Existence of the thermodynamic limit for the Hopfield model

Abstract

We prove that the quenched pressure of the Hopfield model with $K\simαN$ i.i.d. symmetric Rademacher patterns has a thermodynamic limit at every load $α\in(0,\infty)$, finite inverse temperature and uniform external field, with no replica-symmetry, order-parameter concentration or non-condensation hypothesis required. The limit is the supremum over $r$ of limiting pressures of mixed models with $r$ binary patterns in a Gaussian bulk. Via the Hubbard--Stratonovich representation we extract the finitely many patterns whose hidden coordinates exceed $δ\sqrt N$; a collective interpolation replaces the other binary patterns by Gaussian ones with error $O(δ^2)$ uniform in the volume, and convex concentration on the discrete cube controls the disorder-dependent choice of retained patterns. For the mixed models we prove a Gaussian hidden-spin truncation bound uniform in the volume and in an arbitrary Ising potential, and convergence for Gaussian bulks with finitely many site-field types, reducing to results of Ho and of Chen, Ho and Issa on non-convex vector spin glasses with compact spins. For Gaussian patterns we give an alternative proof of the existence result announced by Camilli, Contucci and Mingione, extended to a uniform field and an extra binary pattern. In both cases the pressure has exponentially small deviation probabilities and converges a.s.; internal energy and entropy converge wherever the limit is differentiable. An extra binary Mattis pattern admits a scalar variational reduction. At zero field the Rademacher pressure dominates the Gaussian one with an extra binary pattern, which dominates the pure Gaussian one. No explicit formula is given. The proofs were generated through extensive interaction with the AI models Claude (Opus~5.5) and ChatGPT (Sol~6.1), based on the problem formulation and guidance of the authors.