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{
    "pk": 64889,
    "title": "Self-avoiding walks and multiple context-free languages",
    "subtitle": null,
    "abstract": "Let \\(G\\) be a quasi-transitive, locally finite, connected graph rooted at a vertex \\(o\\), and let \\(c_n(o)\\) be the number of self-avoiding walks of length \\(n\\) on \\(G\\) starting at \\(o\\). We show that if \\(G\\) has only thin ends, then the generating function \\(F_{\\mathrm{SAW},o}(z)=\\sum_{n \\geq 1} c_n(o) z^n\\) is an algebraic function. In particular, the connective constant of such a graph is an algebraic number.\nIf \\(G\\) is deterministically edge-labelled, that is, every (directed) edge carries a label such that no two edges starting at the same vertex have the same label, then the set of all words which can be read along the edges of self-avoiding walks starting at \\(o\\) forms a language denoted by \\(L_{\\mathrm{SAW},o}\\). Assume that the group of label-preserving graph automorphisms acts quasi-transitively. We show that \\(L_{\\mathrm{SAW},o}\\) is a \\(k\\)-multiple context-free language if and only if the size of all ends of \\(G\\) is at most \\(2k\\). Applied to Cayley graphs of finitely generated groups this says that \\(L_{\\mathrm{SAW},o}\\) is multiple context-free if and only if the group is virtually free.\n \nMathematics Subject Classifications: 20F10, 68Q45, 05C25\n \nKeywords: Self avoiding walk, formal language, multiple context free language, Cayley graph, virtually free group",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Self avoiding walk"
        },
        {
            "word": "formal language"
        },
        {
            "word": "multiple context free language"
        },
        {
            "word": "Cayley graph"
        },
        {
            "word": "virtually free group"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8928m3zx",
    "frozenauthors": [
        {
            "first_name": "Florian",
            "middle_name": "",
            "last_name": "Lehner",
            "name_suffix": "",
            "institution": "Department of Mathematics, The University of Auckland, New Zealand",
            "department": ""
        },
        {
            "first_name": "Christian",
            "middle_name": "",
            "last_name": "Lindorfer",
            "name_suffix": "",
            "institution": "Institut für Diskrete Mathematik, Technische Universität Graz, Austria",
            "department": ""
        }
    ],
    "date_submitted": "2023-03-14T16:51:37Z",
    "date_accepted": "2023-03-14T16:51:37Z",
    "date_published": "2023-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64889/galley/49699/download/"
        }
    ]
}