The solution of the Hopfield model with Gaussian patterns
Abstract
We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load α and temperature T, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto N Ising spins coupled to M=αN Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies Tg=1+√α as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below Tg we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.
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BibTeX
@misc{camilli2026solution,
title = {{The solution of the Hopfield model with Gaussian patterns}},
author = {Francesco Camilli and Pierluigi Contucci and Emanuele Mingione},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32618},
eprint = {2610.32618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32618},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The solution of the Hopfield model with Gaussian patterns
AU - Francesco Camilli
AU - Pierluigi Contucci
AU - Emanuele Mingione
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32618
UR - https://arxiv.org/abs/2610.32618
AB - We prove that the free energy of the Hopfield model with Gaussian patterns has a thermodynamic limit at every load α and temperature T, and we give it as an explicit variational principle. The Hubbard--Stratonovich transformation maps the model onto N Ising spins coupled to M=αN Gaussian hidden units, whose radius, the empirical variance of the hidden layer, is a thermodynamic variable with an entropic cost. At fixed radius the model is an Ising--spherical spin glass with Gaussian couplings, which we solve. The principle (i) reproduces the Amit-Gutfreund-Sompolinsky theory as its replica-symmetric restriction, (ii) identifies T_g=1+√α as the rigorous boundary of the annealed region, (iii) selects a unique spin-glass solution of the AGS equations, and (iv) gives the radius an equation of state. Moreover, (v) below T_g we prove that replica symmetry is broken throughout: the instability of the AGS solution reduces to the zero-field de Almeida-Thouless condition of the Sherrington-Kirkpatrick model.
ER -