Fluctuations for block spin Ising models

We analyze the high temperature ﬂuctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet et al. in [2]. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we prove a non-standard CLT for the magnetization.

To define our model, we partition the set {1, . . . , N } for N even into a set S ⊂ {1, . . . , N } with |S| = N 2 and its complement S c . This segmentation induces a partitioning of the binary hypercube {−1, +1} N , N ∈ N, the state space of the Ising block model. For β > 0 and 0 ≤ α ≤ β the model we will consider is defined by the Hamiltonian Here we write i ∼ j, if either i, j ∈ S or i, j ∈ S c and i ∼ j otherwise. This Hamiltonian (1.1) A closely related version of this model has been investigated in [19]. However, the couplings in [19] between the blocks have the same strength of interaction as the couplings within a block. We were informed that a more general version of the model will be studied in [20]. This observation is not only a convenient way to analyze the block spin Ising model, it also makes m an obvious choice to describe its behaviour and its phase transitions. To characterize them, recall that with the above notation for α = β one reobtains the Curie-Weiss or mean-field Ising model at inverse temperature β, i.e. the model on {−1, +1} N given by H CW (σ) = 1 2N i,j σ i σ j and Gibbs measure µ CW N,β (σ) = e −βH CW (σ) . Also, recall ( [11]) that the Curie-Weiss model undergoes a phase transition at β = 1. This phase transition can be described by saying that the distribution of the parameter m = 1 N i σ i (also called the magnetization) weakly converges to the Dirac measure in 0, δ 0 , if β ≤ 1 while it converges to the mixture 1 is the largest solution of the so-called Curie-Weiss equation z = tanh(βz). A similar result was proven for the block spin Ising model in [2] (the authors also allow for negative values of α).

Remark 1.3.
It is well known that in the standard Curie-Weiss a CLT also holds in the presence of an external field [12], i.e. with a Hamiltonian of the form H CW (σ) =

2N
i,j σ i σ j + h i σ i with h > 0. We are firmly convinced that a similar result is true for our model as well. However, we did not try to prove it. Finally, one might ask, whetherin the spirit of [10] -Stein's method may be applied to our situation as well. We consider this a more challenging question, because a multi-dimensional version of Stein's method would be needed. We may consider this problem in a different paper.
On the other hand, if β + α = 2 the fluctuations are no longer Gaussian We will show Theorems 1.2 and 1.4 in Section 3 using a Hubbard-Stratonovich transformation for an appropriate function of m. Before, in Section 2, however, we will give an alternative proof of Theorem 1.1 using the theory of large deviations. This is not only interesting in its own right, but also provides a way to derive limit theorems in more complicated settings, see e.g [21].

An LDP for the vector of block magnetizations
Differing from the line of arguments in [2], Theorem 1.1 can also be shown by proving a large deviation principle (LDP) for 0 ≤ α ≤ β. To this end, we will slightly change our variables and consider the vector v = (v 1 , v 2 ) with v 1 := 1 2 m 1 and v 2 := 1 2 m 2 . We show This implies that the convergence in Theorem 1.1 (for 0 ≤ α ≤ β) is exponentially fast.
Proof. We will prove this theorem in two steps, first we show the LDP, then, how one derives Theorem 1.1 from it.
Step 1: First, note that the case α = 0 is trivial. Then, the system consists of two independent Curie-Weiss models on N 2 spins at temperature β. The LDP for the magnetization in each of the systems is known (cf. e.g. [11]) and transferring these LDPs to the vector v (with independent components) is trivial. We will thus assume that α > 0.
Let us consider the moment generating function of the vector v. To this end let t = (t 1 , t 2 ) ∈ R 2 . Then the moment generating function of v in t is given by where here E denotes the expectation with respect to the a priori measure 1 2 (δ −1 + δ +1 ). This readily yields lim N →∞ . As the right hand side of this expression is finite and differentiable on all of R 2 , by the Gärtner-Ellis Theorem [8, Theorem 2.3.6] this computation implies an LDP for v under the uniform distribution with speed N and rate function . Now the Hamiltonian H N,α,β,S (σ) of our model can also be rewritten in terms of v: This fact, together with the above LDP and the exponential form of the Gibbs measure and the LDP for integrals of exponential functions (see e.g. [ where F v : R 2 → R is given by (2.1). A change of variables yields the desired LDP.
Step 2: are open balls of radius ε > 0 centered around y) we obtain from the upper bound of the LDP that for N large enough. The inf on the right hand side of the inequality is positive. In this sense, v concentrates in the minima of J v exponentially fast. By a change of variables, again, this implies that m concentrates exponentially fast in the (global) minima of J m defined by Note that the vector (0, 0) is always a solution to this system of equations and hence a critical point of F m −J.
We start with the high temperature regime, i.e. we consider β + α < 2. By an easy calculation we find that the Hesse matrix of F m (x, y) −J(x, y) is given by Next, we will see that, in this case, the point (0, 0) is the only solution to the system of equations in (2.2), and hence the global maximum of F m (x) −J(x). To this end, we rewrite the equations in (2.2) as Hence, for |x| < 1 and f (x) := 2 α artanh(x) − 1 2 βx we have This means we are looking for the fixed points of f 2 . We note that for 0 ≤ β ≤ 2, we have for all |x| < 1 The fixed points of f 2 are thus the same as the fixed points of (f −1 ) 2 resp. of (f −1 ). We see that f −1 is a strong contraction for α + β < 2 from (f −1 ) = 1 f and for all y ∈ (−1, 1) for ε > 0 small enough. Thus, by Banach's fixed point theorem, there is a single fixed point, which has to be equal to zero.
Next consider the critical line β + α = 2. Here the arguments are almost the same.
The only difference is, that now 1 f (y) y=0 = 1, while 1 f (y) < 1 for all other y. Hence f −1 is a weak contraction for α + β = 2. However, the magnetizations m 1 and m 2 live on the compact interval [−1, 1], such that we can again conclude that f −1 has the unique fixed point 0.
Now we consider α + β > 2. In this case (0, 0) is still a solution to (2.2). However, in this case, it is either a saddle point or a local minimum of F m −J, because it is not a maximum. Indeed, choose x = y, i.e. This proves the claim.

