Extendable self-avoiding walks

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.