CONDITIONAL PERCOLATION ON ONE-DIMENSIONAL LATTICES

Conditioning independent and identically distributed bond percolation with retention parameter p on a one-dimensional periodic lattice on the event of having a bi-infinite path from -infinity to infinity is shown to make sense, and the resulting model exhibits a Markovian structure that facilitates its analysis. Stochastic monotonicity in p turns out to fail in general for this model, but a weaker monotonicity property does hold: the average edge density is increasing in p.