The percolation phase transition in the Hamming cube

Consider percolation on the Hamming cube 0,1ⁿ at the critical probability p_c (at which the expected cluster size is 2^n/3). It is known that if p=p_c(1+O(2^-n/3), then the largest component is of size roughly 2^2n/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 p_c(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.