{"paper":{"title":"An upper bound on the number of self-avoiding polygons via 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":"2018-08-27T20:43:29Z","abstract_excerpt":"For $d \\geq 2$ and $n \\in \\mathbb{N}$ even, let $p_n = p_n(d)$ denote the number of length $n$ self-avoiding polygons in $\\mathbb{Z}^d$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\\lim_{n \\in 2\\mathbb{N}} p_n^{1/n} \\in (0,\\infty)$ is called the connective constant and denoted by $\\mu$. Madras [J. Statist. Phys. 78 (1995) no. 3--4, 681--699] has shown that $p_n \\mu^{-n} \\leq C n^{-1/2}$ in dimension $d=2$. Here we establish that $p_n \\mu^{-n} \\leq n^{-3/2 + o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of sel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09032","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"}