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**1 - 2**of**2**### Fluctuations of the Free Energy in the High . . .

, 2003

"... 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 in ..."

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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 in [10] 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

### Fluctuations of the Free Energy in the high temperature Hopfield Model Francis Comets ∗†

, 2003

"... 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 in [11 ..."

Abstract
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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 in [11] 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.