Hypercube percolation

We study bond percolation on the Hamming hypercube f0; 1gm around the critical probability pc. It is known that if p D pc.1 C O.2-m=3//, then with high probability the largest connected component Here we show that for any sequence ".m/such that ".m/D o.1/but ϵ.m/percolation on the hypercube at pc.1 C ϵm//has with high probability, where C2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [18].