Fluctuations of the free energy in the high temperature Hopfield model

Francis Comets, Irina Kurkova and José Trashorras

Stochastic Processes and their Applications, 2004, vol. 113, issue 1, 1-35

Abstract: 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.

Keywords: Hopfield; model; Spin; glass; Fluctuations; Martingales (search for similar items in EconPapers)
Date: 2004
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