Two Groups in a Curie–Weiss Model with Heterogeneous Coupling

Abstract

We discuss a Curie-Weiss model with two groups in the critical regime. This is the region where the central limit theorem does not hold any more but the mean magnetization still goes to zero as the number of spins grows. We show that the total magnetization normalized by N 3/4 converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie-Weiss model.

Fluctuation Results for General Block Spin Ising Models

“…This is stated in [16] for the bipartite model. In [24] the authors prove the existence of such a phase transition using the method of moments. Of course, with that method one cannot obtain an exponential speed of convergence as in Corollary 1.2.…”

“…formula (4.1) in [23]. Later, they were rediscovered as interesting models for statistical mechanics systems, see [18], [15], [8], [26], [24], as well as models for social interactions between several groups, e.g. in [17], [1], [28].…”

Exact recovery in block spin Ising models at the critical line

“…Note that Z is of order 1 √ N . This either, follows from Theorem 6 and Remark from the paper by Kirsch and Toth [16] cited at the beginning of Section 3 or from Theorem 3.1 directly: As noted in Remark 3.3 1 N 3/4 i σ i converges to a nondegenerate limit distribution. Since Z ≥ |ζ| as a consequence of β ≥ |α| (α = 1, i.e.…”

“…Indeed, in view of Theorem 3.1 below the main tool in this reference does not work, because at least for α > 0, and α + β = 2 the vector √ N (m 1 , m 2 ) is not asymptotically Gaussian. Moreover, considering Theorem 6 and Remark 7 in [16] one may wonder, whether an exact reconstruction of (S, S c ) on the basis of the correlations between the spins is possible at all. There, the authors show that for α > 0 and α+β = 2 asymptotically:…”

“…The first of these papers uses a very general interaction structure, while the latter investigates the situation in the spirit of social interaction models or statistical physics models on random graphs, see [23], [3], [7], and [18]. A related version of this model has been investigated using the method of moments in [15], [17] and [16]. The article by Berthet et al is motivated by a considerable amount of articles investigating block models in the recent past, see e.g.…”