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

TL;DR

This paper proves that in 2D critical percolation, the largest cluster size in a box is tightly concentrated around a predictable scale, extending known bounds to a more precise probabilistic range.

Contribution

It establishes that the probability of the largest cluster size falling within a specific scaled interval is bounded away from zero, confirming a conjecture related to cluster size distribution.

Findings

Largest cluster size is tightly concentrated around n^2 pi(n)

Probability bounds hold for clusters within a scaled interval

Sublinearity of 1/pi(n) is crucial for the proof

Abstract

We consider (near-)critical percolation on the square lattice. Let 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 M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. 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 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.