Consider percolation on the Hamming cube 0,1n at the critical probability pc (at which the expected cluster size is 2n/3). It is known that if p=pc(1+O(2-n/3), then the largest component is of size roughly 22n/3 with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)2-n/3 and eps(n)=o(1) percolation at pc(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2n.
This result settles a conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer.
Joint work with Remco van der Hofstad.