标题： 该 bunkbed 猜想在$p\uparrow 1$极限下成立 显示英文标题

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

摘要： 设$G=(V,E)$是一个可数图。 Bunkbed图的$G$是乘积图$G \times K_2$，其顶点集为$V\times \{0,1\}$，具有从$G$继承的“水平”边，并且对于每个$w \in V$，还有连接$(w,0)$和$(w,1)$的附加“垂直”边。 Kasteleyn的bunkbed猜想指出，对于每个$u,v \in V$和$p\in [0,1]$，在bunkbed图上的伯努利-$p$键渗流下，顶点$(u,0)$与$(v,0)$相连的可能性至少与与$(v,1)$相连的可能性一样大。 我们证明了在$p \uparrow 1$极限下该猜想成立，即对于每个有限图$G$，存在$\varepsilon(G)>0$，使得 bunkbed 猜想对$p \geqslant 1-\varepsilon(G)$成立。 显示英文摘要

摘要： 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 $\varepsilon(G)>0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon(G)$.