2002

# The lowest crossing in 2D critical percolation

## Publication

### Publication

We study the following problem for critical site percolation on the triangular lattice. Let $A$ and $B$ be sites on a horizontal line $e$ separated by distance $n$. Consider, in the half-plane above $e$, the lowest occupied crossing $R_n$ from the half-line left of $A$ to the half-line right of $B$. We show that the probability that $R_n$ has a site at distance smaller than $m$ from $AB$ is of order $(log (n/m))^{-1}$, uniformly in $1 leq m leq n/2$. Much of our analysis can be carried out for other two-dimensional lattices as well.

Additional Metadata | |
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Interacting random processes; statistical mechanics type models; percolation theory (msc 60K35) | |

Logistics (theme 3), Energy (theme 4) | |

CWI | |

CWI. Probability, Networks and Algorithms [PNA] | |

Organisation | Stochastics |

van den Berg, J, & Járai, A.A. (2002).
The lowest crossing in 2D critical percolation. CWI. Probability, Networks and Algorithms [PNA]. CWI. |