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The bunkbed conjecture holds in the p ↑ 1 limit

Hutchcroft, Tom and Kent, Alexander and Nizić-Nikolac, Petar (2022) The bunkbed conjecture holds in the p ↑ 1 limit. Combinatorics, Probability and Computing . pp. 1-7. ISSN 0963-5483. doi:10.1017/s096354832200027x. (In Press)

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Let G = (V, E) be a countable graph. The Bunkbed graph of G is the product graph G x K₂, which has vertex set V x {0,1} with “horizontal” edges inherited from G and additional “vertical” edges connecting (w,0) and (w,1) for each w ϵ V. Kasteleyn’s Bunkbed conjecture states that for each u, v ϵ V and p ϵ [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 ↑ 1 limit in the sense that for each finite graph G there exists ε(G) > 0 such that the bunkbed conjecture holds for p ⩾ 1 - ε(G).

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Hutchcroft, Tom0000-0003-0061-593X
Additional Information:This paper is the result of an undergraduate summer research project at the University of Cambridge in the summer of 2020, where PNN and AK were mentored by TH. PNN was supported jointly by a Trinity College Summer Studentship (F. J. Woods Fund) and a CMS Summer Studentship, AK was supported by a CMS Summer Research in Mathematics bursary, and TH was supported in part by ERC starting grant 804166 (SPRS). We thank Piet Lammers for helpful comments on a draft.
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Trinity CollegeUNSPECIFIED
European Research Council (ERC)804166
Record Number:CaltechAUTHORS:20230103-818063100.57
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ID Code:118647
Deposited By: Research Services Depository
Deposited On:07 Feb 2023 18:53
Last Modified:07 Feb 2023 18:53

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