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Let $\\mu$ be the connective constant of the lattice and, for any $n \\in [0, \\infty)$, let $\\lambda_c(n)$ be the largest value of $\\lambda$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\\lambda_c(n) =1/\\mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and tha"},"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":"1806.09360","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-06-25T10:14:15Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"fdb7ae00cb81dceeee67f4d6b9bcd6484caf4c3cf2e62f2b2d675e4cf75a91d3","abstract_canon_sha256":"8a3dbd1aaaf9a2104b021eea0a468c320bbe4991c9900a1fd4ca8c566d65341a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:30.132968Z","signature_b64":"5i+O7ToQQybBHEQ28f0L8wQuf9wtYqs+iVT3YgvMnNsvcuGfAYqI0BPaaWren+gNiX8WM9xDr5b7SbnxIrTtCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2a9206249390a989ec5fb5f13c5b09f9648db07005ff8fe570d8c73fb137352","last_reissued_at":"2026-05-18T00:00:30.132414Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:30.132414Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shifted critical threshold in the loop $O(n)$ model at arbitrary small $n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Lorenzo Taggi","submitted_at":"2018-06-25T10:14:15Z","abstract_excerpt":"In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to $${\\lambda^{\\# \\mbox{edges}} n^{\\# \\mbox{loops}},}$$ where $\\lambda, n \\in [0, \\infty)$. Let $\\mu$ be the connective constant of the lattice and, for any $n \\in [0, \\infty)$, let $\\lambda_c(n)$ be the largest value of $\\lambda$ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\\lambda_c(n) =1/\\mu$ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09360","kind":"arxiv","version":4},"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":"1806.09360","created_at":"2026-05-18T00:00:30.132525+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.09360v4","created_at":"2026-05-18T00:00:30.132525+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.09360","created_at":"2026-05-18T00:00:30.132525+00:00"},{"alias_kind":"pith_short_12","alias_value":"6KUSAYSJHEFJ","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"6KUSAYSJHEFJRHWF","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"6KUSAYSJ","created_at":"2026-05-18T12:32:08.215937+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/6KUSAYSJHEFJRHWF7NPRHRNQT6","json":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6.json","graph_json":"https://pith.science/api/pith-number/6KUSAYSJHEFJRHWF7NPRHRNQT6/graph.json","events_json":"https://pith.science/api/pith-number/6KUSAYSJHEFJRHWF7NPRHRNQT6/events.json","paper":"https://pith.science/paper/6KUSAYSJ"},"agent_actions":{"view_html":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6","download_json":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6.json","view_paper":"https://pith.science/paper/6KUSAYSJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.09360&json=true","fetch_graph":"https://pith.science/api/pith-number/6KUSAYSJHEFJRHWF7NPRHRNQT6/graph.json","fetch_events":"https://pith.science/api/pith-number/6KUSAYSJHEFJRHWF7NPRHRNQT6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6/action/storage_attestation","attest_author":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6/action/author_attestation","sign_citation":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6/action/citation_signature","submit_replication":"https://pith.science/pith/6KUSAYSJHEFJRHWF7NPRHRNQT6/action/replication_record"}},"created_at":"2026-05-18T00:00:30.132525+00:00","updated_at":"2026-05-18T00:00:30.132525+00:00"}