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For $T$ near the critical temperature $T_c$, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in powers of $\\delta Q_{ab} = (Q_{ab} - Q_{ab}^{\\rm RS})$ about the Replica Symmetric solution $Q_{ab}^{\\rm RS}$. The expansion is kept up to 4-th order in $\\delta{\\bm Q}$ where a Parisi solution $Q_{ab} = Q(x)$ emerges, but only if one remains close enough to $T_c$.\n  For $T$ near zero we show how to separate contributions from $x\\ll T\\ll 1$ where the Hessian maint"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.1034","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.dis-nn","submitted_at":"2011-08-04T09:48:29Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"564416bd9ac2c2a788d0eb9a6e05e0848bc9429c8622eb6889072c39ce5f03fc","abstract_canon_sha256":"e3126ef68c07a41dcefa9c227a6387c52e675628f41d9a12850ef6f4d5f66908"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:45.562927Z","signature_b64":"5iQZMYnzJbC+jkfzgx856Si7ucYY9xLSDqSMS3PEIf/CKjX47ckMkZfJ6G8Kwbn8vmPyjUeV+Eahwhb8az0qDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50e9d3ec0b567e89c3d4a4075fbce7befd312569af34e47c2d44533147f91812","last_reissued_at":"2026-05-18T04:09:45.562292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:45.562292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Sherrington-Kirkpatrick model near T_c and near T=0","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"cond-mat.dis-nn","authors_text":"A. Crisanti, C. De Dominicis","submitted_at":"2011-08-04T09:48:29Z","abstract_excerpt":"Some recent results concerning the Sherrington-Kirkpatrick model are reported. For $T$ near the critical temperature $T_c$, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in powers of $\\delta Q_{ab} = (Q_{ab} - Q_{ab}^{\\rm RS})$ about the Replica Symmetric solution $Q_{ab}^{\\rm RS}$. The expansion is kept up to 4-th order in $\\delta{\\bm Q}$ where a Parisi solution $Q_{ab} = Q(x)$ emerges, but only if one remains close enough to $T_c$.\n  For $T$ near zero we show how to separate contributions from $x\\ll T\\ll 1$ where the Hessian maint"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.1034","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.1034","created_at":"2026-05-18T04:09:45.562399+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.1034v2","created_at":"2026-05-18T04:09:45.562399+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.1034","created_at":"2026-05-18T04:09:45.562399+00:00"},{"alias_kind":"pith_short_12","alias_value":"KDU5H3ALKZ7I","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"KDU5H3ALKZ7ITQ6U","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"KDU5H3AL","created_at":"2026-05-18T12:26:32.869790+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3","json":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3.json","graph_json":"https://pith.science/api/pith-number/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/graph.json","events_json":"https://pith.science/api/pith-number/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/events.json","paper":"https://pith.science/paper/KDU5H3AL"},"agent_actions":{"view_html":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3","download_json":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3.json","view_paper":"https://pith.science/paper/KDU5H3AL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.1034&json=true","fetch_graph":"https://pith.science/api/pith-number/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/graph.json","fetch_events":"https://pith.science/api/pith-number/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/action/storage_attestation","attest_author":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/action/author_attestation","sign_citation":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/action/citation_signature","submit_replication":"https://pith.science/pith/KDU5H3ALKZ7ITQ6UUQDV7PHHX3/action/replication_record"}},"created_at":"2026-05-18T04:09:45.562399+00:00","updated_at":"2026-05-18T04:09:45.562399+00:00"}