Higher-dimensional stick percolation

We consider two cases of the so-called stick percolation model with sticks of length $L.$ In the first case, the orientation is chosen independently and uniformly, while in the second all sticks are oriented along the same direction. We study their respective critical values $\lambda_c(L)$ of the percolation phase transition, and in particular we investigate the asymptotic behavior of $\lambda_c(L)$ as $L\to \infty$ for both of these cases. In the first case we prove that $\lambda_c(L)\sim L^{-2}$ for any $d\geq 2,$ while in the second we prove that $\lambda_c(L)\sim L^{-1}$ for any $d\geq 2.$