Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .