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

TL;DR

This paper analyzes the distribution of local overlap profiles in a one-dimensional Kac-Hopfield model, revealing that typical configurations are constant and governed by quenched disorder, with detailed large deviation estimates.

Contribution

It provides the first large deviation estimates for local overlaps in a one-dimensional Kac-Hopfield model below the critical temperature.

Findings

Local overlaps are constant and match mean-field values

Profiles are governed by quenched disorder rather than entropy

Estimates on the size of transition regions between equilibrium states

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, $\gamma^{-1}$, 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.