  Considering supercritical Bernoulli percolation on $\mathbb{Z}^d$, Garet and
Marchand [GM09] proved a diffusive concentration for the graph distance. In
this paper, we sharpen this result by establishing the subdiffusive
concentration inequality, which revisits the sublinear bound of the variance
proved by Dembin [Dem22] as a consequence. Our approach is inspired by similar
work in First-passage percolation [BR08, DHS14], combined with new tools to
address the challenge posed by the infinite weight of the model. These tools,
including the notion of effective radius and its properties, enable a simple
one-step renormalization process as a systematic means of managing the effects
of resampling edges.
