{"paper":{"title":"The Hopfield Model with Superlinearly Many Patterns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"James Y. Zhao","submitted_at":"2011-08-24T07:38:56Z","abstract_excerpt":"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 $B$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4771","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}