The gaps between the sizes of large clusters in 2D critical percolation

Abstract

Consider critical bond percolation on a large $2 n \times 2 n$ box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order $n^2 \pi(n)$, where $\pi(n)$ denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera.

Jàrai showed that the differences between the sizes of these clusters is, with high probability, at least of order $\sqrt{n^2 \pi(n)}$. Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. $n^2 \pi(n)$. Our main result is a proof that this is indeed the case.