Quantitative sub-ballisticity of self-avoiding walk on Zᵈ

We prove quantitative sub-ballisticity for the self-avoiding walk on the lattice Zᵈ, d≥ 2. Namely, we obtain an explicit exponential rate of decay for the probability that a self-avoiding walk of length n reaches distance at least $vn$, in terms of $v>0$. As a consequence, we show that the displacement of an n-step self-avoiding walk is at most n/exp(clog^* n) with very high probability. Previously, only a non-quantitative $o(n)$ bound was known from the work of Duminil-Copin and Hammond. Along the way, we show that at criticality the partition function of bridges of length n is on average bounded by C/loglog n, which we believe to be of independent interest. Our approach is based on a quantitative analysis of the original nonconstructive proof of Duminil-Copin and Hammond using ideas from proof theory.