Zebra-percolation on Cayley trees

TL;DR

This paper introduces zebra-percolation on Cayley trees, a new percolation model involving alternating open and closed edges, and identifies its critical thresholds and explicit formula.

Contribution

It defines zebra-percolation, derives its critical thresholds, and provides an explicit formula, contrasting with standard percolation theory.

Findings

Zebra-percolation occurs between two explicit critical values.

Explicit formula for the zebra-percolation function is provided.

Zebra-percolation differs from standard percolation with a single critical point.

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 $p_{{\rm c},1}$ and $p_{{\rm c},2}$ (explicitly given). We provide the specific formula of zebra-percolation function.