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 𝜇𝑛 𝑤, in the sense that there exists 𝐶 > 0 such that 𝜇𝑛 𝑤 ≤ 𝑐𝑛 ≤ 𝐶𝜇𝑛 𝑤 for every 𝑛 ≥ 1.These results generalise earlier work by Li (J. Comb. Theory Ser. A, 2020). The key to this greater generality is that in contrast to Li’s approach, our proof does not require the existence of a special structure that enables the construction of separating patterns. Our results also 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.