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The Hopfield Model with Superlinearly Many Patterns

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arxiv 1108.4771 v3 pith:EULCSLLG submitted 2011-08-24 math.PR

classification math.PR
keywords betamodelalphaenergyexternalfieldfreegamma
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abstract

We study the Hopfield model where the ratio $\alpha$ of patterns to sites grows large. We prove that the free energy with inverse temperature $\beta$ and external field $B$ behaves like $\beta\sqrt\alpha+\gamma$, where $\gamma$ is the limiting free energy of the Sherrington-Kirkpatrick model with inverse temperature $\sqrt2\beta$ and external field $B$.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. CLuP practically achieves $\sim 1.77$ positive and $\sim 0.33$ negative Hopfield model ground state free energy

    cond-mat.dis-nn 2025-07 conditional novelty 6.0 of 10

    CLuP±Hop approximates Hopfield ground state free energies to within about 0.3% using simple gradient descent, backed by the author's fully lifted random duality theory.

  2. Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT

    stat.ML 2025-06 conditional novelty 5.0 of 10

    For the asymmetric binary perceptron, the worst-case local entropy breaks down for constraint density alpha in (0.77, 0.78), matching replica predictions and the range where fast algorithms stop working.

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