The bunkbed conjecture holds in the $p\uparrow 1$ limit

Abstract

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 $\epsilon (G)>0$ such that the bunkbed conjecture holds for $p⩾1-\epsilon (G)$.

7 pages, 5 figures