When does the chaos in the Curie-Weiss model stop to propagate?

Abstract

We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with $N$ spins at inverse temperature $\beta>0$ and subject to an external magnetic field of strength $h\in\mathbb{R}$. Using a different proof technique than in [Ben Arous, Zeitouni; 1999] we confirm the well-known propagation of chaos phenomenon: If $k=k(N)=o(N)$ as $N\to\infty$, then the $k$'th marginal distribution of the Gibbs measure converges to a product measure at $\beta <1$ or $h \neq 0$ and to a mixture of two product measures, if $\beta >1$ and $h =0$. More importantly, we also show that if $k(N)/N\to \alpha\in (0,1]$, this property is lost and we identify a non-zero limit of the total variation distance between the number of positive spins among any $k$-tuple and the corresponding binomial distribution.