Locality of connective constants, I. Transitive graphs

Geoffrey R. Grimmett, Zhongyang Li

Published 2014-11-29Version 1

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. 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 graph height function. It is proved that a broad category of transitive graphs have graph height functions.