A new proof of the bunkbed conjecture in the $p\uparrow 1$ limit

TL;DR

This paper provides a new proof of the bunkbed conjecture for percolation on bunkbed graphs as the percolation probability approaches 1, extending previous results to variable edge probabilities.

Contribution

It introduces a novel proof technique for the bunkbed conjecture in the high-probability limit, accommodating non-uniform edge probabilities.

Findings

Proof of the bunkbed conjecture as p approaches 1

Extension to graphs with non-uniform edge probabilities

Applicable to a broader class of percolation models

Abstract

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 $G^\pm$.