Abstract
AbstractWe study a multi-group version of the mean-field Ising 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, for 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.
Highlights
The Curie–Weiss model is a model of ferromagnetism
For example, [7,17,27,28,32] for other applications. Another area the Curie–Weiss model has found application is the study of random matrices
We show a local limit theorem for the normalised magnetisation vector
Summary
The Curie–Weiss model is a model of ferromagnetism. In its classic form, there is a random vector (X1, . . . , Xn) of binary random variables with values in the set of spin configurations {−1, 1}n. For example, [7,17,27,28,32] for other applications Another area the Curie–Weiss model has found application is the study of random matrices (see [1,12,13,15,16,18]). 2, we define the multi-group Curie–Weiss model for general coupling matrices, see Definition 1. After this definition, we introduce the specific coupling matrices considered in this paper. Our study is constrained to the so-called high-temperature regime as in Definition 2 For this regime, a non-local (or global) central limit theorem has been derived in [22], which we recite in Theorem 3. Our main result of this paper is a local version of Theorem 3 stated in Theorem 4 and proved in Sect. 3, the last section of this paper
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