Self-avoiding walks and amenability

The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $\mu(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy. In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function. In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group.Comment: v2 differs from v1 in the inclusion of further material concerning non-amenable graphs, notably an improved lower bound for the connective constan

Self-avoiding walks and amenability

The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $\mu(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy. In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function. In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group

Cubic graphs and the golden mean

The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality $\mu \ge \phi$ for infinite, transitive, simple, cubic graphs, where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for $\mu$ are presented for transitive cubic graphs with girth either $3$ or $4$, and for certain quasi-transitive cubic graphs.This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. ZL acknowledges support from the Simons Foundation under grant #351813 and the National Science Foundation under grant DMS-1608896