Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach

A layered graph $G^{\times }$ is the Cartesian product of a graph $G=(V,E)$ with the linear graph $Z$, e.g. $Z^{\times }$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p\in [0,1]$ on $G^{\times }$ one intuitively would expect that $P_p((o,0)\leftrightarrow (v,n))\ge P_p((o,0)\leftrightarrow (v,n+1))$ for all $o,v\in V$ and $n\ge 0$. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite $G$ we thus can show that for some $N\ge 0$ the above holds for all $n\ge N$ $o,v\in V$ and $p\in [0,1]$. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.