Worked solution: Gladkov–Pak–Zimin's explicit counterexample disproving the bunkbed conjecture (2024)

For a connected graph $G = (V, E)$ and a chosen subset of transversal vertices $T \subseteq V$, the bunkbed graph $\overline{G}$ consists of two copies $G$ and $G'$ (levels $0$ and $1$) together with a post edge joining $w$ to $w'$ for every $w \in T$. In bunkbed percolation, each edge of $G$ and $G'$ is kept independently with probability $p \in (0,1)$, while every post is always kept. Kasteleyn's 1985 bunkbed conjecture (Gladkov, Pak & Zimin 2024, Conjecture 1.1, attributed to Kasteleyn via van den Berg & Kesten 2001, Remark 5) states that for every such $G$, $T$, and $p$, and every pair of vertices $u, v \in V$, $\Pr^{\text{bb}}_p[u \leftrightarrow v] \ge \Pr^{\text{bb}}_p[u \leftrightarrow v']$ — connecting to $v$'s own level is at least as likely as connecting to $v$'s copy on the other level.

The most commonly cited special case takes $T = V$ (a post at every vertex), but the general statement with an arbitrary transversal set $T$ is what Gladkov, Pak, and Zimin actually address, and it is the version their counterexample refutes. The conjecture had been verified in several special cases — for wheels, complete graphs, complete bipartite graphs, graphs with an automorphism swapping $u$ and $v$, and for one or two transversal vertices — and, remarkably, in the limit $p \to 1$, but a fully general proof or disproof had eluded probabilists for essentially the whole four decades since it was posed.

The rest of this proof builds an explicit graph, transversal set, and pair of vertices for which the inequality fails at $p = \tfrac12$, disproving the conjecture in general.