Locality of connective constants

TL;DR

This paper proves a locality theorem for the connective constant of quasi-transitive graphs, showing that similar local structures imply similar growth rates of self-avoiding walks, using a generalized bridge decomposition method.

Contribution

It establishes a locality result for connective constants of quasi-transitive graphs, extending understanding of how local graph structure influences global self-avoiding walk growth.

Findings

Connective constants are close when graphs agree on large local neighborhoods.

The proof uses a generalized bridge decomposition of self-avoiding walks.

The theorem applies to graphs with a unimodular graph height function.

Abstract

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. We prove a locality theorem 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 (and a further condition is satisfied). The proof exploits a generalized bridge decomposition of self-avoiding walks, which is valid subject to the assumption that the underlying graph is quasi-transitive and possesses a so-called unimodular graph height function.