Self-avoiding Walks and Connective Constants

The connective constant μ(G) of a quasi-transitive graph G is the asymptotic growth rate of the number of selfavoiding walks (SAWs) on G from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph G. • We present upper and lower bounds for μ in terms of the vertex-degree and girth of a transitive graph. • We discuss the question of whether μ ≥ φ for transitive cubic graphs (where φ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). • We present strict inequalities for the connective constants μ(G) of transitive graphs G, as G varies. • As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. • We describe so-called graph height functions within an account of ‘bridges’ for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. • A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. • Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. • The review closes with a brief account of the ‘speed’ of SAW. Date: 19 April 2017. 2010 Mathematics Subject Classification. 05C30, 82B20, 60K35.