Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
Philipp König, Thomas Richthammer
TL;DR
The paper studies monotonicity of Bernoulli percolation on layered graphs by introducing a layer-by-layer Markov chain on infection patterns. It proves that for any finite base graph $G$, there is an $N$ such that for all $n\ge N$, the connectivity probability from the origin to any vertex does not increase with the layer index, and the expected number of infected vertices per layer is nonincreasing as well, for all $p\in(0,1)$. The authors formalize the infection-pattern process $\mathcal X_n$ with transition kernel $\pi_p$, analyze it via state-space reductions and CAS computations, and treat intermediate, small, and large $p$ regimes to establish pattern-monotonicity. This Markov-chain perspective links layered-graph monotonicity to bunkbed and Zd-monotonicity questions and suggests broad applicability to other percolation problems on layered structures.
Abstract
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.