THE CRITICAL CAPACITY OF THE HOPFIELD MODEL
Source: hexagonmath.org/abs/2611.00105v1
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…
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