High-Temperature Increasing Propagation of Chaos and its breakdown for the Hopfield Model

We analyze increasing propagation of chaos in the high temperature regime of a disordered mean-field model, the Hopfield model. We show that for $β<1$ (the true high temperature region) we have increasing propagation of chaos as long as the size of the marginals $k=k(N)$ and the number of patterns $M=M(N)$ satisfies $Mk/N \to 0$. For $M=o(\sqrt N)$ we show that propagation of chaos breaks down for $k/N \to c>0$. At the ciritcal temperature we show that, for $M$ finite, there is increasing propagation of chaos, for $k=o(\sqrt N)$, while we have breakdown of propagation of chaos for $k=c \sqrt N$, for a $c>0$. All these reulst hold in probability in the disorder.

💡 Research Summary

The paper investigates the phenomenon of “increasing propagation of chaos” for the Hopfield model, a disordered mean‑field spin system, in both the high‑temperature regime (inverse temperature β < 1) and at the critical temperature (β = 1). Propagation of chaos means that, as the system size N tends to infinity, any finite collection of spins becomes asymptotically independent and identically distributed according to a Rademacher law (π). “Increasing” refers to allowing the size of the marginal k = k(N) to grow with N, rather than staying fixed.

The authors start by defining the Hopfield model: independent random patterns ξ_i ∈ {−1,+1}^M with zero mean and unit variance, the overlap vector m_N(σ)=N^{−1}∑_{i=1}^N σ_i ξ_i, and the Gibbs measure
 μ_N(σ)=Z_N^{−1} exp