Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach

A layered graph G × is the Cartesian product of a graph G = ( V , E ) with the linear graph Z , e.g. Z × is the 2D square lattice Z 2 . For Bernoulli percolation with parameter p ∈ [ 0 , 1 ] on G × one intuitively would expect that P p ( ( o , 0 ) ↔ ( v , n ) ) ≥ P p ( ( o , 0 ) ↔ ( v , n + 1 ) ) for all o , v ∈ V and n ≥ 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 ≥ 0 the above holds for all n ≥ N o , v ∈ V and p ∈ [ 0 , 1 ] . One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.