Hopfield model thermodynamic limit proven at all loads without replica-symmetry assumptions
_onionesque · x · 2026-11-03
A new arXiv paper by Adriano Barra and Fabrizio Durante proves existence of the thermodynamic limit for the Hopfield model's quenched pressure with K∼αN i.i.d. symmetric Rademacher patterns, at every load α∈(0,∞), finite inverse temperature, and uniform external field — with no replica-symmetry, order-parameter concentration, or non-condensation hypotheses.
The limit is a supremum over mixed models with r binary patterns in a Gaussian bulk. Via a Hubbard–Stratonovich representation, the authors extract the finitely many patterns whose hidden coordinates exceed δ√N, replacing remaining binary patterns with Gaussian ones via collective interpolation with uniform O(δ²) error. For Gaussian patterns they give an alternative proof of the result announced by Camilli, Contucci and Mingione, extended to a uniform field and an extra binary pattern. Pressure deviations are exponentially unlikely and convergence is almost sure; internal energy and entropy converge wherever the limit is differentiable.
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