Zebra-percolation on Cayley trees

Abstract

We consider Bernoulli (bond) percolation with parameter $p$ on the Cayley tree of order $k$. We introduce the notion of zebra-percolation that is percolation by paths of alternating open and closed edges. In contrast with standard percolation with critical threshold at $p_c=1/k$, we show that zebra-percolation occurs between two critical values $p_{\mathrm{c},1}$ and $p_{\mathrm{c},2}$ (explicitly given). We provide the specific formula of zebra-percolation function.

8 pages, 2 figures