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

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