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

TL;DR

The paper analyzes propagation of chaos for the Hopfield disordered mean-field model in high-temperature and near-critical regimes. It introduces a Hubbard–Stratonovich mixture representation, turning the Gibbs measure into a mixture of product measures and reducing chaos questions to stability properties of the random field under the mixing measure. The authors identify precise scaling thresholds: for $\beta<1$ chaos holds when $M/N\to0$ and $kM/N\to0$, but breaks down for macroscopic $k$ when $M=o(\sqrt N)$; at the critical point $\beta=1$, chaos persists for $k=o(N^{1/4})$ in some regimes and breaks down around $k\sim\sqrt N$, with a sharp critical window. These results illuminate how disorder fluctuations govern the transition between asymptotic independence and correlated behavior, and they establish sharp, regime-dependent scaling laws for the validity of mean-field approximations in random Hopfield-type systems.

Abstract

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.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5: Critical-window breakdown at $\beta=1$ (fixed $M$)
• Lemma 2.1: Hubbard--Stratonovich mixture
• proof
• Corollary 2.2
• Lemma 2.3
• proof