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arXiv · 2406.14834

Subdiffusive concentration for the chemical distance in Bernoulli percolation

Van Hao Can·Van Quyet Nguyen
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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.

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Van Hao Can, Van Quyet Nguyen. 2024-06-21. Subdiffusive concentration for the chemical distance in Bernoulli percolation. https://arxiv.org/abs/2406.14834

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