Self-avoiding walk is ballistic on graphs with more than one end

We prove that on any transitive graph $G$ with infinitely many ends, a self-avoiding walk of length $n$ is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that ${\mathbb{P}}_n(d_G(w_0,w_n)\ge cn)\ge 1-e^{-tn}$ for every $n\ge 1$. Furthermore, we show that the number of self-avoiding walks of length $n$ grows asymptotically like ${\mu }_w^n$, in the sense that there exists $C>0$ such that ${\mu }_w^n\le c_n\le C{\mu }_w^n$ for every $n\ge 1$. Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of $G$ which does not fix an end of $G$.