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

TL;DR

This paper studies the limits of chaos propagation in the Curie-Weiss model, showing when the system's spins become correlated or remain independent as the system size grows, with new proof techniques confirming known phenomena and identifying when chaos stops propagating.

Contribution

The paper provides a new proof of the propagation of chaos in the Curie-Weiss model and characterizes the precise conditions under which chaos ceases to propagate as the number of spins increases.

Findings

Propagation of chaos holds for k=o(N) when β<1 or h≠0.

For β>1 and h=0, the system converges to a mixture of two product measures.

Chaos stops propagating when k/N approaches a positive limit, with a non-zero total variation distance.

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.