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

Abstract

We consider long-range percolation in dimension $d\ge 1$, 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 $p_{x,y}=\Vert x-y{\Vert }^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.

9 pages

References in corpus (1)

• On the scaling of the chemical distance in long-range percolation models