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 $\mathit{\beta }>0$ and subject to an external magnetic field of strength $h\in \mathbb{R}$. Using a different proof technique than in Ben Arous and Zeitouni [Ann. Inst. H. Poincaré: Probab. Statist., 35(1): 85–102, 1999] we confirm the well-known propagation of chaos phenomenon: If $k=k(N)=o(N)$ as $N\to \mathrm{\infty }$, then the k’th marginal distribution of the Gibbs measure converges to a product measure at $\mathit{\beta }<1$ or $h\ne 0$ and to a mixture of two product measures, if $\mathit{\beta }>1$ and $h=0$. More importantly, we also show that if $k(N)∕N\to \mathit{\alpha }\in (0,1]$, 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.