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I want to know for bond Bernoulli percolation on $\mathbb{Z}^2$, does it holds that $$ \mathbb{P} \left( (0,0)\longleftrightarrow (0,n) \right) \geq \mathbb{P} \left( (0,0)\longleftrightarrow (k,n) \right) $$ for $k\in [-n,n]$ and $n\in \mathbb{Z}_+$?

I don't find any positive or negetive answer, can anyone give any reference about it?

I'm a beginer in percolation theory, so if the question is in fact trivial, please remind me and I will eliminate the question.

Thanks in advance.

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  • $\begingroup$ I'm not sure about $\mathbb{Z}^2$ specifically, but inequalities of this form are certainly not known to hold on $\mathbb{Z}^d$. Inequalities like this were probably the key motivation behind Kastelyn's recently disproven "bunkbed conjecture", see the introduction to arxiv.org/abs/2110.00282. $\endgroup$
    – tmh
    Commented Oct 12 at 16:53

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