Relying on a recent progress made in studying bilinearly indexed (bli) random processes in \cite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}, the main foundational principles of fully lifted random duality theory (fl RDT) were established in \cite{Stojnicflrdt23}. We here study famous Hopfield models and show that their statistical behavior can be characterized via the fl RDT. Due to a nestedly lifted nature, the resulting characterizations and, therefore, the whole analytical machinery that produces them, become fully operational only if one can successfully conduct underlying numerical evaluations. After conducting such evaluations for both positive and negative Hopfield models, we observe a remarkably fast convergence of the fl RDT mechanism. Namely, for the so-called square case, the fourth decimal precision is achieved already on the third (second non-trivial) level of lifting (3-sfl RDT) for the positive and on the fourth (third non-trivial) level of lifting (4-sfl RDT) for the corresponding negative model. In particular, we obtain the scaled ground state free energy $\approx 1.7788$ for the positive and $\approx 0.3279$ for the negative model.

翻译：基于近期在双线性索引随机过程研究中取得的进展（参见\cite{Stojnicnflgscompyx23,Stojnicsflgscompyx23}），完全提升随机对偶理论（fl RDT）的基本原理已在\cite{Stojnicflrdt23}中建立。本文研究著名的Hopfield模型，并证明其统计行为可通过fl RDT进行刻画。由于嵌套提升特性，所得刻画及生成这些刻画的分析机制，仅在成功进行相应数值评估时方可完全运作。通过对正、负两种Hopfield模型进行此类评估，我们观察到fl RDT机制具有显著快速的收敛性。具体而言，对于所谓的平方情形，正模型在第三（第二个非平凡）提升水平（3-sfl RDT）即可达到四位小数精度，而负模型则在第四（第三个非平凡）提升水平（4-sfl RDT）实现。特别地，我们得到正模型缩放基态自由能约为$1.7788$，负模型约为$0.3279$。