Conditional percolation on one-dimensional lattices

Conditioning i.i.d.\ bond percolation withretention parameter $p$ on a one-dimensionalperiodic lattice on the event of having a bi-infinite path from$-\infty$ to $\infty$ is shown to make sense, and the resulting modelexhibits a Markovian structure that facilitates its analysis. Stochasticmonotonicity in $p$ turns out to fail in general for this model, buta weaker monotonicity property does hold: the average edge density isincreasing in $p$