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Self-avoiding walks and multiple context-free languages

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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.

If \(G\) 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.

Mathematics Subject Classifications: 20F10, 68Q45, 05C25

Keywords: Self avoiding walk, formal language, multiple context free language, Cayley graph, virtually free group

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