The bunkbed conjecture is not robust to generalisation (2406.01790v1)

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.

• The paper demonstrates through counterexamples that the longstanding bunkbed conjecture fails when generalized to site percolation, hypergraphs, and directed graphs.
• Counterexamples are meticulously constructed for conditioned site percolation, hypergraphs, and directed graphs to prove the conjecture's failure in these extended domains.
• While the conjecture fails under generalizations, it is reaffirmed for site percolation on paths, cycles, and wheel graphs, highlighting the structural properties crucial for its validity.

Analysis of the Bunkbed Conjecture and Its Generalizations

The paper "The bunkbed conjecture is not robust to generalisation" by Lawrence Hollom critically examines the longstanding bunkbed conjecture in percolation theory and its proposed generalizations. First introduced by Kasteleyn in 1985, the bunkbed conjecture has piqued the interest of researchers due to its seemingly intuitive nature, yet remains unproven in a general context. This analysis underscores the fragility of the conjecture when extended beyond its traditional bounds, namely into site percolation, hypergraphs, and directed graphs.

Summary of the Bunkbed Conjecture

The original conjecture relates to bond percolation on the product graph $G \Box K_2$. It posits that for any vertices $u,v \in V(G)$, the probability of percolation from $u^{(0)}$ to $v^{(0)}$ is at least as high as that from $u^{(0)}$ to $v^{(1)}$. While intuitively plausible, this conjecture has resisted a general proof despite its longstanding presence in probability theory literature.

Generalizations and Counterexamples

The paper considers three natural generalizations: site percolation, hypergraphs, and directed graphs. The author's primary contribution is demonstrating through meticulously constructed counterexamples that all these generalizations fail to hold true.

1. Site Percolation: For this scenario, the author introduces two models—conditioned and unconditioned. A specific graph (G_2) is constructed as a counterexample for conditioned site percolation (E2^T model). The probability discrepancies between $u^{(0)} \to v^{(0)}$ versus $u^{(0)} \to v^{(1)}$ are meticulously calculated to demonstrate the conjecture's failure.
2. Hypergraphs: Transitioning from graphs to hypergraphs, model $E_4^T$ is scrutinized. A hypergraph (H_4) derived from a hypergraph dual of G_2 and modifications, such as collapsing edges to single vertices and adding additional connections, serves as a decisive counterexample.
3. Directed Graphs: For directed graphs, the paper considers models with specific edge retention conditions. Like hypergraphs, substantial augmentations to the graph structure illustrate the conjecture's shortcomings. The counterexample (D_6) is established by converting hyperedges into directed edges, with edge orientations providing the desired conflict in percolation probabilities.

Positive Results and Restricted Cases

While these generalizations fail, the author reaffirms the conjecture holds under specific conditions. The paper proves the bunkbed conjecture for site percolation on paths, cycles, and wheel graphs remains true. This implies a limitation on the scenarios where the conjecture's extension may be considered applicable, highlighting the conditions and structural properties that preserve the conjecture's integrity.

Implications and Future Directions

The findings have significant theoretical implications, as they challenge previously held notions about the conjecture's robustness. The demonstrated inapplicability to generalized models introduces caution in presuming the conjecture's validity beyond familiar territories of undirected graphs. It casts doubt on simplistic intuitive reasoning and calls for refined techniques that respect the distinct constraints of each structure.

The research opens several pathways for further inquiry:

• Exploring the conjecture's status in random graph models like Erdős–Rényi random graphs.
• Examining small-diameter or highly symmetric graphs for positive affirmation of the conjecture.
• Delving into acyclic directed graphs, probing whether the conjecture's truth may yet emerge in these restricted cases.

In conclusion, while the paper presents envelopes of skepticism for overgeneralization, it also carves out realms ripe for deeper exploration, urging the community to reapproach the bunkbed conjecture with both critical insight and innovation.