Extendable self-avoiding walks

Source abstract

The connective constant \mu of a graph is the exponential growth rate of the number of n -step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward ) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by \mu^F , \mu^B , \mu^{FB} , exist and satisfy \mu=\mu^F=\mu^B=\mu^{FB} for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular.