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A lower bound for the chemical distance in sparse long-range percolation models

Noam Berger
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Abstract

We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.

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Noam Berger. 2004-09-01. A lower bound for the chemical distance in sparse long-range percolation models. https://arxiv.org/abs/math/0409021

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