Remark 2.2.
In the spirit of [19], one can also allow for other sizes of S, i.e. for 0 < γ < 1 we can consider sets S ⊂ {1, . . . , N } with |S| = γN . Here, we assume, for simplicity, γN to be an integer. In this case, the Hamiltonian H N,α,β,S is the same as in (1.1) We believe that a result analogue to Theorem 1.1 can be shown by generalizing the large deviation techniques in the proof of Theorem 2.1. In the same spirit as in Theorem 1.2 and Theorem 1.4, one can also show a Central Limit Theorem for this generalized setting. The technical problems are, however, more demanding. We will return to these questions in a later publication.

Proof of Theorem 1.2 and 1.4
The proofs of Theorems 1.2 and 1.4 rely on the same idea. We will first prove limit theorems for two other parameters, that are closely related to m 1 and m 2 . To this end Note that m 1 = w 1 + w 2 and m 2 = w 1 − w 2 and thus limit theorems for w = (w 1 , w 2 ) will imply limit theorems for m and vice versa. Again, note that the Hamiltonian H N,α,β,S can also be rewritten in terms of the variables w 1 and w 2 resp. in terms ofw 1 andw 2 as Next we will show a Central Limit Theorem for the vectorw := (w 1 ,w 2 ) in the high temperature region 0 ≤ α < β < 2 and α + β < 2. Proof. Our principal strategy consists of computing a suitable Hubbard-Stratonovich transform of our measure of interest (as e.g. in [18]) and expanding it. To this end, let N (0, C) denote a two-dimensional Gaussian distribution with expectation 0 and covariance matrix C given by C = Here, we used K 1 := Now recall the second order Taylor expansion log cosh(x) = 1 2 where the constant in the O(N −1 )-term depends on x and y. However, the convergence is uniform on compact subsets of R 2 . Thus To turn this into a weak convergence statement, we need to control integrals over unbounded sets as well, in particular, we need to treat the case A = R 2 to see that K 3 converges to Hence, for any measurable set A ⊂ R 2 we write where we set Ψ(x, y) := e −Φ(x,y) .
Here for any l > 0 we denote by B(0, l) the ball in R 2 with center in 0 and radius l. Further, we consider numbers R > 0 and r > 0 and we will send R to ∞ and consider r sufficiently small. We will refer to the summands on the right hand of (3.2) as inner region, intermediate region and outer region, respectively. The goal is to see that the inner region contributes all mass to the integral as R → ∞.
Fluctuations for block spin Ising models As already marked above for fixed R > 0 Next, we treat the outer region. Let us rewrite the exponent in this case as AnalyzingΦ we see that it becomes minimal only if ∇Φ = 0 and where we abbreviate c 1 := α + β and c 2 := β − α. This means, we aim to solve This is done in the spirit of the arguments in Section 2. Indeed, denoting by G(x, y) := 1 2 tanh( c1x+c2y we see that its Jacobian is given by This means that ||J G (x, y)|| 1 ≤ max c1 2 , c2 2 = c1 2 < 1, where · 1 denotes the maximum absolute column sum of a matrix. Thus G is a (strict) contraction with fixed point (0, 0) and henceΦ is minimal in (0, 0). Therefore, for every r > 0. inf for any r > 0. Therefore, the outer region is asymptotically negligible.
Let us turn to the intermediate region.
Here we take again a Taylor expansion of the log cosh on an interval [−z 0 , z 0 ], z 0 > 0, around the origin to first order with a Lagrange bound on the remainder: log cosh(z) = z 2 2 + Cz 4 with a constant C that depends on z 0 .

