论文标题
分层图上Bernoulli渗透的单调性能 - 马尔可夫链方法
Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
论文作者
论文摘要
一个分层图$ g^\ times $是图形$ g =（v，e）$的笛卡尔产物，线性图$ z $，例如。 $ z^\ times $是2D方格$ z^2 $。对于$ g^\ times $ in [0,1] $ in $ g^\ times $ in in Intuity上，一个直觉上的$ p \ in [0,1] $ in [0,1]这让人联想到众所周知的繁殖猜想。在这里，我们介绍了上述单调性猜想的方法，该方法利用马尔可夫链逐层构建了渗透模式。因此，在有限$ g $的情况下，我们可以证明，对于某些$ n \ ge 0 $，上述所有$ n \ ge n $ o，v \ in v $ in v $ in [0,1] $。人们可能希望这种马尔可夫链方法对在分层图上伯努利渗透的其他问题可能有用。
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.
