math.PR — Probability
THE CRITICAL CAPACITY OF THE HOPFIELD MODEL
Abstract. We prove that the critical capacity of the Hopfield model at zero temperature lies in [0.138198129, 0.13820106]. The best rigorous lower bound was 0.113, and our lower end exceeds the published one-step and two-step replica symmetry breaking values. The capacity is the supremum of the loads at which the free energy at fixed overlap with one pattern has a well. At every positive temperature, we give a variational formula for this free energy and show that binary patterns and Gaussian rows have the same wells, which characterizes the capacity. The proof builds on the Hamilton–Jacobi solution of the model by Ali, Boursier, Chen, Ho, Issa and Mourrat, and on a
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