Self-avoiding walks and amenability

TL;DR

This paper investigates the relationship between the connective constant of self-avoiding walks and the property of amenability in infinite transitive graphs, revealing new connections and bounds related to graph height functions.

Contribution

It establishes that elementary amenable groups support harmonic graph height functions, contrasts this with non-elementary amenable groups, and provides bounds on connective constants for non-amenable graphs.

Findings

Elementary amenable groups support harmonic graph height functions.

Non-elementary amenable groups like the Grigorchuk group lack graph height functions.

Lower bounds for connective constants are derived for non-amenable graphs with large girth.

Abstract

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.