This yields for
where we used α+β forC r uniformly on B(0, r √ N ). Note thatC r r 2 depends continuously on r and converges to 0 as r → 0. In particular, if r is small enough we have that α+β 2 − α+β But for this choice of r andC r we arrive at − C r r 2 dxdy and the right hand side is an integrable function. Thus for R → ∞ the right hand side as well as the left hand side converges to 0.
Putting the estimates together, we have seen that χ N,α,β converges weakly to the 2-dimensional Gaussian distribution with expectation 0 and covariance matrix Σ , where Σ is given in (3.1). This weak convergence is equivalent to the convergence of the characteristic functions.
Computing the characteristic functions of the Gaussian distribution involved in the above proof, we have therefore shown that the characteristic function ofw in the point Turning this into a weak convergence statement again, we obtain Proof of Theorem 1.2. The proof of Theorem 1.2 is straightforward from the above lemma. As observed we have that m 1 = w 1 + w 2 and m 2 = w 1 − w 2 , thus √ N m 1 = √ N (w 1 + w 2 ) =w 1 +w 2 as well as √ N m 2 = √ N (w 1 − w 2 ) =w 1 −w 2 . Thus Lemma 3.1 gives that m 1 and m 2 are asymptotically normal with mean 0 and variance Moreover, the same considerations together with Lemma 3.1 show that their covariance is given by On the other hand, the proof Lemma 3.1 also inspires the proof of Theorem 1.4.
Indeed, redoing the computations there shows that for α + β = 2 the quadratic term in the first component of χ N,α,β cancels. To this end we have to rescalew 1 to make the second term in the Taylor expansion of log cosh appear (as a matter of fact this is very similar, to what happens in the Curie-Weiss model at its critical temperature β = 1).
Proof of Theorem 1.4. As motivated above we will now consider the vectorŵ = (ŵ 1 ,ŵ 2 ) consisting of the componentŝ This time we will convolute the distribution ofŵ under the Gibbs measure µ N,α,β,S with a two-dimensional Gaussian distribution N (0,Ĉ), whereĈ = 1 √ N 0 0 2 β−α (note that this is well defined since β > α). Computing the density ofχ N,α,β := µ N,α,β (ŵ) −1 * N (0,Ĉ) as in the proof of Lemma 3.1 we obtain building on the fact that now α + β = 2 with the normalizing constantsK 1 ,K 2 , andK 3 chosen similarly to K 1 , K 2 , and K 3 in the proof of Lemma 3.1. Now we expand the log cosh to fourth order: log cosh(z) = z 2 2 − 1 12 z 4 + O(z 6 ). We thus see that the x 2 terms in the exponent cancel and so do the xy-terms (fortunately). For fixed x and y only the x 4 is of vanishing order. The y 2 terms are treated as in the proof of Lemma 3.1. We thus see that with a O(N − 1 2 ) term that depends on x and y. To conclude the convergence ofχ N,α,β (A) we now proceed as in the proof of Lemma 3.1. Here we will only sketch the differences, because many steps are very similar. The exact steps are left to the reader. The main differences to the above proof of Lemma 3.1 is that the inner region is again B(0, R), is given by exp(−d 1 x 4 − d 2 y 2 ) for suitable constants d 1 , d 2 > 0. With these changes we see thatχ N,α,β (A) converges to a 2-dimensional distribution with density proportional to exp − 1 12 with respect to the 2-dimensional Lebesgue measure. However, from here we see that w 1 converges in distribution to a random variable with density proportional to e − 1 12 x 4 since the Gaussian measure we convoluted the first coordinate ofŵ with converges to 0 in probability. Moreover, the same computation as in Lemma 3.1 shows thatŵ 2 =w 2 converges to a normal distribution with mean 0 and variance 2 2−(β−α) .
However, the latter convergence implies that N