Hypercube percolation

We study bond percolation on the Hamming hypercube {0,1} around the critical probability pc . It is known that if p = pc (1+O(2−m/3)), then with high probability the largest connected component C1 is of size Θ(2 ) and that this quantity is non-concentrated. Here we show that for any sequence εm such that εm = o(1) but εm ≫ 2−m/3 percolation on the hypercube at pc (1+εm ) has |C1| = (2+o(1))εm 2 and |C2| = o(εm 2 ) , with high probability, where C2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].

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