Two Groups in a Curie-Weiss Model

We analyse a Curie-Weiss model with two disjoint groups of spins with homogeneous coupling. We show that similarly to the single-group Curie-Weiss model a bivariate law of large numbers holds for the normed sums of both groups’ spin variables. We also show central limit theorem in the high temperature regime.


Introduction
The Curie-Weiss model is probably the easiest model of magnetism which shows a phase transition between a diamagnetic and a ferromagnetic phase. In this model the spins can take values in {−1, 1} (or up/down), each spin interacts with all the others in the same way. More precisely, for finitely many spins (X 1 , X 2 , . . . , X N ) ∈ {−1, 1} N the energy of the spins is given by Consequently, in the 'canonical ensemble' with inverse temperature β ≥ 0 the probability of a spin configuration is given by The quantity is called the (total) magnetization. It is well known (see e. g. Ellis [4] or [10]) that the Curie-Weiss model has a phase transition at β = 1 in the following sense 1 where ⇒ denotes convergence in distribution, δ x the Dirac measure in x.
If β > 1 (5) has exactly three solutions and m(β) is the unique positive one. Equation (4) is a substitute for the law of large numbers for i.i.d. random variables. Moreover, for β < 1 there is a central limit theorem, i. e.
For β = 1 there is no such central limit theorem. In fact, the random variables 1 N 3/4 S N (7) converge in distribution to a limit which is not a normal distribution.
The Curie-Weiss model is also called the Husimi-Temperley model. It was first introduced by Husimi [7] and Temperley [15]. Subsequently it was discussed by Kac [9], Thompson [16], and Ellis [4]. More recently, the Curie-Weiss model has been used in the context of social and political interactions. See e.g. [3] and [11].
We mention that the Curie-Weiss model is also used to describe the behaviour of voters who have the choice to vote 'Yea' (spin=1, say) or 'Nay' (spin=-1) (see [11]).
In this paper, we partition the set of all N Curie-Weiss spins into two disjoint groups X 1 , . . . , X N 1 and Y 1 , . . . , Y N 2 with N 1 + N 2 = N. We let N 1 and N 2 depend on N in such a way that both N 1 and N 2 go to infinity as N does. We consider the asymptotic behaviour of the two-dimensional random variables ⎛ as N goes to infinity and prove results similar to the one-group case considered above. We will use the method of moments which was used in [12] for the one-group case. We rely on the method developed there. Note that all spins are coupled by the same constant β ≥ 0. We could also define the model using a homogeneous coupling matrix For an analysis of heterogeneous coupling matrices see [13]. We prove Above '=⇒' denotes convergence in distribution of the 2-dimensional random variable on the left hand side and m(β) is again the largest solution of (5), in particular m(β) = 0 for β ≤ 1 and m(β) > 0 for β > 1.

Remark 2
If we consider a model without interaction between the groups X i and Y j then the limit in (9) is For β < 1 we also have a central limit theorem. The covariance of the limiting normal distribution depends on the growth rate of N 1 and N 2 . We set and assume that these limits exist.
Theorem 3 (Central Limit Theorem) If β < 1, then where the covariance matrix C is given by In particular, for sublinear growth of either N 1 or N 2 , i. e. if α 1 = 0 or α 2 = 0, the standardized sums in (11) are asymptotically independent.
In the proof of both results we employ the moment method (see e. g. [1] or [10]). In [12] the method of moments is used to prove limit theorems for the (one-group) Curie-Weiss Model.
Thus, to show the convergence in distribution of a sequence (X n , Y n ) of twodimensional random variables to a measure μ on R 2 we prove that for all K, L ∈ N.
Equation (13) implies convergence in distribution if the moments of μ grow only moderately, namely if for some constants A and C and all K, L holds. For the multidimensional case of the moment method we refer to [14].
After publishing the first version of this paper on arXiv, we became aware of the articles [5] and [6] which contain the above results as special cases. The methods used by those authors are different from ours. In [6], the authors show a central limit theorem based on the assumption that a certain function has one or more global minima. The article [5] shows how for two groups, under asymmetric scaling of the sums of spins, a central limit theorem can be proved. We are grateful to Francesca Collet for drawing our attention to the papers [2], [5] and [6].

