{
  "abstract": "A self-avoiding walk (SAW) on a vertex-transitive graph $G$ is a path that visits no vertex more than once; a trail is a path that traverses no edge more than once. The number of $n$-step self-avoiding walks (respectively, trails) starting at a given vertex has order $μ(G)^n$ (respectively, $μ_{\\text{T}}(G)^n$), where $μ(G)$ is called the connective constant and $μ_{\\text{T}}(G)$ is the trail constant. It is immediate that $μ=μ_{\\text{T}}$ if $G$ either has degree $3$ or less, or is a tree. We prove a necessary and sufficient condition for the strict inequality $μ< μ_{\\text{T}}$ if $G$ has degree $4$ or more and is not a tree, and in addition $G$ possesses a so-called height function. This condition is satisfied whenever the height function is transitive. The core of the proof is a pattern theorem for graphs possessing height functions, and this is achieved using an unpublished argument for ${\\Bbb Z}^d$ by Hammersley, reported by Janse van Rensburg et al. (J. Stat. Phys. 85, 1996, 103--130). Some ancillary results are included concerning SAWs on the line graph of $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00816",
  "author_total": 1,
  "authors": [
    {
      "name": "Geoffrey R. Grimmett"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgbffm22wn543.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00816",
  "fulltext_url": null,
  "id": "dgbffm22wn543",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00816",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgbffm22wn543.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.00816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "SAWs, trails, kites, patterns",
  "title_text": "SAWs, trails, kites, patterns",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/dgbffm22wn543",
  "venue": "arXiv",
  "versions": []
}