Local Central Limit Theorem for Multi-Group Curie-Weiss Models

We define a multi-group version of the mean-field spin model, also called Curie-Weiss model. It is known that, in the high temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high temperature regime.


Introduction
The Curie-Weiss model is a model of ferromagnetism. In its classic form it is defined through a probability distribution on the set of spin configurations {−1, 1} n , given by where Z is a normalisation constant that depends on n and β. The parameter β is called the inverse temperature. It induces correlation between individual spins, causing spins to align in the same direction. At low values of β ('high temperature'), the spins are 'nearly independent'. At high values of β ('low temperature'), the spins are strongly correlated. There is a critical value of β = 1, where the collective behaviour of spins changes. This is called a phase transition. The Curie-Weiss model has been well studied, and hence the literature is far too extensive to cite here. The model was first defined by by Husimi [14] and Temperley [24]. Discussions of it can be found in Kac [15], Thompson [25], and Ellis [5]. More recently, the Curie-Weiss model has been used in the context of social and political interactions. See e.g. [3,17,12,22,26]. Another area the Curie-Weiss model has found application is the study of random matrices (see [11,13,18,10,9,8]).
In this article, we deal with a multi-group version of this model. Multi-group versions of the Curie-Weiss model have also been studied recently. Some references are [2,7,1,6,23,20,21,19].
Let there be d ∈ N groups with n λ spins in group λ ∈ {1, . . . , d}, d λ=1 n λ = n. The spin variables are We assume that each of the d groups converges to a fixed proportion of the overall population: so that the α λ sum to 1.
Instead of a single inverse temperature parameter, there is a coupling matrix that describes the spin interactions. We will call this matrix Every spin in group λ interacts with every spin in group µ with a strength given by the coupling constant J λµ . Just as in the single-group model, there is a Hamiltonian function that assigns to each spin configuration a certain energy level: x λi x µj .
As we can see from the definition of H, it suffices to consider symmetric J, for otherwise we can replace J by J+J T 2 , leaving the Hamiltonian unchanged. Definition 1. The Curie-Weiss measure P, which gives the probability of each of the 2 n spin configurations, is defined by where each x λi ∈ {−1, 1} and Z is a normalisation constant which depends on n and J.
We distinguish two different classes of coupling matrices: 1. Homogeneous coupling matrices where all entries are equal to the same constant β ≥ 0.

Heterogeneous coupling matrices
which we assume to be positive definite.
Remark 2. Without the assumption of positive definiteness, the high temperature regime (see below) may be empty. For more details, see [19].
This model has three regimes: The high temperature, the critical, and the low temperature regime. In each regime, the spins behave differently and the limiting distribution for large n is different in each case. For each group λ, we define S λ := n λ i=1 X λi to be the sum of all spins belonging to that group. In this article, we show a local limit theorem for the normalised magnetisation vector in the high temperature regime. If the coupling matrix is homogeneous, then the high temperature regime is characterised by β < 1.
For heterogeneous coupling matrices, the situation is somewhat more complicated. Here, the parameter space is containing all possible combinations of asymptotic relative group sizes (α 1 , . . . , α d ) as in (1) and coupling matrices. We define where 'diag' stands for a diagonal matrix with the entries given between parentheses, and Note that this definition of a multi-group Curie-Weiss model reduces to the classical single-group model if we set d = 1, since then n 1 = n and J = β. See also Remark 11. The parameter space Φ is partitioned into three regimes (for details, see [19]).
We are only concerned with the high temperature regime.
Definition 3. The 'high temperature regime' for heterogeneous coupling matrices is the set of parameters In the high temperature regime, a multivariate central limit theorem holds for the normalised sums of spins in each group. For a proof see e.g. [19].
Theorem 4. In the high temperature regime, we have where N ((0, . . . , 0), C) is a zero-mean multivariate normal distribution with positive definite covariance matrix C, and '=⇒' stands for weak convergence.
Remark 5. In the central limit theorem above, the limiting distribution has the covariance matrix The matrix Σ depends on the class of coupling matrices: We shall write φ C for the density function of N ((0, . . . , 0), C), and we set For a given n ∈ N and group λ, S λ √ n λ takes values on the grid n λ +2Z √ n λ . Hence, the vector S1 √ n1 , . . . , S d √ n d takes values on the grid We show that the central limit theorem above can be strengthened to a multivariate local limit theorem: Theorem 6. In the high temperature regime, the following local limit theorem holds:

