  Let M_n denote the number of sites in the largest cluster in critical site
percolation on the triangular lattice inside a box side length n. We give lower
and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C
x^(2/alpha)) for x > 1 and large n with alpha = 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.
