# Conditional percolation on one-dimensional lattices Journal article, 2009

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 $p$.

Conditional percolation

stochastic domination

Markov chains

one-dimensional lattices

## Author

### Marina Axelson-Fisk

Chalmers, Mathematical Sciences, Mathematical Statistics

University of Gothenburg

### Olle Häggström

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematical Statistics

#### Advances in Applied Probability

0001-8678 (ISSN) 1475-6064 (eISSN)

Vol. 41 4 3395-3415

### Subject Categories

Probability Theory and Statistics

### DOI

10.1239/aap/1261669588

10/7/2017