Subdiffusive concentration for the chemical distance in Bernoulli percolation

Considering supercritical Bernoulli percolation on Zᵈ, 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.