For every epsilon > 0 there is a C such that the percolated hypercube Q_d^p with p >= C/d contains, with high probability, a cycle of length at least (1-epsilon)2^d.
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Nearly spanning cycle in the percolated hypercube
For every epsilon > 0 there is a C such that the percolated hypercube Q_d^p with p >= C/d contains, with high probability, a cycle of length at least (1-epsilon)2^d.