\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs} \usepackage{enumitem} \usepackage{microtype} \usepackage[margin=1.15in]{geometry} \usepackage[colorlinks=true,allcolors=black]{hyperref} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \input{macros} \title{The critical capacity of the Hopfield model} \author{Yuwen Zhou} \dedicatory{\upshape University of British Columbia Okanagan\\ \texttt{zhouyw07@student.ubc.ca}} \date{} \setcounter{tocdepth}{1} \begin{document} \input{abstract} \maketitle \tableofcontents \input{sec1_intro} \input{sec2_gaussian} \input{sec3_reduction} \input{sec4_convex} \input{sec5_zero} \input{sec6_merging} \input{sec7_digits} \input{sec8_open} \appendix \input{appA_certificate} \bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{10} \bibitem{ABCHIM} A.~Ali, J.~Boursier, H.-B. Chen, F.-H. Ho, V.~Issa, and J.-C. Mourrat, \emph{Free energy of the {H}opfield model}, arXiv:2611.03306, 2026. \bibitem{AGS85} D.~J. Amit, H.~Gutfreund, and H.~Sompolinsky, \emph{Storing infinite numbers of patterns in a spin-glass model of neural networks}, Phys. Rev. Lett. \textbf{55} (1985), 1530--1533. \bibitem{AGS87} \bysame, \emph{Statistical mechanics of neural networks near saturation}, Ann. Physics \textbf{173} (1987), 30--67. \bibitem{AC17} A.~Auffinger and W.-K. Chen, \emph{Parisi formula for the ground state energy in the mixed $p$-spin model}, Ann. Probab. \textbf{45} (2017), 4617--4631. \bibitem{BD} A.~Barra and F.~Durante, \emph{Existence of the thermodynamic limit for the {H}opfield model}, arXiv:2610.41938, 2026. \bibitem{BM} J.~Boursier and J.-C. Mourrat, \emph{Free energy of non-convex multi-species {I}sing spin glasses with external fields}, arXiv:2610.18350, 2026. \bibitem{CCM} F.~Camilli, P.~Contucci, and E.~Mingione, \emph{The solution of the {H}opfield model with {G}aussian patterns}, arXiv:2610.32618, 2026. \bibitem{CHI} H.-B. Chen, F.-H. Ho, and V.~Issa, \emph{A formula for the free energy of non-convex vector spin glasses}, arXiv:2610.18744, 2026. \bibitem{CM} H.-B. Chen and J.-C. Mourrat, \emph{Free energy of non-convex multi-species spherical spin glasses}, arXiv:2609.32710, 2026. \bibitem{CAG86} A.~Crisanti, D.~J. Amit, and H.~Gutfreund, \emph{Saturation level of the {H}opfield model for neural network}, Europhys. Lett. \textbf{2} (1986), 337--341. \bibitem{FST01} J.~Feng, M.~Shcherbina, and B.~Tirozzi, \emph{On the critical capacity of the {H}opfield model}, Comm. Math. Phys. \textbf{216} (2001), 139--177. \bibitem{Hopfield82} J.~J. Hopfield, \emph{Neural networks and physical systems with emergent collective computational abilities}, Proc. Natl. Acad. Sci. USA \textbf{79} (1982), 2554--2558. \bibitem{Arb} F.~Johansson, \emph{Arb: efficient arbitrary-precision midpoint-radius interval arithmetic}, IEEE Trans. Comput. \textbf{66} (2017), 1281--1292. \bibitem{Kohring90} G.~A. Kohring, \emph{A high-precision study of the {H}opfield model in the phase of broken replica symmetry}, J. Stat. Phys. \textbf{59} (1990), 1077--1086. \bibitem{Loukianova97} D.~Loukianova, \emph{Lower bounds on the restitution error in the {H}opfield model}, Probab. Theory Related Fields \textbf{107} (1997), 161--176. \bibitem{Newman88} C.~M. Newman, \emph{Memory capacity in neural network models: rigorous lower bounds}, Neural Networks \textbf{1} (1988), 223--238. \bibitem{Rockafellar70} R.~T. Rockafellar, \emph{Convex analysis}, Princeton University Press, 1970. \bibitem{Sion58} M.~Sion, \emph{On general minimax theorems}, Pacific J. Math. \textbf{8} (1958), 171--176. \bibitem{SK94} H.~Steffan and R.~K{\"u}hn, \emph{Replica symmetry breaking in attractor neural network models}, Z. Phys. B \textbf{95} (1994), 249--260. \bibitem{SMK96} T.~Stiefvater, K.-R. M{\"u}ller, and R.~K{\"u}hn, \emph{Averaging and finite-size analysis for disorder: the {H}opfield model}, Phys. A \textbf{232} (1996), 61--73. \bibitem{Stojnic24} M.~Stojnic, \emph{Capacity of the {H}ebbian--{H}opfield network associative memory}, arXiv:2403.01907, 2024. \bibitem{Stojnic26} \bysame, \emph{Capacity of the {H}opfield--{H}ebbian dynamics with correlated patterns}, arXiv:2610.38437, 2026. \bibitem{Talagrand98} M.~Talagrand, \emph{Rigorous results for the {H}opfield model with many patterns}, Probab. Theory Related Fields \textbf{110} (1998), 177--275. \bibitem{TW14} L.~N. Trefethen and J.~A.~C. Weideman, \emph{The exponentially convergent trapezoidal rule}, SIAM Rev. \textbf{56} (2014), 385--458. \end{thebibliography} \end{document}