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We show that the closing probability $\\mathsf{W}_n \\big( \\vert \\vert \\Gamma_n \\vert \\vert = 1 \\big)$ that $\\Gamma$'s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a technique for proving closing probability upper bounds, which originated in [3] and was ma"},"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":"1808.10500","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-30T19:59:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0e1b5d252ad39caddf7d07c667f09b816a7376f7b4a19141c9b2293dec8933d3","abstract_canon_sha256":"57407ab9eda45f8518a93c60e6257ce032a8f0dfa306a319df60b3693042e0bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:12.925228Z","signature_b64":"sGQM6DltiZBLsVbjGel34FH7zx1+/7cGTqwVcOEzJIApmNWf72CeulVag/8JCw8bo4CZZDGZFpypQk1/sG23DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"684e94c718e1a4261edc804bfcbab90c0f65ac9055c33441cfcca83f77a0b376","last_reissued_at":"2026-05-18T00:00:12.924655Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:12.924655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On self-avoiding polygons and walks: the snake method via polygon joining","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alan Hammond","submitted_at":"2018-08-30T19:59:53Z","abstract_excerpt":"For $d \\geq 2$ and $n \\in \\mathbb{N}$, let $\\mathsf{W}_n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\\mathbb{Z}^d$, and write $\\Gamma$ for a $\\mathsf{W}_n$-distributed walk. 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