The bunkbed conjecture holds in the limit

Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$ , which has vertex set $V\times \{0,1\}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$ . Kasteleyn’s Bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$ , the vertex $(u,0)$ is at least as likely to be connected to $(v,0)$ as to $(v,1)$ under Bernoulli- $p$ bond percolation on the bunkbed graph. We prove that the conjecture holds in the $p \uparrow 1$ limit in the sense that for each finite graph $G$ there exists $\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon (G)$ .