2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72345We 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.9 pagesProbability60K35; 82B43, 82B28text