Fluctuations in the Hopfield Model at the Critical Temperature

B. Gentz, M. Lowe

1999, v.5, Issue 4, 423-449

ABSTRACT

We investigate the fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks at the critical temperature $1/\beta_\crit=1$. The number of patterns $M(N)$ is allowed to grow with the number $N$ of spins but the growth rate is subject either to the constraint $M(N)^{7}/N\to 0$ or even to the constraint $M(N)^{13}/N\to 0$, depending on the precise formulation of the result. As the system size $N$ increases, on a set of large probability the distribution of the appropriately scaled order parameter under the Gibbs measure comes arbitrarily close (in a metric which generates the weak topology) to a non-Gaussian measure which depends on the realization of the random patterns. This random measure is given explicitly by its (random) density.

Keywords: Hopfield model,spin glasses,neural networks,random disorder,limit theorems,non-Gaussian fluctuations,critical temperature

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