Abstract

Co-author: Geoffrey Grimmett (University of Cambridge)

A self-avoiding walk (SAW) is a path on a graph that revisits no vertex. The connective constant of a graph is defined to be the exponential growth rate of the number of n-step SAWs with respect to n. We prove that sqrt{d-1} is a universal lower bound for connective constants of any infinite, connected, transitive, simple, d-regular graph. We also prove that the connective constant of a Cayley graph decreases strictly when a new relator is added to the group and increases strictly when a non-trivial word is declared to be a generator. I will also present a locality result regarding to the connective constants proved by defining a linearly increasing harmonic function on Cayley graphs. In particular, the connective constant is local for all solvable groups. Joint work with Geoffrey Grimmett.