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For a finite simple graph $G$, the bunkbed graph $G^\pm$ is defined to be theproduct graph $G\square K_2$. We will label the two copies of a vertex $v\inV(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, statesthat for independent bond percolation on $G^\pm$, percolation from $u_-$ to$v_-$ is at least as likely as percolation from $u_-$ to $v_+$, for any $u,v\inV(G)$. Despite the plausibility of this conjecture, so far the problem in fullgenerality remains open. Recently, Hutchcroft, Nizi\'{c}-Nikolac, and Kent gavea proof of the conjecture in the $p\uparrow 1$ limit. Here we present a newproof of the bunkbed conjecture in this limit, working in the more generalsetting of allowing different probabilities on different edges of $G^\pm$.

A new proof of the bunkbed conjecture in the $p\uparrow 1$ limit