Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice

We prove quantitative sub-ballisticity for the self-avoiding walk on the hexagonal lattice. Namely, we show that with high probability a self-avoiding walk of length $n$ does not exit a ball of radius $O(n\mathrm{/}\log n)$. Previously, only a non-quantitative $o(n)$ bound was known from the work of Duminil-Copin and Hammond \cite{DCH13}. As an important ingredient of the proof we show that at criticality the partition function of bridges of height $T$ decays polynomially fast to $0$ as $T$ tends to infinity, which we believe to be of independent interest.