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

2012/08/20 by Berg, Jacob van den, Conijn, Rene
#(Secondary) #60K35 (Primary) 60C05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1208.4014

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 an² pi(n) and the probability that M_n is larger than bn² 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 an² pi(n) and bn² 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.

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