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Distribution of overlap profiles in the one-dimensional Kac-Hopfield model

Anton Bovier·Veronique Gayrard·Pierre Picco
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Abstract

We study a one-dimensional version of the Hopfield model with long, but finite range interactions below the critical temperature. In the thermodynamic limit we obtain large deviation estimates for the distribution of the ``local'' overlaps, the range of the interaction, $γ^{-1}$, being the large parameter. We show in particular that the local overlaps in a typical Gibbs configuration are constant and equal to one of the mean-field equilibrium values on a scale $o(\g^{-2})$. We also give estimates on the size of typical ``jumps''. i.e. the regions where transitions from one equilibrium value to another take place. Contrary to the situation in the ferromagnetic Kac-model, the structure of the profiles is found to be governed by the quenched disorder rather than by entropy.

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Anton Bovier, Veronique Gayrard, Pierre Picco. 1996-02-06. Distribution of overlap profiles in the one-dimensional Kac-Hopfield model. https://doi.org/10.1007/s002200050112

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