Fluctuations in the Hopfield model at the critical temperature
- Gentz, Barbara
- Löwe, Matthias
2010 Mathematics Subject Classification
- 60K35 60F05 82C32
- Hopfield model, spin glasses, neural networks, random disorder, limit theorems, non-Gaussian fluctuations, critical temperature
We investigate the fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks at the critical temperature 1 / βc = 1. The number of patterns M(N) is allowed to grow with the number N of spins but the growth rate is subject to the constraint M(N)15/N → 0. 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.
- Markov Process. Related Fields 5 (1999), no. 4. 423-449