×

On the size of the largest cluster in 2D critical percolation. (English) Zbl 1320.60161

Summary: 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 (see [C. Borgs et al., Commun. Math. Phys. 224, No. 1, 153–204 (2001; Zbl 1038.82035)]) that for all \(0<a<b\) the probability that \(\mathcal{M}_n\) is smaller than \(an^2\pi(n)\) and the probability that \(\mathcal{M}_n\) is larger than \(bn^2\pi(n)\) are bounded away from 0 as \(n\to \infty\). 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 between \(an^2\pi(n)\) and \(bn^2\pi(n)\). 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation

Citations:

Zbl 1038.82035