Preparation
We have to evaluate terms of the form To shorten notation we set The energy function H and hence the probability measure P are invariant with respect to permutation of indices. Thus we observe that whenever both i 1 , . . . , i K and j 1 , . . . , j L are pairwise distinct. Since where ρ(i) denotes the number of indices i K which occur an odd number of times in i. So, to compute sums of the form (15) we need good estimates of expectations ('correlations') as in (17). Such estimates are provided in [12] (see also [11]) .

Proposition 4
For K + L even we have 1) for β < 1: 3) for β > 1: where m (β) is the unique (strictly) positive solution of t = tanh(β t). For K + L odd: Above a n ≈ b n means lim

Laws of Large Numbers
To prove Theorem 1 we consider We assume β ≤ 1 first. By U K,N 1 we denote the set of those i ∈ W K,N 1 for which the i ν are pairwise distinct.
Note that for i ∈ U K,N 1 and j ∈ W L,N 2 we have by Proposition 4. We single out the following Lemma (where |M| denotes the cardinality of the set M). We estimate (18) For β > 1 we prove in a similar way that which are the moments of the measure 1 2 δ (−m(β),−m(β)) + δ (+m(β),+m(β)) .

The Central Limit Theorem
We start this section by computing the moments of a centred two-dimensional normal distribution. Let (Z 1 , Z 2 ) be normally distributed with mean zero and covariance matrix = σ 2 1σ σ σ 2 2 (21)

Lemma 6
We have and A proof is given in the Appendix.
For this section we assume β < 1. We have to evaluate As for the law of large numbers, many multiindices i, j are negligible. However, this time the selection of the leading terms is more subtle.
For a proof see the Appendix or [12]. Now, we split the sum (24) in four parts. (24) by Proposition 4. Thus It follows that A 2 −→ 0. The proofs for A 3 and A 4 are similar.
We turn to the asymptotic computation of A 1 .
First, we observe that for K + L odd E X(i) Y (j) = 0 by Proposition 4. So, we may assume that K and L are both even or both odd. If K and L are even (odd) then both r and s have to be even (odd) otherwise W 0 K, N 1 (r) = ∅ or W 0 L, N 2 (s) = ∅. In the following we will treat the case K and L even, the other one being similar. So we assume K = 2k and L = 2 and rewrite A 1 : using Proposition 8 we get: where we setβ := β 1−β .
This proves that all moments converge, hence In the following we have to identify the limit measure.
First we consider the case that α 1 = 0 or α 2 = 0. Let's assume α 2 = 0. Then (25) simplifies to The last expression is, according to Lemma 6, the moment E Z 2k 1 Z 2 2 of a bivariate normal distribution with zero mean and covariance matrix This concludes the proof of the Central Limit Theorem 3.
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Appendix
In this appendix we discuss some combinatorics in connection with the moment method and the Theorem of Isserlis. Let us denote by P L (r) the set of all partitions = {π 1 , π 2 , ..., π } of {1, 2, ..., L} with r sets π i with |π i | = 1 and − r sets π j with |π j | = 2. In particular, P L (0) is the set of pair partitions of {1, 2, ..., L}. We show

Lemma 10
Proof The claim is true for L = 1. Suppose P 2L (0) = (2L − 1)!!. Then to build {π 1 , ..., π L+1 } we can match the number 2L+2 with any of the other 2L−1 numbers. Thus, by induction hypothesis, we have (2L − 1)!! choices to build pair partitions from the remaining 2L unmatched elements. Proof If {π 1 , ..., π } ∈ P L (r) then 2 − r = 2( − r) + r = L, so L − r is even. This proves the second assertion of the Proposition. Let L − r be even. Then we have L r choices for the sets π i with |π i | = 1. There remain L − r elements to build pair partitions from. By Lemma 10 this can be done in (L − r − 1)!! ways.

Proof (Proposition 8):
We prove the final assertion of the proposition, the other assertions are easier to prove. There are Q+r 2 different indices to choose from a total of M possible choices. This gives This Corollary follows by summing p k,m (ρ) over all possible ρ. We end this appendix with the proof of Lemma 6.