Connective constants and height functions for Cayley graphs

TL;DR

This paper investigates the existence and properties of height functions on Cayley graphs, establishing conditions for their existence and harmonicity, and applying these results to understand the locality of connective constants in self-avoiding walks.

Contribution

It introduces a necessary and sufficient condition for the existence of group height functions on Cayley graphs and demonstrates their harmonicity and applicability to various classes of groups.

Findings

Existence of harmonic unimodular height functions for many Cayley graphs.

Conditions under which group height functions exist and are harmonic.

Application of height functions to establish locality of connective constants.

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. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph height function termed here a 'group height function'. A necessary and sufficient condition for the existence of a group height function is presented, and may be applied in the context of the bridge constant, and of the locality of connective constants for Cayley graphs. Locality may thereby be established for a variety of infinite groups including those with strictly positive deficiency. It is proved that a large class of Cayley graphs support unimodular graph height functions, that are in addition harmonic on the graph. This implies, for example, the existence of unimodular graph height functions for the Cayley graphs of finitely generated solvable groups. It turns out that graphs with non-unimodular automorphism subgroups also possess graph height functions, but the resulting graph height functions need not be harmonic. Group height functions, as well as the graph height functions of the previous paragraph, are non-constant harmonic functions with linear growth and an additional property of having periodic differences. The existence of such functions on Cayley graphs is a topic of interest beyond their applications in the theory of self-avoiding walks.