Self-avoiding walks and multiple context-free languages

TL;DR

This paper proves that for certain graphs, the generating function of self-avoiding walks is algebraic, and characterizes the language of such walks as multiple context-free depending on the graph's ends, linking graph structure to formal language complexity.

Contribution

It establishes algebraicity of the generating function for self-avoiding walks on graphs with thin ends and characterizes the language of walks as multiple context-free based on the number of ends.

Findings

Generating function is algebraic for graphs with thin ends.

Language of self-avoiding walks is multiple context-free iff the graph has at most two ends.

For Cayley graphs, the language is multiple context-free iff the group is virtually free.

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 $F_{\mathrm{SAW},o}(z)=\sum_{n \geq 0} c_n(o) z^n$ 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 any two edges starting at the same vertex have different labels, 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.