Abstract
AbstractLet
$G=(V,E)$
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
$p \geqslant 1-\varepsilon (G)$
.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
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