Fluctuations of the free energy in the high temperature Hopfield model

We consider the Hopfield model of size N and with p∼tN patterns, in the whole high temperature (paramagnetic) region. Our result is that the partition function has log-normal fluctuations. It is obtained by extending to the present model the method of the interpolating Brownian motions used by Comets (Comm. Math. Phys. 166 (1995) 549–564) for the Sherrington–Kirkpatrick model. We view the load t of the memory as a dynamical parameter, making the partition function a nice stochastic process. Then we write some semi-martingale decomposition for the logarithm of the partition function, and we prove that all the terms in this decomposition converge. In particular, the martingale term converges to a Gaussian martingale.