Subdiffusive concentration for the chemical distance in Bernoulli percolation

TL;DR

This paper improves understanding of the chemical distance in supercritical Bernoulli percolation by establishing subdiffusive concentration inequalities, refining previous diffusive results and introducing new tools for analysis.

Contribution

It introduces subdiffusive concentration bounds for the chemical distance, building on and refining prior diffusive results with novel methods inspired by first-passage percolation.

Findings

Established subdiffusive concentration inequality for chemical distance.

Revisited and strengthened variance bounds from prior work.

Developed new tools like effective radius for analysis.

Abstract

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.