2012-01-01

# On the size of the largest cluster in 2D critical percolation

## Publication

### Publication

*Electronic Communications in Probability , Volume 17*

We consider (near-)critical percolation on the square lattice. Let $\mathcal{M}_{n}$ be the size of the largest open cluster contained in the box $[-n,n]^2$, and let $\pi(n)$ be the probability that there is an open path from $O$ to the boundary of the box. It is well-known that for all $0< a < b$ the probability that $\mathcal{M}_{n}$ is smaller than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are bounded away from $0$ as $n \rightarrow \infty$. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that $\mathcal{M}_{n}$ is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of $1/\pi(n)$ appears to be essential for the argument.

Additional Metadata | |
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Keywords | Critical percolation, Cluster size |

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

Publisher | ims |

Persistent URL | dx.doi.org/10.1214/ECP.v17-2263 |

Journal | Electronic Communications in Probability |

Citation |
van den Berg, J, & Conijn, R.P. (2012). On the size of the largest cluster in 2D critical percolation.
Electronic Communications in Probability, 17. doi:10.1214/ECP.v17-2263 |