The bunkbed conjecture holds in the p\uparrow 1 limit

2021/10/01 by Hutchcroft, Tom, Nizić-Nikolac, Petar, Kent, Alexander
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2110.00282

Abstract

Let G=(V,E) be a countable graph. The Bunkbed graph of G is the product graph G × K₂, which has vertex set V× \0,1\ with "horizontal'' edges inherited from G and additional "vertical'' edges connecting (w,0) and (w,1) for each w ∈ V. Kasteleyn's bunkbed conjecture states that for each u,v ∈ V and p∈ [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 ε(G)>0 such that the bunkbed conjecture holds for p \geqslant 1-ε(G).

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