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