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

TL;DR

This paper investigates how the propagation of chaos behaves in the high-temperature phase of the Hopfield model, revealing conditions under which it increases or breaks down depending on system parameters.

Contribution

It provides a detailed analysis of the propagation of chaos in the Hopfield model at high temperatures, identifying precise thresholds for its increase and breakdown.

Findings

Propagation of chaos increases when Mk/N → 0 for β<1.

Breakdown occurs for M=o(√N) when k/N → c>0.

At critical temperature, chaos propagates for k=o(√N) but breaks down for k=c√N.

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 $\beta<1$ (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.