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 Pₙ(dG(w₀,wₙ)≥ cn)≥ 1-e⁻ᵗⁿ for every n≥ 1. Furthermore, we show that the number of self-avoiding walks of length n grows asymptotically like μwⁿ, in the sense that there exists $C>0$ such that μwⁿ≤ cₙ≤ Cμwⁿ for every n≥ 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.