Dark matter density field of the Millennium run (left) and its complementary pair (right) at z = 0 redshift. The bottom row is a zoom onto the region outlined by the red boxes in the figures in the top row. As the φ(k) phases are shifted by π in the initial condition, a region that collapses to a halo in the original Millennium run tends to expand into a void in the corresponding complementary pair and vice versa, which is similar to the paired simulation method.