具有有限连通性的三体霍普菲尔德模型的统计力学
Statistical-mechanics of a three-body Hopfield model with finite connectivity
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中文总结 AI 辅助
研究具有有限连通性的三体霍普菲尔德模型，用复制对称分析，推导求解自洽方程。三体作用致不连续检索转变等，RS种群动力学低温有捕获行为，模拟支持RS描述，明确多体不连续性和稀疏图无序对关联记忆的影响。
中文摘要 AI 辅助
本文研究了定义在具有有限平均连通性的稀疏随机图上的三体霍普菲尔德模型，采用复制对称（RS）分析方法。通过扩展传统两体有限连通性霍普菲尔德模型的泛函复制框架，推导出自洽方程并通过种群动力学进行数值求解。三体相互作用导致了不连续的检索转变、顺磁和检索解的共存以及独特的旋节线结构。从无信息的有限振幅场分布开始的RS种群动力学在低温下导致类似自旋玻璃的不动点而非检索状态。有限尺寸蒙特卡罗模拟支持了RS对检索分支的描述，并再现了冷却下的捕获行为，同时低温玻璃态下与RS自旋玻璃分支的偏差表明存在复制对称破缺。特征存储负载与平均连通性线性相关。这些结果阐明了多体不连续性和稀疏图无序如何同时影响关联记忆的可及性和容量。
英文摘要
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.
