Bunkbed conjecture (original formulation)

Prove that for any connected graph G=(V,E), any subset of transversal vertices T⊆V, and any percolation parameter 0<p<1, the bunkbed percolation connectivity probability between same-level vertices satisfies P_p[u v] ≥ P_p[u v′] for all u,v∈V, where u′ and v′ denote the corresponding vertices in the second level of the bunkbed graph G×K₂ and percolation is performed only on horizontal edges while all post edges between T and T′ are retained.

Background

The bunkbed conjecture was introduced by Kasteleyn in 1985 and has been regarded as a natural, intuitive, yet resistant problem in percolation theory. It asserts that, in the bunkbed percolation model on G×K₂ with posts fixed on T, connectivity is more likely within the same level than across levels.

The conjecture has been proved in several special cases (e.g., wheels, complete graphs, complete bipartite graphs, and certain symmetric graphs) and in the p↑1 limit, but remained open in general until this paper, which provides an explicit counterexample. The statement here records the original conjectural inequality as stated in the paper.