Statistical-mechanics of a three-body Hopfield model with finite connectivity

Yushi Sugawara; Koji Hukushima

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arXiv · 2607.17234

Statistical-mechanics of a three-body Hopfield model with finite connectivity

Yushi Sugawara·Koji Hukushima

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Abstract

A three-body Hopfield model defined on sparse random graphs with finite mean connectivity is studied using a replica-symmetric (RS) analysis. By extending the functional replica framework for conventional two-body finite-connectivity Hopfield models, self-consistent equations for the local-field distribution are derived and solved numerically by population dynamics. In contrast to two-body sparse Hopfield models, three-body interactions induce a discontinuous retrieval transition, coexistence of paramagnetic and retrieval solutions, and a distinct spinodal structure. Starting the RS population dynamics from an uninformative finite-amplitude field distribution leads to a spin-glass-like fixed point at low temperatures rather than to the retrieval state. This trapping already occurs for a single embedded pattern, where conventional cross-talk noise among stored patterns is absent, indicating that it originates from the combination of sparse connectivity and three-body interactions. Within the RS description, increasing the number of embedded patterns drives a crossing between the retrieval and glassy free-energy branches, making the glassy branch thermodynamically stable at high load. Finite-size Monte Carlo simulations using population annealing support the RS description of the retrieval branch and reproduce the trapping behavior under cooling, while deviations from the RS spin-glass branch point to replica-symmetry breaking in the low-temperature glassy regime. The characteristic storage load scales linearly with the mean connectivity, reflecting the sparse number of couplings. These results clarify how many-body discontinuity and sparse graph disorder simultaneously influence the accessibility and capacity of associative memory.

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Yushi Sugawara, Koji Hukushima. 2026-07-19. Statistical-mechanics of a three-body Hopfield model with finite connectivity. https://arxiv.org/abs/2607.17234

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