The acyclic directed bunkbed conjecture is false

TL;DR

The paper disproves the acyclic directed bunkbed conjecture by constructing a simple acyclic directed graph where the directed bunkbed monotonicity fails at $p=\tfrac{1}{2}$, i.e., $\mathbb{P}_{1/2}(u^- \to v^-) < \mathbb{P}_{1/2}(u^- \to v^+)$. It proceeds in two steps: first a conditioned counterexample on a base graph $G_1$ with posts $T=\{2,5,8\}$ proves the inequality, and then an unconditioned counterexample is obtained by replacing those vertices with gadgets to form $G_2^k$, with a contraction argument linking the conditioned and unconditioned models. The method relies on a careful event decomposition, mirroring/cut-set arguments, and the use of a contraction map to transfer asymptotic behavior from the gadget-augmented graph back to the conditioned base graph. Collectively, the results resolve conjectures posed by Leander and Hollom and reveal that even acyclic directed bunkbed models can violate the proposed monotonicity, highlighting the need for refined structural conditions in percolation on bunkbed constructions.

Abstract

We construct a simple acyclic directed graph for which the Bunkbed Conjecture is false, thereby resolving conjectures posed by Leander and by Hollom.

Figures (2)

• Figure 1: Graph $G_1$.
• Figure 4: Graph $G_1$ with its shaded part corresponding to $\widetilde{\mathcal{E}}_T$.

Theorems & Definitions (9)

• Conjecture 1.1
• Theorem 1.2
• Definition 3.1: Conditioned
• Proposition 3.2
• proof : Proof of \ref{['prop:conditional']}
• Claim 3.3
• proof : Proof of claim.
• Definition 4.1
• proof : Proof of \ref{['thm:main']}