Barbara Gentz, Matthias Loewe
Fluctuations in the Hopfield Model at the critical temperature
(332K, Postscript)
ABSTRACT. We investigate the fluctuations of the order parameter in the
Hopfield model of spin glasses and neural networks at the critical
temperature $1/\beta_c=1$. 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 $M(N)^{15}/N\to 0$. 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.