The bunkbed conjecture is not robust to generalisation

TL;DR

This paper demonstrates that natural extensions of the bunkbed conjecture to site percolation, hypergraphs, and directed graphs are false by providing counterexamples, while also identifying specific conditions where the conjecture still holds.

Contribution

The paper shows that the generalized forms of the bunkbed conjecture are false and identifies conditions under which the original conjecture remains valid.

Findings

Counterexamples for site percolation, hypergraphs, and directed graphs generalizations.

Conditions under which the bunkbed conjecture still holds.

Insights into the limitations of the conjecture's robustness.

Abstract

The bunkbed conjecture, which has featured in the folklore of probability theory since at least 1985, concerns bond percolation on the product graph $G\Box K_2$. We have two copies $G_0$ and $G_1$ of $G$, and if $x^{(0)}$ and $x^{(1)}$ are the copies of a vertex $x\in V(G)$ in $G_0$ and $G_1$ respectively, then edge $x^{(0)}x^{(1)}$ is present. The conjecture states that, for vertices $u,v\in V(G)$, percolation from $u^{(0)}$ to $v^{(0)}$ is at least as likely as percolation from $u^{(0)}$ to $v^{(1)}$. While the conjecture is widely expected to be true, having attracted significant attention, a general proof has not been forthcoming. In this paper we consider three natural generalisations of the bunkbed conjecture; to site percolation, to hypergraphs, and to directed graphs. Our main aim is to show that all these generalisations are false, and to this end we construct a sequence of counterexamples to these statements. However, we also consider under what extra conditions these generalisations might hold, and give some classes of graph for which the bunkbed conjecture for site percolation does hold.