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 -> infinity for both of these cases. In the first case we prove that lambda(c) (L) similar to L(-2 )for any d >= 2, while in the second we prove that lambda(c) (L) similar to L-1 for any d >= 2