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 $\beta <1$ (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 $Mk\mathrm{/}N\to 0$. For $M=o(\sqrt{N})$ we show that propagation of chaos breaks down for $k\mathrm{/}N\to c>0$. 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.