  This work continues the investigation, initiated in a recent work by
Benjamini and Sznitman, of percolative properties of the set of points not
visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high
dimension d. If u>0 is chosen sufficiently small it has been shown that with
overwhelming probability this vacant set contains a unique giant component
containing segments of length c_0 log N for some constant c_0 > 0, and this
component occupies a non-degenerate fraction of the total volume as N tends to
infinity. Within the same setup, we investigate here the complement of the
giant component in the vacant set and show that some components consist of
segments of logarithmic size. In particular, this shows that the choice of a
sufficiently large constant c_0 > 0 is crucial in the definition of the giant
component.
