. Formula (21) implies that our computation on the derivative of ϕ can be completely determined by T i,i , T i,j , and T i in the following manner. Each T i,i is associated ∂V j ∂y (u j (t), r) for some j = i. This observation will be very useful when we control the derivatives of all orders of ψ. Lemma 4 will be used iteratively up to some optimal order. Since on each iteration, equation (21) brings us many terms, we will finally obtain a very huge number of summations. Therefore, in order to make our argument clearer, we formulate the following result.

(28)
Proof. It suffices to prove that Indeed, if (29) holds, then (28) can be deduced by applying (0) and (27) to (29). Let us prove (29) by induction on m. If m = 1, from (26), (29) holds clearly by Suppose that the announced result is true for m − 1 ≥ 1. Let ψ and ψ sn be real-valued smooth functions for every s n ∈ {0, 1} n with n ≤ m + 1 satisfying the assumptions of this lemma. Notice that from (27), we obtain and also by induction hypothesis,  Proof of Theorem 2 : Basically our proof is based on the same idea as the proof of Theorem 1. Recall formulas (17) and (18) for V and ψ, respectively. From Lemmas 1 and 2, V is an infinitely differentiable function and the partial derivatives of all orders of V are of moderate growth. We also note that ψ is infinitely differentiable by applying the same argument as Lemma 4. Recall the definition of E 0 and use Fubini's theorem, ψ(t) is equal to

Proof of Theorem 3
The following proposition is the key to proving Theorem 3.
Proposition 1. Let β 0 < 1/2 and k ∈ N. Suppose that U is an infinitely differentiable function defined on R and the derivatives of all orders of U are of moderate growth. Recall l and r as defined by (9) and (10). Then for any β ≤ β 0 and h, where ξ is a centered Gaussian distribution with variance 1 − q and K depends on β 0 , k and U only.
Recall that q is defined by (2). We also define q − as the unique solution of q − = E tanh 2 (β − z √ q − + h), where β − = (N − 1)/Nβ and z is a standard Gaussian distribution. Notice that the existence and uniqueness of q − are always guaranteed since we only consider the high temperature region, that is, β − < 1/2. Also, recall the quantity γ i from (12).
Lemma 6. There is a constant L > 0 so that for every β < 1/2, h, and N. Let β 0 < 1/2 be fixed. Then for every 1 ≤ i ≤ N, β ≤ β 0 , and h, where K is a constant depending only on β 0 and k.
Proof of Theorem 3 : By symmetry among the sites, it suffices to prove (13) is true when i = N. Recall from (7) and (12), for simplicity, we set l = l N , γ = γ N . We also set where · − is the Gibbs measure with Hamiltonian (6) and inverse temperature β − = (N − 1)/Nβ. From Proposition 1, we know where K is a constant depending on β 0 , k, and U only. The goal of the proof is then to prove that (39) is very close to (13). We divide our estimates into several steps.
Note that β − l − = βl. Therefore, and this quantity is very close to U(l) in the sense that Indeed, the first inequality is true by the use of Jensen's inequality. The second inequality holds by using the mean value theorem and and then applying Lemma 3.