Proof
We first state two auxiliary lemmas.
The following two properties hold: and Proof. This follows from a straightforward modification of the proof Theorem 3.5.2 on page 140 in [4].
The second statement above gives us an upper bound for the characteristic function of a random variable on the grid, which we shall use in our calculations later on. We have the inversion formula, by which we can recover a discrete distribution from its characteristic function:

be a random vector as in the lemma above. Then for
For a proof see e.g. section 3.10 in [4]. For integrable characteristic functions ϕ, we also have an inversion formula: Lemma 9. Let ϕ be the characteristic function of some d-dimensional random vector with density function f , and assume ϕ is integrable. Then the inversion formula allows us to recover the density function f .
We use this inversion formula to show Theorem 6. Let ϕ S n be the characteristic function of S n and ϕ N (C) that of N ((0, . . . , 0), C). We use the symbol E as the expectation with respect to the probability measure P of the underlying probability space.
By the lemma, we have and, therefore, We see that the term (6) converges to 0 as n → ∞, since ϕ N (C) (t) is integrable. We also note that the expression (5) is independent of the point x ∈ L n . If we can show that (5) converges to 0, then we are done. To this end, we see by Theorem 4 that ϕ n S (t) → ϕ N (C) (t) pointwise. Thus, to show that (5) converges to zero is for the most part a matter of finding an appropriate integrable majorant, so that the theorem of dominated convergence can be applied. To construct a suitable majorant, we need to apply some properties of the multivariate Curie-Weiss distribution.
Let the Rademacher distribution with parameter m ∈ R R m be defined on {±1} by the probability of the event {1} equal to 1+m 2 settingm := tanh m.
We use the de Finetti representation of the Curie-Weiss measure (see [19]): Proposition 10. The distribution of the multi-group Curie-Weiss model has the following representation: For any spin configuration (x 11 , . . . , x dn d ), we have where P m is the product measure of Rademacher distributions with parameters m λ for all spins belonging to group λ.
The de Finetti measure µ J,n is defined by the density function In the high temperature regime, µ J,n has an asymptotic concentration property, such that for all δ > 0 there is a D > 0 with the property for large enough n.
Remark 11. If we set d = 1, then the de Finetti density above becomes proportional to exp −n 1 2β m 2 − ln cosh m , m ∈ R.
From this expression, we obtain the usual de Finetti density defined on [−1, 1] proportional to by the substitution t := tanh m. Cf. [16].
Let ϕ R(m) be the characteristic function of a centred Rademacher distribution with parameter m, i.e. the distribution of X − tanh m for a random variable X ∼ R m , and let E m be the expectation under that distribution. Now we deal with expression (5). We pick some 0 < δ < π/2 and partition the Our goal is to show that (5) converges to 0. We do so by showing the result separately over A n and B n .
The following upper bound holds over A n : We calculate an upper bound for the Rademacher characteristic function: The first inequality follows from a Taylor expansion of the exponential function with the remainder term of order three u 2 E m min |u| |X λ1 −m| 3 , |X λ1 −m| 2 , which is smaller or equal u 2 1 −m 2 min |u| 1 +m 2 , 1 as can be verified by direct calculation. The second inequality holds for small enough |u|. The third inequality for any u ∈ R is well-known. Therefore, and we pick some τ ∈ (0, 1) to continue with our calculation: where the second term in the last line follows from Lemma 10. Note that η > 0. It is clear that the first summand in (8) is integrable. For the second summand, we have Let λ d be the Lebesgue measure on R d . We show that the function f on the right hand side is an integrable majorant for all I An (t) exp (−ηn) , n ∈ N: Each summand in the series above can be bounded above by which is summable in k.
As ϕ N (C) (t) is integrable as well, we have found that I An (t) ϕ S n (t) − ϕ N (C) (t) has an integrable majorant. From the global central limit theorem 4, we know that ϕ S n (t) − ϕ N (C) (t) → 0 pointwise as n → ∞, so we conclude that the integral of I An (t) ϕ S n (t) − ϕ N (C) (t) over R d converges to 0 as n → ∞.
With this final upper bound for I Bn (t) |ϕ S n (t)|, we see that as n → ∞, due to the exponential decay of the terms s n and µ J,n R d \ [−τ, τ ] d .