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

Philipp König and Thomas Richthammer

Stochastic Processes and their Applications, 2025, vol. 181, issue C

Abstract: 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 Z2. For Bernoulli percolation with parameter p∈[0,1] on G× one intuitively would expect that Pp((o,0)↔(v,n))≥Pp((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≥No,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.

Keywords: Bernoulli percolation; Spatial monotonicity; Connectivity function; Markov chain (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2024.104549

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