Higher-Dimensional Stick Percolation

Abstract

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 $$\lambda _c(L)$$ λ c ( L ) of the percolation phase transition, and in particular we investigate the asymptotic behavior of $$\lambda _c(L)$$ λ c ( L ) as $$L\rightarrow \infty $$ L → ∞ for both of these cases. In the first case we prove that $$\lambda _c(L)\sim L^{-2}$$ λ c ( L ) ∼ L - 2 for any $$d\ge 2,$$ d ≥ 2 , while in the second we prove that $$\lambda _c(L)\sim L^{-1}$$ λ c ( L ) ∼ L - 1 for any $$d\ge 2.$$ d ≥ 2 .

Continuum Percolation in a Nonstabilizing Environment

“…For example, instead of a fixed connectivity threshold r > 0, random radii can be used to define connected components [MR96,Gou08]. Also in this direction, other local geometries have been used to define edges, see for example [Roy91,BT16,Bro22]. Another line of research is concerned with generalizations towards using other stationary point processes as the underlying set of vertices in the network.…”