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It has been predicted by Nienhuis that for $0\\le n\\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\\tfrac{1}{\\sqrt{2+\\sqrt{2-n}}}$. For $0<n\\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $x\\ge x_c(n)$.\n  In this paper, we prove that for $n\\in [1,2]$ and $x=x_c(n)$","authors_text":"Alexander Glazman, Hugo Duminil-Copin, Ron Peled, Yinon Spinka","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T17:23:23Z","title":"Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.09335","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0057865dbe07fff6afa17e031e3f34b77ebd6e02044cbafb2787cf4420712d86","target":"record","created_at":"2026-07-05T00:57:21Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b3adf4a9c93d91c5a2090ad98812ed72ca07059adb857f692417dc44f6cb06e1","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-28T17:23:23Z","title_canon_sha256":"c0211638dc836b78d81d678cf68cf6ccf6ebe9d919fce57984abcad50699ad08"},"schema_version":"1.0","source":{"id":"1707.09335","kind":"arxiv","version":3}},"canonical_sha256":"99ef2e5b2404bce9b6c8ae589e5ce65dc047fdc2713e630efcc02ff721e0cd35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99ef2e5b2404bce9b6c8ae589e5ce65dc047fdc2713e630efcc02ff721e0cd35","first_computed_at":"2026-07-05T00:57:21.171451Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:57:21.171451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qAhdkXXRS9h2Z9Ez/GCncWlig/868BFRCYj4kLnYeFrDrGqOLhm0ATPHeO0B3ehCvaI7WNjrGmZ0wWinB1i1Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:57:21.171796Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.09335","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0057865dbe07fff6afa17e031e3f34b77ebd6e02044cbafb2787cf4420712d86","sha256:d9f949e45daf6e8d2531954182e8b4a2b49921e873727b0ab64cefdc1e0a2712"],"state_sha256":"96c6ffb5a0ef8da57d3f651bcc15dee9c6e23e74f2f926d8a8a155fc1f179362"}