{"paper":{"title":"On self-avoiding polygons and walks: the snake method via pattern fluctuation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Alan Hammond","submitted_at":"2018-08-29T01:24:32Z","abstract_excerpt":"For $d \\geq 2$ and $n \\in \\mathbb{N}$, let $\\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\\mathbb{Z}^d$, and write $\\Gamma$ for a $\\mathsf{W}_n$-distributed walk. 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 $n^{-1/2 + o(1)}$ in any dimension $d \\geq 2$. The method of proof is a reworking of that in [4], which found a closing probability upper bound of $n^{-1/4 + o(1)}$. A key element of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09597","kind":"arxiv","version":1},"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"}