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# Conditional percolation on one-dimensional lattices

Journal article
Authors Marina Axelson-Fisk Olle Häggström Advances in Applied Probability 41 4 3395-3415 0001-8678 2009 Department of Mathematical Sciences Department of Mathematical Sciences, Mathematical Statistics 3395-3415 en dx.doi.org/10.1239/aap/1261669588 Conditional percolation, stochastic domination, one-dimensional lattices, Markov chains Mathematical statistics

## Abstract

Conditioning i.i.d.\ bond percolation with retention parameter $p$ on a one-dimensional periodic lattice on the event of having a bi-infinite path from $-\infty$ to $\infty$ 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 $p$.

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