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Fluctuations of the free energy in the high temperature Hopfield model

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  • Comets, Francis
  • Kurkova, Irina
  • Trashorras, José

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.

Suggested Citation

  • Comets, Francis & Kurkova, Irina & Trashorras, José, 2004. "Fluctuations of the free energy in the high temperature Hopfield model," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 1-35, September.
    • Handle: RePEc:eee:spapps:v:113:y:2004:i:1:p:1-35
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