Abstract
We consider random walk on a discrete torus
E
of side-length
N
, in sufficiently high dimension
d
. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time
uN^d
. We show that when
u
is chosen small, as
N
tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const
\log N
. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites
N^d
of
E
, and any point of
E
lies within distance
N^\beta
of this component, with
\beta
an arbitrary positive number.