Abstract
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.
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References
Breiman, L.: Probability Addison-Wesley (1968)
Contucci, P., Gallo, I.: Bipartite mean field spin systems. Existence and solution. Math. Phys. Elec. Jou. 14(1), 1–22 (2008)
Contucci, P., Ghirlanda, S.: Modelling society with statistical mechanics: an application to cultural contact and immigration. Qual. Quant. 41, 569–578 (2007)
Ellis, R.: Entropy, large deviations, and statistical mechanics Whiley (1985)
Fedele, M.: Rescaled magnetization for critical bipartite Mean-Fields models. J. Stat. Phys. 155, 223–236 (2014)
Fedele, M., Contucci, P.: Scaling Limits for Multi-species Statistical Mechanics Mean-Field Models. J. Stat. Phys. 144, 1186–1205 (2011)
Husimi, K.: Statistical Mechanics of Condensation. In: Proceedings of the International Conference of Theoretical Physics, pp. 531-533. Science Council of Japan, Tokyo (1953)
Isserlis, L.: On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables. Biometrika 12(1/2), 134–139 (1918)
Kac, M.: Mathematical Mechanisms of Phase Transitions, in Statistical physics: Phase Transitions and Superfluidity, Vol. 1, pp. 241-305 Brandeis University Summer Institute in Theoretical Physics (1968)
Kirsch, W.: A Survey on the Method of Moments, available from http://www.fernuni-hagen.de/stochastik/
Kirsch, W: On Penrose’s Square-root Law and Beyond. Homo Oeconomicus 24(3/4), 357–380 (2007)
Kirsch, W., The Curie-Weiss model – an approach using moments. arXiv:1909.05612
Kirsch, W., Toth, G.: Two Groups in a Curie-Weiss Model with Heterogeneous Coupling, J. Theor. Prob. arXiv:712.08477
Kleiber, C., Stoyanov, J.: Multivariate distributions and the moment problem. J. Multivar. Anal. 113, 7–18 (2013)
Temperley, H. N. V.: The mayer theory of condensation tested against a simple model of the imperfect gas. Proc. Phys. Soc., A 67, 233–238 (1954)
Thompson, C. J.: Mathematical statistical mechanics macmillan (1972)
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Appendix
Appendix
In this appendix we discuss some combinatorics in connection with the moment method and the Theorem of Isserlis. Let us denote by \(\mathcal {P}_{L}(r)\) the set of all partitions π = {π1,π2,...,πℓ} of {1, 2,...,L} with r sets πi with |πi| = 1 and ℓ − r sets πj with |πj| = 2. In particular, \(\mathcal {P}_{L}(0)\) is the set of pair partitions of {1, 2,...,L}. We show
Lemma 10
Proof
The claim is true for L = 1. Suppose \(\mathcal {P}_{2L}(0)=\left (2L-1\right )!!\). Then to build {π1,...,πL+ 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. □
Proposition 11
Let \(L,r\in \mathbb {N}\). If L − r is even, then
and if L − r is odd, then pL(r) = 0.
Proof
If \(\lbrace \pi _{1},...,\pi _{\ell }\rbrace \in \mathcal {P}_{L}(r)\) then 2ℓ − r = 2(ℓ − r) + r = L, so L − r is even. This proves the second assertion of the Proposition. Let L − r be even. Then we have \(\binom {L}{r}\) choices for the sets πi with |πi| = 1. There remain L − r elements to build pair partitions from. By Lemma 10 this can be done in \(\left (L-r-1\right )!!\) ways. □
Proof (Proposition 8):
We prove the final assertion of the proposition, the other assertions are easier to prove. There are \(\frac {Q+r}{2}\) different indices to choose from a total of M possible choices. This gives \(\frac {M!}{\left (M-\frac {Q+r}{2}\right )!}\) possibilities. To distribute them on the Q indices with r single and \(\frac {Q-r}{2}\) double occurrences gives
Multiplying gives the assertion. □
Now, we consider a disjoint partition {1, 2,..., 2L} into two sets K and M with k = |K| and m = |M|, and look for all pair partitions \({\Pi }=\left (\pi _{1},...,\pi _{L}\right )\) such that exactly r of the πi are mixed, i.e. πi ∩ K≠∅ and πi ∩ M≠∅.
Proposition 12
The number pK,M(r) of pair partitions \({\Pi }=\left (\pi _{1},...,\pi _{L}\right )\) of {1, 2,..., 2L} with exactly r mixed πi is given by
if both k − r and m − r are even and non-negative, and
Proof
There are \(\frac {k!}{(k-r)!r!}\) choices for elements of K in the mixed πi and \(\frac {m!}{(m-r)!}\) ways to fill them with elements from M. The choices for the (k − r) pure πi from K and (m − r) pure πi from M are given by Lemma 10. □
Corollary 13
If both k and m are even, then
If both k and m are odd, then
This Corollary follows by summing pk,m(ρ) over all possible ρ. We end this appendix with the proof of Lemma 6.
Proof (Lemma 6):
By the Theorem of Isserlis [8], we have
where
Therefore,
by Proposition 12. Equation (23) can be proved analogously. □
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Kirsch, W., Toth, G. Two Groups in a Curie-Weiss Model. Math Phys Anal Geom 23, 17 (2020). https://doi.org/10.1007/s11040-020-09343-5
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DOI: https://doi.org/10.1007/s11040-020-09343-5