A new proof of the bunkbed conjecture in the $p\uparrow 1$ limit
For a finite simple graph $G$, the bunkbed graph $G^\pm$ is defined to be the product graph $G\square K_2$. We will label the two copies of a vertex $v\in V(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, states that for independent bond percolation on $G^\pm$, percolation from $...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
31.01.2023
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| Subjects | |
| Online Access | Get full text |
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| Summary: | For a finite simple graph $G$, the bunkbed graph $G^\pm$ is defined to be the
product graph $G\square K_2$. We will label the two copies of a vertex $v\in
V(G)$ as $v_-$ and $v_+$. The bunkbed conjecture, posed by Kasteleyn, states
that 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 $u,v\in
V(G)$. Despite the plausibility of this conjecture, so far the problem in full
generality remains open. Recently, Hutchcroft, Nizi\'{c}-Nikolac, and Kent gave
a proof of the conjecture in the $p\uparrow 1$ limit. Here we present a new
proof of the bunkbed conjecture in this limit, working in the more general
setting of allowing different probabilities on different edges of $G^\pm$. |
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| DOI: | 10.48550/arxiv.2302.00031 |