Large deviation bounds for the volume of the largest cluster in 2D critical percolation

Abstract

Let $M_n$ denote the number of sites in the largest cluster in site percolation on the triangular lattice inside a box side length $n$. We give lower and upper bounds on the probability that $M_n / \mathbb{E} M_n > x$ of the form $\exp(-Cx^{2/\alpha_1})$ for $x \geq 1$ and large $n$ with $\alpha_1 = 5/48$ and $C>0$. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions $d > 2$.