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

Part of: Special processes Graph theory

Published online by Cambridge University Press:  14 December 2022

Tom Hutchcroft Open the ORCID record for Tom Hutchcroft [Opens in a new window] ,
Alexander Kent  and
Petar Nizić-Nikolac
Tom Hutchcroft*
Affiliation:
The Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Alexander Kent
Affiliation:
University of Warwick, Coventry, CV4 7AL, UK
Petar Nizić-Nikolac
Affiliation:
University of Cambridge, Cambridge, CB3 0WB, UK
*
*Corresponding author. Email: [email protected]
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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 $\varepsilon (G)\gt 0$ such that the bunkbed conjecture holds for $p \geqslant 1-\varepsilon (G)$ .

Keywords

percolationbunkbedconnectivityprobabilitygraph theory

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 3 , May 2023 , pp. 363 - 369
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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