Free energy of the Hopfield model
Abstract
We identify the limit of the free energy of the Hopfield model with independent symmetric Bernoulli patterns, for every temperature and loading ratio. A Hubbard-Stratonovich transformation reduces the problem to that of identifying the limit free energy of a bipartite spin glass with ± 1 and spherical spins. One key difficulty is that the patterns, which become the coupling coefficients in the bipartite model, are Bernoulli instead of Gaussian, and the limit free energy is sensitive to the law of these coefficients. We show that the model is in fact equivalent to one involving Gaussian patterns and also subject to a finite number of Bernoulli random fields. We then solve this model by adapting the techniques recently introduced to identify the limit free energies of spin glasses with either ± 1 or spherical spins.
Related
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BibTeX
@misc{ali2026free,
title = {{Free energy of the Hopfield model}},
author = {Ayman Ali and Jeanne Boursier and Hong-Bin Chen and Fu-Hsuan Ho and Victor Issa and Jean-Christophe Mourrat},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03306},
eprint = {2611.03306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03306},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Free energy of the Hopfield model
AU - Ayman Ali
AU - Jeanne Boursier
AU - Hong-Bin Chen
AU - Fu-Hsuan Ho
AU - Victor Issa
AU - Jean-Christophe Mourrat
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03306
UR - https://arxiv.org/abs/2611.03306
AB - We identify the limit of the free energy of the Hopfield model with independent symmetric Bernoulli patterns, for every temperature and loading ratio. A Hubbard-Stratonovich transformation reduces the problem to that of identifying the limit free energy of a bipartite spin glass with ± 1 and spherical spins. One key difficulty is that the patterns, which become the coupling coefficients in the bipartite model, are Bernoulli instead of Gaussian, and the limit free energy is sensitive to the law of these coefficients. We show that the model is in fact equivalent to one involving Gaussian patterns and also subject to a finite number of Bernoulli random fields. We then solve this model by adapting the techniques recently introduced to identify the limit free energies of spin glasses with either ± 1 or spherical spins.
ER -